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JNL: ROP PIPS: 295858 TYPE: PAPTS: NEWGEN DATE: 10/6/2009EDITOR: MJ
IOP PUBLISHING
REPORTS ON PROGRESS IN PHYSICS
Rep. Prog. Phys. 72 (2009) 000000 (185pp)UNCORRECTED PROOF
Physics at a future Neutrino Factory and
superbeam facility
The ISS Physics Working Group
Editors: S F King1, K Long2, Y Nagashima3, B L Roberts4and O Yasuda5
1School of Physics and Astronomy, University of Southampton, Highfield, Southampton S017 1BJ, UK
2Department of Physics, Blackett Laboratory, Imperial College London, Exhibition Road, London, SW7 2AZ, UK
3Department of Physics, Osaka University, Toyonaka, Osaka 5600043, Japan
4Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA
5Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 1920397, Japan
A Bandyopadhyay1, S Choubey1, R Gandhi1, S Goswami1, B L Roberts2,
J Bouchez3, I Antoniadis4, J Ellis4, G F Giudice4, T Schwetz4,
S Umasankar5, G Karagiorgi6, A AguilarArevalo6, J M Conrad6,
M H Shaevitz6, S Pascoli7, S Geer8, J E Campagne9, M Rolinec10,
A Blondel11, M Campanelli11, J Kopp12, M Lindner12, J Peltoniemi13,
P J Dornan14, K Long14, T Matsushita14, C Rogers14, Y Uchida14,
M Dracos15, K Whisnant16, D Casper17, MuChun Chen17,64, B Popov18,
J¨Ayst¨ o19, D Marfatia20, Y Okada21, H Sugiyama21, K Jungmann22,
J Lesgourgues23, M Zisman24, M A Tortola25, A Friedland26, S Davidson27,
S Antusch28,65, C Biggio28, A Donini28, E FernandezMartinez28,
B Gavela28, M Maltoni28, J LopezPavon28, S Rigolin28, N Mondal29,
V Palladino30, F Filthaut31, C Albright32,64, A de Gouvea33, Y Kuno34,
Y Nagashima34, M Mezzetto35, S Lola36, P Langacker37, A Baldini38,
H Nunokawa39, D Meloni40, M Diaz41, S F King42, K Zuber43,
A G Akeroyd44, Y Grossman45, Y Farzan46, K Tobe47, Mayumi Aoki48,
H Murayama49,50,51, N Kitazawa52, O Yasuda52, S Petcov53, A Romanino53,
P Chimenti54, A Vacchi54, A Yu Smirnov55, E Couce56,
J J GomezCadenas56, P Hernandez56, M Sorel56, J W F Valle56,
P F Harrison57, C Lunardini58, J K Nelson59, V Barger60, L Everett60,
P Huber60, W Winter61, W Fetscher62and A van der Schaaf63
1HarishChandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
2Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA
3Service de Physique des Particules, CEA – Saclay, F91191 Gif sur Yvette Cedex, France
4Department of Physics, CERN Theory Division, 1211 Geneva 23, Switzerland
5Institute of Mathematical Sciences, Taramani, C.I.T. Campus, Chennai 600113, India
6Department of Physics, Columbia University, New York, NY 10027, USA
7Department of Physics, University of Durham, Ogen Center for Fundamental Physics, South Road,
Durham, DH1 3LE, UK
8Fermilab, Batavia, IL 605100500, USA
9LAL, Universit´ e ParisSud 11, Bˆ atiment 200, F91898 Orsay cedex, France
10PhysikDepartment T30d, Technische Universitaet Muenchen, JamesFranckStrasse, 85748 Garching,
Germany
11Departement de Physique Nucleaire et Corpusculaire (DPNC), Universite de Geneve, Geneve,
Switzerland
12MaxPlanckInstitut f¨ ur Kernphysik, Saupfercheckweg 1, 69117, Heidelberg, Germany
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64Also at: Theoretical Physics Department, Fermilab, Batavia, IL 605100500, USA.
65Alsoat:MaxPlanckInstitutf¨ urPhysik(WernerHeisenbergInstitut), F¨ ohringerRing6, 80805M¨ unchen, Germany.
00344885/09/000000+185$90.00
1
© 2009 IOP Publishing LtdPrinted in the UK
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
13Centre for Underground Physics in Pyh¨ asalmi, Universty of Oulu, Oulu, Finland
14Imperial College London, Blackett Laboratory, Department of Physics, Exhibition Road, London,
SW7 2AZ, UK
15Centre de Recherches Nucleaire, Hautes Energies, Institut National de Physique Nucleaire et des
Particules, Universite Louis Pasteur, Paris, France
16Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
17Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA
18Joint Institute for Nuclear Research, JoliotCurie 6, 141980, Dubna, Moscow Region, Russia
19Jyvaskyla University, Finland
20Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA
21Theory Group, KEK, 11 Oho, Tsukuba, Ibaraki 3050801, Japan
22KVI, Groningen, The Netherlands
23Laboratoire de Physique Theorique, LAPTH, F74941, AnnecyleVieux Cedex, France
24Lawrence Berkeley National Laboratory, Physics Division, 1 Cyclotron Road, 50R4049, Berkeley,
CA 947208153, USA
25CFTP and Departamento de Fisica, Instituto Superior Tecnico. Av. Rovisco Pais, 1049001 Lisboa,
Portugal
26Los Alamos National Lab, PO Box 1663, Los Alamos, NM 87545, USA
27IPN de Lyon, Universit´ e Lyon 1, CNRS, 4 rue Enrico Fermi, Villeurbanne, F69622 cedex France
28Departamento de Fisica Teorica (IFT), Facultad de Ciencias CXVI, Universidad Autonoma de Madrid,
UAMCSIC, Cantoblanco, Madrid 28049, Spain
29Tata Institute of Fundamental Research, School of Natural Sciences, Homi Bhabha Road, Mumbai
400005, India
30Universit` a Federico II and INFN Napoli, Italy
31Katholieke Universiteit Nijmegen, HEFIN  High Energy Physics Inst., PO Box 9010, NL6500 GL
Nijmegen, Netherlands
32Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA
33Northwestern University 633 Clark Street Evanston, IL 60208, USA
34Department of Physics, Osaka University, Toyonaka, Osaka 5600043, Japan
35Dept. di Fisica, Univ. di Padova, via Marzolo 8, I35100 Padua, Italy
36Department of Physics, University of Patras, GR26100 Patras, Greece
37Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
38INFN Pisa, Italy
39Departamento de F´ ısica, Pontif´ ıcia Universidade Cat´ olica Ddo Rio de Janeiro, C. P. 38071, 22452970,
Rio de Janeiro, Brazil
40INFN and Dipto. di Fisica, Universit` a degli Studi di Roma ‘La Sapienza’, P.le A. Moro 2, 00185, Rome,
Italy
41Departamento de Fisica, Universidad Catolica de Chile, Avenida Vicuna Mackenna 4860, Santiago,
Chile
42School of Physics and Astronomy, University of Southampton, Highfield, Southampton S017 1BJ, UK
43Department of Physics and Astronomy, University of Sussex, Falmer, Brighton, Sussex BN1 9QH, UK
44Department of Physics, National Cheng Kung University, Tainan 701, Taiwan
45Department of Physics, Technion, Haifa 32000, Israel
46Institute for Studies in Theoretical Physics and Mathematics, PO Box 193951795, Tehran, Iran
47Department of Physics, Tohoku University, Aobaku, Sendai, Miyagi 9808578, Japan
48Theory Group, Institute of Cosmic Ray Research, University of Tokyo, 515 Kashiwanoha, Kashiwa,
Chiba 2778582, Japan
49Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Japan 277
8568
50Berkeley Center of Theoretical Physics, University of California, Berkeley, CA 94720, USA
51Theoretical Physics Group, Lawrence Berkeley National Laboratory, MS 50A5104, Berkeley,
CA 94720, USA
52Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 1920397, Japan
53International School for Advanced Studies (SISSA/ISAS), via Beirut 24, I34100 Trieste, Italy
54INFN presso il Dipartimento di Fisica dell’Università di Trieste, Via Valerio, 2, I  34127 Trieste, Italia
55Abdus Salam International Centre for Theoretical Physics Strada Costiera 11, I34014 Trieste, Italy
56Instituto de F´ ısica Corpuscular, IFIC, CSIC and Universidad de Valencia, Spain
57Department of Physics, University of Warwick, Coventry, Warwickshire, CV4 7AL, England
58Institute for Nuclear Theory and Department of Physics, University of Washington, Seattle, WA 98195,
USA
59Department of Physics, College of William and Mary, PO Box 8795, Williamsburg, VA 231878795,
USA
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
60DepartmentofPhysics, UniversityofWisconsin, 2506SterlingHall, 1150UniversityAvenue, Madison,
WI 53706, USA
61Universit¨ atW¨ urzburg,LehrstuhlfurTheoretischePhysikII,AmHubland,D97074W¨ urzburg,Germany
62ETH, Z¨ urich, Switzerland
63PhysikInstitut der Universit¨ at Z¨ urich, Switzerland
Received 15 October 2008, in final form 4 January 2009
Published
Online at stacks.iop.org/RoPP/72
Abstract
The conclusions of the Physics Working Group of the international scoping study of a future
Neutrino Factory and superbeam facility (the ISS) are presented. The ISS was carried out by
the international community between NuFact05, (the 7th International Workshop on Neutrino
Factories and superbeams, Laboratori Nazionali di Frascati, Rome, 21–26 June 2005) and
NuFact06 (Ivine, CA, 24–30 August 2006). The physics case for an extensive experimental
programme to understand the properties of the neutrino is presented and the role of
highprecision measurements of neutrino oscillations within this programme is discussed in
detail. The performance of secondgeneration superbeam experiments, betabeam facilities
and the Neutrino Factory are evaluated and a quantitative comparison of the discovery
potential of the three classes of facility is presented. Highprecision studies of the properties of
the muon are complementary to the study of neutrino oscillations. The Neutrino Factory has
the potential to provide extremely intense muon beams and the physics potential of such
beams is discussed in the final section of the report.
(Some figures in this article are in colour only in the electronic version)
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Executive summary
TheinternationalscopingstudyofafutureNeutrinoFactoryandsuperbeamfacility(theISS)wascarriedoutbytheinternational
community between NuFact05, (the 7th International Workshop on Neutrino Factories and superbeams, Laboratori Nazionali
di Frascati, Rome, 21–26 June 2005) and NuFact06 (Ivine, CA, 24–30 August 2006). The physics case for the facility was
evaluated and options for the accelerator complex and the neutrinodetection systems were studied. The principal objective
of the study was to lay the foundations for a full conceptualdesign study of the facility. The plan for the scoping study was
prepared in collaboration by the international community that wished to carry it out: the ECFA/BENE network in Europe,
the Japanese NuFactJ collaboration, the US Neutrino Factory and Muon Collider collaboration and the UK Neutrino Factory
collaboration. STFC’s Rutherford Appleton Laboratory was the host laboratory for the study. The study was directed by a
Programme Committee advised by a Stakeholders Board. The work of the study was carried out by three working groups:
the Physics Group; the Accelerator Group; and the Detector Group. Four plenary meetings at CERN, KEK, RAL and Irvine
were held during the study period; workshops on specific topics were organized by the individual working groups in between
the plenary meetings. The conclusions of the study were presented at NuFact06. This document, which presents the Physics
Group’s conclusions, was prepared as the physics section of the ISS study group. More details of the ISS activities can be found
at http://www.hep.ph.ic.ac.uk/iss/.
Neutrino oscillations are the sole body of experimental evidence for physics beyond the Standard Model of particle physics.
The observed properties of the neutrino—the large flavour mixing and the tiny mass—are believed to be consequences of
phenomena which occur at energies never seen since the Big Bang. Neutrino facilities to pursue the study of oscillation
phenomena are therefore complementary to highenergy colliders and competitive candidates for the next worldclass facilities
for particle physics. Neutrino oscillations also provide a window on important issues in astrophysics and cosmology. Ongoing
and approved experiments utilize intense pion beams (superbeams) to generate neutrinos. They are designed to seek and
measure the third mixing angle θ13of neutrinomixing matrix (the ‘PMNS’ matrix), but will have little or no sensitivity to
matter–antimatter symmetry violation. Several neutrino sources have been conceived to reach high sensitivity and to allow the
range of measurements necessary to remove all ambiguities in the determination of oscillation parameters. The sensitivity of
these facilities is well beyond that of the presently approved neutrino oscillation programme. Studies so far have shown that
the Neutrino Factory, an intense highenergy neutrino source based on a storedmuon beam, gives the best performance over
virtually all of the parameter space; its time scale and cost, however, remain important question marks. Secondgeneration
superbeam experiments using megawatt proton drivers may be an attractive option in certain scenarios. Superbeams have
many components in common with the Neutrino Factory. A betabeam, in which electronneutrinos (or antineutrinos) are
produced from the decay of stored radioactive ion beams, in combination with a secondgeneration superbeam, may be a
competitive option.
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
The role of the ISS Physics Group was to establish the strong physics case for the various proposed facilities and to find
optimum parameters for the accelerator and detector systems from the physics point of view. The first objective of this report,
therefore, is to try to answer the big questions of neutrino physics; questions such as the origin of neutrino mass, the role that
neutrinos played in the birth of the universe and what the properties of the neutrino can tell us about the unification of matter
and force. These questions form the basis for the clarification of the physics cases for various neutrino facilities. Since it is not
(yet) possible to answer these questions in general, studies have concentrated on more specific issues that may lead to answers
to the big questions. In particular, studies have addressed such issues as:
1. The relevance of neutrino physics to the understanding of dark matter and dark energy, the connection between neutrino
mass and leptogenesis and galaxycluster formation;
2. The connection of predictions at the grandunification scale with lowenergy phenomena in the framework of seesaw
mechanism and supersymmetric extensions of the Standard Model and
3. The understanding of flavour and the connection between quarks and leptons, the possible existence of hidden flavour
quantum numbers that may be connected with the small mixings among the quarks, the mass hierarchy of the quarks and
charged leptons and the relationship of these phenomena with the neutrinomass matrix.
The second objective of this report is to review the predictions of the various models for the physics that gives rise to neutrino
oscillationsandtoreviewtheassociatedphenomenologythatisrelevantforprecisionmeasurementsofneutrinooscillations. For
this purpose, we have evaluated the degree to which the various facilities, alone or in combination, can distinguish between the
various models of neutrino mixing and determine optimum parameter sets for these investigations. A class of directlytestable
predictions is afforded by the fact that the GUT and family symmetries result in relationships between the quark and lepton
mixing parameters. These relationships can be cast in the form of sum rules. One example that can be used to discriminate
amongst various models is:
θ?
12≡ θ12− θ13cos(δ),
where θ?
small mixing angle and the cosine of the CPviolating phase) are measured experimentally. Another class of test is afforded by
the investigation of the unitarity of the PMNS matrix. While the quarkmixing (CKM) matrix is constrained to be unitary in
the Standard Model, the PMNS matrix, which originates from physics beyond the Standard Model, may not be exactly unitary;
this is the case, for example, in seesaw models. The third class of the test is the existence of flavourchanging interactions that
might appear at the production point, in the oscillations that occur during propagation, or at the point of detection. The possible
strong correlations between leptonflavour violation and neutrino oscillations are also discussed.
The potential of nonaccelerator, longbaseline neutrino oscillation measurements were also considered.
improvements in the precision with which the solar parameters are known could be made using a new longbaseline reactor
experiment or by using gadolinium loading of the water in the SuperKamiokande detector to increase its sensitivity to solar
neutrinos. A large, underground, magnetizediron detector could be used to improve the precision of the atmospheric mixing
parameters, to determine the octant degeneracy and to search for deviations from maximal atmospheric mixing.
The third, and key, objective of this report is to present the first detailed comparison of the performance of the various facil
ities. Using realistic specifications, we have estimated the likely performance, tried to find optimum combinations of facilities,
baselines and neutrino energies, and attempted to identify some staging scenarios. The cases considered are described in detail
in the main report; only a brief summary is given here. Although the Neutrino Factory can achieve very large data samples with
smallbackgrounds,itoperatesatenergiesconsiderablyhigherthanthefirstoscillationpeak(Emax/GeV = L/564km). Because
of this, at intermediate values of θ13(10−3? sin22θ13? 10−2) the Neutrino Factory with only one goldenchannel (νe→ νµ
and ¯ νe→ ¯ νµ) detector (at, say, 4000km) cannot resolve all parameter degeneracies and the precision of the measurement of
a particular parameter is reduced by correlations among the parameters. These problems can be resolved in one of three ways:
12can be predicted in classes of flavour models while θ12(the solar mixing angle) and θ13cos(δ) (the product of the
Significant
1. Placing a second detector at a different baseline (i.e. varying the ratio L/E);
2. Adding a detector sensitive to the silver channel (νe→ ντ) or
3. Using an improved detector with lower neutrinoenergy threshold and better energy resolution.
Possible configurations for each alternative, alone and in combination, were investigated to find an optimum performance of
the Neutrino Factory. It was shown that a considerable reduction of parent muon energy down to ∼25GeV is feasible without
a significant loss of oscillationphysics output, provided a detector performance improved with respect to the one assumed in
earlier studies can be achieved.
To make direct, quantitative comparisons of the various facilities, the GLoBES package was used. Three representative
superbeam configurations were considered: the SPL, a superbeam directed from CERN to the Modane laboratory; T2HK, an
upgrade of the JPARC neutrino beam illuminating a detector close to Kamioka, and the WBB, a wideband, onaxis beam
from BNL or FNAL to a deep underground laboratory in the US. Each superbeam was assumed to illuminate a megatonclass
water Cherenkov detector. The betabeam options considered were the CERN baseline scheme in which helium and neon ions
are stored with a relativistic γ of 100 and an optimized betabeam for which γ = 350. Two Neutrino Factory options were
considered: a conservative option with a single 50kton detector sited at a baseline of 4000km from a 50GeV Neutrino Factory;
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
and the optimized Neutrino Factory (see the full report) with two detectors, one at a baseline of 4000km and the second at
the magic baseline (∼7500km). The result of the comparisons may be summarized as follows: for the options considered, the
Neutrino Factory has the best discovery reach for sin22θ13followed by the betabeam and the superbeam, while the sin22θ13
reach for resolving the sign of the atmospheric mass difference is mainly controlled by the length of the baseline. For large
values of θ13(sin22θ13? 10−2), the three classes of facility have comparable sensitivity for the discovery of CP violation; the
best precision on individual parameters being achieved at the Neutrino Factory using optimized detectors. The reduction of
systematic uncertainties is the key issue at large θ13; by reducing systematic uncertainties, the superbeam may be favourably
compared with the conservative Neutrino Factory. For intermediate values of θ13(10−3? sin22θ13? 10−2), the superbeams
are outperformed by the betabeam and the Neutrino Factory and the best CP coverage is achieved by the betabeam. For small
values of θ13(sin22θ13? 10−3), the Neutrino Factory outperforms the other options. Note that the comparisons are made using
three performance indicators only (sin22θ13, the sign of mass hierarchy and the CPviolating phase δ). If other physics topics,
such as the search for e,µ,τ flavour anomalies, were to be emphasized, the relative performance may be different.
The final contribution to this report reviews the muon physics that can be performed with the intense muon beams that will
be available at the Neutrino Factory. The study of rare, leptonflavourviolating processes in muon decay, and the search for a
permanentelectricdipolemomentofthemuon,arecomplementarytoprecisionstudiesofneutrinophysics;oftensensitivetothe
sameunderlyingphysics. ThecomplementarityandthepotentialofamuonphysicsprogrammeattheNeutrinoFactoryisinves
tigated. It will be important in the coming years to establish quantitatively the synergy between muon physics and the study of
neutrinooscillationsandtodevelopaplanforthecoexistenceofmuonandneutrinoprogrammesattheNeutrinoFactoryfacility.
A significant amount of conceptual design work and hardware R&D is required before the performance assumed for each
of the facilities can be realized. Therefore, an energetic programme of R&D into the accelerator facilities and the neutrino
detectors must be established with a view to the timely delivery of conceptual design reports for the various facilities.
Contents
1. Introduction
1.1. Neutrino in a nutshell
1.2. Neutrino physics as part of the highenergy
physics programme
1.3. Implications and opportunities
1.4. Precision measurements and sensitive searches
1.5. What the study tried to achieve
2. The standard neutrino model
2.1. Introduction
2.2. Review of the present generation of experiments 15
2.2.1. Solar and reactorneutrino experiments 15
2.2.2.Atmosphericneutrino experiments
2.2.3. Longbaseline
experiments
2.2.4.
0νββ experiments
2.2.5. Recent progress in measurements of
neutrino oscillations
2.3. Completing the picture
2.3.1. Bounds on θ13from approved
experiments
2.4. Degeneracies and correlations
2.4.1. Appearance channels: νe → νµ,ντ
and νµ→ νe
2.4.2.Disappearance channels: νµ→ νµ
2.4.3. A matter of conventions
2.4.4.Disappearance channels: νe→ νe
3. Implications for new physics and cosmology
3.1. The origin of small neutrino mass
3.1.1. Seesaw mechanisms
3.1.2.Supersymmetry and Rparity violation
3.1.3. Extra dimensions
3.1.4.String theory
3.1.5. TeVscale mechanisms for small
neutrino masses
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7
7
8
10
11
12
12
16
neutrinooscillation
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20
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3.2. Unification and flavour
3.2.1. Model survey
3.2.2. Sum rules
3.2.3. Cabibbo haze in lepton mixing
3.3. Leptonflavour violation
3.4. Cosmology
3.4.1. Neutrinos and largescale structure
3.4.2.Leptogenesis
3.4.3.Neutrinos and inflation
4. Effects of new physics beyond the Standard
Neutrino model
4.1. Sterile neutrinos
4.1.1. Theoretical issues
4.1.2.Phenomenologyoflightsterileneutrinos 58
4.1.3.Signatures of heavy sterile neutrinos
4.1.4. Sterile neutrinos and cosmology and
astrophysics
4.1.5.The LSND challenge
4.2. Mass varying neutrinos
4.3. CPT and Lorentzinvariance violation
4.3.1.Direct bounds on CPTV
4.3.2. CPTV/LIV effect on conversion
probability
4.3.3. Future prospects
4.4. Leptonic unitarity triangle and CP violation
4.4.1. Properties of the leptonic triangles
4.4.2. Leptonic triangles and coherence of
neutrino states
4.4.3. The unitarity triangle and oscillation
experiments
4.4.4.Leptonic unitarity triangle and future
experiments
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58
58
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
4.4.5.
4.4.6.
Beyond three neutrinos
Constraints on unitarity
73
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75
4.5. Nonstandard interactions
4.5.1. Nonstandard interactions in
production and detection
4.5.2. Nonstandard interactions in
propagation
4.5.3. Constraints from nonoscillation
neutrino experiments
4.5.4. Oscillation experiments as probes of
the NSI
4.5.5.The role of MINOS
4.5.6.Complementarity of long and short
baseline experiments
5. Performance of proposed future longbaseline
neutrino oscillation facilities
5.1. Introduction
5.1.1. Definition of observables
5.2. The physics potential of superbeams
5.2.1. The superbeam concept
5.2.2.T2K and T2HK
5.2.3. The SPL
5.2.4.NOνA
5.2.5. Wideband superbeam
5.2.6.Physics at a superbeam facility
5.2.7.The waterˇCerenkov detector
5.2.8. Backgrounds and efficiencies
5.2.9. The superbeam performance
5.2.10. The θ13discovery potential
5.2.11. CPviolation discovery potential
5.2.12. Maximal θ23exclusion potential
5.2.13. Sensitivity to the atmospheric
parameters
5.2.14. Sensitivitiestothemasshierarchyand
the θ23octant
5.2.15. Combination with atmospheric
neutrino measurements
5.2.16. Superbeamassociatedwithabetabeam100
5.2.17. Superbeam associated with the
Neutrino Factory
5.3. The physics potential of betabeam facilities
5.3.1. Betabeam setups
5.3.2. The lowenergy betabeam: LEββ
5.3.3. Highenergy betabeams: HEββ
5.3.4. Ion production and ν fluxes
5.3.5.Detector technology
5.3.6. WaterˇCerenkov
5.3.7.NOνAlike detector
5.3.8. Atmospheric backgrounds
5.3.9. Analysis of performance and
optimization
5.3.10. Sensitivity to θ13
5.3.11. Sensitivity to CP violation
5.3.12. Sensitivity to the discrete ambiguities
5.3.13. Measurement of θ13and δ
5.3.14. Towards an optimal betabeam setup
5.3.15. Combination with atmospheric data
5.3.16. An associated superbeam
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5.3.17. Combination of different ions
5.3.18. Higher γ?
5.3.19. Higher fluxes?
5.3.20. Monocromatic ecapture beams
5.4. Optimization and physics potential of a
Neutrino Factory oscillation experiment
5.4.1. The Neutrino Factory setup
5.4.2. Physics potential of the golden channel 117
5.4.3.Solving degeneracies
5.4.4.The optimal Neutrino Factory
5.4.5.Lowenergy Neutrino Factory
5.5. Comparisons
6. The potential of other alternatives
6.1. Solar and reactorneutrino experiments
6.1.1. The Generic pp experiment
6.1.2.The SKGd reactor experiment
6.1.3. The SPMIN reactor experiment
6.2. Atmosphericneutrino experiments
6.2.1.Is the mixing angle θ23maximal?
6.2.2.Resolving the θ23octant ambiguity
6.2.3.Resolving the
neutrinomass hierarchy
6.3. Neutrino mass hierarchy from future
0νββ experiments
6.4. Astrophysical methods of determining the
mixing parameters
6.4.1. General remarks about astrophysical
neutrinos
6.4.2. Unstable neutrinos arriving from
cosmic distances
6.4.3.Stable neutrinos and loss of coherence 146
6.4.4.Summary
7. Muon physics
7.1. Introduction
7.2. The magnetic and electricdipole moments of
the muon
7.3. Search for muonnumber violation
7.3.1. Theoretical considerations
7.3.2.Modelindependent analysis of rare
muon processes
7.4. Experimental prospects
7.4.1.
µ → eγ
7.4.2.
µ+→ e+e+e−
7.4.3.
µ → e conversion
7.4.4. Muonium–antimuonium conversion
7.5. Normal muon decay
7.5.1.Theoretical background
7.5.2.Muonlifetime measurements
7.5.3.Precision measurement of the Michel
parameters
7.5.4. Experimental prospects
7.6. Muonphysics conclusions
Acknowledgments
Appendix A. Origin of the ISS and its Committees
Appendix A.1. Origin
Appendix A.2. Committee
References
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ambiguity in the
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
1. Introduction
1.1. Neutrino in a nutshell
Elusive, mysterious, yet abundant. Neutrinos are elementary
particles, just like the electrons in our bodies. Neutrinos are so
elusive that we do not feel ten trillions of them going through
our body every second. They were discovered 50 years ago,
butstillposemanymysteries,defyingoureffortstounderstand
them due to their elusiveness. Yet neutrinos are the most
numerous matter particles in the universe; there are about a
billion neutrinos for every single atom.
Slowly we began to appreciate the important roles that
neutrinos have played in shaping the universe as we see today.
We already know that stars would not burn without neutrinos.
Neutrinos played an important role in producing the various
chemical elements that we need for daily life. Given that we
(atoms) are completely outnumbered by neutrinos, it is quite
certain that they played even more important roles.
There was a major surprise 8 years ago when we
discovered that neutrinos have a tiny, but nonzero, mass quite
against the expectations of our best theory. This discovery
opened up new important roles for neutrinos. We are all of a
sudden grappling with new exciting questions about neutrinos
that may lead to revolutionary understandings on how the
universe came to be.
Because they have mass, neutrinos may have played an
importantroleinshapingthegalaxies,andeventuallystarsand
planets. Neutrinos may actually be their own antiparticles;
this may be the reason why the Universe did not end up empty
but has atoms in it. Neutrinos seem to be telling us profound
facts about the way matter and forces are unified, and how the
three types of neutrino are related to each other; we have yet
to decipher their message. In addition, neutrinos may actually
be the reason why the universe exists at all.
We are only beginning to understand neutrinos and their
rolesinhowtheuniverseworks. Itwilltakemanyexperimental
approaches to get the full picture.
discussed here will most likely be an essential component of
this programme.
A Neutrino Factory
1.2. Neutrino physics as part of the highenergy physics
programme
Thepresentisaveryinterestingtimeinthefieldoffundamental
physics: over the past four decades, an impressive theoretical
framework, the Standard Model, has been established. The
Standard Model is capable of explaining how nature works
at the smallest, experimentally accessible distance scales;
yet a handful of phenomena seem decisively to elude an
explanation within the Standard Model and are therefore clues
to a more fundamental understanding. These observations
provide the only clues we have that our understanding of
fundamental physics is incomplete. The experimental and
theoreticalpursuitofthesecluesdrivesthehighenergyphysics
programme and is likely to guide the bulk of the research
in this area over the coming decades.
section, the forces currently driving research in fundamental
physics are discussed and the possible interplay between the
In this brief sub
component parts of the research programme are investigated,
the objective being to establish the context within which
the future experimental neutrinophysics programme must be
developed.
