EvtWilsonCoefficients.cpp

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* Copyright 1998-2020 CERN for the benefit of the EvtGen authors       *
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#include "EvtGenModels/EvtWilsonCoefficients.hh"
 
#include "EvtGenBase/EvtConst.hh"
#include "EvtGenBase/EvtReport.hh"
 
#include <cstdlib>
 
double li2spence( double );
 
EvtWilsonCoefficients::EvtWilsonCoefficients()
{
    int i, j;
    double tmpa[8] = { 14. / 23., 16. / 23., 6. / 23., -12. / 23.,
                       0.4086,    -0.4230,   -0.8994,  0.1456 };
    double tmph[8] = { 2.2996,  -1.0880, -3. / 7., -1. / 14.,
                       -0.6494, -0.0380, -0.0186,  -0.0057 };
    double tmpp[8] = { 0,      0,      -80. / 203., 8. / 33.,
                       0.0433, 0.1384, 0.1648,      -0.0073 };
    double tmps[8] = { 0,      0,       -0.2009, -0.3579,
                       0.0490, -0.3616, -0.3554, 0.0072 };
    double tmpq[8] = { 0, 0, 0, 0, 0.0318, 0.0918, -0.2700, 0.0059 };
    double tmpg[8] = { 313063. / 363036., 0,      0,       0,
                       -0.9135,           0.0873, -0.0571, -0.0209 };
    double tmpk[6][8] = {
        { 0, 0, 1. / 2., -1. / 2., 0, 0, 0, 0 },
        { 0, 0, 1. / 2., +1. / 2., 0, 0, 0, 0 },
        { 0, 0, -1. / 14., +1. / 6., 0.0510, -0.1403, -0.0113, 0.0054 },
        { 0, 0, -1. / 14., -1. / 6., 0.0984, +0.1214, +0.0156, 0.0026 },
        { 0, 0, 0, 0, -0.0397, 0.0117, -0.0025, +0.0304 },
        { 0, 0, 0, 0, +0.0335, 0.0239, -0.0462, -0.0112 } };
    double tmpr[2][8] = {
        { 0, 0, +0.8966, -0.1960, -0.2011, 0.1328, -0.0292, -0.1858 },
        { 0, 0, -0.1193, +0.1003, -0.0473, 0.2323, -0.0133, -0.1799 } };
    for ( i = 0; i < 8; i++ ) {
        m_a[i] = tmpa[i];
        m_h[i] = tmph[i];
        m_p[i] = tmpp[i];
        m_s[i] = tmps[i];
        m_q[i] = tmpq[i];
        m_g[i] = tmpg[i];
        for ( j = 0; j < 6; j++ )
            m_k[j][i] = tmpk[j][i];
        for ( j = 0; j < 2; j++ )
            m_r[j][i] = tmpr[j][i];
    }
    m_n_f = 5;
    m_Lambda = 0.2167;
    m_alphaMZ = 0.1187;
    m_mu = 4.8;
    m_M_Z = 91.1876;
    m_M_t = 174.3;
    m_M_W = 80.425;
    m_alphaS = 0;
    m_eta = 0;
    m_sin2W = 0.23120;
    m_ialpha = 137.036;
    m_C1 = m_C2 = m_C3 = m_C4 = m_C5 = m_C6 = m_C7 = m_C7eff0 = m_C8 =
        m_C8eff0 = m_C9 = m_C9tilda = m_C10 = m_C10tilda = m_P0 = 0;
    m_A = m_B = m_C = m_D = m_E = m_F = m_Y = m_Z = m_PE = 0;
    m_ksi = 0;
}
 
void EvtWilsonCoefficients::SetRenormalizationScheme( std::string scheme )
{
    if ( scheme == "NDR" )
        m_ksi = 0;
    else if ( scheme == "HV" )
        m_ksi = 1;
    else {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << "ERROR: EvtWilsonCoefficients knows only NDR and HV schemes !"
            << std::endl;
        ::abort();
    }
}
 
