Corrections to RD and RD* in the BLMSSM

The deviation of the measurement of RD (RD*) from the Standard Model (SM) expectation is 2.3σ (3.1σ). RD (RD*) is the ratio of the branching fraction of B¯→Dτν¯τ ( B¯→D*τν¯τ ) to that of B¯→Dlν¯l ( B¯→D*lν¯l ), where l = e or μ. This anomaly may imply the existence of new physics (NP). In this paper, we restudy this problem in the supersymmetric extension of the Standard Model with local gauged baryon and lepton numbers (BLMSSM), and give one-loop corrections to RD (RD*).


Introduction
The Standard Model (SM) is the most successful particle physics model to date. It gives accurate predictions for a significant number of experiments. However, for some experiments, it cannot give a good explanation. In the last few years, the experimental measurements of R D ( * ) (the ratio of the branching fraction of B → Dτ ν τ (B → D * τ ν τ ) to that of B → Dlν l (B → D * lν l ), where l=e or µ) show deviations from the SM theoretical predictions -these measurements are larger than SM expectations. Therefore, in order to explain these anomalies, it is necessary for us to try some new physics (NP) models.
R D =0.407±0.039±0.024 and R D * =0.304±0.013±0.007 exceed the SM predictions by 2.3σ and 3.1σ respectively. These anomalies have caused physicists to seek a variety of ways to explain the experimental data . Most physicists tend to seek the solutions in NP models. So, various NP models have been used, such as charged Higgs [30][31][32] and lepton flavor violation [33][34][35]. The supersymmetric extension of the SM is a popular choice in various NP models. In fact, theorists have been fond of the minimal supersymmetric model (MSSM) for a long time. However, baryon number (B) should be broken because of the matter-antimatter asymmetry in the Universe. The neutrino oscillation experiments imply that neutrinos have tiny masses, therefore lepton number (L) also needs to be broken. A minimal supersymmetric extension of the SM with local gauged B and L (BLMSSM) [36,37] is more promising. Thus, we try to deal with the anomalies of R D ( * ) in the BLMSSM.
In our work, we use effective field theory to do the theoretical calculation. The effective Lagrangian is described by the four fermion operators and the corresponding Wilson coefficients (WCs). NP contributions with non-zero WCs are possible solutions to the R D ( * ) anomalies [38]. After considering all the 10 independent 6-dimensional operators and calculating the values of the corresponding WCs at one-loop level, we obtain the theoretical values of R D ( * ) in the BLMSSM. This paper is organised as follows. In Section 2, we introduce some content of the BLMSSM. In Section 3, we give the mass matrices of the BLMSSM particles that we use. In Section 4, we write down the needed couplings. In Section 5, we provide the relevant formulae, including observables R D ( * ) and the effective Lagrangian with all the four fermion operators. In Section 6, we show the one-loop Feynman diagrams that can correct R D ( * ) . At the same time, NP contributions of some diagrams are given by WCs. In Section 7, we present our numerical results. Finally, we summarise our findings in Section 8. Some integral formulae are shown in the Appendix. 2 Some content of the BLMSSM As an extension of the MSSM, the BLMSSM includes many new fields [39,40]. The exotic quarks anomaly. The exotic leptons (L 4 ,Ê c 4 ,N c 4 ,L c 5 ,Ê 5 ,N 5 ) are used to cancel the L anomaly. The exotic Higgs su-perfieldsΦ B ,φ B are introduced to break baryon number spontaneously with non-zero vacuum expectation values (VEVs). The exotic Higgs superfieldsΦ L ,φ L are introduced to break lepton number spontaneously with nonzero VEVs. The model introduces the right-handed neutrinos N c R , so we can obtain tiny masses of neutrinos through the see-saw mechanism. The model also includes the superfieldsX to make the exotic quarks unstable.

Mass matrices for some BLMSSM particles
Lepneutralinos are made up of λ L (the superpartner of the new lepton boson), and ψ Φ L and ψ ϕ L (the superpartners of the SU (2) L singlets Φ L and ϕ L ). The mass mixing matrix of lepneutralinos M LN is shown in the basis (iλ L ,ψ Φ L ,ψ ϕ L ) [42][43][44][45]. χ 0 L i (i = 1,2,3) are mass eigenstates of lepneutralinos. The masses of the three lepneutralinos are obtained from diagonalizing M LN by Z N L : The slepton mass squared matrix becomes which is diagonalized by the matrix ZL.
The mass squared matrix of sneutrino Mñ withñ T = (ν,Ñ c ) reads [46] M 2 n (ν * IνJ ), M 2 n (ν IÑ c J ) and M 2 n (Ñ c * IÑ c J ) are: Then the masses of the sneutrinos are obtained by using The up scalar quark mass squared matrix in the BLMSSM is given by which is diagonalized by the matrix ZŨ .
The down scalar quark mass squared matrix in the BLMSSM is given by which is diagonalized by the matrix ZD.
In the basis (ψ ν I L ,ψ N cI R ), the neutrino mass mixing matrix is diagonalized by Z ν [46]: ν α denotes the mass eigenstates of the neutrino fields mixed by the left-handed and right-handed neutrinos.
In this paper, we deal with the neutrinos by an approximation, Z ν ≈1, so the theoretical values at tree level are consistent with those in the SM.

