EvtVubNLO.cpp

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#include "EvtGenModels/EvtVubNLO.hh"
 
#include "EvtGenBase/EvtDiLog.hh"
#include "EvtGenBase/EvtGenKine.hh"
#include "EvtGenBase/EvtPDL.hh"
#include "EvtGenBase/EvtParticle.hh"
#include "EvtGenBase/EvtRandom.hh"
#include "EvtGenBase/EvtReport.hh"
#include "EvtGenBase/EvtVector4R.hh"
 
#include "EvtGenModels/EvtBtoXsgammaFermiUtil.hh"
#include "EvtGenModels/EvtItgPtrFunction.hh"
#include "EvtGenModels/EvtItgSimpsonIntegrator.hh"
#include "EvtGenModels/EvtPFermi.hh"
 
#include <array>
#include <stdlib.h>
#include <string>
 
using std::cout;
using std::endl;
 
EvtVubNLO::~EvtVubNLO()
{
    cout << " max pdf : " << m_gmax << endl;
    cout << " efficiency : " << (float)m_ngood / (float)m_ntot << endl;
}
 
std::string EvtVubNLO::getName() const
{
    return "VUB_NLO";
}
 
EvtDecayBase* EvtVubNLO::clone() const
{
    return new EvtVubNLO;
}
 
void EvtVubNLO::init()
{
    // max pdf
    m_gmax = 0;
    m_ntot = 0;
    m_ngood = 0;
    m_lbar = -1000;
    m_mupi2 = -1000;
 
    // check number of arguments
    int npar = 8;
    if ( getNArg() < npar ) {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "EvtVubNLO generator expected "
            << " at least npar arguments  but found: " << getNArg() << endl;
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "Will terminate execution!" << endl;
        ::abort();
    }
    // this is the shape function parameter
    m_mb = getArg( 0 );
    m_b = getArg( 1 );
    m_lambdaSF = getArg( 2 );    // shape function lambda is different from lambda
    m_mui = 1.5;                 // GeV (scale)
    m_kpar = getArg( 3 );        // 0
    m_idSF = abs( (int)getArg(
        4 ) );    // type of shape function 1: exponential (from Neubert)
    int nbins = abs( (int)getArg( 5 ) );
    m_masses.resize( nbins );
    m_weights.resize( nbins );
 
    // Shape function normalization
    m_mB = 5.28;    // temporary B meson mass for normalization
 
    std::vector<double> sCoeffs( 11 );
    sCoeffs[3] = m_b;
    sCoeffs[4] = m_mb;
    sCoeffs[5] = m_mB;
    sCoeffs[6] = m_idSF;
    sCoeffs[7] = lambda_SF();
    sCoeffs[8] = mu_h();
    sCoeffs[9] = mu_i();
    sCoeffs[10] = 1.;
    m_SFNorm = SFNorm( sCoeffs );    // SF normalization;
 
    cout << " pdf 0.66, 1.32 , 4.32 " << tripleDiff( 0.66, 1.32, 4.32 ) << endl;
    cout << " pdf 0.23,0.37,3.76 " << tripleDiff( 0.23, 0.37, 3.76 ) << endl;
    cout << " pdf 0.97,4.32,4.42 " << tripleDiff( 0.97, 4.32, 4.42 ) << endl;
    cout << " pdf 0.52,1.02,2.01 " << tripleDiff( 0.52, 1.02, 2.01 ) << endl;
    cout << " pdf 1.35,1.39,2.73 " << tripleDiff( 1.35, 1.39, 2.73 ) << endl;
 
