Non-factorisable Contributions of Strong-Penguin Operators in Λ b → Λ ℓ + ℓ − Decays

We investigate for the first time a certain class of non-factorisable contributions of the four-quark operators O 3 − 6 in the weak effective Hamiltonian to the Λ b → Λ ℓ + ℓ − decay amplitude. We focus on the case where a virtual photon is radiated from one of the light constituents of the Λ b baryon, in the kinematic situation of large hadronic recoil with an energetic Λ baryon in the final state. The effect on the suitably defined “non-local form factors” is calculated using the light-cone sum rule approach for a correlator with an interpolating current for the light Λ baryon. We find that this approach requires the introduction of new soft functions that generalise the standard light-cone distribution amplitudes (LCDAs) for the heavy Λ b baryon. We give a heuristic discussion of their properties and a model that relates them to the standard LCDAs. Within this framework, we provide numerical results for the size of the non-local form factors considered.


Introduction
Rare b-quark decays in general, and rare b → sℓ + ℓ − transitions in particular, have received much attention in the past.From a phenomenological point of view, these decays provide a large number of complementary observables that allow us to explore the flavour sector of the Standard Model (SM) and its possible new-physics extensions (for comprehensive reviews, see e.g.Refs.[1,2] and references therein).From a theoretical point of view, the decays of a heavy quark into light degrees of freedom provide a valuable playground to develop and refine calculation methods to address the factorisation of hadronic bound-state effects from short-distance QCD corrections.In particular, one can establish QCD factorisation theorems (see e.g.Refs.[3][4][5] for the pioneering works) for decay amplitudes of exclusive decays with large energy transfer to one or more light hadrons in the final state.Here, the bound-state effects are contained in hadronic transition form factors of local decay currents and light-cone distribution amplitudes (LCDAs) for light and heavy hadrons.Alternatively, it is possible to replace one or the other hadron by suitably chosen interpolating currents and relate the exclusive decay amplitude to the corresponding correlators by dispersion relations, namely the light-cone sum rule (LCSR) method (for a recent review, see Ref. [6]).In both cases, the factorisation of "soft" degrees of freedom in the b-hadron and the energetic degrees of freedom in the final state can be formally achieved by matching onto a soft-collinear effective theory (SCET), see e.g.Refs.[7][8][9] for early applications in b-decays.
In the past, much of the phenomenological study of rare semileptonic b → sℓ + ℓ − transitions has been devoted to exclusive B → Kℓ + ℓ − and B → K * ℓ + ℓ − decays.In particular, a number of "flavour anomalies" (i.e.deviations between experimental measurements and theoretical expectations within the SM) may be due to physics beyond the SM, for recent reviews see Refs.[10,11].However, a correct interpretation of these anomalies requires the control of systematic experimental effects as well as hadronic uncertainties in the theoretical predictions.For this reason, independent cross-checks of b → sℓ + ℓ − transitions with complementary sensitivity to the different short-distance coefficients associated with the low-energy effective operators are desirable.
For instance, the angular analysis of baryonic transitions such as Λ b → Λ(→ pπ)ℓ + ℓ − [12] provides a number of independent observables [13,14] that can be combined with data on mesonic decays in a global fit, see e.g.Refs.[15,16].The most important hadronic input functions in these analyses are the Λ b → Λ (local) transition form factors, which appear after factorising the (local) hadronic and leptonic currents in the semi-leptonic and electromagnetic penguin operators in the weak effective Hamiltonian, see the illustration in the left panel of Fig. 1.Quantitative results for the form factors can be obtained from lattice-QCD studies [17,18] for small recoil energy (large invariant lepton mass squared q 2 ) and LCSRs for large recoil energy, see e.g.Refs.[19][20][21][22].Furthermore, one can exploit unitarity constraints in the form of a "z-expansion", which allows one to interpolate between small and large values of q 2 in a controlled way, see e.g.Ref. [23].Finally, in the heavy-quark limit for the b-quark, the number of independent form factors is reduced from ten to two at small recoil, and to only one at large recoil [21,24,25].Note that the numerically dominant part of the form-factor values at large recoil is due to the so-called "soft-overlap" mechanism, where the energy transfer to the final state cannot be described by a finite number of virtual gluon exchanges (despite the fact that the latter mechanism is parametrically of leading power in QCD factorisation, see the discussion in Ref. [26]).
