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EvtGoityRoberts.cpp
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EvtGoityRoberts.cpp
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/***********************************************************************
* Copyright 1998-2020 CERN for the benefit of the EvtGen authors *
* *
* This file is part of EvtGen. *
* *
* EvtGen is free software: you can redistribute it and/or modify *
* it under the terms of the GNU General Public License as published by *
* the Free Software Foundation, either version 3 of the License, or *
* (at your option) any later version. *
* *
* EvtGen is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
* GNU General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with EvtGen. If not, see <https://www.gnu.org/licenses/>. *
***********************************************************************/
#include "EvtGenModels/EvtGoityRoberts.hh"
#include "EvtGenBase/EvtDiracSpinor.hh"
#include "EvtGenBase/EvtGenKine.hh"
#include "EvtGenBase/EvtPDL.hh"
#include "EvtGenBase/EvtParticle.hh"
#include "EvtGenBase/EvtPatches.hh"
#include "EvtGenBase/EvtReport.hh"
#include "EvtGenBase/EvtTensor4C.hh"
#include "EvtGenBase/EvtVector4C.hh"
#include <stdlib.h>
#include <string>
std::string EvtGoityRoberts::getName()
{
return "GOITY_ROBERTS";
}
EvtDecayBase* EvtGoityRoberts::clone()
{
return new EvtGoityRoberts;
}
void EvtGoityRoberts::init()
{
// check that there are 0 arguments
checkNArg( 0 );
checkNDaug( 4 );
checkSpinParent( EvtSpinType::SCALAR );
checkSpinDaughter( 1, EvtSpinType::SCALAR );
checkSpinDaughter( 2, EvtSpinType::DIRAC );
checkSpinDaughter( 3, EvtSpinType::NEUTRINO );
}
void EvtGoityRoberts::initProbMax()
{
setProbMax( 3000.0 );
}
void EvtGoityRoberts::decay( EvtParticle* p )
{
//added by Lange Jan4,2000
static EvtId DST0 = EvtPDL::getId( "D*0" );
static EvtId DSTB = EvtPDL::getId( "anti-D*0" );
static EvtId DSTP = EvtPDL::getId( "D*+" );
static EvtId DSTM = EvtPDL::getId( "D*-" );
static EvtId D0 = EvtPDL::getId( "D0" );
static EvtId D0B = EvtPDL::getId( "anti-D0" );
static EvtId DP = EvtPDL::getId( "D+" );
static EvtId DM = EvtPDL::getId( "D-" );
EvtId meson = getDaug( 0 );
if ( meson == DST0 || meson == DSTP || meson == DSTM || meson == DSTB ) {
DecayBDstarpilnuGR( p, getDaug( 0 ), getDaug( 2 ), getDaug( 3 ) );
} else {
if ( meson == D0 || meson == DP || meson == DM || meson == D0B ) {
DecayBDpilnuGR( p, getDaug( 0 ), getDaug( 2 ), getDaug( 3 ) );
} else {
EvtGenReport( EVTGEN_ERROR, "EvtGen" )
<< "Wrong daugther in EvtGoityRoberts!\n";
}
}
return;
}
void EvtGoityRoberts::DecayBDstarpilnuGR( EvtParticle* pb, EvtId ndstar,
EvtId nlep, EvtId /*nnu*/ )
{
pb->initializePhaseSpace( getNDaug(), getDaugs() );
//added by Lange Jan4,2000
static EvtId EM = EvtPDL::getId( "e-" );
static EvtId EP = EvtPDL::getId( "e+" );
static EvtId MUM = EvtPDL::getId( "mu-" );
static EvtId MUP = EvtPDL::getId( "mu+" );
EvtParticle *dstar, *pion, *lepton, *neutrino;
// pb->makeDaughters(getNDaug(),getDaugs());
dstar = pb->getDaug( 0 );
pion = pb->getDaug( 1 );
lepton = pb->getDaug( 2 );
neutrino = pb->getDaug( 3 );
EvtVector4C l1, l2, et0, et1, et2;
EvtVector4R v, vp, p4_pi;
double w;
v.set( 1.0, 0.0, 0.0, 0.0 ); //4-velocity of B meson
vp = ( 1.0 / dstar->getP4().mass() ) * dstar->getP4(); //4-velocity of D*
p4_pi = pion->getP4();
w = v * vp; //four velocity transfere.
EvtTensor4C omega;
double mb = EvtPDL::getMeanMass( pb->getId() ); //B mass
double md = EvtPDL::getMeanMass( ndstar ); //D* mass
EvtComplex dmb( 0.0460, -0.5 * 0.00001 ); // B*-B mass splitting ?
