Phenomenology of Λ b → Λ c τ ν ¯ τ using lattice QCD calculations
UUMISS-HEP-2017-01RBRC-1228
Prepared for submission to JHEP
Phenomenology of Λ b → Λ c τ ¯ ν τ using lattice QCDcalculations Alakabha Datta, a,b
Saeed Kamali, a Stefan Meinel, c,d and Ahmed Rashed a,e a Department of Physics and Astronomy, University of Mississippi, Oxford, MS 38677, USA b Department of Physics and Astronomy, University of Hawaii, Honolulu, HI 96826, USA c Department of Physics, University of Arizona, Tucson, AZ 85721, USA d RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA e Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In a recent paper we studied the effect of new-physics operators with differentLorentz structures on the semileptonic Λ b → Λ c τ ¯ ν τ decay. This decay is of interest in lightof the R ( D ( ∗ ) ) puzzle in the semileptonic ¯ B → D ( ∗ ) τ ¯ ν τ decays. In this work we add tensoroperators to extend our previous results and consider both model-independent new physics(NP) and specific classes of models proposed to address the R ( D ( ∗ ) ) puzzle. We show thata measurement of R (Λ c ) = B [Λ b → Λ c τ ¯ ν τ ] / B [Λ b → Λ c (cid:96) ¯ ν (cid:96) ] can strongly constrain the NPparameters of models discussed for the R ( D ( ∗ ) ) puzzle. We use form factors from latticeQCD to calculate all Λ b → Λ c τ ¯ ν τ observables. The Λ b → Λ c tensor form factors had notpreviously been determined in lattice QCD, and we present new lattice results for theseform factors here. a r X i v : . [ h e p - ph ] S e p ontents Λ b → Λ c tensor form factors from lattice QCD 114 Model-independent analysis of individual new-physics couplings 15 R ( D ), R ( D ∗ ), and τ B c R (Λ c ) measurement 19 SU (2) and Leptoquark models 23 A.1 Λ b rest frame 38A.2 Dilepton rest frame 38 A major part of particle physics research is focused on searching for physics beyond thestandard model (SM). In the flavor sector a key property of the SM gauge interactions isthat they are lepton flavor universal. Evidence for violation of this property would be aclear sign of new physics (NP) beyond the SM. In the search for NP, the second and thirdgeneration quarks and leptons are quite special because they are comparatively heavier andare expected to be relatively more sensitive to NP. As an example, in certain versions of thetwo Higgs doublet models (2HDM) the couplings of the new Higgs bosons are proportionalto 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 ¯ B → D ( ∗ ) (cid:96) − ¯ ν (cid:96) have been measured by the BaBar [1], Belle[2, 3] and LHCb [4] Collaborations. It is found that the values of the ratios R ( D ( ∗ ) ) ≡ – 1 – ( ¯ B → D ( ∗ ) τ − ¯ ν τ ) / B ( ¯ B → D ( ∗ ) (cid:96) − ¯ ν (cid:96) ), where (cid:96) = e, µ , considerably exceed their SM pre-dictions.This ratio of branching fractions has certain advantages over the absolute branchingfraction measurement of B → D ( ∗ ) τ ν τ decays, as this is relatively less sensitive to formfactor variations and several systematic uncertainties, such as those on the experimentalefficiency, as well as the dependence on the value of | V cb | , cancel in the ratio.There are lattice QCD predictions for the ratio R ( D ) SM in the Standard Model [5–7]that are in good agreement with one another, R ( D ) SM = 0 . ± .
011 [FNAL / MILC] , (1.1) R ( D ) SM = 0 . ± .
008 [HPQCD] . (1.2)These values are also in good agreement with the phenomenological prediction [8] R ( D ) SM = 0 . ± . , (1.3)which is based on form factors extracted from experimental data for the B → D(cid:96) ¯ ν differ-ential decay rates using heavy-quark effective theory. See also Ref. [9] for a recent analysisof B → D form factors using light-cone sum rules.A calculation of R ( D ∗ ) SM is not yet available from lattice QCD. The phenomenologicalprediction using form factors extracted from B → D ∗ (cid:96) ¯ ν experimental data is [10] R ( D ∗ ) SM = 0 . ± . . (1.4)The averages of R ( D ) and R ( D ∗ ) measurements, evaluated by the Heavy-Flavor Av-eraging Group, are [11] R ( D ) exp = 0 . ± . ± . , (1.5) R ( D ∗ ) exp = 0 . ± . ± . , (1.6)where the first uncertainty is statistical and the second is systematic. R ( D ∗ ) and R ( D )exceed the SM predictions by 3.3 σ and 1.9 σ , respectively. The combined analysis of R ( D ∗ )and R ( D ), taking into account measurement correlations, finds that the deviation is 4 σ from the SM prediction [11, 12].Since lattice QCD results are not yet available for the B → D ∗ form factors at nonzerorecoil and for the B → D tensor form factor, we use the phenomenological form factorsfrom Ref. [8] for both channels in our analysis. For B → D , we have compared thephenomenological results for f and f + to the results obtained from a joint BGL z -expansionfit [13] to the FNAL/MILC lattice QCD results [6] and Babar [14] and Belle experimentaldata [15], 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 theexperimental measurement of R ( D ) obtained with both sets of form factors are practicallyidentical. – 2 –e also construct the ratios of the experimental results (1.5) and (1.6) to the phe-nomenological SM predictions (1.3) and (1.4): R RatioD = R ( D ) exp R ( D ) SM = 1 . ± . , (1.7) R RatioD ∗ = R ( D ∗ ) exp R ( D ∗ ) SM = 1 . ± . . (1.8)There have been numerous analyses examining NP explanations of the R ( D ( ∗ ) ) mea-surements [8, 16–31]. The new physics involves new charged-current interactions. Inthe neutral-current sector, data from b → s(cid:96) + (cid:96) − decays also hint at lepton flavor non-universality – the so called R K puzzle: the LHCb Collaboration has found a 2.6 σ deviationfrom the SM prediction for the ratio R K ≡ B ( B + → K + µ + µ − ) / B ( B + → K + e + e − ) inthe dilepton invariant mass-squared range 1 GeV ≤ q ≤ [32]. There are alsoother, not necessarily lepton-flavor non-universal anomalies in b → s(cid:96) + (cid:96) − decays, mostsignificantly in the B → K ∗ µ + µ − angular observable P (cid:48) [33, 34]. Global fits of the ex-perimental data prefer a negative shift in one of the b → sµ + µ − Wilson coefficients, C [35].Common explanations of the b → cτ − ¯ ν τ and b → sµ + µ − anomalies have been proposed inRefs. [31, 36–40].The underlying quark level transition b → cτ − ¯ ν τ in the R ( D ( ∗ ) ) puzzle can be probedin both B and Λ b decays. Recently, the decay Λ b → Λ c τ ¯ ν τ was discussed in the standardmodel and with new physics in Ref. [41–47]. Λ b → Λ c τ ¯ ν τ decays could be useful to confirmpossible new physics in the R ( D ( ∗ ) ) puzzle and to point to the correct model of new physics.In Ref. [43] the following quantities were calculated within the SM and with variousnew physics operators: R (Λ c ) = B [Λ b → Λ c τ ¯ ν τ ] B [Λ b → Λ c (cid:96) ¯ ν (cid:96) ] (1.9)and B Λ c ( q ) = d Γ[Λ b → Λ c τ ¯ ν τ ] dq d Γ[Λ b → Λ c (cid:96) ¯ ν (cid:96) ] dq , (1.10)where (cid:96) represents µ or e . In this paper we work with the ratio R Ratio Λ c , defined as R Ratio Λ c = R (Λ c ) SM + NP R (Λ c ) SM . (1.11)We also consider the forward-backward asymmetry A F B ( q ) = (cid:82) ( d Γ /dq d cos θ τ ) d cos θ τ − (cid:82) − ( d Γ /dq d cos θ τ ) d cos θ τ d Γ /dq , (1.12)where θ τ is the angle between the momenta of the τ lepton and Λ c baryon in the dileptonrest frame.This paper improves upon the earlier work [43] in several ways:– 3 – We add tensor interactions in the effective Lagrangian. • Instead of a quark model, we use form factors from lattice QCD to calculate allΛ b → Λ c τ ¯ ν τ observable. The vector and axial vector form factors are taken fromRef. [48], and we extend the analysis of Ref. [48] to obtain lattice QCD results forthe tensor form factors as well. • In addition to R (Λ c ) and B Λ c ( q ), we also calculate the forward-backward asymmetry(1.12) in the SM and with new physics. • We include new constraints from the B c lifetime [47, 49, 50] in our analysis. • In addition to analyzing the effects of individual new physics-couplings, we studyspecific models that introduce multiple new-physics couplings simultaneously. Weconsider a 2-Higgs doublet model, models with new vector bosons, and several lepto-quark models.The paper is organized in the following manner: In Sec. 2 we introduce the effective La-grangian to parametrize the NP operators and give the expressions for the decay distribu-tion in terms of helicity amplitudes. In Sec. 3, we present the new lattice QCD results forthe tensor form factors. The model-independent phenomenological analysis of individualnew-physics couplings is discussed in Sec. 4, while explicit models are considered in Sec. 5.We conclude in Sec. 6.