Besides the local form-factor terms, time-ordered products of the hadronic operators in the weak effective Hamiltonian and the electromagnetic QED interaction also appear, leading to generalised hadronic matrix elements, often called "non-local form factors".An important example in b → sℓ + ℓ − transitions is the so-called "charm-loop effect", where a cc pair is produced by a four-quark operator in the weak effective Hamiltonian and then annihilated by a virtual photon that subsequently decays into the charged lepton pair, see the illustration in the middle panel of Fig. 1.This effect restricts the perturbative treatment within QCD factorisation to values of q2 well below the onset of charmonium resonances. 1 Nevertheless, phenomenological information about the lowest-lying cc resonances (J/ψ and ψ ′ ) can be implemented in a dispersive approach, which allows to extrapolate the perturbative results at small (or even negative) values to higher values of q 2 .This has been worked out in some detail for mesonic transitions, see e.g.Refs.[28][29][30].It is important to stress that an adequate knowledge of the size of the non-local form factors in rare b-hadron decays is essential for a reliable estimate of theoretical uncertainties.Only the combination of accurate theoretical predictions and experimental measurements of rare flavour observables allows us to constrain the size of physics beyond the SM in these decays or even establish deviations from SM predictions.
In this work, we are interested in decay topologies like the one shown on the right panel of Fig. 1, for which there is no estimate to date.Here the four-quark operators O 3−6 connect two pairs of quarks in the initial and final baryon, with a virtual photon emitted from one of these quarks dissociating into ℓ + ℓ − .This is analogous to the annihilation topologies in rare B → K ( * ) ℓ + ℓ − decays, and hence we refer to this contribution as "annihilation topologies".From the example shown, it is immediately clear that the third valence quark (plus additional gluons or q q pairs) does not necessarily participate in the hard-scattering process.In the context of QCD factorisation [5] this would correspond to kinematic endpoint configurations which prevent a complete factorisation of the decay amplitude into hadronic LCDAs and short-distance kernels.In the following, we therefore consider the "annihilation topologies" in the framework of LCSRs with b-hadron LCDAs, following the procedures outlined in Refs.[31,32] for mesonic form factors and Refs.[21,22] for baryonic form factors.
The outline of this article is as follows.In Sec. 2 we give our definitions of local and non-local form factors for the relevant operators in the weak effective Hamiltonian.We also introduce our kinematic notations and conventions that we use in the sum-rule calculation.In Sec. 3 we introduce the correlator from which we deduce the contribution of the annihilation topologies by comparing the perturbative calculation in leading-order QCD with the hadronic representation in terms of non-local form factors.We find that the dominant effects are associated with the case where a virtual photon is emitted from one of the light-quarks in the Λ b baryon.In this situation we find that the soft hadronic function describing the bound-state properties of the Λ b baryon is given in terms of a tri-local operator where the two light-quark fields are separated along two different light-cone directions.We derive the general momentum-space projector for this case in terms of generalised LCDAs.Sec. 4 is devoted to the numerical analysis of the sum rule with a careful assessment of parametric uncertainties from various sources.In comparison with the corresponding contribution of the local form-factor terms in the SM, we observe that the leading effect of the annihilation topologies is of similar size as in the mesonic counterpart for B → Kℓ + ℓ − or B → K * ∥ ℓ + ℓ − .In fact, we find O (1%) effects at the amplitude level for transverse γ * polarisation, whereas the effect for longitudinal γ * polarisation turns out to be negligible.After our conclusions in Sec. 5, we provide some additional details about our modelling of the generalised LCDAs and some of the calculation steps in the sum-rule calculation in Apps.A and B.

