I will report on the calculation of the NLO QCD corrections to heavy boson production and decay in association with top quark. In the calculation all resonant as well as non-resonant Feynman diagrams, interferences and off-shell effects are included both for the top quark and W gauge boson. Numerical results will be given at the integrated and differential fiducial level for various factorisation and renormalisation scale choices and different PDF sets. I will discuss the main theoretical uncertainties that are associated with neglected higher order terms in the perturbative expansion and with different parametrisations of the PDFs. Furthermore, I will talk about the size of the off-shell effects and show an explicit comparison to the calculation in the full narrow-width-approximation.
Pushing the reach of NNLO QCD predictions to 2→3 production processes is one of the pillars of precision phenomenology program at the LHC. In this talk we will overview recent results and developments in calculation of two-loop five-point amplitudes contributing towards achieving this goal. We will discuss challenges encountered in advancing the state-of-the-art beyond the class of massless five-point scattering. We will then present a basis set of transcendental functions sufficient to express any planar two-loop five-particle scattering amplitude with one external massive leg. This basis greatly facilitates derivation of compact analytic form of scattering amplitudes, and opens a possibility of their fast and reliable numerical evaluation. Applications for phenomenology of electroweak boson production can be reasonably anticipated in the near future.
The MATRIX framework was originally developed to compute diboson production
at next-to-next-to-leading order (NNLO) accuracy in QCD perturbation theory
by means of the qT-subtraction method, building upon the MUNICH integrator and
the external scattering amplitudes.
In its second official release of last summer, this framework was extended in
several directions, in particular by including NLO EW corrections and leading
N³LO effects from NLO QCD corrections to the loop-induced gluon fusion channels.
In this talk I will present the MATRIX framework with a focus on the new
features added in this latest public version.
Moreover, I will discuss further recent developments built upon the MATRIX machinery:
the application to triphoton production as a first 2->3 scattering process of the
triboson process class;
the extension of the qT subtraction method towards massive coloured final
states and its application on heavy-quark pair production and beyond;
the first complete calculation of mixed NNLO QCD-EW corrections to the
charged- and neutral-current Drell-Yan processes, based on an abelianized version
of this heavy-quark extension;
inclusion of recoil-driven linear power corrections to improve the convergence in
dedicated back-to-back configurations, in particular for the Drell-Yan process.
Recent development of fixed order QCD calculations enables us to achieve next-to-next-to-next-to leading order QCD accuracy for differential observables at the LHC. We present the latest results of neutral and charged current production at the LHC using NNLOJET package plus factorised contributions calculated in SCET.
I will present analytic expressions for the two tennis-court integral families relevant to three-loop $2\to2$ scattering processes involving one massive external particle and massless propagators in terms of Goncharov polylogarithms of up to transcendental weight six. Additionally, I will also present analytic expressions for physical kinematics for the ladder-box family and the two tennis-court families in terms of real-valued polylogarithmic functions, making these results well-suited for phenomenological applications. The full calculation, accompanied by all relevant results in computer-readable form, became recently available in arXiv:[2112.14275].
We present a powerful method to compute one-scale master integrals
using differential equations combined with numerical matching.
The latter is achieved with the help of deep expansions around
regular and singular kinematic points.
As an applications we consider massive three loop form factors.
We show that the method can be applied to systems of
differential equations involving several hundred master integrals.
I will present ideas to compute helicity amplitudes for multiloop/multileg scattering amplitudes directly in the so-called 't Hooft-Veltman scheme, avoiding evanescent structures and ambiguities in the scheme definitions.
I will present results on five-point two-loop planar and non-planar Master Integrals. Based on the Simplified Differential Equations approach, analytic representation in terms of Goncharov polylogarithms, for the planar and hexa-box families, has been obtained. These results are a necessary ingredient for the calculation of many two-loop two-to-three processes of interest at the LHC.
Scattering amplitudes serve as a bridge between theoretical predictions and experimental data and are therefore key objects in QFT. Although much progress has been made in this field, it is still very hard to compute scattering amplitudes at higher orders in perturbation theory. I will describe modern methods for the computation of amplitudes in general field theories and present new results in QCD which confirm the quadrupole contributions to the IR structure of QCD.
We present a novel framework to streamline the calculation of beam functions to next-to-next-to-leading order in perturbation theory. By exploiting the infrared behavior of the collinear splitting functions, we factorize the singularities with suitable phase-space parametrizations and perform the observable-dependent integrations numerically. We have implemented our approach in the publicly available code pySecDec and present the first results for sample beam functions.
I report on the calculation of the beam functions for zero-jettiness at next-to-next-to-next-to-leading order in the strong coupling constant. In the limit of vanishing zero-jettiness scattering cross sections factorise into beam, soft and hard functions as well as the leading-order cross section. Here, the beam functions describe collinear emissions off the initial state partons. If these building blocks are available they can, for example, be used to derive a slicing scheme for colour singlet production. I will present details of the calculation as well as results for beam functions.
