Renormalisation in Quantum Field Theory

Renormalisation in Quantum Mechanics

This lecture provides an introduction to the renormalisation group as applied to scattering of two nonrelativistic particles. As well as forming a framework for constructing effective theories of few-nucleon systems, these ideas also provide a simple example which illustrates general features of the renormalisation group.

Two-loop renormalisation of gauge theories in $4D$ Implicit Regularisation and connections to dimensional methods

We compute the two-loop $β$-function of scalar and spinorial quantum electrodynamics as well as pure Yang-Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using Implicit Regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisa…

Wilsonian renormalisation of CFT correlation functions: Field theory

We examine the precise connection between the exact renormalisation group with local couplings and the renormalisation of correlation functions of composite operators in scale-invariant theories. A geometric description of theory space allows us to select convenient non-linear parametrisations that serve different purposes. First, we identify normal parameters in which the renormalisation group flows take their simplest form; normal correlators are defined by functional differentiation with respect to these parameters. The renormalised correlation functions are given by the continuum limit of …

Single-scale Renormalisation Group Improvement of Multi-scale Effective Potentials

We present a new method for renormalisation group improvement of the effective potential of a quantum field theory with an arbitrary number of scalar fields. The method amounts to solving the renormalisation group equation for the effective potential with the boundary conditions chosen on the hypersurface where quantum corrections vanish. This hypersurface is defined through a suitable choice of a field-dependent value for the renormalisation scale. The method can be applied to any order in perturbation theory and it is a generalisation of the standard procedure valid for the one-field case. I…

$\texttt{RGE++}:$ A $\texttt{C++}$ library to solve renormalisation group equations in quantum field theory

In recent years three-, four- and five-loop beta functions have been computed for various phenomenologically interesting models. However, most of these results have not been implemented in easy to use software packages. $\texttt{RGE++}$ bridges this gap by providing a flexible, template-based, $\texttt{C++}$ library to solve renormalisation group equations. Furthermore, we implement the available beta functions for the Standard Model, the minimal supersymmetric extension of the Standard Model and two-Higgs-doublet models, as well as right-handed neutrino extensions of the former two.

Renormalisation of out-of-equilibrium quantum fields

We consider the initial value problem and its renormalisation in the framework of the two-particle-irreducible (2PI) effective action. We argue that in the case of appropriately chosen self-consistent initial conditions, the counterterms needed to renormalise the system in equilibrium are also sufficient to renormalise its time evolution. In this way we improve on Gaussian initial conditions which have the disadvantage of generically not showing a continuum limit. For a more detailed discussion see arXiv:0809.0496.

Renormalisation group flows for gauge theories in axial gauges

Gauge theories in axial gauges are studied using Exact Renormalisation Group flows. We introduce a background field in the infrared regulator, but not in the gauge fixing, in contrast to the usual background field gauge. It is shown how heat-kernel methods can be used to obtain approximate solutions to the flow and the corresponding Ward identities. Expansion schemes are discussed, which are not applicable in covariant gauges. As an application, we derive the one-loop effective action for covariantly constant field strength, and the one-loop beta-function for arbitrary regulator.

标准模型有效场论至八维的重整化：玻色子相互作用 II

Towards the renormalisation of the Standard Model effective field theory to dimension eight: Bosonic interactions II

本文计算标准模型玻色子有效算符在单圈、v^4/Λ^4 阶的重整化群演化，补充八维项与SM耦合贡献，并考察正定性界对反常规范四线性耦合算符的有效性。

一般Gross-Neveu-Yukawa理论中的四圈重整化群方程

Four Loop Renormalisation Group Equations in general Gross-Neveu-Yukawa Theories

本研究计算了含费米子与标量场的任意可重整化量子场论在$\u0001overline{\text{MS}}$方案下的四圈$\beta$函数与反常维数模板公式，提取了希格斯与顶夸克耦合的新贡献，解决了四圈维数延拓的相关挑战。

来自标度对称性和非重叠对称性的重正化群不变量

Renormalisation Group Invariants from Scaling and Non-overlapping Symmetries

本文展示了标度对称性与非重叠全局对称性的协同作用如何导致多标量势参数中的重正化群不变量，并通过标度不变场方向构造了双线性场算符的全阶RGI，应用于非超对称模型和二重希格斯二重态模型，讨论了规范层级问题。

