Effective theory approach to new physics in b\to u and b\to c leptonic and semileptonic decays.

# Effective theory approach to new physics in b→u and b→c leptonic and semileptonic decays.

Rupak Dutta    Anupama Bhol    Anjan K. Giri Indian Institute of Technology Hyderabad, Hyderabad 502205, India
###### Abstract

Recent measurements of exclusive and decays via the transition process differ from the standard model expectation and, if they persist in future experiments, will be a definite hint of the physics beyond the standard model. Similar hints of new physics have been observed in semileptonic transition processes as well. BABAR measures the ratio of branching fractions of to the corresponding , where represents either an electron or a muon, and finds discrepancy with the standard model expectation. In this context, we consider a most general effective Lagrangian for the and transition processes in the presence of new physics and perform a combined analysis of all the and semi-(leptonic) data to explore various new physics operators and their couplings. We consider various new physics scenarios and give predictions for the and decay branching fractions. We also study the effect of these new physics parameters on the ratio of the branching ratios of to the corresponding decays.

###### pacs:
14.40.Nd, 13.20.He, 13.20.-v

## I Introduction

Although, the standard model (SM) of particle physics can explain almost all the existing data to a very good precision, there are some unknowns which are beyond the scope of the SM. The latest discovery of a Higgs-like particle by CMS CMS () and ATLAS atlas () further confirms the validity of the SM as a low energy effective theory. There are two ways to look for evidence of new physics (NP): direct detection and indirect detection. The Large Hadron Collider (LHC), which is running successfully at CERN, in principle, has the ability to detect new particles that are not within the SM, while, on the other hand the LHCb experiment has the ability to perform indirect searches of NP effects, and since any NP will affect the SM observables, any discrepancy between measurements and the SM expectation will be an indirect evidence of NP beyond the SM.

Recent measurements of and leptonic and semileptonic decays differ from SM expectation. The measured branching ratio of  belle (); babar (); pdg () for the leptonic decay mode is larger than the SM expectation Charles (); ckm (); Bona (). However, the measured branching ratio of  babar-pilnu (); belle-pilnu (); HFAG () for the exclusive semileptonic decays is consistent with the SM prediction. The SM calculation, however, depends on the hadronic quantities such as meson decay constant and transition form factors and the Cabibbo-Kobayashi-Maskawa (CKM) element . The ratio of branching fractions defined by

 Rlπ=τB0τB−B(B−→τ−ν)B(B0→π+l−ν) (1)

is independent of the CKM matrix elements and is measured to be  Fajfer:2012jt (), and there is still more than discrepancy with the SM expectation. More recently, BABAR babar-dstnu () measured the ratio of branching fractions of to the corresponding and found discrepancy with the SM expectation Fajfer (). The measured ratios are

 RD=B(¯B→Dτ−¯ντ)B(¯B→Dl−¯νl)=0.440±0.058±0.042, RD∗=B(¯B→D∗τ−¯ντ)B(¯B→D∗l−¯νl)=0.332±0.024±0.018, (2)

where the first error is statistical and the second one is systematic. For definiteness, we consider , , , and throughout this paper. However, for brevity, we denote all these decay modes as , , , and , respectively.

