Final state interactions and gauge invariant parton distributions

Abstract

Parton distributions contain factorizable final state interaction effects originating from the fast-moving struck quark interacting with the target spectators in deeply inelastic scattering. We show that these interactions give rise to gauge invariance of the transverse momentum-dependent parton distributions. As compared to previous analyses, our study demonstrates the existence of extra scaling contributions from transverse components of the gauge potential at the light-cone infinity. They form a transverse gauge link which is indispensable for restoration of the gauge invariance of parton distributions in the light-cone gauge where the gauge potential does not vanish asymptotically. Our finding helps to explain a number of features observed in a model calculation of structure functions in the light-cone gauge.Keywords: parton distributions, light-cone gauge, final state interactions, dipole scattering PACS numbers: 12.38.Bx, 13.60.Nb 1 Parton model and QCD Hadron structure functions, measurable in deeply inelastic scattering, are genuine physical observables which provide direct access to the microscopic constituents of matter and their intricate interaction dynamics. In the naive parton model [1], the structure function is expressed in terms of a probability density q(x) to find a parton of a specific flavor with a certain fraction x of the parent hadron's momentum. The underlying probabilistic picture for the scattering process relies on the fact that the constituents in a hadron boosted to the infinite momentum frame behave as a collections of noninteracting quanta due to time dilation. This simple and intuitive description of hard reactions has found its firm foundation in rigorous field theoretical approach based on asymptotically free Quantum Chromodynamics (QCD). The result is factorization theorems which separate incoherent contributions responsible for physics of large and small distances involved in hard reactions in a universal and controllable manner: The physical observables such as structure functions are calculated as a convolution of QCD parton distributions in the hadrons and parton scattering cross sections. The parton model result arises as a lowest order term in the expansion in the coupling constant and inverse power of the hard momentum transfer of QCD factorization formulas.The QCD quark distribution follows from the factorization theorem in deeply inelastic scat-whereis the gauge link between the quark fields, which arises from final state interactions between the struck quark and the target spectators. This interaction does not ruin factorization and is in fact much needed to maintain gauge invariance. On the other hand, the presence of this gauge link seems to spoil the interpretation of q(x) as a pure quark distribution, as the bilocal operator in the above expression is not obviously a quark number operator. The probabilistic interpretation is expected to hold only in the light-cone gauge [3,4],since only the physical degrees of freedom remain with this choice. In this spec...

Azimuthal and single spin asymmetries in hard scattering processes

“…As extensively discussed in Ref. [141], this link in the transverse direction is crucial to maintain the gauge invariance of TMD parton distributions under residual gauge transformations; at the same time, it is responsible for the final(initial) state interactions in SIDIS(DY) [138,139]. In nonsingular gauges, like covariant gauges, where the vector potential at infinity vanishes, in Eq.…”

“…The symbol W ξ∞ indicates a Wilson-line operator, also called gauge-link, going out from the point ξ to future infinity. These links are path-ordered exponentials, defined as [140,141] …”

Gluonic pole cross sections and single spin asymmetries in hadron-hadron scattering

“…For instance, the Sivers function f ⊥(1) 1T (x) is proportional to the projection Tr[ / n πΦ G (x, x) ]. The relative minus sign in the Sivers effect in SIDIS and DY (5) originates from the different (link-dependent) signs in the decomposition (8).…”

“…In those cases one can still make a decomposition such as in (8), but with different factors in front of the gluonic pole matrix element:…”

“…In transverse momentum dependent (TMD) correlators the nonlocality is not restricted to a lightlike direction, but rather to a light-front. It turns out that the integration path of the gauge-link between the two parton fields, which in this case involves resumming collinear and transverse gluon interactions [8,9], depends on the hard partonic subprocess. In particular it depends on the color-flow through the subprocess.…”