The simultaneous study of top-down and bottom-up approaches to modular flavor symmetry leads necessarily to the concept of eclectic flavor groups. These are nontrivial products of modular and traditional flavor symmetries that exhibit the phenomenon of local flavor enhancement in moduli space. We develop methods to determine the eclectic flavor groups that can be consistently associated with a given traditional flavor symmetry. Applying these methods to a large family of prominent traditional flavor symmetries, we try to identify potential candidates for realistic eclectic flavor groups and show that they are relatively rare. Model building with finite modular flavor symmetries thus appears to be much more restrictive than previously thought.
A top-down approach to the flavor problem motivated from string theory leads to the concept of eclectic flavor groups that combine traditional and modular flavor symmetries. To make contact with models constructed in the bottom-up approach, we analyze a specific example based on the eclectic flavor group Omega(1) (a nontrivial combination of the traditional flavor group Delta(54) and the finite modular group T') in order to extract general lessons from the eclectic scheme. We observe that this scheme is highly predictive since it severely restricts the possible group representations and modula…
Modular symmetries naturally combine with traditional flavor symmetries and $\mathcal{CP}$, giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional $\mathbb{Z}_2$ orbifold, which is equipped with two modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ associated with two moduli: the Kähler modulus $T$ and the complex structure modulus $U$. The resulting finite modular group is $((S_3\times S_3)\rtimes \mathbb{Z}_4)\times\mathbb{Z}_2$ including mirror symmetry (that exchanges $T$ and $U$) and a generalized $\mathcal{CP}$-…
String theory leads to a flavor scheme where modular symmetries play a crucial role. Together with the traditional flavor symmetries they combine to an eclectic flavor group, which we determine via outer automorphisms of the Narain space group. Unbroken flavor symmetries are subgroups of this eclectic group and their size depends on the location in moduli space. This mechanism of local flavor unification allows a different flavor structure for different sectors of the theory (such as quarks and leptons) and also explains the spontaneous breakdown of flavor- and CP-symmetries (via a motion in m…
In a consistent top-down approach based on orbifold compactifications, modular and traditional flavor symmetries combine nontrivially to the so-called eclectic flavor symmetry. We extend this scheme from two extra dimensions, discussed previously, to the six extra dimensions of string theory. By doing so, new insights on the nature of CP and its spontaneous breaking emerge. Moreover, we identify a new interpretation of R-symmetries as unbroken remnants from modular symmetries that are associated with geometrically stabilized complex structure moduli. Hence, all symmetries (i.e. modular, tradit…
We present a detailed analysis of the eclectic flavor structure of the two-dimensional $\mathbb Z_2$ orbifold with its two unconstrained moduli $T$ and $U$ as well as $\mathrm{SL}(2,\mathbb Z)_T\times \mathrm{SL}(2,\mathbb Z)_U$ modular symmetry. This provides a thorough understanding of mirror symmetry as well as the $R$-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the $(T,U)$ modulus landscape. In view of applications towards the flavor structure of particle physics models, we are l…
Eclectic flavor groups arising from string compactifications combine the power of modular and traditional flavor symmetries to address the flavor puzzle. This top-down scheme determines the representations and modular weights of all matter fields, imposing strict constraints on the structure of the effective potential, which result in controlled corrections. We study the lepton and quark flavor phenomenology of an explicit, potentially realistic example model based on a T6/Z3xZ3 orbifold compactification of the heterotic string that gives rise to an Omega(2) eclectic flavor symmetry. We find t…
The framework of compactified heterotic string theory offers consistent ultraviolet (UV) completions of the Standard Model (SM) of particle physics. In this approach, the existence of flavor symmetries beyond the SM is imperative and the flavor symmetries can be derived from the top-down. Such a derivation uncovers a unified origin of traditional discrete flavor symmetries, discrete modular flavor symmetries, discrete R symmetries of supersymmetry, as well as CP symmetry - altogether called the eclectic flavor symmetry. I will show a specific example of such a top-down derived eclectic flavor …
Modular flavor symmetries provide us with a new, promising approach to the flavor problem. However, in their original formulation the kinetic terms of the standard model fields do not have a preferred form, thus introducing additional parameters, which limit the predictive power of this scheme. In this work, we introduce the scheme of quasi-eclectic flavor symmetries as a simple fix. These symmetries are the direct product of a modular and a traditional flavor symmetry, which are spontaneously broken to a diagonal modular flavor subgroup. This allows us to construct a version of Feruglio's mod…
A top-down approach to the flavor puzzle leads to eclectic flavor groups which include modular and traditional flavor symmetries. Based on examples of semirealistic T2/Z3 orbifold compactifications of heterotic string theory, we discuss the breakdown patterns of the eclectic flavor group via the interplay of vacuum expectation values (vevs) of moduli and flavon fields. This leads to an attractive flavor scheme with various possibilities to obtain "flavor hierarchies" through the alignment of these vevs. Despite the fact that the top-down approach gives strong restrictions for bottom-up flavor …
The framework of compactified heterotic string theory offers consistent ultraviolet (UV) completions of the Standard Model (SM) of particle physics. In this approach, the existence of flavor symmetries beyond the SM is imperative and the flavor symmetries can be derived from the top down. Such a derivation uncovers a unified origin of traditional discrete flavor symmetries, discrete modular flavor symmetries, discrete R symmetries of supersymmetry, as well as charge-parity (CP) symmetry - altogether dubbed the eclectic flavor symmetry. I will illustrate how the eclectic flavor symmetry is unam…
Modular transformations of string theory are shown to play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They include CP transformations and provide a unification of CP with traditional flavor symmetries within the framework of the "eclectic flavor" scheme. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of "Local Flavor Unification", where different sectors of the theory (like quarks and leptons) can be subject to different flavor structures.