Nonzero neutrino masses cannot be explained by the
Standard Model. To allow for massive neutrinos, the Standard
Model must be modified qualitatively.
distinct ways in which the Standard Model can be modified to
accommodate neutrino mass, some of which will be discussed
in detail in the next section.
knowledge of the properties of the neutrino property does not
allow us to choose a particular ‘new Standard Model’ over all
the others. We do not know, for example, whether neutrino
masses are to be interpreted as evidence of new, very light,
fermionic degrees of freedom (as is the case if the neutrinos
areDiracfermions),new,veryheavy,degreesoffreedom(asis
the case if the canonical seesaw mechanism is responsible for
tinyMajorananeutrinomasses)orwhetheramorecomplicated
electroweaksymmetrybreaking sector is required. To make
progress,itisimperativethatnewprobesofneutrinoproperties
be vigorously developed. This is the main driving force of all
experimental endeavours discussed in this study.
According to the Standard Model, the Lagrangian of
nature is invariant under an SU(3)c× SU(2)L× U(1)Ylocal
symmetry, but the quantum numbers of the vacuum are such
thatthisgaugesymmetryisspontaneouslybrokentoSU(3)c×
U(1)EM. The physics responsible for this electroweak
symmetry breaking is not known.
states that electroweak symmetry breaking arises due to the
dynamicsofascalarfield—theHiggsfield. WhiletheStandard
Model explanation for this phenomenon is in (reasonable)
agreement with precision electroweak measurements, the
definitive prediction—the existence of a new, fundamental
scalar boson, the Higgs boson—has yet to be confirmed
experimentally.
Evenifthestandardmechanismofelectroweaksymmetry
breaking is realized in nature, our theoretical understanding of
particle physics strongly hints at the possibility that there are
more degrees of freedom at or slightly above the electroweak
symmetrybreaking scale (around 250GeV). Furthermore, it
is widely anticipated that these new degrees of freedom will
serve as evidence of new organizing principles; examples of
such principles include supersymmetry and the existence of
new dimensions of space.
Thepursuitofthemechanismresponsibleforelectroweak
symmetry breaking is the driving force behind the current
andthefuturehighenergycolliderphysicsprogramme,which
aims at exploring the highenergy frontier.
future, the large hadron collider (LHC) is expected to supplant
the ongoing Tevatron collider as the highest energy particle
accelerator in the world. It is widely expected that the
LHC will reveal the mechanism of electroweak symmetry
breaking and provide evidence of new heavy degrees of
freedom. Anticipating the potential findings of the LHC
basedexperiments,thecolliderphysicscommunityiscurrently
planningahighintensity,highprecision,highenergyelectron
collider—the International Linear Collider (ILC). The ILC
should be able to study in detail the electroweaksymmetry
There are several
Our current experimental
The Standard Model
In the near
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
breaking sector and reveal the properties of the new physics at
the electroweak scale.
Finally, several very different but equally impressive
measurements of the massenergy budget of the Universe have
revealed beyond reasonable doubt the existence of what is
referredtoas‘darkmatter’. Afterseveralyearsofobservation,
it is now clear that the Standard Model does not contain the
degreesoffreedomnecessarytoexplainthedarkmatter. Other
thanitsgravitationalproperties,verylittleisknownaboutdark
matter. It could consist of weakly interacting fundamental
particles, but it may also consist of very heavy, very weakly
interacting states or more exotic objects.
searches for dark matter are currently among the highest
priorities of the fundamental physics research programme.
Experiments that are sensitive to dark matter vary from direct
detection experiments, to neutrino telescopes, to gammaray
observatories. The hope is that the pursuit of the darkmatter
clue will not only reveal the existence of a new form of
matter but will also provide other clues that will allow a more
satisfying understanding of the composition of the universe
and its behaviour in the first instants after the Big Bang to be
developed.
Ontopofthedarkmatterproblem, wearenowfacedwith
a seemingly deeper puzzle—the existence of dark energy. We
arestillfarfromproperlydecodingwhatthispuzzlemeans,and
itisnotclearhowprogresswillbemadetowardsresolvingthis
most mysterious issue. New experiments are being devised to
study in more detail the properties of dark energy. The results
of these experiments may play a large role in modifying our
picture of how nature works at its most fundamental level.
The different probes discussed above not only address
different clues regarding new fundamental physics, but also
complete and complement oneanother.
synergy among the different experiments cannot be over
emphasized.Consider the following examples of such
synergies:
Experimental
The amount of
1. While new physics at the electroweak scale is usually best
studiedusingahighenergycollider,thereareseveralnew
physics phenomena that will only manifest themselves in
neutrino experiments, including new light, veryweakly
coupled degrees of freedom that could be related to dark
matter or dark energy;
2. The knowledge of neutrino properties is essential for
the understanding of certain darkmatter searches (for
example those performed using neutrino telescopes);
3. A highenergy collider may provide the only means of
studyinginanydetailthepropertyofdarkmatterparticles
and
4. A proper understanding of the origin of neutrino mass
can only be obtained after the mechanism of electroweak
symmetry breaking is properly understood.
It is important to bear in mind that we do not know
what the next set of clues will be, or where they will come
from. It may turn out, for example, that neutrino experiments
provide our only handle on grand unification and other types
of very high energy physics, or that astrophysical searches for
the properties of dark energy will reveal a direct window on
quantum gravity. Or it may turn out that collider experiments
willbeabletostudydirectlystringtheoreticaleffects(thismay
be the case if there really are large extra dimensions). Only a
comprehensive pursuit of the questions that we can formulate
today will allow us to reach the next stage in understanding
fundamentalphysics—andaskanewsetofmorefundamental,
deeper questions tomorrow. In this sense, we perceive the
physicsdiscussedheretobeonequalfootingwithotherstudies
of fundamental importance to our field, including the direct
searches for dark matter, satellite missions that will measure
the acceleration of the universe or collider experiments at the
energy frontier. These are all different, complementary ways
of addressing the different questions that we cannot answer
given our current understanding of fundamental physics.
1.3. Implications and opportunities
Fundamentalfermions,areclassifiedinthreegenerations,each
generation containing six quarks (two flavours, three colours)
and two leptons. The measured properties of these particles
exhibit a clear ‘horizontal’ hierarchy in which the mass of
fermionscarryingthesameStandardModelquantumnumbers
increases with generation number. Within a generation, the
fermionpropertiesalsoexhibit‘vertical’patterns,forexample,
the sum of the electric charge of the members of a particular
generation is zero. The quarks come in three colour varieties,
the source of the strong force. Under the weak force, both the
quarksandtheleptonswithinaparticulargenerationtransform
as a doublet. In contrast to the general expectation, the mixing
angles among lepton flavours have turned out to be different
to the quarkmixing angles. Many of the properties of the
neutrino are unique, not shared by the other fundamental
fermions. Firstly, it has neither colour nor electric charge;
hence it is the only fundamental fermion that feels solely the
weak force in addition to the gravitational force. This fact
becomes important when cosmological impact of the neutrino
is discussed. Secondly, neutrino masses are tiny compared
with the masses of all other fundamental fermions. Thirdly,
the neutrino could be a Majorana particle; a fermion which
cannot be distinguished from its own antiparticle66.
The physics of flavour seeks to provide an explanation of
these observed patterns. The vertical patterns noted above can
beexplainedin‘GrandUnifiedTheories’(GUTs)inwhichthe
fermions are assigned to representations of a large symmetry
group such as SO(10). The horizontal, or family patterns,
can be explained by assuming a family symmetry such as
SU(3)family. Some models that incorporate GUT and family
symmetries with supersymmetric extensions come within the
realm of string theories that incorporate extra dimensions.
Understandingthesymmetrystructureseemstobeapromising
strategy to arrive at a description of the physics of flavour.
Neutrino oscillations are a phenomenon in which the
neutrino changes flavour as it propagates. It was predicted by
Pontecorvo[2,3]andMakietal [4]andthefirstclearevidence
for neutrino oscillations was presented by SuperKamiokande
in 1998 in observations of the zenithangle distributions of
66The observation of doublebetadecay processes in which no neutrino is
produced would imply that the neutrino is its own antiparticle [1].
8
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 1. The ratio of the measured to the predicted neutrino flux is plotted as a function of L/E. The antielectronneutrino contribution to
the reactor neutrino flux measured by the KamLAND collaboration [8] is shown. (Figure courtesy of K Inoue).
atmospheric neutrinos [5].
dates back as early as 1970, when the Homestake group
detected a deficit in the solarneutrino flux compared with
that predicted by the Standard Solar Model [6]. The long
standing ‘solar neutrino puzzle’ was finally proved to be a
resultofoscillationsbySNOin2001[7]. Thefirstobservation
of neutrino oscillations from terrestrial neutrino sources was
obtained by KamLAND by measuring the energy spectrum
of neutrinos produced in nuclear reactors [8].
of the KamLAND measurement, shown in figure 1, exhibits
the expected oscillatory behaviour and constitutes compelling
evidence for neutrino oscillations [9].
Neutrinooscillationsoccurbecauseofflavourmixingand
the tiny, but different, masses of the neutrinos.
saw mechanism, the most attractive and promising scheme
to explain the tiny mass, requires the presence of very heavy
Majorana neutrinos. In such models, neutrino oscillations are
a consequence of the physics which pertains at an extremely
large energy scale.Seesaw models are able to explain
the striking difference between the quark and leptonmixing
angles in a natural way. If the heavy Majorana neutrino
is abundant in the early Universe and decays preferentially
into matter leptons in a CP violating process, then the lepton
asymmetry would be converted into a baryon asymmetry a
split second later during the electroweak era. This process is
referred to as ‘leptogenesis’ and is a primary theory to explain
our matter universe. The neutrino is the most abundant of the
matter fermions in the Universe; with a billion neutrinos for
each of the other known matter particles, only the ubiquitous
photon is more abundant. Hence, the tiny neutrino mass could
contribute a nonnegligible fraction of the dark matter and is
knowntoplayanimportantroleintheformationoflargescale
structure in the Universe.
Because of its direct connection with phenomena at
energies never seen since the Big Bang, the precise
determination of the masses and mixing angles of the three
families of neutrino is a unique window onto these early
times and provides a path to the possible unification of all
The first indication, however,
The result
The see
forces. Measurements of neutrino oscillations can be used to
determine the three mixing angles and the CPviolating phase
of the leptonmixing matrix (the PMNS, Pontecorvo–Maki–
Nakagawa–Sakata matrix) and two masssquared differences.
Examining neutrino oscillations is a most direct way to
distinguishbetweenthevariouspossibletheoriesofthephysics
of flavour and to understand the origin of neutrino mass. It is
also a logical place to seek for the origin of the CP violation.
Taking a different perspective, ever since Pauli’s 1931
prediction, the unveiling of the properties of the neutrino has
always heralded a new epoch in the history of elementary
particle physics. Including the neutrino as a player of beta
decay, Fermi formulated the first successful theory of weak
interactions. The absence of righthanded neutrinos has
manifested itself as the ‘V–A’ structure of the weak interaction
and the pursuit of its origin led to the discovery of the
chiral gauge theory which forms the foundation of modern
particle theories. The realization of intense neutrino beams
immediately resulted in the discovery of the neutral current,
establishing the unification of the electroweak interactions.
The ability of neutrino interactions to distinguish flavour and
handedness has been extensively utilized in deep inelastic
interactions to clarify the structure of the nucleon and
to establish the asymptotic freedom of QCD. The recent
discoveryofneutrinooscillationscouldberegardedasanother
epochmaking observation. So far it is the only experimental
evidence for, and a vital clue to, the physics beyond the
Standard Model. Both the mysteries, and the brilliant record,
oftheneutrinocanbeattributedtoitsuniqueandcharacteristic
insensitivity to both the strong and the electromagnetic forces.
Thereareamplereasonstobelievethatthisassetremainsvalid
in uncovering the veils that surround the neutrino. Neutrino
facilities to pursue oscillation phenomena are complementary
to highenergy colliders and are competitive candidates for the
next worldclass facilities for particle physics.
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
1.4. Precision measurements and sensitive searches
Experimentally, there are several different approaches to
elucidate the properties of the neutrino. In this report we
concentratemainlyonacceleratorbasedfacilitiesilluminating
massive, underground detectors. Other complementary means
are also taken into account and are described in section 6.
Among them, reactorbased oscillation experiments may play
a crucial role in untangling the degeneracies inherent in
oscillation measurements. Other techniques include double
beta decay, a unique tool to test the Majorana nature of the
neutrino [1]. Massive, underground detectors, while serving
as a far detector for the oscillation experiments, also have an
important role in their own right as telescopes for neutrino
astronomy and as a possible window on grand unified theories
by way of searching for proton decay.
Theories that purport to explain neutrino oscillations have
consequences for the properties of the charged leptons, such
as flavourchanging process in lepton decay or leptoninduced
reactions. Considering that very intense muon beams will be
available as a byproduct of the Neutrino Factory, it is natural
to include muon physics as an indispensable ingredient of the
study. Section7discussesindetailtheopportunitiesthathigh
statistics studies of the properties of the muon have to offer.
The present generation of neutrinooscillation experi
ments [10–12], reviewed in section 2, are designed to measure
the smallest neutrinomixing parameter if it is not ‘too small’.
Theyutilizeintensepionbeams(superbeams)togenerateneu
trinos. They are designed to seek and measure the third mix
ing angle θ13of the PMNS matrix, but will have little or no
sensitivity to matter–antimatter symmetry violation. Several
neutrino sources, including secondgeneration superbeams,
betabeams and the Neutrino Factory have been envisaged to
reach high sensitivity and redundancy well beyond that which
can be achieved in the presently approved neutrinooscillation
programme. Section 5 reviews the detailed performance of
each of these classes of facility and presents quantitative com
parisons of the physics potential. Their essential features are
briefly introduced below.
The superbeam is a natural extension of the conventional
neutrino beam and the current and approved experiments are
mostly of this type [13–23]. The neutrino beam is produced
through pion and kaon decay and hence these facilities
provide beams in which νµ and ¯ νµ dominate the neutrino
flux. However, these beams also contain νe(¯ νe) from kaon
and muon decay which constitute an irreducible background
for the oscillation signal νµ(¯ νµ) → νe(¯ νe).
the selection of samples of νe(¯ νe) is prone to neutral current
contamination. The principal source of systematic uncertainty
arises from the fact that the spectral shape of the pions and
kaons is not well known. In the secondgeneration super
beam experiments, the emphasis is on large detector mass,
i.e. the collection of large data samples, and on muon and
electron particle identification. This detector solution most
often adopted for secondgeneration superbeam experiments
is the megatonscale water Cherenkov counter. Liquidargon
detectors or large volume scintillator detectors have also been
considered.
In addition,
The betabeam [24], in which electronneutrinos or (anti
neutrinos) are produced from the decay of stored radioactive
ionbeams,providesessentiallybackgroundfreepure‘golden
channel’ (νe → νµ), i.e. ‘the appearance of wrongsign
muons’. Unlike the Neutrino Factory, the betabeam does not
need a magnetized detector, because there is no contamination
from antineutrinos. This allows the betabeam to use a very
massive detector just as the secondgeneration superbeam
does [25,26]. Simultaneous operation of the betabeam and
the secondgeneration superbeam has also been considered
[27]. The disadvantage of the betabeam is lack of the ‘silver
channel’, the νe→ ντ, transition for most of the case studied.
The Neutrino Factory [28,29], an intense highenergy
neutrino source derived from the decay of a storedmuon
beam, has access to all channels of neutrinoflavour transition
including the golden channel.
induced muon–neutrino events requires that the detector be
magnetized. This leads most naturally to the magnetized iron
calorimeter design. Another unique feature of the Neutrino
Factory is the possibility to observe the silver channel. This
can be achieved using either emulsion based detectors or a
magnetized liquidargon timeprojection chamber.
Studies [30–35] so far have shown that the Neutrino
Factory gives the best performance over virtually all of
the parameter space; its time scale and cost remain,
however, important question marks. Superbeams have many
components in common with the Neutrino Factory. A beta
beam may be competitive with the Neutrino Factory in some
parameter space, but, being relatively new in this field, needs
further study to fully explore its capability.
Thereisanimportantissuecommontoallthefacilitiesthat
mustbeborneinmind. Atypicaloscillationexperiment,trying
to determine the small mixing angle θ13and the CPviolating
phase δ, generally suffers from the correlation/degeneracy
problem, described in detail in section 2.4. These correlations
and degeneracies reduce the sensitivity typically by one order
of magnitude over that given by the statistical and systematic
uncertainties. This happens because a canonical oscillation
experiment measures only two transition rates, P(να → νβ)
and P(¯ να → ¯ νβ) and the expression for these probabilities
is a quadratic function of two unknown variables, sin2θ13
and sinδ. Given two measured values at fixed energy, E,
and baseline, L, the solution of the equations has an extra,
fake, (θ13,δ) solution which is referred to as the intrinsic
degeneracy. Our ignorance of the sign of ?m2
degeneracy), andtheindistinguishabilityofθ23fromπ/2−θ23
(theoctantdegeneracy)resultsinatotaleightfolddegeneracy.
The problem can be resolved in one of three ways:
However, to reject beam
23(the sign
1. To place a second detector at different values of L/E;
2. To add a different channel (or to combine data from
a complementary source, for example from a reactor
experiment) and
3. To use an improved detector with lower threshold and
better energy resolution.
Method (3) may be regarded as a variant of (1) from the
physics point of view because it is essentially equivalent to
widening the energy spectrum. This is the reason why the
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
consideration of synergy among the proposed, as well as the
current experiments, is particularly important in oscillation
physics. More often than not, the combination of experiments
of different design can achieve a sensitivity that far exceeds
what a mere statistical gain would suggest.
NOvA [16] expects to enhance its sensitivity by combining
withaproposedreactorexperimentandanupgradedversionof
NOvA [17,18] proposes to put a second detector at a different
offaxis angle (i.e. energy). T2KK [19,20], a variation of
T2HK [13], proposes to split their megaton water detector
in two and place one in Korea, a remedy using two different
baselines (L = 295km and ?1050km). The onaxis WBB
[23,36,37], averylongbaselinewidebandbeamfromFNAL
or BNL to Henderson or Homestake mine in the US, on the
other hand, takes advantage of its wide spectrum to resolve
the problem. It has also been shown that the combination
of atmosphericneutrino data with T2HK [38] or a low
energy betabeam [27] is extremely helpful in resolving the
degeneracies related to the mass hierarchy and the octant
degeneracy. These examples illustrate the importance of
workingtowardstheidentificationofanoptimumcombination
of the various facilities.
For all detector concepts, there are important questions
concerning cost, feasibility and time scales. In addition, there
are design optimizations to be made, e.g. between energy and
angle resolution, optimum baseline length and detector mass.
The study of detector concepts for the near detector stations
will be an important aspect of the neutrino physics because of
its access to many reactions complementary to the oscillation
process.
For the neutrinophysics community to arrive at a
consensusonthebestpossibleneutrinooscillationprogramme
to follow the present generation of experiments requires
a detailed evaluation of the performance and cost of the
various options and of the timescale on which each can be
implemented. Further R&D programmes into the accelerator
systems and the neutrino detectors must be carried out.
For instance,
1.5. What the study tried to achieve
TheroleofthePhysicsGroupoftheISSstudywastoestablish
the strong physics case for the various proposed facilities and
to find the optimum parameters of the accelerator facility and
detector systems from the physics point of view. The first
objective of this report, therefore, is to try to identify the big
questions of neutrino physics such as the origin of neutrino
mass, the role of the neutrino in the birth of the universe, what
the properties of the neutrino can tell us about the unification
of matter and force. These questions lay down the basis for
making the physics case for the various neutrino facilities.
Since it is not (yet) possible to answer these questions in
general, studies have concentrated on more specific issues that
may lead to answers to the big questions. A class of directly
testablepredictionsisaffordedbythefactthatGUTandfamily
symmetries result in relationships between the quark and
leptonmixing parameters; such relationships can be cast in
the form of sum rules.
Thesecondobjectivewastolookforpossiblecluesofnew
physics in a ‘bottomup’ approach. For this purpose, we have
evaluated the degree to which the various facilities, alone or
in combination, can distinguish between the various models
of neutrino mixing and determined optimum parameter sets
for these investigations. One example is to search for the
existenceofasterileneutrino. Althoughtheanomalypresented
by LSND [39] was not confirmed by MiniBooNE [40], the
questionisimportantenoughtobepursuedfurther. Thesecond
example is the unitary triangle: while the CKM matrix in the
quarksectorisconstrainedtobeunitaryintheStandardModel,
thePMNSmatrixoriginatesfromphysicsbeyondtheStandard
Modeland,inseesawmodels,maynotbeexactlyunitary. The
third example is the existence of flavourchanging interactions
that might appear at the production point, in the oscillation
stage or at the detection point. Possible strong correlations
between leptonflavour violation and neutrino oscillations
were also discussed. Other approaches to the determination
of the threeflavour parameters (i.e. nonaccelerator based
measurements) were also considered.
possibility of a new longbaseline reactor experiment and the
loading of the water in the SuperKamiokande detector with
gadolinium to improve the solarneutrino parameters, or an
large, underground, magnetizediron detector to improve the
atmosphericneutrinoparametersandtotestfordeviationfrom
maximal mixing and determine the octant degeneracy were
also discussed.
Thethird, andthekey, objectiveofthisreportistopresent
the first detailed comparison of the performance of the various
facilities. Utilizing realistic specifications, we have estimated
likely performances, tried to find an optimum combination of
facilities, baselinesandneutrinoenergies, andtocomeupwith
some staging scenarios.
Although past studies have shown that the Neutrino
Factory can be considered as an excellent, and perhaps as
an ultimate, facility, many questions remain open.
instance, the performance of the Neutrino Factory at large
θ13(sin22θ13 ? 10−2) where most superbeam experiments
work is only now being studied in detail [41]. A question
that must therefore be asked is: ‘Can the Neutrino Factory
remain competitive if θ13turns out to be large?’. Another
concern is the cost of the accelerator facility and the detector
systems. One estimate [32] in previous studies gives a total
cost of 1500 M$ + 400 M$ × E/20(GeV). Therefore, the
second question is: ‘What is the minimum energy that will
deliver the physics?’. The Neutrino Factory operates at
energies considerably higher than the first oscillation peak
(Emax/GeV = L/564km). Thecanonicaloperatingcondition
in past studies has been to use a parent muon beam of 50GeV
and a 50kton magnetizediron detector at a distance of 3000–
4000km [42]. Because of its operation at high energy with
a single detector, it suffers from the degeneracy problem at
intermediate values of θ13(10−3? sin22θ13? 10−2). It has
been shown that remedies exist through the addition of either
a second detector at the ‘magic’ baseline (L ? 7500km) [43]
or the silver channel [44]. Both of these solutions require the
seconddetector. So,thethirdquestionis‘Canasingledetector
configuration with improved performance do any better, and if
twodetectorsareunavoidable,whichcombinationisthebest?’.
In order to answer those questions, an extensive investigation
For example, the
For
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in the parameter space (Eµ− L) was carried out.
various combinations have been compared with the intention
to identify both a conservative option and an improved set of
detector configurations with possible staging.
Direct, quantitative comparison of the various facilities
is a highlight of the study. The GLoBES package [45,46]
was used. Other codes, Valencia and Madrid, showed good
agreement with GLoBES in a test using a single reference
input. A realistic set of detector specifications and a
precise normalization of neutrino flux and crosssections were
prepared. The comparisons are made for three performance
indicators only (sin22θ13, the sign of mass hierarchy and
the CP violation phase δ). If other physics topics, such as
e,µ−τ flavouranomalysearches,areemphasized,therelative
importance may be different.
The final contribution to this report reviews the muon
physics that can be performed with the intense muon beams
that will be available at the Neutrino Factory. The study of
rare processes in muon decay and muonelectron and muon
nucleon scattering is complementary to precision studies of
neutrino oscillations; often sensitive to the same underlying
physics. The complementarity and the potential of a muon
physics programme at the Neutrino Factory is investigated. It
willbeimportantinthecomingyearstoestablishquantitatively
thequalitativesynergybetweenmuonphysicsandthestudyof
neutrino oscillations.
This report is organized as follows. First, in section 2,
we give a review of the present generation of experiments,
state what is needed to complete the picture and explain
the degeneracy problem. Next we expand upon the physics
motivationfortheneutrinooscillationprogrammeinsections3
and4; section3containsa‘bigpicture’descriptionofneutrino
physics addressing such questions as the origin of neutrino
mass, extra dimensions, flavour symmetry and the role of the
neutrino in unification and in cosmology, while section 4 takes
a phenomenological approach to consider how measurements
of neutrino properties may provide clues to new physics
through studies such as the search for sterile neutrinos, the
investigation of the leptonic unitary triangle and the search
for nonstandard interactions in the oscillation experiments.
Section 5 deals with the physics potential of the proposed
facilities: the superbeam; the betabeam and the Neutrino
Factory. Direct comparison of various facilities is given here.
Alternativeexperimentswhichcancomplementtheoscillation
experiments are described in section 6. The final section 7 is
devoted to muon physics.
Then,
2. The standard neutrino model
2.1. Introduction
The ‘standard neutrinomixing model’ emerged as a result
of the remarkable progress made in the past decade in the
studies of neutrino oscillations. The experiments with solar,
atmospheric and reactor neutrinos [5–7,47–55] have provided
compelling evidence for the existence of neutrino oscillations
driven by nonzero neutrino masses and neutrino mixing.
Evidence for neutrino oscillations were also obtained in the
longbaseline acceleratorneutrino experiments K2K [56,57]
and MINOS [11].
We recall that the idea of neutrino mixing and neutrino
oscillations was formulated in [2–4].
1967 [58] that the existence of νeoscillations would cause
a ‘disappearance’ of solar νeon the way to the Earth. The
hypothesis of solarνeoscillations, which (in one variety or
another) were considered from ∼1970 on as the most natural
explanation of the observed [6,47–50] solarneutrino, νe,
deficit (see, e.g. [59–64]), has been convincingly confirmed in
themeasurementofthesolarneutrinofluxthroughtheneutral
current (NC) reaction on deuterium by the SNO experiment
[7,52–54], and by the first results of the KamLAND
experiment [55]. The combined analysis of the solarneutrino
data obtained by the Homestake, SAGE, GALLEX/GNO,
SuperKamiokande, and the SNO experiments, and of the
KamLAND reactor ¯ νe data [55], established large mixing
angle (LMA), MSW oscillations [60,61] as the dominant
mechanism giving rise to the observed solarνedeficit (see,
e.g. [65]). The Kamiokande experiment [47] provided the first
evidence for oscillations of atmospheric neutrinos, νµand ¯ νµ,
while the data from the SuperKamiokande experiment made
the case for atmosphericneutrino oscillations convincing
[5]. Indications for νoscillations were also reported by the
LSND collaboration [39] but are disfavoured by the recent
MiniBooNE measurement [40].
Compelling confirmation of oscillations in (νµ, ¯ νµ),
and reactor, ¯ νewas provided by L/Edependence observed
by SuperKamiokande [9] and by the spectral distortion
observed by the KamLAND and K2K experiments [8,57].
For the first time the data exhibit directly the effects of
the oscillatory dependence on L/E and E characteristic of
neutrinooscillations in vacuum [66]. As a result of these
developments, the oscillations of solar νe, atmospheric νµand
¯ νµ, accelerator νµ(at L ∼ 250km and L ∼ 730km) and
reactor ¯ νe(at L ∼ 180km), driven by nonzero νmasses and
νmixing, can be considered as practically established.
All existing νoscillation data, except the data of LSND
experiment [39], can be described assuming threeneutrino
mixing in vacuum. Let us recall that in the LSND experiment
indications for ¯ νµ → ¯ νe oscillations with (?m2)LSND ?
1eV2were obtained.The minimal fourneutrinomixing
scheme which could incorporate the LSND indications for ¯ νµ
oscillations is disfavoured by the existing, longbaseline data
[67]andbytherecentMiniBooNEdata[40]. Theνoscillation
explanation of the LSND results is possible assuming five
neutrino mixing [68].
The threeneutrino mixing scheme will be referred to in
what follows as the ‘Standard Neutrino Model’ (SνM). It is
the minimal neutrinomixing model which can account for the
oscillations of solar (νe), atmospheric (νµand ¯ νµ), reactor (¯ νe)
and accelerator (νµ) neutrinos. In the SνM, the (lefthanded)
fields of the flavour neutrinos νe, νµand ντin the expression
for the weak chargedlepton current are linear combinations
of fields of three neutrinos νj, j = 1,2,3, having definite
12
It was predicted in
Page 13
Rep. Prog. Phys. 72 (2009) 000000
mass mj:
A Bandyopadhyay et al
νeL
νµL
ντL
= UPMNS
ν1L
ν2L
ν3L
=
Ue1
Ue2
Ue3
Uµ1
Uµ2
Uµ3
Uτ1
Uτ2
Uτ3
ν1L
ν2L
ν3L
,
(1)
where UPMNS is the Pontecorvo–Maki–Nakagawa–Sakata
(PMNS) neutrinomixing matrix [2–4], UPMNS ≡ U. The
PMNSmixingmatrixcanbeparametrizedbythreeangles,and,
depending on whether the massive neutrinos νjare Dirac or
Majoranaparticles,byoneorthreeCPviolation(CPV)phases
[69–72]. In the standard parametrization (see, e.g. [73]),
UPMNShas the form:
UPMNS
=
×diag(1,eiα/2,eiβ/2),
where cij = cosθij, sij = sinθij, the angles θij = [0,π/2],
δ = [0,2π] is the Dirac CPV phase and α,β are two
Majorana CPviolation phases [69–72].
?m2
responsible for the solarneutrino oscillations. In this case
?m2
squared difference driving the dominant atmosphericneutrino
oscillations, while θ12 = θ?and θ23 = θAare the solar and
atmospheric neutrinomixing angles, respectively. The angle
θ13is the socalled ‘CHOOZ mixing angle’—it is constrained
by the data from the CHOOZ and Palo Verde experiments
[74,75].