EvtComplex dmd( 0.1421, -0.5 * 0.00006 );
// The last two sets of numbers should
// be correctly calculated from the
// dstar and pion charges.
double g = 0.5; // EvtAmplitude proportional to these coupling constants
double alpha3 = 0.690; // See table I in G&R's paper
double alpha1 = -1.430;
double alpha2 = -0.140;
double f0 = 0.093; // The pion decay constants set to 93 MeV
EvtComplex dmt3( 0.563, -0.5 * 0.191 ); // Mass splitting = dmt - iGamma/2
EvtComplex dmt1( 0.392, -0.5 * 1.040 );
EvtComplex dmt2( 0.709, -0.5 * 0.405 );
double betas = 0.285; // magic number for meson wave function ground state
double betap = 0.280; // magic number for meson wave function state "1"
double betad = 0.260; // magic number for meson wave function state "2"
double betasp = betas * betas + betap * betap;
double betasd = betas * betas + betad * betad;
double lambdabar = 0.750; //M(0-,1-) - mQ From Goity&Roberts's code
// Isgur&Wise fct
double xi = exp( lambdabar * lambdabar * ( 1.0 - w * w ) /
( 4 * betas * betas ) );
double xi1 =
-1.0 * sqrt( 2.0 / 3.0 ) *
( lambdabar * lambdabar * ( w * w - 1.0 ) / ( 4 * betas * betas ) ) *
exp( lambdabar * lambdabar * ( 1.0 - w * w ) / ( 4 * betas * betas ) );
double rho1 = sqrt( 1.0 / 2.0 ) * ( lambdabar / betas ) *
pow( ( 2 * betas * betap / ( betasp ) ), 2.5 ) *
exp( lambdabar * lambdabar * ( 1.0 - w * w ) / ( 2 * betasp ) );
double rho2 = sqrt( 1.0 / 8.0 ) *
( lambdabar * lambdabar / ( betas * betas ) ) *
pow( ( 2 * betas * betad / ( betasd ) ), 3.5 ) *
exp( lambdabar * lambdabar * ( 1.0 - w * w ) / ( 2 * betasd ) );
//EvtGenReport(EVTGEN_INFO,"EvtGen") <<"rho's:"<<rho1<<rho2<<endl;
EvtComplex h1nr, h2nr, h3nr, f1nr, f2nr;
EvtComplex f3nr, f4nr, f5nr, f6nr, knr, g1nr, g2nr, g3nr, g4nr, g5nr;
EvtComplex h1r, h2r, h3r, f1r, f2r, f3r, f4r, f5r, f6r, kr, g1r, g2r, g3r,
g4r, g5r;
EvtComplex h1, h2, h3, f1, f2, f3, f4, f5, f6, k, g1, g2, g3, g4, g5;
// Non-resonance part
h1nr = -g * xi * ( p4_pi * v ) /
( f0 * mb * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) );
h2nr = -g * xi / ( f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) );
h3nr = -( g * xi / ( f0 * md ) ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) -
EvtComplex( ( 1.0 + w ) / ( p4_pi * vp ), 0.0 ) );
f1nr = -( g * xi / ( 2 * f0 * mb ) ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) -
1.0 / ( EvtComplex( p4_pi * vp, 0.0 ) + dmd ) );
f2nr = f1nr * mb / md;
f3nr = EvtComplex( 0.0 );
f4nr = EvtComplex( 0.0 );
f5nr = ( g * xi / ( 2 * f0 * mb * md ) ) *
( EvtComplex( 1.0, 0.0 ) +
( p4_pi * v ) / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) );
f6nr = ( g * xi / ( 2 * f0 * mb ) ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) -
EvtComplex( 1.0 / ( p4_pi * vp ), 0.0 ) );
knr = ( g * xi / ( 2 * f0 ) ) *
( ( p4_pi * ( vp - w * v ) ) / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) +
EvtComplex( ( p4_pi * ( v - w * vp ) ) / ( p4_pi * vp ), 0.0 ) );
g1nr = EvtComplex( 0.0 );
g2nr = EvtComplex( 0.0 );
g3nr = EvtComplex( 0.0 );
