Form Factors for Lambda_b -> Lambda Transitions in SCET
IIPPP/11/70DCPT/11/140(incl. erratum)
Form Factors for Λ b → Λ Transitions in SCET
Thorsten Feldmann ‡ , Matthew W Y Yip § IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK
Abstract
We present a systematic discussion of Λ b → Λ transition form factors in the frameworkof soft-collinear effective theory (SCET). The universal soft form factor, which enters thesymmetry relations in the limit of large recoil energy, is calculated from a sum-rule analysisof a suitable SCET correlation function. The same method is applied to derive the leadingcorrections from hard-collinear gluon exchange at first order in the strong coupling constant.We present numerical estimates for form factors and form-factor ratios and their impact ondecay observables in Λ b → Λ µ + µ − decays. ‡ Address after 1 Nov 2011:
Theoretische Elementarteilchenphysik, Naturwissenschaftlich Techn. Fakult¨at,Universit¨at Siegen, 57068 Siegen, Germany , email:[email protected] § email:[email protected] a r X i v : . [ h e p - ph ] O c t ontents Λ b → Λ Form Factors 2 b → Λ µ + µ − Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Λ b → Λ µ + µ −
18B Alternative Form-Factor Parametrizations 19
B.1 Convention by Chen and Geng . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.2 Symmetry-Based Form-Factor Parametrization . . . . . . . . . . . . . . . . . . 20
C Corrections to SCET Symmetry Relations 21
C.1 Hard Vertex Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21C.2 Hard-Collinear Gluon Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 21C.3 Form-Factor Relations to O ( α s ) Accuracy . . . . . . . . . . . . . . . . . . . . . 22 D Light-Cone Distribution Amplitudes 23
D.1 Distribution Amplitudes for the Λ b baryon . . . . . . . . . . . . . . . . . . . . . 23D.2 Simplified Set-Up with Scalar Di-quark . . . . . . . . . . . . . . . . . . . . . . . 25 Motivation
The decays Λ b → Λ (cid:96) + (cid:96) − offer the possibility to study rare semi-leptonic and radiative b → s transitions within the Standard Model (SM) and beyond. The observables in the baryonic tran-sitions provide complementary phenomenological information compared to the correspondingmesonic or inclusive decays, see e.g. [1–9]. The decay Λ b → Λ µ + µ − has been recently measuredby the CDF collaboration [10] with a branching ratio of the order 10 − .Theoretical predictions for the exclusive decay matrix elements require non-perturbativehadronic input. To first approximation, this can be parametrized in terms of baryonic tran-sition form factors for vector, axial-vector and tensor currents. The number of independentform factors drastically reduces in the limit of infinitely heavy b -quark mass, exploiting theapproximate symmetries in heavy-quark effective theory (HQET), see e.g. [11, 12]. Additionalsimplification is expected in the kinematic limit of large recoil energy, E Λ → ∞ , where the num-ber of independent form factors is known to be reduced further [13], and part of the correctionsto this limit should factorize in terms of process-independent hadronic quantities (light-conedistribution amplitudes, LCDAs, [14,15]) and perturbative interaction kernels, in a similar wayas it has been discussed for the analogous mesonic transitions [16]. The non-perturbative cal-culation of the remaining hadronic transition form factors can, for instance, be obtained fromQCD sum rules. In the limit of large recoil energy, a systematic expansion in the heavy-quarkmass is achieved in the framework of soft-collinear effective theory (SCET [17, 18]), where onestudies the spectrum of correlation functions between the decay current and an interpolatingcurrent with the quantum numbers of the light hadron [19] (see also [20]).The aim of this paper is to provide a systematic analysis of Λ b → Λ form factors, startingfrom the symmetry relations in the heavy-quark/large-energy limit. To this end, in the nextsection, we will present a convenient definition of the 10 independent physical form factors,in terms of which the HQET and SCET symmetry relations look particularly simple. In thefollowing section 3, we derive the leading expressions for the universal (“soft”) form factor ξ Λ from a sum-rule analysis of an appropriate correlation function in SCET, involving the LCDAsof the Λ b baryon. The same method is used to calculate the leading correction ∆ ξ Λ to the form-factor symmetry relations that arise from hard-collinear gluon exchange. In contrast to themesonic case, one of the light spectator quarks still does not take part in the hard-scatteringprocess, and therefore the corresponding effect could not be calculated in the framework ofQCD factorization. (A similar discussion has been led for the electromagnetic form factorsof the nucleon in [21].) The sum-rule expressions are analysed numerically in section 4. Wefocus on the theoretical uncertainty related to various hadronic input parameters entering theestimate for the soft form factor ξ Λ . Part of these uncertainties drops out in the ratio ∆ ξ Λ /ξ Λ .We also provide estimates for the partial branching fractions (transverse and longitudinal rate,forward-backward asymmetry) of Λ b → Λ µ + µ − in the large recoil (small q ) limit, beforewe conclude. Finally, in our appendix, we collect the expressions for the double-differentialΛ b → Λ µ + µ − decay rates, and discuss an alternative form-factor basis that is optimized fora systematic discussion of power corrections to the symmetry relations. We also extract thehard vertex corrections to the Λ b → Λ form factors arising from the matching of the decaycurrents from QCD onto SCET, and we identify 5 form-factor relations that are unaffected byshort-distance O ( α s ) corrections. Finally, we summarize the relevant information on baryonLCDAs and briefly comment on a simplified set-up with elementary light di-quark fields in thelight and heavy baryon. 1 Λ b → Λ Form Factors
In the following, we provide some useful definitions for Λ b → Λ form factors that aim to improveprevious definitions, as discussed for instance in [1, 4], in two aspects: (i) the form factors aredefined from a helicity basis, (ii) the form factors are normalized to the limit of point-likehadrons. As a result, our form factor convention leads to rather simple expressions for partialrates, unitarity bounds (cf. [22, 23]) and symmetry relations in the HQET or SCET limit.
The form factors for Λ b → Λ transitions can be parametrized as follows. Starting with thevector and scalar decay currents, we have ( q = s ( x ) and b = b ( x ) denote the light and heavyquark fields in the b → s transitions) (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q γ µ b | Λ b ( p, s ) (cid:105) = ¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:26) f ( q ) ( M Λ b − m Λ ) q µ q + f + ( q ) M Λ b + m Λ s + (cid:18) p µ + p (cid:48) µ − q µ q ( M b − m ) (cid:19) + f ⊥ ( q ) (cid:18) γ µ − m Λ s + p µ − M Λ b s + p (cid:48) µ (cid:19)(cid:27) u Λ b ( p, s ) , (2.1)where we have defined s ± = ( M Λ b ± m Λ ) − q . (2.2)At vanishing momentum transfer, q →
0, one further has the kinematic constraint f (0) = f + (0) . (2.3)The individual form factors are defined in such a way that they correspond to time-like (scalar),longitudinal and transverse polarization with respect to the momentum-transfer q µ for f , f + and f ⊥ , respectively (cf. [22, 23]). The normalization is chosen in such a way that for f , f + , f ⊥ → , one recovers the expression for a transition between point-like baryons, i.e. (cid:104) Λ | ¯ q Γ b | Λ b (cid:105) → ¯ u Λ Γ u Λ b . The form factor f is also obtained from the scalar decay current via the equationsof motion (e.o.m.), (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q b | Λ b ( p, s ) (cid:105) = q µ M b − m q (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q γ µ b | Λ b ( p, s ) (cid:105) = f ( q ) M Λ b − m Λ M b − m q ¯ u Λ ( p (cid:48) , s (cid:48) ) u Λ b ( p, s ) . (2.4)2he expression for the axial-vector and pseudo-scalar currents can be obtained by appropriatelychanging the relative sign between the light and heavy baryon mass, and we thus define (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q γ µ γ b | Λ b ( p, s ) (cid:105) = − ¯ u Λ ( p (cid:48) , s (cid:48) ) γ (cid:26) g ( q ) ( M Λ b + m Λ ) q µ q + g + ( q ) M Λ b − m Λ s − (cid:18) p µ + p (cid:48) µ − q µ q ( M b − m ) (cid:19) + g ⊥ ( q ) (cid:18) γ µ + 2 m Λ s − p µ − M Λ b s − p (cid:48) µ (cid:19)(cid:27) u Λ b ( p, s ) , (2.5)with the kinematic constraint g (0) = g + (0) at q →