Let us recall that the properties of Majorana particles
are very different from those of Dirac particles. A massive
Majorana neutrino χk with mass mk > 0 can be described
(in local quantum field theory) by a 4component, complex
spin1/2 field, χk(x), which satisfies the Majorana condition:
c12c13
s12c13
s13e−iδ
−s12c23− c12s23s13
s12s23− c12c23s13eiδ−c12s23− s12c23s13eiδc23c13
c12c23− s12s23s13eiδ
s23c13
(2)
One can identify
?= ?m2
21> 0 with the neutrino mass squared difference
A = ?m2
31∼= ?m2
32 ? ?m2
21is the neutrino mass
C( ¯ χk(x))T= ξkχk(x),
where C is the charge conjugation matrix. The Majorana
condition is invariant under proper Lorentz transformations.
It reduces by two the number of independent components
in χk(x).
Condition (3) is invariant with respect to U(1) global
gaugetransformationsofthefieldχk(x)carryingaU(1)charge
Q, χk(x) → eiαQχk(x), only if Q = 0. As a result and
in contrast to the Dirac fermions: (i) the Majorana particles
χkcannot carry nonzero additive quantum numbers (lepton
charge, etc) and (ii) the Majorana fields χk(x) cannot ‘absorb’
phases. This is the reason why the PMNS matrix contains
two additional CPviolating phases in the case when the
massive neutrinos νkare Majorana fermions [69], νk≡ χk. It
followsfromtheabovethattheMajorananeutrinofield,χk(x),
describes the two spin states of a spin 1/2, absolutelyneutral
particle, which is identical with its antiparticle, χk ≡ ¯ χk. If
CPinvariance holds, Majorana neutrinos have definite CP
parity, ηCP(χk) = ±i:
UCPχk(x)U−1
ξk2= 1,
(3)
CP= ηCP(χk)γ0χk(x?),ηCP(χk) = ±i.
(4)
It follows from the Majorana condition that the currents:
¯ χk(x)Oiχk(x) :≡ 0, for Oi= γα; σαβ; σαβγ5. This means
that Majorana neutrinos cannot have nonzero U(1) charges
and intrinsic magnetic and electricdipole moments. Dirac
fermions can possess nonzero lepton charge and intrinsic
magnetic and electricdipole moments67.
The existing data allow a determination of ?m2
and of ?m2
e.g. [77–79] and sections 2.2.1 and 2.2.2). For the bestfit
values we have ?m2
?m2
?m2
not fixed by current data. The present atmosphericneutrino
data is essentially insensitive to θ13, satisfying the upper limit
on sin2θ13 obtained in the CHOOZ experiment [80]. The
probabilities of survival of solar νeand reactor ¯ νe, relevant
for the interpretation of the solar neutrino, KamLAND and
CHOOZ neutrinooscillation data, depend on θ13in the case
of interest, ?m2
?
P3ν
?, sin2θ12
A, sin22θ23with a relatively good precision (see,
?= 8.0 × 10−5eV2, sin2θ12 = 0.30,
A = 2.5 × 10−3eV2, sin22θ23 = 1. Thus, ?m2
A. It should be noted, however, that the sign of ?m2
??
Ais
31 ? ?m2
21:
P3ν
KL∼= sin4θ13+ cos4θ13
CHOOZ∼= 1 − sin22θ13sin2?m2
P3ν
1 − sin22θ12sin2?m2
21L
4E
?
,
31L
4E
,
?∼= sin4θ13+ cos4θ13P2ν
where P2ν
?
corresponding to 2ν oscillations driven by ?m2
whichthesolare−numberdensityNeisreplacedbyNecos2θ13
[84], P2ν
?
[81–83] and
?(?m2
21,θ12;Necos2θ13),
is the solar νe survival probability [81–83]
21and θ12, in
= ¯ P2ν
?+ P2ν
? osc, P2ν
? oscbeing an oscillating term
¯ P2ν
?=1
2+?1
2− P??cos2θm
21
2E
12cos2θ12,
(5)
P?=e−2πr0
?m2
sin2θ12− e−2πr0
1 − e−2πr0
?m2
2E
21
?m2
2E
21
.
(6)
Here¯ P2ν
‘double exponential’ jump probability [81–83], r0is the scale
height of the change of Nealong the νtrajectory in the Sun
[81–83,87–89], and θm
in the vacuum limit coincides with θ12. In the LMA solution
region of interest, P2ν
analysis of the solarneutrino, CHOOZ, and KamLAND data,
one finds [77–79]: sin2θ13< 0.040 at 99.73% CL.
It follows from the results described above that the
atmosphericneutrino mixing is close to maximal, θ23∼= π/4,
the solarneutrinomixing angle θ12∼= π/3 and the CHOOZ
angle θ13< π/15. Correspondingly, the pattern of neutrino
?is the average probability [81–83,85,86], P?is the
12is the mixing angle in matter, which
? osc∼= 0 [89]. Performing a combined
67Let us add, finally, that Majorana neutrinos have in addition to the standard
propagator (formed by the neutrino field and its Dirac conjugate field), two
nontrivial nonstandard (Majorana) propagators. If νj(x) in equation (1)
are massive Majorana neutrinos, the process of (ββ)0νdecay, (A,Z) →
(A,Z+2)+e−+e−,forexample,canproceedbyexchangeofvirtualneutrinos
νj due to one of these Majorana propagators. For Dirac fermions, the two
analogousnonstandardpropagatorsareidenticallyequaltozero. Forafurther
detailed discussion of the properties of Majorana neutrinos (fermions) see,
e.g. [62,76].
13
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
mixing is drastically different from that of the quark mixing.
A comprehensive theory of flavour and of neutrino mixing
must be capable of explaining this remarkable difference. The
currenttheoreticalideasaboutthepossibleoriginofthepattern
of neutrino mixing are reviewed in section 3.
As we have seen, the fundamental parameters character
izing the SνM are (i) the three angles θ12, θ23, θ13; (ii) de
pending on the nature of massive neutrinos νj 1 Dirac (δ), or
1Dirac+2Majorana(δ,α,β),CPviolationphasesand(iii)the
threeneutrinomasses,m1,m2,m3. Thismakesnineadditional
parameters in the Standard Model of particle interactions.
Itisconvenienttoexpressthetwolargerneutrinomassesin
terms of the third mass and the measured ?m2
and ?m2
neutrino,K2K,andMINOSdatadonotallowonetodetermine
the sign of ?m2
This implies that, if we identify ?m2
with ?m2
?m2
correspond to two types of νmass spectrum:
?= ?m2
21> 0
A. We have remarked earlier that the atmospheric
A.
A
31(2)in the case of 3neutrino mixing, one can have
31(2)> 0 or ?m2
31(2)< 0. The two possible signs of ?m2
A
• Normal ordering: m1< m2< m3, ?m2
m2(3)= (m2
• Invertedordering68:m3< m1< m2, ?m2
m2= (m2
Depending on the values of the lightest neutrino mass,
min(mj), the neutrinomass spectrum can also be
A= ?m2
31> 0,
1+ ?m2
21(31))
1
2 and
A= ?m2
23− ?m2
32< 0,
21)
3+ ?m2
23)
1
2, m1= (m2
3+ ?m2
1
2.
• Normal
(?m2
• Inverted hierarchy (IH): m3 ? m1 < m2, with m1,2∼=
?m2
• Quasidegenerate (QD): m1∼= m2∼= m3∼= m0, m2
?m2
One of the principal goals of the future studies of neutrino
mixing is to determine the basic parameters of the SνM and to
test its validity.
The possibilities of measuring with high precision
the basic parameters of SνM ?m2
sin22θ23, sin2θ13, of determining sgn(?m2
for the effects of CP violation due to the Dirac phase
δ, in neutrinooscillation experiments, will be discussed in
detail below. It is well known that the neutrinooscillation
experiments are not sensitive to the absolute scale of neutrino
masses. Information on the absolute neutrinomass scale, or
on min(mj), can be derived in3H βdecay experiments and
from cosmological and astrophysical data (see sections 2.2.4
and 3.4.1.2).The most stringent upper bounds on the ¯ νe
mass and on the sum of neutrino masses will be discussed
briefly in sections 2.2.4 and 3.4.2. These bounds lead to the
conclusion that neutrino masses satisfy mj ? 1eV and thus
are much smaller than the masses of the charged leptons and
quarks. A comprehensive theory of neutrino mixing should be
hierarchy
2 ∼ 0.009eV, m3∼= ?m2
(NH):
m1? m2?m3,
A
m2
∼=
?)
11
2 ∼ 0.05eV;
A
1
2 ∼ 0.05eV or
j?
A, m0? 0.10eV.
?, sin2θ? ?m2
31) and of searching
A,
68In the convention we use (called A), the neutrino masses are not ordered
in magnitude according to their index number: ?m2
m3< m1< m2. We can also always number the neutrinos with definite mass
insuchawaythat[90]m1< m2< m3. Inthisconvention(calledB),wehave
in the case of invertedhierarchy spectrum: ?m2
Convention B is used, e.g. in [73,91].
31< 0 corresponds to
?= ?m2
32, ?m2
A= ?m2
31.
able to explain this enormous difference between the neutrino
and chargedfermion masses. The theoretical aspects of the
problem of neutrinomass generation and of the smallness of
neutrino masses are reviewed in section 3.1.
Neutrinooscillation experiments are also insensitive to
the nature, Dirac or Majorana, of massive neutrinos and,
correspondingly, to the two CPviolating, Majorana phases
in the PMNS matrix [69,92] since the latter do not enter into
the expressions for the probabilities for neutrino oscillations.
The only realistic experiments which could verify that the
massive neutrinos νj are Majorana particles are, at present,
the neutrinoless doublebeta ((ββ)0ν) decay experiments.
The physics potential of these experiments is discussed in
section 2.2.4. Even if massive neutrinos are proven to be
Majoranafermions, measuringtheMajoranaCPphaseswould
be extremely challenging. It is quite remarkable, however,
that the Majorana CPviolating phase(s) in the PMNS matrix,
through leptogenesis (see section 3.4.2, may result in the
baryon asymmetry of the Universe [93–95]).
The existing data on neutrino oscillation, as we will see,
allow a determination of ?m2
at 3σ with an uncertainty of approximately ∼12%, ∼24%,
∼28% and ∼15%, respectively. These parameters can, and
very likely will, be measured with much higher accuracy in
the future: the indicated 3σ errors in the determination, for
instance, of ?m2
and10%,aswillbereviewedbelow. ‘Near’futureexperiments
with reactor ¯ νecan improve the current sensitivity to the value
of sin2θ13 by a factor of between 5 and 10. The type of
neutrinomassspectrum,i.e.sgn(?m2
studying the oscillations of neutrinos and antineutrinos, say,
νµ↔ νeand ¯ νµ↔ ¯ νe, in which matter effects are sufficiently
large. If sin22θ13 ? 0.05 and sin2θ23 ? 0.50, information
on sgn(?m2
experiments by investigating the effects of the subdominant
transitions νµ(e) → νe(µ)and ¯ νµ(e) → ¯ νe(µ)of atmospheric
neutrinos which traverse the Earth [99–101]. For νµ(e)(or
¯ νµ(e)) crossing the Earth’s core, new types of resonance
like enhancement of the oscillation probabilities may take
place due to the mantlecore constructiveinterference effect
(neutrino oscillation length resonance (NOLR)) [102–105].
As a consequence of this effect, the corresponding νµ(e)
(or ¯ νµ(e)) transition probabilities can be maximal [103–105].
For ?m2
enhanced, while for ?m2
neutrino transitions ¯ νµ(e) → ¯ νe(µ)takes place, which might
allow determination of sgn(?m2
It should be emphasized that the CP violation in the
lepton sector is one of the most challenging frontiers in
future studies of neutrino mixing. The experimental searches
for CP violation in neutrino oscillations can help answer
fundamental questions about the status of CPsymmetry
in the lepton sector at low energy.
leptonicCP violation at low energies will have farreaching
consequences. It can shed light, in particular, on the possible
origin of the baryon asymmetry of the Universe.
realized recently [93,94], the CP violation necessary for the
generation of the baryon asymmetry can be due exclusively to
?, sin2θ?, ?m2
A, and sin22θ23,
?and sin2θ?, can be reduced to [96–98] 4%
31),canbedeterminedby
31) might be obtained in atmosphericneutrino
31> 0, the neutrino transitions νµ(e) → νe(µ)are
31< 0 the enhancement of anti
31).
The observation of
As was
14
Page 15
Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
the Dirac (and/or Majorana) CPviolating phase in the PMNS
matrix. Thus, there can be a direct relation between low
energy CP violation in the lepton sector, observable, e.g. in
neutrino oscillations, and the matter–antimatter asymmetry
of the Universe. These results underline the importance of
understanding the status of the CPsymmetry in the lepton
sector and, correspondingly, of the experiments aiming to
measure the CHOOZ angle θ13 and of the experimental
searches for CP violation in neutrino oscillations.
2.2. Review of the present generation of experiments
2.2.1. Solar and reactorneutrino experiments.
ments of the solarneutrino flux were the first to indicate that
neutrinos undergo flavour oscillations. The first indications
that the solarneutrino flux was smaller than that predicted by
the Standard Solar models came from Davis’ experiment at
Homestake (USA) [6]. The results of this experiment have
been confirmed by a series of solarneutrino experiments,
the SAGE experiment in Russia [48], the Gallex and GNO
experiments in Italy [106,107], the Kamiokande and Super
Kamiokande(SK)inJapan[51,108]andfinallybytheSudbury
Neutrino Observatory (SNO) in Canada [7,52–54,109]. In
particular, the neutral current (NC) to chargedcurrent (CC)
ratio from the SNO data in 2002 [52] established the presence
ofanactiveneutrinoflavourotherthanνeintheobservedsolar
neutrino flux at the 5.3σ level, putting to rest any doubt about
the existence of flavour oscillations of solar neutrinos. Further
evidence was provided by the statistically powerful NC data
from the salt phase of the SNO experiment [54,109]. The
cumulative result of solarneutrino data, collected from differ
ent experiments over a period of more than four decades, cul
minated in the emergence of the ‘large mixing angle’ (LMA)
solution as the most favoured explanation of the solar neutrino
problem.
Figure 2 shows the confidence level contours in the
?m2
all solarneutrino data combined [77,110,111]. To illustrate
the effect of the results from the saltphase data from SNO, the
figureshowsintherighthandandlefthandpanels,theallowed
areas obtained with and without the saltphase SNO results
respectively. The high statistics NC to CC ratio in SNO salt
data,causestheshrinkingoftheallowedregions. Inparticular,
the upper bound on both ?m2
remarkably.
TheKamLANDreactorantineutrinoexperimentinJapan
[8,55], specifically designed to test the LMA region of the
solar neutrino parameter space, presented its first results in
2002, confirming the LMA solution [55]. The higher statistics
data from this experiment released in 2004 [8] not only
confirmed the observed depletion of the reactor antineutrinos
from the first results [55], but for the first time unambiguously
showed the existence of an L/E dependence in its positron
spectrum, confirming that the observed ¯ νeflavour oscillations
were indeed due to neutrino mass and mixing.
Figure 3 [110,111] shows the impact of the first and
second sets of data from the KamLAND experiment on the
solarneutrino oscillation parameter space. The current 3σ
Measure
21− sin2θ12plane, allowed from the global analysis of
21and sin2θ12is seen to improve
0.1 0.20.3 0.4
sin
2θ12
10
–5
10
–4
10
–3
Solar(BP04) [presalt]
(before sept 2003)
∆m
2
21/eV
2
Solar(BP04) [postsalt]
(after march 2005)
90% CL
95% CL
99% CL
99.73% CL
0.10.20.30.4 0.5
10
–5
10
–4
10
–3
Figure 2. The 90%, 95%, 99% and 99.73% CL contours (for two
degrees of freedom (dof)) show the allowed areas from the global
analysis of the solarneutrino data with (right) and without (left) the
SNO saltphase data. Taken with the kind permission of the
International Journal of Modern Physics from figure 1 in [111].
Copyrighted by World Scientific Publishing Company.
allowed range of ?m2
Bandyopadhyay et al [77,110,111] is given in table 1 along
with their corresponding ‘spread’ defined as
21and sin2θ12obtained in the analysis of
spread =Pmax− Pmin
Pmax+ Pmin
× 100,
(7)
wherePminandPmaxaretheminimumandmaximumallowed
values of the parameter P at 3σ. The allowed regions were
derived on the assumption of CPT invariance and that θ13is
negligible. Also given in table 1 are the bounds on ?m2
sin2θ12that are expected to be obtained when additional data
from the running SNO and KamLAND experiments become
available. For SNO, the analysis assumes that the third and
finalphaseoftheexperimentwillmeasurethesameNCandCC
rates as the salt phase, but with reduced errors of 6% and 5%,
respectively[112]. ForKamLAND,theprospective3kTydata
issimulatedat?m2
a systematic error of 5% is assumed. Better measurement of
chargedcurrent(CC)andneutralcurrent(NC)ratesinSNOis
expectedtoimprovethelimitsonsin2θ12. Thesensitivityofthe
KamLAND experiment to the shape of the reactorinduced ¯ νe
positron spectrum, gives the experiment a tremendous ability
to constrain ?m2
KamLANDisnotassensitivetothemixingangleθ12[98,113].
Theuncertaintyin?m2
3kTy of data from KamLAND. The uncertainty in sin2θ12
is expected to improve after the phaseIII results from SNO
to 18% at 3σ.This would improve to about 16% if the
SNO phaseIII projected results are combined with the 3kTy
simulated data from KamLAND. However, we note that even
with the combined data from phaseIII of SNO and 3kTy
statistics from KamLAND, the uncertainty on sin2θ12would
stay well above the 10–15% level at 3σ.
21and
21= 8.0×10−5eV2andsin2θ12= 0.3and
21. However, as we can see from table 1,
21isexpectedtoreduceto6%at3σ with
15
Page 16
Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
0.20.30.4
sin
2θ12
1.0e–05
6.0e–05
1.1e–04
1.6e–04
∆m
2
21
Global Solar
Solar + KamLAND
162 Ty Data
Solar + KamLAND
766.3 Ty Data
0.20.3 0.4
sin
2θ12
0.20.3 0.40.5
sin
2θ12
Figure 3. The 90%, 95%, 99% and 99.73% CL contours (2 dof) show the allowed areas from the global analysis of the solarneutrino data
(left) and solarneutrino data combined with the first KamLAND results [55] (middle) and second KamLAND results [8] (right). Taken with
the kind permission of the International Journal of Modern Physics from figure 2 in [111]. Copyrighted by World Scientific Publishing
Company.
Table 1. The 3σ allowed ranges (1 dof) and % spread of ?m2
generation of solar neutrino and KamLAND experiments.
21and sin2θ12obtained using current and expected future data from the current
Spread in
?m2
Range of
sin2θ12
Spread in
sin2θ12(%)
Data set usedRange of ?m2
21(eV2)
21(%)
Only solar
Solar + 766.3Ty KL
Solar(SNO3) + 766.3Ty KL
Solar(SNO3) + 3KTy KL
(3.3–18.4) × 10−5
(7.2–9.2) × 10−5
(7.2–9.2) × 10−5
(7.6–8.6) × 10−5
69
12
12
0.24–0.41
0.25–0.39
0.26–0.37
0.26–0.36
26
22
18
166
In our discussion so far, we have assumed the mixing
angle θ13to be zero. If θ13is allowed to vary freely, then
the allowed regions obtained are those shown in figure 4
[114]. Note that this figure shows only the 2σ contours
and uses the confidencelevel definition appropriate for one
degree of freedom. The data do not exclude the possibility
that θ13 = 0. The KamLAND experiment places an upper
bound on the value of θ13by taking into account the neutrino
energy spectrum as well as the absolute rate. By lowering the
value of θ12, the anticorrelation between θ12and θ13can be
used to explain the KamLAND rate data for a wide range of
values of θ13. In contrast, the KamLAND data on the positron
energy spectrum can be explained only for a certain range
ofθ12. Thisimposesanupperlimitontheallowedvalueofθ13.
For the solar neutrinos, the upper limit on θ13comes mainly
from the difference in the θ12–θ13 anticorrelation between
the low and highenergy end of the solarneutrino spectrum.
The tension between the lowenergy solarneutrino data from
SAGE, GALLEX and GNO and the high energy8B data from
SK and SNO results in a reasonably tight upper bound on θ13
[97,115]. Together, the data from solarneutrino experiments
and KamLAND put a rather stringent limit of sin2θ13> 0.05
at 2σ [114].
2.2.2. Atmosphericneutrino experiments.
?m2
angledependenceoftheatmosphericneutrinodataobtainedby
the SuperKamiokande experiment (SK) [5,116]. The results
from the earlier Kamiokande [117,118], MACRO [119,120]
and Soudan2 [121] experiments are in agreement with the
SK data. Figure 5 [116] shows the allowed areas in the ?m2
sin22θ23parameterspace,fromatwogenerationanalysis. The
allowed regions are obtained by fitting both the zenithangle
data [116] and the L/E dependent data [9] from SK.
The values of ?m2
the results from the K2K [57] and MINOS [11] longbaseline
experiments. While K2K has finished its run, MINOS has
declared its first results in the summer of 2006. Both K2K
and MINOS results are consistent with the SK atmospheric
neutrino data, and while the allowed range of values for
sin22θ23is still controlled mainly by the SK atmospheric data,
the results from the longbaseline experiments have an impact
on the allowed range of values for ?m2
Figure 6 [67] shows the projected allowed areas obtained
from a full threegeneration analysis of the global data from
all solar, atmospheric, longbaseline and reactorneutrino
experiments. Filled regions correspond to allowed areas with
The parameters
32(≈?m2
31) and sin2θ23are constrained by the zenith
31–
31and sin22θ23are also constrained by
31.
16
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 4. Three flavour analysis of solar and KamLAND data (both separately and in combination) in the (?m2
The contours show the 2σ allowed regions corresponding to ?χ2= 4. Taken with the kind permission of Progress in Particle and Nuclear
Physics from figure 14 in [114]. Copyrighted by Elsevier Science B.V.
21(≡ δm2), sin2θ12, sin2θ13).
sin22θ
∆m2 (eV2)
Zenith angle analysis
L/E analysis
0.8 0.85 0.9 0.951.0
0.0
1.0
2.0
3.0
4.0
5.0
× 103
Figure 5. The 68% (red lines), 90% (black lines) and 99% (blue
lines) CL (2 dof) allowed oscillation parameter regions obtained in
twogeneration framework by the SK collaboration. The solid lines
are with the analysis of the zenith angle binned data, while the
dashed lines are obtained using the L/E binned analysis. Taken
with the kind permission of the Physical Review from figure 42
in [116]. Copyrighted by the American Physical Society.
thelatestMINOS[11]andSNO[109]results,whilethehollow
regionscorrespondtotheallowedareasobtainedwithoutthese
updates. In the ?χ2versus parameter curves in the figure, the
solid lines are for the full data set, while the dashed lines are
AQ4
without the new SNO [109] and MINOS [11] results. The
impact of the MINOS data on the allowed values of ?m2
is clearly visible. The best fit for ?m2
value compared with that obtained from the SK atmospheric
neutrino data alone. The range of allowed values for ?m2
is also significantly changed.
?m2
considerablyimproved. Thecurrentlimitsonalltheoscillation
parameters can be directly read from this figure.
Figure 7 shows the 90% CL upper limit on θ13 and
how it depends on the different data sets.
from this figure that the bound from the solar + KamLAND
combined analysis is comparable to the one obtained using the
atmospheric + K2K + MINOS results. The 90%(3σ) bounds
(1 dof) on sin2θ13from an analysis of different sets of data
read as [67]:
The bestfit values and allowed range of values of the
oscillation parameters at different C.L. obtained by Maltoni
et al in [67] are shown in table 2.
31
31shifts to a larger
31
While the upper bound on
31is hardly affected, the lower limit on this parameter is
One can note
sin2θ13?
0.033 (0.071)
0.026 (0.054)
(solar + KamLAND),
(CHOOZ + atmospheric
+K2K + MINOS),
(global data).
0.020 (0.040)
(8)
2.2.3. Longbaseline neutrinooscillation experiments.
1962, just a few years after neutrinos were observed directly
In
17
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
105
104
103
102
{∆m2
21, ∆m2
31} [eV2]
0
5
10
15
20
∆χ2
3σ
2σ
0 0.25
{sin2θ12, sin2θ23}
0.50.751
104
103
{∆m2 21, ∆m231} [eV2]
103
102
101
100
sin2θ13
Figure 6. Projections of the allowed regions from the global oscillation data at 90%, 95%, 99%, and 3σ CL for 2 dof for various parameter
combinations. Also shown is ?χ2as a function of the oscillation parameters sin2θ12,sin2θ23,sin2θ13,?m2
to all undisplayed parameters. Dashed lines and empty regions correspond to the global analysis before this update, while solid lines and
coloured regions show our most recent results. Taken with the kind permission of New Journal of Physics from figure 12 in [67].
Copyrighted by Deutsche Physikalische Gesellschaft & Institute of Physics.
21,?m2
31, minimized with respect
102
101
sin2θ13
103
102
∆m2 31 [eV2]
SK+K2K+MINOS
SOL+KAML
+CHOOZ
CHOOZ
GLOBAL
90% CL (2 dof)
Figure 7. 90% CL upper bound on sin2θ13(2 dof) from the
combination of all neutrinooscillation data as a function of ?m2
Taken from figure C2 in [67] (v6).
31.
for the first time using the intense flux generated in a nuclear
reactor [122], the AGS proton accelerator at Brookhaven was
usedtoshowthatasecondgenerationofneutrinosexists[123].
In this experiment, a 15GeV proton beam impinged on a
beryllium target, producing pions, which decayed into muons
and neutrinos. 13.5m of steel separated the volume where
the pions decayed and the spark chambers detected the muons
created by the neutrinos penetrating the steel.
Today, the same fundamental principles are used to study
the phenomenon of neutrino oscillations. The energies of the
Table 2. Bestfit values, 2σ, 3σ and 4σ intervals (1 dof) for the
threeflavour neutrino oscillation parameters from global data
including solar, atmospheric, reactor (KamLAND and CHOOZ) and
accelerator (K2K and MINOS) experiments.
ParameterBest fit2σ
3σ
4σ
?m2
?m2
sin2θ12
sin2θ23
sin2θ13
21(10−5eV2)
31(10−3eV2)
7.9
2.6
0.30
0.50
0.000
7.3–8.5
2.2–3.0
0.26–0.36
0.38–0.63
?0.025
7.1–8.9
2.0–3.2
0.24–0.40
0.34–0.68
?0.040
6.8–9.3
1.8–3.5
0.22–0.44
0.31–0.71
?0.058
neutrinos are fixed at a GeV or more due to the production
mechanism. Therefore, to probe the oscillations first seen
in atmospheric neutrinos, the distances between the neutrino
source and the target have stretched to hundreds of kilometres,
giving rise to their collective name of longbaseline (LBL)
neutrinooscillation experiments.
At the time of writing, two such experiments, K2K and
MINOS,havedemonstratedthatneutrinosdisappearfromtheir
muonneutrino beams in a way that is consistent with neutrino
oscillations. A third LBL beam, providing neutrinos with
energies running up to tens of GeV, has just started operating
from CERN to Gran Sasso. This facility will test whether
the νµdisappearance signals are actually accompanied by
conversions of νµinto ντ, by looking for tau production in
a beam that is originally free of tau neutrinos.
The K2K (KEKtoKamioka) experiment was formally
proposed in 1995 [124], after the first indications of
oscillations were seen in Kamiokande, IMB and SoudanII
atmosphericneutrino data, but before the confirmation by
18
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Eν
rec
GeV
[
events/0.2GeV
0
2
4
6
8
10
12
14
16
18
01 2
345
]
105
106
107
108
109
1010
1011
012345678
Eν [
GeV]
ΦνND [ / cm2 / 0.1 GeV / 1020 POT]
νµ
ν
_
µ
νe
ν
_
e
Near Detector
Figure 8. (Left) The energy spectrum for each type of neutrino at the K2K near detector, estimated by MC simulations. The neutrino beam
consists of 97.3% muonneutrinos. (Right) The 58 fullycontained muonlike singlering events, out of the 112 beamoriginated neutrino
events in K2K. The muon energies and directions can be reconstructed for these events, allowing their parent neutrino energies to be
estimated under the assumption that they are from quasielastic interactions. The solid line is the best fit spectrum with neutrino oscillation
and the dashed line is the expectation without oscillation, both normalized to the number of events seen [10]. Both figures taken with the
kind permission of the Physical Review from figures 6 and 43 in [10]. Copyrighted by the American Physical Society.
SuperKamiokande, and indeed before the completion of the
50kton water Cherenkov detector.
K2K had a baseline of 250km, and the muonneutrino
energy was a GeV or so.The beam was created from a
12GeV proton beam, the hadrons from which were focussed
in a hornshaped electromagnetic volume to increase the beam
intensity. A dedicated detector complex, with a 1kton water
Cherenkov tank, finegrained detectors and a muon ranger,
was located 100m from the end of the piondecay volume,
and measured the beam before it started oscillating on its way
to Kamioka. SuperKamiokande was used as the far detector,
and the first beaminduced neutrino event was observed in the
summer of 1999.
