# Reconstructing SUSY and R-Neutrino Masses in SO(10)

**ABSTRACT** We report on the extrapolation of scalar mass parameters in the lepton sector to reconstruct SO(10) scenarios close to the unification scale. The method is demonstrated for an example in which SO(10) is broken directly to the Standard Model, based on the expected precision from coherent LHC and ILC collider analyses. In addition to the fundamental scalar mass parameters at the unification scale, the mass of the heaviest right-handed neutrino can be estimated in the seesaw scenario.

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arXiv:0809.2047v1 [hep-ph] 11 Sep 2008

Reconstructing SUSY and R-Neutrino Masses in SO(10)

F. Deppisch∗, A. Freitas†,∗∗, W. Porod‡and P.M. Zerwas§,¶

∗School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK

†Department of Physics & Astronomy, University of Pittsburgh, PA 15260, USA

∗∗HEP Division, Argonne National Laboratory, Argonne, IL 60439, USA

‡Inst. Theor. Physik und Astrophysik, Universität Würzburg, D-97074 Würzburg, Germany

§Inst. Theor. Physik E, RWTH Aachen U, D-52056 Aachen, Germany

¶Deutsches Elektronen-SynchrotronDESY, D-22603 Hamburg, Germany

Abstract. We report on the extrapolation of scalar mass parameters in the lepton sector to reconstruct SO(10) scenarios close

to the unification scale. The method is demonstrated for an example in which SO(10) is broken directly to the Standard Model,

based on the expected precision from coherent LHC and ILC collider analyses. In addition to the fundamental scalar mass

parameters at the unification scale, the mass of the heaviest right-handed neutrino can be estimated in the seesaw scenario.

Keywords: Supersymmetry, Grand Unified Theories, Neutrinos

PACS: 12.10.Dm, 12.60.Jv, 14.60.Pq

Theoretical basis

A naturalexplanationoftheverylightneutrinomasses

in relation to the electroweak scale is offered by the see-

saw mechanism [1], which naturally suggests the sym-

metry group SO(10) as the grand unification group [2].

In this report, which summarizes the results of

Ref. [3], we focus on a simple model incorporating

one-step symmetry breaking from SO(10) down to the

Standard Model SM gauge group, SO(10) → SM, at

the GUT scale ΛU ≈ 2 · 1016GeV where the gauge

couplings unify in SUSY. We implicitly assume that a

number of fundamental problems [4] are solved without

interfering strongly with the key points of the present

analysis, i.e. mechanisms leading to doublet-triplet split-

ting and suppressing proton decay. Such solutions may

include extended Higgs sectors or higher-dimensional

Planck-scale suppressed operators.

The SUSY SO(10) model is characterized by the fol-

lowing part of the superpotential which involves matter

fields:

WM=Y1016·16·10+Y10′16·16·10′+Y12616·16·126.

The matter superfields of the three generations belong to

16-dimensional representations of SO(10). Two Higgs-

10 fields generate masses separately for up- and down-

type fermions at the electroweak scale. The masses of

the heavy R-neutrino superfields are generated by a 126-

dimensional Higgs, resulting in a seesaw type I mecha-

nism of light neutrino mass generation.

The superpotential is supplemented by soft SUSY

breaking masses for the fermion and Higgs multiplets,

assumed to be universal at the unification scale,

m16= m10= m10′ = m126= M0.

(1)

However, the breaking of the SO(10) symmetry group to

the SM group generates GUT D-terms DUsuch that the

boundary conditions at the GUT scale read [5]

m2

m2

m2

L= M2

E= M2

νR= M2

0−3DU

0+2DU

0+5DU

(2)

for the slepton L-isodoublet and R-isosinglet fields. The

D-term is of the order of the soft SUSY breaking masses

and we will treat DUas a free parameter.

