Baryon-to-Meson Transition Distribution Amplitudes: Formalism and Models

In specific kinematics, hard exclusive amplitudes may be factorized into a short distance dominated part computable in a perturbative way on the one hand, and universal, confinement related hadronic matrix elements on the other hand. The extension of this description to processes such as backward meson electroproduction and forward meson production in antiproton-nucleon scattering leads to define new hadronic matrix elements of three quark operators on the light cone, the nucleon-to-meson transition distribution amplitudes, which shed a new light on the nucleon structure.


Introduction
Accessing transition distribution amplitudes (TDAs) in specific exclusive reactions is an important goal to progress in our understanding of quark and gluon confinement. The baryon-to-meson TDAs are defined through baryon-meson matrix elements of the nonlocal three quark (antiquark) operators on the light cone (n 2 = 0): O αβγ ρτ χ (λ 1 n, λ 2 n, λ 3 n) = ε c 1 c 2 c 3 Ψ c 1 α ρ (λ 1 n)Ψ c 2 β τ (λ 2 n)Ψ c 3 γ χ (λ 3 n), where α, β, γ stand for the quark (antiquark) flavor indices and ρ, τ , χ denote the Dirac spinor indices. Antisymmetrization is performed over the color group indices c 1,2,3 . Gauge links in (1) are omitted by adopting the light-like gauge A · n = 0. These non-perturbative objects, first studied in [1][2][3], share common features both with baryon distribution amplitudes (DAs) introduced in [4,5] as baryon-to-vacuum matrix elements of the same operators (1) and with generalized parton distributions (GPDs), since the matrix element in question depends on the longitudinal momentum transfer Δ + = ( p M − p N ) · n between a baryon and a meson characterized by the skewness variable ξ = − ( p M − p N )·n ( p M + p N )·n and by a transverse momentum transfer Δ T . For the QCD evolution equations obeyed by baryon-to-meson TDAs one distinguishes the Efremov-Radyushkin-Brodsky-Lepage (ERBL)-like domain in which all three momentum fractions of quarks are The physical picture encoded in baryon-to-meson TDAs is conceptually close to that contained in baryon GPDs and baryon DAs. Baryon-to-meson TDAs characterize partonic correlations inside a baryon and give access to the momentum distribution of the baryonic number inside a baryon. The same operator also defines the nucleon DA, which can be seen as a limiting case of baryon-to-meson TDAs with the meson state replaced by the vacuum. In the language of the Fock state decomposition, baryon-to-meson TDAs are not restricted to the lowest Fock state as DAs. They rather probe the non-minimal Fock components with additional quarkantiquark pair: depending on the particular support region in question (see Fig. 1). Note that this interpretation can be justified only at a very low normalization scale and can be significantly altered at higher scales due to the evolution effects. Similarly to GPDs, by Fourier transforming baryon-to-meson TDAs to the impact parameter space (Δ T → b T ), one obtains additional insight on the baryon structure in the transverse plane. This allows one to perform the femto-photography of hadrons [6] from a new perspective.

