Search

Abstract Discrete flavor symmetries have been an appealing approach for explaining the observed flavor structure, which is not justified in the Standard Model (SM). Typically, these models require a so-called flavon field in order to give rise to the flavor structure upon the breaking of the flavor symmetry by the vacuum expectation value (VEV) of the flavon. Generally, in order to obtain the desired vacuum alignment, a flavon potential that includes additional so-called driving fields is required. On the other hand, allowing the flavor symmetry to be modular leads to a structure where the couplings are all holomorphic functions that depend only on a complex modulus, thus greatly reducing the number of parameters in the model. We show that these elements can be combined to simultaneously explain the flavor structure and dark matter (DM) relic abundance. We present a modular model with flavon vacuum alignment that allows for realistic flavor predictions while providing a successful fermionic DM candidate.

Abstract Only four T 2 / ℤ K $$ {\mathbbm{T}}^2/{\mathbb{Z}}_K $$ orbifold building blocks are admissible in heterotic string compactifications. We investigate the flavor properties of all of these building blocks. In each case, we identify the traditional and modular flavor symmetries, and determine the corresponding representations and (fractional) modular weights of the available massless matter states. The resulting finite flavor symmetries include Abelian and non-Abelian traditional symmetries, discrete R symmetries, as well as the double-covered finite modular groups (S 3 × S 3) ⋊ ℤ 4, T ′, 2D 3 and S 3 × T ′. Our findings provide restrictions for bottom-up model building with consistent ultraviolet embeddings.

Abstract Inspired by the structure of top-down derived models endowed with modular flavor symmetries, we investigate the yet phenomenologically unexplored binary dihedral group 2D 3. After building the vector-valued modular forms in the representations of 2D 3 with small modular weights, we systematically classify all (Dirac and Majorana) mass textures of fermions with fractional modular weights and all possible 2 + 1-family structures. This allows us to explore the parameter space of fermion models based on 2D 3, aiming at a description of both quarks and leptons with a minimal number of parameters and best compatibility with observed data. We consider the separate possibilities of neutrino masses generated by either a type-I seesaw mechanism or the Weinberg operator. We identify a model that, besides fitting all known flavor observables, delivers predictions for six not-yet measured parameters and favors normal-ordered neutrino masses generated by the Weinberg operator. It would be interesting to figure out whether it is possible to embed our model within a top-down scheme, such as $${\mathbb{T}}^{2}/{\mathbb{Z}}_{4}$$ heterotic orbifold compactifications.

Discrete flavor symmetries provide a promising approach to understand the flavor sector of the standard model of particle physics. Top-down (TD) explanations from string theory reveal two different types of such flavor symmetries: traditional and modular flavor symmetries that combine to the eclectic flavor group. There have been many bottom-up (BU) constructions to fit experimental data within this scheme. We compare TD and BU constructions to identify the most promising groups and try to give a unified description. Although there is some progress in joining BU and TD approaches, we point out some gaps that have to be closed with future model building.

Modular flavor symmetries provide us with a very compelling approach to the flavor problem. It has been argued that moduli values close to some special values like τ=i or τ=ω provide us with the best fits to data. We point out that the presence of hidden “matter” fields, needed to uplift symmetric AdS vacua, gives rise to a dynamical mechanism that leads to such values of τ.

Artificial neural networks can be an important tool to improve the search for admissible string compactifications and characterize them. In this paper we construct the heterotic orbiencoder , a general deep autoencoder to study heterotic orbifold models arising from various Abelian orbifold geometries. Our neural network can be easily trained to successfully encode the large parameter space of many orbifold geometries simultaneously, independently of the statistical dissimilarities of their training features. In particular, we show that our autoencoder is capable of compressing with good accuracy the large parameter space of two promising orbifold geometries in just three parameters. Further, most orbifold models with phenomenologically appealing features appear in bounded regions of this small space. Our results hint towards a possible simplification of the classification of (promising) heterotic orbifold models.

