Left-Right SU(4) Vector Leptoquark Model for Flavor Anomalies

Building on our recent proposal to explain the experimental hints of new physics in $B$ meson decays within the framework of Pati-Salam quark-lepton unification, through the interactions of the $(3,1)_{2/3}$ vector leptoquark, we construct a realistic model of this type based on the gauge group ${\rm SU}(4)_L \times {\rm SU}(4)_R \times {\rm SU}(2)_L \times {\rm U}(1)'$ and consistent with all experimental constraints. The key feature of the model is that ${\rm SU}(4)_R$ is broken at a high scale, which suppresses right-handed lepton flavor changing currents at the low scale and evades the stringent bounds from searches for lepton flavor violation. The mass of the leptoquark can be as low as $10 \ {\rm TeV}$ without the need to introduce mixing of quarks or leptons with new vector-like fermions. We provide a comprehensive list of model-independent bounds from low energy processes on the couplings in the effective Hamiltonian that arises from generic leptoquark interactions, and then apply these to the model presented here. We discuss various meson decay channels that can be used to probe the model and we investigate the prospects for discovering the new gauge boson at future colliders.


I. INTRODUCTION
The Standard Model (SM) provides a remarkably successful description of nature at the elementary particle level and, so far, there are only a handful of experimental indications of deviations from its predictions. Perhaps the most significant direct hint of physics beyond the SM are the recently observed anomalies in B meson decays [1,2], which suggest that lepton universality might be violated. Assuming that those anomalies are not a result of experimental systematics, they are best accounted for by the vector leptoquark (3, 1) 2/3 or (3, 3) 2/3 [3]. However, building viable UV complete models involving those particles is challenging, especially in light of very stringent constraints on lepton flavor violation (LFV) from various experimental searches.
The first attempt to construct a vector leptoquark model for the R K ( * ) anomalies was made in [4], where we proposed that the vector leptoquark (3, 1) 2/3 explaining the anomalies might be the gauge boson of a theory with Pati-Salam unification. The conclusion was that the minimal model based on SU(4) × SU(2) L × SU(2) R is not capable of this because of strict bounds on kaon and B meson rare decays [5][6][7][8][9][10]. The underlying problem in that model arises from the interference between left-handed (LH) and right-handed (RH) lepton flavor changing currents. We outlined a possible solution to this: extending the gauge group to SU(4) L × SU(4) R × SU(2) L × U(1) and breaking SU(4) R at a high scale, such that the RH lepton flavor changing currents are suppressed.
A viable realization of this idea is the subject of this paper. We demonstrate that a Pati-Salam gauge leptoquark as light as 10 TeV can explain the R K ( * ) anomalies and remain consistent with all experimental bounds without introducing any mixing of quarks and leptons with new fermions. We discuss in detail the constraints arising from LFV searches and show that the absence of RH lepton flavor changing currents relaxes the bounds considerably. The model is expected to have clean signatures at future colliders, which we investigate in the case of the prospective 100 TeV machine.
In App. C we provide a model-independent analysis of the low energy consequences of a (3, 1) 2/3 vector leptoquark that interacts with both LH and RH fields. We present an extensive list of bounds from flavor physics on generic coupling constants in this model-independent approach; App. C is thus a resource in its own right, of use to researchers interested in any specific model of this type. Appendix D is one such example, where we apply the results of App. C to the specific Pati-Salam model constructed in this work. The calculations in App. C update and extend previous results [5][6][7][8][9][10]. For instance, for B decays we use the most recent lattice results for the form factors [25], which weaken the bounds considerably compared to assuming the nonphysical values f + = f 0 = 1 adopted previously in the literature.

II. THE MODEL
The theory we propose is based on the gauge group The crucial feature of the model is that the subgroup SU(4) R is broken at a much higher scale than SU(4) L , leading to a suppression of RH lepton flavor changing currents.