In the kinematic region where three particles $i$, $j$, $k$ are collinear, the multi-parton scattering amplitudes factorise into a product of a triple collinear splitting function and a multi-parton scattering amplitude with two fewer particles. These triple collinear splitting functions contain both iterated single unresolved contributions, and genuine double unresolved contributions. We make this explicit by rewriting the known triple collinear splitting functions in terms of products of two-particle splitting functions, and a remainder that is explicitly finite when any two of $\{i,j,k\}$ are collinear. We analyse all of the single unresolved singularities present in the remainder.
Over the coming decade, the experimental program at the LHC will reach unprecedented levels of precision. To match this on the theory side, extremely complicated amplitude calculations must be performed. Recently, we have witnessed a boom in analytic calculations of scattering amplitudes, pushed forward by the application of finite fields and Ansatz methodology. These approaches have made possible the computation a number of five point amplitudes at two-loop relevant for NNLO QCD corrections at the LHC. This modern paradigm allows one to efficiently compute both the differential equations that control the master integrals and the decomposition of the amplitudes in terms of said integrals.
In this talk, we discuss the recent application of this approach in the case of two-loop amplitudes for the production of a W boson in association with two jets at the LHC. We briefly describe the construction of the differential equation for the master integrals and thereafter elaborate on the modern reconstruction approach for the calculation of the amplitude coefficients, which makes use of efficient Vandermonde-based sampling and a univariate partial fractions decomposition.
In this talk, I will present recent achievements in cross-section predictions obtained with the sector-improved residue subtraction scheme at NNLO QCD. In particular, I will review results on cross sections for two-to-three processes involving jets and photons, I will also discuss top-quark mass effects in Higgs-boson production, and demonstrate how to include fragmentation our subtraction scheme.
pySecDec is a program to evaluate multi-loop Feynman integrals numerically based on the sector decomposition approach; its new release version 1.5 introduces features significantly improving its performance: automatic adaptive evaluation of weighted sums of integrals (e.g. amplitudes) and asymptotic expansion in kinematic ratios. I'd like to briefly review both, illustrating the expected performance benefits.
We present NLO results in QCD for the production of a Higgs boson in association with a Z-boson in gluon fusion including top-quark mass effects. Our result is obtained by combining virtual corrections evaluated numerically using sector decomposition with virtual corrections obtained in an expansion around small top-quark mass. In addition to the total cross section for this process, we present also differential results including the invariant mass distribution and p_T distributions for the Higgs boson and Z-boson.
We compute the top quark contribution to the two-loop amplitudes for on-shell W/Z boson pair production in gluon fusion. Exact dependence on the top quark mass is retained. For each phase space point the integral reduction is performed numerically and the master integrals are evaluated using the auxiliary mass flow method, allowing fast computation of the amplitude with very high precision.
I present work at two loops on the class of amplitudes with all gluons of
identical helicity in Yang–Mills theory. I show how to
compute their rational terms, the hardest parts, via well-understood
one-loop unitarity techniques.
The detailed study of the Higgs boson is one of the main tasks of contemporary particle physics. Gluon fusion, the main production channel of Higgs bosons at the LHC, has been successfully modelled in QCD up to $\text{N}^3\text{LO}$. To fully exploit this unprecedent theoretical effort, sub-leading contributions, such as electroweak corrections, must be investigated. I will present the analytic calculations of the gluon- and quark-induced Higgs plus jet amplitudes in mixed QCD-electroweak corrections mediated by light quarks up to order $v \alpha^2 \alpha_S^{3/2}$.
Since Feynman integrals in QFT evaluate to special functions and numbers,
it is essential to profit from this knowledge to extract in a simple way
preliminary information of multi-loop scattering amplitudes.
To this end, one can focus on their maximal transcendental weight contribution.
In this talk, I report on a method that uses insights into
the singularity structure of space-time loop integrands,
and complements unitarity-based methods.
I illustrate this method with the application to the two-loop scattering
amplitudes of the Higgs decay into two gluons in the heavy
top-quark mass limit.
Up to now, NNLO QCD calculations of photon production cross sections applied an idealised photon isolation procedure, which differs from the isolation used in experiments. We present first numerical results for NNLO QCD predictions of isolated photon cross sections at the LHC with a realistic cone-based isolation. Photon fragmentation processes are included for the first time at NNLO, by extending the antenna subtraction method to handle infrared-singular parton-photon configurations while retaining the information on the photonic energy inside the collinear parton-photon cluster. We describe how these singularities are subtracted in antenna subtraction using new fragmentation antenna functions and outline their integration.
Scattering amplitudes in perturbative quantum field theory exhibit a rich structure of zeros, poles, and branch cuts, which are best understood as varieties in complexified momentum space. It is also well known that scattering amplitudes in gauge theories admit compact representations in the spinor-helicity formalism. However, obtaining such compact representations is often a non-trivial task due to the many variables subject to constraints, such as momentum conservation and Schouten identities. We build compact spinor-helicity ansätze for the rational coefficients of scattering amplitudes by making manifest their behavior on varieties where they may diverge, which we dub “singular varieties”. Algebraic geometry provides a natural language to describe such geometric varieties in terms of algebraic ideals. For the first time, we systematically identify irreducible singular varieties via primary decompositions of the respective ideals in spinor space, and we introduce a tool from number theory, namely $p$-adic numbers, to evaluate the rational coefficients in proximity to these singular varieties in a stable and efficient manner. In some sense, $p$-adic numbers bridge the gap between finite fields and floating-point numbers by combining the stability of finite fields with a non-trivial absolute value. By the Zariski-Nagata theorem, numerical evaluations of the rational coefficients close to varieties of codimension two lead to constraints on the numerators in terms of membership to a particular type of ideals, the symbolic power. Finally, as a proof-of-concept application, we use these constraints to build compact ansätze for the pentagon-function coefficients of the two-loop $0\rightarrow q\bar q \gamma\gamma\gamma$ helicity amplitudes.
