Efficient Fourier transforms for transverse momentum dependent distributions

Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell–Yan process and back-to-back hadron pair production in e^+ e^- collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate b_⊥-space, the conjugate space of the transverse momentum. The Fourier transform from b_⊥-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional b_⊥ integration reduces to a one-dimensional integration over the magnitude b_⊥. In this paper we develop a fast numerical Hankel transform algorithm for such a b_⊥-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source.

半包容深度非弹性散射（semi-inclusive deep inelastic scattering）中低横向动量的强子产生过程，可通过横向动量依赖（transverse momentum dependent, TMD）因子化进行描述。该形式体系已被广泛用于研究德雷尔-杨过程（Drell–Yan process）以及正负电子对撞中的背对背强子对产生过程。上述过程是提取TMD部分子分布函数与TMD碎裂函数的主要途径，而这两类函数蕴含了核子结构与强子化过程的关键信息。在唯象学研究中，应用最广泛的TMD因子化形式体系之一，是在横向动量的共轭空间——坐标b⊥空间中构建TMD可观测量。从b⊥空间逆傅里叶变换回横向动量空间的过程因存在振荡被积函数而极具复杂性，需要开展严谨且计算量庞大的数值处理，以避免潜在的显著数值误差。在TMD形式体系框架内，方位角依赖项可通过解析积分消去，二维b⊥积分可简化为仅针对b⊥模长的一维积分。本文针对此类b⊥积分开发了一种快速数值汉克尔变换（Hankel transform）算法，可提升所有标准TMD过程计算的数值精度。该算法的程序库已通过Python 2.7、3、C++以及FORTRAN77语言实现，所有程序包均已开源。