Flavor physics in SU(5) GUT with scalar fields in the 45 representation

We study a realistic SU(5) grand unified model, where a 45 representation of scalar fields is added to the Georgi-Glashow model in order to realize the gauge coupling unification and the masses and mixing of quarks and leptons. The gauge coupling unification together with constraints from proton decay implies mass splittings in scalar representations. We assume that an SU(2) triplet component of the 45 scalar, which is called $S_3$ leptoquark, has a TeV-scale mass, and color-sextet and color-octet ones have masses of the order of $10^6$ GeV. We calculate one-loop beta functions for Yukawa couplings in the model, and derive the low-energy values of the $S_3$ Yukawa couplings which are consistent with the grand unification. We provide predictions for lepton-flavor violation and lepton-flavor-universality violation induced by the $S_3$ leptoquark, and find that current and future experiments have a chance to find a footprint of our SU(5) model.


I. INTRODUCTION
The idea of Grand Unification is an attractive candidate for the fundamental theory behind the present understanding of particle physics described by the Standard Model (SM) [1][2][3][4][5][6].Although the SM has been established as a successful effective model at the electroweak (EW) scale by the discovery of the Higgs boson, reaching a deeper understanding of nature is a desire of particle physicists.Interestingly, some properties of the SM suggest the existence of a Grand Unified Theory (GUT) as a high-energy theory beyond the SM.For example, the renormalization group (RG) runnings of the gauge couplings in the SM show a unification tendency at a high scale [4], and the charge quantization of the SM fermions suggests the unification of matter.Once the SM gauge groups for electromagnetic, weak, and strong interactions are unified to a GUT gauge symmetry group, quarks and leptons are consequently unified in a single or a few representations of the GUT group.Various groups, such as SU (5), SO (10), E 6 , etc., have been considered as the GUT gauge symmetry group [7].
The SU (5) is the minimal simple group which contains the SM gauge groups SU(3) C ×SU(2) L ×U(1) Y .The minimal version of the GUT model based on the SU(5) symmetry, called the minimal SU(5) GUT, was originally proposed by Georgi and Glashow [3].In the minimal SU (5) GUT model, the right-handed down quarks and the left-handed lepton doublets are embedded in 5 representations of SU (5), and the left-handed quark doublets, the right-handed up quarks, and the right-handed charged leptons are embedded in 10 representations.The SM Higgs doublet is embedded in a 5 representation of scalars.In addition, the minimal model also contains a 24 representation of scalars, which breaks the SU (5) gauge symmetry to the SM ones.
Although the concept of the minimal SU(5) GUT is beautiful, there are two serious issues that have to be solved to construct a more realistic model.First, the three gauge couplings are not unified at a high-energy scale only with the RG runnings in the SM.In the minimal SU(5) model, there is a grand desert between the EW and the GUT scales, where there is no new contribution to the RG runnings.Second, the measured values of the masses of the charged leptons and the down-type quarks cannot be accommodated with the minimal SU(5) GUT, where they originate from a common Yukawa interaction in the GUT Lagrangian.
The first issue on the gauge coupling unification can be overcome by introducing extra fields in the grand desert, since such fields modify the RG runnings of the gauge couplings.A famous example of this direction is the supersymmetric SU(5) GUT model, in which the superpartners of the SM particles as well as the second Higgs doublet are introduced and the gauge coupling unification occurs at the scale of the order of 10 16 GeV [8][9][10][11][12].In nonsupersymmetric SU (5) GUT models, a single or a few particles in an extra representation of SU (5) are predicted to lie in the grand desert in order to realize the gauge coupling unification [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].
The second issue on the masses of the charged leptons and the down-type quarks can be resolved by introducing a 45 representation of scalar fields to the minimal SU (5) GUT.Similar to the 5 scalar, the 45 scalar couples with the 10 and the 5 fermions since 10⊗ 5 = 5⊕45.This coupling makes modifications in the relation between the charged-lepton and the down-type-quark Yukawa matrices at the GUT scale through the Georgi-Jarlskog mechanism [31].
We combine the above two ideas on the extensions of the minimal SU(5) GUT, and construct a concrete example of a realistic SU (5) GUT model, where the gauge coupling unification and the correct fermion masses are realized simultaneously.This kind of model having the 45 scalar can be found, for example, in Refs.[13, 15, 18, 21-25, 27, 28].
We consider an SU (5) GUT model, where the SM fermions reside in 10 and 5 representations of SU (5), denoted by Ψ 10 i and Ψ5 i with i = 1, 2, 3 being the generation index, and the scalar sector is composed of one 24, one 5, and one 45-dimensional scalar representation, denoted by Σ, Φ 5 , and Φ 45 , respectively.The SU(5)-symmetric renormalizable Lagrangian is given by where V µν is the field strength tensor of the SU(5) gauge bosons, A, B, C = 1, . . ., 5 are SU (5) indices, and L Y and V (Σ, Φ 5 , Φ 45 ) represent the Yukawa interactions and the scalar potential, respectively.The summation over repeated indices is implied.Here the fields Ψ 10 i , Σ, and Φ 45 satisfy the following relations: In general the Yukawa term L Y in Eq. ( 1) consists of the four interactions: where the totally antisymmetric tensor is defined as ϵ 12345 = 1, and Y U 5 and Y U 45 are symmetric and antisymmetric matrices in the generation space, respectively: The explicit expression for the scalar potential V (Σ, Φ 5 , Φ 45 ) is given in Appendix A.