Five and a half years after commissioning, K2K running
ended late in 2004. The final oscillation analysis [10] was
performed using a data set corresponding to 0.922 × 1020
protons on target. The estimated beam spectra for different
neutrino types are shown in figure 8. 112 beamoriginated
neutrino events were observed, where the expected number in
the absence of oscillations was 158.1+9.2
58 were singlering muonlike events fullycontained within
the SuperKamiokande detector. The energies and directions
of the muons in fullycontained events can be reconstructed,
and because of the simple kinematics of the chargedcurrent
quasielastic (CCQE) events that make up much of the cross
section around 1GeV, it is possible to estimate the energy of
the incoming neutrinos. Such a spectrum is shown in figure 8,
for the 58 events, with unoscillated and bestfit oscillated
curves,normalizedtothenumberofeventsseen. Theseresults
supportmaximalmixing,withbestfittwoneutrinooscillation
parameters of sin22θ = 1 and ?m2= 2.8 × 10−3eV2. The
90% CL range for ?m2at sin22θ = 1 is between 1.9 and
3.5 × 10−3eV2.
The Main Injector Neutrino Oscillation Search (MINOS)
experiment was also proposed in 1995, with a neutrino beam
pointed from Fermilab to the Soudan mine in Minnesota, with
−8.6. Of these events,
(GeV)
ν
Reconstructed E
POT
16
Events / GeV / 10
Ratio
05 10 15
1
1.5
10
20
30
40
Data
Fluka05 MC
Full MC Tuning
0510 15
1
1.5
20
40
60
80
05 1015
1
1.5
20
40
60
80
(a)
(b)
(c)
(GeV) (GeV)
νν
Reconstructed EReconstructed E
Events/GeV
0
10
10
20
20
30
30
40
40
50
50
60
60
05 101518 30
Events/GeV
Beam Matrix Unoscillated
NDFit Unoscillated
Beam Matrix Best Fit
NC Background
MINOS Data
Figure 9. (Top) MINOS neutrino beam spectra at the near detector,
for three beam configurations. (Bottom) The final far detector
spectrum and predicted distributions, after the first full year of
MINOS running (1.27 × 1020protons on target) [11]. Two different
methods of neartofar extrapolation are shown for the unoscillated
spectrum. Both figures taken with the kind permission of the
Physical Review Letters from figures 2 and 3 in [11]. Copyrighted
by the American Physical Society.
a baseline of 735km. The beam has a system of movable
focussing horns to allow the beam energy spectrum to be
altered. Three different spectra are shown in the upper plot
in figure 9. Both near and far detectors consist of a steel
19
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
)
23
θ
(2
2
sin
0.20.40.6 0.8 1.0
)
4
/c
2
 (eV
2
m
∆ 
32
1.5
2.0
2.5
3.0
3.5
3 3
10
×
10
×
MINOS Best Fit
MINOS 90% C.L.
MINOS 68% C.L.
K2K 90% C.L.
SK 90% C.L.
SK (L/E) 90% C.L.
4.0
Figure 10. Confidence intervals from the MINOS experiment [11].
Results from K2K [57] and SuperK [9,116] are also shown. Taken
with the kind permission of the Physical Review Letters from
figure 4 in [11]. Copyrighted by the American Physical Society.
and plasticscintillator sandwich structure, the performance of
which was studied in detail in test beam work at CERN [125].
The experiment started running in the spring of 2005, and
within a year had gathered data corresponding to 1.27 × 1020
protons on target. The data are shown in the lower plot in
figure 9. The MINOS results support maximal mixing, with
bestfit parameters of ?m2
sin22θ23> 0.87 at 68% CL. The oscillation parameters from
the K2K and MINOS experiments, together with results from
SuperKamiokande are shown in figure 10. MINOS will run
for 5 years, with the goal of accumulating 16 × 1020protons
on target. This data set should improve our knowledge of
the oscillation parameters substantially. Both the experiments
described here are linked, if only indirectly, to future projects
to make precision measurements of the oscillation parameters
and to probe the third mixing angle. These projects, T2K and
NOνA, are discussed below.
32 = 2.74+0.44
−0.26× 10−3eV2and
2.2.4. 0νββ experiments.
isaDiracoraMajoranafermionisoffundamentalimportance
for understanding the origin of neutrino masses and mixing
(see, e.g. [126]). Let us recall that the neutrinos, νj, with
definitemass,mj,willbeDiracfermionsifparticleinteractions
conserve some additive lepton number, e.g. the total lepton
number L = Le+Lµ+Lτ. If no lepton number is conserved,
the neutrinos will be Majorana fermions (see, e.g. [62]).
The heavy neutrinos are predicted to be Majorana in nature
by the seesaw mechanism [127], which also provides an
attractive explanation of the smallness of neutrino masses and,
through the leptogenesis theory [128], of the observed baryon
asymmetry of the Universe. The observed patterns of neutrino
mixing and of neutrino masssquared differences driving the
solar and the dominant atmosphericneutrino oscillations, can
be related to massive Majorana neutrinos and the existence of
Establishing whether the neutrino
an approximate symmetry in the lepton sector corresponding
to the conservation of the nonstandard lepton number L?=
Le− Lµ− Lτ(see, e.g. [129]).
The only experiments which have the potential of
establishing the Majorana nature of massive neutrinos are the
(ββ)0νdecayexperimentssearchingfortheprocess(A,Z) →
(A,Z + 2) + e−+ e−(for reviews see, e.g. [62,130–134]).
The observation of (ββ)0νdecay and the measurement of the
correspondinghalflifewithsufficientaccuracywouldnotonly
be a proof that total lepton number is not conserved, but might
also provide unique information on: (i) the type of neutrino
mass spectrum; (ii) the absolute scale of neutrino masses and
(iii) the Majorana CPviolating phases in the neutrinomixing
matrix [69–71,73,90,91,135–153].
If the νj are Majorana fermions, obtaining information
about the Majorana CP phases in UPMNSwill be remarkably
difficult [73,135,151,154,155].
supersymmetric theories which include the seesaw neutrino
massgeneration mechanism, the phases α and β can
significantly affect the predictions for the rates of lepton
flavourviolating(LFV)decayssuchasµ → e+γ,τ → µ+γ,
etc (see, e.g. [156–158]).
Under the assumptions of massive, Majorana neutrinos,
threeneutrino mixing, and (ββ)0νdecay being generated
solely through the (V–A) chargedcurrent weak interaction
mediated by the exchange of the three Majorana neutrinos,
the (ββ)0νdecay amplitude has the form (see, e.g. [73,135]):
A(ββ)0ν∼= ?m?M, where M is the corresponding nuclear
matrixelement(NME)whichdoesnotdependontheneutrino
mixing parameters, and
In a large class of
?m? = m1Ue12+ m2Ue22eiα+ m3Ue32eiβ,
istheeffectiveMajoranamassin(ββ)0νdecay,Ue1 = c12c13,
Ue2 = s12c13, Ue3 = s13. In the case of CPinvariance one
has[159–162],η21≡ eiα= ±1,η31≡ eiβ= ±1;η21(31)being
the relative CPparity of Majorana neutrinos ν2(3)and ν1.
Information on the absolute scale of neutrino masses can
be derived in3H βdecay experiments [163,164] and from
cosmologicalandastrophysicaldata. Themoststringentupper
bounds on the ¯ νemass were obtained in the Troitzk [163] and
Mainz [164] experiments:
(9)
m¯ νe< 2.3eVat 95% CL.
(10)
We have m¯ νe∼= m1,2,3in the case of the QD νmass spectrum.
TheKATRINexperiment[164]isplannedtoreachasensitivity
of m¯ νe∼ 0.20eV, i.e. it will probe the region of the QD
spectrum. TheCMBdataoftheWMAPexperiment,combined
withdatafromlargescalestructuresurveys(2dFGRS,SDSS),
lead to a limit on the sum of νjmasses (see, e.g. [165,166]):
?
Data on weak lensing of galaxies, combined with data from
the WMAP and PLANCK experiments, may allow ? to be
determined with an uncertainty of ∼0.04eV [167,168]. It
proves convenient to express [169,170] the three neutrino
j
mj≡ ? < (0.4–1.7)eVat 95% CL.
(11)
20
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
1e05
0.0010.1
mMIN [eV]
0.01
1
<m> [eV]
NH
IH
QD
0.00010.01
1
mMIN [eV]
0.01
1
<m> [eV]
IH
QD
NH
Figure 11. The value of ?m? as a function of min(mj), obtained using (i) the 95% CL allowed ranges of ?m2
(left) and (ii) prospective 2σ uncertainty in ?m?, corresponding to input 1σ experimental errors in ?m2
4% and sin2θ13= 0.010 ± 0.006 (right). The bestfit values and the 2σ ranges used in the analysis are given in equations (2.1)–(2.4)
in [170]. The regions shown in red/grey correspond to violation of CPsymmetry. Taken with the kind permission of Physical Review from
figures 1 and 2 in [170]. Copyrighted by the American Physical Society.
?, ?m2
31and sin2θ?of 2%, 2% and
A, sin2θ?and sin2θ13
?, ?m2
masses in terms of ?m2
oscillation experiments, and the absolute neutrinomass scale
determined by min(mj) [73,132–135]. In both the normal
and the invertedhierarchy, one has: ?m2
m2= (m2
and m3 = (m2
inverted ordering, mMIN = m3, ?m2
m1= (m2
and θ13, ?m? depends on min(mj), Majorana phases α, β and
the type of νmass spectrum.
The problem of obtaining the allowed values of ?m? given
the constraints on the parameters following from neutrino
oscillationdata,and,moregenerally,ofthephysicspotentialof
(ββ)0νdecay experiments, was first studied in [169,170] and
subsequentlyin[132–134]. Detailedanalyseswereperformed
more recently in [151–153,170]. The results are illustrated
in figure 11. The main features of the predictions for ?m?
are [73,91,135,141,142] (figure 11(left)) as follows:
?and ?m2
A, measured in neutrino
?= ?m2
A= ?m2
21> 0,
31> 0
1+?m2
?)
1+ ?m2
1
2. Fornormalordering,?m2
1
2, while if the spectrum is with
A= ?m2
A− ?m2
A)
23> 0 and
A, ?m2
3+ ?m2
?)
1
2. Thus, given ?m2
?, θ?
1. For
?
2. For the IH spectrum, ?m?∼=
sin2 α
?
to the values α = 0; π and
3. For the QD spectrum, ?m?∼= m0(1−sin22θ?sin2 α
m0? ?m? ? m0cos2θ?? 0.03eV, with m0? 0.1eV,
m0< 2.3eV [164] or m0? 0.5eV [165,166].
For the IH (QD) spectrum we have sin2(α/2)∼= (1 − ?m?2/
˜ m2)/ sin22θ?, ˜ m2= ?m2
?m? (and m0for QD spectrum) can allow α to be determined.
Many experiments have searched for (ββ)0νdecay [130].
The best sensitivity was achieved in the HeidelbergMoscow
theNHspectrum,
?m?
∼=

?
?m2
?s2
12+
?m2
As2
13ei(α−β) ? 0.005eV;
?
?m2
A(1 − sin22θ?
2)1/2, thus ?m? ?
?m2
?
?m2
A ? 0.055eV and ?m? ?
Acos2θ?? 0.013eV, the bounds corresponding
2)1/2,
A (m2
0). Thus, a measurement of
76Ge experiment [171]: ?m? < (0.35–1.05)eV (90% CL),
where a factor of 3 uncertainty in the relevant NME (see, e.g.
[172–174]) is taken into account. The IGEX collaboration has
obtained [175]: ?m? < (0.33–1.35)eV (90% CL). A positive
signal at >3σ, corresponding to ?m? = (0.1–0.9)eV, is
claimed to be observed [176]. Two experiments, NEMO3
(with100Mo and82Se) [177] and CUORICINO (with130Te)
[178], designed to reach a sensitivity to ?m? ∼ of ?m? ∼
(0.2–0.3)eV, published first results: ?m? < (0.7–1.2)eV
[177] and ?m? < (0.2–0.9)eV [178] (90% CL), where
estimated uncertainties in the NME are accounted for. Most
importantly, a number of projects aim at sensitivity of
?m? ∼ (0.01–0.05)eV [179]: CUORE (130Te), GERDA
(76Ge), SuperNEMO (100Mo), EXO (136Xe), MAJORANA
(76Ge), MOON (100Mo), XMASS (136Xe), CANDLES (48Ca),
etc. These experiments will probe the region corresponding to
IHandQDspectraandtestthepositiveresultclaimedin[176].
The existence of significant lower bounds on ?m? in the
cases of IH and QD spectra [91], which lie either partially
(IHspectrum)orcompletely(QDspectrum)withintherangeof
sensitivityofthenextgenerationof(ββ)0νdecayexperiments,
is one of the most important features of the predictions of ?m?.
These minimal values are given, up to small corrections, by
?m2
analysis of the solar and reactorneutrino data [77,79,180]
including the latest SNO and KL results: (i) the possibility
of cos2θ?= 0 is excluded at ∼6σ; (ii) the bestfit value of
cos2θ?is cos2θ? = 0.38 and (iii) at 95% CL one has for
sin2θ13= 0 (0.02), cos2θ?? 0.28 (0.28). The quoted results
oncos2θ?togetherwiththerangeofpossiblevaluesof?m2
and m0lead to the significant and robust lower bounds on ?m?
in the cases of the IH and the QD spectrum [91,143–145].
At the same time one can always have ?m? = 0 in the case
of spectrum with (partial) normal hierarchy [141,142]. As
figure 11 indicates, ?m? cannot exceed ∼6meV for the NH
neutrinomass spectrum. This implies that max(?m?) in the
Acos2θ? and m0cos2θ?. According to the combined
A
21
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
caseoftheNHspectrumisconsiderablysmallerthanmin(?m?)
for the IH and the QD spectra. This makes it possible that
information about the type of neutrinomass spectrum may be
obtained from a measurement of ?m? ?= 0 [91]. In particular,
a positive result in the future generation of (ββ)0νdecay
experiments with ?m? > 0.01eV would imply that the NH
spectrumisstronglydisfavoured(ifnotexcluded). Prospective
experimental errors in the values of the oscillation parameters
(figure 11(right)), in ?m? and the sum of neutrino masses, and
the uncertainty in the relevant NME [172–174], can weaken
but do not invalidate these results [141–145,151].
As figure 11 indicates, a measurement of ?m? ? 0.01eV
would either: (i) determine a relatively narrow interval of
possible values of the lightest νmass mMIN or (ii) would
establish an upper limit on mMIN. If an upper limit on ?m?
is experimentally obtained below 0.01eV, this would lead to a
significant upper limit on mMIN.
The possibility of establishing CP violation in the
lepton sector due to Majorana CPV phases has been
studied in [73,135,154,155] and in much greater detail in
[141,142,151].It was found that it is very challenging:
it requires quite accurate measurements of ?m? (and of
m0 for QD spectrum), and holds only for a limited range
of values of the relevant parameters.
[141,142,151],establishingat2σ CPviolationassociatedwith
Majorana neutrinos in the case of QD spectrum requires, for
sin2θ? = 0.31 in particular, a relative experimental error
on the measured value of ?m? and m0smaller than 15%, a
‘theoretical uncertainty’ F ? 1.5 in the value of ?m? due to an
imprecise knowledge of the corresponding NME, and value of
therelevantMajoranaCPVphaseα typicallywithintheranges
of ∼(π/4 − 3π/4) and ∼(5π/4 − 7π/4) (figure 11(right)).
The knowledge of the NMEs with sufficiently small
uncertainty is crucial for obtaining quantitative information
on the neutrinomixing parameters from a measurement of
(ββ)0νdecay halflife. Possible tests of the NME calculations
are discussed in [181].
More specifically
2.2.5.
oscillations.
neutrino oscillations and improving the precision with which
the various parameters are known have been reported since the
ISSconcluded. Comprehensivereviewsoftheseresultscanbe
foundin,forexample,[182,183]. Whileitisnotappropriateto
attempt a complete review here, it is of interest to note, briefly,
the most important developments.
KamLAND has reported results based on a fourfold
increase in exposure and with an improved analysis leading to
a significant reduction in the systematic error [184]. The new
data and analysis yield ?m2
10−5eV2and tan2θ12= 0.56+0.10
MINOS has provided an improved measurement of ?m2
based on 2 years of running and 3.36 × 1020protons on
target [185–187]. The results confirm the neutrinomixing
hypothesis and yield ?m2
OPERA experiment, which took its first data in August 2006,
has been commissioned, recording ∼640 neutrino events from
an exposure corresponding to ∼16 × 1017protons on target.
Recent progress in measurements of neutrino
Several results confirming the hypothesis of
21= 7.58+0.14
−0.07(stat)+0.10
−0.13(stat)+0.15
−0.06(syst).
−0.15(syst) ×
32
32= 2.43 ± 0.13 × 10−3eV2. The
The experiment is now poised to begin the search for direct
evidence of νµ→ ντ.
MiniBooNE has carried out a detailed evaluation of
backgrounds to the νe appearance signal and published
improved results [188–190] that indicate that a twoneutrino
oscillation explanation of the data from Bugey, KARMEN2,
LSND and MiniBooNE is only possible at the 3.94% level.
MiniBooNE is now taking data with an antineutrino beam.
The results of this phase of the experiment will be of interest
since the LSND experiment was also carried out in a ¯ νµbeam.
Progress has also been made in the development of the
reactorneutrino programme and the preparation of the T2K
and NOνA experiments. The interested reader is referred to
[188–190] for further information.
2.3. Completing the picture
The measurements of the neutrinooscillation parameters
reviewedabovehintatnewinteractionspresentatanextremely
large mass scale, ?. In scattering experiments, for example
at hadron or lepton colliders, these new interactions are
suppressed by powers of ?. In contrast, neutrino oscillations
are widely believed to be a direct consequence of the physics
at the large mass scale; hence, measurements of neutrino
oscillations probe physics at a uniquely high mass scale. The
measurementsreviewedabovehaveestablishedthepresenceof
neutrinooscillationsandhavedeterminedanumberofrelevant
parameters. Tocompletethepicture, adedicatedexperimental
programme is required; the elements of this experimental
programme are [191]) as follows:
• Thesearchforneutrinolessdoublebetadecay,toestablish
whether neutrinos are Majorana particles [192,193];
• The determination of the neutrinomass scale by direct
measurement (see, e.g. [194]) or through cosmology (see,
e.g. [195,196]);
• The determination of the neutrinomass hierarchy by
combining neutrinooscillation measurements with the
resultsofdirectneutrinomassmeasurementsandsearches
for 0νββ decay;
• The determination of the small mixing angle θ13through
measurements of the subdominant neutrino oscillations;
• The precise determination of the mixing angle θ23to seek
to establish whether θ23is maximal;
• The search for leptonicCP violation in neutrino
oscillations and
• The search for sterile light neutrinos through the
observation of a third masssquared difference in neutrino
oscillations. The recent measurements from MiniBooNE
[40] disfavour a sterileneutrino interpretation of the
LSND results [197].
2.3.1.
present generation of longbaseline oscillation experiments
(K2K [10] at KEK, MINOS [11] at the NuMI beam and
ICARUS [198] and OPERA [12] at the CNGS beam, see
table 3), are expected to measure sin22θ23and ?m2
precisionof∼10%,if?m2
could, in principle, measure θ13through νµ→ νeoscillations
Bounds on θ13 from approved experiments.
The
31 with a
31 > 10−3eV2. Theseexperiments
22
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Table 3. Main parameters for present longbaseline neutrino beams.
Proton
momentum
(GeV/c)
L
(km)
Eν
(GeV)
potyear−1
(1019) Neutrino facility
KEK PS [10]
FNAL NuMI [201]
CERN CNGS [202]
12 250
735
732
1.5
3
17.4
2
20–34
4.5–7.6
120
400
T2K
sin 2θ
2
13
∆m (eV )
2
23
2
2.5
Opera
MINOS
CHOOZ
excluded
90% C.L.
SuperK allowed
90% C.L.
Figure 12. Expected θ13sensitivity (in vacuum and for δCP= 0) for
MINOS, OPERA and for the next T2K experiment, compared with
the CHOOZ exclusion plot. Taken from [203].
even though they are not optimized for such a measurement.
MINOS is expected to reach a sensitivity of sin2θ13? 0.02 at
a confidence level (CL) of 90% in 5 years [11]. The main
limitation of the MINOS experiment is the poor electron
identification efficiency of the detector. Thanks to the high
density and high granularity of the emulsion cloud chamber
(ECC) structure, the OPERA detector is better suited for
electron detection and can reach sin2θ13? 0.015 at 90% CL
(for ?m2
CNGS beam at nominal intensity [199,200].
The θ13sensitivity of the present LBL experiments
(includingtheT2K,thatwillbediscussedinmoredetailbelow)
is shown in figure 12. The sensitivity of such experiments to
θ13is limited by the power of the proton driver and by the νe
contamination of the beam. In particular, the CNGS beam,
whichhasbeenoptimizedforτ production, hasameanenergy
about ten times larger than the first νµ→ νeoscillation peak
at a baseline of 732km.
Anotherapproachtosearchfornonvanishingθ13(andthe
θ23octant [204]) is to look at ¯ νedisappearance using reactor
neutrinos. The relevant oscillation probability is
31= 2.5 × 10−3eV2), after 5 years’ exposure to the
P(¯ νe→ ¯ νe) ? 1 − sin22θ13sin2
??m2
31L
4E
?
+ ···,
(12)
which does not depend on θ23 or δCP.
relevant for reactorneutrino experiments, the dependence of
At the baselines
the oscillation probability on ?m2
Therefore,thisapproachallowsanunambiguousmeasurement
of θ13free of correlations and degeneracies (see section 2.4),
thoughitrequiresaverypreciseknowledgeoftheabsoluteflux.
TheDoubleCHOOZexperiment[205,206]willemployanear
and far detector, located at baselines of 0.2km and 1.05km,
respectively. Both detectors will be based on gadolinium
loaded liquid scintillator with a fiducial mass of 10.16ton.
Antineutrinos will be detected using the delayed coincidence
of the positron from the inverse βdecay and the photons from
neutron capture. The direct comparison of the event rates in
the two detectors will allow the cancellation of many of the
systematicerrors. After5yearsofdatataking, thisexperiment
will reach a θ13sensitivity of sin2θ13 ? 0.0025 at 90% CL.
Another reactor experiment has been recently proposed in
Japan [207]. This experiment has an expected sensitivity of
sin2θ13? 0.0038 at 90% CL.
Present LBL and reactorneutrino experiments cannot
addresstheotherissuesraisedabove;thebaselinesaretooshort
totakeadvantageofmattereffectsrequiredtoidentifythemass
hierarchy, and they are not designed to look for CP violation.
21and θ12 is negligible.
2.4. Degeneracies and correlations
We will follow [208] to introduce the degeneracy problem.
Other approaches have been proposed in [204,209–213].
2.4.1. Appearance channels: νe → νµ,ντ and νµ → νe.
It was originally pointed out in [214] that a measurement
of the appearance probability P(να → νβ) = Pαβ for a
neutrinooscillation experiment with a fixed baseline (L) and
energy(E)cannotbeusedtodetermineuniquelytheoscillation
parameters. Indeed, taking (¯θ13,¯δ) as the ‘true’ values, the
equation
Pαβ(¯θ13,¯δ) = Pαβ(θ13,δ)
has a continuous number of solutions. The locus of points
in the (θ13,δ) plane satisfying this equation is called an
‘equiprobability’ curve. As can be seen from figure 13(left),
the strong correlation between θ13and δ [215] defines a strip
in the (θ13,δ) plane compatible with Pαβ(¯θ13,¯δ).
Consider now an experiment that can measure both
neutrino (+) and antineutrino (−) appearance oscillation
probabilities, at the same L/E. The system of equations
(13)
P±
αβ(¯θ13,¯δ) = P±
αβ(θ13,δ)
(14)
describestwoequiprobabilitycurves,seefigure13(right). The
system has two solutions: the input pair (¯θ13,¯δ) and a second,
(L/E)dependent, point. The ‘continuum degeneracy’ has
been solved, but a discrete ambiguity in the measurement of
the physical values of θ13and δ is still present; the ‘intrinsic
degeneracy’ or ‘intrinsic clone’ [214].
More information is needed to solve the intrinsic
degeneracy. Thisinformationcanbeobtainedeitherbymaking
independent measurements at different values of L/E or by
making use of independent oscillation channels. The value of
L/E may be varied, for example, by measuring the Neutrino
Factory beam at a number of baselines [214,217], by varying
23
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
Figure 13. Correlation of θ13and δ. (Left) If only neutrinos (or antineutrinos) are measured: continuum degeneracy; (Right) if both
neutrinos (full line) and antineutrinos (dashed line) are measured: twofold degeneracy. Taken with the kind permission of the Nuclear
Physics B Proceedings Supplements from figures 1 and 2 in [216]. Copyrighted by Elsevier Science B.V.
Figure 14. Solving the intrinsic degeneracy using: (Left) same oscillation channel, but two different baselines; (Right) same L/E, but two
different oscillation channels (i.e. golden and silver). Taken with the kind permission of the Nuclear Physics B Proceedings Supplements
from figures 3 and 4 in [216]. Copyrighted by Elsevier Science B.V.
the neutrinobeam energy at a betabeam facility [218], or by
measuring precisely the neutrinoenergy spectrum in a liquid
argon detector [219]. In figure 14(left) it can be seen that
experiments with different baselines have intrinsic clones in
different regions of the (θ13,δ) plane. If the clones are well
separated, the degeneracy can be solved. The equiprobability
curves for the two oscillation channels νe→ νµand νe→ ντ
measured at a particular L/E are shown in figure 14(right).
The figure shows that the intrinsic clones for the two channels
appear in different regions of the parameter space, making it
possible to resolve the intrinsic degeneracy.
Two other sources of ambiguities are also present
[209,220,221]:
• Atmosphericneutrinoexperimentsmeasureνµdisappear
ance or νµ→ ντappearance for which the leading terms
in the expressions for the oscillation probabilities depend
quadratically on ?m2
known [222] and
• At leading order, the oscillation probabilities for νµ,νe
disappearance and νµ → ντ appearance depend upon
sin22θ23. Therefore, only the difference of θ23from 45◦
13, therefore the sign of ?m2
13is not
24
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
(maximal mixing) is known, i.e. it is not known whether
θ23is smaller or greater than 45◦.
As a consequence, future experiments must measure the two
continuous variables θ13 and δ as well as the two discrete
variables:
satm= sign[?m2
These two variables assume the values ±1 depending on the
sign of ?m2
for m2
soct= −1 for θ23> π/4). Therefore, taking into account the
present ignorance on the neutrino masses and mixing matrix,
equation (14) must be rewritten, more precisely, as
23],soct= sign[tan(2θ23)].
(15)
23(satm = 1 for m2
3< m2
3> m2
2and satm = −1
2) and θ23 (soct = 1 for θ23 < π/4 and
P±
αβ(¯θ13,¯δ; ¯ satm, ¯ soct) = P±
αβ(θ13,δ;satm= ¯ satm;soct= ¯ soct),
(16)
where ¯ satmand ¯ socthave been included as input parameters in
addition to¯θ13and¯δ. In equation (16) we have implicitly
assumed that the sign of ?m2
unknown. The following systems of equations should be
considered:
23and the octant for θ23 are
P±
αβ(¯θ13,¯δ; ¯ satm, ¯ soct)
= P±
= P±
= P±
These new sets of equiprobability systems arise when we
equate the measured probability (lhs) with the theoretical
probabilities obtained including one of the three possible
wrong guesses of satmand soct(rhs).
Solvingthefoursystemsofequations(16)–(19)willyield
thetruesolutionplusadditional‘clones’,forminganeightfold
degeneracy [221]. These eight solutions are, respectively, the
following:
αβ(θ13,δ;satm= −¯ satm;soct= ¯ soct)
αβ(θ13,δ;satm= ¯ satm;soct= −¯ soct)
αβ(θ13,δ;satm= −¯ satm;soct= −¯ soct).
(17)
(18)
(19)
• The true solution and its intrinsic clone, obtained solving
the system in equation (16);
• The ?m2
thetrueandintrinsicsolution,obtainedsolvingthesystem
in equation (17);
• The θ23octant clones (hereafter called ‘octant’ clones) of
thetrueandintrinsicsolution,obtainedsolvingthesystem
in equation (18) and
• The ?m2
‘mixed’clones)ofthetrueandintrinsicsolution, obtained
solving the system in equation (19).
23sign clones (hereafter called ‘sign’ clones) of
atmsign θ23octant clones (hereafter called
Notice, however, that transition probabilities are not the
experimentally measured quantities. Experimental results are
given in terms of the number of charged leptons observed in
a specific detector. For the Neutrino Factory ‘golden channel’
(νe → νµ), for example, one counts the number of muons
with charge opposite to the charge of the muons circulating
in the storage ring. If the detector can measure the final state
lepton and hadron energies with enough precision, events can
begroupedinenergybinsofwidth?E. Thenumberofmuons
intheithenergybinfortheinputpair(¯θ13,¯δ),foraparentmuon
energy¯ Eµ, is given by
?dσνµ(¯ νµ)(Eµ,Eν)
⊗d?νe(¯ νe)(Eν,¯ Eµ)
dEν
Ei
where ⊗ stands for a convolution integral, Niis the number of
events in bin i, Eνis the neutrino energy, Eµis the scattered
muonenergy,σνµ(¯ νµ),istheneutrinochargedcurrentscattering
crosssection,and?istheneutrinoflux. Solvingthefollowing
systems of equations, for a given energy bin and fixed input
parameters (¯θ13,¯δ):
Ni
Ni
µ∓(¯θ13,¯δ) =
dEµ
?Ei+?Eµ
⊗ P±
eµ(Eν,¯θ13,¯δ)
,
(20)
AQ5
µ±(¯θ13,¯δ; ¯ satm, ¯ soct)
= Ni
= Ni
= Ni
= Ni
yields the eight solutions corresponding to the ith bin.
