On the anomalous dimension for the transversity distribution

In the Drell-Yan process a quark-antiquark pair is annihilated to a virtual photon, which produces lepton pairs. The corresponding cross section contains a contribution from quarks of different chirality: a quark originating from one nucleon may return to this nucleon in the squared scattering amplitude with its chirality flipped. This is possible because the intermediate photon does not remember the distribution of quark chirality on the scattering nucleon. This gives rise to the leading twist chiral odd distribution function h1, which first appeared in the discussion of the transverse polarized Drell-Yan process [2–5]. One should emphasize that such chiral odd quark distribution functions do not appear in totally inclusive deep inelastic scattering because the quark chirality is conserved at each vertex of the scattering process. The quark helicity distribution g1 has a partonic interpretation in the chiral basis. This is not the case for h1 which gets a partonic interpretation only by switching to the transversity basis [6]. Looking on transversely polarized nucleons h1 is interpreted as the probabilistic asymmetry of finding quarks in eigenstates of the transverse Pauli-Lubanski vector with eigenvalue ±1/2. One should be aware, that g1 looses its partonic interpretation in the transversity basis. There are several attempts to measure the transversity distribution h1, see e.g. [4,7–10]. For general reasons the transversity distribution has to obey restrictions as Soffers inequality [11] for twist-2 distribution functions 2|h1(x)| ≤ f1(x) + g1(x). The anomalous dimension of h1 and the corresponding splitting function can be found in literature [1,12]. This result is found by direct evaluation of the splitting function. Nevertheless the behavior at Bjorken-x equal to 1 remains unfixed in this procedure. In the case of the helicity distribution g1 the behavior for x = 1 is fixed by a corresponding sum rule. For h1 such a sum rule is lacking. So the full h1 splitting function was determined by a general argument, which demands the splitting function of h1 to behave exactly as the splitting function of g1 for x ↑ 1. To our knowledge, no explicit calculation (without additional arguments) of the full splitting function of the transversity distribution exists until now. This paper is organized as follows. First, the forward scattering amplitudes corresponding to the helicity and transversity operator are defined. We derive the leading order anomalous dimension of g1 and h1 using a dispersion relation and show that the behavior of the splitting function for h1 does not match the expected result found in literature in the limit x ↑ 1. We discuss possible explanations and implications of our result.