Higher-order calculations and collider analyses with Mellin-space techniques

拟议的研究重点是微扰量子场论中的高阶计算，特别是量子色动力学（QCD，夸克和胶子相互作用的理论），以及将这种计算有效地应用于分析实验数据，特别是来自目前和即将到来的高能电子-质子和质子-质子对撞机的数据。量子场论是量子力学和狭义相对论的结合，是粒子物理现象学的数学框架，理论研究与实验研究直接相关。即使对于简单的过程，唯象相关的量子场论的预言也不能用目前或可预见的数学方法精确计算。格点理论提供了一条出路，它将时空离散化，并在超级计算机上数值求解由此产生的方程。然而，这种方法不适用于大量的散射可观测量，因此留下微扰理论，即，在一个小参数（耦合常数），如量子电动力学的精细结构常数的系列扩展方面的预测的评价。为了得到定量可靠的预测，这种扩展必须扩展到第一项（首阶）之外。事实上，对于许多重要的过程，甚至需要计算第二项（次领先顺序）以外的贡献。这样的高阶计算也有助于揭示所考虑的理论的一般结构特征，从而也刺激了形式量子场论的研究，并通过“对偶”关系，甚至刺激了弦理论的研究。在拟议研究的时间跨度内，粒子物理学将迈出下一大步，在欧洲核子研究中心的大型强子对撞机（LHC）上开始实验，这是一种质子-质子加速器，具有前所未有的14 TeV的碰撞能量。在这个设施的研究将揭示一些基本问题，如电弱对称性破缺的机制和超对称性的实现，或其他“新物理”，在TeV的规模。在质子对撞机上，只有在对质子中夸克和胶子（“部分子”）的动量分布中编码的质子的非微扰结构进行因式分解后，才能应用微扰理论。这些普适量也不能从晶格理论中获得，但必须从实验数据中拟合，包括来自DESY的320 GeV电子质子对撞机HERA的结果，该对撞机目前正处于高光度数据采集的最后阶段。因此，不仅要对质子-质子截面进行足够精确的计算，而且要对用于确定部分子分布的电子-质子可观测量进行足够精确的计算。此外，由此产生的，通常是非常复杂的高阶结果需要包括在一个高效的数值设置，改善代码到目前为止用于确定部分子分布和碰撞截面。本研究将通过1.进行与对撞机过程相关的高阶QCD计算。这些计算将通过积分变换（导致整数梅林矩）进行，这是申请人及其合作者在过去几年中已经用于开拓三阶计算的技术。2.建立、应用和宣传一个有效的质子对撞机数据数值分析平台，其中包括上述计算和其他小组的相关结果。该平台将基于使用复杂梅林时刻的技术，申请人是国际领先的专家之一。该项目是利物浦大学的一项新举措，将大大加强数学科学理论组和物理系实验组之间的协同作用。

The proposed research focuses on higher-order calculations in perturbative quantum field theory, in particular quantum chromodynamics (QCD, the theory of the interactions of quarks and gluons), and the efficient application of such calculations to analyses of experimental data, especially from present and forthcoming high energy electron-proton and proton-proton colliders. Quantum field theory, the merger of quantum mechanics and the theory of special relativity for pointlike fundamental objects, is the mathematical framework for particle physics phenomenology, the theoretical research with direct relevance to experimental investigations. Even for simple processes, the predictions of phenomenologically relevant quantum field theories cannot be calculated exactly by present or foreseeable mathematical methods. Lattice theory offers a way out by discretizing space-time and numerically solving the resulting equations on supercomputers. This method, however, is not applicable to a large amount of scattering observables, thus leaving perturbation theory, i.e., the evaluation of the predictions in terms of a series expansion in a small parameter (coupling constant) such as the fine-structure constant of quantum electrodynamics. In order to derive quantitatively reliable predictions, this expansion has to be extended beyond the first term (the leading order). In fact, for many important processes even contributions beyond the second term (the next-to-leading order) need to be computed. Such higher-order calculations also help to uncover general structural features of the theory under consideration, thus stimulating also research in formal quantum field theory and, via `duality' relations, even string theory. During the time span of the proposed research, particle physics will take the next big step by starting experimentation at the Large Hadron Collider (LHC) at CERN, a proton-proton accelerator with an unprecedented collision energy of 14 TeV. Research at this facility will shed light on fundamental questions like the mechanism of electroweak symmetry breaking and the realization of supersymmetry, or other `new physics', at the TeV scale. At a proton collider, perturbation theory can be applied only after factorizing the non-perturbative structure of the protons encoded in the momentum distributions of the quarks and gluons (`partons') in the proton. These universal quantities cannot be obtained from lattice theory either, but have to be fitted from experimental data, including results from the 320 GeV electron-proton collider HERA at DESY which is now in its final phase of high-luminosity data taking. Hence it is vital to perform sufficiently accurate calculations not only for proton-proton cross sections, but also the electron-proton observables employed for the determination of the parton distributions. Furthermore the resulting, usually very complex higher-order results need to be included in a highly efficient numerical setup, improving on codes so far used for the determination of parton distributions and collider cross sections. The proposed research will address these issues by 1. performing higher-order QCD calculations relevant to collider processes. These calculations will be performed via an integral transform (leading to integer Mellin-moments), a technique already used in the past years for pioneering third-order calculations by the applicant and his collaborators. 2. setting up, applying and publicizing an efficient platform for the numerical analysis of proton collider data which incorporates the above calculations and relevant results by other groups. This platform will be based on a technique using complex Mellin moments, for which the applicant is one of the leading international experts. This project is a novel initiative at Liverpool University which will considerably strengthen the synergy between the theory group in Mathematical Sciences and the experimental group in the Physics Department.