Due to the large mass of the tau lepton, decay processes with a tau lepton in the final state are more sensitive to some new physics effects than processes with first two generation leptons. These NP, in principle, can enhance the decay rate for these helicity-suppressed decay modes quite significantly from the SM prediction. In Ref. Fajfer (), a thorough investigation of the lowest dimensional effective operators that leads to modifications in the decay amplitudes has been done. Possible NP effects on various observables have been explored. Among all the leptonic and semileptonic decays, decays with a tau lepton in the final state can be an excellent probe of new physics as these are sensitive to non-SM contributions arising from the violation of lepton flavor universality (LFU). A model-independent analysis to identify the new physics models has been explored in Ref. Fajfer:2012jt (). They also look at the possibility of a scalar leptoquark or a vector leptoquark, which can contribute to these decay processes at the tree level and obtain a bound of on the mass of the scalar electroweak triplet leptoquark. Model with composite quarks and leptons also modify these and semileptonic measurements Fajfer:2012jt (). The enhanced production of a tau lepton in leptonic and semileptonic decays can be explained by NP contribution with different models among which the minimal supersymmetric standard model (MSSM) is well motivated and is a charming candidate of NP whose Higgs sector contains the two Higgs doublet model (2HDMs). There are four types of 2HDMS such as type-I, type-II, lepton specific, and flipped Branco (). New particles such as charged Higgs bosons whose coupling is proportional to the masses of particles in the interaction can have significant effect on decay processes having a tau lepton in the final state. In Ref. Hou (), the author uses the 2HDM model of type-II for purely leptonic decays that are sensitive to charged Higgs boson at the tree level. This model, however, cannot explain all the semileptonic measurements simultaneously babar-dstnu (). A lot of studies have been done using the 2HDM of type II and type III models 2HDM (). However, none of the above 2HDMS can accommodate all the existing data on and semi-(leptonic) decays. Recently, a detailed study of a 2HDM of type III with MSSM-like Higgs potential and flavor-violation in the up sector in Ref. Crivellin () has demonstrated that this model can explain the deviation from the SM in , , and simultaneously and predict enhancement in the , , and the decay branching ratios. Also, in Refs. datta (); datta1 (), the authors have used a model independent way to analyse the and data by considering an effective theory for the processes in the presence of NP and obtain bounds on each NP parameter. They consider two different NP scenarios and see the effect of various NP couplings on different observables. This analysis, however, does not include the data. A similar analysis has been performed in Ref. fazio () considering a tensor operator in the effective weak Hamiltonian. Also, in Ref. Crivellin1 (), the author investigates the effects of an effective right handed charged currents on the determination of and from inclusive and exclusive decays. Moreover, the aligned two Higgs doublet model (A2HDM) Celis () and, more recently a non-universal left-right model He () have been explored in order to explain the discrepancies between the measurements and the SM prediction.

The recent measurements suggest the possibility of having new physics in the third generation leptons only. However, more experimental studies are needed to confirm the presence of NP. A thorough investigation of these decays will enable us to have significant constraints on NP scenarios. In this report, we use the most general effective Lagrangian for the semi-(leptonic) transition decays and do a combined analysis of and semi-(leptonic) decay processes where we use constraints from all the existing data related to these decays. It differs considerably from earlier treatments. First, we have introduced the right-handed neutrinos and their interactions for our analysis. Second, we have performed a combined analysis of all the and data. We illustrate four different scenarios of the new physics and the effects of each NP coupling on various observables are shown. We predict the branching ratio of and decay processes in all four different scenarios. We also consider the ratio of branching ratio of to the corresponding decay mode for our analysis.

The paper is organized as follows. In Sec. II, we start with a brief description of the effective Lagrangian for the processes and then present all the relevant formulae of the decay rates for various decay modes in the presence of various NP couplings. We then define several observables in , , and decays. The numerical prediction for various NP couplings and the effects of each NP coupling on various observables are presented in Sec. III. We also discuss the effects of these NP couplings on , , and the ratio for various NP scenarios in this section. We conclude with a summary of our results in Sec. IV. We report the details of the kinematics and various form factors in the Appendix.

## Ii Effective Lagrangian and decay amplitude

The most general effective Lagrangian for in presence of NP, where , can be written as Bhattacharya (); Cirigliano ()

 Leff = −g22M2WVq′b{(1+VL)¯lLγμνL¯q′LγμbL+VR¯lLγμνL¯q′RγμbR (3) +˜VL¯lRγμνR¯q′LγμbL+˜VR¯lRγμνR¯q′RγμbR +SL¯lRνL¯q′RbL+SR¯lRνL¯q′LbR +˜SL¯lLνR¯q′RbL+˜SR¯lLνR¯q′LbR +TL¯lRσμννL¯q′RσμνbL+˜TL¯lLσμννR¯q′LσμνbR}+h.c.,

where g is the weak coupling constant which can be related to the Fermi constant by the relation and is the CKM Matrix elements. The new physics couplings denoted by , , and involve left-handed neutrinos, whereas, the NP couplings denoted by , , and involve right-handed neutrinos. We assume the NP couplings to be real for our analysis. Again, the projection operators are and . We neglect the new physics effects coming from the tensor couplings and for our analysis. With this simplification, we obtain