A somewhat eclectic choice of topics: Review of Flavor in the quark sector of the SM, the appearance of the CKM matrix and determination of its elements, FCNCs, the GIM mechanism. Minimal Flavor Violation. A close look at neutral meson mixing. CP asymmetries. Effective Field Theory in Weak interactions.
The recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with two- and three-dimensional irreducible representations. This paper details the mathematics of the two finite simple groups with such representations, the Icosahedral group A_5, a subgroup of SO(3), and PSL_2(7), a subgroup of SU(3).
The large neutrino mixing angles have generated interest in finite subgroups of SU(3), as clues towards understanding the flavor structure of the Standard Model. In this work, we study the mathematical structure of the simplest non-Abelian subgroup, Delta(3n^2). Using simple mathematical techniques, we derive its conjugacy classes, character table, build its irreducible representations, their Kronecker products, and its invariants.
Many non-Abelian finite subgroups of SU (3) have been used to explain the flavor structure of the Standard Model. In order to systematize and classify successful models, a detailed knowledge of their mathematical structure is necessary. In this paper we shall therefore look closely at the series of finite non-Abelian groups known as Delta(6n^2), its smallest members being S3 (n = 1) and S4 (n = 2). For arbitrary n, we determine the conjugacy classes, the irreducible representations, the Kronecker products as well as the Clebsch-Gordan coefficients.
Flavor symmetry plays a crucial role in the standard model of particle physics but its origin is still unknown. We develop a new method (based on outer automorphisms of the Narain space group) to determine flavor symmetries within compactified string theory. A picture emerges where traditional (discrete) flavor symmetries, CP-like symmetries and modular symmetries (like T-duality) of string theory combine to unified flavor symmetries. The groups depend on the geometry of compact space and the geographical location of fields in the extra dimensions. We observe a phenomenon of "local flavor grou…
A new approach to the generation structure of fermions is proposed. We consider a brane configuration in which the brane intersection yields a two-torus in the extra space. It is assumed that the two-torus is discretized and is given by the torsion points of the elliptic curve over Q . We direct our attention to the arithmetic structure of the elliptic curve with complex multiplication (CM). In our approach the flavor symmetry including the R-parity has its origin in the Galois group of elliptic curves with CM. We study the possible types of the Galois group. The Galois group is shown to be an…
We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarz phases. In particular, we investigate finite modular flavor groups for three-generation modes on magnetized orbifolds. The three-generation modes can be the three-dimensional irreducible representations of covering groups and central extended groups of $Γ_N$ for $N=3,4,5,7,8,16$, that is, covering groups of $Δ(6(N/2)^2)$ for $N=$ even and central extensions of $PSL(2,\mathbb{Z}_{N})$ for $N=$odd with Scherk-Schwarz phases. We also study anomaly behaviors.