double EvtWilsonCoefficients::alphaS( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167 )
{
    // calculate strong coupling constant for n_f flavours and scale mu
    double beta0 = 11. - 2. / 3. * n_f;
    double beta1 = 51. - 19. / 3. * n_f;
    double beta2 = 2857. - 5033. / 9. * n_f + 325. / 27. * n_f * n_f;
    double lnratio = log( mu * mu / Lambda / Lambda );
    double aS = 4. * EvtConst::pi / beta0 / lnratio *
                ( 1. - 2 * beta1 / beta0 / beta0 * log( lnratio ) / lnratio +
                  4 * beta1 * beta1 / beta0 / beta0 / beta0 / beta0 / lnratio /
                      lnratio *
                      ( ( log( lnratio ) - 0.5 ) * ( log( lnratio ) - 0.5 ) +
                        beta2 * beta0 / 8 / beta1 / beta1 - 5. / 4. ) );
    return aS;
}
 
double EvtWilsonCoefficients::Lambda( double alpha = 0.1187, int n_f = 5,
                                      double mu = 91.1876,
                                      double epsilon = 0.00005,
                                      int maxstep = 1000 )
{
    // calculate Lambda matching alphaS using simple iterative method
    int i;
    double difference = 0;
    double Lambda = mu * 0.9999999999;
    double step = -mu / 20;
    for ( i = 0;
          i < maxstep &&
          ( difference = fabs( alphaS( mu, n_f, Lambda ) - alpha ) ) >= epsilon;
          i++ ) {
        EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
            << " Difference of alpha_S from " << alpha << " is " << difference
            << " at Lambda = " << Lambda << std::endl;
        if ( alphaS( mu, n_f, Lambda ) > alpha ) {
            if ( step > 0 )
                step *= -0.4;
            if ( alphaS( mu, n_f, Lambda + step - epsilon ) <
                 alphaS( mu, n_f, Lambda + step ) )
                Lambda += step;
            else
                step *= 0.4;
        } else {
            if ( step < 0 )
                step *= -0.4;
            if ( Lambda + step < mu )
                Lambda += step;
            else
                step *= 0.4;
        }
    }
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
        << " Difference of alpha_S from " << alpha << " is " << difference
        << " at Lambda = " << Lambda << std::endl;
    if ( difference >= epsilon ) {
        EvtGenReport( EVTGEN_ERROR, "EvtGen" )
            << " ERROR: Did not converge Lambda for alpha_s = " << alpha
            << " , difference " << difference << " >= " << epsilon << " after "
            << i << " steps !" << std::endl;
        ::abort();
        return -1;
    } else {
        EvtGenReport( EVTGEN_INFO, "EvtGen" )
            << " For alpha_s = " << alphaS( mu, n_f, Lambda )
            << " was found Lambda = " << Lambda << std::endl;
        return Lambda;
    }
}
 
double EvtWilsonCoefficients::eta( double mu = 4.8, int n_f = 5,
                                   double Lambda = 0.2167, double M_W = 80.425 )
{
    return alphaS( M_W, n_f, Lambda ) / alphaS( mu, n_f, Lambda );
}
 
EvtComplex EvtWilsonCoefficients::C1( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myC1( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC1 += m_k[0][i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    return myC1;
}
 
EvtComplex EvtWilsonCoefficients::C2( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myC2( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC2 += m_k[1][i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    return myC2;
}
 
EvtComplex EvtWilsonCoefficients::C3( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myC3( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC3 += m_k[2][i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    return myC3;
}
 
EvtComplex EvtWilsonCoefficients::C4( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myC4( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC4 += m_k[3][i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    return myC4;
}
 
EvtComplex EvtWilsonCoefficients::C5( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myC5( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC5 += m_k[4][i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    return myC5;
}
 
EvtComplex EvtWilsonCoefficients::C6( double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myC6( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC6 += m_k[5][i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    return myC6;
}
 
EvtComplex EvtWilsonCoefficients::C7( double M_t = 174.3, double M_W = 80.425 )
{
    return EvtComplex( -0.5 * A( M_t * M_t / M_W / M_W ), 0 );
}
 