B. Fermions
The SM fermions q Li , u c Ri , d c Ri , ℓ Li , and e c Ri are embedded into the 10 and 5 representations as where i, k are the generation indices, and the totally antisymmetric tensors are defined as ϵ 12 = ϵ 12 = 1 and ϵ 123 = ϵ 123 = 1.Without loss of generality, one can rotate the basis of Ψ 10 and Ψ 5 as where U 10 and U 5 are arbitrary unitary matrices in the generation space.By using the degrees of freedom associated with the unitary rotations, we can take the basis where the up-type quarks and the charged leptons are in their mass eigenstates: where the mass eigenstates are denoted with a hat, and V CKM is the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the Particle Data Group (PDG) phase convention [49,50].Analogous to the CKM matrix that represents a mismatch of the bases in q L , the unitary matrices V QU , V QE , and V DL are introduced in Ψ 10 and Ψ5 as in Eq. (6).
Field SU(5 where σ a (a = 1, 2, 3) are the Pauli matrices, λ a (a = 1, 2, . . ., 8) the Gell-Mann matrices, and η a (a = 1, 2, . . ., 6) the symmetric matrices defined by The decompositions of Σ, Φ 5 , and Φ 45 are summarized in Table I.Here the scalar H (45) (S ) has the same quantum numbers under the SM gauge group as H (5) (S (5) 1 ).Therefore, they can mix with each other, and we define the mass eigenstates H, H ′ , H C , and S 1 by introducing the mixing angles θ H and θ S and the phases δ H and δ S : H (45) , where c H = cos θ H , s H = sin θ H , c S = cos θ S , and s S = sin θ S .The presence of the two doublet scalars allows us to explain the masses of the down-type quarks and the charged leptons simultaneously.
Owing to the symmetry breaking of SU (5) to the SM gauge groups, mass splitting may occur among the scalar fields embedded in the SU(5) multiplets.At least one SU(2) L -doublet scalar has to be light to break the EW symmetry spontaneously below the TeV scale. 2 We assume that the scalar H is light and corresponds to the SM Higgs doublet.
It is well-known that the SM gauge couplings do not unify only by naive RG running in the SM.The mass splitting of the SU(5) scalar multiplets can improve the situation.We consider the scenario where some of the scalar fields, in addition to H, are much lighter than others.At the energy scale above the mass of an additional light scalar, the scalar contributes to the RG running of the gauge couplings.The gauge coupling unification is realized if an appropriate set of light scalars is considered.We define α 3 (µ), α 2 (µ), and α 1 (µ) as where g s , g, and g ′ are the gauge couplings of SU(3) C , SU(2) L , and U(1) Y , respectively, and µ is the renormalization scale.Our analysis assumes that the SM gauge couplings are unified at the scale M X , i.e., α , and all the scalar masses are not heavier than M X .Then, above the M X scale, all the scalars contribute to the running as complete SU(5) multiplets so that the coupling unification holds above M X .We also make an ansatz that the mass of the X boson is equal to the unification scale M X .Solving the renormalization group equations (RGEs) in Appendix C with the unification assumption, we get the three relations, Input values for the Z-boson mass mZ , the gauge couplings αs(mZ ) and α −1 (mZ ), the weak mixing angle sin 2 θW (mZ ), the quark masses, and the CKM parameters sij and δ, taken from Ref. [50].Other parameters, such as the pole masses of the charged leptons me, mµ, and mτ , are also taken from Ref. [50].
where m Z is the Z-boson mass, ϕ is summed over all the relevant scalars, and the coefficients B SM gi and B ϕ gi are given in Table IV 44 log where the gauge couplings at µ = m Z are evaluated for six active quark flavors [52] with the input values shown in Table II.As a general property, the S 3 contribution improves the gauge coupling unification [18].In the case that only the Higgs boson and S 3 are lighter than M X in the scalar sector, the gauge coupling unification occurs at M X ∼ O(10 14 GeV) with m S3 ∼ O(10 8 GeV).However, M X is severely constrained by proton decay search experiments, since contributions from the GUT gauge-boson exchange generate the dimension-six operators relevant to the proton decay.Then the proton lifetime is naively expected as [7,53] where m p is the mass of proton, and one finds a naive lower bound as by using the experimental lower limit on the lifetime τ (p → π 0 e + ) > 2.4 × 10 34 years [54] and α 2 X ∼ O(10 −3 ).In order to avoid the rapid proton decay by making M X much heavier, we assume that S 6 , S 8 , and Σ 8 , in addition to S 3 , are lighter than the unification scale M X .With their contributions to the RGEs of the gauge couplings, M X can be significantly heavier with keeping the coupling unification.Therefore, we consider a scenario where the masses of S 6 , S 8 , and Σ 8 are below M X .For simplicity, we assume that the other scalar components, except for the SM-like Higgs doublet H, are as heavy as M X .
Let us explain in more detail the masses of the other scalars embedded in the GUT representations.The mass parameter m 2 H associated with the SM-like Higgs doublet H is of the order of the weak scale according to the LHC measurements, while the other scalars associated with the 5 and 45 representations obey the mass relation given in Eq. (A13).We simply choose that Σ 1 , Σ 3 , H ′ , H C , S 1 , and S1 have a common mass M X .As a consequence, the mass of R 2 is determined as , where s H is the sine of the mixing angle defined in Eq. (12).For s 2 H < 1/4, m R2 is lighter than M X .In Fig. 1(a), the contours of M X and m Σ8 are shown in the parameter space of m S3 and m S6 = m S8 .The gauge coupling unification favors rather light S 3 , which can be as light as a TeV scale.The light gray regions are for M X < 3 × 10 15 GeV, which is disfavored by the proton decay search as mentioned above.For example, if we take m S3 = 2 TeV, m S6 = m S8 = m Σ8 = 5.2 × 10 6 GeV, and cot θ H = 50, the gauge coupling unification is realized at M X = 9.7 × 10 16 GeV as shown in Fig. 1(b).
In the phenomenological analysis, we use a benchmark scenario with the following mass spectrum: No Unification < l a t e x i t s h a 1 _ b a s e 6 4 = " O w < l a t e x i t s h a 1 _ b a s e 6 4 = " < l a t e x i t s h a 1 _ b a s e 6 4 = " Y T < l a t e x i t s h a 1 _ b a s e 6 4 = " O w

< l a t e x i t s h a 1 _ b a s e 6 4 = " c H n M x 9 H y / C l 6 M w B 5 S H y K 3 p e 3 w + w = " >
< l a t e x i t s h a 1 _ b a s e 6 4 = " w P 2 c h E i 7 < l a t e x i t s h a 1 _ b a s e 6 4 = " Z V z u 5 g I Y p q l K s W G Z q l L y J l e j u q Q   • The masses of the quarks and leptons, the SM gauge bosons, and the SM-like Higgs boson H are set to be consistent with their measurements; • S 3 has a TeV-scale mass: m S3 ∼ O(10 3 GeV); • S 6 , S 8 , and Σ 8 have intermediate masses, and we set them to an identical scale, i.e., m S6 = m S8 = m Σ8 ≡ M I ∼ O(10 6 GeV); • The other particles including the X bosons have masses of the order of the GUT scale M X .

D. Yukawa couplings
Below the GUT scale, the Yukawa interactions with the scalars H, S 3 , S 6 , and S 8 are given by where the first three terms lead to the fermion mass terms after the Higgs field H acquires a VEV at the EW scale.The 45 scalar plays an essential role in reproducing the masses of the SM fermions.If the 45 scalar is absent, the Yukawa matrices must obey a condition Y E = V T QE Y T D V DL at the GUT scale.This condition conflicts with the low-energy values of the masses of the down-type quarks and the charged leptons.In the current model, this problem is solved by the presence of the Yukawa coupling Y D 45 .Because the SM-like Higgs field H is a mixture of H (5) and H (45) as in Eq. ( 12), the Yukawa matrices Y U , Y D , and Y E are given at the GUT scale by which can lead to realistic Yukawa matrices at the low energy.Moreover, the GUT-scale matching conditions for the other couplings in Eq. ( 19) read as The scalar S 3 couples to a quark and a lepton simultaneously and thus is a leptoquark.The RGEs for these couplings are given in Appendix C. Let us count the physical degrees of freedom in the Yukawa sector.In the general case, there are four Yukawa matrices where ŶU , ŶD , and ŶE represent diagonal matrices in the mass basis.It is noted that an overall phase in V QU and three phases in V QE (and/or V DL ) can be removed by U(1) B , U(1) e , U(1) µ , and U(1) τ transformations.The right-hand sides of Eq. ( 22) then contain nine eigenvalues in ŶU , ŶD , and ŶE , three mixing angles and one phase in V CKM , eight parameters in V QU , and fifteen ones in V QE and V DL .
In general, the scalar S 3 can have two types of Yukawa couplings, Y QQ 3 and Y QL

