R-parity violating solutions to the $R_{D^{(\ast)}}$ anomaly and their GUT-scale unifications

Recently, several $B$-physics experiments report interesting anomalies in the semi-leptonic decays of $B$-mesons, such as the excess in the $R_{D^{(\ast)}}$ measurements. These anomalies seem to suggest intriguing hints of lepton flavor non-universality, and the R-parity violating (RPV) interactions are candidates for explaining this non-universality. In this paper, we discuss the RPV interactions for resolving the $R_{D^{(\ast)}}$ anomaly with the Grand Unified Theory (GUT) assumption. To solve the $R_{D^{(\ast)}}$ anomaly, it is known that large RPV couplings and around $1~{\rm TeV}$ sfermion masses are required. At the same time, large RPV couplings are conducive to realize the bottom-tau Yukawa unification which appears in the GUT models. On the other hand, there are problems for realizing favorable sfermion masses in the constrained minimal supersymetric standard model. To resolve these problems, we show that two non-universalities, the non-universal sfermion masses and the non-universal gaugino masses, are favorable.

As demonstrated in Refs. [16,17], the R-parity violating (RPV) supersymetric (SUSY) models [18,19] are also viable candidates for explaining the R D ( * ) anomaly. In the minimal supersymetric SM (MSSM) with RPV interactions, an exchange of right-handed down-type squarks coupled to quarks and leptons can yield the required four-fermion interactions contributing to R D ( * ) at the tree level, which is similar to the case with leptoquark induced interactions. In Ref. [16], the RPV contributions to R D ( * ) are discussed within a minimal effective RPV-SUSY scenario, and it is shown that O(1) RPV coupling and around 1 TeV sbottom mass are required to satisfy the R D ( * ) measurements, while being consistent with other experimental constraints as well as preserving the gauge coupling unification. In this work, we shall discuss this possibility within the Grand Unified Theory (GUT) framework.
GUTs [20,21] are interesting candidates for physics beyond the SM. In the GUTs, two interesting unifications, the gauge coupling unification and the matter unification, are realized. The matter unification plays an important role in restricting the model parameters. In the MSSM [22], there are enormous SUSY breaking parameters and, in order to make these SUSY breaking parameters restrictive, the constrained MSSM (CMSSM) is introduced as a very well-motivated, realistic and concise SUSY extension of the SM. In the CMSSM [23], the matter unification unifies the SUSY breaking parameters; for example, gaugino masses are unified into one universal gaugino mass. Moreover, for obtaining a realistic GUT framework, the introduction of RPV interactions is beneficial [24]. In the SU (5) GUTs, the down and charged-lepton Yukawa couplings are unified, which is however not easy to be realized in the standard minimal setup. The RPV interactions modify the renormalization group (RG) flow for realizing this unification [24] and, as we will show in this paper, these RPV couplings are also large enough to explain the R D ( * ) anomaly.
In this paper, we will examine the sfermion masses within the GUT framework. As we will show later, models with the CMSSM assumption do not satisfy the R D ( * ) measurements, because a large RPV coupling makes these models already excluded, and the right-handed sbottom mass becomes too heavy to explain the R D ( * ) anomaly. To solve this problem, two non-universalities, the non-universal sfermion masses and the non-universal gaugino masses, which are compatible with the GUT framework, play an important role. Moreover, these non-universalities are motivated to stabilize the electro-weak (EW) scale and will be future signals of our scenario. This paper is organized as follows. In Sec. 2, we explain the relations among the RPV interactions, the R D ( * ) anomaly, and the SU (5) unification. In Sec. 3, we first show the problem encountered with the CMSSM assumption for resolving the R D ( * ) anomaly, and then demonstrate the necessary two non-universalities for figuring out this problem. Our discussion and summary are given in Sec. 4. For convenience, we give all the relevant formulae for the processes used to produce Figure 1 in Appendix A, while the density plots for the sfermion masses at 1 TeV are shown in Appendix B.