g4nr = ( g * xi ) / ( f0 * md * EvtComplex( p4_pi * vp ) );
g5nr = EvtComplex( 0.0 );
// Resonance part (D** removed by hand - alainb)
h1r = -alpha1 * rho1 * ( p4_pi * v ) /
( f0 * mb * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt1 ) ) +
alpha2 * rho2 * ( p4_pi * ( v + 2.0 * w * v - vp ) ) /
( 3 * f0 * mb * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) -
alpha3 * xi1 * ( p4_pi * v ) /
( f0 * mb * md * EvtComplex( p4_pi * v, 0.0 ) + dmt3 );
h2r = -alpha2 * ( 1 + w ) * rho2 /
( 3 * f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) -
alpha3 * xi1 / ( f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) );
h3r = alpha2 * rho2 * ( 1 + w ) /
( 3 * f0 * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) -
alpha3 * xi1 / ( f0 * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) );
f1r = -alpha2 * rho2 * ( w - 1.0 ) /
( 6 * f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) -
alpha3 * xi1 /
( 2 * f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) );
f2r = f1r * mb / md;
f3r = EvtComplex( 0.0 );
f4r = EvtComplex( 0.0 );
f5r = alpha1 * rho1 * ( p4_pi * v ) /
( 2 * f0 * mb * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt1 ) ) +
alpha2 * rho2 * ( p4_pi * ( vp - v / 3.0 - 2.0 / 3.0 * w * v ) ) /
( 2 * f0 * mb * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) +
alpha3 * xi1 * ( p4_pi * v ) /
( 2 * f0 * mb * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) );
f6r = alpha2 * rho2 * ( w - 1.0 ) /
( 6 * f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) +
alpha3 * xi1 /
( 2 * f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) );
kr = -alpha1 * rho1 * ( w - 1.0 ) * ( p4_pi * v ) /
( 2 * f0 * ( EvtComplex( p4_pi * v, 0.0 ) + dmt1 ) ) -
alpha2 * rho2 * ( w - 1.0 ) * ( p4_pi * ( vp - w * v ) ) /
( 3 * f0 * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) +
alpha3 * xi1 * ( p4_pi * ( vp - w * v ) ) /
( 2 * f0 * ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) );
g1r = EvtComplex( 0.0 );
g2r = EvtComplex( 0.0 );
g3r = -g2r;
g4r = 2.0 * alpha2 * rho2 /
( 3 * f0 * md * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) );
g5r = EvtComplex( 0.0 );
//Sum
h1 = h1nr + h1r;
h2 = h2nr + h2r;
h3 = h3nr + h3r;
f1 = f1nr + f1r;
f2 = f2nr + f2r;
f3 = f3nr + f3r;
f4 = f4nr + f4r;
f5 = f5nr + f5r;
f6 = f6nr + f6r;
k = knr + kr;
g1 = g1nr + g1r;
g2 = g2nr + g2r;
g3 = g3nr + g3r;
g4 = g4nr + g4r;
g5 = g5nr + g5r;
EvtTensor4C g_metric;
g_metric.setdiag( 1.0, -1.0, -1.0, -1.0 );
if ( nlep == EM || nlep == MUM ) {
omega =
EvtComplex( 0.0, 0.5 ) *
dual( h1 * mb * md * EvtGenFunctions::directProd( v, vp ) +
h2 * mb * EvtGenFunctions::directProd( v, p4_pi ) +
h3 * md * EvtGenFunctions::directProd( vp, p4_pi ) ) +
f1 * mb * EvtGenFunctions::directProd( v, p4_pi ) +
f2 * md * EvtGenFunctions::directProd( vp, p4_pi ) +
f3 * EvtGenFunctions::directProd( p4_pi, p4_pi ) +
f4 * mb * mb * EvtGenFunctions::directProd( v, v ) +
f5 * mb * md * EvtGenFunctions::directProd( vp, v ) +
f6 * mb * EvtGenFunctions::directProd( p4_pi, v ) + k * g_metric +
