Phenomenology of \Lambda_{b}\to\Lambda_{c}\tau\bar{\nu}_{\tau} using lattice QCD calculations

# Phenomenology of Λb→Λcτ¯ντ using lattice QCD calculations

## Abstract

In a recent paper we studied the effect of new-physics operators with different Lorentz structures on the semileptonic decay. This decay is of interest in light of the puzzle in the semileptonic decays. In this work we add tensor operators to extend our previous results and consider both model-independent new physics (NP) and specific classes of models proposed to address the puzzle. We show that a measurement of can strongly constrain the NP parameters of models discussed for the puzzle. We use form factors from lattice QCD to calculate all observables. The tensor form factors had not previously been determined in lattice QCD, and we present new lattice results for these form factors here.

UMISS-HEP-2017-01

RBRC-1228

## 1 Introduction

A major part of particle physics research is focused on searching for physics beyond the standard model (SM). In the flavor sector a key property of the SM gauge interactions is that they are lepton flavor universal. Evidence for violation of this property would be a clear sign of new physics (NP) beyond the SM. In the search for NP, the second and third generation quarks and leptons are quite special because they are comparatively heavier and are expected to be relatively more sensitive to NP. As an example, in certain versions of the two Higgs doublet models (2HDM) the couplings of the new Higgs bosons are proportional to the masses and so NP effects are more pronounced for the heavier generations. Moreover, the constraints on new physics, especially involving the third generation leptons and quarks, are somewhat weaker allowing for larger new physics effects.

The charged-current decays have been measured by the BaBar Lees:2013uzd (), Belle Huschle:2015rga (); Abdesselam:2016cgx () and LHCb Aaij:2015yra () Collaborations. It is found that the values of the ratios , where , considerably exceed their SM predictions.

This ratio of branching fractions has certain advantages over the absolute branching fraction measurement of decays, as this is relatively less sensitive to form factor variations and several systematic uncertainties, such as those on the experimental efficiency, as well as the dependence on the value of , cancel in the ratio.

There are lattice QCD predictions for the ratio in the Standard Model Bailey:2012jg (); Bailey:2015rga (); Na:2015kha () that are in good agreement with one another,

 R(D)SM = 0.299±0.011[FNAL/MILC], (1) R(D)SM = 0.300±0.008[HPQCD]. (2)

These values are also in good agreement with the phenomenological prediction Sakaki:2013bfa ()

 R(D)SM = 0.305±0.012, (3)

which is based on form factors extracted from experimental data for the differential decay rates using heavy-quark effective theory. See also Ref. Wang:2017jow () for a recent analysis of form factors using light-cone sum rules.

A calculation of is not yet available from lattice QCD. The phenomenological prediction using form factors extracted from experimental data is Fajfer:2012vx ()

 R(D∗)SM = 0.252±0.003. (4)

The averages of and measurements, evaluated by the Heavy-Flavor Averaging Group, are HFAGWinter2016 ()

 R(D)exp = 0.397±0.040±0.028, (5) R(D∗)exp = 0.316±0.016±0.010, (6)

where the first uncertainty is statistical and the second is systematic. and exceed the SM predictions by 3.3 and 1.9, respectively. The combined analysis of and , taking into account measurement correlations, finds that the deviation is 4 from the SM prediction HFAGWinter2016 (); Ricciardi:2016pmh ().

Since lattice QCD results are not yet available for the form factors at nonzero recoil and for the tensor form factor, we use the phenomenological form factors from Ref. Sakaki:2013bfa () for both channels in our analysis. For , we have compared the phenomenological results for and to the results obtained from a joint BGL -expansion fit DeTar () to the FNAL/MILC lattice QCD results Bailey:2015rga () and Babar Aubert:2009ac () and Belle experimental data Glattauer:2015teq (), and we found that the differences between both sets of form factors are below 5% across the entire kinematic range. The constraints on the new-physics couplings from the experimental measurement of obtained with both sets of form factors are practically identical.