unification
First of all, we specify the RPV couplings in the MSSM. The most general renormalizable superpotential consistent with the gauge symmetry and field content of the MSSM is given by [25,26] where L, Q, E c , D c , U c , H d , and H u are the chiral superfields for the MSSM multiplet, and we denote the SU (3) C and SU (2) L fundamental representation indices by α, β, γ = 1, 2, 3 and a, b = 1, 2, respectively, while the generation indices by i, j, k = 1, 2, 3. ab and αβγ , with 12 = 123 = +1, are the totally anti-symmetric tensors for the SU (2) L and SU (3) C gauge groups, respectively. In addition to the R-parity conserving (RPC) couplings (Y e ) ij , (Y d ) ij , (Y u ) ij and µ, the following RPV couplings are introduced in Eq. (2.1): lepton number violating tri-linear couplings (Λ e k ) ij and (Λ d k ) ij , baryon number violating tri-linear couplings (Λ u i ) jk , and lepton number violating bi-linear couplings µ i . As the lepton doublet superfields L i and the Higgs doublet superfield H d have the same gauge and Lorentz quantum numbers in MSSM, the µ i term in Eq. (2.1) can be sent to zero via a rotation in the (L i , H d ) space [18,27], which will be assumed throughout this paper.

RPV interactions contributing to R D ( * )
As the underlying quark-level transition in R D ( * ) involve quarks and leptons, at the tree level we need only consider the Λ d term in Eq. (2.1). Working in the mass eigenstates for the down-type quarks and assuming sfermions are in their mass eigenstates, one obtains from Eq. (2.1) the following effective Lagrangian contributing to the transition d n → u j e i ν m at the tree-level after integrating out the heavy squarks [16,17]: where V CKM is the CKM matrix and md k denotes the mass of the right-handed squark. Specifying to the decays B → D ( * ) ν ( = e, µ, or τ ), one can see that the resulting four-fermion operator has the same chirality structure as in the SM, implying that the RPV contribution to R D ( * ) is simply a rescaling of the SM result.
The contribution of the effective operator in Eq. (2.2) to the R D ( * ) anomaly has been discussed in Refs. [16,17], and it is shown that large values of (Λ d 3 ) 33 ∼ 1 − 2 are required to explain the R D ( * ) anomaly within 1σ level, but such a large (Λ d 3 ) 33 coupling develops a Landau pole below the GUT scale [16]. Along the same line as in Ref. [16], we show in Figure 1 the parameter region in the (mb, (Λ d 3 ) 3j ) plane that satisfy the R D ( * ) measurements as well as the constraints from other relevant processes. For convenience, we have also given in Appendix A all the relevant formulae for the processes used to produce Figure 1.
In our calculation, we have used the latest measurements as well as the updated SM predictions which are summarized in Table 1. In this table, we also summarize differences between the experimental data used in Ref. [16] and here. To obtain the allowed parameter region, we use the following best fit value in the RPV scenario: In Figure 1, the 1σ-, 2σ-, and 3σ-favored regions from the R D ( * ) measurements are shown in dark-green, green, and light-green, respectively. It is clearly seen, especially from the top-right plot in Figure 1, that a realization of the R D ( * ) measurements at the 2σ level is possible. We have also considered the extra constraints from D and τ decays as discussed in Ref. [40], and found that only some τ -decay modes, especially the decay τ → Kν, can provide complementary constraints on the parameter regions allowed by the R D ( * ) measurements, while the leptonic charm decays D → τ ν and D s → τ ν do not lead to any relevant constraints in our scenario.  Table 1.
In Ref. [16], the R D ( * ) anomaly was discussed in a minimal effective SUSY scenario with RPV. In such a natural SUSY scenario, the masses of the third-generation sfermions are below 1 TeV, while the first-and second-generation sfermions are quite heavy and can be thought of being decoupled from the low-energy spectrum [41]. This mass spectrum explains the current experimental results naturally, because the decoupled first-and Table 1: SM predictions and experimental data for R D ( * ) and the other relevant observables.
Differences between the experimental data used in Ref. [16] and here are also shown.