    if ( getNArg() - npar + 2 != int( 2 * m_weights.size() ) ) {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "EvtVubNLO generator expected " << m_weights.size()
            << " masses and weights but found: " << ( getNArg() - npar ) / 2
            << endl;
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "Will terminate execution!" << endl;
        ::abort();
    }
    int j = npar - 2;
    double maxw = 0.;
    for ( unsigned i = 0; i < m_masses.size(); i++ ) {
        m_masses[i] = getArg( j++ );
        if ( i > 0 && m_masses[i] <= m_masses[i - 1] ) {
            EvtGenReport( EVTGEN_ERROR, "EvtGen" )
                << "EvtVubNLO generator expected "
                << " mass bins in ascending order!"
                << "Will terminate execution!" << endl;
            ::abort();
        }
        m_weights[i] = getArg( j++ );
        if ( m_weights[i] < 0 ) {
            EvtGenReport( EVTGEN_ERROR, "EvtGen" )
                << "EvtVubNLO generator expected "
                << " weights >= 0, but found: " << m_weights[i] << endl;
            EvtGenReport( EVTGEN_ERROR, "EvtGen" )
                << "Will terminate execution!" << endl;
            ::abort();
        }
        if ( m_weights[i] > maxw )
            maxw = m_weights[i];
    }
    if ( maxw == 0 ) {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "EvtVubNLO generator expected at least one "
            << " weight > 0, but found none! "
            << "Will terminate execution!" << endl;
        ::abort();
    }
    for ( auto& w : m_weights )
        w /= maxw;
 
    // the maximum dGamma*p2 value depends on alpha_s only:
 
    //  m_dGMax = 0.05;
    m_dGMax = 150.;
 
    // for the Fermi Motion we need a B-Meso\n mass - but it's not critical
    // to get an exact value; in order to stay in the phase space for
    // B+- and B0 use the smaller mass
 
    // check that there are 3 daughters
    checkNDaug( 3 );
}
 
void EvtVubNLO::initProbMax()
{
    noProbMax();
}
 
void EvtVubNLO::decay( EvtParticle* p )
{
    // B+ -> u-bar specflav l+ nu
 
    EvtParticle *xuhad, *lepton, *neutrino;
    EvtVector4R p4;
 
    double pp, pm, pl, ml, El( 0.0 ), Eh( 0.0 ), sh( 0.0 );
 
    p->initializePhaseSpace( getNDaug(), getDaugs() );
 
    xuhad = p->getDaug( 0 );
    lepton = p->getDaug( 1 );
    neutrino = p->getDaug( 2 );
 
    m_mB = p->mass();
    ml = lepton->mass();
 
    bool tryit = true;
 
    while ( tryit ) {
        // pm=(E_H+P_H)
        pm = EvtRandom::Flat( 0., 1 );
        pm = pow( pm, 1. / 3. ) * m_mB;
        // pl=mB-2*El
        pl = EvtRandom::Flat( 0., 1 );
        pl = sqrt( pl ) * pm;
        // pp=(E_H-P_H)
        pp = EvtRandom::Flat( 0., pl );
 
        m_ntot++;
 
        El = ( m_mB - pl ) / 2.;
        Eh = ( pp + pm ) / 2;
        sh = pp * pm;
 
        double pdf( 0. );
        if ( pp < pl && El > ml && sh > m_masses[0] * m_masses[0] &&
             m_mB * m_mB + sh - 2 * m_mB * Eh > ml * ml ) {
            double xran = EvtRandom::Flat( 0, m_dGMax );
            pdf = tripleDiff( pp, pl, pm );    // triple differential distribution
            //      cout <<" P+,P-,Pl,Pdf= "<<pp <<" "<<pm<<" "<<pl<<" "<<pdf<<endl;
            if ( pdf > m_dGMax ) {
                EvtGenReport( EVTGEN_ERROR, "EvtGen" )
                    << "EvtVubNLO pdf above maximum: " << pdf
                    << " P+,P-,Pl,Pdf= " << pp << " " << pm << " " << pl << " "
                    << pdf << endl;
                //::abort();
            }
            if ( pdf >= xran )
                tryit = false;
 
            if ( pdf > m_gmax )
                m_gmax = pdf;
        } else {
            //      cout <<" EvtVubNLO incorrect kinematics  sh= "<<sh<<"EH "<<Eh<<endl;
        }
 
