EWSMTwoLoopQCD Class Reference

A class for \(O(\alpha\alpha_s)\) two-loop corrections to the EW precision observables. More...

#include <EWSMTwoLoopQCD.h>

Detailed Description

A class for \(O(\alpha\alpha_s)\) two-loop corrections to the EW precision observables.

Author

HEPfit Collaboration

Copyright

GNU General Public License

This class handles two-loop QCD contributions of \(O(\alpha\alpha_s)\) to the following quantities, which are relevant to the EW precision observables:

• \(\Delta\alpha_{\mathrm{lept}}(M_Z^2)\)   (with DeltaAlpha_l()),
• \(\Delta\alpha_{\mathrm{top}}(M_Z^2)\)   (with DeltaAlpha_t()),
• \(\Delta\rho\)       (with DeltaRho()),
• \(\Delta r_{\mathrm{rem}}\)   (with DeltaR_rem()),
• \(\delta\rho_{\mathrm{rem}}^{f}\)   (with deltaRho_rem_l() and deltaRho_rem_q()),
• \(\delta\kappa_{\mathrm{rem}}^{f}\)   (with deltaKappa_rem_l() and deltaKappa_rem_q()).

See also the description of EWSM class for their definitions. The above quantities are computed with the auxiliary functions:

• \(\delta_2^{\mathrm{QCD}}\)   (with deltaQCD_2()),
• \(\Delta r^{ud}\)   (with DeltaR_ud()),
• \(\Delta r^{tb}\)   (with DeltaR_tb()),
• \(\Delta \rho^{ud}\)   (with DeltaRho_ud()),
• \(\Delta \rho^{tb}\)   (with DeltaRho_tb()),
• \(\Delta \kappa^{ud}\)   (with DeltaKappa_ud()),
• \(\Delta \kappa^{tb}\)   (with DeltaKappa_tb()),

and

• \(F_1(x)\)   (with F1()),
• \(V_1(r)\)   (with V1()),
• \(V'_1(r)\)   (with V1prime()),
• \(A_1(r)\)   (with A1()),
• \(A'_1(r)\)   (with A1prime()).

The \(O(\alpha\alpha_s)\) two-loop QCD contributions to the vacuum polarization amplitudes of the gauge bosons were calculated in [Djouadi:1987gn], [Djouadi:1987di], [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu], [Kniehl:1992dx] and [Djouadi:1993ss].

Definition at line 55 of file EWSMTwoLoopQCD.h.

Public Member Functions
double	A1 (const double r) const
	The function \(A_1(r)\). More...
double	A1prime (const double r) const
	The derivative of the function \(A_1(r)\). More...
double	DeltaAlpha_l (const double s) const
	Leptonic contribution of \(O(\alpha\alpha_s)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{lept}}^{\alpha\alpha_s}(s)\). More...
double	DeltaAlpha_t (const double s) const
	Top-quark contribution of \(O(\alpha\alpha_s)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{top}}^{\alpha\alpha_s}(s)\). More...
gslpp::complex	deltaKappa_rem_f (const Particle f, const double Mw_i) const
	Remainder contribution of \(O(\alpha\alpha_s)\) to the effective couplings \(\kappa_Z^f\), denoted as \(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha\alpha_s}\). More...
gslpp::complex	DeltaKappa_tb (const double Mw_i) const
	Heavy-quark contribution to \(\kappa_Z^f\), denoted as \(\Delta\kappa^{tb}\). More...
gslpp::complex	DeltaKappa_ud (const double Mw_i) const
	Light-quark contribution to \(\kappa_Z^f\), denoted as \(\Delta\kappa^{ud}\). More...
double	deltaQCD_2 () const
	The function \(\delta^{\mathrm{QCD}}_2\). More...
double	DeltaR_rem (const double Mw_i) const
	Remainder contribution of \(O(\alpha\alpha_s)\) to \(\Delta r\), denoted as \(\Delta r_{\mathrm{rem}}^{\alpha\alpha_s^2}\). More...
double	DeltaR_tb (const double Mw_i) const
	Heavy-quark contribution to \(\Delta r\), denoted as \(\Delta r^{tb}\). More...
double	DeltaR_ud (const double Mw_i) const
	Light-quark contribution to \(\Delta r\), not including \(\Delta\alpha^{l+5q}(M_Z^2)\), denoted as \(\Delta r^{ud}\). More...
double	DeltaRho (const double Mw_i) const
	Leading two-loop QCD contribution of \(O(\alpha\alpha_s)\) to \(\Delta\rho\), denoted as \(\Delta\rho^{\alpha\alpha_s}\). More...
gslpp::complex	deltaRho_rem_f (const Particle f, const double Mw_i) const
	Remainder contribution of \(O(\alpha\alpha_s)\) to the effective couplings \(\rho_Z^f\), denoted as \(\delta\rho_{\mathrm{rem}}^{f,\, \alpha\alpha_s}\). More...
double	DeltaRho_tb (const double Mw_i) const
	Heavy-quark contribution to \(\rho_Z^f\), denoted as \(\Delta\rho^{tb}\). More...
double	DeltaRho_ud (const double Mw_i) const
	Light-quark contribution to \(\rho_Z^f\), denoted as \(\Delta\rho^{ud}\). More...
	EWSMTwoLoopQCD (const EWSMcache &cache_i)
	Constructor. More...
double	F1 (const double x, const double Mw_i) const
	The function \(F_1(x)\). More...
double	V1 (const double r) const
	The function \(V_1(r)\). More...
double	V1prime (const double r) const
	The derivative of the function \(V_1(r)\). More...