Theoretical framework 2.1 Definition of local form factors
In Λ b → Λℓ + ℓ − decays, the naively factorising contributions from the operators in the weak effective Hamiltonian require the knowledge of the local transition form factors for vector and axial-vector b → s currents.Our conventions for these form factors in the helicity basis follow the definition in Ref. [21].For the vector form factor we use where An analogous definition holds for the form factors of axial-vector currents, Throughout this work the spin arguments for the fermion states and Dirac spinors are usually not explicitly shown for simplicity, i.e. u Λ b (p, s) ≡ u Λ b (p).The projections on the vector form factors read and similarly for the axial-vector form factors, where is the metric tensor in the plane transverse to the momentum vectors p and p ′ .

Definition of non-local form factors
In this work, we are interested in hadronic matrix elements of the time-ordered product of the strong-penguin operators in the weak effective Hamiltonian with the electromagnetic current where Q q is the q quark charge.These matrix elements can be decomposed in the same way as the local form factors: Notice that due to the conservation of the electromagnetic current, Lorentz structures proportional to q µ do not appear on the r.h.s of this equation.In the following, we use the term "non-local form factors" for the generalised objects 5) and 5) .These non-local form factors can be isolated by contracting Eq. (2.8) with p µ and g µν ⊥ , respectively.These projections read With these conventions, the non-local contributions to the Λ b → Λℓ + ℓ − decay amplitude can be rewritten in terms of a q2 dependent shift of C 9 .Hence they can be accounted for using the following replacements: Similar relations are known for the mesonic counterpart B → K ( * ) ℓ + ℓ − , see, for instance, Refs.[28,33].

Light-cone vectors and power counting
It is convenient to introduce the following light-cone vectors such that with v µ being the four-velocity of the Λ b baryon, and the transverse metric can simply be written as (2.16) One can decompose any momentum vector in light-cone coordinates and consider the scaling of the individual momentum projections with a small expansion parameter λ ≪ 1.In our case, we take with m b ∼ M Λ b being the mass of the heavy b-quark.To identify the power-counting of the various momenta, we use the short-hand notation where the powers of λ indicate the momentum scaling in units of m b .For the b-quark, which is treated as a quasi-static colour source in the framework of heavy-quark effective theory (HQET), we thus have HQET b-quark : and ∆k µ is referred to as a soft residual momentum.Similarly, the light quarks and gluons in the Λ b bound state have soft momentum scaling: with virtualities k 2 s ∼ λ 4 .In this work, we concentrate on the large-recoil region, wherein the rest frame of Λ b -the energy of the hadronic final state is of the order M Λ b /2.The light constituents of the Λ have collinear momenta, scaling as collinear momenta in Λ baryon : with virtualities k 2 c ∼ λ 4 .Interactions between field modes with soft and collinear momenta are induced by hard-collinear momenta, internal hard-collinear modes : with virtualities k 2 hc ∼ λ 2 .Here, the scaling k ⊥ hc ∼ λ refers to hard-collinear modes in loops (while tree-level interactions would have k ⊥ hc ∼ λ 2 ).Finally, the momentum transfer to the lepton pair is given by and is referred to as anti-hard-collinear (this excludes the kinematic limit q 2 → 0).
3 Calculation of the "annihilation topologies"

Definition of the correlator
The first step in the derivation of the sum rule is the definition of a suitable correlator that allows the extraction of the required matrix element in (2.8).To this end, we replace the light baryon in the final state by an interpolating current, for which we use the same expression as in Ref. [21].In particular, we use the projector to project onto the leading spinor components for a collinear fermion.Here C is the charge conjugation matrix, the indices i, j, k are colour indices, and the quark field u T,i denotes the transpose in Dirac space of the field u i .In the following, the appearance of the charge conjugation matrix is always understood in the chiral representation for Dirac matrices, where and With this, we define the correlator as Here we consider the correlator as a function of the small light-cone projection n − • p ′ for a fixed value of the large light-cone projection As a consequence, the value of q 2 is determined by the value of n + • p ′ and vice versa.
For the perturbative calculation of the correlator, we take the momentum associated to the interpolating current to be hard-collinear : Note that the correlator has an open spinor index inherited from the strange-quark field in the interpolating current J Λ (y).