RPV couplings and GUT-scale unification
From the last subsection, we have seen that O(1) RPV couplings for sbottom masses compatible with the current direct searches are helpful for realizing the R D ( * ) measurements. However, the RPV couplings are strongly constrained by the proton stability [19]: the conservative limit for the combination of the lepton and baryon number violating couplings is |Λ d Λ u | < 10 −11 for any generation indices [43]. Basically, in SUSY SU (5) GUT models with RPV, all the tri-linear RPV couplings are induced from a single term Λ ijk 10 i5j5k , and should therefore preserve the following relation at the GUT scale [20]: To satisfy the conservative limit set by the proton stability, either the RPV couplings become negligibly small or some novel mechanism should be invoked to evade this limit [44,45]. The proton stability constraint is extremely severe when both the lepton and baryon number violating RPV couplings are simultaneously non-zero. For simplicity, we implicitly assume that the proton stability is ensured by some mechanism 1 , and focus only on the Λ d 3 couplings which are crucial for accommodating the R D ( * ) anomaly.
In SU (5) GUT models, the gauge couplings and the Yukawa matrices satisfy their respective unification relations [20]: (2) L , and SU (3) C gauge groups, respectively. Here the hypercharge gauge coupling g 1 includes already the GUT normalization factor. Among the unification relations of Yukawa matrices, the one for the third generation, the bottom-tau Yukawa unification y τ = y b , is particularly promising, because small Yukawa couplings for the first and second generations can be modified at the GUT scale through contributions from higher-dimensional operators [46] and/or higher-dimensional representations of Higgs fields [47]. It is known that the unification relation for the third generation can be satisfied thanks to the RPV contributions [24]. To see the effect of these RPV contributions, we calculate the RG flow for the gauge, Yukawa and RPV couplings. The relevant input parameters involved in our calculation are summarized in Table 2. These parameters at the GUT scale are obtained by the following four steps: First, we use the Mathematica package RunDec [48] for the running and converting of different quark masses, and follow Ref. [49] to translate the lepton pole masses into the running masses, for calculating the MS Yukawa couplings at the M Z scale. Second, we calculate these parameters at 1 TeV from the SM RG flow at the two-loop level [50]. Third, we follow Ref. [51] to translate the MS scheme parameters into the DR scheme ones. Finally, we evaluate these parameters at the GUT scale with the presence of RPV couplings, using the two-loop RG flow for the RPC parameters [52] and the one-loop RG flow for the RPV parameters [26]. In this paper, the GUT scale M GUT is defined by (2.6) In Figure 2, the contribution of the RPV coupling (Λ d 3 ) 33 to R b/τ ≡ y b /y τ is shown for three representative values of tan β, the ratio of the Higgs vacuum expectation values (VEVs). The favored region for (Λ d 3 ) 33 by the R D ( * ) measurements is shown in green when mb = 900 GeV. In the left panel of Figure 2, we make all the RPV couplings except for (Λ d 3 ) 33 zero and, in the right panel, we take (Λ  Yukawa unification, large (Λ d 3 ) 33 is desirable; second, both the ratio R b/τ and the Landau pole depend strongly on the value of tan β. This dependence is due to a tan β dependence of the Yukawa couplings y b and y τ [26]: large tan β makes y b and y τ large through the matching condition of the Yukawa couplings between the SM and the MSSM, and large y b and y τ move the Landau pole for the RPV coupling (Λ d 3 ) 33 closer to the low-energy scale. Therefore, to solve the R D ( * ) anomaly, small tan β is particularly favorable.

Sfermion masses needed for R D ( * ) anomaly
Up to now, we have shown that O(1) RPV couplings are helpful for resolving the R D ( * ) anomaly when the third-generation sfermion masses are around 1 TeV, and O(1) RPV couplings are also favorable for realizing the SU (5) Yukawa relation, the bottom-tau Yukawa unification at the GUT scale. In this section, we proceed to discuss the impact of large RPV couplings on the sfermion masses.
In the beginning, we calculate these contributions in the CMSSM scenario [23]. In this scenario, the SUSY parameters at the GUT scale are defined as where M i , mψ, and A ψ are the gaugino mass, the sfermion mass, and the coupling for the soft tri-linear A term, respectively. We first calculate the sfermion mass at the 1 TeV scale with the SUSY conserving parameters calculated in the last section. For this calculation, we fix tan β = 3, (Λ d 3 ) 31 = −0.2, and (Λ d 3 ) 32 = 0.05 to realize large contributions to R D ( * ) , and m 1/2 = 900 GeV to satisfy the current gluino mass limit M 3 ≥ 2 TeV [53][54][55] Moreover, we fix a 0 = 0 for the sake of simplicity. The RPV coupling Λ d contributes also to the neutrino masses at the one-loop level, and this contribution would be too large to realize the neutrino mass measurements when the RPV couplings are O(1) and the sfermion masses are around O(1 TeV) [18]. To suppress this one-loop contribution, the relation a 0 − µ tan β = 0 is required in the CMSSM scenario [19] 3 , and the parameter µ should be around the Z-boson mass to stabilize the EW scale. Therefore, the parameter a 0 is around the EW scale and can be neglected in this calculation 4 .
In Figure 3, density plots for the sfermion masses mb and ml 3 at the 1 TeV scale are shown as a function of the universal sfermion mass m 0 and the RPV coupling (Λ d 3 ) 33 . White region in the right panel is ruled out by a presence of tachyons. From these figures, we can find two problems to realize the R D ( * ) measurements. The first one is the largeness of the sbottom mass: as shown in the left panel of Figure 3, the sbottom mass is too heavy to satisfy the R D ( * ) measurements. The second one is the presence of tachyons: as shown in the right panel of Figure 3, the slepton becomes tachyonic in the large RPV coupling region. Therefore, to accommodate the R D ( * ) anomaly, these two problems have to be resolved in the CMSSM scenario.
The gaugino contributions to the β-functions for the sfermion masses play an important role in understanding these problems. In the CMSSM, the squark masses are larger than the slepton masses in the low-energy scale. This feature is due to the gaugino contributions. While these contributions make sfermion masses large in the low-energy scale, their effects on the slepton masses are relatively small because of the absence of the gluino contribution, which is the largest among the gaugino contributions. As the gluino mass should be larger than 2 TeV [53][54][55], the gluino contribution is large, making the right-handed sbottom mass becomes larger than 1 TeV. On the other hand, the RPV contributions make the slepton mass small, and overcome the small gaugino contributions, making therefore the squared slepton masses negative.

