High Energy Resummation of Jet Observables

In this paper we investigate the extension of high energy resummation at LLx accuracy to jet observables. In particular, we present the high energy resummed expression of the transverse momentum distribution of the outgoing parton in the general partonic process $g(q) + g(q) \to g(q) + X$. In order to reach this result, several new ideas are introduced and exploited. First we prove that LLx resummation is achieved by dressing with hard radiation an off-shell gluon initiated LO process even if its on-shell limit is vanishing or trivial. Then we present a gauge-invariant framework where these calculations can be performed by using the modern helicity techniques. Finally, we show a possible way to restore gluon indistinguishability in the final state, which is otherwise lost in the resummation procedure, at all orders in $\alpha_s$ at LLx. All partonic channels are then resummed and cross-checked against fixed-order calculations up to $\mathcal{O}(\alpha_s^3)$Comment: 31 pages, 6 figure

Two-loop five-particle scattering amplitudes

I discuss the recent advances in the computation of two-loop scattering amplitudes for five-particle processes. The latter are fundamental ingredients to obtain predictions at the next-to-next-to-leading order (NNLO) in QCD for many interesting LHC processes. I discuss the state-of-the-art technology for computing scattering amplitudes analytically, and present new results relevant for the LHC phenomenology.Comment: Proceedings for Radcor 2023, 7 pages, 1 tabl

Modern analytic methods for scattering amplitudes, with an application to two-loop five-particle processes

The scientific approach to understanding the laws of nature is based on the comparison between theory and experiment. In laypersons' terms, a theory is a set of rules which describe mathematically how we think things work. We use these rules to predict the outcome of a certain experiment, and the comparison against the actual results of the experiment may disprove or uphold the theory. Particle physics is concerned with the tiniest building blocks of the universe - the fundamental particles - and the way they interact. Our best description of fundamental particles is given by the Standard Model of particle physics (SM), which treats particles as oscillations of "quantum fields" permeating the space-time. The spectacular success of the SM at describing the microscopic world is one of humanity's greatest intellectual feats. Yet, this theory fails to address a variety of theoretical concerns and observed phenomena|gravity, to say the most obvious. Understanding the limitations of the SM and constraining its extensions is of primary importance. Scattering amplitudes are the bridge between theory and experiments in Quantum Field Theories (QFTs). Roughly speaking, the amplitude of a scattering process encodes its probability distribution: for a given initial state|say two colliding protons - the scattering amplitude tells us how likely the production of certain other particles is according to the theory. The rules of QFT are however complicated, and scattering amplitudes can only be computed approximately as series in the coupling constants which weigh the interactions. We know - at least in principle - how to compute each term of the series, and including more terms makes the prediction more accurate. The computation however becomes more and more difficult as the order in the couplings - also called the "loop order" - or the number of particles increase. In practice, we need as many terms as is necessary to make the theoretical uncertainty comparable with the experimental one, so that the comparison is statistically significant. Exploiting fully the physics potential of CERN's Large Hadron Collider requires predictions at the Next-to-Next-to-Leading Order (NNLO) in the coupling of the strong interactions. This goal has already been reached for many 2 -> 1 and 2 -> 2 processes. Processes with three particles in the final state are however of great interest, as they would allow for precise measurements of the strong coupling constant and of its scaling, in-depth studies of the Higgs couplings, better background estimates for yet unknown phenomena, and more. The main bottleneck towards NNLO predictions for 2 -> 3 processes is the analytic computation of two-loop five-particle scattering amplitudes. The most difficult part of computing a scattering amplitude is the computation of the Feynman integrals appearing in it. My collaborators and I computed the missing and most complicated set of massless two-loop five-particle Feynman integrals. This opened the doors to the computation of the amplitude for any process involving five massless particles at two-loop order. Such processes feature prominently in the LHC physics program. Cases in point are three-jet, three-photon, and di-photon + jet production. In order to compute these integrals we made use of cutting-edge mathematical techniques, and proposed a new strategy which has already been applied to other difficult problems. Armed with analytic expressions for the Feynman integrals, we tackled the amplitudes. The challenge is one of enormous algebraic complexity. We developed a work ow based on the recent idea of evaluating the rational functions in the intermediate expressions numerically over finite fields. The analytic expression of the final result is then reconstructed by "bootstrapping" an Ansatz or through reconstruction algorithms. Before considering the SM, we tested our approach on the amplitudes in two supersymmetric theories: N = 4 super Yang-Mills theory and N = 8 supergravity. These were the very first complete five-particle scattering amplitudes to be computed analytically at two loops. Although these models do not seek to describe physical particles and forces, they are of great interest. They give precious insights into hidden structures of QFT in general and - thanks to their simplicity and elegance - they are a perfect testing ground for new techniques and ideas which can be later applied to the SM. The successful computation of the supersymmetric amplitudes showed that our technology was mature enough to face the SM. We therefore computed the two-loop amplitude describing the scattering of five positive-helicity gluons in Quantum Chromodynamics (QCD), the part of the SM which describes the strong interactions. Despite the leap in complexity with respect to the supersymmetric theories, we managed to find an extremely compact and elegant analytic expression. Having compact results for the amplitudes is not only a theorist's delight, but is crucial for their use in phenomenology. The simplicity of the expression allowed us to notice that certain parts of the amplitude enjoy an unexpected property: they are invariant under conformal symmetry. We identified the origin of this property in the conformal invariance of the gluonic amplitudes in QCD at one loop, which we proved for any number of gluons. After the publication of the results presented in this thesis there has been a dramatic progress. Several other two-loop five-particle amplitudes have become available analytically, and this has already led to the first theoretical prediction at NNLO in QCD, for three-photon production. Partly using methods similar to those presented in this thesis, many more results are sure to follow in the near future

