Package: coefhl.m
Version: v1.0 (Feb. 2011)
Description:
coefhl is a Mathematica package for the evaluation of
the moments of the heavy-light correlators to three loops.
Authors: Jens Hoff and Matthias Steinhauser
Reference: arXiv:1103.1481v1
(* ---------- *)
The Mathematica function Cbar[case, aOrd, zOrd, xd, OPTIONS]
provides the moments of the heavy-light correlators where
the arguments have the following meaning:
case: "v", "a", "s", "p" for vector, axial-vector, scalar,
pseudo-scalar correlator
aOrd: the correction of O(alpha_s^aOrd) is returned
for aOrd = 0, 1 or 2
zOrd: the moment of z^zOrd is returned: zOrd=-1,...,4
z=q^2/m1^2 where q is the external momentum and m1 is the
heavier of the two quark masses.
xd: (optional parameter) corresponds to x=m2/m1. In case numerical
values are given it is assumed that 0<= xd <= 1.
OPTIONS: It is possible to set the following options (default values):
nl (-> 3) - the number of light flavours,
imu (-> 1) - the ratio of the renormalization scale mu and m1,
warn (-> True) - a flag for suppressing warning messages,
range (-> {0.1,0.5}) - the interval inbetween which the predefined
interpolation is used.
Note: For the optional parameters to work properly xd must be set
explicitely.
At one- and two-loop order the analytic results are used to evaluate
Cbar[case, aOrd, zOrd]. At three-loop order Cbar[case, aOrd, zOrd]
is a piecewise defined function where for x <= x1 and x >= x2
analytical expansions are used. x1=0.1 and x2=0.5 are the default
values which can be modified with the option 'range'.
Interpolation results are available in the region x1 < x < x2
for zOrd = 1,...,4. The interpolation results are only available for
nl=3.
Examples for the usage of Cbar[]:
Cbar["p", 1, 4]
provides the two-loop corrections of the fourth moment of the
pseudo-scalar correlator. One can change the renormalization scale to
mu = 2*m_1 by
Cbar["p", 1, 4, x, imu -> 2].
The one- and three-loop corrections are obtained by changing the
second argument to 0 and 2, respectively.
At three loops the result is returned as a piecewise defined function
as described above. It is possible to access the expressions
valid in each of the three segments by using the command
Refine[]. E.g.,
Refine[Cbar["p", 2, 3], {0 <= x && x <= 0.1}]
Refine[Cbar["p", 2, 3], {0.5 <= x && x <= 1}]
In case the fourth argument of Cbar[] is a number a numerical
evaluation is performed:
Cbar["p", 0, 4, 0.05].
The following commands reproduce the plot for the three-loop
corrections of the third moment of the pseudo-scalar correlator:
fct[x_] = N[Cbar["p", 2, 3, x]]; Plot[fct[x], {x, 0, 1}].
The introduction of the function fct[] guarantees that the
complex mathematical expressions originating from the three-loop
master integrals have to be evaluated only once.