We also construct the ratios of the experimental results (5) and (6) to the phenomenological SM predictions (3) and (4):

 RRatioD = R(D)expR(D)SM=1.30±0.17, (7) RRatioD∗ = R(D∗)expR(D∗)SM=1.25±0.08. (8)

There have been numerous analyses examining NP explanations of the measurements Fajfer:2012jt (); Crivellin:2012ye (); Datta:2012qk (); Becirevic:2012jf (); Deshpande:2012rr (); Celis:2012dk (); Choudhury:2012hn (); Tanaka:2012nw (); Ko:2012sv (); Fan:2013qz (); Biancofiore:2013ki (); Celis:2013jha (); Duraisamy:2013kcw (); Dorsner:2013tla (); Sakaki:2013bfa (); Sakaki:2014sea (); Bhattacharya:2014wla (). The new physics involves new charged-current interactions. In the neutral-current sector, data from decays also hint at lepton flavor non-universality – the so called puzzle: the LHCb Collaboration has found a 2.6 deviation from the SM prediction for the ratio in the dilepton invariant mass-squared range 1 GeV GeV Aaij:2014ora (). There are also other, not necessarily lepton-flavor non-universal anomalies in decays, most significantly in the angular observable Aaij:2013qta (); Aaij:2015oid (). Global fits of the experimental data prefer a negative shift in one of the Wilson coefficients, Blake:2016olu (). Common explanations of the and anomalies have been proposed in Refs. Bhattacharya:2014wla (); Calibbi:2015kma (); Greljo:2015mma (); Boucenna:2016wpr (); Bhattacharya:2016mcc (); Barbieri:2016las ().

The underlying quark level transition in the puzzle can be probed in both and decays. Recently, the decay was discussed in the standard model and with new physics in Ref. Gutsche:2015mxa (); Woloshyn:2014hka (); Shivashankara:2015cta (); Dutta:2015ueb (); Faustov:2016pal (); Li:2016pdv (); Celis:2016azn (). decays could be useful to confirm possible new physics in the puzzle and to point to the correct model of new physics.

In Ref. Shivashankara:2015cta () the following quantities were calculated within the SM and with various new physics operators:

 R(Λc) = B[Λb→Λcτ¯ντ]B[Λb→Λcℓ¯νℓ] (9)

and

 BΛc(q2) = dΓ[Λb→Λcτ¯ντ]dq2dΓ[Λb→Λcℓ¯νℓ]dq2, (10)

where represents or . In this paper we work with the ratio , defined as

 RRatioΛc = R(Λc)SM+NPR(Λc)SM. (11)

We also consider the forward-backward asymmetry

 AFB(q2) =∫10(d2Γ/dq2dcosθτ)dcosθτ−∫0−1(d2Γ/dq2dcosθτ)dcosθτdΓ/dq2, (12)

where is the angle between the momenta of the lepton and baryon in the dilepton rest frame.

This paper improves upon the earlier work Shivashankara:2015cta () in several ways:

• We add tensor interactions in the effective Lagrangian.

• Instead of a quark model, we use form factors from lattice QCD to calculate all observable. The vector and axial vector form factors are taken from Ref. Detmold:2015aaa (), and we extend the analysis of Ref. Detmold:2015aaa () to obtain lattice QCD results for the tensor form factors as well.

• In addition to and , we also calculate the forward-backward asymmetry (12) in the SM and with new physics.

• We include new constraints from the lifetime Li:2016vvp (); Alonso:2016oyd (); Celis:2016azn () in our analysis.

• In addition to analyzing the effects of individual new physics-couplings, we study specific models that introduce multiple new-physics couplings simultaneously. We consider a 2-Higgs doublet model, models with new vector bosons, and several leptoquark models.

