SUSY Breaking Constraints on Modular flavor $S_3$ Invariant $SU(5)$ GUT Model

Modular flavor symmetry can be used to explain the quark and lepton flavor structures. The SUSY partners of quarks and leptons, which share the same superpotential with the quarks and leptons, will also be constrained by the modular flavor structure and show a different flavor(mixing) pattern at the GUT scale. So, in realistic modular flavor models with SUSY completion, constraints from the collider and DM constraints can also be used to constrain the possible values of the modulus parameter. In the first part of this work, we discuss the possibility that the $S_3$ modular symmetry can be preserved by the fixed points of $T^2/Z_N$ orbifold, especially from $T^2/Z_2$. To illustrate the additional constraints from collider etc on modular flavor symmetry models, we take the simplest UV SUSY-completion $S_3$ modular invariance SU(5) GUT model as an example with generalized gravity mediation SUSY breaking mechanism. We find that such constraints can indeed be useful to rule out a large portion of the modulus parameters. Our numerical results show that the UV-completed model can account for both the SM (plus neutrino) flavor structure and the collider, DM constraints. Such discussions can also be applied straightforwardly to other modular flavor symmetry models, such as $A_4$ or $S_4$ models.

and quarks in certain GUT group multiplets, are known to be very predictive. It is therefore very interesting to survey if the modular flavor symmetry can be adopted in the GUT framework. Several works had been proposed to combine GUT with modular flavor symmetry with additional flavon fields. The interesting work of [8], which is based on SU(5) GUT and adopt the minimal non-abelian discrete symmetry Γ 2 S 3 for quark and lepton flavors, can reproduce the Cabibbo- Kobayashi-Maskawa (CKM) and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrices successfully as well as predict the leptonic CP violation phase and the effective mass of the neutrinoless double beta decay with a common modulus parameter. Although reintroducing the flavon fields seems to lost the minimality, such models can still be very predictive.
In previous modular flavor symmetry models, supersymmetry are always adopted. The non-renormalization theorem of SUSY, which states that no new term in the superpotential will be generated in perturbation theory radiatively, can preserve the constraints of modular flavor symmetry on the superpotential in the SUSY limit. Besides, the gauge couplings unification, which can not be exact in SM, can be successfully realized in its SUSY extensions. The SUSY partners of quarks and leptons, which share the same superpotential with the quarks and leptons, can also be constrained by the modular flavor structure and show a different flavor(mixing) pattern at the GUT scale, unlike the ordinary "SUSY breaking universality" inputs at the GUT scale adopted to evade the flavor-changing and CP-violating constraints in the MSSM. However, supersymmetry must be broken in realistic models. Predictions of previous mentioned modular flavor symmetry models are valid only if the SUSY breaking corrections are negligible [2]. So, we need to survey the SUSY breaking effects in modular flavor symmetry models. As the soft SUSY breaking parameters are determined by SUSY breaking mechanism, it is therefore important to combine the SUSY breaking mechanism with modular flavor symmetry models and survey their low energy consequences. Depending on the way the visible sector f eels the SUSY breaking effects in the hidden sector, the SUSY breaking mechanisms can be classified into gravity mediation [9], gauge mediation [10], anomaly mediation [11] scenarios, etc. Soft SUSY breaking parameters from economical generalized gravity mediation with non-renormalizable Kahler potential and superpotential involving high-representation Higgs fields of various GUT group were discussed in [12]. To illustrate the important constraints of SUSY breaking effects on modular GUT models, we UV-complete the modular S 3 SU(5) GUT model [8] as an example with the most economical generalized gravity mediation mechanism, with which the 24 representation Higgs of SU(5) fields in non-renormalizable superpotential trigger both GUT and SUSY breaking. This paper is organized as follows. In Sec 2, we discuss the possibility to preserve S 3 modular flavor symmetry by the fixed points of T 2 /Z 2 . In Sec 3, we UV-complete the modular S 3 SU(5) GUT model with the most economical generalized SUGRA models and obtain the resulting soft SUSY breaking parameters. In Sec 4, we give our numerical results to show the additional constraints on modular flavor symmetry from SUSY breaking. Sec 5 contains our conclusions.

