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.