In the presence of NP, the effective Hamiltonian for the quark-level transition b → cτ − ¯ ν τ can be written in the form [51, 52] H eff = G F V cb √ (cid:110)(cid:104) ¯ cγ µ (1 − γ ) b + g L ¯ cγ µ (1 − γ ) b + g R ¯ cγ µ (1 + γ ) b (cid:105) ¯ τ γ µ (1 − γ ) ν τ + (cid:104) g S ¯ cb + g P ¯ cγ b (cid:105) ¯ τ (1 − γ ) ν τ + (cid:104) g T ¯ cσ µν (1 − γ ) b (cid:105) ¯ τ σ µν (1 − γ ) ν τ + h.c (cid:111) , (2.1)where G F is the Fermi constant, V cb is the Cabibbo-Kobayashi-Maskawa (CKM) matrixelement, and we use σ µν = i [ γ µ , γ ν ] /
2. We consider that the above Hamiltonian is writtenat the m b energy scale.If the effective interaction is written at the cut-ff scale Λ then running down to the m b scale will generate new operators and new contributions, which have been discussed inRefs. [53, 54]. These new contributions can strongly constrain models but to really calculatetheir true impacts we have to consider specific models where there might be cancellationsbetween various terms.The SM effective Hamiltonian corresponds to g L = g R = g S = g P = g T = 0. InEq. (2.1), we have assumed the neutrinos to be always left chiral. In general, with NP the– 4 –eutrino associated with the τ 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. The process under consideration isΛ b ( p Λ b ) → τ − ( p τ ) + ¯ ν τ ( p ¯ ν τ ) + Λ c ( p Λ c ) . The differential decay rate for this process can be represented as [23] d Γ dq d cos θ τ = G F | V cb | π (1 − m τ q ) √ Q + Q − m b (cid:88) λ Λ c (cid:88) λ τ |M λ τ λ Λ c | , (2.2)where q = p Λ b − p Λ c , (2.3) Q ± = ( m Λ b ± m Λ c ) − q , (2.4)and the helicity amplitude M λ τ λ Λ c is written as M λ τ λ Λ c = H SPλ Λ c ,λ τ =0 + (cid:88) λ η λ H V Aλ Λ c ,λ L λ τ λ + (cid:88) λ,λ (cid:48) η λ η λ (cid:48) H ( T ) λ Λ b λ Λ c ,λ,λ (cid:48) L λ τ λ,λ (cid:48) . (2.5)Here, ( λ , λ (cid:48) ) indicate the helicity of the virtual vector boson (see Appendix A), λ Λ c and λ τ are the helicities of the Λ c baryon and τ lepton, respectively, and η λ = 1 for λ = t and η λ = − λ = 0 , ± H SPλ Λ c ,λ =0 = H Sλ Λ c ,λ =0 + H Pλ Λ c ,λ =0 ,H Sλ Λ c ,λ =0 = g S (cid:104) Λ c | ¯ cb | Λ b (cid:105) ,H Pλ Λ c ,λ =0 = g P (cid:104) Λ c | ¯ cγ b | Λ b (cid:105) , (2.6) H V Aλ Λ c ,λ = H Vλ Λ c ,λ − H Aλ Λ c ,λ ,H Vλ Λ c ,λ = (1 + g L + g R ) (cid:15) ∗ µ ( λ ) (cid:104) Λ c | ¯ cγ µ b | Λ b (cid:105) ,H Aλ Λ c ,λ = (1 + g L − g R ) (cid:15) ∗ µ ( λ ) (cid:104) Λ c | ¯ cγ µ γ b | Λ b (cid:105) , (2.7)and H ( T ) λ Λ b λ Λ c ,λ,λ (cid:48) = H ( T λ Λ b λ Λ c ,λ,λ (cid:48) − H ( T λ Λ b λ Λ c ,λ,λ (cid:48) ,H ( T λ Λ b λ Λ c ,λ,λ (cid:48) = g T (cid:15) ∗ µ ( λ ) (cid:15) ∗ ν ( λ (cid:48) ) (cid:104) Λ c | ¯ ciσ µν b | Λ b (cid:105) ,H ( T λ Λ b λ Λ c ,λ,λ (cid:48) = g T (cid:15) ∗ µ ( λ ) (cid:15) ∗ ν ( λ (cid:48) ) (cid:104) Λ c | ¯ ciσ µν γ b | Λ b (cid:105) . (2.8)– 5 –he leptonic amplitudes are defined as L λ τ = (cid:104) τ ¯ ν τ | ¯ τ (1 − γ ) ν τ | (cid:105) ,L λ τ λ = (cid:15) µ ( λ ) (cid:104) τ ¯ ν τ | ¯ τ γ µ (1 − γ ) ν τ | (cid:105) ,L λ τ λ,λ (cid:48) = − i(cid:15) µ ( λ ) (cid:15) ν ( λ (cid:48) ) (cid:104) τ ¯ ν τ | ¯ τ σ µν (1 − γ ) ν τ | (cid:105) . (2.9)Above, (cid:15) µ are the polarization vectors of the virtual vector boson (see Appendix A). Theexplicit expressions for the hadronic and leptonic helicity amplitudes are presented in thefollowing. In this paper, we use the helicity-based definition of the Λ b → Λ c form factors, which wasintroduced in [55]. The matrix elements of the vector and axial vector currents can bewritten in terms of six helicity form factors F + , F ⊥ , F , G + , G ⊥ , and G as follows: (cid:104) Λ c | ¯ cγ µ b | Λ b (cid:105) = ¯ u Λ c (cid:104) F ( q )( m Λ b − m Λ c ) q µ q + F + ( q ) m Λ b + m Λ c Q + ( p µ Λ b + p µ Λ c − ( m b − m c ) q µ q )+ F ⊥ ( q )( γ µ − m Λ c Q + p µ Λ b − m Λ b Q + p µ Λ c ) (cid:105) u Λ b , (2.10) (cid:104) Λ c | ¯ cγ µ γ b | Λ b (cid:105) = − ¯ u Λ c γ (cid:104) G ( q )( m Λ b + m Λ c ) q µ q + G + ( q ) m Λ b − m Λ c Q − ( p µ Λ b + p µ Λ c − ( m b − m c ) q µ q )+ G ⊥ ( q )( γ µ + 2 m Λ c Q − p µ Λ b − m Λ b Q − p µ Λ c ) (cid:105) u Λ b . (2.11)The matrix elements of the scalar and pseudoscalar currents can be obtained from thevector and axial vector matrix elements using the equations of motion: (cid:104) Λ c | ¯ cb | Λ b (cid:105) = q µ m b − m c (cid:104) Λ c | ¯ cγ µ b | Λ b (cid:105) = F ( q ) m Λ b − m Λ c m b − m c ¯ u Λ c u Λ b , (2.12) (cid:104) Λ c | ¯ cγ b | Λ b (cid:105) = q µ m b + m c (cid:104) Λ c | ¯ cγ µ γ b | Λ b (cid:105) = G ( q ) m Λ b + m Λ c m b + m c ¯ u Λ c γ u Λ b . (2.13)In our numerical analysis, we use m b = 4 . m c = 1 . h + , h ⊥ , (cid:101) h + ,– 6 – h ⊥ , (cid:104) Λ c | ¯ ciσ µν b | Λ b (cid:105) = ¯ u Λ c (cid:104) h + ( q ) p µ Λ b p ν Λ c − p ν Λ b p µ Λ c Q + + h ⊥ ( q ) (cid:16) m Λ b + m Λ c q ( q µ γ ν − q ν γ µ ) −
2( 1 q + 1 Q + )( p µ Λ b p ν Λ c − p ν Λ b p µ Λ c ) (cid:17) + (cid:101) h + ( q ) (cid:16) iσ µν − Q − ( m Λ b ( p µ Λ c γ ν − p ν Λ c γ µ ) − m Λ c ( p µ Λ b γ ν − p ν Λ b γ µ ) + p µ Λ b p ν Λ c − p ν Λ b p µ Λ c ) (cid:17) + (cid:101) h ⊥ ( q ) m Λ b − m Λ c q Q − (cid:16) ( m b − m c − q )( γ µ p ν Λ b − γ ν p µ Λ b ) − ( m b − m c + q )( γ µ p ν Λ c − γ ν p µ Λ c ) + 2( m Λ b − m Λ c )( p µ Λ b p ν Λ c − p ν Λ b p µ Λ c ) (cid:17)(cid:105) u Λ b . (2.14)The matrix elements of the current ¯ ciσ µν γ b can be obtained from the above equation byusing the identity σ µν γ = − i (cid:15) µναβ σ αβ . (2.15)In the following, only the non-vanishing helicity amplitudes are given. The scalar andpseudo-scalar helicity amplitudes associated with the new physics scalar and pseudo-scalarinteractions are H SP / , = F g S √ Q + m b − m c ( m Λ b − m Λ c ) − G g P √ Q − m b + m c ( m Λ b + m Λ c ) , (2.16) H SP − / , = F g S √ Q + m b − m c ( m Λ b − m Λ c ) + G g P √ Q − m b + m c ( m Λ b + m Λ c ) . (2.17)The parity-related amplitudes are H Sλ Λ c ,λ NP = H S − λ Λ c , − λ NP ,H Pλ Λ c ,λ NP = − H P − λ Λ c , − λ NP . (2.18)– 7 –or the vector and axial-vector helicity amplitudes, we find H V A / , = F + (1 + g L + g R ) √ Q − (cid:112) q ( m Λ b + m Λ c ) − G + (1 + g L − g R ) √ Q + (cid:112) q ( m Λ b − m Λ c ) , (2.19) H V A / , +1 = − F ⊥ (1 + g L + g R ) (cid:112) Q − + G ⊥ (1 + g L − g R ) (cid:112) Q + , (2.20) H V A / ,t = F (1 + g L + g R ) √ Q + (cid:112) q ( m Λ b − m Λ c ) − G (1 + g L − g R ) √ Q − (cid:112) q ( m Λ b + m Λ c ) , (2.21) H V A − / , = F + (1 + g L + g R ) √ Q − (cid:112) q ( m Λ b + m Λ c )+ G + (1 + g L − g R ) √ Q + (cid:112) q ( m Λ b − m Λ c ) , (2.22) H V A − / , − = − F ⊥ (1 + g L + g R ) (cid:112) Q − − G ⊥ (1 + g L − g R ) (cid:112) Q + , (2.23) H V A − / ,t = F (1 + g L + g R ) √ Q + (cid:112) q ( m Λ b − m Λ c )+ G (1 + g L − g R ) √ Q − (cid:112) q ( m Λ b + m Λ c ) . (2.24)We also have the relations H Vλ Λ c ,λ w = H V − λ Λ c , − λ w ,H Aλ Λ c ,λ w = − H A − λ Λ c , − λ w . (2.25)– 8 –he tensor helicity amplitudes are H ( T ) − / − / ,t, = − g T (cid:104) − h + (cid:112) Q − + (cid:101) h + (cid:112) Q + (cid:105) , (2.26) H ( T )+1 / / ,t, = g T (cid:104) h + (cid:112) Q − + (cid:101) h + (cid:112) Q + (cid:105) , (2.27) H ( T ) − / / ,t, +1 = − g T √ (cid:112) q (cid:104) h ⊥ ( m Λ b + m Λ c ) (cid:112) Q − + (cid:101) h ⊥ ( m Λ b − m Λ c ) (cid:112) Q + (cid:105) , (2.28) H ( T )+1 / − / ,t, − = − g T √ (cid:112) q (cid:104) h ⊥ ( m Λ b + m Λ c ) (cid:112) Q − − (cid:101) h ⊥ ( m Λ b − m Λ c ) (cid:112) Q + (cid:105) , (2.29) H ( T ) − / / , , +1 = − g T √ (cid:112) q (cid:104) h ⊥ ( m Λ b + m Λ c ) (cid:112) Q − + (cid:101) h ⊥ ( m Λ b − m Λ c ) (cid:112) Q + (cid:105) , (2.30) H ( T )+1 / − / , , − = g T √ (cid:112) q (cid:104) h ⊥ ( m Λ b + m Λ c ) (cid:112) Q − − (cid:101) h ⊥ ( m Λ b − m Λ c ) (cid:112) Q + (cid:105) , (2.31) H ( T )+1 / / , +1 , − = − g T (cid:104) h + (cid:112) Q − + (cid:101) h + (cid:112) Q + (cid:105) , (2.32) H ( T ) − / − / , +1 , − = − g T (cid:104) h + (cid:112) Q − − (cid:101) h + (cid:112) Q + (cid:105) . (2.33)The other non-vanishing helicity amplitudes of tensor type are related to the above by H ( T ) λ Λ b λ Λ c ,λ,λ (cid:48) = − H ( T ) λ Λ b λ Λ c ,λ (cid:48) ,λ . (2.34) In the following, we define v = (cid:115) − m τ q . (2.35)The scalar and pseudoscalar leptonic helicity amplitudes are L +1 / = 2 (cid:112) q v, (2.36) L − / = 0 , (2.37)the vector and axial-vector amplitudes are L +1 / ± = ±√ m τ v sin( θ τ ) , (2.38) L +1 / = − m τ v cos ( θ τ ) , (2.39) L +1 / t = 2 m τ v, (2.40) L − / ± = (cid:112) q v (1 ± cos( θ τ )) , (2.41) L − / = 2 (cid:112) q v sin ( θ τ ) , (2.42) L − / t = 0 , (2.43)– 9 –nd the tensor amplitudes are L +1 / , ± = − (cid:112) q v sin( θ τ ) , (2.44) L +1 / ± ,t = ∓ (cid:112) q v sin( θ τ ) , (2.45) L +1 / t, = L +1 / , − = − (cid:112) q v cos( θ τ ) , (2.46) L − / , ± = ∓√ m τ v (1 ± cos( θ τ )) , (2.47) L − / ± ,t = −√ m τ v (1 ± cos( θ τ )) , (2.48) L − / t, = L − / , − = 2 m τ v sin( θ τ ) . (2.49)Here we have the relation L λ τ λ,λ (cid:48) = − L λ τ λ (cid:48) ,λ . (2.50)The angle θ τ is defined as the angle between the momenta of the τ lepton and Λ c baryonin the dilepton rest frame. From the twofold decay distribution (2.2), we obtain the following expression for the dif-ferential decay rate by integrating over cos θ τ : d Γ(Λ b → Λ c τ ¯ ν τ ) dq = G F | V cb | π q √ Q + Q − m b (cid:16) − m τ q (cid:17) (cid:34) A V A + m τ q A V A + 32 A SP +2 (cid:16) m τ q (cid:17) A T + 3 m τ (cid:112) q A V A − SP + 6 m τ (cid:112) q A V A − T (cid:35) , (2.51)where A V A = | H V A / , | + | H V A / , | + | H V A − / , | + | H V A − / , − | ,A V A = | H V A / , | + | H V A / , | + | H V A − / , | + | H V A − / , − | + 3 | H V A / ,t | + 3 | H V A − / ,t | ,A SP = | H SP / , | + | H SP − / , | ,A T = | H ( T )1 / / ,t, + H ( T )1 / / , − , | + | H ( T )1 / − / ,t, − + H ( T )1 / − / , − , | + | H ( T ) − / / , , + H ( T ) − / / ,t, | + | H ( T ) − / − / , − , + H ( T ) − / − / ,t, | ,A V A − SP = Re( H SP ∗ / , H V A / ,t + H SP ∗− / , H V A − / ,t ) ,A V A − T = Re[ H V A ∗ / , ( H ( T )1 / / , − , + H ( T )1 / / ,t, )] + Re[ H V A ∗ / , ( H ( T ) − / / , , + H ( T ) − / / ,t, )]+Re[ H V A ∗− / , ( H ( T ) − / − / , − , + H ( T ) − / − / ,t, )] + Re[ H V A ∗− / , − ( H ( T )1 / − / , − , + H ( T )1 / − / ,t, − )] . (2.52)– 10 –ere, A V A and A V A are the (axial-)vector non-spin-flip and spin-flip terms respectively, A SP and A T are the pure (pseudo-)scalar and tensor terms respectively; and A V A − SP and A V A − T are interference terms. The scalar-tensor interference term is proportional to cos θ τ and vanishes after integration over cos θ τ .For the forward-backward asymmetry (1.12) we have A F B ( q ) = (cid:18) d Γ dq (cid:19) − G F V cb π q √ Q + Q − m b (cid:16) − m τ q (cid:17) (cid:34) B V A + 2 m τ q B V A + 4 m τ q B T +2 m τ (cid:112) q B V A − SP + 4 m τ (cid:112) q B V A − T + 4 B SP − T (cid:35) , (2.53)where B V A = | H V A / , | − | H V A − / , − | ,B V A = Re[ H V A ∗ / ,t H V A / , + H V A ∗− / ,t H V A − / , ] ,B T = | H ( T ) − / / , , + H ( T ) − / / ,t, | − | H ( T )1 / − / , − , + H ( T )1 / − / ,t, − | ,B V A − SP = Re[ H SP ∗ / , H V A / , + H SP ∗− / , H V A − / , ] ,B V A − T = Re[ H V A ∗ / ,t ( H ( T )1 / / , − , + H ( T )1 / / ,t, )] + Re[ H V A ∗ / , ( H ( T ) − / / , , + H ( T ) − / / ,t, )]+ Re[ H V A ∗− / ,t ( H ( T ) − / − / , − , + H ( T ) − / − / ,t, )] − Re[ H V A ∗− / , − ( H ( T )1 / − / , − , + H ( T )1 / − / ,t, − )] ,B SP − T = Re[ H SP ∗ / , ( H ( T )1 / / , − , + H ( T )1 / / ,t, )] + Re[ H SP ∗− / , ( H ( T ) − / − / , − , + H ( T ) − / − / ,t, )] . (2.54)There is no contribution from pure (pseudo-)scalar operators to the forward-backwardasymmetry, but all possible interference terms are present. Λ b → Λ c tensor form factors from lattice QCD This work uses Λ b → Λ c form factors computed in lattice QCD. The vector and axial vectorform factors defined in Eqs. (2.10) and (2.11) are taken from Ref. [48]. For the purposes ofthe present work, one of us (SM) extended the analysis of Ref. [48] to include the tensorform factors defined in Eq. (2.14). The tensor form factors were extracted from the latticeQCD correlation functions using ratios defined as in Ref. [57]. The lattice parameters areidentical to those in Ref. [48], except that for the tensor form factors the “residual matchingfactors” ρ T µν and the O ( a )-improvement coefficients were set to their tree-level values, withappropriately increased estimates for the resulting systematic uncertainties as detailed fur-ther below. Following Ref. [48], two separate fits were performed to the lattice QCD datausing BCL z -expansions [58] augmented with additional terms to describe the dependenceon the lattice spacing and quark masses. The “nominal fit” is used to evaluate the central– 11 – J P m f pole (GeV) h + , h ⊥ − (cid:101) h + , (cid:101) h ⊥ + Table 1 . Values of the pole masses for the tensor form factors.
