arXiv subjects

Ulrich Schubert

Publications and source records attributed to Ulrich Schubert.

15 recordsLinked to original sources

arXiv2022

Master integrals for the NNLO virtual corrections to $μe$ scattering in QED: the planar graphs

We evaluate the master integrals for the two-loop, planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to leading-order in QED. We adopt the method of differential equations and the Magnus exponential series to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at $e^+ e^-$-colliders, as well as for the QCD corrections to $top$-pair production at hadron colliders.

hep-ph↗

arXiv2021

Explainable AI for ML jet taggers using expert variables and layerwise relevance propagation

A framework is presented to extract and understand decision-making information from a deep neural network (DNN) classifier of jet substructure tagging techniques. The general method studied is to provide expert variables that augment inputs ("eXpert AUGmented" variables, or XAUG variables), then apply layerwise relevance propagation (LRP) to networks both with and without XAUG variables. The XAUG variables are concatenated with the intermediate layers after network-specific operations (such as convolution or recurrence), and used in the final layers of the network. The results of comparing networks with and without the addition of XAUG variables show that XAUG variables can be used to interpret classifier behavior, increase discrimination ability when combined with low-level features, and in some cases capture the behavior of the classifier completely. The LRP technique can be used to find relevant information the network is using, and when combined with the XAUG variables, can be used to rank features, allowing one to find a reduced set of features that capture part of the network performance. In the studies presented, adding XAUG variables to low-level DNNs increased the efficiency of classifiers by as much as 30-40\%. In addition to performance improvements, an approach to quantify numerical uncertainties in the training of these DNNs is presented.

physics.data-an↗

arXiv2020

Master Integrals for the mixed QCD-QED corrections to the Drell-Yan production of a massive lepton pair

We showcase the calculation of the master integrals needed for the two loop mixed QCD-QED virtual corrections to the neutral current Drell-Yan process $(q\bar{q}\rightarrow l^+ l^-)$. After establishing a basis of 51 master integrals, we cast the latter into canonical form by using the Magnus algorithm. The dependence on the lepton mass is then expanded such that potentially large logarithmic contributions are kept. After determining all boundary constants, we give the coefficients of the Taylor series around four space-time dimensions in terms of generalized polylogarithms up to weight four.

hep-ph↗

arXiv2020

Top-induced contributions to $H\rightarrow b\bar{b}$ and $H\rightarrow c\bar{c}$ at $\mathcal{O}(α_s^3)$

In this paper we present a fully-differential calculation for the contributions to the partial widths $H\rightarrow b\bar{b}$ and $H\rightarrow c\bar{c}$ that are sensitive to the top quark Yukawa coupling $y_t$ to order $α_s^3$. These contributions first enter at order $α_s^2$ through terms proportional to $y_t y_q$ ($q=b,c$). At order $α_s^3$ corrections to the mixed terms are present as well as a new contribution proportional to $y_t^2$. Our results retain the mass of the final-state quarks throughout, while the top quark is integrated out resulting in an effective field theory (EFT). Our results are implemented into a Monte Carlo code allowing for the application of arbitrary final-state selection cuts. As an example we present differential distributions for observables in the Higgs boson rest frame using the Durham jet clustering algorithm. We find that the total impact of the top-induced (i.e. EFT) pieces is sensitive to the nature of the final-state cuts, particularly b-tagging and c-tagging requirements. For bottom quarks, the EFT pieces contribute to the total width (and differential distributions) at around the percent level. The impact is much bigger for the $H\rightarrow c\bar{c}$ channel, with effects as large as 15%. We show however that their impact can be significantly reduced by the application of jet-tagging selection cuts.

hep-ph↗

arXiv2019

The interplay between SM precision, BSM physics and measurements of $α_{\rm{had}}$ in $μ$-$e$ scattering