Hadronic representation of the correlator
To derive the hadronic dispersive representation of the correlator, we use unitarity, which consists in inserting a complete set of hadronic states between the interpolating current and the four-quark operators in Eq. (3.2).We obtain where the ellipses denote the contribution of the continuum and excited states and we have used our definition of the non-local matrix elements in Eq. (2.8).We define the decay constant of the Λ baryon as in Ref. [21]: which implies that f Λ has mass dimension 2. Using Eq. (3.6) and the projections in Eq. (2.9) yields Here in performing the spin summation, we have used that in our chosen reference frame with p ′ ⊥ = 0 and

OPE analysis of the correlator
For the power counting defined in Sec.2.3, the correlator (3.2) can be calculated using an operator product expansion (OPE).At leading order in the strong coupling, and restricting ourselves to the "annihilation topologies", the OPE analysis of the correlator corresponds to the four types of diagrams shown in Fig. 2. Each diagram comes in two copies, related by isospin symmetry, where the lines labelled by the momenta k 1 (and k 2 ) correspond to u (and d), or vice versa.Throughout we take the light quarks u, d, s to be massless.Notice that the diagram in which the photon is emitted from the spectator quark labelled by k 2 does not contribute to the discontinuity of Π(n − • p ′ ) and can therefore be ignored.In Fig. 2 we also indicate the virtuality of the internal propagators, which follows from the kinematics given above, together with a method-of-regions analysis of the loop integral.As can be seen, diagrams (b)-(d) all contain a hard quark propagator, in addition to a loop with two hard-collinear light-quark propagators.In contrast, diagram (a) contains an additional anti-hard-collinear propagator instead.This leads to an enhancement of diagram (a) compared to the other three diagrams.After integrating out the hard propagators, diagram (a) would thus match onto a correlator in SCET where the quark fields in QCD are trivially replaced by their SCET counterparts.On the other hand, in SCET diagrams (b)-(c) would correspond to another type of correlator in terms of an effective operator involving four quark fields and one additional photon field.In the following, we thus concentrate on diagram (a) since it is expected to give the highest power in the Λ/m b expansion.Moreover, as we discuss in detail below, the appearance of an anti-hard-collinear propagator together with a hard-collinear loop implies that the information about the configuration of the soft momenta k 1 and k 2 in the Λ b bound state is contained in a new type of soft functions that requires a generalisation of the concept of LCDAs for the Λ b .This makes diagram (a) particularly interesting from a conceptual point of view.
By inserting the effective operators and the currents into Eq.(3.2), we obtain the following expression for diagram (a) where colour indices are explicit, and spinor indices within square brackets are contracted.
For concreteness, we have taken the case where the photon is emitted from the u quark.Due to the isospin symmetry, the case where the photon is emitted from the d quark gives the same result except for the replacement where the fermion propagators S F is defined as Realising that the alternative colour contractions in brackets only yield an extra minus sign and performing the trivial integrations of the space-time coordinates, Eq. (3.11) can be written as where we denoted the Fourier transformed quark fields with a tilde.At this step, we perform the tensor reduction for the integration over the loop momentum ℓ 2 .We define with q ′ = p ′ −k 2 and -for the moment -consider the loop integration in D ̸ = 4 dimensions, such that Since the LCSRs are derived using a dispersion relation of the form we only need the discontinuity of the correlator.Thus, only the imaginary part of the integrals A(s) and B(s) is relevant in our calculation, which is finite for D → 4. Performing the algebra, we have ) The imaginary part of these integrals is then easily calculated: Inserting these results in Eq. (3.13), we obtain We observe that at leading power in the expansion parameter λ, the above expression depends on two opposite light-cone projections of the light-quark momenta in the Λ b baryon.The integrals over the loop momentum ℓ 2 depend on while the remaining light-quark propagator connecting the external photon and the effective four-quark operator depends on In the SCET jargon, the associated short-distance dynamics is described by hard-collinear modes, but in different light-cone directions (one in the direction of p ′ , and the other in the direction of q).After performing a Fourier transform to position space, this would correspond to tri-local matrix elements of the form where we have made the spinor indices α, β, δ explicit.These objects define a new type of three-particle LCDAs or, more appropriately, "soft functions" in the context of SCET factorisation (for this, the light-quark fields have to be supplemented with the corresponding soft Wilson lines to render the tri-local operator gauge invariant).Their appearance in the "annihilation topologies" is due to the fact that the two quarks take part in different dynamics: one, associated to the interaction with an anti-collinear photon; and another, associated with the collinear momentum of the interpolating current. 2 Before proceeding with the sum rule, we discuss and classify the new type of soft functions and the connection with the conventional baryon LCDAs.