0, and (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q γ b | Λ b ( p, s ) (cid:105) = q µ M b + m q (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q γ γ µ b | Λ b ( p, s ) (cid:105) = g ( q ) M Λ b + m Λ M b + m q ¯ u Λ ( p (cid:48) , s (cid:48) ) γ u Λ b ( p, s ) . (2.6)Finally, for the tensor and pseudo-tensor current, we write (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q iσ µν q ν b | Λ b ( p, s ) (cid:105) = − ¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:26) h + ( q ) q s + (cid:18) p µ + p (cid:48) µ − q µ q ( M b − m ) (cid:19) +( M Λ b + m Λ ) h ⊥ ( q ) (cid:18) γ µ − m Λ s + p µ − M Λ b s + p (cid:48) µ (cid:19)(cid:27) u Λ b ( p, s ) , (2.7)and (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q iσ µν γ q ν b | Λ b ( p, s ) (cid:105) = − ¯ u Λ ( p (cid:48) , s (cid:48) ) γ (cid:26) ˜ h + ( q ) q s − (cid:18) p µ + p (cid:48) µ − q µ q ( M b − m ) (cid:19) +( M Λ b − m Λ ) ˜ h ⊥ ( q ) (cid:18) γ µ + 2 m Λ s − p µ − M Λ b s − p (cid:48) µ (cid:19)(cid:27) u Λ b ( p, s ) . (2.8)Again, the normalization of the form factors h ⊥ , + , ˜ h ⊥ , + has been fixed by the case of point-likehadrons. This makes 10 independent form factors for the general case, after the e.o.m. havebeen taken into account. In terms of the helicity form factors, the differential decay width forΛ b → Λ µ + µ − takes a particularly simple form, see Appendix A. An alternative parametrization,which is based on the large and small projections of energetic or massive fermion spinors, canbe found in Appendix B.2. For convenience, we summarize in Appendix B.1 the relations of the 10 helicity form factors to the variousform factors defined in [1]. .2 HQET Limit The number of independent Λ b → Λ form factors reduces considerably in the heavy quarklimit, M b → ∞ (see e.g. [12]), when we use the heavy-baryon velocity v µ to project onto thelarge spinor components h ( b ) v = / v h ( b ) v of the heavy b -quark field, (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q Γ b | Λ b ( p, s ) (cid:105) → (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q Γ h ( b ) v | Λ b ( v, s ) (cid:105)(cid:39) ¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:0) A ( v · p (cid:48) ) + / v B ( v · p (cid:48) ) (cid:1) Γ u Λ b ( v, s ) . (2.9)Here Γ is an arbitrary Dirac matrix, and p µ = M Λ b v µ (cid:39) M b v µ . Furthermore, | Λ b ( v, s ) (cid:105) is aheavy-baryon state, and u Λ b ( v, s ) = / v u Λ b ( v, s ) a heavy-baryon spinor in HQET. In the heavy-quark limit, m Λ , v · p (cid:48) (cid:28) M b , the helicity form factors are related to the two HQET form factorsin (2.9) as follows,small recoil: f ( q ) (cid:39) g + ( q ) (cid:39) g ⊥ ( q ) (cid:39) ˜ h + ( q ) (cid:39) ˜ h ⊥ ( q ) (cid:39) A ( v · p (cid:48) ) + B ( v · p (cid:48) ) ,g ( q ) (cid:39) f + ( q ) (cid:39) f ⊥ ( q ) (cid:39) h + ( q ) (cid:39) h ⊥ ( q ) (cid:39) A ( v · p (cid:48) ) − B ( v · p (cid:48) ) , (2.10)with q = M b − M Λ b v · p (cid:48) + m . In the kinematic region of large recoil energy for the Λ baryon in the rest frame of the decayingΛ b , further simplifications arise [13,16]. A formal derivation can be obtained from soft-collineareffective theory (SCET) [17, 18]. To this end, we consider the matrix element of the leadingcurrent involving the collinear quark field ξ ≡ / n − / n + q with two light-like vectors n − = n = 0,satisfying ( n + + n − ) / v and n − · n + = 2. In the large-energy limit, we can further set p (cid:48) µ (cid:39) ( n + p (cid:48) ) n µ − / m Λ →
0. This amounts to (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ ξW Γ Y † h ( b ) v | Λ b ( v, s ) (cid:105) = ¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:0) A ( q ) + / v B ( q ) (cid:1) / n + / n − u Λ b ( v, s )= A ( q ) ¯ u Λ ( p (cid:48) , s (cid:48) ) / n + / n − u Λ b ( v, s ) + B ( q ) ¯ u Λ ( p (cid:48) , s (cid:48) ) / n − u Λ b ( v, s ) . (2.11)Here W ( Y ) are Wilson lines in SCET that render the definition of the form factors invariantunder collinear (soft) gauge transformations. In the following, we will always drop the Wilsonlines (which corresponds to light-cone gauges for collinear and soft gluon fields). Exploitingthe (approximate) equations of motion for ¯ u Λ ( p (cid:48) , s (cid:48) ) / n − (cid:39)
0, this simplifies to (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ ξ Γ h ( b ) v | Λ b ( v, s ) (cid:105) (cid:39) ξ Λ ( n + p (cid:48) ) ¯ u Λ ( p (cid:48) , s (cid:48) ) Γ u Λ b ( v, s ) , (2.12)where ξ Λ ( n + p (cid:48) ) corresponds to A ( v · p (cid:48) ) in (2.10) and defines the so-called “soft” Λ b → Λ formfactor, while the contribution from B ( v · p (cid:48) ) is negligible. In the SCET limit, n + p (cid:48) ∼ M Λ b , allhelicity form factors are thus equal to ξ Λ ( n + p (cid:48) ),large recoil: f ( q ) ≈ f + ( q ) ≈ f ⊥ ( q ) ≈ h + ( q ) ≈ h ⊥ ( q ) ≈ g ( q ) ≈ g + ( q ) ≈ g ⊥ ( q ) ≈ ˜ h + ( q ) ≈ ˜ h ⊥ ( q ) ≈ ξ Λ ( n + p (cid:48) ) , (2.13)with q = M b − M Λ b n + p (cid:48) + m (cid:16) − M Λ b n + p (cid:48) (cid:17) . 4igure 1: Leading diagrams for SCET correlation functions involving the soft form factor ξ Λ (left) andthe form factor ∆ ξ Λ for the hard-scattering corrections in the large-recoil limit. The leading corrections to the form factor relations from hard-collinear gluon exchangecan be described by a form factor term that takes into account the corresponding sub-leadingcurrents in SCET, which contain one additional (transverse) hard-collinear gluon field [18]. Ifwe neglect additional hard vertex corrections for simplicity, the form factors relate to matrixelements of local SCET currents. In the limit M b → ∞ , ( n + p (cid:48) ) → ∞ , these matrix elementcan again be described by a single form factor ∆ ξ Λ , which we define by (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ ξ ˜Γ gA ⊥ µ h ( b ) v | Λ b ( v, s ) (cid:105) ≡ M Λ b ∆ ξ Λ ( n + p (cid:48) ) ¯ u Λ ( p (cid:48) , s (cid:48) ) γ ⊥ µ ˜Γ u Λ b ( v, s ) , (2.14)where the basis of independent Dirac matrices can be reduced to ˜Γ = / n + { , γ ⊥ ν , γ } . Herewe have exploited again that, due to the heavy-quark spin symmetry, the Dirac matrix in theeffective-theory decay current couples trivially to the heavy baryon spinor. The matching ofthe various decay currents in QCD onto the SCET currents is process-independent and can betaken into account by appropriate Wilson coefficients. For convenience, we have summarizedthe relevant results in Appendix C. Our next aim is to obtain non-perturbative estimates for the form factors ξ Λ and ∆ ξ Λ in thelarge-recoil limit, following the analogous calculation as for the B → π ( ρ ) form factors fromSCET sum rules in [19]. The leading diagrams for the calculation of the respective correlationfunctions are shown in Fig. 1. We start with a correlation function, where the Λ baryon in the final state is replaced by aninterpolating current sharing the same quantum numbers. We choose J Λ ( x ) ≡ (cid:15) abc (cid:16) u a ( x ) C γ / n + d b ( x ) (cid:17) s c ( x ) , (3.15)which is normalized by the matrix element (cid:104) | / n ∓ / n ± J Λ (0) | Λ( p (cid:48) , s (cid:48) ) (cid:105) = ( n + p (cid:48) ) f Λ / n ∓ / n ± u Λ ( p (cid:48) , s (cid:48) ) , (3.16)5nd thus corresponds to a leading term in the large-energy limit. A sum-rule estimate [15] gives f Λ (cid:39) (6 . ± . × − GeV for the involved decay constant of the Λ baryon (for comparison,for the nucleon f N (cid:39) . × − GeV has been estimated in [24]). The various light-quarkfields can be decomposed into soft and hard-collinear fields to match the above current ontoSCET. At tree-level, it is sufficient to calculate the correlation function in QCD and performthe appropriate kinematic limits for the propagators.We now define the correlation function between a weak decay current and the interpolatingcurrent J Λ , and consider it as a function of the small (Euclidean) momentum component( n − p (cid:48) ) < p (cid:48)⊥ ≡ n + p (cid:48) ). In order to extract theuniversal soft form factor ξ Λ , we consider the projection of a decay current on the large spinorcomponents for the light and heavy quark fields. We therefore haveΠ Λ ( n − p (cid:48) ) ≡ i (cid:90) d x e ip (cid:48) x (cid:104) | T (cid:20) / n − / n + J Λ ( x ) (cid:20) ¯ s (0) / n + / n − v b (0) (cid:21)(cid:21) | Λ b ( p ) (cid:105) . (3.17)The time-ordered product of the two currents can be calculated in perturbation theory. Theleading diagram just corresponds to the one shown on the left-hand side in Fig. 1, which refersto the situation where the two strange-quark fields are contracted to a propagator, while theup- and down-quark merely act as spectators. Employing the kinematic limits in the QCDdiagram, and performing a Fourier transform such that ω , = ( n − k , ) correspond to thelight-cone momenta of the up- and down-quark, the correlation function at leading order isgiven byΠ Λ ( n − p (cid:48) ) (cid:39) (cid:90) dω dω ω + ω − n − p (cid:48) − i(cid:15) (cid:104) | (cid:15) abc (cid:16) u a ( ω ) C γ / n + d b ( ω ) (cid:17) / n − b cv | Λ b ( v, s ) (cid:105) = f (2)Λ b (cid:90) dω dω ψ ( ω , ω ) ω + ω − n − p (cid:48) − i(cid:15) / n − u Λ b ( v, s )= f (2)Λ b (cid:90) ∞ dω ω (cid:82) du ˜ ψ ( ω, u ) ω − n − p (cid:48) − i(cid:15) / n − u Λ b ( v, s ) . (3.18)To arrive at the second and third line, we have used the momentum-space projector for theheavy Λ b baryon, following from the definition of its light-cone distribution amplitudes asderived in Appendix D. To leading order, the result for the correlation function thus onlyinvolves the sum of the spectator-quark momenta and therefore only requires the partiallyintegrated LCDA φ ( ω ) ≡ ω (cid:90) du ˜ ψ ( ω, u ) . The remaining analysis is then very similar to the B → π, ρ case discussed in [19]. For thehadronic side of the sum rule, the contribution of the Λ baryon to the correlator is given byΠ Λ (cid:12)(cid:12) res . (cid:39) (cid:88) s (cid:48) / n − / n + (cid:104) | J Λ | Λ( p (cid:48) , s (cid:48) ) (cid:105)(cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q / n + / n − Γ h ( b ) v | Λ b ( v ) (cid:105) m − ( p (cid:48) ) = ( n + p (cid:48) ) f Λ ξ Λ ( n + p (cid:48) ) m − ( n + p (cid:48) )( n − p (cid:48) ) (cid:88) s (cid:48) / n − / n + u Λ ( p (cid:48) , s (cid:48) ) ¯ u Λ ( p (cid:48) , s (cid:48) ) / n + / n − u Λ b ( v, s )= ( n + p (cid:48) ) f Λ ξ Λ ( n + p (cid:48) ) m / ( n + p (cid:48) ) − ( n − p (cid:48) ) / n − u Λ b ( v, s ) . (3.19)6omparing the perturbative and hadronic parts of the sum rule, subtracting the continuum(which is modelled by the perturbative result above a threshold parameter ω s ), and performinga Borel transformation in terms of the Borel parameter ω M , we obtain the LO sum rule e − m / ( n + p (cid:48) ) ω