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Numerical Perturbation Theory
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Europe/Zurich
E41.1 (ETHZ)
E41.1
ETHZ
ETH Honggerberg
Stefano-Franscini-Platz 5, 8093 Zürich
Charalampos Anastasiou,
Valentin Jonathan Hirschi
(SLAC)
Description
One-day workshop at ETHZ following the QCD Factorization Symposium and covering presentations from more junior researchers regarding practical implementations and developments of numerical methods to tackle computations in perturbative Quantum Field Theories.
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Morning 1
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1
Applications and challenges in numerical methods for pQFT
I will briefly present a recent application of Local Unitarity to the calculation of the NNLO QCD correction to $\gamma \gamma \rightarrow Q \bar{Q}$.
Local Unitarity is a computational technique achieving the local cancellation of infrared singularities at the integrand level, unlocking the computation of perturbative cross-section fully numerically in momentum space.
This will set me up for giving a more general overview of the challenges that such numerical methods face, especially regarding the various types of singularities requiring regularisation, and what solutions are currently being considered.
Speaker: Valentin Jonathan Hirschi (University of Bern) -
2
The art of constucting and sampling cut integrals
This talk will introduce new Monte Carlo sampling techniques used for the numerical integration of momentum-space loop integrals in the Loop-Tree Duality (LTD) framework.
I will then discuss the factorization property of the Cross-Free-Family (CFF) representation of LTD and how it can be used to conveniently construct three-dimensional integrands of cut diagrams.
Speaker: Mathijs Fraaije (University of Bern) -
3
Localized renormalization and graph-based tooling
The BPHZ formalism provides a systematic framework for subtracting ultraviolet divergences from Feynman graphs. Originally conceived as a purely theoretical construct, it now underpins the ultraviolet regularization strategy for Local Unitarity, as implemented in gammaloop, an upcoming program designed to compute perturbative cross-sections fully numerically in momentum space.
In this talk, I will present our refined variant of the BPHZ subtraction procedure, tailored such that the resulting integrand is directly compatible with the MSbar+OS renormalization schemes. I will also outline the supporting infrastructure, including graph-level manipulations, numerator algebra, and computer algebra system integration that renders the implementation both conceptually transparent and computationally efficient.
Speaker: Lucien Huber (University of Bern)
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1
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10:15
Coffee break
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Morning 2
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4
Off-shell triphoton production via quark loops: Threshold subtraction and multi-channel Monte Carlo
I will give an overview of our approach for the direct numerical integration of two-loop corrections to electroweak production. After subtracting infrared and ultraviolet singularities, we enable numerical integration in momentum space through analytic integration over the energy components of the loop momenta, subtraction of threshold singularities, and the use of importance sampling with the multi-channel Monte Carlo method. These features will be illustrated with examples from our recent calculations of fermionic two-loop corrections to di- and triboson production at the LHC.
Speaker: Dario Kermanschah (ETH Zürich) -
5
Colorless final-state production via $q\bar{q}$ annihilation at two-loops: the fermion loop contributions and beyond
We present recent advances in the numerical integration of two-loop scattering amplitudes for the production of 2 and 3 vector bosons via quark-antiquark annihilation. The numerical integration is performed in loop momentum space in four spacetime dimensions. This is made possible by the simultaneous local subtraction of infrared, ultraviolet, and threshold singularities (more details are given in Dario Kermanschah's presentation). In this talk, I focus on the recent computation of the fermion loop contributions and share ongoing progress toward obtaining fully numerical results for the complete two-loop amplitude.
Speaker: Matilde Vicini (ETH Zürich)
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4
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12:00
Lunch
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Afternoon 1
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6
Factorisable Local Unitarity for Hadronic Cross-Sections
The KLN theorem establishes that infrared divergences in parton‑model diagrams cancel when summed alongside diagrams that account for the simultaneous hard interaction of multiple partons within the same hadron. Meanwhile, it is well established that initial-state infrared poles and logarithms are governed by collinear factorisation. In this talk, I introduce a formalism where an initial-state implementation of KLN works hand in hand with collinear factorisation, providing potential grounds for the application of the Local Unitarity method to the computation of hadronic cross sections.
Speaker: Zeno Capatti (University of Bern) -
7
Loop–Tree Duality at Finite Temperature and Density
In finite-temperature or finite-density QFT, computing perturbative coefficients is notoriously challenging because the thermal bath breaks Lorentz symmetry by singling out the time direction, making standard integration-by-parts and differential-equation techniques inefficient. I will present the generalization of Loop–Tree Duality (LTD) to thermal settings, in which loop-energy (Matsubara) sums are carried out analytically and the remaining integrals over spatial loop momenta are evaluated efficiently with Monte Carlo methods. Since LTD already isolates the energy component of four-vectors, its extension to nonzero temperature and chemical potential is natural and retains much of the structure of the vacuum formulation. I will outline an automated pipeline for the numerical evaluation of thermal multi-loop integrals, including local ultraviolet subtraction, a general formula for Matsubara summation, and benchmarks against known analytical results. Applications span the cold and dense QCD equation of state relevant for neutron-star phenomenology and the computation of thermal effective potentials for cosmological phase transitions, among others.