 Leff = −GF√2Vq′b{GV¯lγμ(1−γ5)νl¯q′γμb−GA¯lγμ(1−γ5)νl¯q′γμγ5b (4) +GS¯l(1−γ5)νl¯q′b−GP¯l(1−γ5)νl¯q′γ5b +˜GV¯lγμ(1+γ5)νl¯q′γμb−˜GA¯lγμ(1+γ5)νl¯q′γμγ5b +˜GS¯l(1+γ5)νl¯q′b−˜GP¯l(1+γ5)νl¯q′γ5b}+h.c.,

where

 GV=1+VL+VR,GA=1+VL−VR, GS=SL+SR,GP=SL−SR, ˜GV=˜VL+˜VR,˜GA=˜VL−˜VR, ˜GS=˜SL+˜SR,˜GP=˜SL−˜SR. (5)

In the SM, and all other NP couplings are zero.

The expressions for , , and decay amplitude depends on nonperturbative hadronic matrix elements that can be expressed in terms of meson decay constants and transition form factors, where denotes a pseudoscalar meson and denotes a vector meson, respectively . The meson decay constant and transition form factors are defined as

 ⟨0|¯q′γμγ5b|B(p)⟩ = −ifBq′pμ, ⟨P(p′)|¯q′γμb|B(p)⟩ = F+(q2)[(p+p′)μ−m2B−m2Pq2qμ]+F0(q2)m2B−m2Pq2qμ, ⟨V(p′,ϵ∗)|¯q′γμb|B(p)⟩ = 2iV(q2)mB+mVεμνρσϵ∗νp′ρpσ, ⟨V(p′,ϵ∗)|¯q′γμγ5b|B(p)⟩ = 2mVA0(q2)ϵ∗.qq2qμ+(mB+mV)A1(q2)[ϵ∗μ−ϵ∗.qq2qμ] (6)

where is the momentum transfer. Again, from Lorentz invariance and parity, we obtain

 ⟨0|¯q′γμb|B(p)⟩=0, ⟨P(p′)|¯q′γμγ5b|B(p)⟩=0, ⟨V(p′,ϵ∗)|¯q′b|B(p)⟩=0. (7)

We use the equation of motion to find the scalar and pseudoscalar matrix elements. That is

 ⟨0|¯q′γ5b|B(p)⟩=im2Bmb(μ)+mq′(μ)fBq′, ⟨P(p′)|¯q′b|B(p)⟩=m2B−m2Pmb(μ)−mq′(μ)F0(q2), ⟨V(p′,ϵ∗)|¯q′γ5b|B(p)⟩=−2mVA0(q2)mb(μ)+mq′(μ)ϵ∗.q, (8)

where, for the form factors, we use the formulae and the input values reported in Ref. Khodjamirian (). Similarly, we follow Refs. Falk (); Caprini (); Sakaki () and employ heavy quark effective theory (HQET) to estimate the and form factors. All the relevant formulae and various input parameters pertinent to our analysis are presented in Appendix. B and in Appendix. C.

Using the effective Lagrangian of Eq. (4) in the presence of NP, the partial decay width of can be expressed as

 Γ(B→lν) = G2F|Vub|28πf2Bm2lmB(1−m2lm2B)2{[GA−m2Bml(mb(μ)+mu(μ))GP]2 (9) +[˜GA−m2Bml(mb(μ)+mu(μ))˜GP]2},

where, in the SM, we have and , so that

 Γ(B→lν)SM = G2F|Vub|28πf2Bm2lmB(1−m2lm2B)2. (10)

It is important to note that the right-handed neutrino couplings denoted by and appear in the decay width quadratically, whereas, the left-handed neutrino couplings denoted by and appear linearly in the decay rates. The linear dependence, arising due to the interference between the SM couplings and the NP couplings, is suppressed for the right-handed neutrino couplings as it is proportional to a small factor and hence is neglected. We now proceed to discuss the and decays.