In this presentation I will develop and demonstrate a method to obtain epsilon factorized
differential equations for elliptic Feynman integrals. This method works by choosing an
integral basis with the property that the period matrix obtained by integrating the basis
over a complete set of integration cycles is diagonal. This method is a generalization of a
similar method known to work for polylogarithmic Feynman integrals. I will demonstrate the
method explicitly for a number of Feynman integral families with an elliptic highest sector.
I will discuss Feynman integrals which depend on more than one elliptic curve and methods to compute them.
Single top quark is mainly produced through the t-channel W boson exchange q + b -> q + t at LHC. This process probes Wtb vertex directly and can be used to measure the CKM matrix element Vtb or constrain the bottom quark PDF. The two-loop non-factorisable contribution is the last missing piece of the NNLO correction. In this talk, I will first talk about its motivation, then I will discuss the calculation procedure and techniques we applied in this work. Finally, I will present some results.
Jet mass measurements for processes involving boosted top quark pair production will be possible at the HL-LHC and may become an important tool for high-precision
measurements of the top quark mass in a well-defined renormalization scheme. Apart from including QCD corrections and resummations, for which well-developed frameworks exist,
eventually also electroweak and finite-lifetime effects need to be accounted for systematically.
For boosted top quark initiated inclusive jets we apply an electroweak Soft-Collinear-Effective-Theory (SCET) framework that allows for a coherent resummation of electroweak Sudakov and rapidity logarithms and finite-lifetime effects together with large logs from QCD. Apart from double top resonant effects, the factorization approach can also account for single-resonant effects which are related to the interference of final states originating from top quark decays and background processes leading to the top decay final state. Concretely we address electroweak effects in inclusive (hemisphere mass) top-dijet production at lepton colliders and also discuss a convenient operator space framework which also allows to account for subleading power effects.
In my talk I would like to summarize our recent work we did
in the computation of the associated W vector boson production
with a massive charm quark at the Next-to-Leading Order accuracy
in QCD at the hadron level. This process plays an important role
in the determination of PDFs for sea quarks hence provides an
important, indirect input to several precision measurements at the
LHC.
We present recent computer algebra tools for Feynman integral.
A special focus will be put on hypergeometric functions and generalization of them
We study integration-by-parts-like relations and differential equations for Feynman integrals in the framework of $\mathcal{D}$-module theory. We leverage the fact that Feynman integrals satisfy a set of PDEs called a GKZ hypergeometric system. This fact allows us to uniquely associate a Feynman integral to an element of a $\mathcal{D}$-module, which can be intepreted as a differential operator in external kinematic variables. We are thereby able to derive relations among integrals by studying relations among $\mathcal{D}$-module elements. In particular, integration-by-parts-like relations follow from reducing higher order differential operators to lower ones, and differential equations for Feynman integrals correspond to Pfaffian systems for a set of basis operators. We apply this philosophy to a couple of simple examples at 1- and 2-loops.
Using methods from algebraic geometry such as Gr\"obner bases, we derive operator solutions to IBP relations that retain the functional dependence on the propagator powers $a_i$. The reduction to master integrals is achieved by applying the operators to the integrals $I(a_i)$. The solution at hand makes a case-by-case bottom-up solution of the system of IBP equations obsolete. In the talk we present the algorithms, give examples and point out interesting future directions.
The high-energy limit of 2->2 scattering amplitudes offers an excellent setting to explore the analytic properties and universal features of gauge theories. At leading logarithmic (LL) accuracy the amplitude is governed by Regge poles in the complex angular momentum plane. Beyond LL, Regge cuts in this plane begin to play an important role. Specifically, the signature-odd amplitude at Next-to-Next-to-Leading logarithmic (NNLL) accuracy presents for the first time both a Regge pole and a Regge cut. Analysing this tower of logarithms and computing it explicitly through four loops we are able to systematically separate between the Regge pole and the Regge cut. We explain how to consistently define the impact factors at two loops, and the Regge trajectory at three loops and we confirm that singularities of the trajectory are given by the cusp anomalous dimension. We also show explicitly that the Regge cut contribution at four loop is non-planar.
