YOUNGST@RS - Loop-the-Loop: Feynman Calculus and its Applications to Gravity and Particle Physics

12 Nov 2024, 09:00 → 14 Nov 2024, 18:00 Europe/Berlin

“Loop-the-Loop: Feynman calculus and its applications to gravity and particle physics” is an online workshop that aims at fostering in-depth discussions and connections among different perspectives in the study of scattering amplitudes.

The workshop is fully online and it takes place on 12-14 November 2024.
Each day of the workshop opens with a review talk delivered by a leading expert in the field and focuses on a different topic.

The three topics of the workshop will be:

• Applied Mathematics for Feynman Calculus
• Scattering Amplitudes in Gravity
• Scattering Amplitudes in Particle Physics
  
  

Additionally, the workshop features two afternoon seminars by leading scientists in the field of gravity and particle physics. 

Organising Committee:

• Giacomo Brunello (University of Padova, IphT-CEA/University of Paris Saclay & INFN-PD)
• Gaia Fontana (University of Zurich)
• Raj Patil (MPI for Gravitational Physics, Potsdam & Humboldt University, Berlin)
• Sid Smith (University of Padova, University of Edinburgh & INFN-PD)

Loop the Loop is also on instagram!

loop-the-loop-poster.png

Tuesday 12 November

1 Welcome

Review: Mathematical methods from scattering amplitudes to modern scientific calculus

abreu_LtL.pdf

2 Mathematical methods from scattering amplitudes to modern scientific calculus

I recall generalised unitarity, integrand decomposition, integration-by-parts identities, differential equations, GKZ-systems and intersection theory, connecting Cauchy's residue theorem, Stokes' theorem and Gauss' linking numbers,
and elaborate on the emerging crucial role played by De Rham co-homology theory, ruling the differential and algebraic properties of the integral functions that appear in the context of fundamental physics.
I will recall recent applications to Feynman and Euler-Mellin integrals in particle physics and cosmology, as well as to quantum mechanics and Wick's theorem in QFT, concluding with the role that Physics Informed Neural Networks may play in the context of (twisted period) integrals evaluation.

Speaker: Pierpaolo Mastrolia

(Pierpaolo Mastrolia) Intersection Theory for Fundamental Interactions.pdf

10:15 Break

Applied Mathematics for Feynman Calculus: Integration-by-parts & Computational Techniques

3 Syzygy method of integration by parts

Integration by parts(IBP) plays an important role in the calculation of Feynman integrals. Numerous Feynman integrals can be reduced to a small amount master integrals, as the simpler integral basis of a given Feynman integral family. Compare with traditional IBP, syzygy method can reduce IBP system redundancy, also make IBP reduction more efficient. I will introduce an IBP package "NeatIBP", which is based on syzygy method. I will also give an example about syzygy IBP with respect to atypical Feynman integrals, such as energy energy correlator integrals.

Speaker: Rourou Ma

(Rourou Ma) Syzygy IBP Method.pdf

4 Two-loop five-point two-mass Feynman integrals

I present the computation of the planar two-loop five-point Feynman integrals with two off-shell external legs. These integrals are relevant, for instance, for the second-order QCD corrections to the production of two heavy vector bosons in association with a jet at hadron colliders. We employ the method of canonical differential equations, boosted by finite-field sampling to tame the algebraic complexity, along with bootstrap techniques to determine the letters.

Speaker: Simone Zoia

zoia_LoopTheLoop.pdf

5 Series expansion approach and application to 2L mixed QCD-EW corrections to Drell-Yan

The calculation of higher order corrections to physical observables is becoming essential for matching the theoretical predictions to the experimental precision at present and future colliders. One of the main bottleneck for their evaluation is the computation of the Feynman integrals appearing in the amplitude, due to the number of loops, external legs or internal masses.
In this talk I will report on recent progresses on the application of semi-analytical methods to this class of problems. In particular, I will present the Mathematica package SeaSyde. The package implements an original algorithm for performing the analytic continuation of the result, which makes it suitable for integrals with internal massive lines and with the masses being in general complex-valued.
As a non-trivial application of this technology, I will present the results for the computation of the two-loop mixed strong-electroweak virtual corrections to the charged current Drell-Yan process. I will show the details of our calculation, performed via semi-analytical methods, with an emphasis on the evaluation of all the relevant two-loop Feynman integrals where the presence of one additional mass, compared to the neutral current case, makes the computation extremely challenging. Finally, I will explain how the semi-analytical approach can be exploited to reconstruct a posteriori the W-mass dependance of the result.

