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Doubly Lopsided Mass Matrices from Supersymmetric SU(N) Unification
Abstract
It is shown that in supersymmetric SU(N) models with N>5 the so-called doubly lopsided mass matrix structure can emerge in a natural way. The non-trivial flavor structure is entirely accounted for by the SU(N) gauge symmetry and supersymmetry, without any flavor symmetry. The hierarchy among the families results directly from a hierarchy of scales in the chain of breaking from SU(N)to the Standard Model group. A simple SU(7) example is presented.
... It was pointed out and discussed in Refs. [3][4][5][6][7] that multiple fermion generations, such as n g = 3 for the SM case, can be embedded non-trivially in GUT groups of SU(7) and beyond. 1 Therefore, it is natural to conjecture that the SM fermion mass hierarchies may originate from the intermediate symmetry breaking scale of some non-minimal GUT with SU(N ≥ 7) [4,5]. Historically, the embedding of the SM generations as well as fermion mass hierarchies were studied in the context of technicolor and extended technicolor models [8][9][10][11][12], where the symmetry breakings are due to the fermion bi-linear condensates. ...
... where we allow the explicit SU(2) F -breaking term in the Yukawa couplings, so that (Y D ) ρσ = Y D δ ρσ . Below, we motivate our Higgs VEV assignments for the viable symmetry breaking from three different aspects, which include A the null results in searching for a second Higgs doublet at the LHC, B the extension to the non-minimal GUTs with n g = 3, e.g., the SU(8) GUT [6,7], C the natural mass generation of the bottom quark and tau lepton with Yukawa couplings of ∼ O(1). ...
... Therefore, the Higgs sector at the EW scale is described by a type-II 2HDM. 7 8 Thus, we name such symmetry breaking pattern as the "fermion-Higgs matching pattern". However, the ongoing probes of the second Higgs doublet at the LHC lack direct evidences for the predicted neutral and charged Higgs bosons from various channels [56][57][58][59][60][61][62][63][64][65][66][67]. ...
We study a toy SU ( 6 ) model with the symmetry breaking pattern of the extended 331 symmetry of SU ( 3 ) c ⊗ SU ( 3 ) W ⊗ U ( 1 ) X . A “fermion-Higgs mismatching” symmetry breaking pattern is proposed for more realistic model building. Within such symmetry breaking pattern, only one Higgs doublet develops vacuum expectation value for the spontaneous electroweak symmetry breaking, and gives tree-level top quark mass. A natural VEV splittings in the 331 breaking Higgs fields gives tree-level masses to both bottom quark and tau lepton. The 125 GeV SM-like Higgs boson discovered at the LHC can have Yukawa couplings to bottom quark and tau lepton as in the SM prediction, and this suggests the 331 symmetry breaking scale to be ∼ O ( 10 ) TeV .
... Since in SU(5) a family can only be assigned to 10+5 (or to the conjugated pair) [11], the unification group must be larger, hence N >5. This idea has led to the supersymmetric SU(7) [12] and the non-supersymmetric SU(8) mo- * albright@fnal.gov † robert.feger@vanderbilt.edu ...
... We present here an SU(12) model found by this scan which is free of any imposed external flavor symmetries. We have now also included the assignment of right-handed neutrinos, which allows the analysis of the full lepton sector as well, which is more ambitious than [12][13][14]. ...
... IV is the major section and presents the SU(12) model in detail: In Sec. IV A we demonstrate how three chiral families arise from SU (12) in our model. After listing the fermion assignments and the Higgs sector in Sec. ...
We present an SU(12) unification model with three light chiral families, avoiding any external flavor symmetries. The hierarchy of quark and lepton masses and mixings is explained by higher dimensional Yukawa interactions involving Higgs bosons that contain SU(5) singlet fields with vacuum expectation values about 50 times smaller than the SU(12) unification scale. The presented model has been analyzed in detail and found to be in very good agreement with the observed quark and lepton masses and mixings.
... In another SUSY SU(7) model, the hierarchies amongst the particle generations are generated by the hierarchy of symmetry breaking scales of the gauge group. Here SU(7) → SU(5) × SU(2) where the SU(2) group breaks into two, relatively smaller scales of the first and second particle generations [37]. ...
