We analyse the structure of Yukawa couplings in local SU(5) F-theory models with $E_8$ enhancement. In this setting the $E_8$ symmetry is broken down to SU(5) by a 7-brane configuration described by T-branes, all the Yukawa couplings are generated in the vicinity of a point and only one family of quarks and leptons is massive at tree-level. The other two families obtain their masses when non-perturbative effects are taken into account, being hierarchically lighter than the third family. However, and contrary to previous results, we find that this hierarchy of fermion masses is not always appropriate to reproduce measured data. We find instead that different T-brane configurations breaking $E_8$ to SU(5) give rise to distinct hierarchical patterns for the holomorphic Yukawa couplings. Only some of these patterns allow to fit the observed fermion masses with reasonable local model parameter values, adding further constraints to the construction of F-theory GUTs. We consider an $E_8$ model where such appropriate hierarchy is realised and compute its physical Yukawas, showing that realistic charged fermions masses can indeed be obtained in this case.
Flux compactifications of $M$ theory on certain $G_2$-manifolds have vacua in which all moduli are determined.These vacua have discrete parameters - the flux quanta. Small changes in these quanta produce small changes in the moduli and these can lead to large changes in the Yukawa couplings between Higgses, quarks and leptons. We argue that a large number of vacua exist with a wide variety of Yukawa hierarchies.
Local SU(5) F-theory models lead naturally to Yukawa couplings for the third generation of quarks and leptons, but inducing Yukawas for the lighter generations has proven elusive. Corrections coming from gauge fluxes fail to generate the required couplings, and naively the same applies to instanton effects. We nevertheless revisit the effect of instantons in F-theory GUT constructions and find that contributions previously ignored in the literature induce the leading non-perturbative corrections to the Yukawa couplings. We apply our results to the case of 10 x 5* x 5* couplings in local SU(5) …
We present a model for the observed hierarchies among the Yukawa couplings of the standard model in the context of an effective low energy theory with an anomalous U(1) symmetry. This symmetry, a generic feature of superstring compactification, is a remnant of the Green-Schwarz anomaly cancellation mechanism. The gauge group is that of the standard model, augmented by X, the anomalous U(1), and two family-dependent phase symmetries $Y^{(1)}$ and $Y^{(2)}$. The correct hierarchies are reproduced only when $sin^2θ_w=3/8$ at the cut-off. To cancel anomalies, right-handed neutrinos and other stand…
We argue that global F-theory compactifications to four dimensions generally exhibit higher rank Yukawa matrices from multiple geometric contributions known as Yukawa points. The holomorphic couplings furthermore have large hierarchies for generic complex structure moduli. Unlike local considerations, the compact setup realizes these features all through geometry, and requires no instanton corrections. As an example, we consider a concrete toy model with $SU(5) \times U(1)$ gauge symmetry. From the geometry, we find two Yukawa points for the ${\bf 10}_{-2} \, \bar{\bf 5}_6 \, \bar{\bf 5}_{-4}$…
We investigate phenomenologically viable four- and five-stack MSSM D-brane quivers which exhibit realistic fermion mass hierarchies. In our analysis, the mass hierarchies arise either from higher order terms containing the VEV's of SM singlets or from D-instanton effects, where the latter utilizes either family splitting or a factorizable Yukawa matrix. Extending the systematic bottom-up analysis of arXiv:0905.3379, we present the only four-stack quiver with a semi-realistic Yukawa texture. Investigation of five-stack MSSM models reveals many more quivers with analogous Yukawa textures, as wel…
This paper explores a recent idea of Nambu to generate hierarchies among Yukawa couplings in the context of effective supergravity and superstrings models. The Yukawa couplings are homogeneous functions of the moduli and a geometrical constraint between them with a crucial role in the Nambu mechanism is found in a class of models of no-scale type. The Yukawas are dynamical variables at low energy to be determined by a minimization process. (Based on the talk given at ICHEP94, Glasgow, july 1994)
The general structure of the matter Kahler metric in the $κ^{2/3}$ expansion of Horava-Witten M-theory with nonstandard embeddings is examined. It is shown that phenomenological models based on this structure can lead to Yukawa and V$_{\rm CKM}$ hierarchies (consistent with all data) without introducing ad hoc small parameters if the 5-branes lie near the distant orbifold plane and the instanton charges of the physical plane vanish. M-theory thus offers an alternate way of describing these hierarchies, different from the conventional models of Yukawa textures.
