We consider the most general effective field theory (EFT) Lagrangian with scalar fields and derivatives, and renormalise it to substantially higher loop order than existing results in the literature. EFT Lagrangians have phenomenological applications, for example by encoding corrections to the Standard Model from unknown new physics. At the same time, scalar EFTs capture the spectrum of Wilson–Fisher conformal field theories (CFTs) in <math altimg='si7.svg'><mrow><mn>4</mn><mo linebreak='goodbreak'>&#8722;</mo><mrow><mi>&#949;</mi></mrow></mrow></math> dimensions. Our results are enabled by a more efficient version of the <italic>R</italic>* method for renormalisation, in which the IR divergences are subtracted via a small-momentum asymptotic expansion. In particular, we renormalise the most general set of composite operators up to engineering dimension six and Lorentz rank two. We exhibit direct applications of our results to Ising ( <italic>Z</italic> <inf>2</inf>), <italic>O</italic>( <italic>n</italic>), and hypercubic ( <italic>S<inf>n</inf></italic>&#8905;( <italic>Z</italic> <inf>2</inf>) <sup><italic>n</italic></sup>) CFTs, relevant for a plethora of real-world critical phenomena. The computed scaling dimensions agree well with known non-perturbative results, and they lead to new predictions where such results do not yet exist. We thereby expand the understanding of generic EFTs and open new possibilities in diverse fields, such as the numerical conformal bootstrap.

Switzerland

Johan Henriksson

United Kingdom

Franz Herzog

Italy

Stefanos R. Kousvos

Taiwan

Jasper Roosmale Nepveu

Multiloop spectra in general scalar EFTs and CFTs

In the limit where partons become collinear to each other, scattering amplitudes factorize into a product of universal, process-independent building blocks and scattering amplitudes involving fewer partons. We compute these universal building blocks — known as splitting amplitudes — for two collinear QCD partons up to third loop order in QCD. Our results describe arbitrary time-like splitting processes. Due to the violation of strict collinear factorization in space-like splitting processes, we specifically present space-like splitting amplitudes for three-parton QCD scattering amplitudes at third loop order. To achieve our results, we perform a collinear expansion of three-loop scattering amplitudes using a new expansion-by-subgraph technology, which is based on the method of regions.

United States

Xin Guan

not found

Yao Ma

Bernhard Mistlberger

Adi Suresh

Splitting amplitudes at N

The method of regions is an approach for developing asymptotic expansions of Feynman Integrals. We focus on expansions in Euclidean signature, where the method of regions can also be formulated as an expansion by subgraph. We show that for such expansions valid around small/large masses and momenta the graph combinatorial operations can be formulated in terms of what we call the asymptotic Hopf algebra. This Hopf algebra is closely related to the motic Hopf algebra underlying the $\textit{R}$ operation, an extension of Bogoliubov’s $\textit{R}$ operation, to subtract both IR and UV divergences of Feynman integrals in the Euclidean. We focus mostly on the leading power, for which the Hopf algebra formulation is simpler. We uncover a close connection between Bogoliubov’s $\textit{R}$ operation in the Connes-Kreimer formulation and the remainder $$ \mathcal{R} $$ of the series expansion, whose Hopf algebraic structure is identically formalised in the corresponding group of characters. While in the Connes-Kreimer formulation the UV counterterm is formalised in terms of a twisted antipode, we show that in the expansion by subgraph a similar role is played by the integrand Taylor operator. To discuss the structure of higher power expansions we introduce a novel Hopf monoid formulation.

Germany

Mrigankamauli Chakraborty

The asymptotic Hopf algebra of Feynman integrals

We present an efficient graphical approach to construct projectors for the tensor reduction of multi-loop Feynman integrals with both Lorentz and spinor indices in $\textit{D}$ dimensions. An ansatz for the projectors is constructed making use of its symmetry properties via an orbit partition formula. The graphical approach allows to identify and enumerate the orbits in each case. For the case without spinor indices we find a 1 to 1 correspondence between orbits and integer partitions describing the cycle structure of certain bi-chord graphs. This leads to compact combinatorial formulae for the projector ansatz. With spinor indices the graph-structure becomes more involved, but the method is equally applicable. Our spinor reduction formulae are based on the antisymmetric basis of $\textit{γ}$ matrices, and make use of their orthogonality property. We also provide a new compact formula to pass into the antisymmetric basis. We compute projectors for vacuum tensor Feynman integrals with up to 32 Lorentz indices and up to 4 spinor indices. We discuss how to employ the projectors in problems with external momenta.

Jae Goode

Anthony Kennedy

Sam Teale

Netherlands

Jos Vermaseren

Tensor reduction for Feynman integrals with Lorentz and spinor indices

Parametric representations of Feynman integrals have a key property: many, frequently all, of the Landau singularities appear as endpoint divergences. This leads to a geometric interpretation of the singularities as faces of Newton polytopes, which facilitates algorithmic evaluation by sector decomposition and asymptotic expansion by the method of regions. Here we identify cases where some singularities appear instead as pinches in parametric space for general kinematics, and we then extend the applicability of sector decomposition and the method of regions algorithms to such integrals, by dissecting the Newton polytope on the singular locus. We focus on 2 → 2 massless scattering, where we show that pinches in parameter space occur starting from three loops in particular nonplanar graphs due to cancellation between terms of opposite sign in the second Symanzik polynomial. While the affected integrals cannot be evaluated by standard sector decomposition, we show how they can be computed by first linearising the graph polynomial and then splitting the integration domain at the singularity, so as to turn it into an endpoint divergence. Furthermore, we demonstrate that obtaining the correct asymptotic expansion of such integrals by the method of regions requires the introduction of new regions, which can be systematically identified as facets of the dissected polytope. In certain instances, these hidden regions exclusively govern the leading power behaviour of the integral. In momentum space, we find that in the on-shell expansion for wide-angle scattering the new regions are characterised by having two or more connected hard subgraphs, while in the Regge limit they are characterised by Glauber modes.