Nominal fit Higher-order fit a h + . ± . . ± . a h + − . ± . − . ± . a h + . ± . a h ⊥ . ± . . ± . a h ⊥ − . ± . − . ± . a h ⊥ . ± . a (cid:101) h ⊥ , (cid:101) h + . ± . . ± . a (cid:101) h + − . ± . − . ± . a (cid:101) h + . ± . a (cid:101) h ⊥ − . ± . − . ± . a (cid:101) h ⊥ . ± . Table 2 . Results for the z -expansion parameters describing the Λ b → Λ c tensor form factors in thephysical limit (in the MS scheme at the renormalization scale µ = m b ). Files containing the valuesand covariances of the parameters of all ten Λ b → Λ c form factors are provided as supplementalmaterial. values and statistical uncertainties of the form factors (and of any observables dependingon the form factors), while the “higher-order fit” is used in conjunction with the nominal fitto evaluate the combined systematic uncertainty associated with the continuum extrapo-lation, chiral extrapolation, z expansion, renormalization, scale setting, b -quark parametertuning, finite volume, and missing isospin symmetry breaking/QED. The procedure forevaluating the systematic uncertainties is given in Eqs. (82)-(84) of Ref. [48]. The renor-malization uncertainty in the tensor form factors is dominated by the use of the tree-levelvalues, ρ T µν = 1, for the residual matching factors in the mostly nonperturbative renor-malization procedure. We estimate the systematic uncertainty in ρ T µν to be 2 times themaximum value of | ρ V µ − | , | ρ A µ − | , which is equal to 0 . µ = m b in the MS scheme. To account forthe renormalization uncertainty in the higher-order fit, we introduced nuisance parametersmultiplying the form factors, with Gaussian priors equal to 1 ± . . . . . . . . . h + (Λ b → Λ c ) . . . . . . . h ⊥ (Λ b → Λ c ) . . . . . . e h + (Λ b → Λ c ) .
55 0 .
60 0 .
65 0 .
70 0 .
75 0 .
80 0 .
85 0 .
90 0 . q /q . . . . . . e h ⊥ (Λ b → Λ c ) Figure 1 . Λ b → Λ c tensor form factors in the high q -region: lattice results and extrapolation tothe physical limit (nominal fit). The bands indicate the statistical uncertainty. The lattice QCDdata sets are labeled as in Ref. [48]. – 13 – . . . . . . . . . h + (Λ b → Λ c ) . . . . . . . . . e h + (Λ b → Λ c ) q (GeV ) . . . . . . . . . h ⊥ (Λ b → Λ c ) q (GeV ) . . . . . . . . . e h ⊥ (Λ b → Λ c ) Figure 2 . Λ b → Λ c tensor form factors in the physical limit, shown in the entire kinematicrange. The form factors are defined in the MS scheme and at µ = m b . The inner bands show thestatistical uncertainty and the outer bands show the total (statistical plus systematic) uncertainty.The procedure for evaluating the uncertainties using the nominal and higher-order fits is given inEqs. (82)-(84) of Ref. [48]. function for a form factor f reduces to the form f ( q ) = 11 − q / ( m f pole ) (cid:2) a f + a f z f ( q ) (cid:3) , (3.1)while the higher-order fit function is given by f HO ( q ) = 11 − q / ( m f pole ) (cid:2) a f , HO + a f , HO z f ( q ) + a f , HO [ z f ( q )] (cid:3) . (3.2)The values of the pole masses are given in Table 1, and the kinematic variables z f aredefined as z f ( q ) = (cid:113) t f + − q − (cid:113) t f + − t (cid:113) t f + − q + (cid:113) t f + − t , (3.3) t = ( m Λ b − m Λ c ) , (3.4) t f + = ( m f pole ) . (3.5)– 14 –s in Ref. [48], in the fits to the lattice data we evaluated the pole masses as am f pole = am (lat) B c + a ∆ f , where am (lat) B c are the lattice QCD results for the pseudoscalar B c mass oneach individual data set, and the splittings ∆ f are fixed to their physical values ∆ h + ,h ⊥ = 56MeV and ∆ (cid:101) h + , (cid:101) h ⊥ = 492 MeV. The form factor results are very insensitive to the choices of∆ f (as expected for poles far above q ). When varying ∆ f by ± z -expansionparameters returned from the fit are found to change in such a way that the changes inthe form factors themselves are below 0 .
2% in the entire semileptonic region.Plots of the lattice QCD data for the tensor form factors, along with the nominal fitfunctions in the physical limit, are shown in Fig. 1. The same fit functions are plotted inthe entire kinematic range in Fig. 2, where also the total (statistical plus systematic) uncer-tainties are shown. The form factor h + has larger uncertainties than the other form factorsbecause of larger excited-state contributions in the lattice QCD correlation functions.The values of the nominal and higher-order fit parameters for the tensor form factorsare given in Table 2. Because of the kinematic constraint (cid:101) h ⊥ ( q ) = (cid:101) h + ( q ) , (3.6)which is at the point z = 0, the form factors (cid:101) h ⊥ and (cid:101) h + share the common parameters a (cid:101) h ⊥ , (cid:101) h + . To evaluate the uncertainties of the form factors and of any observables dependingon the form factors, it is essential to include the (cross-)correlations between all form factorparameters. The full covariance matrices of the nominal and higher-order parameters of allten Λ b → Λ c form factors (vector, axial vector, and tensor) are provided as supplementalfiles. In this section we consider one new-physics coupling at a time. We first compute theconstraints from the existing measurements with mesons, and then study the impact of afuture measurement of R (Λ c ). R ( D ) , R ( D ∗ ) , and τ B c We require the NP couplings to reproduce the measurements (1.7) and (1.8) of R RatioD and R RatioD ∗ within the 3 σ range. The coupling g S ( g P ) only contributes to R RatioD ( R RatioD ∗ ) whilethe other couplings contribute to both channels. If only g L is nonzero, the SM contributiongets rescaled by an overall factor | g L | , so that [31] R RatioD = R RatioD ∗ = R Ratio Λ c = | g L | , (4.1)which is consistent with the present measurements (1.7) and (1.8). Note that in the g L -onlyscenario the forward-backward asymmetry (1.12) is unmodified, A F B = A SMF B .There is also a measurement of the τ polarization by Belle [59] with the result P τ = − . ± . +0 . − . . The uncertainties of this measurement are presently too large to providea significant additional constraint and we therefore do not include P τ in our analysis.– 15 – S only g P only g L only g R only g T only − . . − . − .
044 0 . R (Λ c ) 0 . ± .
009 0 . ± .
010 0 . ± .
014 0 . ± .
011 0 . ± . R Ratio Λ c . ± .
007 1 . ± .
001 1 .
44 1 . ± .
003 1 . ± . − . − . i . − . i . − . i . − . i . − . iR (Λ c ) 0 . ± .
013 0 . ± .
011 0 . ± .
014 0 . ± .
014 0 . ± . R Ratio Λ c . ± .
008 1 . ± .
002 1 .
412 1 . ± .
005 1 . ± . Table 3 . The values of R (Λ c ) and R Ratio Λ c for two example choices (real-valued and complex-valued) of the new-physics couplings. The standard-model value of R (Λ c ) is 0 . ± .
010 [48]. Theuncertainties given are due to the form factor uncertainties.
It was recently pointed out [47, 49, 50] that the measured lifetime of the B c meson, τ B c = 0 . B c → τ − ¯ ν τ decay rate, which yieldsa strong constraint on the g P coupling. According to SM calculations using an operatorproduct expansion [60], only about 5% (for the central value) of the total width of the B c ,Γ B c = 1 /τ B c , can be attributed to purely tauonic and semi-tauonic modes. This can berelaxed as the parameters in the calculations are varied. In our analysis, we use an upperlimit of B ( B c → τ − ¯ ν τ ) ≤
30% to put constraints on the new-physics couplings. We use f B c = 0 . R RatioD , R RatioD ∗ , and τ B c . We see that τ B c puts a strong constraint on g P ,and weak constraints on g L and g R . The tensor coupling g T is strongly constrained by R RatioD ∗ , and only weakly constrained by R RatioD .Example values of the ratios R (Λ c ) and R Ratio Λ c = R (Λ c ) /R (Λ c ) SM for representativeallowed values of the NP couplings are given in Table 3. The standard-model predictionfor R (Λ c ) is 0 . ± .