We perform a systematic analysis of globally consistent D-brane quivers which realize the MSSM, and analyze them with respect to their Yukawa couplings. Often, desired couplings are perturbatively forbidden due to the presence of global U(1) symmetries. We investigate the conditions under which D-brane instantons will induce these missing couplings without generating other phenomenological drawbacks, such as R-parity violating couplings or a mu-term which is too large. Furthermore, we systematically analyze which quivers allow for a mechanism that can account for the small neutrino masses and …
We study the local structure of zero mode wave functions of chiral matter fields in F-theory unification. We solve the differential equations for the zero modes derived from local Higgsing in the 8-dimensional parent action of F-theory 7-branes. The solutions are found as expansions both in powers and derivatives of the magnetic fluxes. Yukawa couplings are given by an overlap integral of the three wave functions involved in the interaction and can be calculated analytically. We provide explicit expressions for these Yukawas to second order both in the flux and derivative expansions and discus…
The Yukawa couplings of the simpler models of D-branes on toroidal orientifolds suffer from the so-called ``rank one'' problem -- there is only a single non-zero mass and no mixing. We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can solve the rank one problem. In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa coupling. In order to do this we provide the necessary quantities for instanton calculus in this class of background.
We construct a four-generation MSSM with rank-4 Yukawa matrices from intersecting D6 branes on a T^6/(Z_2 x Z_2) orientifold. The Yukawa matrices obtained provide an example of Flavor Democracy (FD), where the Yukawa couplings are all nearly equal. Mass hierarchies may then be generated by slight perturbations away from FD. We find that it is possible to obtain hierarchical masses for the quarks and leptons of each generation and mixings between them. In addition, the tree-level gauge couplings are unified at the string scale. Finally, we also construct similar models with one, two, and three …
In F-theory GUT constructions Yukawa couplings necessarily take place at the intersection of three matter curves. For generic geometric configurations this gives rise to problematic Yukawa couplings unable to reproduce the observed hierarchies. We point out that if the U(1)_{B-L}/U(1)_Y flux breaking the SO(10)/SU(5) GUT symmetry is allowed to go through pairs of matter curves with the same GUT representation, the quark/lepton content is redistributed in such a way that all quark and leptons are allowed to have hierarchical Yukawas. This reshuffling of fermions is quite unique and is particula…
The perturbative leading order open string three-point couplings for the Standard Model with hidden USp(6) on fractional D6-branes on T6/Z6' from arXiv:0806.3039 [hep-th], arXiv:0910.0843 [hep-th] are computed. Physical Yukawa couplings consisting of holomorphic Wilsonian superpotential terms times a non-holomorphic prefactor involving the corresponding classical open string Kaehler metrics are given, and mass terms for all non-chiral matter states are derived. The lepton Yukawa interactions are at leading order flavour diagonal, while the quark sector displays a more intricate pattern of mixi…
Warped models, originating with the ideas of Randall and Sundrum, provide a fascinating extension of the standard model with interesting consequences for the LHC. We investigate in detail how string theory realises such models, with emphasis on fermion localisation and the computation of Yukawa couplings. We find, in contrast to the 5d models, that fermions can be localised anywhere in the extra dimension, and that there are new mechanisms to generate exponential hierarchies amongst the Yukawa couplings. We also suggest a way to distinguish these string theory models with data from the LHC.