M ( n + p (cid:48) ) f Λ ξ Λ ( n + p (cid:48) ) = f (2)Λ b (cid:90) ω s dω φ ( ω ) e − ω/ω M , (3.20)which takes the analogous form as for the B → π, ρ case, only that the distribution amplitudefor the spectator anti-quark in the B -meson is replaced by the effective LCDA for the spectatordi-quark in the Λ b baryon.The formal scaling of the (tree-level) result for ξ Λ with the large-energy variable can bederived by further considering the limit ω s,M ∼ Λ . n + p (cid:48) (cid:28) (cid:104) ω (cid:105) , which allows one to expand theLCDA of the Λ b baryon around ω = 0 in the integrand. This yields ξ Λ ( n + p (cid:48) ) ≈ f (2)Λ b ω M φ (cid:48) (0)( n + p (cid:48) ) f Λ e m / ( n + p (cid:48) ) ω M (cid:18) − e − ω s /ω M (1 + ω s ω M ) (cid:19) . (3.21)where φ (cid:48) (0) ∼ /ω with ω ∼ (cid:104) ω (cid:105) being the typical light-come momentum of the light di-quark in the heavy baryon (see Appendix D). In this limit, the soft Λ b → Λ form factor thusscales as 1 / ( n + p (cid:48) ) with the large energy of the final state baryon. Compared to the mesoniccase [19], one encounters an additional factor of 1 / ( n + p (cid:48) ) which physically can be traced backto the phase-space suppression of the additional spectator quark. Technically, the differencebetween the mesonic and baryonic case stems from the fact that the B-meson LCDA φ − B ( ω )does not vanish at the endpoint, while φ ( ω ) vanishes linearly.We should stress that radiative corrections to the sum rule will lead to additional non-analytical dependence of the form factors on ( n + p (cid:48) ) with logarithmically enhanced perturbativecoefficients. Part of these corrections are universal and can be uniquely factorized in terms of:(i) hard vertex corrections absorbed in Wilson coefficients of SCET decay currents, (ii) a jetfunction, absorbing the hard-collinear emissions from the strange-quark propagator in SCET,(iii) the soft evolution of the LCDAs of the Λ b baryon. To this accuracy, we obtain an analogousresult as discussed for the mesonic case [19], F i ( q ) (cid:39) C i ( n + p (cid:48) , µ ) · e m / ( n + p (cid:48) ) ω M f (2)Λ b ( n + p (cid:48) ) f Λ (cid:90) ω s dω (cid:48) e − ω (cid:48) /ω M (cid:40) (cid:20) α s C F π (cid:18) − π + 3 ln (cid:20) µ ω (cid:48) ( n + p (cid:48) ) (cid:21) + 2 ln (cid:20) µ ω (cid:48) ( n + p (cid:48) ) (cid:21)(cid:19)(cid:21) φ ( ω (cid:48) , µ )+ α s C F π ω (cid:48) (cid:90) dω (cid:18) (cid:20) µ ( ω (cid:48) − ω )( n + p (cid:48) ) (cid:21) + 3 (cid:19) φ ( ω (cid:48) , µ ) − φ ( ω, µ ) ω (cid:48) − ω (cid:41) , (3.22)where F i ( q ) denotes a generic form factor with the corresponding Wilson coefficient C i . Theleading (double-logarithmic) µ -dependence cancels between the 3 terms on the right-hand side,thanks to the renormalization-group equations (see e.g. [14, 17, 25–27]), dd ln µ C i ( n + p (cid:48) , µ ) = − α s C F π Γ (1)cusp ln µM b C i ( n + p (cid:48) , µ ) + . . . , (3.23) dd ln µ φ ( ω, µ ) = − α s C F π Γ (1)cusp ln µω φ ( ω, µ ) + . . . (3.24)7ith the cusp-anomalous dimension Γ (1)cusp = 4. Evaluating the terms in curly brackets in (3.22)at a factorization scale of order µ ∼ ω s ( n + p (cid:48) ) and evolving the Wilson coefficients down tothat scale, one achieves the resummation of the leading Sudakov double logarithms.Additional process-dependent corrections to (3.22) arise from hard-collinear gluon exchangebetween the strange quark and the spectator quarks in SCET. As shown in [19], these will leadto logarithmically enhanced terms which are sensitive to the endpoint behaviour of φ ( ω, µ ).The explicit derivation of these terms is left for future work. As explained above, sub-leading currents in the SCET Lagrangian will induce violations of theform-factor symmetry relations in the large recoil limit. Contributions involving hard-collineargluon exchange can be treated perturbatively in SCET correlation functions. The leading effectrequires one to calculate the matrix element in (2.14), whose leading contribution arises fromhard-collinear gluon exchange with one of the two spectator quarks in the baryons, see thecorresponding diagram on the r.h.s. of Fig. 1. From the perspective of QCD factorization,this diagram represents an intermediate (hybrid) case, where some of the constituents undergocalculable short-distance interactions, while the remaining spectator quark remains undisturbedand is thus forced to populate the endpoint region in phase space.In the sum-rule approach, as before, we define a correlation function (in light-cone gauge)Π µ Λ ( n − p (cid:48) ) ≡ i (cid:90) d x e ip (cid:48) x (cid:104) | T (cid:20) / n + / n − J Λ ( x ) (cid:104) ¯ s (0) ˜Γ gA µ ⊥ (0) b (0) (cid:105)(cid:21) | Λ b ( p ) (cid:105) . (3.25)Notice that this time, we have to use the opposite light-cone projector acting on J Λ , as comparedto the correlation function used to extract the soft form factor ξ Λ . It projects on the sub-leadingtransverse momentum in the numerator of the strange-quark propagator which is required fromrotational invariance in the transverse plane. The light-quark momenta in the Λ b baryon againare denoted as k , respectively, with k = ω + k ⊥ , k = ω + k ⊥ , and ω = ω + ω = n − k . Using the momentum-space projector for the LCDAs of Λ b as given in Appendix D, andassuming isospin symmetry of strong interactions, we obtainΠ µ ( n − p (cid:48) ) = − i g s C F f (2)Λ b (cid:90) dω (cid:90) dω × (cid:90) d D l (2 π ) D l ⊥ + ( n + l )( n − l )][ l ⊥ + ( n + l ) ( n − l − ω )][ l ⊥ + ( n + p (cid:48) + n + l )( n − p (cid:48) + n − l − ω )] × tr (cid:104) ˜ M ( k , k ) Cγ / n + (/ k − l/ ) γ µ ⊥ (cid:105) / n + / n − l/ − / k − / k ) ˜Γ u Λ b ( v, s ) , (3.26)where the square bracket around propagator denominators imply a “+ i(cid:15) ” description. TheDirac trace is easily calculated astr (cid:104) ˜ M ( k , k ) Cγ / n + (/ k − l/ ) γ µ ⊥ (cid:105) = − ψ ( ω , ω ) l µ ⊥ + 2( n + l ) (cid:32) G ( ω , ω ) ∂∂k ⊥ µ + H ( ω , ω ) ∂∂k ⊥ µ (cid:33) . (3.27)8his yieldsΠ µ ( n − p (cid:48) ) = i g s C F f (2)Λ b (cid:90) dω (cid:90) dω × (cid:90) d D l (2 π ) D l ⊥ / ( D − ψ ( ω , ω ) + ( n + l ) ( G ( ω , ω ) + H ( ω , ω ))[ l ⊥ + ( n + l )( n − l )][ l ⊥ + ( n + l ) ( n − l − ω )][ l ⊥ + ( n + p (cid:48) + n + l )( n − p (cid:48) + n − l − ω )] × / n + / n − γ µ ⊥ ˜Γ u Λ b ( v, s ) . (3.28)Notice that both terms contribute at the same order in the SCET correlator, since l ⊥ ∼ ( n + l ) ω ∼ m b Λ. However, the contributions from ψ and G will give formerly sub-leadingcontributions to the sum-rule for ω →
0. Performing the integration over ( n − l ) and l ⊥ , theBorel transformation and continuum subtraction, we obtainˆ B Π µ Λ ( ω M ) | subtr . = − α s C F f (2)Λ b π (cid:90) dω (cid:90) dω (cid:90) ω s dω (cid:48) ω M e − ω (cid:48) /ω M × (cid:26) ( ω + ( ω (cid:48) − ω ) θ ( ω − ω (cid:48) )) θ ( ω (cid:48) − ω )4 ω ψ ( ω , ω )+ θ ( ω − ω (cid:48) ) θ ( ω (cid:48) − ω )2 ω ( G ( ω , ω ) + H ( ω , ω )) (cid:27) × / n + / n − γ µ ⊥ ˜Γ u Λ b ( v, s ) , (3.29)In the limit ω s , ω M (cid:28) (cid:104) ω , (cid:105) , the typical momentum of the light quarks in the heavy baryon,the integral can be simplified. Since ω ≤ ω (cid:48) ≤ ω s , we may approximate ω (cid:39) ω of the di-quark compound. In this limit, we haveˆ B Π µ Λ ( ω M ) | subtr . (cid:39) − α s C F π γ µ ⊥ ˜Γ u Λ b ( v, s ) f (2)Λ b (cid:90) ∞ dωω H (0 , ω ) (cid:124) (cid:123)(cid:122) (cid:125) × (cid:16) ω M − e − ω s /ω M ( ω M + ω s ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) , Λ b J Λ (3.30)and the correlation function factorizes, as indicated, into an inverse moment of the heavy-baryon LCDA and a function of the Borel and threshold parameter describing the spectrumof the interpolating current for the light baryon. For the hadronic side of the sum rule, thecontribution of the Λ baryon to the correlator is now given byΠ µ Λ (cid:12)(cid:12) reson . = f Λ m Λ M Λ b ∆ ξ Λ m / ( n + p (cid:48) ) − ( n − p (cid:48) ) γ µ ⊥ ˜Γ u Λ b ( v, s ) , (3.31)9hich leads to the sum rule e − m / ( n + p (cid:48) ) ω M f Λ M Λ b m Λ /ω M ∆ ξ Λ = − α s C F f (2)Λ b π (cid:90) dω (cid:90) dω (cid:90) ω s dω (cid:48) ω M e − ω (cid:48) /ω M × (cid:26) ( ω + ( ω (cid:48) − ω ) θ ( ω − ω (cid:48) )) θ ( ω (cid:48) − ω )4 ω ψ ( ω , ω )+ θ ( ω − ω (cid:48) ) θ ( ω (cid:48) − ω )2 ω ( G ( ω , ω ) + H ( ω , ω )) (cid:27) (3.32) (cid:39) − α s C F π f (2)Λ b (cid:90) ∞ dωω F (0 , ω ) × (cid:16) ω M − e − ω s /ω M ( ω M + ω s ) (cid:17) . (3.33)The correction to the soft form factor, in the large recoil limit, thus scales as∆ ξ Λ /ξ Λ ∼ α s ω m Λ n + p (cid:48) M Λ b , i.e. it formally has the same power-counting in terms of Λ QCD /M b (although, numerically, theratio ω /m Λ is small), but a less pronounced ( n + p (cid:48) ) dependence than the soft form factor.Notice that in the ratio ∆ ξ Λ /ξ Λ , the dependence on the baryon decay constants drops out,while the sensitivity to the sum-rule parameters and the features of the LCDAs of the Λ b baryon remains. In the following section we present some numerical results for the soft Λ b → Λ form factor ξ Λ and the correction from hard-collinear gluon exchange, ∆ ξ Λ , in the large-recoil limit. Thenumerical predictions involve a number of hadronic parameters with respective uncertainties,for which we summarize our default choices in Table 1 for convenience. For the shape of theLCDAs, we use the simple exponential models as summarized in Appendix D.1.Table 1: Summary of hadronic input parameters
Parameter central value remarksthreshold s Λ(1600) ω s ≡ s / ( n + p (cid:48) )Borel M ω M ≡ M / ( n + p (cid:48) )decay constant f Λ · − GeV [15]decay constant f (2)Λ b .
