Abstract
We extend the study of corrections to the eikonal approximation that was initiated in Ref. \cite{Altinoluk:2014oxa} to higher orders. These corrections associated with the finite width of the target are investigated and the gluon propagator in background field is calculated at next-to-next-to-eikonal accuracy. The result is then applied to the single inclusive gluon production cross section at central rapidities and the light-front helicity asymmetry, in pA collisions, in order to analyse these observables beyond the eikonal limit. The next-to-next-to-eikonal corrections to the unpolarized cross section are non-zero and provide the first corrections to the usual $k_\perp$-factorized expression. In contrast, the eikonal and next-to-next-to-eikonal contributions to the helicity asymmetry vanish, while the next-to-eikonal ones are non-zero.
Highlights
The target by a large background field; and (b) the use of the eikonal approximation, in which the constituents of the projectile just experience color rotation upon scattering with the target through picking a Wilson line at a given transverse point but integrated along the light-cone direction of propagation of the projectile, for which the target appears as infinitely Lorentz contracted
These corrections associated with the finite width of the target are investigated and the gluon propagator in background field is calculated at next-to-next-to-eikonal accuracy
The result is applied to the single inclusive gluon production cross section at central rapidities and the single transverse spin asymmetry with a transversely polarized target, in pA collisions, in order to analyze these observables beyond the eikonal limit
Summary
In the large k+ limit at fixed k/k+, the path integral for the gluon propagator becomes increasingly dominated by the classical trajectory z(z+). We take the Fourier transform x → k of the discretized expression (2.12) for the propagator and change variables in order to write the path integral as an integral over deviations with respect to the classical path (2.17), making evident the factorization of the free propagator contribution as in eq (2.16). From the equations (2.16), (2.18) and (2.21), one obtains the discretized path-integral expression for the medium-modification factor Rakb(x+, y+; y) as. In order to understand what happens in the continuum limit N → +∞, let us first consider the case in which only the leading term e an N is kept at each step in eq (2.22), which we call the contribution 0 to Rk(x+, y+; y). That notation will be used frequently in the rest of the paper These results generalise to the leftover contributions to the medium modification factor.
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