cpcChinese Physics CChin. Phys. C1674-1137Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltdcpc_42_11_11310410.1088/1674-1137/42/11/11310442/11/113104PaperParticles and fieldsCorrections to R_{D} and R_{D*} in the BLMSSM*
Supported by the NNSFC (11535002, 11605037, 11705045), the Natural Science Foundation of Hebei Province (A2016201010, A2016201069), the Natural Science Fund of Hebei University (2011JQ05, 2012-242), and Hebei Key Lab of Opto-Electronic Information and Materials, the Midwest Universities Comprehensive Strength Promotion Project
The deviation of the measurement of R_{D} (R_{D*}) from the Standard Model (SM) expectation is 2.3σ (3.1σ). R_{D} (R_{D*}) 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 R_{D} (R_{D*}).
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¯→Dτν¯τ (B¯→D*τν¯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.
The SM expectations for R_{D(*)} are: R_{DSM} = 0.299 ± 0.011 in Ref. [1], R_{DSM} = 0.299 ± 0.003 in Ref. [2], R_{DSM} = 0.300 ± 0.008 in Ref. [3], R_{DSM} = 0.300 ± 0.011 in Ref. [4], R_{DSM} = 0.299 ± 0.003 in Ref. [5], R_{D*SM} = 0.254 ± 0.004 in Ref. [4], R_{D*SM} = 0.257 ± 0.003 in Ref. [5] and R_{D*SM} = 0.252 ± 0.003 in Ref. [6]. The relevant experimental results for R_{D(*)} are listed in Table 1.
The measurements of R_{D(*)}.
observable
experiment
measured value
R_{D}
2012 BaBar
0.440 ± 0.058 ± 0.042 [7, 8]
2015 Belle
0.375 ± 0.064 ± 0.026 [9]
2017 HFAG average
0.407 ± 0.039 ± 0.024 [10]
R_{D*}
2012 BaBar
0.332 ± 0.024 ± 0.018 [7, 8]
2015 Belle
0.293 ± 0.038 ± 0.015 [9]
2015 LHCb
0.336 ± 0.027 ± 0.030 [11]
2016 Belle
0.302 ± 0.030 ± 0.011 [12]
2017 Belle
0.270±0.035−0.025+0.028 [13]
2017 LHCb
0.291 ± 0.019 ± 0.026 ± 0.013 [14]
2017 HFAG average
0.304 ± 0.013 ± 0.007 [10]
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 [15–35]. Most physicists tend to seek the solutions in NP models. So, various NP models have been used, such as charged Higgs [30–32] and lepton flavor violation [33–35]. 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.
Some content of the BLMSSM
As an extension of the MSSM, the BLMSSM includes many new fields [39, 40]. The exotic quarks (Q^4,U^4c,D^4c,Q^5c,U^5,D^5) are used to deal with the B anomaly. The exotic leptons (L^4,E^4c,N^4c,L^5c,E^5,N^5) are used to cancel the L anomaly. The exotic Higgs superfields Φ^B,φ^B are introduced to break baryon number spontaneously with non-zero vacuum expectation values (VEVs). The exotic Higgs superfields Φ^L,φ^L are introduced to break lepton number spontaneously with non-zero VEVs. The model introduces the right-handed neutrinos NRc, so we can obtain tiny masses of neutrinos through the see-saw mechanism. The model also includes the superfields X^ to make the exotic quarks unstable.
The superpotential of the BLMSSM is [41]:
WBLMSSM=WMSSM+WB+WL+WX,WB=λQQ^4Q^5cΦ^B+λUU^4cU^5φ^B+λDD^4cD^5φ^B+μBΦ^Bφ^B+Yu4Q^4H^uU^4c+Yd4Q^4H^dD^4c+Yu5Q^5cH^dU^5+Yd5Q^5cH^uD^5,WL=Ye4L^4H^dE^4c+Yν4L^4H^uN^4c+Ye5L^5cH^uE^5+Yν5L^5cH^dN^5+YνL^H^uN^c+λNcN^cN^cφ^L+μLΦ^Lφ^L,WX=λ1Q^Q^5cX^+λ2U^cU^5X^′+λ3D^cD^5X^′+μXX^X^′,
where WMSSM is the superpotential of the MSSM.