030 GeV [14]LCDA par. ω
300 MeV (our estimate)10 .1 Soft Form Factor
The value for the soft form factor is estimated from the LO sum rule (3.20). We will alsocompare with the approximation (3.21). The default value for the threshold parameter istaken from the position of the next highest b -baryon resonance with I ( J P ) = 0(1 / + ). Forthe relevant LCDAs, we will use the model (D.76) as described in the Appendix. In the softform factor, only the partially integrated function φ ( ω ) appears. In our model, it takes thesimple form φ ( ω ) := ωω e − ω/ω , which is illustrated on the left of Fig. 2.For the default parameter values in Table 1, the soft form factor at maximal recoil isestimated as ξ Λ ( n + p (cid:48) = M Λ b ) (cid:39) .
38 central value, from (3.20),which – within the uncertainties – is consistent with estimates from other methods in [1, 3].We remark in passing, that the authors of [4] estimate the Λ b → Λ form factors with a similarset-up, but without performing the large-recoil limit in SCET explicitly. They quote a rathersmall value g ( q = 0) = 0 . ± .
003 for one of the form factors that, as we understand,should coincide with ξ Λ ( n + p (cid:48) = M Λ b ) in the heavy-quark limit.The dependence of ξ Λ on the LCDA parameter ω is shown on the right of Fig. 2. The energydependence is plotted in Fig. 3. The dependence on the sum-rule parameters (at maximal recoil)is shown in Fig. 4. The following observations can be made: • For values of ω around 300 MeV or smaller, as extracted from the analysis in [14],the approximate formula (3.21) does not yield a reliable estimate, because numerically ω (cid:39) ω s (cid:39) ω M . The respective value of ξ Λ is overestimated by more than a factor 2 inthis case. On the other hand, compared to the mesonic case, one might have expectedlarger values of ω in the baryonic LCDA in the first place. • In any case, the sum-rule result for ξ Λ is very sensitive to the shape of the LCDA ingeneral and the value of ω in particular. Varying ω in a reasonable range between 0 . . ξ Λ . More independent information on theLCDAs of the Λ b baryon and the relevant hadronic parameters is clearly needed to reachreasonable precision in this kind of sum-rule analysis. • For small values of ω , the energy dependence of the form factor follows an approximate1 / ( n + p (cid:48) ) behaviour, rather than a 1 / ( n + p (cid:48) ) behaviour as predicted by (3.21). • The dependence on the Borel parameter is very weak (less than a few percent) andnegligible compared to the other uncertainties. • The dependence on the threshold parameter is almost linear, and the LO sum-rule resultthus depends on the modelling of the continuum contribution to the correlator in anessential way. Varying ω s between 0 .
35 and 0 .
55 GeV, the induced uncertainty for ξ Λ atmaximal recoil amounts to about 10-20%. One should, however, be aware that one may encounter pollution from baryon states with opposite parity,see the recent discussion in [28].
Left: Functional form of the partially integrated LCDA φ ( ω ) for the exponential modeland ω = 300 MeV. Right: Dependence of ξ Λ ( n + p (cid:48) = M Λ b ) on the value of ω . Figure 3:
Dependence of the soft form factor on ( n + p (cid:48) ). Left: energy dependence from the LO sum rule(3.20). Right: comparison of the LO sum rule (3.20) –thick solid line – , with the approximate formula(3.21) – thick dashed line –, and a power-like behaviour with 1 / ( n + p (cid:48) ) (dash-dotted) or 1 / ( n + p (cid:48) ) (dotted). Figure 4:
Dependence of the soft form factor on the sum-rule parameters (for maximal recoil, n + p (cid:48) = M Λ b ). Left: Dependence on the Borel parameter ω M . Right: Dependence on the threshold parameter ω s . b → Λ form factors at large recoil still suffers from sizeable uncertainties, mostly from theΛ b LCDAs and the threshold parameter. The same is true for the energy-dependence of theform factor which varies between a 1 / ( n + p (cid:48) ) behaviour (small values of ω ) and a 1 / ( n + p (cid:48) ) behaviour (large values of ω ). Independent information on the LCDA φ ( ω ) and/or on theΛ b → Λ form factors at intermediate momentum transfer from Lattice QCD would clearly behelpful in this context.
The symmetry relations between the individual Λ b → Λ form factors receive perturbative andnon-perturbative corrections. Let us first consider the corrections from the exchange of onehard-collinear gluon, contributing to the function ∆ ξ Λ as estimated from the sum rule in (3.32).For the default values of the hadronic input parameters, we take the same values as before, seeTable 1. As the default value for the strong coupling constant at a hard-collinear scale, we use α s (cid:39) α s ( µ = 2 GeV) (cid:39) .
3. For the relevant LCDAs, we will again use the exponential modeldiscussed in section D.1. With this, we obtain as our default estimate∆ ξ Λ ( n + p (cid:48) = M Λ b ) (cid:39) − . , ∆ ξ Λ ξ Λ (cid:39) − . . We also find that the ratio ∆ ξ Λ /ξ Λ exhibits a mild linear dependence on the (large) recoil-energy and a pronounced linear dependence on the parameter ω in the exponential modelfor the Λ b LCDAs, see Fig. 5. This is in qualitative agreement with the considerations afterEq. (3.33).The dependence of ∆ ξ Λ on the sum-rule parameters is plotted in Fig. 6. The sensitivityto the Borel parameter ω M , again, is rather weak, while the dependence on the thresholdparameter ω s is somewhat weaker than for the soft form factor ξ Λ . Because of the differentsystematics in (3.20, 3.32) related to the modelling of the continuum and the pollution fromother baryonic resonances, the dependence of the ratio ∆ ξ Λ /ξ Λ on the sum rule parameters isdifficult to estimate numerically. As already emphasized, the dependence on the light and heavydecay constants drops out in the ratio ∆ ξ Λ /ξ Λ . The overall dependence on the renormalizationscale used for the strong coupling constant has to be resolved by calculating higher-orderradiative corrections to ∆ ξ Λ in SCET.The above result can be turned into an estimate for form-factor ratios appearing in physicaldecay observables. As an example, we discuss the ratios h ⊥ f ⊥ , ˜ h ⊥ g ⊥ , appearing in the forward-backward asymmetry for Λ b → Λ µ + µ − , see below. Including theeffect of hard-vertex corrections to O ( α s ) accuracy, we obtain the results shown in Fig. 7,where we have used α s ( m b ) (cid:39) . Form-factor correction ∆ ξ Λ /ξ Λ from the exchange of one hard-collinear gluon from SCETsum rules. Left: Energy dependence from the LO sum rules (3.20, 3.32). Right: Dependence on theparameter ω characterizing the LCDAs of the Λ b baryon. Figure 6:
Dependence on ∆ ξ Λ on the sum-rule parameters (for n + p (cid:48) = M Λ b ). Left: Dependence onthe Borel parameter ω M . Right: Dependence on the threshold parameter ω s . Figure 7:
Form-factor ratios including O ( α s ) corrections from hard (dashed line) and hard plus hard-collinear (solid line) gluon exchange, as a function of the recoil energy ( n + p (cid:48) ): Left: The ratio h ⊥ /f ⊥ .Right: The ratio ˜ h ⊥ /g ⊥ . .3 Λ b → Λ µ + µ − Observables
The general expressions for the double-differential Λ b → Λ µ + µ − decay rate (excluding the non-factorisable contributions, see below) are summarized in Appendix A. Our default values forthe form-factor estimates, in the large-recoil region, yield branching ratios which are slightlyhigher than the central experimental values reported by CDF [10] (and compatible with anindependent theoretical estimate in [3]) within the theoretical and experimental uncertainties,see Fig. 8 (in view of the large hadronic uncertainties, the spectator effects from ∆ ξ Λ representa sub-leading effect and are not included here for simplicity).The functions describing the transverse and longitudinal rate, and the forward-backwardasymmetry become particularly simple in the SCET limit, where all rates are proportional tothe unique form factor ξ Λ ( n + p (cid:48) ), and m Λ (cid:28) M Λ b . To first approximation, the following ratios of observables are thus independent of hadronic form-factor uncertainties, H L ( q ) H T ( q ) (cid:39) q M b · (cid:12)(cid:12)(cid:12) M b C eff9 ( q ) + 2 M b M Λ b C eff7 (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) M b C (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) q C eff9 ( q ) + 2 M b M Λ b C eff7 (cid:12)(cid:12) + | q C | , (4.34)and H A ( q ) H T ( q ) (cid:39) − (cid:104)(cid:0) q C eff9 ( q ) + 2 M b M Λ b C eff7 (cid:1) ∗ q C (cid:105)(cid:12)(cid:12) q C eff9 ( q ) + 2 M b M Λ b C eff7 (cid:12)(cid:12) + | q C | . (4.35)In particular, the leading-order result for the forward-backward asymmetry zero, q , is deter-mined by the same relation between Wilson coefficients,Re (cid:104) q C eff9 ( q ) + 2 M b M Λ b C eff7 (cid:105) q = q (cid:39) , (4.36)as known from the inclusive b → s(cid:96) + (cid:96) − or exclusive B → K ∗ (cid:96) + (cid:96) − decays (see [29] and referencestherein).Our numerical estimates for the ratios H L /H T and H A /H T as a function of q are plottedin Fig. 9, where we compare the SCET limit (4.34,4.35) with the more general result givenFigure 8: Differential branching ratio for Λ b → Λ µ + µ − in units of 10 − as a function of q at largerecoil. The theoretical estimate refers to the SCET limit, and the data points are taken from CDF [10].The (substantial) theoretical uncertainties