Putting emphasis on the relation between rational conformal field theory (RCFT) and algebraic number theory, we consider a brane configuration in which the D-brane intersection is an elliptic curve corresponding to RCFT. A new approach to the generation structure of fermions is proposed in which the flavor symmetry including the R-parity has its origin in the Galois group on elliptic curves with complex multiplication (CM). We study the possible types of the Galois group derived from the torsion points of the elliptic curve with CM. A phenomenologically viable example of the Galois group is pr…
We study the modular invariance in magnetized torus models. Modular invariant flavor model is a recently proposed hypothesis for solving the flavor puzzle, where the flavor symmetry originates from modular invariance. In this framework coupling constants such as Yukawa couplings are also transformed under the flavor symmetry. We show that the low-energy effective theory of magnetized torus models is invariant under a specific subgroup of the modular group. Since Yukawa couplings as well as chiral zero modes transform under the modular group, the above modular subgroup (referred to as modular f…
We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$ that includes the modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ (of the "geometric" moduli $T$ and $U$) as well as mirror symmetry. In addition, string theory provides a candidate for a CP-like symmetry that enhances the Siegel modular group to $\mathrm{GSp}(4,\mathbb{Z})$.
Present data on neutrino masses and mixing favor the highly symmetric tribimaximal neutrino mixing matrix which suggests an underlying flavor symmetry. A systematic study of non-abelian finite groups of order $g \leq 31$ reveals that tribimaximal mixing can be derived not only from the well known flavor symmetry $T \equiv A_4$, the tetrahedral group, but also by using the alternative flavor symmetry X(24) $\equiv SL_2(F_3) \equiv Q_4 \tilde{\times} Z_3$. X(24) does not contain the tetrahedral group as a subgroup, and has the advantage over it as a flavor symmetry that it can not only underwrit…
We determine the vacuum structure and phases of N=1 theories obtained via a mass μfor the adjoint chiral superfield in N=2, SO(n_c) SQCD. For large number of flavors these theories have two groups of vacua. The first exhibits dynamical breaking of flavor symmetry \USp(2n_f) \to U(n_f) and arises as a relevant deformation of a non-trivial superconformal theory. These are in the confined phase. The second group, in an IR-free phase with unbroken flavor symmetry, is produced from a Coulomb branch singularity with Seiberg's dual gauge symmetry. In the large-μregime both groups of vacua are well-de…
We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including $Γ_N$ and $Γ'_N$ as the flavor symmetry. The theory of vector-valued modular forms provide a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-valued mod…
We study the presence of discrete flavor symmetries in D-brane models of particle physics. By analyzing the compact extra dimensions of these models one can determine when such symmetries exist both in the context of intersecting and magnetized D-brane constructions. Our approach allows to distinguish between approximate and exact discrete symmetries, and it can be applied to compactification manifolds with continuous isometries or to manifolds that only contain discrete isometries, like Calabi-Yau three-folds. We analyze in detail the class of rigid D-branes models based on a Z_2 x Z'_2 toroi…
Extra dimension deconstructed on a closed chain has naturally the symmetry of a regular polygon, the dihedral symmetry D_N. We assume that the fields are irreducible representations of the binary dihedral group Q_2N, which is the covering group of D_N. It is found that although the orbifold boundary conditions break the dihedral invariance explicitly, the Q_2N symmetry appears as an intact internal, global flavor symmetry at low energies. A concrete predictive model based on Q_6N$ with an odd N is given.
Multi Higgs doublet models are interesting extensions of the Standard Model that can be related to flavor. The reason is that most flavor models usually involve the presence of several additional scalar fields. In this work we present an analysis that shows that for renormalizable flavor models based on the cyclic group of order $N$, if there is one flavored SU(2) double Higgs per generation, the smallest $N$ that can be used to reproduce the Nearest-Neighbor-Interaction texture for the quark mass matrices is N=5. Results for the Higgs spectrum and consistency under $K - \bar{K}$ mixing in a s…
We study the lepton flavor models, whose flavor symmetries are finite subgroups of the modular group such as $S_3$ and $A_4$. In our models, couplings are also nontrivial representations of these groups and modular functions of the modulus. We study the possibilities that these models realize realistic values of neutrino masses and lepton mixing angles.
