In a renormalizable $SO(10)$ theory, all fermion mass matrices are linear combinations of three fundamental types, $M^{10}, M^{\overline{126}}$, and $M^{120}$, whose superscripts indicate their $SO(10)$ transformation properties. We point out that each of these fundamental mass matrices possesses a natural symmetry that can be used to generate an unbroken horizontal symmetry $\G$, if the natural symmetry is taken to be the residual symmetry. This built-in symmetry is a Coxeter group. If it is finite, it must be one of five groups, $S_4,\ Z_2\x S_4$,\ $Z_2\x A_5$, plus two `rank-4' groups. These symmetries place constraints on the fundamental mass matrices and reduce the number of parameters in an SO(10) fit. Since they are built-in and can be derived theoretically, it is hoped that they impose better constraints than those without a theoretical basis, but that is to be confirmed because there is no attempt to fit the experimental data in this article, except to count the number of free parameters. To illustrate the similarities and differences of various kinds of constraints, a comparison is made with an existing $S_4$ model, and with models possessing the Fritzsch texture.
A general method to derive horizontal symmetry from a mixing matrix is reviewed. The technique has been applied to deduce leptonic symmetry from the tri-bimaximal neutrino mixing matrix and three of its variations. The question of how the quark mixing can be accommodated within the leptonic symmetry group is discussed, including in this connection an example based on the group $D_4$.
New mixing structures between chiral generations of elementary particles at low energy are shown in a vectorlike model with a horizontal symmetry SU(1,1). In this framework the chiral model including odd number chiral generations is realized via the spontaneous symmetry breaking of the horizontal symmetry. It is shown that the Yukawa coupling matrices of chiral generations have naturally hierarchical patterns, and in some cases the overall factors of their Yukawa coupling matrices, e.g. the Yukawa coupling constants of the bottom quark and tau lepton are naturally suppressed.
We extend the nonsupersymmetric SO(10) grand unification theories by adding a horizontal symmetry, which connects the three generations of fermions. Without committing to any specific symmetry group, we investigate the 1-loop renormalization group evolutions of the gauge couplings with one and two intermediate breaking scales. We find that depending on the SO(10) breaking chains, gauge coupling unification is compatible with only a handful of choices of representations of the Higgs bosons under the horizontal symmetry.
A group-theoretical connection between horizontal symmetry $\G$ and fermion mixing is established, and applied to neutrino mixing. The group-theoretical approach is consistent with a dynamical theory based on $U(1)\times \G$, but the dynamical theory can be used to pick out the most stable mixing that purely group-theoretical considerations cannot. A symmetry common to leptons and quarks is also discussed. This higher symmetry picks $A_4$ over $S_4$ to be the preferred symmetry for leptons.
The standard model of particle physics is generalized so as to be furnished with a horizontal symmetry generated by an intermediary algebra between simple Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$. Above a certain high energy scale $\breveΛ$, the horizontal gauge symmetry is postulated to hold so that the basic fermions, quarks and leptons, form its fundamental triplets, and a triplet and singlet of the horizontal gauge fields distinguish generational degrees of freedom. A horizontal scalar triplet is introduced to make the gauge fields super-massive by breaking the horizontal sym…
A generic valon potential invariant under U(1)\x SO(3) is used to determine whether the horizontal symmetry of leptons is $A_4$, $S_4$, or some other finite subgroups of SO(3). Valons in the potential are assigned SO(3) spins 0, 1, and 2, for these are the only ones that can couple to the three generation of fermions with horizontal spin 1. This potential causes a breakdown into three possible phases with three different symmetries. Phase I has an $A_4$ symmetry suitable for leptonic mixing. Phase II has an SO(2) symmetry and phase III has a $Z_2\x Z_2$ symmetry, both capable of describing Cab…
Gauge field theory of a horizontal symmetry of the group G = SU(2) X U(1) is developed so as to generalize the standard model of particle physics. All fermion and scalar fields are assumed to belong to doublets and singlets of the group in high energy regime. Mass matrices with four texture zeros of Dirac and Majorana types are systematically derived. In addition to seven scalar particles, the theory predicts existence of one peculiar vector particle which seems to play important roles in astrophysics and particle physics.