        // reweight the Mx distribution
        if ( !tryit && !m_weights.empty() ) {
            m_ngood++;
            double xran1 = EvtRandom::Flat();
            double m = sqrt( sh );
            unsigned j = 0;
            while ( j < m_masses.size() && m > m_masses[j] )
                j++;
            double w = m_weights[j - 1];
            if ( w < xran1 )
                tryit = true;    // through away this candidate
        }
    }
 
    //  cout <<" max prob "<<gmax<<" " << pp<<" "<<y<<" "<<x<<endl;
 
    // o.k. we have the three kineamtic variables
    // now calculate a flat cos Theta_H [-1,1] distribution of the
    // hadron flight direction w.r.t the B flight direction
    // because the B is a scalar and should decay isotropic.
    // Then chose a flat Phi_H [0,2Pi] w.r.t the B flight direction
    // and and a flat Phi_L [0,2Pi] in the W restframe w.r.t the
    // W flight direction.
 
    double ctH = EvtRandom::Flat( -1, 1 );
    double phH = EvtRandom::Flat( 0, 2 * M_PI );
    double phL = EvtRandom::Flat( 0, 2 * M_PI );
 
    // now compute the four vectors in the B Meson restframe
 
    double ptmp, sttmp;
    // calculate the hadron 4 vector in the B Meson restframe
 
    sttmp = sqrt( 1 - ctH * ctH );
    ptmp = sqrt( Eh * Eh - sh );
    double pHB[4] = { Eh, ptmp * sttmp * cos( phH ), ptmp * sttmp * sin( phH ),
                      ptmp * ctH };
    p4.set( pHB[0], pHB[1], pHB[2], pHB[3] );
    xuhad->init( getDaug( 0 ), p4 );
 
    // calculate the W 4 vector in the B Meson restrframe
 
    double apWB = ptmp;
    double pWB[4] = { m_mB - Eh, -pHB[1], -pHB[2], -pHB[3] };
 
    // first go in the W restframe and calculate the lepton and
    // the neutrino in the W frame
 
    double mW2 = m_mB * m_mB + sh - 2 * m_mB * Eh;
    //  if(mW2<0.1){
    //  cout <<" low Q2! "<<pp<<" "<<epp<<" "<<x<<" "<<y<<endl;
    //}
    double beta = ptmp / pWB[0];
    double gamma = pWB[0] / sqrt( mW2 );
 
    double pLW[4];
 
    ptmp = ( mW2 - ml * ml ) / 2 / sqrt( mW2 );
    pLW[0] = sqrt( ml * ml + ptmp * ptmp );
 
    double ctL = ( El - gamma * pLW[0] ) / beta / gamma / ptmp;
    if ( ctL < -1 )
        ctL = -1;
    if ( ctL > 1 )
        ctL = 1;
    sttmp = sqrt( 1 - ctL * ctL );
 
    // eX' = eZ x eW
    double xW[3] = { -pWB[2], pWB[1], 0 };
    // eZ' = eW
    double zW[3] = { pWB[1] / apWB, pWB[2] / apWB, pWB[3] / apWB };
 
    double lx = sqrt( xW[0] * xW[0] + xW[1] * xW[1] );
    for ( int j = 0; j < 2; j++ )
        xW[j] /= lx;
 
    // eY' = eZ' x eX'
    double yW[3] = { -pWB[1] * pWB[3], -pWB[2] * pWB[3],
                     pWB[1] * pWB[1] + pWB[2] * pWB[2] };
    double ly = sqrt( yW[0] * yW[0] + yW[1] * yW[1] + yW[2] * yW[2] );
    for ( int j = 0; j < 3; j++ )
        yW[j] /= ly;
 
    // p_lep = |p_lep| * (  sin(Theta) * cos(Phi) * eX'
    //                    + sin(Theta) * sin(Phi) * eY'
    //                    + cos(Theta) *            eZ')
    for ( int j = 0; j < 3; j++ )
        pLW[j + 1] = sttmp * cos( phL ) * ptmp * xW[j] +
                     sttmp * sin( phL ) * ptmp * yW[j] + ctL * ptmp * zW[j];
 
    double apLW = ptmp;
 
