arXiv subjects

Claude Duhr

Publications and source records attributed to Claude Duhr.

At least 73 records · Page 4Linked to original sources

arXiv2021

Lepton-pair production at hadron colliders at N$^3$LO in QCD

We compute for the first time the complete corrections at N$^3$LO in the strong coupling constant to the inclusive neutral-current Drell-Yan process including contributions from both photon and $Z$-boson exchange. Our main result is the computation of the QCD corrections to the inclusive production cross section of an axial-vector boson to third order in the strong coupling in a variant of QCD with five massless quark flavours. Since the axial anomaly does not cancel for an odd number of flavours, we also consistently include non-decoupling effects in the top-quark mass through three loops. We perform a phenomenological study of our results, and we present for the first time predictions for the inclusive Drell-Yan process at the LHC at this order in QCD perturbation theory.

hep-ph↗

arXiv2021

Meromorphic modular forms and the three-loop equal-mass banana integral

We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.

hep-th↗

arXiv2021

An analysis of Bayesian estimates for missing higher orders in perturbative calculations

With current high precision collider data, the reliable estimation of theoretical uncertainties due to missing higher orders (MHOs) in perturbation theory has become a pressing issue for collider phenomenology. Traditionally, the size of the MHOs is estimated through scale variation, a simple but ad hoc method without probabilistic interpretation. Bayesian approaches provide a compelling alternative to estimate the size of the MHOs, but it is not clear how to interpret the perturbative scales, like the factorisation and renormalisation scales, in a Bayesian framework. Recently, it was proposed that the scales can be incorporated as hidden parameters into a Bayesian model. In this paper, we thoroughly scrutinise Bayesian approaches to MHO estimation and systematically study the performance of different models on an extensive set of high-order calculations. We extend the framework in two significant ways. First, we define a new model that allows for asymmetric probability distributions. Second, we introduce a prescription to incorporate information on perturbative scales without interpreting them as hidden model parameters. We clarify how the two scale prescriptions bias the result towards specific scale choice, and we discuss and compare different Bayesian MHO estimates among themselves and to the traditional scale variation approach. Finally, we provide a practical prescription of how existing perturbative results at the standard scale variation points can be converted to 68%/95% credibility intervals in the Bayesian approach using the new public code MiHO.

hep-ph↗

arXiv2021

Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives

We provide a comprehensive summary of concepts from Calabi-Yau motives relevant to the computation of multi-loop Feynman integrals. From this we derive several consequences for multi-loop integrals in general, and we illustrate them on the example of multi-loop banana integrals. For example, we show how Griffiths transversality, known from the theory of variation of mixed Hodge structures, leads quite generically to a set of quadratic relations among maximal cut integrals associated to Calabi-Yau motives. These quadratic relations then naturally lead to a compact expression for $l$-loop banana integrals in $D=2$ dimensions in terms of an integral over a period of a Calabi-Yau $(l-1)$-fold. This new integral representation generalizes in a natural way the known representations for $l\le 3$ involving logarithms with square root arguments and iterated integrals of Eisenstein series. In a second part, we show how the results obtained by some of the authors in earlier work can be extended to dimensional regularization. We present a method to obtain the differential equations for banana integrals with an arbitrary number of loops in dimensional regularization without the need to solve integration-by-parts relations. We also present a compact formula for the leading asymptotics of banana integrals with an arbitrary number of loops in the large momentum limit. This generalizes the novel $\widehatΓ$-class introduced by some of the authors to dimensional regularization and provides a convenient boundary condition to solve the differential equations for the banana integrals. As an application, we present for the first time numerical results for equal-mass banana integrals with up to four loops and up to second order in the dimensional regulator.

hep-th↗

arXiv2021

Analytic results for two-loop planar master integrals for Bhabha scattering

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equa- tions with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic mul- tiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.

hep-ph↗

arXiv2021

The diagrammatic coaction beyond one loop

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the $ε$ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

hep-th↗

arXiv2020

Higgs production in bottom-quark fusion to third order in the strong coupling

We present the inclusive cross section at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD for the production of a Higgs boson via bottom-quark fusion. We employ the five-flavour scheme, treating the bottom quark as a massless parton while retaining a non-vanishing Yukawa coupling to the Higgs boson. We find that the dependence of the hadronic cross section on the renormalisation and factorisation scales is substantially reduced. For judicious choices of the scales the perturbative expansion of the cross section shows a convergent behaviour. We present results for the N$^3$LO cross section at various collider energies. In comparison to the cross section obtained from the Santander-matching of the four and five-flavour schemes we predict a slightly higher cross section, though the two predictions are consistent within theoretical uncertainties.