Necessary couplings
In the BLMSSM, due to the superfieldsÑ c , we deduce the corrections to the couplings in the MSSM. The couplings for W -l-ν and W -L-ν read From the interactions of gauge and matter multiplets ig The ν-χ 0 L -ν coupling is We also obtain the χ ± -l-ν coupling and the χ ± -L-ν coupling: The χ 0 -ν-ν coupling in the BLMSSM becomes All the other couplings used are consistent with the MSSM.

Observables
The observable R D ( * ) is defined as , the branching fraction, is given by [4]: where l = e or µ, and ℓ denotes any lepton (e,µ or τ ). q 2 is the invariant mass squared of the lepton-neutrino system, whose integral interval is [m 2 ℓ ,(M B −M D ( * ) ) 2 ]. N , the normalisation factor, is given by Here τ B is the lifetime of the B−meson.
The expressions for a D ℓ and c D ℓ are [4]: The full expressions for a D * ℓ , c D * ℓ and all form factors (F T (q 2 ), F + (q 2 ) and F 0 (q 2 ), etc) are given in Refs. [4,47].

Effective Lagrangian
We use effective field theory to calculate the theoretical values. The effective Lagrangian for the b → cℓν ℓ process is where V cb =0.04, and the full set of operators is [48]: In the SM, C ℓ V L =−C ℓ AL =1 and all the other WCs vanish. In the BLMSSM, we calculate all the WCs at one-loop level to obtain the theoretical values.

Feynman diagrams
In the BLMSSM, the one-loop Feynman diagrams for the lepton sector that can correct the anomalies are shown in Fig. 1 and Fig. 2. 6.1 Penguin-type Feynman diagrams

The WCs
The one-loop Feynman diagrams in Fig. 1(a),(b),(c) and (d) are all UV divergent. Focusing on Fig. 1(a), the three lepneutralinos χ 0 L are new particles in the BLMSSM, and they play very important roles in this decay process. So taking Fig. 1(a) as an example, the non-zero WCs in Eq. (26) are given as follows:  Here, we use the unitary characteristics of the rotation matrices. In Eq. (28), ε is an infinite term, the mass scale κ is introduced in the dimensional regularization, Λ NP is the NP scale, and γ E is the Euler-Mascheroni constant. x i represents , and the concrete form of formula F 21 (x 1 ,x 2 ,x 3 ) is given in the Appendix.
We can see that the infinite terms of the WCs of Fig. 1(a) . Similarly, the infinite terms of the WCs of the following three diagrams ( Fig. 1(b), (c) and (d)) are given as follows: Now we should deal with the UV divergences by renormalization procedures. 6.1.2 The counter term in the on-shell scheme Considering the final state lepton and neutrino are both real particles, we use the on-shell scheme to eliminate the infinite terms. To obtain finite results, the contributions from the counter terms for the vertex l I ν I W − are necessary. The counter term formula for the vertex l I ν I W − is: Following the method in Refs. [49][50][51], we obtain the needed renormalization constants in the BLMSSM. We calculate the Z boson self-energy diagram (loop particles are sneutrinos or sleptons) and get the renormalization constant δm 2 Z : Through calculating the W boson self-energy diagram (with sneutrinos and sleptons in the loop), we can obtain: The renormalization constant of charge is obtained from virtual sleptons: In the same way, we give the renormalization constants δZ ν L I and δZ l L I for neutrinos and leptons respectively: where the vertex couplings are given by The functions F 1 ,F 2 , and F 3 are as follows:

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If x 1 =x 2 , they simplify to Now, the WCs of Fig. 1(counter) read: The corresponding C ℓ(IF ) V L(counter) and C ℓ(IF ) AL(counter) are: It is easy to test that the infinite terms in the sum of Fig. 1(a),(b),(c),(d) and (counter) vanish: . Therefore, the divergences are completely eliminated. Note that the infinite terms in the sum of Fig. 1(a), (b), (c) and (d) can be eliminated by the counter terms. However, a single diagram in Fig. 1(b), (c), (d), such as Fig. 1(b), cannot be counteracted individually in the on-shell scheme.