Einan Gardi

Stephen Jones

Dissecting polytopes: Landau singularities and asymptotic expansions in 2 → 2 scattering

Using the near-detailed-balance distribution function obtained in our recent work, we present a set of covariant gravito-thermal transport equations (i.e. the flow of various charges as linear response to thermodynamical forces) for neutral relativistic gases in a generic stationary spacetime. All relevant tensorial transport coefficients are worked out and are presented using some particular integration functions in <math><mo>(</mo><mi>α</mi><mo>,</mo><mi>ζ</mi><mo>)</mo></math>, where <math><mi>α</mi><mo>=</mo><mo>−</mo><mi>β</mi><mi>μ</mi></math> and <math><mi>ζ</mi><mo>=</mo><mi>β</mi><mi>m</mi></math> is the relativistic coldness, with β being the inverse temperature and μ being the chemical potential. It is shown that the Onsager reciprocal relation holds in the gravito-thermal transport phenomena, and that the heat conductivity and the gravito-conductivity tensors are proportional to each other, with the coefficient of proportionality given by the product of the so-called Lorenz number with the temperature, thus proving a gravitational variant of the Wiedemann-Franz law. It is remarkable that, for strongly degenerate Fermi gases, the Lorenz number takes a universal constant value <math><mi>L</mi><mo>=</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>3</mn></math>, which extends the Wiedemann-Franz law into the Wiedemann-Franz-Lorenz law.

China

Xin Hao

Song Liu

Liu Zhao

Gravito-thermal transports, Onsager reciprocal relation and gravitational Wiedemann-Franz law

We compute the three-loop correction to the universal single-soft emission current for the case of scattering amplitudes with two additional color-charged partons. We present results valid for QCD and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-symmetric Yang-Mills theory. To achieve our results we develop a new integrand expansion technique for scattering amplitudes in the presence of soft emissions. Furthermore, we obtain contributions from single final-state parton matrix elements to the Higgs boson and Drell-Yan production cross section at next-to-next-to-next-to-next-to leading order (N4LO) in perturbative QCD in the threshold limit.

Single-soft emissions for amplitudes with two colored particles at three loops

We have estimated the electrical and thermal conductivity of a hadron resonance gas (HRG) for a time-varying magnetic field, which is also compared with constant and zero magnetic field cases. Considering the exponential decay of electromagnetic fields with time, a kinetic theory framework can provide the microscopic expression of electrical conductivity and thermal conductivity related to baryon current in terms of relaxation and decay times. In the absence of the magnetic field, only a single timescale appears, and in the finite magnetic field case, their expressions carry two timescales—relaxation time and cyclotron time period. Estimating the conductivities for HRG matter in three cases—zero, constant, and time-varying magnetic fields, we have studied the validity of the Wiedemann–Franz law. We noticed that at a high-temperature domain, the ratio saturates at a particular value, which may be considered as Lorenz number of the hadron resonance gas. With respect to the saturation values, the deviation of the Wiedemann–Franz law has been quantified at the low-temperature domain. For the first time, the present work sketches this quantitative deviation of the Wiedemann–Franz law for hadron resonance gas at a constant and a time-varying magnetic field.

India

Kamaljeet Singh

Jayanta Dey

Raghunath Sahoo

Sabyasachi Ghosh

Effect of time-varying electromagnetic field on Wiedemann-Franz law in a hot hadronic matter

We study the mixing of operators under renormalization group flow in quantum theories, and prove a non-renormalization theorem at non-linear order. It dictates zeros up to a certain number of loops in anomalous dimension tensors that control, for example, the mixing of operators at order dimension six squared into dimension eight. We obtain new results at up to three loops for the mass dimension eight anomalous dimension tensor of ϕ 4 theory in D = 4 − 2ε dimensions and verify the zeros predicted by the theorem.

Japan

Weiguang Cao

Tom Melia

Jasper Nepveu

Non-linear non-renormalization theorems

We study the application of the method of regions to Feynman integrals with massless propagators contributing to off-shell Green’s functions in Minkowski spacetime (with non-exceptional momenta) around vanishing external masses,   <math> <msubsup> <mi>p</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>→</mo> <mn>0</mn> </math>  $$ {p}_i^2\to 0 $$ . This on-shell expansion allows us to identify all infrared-sensitive regions at any power, in terms of infrared subgraphs in which a subset of the propagators become collinear to external lightlike momenta and others become soft. We show that each such region can be viewed as a solution to the Landau equations, or equivalently, as a facet in the Newton polytope constructed from the Symanzik graph polynomials. This identification allows us to study the properties of the graph polynomials associated with infrared regions, as well as to construct a graph-finding algorithm for the on-shell expansion, which identifies all regions using exclusively graph-theoretical conditions. We also use the results to investigate the analytic structure of integrals associated with regions in which every connected soft subgraph connects to just two jets. For such regions we prove that multiple on-shell expansions commute. This applies in particular to all regions in Sudakov form-factor diagrams as well as in any planar diagram.

Johannes Schlenk

The on-shell expansion: from Landau equations to the Newton polytope