Quark sivers function at small x: spin-dependent odderon and the sub-eikonal evolution

Abstract

We apply the formalism developed earlier [1, 2] for studying transverse momentum dependent parton distribution functions (TMDs) at small Bjorken x to construct the small-x asymptotics of the quark Sivers function. First, we explicitly construct the complete fundamental “polarized Wilson line” operator to sub-sub-eikonal order: this object can be used to study a variety of quark TMDs at small x. We then express the quark Sivers function in terms of dipole scattering amplitudes containing various components of the “polarized Wilson line” and show that the dominant (eikonal) term which contributes to the quark Sivers function at small x is the spin-dependent odderon, confirming the re- cent results of Dong, Zheng and Zhou [3]. Our conclusion is also similar to the case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou [4] (see also [5]). We also analyze the sub-eikonal corrections to the quark Sivers function using the constructed “polarized Wilson line” operator. We derive new small-x evolution equations re-summing double-logarithmic powers of αs ln2(1/x) with αs the strong coupling constant. We solve the corresponding novel evolution equations in the large-Nc limit, obtaining a sub-eikonal correction to the spin-dependent odderon contribution. We conclude that the quark Sivers function at small x receives contributions from two terms and is given by$$ {f}_{1T}^{\perp q}\left(x,{k}_T^2\right)={C}_O\left(x,{k}_T^2\right)\frac{1}{x}+{C}_1\left({k}_T^2\right){\left(\frac{1}{x}\right)}^0+\cdots $$ f 1 T ⊥ q x k T 2 = C O x k T 2 1 x + C 1 k T 2 1 x 0 + ⋯ with the function CO(x,$$ {k}_T^2 $$ k T 2 ) varying slowly with x and the ellipsis denoting the subasymptotic and sub-sub-eikonal (order-x) corrections.

Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated

“…The amplitude A is normalized such that A = M/(2s) [60], where M is the conventional textbook scattering amplitude and s is the center-of-mass energy squared. Neglecting the quark mass, which does not affect small-x evolution, at the sub-eikonal order the quark S-matrix is [3,64,70]…”

“…When going beyond the eikonal approximation, the degrees of freedom are no longer the light-cone Wilson lines: instead one has to modify the Wilson lines by inserting one or more sub-eikonal operators between segments of Wilson lines [3,33,[62][63][64][65][66][67][68][69][70]. The sub-eikonal operators entering the helicity evolution of [1-3, 31-33, 36, 37] are the gluon field strength operator F 12 or the bi-local quark operator ψ(x 2 )γ + γ 5 ψ(x 1 ).…”

“…However, since F 12 is a local operator, it cannot be used to construct a PDF: hence the mapping of evolution from [1][2][3]33] onto the spin-dependent Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation [73][74][75] in the gluon sector has been problematic [2]. At the same time, F 12 is not the only gluon operator at the sub-eikonal order: there exists another sub-eikonal operator, D i D i , as derived in [30,64,66,70]. Here D µ = ∂ µ − igA µ is the right-acting covariant derivative, D µ = ∂ µ + igA µ is the left-acting covariant derivative, and i = 1, 2 is the transverse index.…”

“…The paper is structured as follows. In Section II we summarize the results of the earlier calculation [70] for highenergy S-matrices of massless quarks and gluons scattering on the background quark and gluon fields at the sub-eikonal accuracy. As we mentioned, these results are consistent with the earlier calculation [64].…”

“…V is done using the background field method [44,77], while in Sec. IV the calculation employs a hybrid formalism developed in [3,33,70] which combines the elements of LCPT [71,72] with the background field method: we refer to it as the light-cone operator treatment (LCOT) method. In Sec.…”