The paper is organized in the following manner: In Sec. 2 we introduce the effective Lagrangian to parametrize the NP operators and give the expressions for the decay distribution in terms of helicity amplitudes. In Sec. 3, we present the new lattice QCD results for the tensor form factors. The model-independent phenomenological analysis of individual new-physics couplings is discussed in Sec. 4, while explicit models are considered in Sec. 5. We conclude in Sec. 6.

## 2 Formalism

### 2.1 Effective Hamiltonian

In the presence of NP, the effective Hamiltonian for the quark-level transition can be written in the form Chen:2005gr (); Bhattacharya:2011qm ()

 Heff = GFVcb√2{[¯cγμ(1−γ5)b+gL¯cγμ(1−γ5)b+gR¯cγμ(1+γ5)b]¯τγμ(1−γ5)ντ +[gS¯cb+gP¯cγ5b]¯τ(1−γ5)ντ+[gT¯cσμν(1−γ5)b]¯τσμν(1−γ5)ντ+h.c},

where is the Fermi constant, is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element, and we use . We consider that the above Hamiltonian is written at the energy scale.

If the effective interaction is written at the cut-ff scale then running down to the scale will generate new operators and new contributions, which have been discussed in Refs. Feruglio:2016gvd (); Feruglio:2017rjo (). These new contributions can strongly constrain models but to really calculate their true impacts we have to consider specific models where there might be cancellations between various terms.

The SM effective Hamiltonian corresponds to . In Eq. (LABEL:eq:Heff), we have assumed the neutrinos to be always left chiral. In general, with NP the neutrino associated with the lepton does not have to carry the same flavor. In the model-independent analysis of individual couplings (Sec. 4) we will not consider this possibility. Specific models will be discussed in Sec. 5.

### 2.2 Decay process

The process under consideration is

 Λb(pΛb)→τ−(pτ)+¯ντ(p¯ντ)+Λc(pΛc).

The differential decay rate for this process can be represented as Tanaka:2012nw ()

 dΓdq2dcosθτ = G2F|Vcb|22048π3(1−m2τq2)√Q+Q−m3Λb∑λΛc∑λτ|MλτλΛc|2, (14)

where

 q = pΛb−pΛc, (15) Q± = (mΛb±mΛc)2−q2, (16)

and the helicity amplitude is written as

 MλτλΛc = HSPλΛc,λτ=0+∑ληλHVAλΛc,λLλτλ+∑λ,λ′ηληλ′H(T)λΛbλΛc,λ,λ′Lλτλ,λ′. (17)

Here, (, ) indicate the helicity of the virtual vector boson (see Appendix A), and are the helicities of the baryon and lepton, respectively, and for and for .

The scalar-type, vector/axial-vector-type, and tensor-type hadronic helicity amplitudes are defined as

 HSPλΛc,λ=0 = HSλΛc,λ=0+HPλΛc,λ=0, HSλΛc,λ=0 = gS⟨Λc|¯cb|Λb⟩, HPλΛc,λ=0 = gP⟨Λc|¯cγ5b|Λb⟩, (18)
 HVAλΛc,λ = HVλΛc,λ−HAλΛc,λ, HVλΛc,λ = (1+gL+gR)ϵ∗μ(λ)⟨Λc|¯cγμb|Λb⟩, HAλΛc,λ = (1+gL−gR)ϵ∗μ(λ)⟨Λc|¯cγμγ5b|Λb⟩, (19)

and

 H(T)λΛbλΛc,λ,λ′ = H(T1)λΛbλΛc,λ,λ′−H(T2)λΛbλΛc,λ,λ′, H(T1)λΛbλΛc,λ,λ′ = gTϵ∗μ(λ)ϵ∗ν(λ′)⟨Λc|¯ciσμνb|Λb⟩, H(T2)λΛbλΛc,λ,λ′ = gTϵ∗μ(λ)ϵ∗ν(λ′)⟨Λc|¯ciσμνγ5b|Λb⟩. (20)

The leptonic amplitudes are defined as

 Lλτ = ⟨τ¯ντ|¯τ(1−γ5)ντ|0⟩, Lλτλ = ϵμ(λ)⟨τ¯ντ|¯τγμ(1−γ5)ντ|0⟩, Lλτλ,λ′ = (21)

Above, are the polarization vectors of the virtual vector boson (see Appendix A). The explicit expressions for the hadronic and leptonic helicity amplitudes are presented in the following.