We investigate Dirac mass matrices derived in the gauge field theory of a horizontal symmetry generated by a central extension of the Pauli algebra. Through numerical analyses of the observed data of the charged fermion masses and the flavor mixing matrix of quarks, values of free parameters in the mass matrices are determined and several empirical relations are found among the Yukawa coupling constants. As one specific feature of the theory, we find different orderings in squared mass eigenvalues for the up and down quark sectors.
We present a model of composite Dark Matter (DM), in which a new QCD-like confining "hypercolor" sector generates naturally stable hyperbaryons as DM candidates and at the same time provides mass to new weakly coupled gauge bosons $H$ that serve as DM mediators, coupling the hyperbaryons to the Standard Model (SM) fermions. By an appropriate choice of the $H$ gauge symmetry as a horizontal $SU(2)_h$ SM flavor symmetry, we show how the $H$ gauge bosons can be identified with the horizontal gauge bosons recently put forward as an explanation for discrepancies in rare $B$-meson decays. We find th…
The group-theoretical method used to derive horizontal symmetry from neutrino mixing is reviewed and expanded. Some misunderstanding in the literature regarding the result is clarified. The method used previously to find vacuum alignments of $S_4$ is applied to compute those of $A_4$ and $S_3$. A study of effective theories based on these three groups shows that in each case there are just enough free parameters to fit all the masses and the remaining mixing parameters. This places constraint on dynamical models because effective theories are just dynamical models with the right-handed fermion…
An extended spin-space model in $7+1$ dimensions is presented that describes the standard-model electroweak quark sector. Up to four generations of massless and massive quarks and two-Higgs doublets derive from the associated representation space, in addition to the W- and Z-vector bosons. Other mass operators are obtained that put restrictions on additional non-Higgs scalars and their vacuum expectation value. After symmetry breaking, the scalar components give rise to a hierarchy effect vertically (within doublets) associated to the Higgs fields, and horizontally (within generations) associa…
We determine the symmetry groups under which the charged-lepton and the Majorana-neutrino mass terms are invariant. We note that those two groups always exist trivially, i.e. independently of the presence of any symmetries in the Lagrangian, and that they always have the same form. Using this insight, we re-evaluate the recent claim that, whenever lepton mixing is tri-bimaximal, S_4 is the minimal unique horizontal-symmetry group of the Lagrangian of the lepton sector, with S_4 being determined by the symmetries of the lepton mass matrices. We discuss two models for tri-bimaximal mixing which …
The largest global symmetry that can be made local in the Standard Model + 3$ν_R$ while being compatible with Pati-Salam unification is $SU(3)_H\times U(1)_{B-L}$. The gauge bosons of this theory would induce flavour effects involving both quarks and leptons, and are a potential candidate to explain the recent reports of lepton universality violation in rare B meson decays. In this letter we characterise this type of models and show how they can accommodate the data and naturally be within reach of direct searches.
We study the contributions of supersymmetric models with a $U(1)$ horizontal symmetry and only spontaneous CP breaking to various lepton flavor observables, such as $μ\to eγ$ and the electron electric dipole moment. We show that both a horizontal symmetry and a lack of explicit CP violation can alleviate the existing bounds from such observables. The undetermined $\mathcal{O}(1)$ coefficients in such mass matrix models muddle the interpretation of the bounds from various flavor observables. To overcome this, we define a new fine-tuning measure for different observables in such setups. This all…
We propose a gauged $U(1)_{H}$ horizontal symmetry around TeV scale that is a subgroup of a $SU(3)_{H}$ horizontal gauge symmetry broken at ${\cal O}(10^{14}) \GeV$. The breaking generates right-handed Majorana neutrino masses through a $SU(3)_H$ sextet scalar. A particular Majorana right-handed neutrino mass matrix explicitly determines the remnant $U(1)_{H}$ at low energy which only couples to $b-s$ and $μ-τ$ in the gauge eigenstate. The dangerous $K-\bar{K}$, $D-\bar{D}$ mixing and $B_s \rightarrow μ^+ μ^-$ are kept to be safe because the relevant couplings are suppressed through high power…
Identical interactions found in the three families of quarks and leptons suggest the presence of a horizontal symmetry. We discuss how such a symmetry can be tested by measuring the decay rates of Higgs into fermion pairs, and the Higgs production cross section. Depending on the details, there is a chance that the decay widths to the bottom-pair and the tau-pair may be down by more than a factor of 3 or more compared to the usual values, and the fusion production cross section of the Higgs also altered. Whatever the outcome, such a test also serves to constraint horizontal symmetry models.