The soft breaking terms ℒsoft of the BLMSSM can be written in the following form [36, 37, 41]:
ℒsoft=ℒsoftMSSM−(mν∼c2)IJN∼Ic*N∼Jc−mQ∼42Q∼4†Q∼4−mU∼42U∼4c*U∼4c−mD∼42D∼4c*D∼4c−mQ∼52Q∼5c†Q∼5c−mU∼52U∼5*U∼5−mD∼52D∼5*D∼5−mL∼42L∼4†L∼4−mν∼42N∼4c*N∼4c−me∼42E∼4c*E∼4c−mL∼52L∼5c†L∼5c−mν∼52N∼5*N∼5−me∼52E∼5*E∼5−mΦB2ΦB*ΦB−mφB2φB*φB−mΦL2ΦL*ΦL−mφL2φL*φL−(MBλBλB+MLλLλL+h.c.)+{Au4Yu4Q∼4HuU∼4c+Ad4Yd4Q∼4HdD∼4c+Au5Yu5Q∼5cHdU∼5+Ad5Yd5Q∼5cHuD∼5+ABQλQQ∼4Q∼5cΦB+ABUλUU∼4cU∼5φB+ABDλDD∼4cD∼5φB+BBμBΦBφB+h.c.}+{Ae4Ye4L∼4HdE∼4c+Aν4Yν4L∼4HuN∼4c+Ae5Ye5L∼5cHuE∼5+Aν5Yν5L∼5cHdN∼5+AνYνL∼HuN∼c+AνcλνcN∼cN∼cφL+BLμLΦLφL+h.c.}+{A1λ1Q∼Q∼5cX+A2λ2U∼cU∼5X′+A3λ3D∼cD∼5X′+BXμXXX′+h.c.}.
The SU(2)_{L} singlets Φ_{L}, φ_{L}, Φ_{B}, φ_{B} and the SU(2)_{L} doublets H_{u}, H_{d} are:
ΦL=12(υL+ΦL0+iPL0),φL=12(υ¯L+φL0+iP¯L0),ΦB=12(υB+ΦB0+iPB0),φB=12(υ¯B+φB0+iP¯B0),Hu=(Hu+12(υu+Hu0+iPu0)),Hd=(12(υd+Hd0+iPd0)Hd−).
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 SU(2)_{L} ⊗ U(1)_{Y} ⊗ U(1)_{B} ⊗ U(1)_{L} breaks down to the electromagnetic symmetry U(1)_{e}.
Mass matrices for some BLMSSM particles
Lepneutralinos are made up of λ_{L} (the superpartner of the new lepton boson), and ψ_{ΦL} and ψ_{ΦL} (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_{ΦL}, ψ_{φL}) [42–45]. χLi0(i=1,2,3) are mass eigenstates of lepneutralinos. The masses of the three lepneutralinos are obtained from diagonalizing M_{LN} by Z_{NL}:
MLN=(2ML2vLgL−2v¯LgL2vLgL0−μL−2v¯LgL−μL0),iλL=ZNL1ikLi0,ψΦL=ZNL2ikLi0,ψφL=ZNL3ikLi0,χLi0=(kLi0k¯Li0).
The slepton mass squared matrix becomes
((ℳL∼2)LL(ℳL∼2)LR(ℳL∼2)LR†(ℳL∼2)RR),
which is diagonalized by the matrix ZL∼. (ℳL∼2)LL,(ℳL∼2)LR and (ℳL∼2)RR are:
(ℳL∼2)LL=(g12−g22)(vd2−vu2)8δIJ+gL2(v¯L2−vL2)δIJ+mlI2δIJ+(mL∼2)IJ,(ℳL∼2)LR=μ*vu2(Yl)IJ−vu2(A′l)IJ+vd2(Al)IJ,(ℳL∼2)RR=g12(vu2−vd2)4δIJ−gL2(v¯L2−vL2)δIJ+mlI2δIJ+(mR∼2)IJ.