    // boost them back in the B Meson restframe
 
    double appLB = beta * gamma * pLW[0] + gamma * ctL * apLW;
 
    ptmp = sqrt( El * El - ml * ml );
    double ctLL = appLB / ptmp;
 
    if ( ctLL > 1 )
        ctLL = 1;
    if ( ctLL < -1 )
        ctLL = -1;
 
    double pLB[4] = { El, 0, 0, 0 };
    double pNB[8] = { pWB[0] - El, 0, 0, 0 };
 
    for ( int j = 1; j < 4; j++ ) {
        pLB[j] = pLW[j] + ( ctLL * ptmp - ctL * apLW ) / apWB * pWB[j];
        pNB[j] = pWB[j] - pLB[j];
    }
 
    p4.set( pLB[0], pLB[1], pLB[2], pLB[3] );
    lepton->init( getDaug( 1 ), p4 );
 
    p4.set( pNB[0], pNB[1], pNB[2], pNB[3] );
    neutrino->init( getDaug( 2 ), p4 );
 
    return;
}
 
double EvtVubNLO::tripleDiff( double pp, double pl, double pm )
{
    std::vector<double> sCoeffs( 11 );
    sCoeffs[0] = pp;
    sCoeffs[1] = pl;
    sCoeffs[2] = pm;
    sCoeffs[3] = m_b;
    sCoeffs[4] = m_mb;
    sCoeffs[5] = m_mB;
    sCoeffs[6] = m_idSF;
    sCoeffs[7] = lambda_SF();
    sCoeffs[8] = mu_h();
    sCoeffs[9] = mu_i();
    sCoeffs[10] = m_SFNorm;    // SF normalization;
 
    double c1 = ( m_mB + pl - pp - pm ) * ( pm - pl );
    double c2 = 2 * ( pl - pp ) * ( pm - pl );
    double c3 = ( m_mB - pm ) * ( pm - pp );
    double aF1 = F10( sCoeffs );
    double aF2 = F20( sCoeffs );
    double aF3 = F30( sCoeffs );
    double td0 = c1 * aF1 + c2 * aF2 + c3 * aF3;
 
    auto func = EvtItgPtrFunction{ &integrand, 0., m_mB, sCoeffs };
    auto jetSF = EvtItgSimpsonIntegrator{ func, 0.01, 25 };
    double smallfrac =
        0.000001;    // stop a bit before the end to avoid problems with numerical integration
    double tdInt = jetSF.evaluate( 0, pp * ( 1 - smallfrac ) );
 
    double SU = U1lo( mu_h(), mu_i() ) *
                pow( ( pm - pp ) / ( m_mB - pp ), alo( mu_h(), mu_i() ) );
    double TD = ( m_mB - pp ) * SU * ( td0 + tdInt );
 
    return TD;
}
 
double EvtVubNLO::integrand( double omega, const std::vector<double>& coeffs )
{
    //double pp=coeffs[0];
    double c1 = ( coeffs[5] + coeffs[1] - coeffs[0] - coeffs[2] ) *
                ( coeffs[2] - coeffs[1] );
    double c2 = 2 * ( coeffs[1] - coeffs[0] ) * ( coeffs[2] - coeffs[1] );
    double c3 = ( coeffs[5] - coeffs[2] ) * ( coeffs[2] - coeffs[0] );
 
    return c1 * F1Int( omega, coeffs ) + c2 * F2Int( omega, coeffs ) +
           c3 * F3Int( omega, coeffs );
}
 
double EvtVubNLO::F10( const std::vector<double>& coeffs )
{
    double pp = coeffs[0];
    double y = ( coeffs[2] - coeffs[0] ) / ( coeffs[5] - coeffs[0] );
    double mui = coeffs[9];
    double muh = coeffs[8];
    double z = 1 - y;
    double result = U1nlo( muh, mui ) / U1lo( muh, mui );
 
    result += anlo( muh, mui ) * log( y );
 