Parametrization and Phenomenological Models for Baryon-to-Meson TDAs
Although baryon-to-meson TDAs can be introduced for all types of baryons and mesons, we start our consideration from the simplest case of nucleon-to-pion (π N ) TDAs. For a given flavor contents (e.g. uud proton-to-π 0 TDA) the parametrization of the leading twist-3 π N TDA involves 8 invariant functions, each depending on the 3 longitudinal momentum fractions x i , the skewness parameter ξ , momentum transfer squared Δ 2 as well as on the factorization scale μ 2 : Here f π = 93 MeV is the pion weak decay constant and f N determines the value of the nucleon wave function at the origin. Throughout this paper we adopt Dirac's "hat" notationv ≡ v μ γ μ ; σ μν = 1 2 [γ μ , γ ν ]; σ vμ ≡ v λ σ λμ ; C is the charge conjugation matrix and U + =pn U( p N , s N ) is the large component of the nucleon spinor. The parametrization (3), originally introduced in Ref. [7], is extremely convenient for the phenomenological applications since in the limit Δ T → 0 just three invariant amplitudes V survive. An alternative parametrization which is better suited to address the symmetry properties of π N TDAs was introduced in [8]. In particular, within this latter parametrization π N TDAs turn to satisfy the polynomiality property of the Mellin moments in x i in its most simple form. The relation between the two parametrizations in presented in the Appendix C of [8].
Working out the physical normalization of baryon-to-meson TDAs and building up consistent phenomenological models for them represents a considerable task. Below we present a short overview of the approaches available at the present moment.
The first attempt to provide a model for π N TDAs relying on chiral dynamics was performed in [7] (see Ref. [8] for the detailed derivation). The soft-pion theorem [9,10] This results in a simple model for π N TDAs summarized in the Erratum to [7] 1 : Despite its obvious drawbacks (like the very narrow validity range limited to the close vicinity of the threshold and lack of an intrinsic Δ 2 -dependence) this simple model for the first time provided a quantitative estimate of the physical normalization for π N TDAs. In particular, the predictions of the revised soft-pion-limit model (4) were recently employed in the first feasibility studies [11] for accessing π N TDAs withPANDA through p p → γ * π 0 . Another simple model for π N TDAs suggested in Ref. [8] accounts for the contribution of the cross-channel nucleon exchange. As one can see from Fig. 2, this model is conceptually similar to the pion exchange model for the polarized nucleon GPDẼ suggested in Ref. [12]. With the use of the π N TDA parametrization (3) the nucleon pole model reads: Here ) ensures the pure ERBL-like support of TDAs and g π N N ≈ 13 is the pion-nucleon phenomenological coupling. This turns to be a consistent model for π N TDAs in the ERBL-like region and satisfies the polynomiality conditions and the appropriate symmetry relations.
By an obvious change of couplings the model (5) can be generalized to the case of other light mesons (η, η , K , ... etc.). Also it is not necessarily limited to the contribution of the cross-channel nucleon exchange. In Ref. [8] the contribution of the cross-channel Δ(1232) into π N TDAs was worked out explicitly. Finally, very recently in Ref. [13] the cross-channel nucleon exchange model was generalized for the case of nucleon-tovector meson TDAs.
Still the aforementioned baryon-to-meson TDA model describes TDAs only within the ERBL-like support region. To get a model defined on the complete support domain one may rely on the spectral representation for baryon-to-meson TDAs in terms of quadruple distributions suggested in Ref. [14]: where Ω i denote three copies of the usual domain in the spectral parameter space. The spectral density f is an arbitrary function of six variables, which are subject to two constraints and therefore effectively is a quadruple distribution.
Similarly to the familiar double distribution representation for GPDs the quadruple distribution representation (6) for TDAs turns to be the most general way to implement the support properties of TDAs as well as the polynomiality property for the x i -Mellin moments, which is a direct consequence of Lorentz invariance (see Ref. [15]).
Contrarily to GPDs, TDAs do not possess the comprehensive forward limit ξ → 0. This complicates the construction of a phenomenological Ansatz for quadruple distributions. However, a partial solution was proposed for the case of π N TDAs. In this case one can rely on the complementary ξ → 1 limit, in which, as already discussed above, π N TDAs V π N 1 , A π N 1 , T π N 1 are constrained by chiral dynamics (see Eq. (4) for ξ = 1). In Ref. [15] we proposed a factorized Ansatz designed as a universal profile function multiplying the nucleon DA combination to which the π N TDA in question reduces in the ξ = 1 limit.
In loose words our Ansatz for quadruple distributions is based on "skewing" the ξ = 1 limit. At this point we are similar to the famous Radyushkin factorized Ansatz for double distributions [16]. In that case rather the ξ = 0 limit, where GPDs reduce to usual parton densities, is "skewed" to provide a non trivial ξ -dependence for GPDs. For practical details we refer the reader to Ref. [15].
Also similarly to the GPD case, in order to satisfy the polynomiality condition in its complete form, the spectral part (6) is to be complemented by a D-term-like contribution defined solely in the ERBL-like region and responsible for the highest possible power of ξ occurring for a given x i -Mellin moment. The simplest possible model for such a D-term is the contribution of the nucleon exchange into π N TDAs (5). Note that we avoid double counting since the nucleon exchange contribution into π N TDAs V dies out in the ξ → 1 limit.
In this way we come to the so-called "two component" model for π N TDAs [15] which contains a spectral part fixed from the ξ → 1 limit and a D-term-like contribution coming from the cross-channel nucleon Reference [15] contains the first attempts of phenomenological application of the "two component" π N TDA model for the description of backward pion electroproduction off nucleons for the typical JLab kinematical conditions. π N TDAs within this model do not nullify at the cross-over trajectories separating the ERBL-like and DGLAP-like TDA support regions. This results in the non-zero contribution into the imaginary part of the relevant elementary amplitude. The observable quantity sensitive to this issue is the transverse target single spin asymmetry (STSA) [17]. The "two component" π N TDA model predicts a sizeable value of STSA for x B typical for JLab@12 GeV conditions.