Scalar sector and symmetry breaking
The Higgs sector contains the scalar representationŝ The scalar potential is given in App. A. The parameters can be chosen such that the fieldsΣ L ,Σ R andΣ develop the vacuum expectation values (vevs), where z > 0. This results in the symmetry breaking pattern The relation between the SM hypercharge Y and the U(1) charge Y is given by where The scalar representations decompose into SM fields aŝ Under the symmetry breaking pattern (5) theĤ d ,Ĥ u fields have (4, 4) → (3 ⊕ 1) ⊗ (3 ⊕ 1); S 1 , S 3 stand for the singlet in 1 ⊗ 1, while S 2 , S 4 are the singlets in3 ⊗ 3. The components ofΣ R ,Σ L ,Σ have masses on the order of the SU(4) R and SU(4) L breaking scales. This is also the natural mass scale for the components ofĤ d andĤ u . However, as shown in App. B, it is possible to fine-tune the parameters of the potential such that only one linear combination of the fields S 1,2,3,4 is light.
In particular, there exists a choice of parameters for which the light state is given by where c u ≈ 1 c d c ν and 1 c e c ν , with the ratio c d : c e ≈ m b : m τ . This reduces the scalar sector of the model to that of the SM at low energies.

Gauge sector
The gauge and kinetic terms are with A = 1, ..., 15 and a = 1, 2, 3. The gauge covariant derivative takes the form where The gauge couplings at the low scale are related to the SM strong and hypercharge couplings via The new gauge bosons are The mass of G is M G = 1 The squared mass matrix for the gauge leptoquarks X L , X R is The leptoquark mass eigenstates can be written as where the mixing angle θ 4 depends on the parameters in Eq. (14). In the limit v R v L and v R v Σ the mixing vanishes, sin θ 4 = 0, and the leptoquark masses become The Z L and Z R squared masses are given by the two nonzero eigenvalues of the matrix

Fermion masses
The Yukawa interactions are where i, j = 1, 2, 3 are family indices and the coefficients "c" are those in Eq. (9). Typically, in theories with quark-lepton unification, the up-type quark and neutrino masses of a given generation are the same at the unification scale, and similarly the down-type quark and charged lepton masses. In our model this is not the case, but since there are only two Yukawa matrices y u and y d , without additional mass contributions the hierarchy of the up-type quark masses is, a priori, the same as for the neutrinos, and the down-type quark mass hierarchy the same as for the charged leptons at the unification scale.
Regarding the up-type quarks and neutrinos, for which the experimentally determined mass hierarchies differ considerably, this is solved by introducing a new scalar representation Φ 10 = (1, 10, 1, −1). If the SM singlet component ofΦ 10 develops a vev v 10 at a high scale, this provides a seesaw mechanism for the neutrinos via the interaction The contribution to the up-type quarks vanishes. Therefore, the up-type quark masses are m u ∼ y u v, whereas the neutrino masses are m ν ∼ (c ν y u v) 2 /(y u v 10 ).
The relative mass hierarchies of the down-type quarks versus charged leptons are not in vast disagreement with experiment. The running of the masses will largely account for m b /m τ . One can also introduce the scalar representa-tionΦ 15 = (15, 1, 1, 0) into the model, with the SM singlet component developing the vev v 15 diag(1, 1, 1, −3). New mass contributions to the down-type quarks and charged leptons would then result from loop processes, parameterized via the effective dimension five interaction y d ijΨ i LĤ dΨ dj RΦ 15 /Λ, and mediated, e.g., by heavy vector-like fermions, leading to additional mass splitting.

Flavor structure
In terms of SM fermion mass eigenstates, the interactions of the vector leptoquarks with quarks and leptons are given by where L u/d , R u/d are unitary mixing matrices. They are where V is the Cabibbo-Kobayashi-Maskawa matrix and U is the Pontecorvo-Maki-Nakagawa-Sakata matrix.

Proton stability
The vector boson (3, 1) 2/3 does not mediate proton decay [4], neither does any of the scalars in our model. In particular, for the scalar (3, 2) 1/6 , which by itself would be problematic [26], gauge invariance forbids tree-level proton decay. In broader terms, the Lagrangian in Eq.
which are simply the SM baryon and lepton number. Proton decay is thus forbidden at all orders in perturbation theory.