    result += C_F( muh ) *
              ( -4 * pow( log( y * coeffs[4] / muh ), 2 ) +
                10 * log( y * coeffs[4] / muh ) - 4 * log( y ) -
                2 * log( y ) / ( 1 - y ) - 4.0 * EvtDiLog::DiLog( z ) -
                pow( EvtConst::pi, 2 ) / 6. - 12 );
 
    result += C_F( mui ) *
              ( 2 * pow( log( y * coeffs[4] * pp / pow( mui, 2 ) ), 2 ) -
                3 * log( y * coeffs[4] * pp / pow( mui, 2 ) ) + 7 -
                pow( EvtConst::pi, 2 ) );
    result *= shapeFunction( pp, coeffs );
    // changes due to SSF
    result += ( -subS( coeffs ) + 2 * subT( coeffs ) +
                ( subU( coeffs ) - subV( coeffs ) ) * ( 1 / y - 1. ) ) /
              ( coeffs[5] - pp );
    result += shapeFunction( pp, coeffs ) / pow( ( coeffs[5] - coeffs[0] ), 2 ) *
              ( -5 * ( lambda1() + 3 * lambda2() ) / 6 +
                2 * ( 2 * lambda1() / 3 - lambda2() ) / pow( y, 2 ) );
    //  result +=  (subS(coeffs)+subT(coeffs)+(subU(coeffs)-subV(coeffs))/y)/(coeffs[5]-pp);
    // this part has been added after Feb '05
 
    //result += shapeFunction(pp,coeffs)/pow((coeffs[5]-coeffs[0]),2)*((lambda1()+3*lambda2())/6+2*(2*lambda1()/3-lambda2())/pow(y,2));
    return result;
}
 
double EvtVubNLO::F1Int( double omega, const std::vector<double>& coeffs )
{
    double pp = coeffs[0];
    double y = ( coeffs[2] - coeffs[0] ) / ( coeffs[5] - coeffs[0] );
    // mubar == mui
    return C_F( coeffs[9] ) *
           ( ( shapeFunction( omega, coeffs ) - shapeFunction( pp, coeffs ) ) *
                 ( 4 * log( y * coeffs[4] * ( pp - omega ) / pow( coeffs[9], 2 ) ) -
                   3 ) /
                 ( pp - omega ) +
             ( g1( y, ( pp - omega ) / ( coeffs[5] - coeffs[0] ) ) /
               ( coeffs[5] - pp ) * shapeFunction( omega, coeffs ) ) );
}
 
double EvtVubNLO::F20( const std::vector<double>& coeffs )
{
    double pp = coeffs[0];
    double y = ( coeffs[2] - coeffs[0] ) / ( coeffs[5] - coeffs[0] );
    double result = C_F( coeffs[8] ) * log( y ) / ( 1 - y ) *
                        shapeFunction( pp, coeffs ) -
                    1 / y *
                        ( subS( coeffs ) + 2 * subT( coeffs ) -
                          ( subT( coeffs ) + subV( coeffs ) ) / y ) /
                        ( coeffs[5] - pp );
    // added after Feb '05
    result += shapeFunction( pp, coeffs ) /
              pow( ( coeffs[5] - coeffs[0] ) * y, 2 ) *
              ( 2 * lambda1() / 3 + 4 * lambda2() -
                y * ( 7 / 6 * lambda1() + 3 * lambda2() ) );
    return result;
}
 
double EvtVubNLO::F2Int( double omega, const std::vector<double>& coeffs )
{
    double pp = coeffs[0];
    double y = ( coeffs[2] - coeffs[0] ) / ( coeffs[5] - coeffs[0] );
    return C_F( coeffs[9] ) *
           g3( y, ( pp - omega ) / ( coeffs[5] - coeffs[0] ) ) *
           shapeFunction( omega, coeffs ) / ( coeffs[5] - pp );
}
 
double EvtVubNLO::F30( const std::vector<double>& coeffs )
{
    double y = ( coeffs[2] - coeffs[0] ) / ( coeffs[5] - coeffs[0] );
    return shapeFunction( coeffs[0], coeffs ) /
           pow( ( coeffs[5] - coeffs[0] ) * y, 2 ) *
           ( -2 * lambda1() / 3 + lambda2() );
}
 