One of the greatest challenges for models of $b \to s$ anomalies is the necessity to produce a large contribution to a quark times a lepton current, $J_q \times J_\ell$, and to avoid accordingly large contributions to flavour-changing $J_q \times J_q$ and $J_\ell \times J_\ell$ amplitudes, which are severely constrained by data. We consider a gauged horizontal symmetry involving the two heaviest generations of all left-handed fermions. In the limit of degenerate masses for the horizontal bosons, and in the absence of mixing between the two heavier generations and the lighter one, such symmetry…
In a supersymmetric SU(5) grand unified model with a horizontal symmetry SU(1,1), we discuss spontaneous generation of generations to produce three chiral generations of quarks and leptons and one generation of higgses by using one structure field with a half-integer spin of SU(1,1) and two structure fields with integer spins. In particular, the colored higgses can disappear without fine-tuning. The difference of the Yukawa coupling matrices between the down-type quarks and charged leptons is discussed. We show that some special SU(1,1) weight assignments include R-parity as a discrete subgrou…
In this talk, I will present the general structure of RPV couplings when a Froggatt-Nielsen horizontal symmetry is responsible for the flavor structure of both the SM and the MSSM. For sub-TeV ({\it natural}) SUSY, lepton number must be an accidental symmetry, while low-energy SUSY is still allowed by baryonic RPV, which lowers the MET signature of superparticles decays. The largest RPV coupling involves the stop, and it is constrained between $10^{-3}$ (from FCNCs) and $10^{-9}$ (from LHC searches).
We present a novel procedure for identifying discrete, leptonic flavour symmetries, given a class of unitary mixing matrices. By creating explicit 3D representations for generators of residual symmetries in both the charged lepton and neutrino sector, we reconstruct large(r) non-abelian flavour groups using the GAP language for computational finite algebra. We use experimental data to construct only those generators that yield acceptable (or preferable) mixing patterns. Such an approach is advantageous because it 1) can reproduce known groups from other 'top-down' scans while elucidating their…
Motivated by the persistence of a large measured top quark forward-backward asymmetry at the Tevatron, we examine a model of non-Abelian flavor gauge symmetry. The exchange of the gauge bosons in the $t$-channel can give a large asymmetry due to the forward Rutherford scattering peak. We address generic constraints on non-Abelian $t$-channel physics models including flavor diagonal resonances and potentially dangerous contributions to inclusive top pair cross sections. We caution on the general difficulty of comparing theoretical predictions for top quark signals to the existing experimental r…
Supersymmetry, local horizontal unification, and a solution to the flavor puzzle
具有局域水平对称性的超对称规范模型会产生由超对称破缺D项引起的大的变味中性流效应。我们展示了U（1）规范对称性的存在如何解决这个问题。然后我们构造了一个以SU（2）_H × U（1）_H为局域水平对称的现实规范模型，并提出U（1）_H因子可以等同于弦紧化引起的反常U（1）。这个模型解释了夸克质量和混合角之间的层次结构，自然地容纳太阳和大气中微子数据，并同时提供了一个解决超对称味问题。如果在当前一轮实验中没有观察到稀有衰变μ--> e γ，则可以排除该模型。
Recent anomalies in $B\to K^{(*)}\ell\ell$ meson decays are consistent with exchange of a heavy $Z'$ vector boson. Here we try to connect such new physics to understanding the origin of flavor, by gauging generation number. Phenomenological and theoretical considerations suggest that the smallest viable flavor symmetry (not including any extra U(1) factors) is chiral ${\rm SU(3)}_L\times{\rm SU(3)}_R$, which acts only on generation indices and does not distinguish between quarks and leptons. Spontaneous breaking of the symmetry gives rise to the standard model Yukawa matrices, and masses for t…
We present a solution to the strong CP problem, which relies on the horizontal gauge symmetry and CP invariance in a full theory. Similar to other Nelson-Barr type solutions, CP violation in both the strong and weak sectors in the Standard model (SM) is attributed to the condensation of complex scalars $Φ$ in the model. The model is differentiated by others in that it explains the hierarchy in quark-Higgs Yukawa coupling in the SM based on a series of sequential breaking of the horizontal $SU(3)_{f}$ gauge symmetry. The experimental constraint $\overlineθ\lesssim10^{-10}$ requires $<\!\!Φ\!\!>…