are not shown. Ratios of observables, H L /H T (left) and H A /H T (right) as a function of q . The dashed lineindicates the SCET limit (4.34, 4.35). The solid line includes the default estimates for the form-factorcorrections from hard gluons, C f i , and hard-collinear gluons, ∆ ξ Λ , as well as the kinematic correctionsof order m Λ /M Λ b . In order to illustrate the (tiny) uncertainty from the variation of ∆ ξ Λ /ξ Λ , we haveinflated the error to an interval [25% , in (A.43) in the Appendix (which, however, still misses the non-factorisable results). In thenumerical analysis, the Wilson coefficients C − are included to leading-logarithmic accuracy,and the Wilson coefficients C , to next-to-leading logarithmic accuracy, with the numericalvalues taken from the analysis in [30]. As one can observe, the inclusion of the kinematiccorrections of order m Λ /M Λ b together with the perturbative corrections to the form-factorrelations leads to a significant effect in the ratio H L /H T above q = 2 GeV , whereas the ratio H A /H T is not very much affected. In particular, we only find a small shift in the value of theforward-backward asymmetry zero, q = (cid:26) . (SCET limit), . (incl. corrections). (4.37)Because of the small imaginary part of the term ( q C eff9 ( q ) + 2 M b M Λ b C eff7 ) in the large-recoilregion, the function H A /H T also develops a pronounced minimum with H A (cid:39) − H T . Again, itsposition is only slightly shifted from q (cid:39) . in the SCET limit, to q (cid:39) . . Noticethat the function ∆ ξ Λ , which describes the spectator corrections to the form factors, enters theabove observables with an additional suppression factor 2 m Λ /M Λ b ∼ ξ Λ /ξ Λ , the considered ratios donot change a lot. The hard vertex corrections from the SCET matching coefficients C f i and thepurely kinematic corrections are thus responsible for the dominant numerical effect, togetherwith the unspecified uncertainties from non-factorizable and power corrections. As already mentioned, a complete next-to-leading order analysis would require one to take into account thenon-factorisable gluon corrections, which is left for future work. Conclusions and Outlook
In this article, we have systematically investigated the form factors entering the baryonicΛ b → Λ (cid:96) + (cid:96) − transitions in the framework of soft-collinear effective theory (SCET). As astarting point, we have introduced an improved form-factor parametrization, which leads tosimple symmetry relations in the limit of heavy b -quark mass m b and/or large recoil-energy E Λ to the Λ baryon, and which yields simple expressions for partial decay widths and decayasymmetries. We have shown that in the large recoil-energy limit, the 10 physical form factorsfor Λ b → Λ transitions reduce to a single “soft” function ξ Λ ( E Λ ), which can be defined as amatrix element of a universal decay current in SCET. The latter has been estimated from asum-rule analysis of an SCET correlation function, where the light Λ baryon is interpolated bya suitable 3-quark current, and the heavy Λ b baryon is described by its light-cone distributionamplitudes (LCDAs). We have studied the energy dependence of the soft form factor, andperformed a critical analysis of the uncertainties arising from the parameters used for thedescription of the hadronic continuum contribution to the sum rule, and for the model of theLCDAs. Compared to the recent measurement of the partially integrated Λ b → Λ µ + µ − rate,we have found agreement within still large experimental and theoretical uncertainties.For phenomenological analyses related to precision tests of the SM or searches for newphysics, it is more convenient to study decay asymmetries, where – to first approximation –the dependence on hadronic form factors drops out in the large recoil-energy limit. In contrastto the analogous mesonic transitions, both, the ratio H L /H T of the longitudinal and trans-verse decay rate, as well as the ratio H A /H T defining the forward-backward asymmetry zeronormalized to the transverse rate, are independent of the hadronic form factors in the SCETlimit. A potentially important source of corrections arises from short-distance gluon exchangebetween the partonic b → s(cid:96) + (cid:96) − transition and the spectator quarks in the baryons. We haveshown that the leading effect can be described by a hadronic matrix element of a particularsub-leading decay current in SCET. In contrast to the mesonic b → s(cid:96) + (cid:96) − transitions, theso-defined correction term ∆ ξ Λ cannot be calculated within the QCD-factorization approach,because one of the two spectator quarks may still populate the kinematic endpoint regionwhere the resulting convolution integrals are ill-defined (in Appendix D.2, we briefly discusshow this could be avoided by switching to a toy model with elementary light di-quark statesin the baryons). Still, the function ∆ ξ Λ can be obtained from a sum-rule analysis of anotherSCET correlation function, and the contributions to the individual transition form factors canbe identified. Numerically, we find that the corrections ∆ ξ Λ /ξ Λ only amount to a few per-cent or less. The corresponding corrections to the decay asymmetries have been estimated aswell, including the effect of α s -corrections to the Wilson coefficients appearing in the matchingof QCD decay currents onto the leading SCET current, and kinematic corrections of order m Λ /M Λ b .Another source of (partially perturbatively calculable) corrections to Λ b → Λ (cid:96) + (cid:96) − decayobservables is related to so-called “non-factorisable” effects which cannot be described in termsof Λ b → Λ form factors. A systematic analysis of these contributions – following the analogouscase of B → K ∗ (cid:96) + (cid:96) − decays in [30] – is left for future work. Finally, sub-leading terms inthe SCET decay currents and SCET interaction terms between soft and collinear fields willlead to power corrections involving sub-leading components of the Λ b wave functions describedby a number of new independent LCDAs. Since, at the moment, only little is known aboutthe partonic structure of the Λ b at sub-leading order, the non-perturbative power corrections17emain an irreducible source of hadronic uncertainties in rare exclusive b -quark decays. Note Added
The symmetry relations between baryonic form factors in the large-recoil limit have also beendiscussed in a related recent paper in [31]. We thank Thomas Mannel and Yu-Ming Wang forsharing their results with us prior to publication. T.F. would also like to thank Yu-Ming Wangfor helpful discussions on the choice of interpolating currents.
Acknowledgements
We thank Yu-Ming Wang for pointing out the inconsistencies in the derivation of the Λ b light-cone projector, which have been corrected in this updated version of the paper. MWYY issupported by a Durham University Doctoral Fellowship. A Differential Decay Widths for Λ b → Λ µ + µ − In this appendix we provide the general formulas for the differential decay widths for theradiative Λ b → Λ µ + µ − transitions in terms of the 10 helicity form factors defined in Sec. 2. Asusual, we consider the center-of-mass frame of the lepton-pair, and define the angle θ betweenthe Λ b baryon and the positively charged lepton. For simplicity, we consider massless leptons,such that q = 2 k (cid:96) + · k (cid:96) − . We then have p Λ b · k (cid:96) ± = M b − m + q ∓ λ cos θ , p Λ · k (cid:96) ± = M b − m − q ∓ λ cos θ . (A.38)Here, λ ≡ √ s + s − = (cid:113) (( M Λ b + m Λ ) − q ) ( M Λ b − m Λ ) − q ) , (A.39)is the usual phase-space factor. If we define d Γ(Λ b → Λ (cid:96) + (cid:96) − ) dq d cos θ ≡ (cid:8) (1 + cos θ ) H T ( q ) + 2 cos θ H A ( q ) + 2(1 − cos θ ) H L ( q ) (cid:9) , (A.40)and neglect non-factorisable contributions, the different contributions to the differential decayrate can be written in terms of the form factors in the helicity basis, H T ( q ) = λ q n π M b (cid:40) s − (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) C eff9 ( q ) f ⊥ + 2 M b ( M Λ b + m Λ ) C eff7 q h ⊥ (cid:12)(cid:12)(cid:12)(cid:12) + | C f ⊥ | (cid:33) + s + (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) C eff9 ( q ) g ⊥ + 2 M b ( M Λ b − m Λ ) C eff7 q ˜ h ⊥ (cid:12)(cid:12)(cid:12)(cid:12) + | C g ⊥ | (cid:33) (cid:41) , (A.41) H A ( q ) = − λ q n π M b Re (cid:34)(cid:18) C eff9 ( q ) f ⊥ + 2 M b ( M Λ b + m Λ ) C eff7 q h ⊥ (cid:19) ∗ ( C g ⊥ )18 (cid:18) C eff9 ( q ) g ⊥ + 2 M b ( M Λ b − m Λ ) C eff7 q ˜ h ⊥ (cid:19) ∗ ( C f ⊥ ) (cid:35) , (A.42) H L ( q ) = λ n π M b (cid:40) s − ( M Λ b + m Λ ) (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) C eff9 ( q ) f + + 2 M b C eff7 M Λ b + m Λ h + (cid:12)(cid:12)(cid:12)(cid:12) + | C f + | (cid:33) + s + ( M Λ b − m Λ ) (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) C eff9 ( q ) g + + 2 M b C eff7 M Λ b − m Λ ˜ h + (cid:12)(cid:12)(cid:12)(cid:12) + | C g + | (cid:33) (cid:41) . (A.43)where n = α G F π | V ts V tb | . (A.44)The functions become particularly simple in the SCET limit, where H T ( q ) (cid:39) λ q n π M b | ξ Λ ( n + p (cid:48) ) | (cid:40) (cid:12)(cid:12)(cid:12)(cid:12) C eff9 ( q ) + 2 M b M Λ b C eff7 q (cid:12)(cid:12)(cid:12)(cid:12) + | C | (cid:41) , (A.45) H A ( q ) (cid:39) − λ q n π M b | ξ Λ ( n + p (cid:48) ) | Re (cid:34)(cid:18) C eff9 ( q ) + 2 M b M Λ b C eff7 q (cid:19) ∗ C (cid:35) , (A.46) H L ( q ) (cid:39) λ n π M Λ b | ξ Λ ( n + p (cid:48) ) | (cid:40) (cid:12)(cid:12)(cid:12)(cid:12) C eff9 ( q ) + 2 M b M Λ b C eff7 (cid:12)(cid:12)(cid:12)(cid:12) + | C | (cid:41) . (A.47) B Alternative Form-Factor Parametrizations