The mass squared matrix of sneutrino ℳn∼ with n∼T=(ν∼,N∼c) reads [46]
(ℳn∼2(ν∼I*ν∼J)ℳn∼2(ν∼IN∼Jc)(ℳn∼2(ν∼IN∼Jc))†ℳn∼2(N∼Ic*N∼Jc)).ℳn∼2(ν∼I*ν∼J), ℳn∼2(ν∼IN∼Jc) and ℳn∼2(N∼Ic*N∼Jc) are:
ℳn∼2(ν∼I*ν∼J)=g12+g228(vd2−vu2)δIJ+gL2(v¯L2−vL2)δIJ+vu22(Yν†Yν)IJ+(mL∼2)IJ,ℳn∼2(ν∼IN∼Jc)=μ*vd2(Yν)IJ−vuv¯L(Yν†λNc)IJ+vu2(AN)IJ(Yν)IJ,ℳn∼2(N∼Ic*N∼Jc)=−gL2(v¯L2−vL2)δIJ+vu22(Yν†Yν)IJ+2v¯L2(λNc†λNc)IJ+(mN∼c2)IJ+μLvL2(λNc)IJ−v¯L2(ANc)IJ(λNc)IJ.
Then the masses of the sneutrinos are obtained by using the formula Zν∼†ℳn∼2Zν∼=diag(mν∼12, mν∼22, mν∼32, mν∼42, mν∼52, mν∼62).
The up scalar quark mass squared matrix in the BLMSSM is given by
((ℳU∼2)LL(ℳU∼2)LR(ℳU∼2)LR†(ℳU∼2)RR),
which is diagonalized by the matrix ZU∼. (ℳU∼2)LL,(ℳU∼2)LR and (ℳU∼2)RR are:
(MU∼2)LL=−e2(vd2−vu2)(1−4cW2)24sW2cW2+vu2Yu22+(KmQ∼2K†)T+gB26(vB2−v¯B2),(MU∼2)RR=e2(vd2−vu2)6cW2+vu2Yu22+mU∼2−gB26(vB2−v¯B2),(MU∼2)LR=−12(vd(A′u+Yuμ*)+vuAu).
The down scalar quark mass squared matrix in the BLMSSM is given by
((ℳD∼2)LL(ℳD∼2)LR(ℳD∼2)LR†(ℳD∼2)RR),
which is diagonalized by the matrix ZD∼. (ℳU∼2)LL,(ℳU∼2)LR and (ℳU∼2)RR are:
(MD∼2)LL=−e2(vd2−vu2)(1+2cW2)24sW2cW2+vd2Yd22+(mQ∼2)T+gB26(vB2−v¯B2),(MD∼2)RR=−e2(vd2−vu2)12cW2+vd2Yd22+mD∼2−gB26(vB2−v¯B2),(MD∼2)LR=12(vu(−Ad′+Ydμ*)+vdAd).
In the basis (ψνLI,ψNRcI), the neutrino mass mixing matrix is diagonalized by Zν [46]:
ZνT(0vu2(Yν)IJvu2(YνT)IJv¯L2(λNc)IJ)Zν=diag(mνα),α=1…6.ψνLI=ZνIαkNα0,ψNRcI=Zν(I+3)αkNα0,να=(kNα0k¯Nα0).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 superfields N∼c, we deduce the corrections to the couplings in the MSSM. The couplings for W-l-ν and W-L∼-ν∼ read
ℒWlν=−e2sWWμ+∑I=13∑α=16ZνIα*ν¯αγμPLlI,ℒWL∼ν∼=−ie2sWWμ−∑I=13∑i,α=16(ZL∼IiZν∼Iα)(L∼i+(∂μ→−∂μ←)ν∼α).
From the interactions of gauge and matter multiplets ig2Tija(λaψjAi*−λ¯aψ¯iAj), the l-χL0-L∼ coupling is deduced here:
ℒlχL0L∼=2gLχ¯Lj0(ZNL1jZL∼IiPL−ZNL1j*ZL∼(I+3)iPR)lIL∼i++h.c.