Λ b soft functions with two light-like separations
The standard definitions of three-particle LCDAs for the Λ b baryon can be found in Ref. [40].
In order to generalise these definitions to our case, it is convenient to follow the procedure of Ref. [41], where the starting point is the decomposition of the hadronic matrix element of a general tri-local operator: Here z 1 and z 2 are space-time points and QCD gauge-links are understood implicitly.The Dirac matrices M (1(2)) contain an even (odd) number of Dirac matrices, respectively.In the following, we focus on the matrix M (2) which is relevant for our sum rule.(The matrix M (1) can be treated in a completely analogous way, see Ref. [41].)Here, the most general Lorentz-covariant decomposition is where t i = v • z i .The standard LCDAs can be obtained by expanding the above expression around the limit z 2 1 = z 2 2 = z 1 • z 2 = 0.However, for the new type of soft functions, we rather have to consider the expansion for the situation etc.
We introduce the LCDAs in momentum space by performing a Fourier transform Therefore, from the Fourier transform of Eq. (3.22), we can construct the momentum-space projector: where the derivatives are understood to act on a hard-scattering kernel that has been Taylorexpanded in transverse momenta, and we define We also introduced the abbreviations Explicit models for these LCDAs are derived in App. A.

Expressing the correlator in terms of soft functions
We can now proceed to express the hadronic matrix element appearing in the leading order result for the correlator (3.19) in terms of the momentum-space projector following from (3.20) and (3.24).To this end, we write the hadronic matrix element in Eq. (3.19) as a Dirac trace: and hence The function f αβ (k ⊥ 2 ) can be read off Eq. (3.19): where we have kept terms of order λ 2 but dropped subsubleading terms of order λ 4 , as well as terms proportional to n α + or n β + which do not contribute in (3.19).To derive Eq. (3.29) we have used that γ T µ = −Cγ µ C −1 , and the additional sign in front of the trace stems from anti-commuting the field operators u α (z 1 ) and d β (z 2 ) when using eq.(3.20) in (3.19).As already mentioned, the projector M (1) does not contribute to the trace, because the Dirac structure Γ has an odd number of Dirac matrices.For later convenience, we also expand the term (/ k 1 − / q) up to O (λ 4 ): Let us first consider the contribution arising from the leading term in f αβ together with the leading term in (/ k 1 − / q).For this we obtain where the upper sign is for the operators O 3,4 , and the lower sign for O 5,6 .For the Levi-Civita symbol we use the convention ϵ 0123 = 1 and the short-hand notation ϵ µνρσ q σ ≡ ϵ µνρ{q} .Note that Eq. (3.32) fixes the Lorentz index µ to be transversal and hence this term only contributes to the LCSR for H This can be further simplified by using that after combining with the Dirac projection of the b-quark field in (3.19), we have which holds in D = 4 dimensions.Therefore only the operators O 5 and O 6 contribute to the LCSR for H ⊥( 5) in the considered order of the calculation.Actually, this result can already be understood by considering the Dirac structure of the original four-quark operators in QCD and projecting the light-quark fields onto the leading collinear or anti-collinear spinor components in SCET that correspond to the topology in Fig. 2(a It follows from this argument, that neither the current-current operators O which have larger Wilson coefficients, but enter with Cabibbo-suppressed CKM factors in b → s transitions -contribute to the baryonic annihilation topologies at leading power.It is worth noting that in the mesonic counterpart, the role of q and q is interchanged in the leading annihilation topology compared to the baryonic case; and for that reason -by the same argument -only the operators O 3 and O 4 (and also O 2 ) contribute at leading power [9].
To proceed in the calculation of the associated non-local form factors H + , we contract Eq. (3.27) with p µ .Since the leading term vanishes, we consider sub-leading terms of O (λ 2 ): where the terms proportional to n β − n ν − and n β + have been dropped, since they vanish once contracted with the Dirac matrix in the last line of Eq. (3.19).We obtain the leading contribution to the OPE calculation of p µ Im Π OPE where we have used that Therefore, in analogy with H