Singular factors originating from the QCD factorisation of scattering amplitudes in soft and collinear limits play a prominent role in both organising and computing high-order perturbative contributions to hard-scattering cross sections. In this talk, I will report on recent work done in collaboration with Stefano Catani. We start from the factorization structure of scattering amplitudes in the collinear limit, and we introduce collinear functions that have a process-independent structure. These collinear functions, which are defined at the fully-differential level, can then be integrated over the appropriate observable-dependent phase space to compute logarithmically-enhanced contributions to the corresponding observable. For transverse-momentum dependent observables, we show how the collinear functions can be defined without introducing what is known as rapidity divergences in the literature. We present the results of explicit computations of the collinear functions up to NNLO.
We present the resummed predictions for inclusive cross section for Drell-Yan and Higgs boson production at next-to-next-to leading logarithmic (\bar{NNLL}) accuracy taking into account both soft-virtual (SV) and next-to SV (NSV) threshold logarithms. We restrict ourselves to resummed contributions only from the diagonal channels for both the processes. We derive the N-dependent coefficients and the N-independent constants in Mellin-N space for our study. Using the minimal prescription we perform the inverse Mellin transformation and match it with the corresponding fixed order results. We report in detail the numerical impact of N-independent part of resummed result and explore the ambiguity involved in exponentiating them for both the processes. In the case of Higgs Boson production, by studying the K factors at different logarithmic accuracy, we find that the perturbative expansion shows better convergence improving the reliability of the prediction at NNLO + \bar{NNLL} accuracy. We also observe that the resummed SV + NSV result improves the renormalisation scale uncertainty at every order in perturbation theory. The uncertainty from the renormalisation scale ranges between (+8.85%, −10.12%) at NNLO whereas it goes down to (+6.54%, −8.32%) at NNLO + \bar{NNLL} accuracy. However, the factorisation scale uncertainty is worsened by the inclusion of these NSV logarithms hinting the importance of resummation beyond NSV terms for Higgs Boson case. For Drell-Yan production, we find that the resummation, taking into account the NSV terms, appreciably increases the cross section while decreasing the sensitivity to renormalisation scale. We observe that, at 13 TeV LHC energies, the SV+NSV resummation at \bar{NLL}(\bar{NNLL}) gives about 8% (2%) corrections respectively to the NLO (NNLO) results for the considered Q range: 150-3500 GeV. In addition, the absence of quark gluon initiated contributions to NSV part in the resummed terms leaves large factorisation scale dependence indicating their importance at NSV level. Finally, we present our predictions for SV+NSV resummed result at different collider energies for both the processes.
Flavor observables are usually computed with the help of the electroweak Hamiltonian which separates the perturbative from the non-perturbative regime. The Wilson coefficients are calculated perturbatively, while matrix elements of the operators require non-perturbative treatment, e.g. through lattice simulations. The resulting necessity to compute the transformation between the different renormalization schemes in the two calculations constitutes an important source of uncertainties. An elegant solution to this problem is provided by the gradient flow formalism, already widely used in lattice simulations, because its composite operators do not require renormalization. In this talk we report on the construction of the electroweak Hamiltonian in the gradient flow formalism through NNLO in QCD.
Recently many of the advanced methods originally developed in the context of collider phenomenology have found important applications to the general relativistic two-body problem.
I will describe how on-shell scattering amplitudes, effective field theory and multi-loop integration methods allow to extract observables for the conservative binary dynamics at fourth post-Minkowskian order and lay out challenges faced in present and future computations challenges.
The worldline formalism shares with string theory the property that it allows one to write down master integrals that effectively combine the contributions of many Faynman diagrams. While at the one-loop level these diagrams differ only by the position of the external legs along a fixed line or loop, at the multiloop level they generally involve different topologies. However, evaluating such master integrals analytically without decomposing them into the individual topologies leads to a difficult and non-standard mathematical challenge. Here, I will summarize recent progress with this problem based on a framework involving a reduction to quantum mechanics on the circle and the relation between inverse derivatives and Bernoulli polynomials.
We present the DIS coefficient functions at four loops in QCD and beyond.
In QCD the anomalous dimensions of gauge invariant operators of twist 2 play a key role, because they control the scale dependence of the parton distribution functions. Notably, flavour singlet operators, such as those associated to the gluon distribution, mix under renormalisation with a set of unphysical operators, also known as aliens. Missing this effect leads to wrong results already for the two-loop anomalous dimensions.
The correct renormalisation of gluonic operators has not been developed systematically beyond two loops yet. This is an important step towards the computation of the scale evolution of flavour singlet parton distributions, which is now required to 4 loops.
Leveraging both the background field method and an enhanced BRST symmetry, we construct the required ghost and alien operator basis up to 4 loops for arbitrary mass dimensions. Furthermore, we extract four-loop anomalous dimensions of the physical operators of lowest dimension.
We compute the two-loop master integrals relevant for the NNLO QCD correction to heavy pseudo-scalar quarkonium production and decay both analytically and numerically. The analytic expressions involve elliptic multiple polylogarithms and iterated integrals of modular forms. We discuss the master integral computation and the form-factors obtained. We briefly discuss their phenomenological importance and present in addition the form-factors for the para-leptonium states.
We report on the calculation of the previously unknown 2- and 3-loop QCD corrections to the flavor precision observable $\Delta \Gamma_{12}^s$ that arises in the mixing of neutral $B_s$ mesons. This project is a crucial step towards the task of reducing the existing perturbative uncertainties of this observable which happen to be much larger than the current experimental error.
