Decay time integrals in neutral meson mixing and their efficient evaluation

TL;DR

This paper presents analytic solutions and efficient numerical routines for decay time integrals involving the error function in neutral meson mixing, improving speed and accuracy over previous methods and integrating into the RooFit package.

Contribution

It introduces new analytic expressions and C++ routines for evaluating complex error function integrals in meson mixing, enhancing computational efficiency and accuracy.

Findings

Derived general analytic solutions for decay time integrals

Developed improved C++ routines for the Faddeeva function

Integrated routines into the RooFit package since ROOT 5.34/08

Abstract

In neutral meson mixing, a certain class of convolution integrals is required whose solution involves the error function $\mathrm{erf}(z)$ of a complex argument $z$. We show the the general shape of the analytic solution of these integrals, and give expressions which allow the normalisation of these expressions for use in probability density functions. Furthermore, we derive expressions which allow a (decay time) acceptance to be included in these integrals, or allow the calculation of moments. We also describe the implementation of numerical routines which allow the numerical evaluation of $w(z)=e^{-z^2}(1-\mathrm{erf}(-iz))$, sometimes also called Faddeeva function, in C++. These new routines improve over the old CERNLIB routine(s) WWERF/CWERF in terms of both speed and accuracy. These new routines are part of the RooFit package, and have been distributed with it since ROOT version 5.34/08.