The natural extension to the SU(5) Georgi-Glashow grand unification model is to enlarge the gauge symmetry group. In this work, the SU(7) symmetry group is examined. The Cartan subalgebra is determined along with their commutation relations. The associated roots and weights of the SU(7) algebra are derived and discussed. The raising and lowering operators are explicitly constructed and presented. Higher dimensional representations are developed by graphical as well as tensorial methods. Applications of the SU(7) Lie group to supersymmetric grand unification as well as applications are discussed.
We study a toy model with the symmetry breaking pattern of the extended 331 symmetry of . A "fermion-Higgs mismatching" symmetry breaking pattern is proposed for more realistic model building. Within such symmetry breaking pattern, only one Higgs doublet develops vacuum expectation value for the spontaneous electroweak symmetry breaking, and gives tree-level top quark mass. A natural VEV splittings in the 331 breaking Higgs fields gives tree-level masses to both bottom quark and tau lepton. The SM-like Higgs boson discovered at the LHC can have Yukawa couplings to bottom quark and tau lepton as in the SM prediction, and this sets the 331 symmetry breaking scale at .
An explicit SUSY SU(12) unification model with three light chiral
families is presented which avoids any external flavor symmetries. The
hierarchy of quark and lepton masses and mixings is explained by higher
dimensional Yukawa interactions involving Higgs bosons containing SU(5)
singlet fields with VEVs appearing at or below the SUSY GUT scale of 2
× 1016 GeV, approximately 50 times smaller than the SU(12)
unification scale. The model has been found to be in good agreement with
the observed quark and lepton masses and mixings, with nearly all
prefactors of \Vbgr(1) in the four Dirac and one Majorana fermion mass
matrices.
It is shown that in SO(10) models where the large solar and atmospheric neutrino angles come from the charged-lepton mass matrix being “lopsided”, there is a characteristic relation between the 13 mixing angle of the neutrinos and the size of the Dirac CP-violating phase in the lepton sector. This is illustrated in a recently proposed realistic and predictive model.
It is shown that natural doublet-triplet splitting can be achieved in a
relatively simple way in supersymmetric SU(N)X U(1) models with N>5.
We present an SU(9) model of family unification with three light chiral
families, and a natural hierarchy of charged fermion masses and mixings. The
existence of singlet right handed neutrions with masses about two orders of
magnitude smaller than the GUT scale, as needed to understand the light
neutrinos masses via the see-saw mechanism, is compelling in our model.
It is shown that the stratified or "doubly lopsided" mass matrix structure that is known to reproduce well the qualitative features of the quark and lepton masses and mixings can arise quite naturally in the context of grand unification based on the groups SU(N) with N > 5. An SU(8) example is constructed with the minimal anomaly-free, three-family set of fermions, in which a realistic flavor structure results without flavor symmetry.
The tiny neutrino masses measured in neutrino oscillation experiments can be naturally explained via a supersymmetric seesaw mechanism. If supersymmetry breaking is mediated by gravity, such seesaw models may predict observable lepton flavor violating effects. In this work, we investigate the lepton flavor violating process μ⃗eγ in supersymmetric seesaw models with a “lopsided” charged lepton mass matrix. Constraints set by the muon anomalous magnetic moment are taken into account. We find that such models generally predict a branching ratio for μ⃗eγ far above the current experimental limit. Thus this process strongly constrains models of this type. Such restrictions suggest a novel form for the charged lepton mass matrix, featuring a “lopsided” structure for elements between both the 2-3 and the 1-3 generations. This structure produces a very small 1-3 mixing and a large 1-2 mixing in the charged lepton sector, leading naturally to a small Br(μ⃗eγ) and to the large mixing angle solution for the solar neutrino problem.
In theories in which different regions of the universe can have different values of certain physical parameters, we would naturally find ourselves in a region where they take values favorable for life. We explore the range of such viable values of the mass parameter in the Higgs potential, μ2. For μ2<0, the requirement that complex elements be formed suggests that the Higgs vacuum expectation value v must have a magnitude less than 5 times its observed value. For μ2>0, baryon stability requires that |μ|≪MP, the Planck mass. Smaller values of |μ2| may or may not be allowed depending on issues of element synthesis and stellar evolution. We conclude that the observed value of μ2 appears reasonably typical of the viable range, and a multiple-domain scenario may provide a plausible explanation for the closeness of the QCD scale and the weak scale.
