Wilson lines and gauge invariant off-shell amplitudes

Abstract

We study matrix elements of Fourier-transformed straight infinite Wilson lines as a way to calculate gauge invariant tree-level amplitudes with off-shell gluons. The off-shell gluons are assigned "polarization vectors" which (in the Feynman gauge) are transverse to their off-shell momenta and define the direction of the corresponding Wilson line operators. The infinite Wilson lines are first regularized to prove the correctness of the method. We have implemented the method in a computer FORM program that can calculate gluonic matrix elements of Wilson line operators automatically. In addition we formulate the Feynman rules that are convenient in certain applications, e.g. proving the Ward identities. Using both the program and the Feynman rules we calculate a few examples, in particular the matrix elements corresponding to gauge invariant $g^{*}g^{*}g^{*}g$ and $g^{*}g^{*}g^{*}g^{*}g$ processes. An immediate application of the approach is in the high energy scattering, as in a special kinematic setup our results reduce to the form directly related to Lipatov's vertices. Thus the results we present can be directly transformed into Lipatov's vertices, in particular into $RRRP$ and $RRRRP$ vertices with arbitrary "orientation" of reggeized gluons. Since the formulation itself is not restricted to high-energy scattering, we also apply the method to a decomposition of an ordinary on-shell amplitude into a set of gauge invariant objects.Comment: 30 pages, quite a few figures. Minor corrections, references updated. Submitted to JHE

Improved TMD factorization for forward dijet production in dilute-dense hadronic collisions

“…In an arbitrary gauge, for internal and external gluon lines, more sophisticated methods have to be used, see e.g. [35,[45][46][47][48].…”

“…[10], such a procedure yields results which are gauge invariant within a subclass of axial gauges with the gauge vector n µ = ap µ p + bp µ A , where a and b are arbitrary complex numbers. There are also methods to calculate gauge invariant off-shell amplitudes in any gauge and choice of polarization vectors [34,35,47,48]. In what follows, we shall use those methods and specifically the results of [35,48].…”

Recursion relations for multi-gluon off-shell amplitudes on the light-front and Wilson lines

“…One complication though in using off-shell matrix elements is the condition of the gauge invariance. Recently, a progress has been made [19][20][21] in the construction of the off-shell amplitudes which do satisfy Ward identities and hence obey gauge invariance. The general method [20] utilizes infinite Wilson line operators corresponding to the off-shell gluons, whose directions are defined by the polarization vectors perpendicular to the momenta of the off-shell gluons.…”

Progress in off-shell amplitudes

“…The on-shell recursion has been implemented in the program AMP4HEF 1 . It numerically evaluates color-ordered helicity amplitudes and matrix elements summed over color and helicity for the processes…”

“…The nonphysical scattering amplitudes are embedded into physical cross sections by the very factorization. Scattering amplitudes involving any number of off-shell external gluons can also be defined in a rigorous manner, as shown in [1]. Such amplitudes are relevant in factorization prescriptions requiring off-shell initial-state partons, like high energy factorization [2,3].…”