Interference Fragmentation Functions and Valence Quark Spin Distributions in the Nucleon∗

We explore further applications of the twist-two quark interference fragmentation functions introduced earlier. We show that semi-inclusive production of two pions in the current fragmentation region in deep inelastic scattering of a longitudinally polarized electron on a longitudinally polarized nucleon can provide a probe of the valence quark spin (or helicity difference) distribution in the nucleon. Submitted to: Physical Review D †Email address: [email protected] ‡Email address: [email protected] §Email address: [email protected] ∗This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818, and by the RIKEN BNL Research Center. Measurements of quark and gluon distributions within hadrons provide us with valuable information about the nonperturbative nature of the quarks and gluons inside the hadrons. In a recent Letter [1], we have studied semi-inclusive production of two pions in the current fragmentation region in deep inelastic scattering on a transversely polarized nucleon. This may provide a practical way to measure the quark transversity distribution in the nucleon, which has proved difficult to access experimentally [2–8]. In this paper, we extend our study to the case of a longitudinally polarized electron beam scattering off a longitudinally polarized nucleon target. We show that the interference between the sand p-wave of the two-pion system around the ρ can provide an asymmetry [Eq. (6)] which is sensitive to the valence quark spin distribution in the nucleon. Note that the asymmetry would vanish by C-invariance if the two pions were in a charge conjugation eigenstate. Hence there is no effect in regions of the ππ mass dominated by a single resonance. Significant effects are possible, however, in the ρ mass region where the sand p-wave production channels are both active and provide exactly the charge conjugation mixing necessary. The asymmetry we obtain throws pions of one charge forward along the fragmentation axis relative to pions of the other charge. If one integrated over the other kinematic variables, whatever result persisted would appear as a difference between the π and π fragmentation functions correlated with the valence quark spin distributions. This effect in single pion fragmentation was proposed and studied some years ago by Frankfurt et al. [9] and Close and Milner [10]. The asymmetry we describe is therefore one particular contribution to this more general effect, with the advantage that it can be characterized in terms of ππ phase shifts and two particle fragmentation functions that appear in other hard processes. Consider the semi-inclusive deep inelastic scattering process: ~e ~ N → eππX. We define the kinematics as follows. The four-momenta of the initial and final electron are k = (E,~k) and k = (E, ~k′), and the nucleon’s momentum is Pμ. The momentum of the virtual photon is q = k−k, and Q = −q = −4EE sin θ/2, where θ is the electron scattering angle. The standard variables in DIS, x = Q/2P · q and y = P · q/P · k, are adopted. We work at low ππ invariant mass, where only the sand p-waves are significant. The σ[(ππ) l=0 ] and ρ[(ππ) l=1 ] resonances are produced in the current fragmentation region with momentum Ph and momentum fraction z = Ph · q/q. The invariant squared mass of the two-pion system is m = (k+ + k−) , with k+ and k− the four-momentum of π + and π, respectively. The decay polar angle in the rest frame of the two-pion system is denoted by Θ. Note that the azimuthal angle φ of the two-pion system does not figure in present analysis and can be integrated out. Following Ref. [1], we use a collinear approximation, i.e., θ ≈ 0 for simplicity, and work only to the leading twist (the complete analysis will be published elsewhere [11]). Invoking the helicity density matrix formalism developed in Refs. [12,8], we factor the process into various basic ingredients expressed as helicity density matrices: 1We recognize that the ππ s-wave is not resonant in the vicinity of the ρ and our analysis does not depend on a resonance approximation. For simplicity we refer to the non-resonant s-wave as the “σ”.