LToolsEvaluate[expr, q] evaluates Passarino-Veltman
functions in expr numerically using LoopTools by T.
Hahn.
In contrast to the default behavior of LoopTools, the function
returns not just the finite part but also the singular pieces
proportional to 1\varepsilon and 1\varepsilon^2. This behavior is controlled
by the option LToolsFullResult.
Notice that the normalization of Passarino-Veltman functions differs between FeynCalc and LoopTools, cf. Section 1.2 of the LoopTools manual. In FeynCalc the overall prefactor is just 1/(i \pi^2), while LoopTools employs 1/(i \pi^{D/2} r_{\Gamma}) with D = 4 -2 \varepsilon and r_{\Gamma} = \Gamma^2 (1 - \varepsilon) \Gamma (1 + \varepsilon) / \Gamma(1-2 \varepsilon).
When the option LToolsFullResult is set to
True, LToolsEvaluate will automatically
account for this difference by multiplying the LoopTools output with
1/\pi^{\varepsilon} r_\Gamma.
However, for LToolsFullResult -> False no such
conversion will occur. This is because the proper conversion between
different \varepsilon-dependent
normalizations requires the knowledge of the poles: when terms
proportional to \varepsilon multiply
the poles, they generate finite contributions. In this sense it is not
recommended to use LToolsEvaluate with
LToolsFullResult set to False, unless you
precisely understand what you are doing.
Overview, LToolsExpandInEpsilon, LToolsFullResult, LToolsImplicitPrefactor, LToolsSetMudim, LToolsSetLambda.
Before using LToolsEvaluate we need to evaluate
LToolsLoadLibray[] to establish a connection with the
Mathlink executable. The value of the option LToolsPath
contains the full path to this file. You might need to adjust it
accordingly if LoopTools is installed in a different directory.
OptionValue[LToolsLoadLibrary, LToolsPath]\text{/home/vs/.Mathematica/Applications/FeynCalc/AddOns/FeynHelpers/ExternalTools/LoopTools/LoopTools}
Notice that LToolsEvaluate can also load the library by
itself on the first run, in case you forget to do so.
LToolsLoadLibrary[]\text{LoopTools library loaded.}
(* ====================================================
FF 2.0, a package to evaluate one-loop integrals
written by G. J. van Oldenborgh, NIKHEF-H, Amsterdam
====================================================
for the algorithms used see preprint NIKHEF-H 89/17,
'New Algorithms for One-loop Integrals', by G.J. van
Oldenborgh and J.A.M. Vermaseren, published in
Zeitschrift fuer Physik C46(1990)425.
====================================================*)The value of the scale \mu can be
set via the option LToolsSetMudim. Evaluating the
PaVe-function A_0(m^2) at
m^2 = 1/10 with \mu^2=20 yields
LToolsEvaluate[A0[m^2], LToolsSetMudim -> 20, InitialSubstitutions -> {m^2 -> 1/10}]0.457637\, +\frac{0.1}{\varepsilon }
Cross-checking this result with Package-X yields the same value
(PaXEvaluate[A0[1/10]] /. ScaleMu^2 -> 20) // N0.457637\, +\frac{0.1}{\varepsilon }
More complicated input is also possible
exp = a FAD[{q, m}, q - p] + b FAD[{q, M, 2}]\frac{a}{\left(q^2-m^2\right).(q-p)^2}+\frac{b}{\left(q^2-M^2\right)^2}
res = LToolsEvaluate[exp, q, InitialSubstitutions -> {m -> 0.12352, M -> 5.14321, SPD[p] -> 0.8813}, LToolsImplicitPrefactor -> 1/(2 Pi)^(4 - 2 Epsilon), LToolsSetMudim -> 23^2]\frac{(0.\, +0.00633257 i) ((1.\, +0. i) a+(1.\, +0. i) b)}{\varepsilon }+(0.0661028\, +0.01955 i) ((0.\, +1. i) a+(0.128951\, +0.436013 i) b)
Compare to Package-X
chk = (PaXEvaluate[exp, q, PaXImplicitPrefactor -> 1/(2 Pi)^(4 - 2 Epsilon)] /. {ScaleMu^2 ->23^2, m -> 0.12352, M -> 5.14321, FCI@SPD[p] -> 0.8813}) // N\frac{(0.\, +0.00633257 i) (a+b)}{\varepsilon }-(0.\, +0.00633257 i) (-14.4075 a-4.94944 b)+(-0.01955-0.0251337 i) a
FCCompareNumbers[res, chk]\text{FCCompareNumbers: Minimal number of significant digits to agree in: }6
\text{FCCompareNumbers: Chop is set to }\;\text{1.$\grave{ }$*${}^{\wedge}$-10}
\text{FCCompareNumbers: No number is set to 0. by Chop at this stage. }
0