2
S 3 modular flavor group preserved by the fixed points of T 2 /Z N Orbifold Discrete family symmetry might come from extra dimensions. It can arise as an accidental symmetry of the orbifold fixed points or be broken by boundary conditions, see [13]. Specific values for the modulus could also be obtained in extra dimensions through orbifolding [14]. We will show that S 3 modular group can be preserved by some of the fixed points from T 2 /Z 2 orbifolding.
We consider the M 4 × T 2 /Z 2 orbifold. The extra dimensional coordinates x 5 , x 6 are combined to be z = x 5 + ix 6 . The torus T 2 is defined by the complex plane C modulo out the lattice with ω 1 , ω 2 the lattice vectors. Consistency of the discrete symmetry on the orbifold T 2 /Z N constrains N = 2, 3, 4, 6 [15,16]. The inomhogeneous modular groupΓ can be generated by which satisfy 3) The action of S, T on the basis vectors are given as If the action of S, T on the orbifold fixed points permutate the fixed points, the fixe points can form a finite representation of typical subgroup of the modular group. We should check the symmetry preserved by the orbifold fixed point. The presentation of the finite modular group Γ 2 Γ /Γ(2) is given as The generator in (2.2) can also be used here. The requirement T 2 = 1 can be imposed as an constraints on the fixed points. For T 2 /Z 2 orbifold with basis vector ω 1 , ω 2 , the fix points can be found to be Orbifold consistency conditions z ∼ −z for Z 2 would not give any constraint on the values of ω 2 /ω 1 . The action of S changes the fixed points as Such an action amounts to the permutation of the fix points by the (2 3) element of permutation group. The action of T changes the fixed points as Such an action amounts to the permutation of the fix points by the ( 3 4) element of permutation group. It is obvious that the action of T is non-trivial and it satisfies T 2 = 1. In fact, it can be checked that under T 2 the fixed points are indeed unchanged We also have 13) which amounts to the (2 3 4) element of permutation group. We have the following relations for the fixed points 14) which is just the S 3 Γ 2 permutation group. So, it can be seen that the fixed points of T 2 /Z 2 can keep the S 3 permutation group.
For T 2 /Z 3 orbifold under the action of Z 3 z ∼ e i 2πm 3 z , (2.15) for m = 0, 1, 2, the orbifold consistency conditions will fix ω 2 /ω 1 = e iπ/3 , e i2π/3 , √ 3e i5π/6 . There are three fix points under the action of Z 3 . Obviously, the residual symmetry preserved by the fixed points is at most Z 2 because the fix point z = 0 will not change under the action of generators on the basis vectors. So we can arrive at the conclusion that the finite modular group S 3 Γ 2 group cannot survive for fixed points set of T 2 /Z 3 .
For T 2 /Z 4 orbifold, which fixed ω 2 /ω 1 = e iπ/2 by orbifold consistency conditions, can have the same type of fixed points as T 2 /Z 2 case because all fixed points of the T 2 /Z 4 orbifold can only appear in the second twisted sector. Similar discussions can be given and the fixed points can be compatible with theΓ 2 S 3 symmetry for ω 2 /ω 1 = e iπ/2 .
There are in total six fixed points for T 2 /Z 6 orbifold. Fixed points from the second twisted sector are the same as that in T 2 /Z 3 while the fixed points from the third twisted sector are the same as that in T 2 /Z 2 . It can be shown that the residual symmetry for the fixed points can at most be S 3 × Z 2 for T 2 /Z 6 orbifold with ω 2 /ω 1 = e iπ/3 , e i2π/3 , √ 3e i5π/6 . The transformation of superfields on orbifold for M 4 × T 2 /Z 2 can be give by (2.16) with the consistency conditions ( 2.17) The N = 2 gauge vector supermultiplets are place in the bulk with opposite parity choices for its (N=1) vector and chiral supermultiplets components to reduce the N = 1 SUSY in 6D(amount to N = 2 in 4D) to N = 1 SUSY in 4D. Unlike ordinary orbifold GUT models which are adopted to solve the doublet-triplet splitting problem, the SU(5) gauge symmetry in our setting is preserved both in the bulk and in the fixed point branes so that the zero modes are still SU(5) symmetric 1 . The supermultiplets of three generation matter contents in SU (5) 10 and5 representation are placed on three different fix points of T 2 /Z 2 (except z = 0). The three generations, located in various fixed points, can keep the S 3 modular flavor symmetry.