010 [48]. We find that large deviations from this value are possiblewith the present mesonic constraints. In Table 4, we present the maximum and minimumallowed values of R Ratio Λ c = R (Λ c ) /R (Λ c ) SM in the presence of each individual new-physicscoupling, and the corresponding values of the coupling at which these occur.Figure 4 shows the effect of representative values of the individual NP couplings on theΛ b → Λ c τ ¯ ν τ differential decay rate (evaluated assuming | V cb | = 0 . B Λ c ( q )[defined in Eq. (1.10)] and A F B ( q ). In all cases, except for the strongly constrained pure g P coupling, substantial deviations from the SM predictions are allowed. We notice that A F B is typically above the SM prediction in the presence of g R or g T , while it is typically belowthe SM prediction in the presence of g S . Hence, it is possible to use A F B to distinguishbetween the different couplings. – 16 –oupling R (Λ c ) max R Ratio Λ c ,max coupling value R (Λ c ) min R Ratio Λ c ,min coupling value g S only 0 .
405 1 .
217 0 .
363 0 .
314 0 . − . g P only 0 .
354 1 .
062 0 .
658 0 .
337 1 .
014 0 . g L only 0 .
495 1 .
486 0 .
094 + 0 . i .
340 1 . − .
070 + 0 . ig R only 0 .
525 1 .
576 0 .
085 + 0 . i .
336 1 . − . g T only 0 .
526 1 .
581 0 .
428 0 .
338 1 . − . Table 4 . The maximum and minimum values of R (Λ c ) and R Ratio Λ c allowed by the mesonic con-straints for each new-physics coupling, and the coupling values at which these extrema are reached. - - - - - [ g s ] I m [ g s ] Only g s present - - - - - - - - [ g P ] I m [ g P ] Only g P present - - - - - - - [ g L ] I m [ g L ] Only g L present - - - - - [ g R ] I m [ g R ] Only g R present - - - - - - - - [ g T ] I m [ g T ] Only g T present Figure 3 . Constraints on the individual new-physics couplings from the measurements of R RatioD , R RatioD ∗ , and τ B c . We require that the couplings reproduce the measurements of R RatioD and R RatioD ∗ in Eqs. (1.7) and (1.8) within 3 σ , and satisfy B ( B c → τ − ¯ ν τ ) ≤ – 17 – M g s =- g s =- - i0.3 q [ GeV ] d / dq [ × - G e V - ] Only g s Present SM g s =- g s =- - i0.3 q [ GeV ] B Only g s Present SM g s =- g s =- - i0.3 - - q [ GeV ] A F B Only g s Present SM g p = g p = - q [ GeV ] d / dq [ × - G e V - ] Only g p Present SM g p = g p = - q [ GeV ] B Only g p Present SM g p = g p = - q [ GeV ] A F B Only g p Present SM g L =- g L = - i0.3 q [ GeV ] d / dq [ × - G e V - ] Only g L Present SM g L =- g L = - i0.3 q [ GeV ] B Only g L Present SM g L =- g L = - i0.3 q [ GeV ] A F B Only g L Present SM g R = - i0.67 g R =- q [ GeV ] d / dq [ × - G e V - ] Only g R Present SM g R = - i0.67 g R =- q [ GeV ] B Only g R Present SM g R = - i0.67 g R =- q [ GeV ] A F B Only g R Present SM g T = - g T = q [ GeV ] d / dq [ × - G e V - ] Only g T Present SM g T = - g T = q [ GeV ] B Only g T Present SM g T = - g T = q [ GeV ] A F B Only g T Present
Figure 4 . The effect of individual new-physics couplings on the Λ b → Λ c τ ¯ ν τ differential decayrate (left), the ratio of the Λ b → Λ c τ ¯ ν τ and Λ b → Λ c (cid:96) ¯ ν (cid:96) differential decay rates (middle), and theΛ b → Λ c τ ¯ ν τ forward-backward asymmetry (right). Each plot shows the observable in the StandardModel and for two representative values of the new-physics coupling (one real-valued choice andone complex-valued choice). The bands indicate the 1 σ uncertainties originating from the Λ b → Λ c form factors. – 18 – .2 Impact of a future R (Λ c ) measurement In this subsection we present the effect of possible future measurements of R (Λ c ) on theNP couplings constraints. We consider two cases, one in which the measured value is nearthe SM prediction and one with measured value far from SM. For the first case we take R Ratio Λ c = 1 ± × .
05, and for the second case R Ratio Λ c = 1 . ± × .
05 (the same centralvalues as R RatioD ). Note that we take the 1 σ uncertainty as 0 .
05. Figures 5 and 6 showthe allowed regions of the parameter space for the first and second case, respectively. Weobserve the following when adding the R Ratio Λ c constraints to the mesonic constraints: • For R (Λ c ) near the SM (Fig. 5), the allowed regions for ( g L , g R , g T ) are reducedsignificantly, the allowed region for g S shrinks only slightly, and the allowed regionfor g P remains the same (as it is dominantly constrained by τ B c ). • For R (Λ c ) far from the SM (Fig. 6), most of the previously allowed region for g S becomes excluded by R (Λ c ). Even more importantly, the g P -only scenario becomesruled out. In this case, R (Λ c ) also provides strong constraints on ( g L , g R , g T ), butthese constraints still overlap with the mesonic constraints.– 19 – - - - - [ g s ] I m [ g s ] Only g s present - - - - - - - - [ g P ] I m [ g P ] Only g P present - - - - - - - [ g L ] I m [ g L ] Only g L present - - - - - [ g R ] I m [ g R ] Only g R present - - - - - - - - [ g T ] I m [ g T ] Only g T present Figure 5 . Constraints on individual new-physics couplings from a possible R (Λ c ) measurement(shown in blue), assuming that R Ratio Λ c = 1 ± × .
05 where the 1 σ uncertainty is 0 .
05. Also shownare the mesonic constraints as in Fig. 3. – 20 – - - - - [ g s ] I m [ g s ] Only g s present - - - - [ g P ] I m [ g P ] Only g P present - - - - - - - [ g L ] I m [ g L ] Only g L present - - - - - [ g R ] I m [ g R ] Only g R present - - - - - - - - [ g T ] I m [ g T ] Only g T present Figure 6 . Constraints on individual new-physics couplings from a possible R (Λ c ) measurement(shown in blue), assuming that R Ratio Λ c = 1 . ± × .
05 where the 1 σ uncertainty is 0 .
05. Alsoshown are the mesonic constraints as in Fig. 3. – 21 –
Explicit models
In this section we will discuss explicit models that can generate the couplings in the effectiveHamiltonian (2.1). We will consider three categories: Two-Higgs-doublet models whichgenerate ( g S , g P ), SU (2) models which generate g L , and leptoquark models which generate( g S , g P , g L , g T ). We do not consider models that generate g R , as in the standard-model-effective-theory picture it is difficult to have a g R coupling that leads to lepton universalityviolation effects [62]. The simplest scalar extensions of the SM are the two-Higgs-doublet models (2HDM). The2HDM of type II is disfavored by experiment [1]. We will consider the Aligned Two-Higgs-Doublet Model (A2HDM) from Ref. [21]. The Lagrangian of the model is L H ± Y = − √ v H + { ¯ u [ ξ d V M d P R − ξ u M u V P L ] d + ξ l ¯ νM l P R l } + h . c ., (5.1)where u , d , and l denote all three generations of up-type quarks, down-type quarks, andcharged leptons, M u and M d are the quark mass matrices, and V is the CKM matrix.Above, ξ f ( f = u, d, l ) are the proportionality parameters in the so-called “Higgs basis”,in which only one scalar doublet acquires a nonzero vacuum expectation value. The cases ξ d = ξ l = − /ξ u = − tan β and ξ u = ξ d = ξ l = cot β correspond to the Type-II and Type-Imodels, respectively. The general effective couplings in Eq. (2.1) read g q u q d lS = g q u q d lR + g q u q d lL ,g q u q d lP = g q u q d lR − g q u q d lL , (5.2)where g q u q d lL = ξ u ξ ∗ l m q u m l M H ± , g q u q d lR = − ξ d ξ ∗ l m q d m l M H ± . (5.3)The scenario in which the ξ u,d,l parameters are universal for all three generations is ruledout [21]. We therefore assume that Eq. (5.3) only gives the couplings for processes involvingthe b quark, while the couplings for the first two generations are considered independently.In this model we find significant deviation from the standard model contribution to thedecay Λ b → Λ c τ ¯ ν τ , but for a more complete analysis RGE evolution should be considered.The RGE evolution of the couplings of the A2HDM has been discussed in Ref. [63]. Thealignment condition, which guarantees the absence of tree-level FCNC processes, is pre-served by the RGE only in the case of the standard type-I, II, X, and Y models which arediscussed in [64]. However, our framework requires non-universal flavor dependent cou-plings and the RGE evolution has not been worked out and is not included in the analysis.Keeping in mind that RGE effects could change the phenomenology of the model, thediscussion of the full numerical analysis of the model is not included in this work.– 22 – .2 SU (2) and Leptoquark models The analysis of the R ( D ( ∗ ) ) and R K anomalies could favor the left-handed operator g L . InRef. [31], it was pointed out that, assuming that the scale of NP is much higher than theweak scale, the g L operator should be invariant under the full SU (3) C × SU (2) L × U (1) Y gauge group. There are two possibilities: O NP = G Λ ( ¯ Q (cid:48) L γ µ Q (cid:48) L )( ¯ L (cid:48) L γ µ L (cid:48) L ) , O NP = G Λ ( ¯ Q (cid:48) L γ µ σ I Q (cid:48) L )( ¯ L (cid:48) L γ µ σ I L (cid:48) L )= G Λ (cid:104)