Baryon-to-Meson TDAs from Backward Meson Electroproduction at JLab
Within the generalized Bjorken limit, in which Q 2 = −q 2 and s = (q + p 1 ) 2 = W 2 are large, x B ≡ Q 2 2 p 1 ·q is fixed and the u-channel momentum transfer squared is small compared to Q 2 and s (|u| ≡ |( p 2 − p 1 ) 2 | Q 2 , s) the amplitude of the hard subprocess of the exclusive electroproduction of mesons off nucleons is supposed to admit a collinear factorized description in terms of nucleon-to-meson TDAs and nucleon DAs, as it is shown on Fig. 3. The small u corresponds to the meson being produced in the near-backward direction in the γ * N center-of-mass system (CMS). This regime, referred as the near-backward kinematics, is complementary to the more conventional near-forward kinematical regime (Q 2 = −q 2 , s -large, x B -fixed, |t| ≡ |( p 2 − p 1 ) 2 | Q 2 , s) in which the factorized description in terms of GPDs and meson DAs applies to hard meson production subprocess.
The details of the formalism for the case of backward electroproduction of light pseudoscalar mesons (π, η) and rough cross section estimates for the kinematics conditions of JLab can be found in Refs. [7,15]. A generalization for the case of vector mesons (ρ, ω, φ) was proposed in [13].
As an example of our predictions, on the right panel of Fig. 3 we present out estimates of the backward γ * p → π + n cross section within the factorized description involving π N TDAs using the cross channel nucleon exchange model (5). The cross section turns out to very sensitive to the form of the input phenomenological solution for nucleon DA. We show the results for Chernyak-Ogloblin-Zhitnitsky (COZ) [18] (long blue dashes); King-Sachrajda (KS) [19] (solid green line); Braun-Lenz-Wittmann next-to-leading-order  [20] (medium orange dashes) and NNLO modification [21] of BLW (short brown dashes). The solutions close to the asymptotic form of the nucleon DA φ(y i ) = 120y 1 y 2 y 3 result in a very small cross section, while those significantly different from the asymptotic form like COZ and KS result in larger cross sections. We refer the reader e.g. to the discussion in Ref. [22] on various phenomenological inputs for nucleon DAs.
A first experimental signal may have been detected at JLab [23] in backward kinematics for e − N → e − π N and for e − N → e − ω N [24]. We expect the accessible kinematical domain to be more adequate to a factorized leading twist analysis in higher energy experiments at JLab@12 GeV.

Study of Baryon-to-Meson TDAs atPANDA
Another tempting possibility to get experimental access to baryon-to-meson TDAs is to consider the time-like counterpart of reaction (7): the nucleon-antinucleon annihilation into a high invariant mass lepton pair in association with a light meson M: The factorization mechanism for (8) suggested in [25,26] applies within two kinematic regimes referred as the near-forward 2 with (s = ( Fig. 4) and the near-backward one, which differs by the obvious change of kinematical variables ( p N → pN , pN → p N , t → u). The charge conjugation invariance results in perfect symmetry between the two kinematical regimes. The suggested reaction mechanism manifests itself through the characteristic forward and backward peaks of the NN → γ * M cross section. The characteristic features of the TDA-based description of (8) are the scaling behavior in 1/q 2 of the cross-section and the specific behavior in cos θ * (by θ * we denote the lepton polar angle defined in the CMS of the lepton pair) resulting from the leading twist dominance of the transverse polarization of the virtual time-like photon. A detailed feasibility study for accessing π N TDAs throughp p → γ * π 0 → e + e − π 0 atPANDA was performed in [11]. As a phenomenological input, the predictions of the simple π N TDA model (4) were employed. The results of this analysis are promising concerning the experimental perspectives for accessing π N TDAs withinPANDA@GSI-FAIR experiment. On Fig. 5 we present the integrated cross section dσ /d Q 2 with |Δ 2 T | ≤ 0.2 GeV 2 forp p → e + e − π 0 as a function of Q 2 for W 2 = 10 GeV 2 and W 2 = 20 GeV 2 within the collinear factorization approach. We employ the cross channel nucleon exchange model (5) for π N TDAs with various phenomenological DAs used as inputs. The mechanism involving nucleon-to-meson TDAs was also proposed in Ref. [27] for the reaction which can also be studied atPANDA alongside with the investigation of the spectrum of charmonium states. Similarly to (8), one can consider the near-forward (see the right panel of Fig. 4) and the near-backward kinematic regimes symmetric due to charge conjugation invariance. On Fig. 6 we show the differential cross section dσ dΔ 2 for pp → J/ψ π 0 as a function of W 2 for exactly forward (backward) pion production (Δ 2 T = 0) and as a function of Δ 2 T for a fixed value of W 2 . The cross channel nucleon exchange model (5) is employed for the relevant π N TDAs. To cope with the strong ∼α 6 s dependence of the cross section we fix the value of α s from the requirement that it allowed to reproduce the experimental value of Γ (J/ψ → pp) within the perturbative QCD description of Ref. [18]. Based on the cross-section estimates presented in Ref. [27] a detailed feasibility study for accessing pp → J/ψπ 0 atPANDA has been performed [11,28,29].

Conclusion
Baryon-to-meson TDAs are new non-perturbative objects which have been designed to help us scrutinize the inner structure of nucleons. Experimentally, one may access TDAs both in the space-like domain with backward electroproduction of mesons at JLab and COMPASS and in the time-like domain in antiproton nucleon annihilation atPANDA. We also expect [30] the time-reversed π → N TDAs to be accessible at J-Parc through the reactions: π − p → J/ψ n ; π − p → γ * n → μ + μ − n. Extracting TDAs from space-like and time-like reactions will be a stringent test of their universality [31], and hence of the factorization property of hard exclusive amplitudes. This hopefully will help us to disentangle details of the complex dynamics of quark and gluon confinement in hadrons.
The special π N TDA case does not exhaust all interesting possibilities, and the vector meson sector should be experimentally accessible as well as the pseudoscalar meson sector [13]. A double handbag description of other processes such as charm meson pair production in proton-antiproton annihilation may also necessitate the introduction of baryon to charmed meson TDAs [32].