Numerical results
For the numerical discussion, the parameters used are: If not otherwise noted, the non-diagonal elements of the parameters used should be zero. The Yukawa couplings of neutrinos Y IJ ν are of the order of 10 −8 ∼10 −6 ; their effects are tiny and can be ignored.
At present, all supersymmetric mass bounds are model-dependent. Based on the PDG [52] data, we consider the limitations on masses of the charginos and neutralinos (the strongest limitations are 345 GeV). In our work, the masses of charginos m χ ± ≃ (1000 ∼ 2000) GeV and the masses of neutralinos m χ 0 ≃ (400 ∼ 2000) GeV, all of which can satisfy the mass bounds. The limits for the sleptons are around 290 GeV ∼ 450 GeV [53], which can be satisfied easily. The masses of squarks in this paper are larger than 1000 GeV, so the limits for squarks are also satisfied. In other words, the parameters given above and the parameter space to be discussed below can all satisfy the mass bounds. To study the impacts of these parameters on R D ( * ) , we use the parameters (m 2 where ξ is a variable. After calculation we obtain Fig. 3. Here, we used the central value of the SM prediction in our calculation. The left-hand diagram shows R D and the right-hand diagram shows R D * .
We know the measurement of R D ( * ) e (which implies l = e in Eq. (20)) is approximately equal to that of R D ( * ) µ (l = µ in Eq. (20)). This is the reason why we set (m 2 To solve the problem of R D ( * ) , we should violate lepton flavour symmetry for generations 1(2) and generation 3. Therefore, we suppose (m 2 L ) 33 =(m 2 R ) 33 =(m 2 L ) 11 . It is easy to see from Fig. 3 that R D ( * ) decreases as ξ increases. Obviously, our results satisfy the decoupling rule. When the sleptons are very heavy, the BLMSSM results are very near the SM predictions. In fact, the SM predictions of R D ( * ) cannot explain the experimental values well, and our goal is to increase the theoretical values. From the numerical analysis, the following relational expression should be set up: (m 2 33 . We need to select a set of reasonable parameters, and finally choose: Up to now, our theoretical values of R D ( * ) are only a little bigger than those of the SM, so we also need to study the effects of other parameters on R D ( * ) .   g L is the coupling constant of the vertexes lχ 0 LL and νχ 0 Lν . As a new parameter in the BLMSSM, g L should affect R D ( * ) , which is of interest. The obtained numerical results are plotted in Fig. 4. The left-hand diagram shows R D and the right-hand diagram shows R D * .
From Fig. 4, we can see that R D and R D * both increase gently with increasing g L . This is easy to understand: larger g L improves the effects from NP. In order to get larger theoretical values of R D ( * ) , we need to choose a larger g L . After considering the reasonableness of the range of parameter g L , we use g L =0. 45. In this case, our numerical results are further improved. All the points in Fig. 5 can make R D (R D * ) reach 0.304 (0.261), and some particular points can bring R D (R D * ) to 0.305 (0.262). The theoretical values are improved, but they are not as big as we expected. However, our results are still better than those in the SM. All of the above discussions only consider the central values in the SM. If we consider the uncertainty of SM predictions R D = 0.299±0.003 [5] and R D * = 0.257±0.003 [5], our theoretical value of R D (R D * ) can reach 0.308 (0.265), when we take the biggest value of the SM prediction.

Summary and future prospects
The SM cannot explain the experimental data for R D ( * ) well, so we hold that SM should be the low energy effective theory of a large model. We think the BLMSSM is promising for testing in the future. Compared with the MSSM, there are new particles and new parameters in the BLMSSM, and the new contributions from these are the keys to solve the anomalies in R D ( * ) . For instance, the three lepneutralinos χ 0 L are new particles in the BLMSSM, and the Feynman diagram with χ 0 L can give new contributions to R D ( * ) .
We find that the parameters (m 2 L ) ii and (m 2 R ) ii influence the theoretical results to some extent, and R D ( * ) e is approximately equal to R D ( * ) µ only if there is a certain relationship between parameters (m 2 L ) ii and (m 2 R ) ii . After that, the effect of parameter g L is important, and we can further raise theoretical values when it takes some appropriate values. Finally, using the central value of the SM prediction we scan the parameter space, and bring the value of R D (R D * ) to 0.305 (0.262). Taking into account the SM uncertainty and adopting the biggest value in the SM, our result for R D (R D * ) can reach 0.308 (0.265).
In this paper, we use effective field theory to compute R D ( * ) in the BLMSSM. The one-loop corrections to R D ( * ) have an effect and the theoretical values can be increased (though they are not big improvements). We notice that the measurements of R D * (see Table 1) are not as large as the original measurements. This suggests that R D * perhaps is not so large. From the trend of experimental measurement, the experimental values of R D ( * ) might be smaller in the future. In fact, without considering this case, the measurement of R D (R D * ) shows 2.3σ (3.1σ) deviation from its SM prediction, and our theoretical values are still better than the predictions given by SM. On the whole, the problem of R D (R D * ) should be further researched both experimentally and theoretically in the future.