Abstract
Transverse momentum dependent (TMD) parton distribution functions (PDFs), TMDs for short, are defined as the Fourier transform of matrix elements of nonlocal combinations of quark and gluon fields. The nonlocality is bridged by gauge links, which for TMDs have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. It is possible, however, to construct sets of universal TMDs of which in a given process particular combinations are needed with calculable, process-dependent, coefficients. This occurs for both T-odd and T-even TMDs, including also the {\it unpolarized} quark and gluon TMDs. This extends the by now well-known example of T-odd TMDs that appear with opposite sign in single-spin azimuthal asymmetries in semi-inclusive deep inelastic scattering or in the Drell-Yan process. In this paper we analyze the cases where TMDs enter multiplied by products of two transverse momenta, which includes besides the $p_T$-broadening observable, also instances with rank two structures. To experimentally demonstrate the process dependence of the latter cases requires measurements of second harmonic azimuthal asymmetries, while the $p_T$-broadening will require measurements of processes beyond semi-inclusive deep inelastic scattering or the Drell-Yan process. Furthermore, we propose specific quantities that will allow for theoretical studies of the process dependence of TMDs using lattice QCD calculations.
Highlights
JHEP08(2015)053 depending on the path of the GL [8,9,10, 22, 24,25,26]
To construct sets of universal Transverse momentum dependent (TMD) of which in a given process particular combinations are needed with calculable, process-dependent, coefficients. This occurs for both T-odd and T-even TMDs, including the unpolarized quark and gluon TMDs. This extends the well-known example of T-odd TMDs that appear with opposite sign in single-spin azimuthal asymmetries in semi-inclusive deep inelastic scattering or in the Drell-Yan process
In the case of tensor combinations it leads to TMD functions in the parametrization that show up in the description of azimuthal asymmetries such as cos(2φ) or sin(2φ), where φ is an appropriate azimuthal angle that can be defined in processes where at least two hadrons are involved, such as semi-inclusive deep inelastic scattering (SIDIS) or Drell-Yan process (DY)
Summary
The quark and gluon TMD correlators in terms of matrix elements of quark fields [1, 2] including the Wilson lines U needed for color gauge invariance of the TMD case [5, 6] are given by [7, 8]. For gluons one has for the leading correlator in a polarized nucleon in general eight GL-dependent functions. We do not discuss any complicating issues associated with scale dependence of the distribution functions and with convergence of the pT integrals. These are addressed briefly in 5 and more extensively in [32, 41, 42]. We make the GL dependence in the TMD functions like f1[U](x, p2T ) for quarks or f1g[U,U′](x, p2T ) for gluons explicit
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