Muon electron scattering experiments such as the proposed MUonE experiment, offer an opportunity for an improved measurement of the Leading Order hadronic running of $α$, denoted $Δα_{\rm{had}}$. Such a measurement could be utilized to reduce the theoretical uncertainty on the prediction of the anomalous magnetic moment of the muon, $g-2$. Currently there is a discrepancy between theory and data for this observable which could potentially be explained by Beyond the Standard Model (BSM) physics. Here we investigate the possible impact of missing Standard Model (SM) higher order corrections and BSM physics on the proposed measurement of $Δα_{\rm{had}}$. In principle either could be indirectly fitted into $Δα_{\rm{had}}$, causing inconsistencies if used in a $g-2$ application. The literature suggests a target of 10 ppm on the cross section for the theoretical accuracy. We assess the validity of this target in detail using a variety of methods, finding that a 1 ppm target is a more conservative estimate to ensure missing higher orders do not dominate the theoretical uncertainty. For the potential BSM contributions we study various models which contribute first at tree- and loop-level. Of particular interest is the impact from dark photon models, which can potentially affect the measurement of $Δα_{\rm{had}}$ at the desired accuracy. At loop-level there exists in general a kinematic suppression adequate to reduce the BSM contributions to a level which can be neglected for the extraction of $Δα_{\rm{had}}$.

hep-ph↗

arXiv2019

Master integrals for the NNLO virtual corrections to $q \bar{q} \rightarrow t \bar{t}$ scattering in QCD: the non-planar graphs

We complete the analytic evaluation of the master integrals for the two-loop non-planar box diagrams contributing to the top-pair production in the quark-initiated channel, at next-to-next-to-leading order in QCD. The integrals are determined from their differential equations, which are cast into a canonical form using the Magnus exponential. The analytic expressions of the Laurent series coefficients of the integrals are expressed as combinations of generalized polylogarithms, which we validate with several numerical checks. We discuss the analytic continuation of the planar and the non-planar master integrals, which contribute to $q {\bar q} \to t {\bar t}$ in QCD, as well as to the companion QED scattering processes $ e e \to μμ$ and $e μ\to e μ$.

hep-ph↗

arXiv2018

Master integrals for the NNLO virtual corrections to $μe$ scattering in QED: the non-planar graphs

We evaluate the master integrals for the two-loop non-planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to-leading order in QED. We adopt the method of differential equations and the Magnus exponential to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as a combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at $e^+e^-$ colliders, as well as for the QCD corrections to top-pair production at hadron colliders. In particular, our results, together with the planar master integrals recently computed, represent the complete set of functions needed for the evaluation of the photonic two-loop virtual next-to-next-to-leading order QED corrections to $μe \to μe$ and $e^+ e^-\toμ^+μ^-$.

hep-ph↗

arXiv2017

The spin-dependent quark beam function at NNLO

We calculate the beam function for longitudinally-polarized quarks through next-to-next-to-leading order (NNLO) in QCD perturbation theory. This is the last missing ingredient needed to apply the factorization theorem for the $N$-jettiness event-shape variable in polarized collisions through the NNLO level. We present all technical details of our derivation. As a by-product of our calculation we provide the first independent check of the previously-obtained unpolarized quark beam function. We anticipate that our result will have phenomenological applications in describing data from polarized collisions.

hep-ph↗

arXiv2017

Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a $W$ pair and to the triple gauge couplings $ZWW$ and $γ^*WW$

We compute the two-loop master integrals required for the leading QCD corrections to the interaction vertex of a massive neutral boson $X^0$, e.g. $H,Z$ or $γ^{*}$, with a pair of $W$ bosons, mediated by a $SU(2)_L$ quark doublet composed of one massive and one massless flavor. All the external legs are allowed to have arbitrary invariant masses. The Magnus exponential is employed to identify a set of master integrals that, around $d=4$ space-time dimensions, obey a canonical system of differential equations. The canonical master integrals are given as a Taylor series in $ε= (4-d)/2$, up to order four, with coefficients written as combination of Goncharov polylogarithms, respectively up to weight four. In the context of the Standard Model, our results are relevant for the mixed EW-QCD corrections to the Higgs decay to a $W$ pair, as well as to the production channels obtained by crossing, and to the triple gauge boson vertices $ZWW$ and $γ^*WW$.