The ν−χL0-ν∼ coupling is
ℒνχL0ν∼=[2gLZNL1iZνIαZν∼Jj*δIJ−(ZNL3i(λNcIJ+λNcJI)+2gLZNL1iδIJ)×Zν(I+3)αZν∼(J+3)j*]χ¯Li0PLναν∼j*+h.c.
We also obtain the χ±−l−ν∼ coupling and the χ±−L∼−ν coupling:
ℒχ±lν∼=−∑I,J=13∑α=16χ¯j−(YlIJZ−2j*(Zν∼Iα)*PR+[esWZ+1j(Zν∼Iα)*+YνIJZ+2j(Zν∼(I+3)α)*]PL)lJν∼α*+h.c.ℒχ±L∼ν=−∑I,J=13∑i=12∑j,α=16χ¯i+(YνIJZ+2i*ZL∼IjZν(J+3)α*PR+[esWZ−1iZL∼Ij+YlIJZ−2iZL∼(I+3)j]ZνJαPL)ναL∼j++h.c.
The χ0−ν∼−ν coupling in the BLMSSM becomes
ℒχ0ν∼ν=[ZνIαZν∼Jj*e2sWcW(ZN1isW−ZN2icW)+YνIJ2ZN4i(ZνIαZν∼(J+3)j*+Zν(I+3)αZν∼Jj*)]×χ¯i0PLναν∼j*+h.c.
All the other couplings used are consistent with the MSSM.
FormulaeObservables
The observable R_{D(*)} is defined as
RD(*)=ℬτD(*)ℬlD(*)=ℬ(B¯→D(*)τν¯τ)ℬ(B¯→D(*)lν¯l).ℬℓD(*), the branching fraction, is given by [4]:
ℬℓD(*)=∫N|pD(*)|(2aℓD(*)+23cℓD(*))dq2,
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 [mℓ2,(MB−MD(*))2]. N, the normalisation factor, is given by
N=τBGF2|Vcb|2q2256π3MB2(1−mℓ2q2)2.
Here τ_{B} is the lifetime of the B–meson. GF=2e2/8mW2sW2 is the Fermi coupling constant. |p_{D(*)}|, the absolute value of the D^{(*)}–meson momentum, is given by
|pD(*)|=(MB2)2+(MD(*)2)2+(q2)2−2(MB2MD(*)2+MD(*)2q2+q2MB2)2MB.
The expressions for aℓD and cℓD are [4]:
aℓD=8{MB2|pD|2q2(|CVLℓ|2+|CVRℓ|2)F+2+(MB2−MD2)24(mb−mc)2(|CSLℓ|2+|CSRℓ|2)F02+mℓ[(MB2−MD2)22q2(mb−mc)(ℛ(CSLℓCVLℓ*)+ℛ(CSRℓCVRℓ*))F02+4MB2|pD|2q2(MB+MD)(ℛ(CTLℓCVLℓ*)+ℛ(CTRℓCVRℓ*))F+FT]+mℓ2[(MB2−MD2)24q4(|CVLℓ|2+|CVRℓ|2)F02+4|pD|2MB2q2(MB+MD)2(|CTLℓ|2+|CTRℓ|2)FT2]},cℓD=8{4MB2|pD|2(MB+MD)2(|CTLℓ|2+|CTRℓ|2)FT2−MB2|pD|2q2(|CVLℓ|2+|CVRℓ|2)F+2+mℓ2[|pD|2MB2q4(|CVLℓ|2+|CVRℓ|2)F+2−4|pD|2MB2(MB+MD)2q2(|CTLℓ|2+|CTRℓ|2)FT2]}.
The full expressions for aℓD*, cℓD* 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
ℒeffb→cℓν¯ℓ=2GFVcb(CVLℓOVLℓ+CVRℓOVRℓ+CALℓOALℓ+CARℓOARℓ+CSLℓOSLℓ+CSRℓOSRℓ+CPLℓOPLℓ+CPRℓOPRℓ+CTLℓOTLℓ+CTRℓOTRℓ),
where V_{cb} = 0.04, and the full set of operators is [48]:
OVLℓ=[c¯γμb][ℓ¯γμPLνℓ],OVRℓ=[c¯γμb][ℓ¯γμPRνℓ],OALℓ=[c¯γμγ5b][ℓ¯γμPLνℓ]OARℓ=[c¯γμγ5b][ℓ¯γμPRνℓ],OSLℓ=[c¯b][ℓ¯PLνℓ],OSRℓ=[c¯b][ℓ¯PRνℓ],OPLℓ=[c¯γ5b][ℓ¯PLνℓ],OPRℓ=[c¯γ5b][ℓ¯PRνℓ],OTLℓ=[c¯σμνb][ℓ¯σμνPLνℓ],OTRℓ=[c¯σμνb][ℓ¯σμνPRνℓ].