The neutrino sector

The heavy right-handedneutrino masses are related to

the light neutrino masses by the Yukawa matrix Yν in

the seesaw mechanism. Neglecting higher-order effects

in the calculation of the Majorana neutrino mass matrix,

it follows from the Higgs-10 SO(10) relation

Yν=Yu

(3)

betweentheneutrinoandup-typequarkYukawamatrices

that Yν≈ diag(mu,mc,mt)/vu holds approximately for

the neutrino Yukawa matrix at the GUT scale; vu=

vsinβ, with v and tanβ being the familiar vacuum and

mixingparametersinthe Higgssector.Quarkmixingand

RG runningeffects in the neutrinosector are neglectedin

theanalyticalapproachbutproperlytakenintoaccountin

the numerical analysis.

Page 2

10?1

10?2

10?3

10?4

mΝ1?eV

104

106

108

1010

1012

1014

1016

MΝRi?GeV

MΝR1

MΝR2

MΝR3

FIGURE 1.

tions of the lightest neutrino mass mν1. The dashed (blue) lines

assume perfect Yukawa unification, Eq. (3). The solid (black)

lines indicate shifts of the νRmasses if the Yukawa identity

Eq. (3) is violated by an additional contribution ∼ 100 MeV/v.

Masses of right-handed neutrinos MνRias func-

The effectivemass matrixof the light neutrinosis con-

strained by the results of the neutrino oscillation experi-

ments:

mν=U∗

MNS·diag(mν1,mν2,mν3)·U†

MNS.

(4)

We will assume the normal hierarchy for the light neu-

trino masses mνi, and for the MNS mixing matrix the

tri-bimaximal form. From the seesaw relation

MνR=Yνm−1

νYT

ν·v2

u,

(5)

the heavy Majorana R-neutrino mass matrix MνRcan

now be calculated. Solving Eq. (5) for the eigenvalues

MνRi(i = 1,2,3), the heavy Majorana masses are deter-

mined by the up-quark masses mu,c,t at the GUT scale

and the lightest neutrino mass mν1, assuming best fit

values for the light neutrino squared mass differences.

The mass spectrum of the R-neutrinos is strongly hi-

erarchical in this scenario, MνR3: MνR2: MνR1∼ m2

m2

like RG running effects, is displayed in Fig. 1 for a

wide range of mν1values. Modifying the relation be-

tween the neutrinoand up-typequarkYukawacouplings,

Eqs. (3), ad-hoc by a small additional term, Yν= Yu+

O(100 MeV)/v, associated potentiallywith a morecom-

plex Higgs scenario, Planck-scale suppressed contribu-

tions or non-perturbative effects, the first generation R-

neutrinomassesis liftedtoO(1010GeV),whiletheheav-

iersecond/thirdgenerationmasses remainpracticallyun-

changed.

t:

c:m2

u. The numericalevaluation,includingrefinements

The scalar sector

To leading order,the solutions of the RG equations for

the masses of the scalar selectrons, can be expressed in

TABLE 1.

neutrino parameters in an example scenario point

(µ > 0).

Reconstruction of SO(10), SUSY and

Parameter

M0[GeV]

M1/2[GeV]

A0[GeV]

tanβ

√−DU[GeV]

ΛU[GeV]

MνR3[GeV]

mν1[eV]

Ideal

90.0

250.0

-640.0

10.0

30.0

2.16·1016

7.2·1014

3.5·10−3

1σ Error

0.25

0.4

13.0

1.0

0.9

0.02·1016

[4.8,11]·1014

[1.6,6.7]·10−3

terms of the universal scalar mass M0, the gaugino mass

M1/2, and the GUT and electroweak D-terms, DU and

DEW= M2

Z/2 cos2β, respectively1,

m2

m2

˜ eR= M2

˜ eL= M2

0+ DU+αRM2

0−3DU+αLM2

1/2−6

1/2+3

5S′−2s2

5S′−c2WDEW. (7)

WDEW, (6)

The coefficients αLand αRare determined by the gaug-

ino/gauge boson loops in the RG evolution from the

SUSY scale

˜ M [6] to the unification scale, and the

numerical evaluation yields αR≈ 0.15 and αL≈ 0.5.