In this talk I will discuss the computation of the top quark mass
dependence of NNLO double Higgs boson production, including both the real
and virtual contributions. The dependence is computed in the large top
quark mass limit, which gives a good approximation of the exact cross
section below the top quark production threshold.
We discuss the complete computation of the mixed QCD–EW corrections to lepton- pair production via the Drell-Yan mechanism. We present results for fiducial cross sections and differential distributions to both the neutral current- and charged current- process. In particular, for the neutral current case, we report on the first result at this order that is valid in the entire region of dilepton invariant masses.
We calculate the two-loop QCD corrections to Higgs boson pair production
in gluon fusion within Standard Model Effective Field Theory (SMEFT),
including also squared dimension-6 operators and double insertions of
operators. The different options to truncate the EFT expansion are
contrasted to a non-linear EFT approach (HEFT) and the effects are
illustrated with several phenomenological examples.
In my talk I will present recent advances in our ability to expand perturbative scattering cross sections around collinear limits. This technique allows to connect universal ingredients like Beam Functions and Fragmentation Functions with technology used in the computation fixed order cross section. Furthermore, collinear expansions may serve as a powerful approximation of particle physics cross sections and I will demonstrate some examples.
We pursue a no-compromise approach to the gamma5 problem of dimensional regularization. gamma5 is treated in the 't Hooft/Veltman/Breitenlohner/Maison scheme, which is mathematically rigorous but which breaks gauge invariance. As a result, a correct renormalization procedure based on this scheme involves three specific kinds of counterterms: cancelling UV singularities requires (1) counterterms which do not correspond to field or parameter renormalization, (2) evanescent counterterms, and (3) the breaking of gauge invariance necessitates finite symmetry-restoring counterterms. We determine the full structure of all these counterterms at the two-loop level, focusing on the example of a chiral abelian gauge theory. We explain the methodology, which is based on the quantum action principle and a direct computation of the breaking of Slavnov-Taylor identities, and we provide illustrations and checks based on well-known Ward identities.
The dynamics of binary black hole and neutron star systems in the early inspiral phase are well described by a post-Newtonian expansion in small velocities and weak coupling. Adopting nonrelativistic effective field theory techniques known from quarkonium systems this expansion can be formulated in terms of multiloop Feynman diagrams. I discuss results for the complete fifth and partial sixth order corrections.
Collider observables involving heavy states are subject to large logarithmic terms near threshold, which must be summed to all orders in perturbation theory to obtain sensible results. Relatively recently, this resummation has been extended to next-to-leading power in the threshold variable, using diagrammatic and effective field theory techniques. In this talk I will present the state of the art and discuss current limitations, related to the appearance of divergent convolutions, which prevent the application of factorization methods known from leading power resummation. To this end I will focus on partonic channels that turn on only at next-to-leading power, in deep inelastic scattering, Drell-Yan and Higgs boson production. I will illustrate how the study of these channels gives us a better understanding of the origin of such singularities. Furthermore, I will show that an explicit all-order form for the leading logarithmic partonic cross section can be obtained by employing d-dimensional consistency relations, derived from requiring 1/ϵ pole cancellations in dimensional regularization between momentum regions.
High precision calculations in perturbative QFT often require evaluation of big collection of Feynman integrals. Complexity of this task can be greatly reduced via the usage of linear identities among Feynman integrals. Based on mathematical theory of intersection numbers, recently a new method for derivation of such identities and decomposition of Feynman integrals was introduced and applied to many non-trivial examples.
In this talk we will discuss the latest developments in algorithms for the evaluation of intersection numbers, and their application to the reduction of Feynman integrals.
We extend available $\cal{O}(\alpha_s^4)$ results for two main ingredients of
the Crewther-Broadhurst-Kataev (CBK) relation, namely the Adler $D$-function
and the Bjorken polarized sum rule $C^{Bjp}$ by including
contributions from any number of (in general different) fermion
representations. The results are used to confirm the CBK relation and get new
information on {β}-expansion for the $D$-function and $C^{Bjp}$.
We developed a new framework to address the renormalization
issues of the singlet twist-two operators. As a first application, we apply our method to compute the three-loop singlet splitting functions
We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD.
In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current.
We compute the tree-level current for triple soft-gluon emission from both massless
and massive hard partons.
The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations.
We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current.
The radiation of one and two soft gluons leads to colour dipole correlations.
Triple soft-gluon radiation produces in addition colour quadrupole correlations
between the hard partons.
We examine the soft and collinear singularities of the squared current in various
energy ordered and angular ordered regions.
We discuss some features of soft radiation to all-loop orders for processes with two and three hard partons.
Considering triple soft-gluon radiation from three hard partons, colour quadrupole interactions break the Casimir scaling symmetry between
quarks and gluons.
We also present some results on the radiation of four soft gluons from two hard partons,
and we discuss the colour monster contribution and its relation with the violation
(and generalization) of Casimir scaling.
We also compute the first correction of ${\cal O}(1/N_c^2)$ to the eikonal formula for multiple soft-gluon radiation with strong energy ordering from two hard gluons.