Derivation of light-cone sum rules
Following the usual procedure to derive a LCSR -see e.g Ref. [42] -we match the OPE calculation of the correlator Π µ of Eqs.(3.33) and (3.39) onto the corresponding hadronic representations of Eqs.(3.7) and (3.8).The contribution of the continuum and excited states is removed by using the semi-global quark-hadron duality approximation.In practice, we assume that the second line of Eqs.(3.7)-(3.8) is equal to the dispersive integral in the OPE calculation above the effective threshold s 0 , whose value is discussed in Sec. 4. We perform a Borel transform with respect to the variable (n − • p ′ ) to further suppress the continuum and excited states contribution.This reduces the systematic error due to the quark-hadron duality approximation.Performing a Borel transform in our case consists in replacing where ω M is the associated Borel parameter and K does not depend on (n − • p ′ ).The bulky formulae resulting from the OPE calculation with the hadronic representation are collected in App.B. In our calculation, this matching implies that (taking into account Eqs. +(5) = −H +( 5) , H +(5) = 0 , H up to higher order corrections.Hence, in the limit M Λ b ≫ m Λ , the last of these identities becomes +5 .Thus, it is sufficient to present the LCSRs for, e.g., only H ⊥ and H + .These LCSRs read and where σ (0) ≡ s (0) /(n + • p ′ ).The integrals can be performed analytically, assuming the models for the LCDAs given in App. A. These formulae are given in App.B. The LCSRs for the non-local form factors H (i) ⊥( 5) and H (i) +( 5) can be compared with the LCSR for the local form factor ξ Λ derived in Ref. [21].In the large recoil limit, i.e. (n + • p ′ ) ∼ M Λ b , ξ Λ is equal to each of the helicity form factors: The LCSR for ξ Λ reads where ϕ 4 is one the standard LCDAs partially integrated (see Ref. [21] for its definition).
Comparing this sum rule with the one for H ⊥ , we observe that they contribute at the same power of λ 2 .The factor of 192π due to the loop in Eq. (3.44) is compensated in the decay amplitude by the factor −16π 2 2M 2 Λ b q 2 appearing in Eqs.(2.10)- (2.13).Therefore the suppression of the "annihilation topologies" is only due the small Wilson coefficients of the operators O 3−6 .It is also important to stress that, while local form factors are real-valued, the non-local form factors are generally complex-valued.This is evident in our LCSRs, as there is a pole in the integration path of dω 1 .From a phenomenological point of view, this imaginary part is due to uū and d d hadronic states going on shell.In other words, H (i) ⊥ (5) and 5) have a branch cut on the real positive axis starting at q 2 = 4m 2 π .It is also interesting to compare our results for Λ b → Λℓ + ℓ − decays calculated using LCSRs with the corresponding results for B → K ( * ) ℓ + ℓ − calculated using QCD factorisation, i.e. the annihilation topologies [33,43].Confronting our Eqs.(3.44)-(3.45)with Eq. ( 18) of Ref. [33], we find that the two hard scattering kernels have a very similar structure, with a pole appearing in the denominator for any q 2 > 0. Another analogy concerns the fact that in both the mesonic and baryonic cases the leading contribution comes from the diagram where 1/ω 0 3.4 ± 1.6 MeV −1 [21,22] Table 1: Input parameters used to evaluate the LCSRs.The lattice QCD calculation of f Λ in Ref. [44] is compatible with an older sum-rule estimate in Ref. [46].Notice that the normalisation convention in Ref. [44] differs from that used by us and in Ref. [46] by a SU (3) Clebsch-Gordan the photon is emitted from the spectator quark in the b hadron.However, as mentioned above, in the mesonic case the contributing operators are O 3 and O 4 , while in the baryonic case they are O 5 and O 6 .Also, in the baryonic case only the transverse γ * polarisation contributes at leading power, while in the mesonic case the dominant annihilation effect appears for longitudinal γ * polarisation.