Speaker: Tommaso Armadillo

(Tommaso Armadillo) Series expansion approach and application to 2L mixed corrections to Drell-Yan.pdf

12:00 Lunch Break

Applied Mathematics for Feynman Calculus: Intersection Theory

6 Intersecting companion matrices for Feynman integrals

Twisted period integrals play an essential role in theoretical physics and
mathematics, residing in a finite-dimensional vector space with an inner
product known as the intersection number. In this talk, we explore the emerging
tensor structures in intersection numbers within the fibration-based evaluation
scheme. By introducing companion matrices, we reformulate the computation of
intersection numbers using matrix-valued operators. Our algorithm
enables a complete decomposition of two-loop five-point massless functions,
marking a significant advance in projecting Feynman integrals to master
integrals via intersection numbers.

Speaker: Vsevolod Chestnov

talk.pdf

7 Twisted Cohomology & Canonical Differential Equations

Interpreting Feynman integrals as periods of (relative) twisted cohomology groups has lead to many interesting and fruitful insight. Recently we explored, what one can learn from the twisted Riemann bilinear relations that these periods satisfy. This led to further insights on a notion of self-duality for maximal cuts and specifically, the form of the intersection matrix for a canonical basis choice. I will also explain, how these insights can be used to construct a canonical basis for a hyperelliptic integral family.

Speaker: Franziska Porkert

FP_Loop_the_loop.pdf

8 Recent advancements and hidden structures in intersection numbers for Feynman Integrals

The intersection number is an inner product that can be defined on the vector space of Feynman Integrals, allowing one to perform integral reductions without using IBPs. In this talk I will review the main ideas behind intersection theory, and present some new computational and advancements in the field. Specifically, I will showcase a prescription for choosing orthogonal bases of differential n-forms belonging to quadratic twisted period integrals, as well as a new closed formula to evaluate intersection numbers beyond dlog forms. These findings allow us to systematically construct orthonormal bases between twisted periods for all one-loop Feynman integrals.

Speaker: Giulio Crisanti

GC_loop_the_loop.pdf

15:00 Break

Applied Mathematics for Feynman Calculus: Analytic Structure of Feynman Integrals

9 Systematically evaluate cosmological correlators by IBP and differential equations

We generalize the Integration-By-Parts (IBP) and IBP-based differential equations method to general cosmological correlator, including massive correlator and time-derivative interactions cases, and find the factorization property from the perspective of IBP. By our selection of master integrals, we find uniform formulas for iterative reduction and dlog-form differential equations of arbitrary tree-level cosmological correlators. Solving the dlog-form differential equations by power series expansion, using blow-up of differential equations and the factorization property, we get the homogeneous part solutions of arbitrary tree-level cosmological correlators. We will also provide an example of solving a non-homogeneous solution, which is straightforward as well. Finally, we will discuss the possibility of applying these techniques in multi-loop calculation in flat QFT.

Speaker: Jiaqi Chen

2411 dS IBP&DE - EN Jiaqi Chen.pdf

10 Minimal Cuts and Genealogical Constraints on Feynman Integrals

We introduce an efficient method for deriving hierarchical constraints on the discontinuities of individual Feynman integrals. This method can be applied at any loop order and particle multiplicity, and to any configuration of massive or massless virtual particles. The resulting constraints hold to all orders in dimensional regularization, and complement the extended Steinmann relations -- which restrict adjacent sequential discontinuities -- by disallowing ordered pairs of discontinuities from appearing even when separated by (any number of) other discontinuities. We focus on a preferred class of hierarchical constraints, which we refer to as genealogical constraints, that govern what singularities can follow from certain minimal cuts that act as the primogenitors of the discontinuities that appear in Feynman integrals. While deriving the full set of hierarchical constraints on a given Feynman integral generally requires identifying all solutions to the (blown up) Landau equations, these genealogical constraints can be worked out with only minimal information about what singularities may appear. We illustrate the power of this new method in examples at one, two, and three loops, and provide evidence that genealogical constraints restrict the analytic structure of Feynman integrals significantly more than the extended Steinmann relations.

Speaker: Maria Polackova

11 Progress in Landau Analysis

We review a recent recursive unitary-based workflow that opens the path further to a systematic treatment of Landau singularities for n-point scattering amplitudes in perturbation theory. For two-particle cut reducible diagrams, our method allows for fast computations of Landau singularities beyond the current state-of-the-art, including examples relevant to two- and higher-loop massive Standard Model processes involving QCD and electroweak particles. If time permits, I will also briefly discuss ongoing work, addressing questions related to alpha-positivity and extensions to diagrams beyond two-particle cuts.