Soft SUSY breaking parameters from S modular invariance in SU(5) GUT
It is possible to explain both the quark and lepton flavors by a finite modular group with a common modulus parameter τ , for example in [8]. The choices of τ will be constrained by the flavor structure of the quark and lepton sectors. On the other hand, as supersymmetry is necessary to keep the form of the superpotential by the non-renormalizable theorem, the SUSY breaking effects should also be taken into account in realistic modular flavor models. Predictions of the modular flavor symmetry models are valid only if the SUSY breaking corrections are negligible [2]. With proper SUSY breaking mechanism, the SUSY partners of quarks and leptons, which share the same superpotential with the quarks and leptons, could also be constrained by the modular flavor structure. Therefore, some regions of the allowed τ by flavor consideration can be favored or disfavored by additional LHC or DM constraints. We will take the simplest realistic S 3 modular flavor model mentioned above as an example to illustrate the main features. Besides, to generate the soft SUSY breaking spectrum in this model, the economical generalized gravity mediation SUSY breaking mechanism [12] is adopted so that no additional SUSY breaking spurions are needed. Under the modular transformation the superfield φ (I) with modular weight −n I,φ transform in a representation ρ (I) of the quotient group Γ N as follow Table 1. In order to compare the constraints from SUSY and modular flavor symmetry, we choose the same charge assignments of SU (5), S 3 and modular weight for superfields (and modular forms) as that from ref. ([8]). The subscript i of F i and T i denotes the i-th family.
So the modular invariant Kahler potential can take the following form Based on the modular weight choices of the various SU (5) multiplets, which are given in Table 1 with the third generation being special, the Kahler potential takes the following form 2 It is obvious from the Kahler potential that the kinetic terms are not canonical. So we need to rescale the fields by a wavefunction normalization factor Z φ to obtain the proper forms of the physical spectrums. From the Kahler potential, we can obtain the normalization factor The value of v 24 /Λ was set to be 0.3 in [8] to generate additional contributions to the flavor structure. Here we neglect the subleading terms proportional to v 2 24 /Λ 2 ∼ 0.1 in the normalization factors.
To adopt the generalized gravity mediation scenario, the 24 representation Higgs of SU (5) is assumed to trigger both the SU(5) gauge symmetry breaking and the SUSY breaking. The lowest and F-component VEVs of the adjoint Higgs of SU (5) can take the following form (3.6) 2 We do not include the heavy right-handed neutrinos N c L;1,2,3 superfields in the Kahler potential because they will be integrated out at the low energy effective theory. They will not affect the low energy soft SUSY breaking spectrum.
Modular invariance also constrain to some extent the non-renormalizable terms involving the 24 representation Higgs in the Kahler potential although additional terms involving the Y i , Y i etc can still be consistent with the modular flavor symmetry of the model. As noted in [17], the presence of such additional terms will reduce the predictive power of these modular group constructions. So we simply assume that proper mechanism will forbidden other terms to appear in the non-renormalizable Kahler potential (3.7). We should note that the normalization factors will eventually be canceled out in the final expressions although the kinetic terms are not canonical. So the (universal) soft SUSY breaking masses for the sfermions (at the GUT scale) are given as To introduce the higher-dimensional operators involving 24 representation Higgs fields, we need to know the decompositions of the tensor products of various SU (5) representations for the Yukawa coupling terms [12] 10 ⊗ 10 ⊗ 5 = (5 ⊕ 45 ⊕ 50) ⊗ 5 We can see that there are two types of contraction of Yukawa with adjoint representation Higgs Φ 24 for both Yukawa couplings. The most general discussions on soft SUSY parameters related to the introduction of higher-dimensional operators can be seen in our previous works. Although not shown explicitly, the contraction rules within [8] adopt the following contractions Modular invariance also constrain stringently the form of the superpotential. The modular weight choices and the modular flavor group representation choices can fix the form of the superpotential except the non-renormalizable terms involving Φ 24 . To compare the constraints from SUSY and contraints from the SM flavor structures, we adopt the same form of non-renormalizable terms involving Φ 24 as that in [8].
