Scalar leptoquarks from GUT to accommodate the $B$-physics anomalies

We address the $B$-physics anomalies within a two scalar leptoquark model. The low-energy flavor structure of our set-up originates from two $SU(5)$ operators that relate Yukawa couplings of the two leptoquarks. The proposed scenario has a UV completion, can accommodate all measured lepton flavor universality ratios in $B$-meson decays, is consistent with related flavor observables, and is compatible with direct searches at the LHC. We provide prospects for future discoveries of the two light leptoquarks at the LHC and predict several yet-to-be-measured flavor observables.

New Physics (NP) explanations of the B-physics anomalies suggest a presence of one or more TeV scale mediators which couple to left-handed currents with predominantly third generation fermions [12][13][14][15][16]. Among the most prominent NP candidates are leptoquarks (LQs). It turns out that a single scalar LQ cannot provide solution to the both B-physics anomalies. The scalar LQ that can explain R K * is not suitable for accommodating R D * and vice versa. By using an effective theory approach, it was shown in Ref. [13] that of all possible single mediators only one particular vector LQ can generate suitable V −A operators for the anomalies and satisfy both low-energy and high-p T constraints. The construction of ultra-violet (UV) complete models for these scenarios became a challenge that was addressed in Refs. [17][18][19][20][21][22][23]. Another possible approach is to consider models with several mediators. The low energy V −A structure can be generated by integrating out two scalar LQs [13,24,25]. Incidentally, one can explore two scalar LQs even when they are known to generate operators with Lorentz structures other than V −A for R D ( * ) .
In this Letter we propose a UV complete model based on SU (5) Grand Unified Theory (GUT) with two light scalar LQs that can address both anomalies. The LQs in question are R 2 (3, 2, 7/6) and S 3 (3, 3, 1/3), where we specify their representation under the SM gauge group SU (3) c × SU (2) L × U (1) Y . At low energies, R 2 generates a combination of scalar and tensor effective operators that accommodate R D ( * ) , while S 3 generates a V − A operator which accommodates R K ( * ) . In our set-up, since the Yukawa interactions have a common SU (5) origin, both LQs share one Yukawa matrix. If we take into account all relevant flavor constraints we find that the preferred region in the parameter space is compatible with direct searches at the LHC. Furthermore, if we demand perturbativity of all the couplings to the GUT scale, we find that the mass of R 2 needs to be around 1 TeV. In the following we present our set-up, outline the UV completion, discuss the low-energy phenomenology and LHC signatures.