III. FLAVOR ANOMALIES
In this section we discuss how the vector leptoquark of SU(4) L can explain the recent hints of physics beyond the SM in B meson decays, i.e., the deficit in the ratios with respect to SM predictions [1,2]. For an analysis of the anomalies at the effective operator level see [27][28][29][30][31][32].
To describe the decays in Eq. (23) quantitatively, it is convenient to start out from the effective Lagrangian for flavor changing neutral current processes with a b → s transition. Up to four-quark operators, it can be written as The operators O ij 7 and O ij 8 correspond to electromagnetic and chromomagnetic moment transitions, the O and Tensor operators were neglected since they cannot arise from short distance new physics with SM linearly realized [33].
The R K ( * ) anomalies are best fit by [27] Re (∆C µµ and with the contributions to other Wilson coefficients being small. In our model, the vector leptoquarks X 1 , X 2 modify the coefficients by Guided by the tightness of the bounds from LFV searches (discussed in Sec. IV and App. C), we assume that SU(4) R is broken at a much higher scale than SU(4) This suppresses RH lepton flavor changing currents and results in the contributions to the Wilson coefficients other than ∆C ij 9,10 being small. The condition in Eq. (27) becomes
Implications of those constraints for Pati-Salam unification have been considered in the literature [5][6][7][8][9][10], but focused on models in which the vector leptoquark (3, 1) 2/3 couples to both LH and RH fermion fields with similar strength. The conclusion of those analyzes, updated with the most recent experimental bounds [48][49][50], is that the leptoquark mass has to be 90 TeV [10]. In addition, constraints from searches for µ → e γ when both LH and RH leptoquark interactions are present can push this limit much higher due to the bottom quark mass enhancement of the one-loop diagram (see App. C and also [55] for a discussion of a similar effect in scalar leptoquark models). Such a heavy leptoquark would not explain the R K ( * ) anomalies, since the required relation, analogous to the one in Eq. (30), could not be satisfied for a perturbative gauge coupling and unitary mixing matrices.
In our model, for a sufficiently high scale of SU(4) R breaking, the constraints arising from the presence of leptoquark RH couplings to fermions are eliminated and the remaining bounds on LH interactions can be satisfied for a significantly lower leptoquark mass. The tightest limits are listed in the appendix, for arbitrary LH and RH leptoquark interactions in App. C and for the case of just LH interactions in App. D.
If the mixing matrix entries L d 11 , L d 12 are O(1), the limits from searches for K 0 L → e ± µ ∓ and µ − e conversion a priori push the leptoquark mass up to hundreds of TeV in our model (thousands of TeV for models in which both LH and RH leptoquark interactions are present, due to the enhancement of the scalar current contribution, see App. C). The bounds, however, are satisfied for a much lighter leptoquark provided L d 11 , L d 12 1. Unitarity then implies that L d 13 ≈ 1 and L d 23 , L d 33 1, therefore L d takes the form Note that the suppression of RH flavor changing currents in our model implies that there are no significant bounds from π 0 → νν or K 0 L → νν. The remaining entries of L d are subject to further constraints, mainly from B meson and τ decays. If both LH and RH leptoquark interactions were present, the B 0 → µ + µ − decay would provide the most stringent bound. However, with only LH interactions the tightest limits arise from searches for B + → K + e ± µ ∓ . We calculated the corresponding branching fractions (see App. C) using the most recent lattice results for the form factors [25] based on the Bourrely-Caprini-Lellouch parameterization [56], which relaxes the bounds considerably compared to taking the nonphysical values f + = f 0 = 1 [9].
The resulting bound on M X L is minimized for θ ≈ π/4 and requires merely M X L /g L 9.2 TeV. Given the relation between the gauge couplings in Eq. (12) and assuming g R ≈ √ 3π (close to the perturbative limit) implies g L ≈ 1.06 g s , where g s ≈ 0.96 is the strong coupling constant at 10 TeV. This leads to the constraint (If one chose instead g L = g R = √ 2 g s , this would result in the constraint M X L 14 TeV.) Saturating the bound in Eq. (32), the condition in Eq. (30) for explaining the R K ( * ) anomalies is fulfilled if cos(φ 1 + φ 2 ) ≈ 0.18. We also note that for M X L ≈ 10 TeV one could have |δ i | ∼ 0.02, so the matrix L d in Eq. (31) does not need to be highly tuned.
Finally, let us note that all loop-level constraints, including K−K, B−B, B s −B s mixing, radiative decays µ → e γ (see App. C), τ → e γ, anomalous magnetic and electric moments of leptons, Z → b b and others [57] are satisfied due to the unitarity of L d and the leptoquark mass being 10 TeV.