Speaker: Kaapo Elmo Ilmari Seppänen (University of Bern) -
8
Soft Factorisation and Exponentiation from Schwinger-Space Geometry
In this talk I will report on a recent analysis of soft divergences of entire scattering amplitudes within the Schwinger-parameter representation of Feynman integrals. Using tools from tropical geometry and graph theory, I will show how soft and hard contributions factorise directly in parameter space and how “worldline” Schwinger variables make exponentiation of soft divergences manifest at the integrand level.
Speaker: Giulio Gambuti (University of Oxford)
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6
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14:45
Coffee Break
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Afternoon 2
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9
bbH production: a theoretical laboratory for flavour-scheme studies
We present NNLO QCD predictions for Higgs boson production in association with a bottom-quark pair, matched to parton showers. This process provides a unique testing ground for precision QCD and flavour-scheme studies, as the presence of bottom quarks makes it suitable for investigating mass effects and factorisation schemes. In the collinear heavy-quark factorisation approach (5FS), logarithmically enhanced terms in the bottom-quark mass are resummed to all orders, while in the decoupling scheme (4FS) power-suppressed mass effects are fully retained. We compare NNLO+PS predictions obtained in the two schemes and discuss strategies to combine them consistently at the fully differential level for fixed-order predictions in a general-mass variable flavour number scheme (GM-VFNS).
Speaker: Christian Biello (Max-Planck Institute for Physics) -
10
Locally finite two-loop amplitudes for photoproduction in quark annihilation
Precise predictions for increasingly complex processes at higher perturbative orders call for a process-independent algorithm that can numerically integrate multi-loop amplitudes directly in momentum space. However, Feynman amplitudes are plagued by infrared and ultraviolet singularities which must be removed at the integrand level before numerical integration can be performed. In recent years, substantial progress has been made toward a framework that constructs locally finite two-loop amplitudes for a broad class of processes by making the factorization of infrared singularities manifest at the integrand level. In a previous work, two-loop QCD amplitudes for colorless, off-shell production processes initiated by quark-antiquark annihilation was rendered locally finite. In this work, we present an extension of this approach to processes involving real photons in the final state. Such final states introduce new transient singularities that vanish upon integration but still require local subtraction. We outline the methods developed to remove these singularities, preparing the amplitude for numerical integration. This work is a significant step toward applying factorization-based IR subtraction methods to a wider range of processes, particularly those involving final-state jets.
Speaker: Roshni Sahoo -
11
Locally finite QCD amplitudes and cross sections
In this talk I will present a framework for local subtractions of infrared singularities in hadron collider cross sections and amplitudes. The construction is guided by infrared factorization and leads to integrands in loop momentum and phase space that are suitable for integration via numerical methods.
I will focus on electroweak production via quark–antiquark annihilation at NNLO. To achieve infrared factorization for the two-loop amplitude at the local level, a careful construction of the integrand is necessary. Modifications are introduced to cure “loop polarizations” and power-like divergences, which arise in collinear limits from loop corrections to initial-state vertices and obstruct local factorization. The tools developed for amplitude integrands also guide the local subtractions at the cross section level, where initial-state singularities are factorized locally at the level of partonic integrands, while final-state singularities cancel in the sum of real and virtual contributions. All required subtraction terms are predominantly constructed from lower-order building blocks with known analytic integrals.
Speaker: Julia Karlen (ETH Zurich) -
12
Numerical Evaluation of the Two-Loop Dark Matter Power Spectrum in the EFTofLSS
This talk focuses on the application of modern numerical perturbative methods, familiar from collider physics, to the precision frontier of cosmology. To match the accuracy of modern cosmological surveys, theoretical predictions from the Effective Field Theory of Large-Scale Structure (EFTofLSS) must be pushed to the two-loop level, a task for which purely numerical techniques are essential due to the absence of known analytic master integrals. I will present a robust numerical framework for computing these two-loop corrections. Its efficiency for cosmological parameter scans is achieved by decoupling the cosmology dependence from the loop integrations. We ensure the stability of the multi-dimensional Monte Carlo integration by treating UV and IR singularities locally, at the integrand level. The core of the method is to pre-compute a set of universal tensor integrals, which are then combined to produce predictions for any specific cosmology. This framework allows us to compute the renormalized two-loop dark matter power spectrum, which requires 8 counterterms. Our work demonstrates a successful application of numerical perturbative techniques to cosmology, enabling precision parameter inference and establishing the foundation for two-loop computations of other observables, like the galaxy power spectrum.
Speaker: Andrea Favorito (ETH Zürich)
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9
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