hep-ph↗

arXiv2020

The Drell-Yan cross section to third order in the strong coupling constant

We present phenomenological results for the inclusive cross section for the production of a lepton-pair via virtual photon exchange at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. In line with the case of Higgs production, we find that the hadronic cross section receives corrections at the percent level, and the residual dependence on the perturbative scales is reduced. However, unlike in the Higgs case, we observe that the uncertainty band derived from scale variation is no longer contained in the band of the previous order.

hep-ph↗

arXiv2020

Charged Current Drell-Yan Production at N3LO

We present the production cross section for a lepton-neutrino pair at the Large Hadron Collider computed at next-to-next-to-next-to leading order (N3LO) in QCD perturbation theory. We compute the partonic coefficient functions of a virtual $W^{\pm}$ boson at this order. We then use these analytic functions to study the progression of the perturbative series in different observables. In particular, we investigate the impact of the newly obtained corrections on the inclusive production cross section of $W^{\pm}$ bosons, as well as on the ratios of the production cross sections for $W^+$, $W^-$ and/or a virtual photon. Finally, we present N3LO predictions for the charge asymmetry at the LHC.

hep-ph↗

arXiv2020

Tree-level splitting amplitudes for a gluon into four collinear partons

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next- to-next-to-next-to-leading order (N$^3$LO) in the strong coupling constant. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.

hep-ph↗

arXiv2020

A double integral of dlog forms which is not polylogarithmic

Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.

hep-th↗

arXiv2020

Higgs production in bottom quark fusion: Matching the 4- and 5-flavour schemes to third order in the strong coupling

We present analytic results for the partonic cross sections for the production of a Higgs boson via the fusion of two bottom quarks at N$^3$LO in QCD perturbation theory in the five-flavour scheme. We combine this perturbative result with NLO accurate predictions in the four-flavour scheme that include the full bottom quark mass dependence by appropriately removing any double-counting stemming from contributions included in both predictions. We thereby obtain state-of-the-art predictions for the inclusive production probability of a Higgs boson via bottom quark fusion at hadron colliders.

hep-ph↗

arXiv2020

Tree-level splitting amplitudes for a quark into four collinear partons

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a quark parent in the limit where four partons become collinear to each other. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next-to-next-to-next-to-leading order (${\rm N^3LO}$) in the strong coupling constant. Further, we consider the iterated limit when $m'$ massless partons become collinear to each other within a bigger set of $m$ collinear partons, as well as the limits when one gluon or a $q\bar{q}$ pair or two gluons become soft within a set of $m$ collinear partons.

hep-ph↗

arXiv2019

Diagrammatic Coaction of Two-Loop Feynman Integrals

It is known that one-loop Feynman integrals possess an algebraic structure encoding some of their analytic properties called the coaction, which can be written in terms of Feynman integrals and their cuts. This diagrammatic coaction, and the coaction on other classes of integrals such as hypergeometric functions, may be expressed using suitable bases of differential forms and integration contours. This provides a useful framework for computing coactions of Feynman integrals expressed using the hypergeometric functions. We will discuss examples where this technique has been used in the calculation of two-loop diagrammatic coactions.

hep-th↗

arXiv2019

Generalized hypergeometric functions and intersection theory for Feynman integrals

Feynman integrals that have been evaluated in dimensional regularization can be written in terms of generalized hypergeometric functions. It is well known that properties of these functions are revealed in the framework of intersection theory. We propose a new application of intersection theory to construct a coaction on generalized hypergeometric functions. When applied to dimensionally regularized Feynman integrals, this coaction reproduces the coaction on multiple polylogarithms order by order in the parameter of dimensional regularization.

hep-th↗

arXiv2019

Three-loop contributions to the $ρ$ parameter and iterated integrals of modular forms

We compute fully analytic results for the three-loop diagrams involving two different massive quark flavours contributing to the $ρ$ parameter in the Standard Model. We find that the results involve exactly the same class of functions that appears in the well-known sunrise and banana graphs, namely elliptic polylogarithms and iterated integrals of modular forms. Using recent developments in the understanding of these functions, we analytically continue all the iterated integrals of modular forms to all regions of the parameter space, and in each region we obtain manifestly real and fast-converging series expansions for these functions.

hep-th↗

arXiv2019

Algorithms and tools for iterated Eisenstein integrals

We present algorithms to work with iterated Eisenstein integrals that have recently appeared in the computation of multi-loop Feynman integrals. These algorithms allow one to analytically continue these integrals to all regions of the parameter space, and to obtain fast converging series representations in each region. We illustrate our approach on the examples of hypergeometric functions that evaluate to iterated Eisenstein integrals as well as the well-known sunrise graph.

hep-th↗

arXiv2019

From positive geometries to a coaction on hypergeometric functions

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally-regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter $ε$. We show that the coaction defined on this class of integral is consistent, upon expansion in $ε$, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric ${}_{p+1}F_p$ and Appell functions.

hep-th↗