Non-universal sfermion masses
Non-universal sfermion masses are helpful for resolving the first problem, i.e. the largeness of the sbottom mass. In this paper, the non-universal sfermion masses are specified as When r m 1, the natural SUSY scenario is realized, where the third generation sfermion masses become much lighter than that of the first two generations. With the two-loop β-functions for the sfermion masses [52], the resulting sfermion masses, especially the squark masses, become small. In particular, contributions from the trace term Tr m 2 ψ = (1 + 2r 2 m )m 2 0 have a much stronger effect on the third-generation squark masses. In the left panel of Figure 4, the density plots for mb at the 1 TeV scale are shown as a function of the sfermion mass m 0 and the RPV coupling (Λ d 3 ) 33 , when r m = 20. It can be seen that, owing to the non-universality of sfermion masses, the right-handed sbottom mass can now be around 1 TeV, together with O(1) RPV couplings.

Non-universal gaugino masses
The gaugino contributions to the slepton masses are enlarged when the bino mass becomes large. In the GUT models, while universal gaugino masses are usually assumed, nonuniversal gaugino masses are also feasible [62]. In the SU (5) GUT models, the tensor product of the adjoint fields is decomposed as (24 ⊗ 24) (3.5) In the CMSSM, we assume that this tensor product becomes singlet. However, when it is not singlet, non-universal gaugino masses are achieved by the F -term breaking of the non-universal sfermion masses non-universal gaugino masses Figure 4: The captions are the same as those in Figure 3, but now with non-universal sfermion masses (left) and non-universal gaugino masses (right).
VEVs of the non-singlet scalar fields. When the F -term breaking is assumed, the VEV of the dimensional-75 scalar field would induce nonzero gaugino masses, with the mass relation at the GUT scale given by Therefore, the bino mass can be large. The same observation is also applied to the case with the dimensional-200 scalar field. In the right panel of Figure 4, the density plots for ml 3 at the 1 TeV scale are shown as a function of the universal sfermion mass m 0 and the RPV coupling (Λ d 3 ) 33 , when M 1 : M 2 : M 3 = −5 : 3 : 1 at the GUT scale. Owing to the non-universality of gaugino masses, the tachyonic region disappear. Therefore, this non-universality is helpful for resolving the second problem, i.e. the presence of tachyons.