We propose, within a perturbative string theory example, a cosmological way to generate a large hierarchy between the observed Planck mass and the fundamental string scale. Time evolution results in three large space dimensions, one additional dimension transverse to our world and five small internal dimensions with a very slow time evolution. The evolution of the string coupling and internal space generate a large Planck mass. However due to an exact compensation between the time evolution of the internal space and that of the string coupling, the gauge and Yukawa couplings on our Universe ar…
We study the deformation induced by fluxes and instanton effects on Yukawa couplings involving 7-brane intersections in local F-theory constructions. In the absence of non-perturbative effects, holomorphic Yukawa couplings do not depend on open string fluxes. On the other hand instanton effects (or gaugino condensation on distant 7-branes) do induce corrections to the Yukawas. The leading order effect may also be captured by the presence of closed string (1,2) IASD fluxes, which give rise to a non-commutative structure. We check that even in the presence of these non-perturbative effects the h…
We describe the construction of string theory models with semirealistic spectrum in a sector of (anti) D3-branes located at an orbifold singularity at the bottom of a highly warped throat geometry, which is a generalisation of the Klebanov-Strassler deformed conifold. These models realise the Randall-Sundrum proposal to naturally generate the Planck/electroweak hierarchy in a concrete string theory embedding, and yielding interesting chiral open string spectra. We describe examples with Standard Model gauge group (or left-right symmetric extensions) and three families of SM fermions, with corr…
A unification of left-right $\rm{SU}(3)_\rm{L}\times \rm{SU}(3)_\rm{R}$, colour $\rm{SU}(3)_\rm{C}$ and family $\rm{SU}(3)_\rm{F}$ symmetries in a maximal rank-8 subgroup of ${\rm{E}}_8$ is proposed as a landmark for future explorations beyond the Standard Model (SM). We discuss the implications of this scheme in a supersymmetric (SUSY) model based on the trinification gauge $\left[\rm{SU}(3)\right]^3$ and global $\rm{SU}(3)_\rm{F}$ family symmetries. Among the key properties of this model are the unification of SM Higgs and lepton sectors, a common Yukawa coupling for chiral fermions, the abs…
We show that a special type of colliding 7-brane configuration of a codimension-two singularity realizes not only a six-dimensional spectrum with exactly the same quantum numbers as that of the three-generation E7/(SU(5) x U(1)^3) coset family unification model, but also the three sets of nonchiral singlet pairs with precisely the correct U(1) charges needed for explaining the Yukawa hierarchies and large lepton-flavor mixings in a well-known seesaw scenario.
We study renormalization-group flows in Yukawa theories with massless fermions, including determination of fixed points and curves that separate regions of different flow behavior. We assess the reliability of perturbative calculations for various values of Yukawa coupling $y$ and quartic scalar coupling $λ$ by comparing the properties of flows obtained with the beta functions of these couplings calculated to different orders in the loop expansion. The results provide a determination of the region in $y$ and $λ$ where calculations up to two loops can yield reasonably reliable results.
In the Minimal Supersymmetric Standard Model (MSSM), the scale $M_{SUSY}$ of soft supersymmetry breaking is usually {\em assumed} to be of the order of the electroweak scale. We reconsider here the possibility of treating $M_{SUSY}$ as a dynamical variable. Its expectation value should be determined by minimizing the vacuum energy, after including MSSM quantum corrections. We point out the crucial role of the cosmological term for a dynamical generation of the desired hierarchies $m_Z, M_{SUSY} << M_P$. Inspired by four-dimensional superstring models, we also consider the Yukawa couplings as d…
We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point $τ=i$, where modular invariant theories possess a residual $Z_4$ invariance. In this region the breaking of $Z_4$ can be fully described by the spurion $ε\approx τ- i$, that flips its sign under $Z_4$. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the $Z_4$ symmetry at $τ=i$, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of $|ε|$. We present examples where this mechanism is a key …
A pair of complex-conjugate fixed points that lie close to the real axis generates a large mass hierarchy in the real renormalization group flow that passes in between them. We show that pairs of complex fixed points that are close to the real axis and to one another generate multiple hierarchies, some of which can be parametrically enhanced. We illustrate this effect at weak coupling with field-theory examples, and at strong coupling using holography. We also construct complex flows between complex fixed points, including flows that violate the $c$-theorem.
${Z}_2$-Yukawa-QCD models are a minimalistic model class with a Yukawa and a QCD-like gauge sector that exhibits a regime with asymptotic freedom in all its marginal couplings in standard perturbation theory. We discover the existence of further asymptotically free trajectories for these models by exploiting generalized boundary conditions. We construct such trajectories as quasi-fixed points for the Higgs potential within different approximation schemes. We substantiate our findings first in an effective-field-theory approach, and obtain a comprehensive picture using the functional renormaliz…
We investigate the a theorem for nonsupersymmetric gauge-Yukawa theories beyond the leading order in perturbation theory. The exploration is first performed in a model-independent manner and then applied to a specific relevant example. Here, a rich fixed point structure appears including the presence of a merging phenomenon between non-trivial fixed points for which the a theorem has not been tested so far.