Speaker: Mathieu Giroux

(Mathieu Giroux) Progress in Landau Analysis.pdf

17:00 Break

Afternoon Seminar: Finite Feynman Integrals

12 Finite Feynman Integrals

Speaker: David Kosower

Wednesday 13 November

Review: Classical black-hole scattering from worldline quantum field theory at highest precision

abreu_LtL.pdf

13 Classical black-hole scattering from worldline quantum field theory at highest precision

Predicting the outcome of scattering processes of elementary particles in colliders is the central
achievement of relativistic quantum field theory applied to the fundamental (non-gravitational)
interactions of nature. While the gravitational interactions are too minuscule to be observed
in the microcosm, they dominate the interactions at large scales. As such the inspiral and merger of
black holes and neutron stars in our universe are now routinely observed by gravitational wave
detectors. The need for high precision theory predictions of the emitted gravitational waveforms
has opened a new window for the application of perturbative quantum field theory techniques to the domain
of gravity. In my talk I will review the worldline quantum field theory formalism that presents a highly
efficient means to compute scattering observables up to very high orders in the post-Minkowskian expansion,
the present state of the art being at the 5PM order. If time permits I will also discuss the inclusion of
Spin through a supersymmetric particle formalism.

Speaker: Jan Plefka

10:15 Break

Scattering Amplitudes in Gravity: Extreme Mass Ratio Systems

14 Solving Einstein equation using recursions

I’ll discuss solving the perturbative Einstein equation using recursions and the iterative structure of the loop integrals. Using this method, I’ll show how to derive the Schwarzschild metric to all orders in Newton's constant. It is a first all-order perturbative computation in Einstein gravity with a matter source. I'll also discuss the generalization to the binary black holes.

Speaker: Kanghoon Lee

(Kanghoon Lee) Solving Einstein Equations using Recursions.pdf

15 Schwarzschild geodesics from Scattering Amplitudes to all orders in G

In this talk, we will discuss the application of the EFT framework to the study of the gravitational two-body problem in the Self-Force (SF) expansion. After reviewing some basic ingredients of worldline EFT formalism, we will introduce an SF-EFT framework. Finally, a complete computation at 0SF starting from flat background will be presented, resulting from an explicit resummation of infinite “loop” contributions in G.

Speaker: Stavros Mougiakakos

Mainz 131124.pdf

16 Effective field theory in a mass ratio expansion

Field theoretic approaches, perfectly tailored for computations perturbative in Newton’s constant, have had great success in producing state-of-the-art results for the gravitational two-body problem. On the other hand, metrics such as the Schwarzschild metric and geodesic trajectories in these spacetimes carry information to all orders in the gravitational constant and can be interpreted as resummations of infinite classes of Feynman diagrams. In this talk, I will describe recent work that seeks to leverage the all-orders-in-G information encoded in classical solutions to simplify post-Minkowskian computations. Dynamics beyond the leading order in mass ratio are described by an effective field theory with a systematic mass ratio expansion. The EFT naturally provides “recoil operators” that encode the back-reaction caused by the effect of the test body on the background.

Speaker: Nabha Shah

(Nabha Shah) Effective Field Theory in a Mass Ratio Expansion.pdf

12:00 Lunch Break

Scattering Amplitudes in Gravity: Post-Minkowskian Integrals

17 How to compute Post-Minkowskian integrals

The Post-Minkowskian EFT (PMEFT) description of the gravitational two-body problem features a set of Feynman-like integrals that differ from those usually seen in particle physics. These defining properties require a retooling of the highly developed particle physics integration machinery. Some of these differences make your life difficult while others allow you to cut some corners or optimize/specialize a given tool. I will discuss how these properties feature in our integration toolchain and their importance to overcome the computational bottlenecks on our journey into a high-precision gravitational wave era.

Speaker: Gregor Kälin

18 Analytic Waveforms from Scattering Amplitudes

The central theme of the talk is the direct computation of gravitational waveform produced by the scattering of two black holes through scattering-amplitude methods in the post-Minkowskian expansion. I will discuss how the results of Scattering Amplitude computations match the analytic results from General Relativity in the small-velocity expansion and a novel computational strategy to obtain analytic results to all orders in velocity.