In this paper, we use the helicity-based definition of the form factors, which was introduced in Feldmann:2011xf (). The matrix elements of the vector and axial vector currents can be written in terms of six helicity form factors , , , , , and as follows:

 ⟨Λc|¯cγμb|Λb⟩ = ¯uΛc[F0(q2)(mΛb−mΛc)qμq2 (22) +F+(q2)mΛb+mΛcQ+(pμΛb+pμΛc−(m2Λb−m2Λc)qμq2) +F⊥(q2)(γμ−2mΛcQ+pμΛb−2mΛbQ+pμΛc)]uΛb, = −¯uΛcγ5[G0(q2)(mΛb+mΛc)qμq2 (23) +G+(q2)mΛb−mΛcQ−(pμΛb+pμΛc−(m2Λb−m2Λc)qμq2) +G⊥(q2)(γμ+2mΛcQ−pμΛb−2mΛbQ−pμΛc)]uΛb.

The matrix elements of the scalar and pseudoscalar currents can be obtained from the vector and axial vector matrix elements using the equations of motion:

 ⟨Λc|¯cb|Λb⟩ = qμmb−mc⟨Λc|¯cγμb|Λb⟩ (24) = F0(q2)mΛb−mΛcmb−mc¯uΛcuΛb, ⟨Λc|¯cγ5b|Λb⟩ = qμmb+mc⟨Λc|¯cγμγ5b|Λb⟩ (25) = G0(q2)mΛb+mΛcmb+mc¯uΛcγ5uΛb.

In our numerical analysis, we use GeV, GeV Olive:2016xmw (). The matrix elements of the tensor currents can be written in terms of four form factors , , , ,

 ⟨Λc|¯ciσμνb|Λb⟩=¯uΛc[2h+(q2)pμΛbpνΛc−pνΛbpμΛcQ+ +h⊥(q2)(mΛb+mΛcq2(qμγν−qνγμ)−2(1q2+1Q+)(pμΛbpνΛc−pνΛbpμΛc)) +˜h+(q2)(iσμν−2Q−(mΛb(pμΛcγν−pνΛcγμ) −mΛc(pμΛbγν−pνΛbγμ)+pμΛbpνΛc−pνΛbpμΛc)) +˜h⊥(q2)mΛb−mΛcq2Q−((m2Λb−m2Λc−q2)(γμpνΛb−γνpμΛb) −(m2Λb−m2Λc+q2)(γμpνΛc−γνpμΛc)+2(mΛb−mΛc)(pμΛbpνΛc−pνΛbpμΛc))]uΛb.

The matrix elements of the current can be obtained from the above equation by using the identity

 σμνγ5=−i2ϵμναβσαβ. (27)

In the following, only the non-vanishing helicity amplitudes are given. The scalar and pseudo-scalar helicity amplitudes associated with the new physics scalar and pseudo-scalar interactions are

 HSP1/2,0 = F0gS√Q+mb−mc(mΛb−mΛc)−G0gP√Q−mb+mc(mΛb+mΛc), (28) HSP−1/2,0 = F0gS√Q+mb−mc(mΛb−mΛc)+G0gP√Q−mb+mc(mΛb+mΛc). (29)

The parity-related amplitudes are

 HSλΛc,λNP = HS−λΛc,−λNP, HPλΛc,λNP = −HP−λΛc,−λNP. (30)