We consider the class of inclusive hadron collider processes in which one or more hard jets are produced, possibly accompanied by colourless particles (such as Higgs boson(s), vector boson(s) with their leptonic decays, and so forth). We propose a new variable that smoothly captures the N+1 to N jet transition. This variable, that we dub ktness, represents an effective transverse momentum controlling the singularities of the cross section when the additional jet is unresolved. The ktness variable offers novel opportunities to perform higher-order QCD calculations by using non-local subtraction schemes. We study the singular behavior of the N+1-jet cross section when ktness->0 and, as a phenomenological application, we use the ensuing results to evaluate next-to-leading order corrections to H+jet and Z+2 jet production at the LHC. We show that kTness performs extremely well as a resolution variable and appears to be very stable with respect to hadronization and multiple-parton interactions.
Processes with identified hadrons require the introduction of fragmentation functions to describe the hadronisation of a quark or a gluon into the observed hadron particle. Such identified particles in the final state make the treatment of infrared divergences more subtle, because of additional collinear divergences to be handled. We extend the antenna subtraction method to include hadron fragmentation processes up to next-to-next-to-leading order (NNLO) in QCD in $e^+e^+$ collisions. To this end, we introduce new double-real and real-virtual fragmentation antenna functions in the final-final kinematics, with associated phase space mappings. We present results for the antenna functions, for the master integrals required to integrate them over the relevant phase space and finally for the integrated antennae themselves. Our results are cross-checked against the known NNLO coefficient functions for identified hadron production in $e^+e^+$ annihilation.
In the past decade, antenna subtraction has been used to compute NNLO QCD corrections to a series of phenomenologically relevant processes. However, the application of this method proceeds in a process-dependent way, with each new calculation requiring a significant amount of work. Moreover, in the present formulation, the antenna subtraction method can not handle systematically incoherent interferences between different color orderings. In this talk we present an improved version of antenna subtraction which aims at achieving an automated and process-independent generation of the subtraction terms required for a NNLO calculation, as well as at overcoming the intrinsic limitations present in the traditional formulation. In this new approach, a set of integrated dipoles is used to reproduce the known infrared singularity structure of one- and two-loop amplitudes in color space. The real-virtual and double-real subtraction terms are subsequently generated inferring their structure from the corresponding integrated subtraction terms. We demonstrate the applicability of this method on the leading-color contribution to gluon-induced three-jet production.
The QCD energy-momentum tensor exhibits the well-known property of trace anomaly. The anomalous contribution can be distributed among the quark and gluon parts. Although the total energy-momentum tensor remains unrenormalized owing to the conservation of energy and momentum, the individual components do go through ultraviolet renormalization. We perform this renormalization at four-loop level. As a spin-off, the phenomenological consequences of our result concerning the anomaly induced mass structure of hadrons are discussed.
In this talk, I will present an exact result for the three-loop QCD correction to the axial component of the singlet quark form factor mediated by top quarks.
The result is obtained via a numerical solution of a system of differential equation for the occurring master integrals.
Furthermore, the Wilson coefficient in front of the axial current of massless quarks in the low energy effective QCD is extracted, which can be conveniently used to approximate the leading behavior of the singlet contribution in the large top mass limit.
I will highlight the latest developments in computing higher-order
scattering amplitudes with massive contributions. The contributing Feynman
integrals often lead to special classes of functions, for example,
functions associated with elliptic curves. With the presence of more
scales in the amplitudes, it becomes imperative to have a better
understanding of the contributing Feynman integrals and using current cutting-edge technologies to tackle the growth in the analytic complexity. In particular, I will start with discussing two-loop scattering amplitudes for top-quark pair production and end with shedding some light on scattering amplitudes for five-point processes.
We calculate the two-loop coefficient functions for the vector and axial-vector flavor-nonsinglet contribution to deeply-virtual Compton scattering (DVCS) using the approach based on conformal symmetry. We present the analytic expressions for the coefficient functions in momentum fraction space. The calculated NNLO corrections prove to be rather large and have to be taken into account.
In the context of infrared subtraction algorithms beyond next-to-leading order, it
becomes necessary to consider multiple infrared limits of scattering amplitudes, in which
several particles become soft or collinear in a strongly ordered sequence. We study these
limits from the point of view of infrared factorisation, and we provide general definitions of
strongly-ordered soft and collinear kernels in terms of gauge-invariant operator matrix
elements. With these definitions in hand, it is possible to construct local subtraction
counterterms for strongly ordered congurations. These are building blocks of infrared-finite
soft and collinear cross sections, therefore, upon integration, they cancel virtual poles by
construction. We test these ideas at tree level for multiple emissions, and at one
loop for single emission, which is sufficient for NNLO subtraction.