3
, and the combination of these couplings leads to baryon-number-violating dimension-six operators, which cause too fast proton decay.For example, the bound from p → π 0 e + is estimated as Other components in Y U 45 also have to be highly suppressed to avoid the constraints from the proton decay.As explained in Appendix C, the coupling Y QQ 3 in Eq. ( 19) is forbidden in the whole range of the renormalization scale by an accidental global symmetry is once set to be zero at the GUT scale.Therefore, in the following, we make an ansatz that Y U 45 = 0 at the GUT scale.We here show a parametrization for the mixing matrices V QU , V QE , and V DL .According to the matching condition in Eq. ( 22), the ansatz Y U 45 = 0 at the GUT scale requires that V QU should be a diagonal phase matrix: The other two matrices V DL and V QE can be parametrized as where VQE and VDL are the 3 × 3 unitary matrices parametrized by three angles and one phase as the CKM matrix, and V CKM is extracted in V QE .We define the coupling Ȳ QL 3 in the mass basis of the down-type quarks and the charged leptons: where VQE = V † CKM V QE .The mixings in VQE (V DL ) cause flavor transitions between different generations of the down-type quarks (the charged leptons).To suppress dangerous contributions to flavor-changing processes associated with the first generation [44,47], such as K → πν ν and µ − → e − γ, we assume that VQE and VDL have only the mixing between the second and the third generations at the GUT scale: where the mixing angles θ QE and θ DL are varied from 0 to π/2.The three Yukawa matrices Y 10 , Y 5 , and Y D 45 are then determined at the GUT scale by the thirteen input parameters in addition to ŶU , ŶD , ŶE , and V CKM , i.e., the two mixing angles θ QE and θ DL , the nine phases in V QU , V QE , and V DL , and the two parameters s H = sin θ H and δ H in the Higgs sector.
The Y QL 3 term in Eq. ( 19) is decomposed in terms of the fields in the EW broken phase as follows [36]: where the hatted quark and lepton fields represent the mass eigenstates as in Eq. ( 8), and S Q 3 denotes a charge eigenstate with charge Q defined in the matrix form

III. PHENOMENOLOGICAL ANALYSIS A. Input parameters
We study low-energy phenomenology of the SU(5) GUT model proposed in the last section, where there is an S 3 leptoquark with a TeV-scale mass.As explained in Sec.II D the S 3 leptoquark has the Yukawa couplings with the left-handed quarks and the left-handed leptons, which lead to rich flavor phenomenology at the low-energy scale.In particular, the S 3 couplings generate processes with lepton-flavor violation (LFV) and lepton-flavor-universality violation (LFUV), while such exotic flavor processes are severely constrained by experiments.Our aim is to investigate whether current and future flavor experiments have a potential to explore our GUT-inspired scenario.The S 3 Yukawa matrix Y QL  20), which could help to explain the observed masses of the down-type quarks and the charged leptons.Thus, nontrivial correlations are expected among flavor observables where the S 3 leptoquark contributes.
The parameters in the GUT model, such as the Yukawa couplings Y U 5 , Y D 5 , and Y D 45 and the mixing matrices V QE and V DL , are constrained by the low-energy values of the SM fermion masses and the CKM matrix elements.We use the fermion masses and the CKM matrix elements listed in Table II as inputs, and calculate the running masses at the EW scale by taking into account QCD corrections for quarks with RunDec [55,56] and one-loop QED corrections for charged leptons [52].The masses and the CKM matrix elements as well as the gauge couplings at the EW scale are then evolved up to the GUT scale with the one-loop RGEs in Appendix C, where the Yukawa couplings Y QL  20) and ( 21), and we then perform the RG evolution from the GUT scale to the low scale.together with the original values of the SM fermion masses and the CKM elements until the difference in the masses and the CKM elements becomes small enough.In this way we can determine a set of the GUT parameters that are consistent with the low-energy values of the SM fermion masses, the CKM matrix elements, and the gauge couplings.
We fix the mass of the S 3 leptoquark to be m S3 = 2 TeV to avoid constraints from high-p T searches at the LHC [57].In addition, there are the thirteen arbitrary parameters: the three mixing angles θ QE , θ DL , and θ H , and the ten phases , and δ H .In general the Yukawa couplings ( Ȳ QL

3
) ij in Eq. ( 27) become larger for a smaller Higgs mixing angle θ H .We choose cot θ H = 50 as a benchmark scenario, while the other parameters are varied arbitrarily in their physical domain.The S 3 contributions to the flavor observables considered below are reduced by taking a heavier m S3 and/or a smaller cot θ H .

3
) ij of the S 3 leptoquark are constrained by the GUT relation in Eq. ( 27) to accommodate with the measured masses of the down-type quarks and the charged leptons at the low-energy scale.The RG effects from the GUT scale M X to the S 3 mass scale m S3 are shown in Fig. 2. It is noted that the magnitudes of the couplings typically enhance at the lower scale.In particular, the 22 coupling is increased by about a factor of 2, receiving one-loop corrections with the other couplings.Therefore, the inclusion of the RG evolution is essential to study low-energy phenomenology associated with the 22 coupling, such as b → sµ + µ − processes.
According to Fig. 2, the couplings with the second-generation fermions are typically smaller than those with the third-generation ones: On the other hand, the couplings with the first-generation fermions are negligibly small due to our ignorance of the corresponding mixings in Eq. ( 28).

C. Matching onto low-energy theory
The gauge couplings and the Yukawa couplings in Eq. ( 19) at the GUT scale are evolved down to the mass scale m S3 using the RGEs given in Appendix C, where S 6 , S 8 , and Σ 8 are decoupled at the intermediate scale M I .The leptoquark S 3 is then decoupled at the scale m S3 , and the theory is matched onto the Standard Model Effective Field Theory (SMEFT).The corresponding tree-level matching conditions are presented in Refs.[58,59], while the one-loop ones are calculated in Ref. [60].In addition, the one-loop anomalous dimensions in the SMEFT are found in Refs.[61][62][63].
We adopt the dimension-six SMEFT operators in the so-called Warsaw basis [64], where the Lagrangian in the SMEFT is given by the sum of the renormalizable SM Lagrangian and terms with higher-dimensional operators O i : At the tree level, only the semileptonic operators [O ℓq ] ijkl = ( lLi γ µ ℓ Lj )(q Lk γ µ q Ll ) and [O (3) ℓq ] ijkl = ( lLi γ µ σ a ℓ Lj )(q Lk γ µ σ a q Ll ) are generated by integrating out the S 3 leptoquark, where the corresponding Feynman diagram above the S 3 mass scale is presented in Fig. 3(a), and that below the S 3 mass scale, i.e., in the SMEFT, is in Fig. 3(b).The tree-level matching conditions for the semileptonic operators are given by where Y QL 3 in the right-hand side is the S 3 Yukawa coupling at the S 3 mass scale, obtained from the coupling at the GUT scale in Eq. ( 21) applying the RG evolution.We also define the coefficients C(1,3) ℓq in the mass basis of the down-type quarks and the charged leptons, and their matching conditions read as where