The A u trilinear couplings can be obtained by substituting the F-component VEV of Φ 24 into the superpotential involving 10 − 10 − 5 H couplings of SU(5) 3 with the production rule of 2 representation of S 3 given as 3.13) and the expressions of Y 1 , Y 2 etc given in the appendix. The trilinear couplings are therefore given by TheÃ d trilinear couplings can be obtained from the 10 −5 −5 H coupling after substituting the F-term VEV of Φ 24 with the contraction given in (3.11). It reads The trilinear couplings for the lepton Yukawa couplings can be similarly obtained to be Due to the fact that the superfields in the superpotential are not canonically normalized, proper wavefunction factors should be divided from the previous expressions of trilinear terms to get the physical ones. So, the genuine trilinear soft terms are given as (3.18) with Z φ given in (3.5).
Unlike the KKLT realization, in which gauge fields origin from the D3/D7 brane, the gauge fields here need not transform under modular transformations. The gaugino masses are generated by non-renormalizable term involving the Φ 24 Higgs.

Numerical Results
This minimal UV-completion S 3 modular flavor SUSY GUT model has three additional SUSY breaking related free parameters c 0 , c 1 , F 24 /Λ in addition to the modulus parameter τ and other parameters chosen in the S 3 modular SU(5) GUT model [8] to generate the flavor structure for quarks and leptons. As noted previously, constrains on the superpotential by the modular flavor structure will lead to "non-universality" of the SUSY breaking parameters for the SUSY partners of quarks and leptons at the input scale, which share the same superpotential with the quarks and the leptons. Therefore, a different flavor pattern at the EW scale (in contrast to "universality" soft SUSY breaking inputs) will be generated and low energy flavor constraints etc will impose non-trivial constraints on the input parameters. So, it is interesting to survey if this SUSY completion model can survive the low energy (SUSY) flavor, collider as well as the DM constraints. The package SPheno4.0.4 [18] is used to scan the parameters spaces of with the model file generated by SARAH4.14.3 [19]. The DM relic density and other collider constrains are obtained with the package micrOMEGA5.2.4 [20]. We use the package HiggsBounds5.3.2 [21] to check whether or not the predictions of the survived points can pass the exclusion bounds obtained from the Higgs searches by the colliders. We impose the following constraints in our numerical studies: • (I) The conservative lower bounds from current LHC constraints on SUSY particles [22,23], the Z-boson invisible decay and the LEP search for sparticles [24] as well as Higgs mass • (II) Constraints from various flavor constraints, such as the rare B mesons decay [25] and lepton flavor violation(LFV) bounds [26][27][28][29]: • (III) The relic density of dark matter(DM) should satisfy the Planck data Ω DM = 0.1199 ± 0.0027 [30] in combination with the WMAP data [31](with a 10% theoretical uncertainty). Mh(GeV) Figure 1. In the left panel, we show the allowed range of Im(τ ) versus Re(τ ) that can satisfy the SM flavor structure, the collider and DM constraints. The best-fit region of quark flavor structure in [8] is labeled with blue color and the best-fit regions for both quark and lepton flavor structures with neutrino mass pattern either in IH (NH) is labeled with red color (cyan color), respectively. In the right panel, we show the survived range of c 0 , c 1 parameter with the corresponding Higgs masses.
We have the following discussions related to our numerical results: • It can be seen from the left-panel of fig.1 that the constraints from (I) to (III) can rule out much of the allowed parameter space of (τ ) versus (τ ), especially a large portion of the best-fit region of the SM(plus neutrino) flavor structure in [8]. On the other hand, it is interesting to note that the survived region of (τ ) versus (τ ) can still have a overlap with such best-fit region, which means that the UV-completed model can account for both the SM (plus neutrino) flavor structure and the collider, DM constraints.
A tiny fraction of allowed parameter space can lie within the best-fit region for quark flavor structure, which is label with blue color. The best-fit regions for both quarks and leptons flavor structures with neutrino mass pattern either in invert hierarchy(IH) (label with red color) or normal hierarchy(NH) (label with cyan color) can also be consistent with the collider and DM constraints. As the best-fit region for IH case has large overlap with the best-fit region for NH case, constraints from the the collider and DM still not show any preference between the two cases. If more experimental data for the SM (plus neutrino) flavor structure can be released in the future to separate the best-fit region for NH case and IH case, SUSY breaking consideration can possibly show which type of neutrino mass hierarchy is more preferable.