We calculate non-singlet quark operator matrix elements of deep-inelastic scattering in the chiral limit including operators with total derivatives. This extends previous calculations with zero-momentum transfer through the operator vertex which provides the well-known anomalous dimensions for the evolution of parton distributions, as well as calculations in off-forward kinematics utilizing conformal symmetry. Non-vanishing momentum-flow through the operator vertex leads to mixing with total derivative operators under renormalization. In the limit of a large number of quark flavors $n_f$ and for low moments in full QCD, we determine the anomalous dimension matrix to fifth order in the perturbative expansion in the strong coupling $\alpha_s$ in the $\overline{\mbox{MS}}$-scheme. We exploit consistency relations for the anomalous dimension matrix which follow from the renormalization structure of the operators, combined with a direct calculation of the relevant diagrams up to fourth order.
The Drell-Yan process continues to play an import role in putting to test the Standard Model (SM) and possibly revealing physics beyond it. In particular, investigating the high invariant mass region can be used to constrain heavy New Physics. To achieve this goal, high-precision theoretical predictions within the SM are needed. In this talk, we will focus on mixed QCD-electroweak corrections, which are expected to reach the percent level at high invariant masses. A critical aspect of the calculation is the extraction of soft and collinear singularities from real emission contributions. We will discuss some aspects of their treatment, and comment on the phenomenological outcomes of our studies.
We present the mixed QCD-EW two-loop virtual amplitudes for the neutral current Drell-Yan production, one of the bottlenecks for the complete calculation of the NNLO mixed QCD-EW corrections. We present the computational details and the first steps towards their automation.
We describe the evaluation of all the relevant two-loop Feynman integrals using analytical and semi-analytical methods, the subtraction of the universal infrared singularities and present the numerical evaluation of the finite remainder.
In BSM models often many choices for the renormalization scheme (RS) for the BSM particles (masses and mixings) are possible. Several of them lead to numerically unstable results. However, for a given parameter point in a BSM model it is a priori not known which RS leads to stable results, which makes the implementation of higher-order corrections in BSM models into automated codes complicated. We present a new and simple method to test the RS's for BSM parameter points, and to determine which one leads to stable results. This will facilitate the implementation of higher-order corrections to BSM processes into fully automated codes.
Obtaining precise theoretical predictions for both production and decay processes of heavy new particles is of great importance to constrain the allowed parameter space of BSM models and to properly assess the sensitivity for discoveries and for discriminating between different possible BSM scenarios.
In this context, it is well known that large logarithmic corrections can appear in the presence of widely separated mass scales. In this talk, I will point out the existence of possible large, Sudakov-like, logarithms in external-leg corrections of heavy scalars. In constrast with usual Sudakov logarithms, these can furthermore potentially be enhanced by large trilinear couplings. I will show that such large logarithms are associated with infrared singularities and examine several techniques to address these. In addition to a discussion at one loop, I will also present the derivation of the two-loop corrections containing this type of large logarithms, pointing out in this context the importance of adopting an on-shell renormalisation scheme. I will illustrate these calculations and results for a simple scalar toy model as well as for several decay processes involving heavy scalars in the Minimal Supersymmetric Standard Model (MSSM) and the singlet-extended Two-Higgs-Doublet Model (N2HDM).
Although the advanced concepts and techniques of perturbative QFT still apply to models with extended Higgs sectors, new issues arise, such as the question about phenomenoligically sound renormalization and input-parameter schemes, which fulfill desirable theoretical requirements (gauge invariance, perturbative stability, symmetries, simplicity, etc.), which are prerequisites for a confrontation of theory and experiment at the precision level. Moreover, there are non-trivial problems in the renormalization of vacuum expectation values (tadoles) if MSbar-renormalized parameters related to masses (running masses and Higgs mixing angles) are used, even in the Standard Model (SM).
In this talk these issues are discussed and exemplified for the Two-Higgs-Doublet Model and for a Singlet Extensions of the SM in particular. Old and new renormalization schemes are discussed and illustrated in applications to some Higgs-boson production and decay processes. In particular, a new concept for the renormalization of vacuum expectation values is introduced that is based on non-linear representations of scalar sectors.
In this talk we discuss new developments in the open-source Feynman
integral reduction tool Kira.
We explore algorithmic improvements like the construction of
block-triangular systems for more efficient reductions and new features
to help in the calculation of master integrals through differential
equations.
Kira's ability to solve homogeneous linear systems of equations can also
be applied to countless other problems apart from the reduction of
Feynman integrals.
We thus present new features to facilitate such applications.
Rational terms are a key ingredient for the automation of loop calculations through numerical methods. Nowadays widely-used automated NLO tools on the market usually construct their numerators of loop integrands in 4 dimensions, while the missing (D-4) part is reconstructed through the rational terms separately. Recently, several progresses [1-3] have been made to formulate and compute the rational terms at two loop order, which opens the door to the numerical automation of NNLO virtual corrections on various non-trivial LHC processes.
In this talk, I will first review our established formalism [1-3] for the two-loop UV rational terms in a modified R-opeartion approach. In particular, I will discuss the reconstruction of D-dimensional two-loop renormalised amplitudes with 4-dimensional loop numerators and a finite number of local UV rational counterterms in any renormalisable theories, and show the results of QCD rational terms in the full Standard Model.
I will further discuss our most recent progress [4] on the two-loop IR rational terms in massive QED amplitudes. On this aspect, I will present the strategy and key ingredients to resolve the IR rational terms by discovering interesting cancellation mechanisms.
