Corrections to $R_{D}$ and $R_{D^{*}}$ in the BLMSSM

The deviation of the measurement of $R_{D}$ ($R_{D^{*}}$) from the Standard Model (SM) expectation is $2.3\sigma$ ($3.1\sigma$). $R_{D}$ ($R_{D^{*}}$) is the ratio of the branching fraction of $\overline{B} \rightarrow D\tau\overline{\nu}_{\tau}$ ($\overline{B} \rightarrow D^{*}\tau\overline{\nu}_{\tau}$) to that of $\overline{B} \rightarrow Dl\overline{\nu}_{l}$ ($\overline{B} \rightarrow D^{*}l\overline{\nu}_{l}$), where $l = e$ or $\mu$. 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 $R_{D}$ ($R_{D^{*}}$).

VIII. Summary and future prospects 19 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 predictionsthese 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.
In our work, we use effective field theory to do the theoretical calculation. The effec- 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 II, we introduce some content of the BLMSSM. In section III, we give the mass matrices of the BLMSSM particles that we use.
In section IV, we write down the needed couplings. In section V, we provide the relevant formulae, including observables R D ( * ) and the effective Lagrangian with all the four fermion operators. In section VI, 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 VII, we present our numerical results. Finally, we summarise our findings in section VIII. Some integral formulae are shown in the Appendix.

II. SOME CONTENT OF THE BLMSSM
As an extension of the MSSM, the BLMSSM includes many new fields [39,40]. The exotic quarks (Q 4 ,Û c 4 ,D c 4 ,Q c 5 ,Û 5 ,D 5 ) are used to deal with the B 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 superfieldŝ Φ B ,φ B are introduced to break baryon number spontaneously with nonzero vacuum expectation values (VEVs). The exotic Higgs superfieldsΦ L ,φ L are introduced to break lepton number spontaneously with non-zero 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.
The superpotential of the BLMSSM is [41]: The SU(2) L singlets Φ L , ϕ L , Φ B , ϕ B and the SU(2) L doublets H u , H d are: The SU(2) L singlets Φ L , ϕ L , Φ B , ϕ B and the SU(2) L doublets H u , H d should obtain non-zero VEVs υ L , υ L , υ B , υ B and υ u , υ d respectively. Therefore, the local gauge symmetry

III. 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 The mass squared matrix of sneutrino Mñ withñ T = (ν,Ñ c ) reads [46]    Then the masses of the sneutrinos are obtained by using the formula Z † ν M 2 n Zν = diag(m 2 ν 1 , m 2 ν 2 , m 2 ν 3 , m 2 ν 4 , m 2 ν 5 , m 2 ν 6 ). 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 righthanded 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.

IV. 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.

A. 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 . N , the normalisation factor, is given by Here τ B is the lifetime of the B−meson. G F = √ 2e 2 /8m 2 W s 2 W is the Fermi coupling constant. |p D ( * ) |, the absolute value of the D ( * ) −meson momentum, is given by 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].

B. Effective Lagrangian
We use effective field theory to calculate the theoretical values. The effective Lagrangian 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.

VI. 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. A. Penguin-type Feynman diagrams

The WCs
The one-loop Feynman diagrams FIG. 1 (a) 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 N P is the NP scale, and γ E is the Euler-Mascheroni constant.
, 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.

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 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: If x 1 = x 2 , they simplify to Now, the WCs of FIG. 1 (counter) read: The corresponding C It is easy to test that the infinite terms in the sum of FIG. 1 (a),(b),(c),(d) and (counter) 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.

VII. NUMERICAL RESULTS
For the numerical discussion, the parameters used are: 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 345GeV). 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 290GeV ∼ 450GeV [53], which can be satisfied easily. The masses of squarks in this paper are larger than 1000GeV, 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. 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.

VIII. SUMMARY AND FUTURE PROSPECTS
The SM cannot well 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 more 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 oneloop 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   I) 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. , ,