• The allowed range of c 0 , c 1 is shown in the right-panel of fig.1 with the corresponding Higgs mass. It is obvious from the figure that the discovered 125 GeV Higgs can also be accommodated in this SUSY UV-completion framework. From the right panel of fig.1, we can see that the predicted Higgs mass is always below 125 GeV. Taking into account the theoretical uncertainty in calculating the Higgs mass, which is about 0.5 ∼ 2 GeV, the best-fit region of (τ ) versus (τ ) for flavor structures can also marginally accommodate the 125 GeV Higgs mass.
• The survived region for the mass parameter F 24 /Λ, which is the only new mass parameter that determines the soft SUSY breaking scale at the GUT scale, is constrained to lie almost between 1.0 TeV to 4.8 TeV (see the upper-left panel of fig.2). The values of F 24 /Λ and c 0 , c 1 can determine the whole soft SUSY spectrum. Our numerical results indicate that the gluino masses are predicted to lie between 2.85 TeV to 3.92 TeV. The lighter stop,t 1 , are predicted to lie between 1.85 TeV to 2.78 TeV. Both of them can be possibly tested on the near future upgraded LHC. Because new flavor mixing and CP violating structures will appear in the squark and slepton sector (in contrast to the "universality " soft SUSY breaking inputs), some of the low energy flavor constraints will become important. The branch ratios of typical SUSY-sensitive B meson and LFV decay modes are shown in the lower panels of fig.2, which again can be checked in the near-future collider experiments.
• From our numerical results, it can also be seen that correct DM relic abundance can be obtained and the DM candidate is always the lightest neutralino, which is mostly bino-like(see the left-panel of fig.3). As pure bino-like DM can not annihilate into W or Z gauge bosons, they can only annihilate into standard model fermions via t-channel sfermions exchange. So, unless bino is lighter than 300 GeV, pure binolike DM will always lead to overabundance of DM relic density. On the other hand, coannihilation with stau can reduce the DM abundance efficiently. We checked that the bino-stau coannihilation effect is very crucial to obtain correct DM relic density. The spin-independent(SI) DM-nucleon cross section for DM direct detection is shown in the right-panel of fig.3. We can see that most of the allowed parameter space will survived the current DM direct detection bounds, such as LUX [32],XENON1T [33] and PANDAX [34]. Besdies, they can even survive the near-future LUX-ZEPLIN [35] experimental bounds. So, no DM signal is anticipated to be found in the near-future DM direct detection experiments.

Conclusions
Modular flavor symmetry can be used to explain the quark and lepton flavor structure. The SUSY partners of quarks and leptons, which share the same superpotential with the quarks and leptons, will also be constrained by the modular flavor structure and show a different flavor(mixing) pattern at the GUT scale. So, in realistic modular flavor models with SUSY completion, constraints from the collider and DM constraints can also be used to constrain the possible values of the modulus parameter. In the first part of this work, we discuss the possibility that the S 3 modular symmetry can be preserved by the fixed points of T 2 /Z N orbifold, especially from T 2 /Z 2 . To illustrate the additional constraints from collider etc on modular flavor symmetry models, we take the simplest UV SUSY-completion S 3 modular invariance SU(5) GUT model as an example with generalized gravity mediation SUSY breaking mechanism. We find that such constraints can indeed be useful to rule out a large portion of the modulus parameters. Our numerical results show that the UV-completed model can account for both the SM (plus neutrino) flavor structure and the collider, DM constraints. Such discussions can also be applied straightforwardly to other modular flavor symmetry models, such as A 4 or S 4 models.
A Expressions of the modular forms for S 3 modular group For completeness, we collect the expressions of modular forms with different modular weight from [8]. The modular forms of weight 2 corresponding to the S 3 doublet are written as The modular forms of weight 4, that is Y (4) , can be constructed by using the tensor product of the two doublets (Y 1 (τ ), Y 2 (τ )) T 1 : Y The S 3 singlet 1 modular form of the weight 4 vanishes.