In the SM, CVLℓ=−CALℓ=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.
The penguin-type Feynman diagrams that can correct R_{D(*)} in the BLMSSM.
The box-type Feynman diagrams that can correct R_{D(*)} in the BLMSSM.
Penguin-type Feynman diagramsThe 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 χL0 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:
CVL(a)ℓ=[∑β,s=16∑j=13ℬ1ℓsjA2βℓjA3βsA4mW2164π2(ΔUV+1−F21(xχLj0,xL∼s,xν∼β))]/(2GFVcb)=gL216π2ΔUV+finiteterms,CAL(a)ℓ=−CVL(a)ℓ.
Here, we use the unitary characteristics of the rotation matrices. In Eq. (28),
ℬ1ℓsj=2gLZNL1j*ZL∼ℓs*,A2βℓj=∑I=13(2gLZNL1jZνIℓZν∼Iβ−(2λNcIIZNL3j+2gLZNL1j)×Zν(I+3)ℓZν∼(I+3)β*),A3βs=−∑I=13(e2sWZν∼IβZL∼Is),A4=−e2sWVcb.ΔUV=1/ϵ+ln(4πκ2/ΛNP2)−γE,1ε 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 mi2ΛNP2, 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) are CVL(a)ℓ(IF)=gL216π2ΔUV and CAL(a)ℓ(IF)=−CVL(a)ℓ(IF). Similarly, the infinite terms of the WCs of the following three diagrams (Fig. 1(b), (c) and (d)) are given as follows:
CVL(b)ℓ(IF)=e264π2sW2cW2(1−2cW2)ΔUV,CAL(b)ℓ(IF)=−CVL(b)ℓ,CVL(c)ℓ(IF)=e232π2sW2ΔUV,CAL(c)ℓ(IF)=−CVL(c)ℓ,CVL(d)ℓ(IF)=(e232π2sW2+(Ylℓ)232π2)ΔUV,CAL(d)ℓ(IF)=−CVL(d)ℓ.
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 lI¯νIW− are necessary. The counter term formula for the vertex lI¯νIW− is:
δVlI¯νIW−μ(OS)=−ie22sW(δmZ2mZ2−δmZ2−δmW2mZ2−mW2+2δe+δZLlI+δZLνI+δZWW)γμPL,
Following the method in Refs. [49–51], 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 δmZ2:
δmZ2mZ2=[e232π2sW2cW2(1−2sW2)2+e216π2cW2]ΔUV−e2sW2cW2{14∑j=16F1(xν∼j,xν∼j)+∑α,β=16|(G)αβ|2F1(xL∼α,xL∼β)}.
Through calculating the W boson self-energy diagram (with sneutrinos and sleptons in the loop), we can obtain:
δmW2mZ2=e2cW232π2sW2ΔUV−e2cW22sW2∑i=16∑α=16|(η)iα|2F1(xν∼α,xL∼i),δZWW=−e232π2sW2ΔUV+e22sw2∑i=16∑α=16|(η)iα|2F1(xν∼α,xL∼i).
The renormalization constant of charge is obtained from virtual sleptons:
δe=e216π2ΔUV−12e2∑i=16F1(xL∼i,xL∼i).