We initiate the study of the generalized global flavor symmetries of the Standard Model. The presence of nonzero triangle diagrams between the $U(3)^5$ flavor currents and the $U(1)_Y$ hypercharge current intertwines them in the form of a higher-group which mixes the zero-form flavor symmetries with the one-form magnetic hypercharge symmetry. This higher symmetry structure greatly restricts the possible flavor symmetries that may remain unbroken in any ultraviolet completion that includes magnetic monopoles. In the context of unification, this implies tight constraints on the combinations of f…
Modular transformations of string theory (including the well-known stringy dualities) play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They are at the origin of CP transformations and provide a unification of CP with traditional flavor symmetries. Here, we present a novel, fully systematic method to reliably compute the unified flavor symmetry of the low-energy effective theory, including enhancements from the modular transformations of string theory. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of "Local Flavor…
We show that non-Abelian discrete symmetries in orbifold string models have a gauge origin. This can be understood when looking at the vicinity of a symmetry enhanced point in moduli space. At such an enhanced point, orbifold fixed points are characterized by an enhanced gauge symmetry. This gauge symmetry can be broken to a discrete subgroup by a nontrivial vacuum expectation value of the Kähler modulus $T$. Using this mechanism it is shown that the $Δ(54)$ non-Abelian discrete symmetry group originates from a $SU(3)$ gauge symmetry, whereas the $D_4$ symmetry group is obtained from a $SU(2)$…
The exact NSVZ $β$-function is obtained for ${\cal N}=1$ SQED with $N_f$ flavors in all orders of the perturbation theory, if the renormalization group functions are defined in terms of the bare coupling constant and the theory is regularized by higher derivatives. However, if the renormalization group functions are defined in terms of the renormalized coupling constant, the NSVZ relation between the $β$-function and the anomalous dimension of the matter superfields is valid only in a certain (NSVZ) scheme. We prove that for ${\cal N}=1$ SQED with $N_f$ flavors the NSVZ relation is valid for t…
A Z_2 orbifold compactification of the heterotic string is considered. The resulting 6D GUT groups can be SO(16) or E_7 times SU(2) plus some hidden sector groups. The N=4 supersymmetry is reduced to N=2. In particular, the SO(16) 6D model with one spinor representation 128 can reduce to the previous 5D SO(16) or SO(14) family unification models after compactifying the sixth dimension. To obtain one spinor, we have to take into account the left-over center of SO(16). We also comment on the E_7 times SU(2) model.
We investigate the renormalization group flow of the field-dependent Yukawa coupling in the framework of the three flavor quark-meson model. In a conventional perturbative calculation, given that the field rescaling is trivial, the Yukawa coupling does not get renormalized at the one-loop level if it is coupled to an equal number of scalar and pseudoscalar fields. Its field-dependent version, however, does flow with respect to the scale. Using the functional renormalization group technique, we show that it is highly nontrivial how to extract the actual flow of the Yukawa coupling as there are …
We extend the work of Carone, Chaurasia and Vasquez on non-supersymmetric models of flavor based on the double tetrahedral group. Three issues are addressed: (1) the sector of flavor-symmetry-breaking fields is simplified and their potential studied explicitly, (2) a flavorful axion is introduced to solve the strong CP problem and (3) the model is extended to include the neutrino sector. We show how the model can accommodate the strong hierarchies manifest in the charged fermion Yukawa matrices, while predicting a qualitatively different form for the light neutrino mass matrix that is consiste…
A nonabelian finite flavor group $G \subset SO(3)$ can have double covering $G^{'} \subset SU(2)$ such that $G \not\subset G^{'}$. This situation is not contradictory, but quite natural, and we give explicit examples such as $G=D_n, G^{'}=Q_{2n}$ and $G=T, G^{'}=T^{'}$. This observation can be crucial in particle theory model building.
We discuss the possibility of obtaining a non-abelian discrete flavor symmetry from an underlying continuous, possibly gauged, flavor symmetry SU(2) or SU(3) through spontaneous symmetry breaking. We consider all possible cases, where the continuous symmetry is broken by small representations. "Small" representations are these which couple at leading order to the Standard Model fermions transforming as two- or three-dimensional representations of the flavor group. We find that, given this limited representation content, the only non-abelian discrete group which can arise as a residual symmetry…
The interplay of mesonic fluctuations with an axial $U(1)_A$-symmetry breaking and resulting effects on the location of a possibly existing critical endpoint in the QCD phase diagram are investigated in a framework of the functional renormalization group within a $N_f = 2 + 1$ flavor quark-meson model truncation. The axial $U(1)_A$-symmetry breaking is imposed by a mesonic Kobayashi-Maskawa-'t Hooft determinant. The quark mass sensitivity of the chiral phase transition with and without the $U(1)_A$-symmetry breaking is studied.
The $SU(2)_A \times U(2)_V$-symmetric chiral linear sigma model in the presence of the axial anomaly is studied in the local-potential approximation of the Functional Renormalization Group (FRG). The renormalization group (RG) flow is investigated in a truncation which reproduces recent results for the $U(2)_A \times U(2)_V$-symmetric model in the limit of vanishing axial anomaly strength. We search for the conjectured O(4) fixed point in the presence of the $U(1)_A$ anomaly and analyze its stability properties.