The existence of unsolved degeneracies results in a loss
of sensitivity to the unknowns θ13,δ,satm(see below). The
best way to solve the degeneracies is to perform a set
of complementary measurements; experiments must have
different baselines, good energy resolution and access to
different channels. There is no ‘synergy’ in experiments at
the same L/E measuring the same channel [223]. A method
to look for optimal combinations of measurements based on
solving the set of systems of equations (21)–(24) has been
presented in [208]. Most of the previous considerations also
apply to the Tconjugated transition νµ→ νeand to νe→ ντ
(the Neutrino Factory ‘silver channel’).
µ±(θ13,δ;satm= ¯ satm,soct= ¯ soct)
µ±(θ13,δ;satm= ¯ satm,soct= −¯ soct)
µ±(θ13,δ;satm= −¯ satm,soct= ¯ soct)
µ±(θ13,δ;satm= −¯ satm,soct= −¯ soct),
(21)
(22)
(23)
(24)
2.4.2. Disappearance channels: νµ→ νµ.
measurementoftheatmosphericparametersθ23and?m2
bemadeviatheνµdisappearancechannelusingaconventional
neutrino beam or the Neutrino Factory.
that this kind of measurement will reduce the error on the
atmospheric mass difference to less than 10% with a few years
of data if ?m2
error on the atmospheric angle depends on the value of θ23
itself, the smallest error being achieved for large, but non
maximal, mixing [225]. It is interesting to study in detail
the parameter correlations and degeneracies that affect this
measurement and that can induce large uncertainties. The
vacuumoscillation probability expanded to the second order
in the small parameters θ13and (?12L/E) [222] is
P(νµ→ νµ) = 1 − [sin22θ23− s2
×sin2
22
??12L
×[c4
An independent
23can
It is expected
23? 2.2 × 10−3eV2[224].The expected
23sin22θ13cos2θ23]
??23L
?
−
??12L
?
[s2
12sin22θ23+˜ Js2
23cosδ]
×sin(?23L) −
2
?2
23sin22θ12+ s2
12sin22θ23cos(?23L)],
(25)
25
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 15. The sign degeneracy at T2KI; (left) θ23= 45◦; (right) θ23= 41.5◦. Taken with the kind permission of the Nuclear Physics B
from figure 4 in [227]. Copyrighted by Elsevier Science B.V.
where
?m2
comes from the first term in the first parentheses which is
symmetric under θ23→ π/2 − θ23. This symmetry is lifted
by the other terms which introduce a mild CPconserving δ
dependence, albeit through subleading effects which are very
difficult to isolate.
Since the satm= sign(?m2
equations must be solved:
˜ J
=
cosθ13sin2θ12sin2θ13sin2θ23 and ?23
23/2E, ?12 = ?m2
=
12/2E. The dominant contribution
23) is unknown, two systems of
N±
µµ(¯θ23,?m2
atm; ¯ satm) = N±
µµ(θ23,?m2
23; ¯ satm)
(26)
and
N±
µµ(¯θ23,?m2
where ¯ satmis the physical mass hierarchy. For nonmaximal
¯θ23,fourdifferentsolutionsareobtained. For?m2
equation (26) yields two solutions, the input value θ23=¯θ23
and θ23 ? π/2 −¯θ23. The second solution is not exactly
θ23 = π/2 −¯θ23 due to the small θ23octant asymmetry.
Two more solutions from equation (27) at a different value
of ?m2
thatchangingthesignof?m2
a change that must be compensated with an increase in ?m2
to give P±
The result of a fit to the disappearancechannel data at
the T2K phase I experiment is shown in figure 15 for three
different values of the atmospheric mass difference ?m2
(2.2,2.5,2.8)×10−3eV2. Fixedvaluesofthesolarparameters
have been used, ?m2
maximal mixing, θ23= 45◦, figure 15(left), two solutions are
found at 90% CL when both choices of satmare considered.
On the other hand, using a nonmaximal atmospheric angle
θ23 = 41.5◦(sin2θ23 = 0.44) four degenerate solutions are
found, figure 15(right). In general, a twofold or fourfold
degeneracy must be discussed in the disappearance channel.
Notice how the disappearance sign clones appear at a
value of ?m2
atm; ¯ satm) = N±
µµ(θ23,?m2
23;−¯ satm),
(27)
23 ∼ ?m2
atm
23 are also present [226]. In equation (25) we can see
23makesthesecondtermpositive,
23
µµ(?m2
atm; ¯ satm) = P±
µµ(?m2
23;−¯ satm).
23=
12= 8.2 × 10−5eV2; θ12 = 33◦. For
23 higher than the input value. This is expected
from equation (25); the shift in the vertical axis is a function
of θ13 and δ which, in this case, has been kept fixed at
θ13 = 0◦= δ. The degeneracy can be softened or solved
by using detectors at baselines long enough that matter effects
can be exploited [227].
2.4.3. A matter of conventions.
a short parentheses to address a problem that arose recently
concerning the ‘physical’ meaning of the variables used to
fit the ‘atmospheric’ mass difference, ?m2
of all, that the experimentally measured solarmass difference
?m2
parameter ?m2
experimentallymeasuredatmosphericmassdifference?m2
Since the subleading solar effects are, at present, barely
seen in atmosphericneutrino experiments we can define the
threefamily parameter to be used in the fits in a number of
ways: by using ?m2
?m2= (?m2
data will be obtained with either choice. When measurements
of the atmospheric masssquared difference with a precision
at the level of 10−4eV2are available, however, the different
choices of the fitting parameter will give different results.
This effect can be observed in figure 16, where the three
choicesintroducedabovearecompared. Thethreepanelsshow
the 90% CL contours resulting from a fit to the experimental
data corresponding to the input value, ?m2
in normal hierarchy, but fitted using in turn ?m2
(middle) and ?m2(right). It can be seen that the contour
corresponding to the normal hierarchy, satm= ¯ satm, is always
located around the input value. On the other hand, the contour
obtained for the inverted hierarchy is located above, below or
on top of the input value depending on the choice of fitting
variable. This is a consequence of the fact that the difference
between each of the possible choices is O(?m2
For threefamily mixing, three ‘frequencies’ can be
defined, the shortest being the solaroscillation frequency
It is useful to open here
atm. Notice, first
SOLcan be unambiguously identified with the threefamily
12= m2
2− m2
1. This is not true for the
atm.
23= m2
13)/2 [228]. A good description of the
3− m2
2; ?m2
13= m2
3− m2
1or
23+ ?m2
atm= 2.5 × 10−3,
23(left), ?m2
13
12).
26
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 16. Different choices of the threefamily ‘atmospheric’ mass difference; (left) ?m2
kind permission of the Nuclear Physics B from figure 5 in [227]. Copyrighted by Elsevier Science B.V.
23; (middle) ?m2
13; (right) ?m2. Taken with the
(unambiguously related to the mass difference ?m2
the case of the normal hierarchy, the middle frequency is
related to ?m2
the inverted hierarchy these two frequencies are interchanged
and the middle frequency will be related to ?m2
?m2
of the normal and inverted hierarchies should be presented
using variables which maintain the ordering of the oscillation
frequencies.
12).In
23and the longest one to ?m2
13. In the case of
31and not to
32. For this reason, it has been suggested that the analysis
2.4.4. Disappearance channels: νe→ νe.
Neutrino Factory can exploit the νedisappearance channel to
measurethesolarparameters?m2
free environment. The νedisappearance probability does not
depend on δ or on θ23. The θ13measurement is, therefore, not
affected by (θ13− δ) correlations or the soctambiguity. The
νe → νematteroscillation probability, expanded at second
order in the small parameters θ13and (?m2
??23
??12
where ?23
=
√2GFNe, and B∓ = A ∓ ?23. This equation describes
reasonably well the behaviour of the transition probability in
the energy range covered by the betabeam facilities presently
considered. Two sources of ambiguities are still present in
νedisappearance measurements, satm(for large values of θ13,
i.e. in the ‘atmospheric’ region) and the θ13–θ12correlation
(for small values of θ13, i.e. in the ‘solar’ region). A beta
beam could in principle improve the precision with which
the solar parameters are known through νe disappearance
measurements. This is not the case for a betabeam facility
in which the neutrino energy of ∼100MeV is matched to a
baseline of ∼100km. For such a facility, at large θ13, the
second term in equation (28) dominates over the last term.
On the other hand, for small θ13the statistics is too low to
improve upon the present uncertainties on θ12and ?m2
that the energy and baseline of the lowγ betabeam has not
A betabeam or
12,θ12orθ13inadegeneracy
12L/E), is [229]:
?
P±
ee= 1 −
B∓
?2
sin2(2θ13)sin2
?B∓L
?AL
=
2
?
−
A
?2
sin2(2θ12) sin2
2
,
(28)
?m2
23/2E, ?12
?m2
12/2E, A
=
12(note
been chosen to perform this task). It has been shown that if
systematic errors cannot be controlled to better than at 5%,
the betabeam disappearance channel does not improve the
CHOOZ bound on θ13[226].
Equation (28) can also be applied to reactorneutrino
experiments which aim at a precise measurement of θ13in
a ‘degeneracyfree’ regime.
energy of a reactor experiment (e.g. L = 1.05km and
?Eν? = 4MeV for the DoubleCHOOZ proposal [205,206])
we can safely consider antineutrino propagation in vacuum.
As a consequence, no sensitivity to satmis expected at these
experiments,sinceB∓→ ?23for?23? A. Itisverydifficult
for reactor experiments to test small values of θ13, and thus the
θ13–θ12correlation (significant only in the ‘solar’ region) can
also be neglected.
For the typical baseline and
3. Implications for new physics and cosmology
Neutrino mass is the first example of physics beyond the
Standard Model. The extreme smallness of neutrino masses,
compared with chargedfermion masses, and the large mixing
angles, are both mysteries that make more acute the flavour
problem in the Standard Model: why are there three families
of quarks and leptons with the masses and mixings that are
observed? Although there are many ideas concerning the
underlying mechanism by which neutrino mass is generated,
at present none of the proposed mechanisms have any
experimental foundation; to make real progress more data is
required. Theneutrinomassesandmixingsareasfundamental
as those of the quarks, yet the precision with which the
neutrinomixing parameters are known is very poor when
compared with the precision of the quark parameters. Some
of the neutrino parameters, such as the reactor angle and the
CPviolating phase, have yet to be measured, and the sign
of the atmospheric masssquared difference is undetermined.
If neutrino are Dirac fermions, then neutrino masses may
arise in a manner similar to that which generates the masses
of the other charged fundamental fermions.
neutrinos are Majorana particles, then the massgeneration
mechanism may be quite different. These issues, which have
However, if
27
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
profoundimplicationsforparticlephysicsandcosmology,will
be discussed in detail in this section.
3.1. The origin of small neutrino mass
This section will address the implications of seesaw
mechanisms, supersymmetry and Rparity violation, extra
dimensions,stringtheoryandTeVscalemechanismsforsmall
neutrino masses on the properties of the neutrinos.
3.1.1. Seesaw mechanisms.
alreadycontainsquitestronghierarchieswiththeelectronmass
being a few million times smaller than the topquark mass.
Neutrino masses are also very small compared with charged
fermion masses, with the atmospheric neutrino mass being
a few million times smaller than the electron mass. Such
severe fermion mass hierarchies demand some explanation.
One simple approach is based on the seesaw mechanism and
its generalization to include the chargedfermion masses by
Froggatt and Nielsen [230].
couplings are of order unity, but lowest order Yukawa
couplings to Higgs fields are forbidden by some symmetry;
neutrino masses are further suppressed by the fact that right
handed neutrinos are very heavy.
couplings and small Majorana masses are then generated at
higher order, suppressed by ratios of vacuum expectation
values (vevs) to heavy field masses. The seesaw mechanism
thus provides a convincing explanation for the smallness of
neutrino masses. Here we review its simplest form, the type I
seesaw mechanism and its generalization to the type II see
saw mechanism.
Before discussing the seesaw mechanism, the different
types of neutrino mass that are possible will be reviewed. So
far we have been assuming that neutrino masses are Majorana
masses of the form
mν
The chargedfermion spectrum
The idea is that all Yukawa
Small effective Yukawa
LLνLνC
L,
(29)
where νL is a lefthanded neutrino field and νC
CPconjugate of a lefthanded neutrino field, in other words
a righthanded antineutrino field. Majorana masses imply
leptonnumber violation. Note that leptonnumber violation
is forbidden by gauge invariance at the renormalization level
in extensions of the Standard Model in which the Higgs
sector only contains doublets.
the seesaw mechanism assumes that Majorana mass terms
are generated through the interactions of the righthanded
neutrinos [127,231,232].
If we introduce righthanded neutrino fields then there are
two sorts of additional neutrino mass terms that are possible:
additional Majorana masses of the form
Lis the
The simplest version of
MRRνRνC
R+ hermitian conjugate,
(30)
where νR is a righthanded neutrino field, νC
CPconjugate of a righthanded neutrino field, in other words
a lefthanded antineutrino field; and Dirac masses of the form
Ris the
mν
LRνLνR+ hermitian conjugate.
(31)
Such Dirac mass terms conserve lepton number and are not
forbidden by electriccharge conservation.
Once this is done, the types of neutrino mass described
in equations (30) and (31) (but not equation (29) since we do
not assume direct mass terms, e.g. from Higgs triplets, at this
stage) are permitted, and we have the mass matrix:
?
Since the righthanded neutrinos are electroweak singlets, the
Majorana masses of the righthanded neutrinos, MRR, may be
orders of magnitude larger than the electroweak scale. In the
approximation that MRR? mν
may be diagonalized to yield effective Majorana masses of the
type in equation (29):
νL
νC
R
??
0
mν
LR
mνT
LR
MRR
??
νC
L
νR
?
+ hermitian conjugate. (32)
LRthe matrix in equation (32)
mν
LL= −mν
LRM−1
RRmνT
LR.
(33)
TheeffectivelefthandedMajoranamasses, mν
suppressedbytheheavyscale, MRR. Inaonefamilyexample,
if we take mν
10−3eV, which looks good for solar neutrinos. Atmospheric
neutrino masses would require a righthanded neutrino with a
mass below the GUT scale.
With three lefthanded neutrinos and three righthanded
neutrinos the Dirac masses, mν
and the heavy Majorana masses, MRR, form a separate 3 × 3
(complex, symmetric) matrix. The light effective Majorana
masses mν
continuetobegivenbyequation(33),whichisnowinterpreted
as a matrix product. From a modelbuilding perspective the
fundamental parameters which must be input into the see
saw mechanism are the Dirac mass matrix mν
righthanded neutrino Majorana mass matrix MRR. The light
effective lefthanded Majorana mass matrix mν
output according to the seesaw formula in equation (33).
The version of the seesaw mechanism discussed so far
is sometimes called the type I seesaw mechanism.
the simplest version of the seesaw mechanism and can be
thoughtofasresultingfromintegratingoutheavyrighthanded
neutrinos to produce the effective dimension5 neutrino mass
operator:
−1
where the dot indicates the SU(2)Linvariant product, and
LL, arenaturally
LR= MWand MRR= MGUT, then we find mν
LL∼
LR, are a 3×3 (complex) matrix
LLare also a 3 × 3 (complex symmetric) matrix and
LRand the heavy
LLarises as an
It is
4(Hu· LT)κ(Hu· L),
(34)
κ = 2YνM−1
RRYT
ν,
(35)
withYνbeingtheneutrinoYukawacouplingsandmν
with vu= ?Hu?. The type I seesaw mechanism is illustrated
diagrammatically in figure 17.
LR= Yνvu
Figure 17. Diagram illustrating the type I seesaw mechanism.
28
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
Inmodelswithaleft–rightsymmetricparticlecontentsuch
as minimal left–right symmetric models, PatiSalam models,
orGrandUnifiedTheories(GUTs)basedonSO(10),thetypeI
seesaw mechanism is often generalized to a type II seesaw
(see,e.g.[70,233–236]),whereanadditionaldirectmassterm,
mII
With such an additional direct mass term, the general
neutrinomass matrix is given by
??
Under the assumption that the mass eigenvalues MRiof MRR
areverylargecomparedwiththecomponentsofmII
the mass matrix can approximately be diagonalized yielding
effective Majorana masses:
LL, for the light neutrinos is present.
?
νL
νC
R
mII
mνT
LL
mν
LR
LR
MRR
??
νC
L
νR
?
.
(36)
LLandmLR,
mν
LL≈ mII
LL+ mI
LL,
(37)
with
mI
LL≈ −mν
LRM−1
RRmνT
LR,
(38)
for the light neutrinos.
The direct mass term, mII
small contribution to the lightneutrino masses if it stems, e.g.
fromaseesawsuppressedinducedvacuumexpectationvalue.
We will refer to the general case, where both possibilities
are allowed, as the II seesaw mechanism.
type II contribution by generating the dimension5 operator
in equation (34) via the exchange of heavy Higgs triplets of
SU(2)Lis illustrated diagrammatically in figure 18.
LL, can also provide a naturally
Realizing the
AQ6
3.1.2.
example of the origin of small neutrino masses is Rparity
violating supersymmetry (SUSY) (for a review see [237]).
Here, the lefthanded neutrinos mix with neutralinos after
SUSY breaking, leading to small, loop suppressed, Majorana
masses. The masses depend on the SUSY mass spectrum.
Should SUSY be discovered, and the mass spectrum
determined, at highenergy colliders, the theory could be used
to predict the Majorana masses.
In any supersymmetric extension of the Standard Model
it is possible to introduce interactions that break Rparity,
defined as R = (−1)3B+L+2S[238], where L, B and S are
the lepton number, baryon number and spin, respectively. The
Supersymmetry and Rparity violation.
Another
Figure 18. Diagram leading to a type II contribution mII
neutrinomass matrix via an induced vev of the neutral component
of a triplet Higgs ?.
LLto the
interactions that can contribute to the neutrino masses must
also violate lepton number, and are given by [239]:
?1
The trilinear RParity violating (TRpV) parameters λijkand
λ?
number keeping baryon number conserved.
numberviolatinginteractions(oftheform1
be included, leading to proton decay. The present limit on
the lifetime of the proton [240] leads to stringent constraints
on products of λ couplings, although such constraints can be
relaxed in the case of Split Supersymmetry [241].
The bilinear RParity violating (BRpV) parameters, ?i,
induce sneutrino vacuum expectation values vi, as well
as mixing between particles and sparticles.
neutrinos mix with neutralinos forming a set of seven neutral
fermions F0
treelevel neutrinomass matrix [242]:
M(0)
ν
= −m · M−1
where Mχ0 is the Minimal Supersymmetric Standard Model
(MSSM) (for a review see [243]) neutralino mass matrix
and the parameters ?i ≡ µvi+ ?ivdare proportional to the
sneutrino vevs in the basis where the ?i terms are rotated
away from the superpotential. Note that if this is done BRpV
reappears in the soft terms [244].
Thetreelevelneutrinomassmatrixhasonlyonenonzero
eigenvalue, equal to the trace of the matrix in equation (40),
and therefore proportional to ? ?2. If the above treelevel
contribution dominates over oneloop graphs, the square of
this eigenvalue would be equal to the atmospheric mass
squared difference, ?m2
reactor angles would be given by tan2θ(0)
tan2θ(0)
13
≈ ?2
corrections, the solar masssquared difference and the solar
angle remain undetermined.
Once the oneloop corrections are included [245] the
symmetry of the neutrinomass matrix in equation (40) is
brokenandthusthesolarmasssquareddifferenceisgenerated
radiatively. The oneloop corrected neutrinomass matrix has
the general form:
WRpV= εab
2λijk? La
i? Lb
j? Rk+ λ?
ijk? La
i?
Qb
j? Dk+ ?i? La
i?
Hb
u
?.
(39)
ijkare dimensionless Yukawa couplings that violate lepton
The baryon
2λ??UDD)canalso
In particular,
i. A lowenergy seesaw mechanism induces the
χ0· mT
=M1g2+ M2g?2
4det(Mχ0)
?2
1
?1?2
?2
?1?3
?1?2
2
?2?3
?2
?1?3
?2?3
3
,
(40)
31≈ m(0)2
3
, and the atmospheric and
≈ ?2
3) respectively. Without oneloop
23
2/?2
3and
1/(?2
2+ ?2
Mν
ij= A?i?j+ B(?i?j+ ?j?i) + C?i?j,
where A(0)
= (g2M1 + g?2M2)/4det(Mχ0) is the only
nonzero coefficient at treelevel. In BRpV most particles
contribute in loops to the neutrinomass matrix. An important
loop is the one involving bottom quarks and squarks, which
is shown in figure 19.The external arrows represent the
flow of lepton number, while the internal ones show the flow
of the bottomquark electric charge, and the cross signals a
mass insertion. The complete dashed line represents a single
scalar propagator corresponding to the heavy bottom squark
˜b2, with the full circles pictorially showing the component of
(41)
29
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 19. Pictorial representation of the bottomsbottom loops
contributing to the neutrinomass matrix, with Rp violated bilinearly
in the open circles.
this mass eigenstate in left and right sbottoms. The external
lines are the neutrino states which define the basis used to
write the neutrinomass matrix in equation (40). The open
circles pictorially represent the component of these neutrinos
inhiggsinosandindicatestheplacewhereRParityisviolated.
Asimilargraphwiththelightsbottom,˜b1,isobtainedreplacing
c˜b→ −s˜band s˜b→ c˜b.
contributes to the coefficient C in the following way:
The sum of these two graphs
C(˜b)= −Ncmb
16π2µ2h2
bsin(2θ˜b) ?B
˜b1˜b2
0
,
(42)
where Nc= 3 is the number of colours, and we have defined
?B
0
≡ B0(0;m2
˜b1˜b2
b,m2
˜b2)− B0(0;m2
b,m2
˜b1)≈−ln(m2
˜b2/m2
˜b1).
(43)
The result in equation (42) can be understood with the help of
the graph presented above. It is proportional to the bottom
quark mass due to the mass insertion, and to the square of
the bottom Yukawa coupling due to the vertices. The sbottom
mixing contributes with the factor sin(2θ˜b), and the higgsino
neutrino mixing accounts for the factor ?i?j/µ2, where the ?
parameters have been factored out from C. The contribution
is finite because Veltman functions [246] from˜b2 and˜b1
are subtracted from each other. The contribution to the B
parameter can be obtained using B(˜b)= −a3µC(˜b), with
a3= vu(g2M1+ g?2M2)/4det(Mχ0), as can be inferred from
the neutralinoneutrino mixing shown in the graph. There is
also a contribution A˜b, but it is in general a small correction
to A(0).
There are similar loops with charged scalars S+
Higgs bosons mixing with charged sleptons [247]) together
with charged fermions F+
leptons [248]). Among these are the charged Higgs and
stau contributions which have the same form as that given in
equation(42)withthereplacementsb → τ,˜b → ˜ τ andtaking
Nc= 1 . There are also loops with neutral scalars S0
Higgs bosons mixing with sneutrinos [249]) together with the
neutral fermions F0
BRpV can successfully be embedded in supergravity
[250], although with nonuniversal ?iterms at the GUT scale
(as well as bilinear soft terms Bi, associated with ?i). By
definition, thecoefficientsA, B andC inequation(41)depend
exclusively on the universal scalar mass m0, gaugino mass
M1/2and trilinear parameter A0at the GUT scale, and the
values of tanβ and µ at the weak scale. In figure 20 we
see the region of the m0–M1/2plane consistent with neutrino
i(charged
j(charginos mixing with charged
i(neutral
jmentioned above.
Figure 20. Region of parameter space where solutions satisfy all
experimental constraints. Adapted with the kind permission of the
European Physical Journal from figure 6 in [251]. Copyrighted by
Springer Berlin/Heidelberg.
experimental data, for fixed values of the BRpV parameters
?1= −0.0004, ?2= 0.052, ?3= 0.051GeV and ?1= 0.022,
?2 = 0.0003, ?3 = 0.039GeV2[251]. In this scenario,
the solar masssquared difference strongly limits the universal
gaugino mass from above and below. Large values of the
universal scalar mass are limited mainly by the atmospheric
masssquared difference.
This model can be tested at colliders, and the main signal
thatdifferentiatesitfromtheMSSMisthedecayofthelightest
neutralino which decays only in RpV modes. In the scenario
of figure 20 the neutralino mass is 99GeV and decays to an
onshell W, satisfying
B(χ0
B(χ0
1→ We)
1→ Wµ)=?2
1
?2
2
.
(44)
Such ratios can be directly related to neutrinomixing
angles [252].
Other scenarios have been studied,
anomaly mediated supersymmetry breaking (AMSB) [254],
Gauge Mediated Supersymmetry Breaking [255] and Split
Supersymmetry[256]. InthecaseofAMSBweseeinfigure21
how the solar and atmospheric masssquared differences
depend on the universal scalar and gaugino masses, for fixed
values of the BRpV parameters ?1= −0.015, ?2= −0.018,
?3 = 0.011GeV, and ?1 = −0.03, ?2 = −0.09, ?3 =
−0.09GeV2.
TRpV interactions do not contribute to neutrino masses at
treelevel [257]. The oneloop contributions to these diagrams
are given by the diagrams shown in figure 22. The convention
for the graphs is the same as before. Analogous graphs are
obtained for the light scalars˜dn
c˜dn→ −s˜dnand s˜dn→ c˜dn. The mixing angles are
2(M˜d 2
LR)n
M˜d 2
Rn
for example
1and˜ln
1with the replacement
sin(2θ˜dn) =
Ln− M˜d 2
,
sin(2θ˜ln) =
2(M˜l 2
M˜l 2
LR)n
Ln− M˜l 2
Rn
.
(45)
30
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
Figure 21. Atmospheric and solar masssquared differences as a function of SUSY masses in AMSB. Taken with the kind permission of the
Physical Review from figures 1 and 3 in [253]. Copyrighted by the American Physical Society.
Figure 22. Pictorial representation of the fermionsfermion loops contributing to the neutrinomass matrix, with Rp violated trilinearly in
the open circles.
The contribution to the neutrinomass matrix due to TRpV is
?
×sin(2θ˜dn)mdk?B0(0;m2
Note that the contribution to the neutrinomass matrix is
symmetric in the indices i and j [258]. A similar contribution
holds for leptons and sleptons inside the loop, replacing λ?by
λ couplings. In the approximation where only particles of the
third generation contribute inside the loops, the shift to the
neutrinomass matrix from TRpV is
(?Mν
ij)λ?=
Nc
16π2
kn
(λ?
iknλ?
jnk+ λ?
inkλ?
jkn)
dk;m2
˜dn).
(46)
(?Mν
ij)TRpV= Dλ?
i33λ?
j33+ Eλi33λj33,
(47)
which can be added to equation (41). In this way, BRpV and
TRpV, together or separated, can explain the neutrino masses
and oscillations observed in experiments.
3.1.3. Extra dimensions.
account for small neutrino masses is to ascribe them to the
violation of lepton number by adding to the SM an effective
dimensionfive operator O = λLHLH [259] (see figure 23).
The favourite scenario realizing this idea is the ‘seesaw’
The basic gaugetheoretic way to
Figure 23. Dimension five operator responsible for neutrino mass.
L denotes any of the three lepton doublets and H is the SM scalar
doublet.
mechanism, which requires the presence of singlet ‘right’
handed neutrinos, which mix with the ordinary SU(2) doublet
‘left’handed neutrinos [260]. The suppression of the neutrino
masses results from the structure of the full mass matrix
[70,261]. In the simplest versions of this mechanism the mass
of the extra states should be about ten orders of magnitude
larger than the electroweak scale.
Recently, there have been a number of attempts to explain
neutrinooscillationsintheorieswithlargeinternaldimensions
and a low fundamental scale [262–266].
perturbativelycalculableframeworkistypeIstringtheorywith
A convenient,
31
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
Dbranes. The SM is then localized on a stack of Dbranes,
transverse to some large extra dimensions, where gravity
propagates. Dbrane models offer a novel scenario to account
for neutrino masses [267–272]; righthanded neutrinos are
assumed to propagate in the bulk while lefthanded neutrinos,
being a part of the lepton doublet, live on the SM branes. As a
result, the Dirac neutrino mass is naturally suppressed by the
bulk volume. Adjusting this volume, so that the string scale
lies in the TeV range, leads to tiny neutrino masses compatible
with current experimental data.
Indeed, the relation between the string scale, Ms, and the
fourdimensional Planck mass, MP, is
M2
P=
8
g4VbM2
s,
(48)
where g is the SM gauge coupling and Vbis the volume of the
bulk in string units. The simplest way to introduce a right
handed neutrino is to identify it with an open string excitation
onsome(stackof)brane(s)extendedinthebulk. Moreoverthe
SM Higgs and lepton doublets must come from open strings
stretched between the SM and bulk branes and thus, living at
their intersection, they couple to the bulk neutrino state. More
precisely, its kinetic term is
?
m
+m
RνRmνc
where the sum is extended over all KaluzaKlein (KK)
excitations, denoted collectively by m.
assumed a toroidal compactification for n extra dimensions
of common radius R, with Vb = (2πR)nin string units.