B.1 Convention by Chen and Geng
The form factors in [1], which have been commonly used in the recent literature, are related toours as follows. For the vector form factors, we obtain f = f + q M Λ b − m Λ f , f + = f − q M Λ b + m Λ f , f ⊥ = f − ( M Λ b + m Λ ) f . (B.48)Similarly, for the axial-vector form factors, one gets g = g − q M Λ b + m Λ g , g + = g + q M Λ b − m Λ g , g ⊥ = g + ( M Λ b − m Λ ) g . (B.49)The tensor and pseudo-tensor form factors are related by h + = f T − M Λ b + m Λ q f T , h ⊥ = f T − M Λ b + m Λ f T , (B.50)and ˜ h + = g T + M Λ b − m Λ q g T , ˜ h ⊥ = g T + 1 M Λ b − m Λ g T . (B.51)19 .2 Symmetry-Based Form-Factor Parametrization An alternative parametrization considers the different projections of the decay current in theheavy-quark and/or large-energy limit, respectively. On the heavy-quark side, we consider theheavy-baryon velocity v µ = p µ /M Λ b such that / v u Λ b ( p ) = u Λ b ( p ). Also taking into account theprojections on the light-quark side (using parity invariance of strong interactions), we end upwith the general expression (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ q Γ b | Λ b ( p, s ) (cid:105) = ξ ( ± ) ij ( v, p (cid:48) ) ¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:26) Γ i / n ± / n ∓ j (cid:27) u Λ b ( p, s ) (B.52)where the basis of Dirac matrices can be chosen asΓ i = { , γ , γ α ⊥ } , Γ j = { , γ , (cid:126)γ µ , (cid:126)γ µ γ } , (B.53)and γ α ⊥ = γ α − / n + n α − − / n − n α + , while (cid:126)γ µ = γ µ − / v v µ etc. Here and in the following, we considera frame where / p (cid:48)⊥ = 0 and v µ = ( n µ − + n µ + ) /
2. The non-vanishing form factors are ξ ( ± )11 ( v, p (cid:48) ) ≡ A ( ± ) ( v · p (cid:48) ) ∼ O (1) , ξ ( ± )13 ( v, p (cid:48) ) ≡ p (cid:48) µ v · p (cid:48) B ( ± ) ( v · p (cid:48) ) ∼ O ( (cid:15) ) ,ξ ( ± )22 ( v, p (cid:48) ) ≡ C ( ± ) ( v · p (cid:48) ) ∼ O ( (cid:15) ) , ξ ( ± )24 ( v, p (cid:48) ) ≡ p (cid:48) µ v · p (cid:48) D ( ± ) ( v · p (cid:48) ) ∼ O ( (cid:15) ) ,ξ ( ± )33 ( v, p (cid:48) ) ≡ δ µα E ( ± ) ( v · p (cid:48) ) ∼ O ( (cid:15) ) ,ξ ( ± )34 ( v, p (cid:48) ) ≡ i(cid:15) αµρσ v ρ p (cid:48) σ v · p (cid:48) F ( ± ) ( v · p (cid:48) ) ∼ O ( (cid:15) ) , (B.54)From the above 12 form factors, again, only 10 are independent, after the e.o.m. constraintshave been taken into account. Here, the indicated suppression of the form factors with (cid:15) = Λ /M refers to the violation of the heavy-quark spin symmetry. In addition, in the large recoil limit thecontributions from the form factors with an index “ − ” are additionally suppressed. Therefore,we may neglect the 5 form factors B ( − ) through F ( − ) , which is a good approximation, because • In the HQET limit, v · p (cid:48) ∼ O ( m Λ ), their contribution is suppressed at least by a factorΛ /M . • In the SCET limit, n + p (cid:48) ∼ O ( M Λ b ), their contribution is suppressed by at least a factor(Λ /M ) (for non-factorizable effects) or α s (for factorizable effects, see below).We thus end up with a rather efficient description which combines the symmetry constraintsin both cases and allows one to systematically take into account sub-leading corrections inthe large-recoil limit, which are partially calculable in the framework of QCD factorization orlight-cone sum rules. In this approximation, the 10 physical helicity form factors are relatedby 5 equations (for vanishing light quark masses, m s → f = M Λ b + m Λ M Λ b − m Λ n + p (cid:48) − m Λ n + p (cid:48) + m Λ f + + M Λ b − n + p (cid:48) M Λ b − m Λ (cid:18) g ⊥ − n + p (cid:48) − m Λ n + p (cid:48) + m Λ f ⊥ (cid:19) ,g = M Λ b − m Λ M Λ b + m Λ n + p (cid:48) + m Λ n + p (cid:48) − m Λ g + + M Λ b − n + p (cid:48) M Λ b + m Λ (cid:18) f ⊥ − n + p (cid:48) + m Λ n + p (cid:48) − m Λ g ⊥ (cid:19) , ˜ h ⊥ = M Λ b + m Λ M Λ b − m Λ n + p (cid:48) − m Λ n + p (cid:48) + m Λ h ⊥ + M Λ b − n + p (cid:48) M Λ b − m Λ (cid:18) g ⊥ − n + p (cid:48) − m Λ n + p (cid:48) + m Λ f ⊥ (cid:19) , (B.55)20nd h + = M Λ b + m Λ M b f + + n + p (cid:48) − Λ M b (cid:18) f ⊥ − n + p (cid:48) + m Λ n + p (cid:48) − m Λ g ⊥ (cid:19) , ˜ h + = M Λ b − m Λ M b g + + n + p (cid:48) − Λ M b (cid:18) g ⊥ − n + p (cid:48) − m Λ n + p (cid:48) + m Λ f ⊥ (cid:19) . (B.56) C Corrections to SCET Symmetry Relations
C.1 Hard Vertex Corrections
The hard vertex corrections to the individual QCD decay currents have been discussed before[16, 17]. From the general 1-loop result in Eq. (28) in [16] we can deduce the corrections to theindividual form factors in the helicity basis, f i = C f i ξ Λ + . . . . Defining the renormalizationscheme through C f + = C g + ≡
1, this leads to C f = C g = 1 + α s C F π − L ) , C f ⊥ = C g ⊥ = 1 + α s C F π L , (C.57)and C h + = C ˜ h + = 1 + α s C F π (cid:18) ln M b µ − − L ) (cid:19) , C h ⊥ = C ˜ h ⊥ = 1 + α s C F π (cid:18) ln M b µ − (cid:19) , (C.58)with the abbreviation L ≡ − M b − q q ln (cid:18) − q M b (cid:19) . C.2 Hard-Collinear Gluon Exchange
We consider the tree-level matching (in light-cone gauge), following [16]¯ q Γ Q v (cid:39) ¯ ξ ˜Γ h v − n + p (cid:48) ¯ ξ g / A ⊥ / n + h v − M b ¯ ξ Γ / n − g / A ⊥ h v + . . . (C.59)The hard-scattering contributions to the individual form factors in the large-recoil limit definedabove can then be identified by means of (2.14) and setting m Λ → M Λ b → M b ≡ M .This is equivalent to using A ( − ) (cid:39) − Mm Λ ∆ ξ Λ , E (+) = F (+) = 12 ∆ ξ Λ (C.60)in (B.52). For the scalar and vector form factors, this yields f ( q ) (cid:39) C f ξ Λ ( n + p (cid:48) ) − Mn + p (cid:48) ∆ ξ Λ ( n + p (cid:48) ) ,f + ( q ) (cid:39) C f + ξ Λ ( n + p (cid:48) ) − (cid:18) − Mn + p (cid:48) (cid:19) ∆ ξ Λ ( n + p (cid:48) ) ,f ⊥ ( q ) (cid:39) C f ⊥ ξ Λ ( n + p (cid:48) ) + 2 Mn + p (cid:48) ∆ ξ Λ ( n + p (cid:48) ) , (C.61)21here C i = C i ( µ, n + p (cid:48) ) denote the hard vertex coefficients as derived above. Similar relationscan be obtained for the axial-vector and tensor form factors, g ( q ) (cid:39) C g ξ Λ ( n + p (cid:48) ) + 2 Mn + p (cid:48) ∆ ξ Λ ( n + p (cid:48) ) ,g + ( q ) (cid:39) C g + ξ Λ ( n + p (cid:48) ) + 2 (cid:18) − Mn + p (cid:48) (cid:19) ∆ ξ Λ ( n + p (cid:48) ) ,g ⊥ ( q ) (cid:39) C g ⊥ ξ Λ ( n + p (cid:48) ) − Mn + p (cid:48) ∆ ξ Λ ( n + p (cid:48) ) , (C.62)and h + ( q ) (cid:39) C h + ξ Λ ( n + p (cid:48) ) + 2 Mn + p (cid:48) ∆ ξ Λ ( n + p (cid:48) ) ,h ⊥ ( q ) (cid:39) C h ⊥ ξ Λ ( n + p (cid:48) ) − (cid:18) − Mn + p (cid:48) (cid:19) ∆ ξ Λ ( n + p (cid:48) ) , (C.63)and ˜ h + ( q ) (cid:39) C ˜ h + ξ Λ ( n + p (cid:48) ) − Mn + p (cid:48) ∆ ξ Λ ( n + p (cid:48) ) , ˜ h ⊥ ( q ) (cid:39) C ˜ h ⊥ ξ Λ ( n + p (cid:48) ) + 2 (cid:18) − Mn + p (cid:48) (cid:19) ∆ ξ Λ ( n + p (cid:48) ) . (C.64) C.3 Form-Factor Relations to O ( α s ) Accuracy
To first order in the strong coupling constant, the hard vertex corrections and the spectatorscattering corrections only provide 5 independent Dirac structures. As a consequence, afterinclusion of O ( α s ) corrections, from the 10 helicity form factors only 5 are still linearly inde-pendent. The 5 symmetry relations which are unaffected by O ( α s ) radiative corrections canbe summarized as f + h + g + ˜ h + = f ⊥ − h + g ⊥ − ˜ h + = f + + h + − h ⊥ g + + ˜ h + − h ⊥ = − f + − f ⊥ ) + h + − h ⊥ g + − g ⊥ ) + ˜ h + − ˜ h ⊥ = − M − q M h + − ˜ h + f + − g + = 1 . (C.65) A similar effect was observed for B → V = ρ, K ∗ . . . transitions, where among the 7 physical form factors 2symmetry relations remain at O ( α s ) [16]. Symmetry arguments based on the helicity conservation of the lightquark in short-distance interactions can be found in [32]. For B -meson decays into light pseudoscalars no suchrelations remain, because there are only 3 physical form factors to start with in the first place. Light-Cone Distribution Amplitudes
Light-cone distribution amplitudes (LCDAs) are introduced as matrix elements of non-localQCD light-ray operators between the considered baryon states and the vacuum.
D.1 Distribution Amplitudes for the Λ b baryon For the heavy Λ b baryon, we follow the definitions in [14] and consider the following twoprojections (two others are not shown), (cid:15) abc (cid:104) | (cid:16) u a ( t n − ) C γ / n − d b ( t n − ) (cid:17) h cv (0) | Λ b ( v, s ) (cid:105) = f (2)Λ b Ψ ( t , t ) u Λ b ( v, s ) ,(cid:15) abc (cid:104) | (cid:16) u a ( t n − ) C γ / n + d b ( t n − ) (cid:17) h cv (0) | Λ b ( v, s ) (cid:105) = f (2)Λ b Ψ ( t , t ) u Λ b ( v, s ) . (D.66)The so-defined LCDAs in position space have a Fourier expansion,Ψ i ( t , t ) = (cid:90) ∞ dω (cid:90) ∞ dω e − i ( t ω + t ω ) ψ i ( ω , ω )= (cid:90) ∞ dω ω (cid:90) du e − iω ( t u + t ¯ u ) ˜ ψ i ( ω, u ) . (D.67)Here, the first alternative refers to a function of the two light-cone momenta ω , = ( n − k , )of the two light quarks in the heavy baryon, while the second alternative considers the totallight-cone momentum ω = ω + ω and the momentum fractions u = ω /ω , ¯ u = 1 − u = ω /ω (notice the additional factor of ω in the Fourier integral in the latter case). The normalizationfactors f (1 , b have mass-dimension 3 and are scale-dependent. For numerical estimates, wewill use f ( i )Λ b (cid:39) . ± .