EvtComplex EvtWilsonCoefficients::C8( double M_t = 174.3, double M_W = 80.425 )
{
    return EvtComplex( -0.5 * F( M_t * M_t / M_W / M_W ), 0 );
}
 
EvtComplex EvtWilsonCoefficients::C7eff0( double mu = 4.8, int n_f = 5,
                                          double Lambda = 0.2167,
                                          double M_t = 174.3, double M_W = 80.425 )
{
    int i;
    EvtComplex myC7eff( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC7eff += m_h[i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    myC7eff *= C2( mu, n_f, Lambda, M_W );
    myC7eff += pow( eta( mu, n_f, Lambda, M_W ), 16. / 23. ) * C7( M_t, M_W );
    myC7eff += 8. / 3. *
               ( pow( eta( mu, n_f, Lambda, M_W ), 14. / 23. ) -
                 pow( eta( mu, n_f, Lambda, M_W ), 16. / 23. ) ) *
               C8( M_t, M_W );
    return myC7eff;
}
 
EvtComplex EvtWilsonCoefficients::C8eff0( double mu = 4.8, int n_f = 5,
                                          double Lambda = 0.2167,
                                          double M_t = 174.3, double M_W = 80.425 )
{
    int i;
    EvtComplex myC8eff( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myC8eff += m_g[i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] );
    myC8eff += pow( eta( mu, n_f, Lambda, M_W ), 14. / 23. ) * C8( M_t, M_W );
    return myC8eff;
}
 
EvtComplex EvtWilsonCoefficients::C10tilda( double sin2W = 0.23120,
                                            double M_t = 174.3,
                                            double M_W = 80.425 )
{
    return EvtComplex( -Y( M_t * M_t / M_W / M_W ) / sin2W, 0 );
}
 
EvtComplex EvtWilsonCoefficients::C10( double sin2W = 0.23120,
                                       double M_t = 174.3, double M_W = 80.425,
                                       double ialpha = 137.036 )
{
    return ( 1. / 2 / EvtConst::pi / ialpha * C10tilda( sin2W, M_t, M_W ) );
}
 
double EvtWilsonCoefficients::A( double x )
{
    return ( x * ( 8 * x * x + 5 * x - 7 ) / 12 / pow( x - 1, 3 ) +
             x * x * ( 2 - 3 * x ) * log( x ) / 2 / pow( x - 1, 4 ) );
}
 
double EvtWilsonCoefficients::B( double x )
{
    return ( x / 4 / ( 1 - x ) + x / 4 / ( x - 1 ) / ( x - 1 ) * log( x ) );
}
 
double EvtWilsonCoefficients::C( double x )
{
    return ( x * ( x - 6 ) / 8 / ( x - 1 ) +
             x * ( 3 * x + 2 ) / 8 / ( x - 1 ) / ( x - 1 ) * log( x ) );
}
 
double EvtWilsonCoefficients::D( double x )
{
    return ( ( -19 * x * x * x + 25 * x * x ) / 36 / pow( x - 1, 3 ) +
             x * x * ( 5 * x * x - 2 * x - 6 ) / 18 / pow( x - 1, 4 ) * log( x ) -
             4. / 9 * log( x ) );
}
 
double EvtWilsonCoefficients::E( double x )
{
    return ( x * ( 18 - 11 * x - x * x ) / 12 / pow( 1 - x, 3 ) +
             x * x * ( 15 - 16 * x + 4 * x * x ) / 6 / pow( 1 - x, 4 ) * log( x ) -
             2. / 3 * log( x ) );
}
 
double EvtWilsonCoefficients::F( double x )
{
    return ( x * ( x * x - 5 * x - 2 ) / 4 / pow( x - 1, 3 ) +
             3 * x * x / 2 / pow( x - 1, 4 ) * log( x ) );
}
 
double EvtWilsonCoefficients::Y( double x )
{
    return ( C( x ) - B( x ) );
}
 
double EvtWilsonCoefficients::Z( double x )
{
    return ( C( x ) + 1. / 4 * D( x ) );
}
 