2( ¯ Q (cid:48) iL γ µ Q (cid:48) jL )( ¯ L (cid:48) jL γ µ L (cid:48) iL ) − ( ¯ Q (cid:48) L γ µ Q (cid:48) L )( ¯ L (cid:48) L γ µ L (cid:48) L ) (cid:105) , (5.4)where G and G are both O (1), and the σ I are the Pauli matrices. Here Q (cid:48) ≡ ( t (cid:48) , b (cid:48) ) T and L (cid:48) ≡ ( ν (cid:48) τ , τ (cid:48) ) T . The key point is that O NP contains both neutral-current (NC) andcharged-current (CC) interactions. The NC and CC pieces can be used to respectivelyexplain the R K and R ( D ( ∗ ) ) puzzles. In the following, we briefly review the literature onmodels of this type.In Ref. [36], UV completions that can give rise to O NP , [Eq. (5.4)], were discussed.One among the four possibilities for the underlying NP model is a vector boson (VB) thattransforms as ( , ,
0) under SU (3) C × SU (2) L × U (1) Y , as in the SM.Concrete VB models were discussed in Ref. [37, 38] and the simplest VB model wasconsidered in Ref. [39]. We refer to the VBs as V = W (cid:48) , Z (cid:48) . In the gauge basis, theLagrangian describing the couplings of the VBs to left-handed third-generation fermions is∆ L V = g qV (cid:16) Q (cid:48) L γ µ σ I Q (cid:48) L (cid:17) V Iµ + g (cid:96)V (cid:16) L (cid:48) L γ µ σ I L (cid:48) L (cid:17) V Iµ , (5.5)where σ I ( I = 1 , ,
3) are the Pauli matrices. Once the heavy VB is integrated out, oneobtains the following effective Lagrangian, relevant for b → s(cid:96) + (cid:96) − , b → cτ − ¯ ν and b → sν ¯ ν decays: L eff V = − g qV g (cid:96)V m V (cid:16) Q (cid:48) L γ µ σ I Q (cid:48) L (cid:17) (cid:16) L (cid:48) L γ µ σ I L (cid:48) L (cid:17) . (5.6)One can study the phenomenology of the model with an ansatz for the mixing matrices.The assumption of Ref. [36, 39] is that the transformations D and L involve only thesecond and third generations. The key observation in Ref. [39] is the Z (cid:48) interaction alsocontributes to B s mixing and the model becomes highly constrained. If fact only a fewpercent deviation from the SM is allowed in the R ( D ( ∗ ) ) observables. For this reason, we donot present a detailed numerical analysis of the SU (2) models for the Λ b → Λ c τ ¯ ν τ decay.We next move to leptoquark models. In Ref. [65], several leptoquark models are con-sidered that generate scalar, vector, and tensor operators. The SU (3) × SU (2) × U (1)quantum numbers of these models are summarized in Table 5. We can group the lepto-quarks as vector or scalar leptoquarks. These leptoquarks can in turn be SU (2) singlets,doublets, or triplets. – 23 –pin SU (3) c SU (2) L U (1) Y = Q − T S ∗ / S ∗ / R / V ∗ / U / U / Table 5 . Quantum numbers of scalar and vector leptoquarks.
The Lagrangians for the various leptoquarks are L LQ = L LQ V + L LQ S , (5.7) L LQ V = (cid:16) h ij L ¯ Q iL γ µ L jL + h ij R ¯ d iR γ µ (cid:96) jR (cid:17) U µ + h ij L ¯ Q iL σ γ µ L jL U µ + (cid:16) g ij L ¯ d c,iR γ µ L jL + g ij R ¯ Q c,iL γ µ (cid:96) jR (cid:17) V µ + h.c. (5.8) L LQ S = (cid:16) g ij L ¯ Q c,jL iσ L jL + g ij R ¯ u c,iR (cid:96) jR (cid:17) S + g ij L ¯ Q c,iL iσ σ L jL S + (cid:16) h ij L ¯ u iR L jL + h ij R ¯ Q iL iσ (cid:96) jR (cid:17) R + h.c. , (5.9)where h ij and g ij are dimensionless couplings, S , S , and R are the scalar leptoquarkbosons, U µ , U µ , and V µ are the vector leptoquark bosons, and the index i ( j ) indicatesthe generation of quarks (leptons).The leptoquark Lagrangian generates the following couplings in Eq. (2.1): g S ( µ b ) = √ G F V cb ( C S ( µ b ) + C S ( µ b )) , (5.10) g P ( µ b ) = √ G F V cb ( C S ( µ b ) − C S ( µ b )) , (5.11) g L = √ G F V cb C l V , (5.12) g R = √ G F V cb C l V , (5.13) g T ( µ b ) = √ G F V cb C T ( µ b ) , (5.14)– 24 –here the Wilson coefficients in the leptoquark models are given by C SM = 2 √ G F V cb , (5.15) C l V = (cid:88) k =1 V k (cid:34) g kl L g ∗ L M S − g kl L g ∗ L M S + h l L h k ∗ L M U − h l L h k ∗ L M U (cid:35) , (5.16) C l V = 0 , (5.17) C l S = (cid:88) k =1 V k (cid:34) − g kl L g ∗ R M V − h l L h k ∗ R M U (cid:35) , (5.18) C l S = (cid:88) k =1 V k (cid:34) − g kl L g ∗ R M S − h l L h k ∗ R M R (cid:35) , (5.19) C l T = (cid:88) k =1 V k (cid:34) g kl L g ∗ R M S − h l L h k ∗ R M R (cid:35) . (5.20)These Wilson coefficients are defined at the energy scale µ = M X , where X represents aleptoquark. Above, V k denotes the relevant CKM matrix element, where the 3 correspondsto the bottom quark. In the following, we neglect the CKM-suppressed contributions from k = 1 and k = 2 in the sums. Because the neutrino is not observed, we have l = 1 , , l = 3 but not for l = 1 ,
2; hence, theconstraints for different l will be different.The renormalization-group running of the scalar and tensor Wilson coefficients from µ = M X to µ = µ b , where µ b is the mass scale of the bottom quark, is given by C S , ( µ b ) = (cid:20) α s ( m t ) α s ( µ b ) (cid:21) − (cid:20) α s ( m LQ ) α s ( m t ) (cid:21) − C S , ( m LQ ) , (5.21) C T ( µ b ) = (cid:20) α s ( m t ) α s ( µ b ) (cid:21) (cid:20) α s ( m LQ ) α s ( m t ) (cid:21) C T ( m LQ ) , (5.22)where α s ( µ ) is the QCD coupling at scale µ . Because the anomalous dimensions of the vec-tor and axial-vector currents are zero, the Wilson coefficients for V , are scale-independent.The different leptoquarks produce different effective operators as summarized below: • The S leptoquark with nonzero ( g L , g ∗ R ) generates C l V , C l S , and C l T , with therelation C l S = − C l T . • The R leptoquark with ( h L , h ∗ R ) generates C l S and C l T with the relation C l S =4 C l T . • The V leptoquark generates C l S and is tightly constrained, so we do not considerthis model. • The U leptoquark with nonzero ( g L , g ∗ R ) generates C l S and C l V . • The S and U leptoquarks with nonzero values of ( g L , g ∗ L ) and ( h L , h ∗ L ) generate C l V . – 25 –he leptoquark couplings can also be constrained using b → sν ¯ ν decays. As pointed out inRef. [39], the exclusive decays ¯ B → Kν ¯ ν and ¯ B → K ∗ ν ¯ ν provide more stringent boundsthan the inclusive mode B → X s ν ¯ ν . The U and R leptoquarks do not contribute to b → sν ¯ ν , while the left-handed couplings of S , S , and U do. (The V leptoquarkalso contributes to b → sν ¯ ν , but we do not consider this model.) The BaBar and BelleCollaborations give the following 90% C.L. upper limits [66, 67]: B ( B + → K + ν ¯ ν ) ≤ . × − , B ( B + → K ∗ + ν ¯ ν ) ≤ . × − , B ( B → K ∗ ν ¯ ν ) ≤ . × − . (5.23)In Ref. [68], these are compared with the SM predictions B SM K ≡ B ( B → Kν ¯ ν ) SM = (3 . ± . ± . × − , B SM K ∗ ≡ B ( B → K ∗ ν ¯ ν ) SM = (9 . ± . ± . × − . (5.24)Taking into account the theoretical uncertainties [68], the 90% C.L. upper bounds onthe NP contributions are B SM+NP K B SM K ≤ . , B SM+NP K ∗ B SM K ∗ ≤ . . (5.25)Following Ref. [8], the b → sν j ¯ ν i process can be described by the effective Hamiltonian H eff = 4 G F √ V tb V ∗ ts (cid:104)(cid:16) δ ij C (SM) L + C ijL (cid:17) O ijL + C ijR O ijR (cid:105) , (5.26)where the left-handed and right-handed operators are defined as O ijL =(¯ s L γ µ b L )(¯ ν jL γ µ ν iL ) ,O ijR =(¯ s R γ µ b R )(¯ ν jL γ µ ν iL ) . (5.27)The SM Wilson coefficient C (SM) L receives contributions from box and Z -penguin diagrams,which yield C (SM) L = α π sin θ W X ( m t /M W ) , (5.28)where the loop function X ( x t ) can be found e.g. in Ref. [69]. The leptoquarks that weconsider produce contributions to C ijL which, to leading order, are equal to [8] C ijL = − √ G F V tb V ∗ ts g i L g j ∗ L M S / + g i L g j ∗ L M S / − h i L h j ∗ L M U − / . (5.29a)– 26 –e obtain common coefficients for b → cτ ¯ ν l and b → sν τ ¯ ν l processes, C l L = − √ G F V tb V ∗ ts g l L g ∗ L M S / + g l L g ∗ L M S / − h l L h ∗ L M U − / . (5.30a)Hence, for l = 3 we obtain B SM+NP K B SM K = B SM+NP K ∗ B SM K ∗ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C (SM) L + C L C (SM) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5.31)while for l = 1 , B SM+NP K B SM K = B SM+NP K ∗ B SM K ∗ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C l L C (SM) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.32)When considering nonzero values only for one coupling at a time ( l = 1 , , R RatioD , R RatioD ∗ , τ B c , and B ( B → K ( ∗ ) ν ¯ ν ) yield the constraintsshown in Figures 7, 8, and 9. The cases with g i L g ∗ L in the S model, g i L g ∗ L in the S model, and h i L h ∗ L in the U model are ruled out for i = 1 , R Ratio Λ c − R RatioD and R Ratio Λ c − R RatioD ∗ planes are shown in Fig. 12. Sincethe S and U leptoquarks produce only the vector coupling g L , all ratios get rescaled bythe common factor of | g L | . The S and U models are tighly constrained and onlysmall effects are allowed. The other leptoquark models can produce substantial effects in R Ratio Λ c , with varying degrees of correlation between the mesonic and baryonic observables.The values of R (Λ c ) and R Ratio Λ c for two typical allowed combinations of the cou-plings in each model are given in Table 6. In Fig. 13, we present plots of the observables( d Γ /dq , B Λ c , A F B ) for the same values of the couplings.– 27 –odel Case Couplings R (Λ c ) R Ratio Λ c S g L g ∗ R = 0 .