3
. The SMEFT coefficients in Eqs. ( 32) and ( 33) are evolved down to the EW scale, at which the SMEFT is matched onto the low-energy effective field theory (LEFT) [65] by integrating out the EW gauge bosons, the Higgs boson, and the top quark.The LEFT operators used in our phenomenological analysis are listed in Eq. (D2).The tree-level matching conditions for the coefficients L i in the LEFT Lagrangian of Eq. (D1) can be found in Refs.[65,66], while the one-loop ones are calculated in Ref. [67].Moreover, the RGEs for L i are calculated at the one-loop level in Refs.[68,69].We decompose L i into the sum of SM and new physics (NP) contributions as L i = L SM i + L NP i .In the current model only the semileptonic operators with the left-handed fermions are generated through the tree-level matching.For example, we have the following coefficients at the weak scale µ = m Z : where V wk denotes a CKM matrix element.
In our numerical analysis, we also include one-loop corrections to the matching onto the SMEFT, the RG evolution from µ = m S3 to µ = m Z , the matching onto the LEFT, and the RG evolution from µ = m Z to the lower energy scale.Let us consider the LEFT coefficient L V,LL ed for b → s processes as an example.Solving the RGEs in the leading-logarithmic approximation, the coefficient L V,LL ed is given at the bottom scale µ = m b by where c W = cos θ W is the cosine of the Weinberg angle, y t represents the SM Yukawa coupling of the top quark, x t = m 2 t /m 2 W with m t and m W being the masses of the top quark and the W boson, N c = 3 is the number of colors, α is the electromagnetic coupling, and I ed (x) is the loop function defined by In Eq. ( 37), the S 3 couplings Y QL 3 and Ȳ QL 3 should be understood as those evaluated at the S 3 mass scale.The one-loop expressions for the other LEFT coefficients relevant to our analysis are given in Appendix D.
It is convenient to convert the LEFT coefficients of the b → s semileptonic operators into the coefficients in the weak Hamiltonian [70]: where G F is the Fermi constant, and the NP contributions to the coefficients at the scale µ are related to the LEFT ones as The argument µ is omitted in Eq. ( 42

D. Constraints
In the current model, the S 3 leptoquark has sizable couplings to quarks and leptons in the second and third generations.Strong constraints on the parameter space of the model come from the mass difference of B s and Bs mesons denoted by ∆M s , the branching ratios for the B → K ( * ) ν ν decays, the LFUV tests in the B → K ( * ) ℓ + ℓ − (ℓ = e, µ) decays, and the branching ratio for the B s → µ + µ − decay.NP contributions to ∆M s are generated at the one-loop level, while those to the others are at the tree level.The current experimental data for these observables are summarized in Table III together with other relevant observables.For the B → K ( * ) ℓ + ℓ − decays, we do not consider their branching ratios and the angular observables that exhibit some tensions with the SM [95], since they suffer from hadronic uncertainties [96][97][98][99][100][101].
For the mass difference ∆M s , we utilize the following formula that is normalized to the SM value: where the SM loop contribution R loop SM = (1.310± 0.010) × 10 −3 and the SM prediction ∆M SM s = (18.4+0.7 −1.2 ) ps −1 are evaluated in Ref. [45].Our analysis includes the theoretical uncertainty in ∆M SM s , which is much larger than the experimental one.In the current model, the LEFT coefficient [L V,LL dd (m b )] NP 2323 , given in Eq. (D10), is generated at the one-loop level.Contributions from other coefficients with the right-handed quarks are suppressed by the small quark B(Z → µ ∓ τ ± ) < 6.5 × 10 −6 (95%) [93] O(10 −9 ) [94] masses and neglected here.We use the PDG average of the measurements for ∆M s [74], which gives a constraint on the product of the S 3 Yukawa couplings ( Ȳ QL 3 Because of the hierarchy in the magnitudes of the couplings, the product is dominated by ( Ȳ QL ) 32 is also constrained from the branching ratios for B → K ( * ) ν ν, which are calculated as where the SM coefficient is given by C

and the SM predictions are B(B
and τ B 0 being the lifetimes of B mesons [102].The NP contribution C NP L is defined by Eq. ( 42), where the one-loop expression of the LEFT coefficient [L V,LL νd ] NP ij23 is given in Eq. (D3).We select B 0 → K * 0 ν ν as a representative of the B → K ( * ) ν ν processes in our numerical analysis, where the use of the other processes gives similar results. 3The upper limit on B(B 0 → K * 0 ν ν) is reported from the Belle experiment [77], and provides a constraint on ( Ȳ QL 3 3 Very recently the Belle II collaboration has reported the first evidence of the B + → K + ν ν decay as B(B + → K + ν ν) = (2.4 ± 0.5 +0.5 −0.4 ) × 10 −5 [103].We do not take into account it in our analysis.In the left plot of Fig. 4, we present constraints in the plane of ∆M s /∆M SM s and B(B 0 → K * 0 ν ν), where the gray region is obtained with the model parameters that are consistent with the low-energy values of the gauge couplings, the fermion masses, and the CKM matrix elements.Here and hereafter, we take m S3 = 2 TeV and cot θ H = 50 as well as the input parameters in Table II.A large portion of the parameter space is excluded by the measurement of ∆M s (magenta vertical bands) [74] and by the upper limit for B(B 0 → K * 0 ν ν) (black horizontal dashed line) [77], where the two bands for ∆M s correspond to the one-sigma and two-sigma regions.
Moreover, the measurements for the b → sµ + µ − processes listed in Table III provide constraints on the product of the Yukawa couplings ( Ȳ QL *

3
) 22 ( Ȳ QL 3 ) 32 .In particular, experimental searches for the violation of the lepton-flavoruniversality (LFU) in b → s semileptonic decays provide severe constraints on our scenario.The LFU ratios R H (H = K + , K * 0 ) are defined by where q 2 min and q 2 max are given in units of GeV 2 .For example, approximate formulas for the region of 1.1 GeV 2 < q 2 < 6.0 GeV 2 are given in Ref. [104]: where These LFU ratios are calculated very accurately in the SM, where the hadronic uncertainty is highly canceled by considering the ratios [105], and the QED correction provides a positive contribution to the ratios about less than 3% for 1 GeV 2 < q 2 < 6 GeV 2 [106,107].The above approximate formulas are derived by neglecting the QED corrections.The theoretical uncertainties are negligible in our study.The recent measurements at LHCb [72] listed in Table III are compatible with the SM predictions.We adopt only R K [1.1, 6.0] and R K * [1.1, 6.0] as constraints, since the ratios in the low q 2 regions R K [0.1, 1.1] and R K * [0.1, 1.1] have larger experimental uncertainties.In addition, we also consider the branching ratio for the leptonic decay B s → µ + µ − , which is written simply with the NP contribution to C 10A : .It is noted that a nonvanishing decay width difference ∆Γ s of the B s system has to be taken into account when comparing the theoretical value calculated using Eq. ( 48) with the experimental data in Table III, since the time dependence of the decay rate is integrated in the experiment [110,111].This gives only a minor effect on our numerical analysis.In the current model, The magenta region can satisfy the experimental bounds from ∆M s within two sigma and B(B 0 → K * 0 ν ν) at 90% C.L., while the cyan region can satisfy further R K + [1.1, 6.0], R K * 0 [1.1, 6.0], and B(B s → µ + µ − ) within two sigma.These regions are overlaid on top of the gray one, which corresponds to that in the left plot.
We also present allowed regions for the products of the S 3 Yukawa couplings at the S 3 mass scale in Fig. 5.Here the cyan regions show the parameter points that are consistent with ∆M s , R K + [1.1, 6.0], R K * 0 [1.1, 6.0], and B(B s → µ + µ − ) within two sigma and B(B 0 → K * 0 ν ν) at 90% confidence level (C.L.).It is noted that the cyan regions are overlaid on top of the gray regions that correspond to those in Fig. 4. The magnitudes of the products in the upper row of Fig. 5 are smaller than those in the lower row because of the hierarchy given in Eq. (31).The product ( Ȳ QL * 3 ) 33 is by ∆M s and B(B 0 → K * 0 ν ν).The other products are less constrained by these observables.