Abstract 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/ℤ3 × ℤ3 orbifold compactification of the heterotic string that gives rise to an Ω(2) eclectic flavor symmetry. We find that the interplay of flavon alignment and the localization of the modulus in the vicinity of a symmetry-enhanced point leads to naturally protected fermion mass hierarchies, favoring normal-ordered neutrino masses arising from a see-saw mechanism. We show that our model can reproduce all observables in the lepton sector with a small number of parameters and deliver predictions for so far undetermined neutrino observables. Furthermore, we extend the fit to quarks and find that Kähler corrections are instrumental in obtaining a successful simultaneous fit to the quark and lepton sectors.

The neutrino sector offers one of the most sensitive probes of new physics beyond the Standard Model of Particle Physics (SM). The mechanism of neutrino mass generation is still unknown. The observed suppression of neutrino masses hints at a large scale, conceivably of the order of the scale of a rand unified theory (GUT), which is a unique feature of neutrinos that is not shared by the charged fermions. The origin of neutrino masses and mixing is part of the outstanding puzzle of fermion masses and mixings, which is not explained ab initio in the SM. Flavor model building for both quark and lepton sectors is important in order to gain a better understanding of the origin of the structure of mass hierarchy and flavor mixing, which constitute the dominant fraction of the SM parameters. Recent activities in neutrino flavor model building based on non-Abelian discrete flavor symmetries and modular flavor symmetries have been shown to be a promising direction to explore. The emerging models provide a framework that has a significantly reduced number of undetermined parameters in the flavor sector. In addition, such a framework affords a novel origin of CP violation from group theory due to the intimate connection between physical CP transformation and group theoretical properties of non-Abelian discrete groups. Model building based on non-Abelian discrete flavor symmetries and their modular variants enables the particle physics community to interpret the current and anticipated upcoming data from neutrino experiments. Non-Abelian discrete flavor symmetries and their modular variants can result from compactification of a higher-dimensional theory. Pursuit of flavor model building based on such frameworks thus also provides the connection to possible UV completions: in particular, to string theory. We emphasize the importance of constructing models in which the uncertainties of theoretical predictions are smaller than, or at most compatible with, the error bars of measurements in neutrino experiments. While there exist proof-of-principle versions of bottom-up models in which the theoretical uncertainties are under control, it is remarkable that the key ingredients of such constructions were discovered first in top-down model building. We outline how a successful unification of bottom-up and top-down ideas and techniques may guide us towards a new era of precision flavor model building in which future experimental results can give us crucial insights into the UV completion of the SM.

Abstract Nontrivial strong dynamics often leads to the appearance of chiral composites. In phenomenological applications, these can either play the role of Standard Model particles or lift chiral exotics by partnering with them in mass terms. As a consequence, the RG flow may change the effective number of chiral generations, a phenomenon we call generation flow. We provide explicit constructions of globally consistent string models exhibiting generation flow. Since such constructions were misclassified in the traditional model searches, our results imply that more care than usually appreciated has to be taken when scanning string compactifications for realistic models.

Abstract We present a detailed analysis of the eclectic flavor structure of the two-dimensional ℤ2 orbifold with its two unconstrained moduli T and U as well as SL(2, ℤ) T × SL(2, 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 led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, CP $$ \mathcal{CP} $$ and R-symmetries on the superpotential and Kähler potential of the theory.

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

Abstract Modular symmetries naturally combine with traditional flavor symmetries and CP $$ \mathcal{CP} $$ , giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ2 orbifold, which is equipped with two modular symmetries SL(2, ℤ) T and SL(2, ℤ) U associated with two moduli: the Kähler modulus T and the complex structure modulus U. The resulting finite modular group is ((S 3 × S 3) ⋊ ℤ4) × ℤ2 including mirror symmetry (that exchanges T and U) and a generalized CP $$ \mathcal{CP} $$ -transformation. Together with the traditional flavor symmetry (D 8 × D 8)/ℤ2, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and T = U = exp π i 3 $$ \left\langle T\right\rangle =\left\langle U\right\rangle =\exp \left(\frac{\pi \mathrm{i}}{3}\right) $$ . This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.

