EvtVubdGamma.cc

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//-----------------------------------------------------------------------
// File and Version Information:
//
// Copyright Information: See EvtGen/COPYRIGHT
//
//
// Description:
//    3                                         2                2
//   d Gamma                 /   _      _ _2  mb           _2  mb
//  ---------- = 12 Gamma   | (1+x-z)(z-x-p ) -- W  + (1-z+p ) -- W 
//         _ 2           0  \                  2  1             2  2
//  dx dz dp                                   2
//                                _   _  _2  mb                 2    \ 
//                             + [x(z-x)-p ] -- (W + 2mb W  + mb W ) |
//                                            4   3       4       5 /
//
//   with 
//        2 E           2
//           l    _2   p        2 v.p    _
//   x = ------ , p = --- , z = ------ , x = 1-x
//         mb           2         mb
//                    mb 
//
//   the triple differential decay rate according to
//   hep-ph/9905351 v2
//   
// Environment:
//      Software developed for the BaBar Detector at the SLAC B-Factory.
//
// Author List:
//      Sven Menke
//
//-----------------------------------------------------------------------
//-----------------------
// This Class's Header --
//-----------------------
#include "EvtGenBase/EvtPatches.hh"
#include "EvtGenModels/EvtVubdGamma.hh"
 
//---------------
// C Headers --
//---------------
#include <math.h>
 
extern "C" {
  double ddilog_(const double *);
}
//----------------
// Constructors --
//----------------
 
EvtVubdGamma::EvtVubdGamma(const double &alphas)
{
  _alphas = alphas;
 
  // the range for the delta distribution in p2 is from _epsilon1 to 
  // _epsilon2. It was checked with the single differential formulae
  // in the paper that these values are small enough to imitate p2 = 0
  // for the regular terms.
  // The ()* distributions, however need further treatment. In order to
  // generate the correct spectrum in z a threshold need to be computed 
  // from the desired value of the coupling alphas. The idea is that 
  // for z=1 p2=0 is not allowed and therefore the part of dGamma proportional
  // to delta(p2) should go to 0 for z->1.
  // Using equation (3.1) and (3.2) it is possible to find the correct value
  // for log(_epsilon3) from this requirement.
 
  _epsilon1 = 1e-10;
  _epsilon2 = 1e-5;
  if ( alphas > 0 ) {
    double lne3 = 9./16.-2*M_PI*M_PI/3.+6*M_PI/4/alphas;
    if ( lne3 > 0 ) 
      lne3 = -7./4. - sqrt(lne3);
    else
      lne3 = -7./4.;
    _epsilon3 = exp(lne3);
  }
  else
    _epsilon3 = 1;
}
 
//--------------
// Destructor --
//--------------
 
EvtVubdGamma::~EvtVubdGamma( )
{
}
 
//-----------
// Methods --
//-----------
 
double EvtVubdGamma::getdGdxdzdp(const double &x, const double &z, const double &p2)
{
  // check phase space
  
  double xb = (1-x);
 
  if ( x < 0 || x > 1 || z < xb || z > (1+xb) ) 
    return 0;
  
  //not used
  //double mx = (0>z-1?0:z-1);
 
  double p2min = (0>z-1.?0:z-1.);
  double p2max = (1.-x)*(z-1.+x);
 
  if (p2 < p2min || p2 > p2max) 
    return 0;
 
  //  // check the phase space
  //  return 1.;
 
  double dG;
 
  if ( p2 >_epsilon1 && p2< _epsilon2) {
    
    double W1      = getW1delta(x,z);
    double W4plus5 = getW4plus5delta(x,z);
    
    dG = 12. * delta(p2,p2min,p2max) * ((1.+xb-z) * (z-xb) * W1
                    + xb*(z-xb) * (W4plus5));
  }
  else {
 
    double W1 = getW1nodelta(x,z,p2);
    double W2 = getW2nodelta(x,z,p2);
    double W3 = getW3nodelta(x,z,p2);
    double W4 = getW4nodelta(x,z,p2);
    double W5 = getW5nodelta(x,z,p2);
 
    dG = 12. * ((1.+xb-z) * (z-xb-p2) * W1 
        + (1.-z+p2) * W2
        + (xb*(z-xb)-p2) * (W3+W4+W5));
  }
  return dG;
}
 
double EvtVubdGamma::delta(const double &x, const double &xmin, const double &xmax)
{
  if ( xmin > 0 || xmax < 0 ) return 0.;
  if ( _epsilon1 < x && x < _epsilon2 ) return 1./(_epsilon2-_epsilon1); 
  return 0.0; 
}
 
double EvtVubdGamma::getW1delta(const double &x, const double &z)
{
  double mz = 1.-z;
  //double p2min = (0>z-1.?0:z-1.);
  //double p2max = (1.-x)*(z-1.+x);
 
  double lz;
  if (z == 1) lz = -1.;
  else        lz = log(z)/(1.-z);
  