Speaker: Stefano De Angelis

19 Black hole scattering at fifth Post-Minkowskian and first self force order

I will discuss the calculation of the impulse at fifth Post Minkowksian order at first self force order including both conservative and dissipative effects in the Worldline Quantum Field Theory formalism. The difficulty in this 4 loop calculation lies in the appearing Feynman integrals. On the one hand the IBP reduction is a major technical obstacle to overcome. On the other hand periods over a Calabi-Yau threefold contribute for a first time to a physical observable. I will discuss how to effectively handle this new type of functions using the method of differential equations.

Speaker: Benjamin Sauer

(Benjamin Sauer) Black Hole Scattering at fifth PM and first self-force order.pdf

15:00 Break

Scattering Amplitudes in Gravity: Cosmological Correlators

20 Differential Equations for Cosmological Correlators

In this talk, I will discuss differential equations for cosmological correlators in FRW spacetimes. These correlators have an integral representation in boundary kinematic space, which can be obtained from a finite set of master integrals that satisfy interesting differential equations. I will describe a graphical representation of these differential equations and show how they can be derived from simple graphical rules. This allows us to reformulate bulk time evolution as energy flow on boundary kinematics, providing a conceptually new way of understanding cosmological correlators.

Speaker: Hayden Lee

(Hayden Lee) Differential Equations for Cosmological Correlators.pdf

21 A physical basis for cosmological correlators from cuts

Significant progress has been made in our understanding of the analytic structure of FRW wavefunction coefficients, facilitated by the development of efficient algorithms to derive the differential equations (DEQ) they satisfy. Moreover, recent findings indicate that the twisted cohomology framework tied to the hyperplane arrangement of these twisted integrals overestimates the number of master integrals required for defining the physical DEQ system. In this talk, we show that the associated dual cohomology is automatically organized in a way that is ideal for understanding and exploiting the cut/residue structure of the FRW wavefunction coefficients. Specifically, the hyperplane arrangement of these FRW integrals is non-generic due to linear relations obeyed by the hyperplane polynomials. Utilizing this understanding, we develop a systematic approach to organize compatible sequential residues, which dictate the physical basis for any graph. The talk will focus on the reduction from 25 to 16 physical integrals in the case of the 3-site tree graph, with a brief discussion on extending this analysis to arbitrary n-site, ℓ-loop graphs via universal graphical rules.

Speaker: Shounak De

MITP Talk (Shounak).pdf

22 On the loop structure of the wavefunction of the universe

In this talk we discuss a systematic analysis to extract the asymptotic behaviour of perturbative contributions to observables in power-law FRW cosmologies, indistinctly the cosmological wavefunction and correlators, as well as one loop corrections to the wavefunction. The perturbative contributions to an observable can be expressed as an integral of the canonical function associated to cosmological polytopes. We discuss how the asymptotic behaviour of these integrals is governed by a special class of nestohedra living in the graph-weight space, both at tree and loop level. We will show the realization of these nestohedra and how they can inform us about all the possible directions (IR and UV), where the integral can diverge as well as their degree of divergence. Then, we will show how this combinatorial formulation makes straightforward the application of sector decomposition for extracting divergences from the integral.
Finally, we will discuss corrections to the one loop wavefunction.

Speaker: Francisco Vazão

(Francisco Vazao) On one-loop corrections to the Bunch-Davies wavefunction of the universe.pdf

17:00 Break

Afternoon Seminar: Scattering Amplitudes, Post-Minkowskian Gravity and Binary Black Holes

23 Scattering Amplitudes, Post-Minkowskian Gravity and Binary Black Holes

Gravitational wave signals from coalescing binary black holes are detected, and analyzed, by using large banks of template waveforms. The construction of these templates makes an essential use of the analytical knowledge of the motion and radiation of gravitationally interacting binary systems. We shall review some of the results concerning the dynamics and the gravitational-wave emission of scattering black holes recently obtained by scattering-amplitude computations, and/or by classical post-Minkowskian gravity computations.

Speaker: Thibault Damour

Thursday 14 November

Review: An overview of new(ish) tools for the calculation of scattering amplitudes

abreu_LtL.pdf

24 An overview of new(ish) tools for the calculation of scattering amplitudes

Speaker: Samuel Abreu

10:15 Break

Scattering Amplitudes in Particle Physics: Amplitudes & Elliptics

25 Two-loop massive corrections for diphoton production in gluon fusion

We consider the two-loop helicity amplitudes for photon pair
production in gluon fusion with a loop of heavy quark. We adopt a
procedure to obtain an $\epsilon$-factorised form for the full system of
MIs, also for those whose analytic structure involves elliptic functions. We discuss the
simplifications at amplitude and integrand level that can be achieved
by expressing the amplitude in terms of iterated integrals.