Non-universal sfermion and gaugino masses
In the subsections 3.1 and 3.2, we have shown that the non-universal sfermion and gaugino masses are helpful for resolving the two problems encountered in the CMSSM scenario, to realize the R D ( * ) measurements. In this subsection, we discuss the sfermion masses when these two non-universalities are assumed.
In Figure 5, density plots for mb and ml 3 at the 1 TeV scale are shown as a function of the sfermion mass m 0 and the RPV coupling (Λ d 3 ) 33 . For this calculation, we assume r m = 20 and M 1 : M 2 : M 3 = −5 : 3 : 1 at the GUT scale. From this figure, we can see that, thanks to these two non-universalities, large RPV couplings and around 1 TeV right-handed sbottom mass can be realized simultaneously. In Appendix B, density plots for the other sfermion masses mq 3 , mt, mτ at the 1 TeV scale are also shown.
From the above discussions, we can conclude model-independently that, among the three tri-linear RPV couplings, Λ d must be large enough to reproduce the ratio R D ( * ) Figure 5: The captions are the same as those in Figure 3, but now with non-universal sfermion and gaugino masses simultaneously. and, at the same time, Λ u must be zero in order to realize the proton decay constraint. This leaves with us two remaining choices for Λ e , being either zero or sizable. Up to now our discussions are restricted to the case with zero Λ e . The choice with sizable Λ e can be realized through the matter-Higgs mixing mechanism [44] and will also affect the RG flows for gauge couplings, Yukawa couplings and squared sfermion masses. Especially, large Λ e would make the squared sfermion masses for the left-handed lepton doublets negative, in the same way as large Λ d does for the right-handed sleptons. Such a problem can, however, be solved by the non-universal gaugino mass too.

Discussion and summary
In this paper, motivated by the excess in the R D ( * ) measurements by the B-physics experiments, we have investigated the RPV interactions for resolving the R D ( * ) anomaly with the GUT assumption.
It has been found that, to resolve the R D ( * ) anomaly, O(1) RPV coupling and around 1 TeV sbottom mass are required. It has also been shown that this large RPV coupling is conducive to realize the bottom-tau Yukawa unification that appears in the SU (5) GUT models and, to realize this unification, small tan β is particularly favored. On the other hand, problems appear for realizing the favorable sfermion masses: large RPV couplings would make the sleptons tachyonic, and around 1 TeV sbottom mass is not acceptable when the CMSSM relations for the SUSY parameters are assumed. For solving these problems, two non-universalities, the non-universal sfermion masses and the non-universal gaugino masses, have been found especially favorable. These non-universalities are also motivated for solving the EW-scale stabilization problem, and can be naturally introduced within the GUT assumption.
The non-universalities of sfermion and gaugino masses can be examined in the future; especially when the third-generation squark masses are around 1 TeV, chromo-electric dipole moments can be a strong signal candidate, although they depend strongly on the quark mixings [63]. Another consequence for our scenario is the relation for suppressing the dangerous one-loop contribution to the neutrino masses, a 0 − µ tan β = 0. This suppression means that the mass insertion parameter δ d LR is zero and, therefore, the SUSY flavor violating contribution from this mass insertion parameter is strongly suppressed. These topics are beyond the scope of this paper and will be explored in the future.
A Relevant formulae for the processes used to produce Figure 1 As we find that there is a global factor 1/2 (−1/2) missing for the NP terms in Eqs. (9) and (12) (Eqs. (20) and (21)) of Ref. [16], which we are closely following, we decide to give in this appendix all the relevant formulae for the processes used to produce Figure 1. At low energies, the effective Lagrangian for d j → u n τ ν τ transition is given by where C nj V L encodes all the contribution from the RPV couplings. Matching Eq. (2.2) onto Eq. (A.1), one gets with v = 246 GeV being the Higgs VEV. The branching ratio of the decay governed by d j → u n τ ν τ transition can then be written as B/B SM = |1 + C nj V L | 2 ; explicitly, we find all of which have to satisfy the constraints summarized in Table 1.
The formulas for other processes are given, respectively, by with the SM loop function X t = 1.469 ± 0.017 [32]. For the corrections to the Z and W couplings to leptons, we have where the loop functions are given by f Z (x) = 1 x−1 − log(x) (x−1) 2 , f W (x) = 1 x−1 − (2−x) log(x) (x−1) 2 [66].

B Sfermion masses in the case with non-universal sfermion and gaugino masses
In Figure 6, density plots for mq 3 , mt, and mτ at 1 TeV are shown as a function of the sfermion mass m 0 and the RPV coupling (Λ d 3 ) 33 . For this calculation, r m = 20 and M 1 : M 2 : M 3 = −5 : 3 : 1 at the GUT scale have been assumed. From these plots, we can see that, with the simultaneous presence of non-universal sfermion and gaugino masses, parameter regions with large RPV couplings and around 1 TeV sbottom mass, required for resolving the R D ( * ) anomaly, can be naturally obtained in our scenario, avoiding at the same time the presence of any tachyons.