E. Predictions
The S 3 leptoquark can generate various LFV and LFUV with the second-and third-generation fermions.Under the constraints studied in Sec.III D, we here consider the following observables: R(D and B(Z → µ ∓ τ ± ).The first six observables receive tree-level contributions, while the rest are induced at the one-loop level.Figures 6  and 7 show predictions for these observables in the current model.Here we only consider flavor-changing-neutral-FIG.6. Predictions for R(D) and R(D * ) (denoted by the red points) with the HFLAV average of their experimental measurements at the levels of one sigma, two sigma, and three sigma (denoted by the orange ellipses) and the SM values (denoted by the black cross).Theoretical uncertainties associated with the SM errors are not included in the predictions.

V1
is defined through the effective Lagrangian, where we use the tree-level result for the LEFT coefficient [L V,LL νedu (m b )] NP 3332 given in Eq. (36).We keep only the 33 component of C NP V1 in Eq. ( 49), since the dominant NP contributions arise in the 23, 32, and 33 ones in the current model and only the 33 one has an interference with the SM contribution.We use the average of the experimental data by the Heavy Flavor Averaging Group (HFLAV) [79].Here the b → cτ − ν transition is dominated by the contribution from the product ( Ȳ QL

3
) 33 , which also contributes to b → sν ν and ∆M s .It is known that the S 3 contribution that explains the b → c anomaly is severely constrained by the b → sν ν processes and ∆M s [40].Consequently, the S 3 contribution does not alter R(D ( * ) ) significantly, and thus the resolution of the R(D ( * ) ) anomaly requires an extension of the model [112,113].We do not consider such a possibility in the current paper.
Next, let us consider decay processes involving b → sτ + τ − transition.The studies of NP contributions to this transition are found, for example, in Refs.[114,115].In the current model, the contributions to the b → sτ + τ − leptonic and semileptonic decays arise at the tree level through the product ( Ȳ QL 3 48), the leptonic mode receives NP contribution to C 10A : where the SM prediction is B(B s → τ + τ − ) SM = (7.73 ± 0.49) × 10 −7 [108].Moreover, the branching ratios of the semileptonic modes in the large q 2 region are calculated in Ref. [115]: [15,22] = 10 −7 1.20 + 0.15 Re B(B → K * τ + τ − ) [15,19] which are the averages of the charged and the neutral modes.The predicted branching ratios in the SM are of O(10 −7 ) [115].The branching ratios for these leptonic and semileptonic modes can largely deviate from their SM values.Figure 7(a) shows that B(B s → τ + τ − ) can be as large as O(10 −5 ), which is an order of magnitude smaller than the future sensitivity at LHCb with 300 fb −1 [73].Similarly, the predictions for B(B → K ( * ) τ + τ − ) in the large q 2 region can be enhanced by an order of magnitude, but it is still much smaller than the future sensitivity at Belle II with 50 fb −1 [76].
We also study the LFV processes b → sµ + τ − and b → sµ − τ + , which are generated through the products of the S 3 Yukawa couplings ( Ȳ QL ) 33 | holds typically.At the LHC experiments, the branching ratio for the leptonic decay is measured as a sum of the two channels B s → µ − τ + and B s → µ + τ − .The corresponding theoretical formula is given by [116] where m τ and τ τ are the mass and the lifetime of τ lepton, m Bs , τ Bs , and f Bs are the mass, the lifetime, and the decay constant of B s meson, and the muon mass is neglected.As shown in Fig. 7(a), the prediction on B(B s → µ ∓ τ ± ) can be as large as O(10 −5 ), which may be probed by the LHCb measurement with 300 fb −1 [73].For the semileptonic channels, approximate numerical formulas are given by [117] B( It is noted that ) 33 |, respectively.We here present results on B + → K + µ ∓ τ ± , since future sensitivities at Belle II can be found for these processes in Ref. [76].As shown in Fig. 7(b), B(B + → K + µ − τ + ) can be large enough to be observed at Belle II with 50 ab −1 , while B(B + → K + µ + τ − ) is out of the reach of Belle II.A part of the parameter space is already excluded by the current measurement of B(B + → K + µ − τ + ) at Belle, but it does not alter the other predictions in Fig. 7 except for that of B(B s → µ ∓ τ ± ). ) 32 |.The current upper limit on B(B + → K + µ − τ + ) directly leads to the limit on B(B s → µ ∓ τ ± ).These correlations among the b → sµ + τ − and b → sµ − τ + observables can be explored by the combination of the Belle II and the LHCb measurements.
Besides, we consider the LFV decays of heavy quarkonia, Υ(nS) → µ ∓ τ ± (n = 1, 2, 3).The branching ratios for these processes are given by [118][119][120] where m Υ(nS) is the mass of Υ(nS), and the charged-lepton masses are neglected.From the bottom-right plot in Fig. 5, we estimate the magnitude of the LEFT coupling as . Therefore, the branching ratios are as large as O(10 −11 ), which are too small to be measured at current and near-future experiments.
In the current model, the muon anomalous magnetic moment (known as the muon g − 2) is generated through the product (Y QL .We find that this contribution is too small to explain the long-standing tension between the measured value and the SM prediction of the muon g − 2 [124,125]. The S 3 leptoquark also affects W -boson and Z-boson couplings with the SM fermions.We evaluate them with the one-loop expressions in Ref. [126], which include radiative corrections beyond the leading-logarithmic approximation.The effects on the W -boson couplings are not significant to be measured at the current and planned future experiments.We here present only the result for B(Z → µ ∓ τ ± ), which is calculated with the formulas given in Appendix E. Figure 7(f) shows a strong correlation between B(Z → µ ∓ τ ± ) and B(τ − → µ − γ).In our scenario, the B(Z → µ ∓ τ ± ) can be as large as O(10 −9 ).The present experimental bounds are given by the LEP experiment as B(Z → µ ∓ τ ± ) < 1.2 × 10 −5 [127] and the LHC experiment as B(Z → µ ∓ τ ± ) < 6.5 × 10 −6 [93].On the other hand, the FCC-ee experiment has a sensitivity to O(10 −9 ) [94].In the case that B(Z → µ ∓ τ ± ) is enhanced enough, B(τ − → µ − γ) is also significantly enhanced.