005 GeV . The LCDAs ψ i ( ω , ω ) in momentum space have mass-dimension ( −
2) and are scale-dependent, too. More details can be found in [14].The above definitions can be converted into momentum-space representations for the Λ b distribution amplitudes, following the analogous procedure that has been explained in detailfor the B -meson LCDA in [16]. Taking an arbitrary light-like vector y µ and defining t = v · y ,we can write the most general Lorentz decomposition in the heavy-quark limit, (cid:15) abc (cid:104) | (cid:16) u a ( τ y ) Cγ γ µ d b ( τ y ) (cid:17) h cv (0) | Λ b ( v, s ) (cid:105) = f (2)Λ b (cid:18) v µ Ψ ( τ , τ ) + Ψ ( τ , τ ) − Ψ ( τ , τ )2 t y µ (cid:19) u Λ b ( v, s ) . (D.68)This can be turned into (cid:15) abc (cid:104) | (cid:16) u aα ( τ y ) d bβ ( τ y ) (cid:17) h cv (0) | Λ b ( v, s ) (cid:105) = f (2)Λ b u Λ b ( v, s ) (cid:20)(cid:18) / v Ψ ( τ , τ ) + Ψ ( τ , τ ) − Ψ ( τ , τ )2 t / y (cid:19) γ C − (cid:21) βα + 2 more terms. (D.69)In the convolution with hard-scattering kernels that have a power expansion in the trans-verse momenta k ⊥ and k ⊥ of the two light quarks in the Λ b baryon, and which have a23orresponding sub-sub-leading dependence on ( n + k i ), the most general momentum-space pro-jector f (2)Λ b u Λ b ( v, s ) ˜ M ( k , k ) βα (cid:12)(cid:12)(cid:12) k i = ω i n + / (D.70)reads [33]˜ M ( k , k ) = (cid:18) / n + ψ ( ω , ω ) + / n − ψ ( ω , ω ) − γ ⊥ µ (cid:90) ω dη (cid:16) ψ (1)42 ( η , ω ) − ψ X ( η , ω ) (cid:17) / n + / n − ∂∂k ⊥ µ − γ ⊥ µ (cid:90) ω dη (cid:16) ψ (1)42 ( η , ω ) + ψ X ( η , ω ) (cid:17) / n − / n + ∂∂k ⊥ µ − γ ⊥ µ (cid:90) ω dη (cid:16) ψ (2)42 ( ω , η ) − ψ X ( ω , η ) (cid:17) / n − / n + ∂∂k ⊥ µ − γ ⊥ µ (cid:90) ω dη (cid:16) ψ (2)42 ( ω , η ) + ψ X ( ω , η ) (cid:17) / n + / n − ∂∂k ⊥ µ (cid:33) γ C − + 2 more terms. (D.71)Here, ψ (2)42 ( ω , ω ) = ψ (1)42 ( ω , ω ) and ψ X ( ω , ω ) = ψ X ( ω , ω ) and ψ (1)42 ( ω , ω ) + ψ (2)42 ( ω , ω ) = ψ ( ω , ω ) − ψ ( ω , ω ) . (D.72)From this we see that ψ , play the analogous role as φ B ± for the B -meson. The asymmetriccombination of ψ (1)42 and ψ (2)42 , as well as ψ X do not contribute in the collinear limit (D.69).However, they do contribute to the correlator used for the sum-rule estimate of ∆ ξ Λ . They alsoallow one to derive approximate Wandzura-Wilczek relations from the equations of motion,/ k ˜ M ( k , k ) (cid:39) ˜ M ( k , k )/ k (cid:39) , (D.73)in the limit of vanishing LCDAs with n > v · k , ), where k , are the on-shell momenta for the lightquarks in the Λ b , f ( v · k , v · k ) := 1 ω e − ( v · k + v · k ) /ω . (D.74)where ω ∼ Λ had is a measure for the typical momentum of the di-quark. We then may use ψ ( ω , ω ) = (cid:90) ∞ dk ⊥ dk ⊥ f (cid:18) ω + k ⊥ ω , ω + k ⊥ ω (cid:19) = ω ω e − ( ω + ω ) /ω ω , (D.75) ψ ( ω , ω ) = (cid:90) ∞ dk ⊥ dk ⊥ k ⊥ ω k ⊥ ω f (cid:18) ω + k ⊥ ω , ω + k ⊥ ω (cid:19) = e − ( ω + ω ) /ω ω , (D.76)24here the pre-factors in the integrand of the second line correspond to the ratios ( n + k i ) / ( n − k i )taking into account that ψ and ψ change their role when switching n + ↔ n − . Also (see [33]for details), ψ (1)42 ( ω , ω ) = ∂∂ω (cid:90) ∞ dk ⊥ dk ⊥ k ⊥ ω (cid:18) k ⊥ ω (cid:19) f (cid:18) ω + k ⊥ ω , ω + k ⊥ ω (cid:19) = ( ω − ω )( ω + ω )2 ω e − ( ω + ω ) /ω , (D.77) ψ X ( ω , ω ) = − ∂∂ω (cid:90) ∞ dk ⊥ dk ⊥ k ⊥ ω (cid:18) − k ⊥ ω (cid:19) f (cid:18) ω + k ⊥ ω , ω + k ⊥ ω (cid:19) = ( ω − ω )( ω − ω )2 ω e − ( ω + ω ) /ω . (D.78)For later use, we also introduce the abbreviations G ( ω , ω ) = (cid:90) ω dη (cid:16) ψ (1)42 ( η , ω ) − ψ X ( η , ω ) (cid:17) → ω ω ω e − ( ω + ω ) /ω , (D.79) H ( ω , ω ) = (cid:90) ω dη (cid:16) ψ (2)42 ( ω , η ) + ψ X ( ω , η ) (cid:17) → ω ω e − ( ω + ω ) /ω . (D.80)The parameter ω has been estimated in [14] from a sum-rule analysis of ˜ ψ ( ω, u ) (alsoincluding corrections from higher-order Gegenbauer polynomials as a function of (2 u − D.2 Simplified Set-Up with Scalar Di-quark
For a simplified picture, one may also approximate the dynamics of the two light quarks in theΛ b baryon by an elementary scalar di-quark field ϕ a ( x ) in the ¯3 representation of SU (3) C . Inthe HQET limit, the Λ b baryon could then be described by a single LCDA, defined as ( t = v · z ) (cid:104) | ϕ a ( z ) h av (0) | Λ b ( v, s ) (cid:105) = ˆ f Λ b Ψ Λ b ( t ) u Λ b ( v, s ) , (D.81)and Ψ Λ b ( t ) = (cid:90) ∞ dω e − itω φ Λ b ( ω ) . (D.82)Here ˆ f Λ b has mass-dimension +1, and ψ Λ b ( ω ) has mass-dimension −
1. The momentum-spaceprojector in this case simply reads ˆ f Λ b φ Λ b ( ω ) u Λ b ( v, s ) . (D.83)Similarly, the Λ baryon can be approximately described by two LCDAs, defined as (cid:104) Λ( p (cid:48) , s (cid:48) ) | ¯ s a ( z ) ϕ a † (0) | (cid:105) = (cid:90) du e iu p (cid:48) · z ¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:16) ˆ f (1)Λ φ (1)Λ ( u ) − σ µν p (cid:48) µ z ν ˆ f (2)Λ φ (2)Λ ( u ) (cid:17) (D.84)which corresponds to a momentum-space projector¯ u Λ ( p (cid:48) , s (cid:48) ) (cid:18) ˆ f (1)Λ φ (1)Λ ( u ) − i σ µν ˆ f (2)Λ (cid:26) n µ − n ν + φ (2)Λ (cid:48) ( u ) − p (cid:48) µ ∂∂k ⊥ ν φ (2)Λ ( u ) (cid:27)(cid:19) . (D.85)25 oft form factor from simplified set-up: We may use the di-quark approximation as atoy model, to obtain alternative expressions for the transition form factors from SCET sumrules. To this end, we consider a correlation function involving the interpolating currentˆ J Λ ( x ) = ϕ a ( x ) s a ( x ) (D.86)with (cid:104) | ˆ J Λ (0) | Λ( p (cid:48) , s (cid:48) ) (cid:105) = ˆ f Λ u Λ ( p (cid:48) , s (cid:48) ) . (D.87)The remaining calculation is analogous to the realistic case considered in Sec 3.1, and yieldsthe LO sum rule e − m / ( n + p (cid:48) )ˆ ω M ˆ f Λ ξ Λ ( n + p (cid:48) ) = ˆ f Λ b (cid:90) ˆ ω s dω φ Λ b ( ω ) e − ω/ ˆ ω M , (D.88)with an according new threshold parameter ˆ ω s and Borel parameter ˆ ω M . Hard-collinear gluon correction from simplified set-up:
In the simplified toy model,as before, we define the correlation function using the interpolating current in (D.86),ˆΠ µ Λ ( n − p (cid:48) ) ≡ i (cid:90) d x e ip (cid:48) x (cid:104) | T (cid:20) / n + / n − J Λ ( x ) (cid:104) ¯ s (0) ˜Γ gA µ ⊥ (0) b (0) (cid:105)(cid:21) | Λ b ( p ) (cid:105) . (D.89)Evaluating the Feynman diagram (using scalar QCD for the di-quark in the ¯3 representation),we obtainˆΠ µ Λ ( n − p (cid:48) ) = − ig s C F ˆ f Λ b (cid:90) ∞ dω φ Λ b ( ω ) (cid:90) d D l (2 π ) D (cid:2) l ⊥ + ( n + l )( n − l ) (cid:3) l/ ⊥ l µ ⊥ ˜Γ u Λ b ( v, s ) × (cid:2) l ⊥ + ( n + l )( n − l − ω ) (cid:3) (cid:2) l ⊥ + ( n + l + n + p (cid:48) )( n − l + n − p (cid:48) − ω ) (cid:3) . (D.90)The correlator can be calculated as before, leading toˆΠ µ Λ ( n − p (cid:48) ) = − g s C F ˆ f Λ b γ µ ⊥ ˜Γ u Λ b ( v, s ) (cid:90) ∞ dω φ Λ b ( ω ) 12 (cid:90) dz × (cid:90) d D − l ⊥ (2 π ) D − (1 − z ) l ⊥ / ( D − l ⊥ − z (1 − z ) ( n + p (cid:48) )( − n − p (cid:48) )] [ l ⊥ − z (1 − z ) ( n + p (cid:48) ) ( ω − n − p (cid:48) )] , (D.91)In this case, the integral over transverse momenta is UV divergent and needs to be regularized,as indicated. However, the divergence only