We follow the helicity methods of Refs. Korner (); Kadeer () for the and semileptonic decays. The differential decay distribution can be written as

 dΓdq2dcosθl = G2F|Vq′b|2|→p(P,V)|29π3m2B(1−m2lq2)LμνHμν, (11)

where and are the usual leptonic and hadronic tensors, respectively. Here, is the angle between the meson and the lepton three momentum vector in the rest frame. The three momentum vector is defined as , where . The resulting differential decay distribution for in terms of the helicity amplitudes , , and is

 dΓdq2dcosθl = 2N|→pP|{H20sin2θl(G2V+˜G2V)+m2lq2[H0GVcosθl−(HtGV+√q2mlHSGS)]2 (12) +m2lq2[H0˜GVcosθl−(Ht˜GV+√q2mlHS˜GS)]2},

where

 N=G2F|Vq′b|2q2256π3m2B(1−m2lq2)2, H0=2mB|→pP|√q2F+(q2), Ht=m2B−m2P√q2F0(q2), HS=m2B−m2Pmb(μ)−mq′(μ)F0(q2). (13)

The details of the helicity amplitudes calculation are given in Appendix. A. We refer to Refs. Korner (); Kadeer () for all omitted details. We determine the differential decay rate by performing the integration, i.e,

 dΓPdq2 = 8N|→pP|3{H20(G2V+˜G2V)(1+m2l2q2) (14)

where, in the SM, and all other couplings are zero. One obtains

 (dΓPdq2)SM = (15)

Our formulae for the differential branching ratio in the presence of NP couplings in Eq. (12) and Eq. (14) differ slightly from those given in Ref. datta (). The term containing and is positive in Eq. (12) and Eq. (14), whereas, it is negative in Ref. datta (). Although, the SM formula is same, the numerical differences may not be negligible once the NP couplings and are introduced. It is worth mentioning that, for , the term containing can be safely ignored. However, same is not true for the decay mode as the mass of lepton is quite large and one cannot neglect the term from the decay amplitude. We assume that the NP affects the third generation lepton only.

Similarly, the differential decay distribution for in terms of the helicity amplitudes , , , , and is

 dΓdq2dcosθl = N|→pV|{2A20sin2θl(G2A+˜G2A)+(1+cos2θl)[A2∥(G2A+˜G2A)+A2⊥(G2V+˜G2V)] (16) +2m2lq2[{A0GAcosθl−(AtGA+√q2mlAPGP)}2 +{A0˜GAcosθl−(At˜GA+√q2mlAP˜GP)}2]},

where

 A0=12mV√q2[(m2B−m2V−q2)(mB+mV)A1(q2)−4M2B|→pV|2mB+mVA2(q2)], A∥=2(mB+mV)A1(q2)√2, A⊥=−4mBV(q2)|→pV|√2(mB+mV), At=2mB|→pV|A0(q2)√q2, AP=−2mB|→pV|A0(q2)(mb(μ)+mc(μ)). (17)

We perform the integration and obtain the differential decay rate , that is

 dΓVdq2 = (18)

where

 A2AV=A20G2A+A2∥G2A+A2⊥G2V, ˜A2AV=A20˜G2A+A2∥˜G2A+A2⊥˜G2V, AtP=AtGA+√q2mlAPGP, ˜AtP=At˜GA+√q2mlAP˜GP. (19)

In the SM, and all other NP couplings are zero. We obtain

 (dΓVdq2)SM = 8N|→pV|3{(A20+A2||+A2⊥)(1+m2l2q2)+3m2l2q2A2t}. (20)

We want to mention that our formulae for the differential decay width in Eq. (16) and Eq. (18) differ slightly from those reported in Ref. datta (). Our formulae, however, agree with those reported in Ref. Fajfer (). In Eq. (16), we have instead of reported in Ref. datta (). Again, note that our definition of , different from that of  datta (), leads to a sign discrepancy in . Depending on the NP couplings and , the numerical estimates might differ from Ref. datta ().

