Forest Formulas for the LHC

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，这是本世纪最伟大的科学事件之一。然而，大型强子对撞机的发现潜力并没有耗尽，因为碰撞正在以人类以前从未达到的能量以越来越快的速度发生。随着未来15年积累的数据越来越多，我们将以前所未有的精度进行测量。这些数据将对粒子物理的标准模型(SM)进行严格的新测试。尽管SM的成功是迄今为止粒子物理学最伟大的成就，但它也给物理学家带来了许多谜团。例如，SM没有解释观测到的物质-反物质不对称性，或者宇宙中暗物质和暗能量的性质。为了克服这些问题，人们提出了新的模型，这些模型具有超对称性或额外维度等异国情调。到目前为止，这些模型都不能在实验中检测到，但超越SM(BSM)物理仍然可以在LHC目前探索的能量下被探测到。为了区分新物理和SM，理论计算必须与实验测量的精度相匹配。这是一个巨大的挑战，因为在量子场论中不可能精确地计算一般的可观测数据。相反，理论物理学家求助于所谓的微扰展开；这是一种系统的方法，将描述散射率的复杂函数展开为相互作用强度的一系列，其中每个连续项都小于前面的项。通过在这种展开式中计算足够的项，就可以得到越来越可靠的结果。特别是在控制质子组成夸克和胶子动力学的量子色动力学(QCD)中，这种展开式的收敛相对较慢，在某些情况下需要计算三到四项。这种方法的问题是，以这种扩展的个别术语出现的费曼图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)和紫外线(UV)发散(奇点)，这些发散来自长距离和短距离。虽然紫外发散的问题在半个世纪前已经通过重正化过程得到了解决，但红外发散的情况却截然不同。计算QCD中的高阶效应需要结合两个独立的贡献：真实修正(由于可观测粒子的发射)和虚拟(环路或量子)修正。虽然众所周知，实际发射校正的发散与虚拟校正的发散相抵消，但抵消只有在执行了所有不同的回路和相空间积分之后才会发生。Bogoliubov-Parasiuk-Hepp-Zimmermann(BPHZ)方案给出了一种严格的重正化方法，也称为“森林公式”，其中术语森林指的是嵌套或不相交的发散子图的集合。这个项目的关键思想是开发和使用“广义森林公式”来减去IR发散。虽然这一命题远非微不足道，但我最近在研究中取得的突破已经证明了这一概念，获得了大量其他方法无法实现的新结果。这种方法的未来潜力是巨大的，因为它为计算量的新方法打开了大门，这些方法是提高当前理论预测精度所迫切需要的，例如控制质子中部分子的能量相关性的4圈分裂函数，以及控制SM EFT中耦合参数的能量相关性的2圈反常维度，SM EFT是BSM物理的一个通用模型无关框架。

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 with an increasing rate at energies never achieved before by mankind. With more and more data being accumulated over the next 15 years we will obtain measurements at unprecedented levels of precision. This data will put stringent new tests on the 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 nature 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 (BSM) physics may still be detected 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 impossible to calculate general observables exactly in quantum field theory. 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. Calculating higher-order effects in QCD requires the combination 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 develop and use "generalised forest formulas" for the subtraction of IR divergences. While this proposition is far from trivial, recent breakthroughs which I have made in my recent research have already proven the concept, obtaining a plethora of new results which could not have been achieved by other means. The future potential of this approach is great, as it opens the door for new ways of calculating quantities, which are desperately needed to improve the precision of current theory predictions, such as the 4-loop splitting functions, which govern the energy dependence of partons in the proton, and the 2-loop anomalous dimensions which govern the energy dependence of coupling parameters in the SM EFT, a general model-independent framework to BSM physics.