EvtComplex EvtWilsonCoefficients::C9( int ksi = 0, double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167,
                                      double sin2W = 0.23120,
                                      double M_t = 174.3, double M_W = 80.425,
                                      double ialpha = 137.036 )
{
    return ( 1. / 2 / EvtConst::pi / ialpha *
             C9tilda( ksi, mu, n_f, Lambda, sin2W, M_t, M_W ) );
}
 
EvtComplex EvtWilsonCoefficients::C9tilda( int ksi = 0, double mu = 4.8,
                                           int n_f = 5, double Lambda = 0.2167,
                                           double sin2W = 0.23120,
                                           double M_t = 174.3,
                                           double M_W = 80.425 )
{
    return ( P0( ksi, mu, n_f, Lambda, M_W ) +
             Y( M_t * M_t / M_W / M_W ) / sin2W - 4 * Z( M_t * M_t / M_W / M_W ) +
             PE( mu, n_f, Lambda, M_W ) * E( M_t * M_t / M_W / M_W ) );
}
 
EvtComplex EvtWilsonCoefficients::P0( int ksi = 0, double mu = 4.8, int n_f = 5,
                                      double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    EvtComplex myP0( 0, 0 );
    for ( i = 0; i < 8; i++ )
        myP0 += m_p[i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] + 1 );
    myP0 = EvtConst::pi / alphaS( M_W, n_f, Lambda ) * ( -0.1875 + myP0 );
    myP0 += 1.2468 -
            ksi * 4. / 9. *
                ( 3 * C1( mu, n_f, Lambda, M_W ) + C2( mu, n_f, Lambda, M_W ) -
                  C3( mu, n_f, Lambda, M_W ) - 3 * C4( mu, n_f, Lambda, M_W ) );
    for ( i = 0; i < 8; i++ )
        myP0 += pow( eta( mu, n_f, Lambda, M_W ), m_a[i] ) *
                ( m_r[ksi][i] + m_s[i] * eta( mu, n_f, Lambda, M_W ) );
    return myP0;
}
 
double EvtWilsonCoefficients::PE( double mu = 4.8, int n_f = 5,
                                  double Lambda = 0.2167, double M_W = 80.425 )
{
    int i;
    double myPE = 0.1405;
    for ( i = 0; i < 8; i++ )
        myPE += m_q[i] * pow( eta( mu, n_f, Lambda, M_W ), m_a[i] + 1 );
    return myPE;
}
 