double EvtVubNLO::F3Int( double omega, const std::vector<double>& coeffs )
{
    double pp = coeffs[0];
    double y = ( coeffs[2] - coeffs[0] ) / ( coeffs[5] - coeffs[0] );
    return C_F( coeffs[9] ) *
           g3( y, ( pp - omega ) / ( coeffs[5] - coeffs[0] ) ) / 2 *
           shapeFunction( omega, coeffs ) / ( coeffs[2] - coeffs[0] );
}
 
double EvtVubNLO::g1( double y, double x )
{
    double result = ( y * ( -9 + 10 * y ) + x * x * ( -12 + 13 * y ) +
                      2 * x * ( -8 + 6 * y + 3 * y * y ) ) /
                    y / pow( 1 + x, 2 ) / ( x + y );
    result -= 4 * log( ( 1 + 1 / x ) * y ) / x;
    result -= 2 * log( 1 + y / x ) *
              ( 3 * pow( x, 4 ) * ( -2 + y ) - 2 * pow( y, 3 ) -
                4 * pow( x, 3 ) * ( 2 + y ) - 2 * x * y * y * ( 4 + y ) -
                x * x * y * ( 12 + 4 * y + y * y ) ) /
              x / pow( ( 1 + x ) * y, 2 ) / ( x + y );
    return result;
}
 
double EvtVubNLO::g2( double y, double x )
{
    double result = y * ( 10 * pow( x, 4 ) + y * y + 3 * x * x * y * ( 10 + y ) +
                          pow( x, 3 ) * ( 12 + 19 * y ) +
                          x * y * ( 8 + 4 * y + y * y ) );
    result -= 2 * x * log( 1 + y / x ) *
              ( 5 * pow( x, 4 ) + 2 * y * y + 6 * pow( x, 3 ) * ( 1 + 2 * y ) +
                4 * y * x * ( 1 + 2 * y ) + x * x * y * ( 18 + 5 * y ) );
    result *= 2 / ( pow( y * ( 1 + x ), 2 ) * y * ( x + y ) );
    return result;
}
 
double EvtVubNLO::g3( double y, double x )
{
    double result = ( 2 * pow( y, 3 ) * ( -11 + 2 * y ) -
                      10 * pow( x, 4 ) * ( 6 - 6 * y + y * y ) +
                      x * y * y * ( -94 + 29 * y + 2 * y * y ) +
                      2 * x * x * y * ( -72 + 18 * y + 13 * y * y ) -
                      pow( x, 3 ) *
                          ( 72 + 42 * y - 70 * y * y + 3 * pow( y, 3 ) ) ) /
                    ( pow( y * ( 1 + x ), 2 ) * y * ( x + y ) );
    result += 2 * log( 1 + y / x ) *
              ( -6 * x * pow( y, 3 ) * ( -5 + y ) + 4 * pow( y, 4 ) +
                5 * pow( x, 5 ) * ( 6 - 6 * y + y * y ) -
                4 * pow( x * y, 2 ) * ( -20 + 6 * y + y * y ) +
                pow( x, 3 ) * y * ( 90 - 10 * y - 28 * y * y + pow( y, 3 ) ) +
                pow( x, 4 ) * ( 36 + 36 * y - 50 * y * y + 4 * pow( y, 3 ) ) ) /
              ( pow( ( 1 + x ) * y * y, 2 ) * ( x + y ) );
    return result;
}
 