II. SET-UP
The interactions of R 2 and S 3 with the SM fermions are are the charge (and mass) eigenstates with charge Q. We define the mass eigenstates u L,R = U L,R u L,R , d L,R = D L,R d L,R , L,R = E L,R L,R , and are the CKM and PMNS matrices, respectively.
We adopt the following features for the Yukawa matrices and assume SU (5) embedding In the simplest SU (5) scenario that can accommodate light R 2 and S 3 and (re)produce the associated flavor structure of Eqs. To generate all three operators of Eq. (1) it is sufficient to introduce a ij 10 i 5 j 45, and b ij 10 i 10 j 50, where a and b(= b T ) are 3×3 matrices in generation space. The former contraction couples R 2 ∈ 45 (S 3 ∈ 45) with the right-handed up-type quarks (quark doublets) and leptonic doublets, while the latter generates couplings of R 2 ∈ 50 with the quark doublets and right-handed charged leptons. To break SU (5) down to the SM gauge group we can use 24 [26] or 75 [27,28]. This allows us to write either m 45 50 24 or m 45 50 75, where m is a dimensionful parameter. The two R 2 's that reside in 45 and 50 mix through either of these two contractions allowing us to end up with two light scalars, i.e., R 2 and S 3 , and one heavy R 2 state that completely decouples from the low-energy spectrum for large values of m. The relevant lagrangian after the SU (5) breaking, in the mass eigenstate basis of the two light LQs, is where we define the mixing angle between the two R 2 's to be φ. The primed unitary transformations in Eq. (5), i.e., V , E R , U R , U , as well as Yukawa matrices a and b are defined at the GUT scale. It is now trivial to compare Eq. (2) with Eq. (5) to obtain the following identification after renormalization group running from the GUT scale down to the electroweak scale: Our particular ansatz given in Eqs. (3) and (4) is consistent with this identification if we take both R 2 states to mix maximally, i.e., s φ = c φ = 1/ √ 2. Clearly, the two SU (5) operators proportional to a and b suffice to generate the three operators of Eq. (1) associated with the Yukawa matrices Y , Y L , and Y R .
Perturbativity We advocate the case that the low-energy Yukawa couplings have an SU (5) origin. We implement the low-energy lagrangian of Eq. (1) in SARAH-4.12.3 [29] and obtain one-and two-loop beta function coefficients to accomplish the renormalization group running from the electroweak to the GUT scale which we set at 5 × 10 15 GeV. The lowenergy Yukawas that we use as input are the ones presented in Eq. (4) and we scan over y bτ R , y cµ L , and y cτ L to identify the region of parameter space for which all Yukawa couplings remain below √ 4π up to the GUT scale. We find, for example, that the most relevant Yukawa coupling contributions for the running of y bτ R are where y t is the top Yukawa coupling. Proton decay LQs are commonly associated with proton decay. It is thus important to address the issue of matter stability. R 2 cannot mediate proton decay at tree-level either directly or through mixing via one or two Higgs scalars in our set-up. It is an innocuous field with regard to the issue of matter stability. It can also be arranged that S 3 does not contribute towards proton decay. One prerequisite for this to happen is the absence (or suppression) of the contraction c ij 10 i 10 j 45 that couples S 3 to two quark doublets [30]. The other prerequisite is that S 3 does not mix with any other LQ with diquark couplings. Both prerequisites can be simultaneously satisfied in a generic SU (5) framework [31]. It is thus possible to have light R 2 and S 3 without any conflict with the stringent experimental limits on matter stability.