Two-loop QCD corrections to WbbˉWb\bar{b} production at hadron colliders

We present an analytic computation of the two-loop QCD corrections to $u\bar{d}\to W^+b\bar{b}$ for an on-shell $W$-boson using the leading colour and massless bottom quark approximations. We perform an integration-by-parts reduction of the unpolarised squared matrix element using finite field reconstruction techniques and identify an independent basis of special functions that allows an analytic subtraction of the infrared and ultraviolet poles. This basis is valid for all planar topologies for five-particle scattering with an off-shell leg.Comment: 7 pages, 3 figures, 1 table. v2 references added. v3 correction to the parity odd contribution. v4 fix normalisation of parity odd contributio

Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED

We compute the two-loop QED helicity amplitudes for the scattering of a lepton pair with an off-shell and an on-shell photon, $0\to\ell\bar\ell\gamma\gamma^*$, using the approximation of massless leptons. We express all master integrals relevant for the scattering of four massless particles with a single external off-shell leg up to two loops in a basis of algebraically independent multiple polylogarithms, which guarantees an efficient numerical evaluation and compact analytic representations of the amplitudes. Analytic forms of the amplitudes are reconstructed from numerical evaluations over finite fields. Our results complete the amplitude-level ingredients contributing to the N$^3$LO predictions of electron-muon scattering $e\mu\to e\mu$, which are required to meet the precision goal of the future MUonE experiment.Comment: 16 + 13 pages, 2 figures, 2 table

Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED

We compute the two-loop QED helicity amplitudes for the scattering of a lepton pair with an off-shell and an on-shell photon, 0 → ℓℓ¯γγ*, using the approximation of massless leptons. We express all master integrals relevant for the scattering of four massless particles with a single external off-shell leg up to two loops in a basis of algebraically independent multiple polylogarithm, which guarantees an efficient numerical evaluation and compact analytic representations of the amplitudes. Analytic forms of the amplitudes are reconstructed from numerical evaluations over finite fields. Our results complete the amplitude-level ingredients contributing to the N3LO predictions of electron-muon scattering eμ → eμ, which are required to meet the precision goal of the future MUonE experiment

Two-loop leading-colour QCD helicity amplitudes for Higgs boson production in association with a bottom-quark pair at the LHC

We compute the two-loop QCD helicity amplitudes for the production of a Higgs boson in association with a bottom quark pair at a hadron collider. We take the approximations of leading colour and work in the five flavour scheme, where the bottom quarks are massless while the Yukawa coupling is non-zero. We extract analytic expressions from multiple numerical evaluations over finite fields and present the results in terms of an independent set of special functions that can be reliably evaluated over the full phase space

Flavour anti-kTk_\text{T} algorithm applied to WbbˉWb\bar{b} production at the LHC

We apply the recently proposed flavoured anti-$k_{\text{T}}$ jet algorithm to $Wb\bar{b}$ production at the Large Hadron Collider at $\sqrt{s}=8$ TeV. We present results for the total cross section and differential distributions at the next-to-next-to-leading order (NNLO) in QCD. We discuss the effects of the remaining parametric freedom in the flavoured anti-$k_{\text{T}}$ prescription, and compare it against the standard flavour-$k_{\text{T}}$ algorithm. We compare the total cross section results against the CMS data, finding good agreement. The NNLO QCD corrections are significant, and their inclusion substantially improves the agreement with the data.Comment: 12 Figures, 19 page

DD-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare $D$-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic $D$-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.Comment: 35 pages, 2 figures, 2 appendices; comments welcom