EvtComplex( 0.0, 0.5 ) *
EvtGenFunctions::directProd(
dual( EvtGenFunctions::directProd( vp, p4_pi ) ).cont2( v ),
( g1 * p4_pi + g2 * mb * v ) ) +
EvtComplex( 0.0, 0.5 ) *
EvtGenFunctions::directProd(
( g3 * mb * v + g4 * md * vp + g5 * p4_pi ),
dual( EvtGenFunctions::directProd( vp, p4_pi ) ).cont2( v ) );
l1 = EvtLeptonVACurrent( lepton->spParent( 0 ),
neutrino->spParentNeutrino() );
l2 = EvtLeptonVACurrent( lepton->spParent( 1 ),
neutrino->spParentNeutrino() );
} else {
if ( nlep == EP || nlep == MUP ) {
omega =
EvtComplex( 0.0, -0.5 ) *
dual( h1 * mb * md * EvtGenFunctions::directProd( v, vp ) +
h2 * mb * EvtGenFunctions::directProd( v, p4_pi ) +
h3 * md * EvtGenFunctions::directProd( vp, p4_pi ) ) +
f1 * mb * EvtGenFunctions::directProd( v, p4_pi ) +
f2 * md * EvtGenFunctions::directProd( vp, p4_pi ) +
f3 * EvtGenFunctions::directProd( p4_pi, p4_pi ) +
f4 * mb * mb * EvtGenFunctions::directProd( v, v ) +
f5 * mb * md * EvtGenFunctions::directProd( vp, v ) +
f6 * mb * EvtGenFunctions::directProd( p4_pi, v ) + k * g_metric +
EvtComplex( 0.0, -0.5 ) *
EvtGenFunctions::directProd(
dual( EvtGenFunctions::directProd( vp, p4_pi ) ).cont2( v ),
( g1 * p4_pi + g2 * mb * v ) ) +
EvtComplex( 0.0, -0.5 ) *
EvtGenFunctions::directProd(
( g3 * mb * v + g4 * md * vp + g5 * p4_pi ),
dual( EvtGenFunctions::directProd( vp, p4_pi ) ).cont2( v ) );
l1 = EvtLeptonVACurrent( neutrino->spParentNeutrino(),
lepton->spParent( 0 ) );
l2 = EvtLeptonVACurrent( neutrino->spParentNeutrino(),
lepton->spParent( 1 ) );
} else {
EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
<< "42387dfs878w wrong lepton number\n";
}
}
et0 = omega.cont2( dstar->epsParent( 0 ).conj() );
et1 = omega.cont2( dstar->epsParent( 1 ).conj() );
et2 = omega.cont2( dstar->epsParent( 2 ).conj() );
vertex( 0, 0, l1.cont( et0 ) );
vertex( 0, 1, l2.cont( et0 ) );
vertex( 1, 0, l1.cont( et1 ) );
vertex( 1, 1, l2.cont( et1 ) );
vertex( 2, 0, l1.cont( et2 ) );
vertex( 2, 1, l2.cont( et2 ) );
return;
}
void EvtGoityRoberts::DecayBDpilnuGR( EvtParticle* pb, EvtId nd, EvtId nlep,
EvtId /*nnu*/ )
{
//added by Lange Jan4,2000
static EvtId EM = EvtPDL::getId( "e-" );
static EvtId EP = EvtPDL::getId( "e+" );
static EvtId MUM = EvtPDL::getId( "mu-" );
static EvtId MUP = EvtPDL::getId( "mu+" );
EvtParticle *d, *pion, *lepton, *neutrino;
pb->initializePhaseSpace( getNDaug(), getDaugs() );
d = pb->getDaug( 0 );
pion = pb->getDaug( 1 );
lepton = pb->getDaug( 2 );
neutrino = pb->getDaug( 3 );
EvtVector4C l1, l2, et0, et1, et2;
EvtVector4R v, vp, p4_pi;
double w;
v.set( 1.0, 0.0, 0.0, 0.0 ); //4-velocity of B meson
vp = ( 1.0 / d->getP4().mass() ) * d->getP4(); //4-velocity of D
p4_pi = pion->getP4(); //4-momentum of pion
w = v * vp; //four velocity transfer.
double mb = EvtPDL::getMeanMass( pb->getId() ); //B mass
double md = EvtPDL::getMeanMass( nd ); //D* mass
EvtComplex dmb( 0.0460, -0.5 * 0.00001 ); //B mass splitting ?
//The last two numbers should be
//correctly calculated from the
//dstar and pion particle number.