III. LOW-ENERGY PHENOMENOLOGY
Charged-current decays The relevant effective lagrangian for (semi-)leptonic decays is where neutrinos are in the flavor basis. The effective Wilson coefficients of d → u ν are related to the LQ couplings at the matching scale Λ(≈ 1 TeV) via the expressions From the above equations we learn that the only transitions affected by R 2 in our scenario are b → cτν . S 3 , on the other hand, contributes to processes involving u, c, s, b, and , = µ, τ , but gives a negligibly small contribution to R D ( * ) .
To compute R D we employ the B → D form factors calculated using the lattice QCD [4,32], resulting in prediction R SM D = 0.29(1) which is ≈ 2 σ below the experimental average R exp D = 0.41(5) [33][34][35]. On the other hand, the B → D * form factors have never been computed on the lattice at nonzero recoil. Thus, for R D * we consider the leading form factors extracted from the B → D * lν (l = e, µ) spectra [36], which are combined with the ratios A 0 (q 2 )/A 1 (q 2 ) and T 1−3 (q 2 )/A 1 (q 2 ) computed in Ref. [6]. We obtain the value R SM D * = 0.257(3) which is ≈ 3 σ below the experimental average R exp D * = 0.30(2) [36]. Moreover, to confront the scalar (tensor) effective coefficients in Eq. (7) with low-energy data, we account for the SM running from the matching scale µ = Λ down to µ = m b , while the vector coefficient is not renormalized by QCD [37].
Neutral-current decays The standard left-handed effective Hamiltonian for the b → s (semi-)leptonic transition can be written as where λ t = V tb V * ts . The relevant operators for our discussion are In our set-up, only S 3 contributes at tree-level via [38] δC µµ 9 = −δC µµ 10 = πv 2 λ t α em In the second line we explicitly write the dependence of δC µµ 9 on sin 2θ. This angle allows one to vary R K ( * ) , independently of R D ( * ) . Since we consider a scenario with relatively small Yukawa couplings, it is a very good approximation to neglect loop-induced contributions of R 2 (and S 3 ) to this transition. For a different set-up, see Ref. [43]. The 1 σ interval C µµ 9 = −C µµ 10 ∈ (−0.85, −0.50) is obtained by performing a fit to the clean b → sll observables, namely, R K , R K * , and B(B s → µµ) [44,45].
Contributions to the left-handed current operators in b → sll transition unavoidably imply contributions to B → K ( * ) νν decays which are well constrained by experiments. These decays are governed by where C ij L = δ ij C SM L + δC ij L is the Wilson coefficient which includes the SM contribution C SM L = −6.38 (6) [46] and the contribution δC ij L from NP. Similarly as in the b → sll transition, the only tree-level contribution to b → sνν comes from the S 3 state and reads [38] These effective coefficients modify the ratios R In Sec. IV we confront the predictions of R ( * ) νν with experimental bounds R νν < 3.9 and R * νν < 2.7 [47]. Further flavor constraints Our low-energy fit also includes constraints which will be more extensively discussed in a future publication. These are (i) the B s −B s mixing amplitude, which is shifted by the S 3 box-diagram, proportional to sin 2 2θ (y cµ L ) 2 + (y cτ L ) 2 2 /m 2 S3 , (ii) the experimental limits B(τ → µφ) ∼ cos 4 θ(y cµ L y cτ L ) 2 /m 4 S3 , which is bounded to remain below 8.4×10 −8 , and B(τ → µγ) exp < 4.4×10 −8 [48], (iii) the muon g − 2, which shows ≈ 3.6 σ discrepancy with respect to the SM [49] but receives only small contribution in our set-up, and (iv) the Z-boson couplings to leptons measured at LEP [50], which are modified at loop level by both R 2 and S 3 . Finally, we have also checked that our model is compatible with measured D −D mixing parameters.

IV. NUMERICAL RESULTS
We perform a fit to the observables listed above by varying the parameters y bτ R , y cµ L , y cτ L and θ, which were introduced in Sec. II. The fit requires S 3 to be more massive than R 2 . The masses m R2 and m S3 are set to the lowest values allowed by projected LHC constraints, namely, m R2 = 800 GeV and m S3 = 2 TeV, as we discuss later on. Note that in our flavor fit we obtain two solutions corresponding to small (θ ∼ 0) and large (|θ| ∼ π/2) mixing angles. These two solutions successfully suppress the key constraints, such as R K ( * ) and ∆m s since they are proportional to sin 2θ. Further inclusion of B(τ → µφ) ∝ cos 4 θ in the fit selects the solution with |θ| ≈ π/2 as the only viable one. The results of our fit in the g S L complex plane are shown in Fig. 1 to 1 σ and 2 σ accuracies. The SM point is excluded with 3.8 σ significance, while the best fit point provides a perfect agreement with R D ( * ) and R K ( * ) . Interestingly, a simultaneous explanation of R D and R D * requires complex g S L , which is why we consider complex y bτ R [51,52]. Note that the phase in y bτ R causes no observable CP violating effects. The best fit point is consistent with the LHC constraints superimposed on the same plot. A purely imaginary solution is: An important prediction of our scenario is that B(B → Kµτ ) is bounded from above and below, as illustrated in Fig. 2. At 1 σ we obtain This value is smaller than the current B(B → Kµτ ) exp < 4.8 × 10 −5 [53], which can certainly be improved by LHCb and Belle-II. Note that our prediction can easily be translated into similar modes via relations B(B → K * µτ ) ≈ 1.9 × B(B → Kµτ ) and B(B s → µτ ) ≈ 0.9×B(B → Kµτ ) [54][55][56]. Another important prediction of our set-up is a > ∼ 50% enhancement of B(B → K ( * ) νν), which can be tested in the near future at Belle-II. Remarkably, these two observables are highly correlated as depicted in Fig. 2. Furthermore, we predict a lower bound on B(τ → µγ), which lies just below the Finally, our description of the B-physics anomalies, and most particularly R D ( * ) , strongly depends on the assumption that the LQ states are not too far from the TeV scale. Thus, these particles are necessarily accessible at the LHC, yielding also predictions for the direct searches which we discuss next. SM for the 1 σ (red) and 2 σ (orange) regions of Fig. 1. The black line denotes the current experimental limit, R * νν < 2.7 [47].