332 + 0 . i , g i L g ∗ R = 0 . − . i , g L g ∗ L = 0 . − . i , g i L g ∗ L = − . − . i . ± .
011 1 . ± . S g L g ∗ R = 0 . − . i , g i L g ∗ R = − .
05 + 0 . i , g L g ∗ L = 0 . − . i , g i L g ∗ L = 0 .
018 + 0 . i . ± .
020 1 . ± . R h L h ∗ R = 0 . − . i , h i L h ∗ R = − . − . i . ± .
016 1 . ± . R h L h ∗ R = 0 . − . i , h i L h ∗ R = 0 . − . i . ± .
018 1 . ± . U h L h ∗ R = − . − . i , h i L h ∗ R = 0 .
049 + 0 . i , h L h ∗ L = − .
468 + 0 . i , h i L h ∗ L = 1 .
116 + 0 . i . ± .
019 1 . ± . U h L h ∗ R = − .
059 + 0 . i , h i L h ∗ R = 0 .
234 + 0 . i , h L h ∗ L = − .
002 + 0 . i , h i L h ∗ L = − .
135 + 0 . i . ± .
018 1 . ± . S g L g ∗ L = − .
035 + 0 . i , g i L g ∗ L = 0 .
061 + 0 . i . ± .
010 1 . S g L g ∗ L = − . − . i , g i L g ∗ L = − . − . i . ± .
011 1 . U h L h ∗ L = − . − . i , h i L h ∗ L = 0 .
003 + 0 . i . ± .
011 1 . U h L h ∗ L = − . − . i , h i L h ∗ L = 0 . − . i . ± .
010 1 . Table 6 . The values of the R (Λ c ) and R Ratio Λ c ratios for two representative cases of the couplings ofthe different leptoquark models. Above, the index i = 1 , R (Λ c ) = 0 . ± .
010 [48]. The uncertainties given aredue to the Λ b → Λ c form factor uncertainties. – 28 – - - - - [ g L g R * ] I m [ g L g R * ] Only S present - - - - - - [ g L i g R * ] I m [ g L i g R * ] Only S present - - - - [ g L g L * ] I m [ g L g L * ] Only S present - - - - [ g L i g L * ] I m [ g L i g L * ] Only S present - - - - - [ h L h R * ] I m [ h L h R * ] Only R present - - - - - - [ h L i h R * ] I m [ h L i h R * ] Only R present Figure 7 . Constraints on the S and R leptoquark models when considering one coupling at a time.Here, i = 1 , R RatioD and R RatioD ∗ in Eqs. (1.7) and (1.8) within 3 σ , satisfy B ( B c → τ − ¯ ν τ ) ≤ B ( B → K ( ∗ ) ν ¯ ν ) at 90% C.L. The allowed regions ofthe parameter space when combining all constraints are highlighted with a black mesh. – 29 – - - - - - [ h L h L * ] I m [ h L h L * ] Only U present - - - - [ h L i h L * ] I m [ h L i h L * ] Only U present - - - - [ h L h R * ] I m [ h L h R * ] Only U present - - - - [ h L i h R * ] I m [ h L i h R * ] Only U present Figure 8 . Constraints on the U leptoquark model when considering one coupling at a time. Here, i = 1 , R RatioD and R RatioD ∗ in Eqs. (1.7) and (1.8) within 3 σ and satisfy B ( B c → τ − ¯ ν τ ) ≤ – 30 – - - - [ g L g L * ] I m [ g L g L * ] Only S present - - - - [ g L i g L * ] I m [ g L i g L * ] Only S present - - - - [ h L h L * ] I m [ h L h L * ] Only U present - - - - [ h L i h L * ] I m [ h L i h L * ] Only U present Figure 9 . Constraints on the S and U leptoquark models when considering one coupling ata time. Here, i = 1 , R RatioD and R RatioD ∗ in Eqs. (1.7) and (1.8) within 3 σ , satisfy B ( B c → τ − ¯ ν τ ) ≤ B ( B → K ( ∗ ) ν ¯ ν ) at 90% C.L. Theallowed regions of the parameter space when combining all constraints are highlighted with a blackmesh. – 31 – - - - [ h L h R * ] I m [ h L h R * ] R Leptoquark - - - - [ h L h R * ] I m [ h L h R * ] R Leptoquark - - - - [ h L h R * ] I m [ h L h R * ] R Leptoquark - - - - - - - [ g L g L * ] I m [ g L g L * ] S Leptoquark - - [ g L g L * ] I m [ g L g L * ] S Leptoquark - - - - - [ g L g L * ] I m [ g L g L * ] S Leptoquark - - - - - - - - [ h L h L * ] I m [ h L h L * ] U Leptoquark - - - - [ h L h L * ] I m [ h L h L * ] U Leptoquark - - - - - - [ h L h L * ] I m [ h L h L * ] U Leptoquark
Figure 10 . Allowed regions for the couplings of the R , S , and U leptoquark models in thecase that all relevant couplings in each model are included simultaneously. We require that thecouplings reproduce the measurements of R RatioD and R RatioD ∗ in Eqs. (1.7) and (1.8) within 3 σ ,satisfy B ( B c → τ − ¯ ν τ ) ≤ B ( B → K ( ∗ ) ν ¯ ν ) at90% C.L (the latter is only relevant for the left-handed couplings in the S and U models). – 32 – - - [ g L g R * ] I m [ g L g R * ] S Leptoquark - - - [ g L g R * ] I m [ g L g R * ] S Leptoquark - - - [ g L g R * ] I m [ g L g R * ] S Leptoquark0.00 0.05 0.10 0.15 - [ g L g L * ] I m [ g L g L * ] S Leptoquark - - - - [ g L g L * ] I m [ g L g L * ] S Leptoquark - - - - [ g L g L * ] I m [ g L g L * ] S Leptoquark - - - - - [ h L h R * ] I m [ h L h R * ] U Leptoquark - - - - - [ h L h R * ] I m [ h L h R * ] U Leptoquark - - - - - [ h L h R * ] I m [ h L h R * ] U Leptoquark - - - - - - - - [ h L h L * ] I m [ h L h L * ] U Leptoquark - - - - [ h L h L * ] I m [ h L h L * ] U Leptoquark - - - - [ h L h L * ] I m [ h L h L * ] U Leptoquark
Figure 11 . Allowed regions for the couplings of the S and U leptoquark models in the case thatall relevant couplings in each model are included simultaneously. We require that the couplingsreproduce the measurements of R RatioD and R RatioD ∗ in Eqs. (1.7) and (1.8) within 3 σ , satisfy B ( B c → τ − ¯ ν τ ) ≤ B ( B → K ( ∗ ) ν ¯ ν ) at 90% C.L (the latteris only relevant for the left-handed couplings in the S model). – 33 – .05 1.10 1.15 1.20 1.25 1.30 1.351.01.21.41.61.8 R Λ C Ratio R D R a t i o S Leptoquark R Λ C Ratio R D * R a t i o S Leptoquark R Λ C Ratio R D R a t i o R Leptoquark R Λ C Ratio R D * R a t i o R Leptoquark R Λ C Ratio R D R a t i o U Leptoquark R Λ C Ratio R D * R a t i o U Leptoquark R Λ C Ratio R D R a t i o S Leptoquark R Λ C Ratio R D * R a t i o S Leptoquark R Λ C Ratio R D R a t i o U Leptoquark R Λ C Ratio R D * R a t i o U Leptoquark
Figure 12 . The allowed regions in the R Ratio Λ c − R RatioD and R Ratio Λ c − R RatioD ∗ planes for eachleptoquark model, given the allowed regions for the couplings from Figs. 10 and 11. – 34 – igure 13 . The effects of the different leptoquark models on the Λ b → Λ c τ ¯ ν τ differential decay rate(left), the ratio of the Λ b → Λ c τ ¯ ν τ and Λ b → Λ c (cid:96) ¯ ν (cid:96) differential decay rates (middle), and the Λ b → Λ c τ ¯ ν τ forward-backward asymmetry (right), for two representative choices of the couplings. Thered and blue curves correspond to the couplings from Cases 1 and 2 in Table 6, respectively, whilethe green curves correspond to the Standard Model. Because the S and U leptoquarks produceonly the vector coupling g L , the forward-backward asymmetry remains equal to the Standard Modelin those cases. The bands indicate the 1 σ uncertainties originating from the Λ b → Λ c form factors. – 35 – Conclusions