An alternative supersymmetric SO(10) grand unification model with lopsided fermion mass matrices is introduced. It generates a large solar-neutrino mixing angle through the neutrinos' Dirac mass matrix constrained by the SO(10) group structure, avoiding the fine-tuning required in the Majorana mass matrix of right-handed neutrinos. The model fits well the known data on masses and mixings of quarks and leptons, and predicts a sizable lepton mixing sin22θ13≃0.074, which is significantly larger than that of the original lopsided model.
One of the puzzles of the Standard Model is why the mass parameter which
determines the scale of the Weak interactions is closer to the scale of QCD
than to the Grand Unification or Planck scales. We discuss a novel approach to
this problem which is possible in theories in which different regions of the
universe can have different values of the physical parameters. In such a
situation, we would naturally find ourselves in a region which has parameters
favorable for life. We explore the whole range of values of the mass parameter
in the Higgs potential, , from to and find that there
is only a narrow window, overlapping with the observed value, in which life is
likely to be possible. The observed value of is fairly typical of the
values in this range. Thus multiple domain theories in which varies
among domains may give a promising approach to solving the fine tunign problem
and explaining the closeness of the QCD scale and the Weak scale.
We point out that the correlation between the large atmospheric neutrino mixing and the transition, if exists, comes from the lopsided flavor structure in SO(10) grand unification theory. We suggest testing the correlation by studying the deviation of and from the standard model predictions along with the constraints from and in a realistic SO(10) model with lopsided flavor structure. We find a specific correlation between and that is intrinsic to the lopsided structure and discuss how to confirm or rule out this flavor pattern by the more accurate measurements of these CP violation quantities in B decay.
Lopsided structure in mass matrices of down quarks and leptons gives a simple explanation for the observed large angles of neutrino mixings. We realize such mass matrices by the Froggatt-Nielsen mechanism in the framework of supersymmetric SO(10) grand unified theory (GUT). It is shown that the model can reproduce the successful mass matrices which have been obtained in SU(5) models. Cosmological implication of the model is also discussed. We show that the hybrid inflation occurs naturally in the model and it offers non-thermal leptogenesis by decays of the next-to-lightest right-handed neutrinos. The present baryon asymmetry is explained by just the oscillation mass scale in the atmospheric neutrinos. Comment: 12 pages
We consider the Froggatt-Nielsen (FN) mechanism reproducing the observed mass hierarchies and mixing angles for quarks and leptons. The large - mixing suggested from recent Superkamiokande experiments on the atmospheric neutrinos implies lopsided FN U(1) charges for the lepton doublets. There are two possible charge assignments to generate the large - mixing. We point out that the two models with different charge assignments have distinct low-energy predictions and hence they are distinguishable in future neutrino experiments on such as CP violation and decay. Comment: 14 pages,11 figures, Several discussions are sophisticated
We argue that the observed quark and lepton masses are related at momenta larger than 1015 GeV as follows: mb = mτ, mμ = 3ms and me = md/3. We construct a model in which these factors of three arise naturally — because quarks come in three colors.
A structure is proposed for the mass matrices of the quarks and leptons that arises in a natural way from the assumption that the breaking of SO(10) gauge symmetry is achieved by the smallest possible set of vacuum expectation values. This structure explains well many features of the observed spectrum of quarks and leptons. It reproduces the Georgi-Jarlskog mass relations and leads to a charm quark mass in reasonable agreement with data. It also predicts a large mixing angle between {nu}{sub {mu}} and {nu}{sub {tau}} , as suggested by atmospheric neutrino data. The mixing angles of the electron neutrino are predicted to be small. {copyright} {ital 1998} {ital The American Physical Society}
We study a coset-space unification model for families based on E-7/SU(5) x U(1)(3). We find that qualitative structure of quark and lepton mass matrices in this model describes very well the observation. We stress, in particular, that the large mixing angle, sin(2)2 theta(v mu v tau) similar or equal to 1, required for the atmospheric neutrino oscillation reported by the SuperKamiokande collaboration, is naturally obtained, which is a consequence of unparallel family structure in the present coset-space unification.