In the same way, we give the renormalization constants δZLνI and δZLlI for neutrinos and leptons respectively:
δZLνI=−e232π2sW2(12cW2+1+(sWYlIe)2+(2sWgLe)2)ΔUV−e22sW2cW2∑i=14∑α=16|(ζI)αi|2F2(xν∼α,xχi0)−e2sW2∑i=12∑α=16|(PI)αi|2F2(xL∼α,xχi−)−e22sW2cw2∑i=13∑α=16|(ζ′I)αi|2F2(xν∼α,xχLi0),δZLlI=−e232π2sW2(12cW2+1+(sWYlIe)2+(2sWgLe)2)ΔUV−e2sW2∑α=16∑i=12{|(ℬi)Iα|2F2+xeI[|(ℬi)Iα|2+|(Ai)Iα|2+2Re[(Ai†)Iα(ℬi)Iα]]F3}(xν∼α,xχi−)−e2∑j=14∑i=16{xeI[|(DI)ij|22sW2+2sWRe[(CI)ij†(DI)ij]+|(CI)ij|2]F3+12sW2|(DI)ij|2F2}(xL∼i,xχj0)−e2∑j=13∑i=16{xeI[|(D′I)ij|22sW2+2sWRe[(C′I)ij†(D′I)ij]+|(C′I)ij|2]F3+12sW2|(D′I)ij|2F2}(xL∼i,xχLj0),
where the vertex couplings are given by
(Ai)Iα=YlIsWeZ−2i*Zν∼Iα*,(ℬi)Iα=Z+1iZν∼Iα*,(η)iα=Zν∼JαZL∼Ji,(PI)αi=−YlJsWeZνJIZL∼(J+3)α*Z−2i*−ZνJIZL∼Jα*Z−1i*,(ζI)αi=Zν∼Jα*ZνJI(ZN1isW−ZN2icW),(CI)ij=−2cWZL∼(I+3)iZN1j*+YlIeZL∼IiZN3j*,(DI)ij=ZL∼IicW(ZN1jsW+ZN2jcW)+2sWYlIeZL∼(I+3)iZN3j,(G)αβ=12ZL∼JαZL∼Jβ*−sW2δαβ,(C′I)ij=−2gLeZNL1j*ZL(I+3)i,(D′I)ij=2gLsWeZNL1jZLIi,(ζ′I)αi=2sWcWe(2gLZNL1iZνJIZν∼Jα*−[2λNcJJZNL3i+2gLZNL1i]Zν(J+3)IZν∼(J+3)α*).
The functions F_{1}, F_{2}, and F_{3} are as follows:
F1(x1,x2)=1288π2(x1−x2)3[6(x1−3x2)x12lnx1+6(3x1−x2)x22lnx2−(x1−x2)(5x12−22x1x2+5x22)],F2(x1,x2)=(2x2−x1)(x2−x1+x1lnx1)−x22lnx232π2(x1−x2)2,F3(x1,x2)=x12+2x1x2(lnx2−lnx1)−x2232π2(x1−x2)3.
If x_{1} = x_{2}, they simplify to
F1(x1,x2)=lnx148π2,F2(x1,x2)=−lnx132π2+164π2,F3(x1,x2)=196π2x1.
Now, the WCs of Fig. 1(counter) read:
CVLℓ(counter)=12(δmZ2mZ2−δmZ2−δmW2mZ2−mW2+2δe+δZLlℓ+δZLνℓ+δZWW),CALℓ(counter)=−CVLℓ(counter).
The corresponding CVL(counter)ℓ(IF) and CAL(counter)ℓ(IF) are:
CVL(counter)ℓ(IF)=12{[e232π2sW2cW2(1−2sW2)2+e216π2cW2]ΔUV−[(e232π2sW2cW2(1−2sW2)2+e216π2cW2)ΔUV−e2cW232π2sW2ΔUV]mZ2mZ2−mW2+e28π2ΔUV−e232π2sW2ΔUV+−e216π2sW2[12cW2+1+(sWYlℓe)2+(2sWgLe)2]ΔUV},CAL(counter)ℓ(IF)=−CVL(counter)ℓ(IF).
It is easy to test that the infinite terms in the sum of Fig. 1(a),(b),(c),(d) and (counter) vanish: CVLℓ(IF)=CVL(a)ℓ(IF)+CVL(b)ℓ(IF)+CVL(c)ℓ(IF)+CVL(d)ℓ(IF)+CVL(counter)ℓ(IF)=0, similarly, CALℓ(IF)=0. 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.