/* old version (before Feb 05 notebook from NNeubert
 
double
EvtVubNLO::F1Int(double omega,const std::vector<double> &coeffs){
  double pp=coeffs[0];
  double y=(coeffs[2]-coeffs[0])/(coeffs[5]-coeffs[0]);
  // mubar == mui
  return C_F(coeffs[9])*(
                         (shapeFunction(omega,coeffs)-shapeFunction(pp,coeffs))*(4*log(y*coeffs[4]*(pp-omega)/pow(coeffs[9],2))-3)/(pp-omega)-
                         (1./y/(coeffs[5]-pp)*shapeFunction(omega,coeffs)*(5-6*y+4*(3-y)*log((pp-omega)/y/coeffs[4])))
                         );
}
 
 
double
EvtVubNLO::F2Int(double omega,const std::vector<double> &coeffs){
  double pp=coeffs[0];
  double y=(coeffs[2]-coeffs[0])/(coeffs[5]-coeffs[0]);
  return C_F(coeffs[9])*shapeFunction(omega,coeffs)*(2-11/y-4/y*log((pp-omega)/y/coeffs[4]))/(coeffs[5]-pp);
}
 
double
EvtVubNLO::F3(const std::vector<double> &coeffs){
  return C_F(coeffs[9])*shapeFunction(omega,coeffs)/(coeffs[2]-coeffs[0]);
}
*/
 
double EvtVubNLO::SFNorm( const std::vector<double>& /*coeffs*/ )
{
    double omega0 = 1.68;    //normalization scale (mB-2*1.8)
    if ( m_idSF == 1 ) {     // exponential SF
        return M0( mu_i(), omega0 ) * pow( m_b, m_b ) / lambda_SF() /
               ( Gamma( m_b ) - Gamma( m_b, m_b * omega0 / lambda_SF() ) );
    } else if ( m_idSF == 2 ) {    // Gaussian SF
        double c = cGaus( m_b );
        return M0( mu_i(), omega0 ) * 2 / lambda_SF() /
               pow( c, -( 1 + m_b ) / 2. ) /
               ( Gamma( ( 1 + m_b ) / 2 ) -
                 Gamma( ( 1 + m_b ) / 2, pow( omega0 / lambda_SF(), 2 ) * c ) );
    } else {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" ) << "unknown SF " << m_idSF << endl;
        return -1;
    }
}
 
double EvtVubNLO::shapeFunction( double omega, const std::vector<double>& sCoeffs )
{
    if ( sCoeffs[6] == 1 ) {
        return sCoeffs[10] * expShapeFunction( omega, sCoeffs );
    } else if ( sCoeffs[6] == 2 ) {
        return sCoeffs[10] * gausShapeFunction( omega, sCoeffs );
    } else {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "EvtVubNLO : unknown shape function # " << sCoeffs[6] << endl;
    }
    return -1.;
}
 
// SSF
double EvtVubNLO::subS( const std::vector<double>& c )
{
    return ( lambda_bar( 1.68 ) - c[0] ) * shapeFunction( c[0], c );
}
double EvtVubNLO::subT( const std::vector<double>& c )
{
    return -3 * lambda2() * subS( c ) / mu_pi2( 1.68 );
}
double EvtVubNLO::subU( const std::vector<double>& c )
{
    return -2 * subS( c );
}
double EvtVubNLO::subV( const std::vector<double>& c )
{
    return -subT( c );
}
 
double EvtVubNLO::lambda_bar( double omega0 )
{
    if ( m_lbar < 0 ) {
        if ( m_idSF == 1 ) {    // exponential SF
            double rat = omega0 * m_b / lambda_SF();
            m_lbar = lambda_SF() / m_b *
                     ( Gamma( 1 + m_b ) - Gamma( 1 + m_b, rat ) ) /
                     ( Gamma( m_b ) - Gamma( m_b, rat ) );
        } else if ( m_idSF == 2 ) {    // Gaussian SF
            double c = cGaus( m_b );
            m_lbar =
                lambda_SF() *
                ( Gamma( 1 + m_b / 2 ) -
                  Gamma( 1 + m_b / 2, pow( omega0 / lambda_SF(), 2 ) * c ) ) /
                ( Gamma( ( 1 + m_b ) / 2 ) -
                  Gamma( ( 1 + m_b ) / 2, pow( omega0 / lambda_SF(), 2 ) * c ) ) /
                sqrt( c );
        }
    }
    return m_lbar;
}
 