IV. SUMMARY
We have constructed a realistic GUT model which addresses two serious issues in the minimal SU(5) GUT: the realization of the gauge coupling unification and that of the flavor structures in the down-type-quark and the chargedlepton sectors.By introducing a 45-dimensional scalar representation Φ 45 to the minimal SU(5) GUT, the Yukawa matrices of the down-type quarks and the charged leptons are reproduced correctly by the Georgi-Jarlskog mechanism.In addition, we have shown that the three gauge couplings can be unified through the RG running under the constraint from proton decay, if S 3 , S 6 , and S 8 in Φ 45 and Σ 8 in the 24-dimensional scalar representation Σ lie much below the GUT scale.In particular, the mass of S 3 , which is a scalar leptoquark, can be of the order of TeV.
The Yukawa couplings of the S 3 leptoquark at the low-energy scale is constrained by the matching condition at the GUT scale in Eq. (22).In our scenario, the S 3 leptoquark couples strongly to the SM fermions in the second and third generations, where the magnitudes of the couplings obey the hierarchy shown in Eq. (31) and Fig. 2. In particular, the coupling ( Ȳ QL We have investigated flavor phenomenology in this realistic GUT scenario with the S 3 leptoquark at the TeV scale.We have derived constraints on the S 3 Yukawa couplings from ∆M s , B(B → K ( * ) ν ν), R K ( * ) , and B(B s → µ + µ − ), where the results are shown in Fig. 5.We have then calculated various decays of B mesons, Υ(nS), tau lepton, and Z boson.In the current model, the R(D ( * ) ) anomaly cannot be explained by the S 3 contribution due to the strong constraints from ∆M s and B(B → K ( * ) ν ν).The LFV processes B s → µ ∓ τ ± , B + → K + µ − τ + , and τ − → µ − ϕ may be observed at Belle II with 50 ab −1 and LHCb with 300 fb −1 .It is noted that B(B + → K + µ + τ − ) cannot reach the future sensitivity at Belle II unlike B(B + → K + µ − τ + ).Therefore, the observation of B + → K + µ − τ + together with the nonobservation of B + → K + µ + τ − is a clear signal of the current model.On the other hand, it is rather hard to observe the other processes τ − → µ − γ, τ − → µ − µ + µ − , and Z → µ ∓ τ ± , and much more data are needed for their observations.
In general, it is challenging to probe a GUT model, since the unification occurs at a very high-energy scale.The proton decay is a direct probe for GUT, but it has not been observed yet.We have provided a well-motivated benchmark scenario which may be able to be probed by the precise measurements of the flavor observables at the Belle II and LHCb experiments.Besides, the S 3 leptoquark at the TeV scale can be directly searched for at the current and future hadron-collider experiments.We thus conclude that the precise flavor measurements as well as the direct searches for the S 3 leptoquark play complementary roles to the searches for proton decay in probing our GUT scenario.
when the condition 2m 2  24 + 4(30λ (2) 24 )v 2 24 − 3χ 24 v 24 = 0 is satisfied [128]. 4From the potential, the squared masses of the component fields in the scalars Σ, Φ 5 , and Φ 45 can be read at the tree level as where the following combinations of the parameters are introduced: and the rotation matrices R H and R S are given as in Eq. ( 12): The masses of Σ 1 , Σ 3 , and Σ 8 can be freely chosen, since V 24 in Eq. (A8) contains a sufficient number of parameters.
On the other hand, the masses of the other scalars are constrained by the following relation: Appendix B: Matching conditions at the GUT scale Below the GUT scale, the Yukawa interactions are given in terms of the component fields in Eqs. ( 6), (9), and (10) as follows: The minimization of the scalar potential for the 45 representation is studied in Refs.[129][130][131].
Appendix C: Renormalization group equations The scale dependence of the gauge couplings is governed by the RGEs, where g i = g s , g and g ′ , and β gi denotes the corresponding beta function.The one-loop contributions to the beta functions are given by where the one-loop beta functions can generally be written as [132][133][134][135]   Below we list explicit formulas for the Yukawa couplings defined in Eq. (B1): (C5) • Gauge-boson-loop contributions: where C i 2 (ψ c ) = C i 2 (ψ).• Self-energy contributions to the fermions: where • Self-energy contributions to the scalars: where Notice that the coupling Y QQ  is set to vanish.The latter condition is satisfied if Y U 45 = 0 at the GUT scale in the tree-level approximation.The appropriate assignment of the baryon and lepton numbers in the above case is listed in Table V.

Appendix D: LEFT Lagrangian
The LEFT Lagrangian is given by where L QCD+QED is the renormalizable QCD and QED Lagrangian with the SM fermions except for top quark.The LEFT operators Q i up to dimension six are classified in Ref. [65].The operators relevant to the current study are where there also exist the Hermitian conjugates of the non-self-conjugate operators.The Wilson coefficients for these operators are calculated as follows.
1.The S 3 field is integrated out at the S 3 mass scale, and the model is matched onto the SMEFT at the one-loop level.The one-loop matching formulas from a model with the S 1 and S 3 leptoquarks to the SMEFT are listed in Refs.[58][59][60].
2. The RG running effects of the SMEFT operators are taken into account.The anomalous dimensions for the dimension-six operators in the SMEFT are listed in Refs.[61][62][63].
3. The SMEFT is matched onto the LEFT at the weak scale.The one-loop matching formulas are listed in Refs.[65][66][67].
4. The RG effects in the LEFT are taken into account.The corresponding anomalous dimensions are given in Refs.[68,69].
The coefficients in the LEFT Lagrangian at the relevant scale for the process under consideration are given in the leading-logarithmic approximation by Similar one-loop expressions for the low-energy coefficients can also be found in Refs.[138][139][140][141].

2 ϵ
αβ ϵ γδ H(45)δ , y y e 3 b N n t k d u 2 G P 7 P 3 X W n 5 Q o + G l R r P a 1 H J H 7 j 8 a y r / + q z J p F t j 7V P 3 p W W A H C 4 F X n b w 7 A d O 4 h d b U V w + P n / O L u V F / j J 2 z J / J / x h 7 Y L d 3 A q r 5 o F 1 m e O 0 G U P i D 9 / b l / g u J U K j 2 X m s 3 O J D N L 4 V d 0 Y h g j m K D 3 n k c G q 1 h D g c 6 t 4 RR X u I 6 8 S Q l p X J p s p k q R U B P H l 5 C m P g B t U J H n < / l a t e x i t > m S3[GeV]

3 <
y 6 P C n M N I 9 / c F q o i w t s h Z g j M U Z o 9 5 P O B V L m B N X j d q u r 5 a 4 1 M M 7 g 4 r x 5 C g Z 7 q n O j 3 R A 7 3 S x 6 + 1 K n 6 N h p c y z 2 p T K + z d g d P h z f d / V Q W e P R x + q f 7 0 7 G E f C 7 5 X n b 3 b P t O 4 h d b U l 0 7 O 6 5 u L G 4 n K B N 1 Q j f 1 f 0 w s 9 8 g 3 M 0 p t 2 u y 4 2 r h D h D 5 B / P n c r S E 8 l 5 b n k 7 P p M f H k p + I o w R j G O S X 7 v e S x j F W t I 8 b l H O M M F L k M 1 K S Y N S y P N V C k U a I b w L a T 4 J 5 u A i 6 I = < / l a t e x i t > 10 l a t e x i t s h a 1 _ b a s e 6 4 = " f q y 9 a O 6 N Y 0 1 b y 6 P C n M N I 9 / c F q o i w t s h Z g j M U Z o 9 5 P O B V L m B N X j d q u r 5 a 4 1 M M 7 g 4 r x 5 C g Z 7 q n O j 3 R A 7 3 S x 6 + 1 K n 6 N h p c y z 2 p T K + z d g d P h z f d / V Q W e P R x + q f 7 0 7 G E f C 7 5 X n b 3 b P t O 4 h d b U l 0 7 O 6 5 u L G 4 n K B N 1 Q j f 1 f 0 w s 9 8 g 3 M 0 p t 2 u y 4 2 r h D h D 5 B / P n c r S E 8 l 5 b n k 7 P p M f H k p + I o w R j G O S X 7 v e S x j F W t I 8 b l H O M M F L k M 1 K S Y N S y P N V C k U a I b w L a T 4 J 5 u A i 6 I = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 8 e 9 M 2 M A W t G M 0 j m M c l n x B g D J P x 8 5 5 y C D Y z O 2 g x p z D S P f 3 B e o I s 7 b C W Y I z F G Z L P B Z 5 l Q 9 Y k 9 f N m q 6 v 1 v g U g 7 v D y h i m 6 I n u 6 J U e 6 Z 5 e 6 O P X W j W / R t P L E c 9 q