Box-type Feynman diagrams
Taking Fig. 2(e) as an example, the corresponding WCs are given as follows:
CVL(e)ℓ=∑β,s=16∑i=12∑j=14[ℬ1βℓiℬ2isA3sjA4βℓjmχj0mχiΛNP4164π2F11(xν∼β,xχj0,xU∼s,xχi)−ℬ1βℓiA2isℬ3sjA4βℓjΛNP21128π2F21(xν∼β,xχj0,xU∼s,xχi)]/(2GFVcb),CAL(e)ℓ=∑β,s=16∑i=12∑j=14[ℬ1βℓiℬ2isA3sjA4βℓjmχj0mχiΛNP4164π2F11(xν∼β,xχj0,xU∼s,xχi)+ℬ1βℓiA2isℬ3sjA4βℓjΛNP21128π2F21(xν∼β,xχj0,xU∼s,xχi)]/(2GFVcb),CSL(e)ℓ=∑β,s=16∑i=12∑j=14[A1βℓiA2isA3sjA4βℓjmχj0mχiΛNP4164π2F11(xν∼β,xχj0,xU∼s,xχi)+A1βℓiℬ2isℬ3sjA4βℓjΛNP2164π2F21(xν∼β,xχj0,xU∼s,xχi)]/(2GFVcb),CTL(e)ℓ=[∑β,s=16∑i=12∑j=14A1βℓiA2isA3sjA4βℓjmχj0mχiΛNP41128π2F11(xν∼β,xχj0,xU∼s,xχi)]/(2GFVcb),CPL(e)ℓ=∑β,s=16∑i=12∑j=14[−(A1βℓiA2is−ℬ1βℓiℬ2is)A3sjA4βℓjmχj0mχiΛNP41128π2F11(xν∼β,xχj0,xU∼s,xχi)+A1βℓiℬ2isℬ3sjA4βℓjΛNP2164π2F21(xν∼β,xχj0,xU∼s,xχi)]/(2GFVcb),CPR(e)ℓ=[∑β,s=16∑i=12∑j=14−(A1βℓiA2is+ℬ1βℓiℬ2is)A3sjA4βℓjmχj0mχiΛNP41128π2F11(xν∼β,xχj0,xU∼s,xχi)]/(2GFVcb),
and all the other WCs in Eq. (26) vanish. In Eqs. (41–46),
A1βℓi=−YlℓZ−2iZν∼ℓβ,A2is=∑I=13(−esWZU∼Is*Z+1i+YuIZU∼(I+3)s*Z+2i)VI3,A3sj=22e3cWZU∼5sZN1j−Yu2ZU∼2sZN4j,A4βℓj=∑I=13{Zν∼Iβ*ZνIℓe2sWcW(ZN1jsW−ZN2jcW)+∑J=13(YνIJ2ZN4j(ZνIℓZν∼(J+3)β*+Zν(I+3)ℓZν∼Jβ*))},ℬ1βℓi=−esWZ+1i*Zν∼ℓβ−∑I=13(YνℓIZ+2i*Zν∼(I+3)β),ℬ2is=−∑I=13(Yd3ZU∼Is*Z−2i*)VI3,ℬ3sj=−e2sWcWZU∼2s(13ZN1j*sW+ZN2j*cW)−Yu2ZU∼5sZN4j*.
The formulae F_{11}(x_{1},x_{2},x_{3},x_{4}) and F_{21}(x_{1},x_{2},x_{3},x_{4}) are given in the Appendix.
Numerical results
For the numerical discussion, the parameters used are:
where i = 1 … 3. If not otherwise noted, the non-diagonal elements of the parameters used should be zero. The Yukawa couplings of neutrinos YνIJ 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_{χ0} ≃ (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.
We now focus on the effects of parameters mL∼2 (or mR∼2) on R_{D(*)}. First, we set the parameters as follows: tanβ = 10, m_{2} = μ = 1200 GeV, g_{L} = 0.1, and (mL∼2)33=(mR∼2)33=3×108GeV2.
To study the impacts of these parameters on R_{D(*)}, we use the parameters (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22=3×10ξ GeV^{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*}.
