Checking the Burkardt sum rule for the Sivers function by model calculations

Abstract

The Sivers mechanism gives rise to a non-zero average transverse momentum of partons inside a transversely polarized target. According to a sum rule of Burkardt this transverse momentum vanishes when summing over all partons. We explicitly check the Burkardt sum rule by means of one-loop calculations for a quark target in perturbative QCD and for a diquark spectator model of the nucleon. In both cases the sum rule is fulfilled.

Figures (2)

• Figure 1: One-loop diagrams relevant to the calculation of the Sivers function for the quark (diagram (a)) and the scalar diquark (diagram (b)) in the diquark model. The Hermitian conjugate diagrams (h.c.) are not shown. We are dealing with cut diagrams indicated by the thin dashed line. In diagram (a) for instance the diquark (dashed line) interacts through the exchange of a photon with the eikonalized quark (double line).
• Figure 2: One-loop diagrams relevant to the calculation of the Sivers function for the quark (diagram (a)) and the gluon (diagram (b)) in the quark target model. The symbol '$\otimes$' in diagram (b) indicates a Feynman rule for the field strength tensor in (\ref{['e:gluon']}) of the form $-i(p^\mu g^{\nu\rho} - p^{\nu}g^{\mu\rho}) \delta_{ab}$ where the indices $\mu, \, \nu, \, a$ correspond to the indices of $F_a^{\mu\nu}$ and $\rho, \, b$ to those of the gluon in the figure (see also Ref. collins_81c).