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Testing and improving the numerical accuracy of the NLO predictions

R. Pittau

TL;DR

Problem: numerical inaccuracies in NLO predictions using $OPP$/Generalized Unitarity. Approach: an event-by-event precision estimator $E^A=|A-A'|/|A|$ based on an independent reconstruction of the 1-loop amplitude via a mass-shift trick, plus a rescue procedure that re-fits at higher precision while keeping $N(q)$ in double. Key contributions: a practical method to test and improve numerical accuracy, a faster determination of the $R_1$ term, and numerical validation showing most unstable points can be rescued with modest extra cost. Impact: provides robust, scalable numerical stability checks for NLO codes using modern unitarity methods.

Abstract

I present a new and reliable method to test the numerical accuracy of NLO calculations based on modern OPP/Generalized Unitarity techniques. A convenient solution to rescue most of the detected numerically inaccurate points is also proposed.

Testing and improving the numerical accuracy of the NLO predictions

TL;DR

Problem: numerical inaccuracies in NLO predictions using /Generalized Unitarity. Approach: an event-by-event precision estimator based on an independent reconstruction of the 1-loop amplitude via a mass-shift trick, plus a rescue procedure that re-fits at higher precision while keeping in double. Key contributions: a practical method to test and improve numerical accuracy, a faster determination of the term, and numerical validation showing most unstable points can be rescued with modest extra cost. Impact: provides robust, scalable numerical stability checks for NLO codes using modern unitarity methods.

Abstract

I present a new and reliable method to test the numerical accuracy of NLO calculations based on modern OPP/Generalized Unitarity techniques. A convenient solution to rescue most of the detected numerically inaccurate points is also proposed.

Paper Structure

This paper contains 7 sections, 30 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The method
  3. Rescuing the inaccurate points
  4. Testing the method
  5. Conclusions
  6. Acknowledgments
  7. An alternative determination of $d^{\,(2m-4)}$

Figures (3)

  • Figure 1: Distribution of the ratio between the true precision $P_d= |A_d-A_e|/|A_e|$ and two different precision estimators. The dashed histogram refers to the estimator at the numerator level $E^{X}_d= E^{N}_d= |N_d-N_{d,rec}|/|N_d|$, the solid one to the estimator at the amplitude level $E^{X}_d= E^{A}_d=|A_{d}-A^\prime_{d}|/|A_{d}|$.
  • Figure 2: Distribution of the true precision variables $P_x=|A_x-A_e|/|A_e|$ (see text). The dashed histograms refers to the double precision result ($P_x=P_d$), the solid histogram to the case with fitting procedure carried out in multi-precision, but numerator function computed in double precision ($P_x=P_m$).
  • Figure 3: The tails of the distributions of the true precision variable $P_d$, with an additional constraint on the value of the precision estimators $E^{A}_d$, $E^{A}_m$, $E^{N}_d$ and $E^N_m= |N_d-N_{m,rec}|/|N_d|$ (see text). In the solid histograms, when $E^{A}_d > E_{\rm lim}$, a rescue of the point is performed by re-fitting the 1-loop coefficients in multi-precision (with numerator functions kept in double precision) and, if also $E^{A}_m > E_{\rm lim}$, the event is discarded. In the dashed histograms, the same procedure is applied, but using the estimators $E^{N}_d$ and $E^{N}_m$ instead.