R_{D} (left) and R_{D*} (right) versus ξ with (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22=3×10ξ GeV^{2}, and (mL∼2)33=(mR∼2)33=3×108 GeV^{2}
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 (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22. To solve the problem of R_{D(*)}, we should violate lepton flavour symmetry for generations 1(2) and generation 3. Therefore, we suppose (mL∼2)33=(mR∼2)33≠(mL∼2)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: (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22<(mL∼2)33=(mR∼2)33. We need to select a set of reasonable parameters, and finally choose: (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22=5.5×105GeV2, and (mL∼2)33=(mR∼2)33=3×108GeV2. 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(*)}.
Effect of parameter <italic>g<sub>L</sub></italic> on <italic>R</italic><sub><italic>D</italic><sup>(*)</sup></sub>
Based on the above analysis, we use the following parameters:
tan β = 10, m_{2} = μ = 1200 GeV, (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22=5.5×105GeV2, and (mL∼2)33=(mR∼2)33=3×108GeV2.
g_{L} is the coupling constant of the vertexes lχL0L∼ and νχL0ν∼. 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*}.
R_{D} (left) and R_{D*} (right) versus g_{L}.
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.
Effects of parameters tan<italic>β</italic>, <italic>m</italic><sub>2</sub> and <italic>μ</italic> on <italic>R</italic><sub><italic>D</italic><sup>(*)</sup></sub>
We also research the effects of parameters tanβ, m_{2} and μ on R_{D(*)}. With the supposition g_{L} = 0.45, (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22=5.5×105 GeV^{2}, (mL∼2)33=(mR∼2)33=3×108 GeV^{2}, and m_{2} = μ = M_{ξ}, we scan the parameters of M_{ξ} versus tanβ in Fig. 5.
The allowed parameters in the plane of M_{ξ} versus tanβ with g_{L} = 0.45, (mL∼2)11=(mR∼2)11=(mL∼2)22=(mR∼2)22=5.5×105 GeV^{2}, (mL∼2)33=(mR∼2)33=3×108 GeV^{2}.
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 χL0 are new particles in the BLMSSM, and the Feynman diagram with χL0 can give new contributions to R_{D(*)}.
We find that the parameters (mL∼2)ii and (mR∼2)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 (mL∼2)ii and (mR∼2)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.
Appendix
The formulae for the one-loop integral are:
(2πκ)4−Diπ2∫dDp1(p2−m12)(p2−m22)(p2−m32)=−1ΛNP2F11(x1,x2,x3),(2πκ)4−Diπ2∫dDpp2(p2−m12)(p2−m22)(p2−m32)=1ε−γE+ln(4πκ2/ΛNP2)+1−F21(x1,x2,x3),(2πκ)4−Diπ2∫dDp1(p2−m12)(p2−m22)(p2−m32)(p2−m42)=−1ΛNP4F11(x1,x2,x3,x4),(2πκ)4−Diπ2∫dDpp2(p2−m12)(p2−m22)(p2−m32)(p2−m42)=−1ΛNP2F21(x1,x2,x3,x4),F11(x1,x2,x3)=x1lnx1(x1−x2)(x1−x3)+x2lnx2(x2−x1)(x2−x3)+x3lnx3(x3−x1)(x3−x2),F21(x1,x2,x3)=x12lnx1(x1−x2)(x1−x3)+x22lnx2(x2−x1)(x2−x3)+x32lnx3(x3−x1)(x3−x2),F11(x1,x2,x3,x4)=x1lnx1(x1−x2)(x1−x3)(x1−x4)+x2lnx2(x2−x1)(x2−x3)(x2−x4)+x3lnx3(x3−x1)(x3−x2)(x3−x4)+x4lnx4(x4−x1)(x4−x2)(x4−x3),F21(x1,x2,x3,x4)=x12lnx1(x1−x2)(x1−x3)(x1−x4)+x22lnx2(x2−x1)(x2−x3)(x2−x4)+x32lnx3(x3−x1)(x3−x2)(x3−x4)+x42lnx4(x4−x1)(x4−x2)(x4−x3).