void EvtWilsonCoefficients::CalculateAllCoefficients()
{
    m_Lambda = Lambda( m_alphaMZ, m_n_f, m_M_Z );
    m_C1 = C1( m_mu, m_n_f, m_Lambda, m_M_W );
    m_C2 = C2( m_mu, m_n_f, m_Lambda, m_M_W );
    m_C3 = C3( m_mu, m_n_f, m_Lambda, m_M_W );
    m_C4 = C4( m_mu, m_n_f, m_Lambda, m_M_W );
    m_C5 = C5( m_mu, m_n_f, m_Lambda, m_M_W );
    m_C6 = C6( m_mu, m_n_f, m_Lambda, m_M_W );
    m_C7 = C7( m_M_t, m_M_W );
    m_C8 = C8( m_M_t, m_M_W );
    m_C7eff0 = C7eff0( m_mu, m_n_f, m_Lambda, m_M_t, m_M_W );
    m_C8eff0 = C8eff0( m_mu, m_n_f, m_Lambda, m_M_t, m_M_W );
    m_C10tilda = C10tilda( m_sin2W, m_M_t, m_M_W );
    m_C10 = C10( m_sin2W, m_M_t, m_M_W, m_ialpha );
    m_A = A( m_M_t * m_M_t / m_M_W / m_M_W );
    m_B = B( m_M_t * m_M_t / m_M_W / m_M_W );
    m_C = C( m_M_t * m_M_t / m_M_W / m_M_W );
    m_D = D( m_M_t * m_M_t / m_M_W / m_M_W );
    m_E = E( m_M_t * m_M_t / m_M_W / m_M_W );
    m_F = F( m_M_t * m_M_t / m_M_W / m_M_W );
    m_Y = Y( m_M_t * m_M_t / m_M_W / m_M_W );
    m_Z = Z( m_M_t * m_M_t / m_M_W / m_M_W );
    m_C9 = C9( m_ksi, m_mu, m_n_f, m_Lambda, m_sin2W, m_M_t, m_M_W, m_ialpha );
    m_C9tilda = C9tilda( m_ksi, m_mu, m_n_f, m_Lambda, m_sin2W, m_M_t, m_M_W );
    m_P0 = P0( m_ksi, m_mu, m_n_f, m_Lambda, m_M_W );
    m_PE = PE( m_mu, m_n_f, m_Lambda, m_M_W );
    m_alphaS = alphaS( m_mu, m_n_f, m_Lambda );
    m_eta = eta( m_mu, m_n_f, m_Lambda, m_M_W );
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " +---------------------------------------" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Table of Wilson coeficients:" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " +---------------------------------------" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C1     =  " << m_C1 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C2     =  " << m_C2 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C3     =  " << m_C3 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C4     =  " << m_C4 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C5     =  " << m_C5 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C6     =  " << m_C6 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C7     =  " << m_C7 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | C7eff0 =  " << m_C7eff0 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C8     =  " << m_C8 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | C8eff0 =  " << m_C8eff0 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | C9     =  " << m_C9 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | C10    =  " << m_C10 << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " +---------------------------------------" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" ) << " | Other constants:" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " +---------------------------------------" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Scale = " << m_mu << " GeV" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Number of effective flavors = " << m_n_f << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Corresponding to aS(M_Z)"
        << "=" << m_alphaMZ << " Lambda = " << m_Lambda << " GeV" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Strong coupling constant = " << m_alphaS << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Electromagnetic constant = 1/" << m_ialpha << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Top mass = " << m_M_t << " GeV" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | W-boson mass = " << m_M_W << " GeV" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Z-boson mass = " << m_M_Z << " GeV" << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " | Sinus squared of Weinberg angle = " << m_sin2W << std::endl;
    EvtGenReport( EVTGEN_INFO, "EvtGen" )
        << " +---------------------------------------" << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
        << " | Intermediate functions:" << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
        << " +---------------------------------------" << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | A    = " << m_A << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | B    = " << m_B << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | C    = " << m_C << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | D    = " << m_D << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | E    = " << m_E << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | F    = " << m_F << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | Y    = " << m_Y << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | Z    = " << m_Z << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | eta  = " << m_eta << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
        << " | C9~  = " << m_C9tilda << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
        << " | C10~ = " << m_C10tilda << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | P0   = " << m_P0 << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" ) << " | PE   = " << m_PE << std::endl;
    EvtGenReport( EVTGEN_DEBUG, "EvtGen" )
        << " +--------------------------------------" << std::endl;
}
 
EvtComplex EvtWilsonCoefficients::hzs( double z, double shat, double mu = 4.8,
                                       double M_b = 4.8 )
{
    EvtComplex i1( 0, 1 );
    double x = 4. * z * z / shat;
    if ( x == 0 )
        return ( 8. / 27. - 8. / 9. * log( M_b / mu ) - 4. / 9. * log( shat ) +
                 4. / 9. * i1 * EvtConst::pi );
    else if ( x > 1 )
        return ( 8. / 27. - 8. / 9. * log( M_b / mu ) - 8. / 9. * log( z ) +
                 4. / 9. * x -
                 2. / 9. * ( 2. + x ) * sqrt( x - 1. ) * 2 *
                     atan( 1. / sqrt( x - 1. ) ) );
    else
        return (
            8. / 27. - 8. / 9. * log( M_b / mu ) - 8. / 9. * log( z ) +
            4. / 9. * x -
            2. / 9. * ( 2. + x ) * sqrt( 1. - x ) *
                ( log( fabs( sqrt( 1. - x ) + 1 ) / fabs( sqrt( 1. - x ) - 1 ) ) -
                  i1 * EvtConst::pi ) );
}
 