Abstract 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 non-trivial 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.

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 model with the Kähler terms under control. At the same time, the starting point is reminiscent of what one obtains from explicit string models.

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond SL(2,Z). In the simplest case with one Wilson line Z, Kähler modulus T and complex structure modulus U, we are led to the Siegel modular group Sp(4,Z). It includes SL(2,Z)T×SL(2,Z)U as well as Z2 mirror symmetry, which interchanges T and U. Possible applications to flavor physics of the Standard Model require the study of orbifolds of Sp(4,Z) to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of Sp(4,Z). Some cases correspond to symmetric orbifolds that extend previously discussed cases of SL(2,Z). Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of Sp(4,Z) reveal exciting new aspects of modular symmetries with promising applications to flavor model building.

Additional Abelian gauge interactions are generic to string compactifications. In heterotic string models, gauge coupling unification of such forces and other gauge interactions is natural due to their common origin. In this letter we study systematically the 1-loop running of the coupling constants in effective vacua emerging from Z8 heterotic orbifold compactifications that provide the matter spectrum of the MSSM plus some vectorlike exotics, restricting to vacua that yield a non-anomalous U(1)′ symmetry, gauge coupling unification and the observed values of known gauge couplings. We determine the low-energy value of the U(1)′ coupling constant for different scales of supersymmetry breakdown. We find that the U(1)′ coupling constant is quite restricted in string models to lie in the range 0.46–0.7 for low-scale supersymmetry or 0.44–0.6 in other cases. We argue that the phenomenology of these string vacua should be further explored to solve some extant issues, such as the stability of the Higgs vacuum.

Abstract It is well-known that the space group selection rule constrains the interactions of closed strings on orbifolds. For some examples, this rule has been described by an effective Abelian symmetry that combines with a permutation symmetry to a non-Abelian flavor symmetry like D 4 or Δ(54). However, the general case of the effective Abelian symmetries was not yet fully understood. In this work, we formalize the computation of the Abelian symmetry that results from the space group selection rule by imposing two conditions only: (i) well-defined discrete charges and (ii) their conservation. The resulting symmetry, which we call the space group flavor symmetry D S , is uniquely specified by the Abelianization of the space group. For all Abelian orbifolds with N = 1 $$ \mathcal{N}=1 $$ supersymmetry we compute D S and identify new cases, for example, where D S contains a ℤ 2 dark matter-parity with charges 0 and 1 for massless and massive strings, respectively.

We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group Sp(4,Z) that includes the modular symmetries SL(2,Z)T and SL(2,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 GSp(4,Z).

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 moduli space). We derive the modular groups, including CP and R-symmetries, for different sub-sectors of six-dimensional string compactifications and determine the general properties of the allowed flavor groups from this top-down perspective. It leads to a very predictive flavor scheme that should be confronted with the variety of existing bottom-up constructions of flavor symmetry in order to clarify which of them could have a consistent top-down completion.

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, traditional flavor, CP and R) share a common origin in string theory: on a technical level, they are given by outer automorphisms of the Narain space group. The eclectic top-down approach leads to a very restrictive scheme with high predictive power. It remains a challenge to connect this with existing bottom-up constructions of modular flavor symmetry.

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 Ω(1) (a nontrivial combination of the traditional flavor group Δ(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 modular weights of matter fields. Thereby, it controls the structure of the Kähler potential and the superpotential, which we discuss explicitly. In particular, both Kähler potential and superpotential are shown to transform nontrivially, but combine to an invariant action. Finally, we find that discrete R-symmetries are intrinsic to eclectic flavor groups.

Models with modular flavor symmetries have been thought to be highly predictive. We point out that these predictions are subject to corrections from non–holomorphic terms in the Lagrangean. Specifically, in the models discussed in the literature, the Kähler potential is not fixed by the symmetries, for instance. The most general Kähler potential consistent with the symmetries of the model contains additional terms with additional parameters, which reduce the predictive power of these constructions. We also comment on potential ways of how one may conceivably retain the predictivity.