double EvtVubNLO::mu_pi2( double omega0 )
{
    if ( m_mupi2 < 0 ) {
        if ( m_idSF == 1 ) {    // exponential SF
            double rat = omega0 * m_b / lambda_SF();
            m_mupi2 = 3 * ( pow( lambda_SF() / m_b, 2 ) *
                                ( Gamma( 2 + m_b ) - Gamma( 2 + m_b, rat ) ) /
                                ( Gamma( m_b ) - Gamma( m_b, rat ) ) -
                            pow( lambda_bar( omega0 ), 2 ) );
        } else if ( m_idSF == 2 ) {    // Gaussian SF
            double c = cGaus( m_b );
            double m1 = Gamma( ( 3 + m_b ) / 2 ) -
                        Gamma( ( 3 + m_b ) / 2,
                               pow( omega0 / lambda_SF(), 2 ) * c );
            double m2 = Gamma( 1 + m_b / 2 ) -
                        Gamma( 1 + m_b / 2, pow( omega0 / lambda_SF(), 2 ) * c );
            double m3 = Gamma( ( 1 + m_b ) / 2 ) -
                        Gamma( ( 1 + m_b ) / 2,
                               pow( omega0 / lambda_SF(), 2 ) * c );
            m_mupi2 = 3 * pow( lambda_SF(), 2 ) *
                      ( m1 / m3 - pow( m2 / m3, 2 ) ) / c;
        }
    }
    return m_mupi2;
}
 
double EvtVubNLO::M0( double mui, double omega0 )
{
    double mf = omega0 - lambda_bar( omega0 );
    return 1 + 4 * C_F( mui ) *
                   ( -pow( log( mf / mui ), 2 ) - log( mf / mui ) -
                     pow( EvtConst::pi / 2, 2 ) / 6. +
                     mu_pi2( omega0 ) / 3 / pow( mf, 2 ) *
                         ( log( mf / mui ) - 0.5 ) );
}
 
double EvtVubNLO::alphas( double mu )
{
    double Lambda4 = 0.302932;
    double lg = 2 * log( mu / Lambda4 );
    return 4 * EvtConst::pi / lg / beta0() *
           ( 1 - beta1() * log( lg ) / pow( beta0(), 2 ) / lg +
             pow( beta1() / lg, 2 ) / pow( beta0(), 4 ) *
                 ( pow( log( lg ) - 0.5, 2 ) - 1.25 +
                   beta2() * beta0() / pow( beta1(), 2 ) ) );
}
 
double EvtVubNLO::gausShapeFunction( double omega,
                                     const std::vector<double>& sCoeffs )
{
    double b = sCoeffs[3];
    double l = sCoeffs[7];
    double wL = omega / l;
 
    return pow( wL, b ) * exp( -cGaus( b ) * wL * wL );
}
 
double EvtVubNLO::expShapeFunction( double omega,
                                    const std::vector<double>& sCoeffs )
{
    double b = sCoeffs[3];
    double l = sCoeffs[7];
    double wL = omega / l;
 
    return pow( wL, b - 1 ) * exp( -b * wL );
}
 
double EvtVubNLO::Gamma( double z )
{
    std::array<double, 6> gammaCoeffs{
        76.18009172947146,  -86.50532032941677,    24.01409824083091,
        -1.231739572450155, 0.1208650973866179e-2, -0.5395239384953e-5 };
 
    //Lifted from Numerical Recipies in C
    double y = z;
    double x = z;
 
    double tmp = x + 5.5;
    tmp = tmp - ( x + 0.5 ) * log( tmp );
    double ser = 1.000000000190015;
 
    for ( const auto& gammaCoeff : gammaCoeffs ) {
        y += 1.0;
        ser += gammaCoeff / y;
    }
 
    return exp( -tmp + log( 2.5066282746310005 * ser / x ) );
}
 
double EvtVubNLO::Gamma( double z, double tmin )
{
    std::vector<double> c( 1 );
    c[0] = z;
    auto func = EvtItgPtrFunction{ &dgamma, tmin, 100., c };
    auto jetSF = EvtItgSimpsonIntegrator{ func, 0.001 };
    return jetSF.evaluate( tmin, 100. );
}