Forest Formulas for the LHC

世纪最伟大的科学事件之一是欧洲核子研究中心的大型强子对撞机（LHC）发现了希格斯玻色子。然而，大型强子对撞机的发现潜力绝没有被耗尽，因为碰撞现在正在以人类以前从未达到的能量发生。随着未来15年积累的数据越来越多，我们将获得以前无法达到的精确度。这些数据将对粒子物理学的流行标准模型（SM）进行严格的新测试。虽然SM的成功是粒子物理学迄今为止最伟大的成就，但它也给物理学家带来了许多谜团。例如，SM不能解释观测到的物质-反物质不对称性，或者宇宙中暗物质和暗能量的丰度。为了克服这些问题，人们提出了新的模型，这些模型的特点是像超对称性或额外维度这样的奇异思想。到目前为止，这些模型都没有在实验中被检测到，但超越SM的物理学可能还隐藏在LHC目前探索的能量中。为了区分新物理学和SM，理论计算必须与实验测量的准确性相匹配。这带来了巨大的挑战，因为在量子场论（SM所依据的理论框架）中仍然不可能精确计算一般的可观测量。相反，理论物理学家诉诸所谓的微扰展开;这是一种系统的方法来展开复杂的函数，描述散射率，在相互作用强度的一系列中，其中每一个连续的项都小于前一个。通过在这种展开式中计算足够多的项，人们可以获得越来越可靠的结果。特别是在量子色动力学（QCD）中，它控制着质子的夸克和胶子的动力学，这种展开的收敛相对较慢，在某些情况下需要三到四项的计算。这种方法的问题是，费曼图，出现在这种扩展的各个方面，在数量和复杂性上都迅速增加。更糟糕的是，这些费曼图还包含复杂的红外（IR）和紫外（UV）发散（奇点），它们是长距离和短距离的起源。虽然紫外发散的问题已经在半个世纪前通过重正化过程解决了，但红外发散的情况却大不相同。为了计算QCD中的高阶效应，需要计算两个独立的贡献：真实的修正（由于可观测粒子的发射）和虚拟（环或量子）修正。虽然众所周知，真实的辐射修正的发散与虚修正的发散相抵消，但这种抵消只发生在所有不同的回路积分和相空间积分完成之后。Bogoliubov-Parasiuk-Hepp-Zimmermann（BPHZ）方案也被称为“森林公式”，其中森林指的是嵌套或不相交的发散子图的集合。这个项目的核心思想是建立和使用一个“广义森林公式”的麻烦IR分歧的减法。虽然这一命题远非微不足道，但我在最近的研究中取得的一项突破现在为它的正确性提供了有力的证据。这种方法的未来潜力是巨大的，因为它承诺在原则上的一般解决方案计算散射率的任意数量的最终状态的粒子和任意订单的微扰膨胀。拟议研究的一个重要目标是在一个专用的代码库中实现这一想法，并将其应用于对LHC至关重要的高阶QCD修正的计算中;例如在微扰膨胀中分别产生第4阶和第3阶的两个和三个喷流。

One of the greatest scientific events of the century is the discovery of the Higgs boson by CERN's Large Hadron Collider (LHC). Yet the LHCs discovery potential has by no means been exhausted as collisions are now happening at energies never achieved before by mankind. With more and more data being accumulated over the next 15 years we will obtain measurements at previously unreached levels of precision. This data will put stringent new tests on the prevalent Standard Model (SM) of particle physics. While the success of the SM is the greatest achievement of particle physics to date, it also poses many mysteries to physicists. For instance, the SM does not explain the observed matter-antimatter asymmetry, or the abundances of dark matter and dark energy in the universe. To overcome these problems new models, featuring as exotic ideas as supersymmetry or extra dimensions, have been proposed. So far none of these models could be detected in experiments, but beyond-the-SM physics may yet be hiding at the energy currently explored by the LHC.To distinguish new physics from the SM, theoretical calculations must match the accuracy of the experimental measurements. This poses a tremendous challenge since it is still impossible to calculate general observables exactly in quantum field theory (the theoretical framework upon which the SM stands). Instead, theoretical physicists resort to what is called the perturbative expansion; this is a systematic way to expand the complicated functions, which describe the scattering rates, in a series in the interaction strength, where each successive term is smaller than the preceding. By calculating enough terms in this expansion one can thus obtain increasingly reliable results. Especially in quantum chromodynamics (QCD), which governs the dynamics of the constituent quarks and gluons of the proton, the convergence of this expansion is relatively slow and in certain cases computations with three or four terms are required. The problem with this approach is that the Feynman diagrams, which appear in the individual terms of this expansion, rapidly increase in both number and complexity. To make matters worse these Feynman diagrams also contain complicated infrared (IR) and ultraviolet (UV) divergences (singularities) which are of long- and short-distance origin.While the problem of UV divergences has been solved already half a century ago by the procedure of renormalisation the situation is very different for the IR divergences. To calculate the higher order effects in QCD requires the calculation of two separate contributions: real corrections (due to emissions of observable particles) and virtual (loop or quantum) corrections. While it is well known that the divergences of the real emission corrections cancel with those of the virtual corrections, the cancellations only happen after all the different loop and phase-space integrals have been performed.A rigorous approach to renormalisation is given by the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) scheme also known as the "forest formula", where the term forest refers to sets of nested or disjoint divergent subgraphs. The key idea of this project is to establish and use a "generalised forest formula" for the subtraction of the troublesome IR divergences. While this proposition is far from trivial, a breakthrough which I have made in my recent research now gives strong evidence for its correctness. The future potential of this approach is great as it promises an in-principle general solution for calculating scattering rates of an arbitrary number of final-state particles and arbitrary orders in the perturbative expansion. One important objective of the proposed research is to implement this idea in a dedicated code-library and apply it in the calculations of higher-order QCD corrections of key importance for the LHC; such as the production of two and three jets at the respective 4th and 3rd order in the perturbative expansion.