We show the specific unusual features on chiral gauge anomalies cancellation in the minimal, necessarily 3-3-1, and the largest 3-4-1 weak isospin chiral gauge semisimple group leptoquark-bilepton extensions of the 3-2-1 conventional standard model of nuclear and electromagnetic interactions. In such models a natural explanation for the fundamental question of fermion generation replication arises from the self-consistency of a local gauge quantum field theory, which constrains the number of the QFD fermion families to the QCD color charges.
We have studied the lepton and quark mixing patterns which can be derived from the dihedral group $D_n$ in combination with CP symmetry. The left-handed lepton and quark doublets are assigned to the direct sum of a singlet and a doublet of $D_n$. A unified description of the observed structure of the quark and lepton mixing can be achieved if the flavor group $D_n$ and CP are broken to $Z_2\times CP$ in neutrino, charged lepton, up quark and down quark sectors, and the minimal group is $D_{14}$. We also consider another scenario in which the residual symmetry of the charged lepton and up quark…
The quadratic Casimir operator of the special unitary $SU(N)$ group is used to construct projection operators, which can decompose any of its reducible finite-dimensional representation spaces contained in the tensor product of two and three adjoint spaces into irreducible components. Although the method is general enough, it is specialized to the $SU(2N_f) \to SU(2)\otimes SU(N_f)$ spin-flavor symmetry group, which emerges in the baryon sector of QCD in the large-$N_c$ limit, where $N_f$ and $N_c$ are the numbers of light quark flavors and color charges, respectively. The approach leads to th…
We study the renormalization group equations (RGEs) of the neutrino parameters in models of Minimal Lepton Flavor Violation. In such models, the RGEs can be described in terms of flavor spurions, such that only the coefficients depend on the specific model. We explicitly demonstrate this method for the SM and MSSM for both Type-I and Type-III seesaw models. For that purpose, the RGEs of neutrino parameters in the MSSM Type-III seesaw have been computed. We have extended this method to get the evolution equations at second order. The implications for leptogenesis are also discussed.
We provide the isoscalar factors of the SU(3) Clebsch-Gordan series 8x35 which are extensions of the previous works of de Swart, McNamee and Chilton and play practical roles in current ongoing strange flavor hadron physics research. To this end, we pedagogically study the SU(3) Lie algebra, its spin symmetries, and its eigenvalues for irreducible representations. We also evaluate the values of the Wigner D functions related to the isoscalar factors; these functions are immediately applicable to strange flavor hadron phenomenology. Exploiting these SU(3) group properties associated with the spi…
We consider a class of flavor models proposed by Aranda, Carone and Lebed, relaxing the assumption of supersymmetry and allowing the flavor scale to float anywhere between the weak and Planck scales. We perform global fits to the charged fermion masses and CKM angles, and consider the dependence of the results on the unknown mass scale of the flavor sector. We find that the typical Yukawa textures in these models provide a good description of the data over a wide range of flavor scales, with a preference for those that approach the lower bounds allowed by flavor-changing-neutral-current constr…
In the absence of low-energy supersymmetry, a multiplicity of weak-scale Higgs doublets would require additional fine-tunings unless they formed an irreducible multiplet of a non-abelian symmetry. Remnants of such symmetry typically render some Higgs fields stable, giving several dark matter particles of various masses. The non-abelian symmetry also typically gives simple, testable mass relations.
In the present paper, we carry out a systematic study of the flavor invariants and their renormalization-group equations (RGEs) in the leptonic sector with three generations of charged leptons and massive Majorana neutrinos. First, following the approach of the Hilbert series from the invariant theory, we show that there are 34 basic flavor invariants in the generating set, among which 19 invariants are CP-even and the others are CP-odd. Any flavor invariants can be expressed as the polynomials of those 34 basic invariants in the generating set. Second, we explicitly construct all the basic in…
The Superkamiokande experiment suggests the large flavor mixing between nu_mu and nu_tau. We show that the mixing angle receives significant corrections from the renormalization group equation (RGE) when both the second and the third generation neutrino masses are larger than O(0.1eV). This means that the mixing angle must be small at the decoupling scale of right-handed neutrinos in the model containing a sterile neutrino nu_s with the mass spectrum of m_nu_s = m_nu_e << m_nu_mu = m_nu_tau.