double g = 0.5; // Amplitude proportional to these coupling constants
double alpha3 = 0.690; // See table I in G&R's paper
double alpha1 = -1.430;
double alpha2 = -0.140;
double f0 = 0.093; // The pion decay constant set to 93 MeV
EvtComplex dmt3( 0.563, -0.5 * 0.191 ); // Mass splitting = dmt - iGamma/2
EvtComplex dmt1( 0.392, -0.5 * 1.040 );
EvtComplex dmt2( 0.709, -0.5 * 0.405 );
double betas = 0.285; // magic number for meson wave function ground state
double betap = 0.280; // magic number for meson wave function state "1"
double betad = 0.260; // magic number for meson wave function state "2"
double betasp = betas * betas + betap * betap;
double betasd = betas * betas + betad * betad;
double lambdabar = 0.750; //M(0-,1-) - mQ From Goity&Roberts's code
// Isgur&Wise fct
double xi = exp( lambdabar * lambdabar * ( 1.0 - w * w ) /
( 4 * betas * betas ) );
double xi1 =
-1.0 * sqrt( 2.0 / 3.0 ) *
( lambdabar * lambdabar * ( w * w - 1.0 ) / ( 4 * betas * betas ) ) *
exp( lambdabar * lambdabar * ( 1.0 - w * w ) / ( 4 * betas * betas ) );
double rho1 = sqrt( 1.0 / 2.0 ) * ( lambdabar / betas ) *
pow( ( 2 * betas * betap / ( betasp ) ), 2.5 ) *
exp( lambdabar * lambdabar * ( 1.0 - w * w ) / ( 2 * betasp ) );
double rho2 = sqrt( 1.0 / 8.0 ) *
( lambdabar * lambdabar / ( betas * betas ) ) *
pow( ( 2 * betas * betad / ( betasd ) ), 3.5 ) *
exp( lambdabar * lambdabar * ( 1.0 - w * w ) / ( 2 * betasd ) );
EvtComplex h, a1, a2, a3;
EvtComplex hnr, a1nr, a2nr, a3nr;
EvtComplex hr, a1r, a2r, a3r;
// Non-resonance part (D* and D** removed by hand - alainb)
hnr = g * xi * ( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) ) /
( 2 * f0 * mb * md );
a1nr = -1.0 * g * xi * ( 1 + w ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) ) / ( 2 * f0 );
a2nr = g * xi *
( ( p4_pi * ( v + vp ) ) / ( EvtComplex( p4_pi * v, 0.0 ) + dmb ) ) /
( 2 * f0 * mb );
a3nr = EvtComplex( 0.0, 0.0 );
// Resonance part (D** remove by hand - alainb)
hr = alpha2 * rho2 * ( w - 1 ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) /
( 6 * f0 * mb * md ) +
alpha3 * xi1 * ( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) ) /
( 2 * f0 * mb * md );
a1r = -1.0 * alpha2 * rho2 * ( w * w - 1 ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) / ( 6 * f0 ) -
alpha3 * xi1 * ( 1 + w ) *
( 1.0 / ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) ) / ( 2 * f0 );
a2r = alpha1 * rho1 *
( ( p4_pi * v ) / ( EvtComplex( p4_pi * v, 0.0 ) + dmt1 ) ) /
( 2 * f0 * mb ) +
alpha2 * rho2 *
( 0.5 * p4_pi * ( w * vp - v ) + p4_pi * ( vp - w * v ) ) /
( 3 * f0 * mb * ( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) +
alpha3 * xi1 *
( ( p4_pi * ( v + vp ) ) / ( EvtComplex( p4_pi * v, 0.0 ) + dmt3 ) ) /
( 2 * f0 * mb );
a3r = -1.0 * alpha1 * rho1 *
( ( p4_pi * v ) / ( EvtComplex( p4_pi * v, 0.0 ) + dmt1 ) ) /
( 2 * f0 * md ) -
alpha2 * rho2 *
( ( p4_pi * ( vp - w * v ) ) /
( EvtComplex( p4_pi * v, 0.0 ) + dmt2 ) ) /
( 2 * f0 * md );
// Sum
h = hnr + hr;
a1 = a1nr + a1r;
a2 = a2nr + a2r;
a3 = a3nr + a3r;
EvtVector4C omega;
if ( nlep == EM || nlep == MUM ) {
omega = EvtComplex( 0.0, -1.0 ) * h * mb * md *
dual( EvtGenFunctions::directProd( vp, p4_pi ) ).cont2( v ) +
a1 * p4_pi + a2 * mb * v + a3 * md * vp;
l1 = EvtLeptonVACurrent( lepton->spParent( 0 ),
neutrino->spParentNeutrino() );
l2 = EvtLeptonVACurrent( lepton->spParent( 1 ),
neutrino->spParentNeutrino() );
} else {
if ( nlep == EP || nlep == MUP ) {
omega = EvtComplex( 0.0, 1.0 ) * h * mb * md *
dual( EvtGenFunctions::directProd( vp, p4_pi ) ).cont2( v ) +
a1 * p4_pi + a2 * mb * v + a3 * md * vp;
l1 = EvtLeptonVACurrent( neutrino->spParentNeutrino(),
lepton->spParent( 0 ) );
l2 = EvtLeptonVACurrent( neutrino->spParentNeutrino(),
lepton->spParent( 1 ) );
} else {
EvtGenReport( EVTGEN_ERROR, "EvtGen" )
<< "42387dfs878w wrong lepton number\n";
}
}
vertex( 0, l1 * omega );
vertex( 1, l2 * omega );
return;
}

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