The two states νR and νc
fourdimensional (4d) components of the higherdimensional
spinor. The zero mode νR0will be identified with the right
handedneutrino state, while νc
the spectrum by an orbifold projection and is not relevant for
our purposes.
The interaction of the bulk neutrino with the localized
Higgs and lepton doublets reads:
?
where it has been assumed that the SM brane stack is localized
at the origin of the bulk and the coupling, λ, is in general of
order g2(λ is equal to g2in the simplest 3brane realization
of the SM). By expanding νRin KK modes, one gets the mass
terms:
Smass=
Skin= Vb
d4x
?
?
¯ νRm/ ∂νRm+ ¯ νc
?
Rm/ ∂νc
Rm
Rm+ c.c.,
(49)
For simplicity we
Rcorrespond to the left and right
R0may be projected out from
Sint= λ
d4xH(x)L(x)νR(x,y = 0),
(50)
λv
√Vb
?
m
νLνRm,
(51)
where v is the Higgs expectation value, ?H? = v.
that the apparent mixing of νLwith all KK excitations can be
neglected since its strength (51) is much smaller than the KK
mass gv/Rn/2? 1/R, or equivalently gv ? Rn/2−1, which
is valid for any n ? 2. As a result, the righthanded neutrino
is essentially the zero mode νR0and taking into account the
Note
normalization of its kinetic term (49), one obtains a Dirac
neutrino mass, mννLνRm, with
mν?
λv
√Vb
?
√8λ
g2vMs
Mp,
(52)
which is of the order of 10−3–10−2eV for Ms∼ 1–10TeV.
The extradimensional neutrinomass suppression mech
anism described above can be destabilized by the presence
of a large Majorana neutrinomass term.
absence of any protecting symmetry, the leptonnumber
violating dimension5 effective operator in figure 23 will be
present. This would lead, in the case of TeVstringscale
models, to an unacceptable Majorana mass term of the order
of a few GeV. Even if we manage to eliminate this operator
in some particular model, higher order operators would also
give unacceptably large contributions, since in lowscale grav
ity models the ratio between the Higgs vacuum expectation
value and the string scale is of order O(1/10–1/100).
An elegant way to avoid this problem was suggested
in reference [273]. It consists of assuming that the bulk
sector, where the SM singlet states live, is eightdimensional.
There is, however, a general theorem that states that in
eight dimensions there can be no massive Majorana spinor
[274–277]. Moreover, further unwanted large Lviolating
contributions to neutrino masses could be prevented by
imposing leptonnumber conservation leaving only the Dirac
mass(52). Indeed,leptonnumberoftenarisesasananomalous
abelian gauge symmetry associated with the U(1)bof the bulk
(stack of) brane(s), possibly in a linear combination with other
U(1)’s [278,279]. The anomaly is cancelled by shifting an
axion field from the closed string (Ramond–Ramond) sector
[280,281]. As a result, the gauge boson becomes massive,
while lepton number remains unbroken as an effective global
symmetry in perturbation theory [282]. The gauge coupling,
gb, of the bulk U(1)bgauge boson is extremely small since it
is suppressed by the volume of the bulk Vb:
Indeed, in the
1
g2
b
=
1
g2Vb=g2
8
M2
M2
P
s
,
(53)
where in the second equality we used equation (48). It follows
that gb ? 10−16–10−14for Ms ∼ 1–10TeV. Such a theory
would lead to light Dirac neutrino masses, in contrast with
general fourdimensional gaugetheoretic expectations which
lead to Majorana neutrinos [260].
If the U(1)bgauge boson is light, it would be copiously
produced in stellar processes, leading to supernova cooling
through energy loss in the bulk of extra dimensions. There
are strong constraints coming from supernova observations.
Note that the corresponding process is much stronger than
the production of gravitons because of the nonderivative
coupling of the gaugeboson interaction [268]. In fact, for the
case of n large transverse dimensions of common radius R,
satisfying mA, R−1? T with mA the gauge boson mass
and T the supernova temperature, the production rate, PA, is
proportional to
PA∼ g2
b× [R(T − mA)]n×
1
T2?Tn−2
Mn
s
,
(54)
32
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
where the factor [R(T − mA)]ncounts the number of KK
excitations of the U(1)bgauge boson with mass less than T.
This rate can be compared with the corresponding graviton
production:
PG∼
1
M2
P
× (RT)n?
Tn
Mn+2
s
,
(55)
showing that for n = 2 (sub)millimetre extra dimensions, it
is unacceptably large, unless the bulk gauge boson acquires a
massmA? 10MeV.Forn ? 3,thesupernovaboundbecomes
muchweakerandmAmaybemuchsmaller[283]. Suchalight
gaugebosoncanmediateshortdistanceforceswithintherange
of tabletop experiments that test Newton’s law at very short
distances [284–287].
Thesetheoriesalsoleadtonovelwaystogenerateneutrino
oscillations. The interaction term in equation (50) involves
in general all lefthanded neutrinos and additional Higgs
doublets:
3
?
i=1
λiLiHiνR→
3
?
i=1
λiviνiLνR,
(56)
where i is a generation index and for each generation i, Hi
is one of the possible available Higgs doublets Hd or Hu,
providing also masses to down or to up quarks, with vi= ?Hi?
the corresponding vev. The above couplings give mass to one
linearcombinationoftheweakeigenstatesνiL, whiletheother
two remain massless. The mass is given by equation (52)
??3
of these states with the ordinary neutrinos may have an impact
upon neutrino oscillations.
with λv replaced by
being a bulk state, has a tower of KK excitations. The mixing
i=1λ2
iv2
i. The righthanded neutrino,
3.1.3.1. The effect of extra dimensions.
features of the data on neutrino oscillations that are relevant
for the present discussion are the following:
The most important
1. Theexistenceofspectraldistortionsindicativeofneutrino
oscillations;
2. The solar mixing angle is large but significantly non
maximal;
3. The atmospheric bestfit mixing angle is maximal and
4. Both solar and atmospheric oscillation data strongly as
well as the recent MiniBooNE data [40] disfavour the
presence of sterileneutrino states in the channel to which
the relevant neutrino is oscillating.
There are several discussions in the literature [267–272]
regarding neutrino masses and oscillations in the context of
extra dimensions. Most of these discussions are restricted to
the case of an effectively onedimensional bulk. This simple
onedimensional bulk picture is not realistic [288], as it is at
odds with the current global status of neutrinooscillation data
given in [67] and described above. Indeed, such a picture
violates at least one of the four points mentioned above. In
addition there is also a serious theoretical problem, since
onedimensional propagation of massless bulk states gives
rise to linearly growing fluctuations which, in general, yield
large corrections to all couplings of the effective field theory,
destabilizing the hierarchy [289].
In the case of a twodimensional bulk the situation
is significantly improved [279].
structure to describe both solar and atmospheric oscillations
byintroducingasinglebulkneutrinopair,usingessentiallythe
two lowest frequencies of the neutrinomass matrix: the mass
of the zero mode (equation (52)), arising via the electroweak
Higgs phenomenon, which is suppressed by the volume of the
bulk, and the mass of the first KK excitation. The former is
used to reproduce the solarneutrino data. The latter is used to
explainatmosphericneutrinooscillations,whichhaveahigher
oscillationfrequency,withanamplitudewhichisenhanceddue
to logarithmic corrections of the twodimensional bulk [289].
One can see, however, that at least condition (4) above is
violated, as there is a significant sterile component at least
in one of the channels of neutrino conversion, corresponding
to the KK excitations of the bulk righthanded neutrino,
and this is highly disfavoured by the global fits of neutrino
oscillations [67].
One way out is to introduce three bulk neutrinos and
explain the observed neutrino oscillations in the traditional
way [290].In this case, νR in equation (56) would carry
a generation index i and all lefthanded neutrinos would
acquire Diractype masses with the zero modes of the bulk
states. Moreover, the effect of KK mixing can be suppressed
by appropriately decreasing the size of the extra dimensions
and thus increasing the value of the string scale.
in this limit one would obtain the generic case of three
Dirac neutrinos, and the leptonmixing matrix depends on
precisely three angles and one CP phase, as the quarkmixing
matrix. Correspondingly, the oscillation pattern is ‘generic’
withoutspecialpredictions. HavingDiracinsteadofMajorana
neutrinos can be experimentally tested by searching for the
existence of processes like 0νββ.
On the other hand, ‘extradimensional’ signatures may be
present in oscillations at a subleading level, as nonstandard
interactions (see [260] for a short discussion). The Neutrino
Factory will provide an interesting laboratory to probe for the
possible presence of such effects.
Indeed, there is enough
Thus,
3.1.4. String theory.
the implications of superstring theories for neutrino masses.
However, it is known that some of the ingredients employed
in Grand Unified Theories and other fourdimensional models
maybedifficulttoimplementinknowntypesofconstructions.
For example, the chiral supermultiplets that survive in the
effective fourdimensional field theory are generally bi
fundamental in two of the gaugegroup factors (including
the case of fundamental under one factor and charged under
a U(1)) for lowestlevel heterotic constructions; or either
bifundamental, adjoint, antisymmetric or symmetric for
intersecting brane constructions. This makes it difficult to
break the GUT symmetry, and even more so to find the high
dimensionalHiggsrepresentations(suchasthe126ofSO(10))
usually employed in GUT models for neutrino and other
There has been relatively little work on
33
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
fermion masses. Thus, it may be difficult to embed directly
many of the models, especially GUT models involving high
dimensional representations rather than higherdimensional
operators, in a string framework. Perhaps more likely is that
the underlying string theory breaks directly to an effective
fourdimensional theory including the Standard Model and
perhaps other group factors [291]. Some of the aspects of
grand unification, especially in the gauge sector, may be
maintainedinsuchconstructions. However,theGUTrelations
forYukawacouplingsareoftennotretained[292–294]because
the matter multiplets of the effective theory may have a
complicated origin in terms of the underlying string states.
Another difference is that Yukawa couplings in stringderived
models may be absent due to symmetries in the underlying
string construction, even though they are not forbidden by any
obvioussymmetriesofthefourdimensionaltheory,contraryto
the assumptions in many nonstring models. Finally, higher
dimensional operators, suppressed by inverse powers of the
Planck scale, are common.
Much activity on neutrino masses in string theory
occurred following the first superstring revolution.
particular, a number of authors considered the implications
of an E6 subgroup of the heterotic E8× E8 construction
[292,295–297].Assuming that the matter content of the
effective theory involves three 27s, one can avoid neutrino
masses altogether by finetuned assumptions concerning
the Yukawa couplings [292].
implement a canonical type I seesaw.
two Standard Model singlets, which are candidates for right
handed neutrinos, and for a field which could generate a large
Majorana mass for the righthanded neutrinos if it acquires a
largevacuumexpectationvalueandhasanappropriatetrilinear
coupling to the neutrinos. However, there are no such allowed
trilinear couplings involving three 27s (this is a reflection of
the fact that the 27 does not contain a 126 of the SO(10)
subgroup). E6stringinspired models were constructed to get
around this problem by invoking additional fields not in the
27 [294,298] or higherdimensional operators [297], typically
leading to extended versions of the seesaw model involving
fields with masses or vevs at the TeV scale.
Similarly, more recent heterotic and intersecting brane
constructions,e.g.involvingorbifoldsandtwistedsectors,may
well have the necessary fields for a type I seesaw, but it is
again required that the necessary Dirac Yukawa couplings and
Majorana masses for the righthanded neutrinos be present
simultaneously.Dirac couplings need not emerge at the
renormalizable level, but can be of the form
In
However, it is difficult to
Each 27 contains
?S?
1···S?
d−3?NLHu/Md−3
PL,
(57)
where the S?
expectation values (d = 3 corresponds to a renormalizable
operator). Similarly, Majorana masses can be generated by
the operators:
iare Standard Model singlets which acquire large
?S1···Sn−2?NN/Mn−3
PL.
(58)
Whether such couplings are present at the appropriate
orders depends on the underlying string symmetries and
selection rules, which are often very restrictive.
also necessary for the relevant S and S?fields to acquire
the large expectation values that are needed, presumably
without breaking supersymmetry at a large scale. Possible
mechanisms involve approximately flat directions of the
potential, e.g. associated with an additional U(1)?gauge
symmetry [299,300], string threshold corrections [301–303]
or hidden sector condensates [304].
There have been surprisingly few investigations of
neutrino masses in explicit semirealistic string constructions.
It is difficult to obtain canonical Majorana masses in
intersecting brane constructions [305] because there are no
interactions involving the same intersection twice.
detailed studies [278,279] of nonsupersymmetric models
with a low string scale concluded that lepton number was
conserved, though a small Dirac mass might emerge from a
large internal dimension. Large enough internal dimensions
for the supersymmetric case may be difficult to achieve, at
least for simple toroidal orbifolds.
There are also difficulties for heterotic models. An early
study of Z3 orbifolds yielded no canonical Dirac neutrino
Yukawa couplings [295] at low order. Detailed analyses of
freefermionic models and their flat directions were carried
out in [304,306] and [307,308]. Both studies concluded that
small Majorana masses could be generated if one made some
assumptionsaboutdynamicsinthehiddensector. In[304,306]
themasseswereassociatedwithanextendedseesawinvolving
a low mass scale. The seesaw found in [307,308] was of the
canonical type I type, but in detail it was rather different from
GUTtype models. A seesaw was also claimed in a heterotic
Z3orbifoldmodelwithE6breakingtoSU(3)×SU(3)×SU(3)
[309]. A recent study of Z6 orbifold constructions found
Majoranatype operators [310], but (to the order studied) the
Si fields did not have the required expectation values when
Rparity is conserved.
In [311] a large class of vacua of the bosonic Z3
orbifoldwereanalysedwithemphasisontheneutrinosectorto
determine whether the minimal type I seesaw is common, or
ifnottofindpossibleguidancetomodelbuilding,andpossibly
to get clues concerning textures and mixing if examples
were found. Several examples from each of 20 patterns of
vacua were studied, and the nonzero superpotential terms
through degree 9 determined. There were a huge number
of Dflat directions, with the number reduced greatly by the
Fflatness condition. Only two of the patterns had Majorana
mass operators, while none had simultaneous Dirac operators
of low enough degree to allow neutrino masses larger than
10−5eV. (One apparently successful model was ruined by
offdiagonal Majorana mass terms.) It is not clear whether
this failure to obtain a minimal seesaw is a feature of the
particular class of construction, or whether it is suggesting
that string constraints and selection rules might make string
vacua with minimal seesaws rare. Systematic analyses of the
neutrino sector of other classes of constructions would be very
useful.
There are other possibilities for obtaining small neutrino
masses in string constructions, such as extended seesaws
[304,306] and small Dirac masses from higher dimension
It is
Two
34
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
operators [300]. Small Dirac neutrino masses in models with
anisotropiccompactificationsmotivatedbytypeIstrings[312]
have been discussed recently in [313].
embedding type II seesaw ideas (involving Higgs triplets)
in heterotic string constructions was considered in [314]. It
is possible to obtain a Higgs triplet of SU(2) with non
zero hypercharge in a higher level construction (in which
SU(2) × SU(2) is broken to a diagonal subgroup). In this
case, because of the underlying SU(2) × SU(2) symmetry
the Majorana mass matrix for the light neutrinos should
involve only offdiagonal elements (often with one of the
three offdiagonal elements small or vanishing). This leads to
phenomenological consequences very different from those of
tripletmodelsthathavebeenmotivatedbygrandunificationor
bottomupconsiderations,includinganinvertedhierarchy,two
large mixings, a value of Ue3induced from the chargedlepton
mixings that is close to the current experimental lower limit,
and an observable neutrinoless doublebeta decay rate. This
string version of the triplet model is a topdown motivation for
theLe−Lµ−Lτconservingmodelsthathavepreviouslybeen
considered from a bottomup point of view [315], but has the
advantage of allowing small mixings from the chargedlepton
sector. A recent study indicates that it may also be possible to
generate a type II seesaw in intersecting D6brane models
involving SU(5) grand unification, although the examples
constructed are not very realistic [316].
These comments indicate that string constructions may
be very different from traditional grand unification or bottom
up constructions, mainly because of the additional string
constraints and symmetries encountered.
minimal seesaw (though perhaps with noncanonical family
structure)areundoubtedlypresentamongstthelargelandscape
ofstringvacua,thoughperhapstheyarerare. Onepointofview
istosimplyfocusonthesearchforsuchstringvacua. However,
another is to keep an open mind about other possibilities that
may appear less elegant from the bottomup point of view but
which may occur more frequently in the landscape.
The possibility of
Versions of the
3.1.5.
Neutrino mass may arise in a class of nonSUSY models
via L = 2 scalarlepton–lepton Yukawa interactions. The
Lagrangian can be written generically as follows:
TeVscale mechanisms for small neutrino masses.
−Lyuk= fijH++lilj+ gijH+liνj+ hijH0νiνj
+hermitian conjugate.
(59)
Here H±±,H±and H0are doublycharged, singlycharged
and neutral scalars, respectively, which originate from an
SU(2)L,Risospin singlets (I = 0) or triplets (I = 1). Each
scalar is assigned L = 2.
neutrinos (ν) may be of either chirality. Four examples of
modelswhichutilizevarioustermsinLyuktogenerateneutrino
mass are listed below:
The charged leptons (l±) and
• The left–right symmetric model: TeV scale breaking
of SU(2)R via the righthanded scalar triplet vacuum
expectation value which gives rise to a TeV scale seesaw
mechanism [317];
• Higgs triplet model: Treelevel neutrino mass for the
observed neutrinos proportional to SU(2)Ltriplet scalar
vev (no righthanded neutrino) [318];
• Zee model: Radiative neutrino mass at 1loop via SU(2)L
singlet scalar H±[319] and
• Babumodel:Radiativeneutrinomassat2loopviaSU(2)L
singlet scalars H±±and H±[320].
All the above models can provide TeVscale mechanisms
of neutrinomass generation consistent with current neutrino
oscillation experiments.New particle discovery (e.g.
Z?,W?,H±±) at the large hadron collider (LHC) is also
a possibility if MZ?,MW?
< 3–4TeV, MH±±
Precision measurements of the neutrinomass matrix at a
Neutrino Factory would provide valuable information on
the Yukawa couplings f,g,h. Such couplings also induce
leptonflavour violating (LFV) decays (e.g. µ → eee,µ →
eγ) [62,321], which might also form part of the research
programme at a Neutrino Factory. Importantly, any signal
for µ → eγ from the MEG experiment can be interpreted
in the above models.The first pair of models above can
accommodate any value of sinθ13and any of the currently
allowed mass hierarchies, normal (NH), inverted (IH) and
degenerate (DG). The second pair of models above are
more predictive for sinθ13and accommodate specific neutrino
mass hierarchies. A distinctive feature of all the models is
the synergy between precision measurements of oscillation
parameters (at a Neutrino Factory), LFV decays of µ and τ
and direct searches for the L = 2 scalars, all of which involve
the couplings f,g,h.
< 1TeV.
Left–right symmetric model.
model[322]isanextensionoftheStandardModelbasedonthe
gaugegroupSU(2)R⊗SU(2)L⊗U(1)B–L. TheLRsymmetric
model has many virtues, e.g.
The left–right (LR) symmetric
• The restoration of parity as an original symmetry of the
LagrangianwhichisbrokenspontaneouslybyaHiggsvev
and
• The replacement of the arbitrary SM hypercharge Y by
the theoretically more attractive B–L.
Although the Higgs sector is arbitrary, a theoretically and
phenomenologicallyappealingwaytobreaktheSU(2)Rgauge
symmetryisbyinvokingHiggsisospintripletrepresentations.
Such a choice conveniently allows the implementation of a
lowenergy seesaw mechanism for neutrino masses. A right
handed neutrino is required by the SU(2)Rgauge group and
leptons are assigned to multiplets with quantum numbers
(TL,TR, B–L):
?
l?
i
L
?
l?
i
R
LiL=
ν?
i
?
?
: (1/2 : 0 : −1),
LiR=
ν?
i
: (0 : 1/2 : −1).
(60)
Here i = 1,2,3 denotes generation number. The Higgs sector
consists of a bidoublet Higgs field, ?, and two triplet Higgs
35
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
fields, ?Land ?R:
? =
?
?
?
φ0
φ−
1φ+
1φ0
L/√2δ++
L− δL+/√2
δ+
δ0
2
2
?
: (1/2 : 1/2 : 0),
?L=
δ+
L
δ0
?
?
: (1 : 0 : 2),
(61)
?R=
R/√2δ++
R− δR+/√2
R
: (0 : 1 : 2).
The vevs for these fields are as follows:
?
0
κ2
?0
The gauge groups SU(2)R and U(1)B–L are spontaneously
broken at the scale vR. Phenomenological considerations
require vR ? κ =
not play a role in the breaking of the gauge symmetries and
is constrained to be small (vL < 8GeV) in order to comply
with the measurement of ρ = M2
Lagrangian responsible for generating neutrino mass is as
follows:
−L =¯LL(yD? + ˜ yD˜?)LR+ iyM(LT
+ LT
where yMis a 3 × 3 Majoranatype Yukawa coupling matrix.
ExpandingthetermsproportionaltoyMresultsinaLagrangian
of the form of equation (59) with yM= f =
6×6 mass matrix for the neutrinos can be written in the block
form
?
Each entry is given by
1
√2(yDκ1+ ˜ yDκ2);
??? =
κ1
0
?
1
√2,
??L? =
?
00
0
vL
?
1
√2,
??R? =
0
0
vR
?
1
√2.
(62)
?
κ2
1+ κ2
2∼ 2MW1/g. The vev vLdoes
Zcos2θW/M2
W∼ 1. The
LCτ2?LLL
RCτ2?RLR) + hermitian conjugate,
(63)
√2g = h. The
MLR
ν
=
ML
mT
mD
D
MR
?
.
(64)
mD=
MR=
√2hvR;
ML=
√2hvL.
(65)
The neutrinomass matrix is diagonalized by a 6 × 6 unitary
matrix V as VTMνV
= Mdiag
M2,M3), where miand Miare the masses for neutrinomass
eigenstates. The small neutrino masses miare generated by
the Type II seesaw mechanism. Obtaining eV scale neutrino
masses with h = O(0.1–1) requires ML(and consequently vL)
to be at the eV scale. In LR model phenomenology, with
vR ∼ TeV, it is customary to arrange the Higgs potential
such that vL = 0 [323]. In this case the masses of the light
neutrinos arise from the Type I seesaw mechanism and are
approximately mi ∼ m2
energy (∼O(1–10)TeV) scale for the righthanded Majorana
neutrinos, the Dirac mass term, mD, should be O (MeV),
which for κ2∼ 0 corresponds to yD∼ 10−6(i.e. comparable
in magnitude to the electron Yukawa coupling).
model with vRof order a TeV predicts leptonflavour violating
(LFV) decays of the muon and tau mediated by H±±with a
ν
= diag(m1,m2,m3,M1,
D/MR. In order to realize the low
The LR
rate ∼hh2/M4
searches at the LHC.
H±± [324], and a rich phenomenology in direct
Higgs triplet model.
[70,325]69a single I = 1, Y = 2 complex SU(2)Ltriplet
?L(see equation (62)) is added to the SM with the Yukawa
coupling:
In the Higgs triplet model (HTM)
iyM(LT
LCτ2?LLL) + hermitian conjugate.
(66)
Expanding equation (66) results in equation (59) with yM=
f =
thelightneutrinosreceiveaMajoranamassproportionaltothe
lefthanded triplet vev (vL) leading to the following neutrino
mass matrix:
MHTM
The presence of a trilinear coupling µ?Tiτ2?†
is the SM Higgs doublet with vev v) in the Higgs potential
ensures a nonzero vL ∼ µv2/M2, where M is the mass of
the triplet scalars. Taking M to be at the TeV scale results
in vL ∼ µ.
HTM does not provide predictions for the elements of Mν
but instead accommodates the observed values (as does the
LR model). However, combining accurate measurements of
the neutrino oscillation parameters with any signals in LFV
processes involving the muon or the tau [326] and/or direct
observation of H±±[327] would enable this mechanism of
neutrino mass generation to be tested. From equation (67) hij
is directly related to the neutrino masses and mixing angles as
follows:
√2g = h. No righthanded neutrino is introduced, and
ν
=
√2vLhij.
(67)
L? (where ?
From equation (67) it is apparent that the
hij=
1
√2vL
VPMNSdiag(m1,m2,m3)VT
PMNS.
(68)
Observation of LFV decays of the muon for example at MEG
and/orofthetau(ataSuperBFactory)togetherwithdiscovery
of H±±(at LHC) would permit measurements of hij.
Neutrino Factory would greatly reduce the experimental error
in the righthand side of equation (68) and allow the above
identity in the HTM to be checked precisely.
A
One loop radiative mechanism via a singlycharged, singlet
scalar(Zeemodel).
Asinglycharged, singletscalarisadded
to the two Higgs doublet model (2HDM) extension of the SM.
Neutrino mass is generated radiatively via a 1loop diagram
figure 24(a) in which the mixing between the charged singlet
and doublet scalars (proportional to a trilinear coupling µ) is
crucial [319]. The relevant Lagrangian is
LZee= gab(LTi
aLCLj
bL)?ijH++
?
i=1,2
ykLLHilR
+ hermitian conjugate,
(69)
69The model of [318] contains a triplet Majoron and was excluded by LEP
data.A viable extension of the HTM which contains a singlet Majoron
(referred to as the ‘123’ model) was introduced in [261].
36
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 24. Diagram for neutrinomass generation in (a) Zee model
and (b) Babu model.
where yk is the Yukawa coupling of the doublet Hk to
the leptons. If only one of the Higgs doublets couples to
leptons (referred to as the ‘minimal Zee model’) the resulting
neutrinomass matrix is symmetric with vanishing diagonal
elements:
where [321]
MZee
ν
=
0
meµ
0
meτ
meµ
mµτ
0
meτ
mµτ
,
(70)
mij= gij(m2
lj− m2
li)µF
1
16π2
1
m2
S1− m2
S2
lnm2
m2
S1
S2
(71)
and mliare the charged lepton masses, MSiare the charged
scalar masses and F = cotβ(tanβ) for Type I (II) couplings
of the doublets to the leptons. The above mass matrix predicts
thesolarangletobealmostmaximal, whichisnowruledoutat
the6σ level(seesection2.2.1). However,allowingbothHiggs
doublets to couple to the leptons (the ‘general Zee model’)
leads to nonzero diagonal elements in MZee
maximal solar angle can then be accommodated, sinθ13?= 0
is expected and an inverted hierarchical neutrino mass pattern
is predicted.
ν
[328]. The non
Two loop radiative mechanism via singly and doublycharged,
singletscalars(Babumodel).
H±±and H±are added to the SM Lagrangian [320] with the
following Yukawa couplings:
SU(2)Lsingletchargedscalars
L = fab(lT
+ hermitian conjugate.
aRClbR)H+++ gab(LTi
aLCLj
bL)?ijH+
(72)
No righthanded neutrino is introduced. A Majorana mass for
the light neutrinos arises at the twoloop level (figure 24(b))
in which the lepton number violating trilinear coupling
µH±H±H±±plays a crucial role. The explicit form for Mν
is as follows:
MBabu
ν
= ζ ×
?2ωττ+ 2???ωµτ+ ??2ωµµ, ?ωττ+ ??ωµτ− ???ωeτ
−??2ωeµ,
ωττ− 2??ωeτ+ ??2ωee,
−?ωττ− ??ωµµ− ?2ωeτ
−???ωeµ
−ωµτ− ?ωeτ+ ??ωeµ
+???ωee
ωµµ+ 2?ωeµ+ ?2ωee
·
··
,
(73)
where ? = geτ/gµτ, ??= geµ/gµτ, ωab= fabmamb(ma,mb
are chargedlepton masses) and ζ is given by
ζ =
8µg2
(16π2)2m2
µτ˜I
H±.
(74)
Here˜I is a dimensionless quantity of O(1) originating from
the loop integration. The expression for Mν involves nine
arbitrary couplings. Since the model predicts one massless
neutrino(atthetwolooplevel),quasidegenerateneutrinosare
not permitted and only normalhierarchy (NH) and inverted
hierarchy (IH) mass patterns can be accommodated.
g couplings (contained in ? and ??) are directly related to the
elements of Mν, and thus would be obtained precisely at a
Neutrino Factory. In the scenario of NH, ? ≈ ??≈ tanθ12/√2
and sinθ13is close to zero. Since ?,??< 1 one may neglect
those terms in Mνwhich are proportional to the electron mass
(i.e. ωee, ωeµ, ωeτ). This simplification leads to the following
prediction: fµµ: fµτ: fττ≈ 1 : mµ/mτ: (mµ/mτ)2. In the
case of IH, large values are required for ?, ??(>5), and thus
neglecting ωee, ωeµ, ωeτin Mνmay not be entirely justified.
However, ifsuchtermsareneglectedthentheaboveprediction
for the ratio of fµµ : fµτ : fττ also holds approximately
for the case of IH. A lower bound on s13 > 0.05 can also
be derived. If the twoloop diagram is solely responsible for
the generation of the neutrinomass matrix the Babu model
requires g,fµµ∼ 10−2. Such relatively large couplings may
lead to observable rates for LFV decays of muons and taus.
The
3.2. Unification and flavour
A survey of the theoretical models that have been developed
to explain the physics of flavour is presented in this section.
Measurables that can be used to distinguish between the
various models is also presented. These measurables include
the mixing angles themselves and combinations of mixing
angles; the latter are referred to as ‘sum rules’. This section
also contains a discussion of leptonflavour violation.
