Loop Momentum Space Research Articles

Feynman integrals from positivity constraints

We explore inequality constraints as a new tool for numerically evaluating Feynman integrals. A convergent Feynman integral is non-negative if the integrand is non-negative in either loop momentum space or Feynman parameter space. Applying various identities, all such integrals can be reduced to linear sums of a small set of master integrals, leading to infinitely many linear constraints on the values of the master integrals. The constraints can be solved as a semidefinite programming problem in mathematical optimization, producing rigorous two-sided bounds for the integrals which are observed to converge rapidly as more constraints are included, enabling high-precision determination of the integrals. Positivity constraints can also be formulated for the ϵ expansion terms in dimensional regularization and reveal hidden consistency relations between terms at different orders in ϵ. We introduce the main methods using one-loop bubble integrals, then present a nontrivial example of three-loop banana integrals with unequal masses, where 11 top-level master integrals are evaluated to high precision.

From infinity to four dimensions: higher residue pairings and Feynman integrals

We study a surprising phenomenon in which Feynman integrals in D = 4 − 2ε space-time dimensions as ε → 0 can be fully characterized by their behavior in the opposite limit, ε → ∞. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on ε and are known to be governed by intersection numbers of twisted forms. They give rise to differential equations that can be obtained exactly as a truncating expansion in either ε or 1/ε. We use the latter for explicit computations, which are performed by expanding intersection numbers in terms of Saito’s higher residue pairings (previously used in the context of topological Landau-Ginzburg models and mirror symmetry). These pairings localize on critical points of a certain Morse function, which correspond to regions in the loop-momentum space that were previously thought to govern only the large-D physics. The results of this work leverage recent understanding of an analogous situation for moduli spaces of curves, where the α′ → 0 and α′ → ∞ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

The loop-tree duality at NLO and beyond

We review the recent developments of the loop-tree duality method, focussing our discussion on analysing the singular behaviour of the loop integrand of the dual representation of one-loop integrals and scattering amplitudes. We show that within the loop-tree duality method there is a partial cancellation of singularities at the integrand level among the different components of the corresponding dual representation. The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to the phase-space of real corrections to cancel the soft and collinear divergences.

On the singular behaviour of scattering amplitudes in quantum field theory

We analyse the singular behaviour of one-loop integrals and scattering amplitudes in the framework of the loop--tree duality approach. We show that there is a partial cancellation of singularities at the loop integrand level among the different components of the corresponding dual representation that can be interpreted in terms of causality. The remaining threshold and infrared singularities are restricted to a finite region of the loop momentum space, which is of the size of the external momenta and can be mapped to the phase-space of real corrections to cancel the soft and collinear divergences.

Direct numerical integration for multi-loop integrals

We present a method to construct a suitable contour deformation in loop momentum space for multi-loop integrals. This contour deformation can be used to perform the integration for multi-loop integrals numerically. The integration can be performed directly in loop momentum space without the introduction of Feynman or Schwinger parameters. The method can be applied to finite multi-loop integrals and to divergent multi-loop integrals with suitable subtraction terms. The algorithm extends techniques from the one-loop case to the multi-loop case. Examples at two and three loops are discussed explicitly.

Direct contour deformation with arbitrary masses in the loop

We present a method, which constructs a suitable deformation vector in loop momentum space, when the loop integration is done numerically with the help of the subtraction method. The method presented here extends previously discussed techniques from the massless case to the general case of arbitrary masses in the loop.

Efficiency improvements for the numerical computation of NLO corrections

In this paper we discuss techniques, which lead to a significant improvement of the efficiency of the Monte Carlo integration, when one-loop QCD amplitudes are calculated numerically with the help of the subtraction method and contour deformation. The techniques discussed are: holomorphic and non-holomorphic division into sub-channels, optimisation of the integration contour, improvement of the ultraviolet subtraction terms, importance sampling and antithetic variates in loop momentum space, recurrence relations.

Scattering amplitudes in high energy QCD

We develop a new systematic procedure for the Regge limit in perturbative QCD to arbitrary logarithmic order. The formalism relies on the IR structure and the gauge symmetry of the theory. We identify leading regions in loop momentum space responsible for the singular structure of the amplitudes and perform power counting to determine the strength of these divergences. Using a factorization procedure introduced by Sen, we derive a sum of convolutions in transverse momentum space over soft and jet functions, which approximate the amplitude up to power-suppressed corrections. A set of evolution equations generalizing the BFKL equation and controlling the high energy behavior of the amplitudes to arbitrary logarithmic accuracy is derived. The general method is illustrated in the case of leading logarithmic gluon reggeization and BFKL equation.