hep-ph↗

arXiv2016

Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering

We present the calculation of the master integrals needed for the two-loop QCDxEW corrections to $ q + \bar{q} \to l^- + l^+$ and $ q + \bar{q}' \to l^- + \overlineν \, , $ for massless external particles. We treat W and Z bosons as degenerate in mass. We identify three types of diagrams, according to the presence of massive internal lines: the no-mass type, the one-mass type, and the two-mass type, where all massive propagators, when occurring, contain the same mass value. We find a basis of 49 master integrals and evaluate them with the method of the differential equations. The Magnus exponential is employed to choose a set of master integrals that obeys a canonical system of differential equations. Boundary conditions are found either by matching the solutions onto simpler integrals in special kinematic configurations, or by requiring the regularity of the solution at pseudo-thresholds. The canonical master integrals are finally given as Taylor series around d=4 space-time dimensions, up to order four, with coefficients given in terms of iterated integrals, respectively up to weight four.

hep-ph↗

arXiv2015

Off-shell Currents and Color-Kinematics Duality

We elaborate on the color-kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process $gg\to ss, q\bar q, gg$, and show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a color-kinematics violating term due to the contributions of sub-graphs only. Such anomaly vanishes when the four particles connected by the Jacobi relation are on their mass shell with vanishing squared momenta, being either external or cut particles, where the validity of the color-kinematics duality is recovered. We discuss the role of the off-shell decomposition in the direct construction of higher-multiplicity numerators satisfying color-kinematics identity in four as well as in $d$ dimensions, for the latter employing the Four Dimensional Formalism variant of the Four Dimensional Helicity scheme. We provide explicit examples for the QCD process $gg\to q\bar{q}g$.

hep-ph↗

arXiv2014

Scattering Amplitudes in Gauge Theories

This thesis is focused on the development of new mathematical methods for computing multi-loop scattering amplitudes in gauge theories. In this work we combine, for the first time, the unitarity-based construction for integrands, and the recently introduced integrand-reduction through multivariate polynomial division. After discussing the generic features of this novel reduction algorithm, we will apply it to the one- and two-loop five-point amplitudes in ${\cal N}=4$ sYM. The integrands of the multiple-cuts are generated from products of tree-level amplitudes within the super-amplitudes formalism. The corresponding expressions will be used for the analytic reconstruction of the polynomial residues. Their parametric form is known a priori, as derived by means of successive polynomial divisions using the Gröbner basis associated to the on-shell denominators. The integrand reduction method will be exploited to investigate the color-kinematic duality for multi-loop ${\cal N}=4$ sYM scattering amplitudes. Our analysis yields a suggestive, systematic way to generate graphs which automatically satisfy the color-kinematic dualities. Finally, we will extract the leading ultra-violet divergences of five-point one- and two-loop amplitudes in ${\cal N}=4$ sYM, which represent a paradigmatic example for studying the UV behavior of supersymmetric amplitudes.

hep-ph↗

arXiv2014

Three-loop master integrals for ladder-box diagrams with one massive leg

The three-loop master integrals for ladder-box diagrams with one massive leg are computed from an eighty-five by eighty-five system of differential equations, solved by means of Magnus exponential. The results of the considered box-type integrals, as well as of the tower of vertex- and bubble-type master integrals associated to subtopologies, are given as a Taylor series expansion in the dimensional regulator parameter epsilon = (4-d)/2. The coefficients of the series are expressed in terms of uniform weight combinations of multiple polylogarithms and transcendental constants up to weight six. The considered integrals enter the next-to-next-to-next-to-leading order virtual corrections to scattering processes like the three-jet production mediated by vector boson decay, V* -> jjj, as well as the Higgs plus one-jet production in gluon fusion, pp -> Hj.

hep-ph↗

arXiv2014

Magnus and Dyson Series for Master Integrals

We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 -> 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.

hep-ph↗

arXiv2013

Multi-loop Integrand Reduction via Multivariate Polynomial Division

We present recent developments on the topic of the integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the corresponding integrands. This approach has already been successfully applied and automated at one loop, and recently extended to higher loops. We describe a coherent framework based on simple concepts of algebraic geometry, such as multivariate polynomial division, which can be used in order to obtain the integrand decomposition of any amplitude at any loop order. In the one-loop case, we discuss an improved reduction algorithm, based on the application of the Laurent series expansion to the integrands, which has been implemented in the semi-numerical library Ninja. At two loops, we present the reduction of five-point amplitudes in N=4 SYM, with a unitarity-based construction of the integrand. We also describe the multi-loop divide-and-conquer approach, which can always be used to find the integrand decomposition of any Feynman graph, regardless of the form and the complexity of the integrand, with purely algebraic operations.

hep-ph↗