V. LHC PHENOMENOLOGY
Direct searches at the LHC can play an important role in constraining LQ model(s) aiming to explain the R D ( * ) and R K ( * ) anomalies. In the following we show that the benchmark masses m R2 = 800 GeV and m S3 = 2 TeV are currently allowed by the high-p T and direct search experiments at the LHC and present exclusion limits for a projected LHC luminosity of 100 fb −1 of data.
High-p T di-tau tails The dominant NP contributions to qq → τ τ production, in view of the flavor structure of Eq. (4), come from the t-channel exchange of R 2 states in charm and bottom annihilation, respectively. Similar contributions from S 3 depend on the value of the mixing angle θ. As discussed in Sec. IV, the low-energy fit prefers |θ| ≈ π/2. In this case an almost exact flavor alignment takes place between τ and the third quark generation, meaning that only the exchange of S from initial bb collisions contributes to τ τ production. Following Ref. [57], we confront this scenario with data by recasting the most recent search by ATLAS [58] at 13 TeV and 36.1 fb −1 for a Z → τ hadτhad heavy resonance in the high-mass tails. Our results for the 95% C.L. limits in the y cτ L -(y bτ R /i) plane are given by the red exclusion region in Fig. 3 for the two benchmark masses, |θ| ≈ π/2, and the LHC luminosity of 100 fb −1 . 3. Summary of the LHC limits for each LQ process at a projected luminosity of 100 fb −1 for mR 2 = 800 GeV, mS 3 = 2 TeV, and |θ| ≈ π/2. The red region corresponds to the exclusion limit from the high-pT di-tau search by ATLAS [58], while the green and turquoise exclusion regions come from LQ pair production searches by CMS [59][60][61]. The region above the solid black contour represents values of the couplings that become non-perturbative at the GUT scale. The region inside the yellow contour corresponds to the 1 σ fit to the low-energy observables.
Leptoquark pair production For the benchmark masses, bounds from pair-produced LQs can only be derived for R 2 . The dominant decay channels for each charged eigenstate are R 2 3 2 → τ b, νc and R 2 ) * R 2 3 2 , we use CMS results from the search [59] for bbττ final states and the multi-jet plus missing energy search [60] for ccνν final states, i.e., jj plus missing energy signature. The 95% C.L. exclusion limits are given by the light green and turquoise regions in Fig. 3 for a luminosity of 100 fb −1 . As for the pair production of R 5 3 2 states, we employ the search by CMS [61] targeting ttττ final states. Results from this search are given by the dark green exclusion region in Fig. 3.

VI. CONCLUSIONS
We propose a two scalar LQ extension of the SM that can accommodate all measured LFU ratios in B-meson decays and related flavor observables, while being compatible with direct search constraints at the LHC. The extension has an SU (5) origin that relates Yukawa couplings of the two LQs through a mixing angle and all Yukawas remain perturbative up to the unification scale. We provide prospects for future discoveries of the two light LQs at the LHC and spell out predictions for several yet-to-be-measured flavor observables. In particular, we predict and correlate B(B → Kµτ ) with B(B → K ( * ) νν). We also predict a lower bound for B(τ → µγ) which is just below the current experimental limit.