Generalized Loop Space and Evolution of the Light-Like Wilson Loops
Equations of motion for the light-like QCD Wilson loops are studied in the generalized loop space (GLS) setting. To this end, the classically conformal-invariant non-local variations of the cusped Wilson exponentials lying (partially) on the light-cone are formulated in terms of the Fréchet derivative. The rapidity and renormalization-group behaviour of the gauge-invariant quantum correlation functions (in particular, the three-dimensional parton densities) are demonstrated to be connected to certain geometrical properties of the Wilson loops defined in the GLS.
Keywords: Wilson lines and loops; generalised loop space; QCD factorization.
The QCD factorization approach to the analysis of the semi-inclusive high-energy processes entails the introduction of transverse-momentum dependent parton densities (TMD), which generalise the collinear (integrated) PDFs and contain essential information about three-dimensional intrinsic structure of the nucleon [?]. In Ref. [?] the following factorization scheme (valid in the large Bjorken- regime) for a generic transverse-distance dependent quark distribution function
has been proposed
where the -independent jet function describes the incoming (collinear) partons and the soft function can be defined as the Fourier transform of an element of the generalized loop space
Here the and stand for the rapidity and ultra-violet regulators, respectively. The Wilson loop reads
and appears to be a cusped configuration consisting of two off-light-cone with , and two light-like with Wilson lines. The quark jet effects in , therefore, are separated from the soft function , which accumulates information about the intrinsic -structure of the nucleon in the large- domain, the latter being available at the planned EIC and Jefferson Lab, see Ref. [?] and Refs. therein.
As it follows from the factorization formula, Eq. (1), the soft function contains the rapidity as well as ultraviolet singularities of the TMD distribution (1). The complex structure of the UV and light-cone (rapidity) divergences and their crucial effects on the evolution of TMDs with the emphasis on the gauge invariance and the properties of the anomalous dimensions has been studied in detail in Refs. [?]. On the other hand, as an element of the GLS, the Wilson loop obeys the integro-differential equations of motion, which prescribe the behaviour of the quantum correlation functions containing with respect to the shape variations of the underlying paths [?]. Let us show that the connection between certain (diffeomorphism-invariant) transformations in the GLS and classically conformal invariant shape transformations allows one to simplify the calculation of the evolution kernels for the TMD (1). In particular, the rapidity differential operators can be represented in terms of the Fréchet differentials enabling the derivation of the full set of the evolution equations.
To this end, let us consider the generic quadrilateral contouraaaThe singularities and renormalization properties of the light-like Wilson polygons have been introduced and extensively studied in Refs. [?]., Fig. 1, with the sides given by the vectors
One can introduce a class of the path transformations generated by the differential operators [?]
It is easy to see that the rapidities associated with the light-like vectors , being formally infinite, can be regulated as follows:
Hence, the differential operators (7) are related to the logarithmic rapidity derivative via
We conjecture then that the rapidity evolution of a correlation function with light-like cusped Wilson loops corresponds to a shape-transformation law of a specific class of elements of the GLS. To reveal this law, we address the shape-transformations in the GLS by means of the so-called Fréchet derivative [?].
By definition, the logarithmic Fréchet derivative associated with a given vector reads
for the Wilson exponential evaluated along a given trajectory , where
2 Calculation of the leading-order contributions
Expand Eq. (10) to the leading non-trivial order:
We assume that the gluon propagator in the Feynman gauge reads in the coordinate space
Let us consider first the generating vector
Computation for the vector
runs similarly. The contributions from the wedge product
can be described as follows:
The sides : and : , by the asymmetry of the wedge product and the fact that the vectors are parallel;
The side : , by the (anti-)linearity of the wedge product;
The side : , since the vector field equals zero along this part of the path.
We have, therefore, the following combinations of the gluon propagators to be evaluated:
2.1 term with
Straightforward computation yields
It is worth noticing that the same result can be obtained by applying the derivative to the original integral
2.2 term with
This term is trivially zero since it represents the self-energy of a light-like WIlson line.
2.3 term with
that is equal to the result of the differentiation .
2.4 term with
We introduce the parametrization
and split up the calculations into the two terms and . The first term then returns
while the second term can be shown to give zero [?].
The same procedure applies to the generating vector
with the point being attached to the side .
Therefore, we obtain
Taking into account the renormalization properties of the light-like Wilson quadrilateral loop [?], we conclude that
where denotes the light-cone cusp anomalous dimension [?, ?] and the summation over the number of cusps is assumed.
The logarithmic Fréchet derivative interpreted as a diffeomorphism-induced differential operator associated with a generating vector field is shown to be equivalent to the non-local infinitesimal shape-transformation introduced in Ref. [?].
Therefore, a specific class of the motions in the GLS, referred as the classically conformal-invariant transformation, can be introduced in terms of the Fréchet derivative. Because diffeomorphisms do not produce new cusps, the number of cusps is diffeomorphism-invariant. We conjecture that the light-like cusped Wilson loops possessing different number of cusps correspond to different physical objects obeying different evolution laws.
Application of the developed formalism to the derivation of the evolution equations for the TMDs on the light-cone is a subject of ongoing investigation and will be reported elsewhere.
We thank Frederik Van der Veken and Pieter Taels for long-term fruitful collaboration and inspiring discussions.
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