SMEFT中夸克和胶子场在壳方案下的双圈重整化

Two-loop renormalisation of quark and gluon fields in the SMEFT in the on-shell scheme

本文计算了SMEFT中CP守恒维度六算子对夸克和胶子场双圈重整化常数的贡献，包括顶夸克色磁算子和三重胶子算子，并分析了四夸克算子对强耦合常数运行的影响。

Schwinger's proper time and worldline holographic renormalisation

Worldline holography states that within the framework of the worldline approach to quantum field theory, sources of a quantum field theory over Mink$_4$ naturally form a field theory over AdS$_5$ {\sl to all orders} in the elementary fields and in the sources of arbitrary spin. (Such correspondences are also available for other pairs of spacetimes, not only Mink$_4\leftrightarrow\mathrm{AdS}_5$.) Schwinger's proper time of the worldline formalism is automatically grouped with the physical four spacetime dimensions into an AdS$_5$ geometry. We show that the worldline holographic effective actio…

四轻子SMEFT算符在夸克场存在时的双环重整化

Two-loop renormalisation of the quark fields in the presence of four-fermion SMEFT operators

该研究计算了四代第三代夸克的维度六SMEFT算符对夸克场和质量的双环重整化贡献，采用两种方案并讨论了方案选择对结果的影响。

SMEFT中夸克和胶子场的双圈重整化

Two-loop renormalisation of quark and gluon fields in the SMEFT

本文采用背景场方法计算了六维SMEFT中CP守恒算符对夸克和胶子场双圈重整化常数的贡献，提取了对顶质量和强耦合常数双圈跑动的影响，并给出了所有相关算符的解析结果。

Inflationary Potentials from the Exact Renormalisation Group

We show that an inflationary slow-roll potential can be derived as an IR limit of the non-perturbative exact renormalisation group equation for a scalar field within the mean-field approximation. The result follows without having to specify a Lagrangian for the UV theory at the Planck scale. We assume that the theory contains a scalar mode with suppressed coupling to other UV fields, and that higher derivative couplings are suppressed. The resulting effective potential gives rise to slow-roll inflation, which is fully consistent with the recent observations. As an example of how the proposed r…

Charges from Dressed Matter: Physics and Renormalisation

Gauge theories are characterised by long range interactions. Neglecting these interactions at large times, and identifying the Lagrangian matter fields with the asymptotic physical fields, leads to the infra-red problem. In this paper we study the perturbative applications of a construction of physical charges in QED, where the matter fields are combined with the associated electromagnetic clouds. This has been formally shown, in a companion paper, to include these asymptotic interactions. It is explicitly demonstrated that the on-shell Green's functions and S-matrix elements describing these …

Renormalisation and power counting in effective field theories for nucleon-nucleon scattering

The renormalisation of NN scattering in theories with zero-range interactions is examined using a cut-off regularisation and taking the cut-off to infinity. Inclusion of contact interactions that depend on energy as well as momentum allows the effective range to be fitted to experiment with any desired sign or magnitude. However, power counting breaks down: terms of different orders in the potential can contribute to the scattering amplitude at the same order.

大N极限下的费米子与重整化群

Fermions and the Renormalisation Group at Large N

本文在大N极限下用泛函重整化群研究费米子理论，证明其有效作用量具有准定域精确解，并给出U(N)对称理论的定域势流方程，确定了各类相互作用下的共形不动点与标度维度。

暴胀动力学的精确重整化群演化：调和α吸引子与ACT数据

Exact Renormalisation Group Evolution of the Inflation Dynamics: Reconciling $α$-Attractors with ACT

本文提出基于泛函重整化群的非微扰框架，推导Friedmann-RG耦合流方程，发现量子修正使α吸引子暴胀轨迹失稳并提前结束慢滚，从而与最新ACT数据完全吻合。

标准模型有效场论中 $pp\ o t\ar{t}h$ 的重整化群跑动效应

Renormalisation group running effects in $pp\to t\bar{t}h$ in the Standard Model Effective Field Theory