10 4 <
l a t e x i t s h a 1 _ b a s e 6 4 = " d 7 M C Y X T 8 u 7 q D T u E g / b 3 r z 4 P O 8 0 w= " > A A A C a X i c h V H L S s N A F D 2 N 7 / p o 1 Y 3 o p l g U V + V G f O F C C m 5 c q r W t 4 K M k c a y x a R K S t K C l P + D K n a g r B R H x M 9 z 4 A y 7 6 C d J l B T c u v E 0 D o k W 9 w 8 y c O X P P n T M z q m 3 o r k d U C 0 k d n V 3 d P b 1 9 4 f 6 B w a F I d H g k 4 1 o l R x N p z T I s Z 1 t V X G H o p k h 7 u m e I b d s R S l E 1 R F Y t r D b 3 s 2 X h u L p l b n k n t t g r K n l T P 9 Q 1 x W M q I 9 N + Z b 6 a i 8 Y p Q X 7 E 2 o E c g D i C W L e i 9 9 j F A S x o K K E I A R M e Y w M K X G 4 7 k E G w m d t D h T m H k e 7 v C 1 Q R Z m 2 J s w R n K M w W e M zz a i d g T V 4 3 a 7 q + W u N T D O 4 O K 2 O Y o h d 6 o A Y 9 0 y O 9 0 s e v t S p + j a a X E 5 7 V l l b d z e 7 2 y Y 0 / o C T m + B E I i J + h o s / 4 O A n S I 8 k L g 7 e b j c R G r y T m X n m m f d 5 5 5 k Z 1 T Z 0 1 y N 6 D k k d n V 3 d P b 1 9 4 f 6 B w a F I N D a c c 6 2 q o

10 6 <
l a t e x i t s h a 1 _ b a s e 64 = " H M f G b 2 9 W h p B E 4 n d j L 2 S 1 Z A I Q m u U = " > A A A C a X i c h V F N L w N B G H 6 6 v u u j L R f h 0 m i I U / O u o O I g E h d H L a 0 m R b O 7 R q 1 ud z e 7 2 y Y 0 / o C T m + B E I i J + h o s / 4 O A n S I 8 k L g 7 e b j c R G r y T m X n m m f d 5 5 5 k Z 1 T Z 0 1 y N 6 D k k d n V 3 d P b 1 9 4 f 6 B w a F I N D a c c 6 2 q o 4 3 a 7 q + W u N T D O 4 O K + O Y p C e 6 o 1 d 6 p H t 6 o Y 9 f a 9 X 9 G k 0 v h z y r L a 2 w i 5 G T 0 f X 3 f 1 U V n j 3 s f 6 n + 9 O x h D y n f q 8 7 e b Z 9 p 3 k J r 6 W t H Z 6 / r i 5 n J + h R d U 4 P 9 X 9 E z P f A N z N q b d p M W m U u E + Q P k n 8 / d D n I z S X k + O Z e e T S w v B V / R i 3 F M Y J r f e w H L W M U a s n z u A U 5 x j o t Q Q 4 p J o 9 J Y K 1 U K B Z o R f A s p 8 Q m l h Y u n < / l a t e x i t > 10 8

8 7 5
X n b 3 b P t O 4 h d b U l 0 7 O 6 5 u L G 4 n K B N 1 Q j f 1 f 0 w s 9 8 g 3 M 0 p t 2 u y 4 2 r h D h D 5 B / P n c r S E 8 l 5 d n k z P p 0 f H k p + I o w R j G O S X 7 v O S x j F W t I 8 b l H O M M F L k M 1 K S Y N S y P N V C k U a I b w L a T 4 J 6 e G i 6 g = < / l a t e x i t > 10 9 d z e 7 2 y Y 0 / o C T m + B E I i J + h o s / 4 O A n S I 8 k L g 7 e b j c R G r y T m X n m m f d 5 5 5 k Z 1 T Z 0 1 y N 6 D k k d n V 3 d P b 1 9 4 f 6 B w a F I N D a c c 6 2 q o y 6 P C n M N I 9 / c F q o i w t s h Z g j M U Z o 9 5 P O B V L m B N X j d q u r 5 a 4 1 M M 7 g 4 r x 5 C g Z 7 q n O j 3 R A 7 3 S x 6 + 1 K n 6 N h p c y z 2 p T K + z d g d P h z f d / V Q W e P R x + q f 7 0 7 G E f 8 7 5 X n b 3 b P t O 4 h d b U l 0 7 O 6 5 u L G 4 n K B N 1 Q j f 1 f 0 w s 9 8 g 3 M 0 p t 2 u y 4 2 r h D h D 5 B / P n c r S E 8 l 5 d n k z P p 0 f H k p + I o w R j G O S X 7 v O S x j F W t I 8 b l H O M M F L k M 1 K S Y N S y P N V C k U a I b w L a T 4 J 6 e G i 6 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 8 y v X m D f j G U a Z r / j F 2 U 8 p e / p z W 6 O b 7 / + q S j x 7 O P x S / e n Z Q x F L v l e d v d s + 0 7 y F 1 t J X j s 9 e N 5 c 3 p m r T d E n P 7 P + C H u m O b 2 B W 3 r S r t N h o I M I f I P 9 8 7 n a s 9 / V C t 0 y 4 m / R 4 p R 5 B m D + y K v b B 7 d s 2 e 2 P u v v Y K w R 8 P L A W W t q e W u k j g e y r 3 9 q 7 I o C + x 9 q v 7 0 L L C D u d C r Q d 7 d k G m c Q m / q a 4 c n L 7 n s e j o Y Y + f s m f y f s U d 2 R y e w a 6 / 6 x R p f P 0 W c H k D + f t 0 / Q W E y I 8 9 k p t e m U o s L 0 9 B X F Z w 4 8 L b 6 Y C o q D c k O T m 5 5 + Y k 0 R x D e D 5 j T x G p r b 2 j s y v a H e v p 7 e u P y w O D 2 5 5 d c X W e 1 2 3 D d o u a 6 n F D W D z v C 9 / g R c f l q q 8 b i h W 7 z g 6 7 7 B i 4 7 L F V M 1 + L Z 6 t N p c 3 6 5 w 1 9 N t a 9 O v O n z P V A 4 s v a R r i k + U L C Z M O c j L 8 7 W l M G d q S + t y U R Z T L M 3 C S P 4 E U g R S i C J r i 9 f Y x T 5 s a C j D B I c F n 7 A B B R 5 9 O 5 D A 4 B C 3 h 4 A 4 l 5 A e r n P U E C d t m a o 4 V S j E H t F 4 Q L O d i L V o 3 u z p h W q N d j H o d 0 m Z x D h 7 Y H X W Y P f s l j 2 x 9 1 9 7 B W G P p p c q Z b W l 5 Y 4 8 e D K S f / 1 X Z V L 2 c f i p + t O z j x I y o V e d v D s h 0 z y F 1 t J X j k 8 b + c X c e D D B L t k z + b 9 g j + y O T m B V X r S r D Z 4 7 Q 5 w e Q P p + 3 T / B 1 k x a m k / P b c y m V p a j p + j G K M Y w R f e 9 g B W s I Y s C 7 V v F O W 5 Q j 7 0 J S W F S m G 6 V C r F I k 8 C X E G Y + A C I b k U M = < / l a t e x i t > m S 6 = m S 8 = M X FIG. 1.(a) The contours of MX (solid) and mΣ 8 (dashed) in the unit of GeV for realizing the coupling unification in the plane of mS 3 and mS 6 = mS 8 for cot θH = 50.In the blue shaded region, the gauge coupling unification does not occur by RG running.The light gray region is disfavored by the proton decay experiments because MX is too small.The green dot-dashed line corresponds to the case with mS 6 = mS 8 = mΣ 8 .(b) RG runnings of the gauge couplings for mS 3 = 2 TeV, mS 6 = mS 8 = mΣ 8 ≡ MI = 5.2 × 10 6 GeV, and cot θH = 50.
and Y D 45 in the GUT Lagrangian.Since Y U 5 and Y U 45 are symmetric and antisymmetric matrices, respectively, the four matrices contain 54 parameters in total.By the redefinitions of the fermion fields by U 10 and U 5 in Eq. (7), 18 degrees of freedom out of the 54 can be eliminated.Thus there remain 36 physical parameters in the Yukawa matrices.Taking the basis where the up-type quarks and the charged leptons are their mass eigenstates, the Yukawa matrices Y U 5 , Y U 45 , Y D 5 , and Y D 45 are written as