For the vector and axial-vector helicity amplitudes, we find

 HVA1/2,0 = F+(1+gL+gR)√Q−√q2(mΛb+mΛc) (31) −G+(1+gL−gR)√Q+√q2(mΛb−mΛc), HVA1/2,+1 = −F⊥(1+gL+gR)√2Q−+G⊥(1+gL−gR)√2Q+, (32) HVA1/2,t = F0(1+gL+gR)√Q+√q2(mΛb−mΛc) (33) −G0(1+gL−gR)√Q−√q2(mΛb+mΛc), HVA−1/2,0 = F+(1+gL+gR)√Q−√q2(mΛb+mΛc) (34) +G+(1+gL−gR)√Q+√q2(mΛb−mΛc), HVA−1/2,−1 = −F⊥(1+gL+gR)√2Q−−G⊥(1+gL−gR)√2Q+, (35) HVA−1/2,t = F0(1+gL+gR)√Q+√q2(mΛb−mΛc) (36) +G0(1+gL−gR)√Q−√q2(mΛb+mΛc).

We also have the relations

 HVλΛc,λw = HV−λΛc,−λw, HAλΛc,λw = −HA−λΛc,−λw. (37)

The tensor helicity amplitudes are

 H(T)−1/2−1/2,t,0 = (38) H(T)+1/2+1/2,t,0 = (39) H(T)−1/2+1/2,t,+1 = −gT√2√q2[h⊥(mΛb+mΛc)√Q−+˜h⊥(mΛb−mΛc)√Q+], (40) H(T)+1/2−1/2,t,−1 = −gT√2√q2[h⊥(mΛb+mΛc)√Q−−˜h⊥(mΛb−mΛc)√Q+], (41) H(T)−1/2+1/2,0,+1 = −gT√2√q2[h⊥(mΛb+mΛc)√Q−+˜h⊥(mΛb−mΛc)√Q+], (42) H(T)+1/2−1/2,0,−1 = gT√2√q2[h⊥(mΛb+mΛc)√Q−−˜h⊥(mΛb−mΛc)√Q+], (43) H(T)+1/2+1/2,+1,−1 = (44) H(T)−1/2−1/2,+1,−1 = (45)

The other non-vanishing helicity amplitudes of tensor type are related to the above by

 H(T)λΛbλΛc,λ,λ′=−H(T)λΛbλΛc,λ′,λ. (46)

#### Leptonic helicity amplitudes

In the following, we define

 v=√1−m2τq2. (47)

The scalar and pseudoscalar leptonic helicity amplitudes are

 L+1/2= 2√q2v, (48) L−1/2= 0, (49)

the vector and axial-vector amplitudes are

 L+1/2±1 = ±√2mτvsin(θτ), (50) L+1/20 = −2mτvcos(θτ), (51) L+1/2t = 2mτv, (52) L−1/2±1 = √2q2v(1±cos(θτ)), (53) L−1/20 = 2√q2vsin(θτ), (54) L−1/2t = 0, (55)

and the tensor amplitudes are

 L+1/20,±1 = −√2q2vsin(θτ), (56) L+1/2±1,t = ∓√2q2vsin(θτ), (57) L+1/2t,0 = L+1/2+1,−1=−2√q2vcos(θτ), (58) L−1/20,±1 = ∓√2mτv(1±cos(θτ)), (59) L−1/2±1,t = −√2mτv(1±cos(θτ)), (60) L−1/2t,0 = L−1/2+1,−1=2mτvsin(θτ). (61)

Here we have the relation

 Lλτλ,λ′=−Lλτλ′,λ. (62)

The angle is defined as the angle between the momenta of the lepton and baryon in the dilepton rest frame.

### 2.3 Differential decay rate and forward-backward asymmetry

From the twofold decay distribution (14), we obtain the following expression for the differential decay rate by integrating over :

 dΓ(Λb→Λcτ¯ντ)dq2 = (63) +2(1+2m2τq2)AT4+3mτ√q2AVA−SP5+6mτ√q2AVA−T6],

where

 AVA1=