[1] with S. Pozzorini and M. Zoller, "Rational terms of UV origin at two loops", JHEP 05 (2020) 077.
[2] with J.-N. Lang et al., "Two-loop rational terms in Yang-Mill theories", JHEP 10 (2020) 016.
[3] with J.-N. Lang et al., "Two-loop rational terms for spontaneously broken theories", JHEP 01 (2022) 105.
[4] with S. Pozzorini, "Two-loop IR rational terms in massive QED", to appear.
Towards two-loop automation in OpenLoops
Numerical tools, such as OpenLoops, provide NLO scattering amplitudes for a very wide range of hard scattering amplitudes in a fully automated way. In order to match the numerical precision of current and future experiments, however, the higher precision of NNLO calculations is essential, and their automation in a similar tool highly desirable.
In our approach, D-dimensional two-loop amplitudes are decomposed into Feynman integrals with four-dimensional numerators as well as (D-4)-dimensional remainders, which can contribute to the finite result through the interaction with the poles of Feynman integrals.
The integrals with four-dimensional numerators are then expressed as tensor integrals in the loop momenta contracted with tensor coefficients. The necessary building blocks for a full calculation in this framework are hence the treatment of (D-4)-dimensional numerator parts, the numerical construction of the tensor integral coefficients, the tensor integral reduction and the evaluation of the master integrals.
As the first important step in this approach, we present a method to reconstruct the contributions stemming from the interplay of (D-4)-dimensional numerator parts and UV poles through universal rational counterterms, as well as the computation of the full set of rational terms of UV origin for QCD and QED corrections to the SM.
The second building block we present in this talk, is a new and completely generic algorithm for the efficient and numerically stable construction of the tensor coefficients to the four-dimensional two-loop Feynman integrals. This algorithm is fully implemented in the OpenLoops framework.
The determination of the hot QCD pressure has a long history, and has – due to its phenomenological relevance in cosmology, astrophysics and heavy-ion collisions — spawned a number of important theoretical advances in perturbative thermal field theory applicable to equilibrium thermodynamics.
In particular, the long-standing infrared problem that obstructs the perturbative series has been overcome by a systematic use of dimensionally reduced effective theories, essentially mapping the problem of determining a full physical leading-order determination of the pressure to an extremely tough, but in principle doable, four-loop perturbative calculation in finite-temperature Yang-Mills theory.
We present advances in organizing this challenging calculation, by classifying and filtering the distinct contributions, and push ahead systematic simplifications of the remaining core sum-integral structures taking into account systems of linear relations that originate from symmetry- as well as integration-by-parts- relations. This will eventually allow us to gauge the grade of difficulty of a full determination of the physical leading-order QCD pressure, by analytic means.
The diagrammatic coaction underpins the analytic structure of Feynman integrals, their cuts and the differential equations they admit. The coaction maps any diagram into a tensor product of its pinches and cuts. These correspond respectively to differential forms defining master integrands, and integration contours which place a subset of the propagators on shell. In a canonical basis these forms and contours are dual to each other. In this talk I review our present understanding of this algebraic structure and its manifestation for dimensionally-regularised Feynman integrals that are expandable to polylogarithms. Using one- and two-loop integral examples, I will explain the duality between forms and contours and correspondence between the local coaction acting on the Laurent coefficients in the dimensional regulator and the global coaction acting on generalised hypergeometric functions. Finally, I will explain some of the salient differences between the one- and two-loop cases.
In a recent work the discrepancy between the QCD perturbation series for the inclusive hadronic tau decay rate computed in the CIPT and FOPT approaches was suggested to be related to a different sensitivity of the two approaches to quartic momenta in the infrared limit. If we make the assumption that the known perturbative correctionsfor the QCD Adler function at the 5-loop level are already governed significantly by the asymptotic behavior related to the gluon condensate renormalon, we can define a renormalon-free scheme for the gluon condensate which is similar to using short-distance quark mass scheme instead of the pole mass. We show how such a scheme can be setup in a concrete way in QCD, and we demonstrate that as a result the discrepancy between CIPT and FOPT is indeed significantly reduced. We use the new scheme in a new measurement of the strong coupling constant.
In this talk I will present the recent calculations in perturbative QCD to third order in $\alpha_s$ of the total rate of inclusive semileptonic $B$ decays and the relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of the heavy quarks. Our results have led to a 34% uncertainty reduction in the value of inclusive $|V_{cb}|$. I will furthermore present the $\alpha^3$ correction to the muon decay which will be important input for future reduction of the uncertainty of the Fermi coupling constant $G_F$.
In this talk, I present the first complete calculation of
form factors to four-loop order in massless QCD. Our results
for the qqA, ggH and bbH vertices confirm the predicted
structure of infrared poles and allow to extract the cusp
and collinear anomalous dimensions. Further, I will discuss
the calculation of three-loop amplitudes for quark and gluon
scattering. We verify predictions for the color quadrupole
contributions to the infrared poles of multi-leg amplitudes
in three-loop QCD.
or "What becomes of vortices when they grow giant"