Numerical results
We provide numerical results for the non-local form factors H ⊥( 5) and H +(5) using the LCSRs in Eqs.(3.44) and (3.45) and also taking into account the identities (3.43) (see also App.B for the integrated LCSRs).These LCSRs are evaluated with the inputs listed in Tab. 1 and the LCDAs models obtained in App. A. As there is no independent estimate of the LCDA parameter ω 0 , we vary its inverse in the interval [1.8, 5.0] MeV −1 .This very conservative interval contains with margin the estimates of Refs.[21,22].The interval for the Borel parameter ω M is chosen in such a way that this parameter is both sufficiently large to suppress higher power corrections in the OPE and sufficiently small to ensure that the contribution of the continuum and excited states is subleading compared to that of the Λ baryon.We use the same central value of Ref. [21] and vary it within ±0.5 GeV 2 .We have checked that our LCSRs are stable in this interval for the Borel parameter.As in Ref. [21], we choose s 0 to be equal to the mass of the next resonance with the same quantum numbers of the Λ baryon.All parameters in Tab. 1 are assumed to be Gaussian distributed, except for the Borel parameter for which we take a flat distribution.
The LCSRs in Eqs.(3.44) and (3.45) can be used for values of q 2 such that the energy of the Λ baryon is of the order of m Λ b /2 in the Λ b baryon rest frame.To avoid large violations of the quark-hadron duality, we also take q 2 larger than the narrow vector resonances such as the ρ and ω mesons.We can therefore evaluate our LCSRs in the range 2 GeV 2 ≲ q 2 ≲ 6 GeV 2 .

Conclusions
In this work, we have studied the non-local contributions of the strong penguin operators in Λ b → Λℓ + ℓ + decays, where a virtual photon is radiated from one of the light quarks.We refer to this situation as "annihilation topologies" because of the analogy with annihilation in B → K ( * ) ℓ + ℓ + decays.Their contribution to the corresponding non-local form factors is calculated using light-cone sum rules (LCSRs) with Λ b light-cone distribution amplitudes (LCDAs).More precisely, we find that -at leading power -the hard-scattering kernel entering the factorisation formula for the underlying correlator depends on opposite lightcone projections of the two light-quark momenta in the Λ b baryon.This implies that in this case the required hadronic information about the Λ b bound state is contained in a new type of soft functions that generalise the standard LCDAs which are known in the literature and used, for instance, in local form-factor calculations.In order to evaluate the LCSRs, we have constructed a model for these new soft functions that links them to the standard Λ b LCDAs.On this basis, we have presented the result from the leading-order LCSRs for the annihilation contribution to the non-local form factors in analytical form.Here we have focused on the leading annihilation topology, where the virtual photon is radiated from one of the light (soft) quarks in the Λ b baryon.Numerical predictions are presented in the form of a q 2 -dependent shift ∆C 9 of the Wilson coefficient C 9 , where q 2 is the invariant mass of the lepton pair.In the considered range q 2 ∈ [2, 6], we observe that |∆C 9,⊥ |/C 9 ∼ O (1%) and Im∆C 9 ̸ = 0, and hence this effect should not be neglected in precision analyses of Λ b → Λℓ + ℓ + observables.
Our findings show a number of analogies with the annihilation topologies in B → K ( * ) ℓ + ℓ + decays.For instance, in both cases ∆C 9 features a very similar q 2 dependence and is of similar numerical size, which can be traced back to the functional form of the intermediate hardcollinear light-quark propagator folded with the modelled shape of the (generalised) LCDAs.A major difference is that in the mesonic case the operators of the weak effective Hamiltonian that contribute at leading order are O 3 and O 4 (with left-handed q q currents), while in the baryonic case they are O 5 and O 6 (with right-handed q q currents).This can be traced back to the Dirac structure of the penguin operators that results from replacing the light quark fields by their leading spinor components in soft-collinear effective theory.
We re-emphasise that in our analysis we only considered one particular non-factorising decay topology, and it is left for future work to perform similar investigations for the subleading annihilation topologies, but also to study related topologies where a quark loop originating from the 4-quark operators or the chromomagnetic penguin operator connects to one of the light quarks from the Λ b bound state by hard-collinear gluon exchange (similar topologies had been studied for mesonic transition in the past).While in both cases, we expect sub-leading numerical effects, verification by explicit calculation would be desirable.
To conclude, the inclusion of genuinely non-factorising contributions from hadronic operators in Λ b → Λℓ + ℓ − decays at large recoil are important, not only to improve the accuracy of SM predictions for physical observables and to sharpen the current constraints on physics beyond the SM, but also as a laboratory to test our understanding and further deepen our knowledge of non-perturbative QCD effects in exclusive baryonic reactions.