3.2.1. Model survey.
lated forms of the Yukawa matrices, one must appeal to some
sort of Family symmetry, GFamily. In the framework of the
seesaw mechanism, new physics beyond the Standard Model
is required to cause leptonnumber conservation to be violated
and to generate righthanded neutrino masses at around the
GUT scale. This is exciting since it implies that the origin of
neutrino masses is related to a GUT symmetry group GGUT,
whichunifiesthefermionswithinafamily. Puttingtheseideas
together leads to the development of a framework for physics
To understand the origin of the postu
37
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
beyond the SM which is based on N = 1 supersymmetry with
commutingGUTandFamilysymmetrygroups,GGUT×GFAM.
TherearemanypossiblecandidateGUTandFamilysymmetry
groups. Unfortunately the model dependence does not end
there; the details of the symmetrybreaking vacuum plays a
crucial role in specifying the model and in determining the
masses and mixing angles. These models may be classified
according to the particular GUT and Family symmetry that is
assumed.
It may be possible to use precise measurements of the
oscillationparameterstodistinguishbetweendifferentmodels.
Asurveyofover60neutrinomassmodelshasbeenperformed.
The survey included the following:
• Modelswithassumptionsaboutthestructureofthemixing
matrix (’texture’ assumptions);
• Models based on lepton symmetries such as A4, S3or
Le–Lµ–Lτand
• ModelsbasedonGUTsymmetriessuchasSU(5),flipped
SU(5), SO(10), E6or E8× E8.
These models are reviewed briefly below with emphasis on
how the different predictions arise from different symmetry
breaking patterns. A detailed, tabulated summary of the
predictions for all three angles with references to models that
have been included in our survey can be found in [329].
Modelswithleptonsymmetriesbasedonµ–τ symmetry.
maximal (or near maximal) mixing observed in atmospheric
neutrinos strongly suggests a µ–τ symmetry in the neutrino
mass matrix. There are two ways to realize the µ–τ symmetry
whichgiverisetomaximalmixingintheatmosphericneutrino
sector, θ23=π
form
which gives rise to the normal mass hierarchy. In this case,
when the µ–τ symmetry is exact, the 1–3 mixing angle
vanishes, sinθ13 = 0. In addition, the mass splitting in the
solar neutrino sector vanishes, ?m2
?m2
small parameters of the order of O(? ? 1),
?
2
d?
The
4[330]. The first possibility is of the following
,
Mν?
0
0
0
0
1
1
0
1
1
(75)
12= 0. Nonvanishing
12can be generated in a µ–τ symmetric way by adding
Mν?
?m2
13
c?d?
1 + ?
−1
d?
−1
1 + ?
d?
,
(76)
where the coefficients c and d are of order 1. This leads to
θ13= 0,θ23=π
4,
tan2θ12?
2√2d
(1 − c),
13and ?m2
(77)
and the parameter ? is fixed by the ratio of ?m2
12as
? =
4
1 + c +
?
(c − 1)2+ 8d2
?
?m2
?m2
12
13
.
(78)
In order to generate nonzero θ13, the µ–τ symmetry has
to be broken. How the symmetry breaking occurs dictates
Table 4. Predictions for θ13and for the deviation (θ23− π/4) in
models with softly broken µ–τ symmetry for different
symmetrybreaking directions. This table is taken from [330].
Symmetry breaking
θ13
θ23−π
0
?8◦
?4◦
Large
4
None
µ–τ sector only
esector only
0
∼?m2
∼
∼
12/?m2
?m2
13
?
12/?m2
13
Dynamical
?
?m2
12/?m2
13
the size of the θ13angle. The µ–τ symmetry breaking also
causes θ23 to differ from
maximal. The breaking of the µ–τ symmetry can generally be
parametrized as
where the parameter a is of order unity. If the breaking is
introduced in the esector, that is, a = 1, b ?= d, one then has
?
?m2
13
π
4, i.e. the mixing is no longer
Mν?
?
?m2
13
2
c? d?b?
−1
1 + ?
d?
1 + a?
−1
b?
,
(79)
? =
4
√1 + 8d2
?m2
12
,
tan2θ12?2(b + d)
(1 − c)
(80)
and a nonvanishing θ13angle:
θ13= (b − d)
?
?m2
?m2
12
13
.
(81)
A nonvanishing deviation of the atmospheric mixing angle
from
breaking of the µ–τ symmetry can also be introduced in the
µ–τ sector. This is characterized by a ?= 1 and b = d. In this
case, the parameter ? is related to ?m2
π
4can exist with magnitude
π
4− θ23 ∼ O(?2). The
12and ?m2
13by
? =
4
c +1
2(1 + a) +
?
(c −1
2(1 − c))2+ 8d2
?
?m2
?m2
12
13
.
(82)
Thus, thepredictionsforsinθ13andπ/4−θ23stronglydepend
on the symmetrybreaking pattern. Table 4 summarizes the
predictions for θ13 and for
breaking scenarios.
The inverted mass hierarchy can be obtained when the
neutrinomass matrix is of the form
This mass matrix has an enhanced Le–Lµ–Lτ symmetry
[331,332] and is a special case of the following mass matrix:
π
4− θ23 for various symmetry
Mν?
0
1
1
1
0
0
1
0
0
.
(83)
Mν?
?
?m2
13
0 sinθ
0
0
cosθ
0
0
sinθ
cosθ
.
(84)
38
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
In the exact Le–Lµ–Lτ symmetric limit, this leads to the
following predictions [331]:
θ12=π
?m2
12= 0,θ13= 0,
4,
sin22θ23= sin22θ.
(85)
Since θ12 ?=
broken. The soft breaking of the Le–Lµ–Lτsymmetry can be
introducedbyaddingsmalle–e, µ–µ, µ–τ andτ–τ couplings:
For nonzero x, y and d, one has
??m2
Thebreakingoftheµ−τ symmetrycanariseintheµ–τ sector,
i.e. cosθ = sinθ = 1/√2 and x ?= y, which leads to
θ13=1
?m2
π
4, the Le–Lµ–Lτ symmetry has to be softly
Mν?
?
?m2
13
z
sinθ
cosθ
sinθ
cosθ
yd
dx
,
x,y,d ? 1.
(86)
sin22θ12? 1 −
12
4?m2
13
− z
?2
.
(87)
2(x − y),
?m2
12
13
= 2(x + y + z + d).
(88)
The breaking of the µ–τ symmetry can also be introduced
in the esector by having cosθ ?= sinθ and x = y. This
leads to θ13? −d cos2θ23. In the invertedhierarchy case, the
correlations among the neutrinomixing angles is not as strong
as in the normalhierarchy case.
SingleRH neutrino dominance.
nance (SRND), proposed in [333], can be implemented in
manyclassesofmodel;itisthereforeamechanismratherthana
model. SRNDprovidesanaturalwaytogeneratelargemixing
angles. In the simplified case, with only the second and third
families, the Dirac neutrinomass matrix and RH Majorana
neutrinomass matrix are generally of the form
in the basis where the RH Majorana neutrinomass matrix is
diagonal. The effective light neutrinomass matrix is then
given by
SingleRH neutrino domi
MD=
·
·
·
·
a
·
b
cd
,
MR=
·
·
·
··
0
x
0
y
,
(89)
mν= −MD· M−1
R· MT
D=
·
·
·
+b2
·
a2
xy
ac
x
c2
+bd
y
·
ac
x
+bd
yx
+d2
y
.
(90)
If one RH neutrino dominates, that is, if y ? x, then the
subdeterminant in the µ–τ block is roughly of the order
∼m2· m3. The normal hierarchy is obtained for m2 ? m3.
The atmospheric mixing angle is roughly given by tanθ23∼
(a/c). For a ∼ c, large mixing angles can arise naturally.
The twofamily case can be generalized to the threefamily
case when sequential dominance with three RH neutrinos is
implemented [334].
Models with GUT symmetries.
on SO(10) accommodate all 16 fermions (including the
righthanded neutrinos) in a single spinor representation.
Furthermore, SO(10) provides a framework in which the see
saw mechanism arises naturally. Models based on SO(10)
combined with a continuous, or discrete, flavour symmetry
grouphavebeenconstructedtounderstandtheflavourproblem,
especially the small neutrino masses and the large leptonic
mixing angles. These models can be classified according to
the family symmetry that is implemented as well as the Higgs
representations introduced in the model. For reviews, see,
for example, [335]. Phenomenologically, the resulting mass
matrices can be either symmetric, lopsided or asymmetric.
Due to the product rule, 16⊗16 = 10⊕120a⊕126s, the
only Higgs particles that can couple to the matter fields at tree
level are in the 10, 120, and 126 representations of SO(10).
The Yukawa matrices involving the 10 and 126 are symmetric
underinterchangeoffamilyindices,whilethematrixinvolving
the 120 is antisymmetric. The Majorana mass term for the
RH neutrinos can arise either from a renormalizable operator
involving the 126, or from a nonrenormalizable operator that
involves the 16s. The case of 126 has the advantage that
Rparity is preserved automatically.
Two large mixing angles in the leptonic sector may arise
in two ways:
Grand unified theories based
1. Symmetric mass textures.
if SO(10) is broken through the left–right symmetry
breaking route. In this case, both the large solar mixing
angle and the maximal atmospheric mixing angle come
from the effective neutrinomass matrix. A characteristic
of this class of models is that the predicted value for the
Ueν3elementtendstobelargerthanthevaluepredictedby
models in class (ii) below. This GUTsymmetrybreaking
patterngivesrisetothefollowingrelationsamongvarious
mass matrices:
This scenario is realized
Mu= MνD,Md= Me,
(91)
up to some calculable, grouptheoretical factors which
are useful in obtaining the Jarlskog relations among
masses for the charged leptons and downtype quarks
when combined with family symmetries.
of Ue3is predicted to be large, close to the sensitivity
of current experiments. The prediction for the rate of
µ → eγ is about two orders of magnitude below the
current experimental bound.
In a particular model constructed by Chen and
Mahanthappa [336], the Higgs sector contains fields
in 10, 45, 54, 126 representations, with the 10 and
126 breaking the electroweak symmetry and generating
fermions masses and the 45, 54, 126 breaking the SO(10)
GUT symmetry. The mass hierarchy can arise if there is
an SU(2)Hsymmetry acting nontrivially on the first two
generations such that the first two generations transform
as a doublet and the third generation transforms as a
singlet under SU(2)H, which breaks down in two steps,
SU(2)
→ U(1)
mass hierarchy is generated by the Froggatt–Nielsen
The value
?M ??M
→ ‘nothing’, ??? ? ? 1. The
39
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
mechanism[230]. TheresultingmassmatricesattheGUT
scale are given by
Therighthandedneutrinomassmatrixisofthesameform
as MνLR:
Notethat,sincethe126dimensionalHiggsrepresentation
is used to generate the heavy Majorana neutrinomass
terms, Rparity is preserved at all energies. The effective
neutrinomass matrix is
Mu,νLR=
0
0
0
?10+
?10+
?10+
2???
3??
1?
?10+
?10+
r2??
4??
3??
?10+
0
0
r2??
2???
=
0
r4??
1
?
MU,
?10−
(1,−3)?126
(92)
Md,e=
0
5???
0
?10−
5???
0
−??
0
0
?10−
1?
=
0
??
0
??
0
0
1
(1,−3)p?
0
MD.
(93)
MνRR=
00
?126
?126
?126
?0
2?δ1
?0
2?δ3
?0
1?
0
?126
?126
MR.
?0
2?δ2
?0
2?δ3
?126
0
0
?0
2?δ1
0
=
δ1
δ2
δ3
1
δ1
δ3
(94)
MνLL= MT
νLRM−1
νRRMνLR=
0
0
0
1
t
1 + t?
1
t
1 + t?
d2v2
u
MR
,
(95)
and causes the atmospheric mixing angle to be maximal
and the solar mixing angle to be large. The form of the
neutrinomass matrix in this model is invariant under the
seesawmechanism. ThevalueofUe3isrelatedtotheratio
∼
sensitivity of current experiments. The prediction for the
rate of µ → eγ is about two orders of magnitude below
the current experimental bound.
2. Lopsided mass textures for charged fermions.
scenario, the large atmospheric mixing angle comes from
the unitary matrix that diagonalizes the chargedlepton
mass matrix. This scenario is realized in models with
SU(5) as the intermediate symmetry which gives rise to
the socalled ‘lopsided’ mass textures, due to the SU(5)
relation:
Me= MT
Due to the lopsided nature of Me and Md, the large
atmospheric neutrino mixing is related to the large
?
?m2
12/?m2
13, which is predicted to be close to the
In this
d.
(96)
mixing in the (23) sector of the RH chargedlepton
diagonalization matrix, instead of Vcb. Thus it explains
why Vcb is small while Uµν3is large. The large solar
mixing angle comes from the diagonalization matrix for
the neutrinomass matrix. Because the two large mixing
anglescomefromdifferentsources, theconstraintonUeν3
is not as strong as in class (1). In fact, the prediction for
Ueν3in this class of models tends to be quite small. On
the other hand, this mechanism also predicts an enhanced
decayratefortheflavourviolatingprocessµ → eγ which
is close to the current experimental limit. As Rparity is
broken by the vev of the 16 dimensional Higgs, a separate
‘matterparity’mustbeimposedtodistinguishtheparticles
from their SUSY partners.
In a particular model constructed by Albright and
Barr [337], the Higgs sector of the model contains
Higgs particles in the 10,16,45, with ?16H1? breaking
SO(10) down to SU(5) and ?16H2? breaking the EW
symmetry. The lopsided textures arise due to the operator
λ(16i16H1)(16j16H2) which gives rise to mass terms for
the charged leptons and down quarks which satisfy the
SU(5) relation Md = MT
included, the lopsided structure of Meresults, provided
the coupling σ is of order 1:
The large mixing in Ue,Lleads to the large atmospheric
mixing angle. Meanwhile, because large mixing in Ue,L
corresponds to large mixing in Ud,R, the CKM mixing
angles remain small. A unique prediction of the lopsided
models is the relatively large branching ratio for LFV
processes, e.g. µ → eγ. By considering a RH Majorana
neutrinomass term of the following form, a large solar
mixing angle can arise for some choice of the parameters
inMνRR,leadingtoalargevalueforthesolarmixingangle:
Models with renormalizationgroup enhancements.
possible to obtain large neutrinomixing angles through
renormalizationgroup evolution. Assuming that the CKM
matrix and the leptonicmixing matrix are identical at the
e. When other operators are
Mu,νLR=
η
η
0
0
0
0
0
(1/3,1)?
1
· md,
δ?eiφ
−?
1
−(1/3,1)?
δ
0
−?/3
η
· mu
(97)
Md=
δ?eiφ
σ + ?/3
1
δ
δ?eiφ
Me=
δ
0
δ
δ?eiφ
σ + ?
· md.
(98)
MνRR=
c2η2
−b?η
aη
−b?η
?2
−?
−?
0
2?
aη
−?
1
· ?R,
m2
Meff
ν
=
00
2?
1
−?
0
u
?R.
(99)
It is
40
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 25. Histogram of the number of models for each sin2θ13. The diagram on the left includes all 60 models, while the diagram on the
right includes only those that give predictions for all three leptonicmixing angles. Taken with the kind permission of the Physical Review
from figures 1 and 2 in [329]. Copyrighted by the American Physical Society.
GUT scale, which is a natural consequence of quark
lepton unification, two large neutrinomixing angles can be
generatedbyrenormalizationgroupevolution[338]. Theonly
requirement for this mechanism to work is that the masses of
thethreeneutrinosarenearlydegenerate(m3? m2? m1)and
havethesameCPparity. Theonelooprenormalizationgroup
equation(RGE)oftheeffectivelefthandedMajorananeutrino
mass operator is given by
dmν
dt
= −{κumν+ mνP + PTmν},
(100)
wheret ≡ lnµandµistheenergyscale. IntheMSSM,P and
κuare given by
1
32π2
32π2
≡ diag(0,0,Pτ);
1
16π2
sin2β
?6
respectively, where g2
1=
constant, YuandYearethe3×3Yukawacouplingmatricesfor
the up quarks and charged leptons, respectively, and htand hτ
are the t and τYukawa couplings. One can then follow the
‘diagonalizeandrun’procedureandobtaintheRGEsatscales
between MR? µ ? MSUSYfor the mass eigenvalues and the
three mixing angles, assuming CPviolating phases vanish:
dmi
dt
ds23
dt
ds13
dt
P = −
Y†
cos2β? −
eYe
1
h2
τ
cos2βdiag(0,0,1)
(101)
κu=
?6
5g2
1+ 6g2
2− 6Tr(Y†
uYu)
?
?
1
16π2
5g2
1+ 6g2
2− 6
h2
t
sin2β
?
;
(102)
5
3g2
Yis the U(1) gauge coupling
= −4PτmiU2
τνi− miκu,(i = 1,2,3);
(103)
= −2Pτc2
23(−s12Uτν1∇31+ c12Uτν2∇32);
(104)
= −2Pτc23c2
13(c12Uτν1∇31+ s12Uτν2∇32);
(105)
ds12
dt
= −2Pτc12(c23s13s12Uτν1∇31
−c23s13c12Uτν2∇32+ Uτν1Uτν2∇21);
where ∇ij ≡ (mi+ mj)/(mi− mj). Because the leptonic
mixing matrix is identical to the CKM matrix, we have, at
the GUT scale, the following initial conditions, s0
s0
parameter. Whenthemassesmiandmjarenearlydegenerate,
∇ij approaches infinity. Thus it drives the mixing angles to
become large. Starting with the values of (m0
(0.2983,0.2997,0.3383)eV at the GUT scale, the solutions
at the weak scale for the masses are (m1,m2,m3)
(0.2410,0.2411,0.2435)eV, which correspond to ?m2
1.1 × 10−3eV2and ?m2
angles predicted at the weak scale are sin22θ23 = 0.99,
sin22θ12 = 0.87 and sinθ13 = 0.08. Because the masses
are larger than 0.1eV, they are testable at the present searches
for the neutrinoless doublebeta decay.
(106)
12? λ,
23? O(λ2) and s0
13? O(λ3), where λ is the Wolfenstein
1,m0
2,m0
3) =
=
13=
?= 4.8 × 10−5eV2. The mixing
Predictions for the oscillation parameters.
there are 30 models based on SO(10), six models that utilize
singleRHneutrino dominance mechanism, five based on Le–
Lµ–Lτ symmetry, ten based on S3 symmetry, three on A4
symmetry,oneonSO(3)symmetryandthreebasedontexture
zero assumptions. The predictions of these models for sin2θ13
are summarized in figures 25 and 26. In some models, a range
of values (rather than a single value) is given for θ13. If these
values range over N bins for sin2θ13in a particular model, a
weightof1/N isassignedforeachbin. Asaresult,noninteger
valuesforthenumberofmodelsforsomevaluesofsin2θ13can
arise.
Figure25showsthehistogramofthenumberofmodelsfor
eachsin2θ13includingallsixtymodelsandoneincludingonly
models that predict all three mixing angles. An observation
one can draw immediately is that the predictions of SO(10)
models are larger than 10−4, and the median value is roughly
∼10−2. Furthermore,sin2θ13< 10−4canonlyariseinmodels
In the literature,
41
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
based on leptonic symmetries. However, these models are not
as predictive as the GUT models, due to the uncertainty in the
chargedlepton mixing matrix. In this case, to measure θ13
will require a neutrino superbeam or the Neutrino Factory. In
table 5 the reach of future experiments is summarized.
In figure 26, histograms of the number of models for each
sin2θ13valueareshownforbothnormalandinvertedneutrino
mass hierarchies. From these two diagrams, one finds that
there are more models that predict the normal hierarchy than
the inverted hierarchy. This could merely be a result of the
theorists’ prejudice for the model. What is more important
is the correlation between the type of the hierarchy and the
predicted values for θ13. In the normalhierarchy case, the
predicted values tend to be larger, while in the inverted case,
the distribution is quite uniform. The normal hierarchy arises
in SO(10) models with type I seesaw, models with single
RHneutrino dominance, and models based on SO(3) and
A4lepton symmetries, while the inverted hierarchy arises in
models based on Le–Lµ–Lτ, S3, and S4lepton symmetries.
In conclusion, predictions for θ13 range from zero to
the current experimental limit. For models based on GUT
symmetries, the normal mass hierarchy can be generated
naturally. The inverted hierarchy may also be obtained in
these models with a type II seesaw, even though some fine
tuning is needed. Predictions for θ13in these models tend
to be large, with a median value sin2θ13 ∼ 0.01. On the
Table 5. A summary of the current experimental limit on θ13and the
reach of future experiments.
sin22θ13
sinθ13
Current limit
Reactor
Conventional beam
Superbeam
Neutrino Factory
10−1
10−2
10−2
3 × 10−3
(5–50) × 10−5
0.16
0.05
0.05
2.7 × 10−2
(3.5–11) × 10−3
Figure 26. Histogram of the number of models for each sin2θ13. The diagram on the left includes models that predict normal mass
hierarchy, while the diagram on the right includes models that predict inverted mass hierarchy. Taken with the kind permission of the
Physical Review from figure 3 in [329]. Copyrighted by the American Physical Society.
other hand, models based on leptonic symmetries can give
rise to inverted hierarchies and the predictions for θ13can be
quite small. Therefore, models based on lepton symmetries
will be favoured if θ13turns out to be tiny and the inverted
hierarchy is observed. However, if θ13turns out to be large,
the two different classes would not be distinguishable.
precise measurement for the deviation of θ23from π/4 can
also be crucial for distinguishing different models. This is
especiallytrueformodelsbasedonleptonsymmetriesinwhich
the deviation strongly depends on how the symmetry breaking
is introduced into the models. Precision measurements are
thus indispensable in order to distinguish different classes of
models.
A
3.2.2. Sum rules.
various models of neutrino masses have been reviewed. Many
particularly attractive classes of models lead to interesting
predictions for the neutrinomass matrix mν, such as for
instance tribimaximal or bimaximal mixing. Measurements
of neutrino oscillation determine matrix elements of the
neutrinomixing matrix, UPMNS, which may be written as the
product of VνL, that diagonalizes the neutrinomass matrix and
VeL, which diagonalizes the chargedlepton mass matrix, i.e.
UPMNS= VeLV†
models are hidden due to the presence of the chargedlepton
corrections. In many cases it can be shown that a combination
of the measurable parameters θ12, θ13, and δ can be combined
to yield a prediction for the 1–2 mixing of the neutrino
mass matrix [339,340], i.e. to arcsin(1
and
familysymmetrymodelbasedontheseesawmechanismwith
sequential dominance that predicts tribimaximal mixing via
vacuum alignment, such a ‘sum rule’ has been obtained in
[339]. In [340], it has been shown that neutrino sum rules are
not limited to one particular model, but apply to large classes
of models under very general assumptions, to be specified
In the previous section, the predictions of
νL. Often, the essential predictions of flavour
√3) for tribimaximal
In an SO(3)
π
4for bimaximal mixing, for example.
42
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
below. Examples for sum rules with theory predictions of tri
bimaximal and bimaximal neutrino mixing, respectively, are
[339–341]:
θ12− θ13cos(δ) ≈ arcsin
1
√3
(107)
θ12− θ13cos(δ) ≈π
4.
(108)
Neutrino sum rules [339,340] are thus a means of exploring
the structure of the neutrinomass matrix in the presence of
chargedlepton corrections and of testing whole classes of
models. The sum rules, such as those of equations (107) and
(108), can only be tested to highenough precision in the most
accurate experimental facilities such as the Neutrino Factory.
Chargedlepton corrections and sum rules.
useofsumrulesintestingtheoriesoftheneutrinomassmatrix
in the presence of chargedlepton corrections, consider two
examples, bimaximal [342] and tribimaximal [343] neutrino
mixing, where the predicted neutrinomixing angles are
To illustrate the
θν
12= π/4,
for bimaximal neutrino mixing
θν
12= π/4,θν
13= 0
and(109)
θν
12= arcsin
?1
√3
?
,θν
12= π/4,θν
13= 0
for tribimaximal neutrino mixing.
A similar, but physically different form, was proposed earlier
[344]. The leptonicmixing matrix is the product of VνL
and VeL, and therefore corrections to the predictions for
the neutrinomixing angles given in equations (109) arising
from the chargedlepton mixing matrix must be evaluated
to obtain estimates of the mixing angles that are accessible
experimentally.
The chargedlepton corrections can be evaluated if it is
assumed that the chargedlepton mixing matrix has a CKM
like structure, i.e. the chargedlepton mixing angles θe
small and dominated by a 1–2 mixing θe
many generic classes of flavour model in the context of GUTs
in which quarks and leptons are assigned to representations of
the unified gauge symmetries [339,345,346]. For θν
which is the case in the examples mentioned above, such
chargedleptoncorrectionsleadtothefollowingPMNSmixing
angles [340]:
θ23≈ θν
θ13≈ sin(θν
θ12≈ θν
The quantity δ which appears on the righthand side of
equation (110c) is the Dirac CP phase observable in neutrino
oscillations. For bimaximal and tribimaximal mixing, this
impliesthatθ23≈ π/4andleadstothepredictionθ13≈
ijare
12. This is the case in
13= 0,
23,
(110a)
23)θe
12,
(110b)
12+ cos(θν
23)θe
12cos(δ).
(110c)
1
√2θe
12.
Substitutingtheexpressionsforθ13andθ23intoequation(110c)
results in the following sum rules [339–341]:
θ12− θ13cos(δ) ≈ θν
Therefore, in the case of bimaximal or tribimaximal
neutrinomixing, precisemeasurementsoftheleptonicmixing
parameters θ13, θ12 and δ allow the prediction for θν
equation (111) to be tested without assuming any particular
value for θe
More generally, if it is assumed that θν
θe
12
=
π
4
for bimaximal neutrino mixing,
arcsin
?1
√3
?
for tribimaximal neutrino mixing.
(111)
12in
12.
13≈ 0, θe
13≈ 0 and
23≈ 0, and assuming θ23≈ π/4, then [340]
θ12− θ13cos(δ) ≈ θν
(θν
12
12from ‘mνtheory black box’) and
1
√2θe
(θe
(112a)
θ13≈
12
12from ‘GUT black box’).
A measurement of the combination of PMNS parameters
(112b)
θ?
12≡ θ12− θ13cos(δ)
(113)
canbeusedtoconstraintheneutrinomixingθν
sum rule in equation (112a). In many unified flavour models,
the Cabibbo angle, θC is related to θe
therefore, can be used to relate θ13to θC. Hence, a precise
measurementofθ13maybeusedtotestsuchGUTpredictions.
12bymeansofthe
12; equation (112b),
Sum rules and sensitivities of future experiments.
to be used to discriminate between the various models,
precise, independent measurements of θ12and on θ13cos(δ)
are required (for more details see [347]). θ12can be measured
using solar neutrinos or using the neutrinos generated in
nuclear reactors; a comparison of these options indicates
that the best precision on is obtained using the latter [97].
An experiment optimized for the measurement of θ12, the
‘Survival Probability MINimum’ (SPMIN) experiment, has
been proposed [97]. In this experiment a single detector is
placed at a baseline of ∼60km so that the first oscillation
minimum is right in the middle of the neutrinoenergy
spectrum. The dependence of the 2σ error on θ12 on the
exposure in units of GWktonyear is shown in figure 27.
The following systematic uncertainties were considered:
normalization, 5%; beam tilt, 2% energy scale, 0.5%,
reactor power, 2% and burnup, 2%.
these systematic uncertainties are as large as the statistical
uncertainty. Thefigurealsoshowstheperformancethatwould
be obtained if the water in the SuperKamiokande detector
were doped with gadolinium to make the detector sensitive to
neutrinos from the nuclear reactors in Japan [96]. Another
alternative, LENA, a 40kton liquid scintillator detector that
has been proposed for the Frejus laboratory in France, would
be sensitive to neutrinos produced in the French nuclear
For θ?
12
At large exposures
43
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
reactors [348]. These experiments would yield 2σ errors on
θ12of 2.6◦and 1.35◦, respectively. The SPMIN experiment
hasagreatersensitivitythaneitheroftheseproposalssincethe
baseline has been chosen to be optimal.
Longbaseline experiments, which are sensitive to δ and
θ13but have little sensitivity to θ12, must be used to determine
θ?
estimated under the assumption that θ12has been measured
in a reactor experiment. Three cases have been considered
corresponding to 2σ errors on θ12of =0.75◦, 1.35◦and 2.6◦,
respectively. For comparison, note that the current error on
θ12is 5.6◦[67]. To estimate the precision on the quantity θ?
the general procedure described in [349] has been followed.
The analysis therefore includes the uncertainties on θ13and δ,
including correlations, as well as the uncertainties on θ12,
?m2
12. The precision with which θ?
12can be determined has been
12
21, θ23, ?m2
31and the matter density. The inclusion of
0
50
100
150
200
Luminosity [GW kton yr]
0
0.5
1
1.5
2
2.5
3
2σ error on θ12 [º]
SKGd (5 yr)
LENA @ Frejus (5 yr)
SPMIN (systematics incl.)
SPMIN (statististics only)
Figure 27. The 2σ error on θ12as a function of the exposure for a
socalled SPMIN experiment.
Figure 28. The 3σ allowed interval for the combination of physical parameters θ?
function of the true value of δ for sin22θ13= 10−1. The lefthand panel is for SPL, whereas the middle one is for T2HK and the righthand
one for WBB. The dashed lines are for the sgn?m2
red—1.35◦and green—0.75◦. For the true value of θ12, sin2θ12= 0.3 (θ12= 33.12◦) has been used. The horizontal lines show the case of
bimaximal and tribimaximal neutrino mixing. Taken with the kind permission of the Journal of High Energy Physics from figure 2 in [347].