It is shown that, in a model in which up-type and down-type fermions acquire mass from different Higgs doublets, the anthropic tuning of the Higgs mass parameters can explain the fact that the observed masses of the d and u quarks are nearly the same with d slightly heavier. If Yukawa couplings are assumed not to scan (vary among domains), this would also help explain why t is much heavier than b. It is also pointed out that the existence of dark matter invalidates some earlier anthropic arguments against the viability of domains where the standard model Higgs has positive mu2, but makes other even stronger arguments possible.
We show that the existence of a hierarchy of masses among light fermions, with typical mass ratios (..cap alpha../..pi..) (where ..cap alpha.. is the unified gauge coupling at the unification mass scale), is a natural feature of a certain class of grand unification schemes that have recently been discussed.
We advocate a new approach to study models of fermion masses and mixings, namely, the anarchy proposed by Hall, Murayama, and Weiner. In this approach, we scan the O(1) coefficients randomly. We argue that this is the correct approach when the fundamental theory is sufficiently complicated. Assuming that there is no physical distinction among three generations of neutrinos, the probability distributions in Maki-Nakagawa-Sakata mixing angles can be predicted independent of the choice of the measure. This is because the mixing angles are distributed according to the Haar measure of the Lie groups whose elements diagonalize the mass matrices. The near-maximal mixings, as observed in the atmospheric neutrino data and as required in the large mixing angle solution to the solar neutrino problem, are highly probable. A small hierarchy between Δm2 for the atmospheric and the solar neutrinos is obtained very easily; the complex seesaw case gives a hierarchy of a factor of 20 as the most probable one, even though this conclusion is more measure dependent. Ue3 has to be just below the current limit from the CHOOZ experiment. The CP-violating parameter sinδ is preferred to be maximal. We present a simple SU(5)-like extension of anarchy to the charged lepton and quark sectors that works well phenomenologically.
A class of predictive SO(10) grand unified theories with highly asymmetric mass matrices, known as lopsided textures, which was developed to accommodate the observed mixing in the neutrino sector, can effectively determine the rate for charged lepton flavor violation (LFV), and in particular the branching ratio for μ⃗eγ. Assuming that the supersymmetric GUT breaks directly to the constrained minimal supersymmetric standard model (CMSSM), we find that in light of the combined constraints on the CMSSM parameters from direct searches and from the WMAP satellite observations, the resulting predicted rate for μ⃗eγ in this model class can be within the current experimental bounds for low tanβ, but that the next generation of μ⃗eγ experiments would effectively rule out this model class if LFV is not detected.
We analyze in detail lepton-flavor violation (LFV) in the charged-lepton sector, such as μ→eγ and τ→μγ within the framework of supersymmetric models with lopsided Froggatt–Nielsen charges. We show that the present experimental limits on the LFV processes already exclude some of the models. The proposed future search for LFV, especially in muon processes, can provide a significant probe to this framework.
Current solar and atmospheric neutrino oscillation data seem to favor a bimaximal pattern for neutrino mixings where the matrix elements Ue2 and Uμ3 are of order one, while Ue3 is much smaller. We show that such a pattern can be obtained quite easily in theories with “lopsided” mass matrices for the charged leptons and the down type quarks. A relation connecting the solar and atmospheric neutrino mixing angles is derived, tan2θatm≃1+tan2θsol, which predicts sin22θatm≃0.97 corresponding to the best fit LMA solution for solar neutrinos. Predictive schemes in SO(10) realizing these ideas are presented. A new class of SO(10) models with lopsided mass matrices is found which makes use of an adjoint VEV along the I3R direction, rather than the traditional B−L direction.
CP violation in hadronic B decays is studied in a definite and well-motivated framework of flavour physics and supersymmetry. Possible deviations from the standard model both in mixing and in decay amplitudes are discussed. An attempt is made to describe an experimental strategy for looking at these deviations and for measuring the relevant parameters.
