Efficiency improvements for the numerical computation of NLO corrections

Abstract

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.

Numerical implementation of the loop–tree duality method

“…In particular, infrared singularities of the virtual contribution can be subtracted by using appropriate semi-analytical terms and combine them with the ones stemming from the real corrections to produce finite results [1]. Purely numerical approaches to the integration of loop momenta have been discussed extensively in the literature [2][3][4][5][6][7][8][9][10][11][12][13][14]. The generation of amplitudes and calculation of cross sections at one loop has seen great progress in recent years and algorithmic calculations at NLO have been automated are now considered standardized, based on purely numerical [15][16][17] and a mix of analytical and numerical approaches [18][19][20].…”

The Loop-Tree Duality: Progress Report

“…(28)- (30) in Eq. (27) we directly obtain the duality relation tween one-loop integrals and phase-space integrals:…”

Direct contour deformation with arbitrary masses in the loop

“…Usually the integrand is the integrand of a primitive bare one-loop amplitude minus subtraction terms for the soft, collinear and ultraviolet singularities. Suitable subtraction terms can be found in the literature [11,12,14]. Since the integral is finite, it can be computed in four dimensions.…”

“…Within the numerical approach the ideas of the subtraction method and Monte Carlo integration are carried over to the virtual part. One subtracts from the one-loop amplitude suitable approximation terms for the soft, collinear and ultraviolet singularities [11][12][13][14][15]. The difference is integrable and the integration over the loop momentum can be combined with the integration over the phase space of the final state particles in one Monte Carlo integration.…”