Searcharxiv⌕ Search

arXiv subjects

William J. Torres Bobadilla

Publications and source records attributed to William J. Torres Bobadilla.

At least 37 records · Page 2Linked to original sources
arXiv2022

Two-loop scattering amplitude for heavy-quark pair production through light-quark annihilation in QCD

We present the first full analytic evaluation of the scattering amplitude for the process $q {\bar q} \to Q {\bar Q}$ up-to two loops in Quantum Chromodynamics, for a massless $(q)$ and a massive $(Q)$ quark flavour. The interference terms of the one- and two-loop amplitudes with the Born amplitude, decomposed in terms of gauge invariant form factors depending on the colour and flavour structure, are analytically calculated by keeping complete dependence on the squared center-of-mass energy, the squared momentum transfer, and the heavy-quark mass. The results are expressed as Laurent series around four space-time dimensions, with coefficients given in terms of generalised polylogarithms and transcendental constants up-to weight four. Our results validate the known, purely numerical calculations of the squared amplitude, and extend the analytic knowledge, previously limited to a subset of form factors, to their whole set, coming from both planar and non-planar diagrams, up-to the second order corrections in the strong coupling constant.

hep-ph↗
arXiv2022

Physical Representations for Scattering Amplitudes and the Wavefunction of the Universe

The way we organise perturbation theory is of fundamental importance both for computing the observables of relevance and for extracting fundamental physics out of them. If on one hand the different ways in which the perturbative observables can be written make manifest different features (e.g. symmetries as well as principles such as unitarity, causality and locality), on the other hand precisely demanding that some concrete features are manifest lead to different ways of organising perturbation theory. In the context of flat-space scattering amplitudes, a number of them are already known and exploited, while much less is known for cosmological observables. In the present work, we show how to systematically write down both the wavefunction of the universe and the flat-space scattering amplitudes, in such a way that they manifestly show physical poles only. We make use of the invariant definition of such observables in terms of {\it cosmological polytopes} and their {\it scattering facet}. In particular, we show that such representations correspond to triangulations of such objects through hyperplanes identified by the intersection of their facets outside of them. All possible triangulations of this type generate the different representations. This allows us to provide a general proof for the conjectured all-loop causal representation of scattering amplitudes. Importantly, all such representations can be viewed as making explicit a subset of compatible singularities, and our construction provides a way to extend Steinmann relations to higher codimension singularities for both the flat-space scattering amplitudes and the cosmological wavefunction.

hep-th↗
arXiv2021

Maximal transcendental weight contribution of scattering amplitudes

Feynman integrals in quantum field theory evaluate to special functions and numbers that are usefully described by the notion of transcendental weight. In this paper, we propose a way of projecting a given dimensionally-regularised Feynman integral, for example contributing to a scattering amplitudes, onto its maximal weight part. The method uses insights into the singularity structure of space-time loop integrands, and is complementary to usual generalised unitarity approaches. We describe the method and give a proof-of-principle application to the two-loop scattering amplitudes $gg \to H$ in the heavy top-quark mass limit, which involves both planar and non-planar Feynman integrals. We also comment on further possible applications and discuss subtleties related to evanescent integrand terms.

hep-th↗
arXiv2021

Causal representation and numerical evaluation of multi-loop Feynman integrals

The loop-tree duality (LTD) has become a novelty alternative to bootstrap the numerical evaluation of multi-loop scattering amplitudes. It has indeed been found that Feynman integrands, after the application of LTD, display a representation containing only physical information, the so-called causal representation. In this talk, we discuss the all-loop causal representation of multi-loop Feynman integrands, recently found in terms of features that describe a loop topology, vertices and edges. Likewise, in order to elucidate the numerical stability in LTD integrands, we present applications that involve numerical evaluations of two-loop planar and non-planar triangles with presence of several kinematic invariants.

hep-ph↗
arXiv2021

Four-loop scattering amplitudes journey into the forest

We present an overview of the analysis of the multiloop topologies that appear for the first time at four loops and the assembly of them in a general expression, the N$^4$MLT universal topology. Based on the fact that the Loop-Tree Duality enables to open any scattering amplitude in terms of convolutions of known subtopologies, we go through the dual representation of the universal N$^4$MLT topology and the manifestly causal representation. Additionally, we expose the application of a quantum algorithm as an alternative methodology to identify the causal singular configurations of multiloop Feynman diagrams.

