横向动量相关分布的有效傅里叶变换
Efficient Fourier Transforms for Transverse Momentum Dependent Distributions
摘要：
半包容深度非弹性散射中低横动量强子的产生可用横动量相关（TMD）因子分解来描述。该方法也被广泛用于研究$e^+e^-$碰撞中的Drell-Yan过程和背靠背强子对产生。这些过程是提取TMD部分子分布函数和TMD碎裂函数的主要过程，它们编码了关于核子结构和强子化的重要信息。唯象学中最广泛使用的TMD因式分解形式之一是在坐标$b_\perp$ -空间（横动量的共轭空间）中表示TMD观测量。从$b_\perp$ -空间到横向动量空间的傅立叶变换由于振荡被积函数而足够复杂，因此需要仔细和计算密集的数值处理，以避免潜在的大数值误差。在TMD形式体系内，方位角依赖性被解析地积分，并且二维$b_\perp$积分减少到幅度$b_\perp$上的一维积分。在本文中，我们开发了一个快速的数值汉克尔变换算法，这样的$b_\perp$ -积分，提高了数值精度的TMD计算在所有标准的过程。该算法的库在Python 2.7和3、C++以及FORTRAN 77中实现。所有软件包都是开源的。
英文摘要：
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 $b_\perp$-space, the conjugate space of the transverse momentum. The Fourier transform from $b_\perp$-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 $b_\perp$ integration reduces to a one-dimensional integration over the magnitude $b_\perp$. In this paper we develop a fast numerical Hankel transform algorithm for such a $b_\perp$-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.
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