V. COLLIDER PHENOMENOLOGY
The aim of this limited phenomenological analysis is to simply demonstrate that the leptoquark X L in our model accounting for the flavor anomalies can be searched for at the next generation collider. Focussing on the proposed 100 TeV Future Circular Collider (FCC), we find that one of the best signature to look for is provided by the single leptoquark production process In an in-depth analysis one could also investigate final states involving other leptons, which for the case of neutrinos would lead to missing energy signatures. Pair production of 10 TeV leptoquarks is suppressed even at a 100 TeV collider.
To simulate the SM background and the leptoquark signal for the process (33) we used MadGraph 5 [58] (version 2.6.3) with the default cuts apart from the lower cut on the transverse momentum of jets and leptons, which was set to 300 GeV. The leptoquark model file for MadGraph was implemented using FeynRules [59] (version 2.3.32). Figure 1 plots the number of background (B) and signal (S) events for a leptoquark mass 10, 12 and 14 TeV expected within the first year of FCC running (estimated to be 250 fb −1 of data [60]) as a function of the invariant mass of the highest transverse momentum jet j and µ + . Implementing the invariant mass cut | M jµ + −M X L | < Γ X , where Γ X is the width of the leptoquark, the significance of the signal, S/ √ B, is very high: 19 σ for M X L = 10 TeV, 6.7 σ for 12 TeV and 4.5 σ for 14 TeV. More sophisticated cuts may make the search more efficient. A detailed analysis of the X L vector leptoquark collider phenomenology is beyond the scope of this paper.
Were the B decay anomalies in R K and R K * confirmed and established, inspection of Eq. (30) indicates this model can be ruled out at a future 100 TeV high luminosity hadron collider. Not only does the right-hand side of Eq. (30) provide an upper bound on the mass of the vector leptoquark, but Eq. (12) shows the strength of the coupling constant g L is bounded from below, and therefore the height of the resonant signal in Fig. 1 is bounded from below.

VI. CONCLUSIONS
We have constructed a new model to account for the recently observed anomalies in B meson decays set within the framework of Pati-Salam unification. The theory avoids all experimental bounds without introducing any vector-like fields mixing with the Standard Model fermions. This was achieved by suppressing the leptoquark right-handed interactions by associating them with a symmetry broken at a high scale, which eliminates the most stringent constraints arising from the simultaneous presence of left-and right-handed lepton flavor changing currents. In some regions of parameter space the mass of the leptoquark can be as low as 10 TeV while remaining consistent with all experimental data.
The tightest constraints on the model come from the experimental limits on rare kaon, B meson and τ decays, as well as µ−e conversion. In the appendix we presented general modelindependent formulae for the various decay rates and listed the corresponding bounds. Those results can be used to read off the constraints on any model with one or more (3, 1) 2/3 vector leptoquarks with arbitrary left-and right-handed interactions with Standard Model quarks and leptons.
In our analysis we chose parameters to explain the R K ( * ) flavor anomalies. Although the vector leptoquark (3, 1) 2/3 in our model is too heavy to account also for the R D ( * ) anomalies, it has been shown [61] that the scalar leptoquark (3, 2) 1/6 might be a good candidate for that. This leptoquark appears in the scalar sector of our model and can be made sufficiently light. It would be interesting to investigate this in more detail.
Currently, there exist many models that account for the hints of lepton universality violation in B meson decays. If these anomalies are established, new physics must emerge at a scale similar to that of the mass of the "left-handed" leptoquark in our model. We have demonstrated that simple kinematic cuts can isolate clearly observable signals with 250 fb −1 of accumulated data at a 100 TeV pp collider. Further analysis is badly required to determine whether such apparatus could distinguished among the many proposed models.

Acknowledgments
This research was supported in part by the DOE Grant No. DE-SC0009919. The scalar potential of the model is given by where we have adopted the notation: Let us consider Σ L , Σ R and Σ . Via a suitable SU(4) L and SU(4) R transformation, it is possible to bring Σ L and Σ R to the form where v L and v R are real and positive.
The remaining SU(3) invariance can be utilized to obtain To argue that Σ can be brought to the diagonal form as in Eq. (4), it is sufficient to consider the potential terms |Σ| 2 , a 2 , a 3 , d). In addition, the terms imply that the minimum occurs at a 1 = a 2 = a 3 . Finally, we are free to choose κ to be real and negative, which through an appropriate redefinition ofΣ leads to real d > 0; therefore with z being real and positive. Note that only one of the parameters λ 345 , λ 345 and λ 345 can be made real by a field redefinition. If any of the other two has a nonzero imaginary part, the scalar potential is CP -violating. A rigorous minimization procedure is beyond the scope of this work.