3
in our scenario cannot have an arbitrary structure unlike that in phenomenological leptoquark models where S 3 is introduced by hand.The coupling Y QL 3 originates from Y D 45 in the GUT Lagrangian, and Y D 45 also contribute to the SM Yukawa couplings Y D and Y E as in Eq. (

3 , Y DU 6 , 3 , Y DU 6 ,
and Y DQ 8 are neglected at this stage.At the GUT scale we calculate the couplings Y U 5 , Y D 5 , Y D 45 with Eq. (22) by inputting V QE and V DL , δ H , and s H .The couplings Y U , Y D , Y E , Y QL and Y DQ 8 are calculated at the GUT scale with Eqs. (

FIG. 2 . 3 , Y DU 6 , and Y DQ 8 . 3 , Y DU 6 ,
FIG. 2. Comparisons of the Yukawa couplings of the S3 leptoquark at the GUT scale µ = MX and at the S3 mass scale µ = mS 3 , where they are identical to each other on the dotted lines.

FIG. 3 .
FIG. 3. Diagrams for the tree-level matching at the S3 mass scale.Corresponding diagrams (a) in the model above the S3 mass scale, and (b) in the SMEFT below the S3 mass scale.

FIG. 4 .
FIG. 4. Left: constraints from ∆Ms/∆M SM s and B(B 0 → K * 0 ν ν).The gray region represents the predictions which are consistent with the low-energy values of the gauge couplings and the fermion masses and mixing.The vertical bands in magenta correspond to the experimental measurements at the one and two sigma ranges, and the horizontal lines are the 90% upper limit at Belle (black dashed line) and the SM prediction (blue solid line).Right: constraints on Re[(C NP 9V )22] and Re[(C NP 10A )22] at the m b scale, where the oblique dotted line represents Re[(C NP 9V )22] = −Re[(C NP 10A )22].The magenta region can satisfy the experimental bounds from ∆Ms and B(B 0 → K * 0 ν ν), while the cyan region can satisfy further with R K + [1.1, 6.0], R K * 0 [1.1, 6.0], and B(Bs → µ + µ − ).These regions are overlaid on top of the gray one, which corresponds to that in the left plot.

FIG. 5 .
FIG. 5. Allowed region for the products of the Yukawa couplings of the S3 leptoquark at the S3 mass scale, where the cyan region shows the parameter points that are consistent with ∆Ms, R K + [1.1, 6.0], R K * 0 [1.1, 6.0], and B(Bs → µ + µ − ) within two sigma and B(B 0 → K * 0 ν ν) at 90% C.L..The cyan regions are overlaid on top of the gray ones, which correspond to those in Fig. 4.

FIG. 7 .
FIG. 7. Predictions on relevant flavor processes, where the colored regions satisfy the experimental bounds from R K + [1.1, 6.0], R K * 0 [1.1, 6.0], B(Bs → µ + µ − ), and ∆Ms within two sigma and B(B 0 → K * 0 ν ν) at 90% C.L..The red and black dashed lines show the present upper bound on each processes by LHC experiments and B factories, respectively, and the red and black dotted lines show the sensitivities expected at the LHCb with 300 fb −1 and the e + e − experiments (such as the Belle II with 50 ab −1 and FCC-ee), respectively.The cyan regions in (a), (b), and (c) are excluded by the upper limit on B(B + → K + µ − τ + ) at Belle.

3 )
33 , respectively.Because of the hierarchy in the magnitudes of the S 3 Yukawa couplings presented in Eq. (31) and Fig.5, the relation |( Ȳ QL 3

8 ) T Y DU * 6 Y U Q 8 ,Γ Y U Q 8 = 8 Y 8 YC− (Y DU 1 ) 8 Y DQ † 8 Y DQ 8 ,
− Y U Y DQ † D − Y ′ U Y DQ † 8 Y ′ D − (Y DU C ) T Y DQ * QQTABLE V. Assignment of the baryon and lepton numbers to the scalars, where the Yukawa interactions in Eq. (B1) are invariant under U(1)B and U(1)L separately if HC , S1, and S1 decouple and Y QQ 3 is set to vanish at the GUT scale.

3
is shown to be vanishing in the whole range of the renormalization scale below the GUT scale if Y QQ 3 = 0 and H C , S 1 , and S1 decouple at the GUT scale.In fact, the Yukawa interaction Lagrangian in Eq. (B1) becomes invariant under U(1) B and U(1) L separately if the terms with H C , S 1 , and S1 are removed and if Y QQ

TABLE III .
Current measurements and future experimental sensitivities of flavor observables.The first column represents the corresponding transition, and the second column shows the dominant coupling that induces the transition, where Loop denotes a loop-level transition.

TABLE IV .
B SM g i and B ϕ g i for the RGEs of the gauge couplings.
and Y DQ 8 .In the beta functions, the coupling Y ϕ ψψ ′ should be understood as Y ϕ ψψ ′ θ(m ϕ − µ) by considering the decoupling of heavy particles, and for ϕ = H, H ′ , H C and S 1 , the term Y ϕ ψψ ′ Θ(ϕ) is replaced as