A general operator expansion is presented for quark and lepton mass matrices in unified theories based on a U(2) flavor symmetry, with breaking parameter of order Vcb ≈ ms/mb ≈ √mc/mt. While solving the supersymmetric flavor-changing problem, a general form for the Yukawa couplings follows, leading to nine relations among the fermion masses and mixings, five of which are precise. The combination of grand unified and U(2) symmetries provides a symmetry understanding for the anomalously small values of mu/mc and mc/mt. A fit to the fermion mass data leads to a prediction for the angles of the CKM unitarity triangle, which will allow a significant test of these unified U(2) theories. A particular SO(10) model provides a simple realization of the general operator expansion. The lighter generation masses and the non-trivial structure of the CKM matrix are generated from the exchange of a single U(2) doublet of heavy vector generations. This model suggests that CP is spontaneously broken at the unification scale—in which case there is a further reduction in the number of free parameters.
In a generic supersymmetric extension of the Standard Model, whether unified or not, a simple and well motivated global U(2) symmetry, acting on the lightest two generations, completely solves the flavour changing problem and necessarily leads to a predictive texture for the Yukawa couplings.
I describe a set of guidelines for the construction of grand unified theories which incorporate multiple flavors. These theories contain the standard SU(5) theory as a subgroup, but their gauge structure also includes non-trivial flavor group structure in the same simple group. I find an example based on an SU(11) gauge symmetry in which six flavors of quarks appear naturally.
We present a supersymmetric standard model with three gauged Abelian symmetries, of a type commonly found in superstrings. One is anomalous, the other two are family symmetries. It has a vacuum in which only these symmetries are broken by stringy effects. It reproduces all observed quark and charged lepton Yukawa hierarchies, and the value of the Weinberg angle. It predicts three massive neutrinos, with mixing that can explain both the small angle MSW effect, and the atmospheric neutrino anomaly. The Cabibbo angle is expressed in terms of the gauge couplings at unification. It conserves R-parity, and proton decay is close to experimental bounds. Comment: 26 pages
It has recently been shown how to break SO(10) down to the Standard Model in a realistic way with only one adjoint Higgs. The expectation value of this adjoint must point in the B-L direction. This has consequences for the possible form of the quark and lepton mass matrices. These consequences are explored in this paper, and it is found that one is naturally led to consider a particular form for the masses of the heavier generations. This form implies typically that there should be large (nearly maximal) mixing of the mu- and tau-neutrinos. An explanation that does not involve large tan beta also emerges for the fact that b and tau are light compared to the top quark. Comment: 20 pages, LaTeX, clarification of statements about multiple adjoint Higgs fields in the context of superstring theory
Typically in unified theories the neutrino mixing angles, like the
Cabibbo-Kobayashi-Maskawa (CKM) angles of the quarks, are related to the small
mass ratios between fermions of different generations and are therefore quite
small. A new approach for explaining the intergenerational mass hierarchies is
proposed here which, while giving small CKM angles, naturally leads to neutrino
angles of order unity. Such large mixing angles may be required for a
resolution of the atmospheric neutrino anomaly and may also be relevant for the
solar neutrino puzzle. The mechanism presented here provides a framework in
which novel approaches to the fermion mass question can arise. In particular,
within this framework a variant of the texture idea allows highly predictive
models to be constructed, an illustrative example of which is given. It is
shown how the neutrino mixing angles may be completely determined in such
schemes.
There are four puzzling questions about by the magnitudes of neutrino mixings and mass splittings. A brief sketch is given of the various kinds of models of neutrino masses and how they answer these questions. Special attention is given to so-called "lopsided" models.
We analyze in detail lepton-flavor violation (LFV) in the charged-lepton sector such as , , and the conversion in nuclei, within the framework of supersymmetric models with lopsided Froggatt--Nielsen charges, in which the large mixing in the neutrino sector as well as small mixings in the quark sector can be naturally accommodated. We show that the present experimental limits on the LFV processes already exclude some of the models. The future proposed search for LFV, especially in muon processes, can provide a significant probe to this framework. We also stress the importance of the measurement of in neutrino experiments, and the fact that the KamLAND experiment could play a significant role to test a certain class of models. Comment: 33 pages, 20 figures
It is shown that the stratified or "doubly lopsided" mass matrix structure that is known to reproduce well the qualitative features of the quark and lepton masses and mixings can arise quite naturally in the context of grand unification based on the groups SU(N) with N > 5. An SU(8) example is constructed with the minimal anomaly-free, three-family set of fermions, in which a realistic flavor structure results without flavor symmetry.
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