Appendix B: Scalar masses
To show that Eq. (9) can be satisfied, it is again sufficient to consider only a few terms in the scalar potential. In terms of hard masses, the relevant part of the Lagrangian is This results in the masses for the color octets and triplets, The mass squared matrix for the fields S 1,2,3,4 is where We have verified that there exists a class of solutions with only one linear combination of the four scalars being light. To reproduce the SM fermion masses while keeping the Yukawas perturbative, it is sufficient to have the light mass eigenstate, identified with the SM Higgs, given by where c u ≈ 1 c d c ν and 1 c e c ν , with the ratio c d : c e ≈ m b : m τ .

Appendix C: Flavor constraints: Model-independent analysis
The general form of the Lagrangian describing interactions of vector leptoquarks (3, 1) 2/3 with fermions is given by where the field X (α) µ corresponds to a leptoquark with mass M α . The resulting contributions to rare processes are listed below, along with the most severe experimental bounds.
The numerical values for particle masses and lifetimes were adopted from PDG [62]. The single-particle state normalization chosen is and the decay constant f M for a meson consisting of quarks/antiquarks q 1 , q 2 is defined via Values of the meson decay constants, obtained from averaging the lattice results, were also taken from PDG, (1) Neutral meson decays to two charged leptons The leptoquark contribution to the decay of a meson M with mass m M to two charged leptons, l + i with mass m i and l − j with mass m j , is given by where In Eq. (C6) the quark masses m q and the factor Q depend on the energy scale, m q = m q (µ) and Q = Q(µ), with Q(µ) given by the formula 12 25 , (C7) applicable for m b > µ > m c . The coupling constant α is calculated from where N f is the number of quark flavors at a given scale, by matching The ratio Q(µ)/m q (µ) is a renormalization group invariant. Adopting the PDG values for the quark masses at µ = 2 GeV and for the strong coupling constant at µ = M Z [62], the value of Q depends only on the leptoquark mass scale through As evident from Eq. (C6), the constraints on the leptoquark contribution to the branching fraction of kaon and B meson decays are much weaker when the leptoquarks have only LH or only RH interactions with SM fermions, as opposed to models with both LH and RH interactions. The bounds on the branching fraction are milder by a factor of which is reflected by the much weaker constraints on the leptoquark mass in our model compared to generic leptoquark models (see App. D). For the majority of decays considered here only the upper bound on the rate was experimentally established. However, in the four cases: K 0 L → e + e − , K 0 L → µ + µ − , B 0 → µ + µ − and B 0 s → µ + µ − nonzero rates have been measured. For those particular decays not only the pure leptoquark contribution is relevant, but also the interference effects with the SM short-distance (SD) contribution. This can be taken into account by making the following substitution in the expressions for A ij and B ij in Eq. (C6), where the +/− depends on the decay considered and corresponds to the SM short-distance amplitude for M → l + i l − j being negative/positive. The leptoquark-induced contribution is then obtained by subtracting off the pure SM part.

(a) Neutral kaon decays
The decays K 0 L → e ± µ ∓ are absent in the SM and the constraint on the leptoquark mass is derived directly from the experimental bound on the branching fraction, Br X ∆Br. The rates for K 0 L → e + e − , µ + µ − were measured [37,62]. They are dominated by long-distance SM effects [63,64]. For K 0 L → e + e − the experimental branching fraction 8.7 +5.7 −4.1 × 10 −12 [37] agrees well with the SM long-distance estimate of (9.0 ± 0.5) × 10 −12 [63]. In that case we use the experimental uncertainty for the measured branching fraction as the upper bound for the leptoquark contribution. For K 0 L → µ + µ − the measured branching fraction is (6.84 ± 0.11) × 10 −9 [39], but it was shown that the short-distance SM contribution is only 0.9 × 10 −9 [63], whereas the upper bound on the total shortdistance contribution is 2.5 × 10 −9 [64].
The constraints below reflect the most conservative bound on the leptoquark mass obtained using Eq. (C5). The branching fractions were left in explicitly for easier use of the formulae given future experimental improvements.
• A where A is given by A

(2) Charged meson decays to a charged lepton and neutrino
Decays of mesons to a charged lepton and a neutrino exist in the SM. The leading order leptoquark contribution comes from interference effects. The theoretical uncertainty in the SM calculation is reduced by taking ratios of decay rates, For the case of Dirac neutrinos, whereas for Majorana neutrinos the only nonzero terms are, The tightest bounds of this type originate from measurements of the branching fraction ratios