We explore whether perturbative interacting fixed points in matter systems can persist under the impact of quantum gravity. We first focus on semi-simple gauge theories and show that the leading order gravity contribution evaluated within the functional Renormalization Group framework preserves the perturbative fixed-point structure in these models discovered in [1]. We highlight that the quantum-gravity contribution alters the scaling dimension of the gauge coupling, such that the system exhibits an effective dimensional reduction. We secondly explore the effect of metric fluctuations on asym…
Generating the fermion mass hierarchy at the TeV scale
Institute for Advanced Study（高等研究院）；Princeton University（普林斯顿大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present the phase diagram and associated fixed points for a wide class of Gauge-Yukawa theories in $d=4+ε$ dimensions. The theories we investigate involve non-abelian gauge fields, fermions and scalars in the Veneziano-Witten limit. The analysis is performed in steps, we start with QCD$_d$ and then we add Yukawa interactions and scalars which we study at next-to- and next-to-next-to-leading order. Interacting infrared fixed points naturally emerge in dimensions lower than four while ultraviolet ones appear above four. We also analyse the stability of the scalar potential for the discovered …
We discuss corrections to the Yukawa matrices of the Standard Model (SM) fermions in intersecting D-brane models due to four-point interactions. Recently, an intersecting D-brane model has been found where it is possible to obtain correct masses and mixings for all quarks as well as the tau lepton. However, the masses for the first two charged leptons come close to the right values but are not quite correct. Since the electron and muon are quite light, it is likely that there are additional corrections to their masses which cannot be neglected. With this in mind, we consider contributions to t…
We discuss integrable aspects of the logarithmic contribution of the partition function of cosmological critical topologically massive gravity. On one hand, written in terms of Bell polynomials which describe the statistics of set partitions, the partition function of the logarithmic fields is a generating function of the potential Burgers hierarchy. On the other hand, the polynomial variables are solutions of the Kadomtsev-Petviashvili equation, and the partition function is a KP $τ$ function, making more precise the solitonic nature of the logarithmic fields being counted. We show that the p…
Various equivariant intersection numbers on Hilbert schemes of points on the affine plane are computed, some of which are organized into tau-functions of 2-Toda hierarchies. A correspondence between the equivariant intersection on Hilbert schemes and stationary Gromov-Witten theory is established.
Multi-critical behavior of interacting fermions in graphene's honeycomb lattice is presented. In particular, we considered the spin triplet insulating orders, where the spin rotational symmetry of the order parameter is explicitly broken. By casting the problem in terms of Gross-Neveu-Yukawa theory we show that such symmetry-breaking terms are irrelevant near the metal-insulator critical point. A finite Yukawa coupling among bosons and fermions improves the stability of such critical point against the symmetry-breaking perturbations. Physical sources of such symmetry-breaking terms are pointed…
We point out that a common feature of integrable hierarchies presenting soliton solutions is the existence of some special ``vacuum solutions'' such that the Lax operators evaluated on them, lie in some abelian subalgebra of the associated Kac-Moody algebra. The soliton solutions are constructed out of those ``vacuum solitons'' by the dressing transformation procedure.
Based upon the mathematical formulas of Lattice gauge theory and non-commutative geometry differential calculus, we developed an approach of generalized gauge theory on a product of the spacetime lattice and the two discrete points(or a $Z_2$ discrete group). We introduce a differentiation for non-nearest-neighbour points and find that this differentiation may lead to the introduction of Wilson term in the free fermion Lagrangian on lattice. The Wilson-Yukawa chiral model on lattice is constructed by the generalized gauge theory and a toy model and Smit-Swift model are studied.
We study the prethermal dynamics of the Gross-Neveu-Yukawa quantum field theory, suddenly quenched in the vicinity of a quantum critical point. We find that the universal prethermal dynamics is controlled by two fixed points depending on the size of the quench. Besides the usual equilibrium chiral Ising fixed point for a shallow quench, a dynamical chiral Ising fixed point is identified for a deep quench. Intriguingly, the latter is a non-thermal fixed point without any equilibrium counterpart due to the participation of gapless fermionic fields. We also find that in the scaling regime control…
