A Perturbative QCD Approach to \(\varvec{\pi ^{-}~p \rightarrow D^{-}~{\varLambda }_{c}^{+}}\)

Few-Body Syst A Perturbative QCD Approach to π − p → D− Λ+c Stefan Kofler Wolfgang Schweiger We employ the generalized parton picture to analyze the reaction π − p → D− Λc+. Thereby it is assumed that the process amplitude factorizes into one for the perturbatively calculable subprocess u¯ u → c¯ c and hadronic matrix elements that can be parameterized in terms of generalized parton distributions for the π − → D− and p → Λc+ transitions, respectively. Representing these parton distributions in terms of valencequark light-cone wave functions for π , D, p and Λc allows us to make numerical predictions for unpolarized differential and integrated cross sections as well as spin observables. In the kinematical region where this approach is supposed to work, i.e. s 20 GeV2 and in the forward hemisphere, the resulting cross sections are of the order of nb. This is a finding that could be of interest in view of plans to measure π − p → D− Λc+, e.g., at J-PARC or COMPASS. Exclusive production of charmed hadrons is still a very controversial topic. Experimental data are very scarce and theoretical predictions differ by orders of magnitude, depending on the approach used. From general scaling considerations [1] one expects, e.g., that the p¯ p → Λ¯c− Λc+ cross section is suppressed by at least two to three orders of magnitude as compared to the p¯ p → Λ¯ Λ cross section. This means that one probably has to deal with cross sections of the order of nb, a challenge which nevertheless seems to be experimentally treatable, as the measurement of e+e− → Λc+ Λ¯c− cross sections has shown [2]. A considerable improvement of the experimental situation on pair production of charmed hadrons is to be expected from the P¯ ANDA detector at FAIR [3]. Another class of reactions for which experimental data may become available even in the near future is the pion-induced exclusive production of charmed hadrons as planned, e.g., at J-PARC [4]. In the present contribution we are going to present a theoretical analysis of π − p → D− Λc+ based on the generalized parton picture. Assuming the intrinsic charm of the p (and the π ) to be negligible, the charmed hadrons in the final state are produced via a handbag-type mechanism (see Fig. 1). The blobs in Fig. 1 indicate soft hadronic matrix elements that are parameterized in terms of generalized parton distributions (GPDs), the c¯ c pair is produced perturbatively with the c-quark mass mc acting as a hard scale. A model for the p → Λc+ GPDs is available from foregoing work on p¯ p → Λ¯c− Λc+ [5]. The new ingredients are the π − → Dtransition GPDs which we model in analogy to Ref. [5] as overlap of valence-quark light-cone wave functions 1 Motivation for π − and D−. With these models for the p → Λc+ and π − → D− GPDs we will estimate the contribution of our handbag-type mechanism to the π − p → D− Λc+ cross section. In Sect. 2 we will sketch the steps and assumptions which finally give us a factorized form of the hadronic π − p → D− Λc+ scattering amplitude. Here also the model wave functions leading to the GPDs used for the numerical calculations are presented. Section 3 contains the numerical predictions and a short discussion of competing production mechanisms based on hadrondynamics and our conclusions. 2 Double-Handbag Amplitude, GPDs and Transition Form Factors We consider π − p → D− Λc+ in a symmetric CM frame. This means that the transverse component of the momentum transfer = ( p − p) = (q − q ) is symmetrically shared between the particles and the 3-vector part of the average momentum p¯ = ( p + p )/2 is aligned along the z-axis (for assignments of momenta see Fig. 1). Expressed in light-cone coordinates the particle four-momenta can then be written as p = (1 + ξ ) p¯+, p = (1 − ξ ) p¯+, m2(2p1++ ξ )2⊥p¯/+4 , − 2⊥ , M Λ2c + 2 /4 ⊥ 2(1 − ξ ) p¯+ , ⊥ 2 , q = q = m2π + 2⊥/4 2(1 + η)q¯ − M D2 + 2⊥/4 2(1 − η)q¯ − , (1 + η)q¯ −, 2 ⊥ , , (1 − η)q¯ −, − 2⊥ , where we have introduced the skewness parameter ξ = − +/2 p¯+. The minus component of the average momentum q¯ − = (q− + q −)/2 and the skewness parameter η = −/2q¯ − have been introduced for convenience, but are determined by ξ , p¯+ and 2⊥. The hadronic amplitude as depicted in Fig. 1 can then be written in the form M = d4k1avθ kav+ 1 d4z1 eik1avz1 (2π )4 d4k2avθ kav− 2 d4z2 eik2avz2 (2π )4 × Λc+: p μ |T Ψ¯ c(−z1/2)Ψ u (z1/2)| p : pμ H k1, k2; k1, k2 D−: q ν |T Ψ¯ u (z2/2)Ψ c(−z2/2)|π −: qν , (1) (2) with H˜ denoting the perturbatively calculable kernel that represents the partonic subprocess u¯ u → g → c¯ c. Here two of the four integrations over the quark 4-momenta (and the corresponding Fourier transforms) have already been eliminated by introducing average quark momenta kiav = (ki + ki )/2 and the fact that the momentum transfer on hadron and parton level should be the same, i.e. (k1 − k1) = ( p − p ) and k2 − k2 = (q − q ) (a consequence of translational invariance). The further analysis of M makes use of the collinear approxim (...truncated)