The universal gaugino mass parameter M1/2can be pre-

determined in the chargino/neutralino sector. The non-

universal initial conditions in the evolution due to the D-

terms generate the small generation-independentcorrec-

tions S′, cf. Ref. [3].

The masses of the staus are shifted relative to the

masses of the first two generationsby two terms [3, 7, 8]:

m2

m2

˜ τR−m2

˜ τL−m2

˜ eR

=

m2

m2

τ−2∆τ,

τ− ∆τ−∆ντ.

(8)

˜ eL

=

(9)

The shifts ∆τ and ∆ντ, generated by loops involving

chargedlepton and neutrino superfields, respectively, are

given to leading order by

∆τ

≈

m2

τ

8π2v2

m2

8π2v2u

d

?3M2

0+A2

0

?logΛ2

U

˜ M2,

(10)

∆ντ

≈

t

?3M2

0+A2

0

?logΛ2

U

M2

νR3

,

(11)

with the universal trilinear coupling A0defined at the

GUT scale.

Anticipating high-precision measurements at future

colliders, such an SO(10) model can be investigated in

1In the following we omit the discussion of sneutrino and Higgs

masses and their role in reconstructing GUT scale parameters, but

their effect is included in the numerical results. For further details see

Ref. [3].

Page 3

103

105

107

109

1011

1013

1015

1017

Μ?GeV

0

1

2

3

mSSB

2

??104GeV2?

mL3

2

mL1

2

mE3

2

mHu

2

??

MΝR3

FIGURE 2.

slepton and Higgs mass parameters [DU= 0].

Evolution of the first and third generation L,R

central facets. As a concrete example, we study a sce-

nario with SUSY parameters close to SPS1a/a′[6, 9] de-

finedinTable1,compatiblewiththelow-energyandcos-

mological data.

Themeasurementofthe sleptonmasses ofthefirst two

generations allows us to extract the common sfermion

parameter m16= M0 as well as the D-term DU, cf.

Eqs. (6/7).Includingthe completeone-loopand the lead-

ingtwo-loopcorrections,theevolutionofthe scalar mass

parameters is displayed in Fig. 2 with the D-term set to

zero for illustration. By extrapolating the measurements

of slepton, Higgs and gauginomasses with the estimated

experimental errors from the results of Refs. [6, 8, 10],

the high-scale parameters can be reconstructed with the

precision shown in Tab. 1. The RG evolution equations

are evaluated to 2-loop order by means of the SPheno

program[11]. The results indicate that the high-scale pa-

rameters M0and DU, driven by the slepton analysis, can

be reconstructed at per-mill to per-cent accuracy.

The right-handedneutrino affects the logarithmicevo-

lution of the mass parameter m2

tion in the most direct form.Thecharacteristic difference

in the evolution between m2

Fig. 2. The position of the kink at MνR3can be derived

from the intersection of the parameter ∆ντas a function

of MνR3, Eq. (11), with its measuredvalue extracted from

the slepton masses. Based on this estimate of MνR3, the

value of the lightest neutrino mass is then determined

via the seesaw mechanism, cf. Fig. 1. The reconstructed

values of MνR3and mν1in the SUSY scenario defined

above,are shown in Tab. 1, along with their uncertainties

from propagating expected experimental errors.

L3of the third genera-

L3and m2

L1is exemplified in

Conclusion

If the roots of physics are located near the Planck

scale, experimental methods must be devised to explore

the high-scalephysicsscenarioincludingthe grandunifi-

cation of the Standard Model interactions. In this report,

we have demonstratedhow this goal can be achievedin a

simplified SO(10) model with direct breaking to the SM

gauge group.As naturally expected,the analysis of more

complicated models will require a larger set of assump-

tions before the analysis can be performed, constrained

however by additional observables like charged lepton

masses, etc. Such an extended scheme is exemplified in

a two-step breaking scenario SO(10) → SU(5) → SM

analyzed in Ref. [3]. The example described in this re-

port has proved nevertheless that renormalization-group

extrapolations based on high-precision results expected

from Terascale experiments can provide essential ele-

ments for the reconstruction of the physics scenario near

the GUT scale.

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