(b) Charged kaon decays
In this case, R(K + ) = (2.493 ± 0.031) × 10 −5 [42] , R(K + ) SM = (2.477 ± 0.001) × 10 −5 [65] , which results in (3) Charged meson three-body decays to a meson and charged leptons When the leptoquark has both LH and RH interactions with SM fermions, the three-body meson decays are less restrictive than the two-body decays. However, in the case of our model, with predominantly LH interactions, the bounds arising from B + → K + e ± µ ∓ impose the most severe constraints on the leptoquark mass. The corresponding decay rate is expressed in terms of the form factors f + (q 2 ) and f 0 (q 2 ) defined via where the four-momentum transfer q = p − p and the meson squared mass difference ∆M 2 = m 2 M − m 2 M . The contribution to the decay rate mediated by leptoquarks is where The form factors f + (q 2 ) and f 0 (q 2 ) are calculated using lattice methods. For the K → π form factor we use the linear fit given in [66]. For the B → π and B → K form factors we adopt the results of [67] and [25], respectively, where the interpolating functions were obtained using the Bourrely-Caprini-Lellouch parameterization [56].
The B → π and B → K form factors are given by where In the B → π case [67]: whereas for B → K [25]: The resulting constraints on B + decays are much weaker than the corresponding bounds presented in [9]. This is due to the fact that the calculation in [9] assumed f + (q 2 ) = f 0 (q 2 ) = 1. This assumption for the B → π and B → K form factors is quite far from their actual shape.

(5) Radiative charged lepton decay
The vector leptoquark contribution to the process l i → l j γ is induced at the loop level. Unlike for scalar leptoquarks, in the case of vector leptoquarks this effect cannot be computed in the general case, since the result is infinite and requires arbitrary subtractions that are well-defined only in a UV complete model. We parameterize our ignorance of this UV completion with the coefficients c LR and c RL , where k = 1, 2, 3 and we expect c LR and c RL to be O(1), with their values dependent on the UV details of the model. The ellipsis denotes interference and mass-suppressed terms.
If the matrices f ij are proportional to unitary matrices, the terms in the first line of Eq. (C51) vanish. The experimental bounds, neglecting higher order terms, become, In our model the leading order terms contributing to l + i → l + j γ are O(m 2 b /M 2 X L ) and the resulting constraints are negligible compared to tree-level bounds.
The effective Hamiltonian for the l + i → l + j conversion consists of the dipole transition part corresponding to l + i → l + j γ and terms arising from integrating out the heavy vector leptoquarks, i.e.

H eff
where m = 1, 2. The steps required to match the effective Hamiltonian (C55) to the Hamiltonian at the nucleon level and compute the conversion rate are provided in [70,71]. The tightest experimental constraint from l + i − l + j conversion arises from µ − e conversion on gold [54]. Since the resulting bound on the dipole transition contribution is less restrictive than the constraint from µ → e γ in Eq. (C52), we concentrate only on the second part of the Hamiltonian (C55). Following [70], the µ − e conversion rate is then given by with similar relations obtained upon switching (L ↔ R). The numerical coefficients were adopted from [72]. For the 197 79 Au nucleus, which provides the most stringent bound, the parameters in Eq. (C56) are, V p = 0.0974 , V n = 0.146 , S p = 0.0614 , S n = 0.0918 , and are the result of the calculation using "method 1" in Sec. III A of [70]. The best bound on µ − e conversion is [54], Γ(µ → e in Au) Γ(µ capture in Au) < 7 × 10 −13 .
The constraints on general (3, 1) 2/3 leptoquark models are derived by inserting Eq. (C56) into (C58) and using the total µ − capture rate in 197 79 Au, Γ(µ capture in Au) = 8.6 × 10 −18 GeV [73]. In the case of our model, with just LH leptoquark couplings, the constraint simplifies to Finally, let us note that the bounds on generic leptoquark models were considered in [5][6][7][8][9][10]. Our formulae reproduce those results up to the difference in the adopted values of quark masses, meson decay constants and form factors used.
Appendix D: Flavor constraints: SU(4) L × SU(4) R model In our model X (1) ≡ X 1 and X (2) ≡ X 2 given by Eq. (15), therefore the coefficients in Eq. (C1) are Constraints on the model parameters are obtained by substituting the expressions in Eq. (D1) into the bounds derived in App. C. In the limit v R v L and v R v Σ , for which sin θ 4 0, X 1 = X L and X 2 = X R , one arrives at the constraints listed below. The numbering scheme indicates which equation in App. C a given constraint originated from.