double EvtWilsonCoefficients::fz( double z )
{
    return ( 1. - 8. * z * z + 8. * pow( z, 6. ) - pow( z, 8. ) -
             24. * pow( z, 4. ) * log( z ) );
}
 
double EvtWilsonCoefficients::kappa( double z, double alpha_S )
{
    return ( 1. - 2. * alpha_S / 3. / EvtConst::pi *
                      ( ( EvtConst::pi * EvtConst::pi - 31. / 4. ) *
                            ( 1. - z ) * ( 1. - z ) +
                        1.5 ) );
}
 
double EvtWilsonCoefficients::etatilda( double shat, double alpha_S )
{
    return ( 1. + alpha_S / EvtConst::pi * omega( shat ) );
}
 
double EvtWilsonCoefficients::omega( double shat )
{
    double o = 0;
    o -= ( 2. / 9. ) * EvtConst::pi * EvtConst::pi;
    o -= ( 4. / 3. ) * li2spence( shat );
    o -= ( 2. / 3. ) * log( shat ) * log( 1. - shat );
    o -= log( 1. - shat ) * ( 5. + 4. * shat ) / ( 3. + 6. * shat );
    o -= log( shat ) * 2. * shat * ( 1. + shat ) * ( 1. - 2. * shat ) / 3. /
         ( 1. - shat ) / ( 1. - shat ) / ( 1. + 2. * shat );
    o += ( 5. + 9. * shat - 6. * shat * shat ) / 6. / ( 1. - shat ) /
         ( 1. + 2. * shat );
    return o;
}
 
EvtComplex EvtWilsonCoefficients::C9efftilda( double z, double shat,
                                              double alpha_S, EvtComplex c1,
                                              EvtComplex c2, EvtComplex c3,
                                              EvtComplex c4, EvtComplex c5,
                                              EvtComplex c6, EvtComplex c9tilda,
                                              int ksi = 0 )
{
    EvtComplex c( 0, 0 );
    c += ( c9tilda + ksi * 4. / 9. * ( 3. * c1 + c2 - c3 - 3. * c4 ) ) *
         etatilda( shat, alpha_S );
    c += hzs( z, shat ) * ( 3. * c1 + c2 + 3. * c3 + c4 + 3. * c5 + c6 );
    c -= 0.5 * hzs( 1, shat ) * ( 4. * c3 + 4. * c4 + 3. * c5 + c6 );
    c -= 0.5 * hzs( 0, shat ) * ( c3 + 3. * c4 );
    c += 2. / 9. * ( 3. * c3 + c4 + 3. * c5 + c6 );
    return c;
}
 
EvtComplex EvtWilsonCoefficients::C7b2sg( double alpha_S, double et,
                                          EvtComplex c2, double M_t = 174.3,
                                          double M_W = 80.425 )
{
    EvtComplex i1( 0, 1 );
    return ( i1 * alpha_S *
             ( 2. / 9. * pow( et, 14. / 23. ) *
                   ( 0.5 * F( M_t * M_t / M_W / M_W ) - 0.1687 ) -
               0.03 * c2 ) );
}
 
EvtComplex EvtWilsonCoefficients::Yld( double q2, double* ki, double* Gi,
                                       double* Mi, int ni, EvtComplex c1,
                                       EvtComplex c2, EvtComplex c3,
                                       EvtComplex c4, EvtComplex c5,
                                       EvtComplex c6, double ialpha = 137.036 )
{
    EvtComplex i1( 0, 1 );
    EvtComplex y( 0, 0 );
    int i;
    for ( i = 0; i < ni; i++ )
        y += ki[i] * Gi[i] * Mi[i] / ( q2 - Mi[i] * Mi[i] - i1 * Mi[i] * Gi[i] );
    return ( -3. * ialpha * ialpha * y * EvtConst::pi *
             ( 3. * c1 + c2 + 3. * c3 + c4 + 3. * c5 + c6 ) );
}