Speaker: Federico Coro

26 ε-form for Feynman integrals with Calabi-Yau geometry

ε-factorized differential equations are needed to solve Feynman integrals order by order in ε. In this talk, I will first introduce geometries beyond multiple polylogarithms, and then discuss how to derive an ε-factorized differential equation for integrals with an underlying Calabi-Yau geometry dependent on one or more parameters. I will show some examples where we tested our method and I will finally conclude with some outlooks.

Speaker: Sara Maggio

eps-form for Feynman integrals with CY-geometry.key

27 Towards two-loop QCD corrections to pp → tt̄j

I discuss the status of the computation of two-loop QCD corrections to top quark pair
production in association with a jet at hadron colliders, which represents a bottleneck
in achieving more precise predictions for this process. I briefly introduce cutting-edge
computational techniques that tackle the extreme algebraic and analytic complexities
of two-loop multi-scale helicity amplitudes involving elliptic functions. I present first
evaluations of the helicity amplitudes for the process gg→tt̄j . Notably, ultraviolet
(UV) and infrared (IR) poles are analytically identified and the finite remainder directly
extracted through the introduction of a basis of special functions.

Speaker: Colomba Brancaccio

CB_LoopTheLoop.pdf

12:00 Lunch Break

Scattering Amplitudes in Particle Physics: Differential Equations

28 A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses

In this talk, we present analytic results in terms of real-valued multiple polylogarithms up to order six in the expansion around the dimensional regulator and on physical kinematics for four integral families that are relevant to the production of two off-shell vector bosons with different masses. Our study consists of a ladder-box, a tennis-court, and two reducible ladder-box-like families.

Speaker: Dhimiter Canko

Canko_LTL.pdf

29 Computing form factors in N=4 sYM for Higgs plus jet(s) production

In high precision era of Higgs physics, it is crucial to compute the amplitudes in higher order so that we can test our Standard Model. The maximally supersymmetric Yang-Mills theory is a nice playground having more simplification while it still retains footprint of QCD. In this talk, I will discuss mainly about the techniques to compute the three-point three-loop form factors in N=4 sYM and their analytic properties. Besides, I will discuss the application to Higgs plus jet productions.

Speaker: Jungwon Lim

30 Tackling Apparent Singularities in Calabi-Yau Feynman Integrals

It is well-known that Calabi-Yau geometries make a prominent appearance in the evaluation of Feynman integrals. Going beyond the simplest cases of Calabi-Yau integrals—-specifically the Banana integrals—-I will examine features that require extensions of established methods, particularly in the presence of apparent singularities. I will illustrate these methods through the derivation of an epsilon-factorized differential equation for a four-loop integral connected to the 5PM correction in black hole scattering, which is linked to a Calabi-Yau 3-fold geometry.

Speaker: Sebastian Pögel

slides_Poegel.pdf

15:00 Break

Scattering Amplitudes in Particle Physics: Infrared Subtraction

31 Subtracting IR singularities with a fully general analytic algorithm in massless QCD

This talk presents the construction of the NNLO subtraction formula for the cancellation of IR singularities obtained within the framework of Local Analytic Sector Subtraction. Such general and automated program has been completed for the treatment of unresolved radiation in processes featuring any arbitrarily-populated massless final state. The final outcome is a compact and analytic expression suitable for direct numerical implementation, thus paving the way to the production of relevant phenomenological results.

Speaker: Gloria Bertolotti

LTL_Bertolotti.pdf

32 Numerical Integration of the $N_f$ Virtual Corrections to Triboson Production at NNLO

I will present our recent computation of previously inaccessible $N_f$
virtual contributions at NNLO for the production of three massive vector
bosons. We perform the numerical integration by simultaneously
addressing infrared, ultraviolet, and threshold singularities through
local subtractions in momentum space. I will outline the ongoing work
toward obtaining the complete NNLO virtual contribution.

Speaker: Matilde Vicini

loop_the_loop_2024_matilde_vicini.pdf

33 Nested soft-collinear infrared subtraction

In this talk, I will present the nested soft-collinear approach to the long-standing issue of infrared subtraction at next-to-next-to-leading order in QCD. At this order in perturbation theory, the structure of infrared singularities is rather clear. However, a fully general method for combining virtual and real emission singularities yielding predictions for physical observables is only now starting to be available for generic processes. I will delve into the nested soft-collinear approach from its first implementations for color singlet production to its most recent efforts toward a generalization to more complex processes.

Speaker: Federica Devoto

LoopTheLoop.pdf

34 Conclusions

Loop_the_Loop_Conclusion.key