The baryonic decay Λ b → Λ c τ ¯ ν τ has the potential to shed new light on the R ( D ( ∗ ) )puzzle. Here, we studied the phenomenology of Λ b → Λ c τ ¯ ν τ in the presence of new-physics couplings with all relevant Dirac structures. In contrast to the mesonic decays,the Λ b → Λ c form factors have not yet been determined from experimental data, and it iseven more important to use form factors from lattice QCD. Here, we presented new latticeQCD results for the Λ b → Λ c tensor form factors, extending the analysis of Ref. [48].The parameters and covariance matrices of the complete set of Λ b → Λ c form factors areprovided as supplemental material.In the first part of our phenomenological analysis, we considered individual new-physicscouplings in the effective Hamiltonian in a model-independent way. After constraining thesecouplings using the R ( D ( ∗ ) ) measurements and the B c lifetime, we calculated the effectsof the NP couplings in Λ b → Λ c τ ¯ ν τ decays, focusing on the observables R (Λ c ), B Λ c ( q ),and A F B ( q ). Measurements of these observables can help in distinguishing among thedifferent NP operators. For instance, the forward-backward asymmetry A F B ( q ) tends tobe mostly above the SM value in the presence of right-handed ( g R ) or tensor ( g T ) couplings,but is lower than the SM value for most allowed values of the scalar ( g S ) coupling. Toillustrate the impact of a future R (Λ c ) measurement, we presented the constraints on allcouplings resulting from two possible ranges of R (Λ c ). The baryonic decay can tightlyconstrain all of the couplings g L , g R , g S , g P , and g T . For example, we have shown that if R Ratio Λ c = R (Λ c ) /R (Λ c ) SM is observed to have a value around 1.3, the scenario with only g P becomes ruled out by the combined constraints from R (Λ c ) and τ B c .In the second part of our phenomenological analysis, we considered explicit models inwhich multiple NP operators are present. For the two-Higgs-doublet model we found sig-nificant contribution to Λ b → Λ c τ ¯ ν τ . However, the full numerical analysis was not includedin this work as we did not consider RGE evolution which could impact the phenomenologyof the model. Models with SU (2) gauge symmetry generally cannot produce large effectsin b → cτ ¯ ν τ transitions without violating bounds from other observables such as B s mix-ing, and we therefore did not present their effects on Λ b → Λ c τ ¯ ν τ . On the other hand,we have demonstrated that some of the leptoquark models can produce large effects inthe Λ b → Λ c τ ¯ ν τ observables, in particular through scalar and tensor couplings. We havepresented correlation plots of R RatioD and R RatioD ∗ versus R Ratio Λ c , which may be helpful indiscriminating among the various models. Acknowledgments : We thank Shanmuka Shivashankara for early work on this project.This work was financially supported by the National Science Foundation under Grant Nos.PHY-1414345 (AD and AR) and PHY-1520996 (SM). SM is also supported by the RHICPhysics Fellow Program of the RIKEN BNL Research Center. AD acknowledges the hospi-tality of the Department of Physics and Astronomy, University of Hawaii, where part of thework was done. The lattice QCD calculations were carried out using high-performance com-puting resources provided by XSEDE (supported by National Science Foundation Grant– 36 –o. OCI-1053575) and NERSC (supported by U.S. Department of Energy Grant No. DE-AC02-05CH11231). – 37 –
Helicity spinors and polarization vectors
In this appendix, we give explicit expressions for the spinors and polarization vectors usedto calculate the helicity amplitudes for the decay Λ b → Λ c τ ¯ ν τ . A.1 Λ b rest frame To calculate the hadronic helicity amplitudes, we work in the Λ b rest frame and take thethree-momentum of the Λ c along the + z direction and the three-momentum of the virtualvector boson along the − z direction. The baryon spinors are then given by [70]¯ u ( ± , p Λ c ) = (cid:112) E Λ c + m Λ c (cid:18) χ †± , ∓| p Λ c | E Λ c + m Λ c χ †± (cid:19) ,u ( ± , p Λ b ) = (cid:112) m Λ b (cid:32) χ ± (cid:33) , (A.1)where χ + = (cid:32) (cid:33) and χ − = (cid:32) (cid:33) are the usual Pauli two-spinors. The polarizationvectors of the virtual vector boson are [70] (cid:15) µ ∗ ( t ) = 1 (cid:112) q ( q ; 0 , , −| q | ) ,(cid:15) µ ∗ ( ±
1) = 1 √ ± , − i, ,(cid:15) µ ∗ (0) = 1 (cid:112) q ( | q | ; 0 , , − q ) , (A.2)where q µ = ( q ; 0 , , −| q | ) is the four-momentum of the virtual vector boson in the Λ b restframe. We have q = 12 m Λ b ( m b − m c + q ) , (A.3) | q | = | p Λ c | = 12 m Λ b (cid:112) Q + Q − , (A.4)where Q ± = ( m Λ b ± m Λ c ) − q . (A.5) A.2 Dilepton rest frame
In the calculation of the lepton helicity amplitudes, we work in the rest frame of the virtualvector boson boson, which is equal to the rest frame of the τ ¯ ν τ dilepton system. We definethe angle θ τ as the angle between the three-momenta of the τ and the Λ c in this frame.The lepton spinors for p τ pointing in the + z direction and p ¯ ν τ pointing in the − z direction are ¯ u τ ( ± , p τ ) = (cid:112) E τ + m τ (cid:18) χ †± , ∓| p τ | E τ + m τ χ †± (cid:19) ,v ¯ ν τ ( , p ¯ ν τ ) = (cid:112) E ν (cid:32) χ + − χ + (cid:33) . (A.6)– 38 –e then rotate about the y axis by the angle θ τ so that after the rotation, the three-momentum of the Λ c points in the + z direction. The two-spinors transform as χ (cid:48)± = e − iθ τ σ / χ ± = (cid:32) cos( θ τ / − sin( θ τ / θ τ /
2) cos( θ τ / (cid:33) χ ± , (A.7)and χ (cid:48)†± = χ †± (cid:32) cos( θ τ /
2) sin( θ τ / − sin( θ τ /
2) cos( θ τ / (cid:33) , (A.8)and the full lepton spinors after the rotation are¯ u τ (+ , p τ ) = (cid:112) E τ + m τ (cid:18) cos( θ τ / , sin( θ τ / , −| p τ | E τ + m τ cos( θ τ / , −| p τ | E τ + m τ sin( θ τ / (cid:19) , ¯ u τ ( − , p τ ) = (cid:112) E τ + m τ (cid:18) − sin( θ τ / , cos( θ τ / , −| p τ | E τ + m τ sin( θ τ / , | p τ | E τ + m τ cos( θ τ / (cid:19) ,v ¯ ν τ ( , p ¯ ν τ ) = (cid:112) E ν cos( θ τ / θ τ / − cos( θ τ / − sin( θ τ / . (A.9)The polarization vectors of the virtual vector boson in this frame are (cid:15) µ ∗ ( t ) = (1; 0 , , ,(cid:15) µ ∗ ( ±
1) = 1 √ ± , − i, ,(cid:15) µ ∗ (0) = (0; 0 , , − . (A.10)The three-momentum and energy of the τ lepton in this frame can be written as | p τ | = (cid:112) q v / ,E τ = | p τ | + m τ / (cid:112) q , (A.11)where v = (cid:115) − m τ q . (A.12)– 39 – eferences [1] BaBar collaboration, J. P. Lees et al.,
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