influences the real part, while the imaginary partgives a similar result as before, leading toˆ B ˆΠ µ Λ (ˆ ω M ) | subtr . = − α s C F π ˆ f (2)Λ b γ µ ⊥ ˜Γ u Λ b ( v, s ) (cid:90) ˆ ω s dω (cid:48) ˆ ω M e − ω (cid:48) / ˆ ω M (cid:90) ∞ dω × φ Λ b ( ω ) (cid:18) θ ( ω (cid:48) − ω ) + ω (cid:48) ω θ ( ω − ω (cid:48) ) (cid:19) . (D.92)26n the formal limit ˆ ω s,M (cid:28) ω , this factorizes again, according toˆ B ˆΠ µ Λ (ˆ ω M ) | subtr . (cid:39) − α s C F π ˆ f Λ b γ µ ⊥ ˜Γ u Λ b ( v, s ) (cid:90) ∞ dωω φ Λ b ( ω ) (cid:16) ˆ ω M − e − ˆ ω s / ˆ ω M (ˆ ω M + ˆ ω s ) (cid:17) , (D.93)showing the same dependence on the sum-rule parameters as before in (3.33). For the hadronicside of the sum rule, in the simplified set-up, we now findˆΠ µ Λ (cid:12)(cid:12) reson . = ˆ f Λ M Λ b ∆ ξ Λ m − ( n + p (cid:48) )( n − p (cid:48) ) / n + / n − p (cid:48) + m Λ ) γ µ ⊥ ˜Γ u Λ b ( v, s ) (cid:39) n + p (cid:48) ˆ f Λ m Λ M Λ b ∆ ξ Λ m / ( n + p (cid:48) ) − ( n − p (cid:48) ) γ µ ⊥ ˜Γ u Λ b ( v, s ) , (D.94)leading to the sum rule e − m / ( n + p (cid:48) )ˆ ω M ˆ f Λ M Λ b m Λ ˆ ω M n + p (cid:48) ∆ ξ Λ = − α s C F π ˆ f Λ b (cid:90) ˆ ω s dω (cid:48) ˆ ω M e − ω (cid:48) / ˆ ω M (cid:90) ∞ dω φ Λ b ( ω ) (cid:18) θ ( ω (cid:48) − ω ) + ω (cid:48) ω θ ( ω − ω (cid:48) ) (cid:19) . (D.95)In the simplified picture, the hard-collinear correction term ∆ ξ Λ could also be obtained fromthe QCD factorization approach, in complete analogy to the mesonic case discussed in [16].This will lead to an (endpoint-converging) convolution of a hard-scattering kernel and the aboveLCDAs for light and heavy baryons in the di-quark approximation. In the heavy-mass limit,the above sum-rule expression can then be interpreted as a particular model for the light-conewave function of the Λ baryon, in a similar way as it has been discussed for the mesonic formfactors in [19]. References [1] C. H. Chen and C. Q. Geng, “Baryonic rare decays of Λ b → Λ (cid:96) + (cid:96) − ,” Phys. Rev. D (2001) 074001 [arXiv:hep-ph/0106193]; “Rare Λ b → Λ (cid:96) + (cid:96) + decays with polarized Λ,”Phys. Rev. D (2001) 114024 [arXiv:hep-ph/0101171]; “Lepton asymmetries in heavybaryon decays of Λ b → Λ (cid:96) + (cid:96) − ,” Phys. Lett. B (2001) 327 [arXiv:hep-ph/0101201].C. H. Chen, C. Q. Geng and J. N. Ng, “T violation in Λ b → Λ (cid:96) + (cid:96) − decays with polarizedΛ,” Phys. Rev. D (2002) 091502 [arXiv:hep-ph/0202103].[2] Y. -L. Liu, L. -F. Gan, M. -Q. Huang, “The exclusive rare decay b → sγ of heavy b-Baryons,” Phys. Rev. D83 (2011) 054007. [arXiv:1103.0081 [hep-ph]].[3] T. M. Aliev, K. Azizi, M. Savci, “Analysis of the Λ b → Λ (cid:96) + (cid:96) − decay in QCD,” Phys.Rev. D81 (2010) 056006. [arXiv:1001.0227 [hep-ph]]. T. M. Aliev and M. Savci, “Po-larization effects in exclusive semileptonic Λ b → Λ (cid:96) + (cid:96) − decay,” JHEP (2006) 001[arXiv:hep-ph/0507324]. T. M. Aliev, M. Savci and B. B. Sirvanli, “Double-lepton po-larization asymmetries in Λ b → Λ (cid:96) + (cid:96) − decay in universal extra dimension model,” Eur.Phys. J. C (2007) 375 [arXiv:hep-ph/0608143].274] Y. -M. Wang, Y. -L. Shen, C. -D. Lu, “Λ b → p, Λ transition form factors from QCD light-cone sum rules,” Phys. Rev.
D80 (2009) 074012. [arXiv:0907.4008 [hep-ph]]. Y. m. Wang,Y. Li and C. D. Lu, “Rare Decays of Λ b → Λ γ and Λ b → Λ (cid:96) + (cid:96) − in the Light-cone SumRules,” Eur. Phys. J. C (2009) 861 [arXiv:0804.0648 [hep-ph]].[5] G. Hiller and A. Kagan, “Probing for new physics in polarized Λ b decays at the Z,” Phys.Rev. D (2002) 074038 [arXiv:hep-ph/0108074].[6] L. Oliver, J. C. Raynal and R. Sinha, “Note on new interesting baryon channels to measurethe photon polarization in b → sγ ,” Phys. Rev. D (2010) 117502 [arXiv:1007.3632 [hep-ph]].[7] P. Colangelo, F. De Fazio, R. Ferrandes and T. N. Pham, “FCNC B s and Λ b transitions:Standard model versus a single universal extra dimension scenario,” Phys. Rev. D (2008) 055019 [arXiv:0709.2817 [hep-ph]].[8] G. Hiller, M. Knecht, F. Legger and T. Schietinger, “Photon polarization from helicitysuppression in radiative decays of polarized Λ b to spin-3/2 baryons,” Phys. Lett. B (2007) 152 [arXiv:hep-ph/0702191].[9] L. Mott, W. Roberts, “Rare dileptonic decays of Λ b in a quark model,” [arXiv:1108.6129[nucl-th]].[10] T. Aaltonen et al. [ CDF Collaboration ], “Observation of the Baryonic Flavor-ChangingNeutral Current Decay Λ b → Λ µ + µ − ,” arXiv:1107.3753 [hep-ex].[11] T. Mannel, W. Roberts, Z. Ryzak, “Baryons in the heavy quark effective theory,” Nucl.Phys. B355 (1991) 38-53.[12] T. Mannel, S. Recksiegel, “Flavor changing neutral current decays of heavy baryons: TheCase Λ b → Λ γ ,” J. Phys. G24 (1998) 979-990. [hep-ph/9701399].[13] J. Charles, A. Le Yaouanc, L. Oliver et al. , “Heavy to light form-factors in the heavy massto large energy limit of QCD,” Phys. Rev.
D60 (1999) 014001. [hep-ph/9812358].[14] P. Ball, V. M. Braun, E. Gardi, “Distribution Amplitudes of the Λ b Baryon in QCD,”Phys. Lett.
B665 (2008) 197-204. [arXiv:0804.2424 [hep-ph]].[15] Y. -L. Liu, M. -Q. Huang, “Distribution amplitudes of Σ and Λ and their electromagneticform factors,” Nucl. Phys.
A821 (2009) 80-105. [arXiv:0811.1812 [hep-ph]].[16] M. Beneke, Th. Feldmann, “Symmetry breaking corrections to heavy-to-light B -mesonform factors at large recoil,” Nucl. Phys. B592 , 3-34 (2001). [hep-ph/0008255].[17] C. W. Bauer, S. Fleming, D. Pirjol, I. W. Stewart, “An Effective field theory for collinearand soft gluons: Heavy to light decays,” Phys. Rev.
D63 (2001) 114020. [hep-ph/0011336].[18] M. Beneke, A. P. Chapovsky, M. Diehl, Th. Feldmann, “Soft collinear effective theoryand heavy to light currents beyond leading power,” Nucl. Phys.
B643 (2002) 431-476.[hep-ph/0206152]; M. Beneke, Th. Feldmann, “Factorization of heavy-to-light form factorsin soft collinear effective theory,” Nucl. Phys.
B685 (2004) 249-296. [hep-ph/0311335].2819] F. De Fazio, Th. Feldmann, T. Hurth, “Light-cone sum rules in soft-collinear effectivetheory,” Nucl. Phys.
B733 (2006) 1-30. [hep-ph/0504088]; “SCET sum rules for B → P and B → V transition form factors,” JHEP (2008) 031. [arXiv:0711.3999 [hep-ph]].[20] A. Khodjamirian, T. Mannel, N. Offen, “Form-factors from light-cone sum rules withB-meson distribution amplitudes,” Phys. Rev. D75 (2007) 054013. [hep-ph/0611193];S. Faller, A. Khodjamirian, C. .Klein, T. .Mannel, “ B → D ( ∗ ) Form Factors from QCDLight-Cone Sum Rules,” Eur. Phys. J.
C60 (2009) 603-615. [arXiv:0809.0222 [hep-ph]].[21] N. Kivel, M. Vanderhaeghen, “Soft spectator scattering in the nucleon form factors atlarge Q within the SCET approach,” Phys. Rev. D83 (2011) 093005. [arXiv:1010.5314[hep-ph]].[22] C. G. Boyd, M. J. Savage, “Analyticity, shapes of semileptonic form-factors, and ¯ B → π(cid:96) ¯ ν ,” Phys. Rev. D56 (1997) 303-311. [hep-ph/9702300].[23] A. Bharucha, Th. Feldmann, M. Wick, “Theoretical and Phenomenological Constraintson Form Factors for Radiative and Semi-Leptonic B-Meson Decays,” JHEP (2010)090. [arXiv:1004.3249 [hep-ph]].[24] V. Braun, R. J. Fries, N. Mahnke et al. , “Higher twist distribution amplitudes of thenucleon in QCD,” Nucl. Phys.