  // ddilog_(&z) is actually the dilog of (1-z) in maple,
  // also in Neuberts paper the limit dilog(1) = pi^2/6 is used 
  // this corresponds to maple's dilog(0), so
  // I take ddilog_(&mz) where mz=1-z in order to satisfy Neubert's definition
  // and to compare with Maple the argument in maple should be (1-mz) ...
 
  double dl = 4.*ddilog_(&mz) + 4.*pow(M_PI,2)/3.;
 
  double w = -(8.*pow(log(z),2) - 10.*log(z) + 2.*lz + dl + 5.)
    + (8.*log(z)-7.)*log(_epsilon3) - 2.*pow(log(_epsilon3),2); 
 
  return (1. + w*_alphas/3./M_PI);
}
 
double EvtVubdGamma::getW1nodelta(const double &x, const double &z, const double &p2)
{
 
  double z2 = z*z;
  double t2 = 1.-4.*p2/z2;
  double t = sqrt(t2);
  
  double w = 0;
  if ( p2 > _epsilon2 ) 
    w += 4./p2*(log((1.+t)/(1.-t))/t + log(p2/z2))  
      + 1. - (8.-z)*(2.-z)/z2/t2 
      + ((2.-z)/2./z+(8.-z)*(2.-z)/2./z2/t2)*log((1.+t)/(1.-t))/t;
  if ( p2 > _epsilon3 )
    w += (8.*log(z)-7.)/p2 - 4.*log(p2)/p2;
 
  return w*_alphas/3./M_PI;
}
 
double EvtVubdGamma::getW2nodelta(const double &x, const double &z, const double &p2)
{
 
  double z2 = z*z;
  double t2 = 1.-4.*p2/z2;
  double t = sqrt(t2);
  double w11 = (32.-8.*z+z2)/4./z/t2;
 
  double w = 0;
  if ( p2 > _epsilon2 ) 
    w -=  (z*t2/8. + (4.-z)/4. + w11/2.)*log((1.+t)/(1.-t))/t;
  if ( p2 > _epsilon2 ) 
    w += (8.-z)/4. + w11;
 
  return (w*_alphas/3./M_PI);
}
 
double EvtVubdGamma::getW3nodelta(const double &x, const double &z, const double &p2)
{
  double z2 = z*z;
  double t2 = 1.-4.*p2/z2;
  double t4 = t2*t2;
  double t = sqrt(t2);
 
  double w = 0;
 
  if ( p2 > _epsilon2 ) 
    w += (z*t2/16. + 5.*(4.-z)/16. - (64.+56.*z-7.*z2)/16./z/t2 
      + 3.*(12.-z)/16./t4) * log((1.+t)/(1.-t))/t;
  if ( p2 > _epsilon2 ) 
    w += -(8.-3.*z)/8. + (32.+22.*z-3.*z2)/4./z/t2 - 3.*(12.-z)/8./t4;
 
  return (w*_alphas/3./M_PI);
}
 
double EvtVubdGamma::getW4nodelta(const double &x, const double &z, const double &p2)
{
  double z2 = z*z;
  double t2 = 1.-4.*p2/z2;
  double t4 = t2*t2;
  double t = sqrt(t2);
 
  double w = 0;
 
  if ( p2 > _epsilon2 ) 
    w -= ((8.-3.*z)/4./z - (22.-3.*z)/2./z/t2 + 3.*(12.-z)/4./z/t4) 
      * log((1.+t)/(1.-t))/t;
  if ( p2 > _epsilon2 ) 
    w += -1. - (32.-5.*z)/2./z/t2 + 3.*(12.-z)/2./z/t4 ;
 
  return w*_alphas/3./M_PI;
}
 
double EvtVubdGamma::getW4plus5delta(const double &x, const double &z)
{
 
  double w = 0;
 
  if ( z == 1 )
    w = -2;
  else
    w = 2.*log(z)/(1.-z);
 
  return (w*_alphas/3./M_PI);
}
 
double EvtVubdGamma::getW5nodelta(const double &x, const double &z, const double &p2)
{
  double z2 = z*z;
  double t2 = 1.-4.*p2/z2;
  double t4 = t2*t2;
  double t = sqrt(t2);
 
  double w = 0;
  if ( p2 > _epsilon2 )
    w += (1./4./z - (2.-z)/2./z2/t2 + 3.*(12.-z)/4./z2/t4) 
      * log((1.+t)/(1.-t))/t;
  if ( p2 > _epsilon2 )
    w += -(8.+z)/2./z2/t2 - 3.*(12.-z)/2./z2/t4;
 
 
  return (w*_alphas/3./M_PI);
}