hep-ph↗
arXiv2021

Lotty -- The loop-tree duality automation

Elaborating on the novel formulation of the loop-tree duality, we introduce the Mathematica package Lotty that automates the latter at multi-loop level. By studying the features of Lotty and recalling former studies, we discuss that the representation of any multi-loop amplitude can be brought in a form, at integrand level, that only displays physical information, which we refer to as the causal representation of multi-loop Feynman integrands. In order to elucidate the role of Lotty in this automation, we recall results obtained for the calculation of the dual representation of integrands up-to four loops. Likewise, within Lotty framework, we provide support to the all-loop causal representation recently conjectured by the same author. The numerical stability of the integrands generated by Lotty is studied in two-loop planar and non-planar topologies, where a numerical integration is performed and compared with known results.

hep-ph↗
arXiv2021

Causal representation of multi-loop Feynman integrands within the loop-tree duality

The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.

hep-ph↗
arXiv2021

Loop-tree duality from vertices and edges

The causal representation of multi-loop scattering amplitudes, obtained from the application of the loop-tree duality formalism, comprehensively elucidates, at integrand level, the behaviour of only physical singularities. This representation is found to manifest compact expressions for multi-loop topologies that have the same number of \textit{vertices}. Interestingly, integrands considered in former studies, with up-to six vertices and $L$ internal lines, display the same structure of up-to four-loop ones. The former is an insight that there should be a correspondence between vertices and the collection of internal lines, \textit{edges}, that characterise a multi-loop topology. By virtue of this relation, in this paper, we embrace an approach to properly classify multi-loop topologies according to vertices and edges. Differently from former studies, we consider the most general topologies, by connecting vertices and edges in all possible ways. Likewise, we provide a procedure to generate causal representation of multi-loop topologies by considering the structure of causal propagators. Explicit causal representations of loop topologies with up-to nine vertices are provided.

hep-ph↗
arXiv2021

A stroll through the loop-tree duality

The Loop-Tree Duality (LTD) theorem is an innovative technique to deal with multi-loop scattering amplitudes, leading to integrand-level representations over an Euclidean space. In this article, we review the last developments concerning this framework, focusing on the manifestly causal representation of multi-loop Feynman integrals and scattering amplitudes, and the definition of dual local counter-terms to cancel infrared singularities.

hep-ph↗
arXiv2021

Efficient resummation of high post-Newtonian contributions to the binding energy

A factorisation property of Feynman diagrams in the context the Effective Field Theory approach to the compact binary problem has been recently employed to efficiently determine the static sector of the potential at fifth post-Newtonian (5PN) order. We extend this procedure to the case of non-static diagrams and we use it to fix, by means of elementary algebraic manipulations, the value of more than one thousand diagrams at 5PN order, that is a substantial fraction of the diagrams needed to fully determine the dynamics at 5PN. This procedure addresses the redundancy problem that plagues the computation of the binding energy with respect to more "efficient" observables like the scattering angle, thus making the EFT approach in harmonic gauge at least as scalable as the others methods.

gr-qc↗
arXiv2021

Universal opening of four-loop scattering amplitudes to trees

The perturbative approach to quantum field theories has made it possible to obtain incredibly accurate theoretical predictions in high-energy physics. Although various techniques have been developed to boost the efficiency of these calculations, some ingredients remain specially challenging. This is the case of multiloop scattering amplitudes that constitute a hard bottleneck to solve. In this paper, we delve into the application of a disruptive technique based on the loop-tree duality theorem, which is aimed at an efficient computation of such objects by opening the loops to nondisjoint trees. We study the multiloop topologies that first appear at four loops and assemble them in a clever and general expression, the N$^4$MLT {\it universal topology}. This general expression enables to open any scattering amplitude of up to four loops, and also describes a subset of higher order configurations to all orders. These results confirm the conjecture of a factorized opening in terms of simpler known subtopologies, which also determines how the causal structure of the entire loop amplitude is characterized by the causal structure of its subtopologies. In addition, we confirm that the loop-tree duality representation of the N$^4$MLT universal topology is manifestly free of noncausal thresholds, thus pointing towards a remarkably more stable numerical implementation of multiloop scattering amplitudes.