We define some physical observables such as differential branching ratio DBR, the ratio of branching fractions , and the forward-backward asymmetry .

 DBR(q2)=(dΓdq2)/Γtot,R(q2)=DBR(q2)(B→(P,V)τν)DBR(q2)(B→(P,V)lν), [AFB](P,V)(q2)=(∫0−1−∫10)dcosθldΓ(P,V)dq2dcosθldΓ(P,V)dq2. (21)

For decay mode, the forward-backward asymmetry in the presence of NP is

 APFB(q2) = 3m2l2q2H0GV[(HtGV+√q2mlHSGS)+(Ht˜GV+√q2mlHS˜GS)]H20(G2V+˜G2V)(1+m2l2q2)+3m2l2q2[(HtGV+√q2mlHSGS)2+(Ht˜GV+√q2mlHS˜GS)2],

where, in the SM, and all other couplings are zero. We obtain

 (APFB)SM(q2) = (23)

Similarly, for decay mode, in the presence of NP

 AVFB(q2) = 32A∥A⊥(GAGV−˜GA˜GV)+m2lq2A0GA[AtGA−√q2mlAPGP+At˜GA−√q2mlAP˜GP]A2AV+m2l2q2[A2AV+3A2tP]+˜A2AV+m2l2q2[˜A2AV+3˜A2tP].

In the SM, while all other NP couplings are zero. Thus we obtain

 (AVFB)SM(q2) = 32A∥A⊥+m2lq2A0At{(A20+A2||+A2⊥)(1+m2l2q2)+3m2l2q2A2t}. (25)

We see that, in the SM, for the light leptons , the forward-backward asymmetry is vanishingly small due to the term for the decay modes. However, for , the first term will contribute and we will get a nonzero value for the forward-backward asymmetry. Any non-zero value of the parameter for the decay modes will be a hint of NP in all generation leptons. We, however, ignore the NP effects in the case of . We strictly assume that only third generation leptons get modified due to NP couplings.

We wish to determine various NP effects in a model independent way. The theoretical uncertainties in the calculation of the decay branching fractions come from various input parameters. first, there are uncertainties associated with well-known input parameters such as quark masses, meson masses, and lifetime of the mesons. We ignore these uncertainties as these are not important for our analysis. Second, there are uncertainties that are associated with not so well-known hadronic input parameters such as form factors, decay constants, and the CKM elements. In order to realize the effect of the above-mentioned uncertainties on various observables, we use a random number generator and perform a random scan of all the allowed hadronic as well as the CKM elements. In our random scan of the theoretical parameter space, we vary all the hadronic inputs such as form factors, decay constants, and CKM elements within from their central values. In order to determine the allowed NP parameter space, we impose the experimental constraints coming from the measured ratio of branching fractions , , and simultaneously. This is to ensure that the resulting NP parameter space can simultaneously accommodate all the existing data on and leptonic and semileptonic decays. We impose the experimental constraints in such a way that we ignore those theoretical models that are not compatible within of the experimental constraints for the random scan.

## Iii Results and discussion

For definiteness, we summarize the input parameters for our numerical analysis. We use the following inputs from Ref. pdg ().

 mb=4.18GeV,mc=1.275GeV,mπ=0.13957GeV, mB−=5.27925GeV,mB0=5.27955GeV,mBc=6.277GeV, mD0=1.86486GeV,mD∗0=2.00698GeV,τB0=1.519×10−12Sec, τB−=1.641×10−12Sec,τBc=0.453×10−12Sec, (26)

where and denote the running and quark masses in scheme. We employ a renormalization scale for which the strong coupling constant . Using the two-loop expression for the running quark mass Buchalla:1995vs (), we find . Thus, the coefficients , , , and are defined at the scale . The error associated with the quark masses, meson masses, and the mean lifetime of mesons is not important and we ignore them in our analysis. In Table 1 and Table 2, we present the most important theoretical and experimental inputs with their uncertainties that are used for our random scan.