Copyrighted by SISSA.
12= θ12− θ13cos(δ) (defined in equation (113)) as a
31degenerate solution. The colours indicate different errors on θ12: blue—2.8◦,
the correlation between θ13and δ is crucial since the relevant
oscillationprobabilitycontainstermswhichgoasθ13sinδ and
θ13cosδ. However, the L/E dependence of these two terms
is different and therefore experiments covering different L/E
ranges may have very different sensitivities to θ?
reasons the accuracy on the combination θ13cosδ may be very
different from the precision with which either θ13or cosδ can
be determined individually.
Numerical estimates of the precision with which θ?
bedeterminedweremadeusingtheassumptionsforthevarious
oscillation parameters defined in section 5. The calculations
are performed with GLoBES [45,46]. The cases considered
are(seesection5.2):T2HK—anupgradeoftheJapanesesuper
beam programme; SPL to Frejus—a European, CERN based
superbeam facility; WBB—a US experiment employing a
wide band neutrino beam; a conservative Neutrino Factory
(NFC) and an optimistic Neutrino Factory NFO (as defined
in section 5.4) and a γ = 350 βbeam (BB350) as described
in [350] (see section 5.3).
Figure28showsthe3σ allowedintervalinθ?
ofthetruevalueofδforsin22θ13= 10−1. Theplotshowsthree
differentexperimentsfromlefttoright:SPL,T2HKandWBB.
All three have good sensitivity to θ?
masshierarchydegenerate solutions (dashed lines) limits the
usefulness of SPL and T2HK severely. These experiments are
not able to distinguish between bimaximal and tribimaximal
mixing (horizontal lines). This problem is absent for WBB for
which the accuracy on θ?
Figure 29 shows the results for: BB350, NFC and NFO.
Each of these experiments is unaffected by the masshierarchy
degeneracyproblemmentionedaboveforthelargevalueofθ13
considered. NFO offers the best sensitivity. The conservative
Neutrino Factory option compares well to BB350, whereas
the performance on δ and θ13 individually is much worse
than for BB350 (see also section 5.4). The reason for this
is that an experiment for which events are centred around
the first oscillation maximum, such as a βbeam or a super
beam, is sensitive mainly to the θ13sinδ term. The Neutrino
Factory, however, produces the bulk of the events above the
12. For these
12can
12asafunction
12. The presence of the
12is also somewhat better.
44
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
Figure 29. The 3σ allowed interval for the combination of physical parameters θ?
function of the true value of δ for sin22θ13= 10−1. The lefthand panel is for BB350, whereas the middle one is for NFC and the right hand
one for NFO. The dashed lines are for the sgn?m2
red—1.35◦and green—0.75◦. For the true value of θ12, sin2θ12= 0.3 (θ12= 33.12◦) has been used. The horizontal lines show the case of
bimaximal and tribimaximal neutrino mixing. Taken with the kind permission of the Journal of High Energy Physics from figure 2 in [347].
Copyrighted by SISSA.
12= θ12− θ13cos(δ) (defined in equation (113)) as a
31degenerate solution. The colours indicate different errors on θ12: blue—2.8◦,
Figure 30. The 3σ error in degrees for θ?
value of δ for sin22θ13= 10−1. The different coloured lines are for
different experiments as given in the legend. The sgn?m2
degenerate solution has been omitted. The error on θ12is 0.75◦.
12as a function of the true
31
first oscillation maximum and thus is much more sensitive to
the θ13cosδ term.
So far, results for large θ13 only have been shown.
However, the relative performance of the various options does
not change very much with θ13.
options considered except the Neutrino Factory suffers from
the masshierarchy degeneracy problem if θ13 is too small.
For intermediate values of sin22θ13 ? 10−2the accuracy
of the measurement of θ12is the dominating factor and the
performance of the various experiments is similar if the mass
hierarchy problem is ignored.
in the plots is θ12 = 33.12◦(sin2θ12 = 0.3). For larger
(smaller) values of true θ12, the bands and islands in figures 28
and 29 are shifted up (down) accordingly. The performance
of all experiments at large sin22θ13= 10−1is summarized in
figure 30. An interesting observation from this figure is that
In contrast, each of the
The true value of θ12 used
theWBBperformssecondonlytoNFO.TheNFisparticularly
well suited to the determination of the combination θ13cosδ,
making this the machine of choice for testing the sum rule,
even for large θ13.
3.2.3. Cabibbo haze in lepton mixing.
an explanation of the physics of flavour, a phenomenological
approach was advocated recently in which parametrizations
of the leptonmixing matrix were developed as an expansion
in λ ≡ sinθc
parametrization of quark mixing [351–353]. In addition to
its practical advantages for phenomenology, the Wolfenstein
parametrization hints at a guiding principle for flavour theory
by providing a framework for examining quark mixing in
the λ → 0 limit. Quarklepton unification implies that if
Cabibbosized perturbations are present in the quark sector,
such perturbations will also be manifest in the lepton sector.
Due to the presence of large angles, however, the lepton
mixing matrix is unknown in the λ → 0 limit (unlike the
quark mixings, which vanish). Hence, if the limit of zero
Cabibbo angle is meaningful for theory, there is a ‘Cabibbo
haze’ in lepton mixing, in which the initial or ‘bare’ values of
the mixings are screened by Cabibbosized effects.
Cabibbo effects therefore represent deviations from bare
mixings. They can be deviations from zero mixing (as in
the quark sector); in this approach such effects are likely
to represent the dominant source of θ13.
(and possibly θ13), Cabibbosized perturbations represent
deviations from (presumably large) nonzero initial values.
Parametrizationsarecategorizedaccordingtothebaremixings
and the structure of the allowed perturbations. Perturbations
which are linear in λ yield shifts of ?θc? 13◦, while O(λ2)
shifts are ∼ 3◦. CPviolating phases can enter the O(λ) shifts
but may only occur at subleading order, in which case the
effectivephaseissuppressedandthesizeofθ13doesnotdictate
the size of CPviolating observables.
One aim of this approach is to obtain an efficient
parametrization of the leptonmixing matrix in analogy to
the Wolfenstein parametrization for the quarkmixing matrix.
As a step toward
? 0.22 in analogy with Wolfenstein’s
For θ23 and θ12
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
However, current data are clearly consistent with many
possibleWolfensteinlikeparametrizations. Onereasonisthat
there is a wide range of possible baremixing parameters and
Cabibbo shifts, though some particular values may be singled
out by wellmotivated flavour theories. Another reason is the
current precision of the data. Recast in terms of the Cabibbo
angle, the error bar on θ12is of O(λ2), while the uncertainties
in θ23and θ13are of O(λ). Although it is not possible to
single out a particular parametrization, the approach provides
an organizing principle for categorizing the many topdown
flavour models based on a λ expansion. The approach also
provides a useful framework in which to interpret the results
of future experiments, such as the programme to measure θ13.
Future facilities are expected to reach the O(λ2) range, which
will yield important insight into the nature of lepton mixing in
the λ → 0 limit.
The classification scheme proceeds as follows. Recall
that the Wolfenstein parametrization is based on the idea that
the hierarchical quarkmixing angles can be understood as a λ
expansion, with
UCKM= 1 + O(λ).
In the lepton sector, a similar parametrization requires a λ
expansion of the form
(114)
UPMNS= W + O(λ).
(115)
The starting matrix W, which is dictated by the (unknown)
underlying flavour theory, is then perturbed multiplicatively
by a unitary matrix V(λ), which in turn is assumed to have a λ
expansion:
V(λ) = 1 + O(λ).
For the quarks, the starting matrix is the identity matrix and
the perturbation matrix takes the Wolfenstein form. For the
leptons, the structure of the allowed perturbations depend on
the details of W. Due to Cabibbo haze, W can take different
forms which are characterized by the number of large angles.
For simplicity, attention will be restricted here to the best
motivated scenario, in which the bare solar and atmospheric
mixings, η12and η23, are nonzero and the bare θ13vanishes
(see [352] for a more general analysis). In this case W is of
the form
where P is a diagonal phase matrix of the form
whichencodesthetwophysicalMajoranaCPviolatingphases
α12≡ α1− α2and α23≡ α2− α3.
(116)
W = R1(η23)R3(η12) ≡
1
0
0
00
cosη23
−sinη23
0
0
1
sinη23
cosη23
×
cosη12
−sinη12
0
sinη12
cosη12
0
P,
(117)
P =
eiα1
0
0
00
0
eiα2
0eiα3
,
(118)
Unlike the quark sector, generically the perturbations do
not commute with the starting matrix:
[W, V(λ)] ?= 0.
(119)
Hence, there are several possible implementations of Cabibbo
shifts:
• RightCabibboshifts. Theperturbationscanbeintroduced
as a multiplication of V(λ) on the right:
UPMNS= WV(λ);
• Left Cabibbo shifts.
implemented as a multiplication of V(λ) on the left:
UPMNS= V(λ)W;
• Middle Cabibbo shifts.
sandwiched between the rotation matrices of W:
UPMNS= R1V(λ)R3P
or
UPMNS= R1V(λ)R3P.
To see that this encompasses all possibilities, recall that the
assumption of Cabibbo haze is that the leptonmixing matrix
has a λ expansion:
(120)
The perturbations can be
(121)
The perturbations can be
(122)
(123)
UPMNS(λ) =
∞
?
n=0
λnWn,
(124)
in which W0≡ W. This can be expressed as a right Cabibbo
shift:
UPMNS(λ) = W
∞
?
n=0
λn(W−1Wn) ≡ WV(λ),
(125)
with V =?∞
UPMNS(λ) = R1R3V(λ);
= R1(R3V(λ)R−1
The generalization to left shifts is straightforward. Note that
since V, by assumption, is given by
n=0λn(W−1Wn). It can also be expressed as a
middle Cabibbo shift (dropping P for simplicity):
3)R3≡ R1V?(λ)R3.
(126)
V(λ) = 1 +
∞
?
i=1
λnVn,
(127)
V?can also be written in an analogous form:
V?(λ) = R3VR−1
3
= 1 +
∞
?
i=1
λn(R3VnR−1
3).
(128)
Hence, the decomposition into right, left or middle shifts is
meaningful for a specific choice of V. To leading order in λ,
V is assumed to be
V =
1
a1λ
1
−b∗
c1λ
−a∗
−c∗
1λb1λ
1
1λ
1λ
+ O(λ2),
(129)
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
which encompasses the Wolfenstein form (a1
b1= c1= 0 and higher order terms b2 = A and c3 =
A(ρ −1
generalperturbations. Finally,astheshiftsinthemixingangles
are clearly dominated by perturbations linear in λ, it is useful
to categorize models further as single, double or triple shifts
according to the number of such O(λ) perturbations in V.
Given these ingredients, a systematic classification of
possible models was presented in [351,352], to which the
reader is referred for further details. Here attention will be
focused on one subset of examples. It is straightforward to
obtain the following general results for the O(λ) shifts in the
mixing angles (including phases):
=
1,
2− iη), in selfevident notation), and allows for more
• Right shifts:
θ12= η12+ λa1cos(α12+ φa1),
θ23= η23+ λ(cosη12b1cos(α23+ φb1)
− sinη12c1cos(α12− α23+ φc1)),
θ13= λb1eiα23sinη12+ c1ei(α12−α23)cosη12.
• Left shifts:
θ12= η12+ λ(cosη23a1cosφa1− sinη23c1cosφc1),
(130)
(131)
(132)
(133)
θ23= η23+ λb1cosφb1,
θ13= λsinη23a1+ cosη23c1.
• Middle shifts:
θ12= η12+ λa1cosφa1,
θ23= η23+ λb1cosφb1,
θ13= λc1.
Each scenario displays distinct correlations between the
Cabibbo shifts of the mixing angles. Note that certain shifts
are sized by factors dependent on the baremixing parameters.
In addition, the shifts in the mixing angles depend on the
Majorana phases α12, α23 only in the right Cabibbo shift
scenario. The reason is that generically
(134)
(135)
(136)
(137)
(138)
[P,V(λ)] ?= 0,
(139)
and hence the right shifts can be rewritten as follows:
UPMNS= WPV
= W(PVP−1)P ≡ WVMP.
VMcan be obtained from V through the replacements ai →
aieiα12, bi→ bieiα23and ci→ ciei(α12−α23).
How might certain examples emerge from the viewpoint
of flavour theory? One class of examples occur within grand
unifiedmodelsinwhichthefermionDiracmassmatricesobey
SU(5) and SO(10). GUT relations based on the simplest
Higgs structures and the downquark mass matrix is further
assumed to be symmetric, such that Md = MT
Mu ∼ Mν. In such models, the quark and leptonmixing
matrices are related [354–357]:
(140)
d∼ Meand
UPMNS= U†
CKMF,
(141)
where F is a matrix which encodes the effects of the neutrino
seesaw; in these models, F must contain two large angles.
In the language of this classification scheme, this scenario is
an example of a left Cabibbo singleshift model, in which
F plays the role of W and V takes the form of U†
Otherpossibleexamplesincludemodelsbasedonquarklepton
complementarity, in which case W is a bimaximalmixing
matrixandV hasa1?= 0,b1= 0,andc1mayormaynotvanish
dependingonthedetailsofthemodel. Differentpredictionsfor
θ13are implied in these cases depending on whether the model
is a right, left or middle Cabibbo shift model. Tribimaximal
mixing scenarios are models in which W takes on the standard
tribimaximalform, andV hasa1= b1= 0andc1mayormay
not be zero, with a range of predictions for θ13depending on
the shift scenario.
Turning now to the issue of CP violation,
parametrizations also display different predictions for the
leptonic Dirac and Majorana phases, depending on the details
of how and whether phases enter W and V. Here, only Dirac
type CP violation is considered (as CPviolating observables
sourced by Majorana phases are helicity suppressed and thus
difficult to observe). For models in which W has two
large angles (the reader is once again referred to [352] for
a more general discussion), the invariant measure of Dirac CP
violation,
CKM.
the
JCP= Im(UαiUβjU∗
βiU∗
αj) ? sin2θ12sin2θ23sin2θ13sinδ,
(142)
vanishes in the λ → 0 limit, and a nonzero value can be
generated in two ways:
• Complex V(λ). V(λ) can be the source of CPviolating
phases,whichcanbeO(1)(asinthequarksector). Models
can be categorized in terms of whether CP violation
enters at leading or higher order in λ, and whether the
effective leptonic phase is predicted to be O(1) or further
suppressed;
• BareMajoranaphases. Majoranaphasescanalsoprovide
a source for Dirac CP violation once the Cabibbosized
perturbations are switched on.
Cabibbo shifts, this does not occur. However, for right
Cabibbo shifts it does, as such shifts encode P through
the modification V → VM.
Consider equation (141) as an illustrative example. If V is of
the Wolfenstein form (complex O(λ3) terms), JCPis
JCP=1
Note that in this model, the shifts in the angles are given to
O(λ2) by
θ12= η12− λcosη23,
θ23= η23− λ2?A +1
and
θ13= −λsinη23.
The effective leptonic phase is δ ∼ O(λ2), in contrast to
the O(1) CKM phase. This suppression occurs because the
phases in V arise in subdominant contributions to the mixing
For left and middle
4Aλ3ηcosη23sin2η23sin2η12.
(143)
(144)
4sin2η23
?
(145)
(146)
47
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
angles. Models with this feature demonstrate that while the
magnitudeofθ13isclearlycorrelatedwiththeprospectsforthe
observability of leptonsector CP violation, it is not the whole
storybecausetheCPviolatingphaseitselfmaybesuppressed.
Insummary, wearebeginningtoreadthenewleptondata,
but there is much work to do before a satisfactory and credible
theoryofflavourisproposed. Inthemeantime, itisillustrative
to examine the lepton sector through the lens of quarklepton
unification, and investigate parametrizations of the lepton
mixing matrix which include Cabibbosized effects.
approach emphasizes the need for precision measurements,
as present data are insufficient for singling out a particular
parametrization. Should the limit of zero Cabibbo mixing
prove to be meaningful for theory, with improved data we may
be able to see flavour patterns through the Cabibbo haze.
The
3.3. Leptonflavour violation
Searching for leptonflavour violation in chargedlepton
decays is an important way to look for new physics beyond
the Standard Model [358].Since the early days of muon
experiments, processes such as µ → eγ have been searched
for, and the absence of such processes has lead us to consider
the separate conservation of electron and muon numbers. The
discoveryoftwoflavoursofneutrinoin1962atBNLindicated
that leptonflavour conservation is indeed realized in nature to
a good degree of accuracy.
The situation has changed since the discovery of the
neutrinooscillations. Theseparateconservationofeachlepton
number individually is likely violated.
flavour violation can be observed in chargedlepton processes
depending on how neutrino mass is generated. In the simple
Dirac neutrino, or the seesaw, framework, leptonflavour
violating processes in muon decays are suppressed by more
than 20 orders of magnitude below the present experimental
upper bounds. On the other hand, leptonflavour violation
becomes large, if some new particles or interactions exists
at the TeV scale. Therefore, searching for leptonflavour
violation in muon and tau decay processes provides important
information on the origin of the neutrino mass.
However, lepton
Leptonflavour violation in threemuon processes.
the various leptonflavourviolating processes, threemuon
processes, µ → eγ, µ → 3e and µ–e conversion in muonic
atoms, are particularly important. The current experimental
upper bound for µ → eγ [359] is at the 10−11level
and about one order of magnitude smaller for the other
two processes [360,361].Although the µ–econversion
process has the smallest upper bound, the process which
imposesthestrongestconstraintsonthetheoreticalparameters
depends on the model under consideration. Muonium–anti
muonium conversion is another process which violates the
conservation of electron and muon numbers but conserves
the total lepton number.This process is sensitive to new
physics which changes the muon and electron numbers by two
units. Upper bounds on the branching ratios of taulepton
flavour violating processes have been improved recently at
KEK and the SLACBfactory experiments, and have reached
Among
the level of 10−7and below depending on the decay mode
in question [362–370].Generally speaking, threemuon
processesputstringentconstraintsonmodelsthatyieldlepton
flavourviolationandthestudyofcorrelationsamongthevarous
processes is useful to identify the correct model.
Innearfuture,theMEGexperimentisexpectedtoimprove
the searchlimit on the µ → eγ process by more than two
orders of magnitude. If leptonflavour violation is discovered,
the next steps will be to discover the nature of leptonflavour
violation and to distinguish between the different models. The
following techniques can be used to do this:
• The ratio of the branching ratios of µ → 3e (µ–e
conversion) and µ → eγ depends on what kinds
of operator are responsible for leptonflavour violation.
In particular, if all three processes are generated by
the same photonicdipoletype operator, the following
relations hold:
B(µ+→ e+e+e−) ∼ 6 × 10−3B(µ → eγ),
σ(µ−Ti → e−Ti)
σ(µ−Ti → capture)∼ 4 × 10−3B(µ → eγ).
This is a good approximation, for example, for most
supersymmetric models. On the other hand, if lepton
flavour violation is generated by treelevel processes,
µ → 3e and/or µ–e conversion, the branching fractions
could be much larger than that of µ → eγ;
• Angular distributions in polarized muon decays provide
information on the chiral and CP structures of lepton
flavourviolating operators [371]. For the µ → eγ search
with a polarized µ+, the µ+→ e+
operators are distinguished by the angular distribution
of the positronmomentum direction with respect to the
initial muonpolarization direction. The chiral structure
carrys information on the origin of the leptonflavour
violating interaction.In supersymmetric models, for
example, the chirality depends on whether the flavour
mixing exists in the right or lefthanded slepton sector,
and this distinction could provide very important clues to
the interaction at the GUT scale and
• In the µ–e conversion search, branchingratio mea
surements of different atoms provides one means of
discriminating between the different operators [372]. The
atomicnumber dependence of the µ–econversion rate
differs for different types of quarklevel operators. For
example, we can distinguish scalar, vector and photon
dipole type operators by comparing branching fractions
measured using different nuclei, for example a low
atomicnumber nucleus such as aluminium and a heavy
nucleus such as lead.
(147)
(148)
Lγ and µ+→ eRγ
These techniques would provide information on different
aspects of leptonflavour violating interactions, and are the
basicstepstorequiredtoclarifythenatureofnewinteractions.
Supersymmetry and muon leptonflavourviolating processes.
Among the newphysics models explored by searches
for leptonflavour violation, supersymmetry is the most
48
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Rep. Prog. Phys. 72 (2009) 000000 A Bandyopadhyay et al
Figure 31. µ → eγ branching ratios for SU(5) and SO(10) SUSY
GUT. Adapted with the permission of the Reviews of Modern
Physics from figures 8 and 13 in [358]. Copyrighted by the
American Physical Society.
important. Since supersymmetry requires the introduction of
a supersymmetric partner for each particle in the Standard
Model, sleptons should exist.
depend on supersymmetrybreaking terms, which do not
have an a priori relation with lepton mass terms.
fact, the flavour mixing in the sleptonmass matrix is
stronglyconstrainedbytheleptonflavourviolatingprocesses.
This is a part of the flavour problem in supersymmetric
models; some mechanism is needed to suppress flavour
changing neutralcurrent processes in the quark and the lepton
sectors. A solution to this problem is one of the necessary
conditions for a realistic supersymmetric model, and a variety
of supersymmetrybreaking mechanisms are proposed.
principle, we will be able to identify the correct scenario
by looking at the superparticle mass spectrum in energy
frontier experiments at the LHC and the International Linear
Collider.
Searches for leptonflavourviolating processes have a
role to play in the determination of the offdiagonal elements
of the sleptonmass matrix.
elements is particularly important because these elements
carry information at very high energy scales such as the
GUT scale and the seesaw neutrino scales [373,374]. Even
if we take a scenario where offdiagonal slepton terms are
absent at the Planck scale, renormalization effects due to large
Yukawa coupling constants can induce sizable offdiagonal
terms. In SUSYGUT models, the large top Yukawa coupling
constant a source of leptonflavour violation because quarks
and leptons are connected to each other above the GUT scale
[375,376]. A typical example is shown in figure 31 for SU(5)
and S0(10) SUSY GUTs. The branching ratio is expected
to be close to the current experimental upper limit for the
SO(10) case.
In the supersymmetric seesaw model, a potentially large
YukawacouplingisprovidedbytheneutrinoYukawacoupling
Mass terms for the slepton
In
In
The determination of these
constants. The offdiagonal term in the lefthanded slepton
mass matrix is given by
(m2
˜lL)ij? −
where MP and MR are the Planck mass and the right
handed neutrino mass, respectively, m0is the universal scalar
mass, A0is the universal triplescalarcoupling constant for
supersymmetrybreaking terms and yνis the neutrino Yukawa
coupling constant. Since the seesaw relation suggests that the
Yukawa coupling is proportional to the square root of MR, the
leptonflavour violating branching ratio is proportional to M2
Although the flavour structure of yνis not directly related to
the flavour mixing in the PMNS matrix, it is natural to expect
sizable offdiagonal elements from the large neutrino mixing.
Infact,theµ → eγ branchingratiocanreachtheexperimental
bound for MR= 0(1013)–0(1014)GeV [377–379].
There is an interesting special case which can be
realized for a larger value of the ratio of the two Higgs
vacuum expectation values (tanβ) [380–382]. In this case,
supersymmetric loop corrections to the Higgslepton vertex
can generate a large leptonflavourviolating coupling. As
a result, heavy Higgsboson exchange diagrams can be
dominant, andtheµ–econversionprocessisenhancedrelative
to the µ → eγ process [383].
figure 32. For a smaller heavyHiggsboson mass, the two
branching ratios can be more similar. For the same parameter
space, we can confirm that the dominant operator is of the
scalar type from the atomicnumber dependence of the µ–e
conversion rate.
1
8π2(yν)∗
ki(yν)kjm2
0(3 + A02)ln
?MP
MR
?
, (149)
R.
An example is shown in
Other theoretical models.
models that predict sizable rates for muon leptonflavour
violating processes [358].
flavour violation is related to the physics of neutrinomass
generation, namely the interaction responsible for the neutrino
mixings also induces leptonflavour violation.
case for the supersymmetric seesaw model discussed above.
Other examples are the Zee model [384], Diractype bulk
neutrinos in the warped extra dimension [385], the triplet
Higgsmodel[326,386]andthenonsupersymmetricleft–right
symmetric model [324,387,388]. Supersymmetric, with R
parity violation, can be considered to be in this category,
sinceneutrinomassescanbegeneratedfromRparityviolating
couplings [389]. Since each model introduces leptonflavour
violation in a different way, the phenomenological features
canbequitedifferentandmeasurementswillprovideimportant
cluestoidentifythecorrectmodelofneutrinomassgeneration.
ThetripletHiggsmodelprovidesasimplewaytogenerate
neutrinomassesfromasmalltripletvacuumexpectationvalue.
In this model, the triplet Higgs and lepton coupling generating
neutrino mass also induces a doubly charged Higgs boson
and lepton coupling. The neutrinomixing matrix has a direct
relationwiththedoublychargedHiggsbosoncoupling. Since
thedoublychargedHiggsbosongivesatreelevelcontribution
to the µ → 3e process, this can dominate over the other two
processes. On the other hand, the µ → eγ and the µ–e
conversion branching ratios become similar.
There are many newphysics
In many cases, the lepton
This is the
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Rep. Prog. Phys. 72 (2009) 000000A Bandyopadhyay et al
10−15
10−14
10−13
10−12
10−11
200400 600
mH0 (GeV)
8001000 1200 1400
B(µAl→eAl)
tanβ = 60
m0=M1/2=2000 GeV
MN = 1014 GeV
µ > 0
µ < 0
10−13
10−12
10−11
200400 600
mH0 (GeV)
800 1000 1200 1400
B(µ→eγ)
tanβ = 60
m0=M1/2=2000 GeV
MN = 1014 GeV
µ > 0
µ < 0
(a)(b)
Figure 32. µ–e conversion branching ratio in aluminium nucleus and µ → eγ branching ratio as a function heavy CP even Higgs boson
mass in the supersymmetric seesaw model. Taken with the kind permission of the Physical Letters from figure 1 in [383]. Copyrighted by
Elsevier B.V.
The left–right symmetric model also has the tripletHiggs
field. In this case, however, neutrino masses can be generated
by the seesaw mechanism. The righthanded neutrinomass
term arises in association with SU(2)L× SU(2)R× U(1)B–L
symmetry breaking to the Standard Model gauge groups. If
this scale is close to the TeV scale, observable leptonflavour
violating effects are generated through the doubly charged
Higgs boson and lepton couplings. Unlike the tripletHiggs
model, the relationship between neutrino mixing and lepton
flavour violation is not straightforward. A generic feature is
that the µ → 3e branching ratio is larger by two orders of
magnitudecomparedwiththeµ → eγ andtheµ–econversion
branching ratios.
In this way, muon leptonflavourviolating processes
provide one way to explore physics beyond the Standard
Model. This is particularly important because neutrino
oscillationsareclearevidenceofnewphysics,andtheoriginof
neutrino masses is still unknown. There are various scenarios
for neutrinomass generation, each with different features that
maygiverisetoobservablesignalsforleptonflavourviolation
in chargedlepton processes.
µ → eγ,µ → 3eandµ–econversionisimportantiftheorigin
of flavour mixing in the lepton sector is to be determined.
The experimental pursuit of
3.4. Cosmology
3.4.1. Neutrinos and largescale structure.
of cosmological perturbations—such as cosmic microwave
background (CMB) anisotropies or the largescale density
perturbations reconstructed, e.g. from the galaxy distribution
in the Universe—are known to provide good measurements of
many cosmological parameters. For instance, the spectrum of
cosmological perturbations is very sensitive to the abundance
of ultrarelativistic particles in the early Universe. This can
be used to make a good estimate of the number of neutrinos
The observation
which were in thermal equilibrium at that time, parametrized
by an effective number, Neff. The standard scenario with three
neutrinoflavoursandnootherrelativisticrelicsintheUniverse
(apart from photons) corresponds to Neff= 3, while scenarios
withonelightsterileneutrinooriginallyinthermalequilibrium
corresponds to Neff = 4; relaxing the thermal equilibrium
assumption, the last scenario would give 3 < Neff ? 4.
Current cosmological bounds give Neff = 3.8+2.0
[390–396], which is compatible with the standard scenario,
but also with the presence of extra relativistic relics. Future
experiments are expected to reach a 1σ sensitivity of 0.3 in
approximately five years from now, and should be able to
confirm the standard Neff = 3 cosmological scenario with
better accuracy than Big Bang nucleosynthesis bounds. In
the rest of this section, it will be assumed, for simplicity,
that Neff= 3.
Neutrino masses are more difficult to measure than Neff
because they are too small to contribute more than ∼1%
of the current energy density of the Universe. Fortunately,
the formation of structures (galaxies and clusters) during
the matterdominated epoch is quite sensitive even to small
neutrino masses.
−1.6at 2σ
3.4.1.1.
theoretical predictions.
depends very much on the velocity dispersion of the
components contributing to the matter of the Universe (for
a review, see [397]).If all nonrelativistic components
(such as baryons and Cold Dark Matter, CDM) have a
very small velocity dispersion the process of gravitational
collapse reaches its maximal efficiency.
energy) density contrast starts from very small values in the
early Universe, with Fourier modes δk = [δρk/ ¯ ρ] of order
10−5. On wavelengths corresponding today to the largescale
structure (LSS) of the Universe, the density contrast starts
Impact of neutrinos on structure formation:
The process of galaxy formation
The matter (or
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