本文以 $pp \ o t\ar{t}h$ 过程为例，研究了标准模型有效场论中 Wilson 系数的重整化群跑动效应，同时考虑强相互作用与顶点 Yukawa 跑动效应，并对比了动力学与固定重整化标度在不同高维算符情景下的差异。

Renormalisation running of masses and mixings in UED models

We review the Universal Extra-Dimensional Model compactified on a S1/Z2 orbifold, and the renormalisation group evolution of quark and lepton masses, mixing angles and phases both in the UED extension of the Standard Model and of the Minimal Supersymmetric Standard Model. We consider two typical scenarios: all matter fields propagating in the bulk, and matter fields constrained to the brane. The resulting renormalisation group evolution equations in these scenarios are compared with the existing results in the literature, together with their implications.

Coupled scalar fields in a flat FRW universe: renormalisation

We study the non-equilibrium dynamics of a system of coupled scalar fields in a Friedmann-Robertson-Walker (FRW) universe. We consider the evolution of spatially homogeneous "classical" fields and of their quantum fluctuations including the quantum backreaction in the one-loop approximation. We discuss in particular the dimensional regularisation of the coupled system and a special subtraction procedure in order to obtain the renormalised equations of motion and the renormalised energy-momentum tensor and ensure that the energy is well-defined and covariantly conserved. These results represent…

洛伦兹量子引力中自洽的引力子谱函数

Self-consistent graviton spectral function in Lorentzian quantum gravity

本研究在物理质壳重整化方案下，利用引力谱重整化群方法首次完成量子引力中引力子谱函数的完全自洽计算，得到带无质量单引力子峰的正谱函数，验证其满足渐近态求和规则，为散射过程和渐近安全引力的幺正性研究提供支撑。

从功能重整化群方法研究θ真空

The $θ$-vacuum from functional renormalisation

本文通过功能重整化群方法研究具有U(1)对称性的量子系统，探讨真空能量结构和拓扑Susceptibility，并利用复化流方程计算拓扑扇区的有效势。

Holographic Renormalisation and the Electroweak Precision Parameters

We study the effects of holographic renormalisation on an AdS/QCD inspired description of dynamical electroweak symmetry breaking. Our model is a 5D slice of AdS_5 geometry containing a bulk scalar and SU(2) times SU(2) gauge fields. The scalar field obtains a VEV which represents a condensate that triggers electroweak symmetry breaking. Fermion fields are constrained to live on the UV brane and do not propagate in the bulk. The two-point functions are holographically renormalised through the addition of boundary counterterms. Measurable quantities are then expressed in terms of well defined p…

Fixed points of quantum gravity in extra dimensions

We study quantum gravity in more than four dimensions with renormalisation group methods. We find a non-trivial ultraviolet fixed point in the Einstein-Hilbert action. The fixed point connects with the perturbative infrared domain through finite renormalisation group trajectories. We show that our results for fixed points and related scaling exponents are stable. If this picture persists at higher order, quantum gravity in the metric field is asymptotically safe. We discuss signatures of the gravitational fixed point in models with low-scale gravity and compact extra dimensions.

Towards the renormalisation of the Standard Model effective field theory to dimension eight: Bosonic interactions I

We compute the one-loop renormalisation group running of the bosonic Standard Model effective operators to order $v^4/Λ^4$, with $v\sim 246$ GeV being the electroweak scale and $Λ$ the unknown new physics threshold. We concentrate on the effects triggered by pairs of the leading dimension-six interactions, namely those that can arise at tree level in weakly-coupled ultraviolet completions of the Standard Model. We highlight some interesting consequences, including the interplay between positivity bounds and the form of the anomalous dimensions; the non renormalisation of the $S$ and $U$ parame…

德西特时空中红外发散的重整化与全息对偶

Renormalisation of IR divergences and holography in de Sitter

本文针对德西特时空中树级 in-in 晚期关联函数的红外发散，提出通过在未来边界添加局域抵消项进行重整化的方法，并建立全息公式将其映射为对偶 CFT 的紫外重整化，为 dS/CFT 对偶提供结构性支持。