New approach to the Sivers effect in the collinear twist-3 formalism

The single-transverse spin asymmetry for hadron production in transversely polarized proton scattering receives a major contribution from the Sivers effect, which can be systematically described within the collinear twist-3 factorization framework in various processes. The conventional method in the evaluation of the Sivers effect known as pole calculation is technically quite different from nonpole calculation, which is another method used in evaluating the final state twist-3 effect. In this paper, we extend the nonpole technique to the Sivers effect and show consistency with the conventional method through an explicit calculation of the $\mathcal{O}({\alpha }_s)$ correction in semi-inclusive deep inelastic scattering. As a result, we clarify that the conventional pole calculation is implicitly using the equation of motion and the Lorentz invariant relations whose importance became widely known in the nonpole calculation. We also clarify some technical advantages in using the new nonpole method.

Figure 1

Diagrammatic description of Eq. (7).

Figure 2

(a) The diagrams for the SGP contribution. The separation of the pole (8) is carried out for the red barred propagators. The complex conjugate diagrams also need to be considered. (b) The diagrams for the SFP contribution. The gluon line with momentum $k_2-k_1$ attaches to each black dot. (c) The diagrams for the HP contribution.

Figure 3

Diagrammatic description of Eq. (25).

Figure 4

The diagrams for $H_{ji,\alpha }(k_1,k_2)$. The gluon line with momentum $k_2-k_1$ attaches to each black dot.

Figure 5

Typical diagrams including $x^{\prime }$-dependent propagators. Calculation for the propagators (1)–(4) are shown in Eq. (47).