hep-ph↗
arXiv2021

On the interplay between the loop-tree duality and helicity amplitudes

The spinor-helicity formalism has proven to be very efficient in the calculation of scattering amplitudes in quantum field theory, while the loop tree duality (LTD) representation of multi-loop integrals exhibits appealing and interesting advantages with respect to other approaches. In view of the most recent developments in LTD, we exploit the synergies with the spinor-helicity formalism to analyse illustrative one- and two-loop scattering processes. We focus our discussion on the local UV renormalisation of IR and UV finite amplitudes and present a fully automated numerical implementation that provides efficient expressions which are integrable directly in four space-time dimensions.

hep-ph↗
arXiv2020

Mathematical properties of nested residues and their application to multi-loop scattering amplitudes

The computation of multi-loop multi-leg scattering amplitudes plays a key role to improve the precision of theoretical predictions for particle physics at high-energy colliders. In this work, we focus on the mathematical properties of the novel integrand-level representation of Feynman integrals, which is based on the Loop-Tree Duality (LTD). We explore the behaviour of the multi-loop iterated residues and explicitly show, by developing a general formal proof for the first time, that contributions associated to displaced poles are cancelled out. The remaining residues, called nested residues as originally introduced in Ref. \cite{Verdugo:2020kzh}, encode the relevant physical information and are naturally mapped onto physical configurations associated to nondisjoint on-shell states. By going further on the mathematical structure of the nested residues, we prove that unphysical singularities vanish, and show how the final expressions can be written by using only causal denominators. In this way, we provide a mathematical proof for the all-loop formulae presented in Ref. \cite{Aguilera-Verdugo:2020kzc}.

hep-ph↗
arXiv2020

Open loop amplitudes and causality to all orders and powers from the loop-tree duality

Multiloop scattering amplitudes describing the quantum fluctuations at high-energy scattering processes are the main bottleneck in perturbative quantum field theory. The loop-tree duality is a novel method aimed at overcoming this bottleneck by opening the loop amplitudes into trees and combining them at integrand level with the real-emission matrix elements. In this Letter, we generalize the loop-tree duality to all orders in the perturbative expansion by using the complex Lorentz-covariant prescription of the original one-loop formulation. We introduce a series of mutiloop topologies with arbitrary internal configurations and derive very compact and factorizable expressions of their open-to-trees representation in the loop-tree duality formalism. Furthermore, these expressions are entirely independent at integrand level of the initial assignments of momentum flows in the Feynman representation and remarkably free of noncausal singularities. These properties, that we conjecture to hold to other topologies at all orders, provide integrand representations of scattering amplitudes that exhibit manifest causal singular structures and better numerical stability than in other representations.

hep-ph↗
arXiv2019

Static two-body potential at fifth post-Newtonian order

We determine the gravitational interaction between two compact bodies up to the sixth power in Newton's constant GN, in the static limit. This result is achieved within the effective field theory approach to General Relativity, and exploits a manifest factorization property of static diagrams which allows to derive static post Newtonian (PN) contributions of (2n+1)-order in terms of lower order ones. We recompute in this fashion the 1PN and 3PN static potential, and present the novel 5PN contribution.

gr-qc↗
arXiv2019

Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the loop-tree duality at higher orders

We present the first comprehensive analysis of the unitarity thresholds and anomalous thresholds of scattering amplitudes at two loops and beyond based on the loop-tree duality, and show how non-causal unphysical thresholds are locally cancelled in an efficient way when the forest of all the dual on-shell cuts is considered as one. We also prove that soft and collinear singularities at two loops and beyond are restricted to a compact region of the loop three-momenta, which is a necessary condition for implementing a local cancellation of loop infrared singularities with the ones appearing in real emission; without relying on a subtraction formalism.

hep-ph↗
arXiv2019

Universal four-dimensional representation of $H \to γγ$ at two loops through the Loop-Tree Duality

We extend useful properties of the $H\toγγ$ unintegrated dual amplitudes from one- to two-loop level, using the Loop-Tree Duality formalism. In particular, we show that the universality of the functional form -- regardless of the nature of the internal particle -- still holds at this order. We also present an algorithmic way to renormalise two-loop amplitudes, by locally cancelling the ultraviolet singularities at integrand level, thus allowing a full four-dimensional numerical implementation of the method. Our results are compared with analytic expressions already available in the literature, finding a perfect numerical agreement. The success of this computation plays a crucial role for the development of a fully local four-dimensional framework to compute physical observables at Next-to-Next-to Leading order and beyond.

hep-ph↗