B Further details on the LCSRs
In the following, we provide a few intermediate steps in the derivation of the LCSRs in Sec.3.6, i.e. the matching of the OPE calculation Π OPE µ of Eqs.(3.33) and (3.39) onto the respective hadronic representations of Eqs.(3.7) and (3.8).This matching yields the LCSRs for the non-local form factors H (i) ⊥( 5) and H (i) + (5) .After applying quark-hadron duality and performing the Borel transform, the resulting LCSRs read and Using the model for the LCDAs presented in App.A, it is possible to perform all integrations over light-cone momenta analytically.Here, one has to take into account that, the integration over dω 1 contains a singularity on the integration path for ϵ → 0. These integrals can be performed by Cauchy's theorem, along the same lines as for the annihilation in B → K ( * ) ℓ + ℓ − [33], leading to ω 0 e  where Ei(x) is the exponential integral function.

Figure 1 :
Figure 1: Illustration of different decay topologies for Λ b → Λℓ + ℓ − : Left: local form-factor contribution for semi-leptonic operators O 9 or O 10 .Center: Non-local contribution from virtual photon radiation off a charm (or light) quark loop.Right: example for a non-local contribution of the fourquark operators O 3−6 in Λ b → Λℓ + ℓ − decays.From the analogy to the corresponding terminology in B → K ( * ) ℓ + ℓ − decays, this is referred to as an "annihilation topology".-In each diagram, operators from the weak effective Hamiltonian are indicated by a black square, the b-quark is indicated by a double line, and the virtual photon subsequently dissociates into ℓ + ℓ − (not shown).

Figure 2 :
Figure 2: Leading-order annihilation diagrams contributing to the correlator in Eq. (3.2).In each diagram, the lines labelled with momentum k 1 and k 2 should be identified with up-and down-quark, or vice versa.The b-quark is indicated by a light blue line; the strange quark by a magenta line.In addition, (anti-)hard-collinear propagators are indicated by dotted lines, and hard propagators by thick lines.The double line denotes the b-quark in HQET.
since we neglect the light quark masses.The projector P L (P R ) acting on the up quark field is for the case of O 3 and O 4 (O 5 and O 6 ) operators.In addition, the colour indices without (with) parenthesis refer to the case of O 3 and O 5 (O 4 and O 6 ) operators, respectively.Performing the Wick contractions corresponding to the diagram Fig. 2(a), we get

⊥( 5 )
. We can now easily obtain the leading contribution to the OPE calculation of Im Π OPE µ by plugging Eqs.(3.27) and (3.32) into Eq.(3.19): ). Ignoring the colour indices for the moment, one has by virtue of Eqs.(3.34)  and(3.35)

⊥( 5 )
and as a consequence of Eqs.(3.36) and (3.37),only the operators O 5 and O 6 contribute to the LCSR for H