Private Attributes
const EWSMcache &	cache
	A reference to an object of type EWSMcache. More...

Constructor & Destructor Documentation

EWSMTwoLoopQCD()

EWSMTwoLoopQCD::EWSMTwoLoopQCD

(

const EWSMcache &

cache_i

)

Constructor.

Parameters

[in]

cache_i

a reference to an object of type EWSMcache

Definition at line 12 of file EWSMTwoLoopQCD.cpp.

: cache(cache_i)
{
}

Member Function Documentation

A1()

double EWSMTwoLoopQCD::A1

(

const double

r

)

const

The function \(A_1(r)\).

The expression for \(A_1(r)\) can be found in [Kniehl:1989yc] and [Halzen:1990je]. See also Chapter 8 of [Bardin:1999ak].

Parameters

[in]

r

the ratio \(r=s/(4m_t^2)\)

Returns

\(A_1(r)\)

Attention

This function is valid for \(0 \leq r < 1\).

Definition at line 163 of file EWSMTwoLoopQCD.cpp.

{
    if (r < 0.0 || r >= 1.0) throw std::runtime_error("r is out of range in EWSMTwoLoopQCD::A1");
 
    /* Zeta functions */
    double zeta_2 = cache.getZeta2();
    double zeta_3 = cache.getZeta3();
 
    if (r == 0.0) return (3.0 * (7.0 / 4.0 - zeta_2 - 2.0 * zeta_3));
 
    double Mz = cache.getSM().getMz();
    double Mt = cache.getSM().getMtpole();
 
    /* Logarithms etc */
    double Phi, gamma, h;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Phi = cache.Phi_QCD2();
        gamma = cache.gamma_QCD2();
        h = cache.h_QCD2();
    } else {
        Phi = asin(sqrt(r));
        gamma = log(2.0 * sqrt(r));
        h = log(2.0 * sqrt(1.0 - r));
    }
 
    /* Clausen functions */
    double Cl3_2Phi, Cl3_4Phi, Cl2_2Phi, Cl2_4Phi;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Cl3_2Phi = cache.Cl3_2Phi();
        Cl3_4Phi = cache.Cl3_4Phi();
        Cl2_2Phi = cache.Cl2_2Phi();
        Cl2_4Phi = cache.Cl2_4Phi();
    } else {
        ClausenFunctions* myClausen;
        myClausen = new ClausenFunctions();
        Cl3_2Phi = myClausen->Cl3(2.0 * Phi);
        Cl3_4Phi = myClausen->Cl3(4.0 * Phi);
        Cl2_2Phi = myClausen->Cl2(2.0 * Phi);
        Cl2_4Phi = myClausen->Cl2(4.0 * Phi);
        delete myClausen;
    }
 
    double A1;
    A1 = 4.0 * (r - 3.0 / 2.0 + 1.0 / 2.0 / r)*(2.0 * Cl3_2Phi - Cl3_4Phi
            + 8.0 / 3.0 * Phi * (Cl2_2Phi - Cl2_4Phi) - 4.0 / 3.0 * Phi * Phi * (gamma + 2.0 * h))
            + sqrt(1.0 / r - 1.0)*(8.0 / 3.0 * (r - 1.0)
            *(-Cl2_2Phi + Cl2_4Phi + 2.0 * Phi * (gamma + 2.0 * h))
            - 2.0 * Phi * (r - 3.0 + 1.0 / 4.0 / r))
            + 8.0 * Phi * Phi * (r - 11.0 / 12.0 + 5.0 / 48.0 / r + 1.0 / 32.0 / r / r)
            - 3.0 * zeta_2 + 13.0 / 6.0 + (-2.0 * zeta_3 + 1.0 / 4.0) / r;
    return A1;
}

A1prime()

double EWSMTwoLoopQCD::A1prime

(

const double

r

)

const

The derivative of the function \(A_1(r)\).

The expression for \(A'_1(r)\) has been derived from \(A_1(r)\) in [Kniehl:1989yc] and [Halzen:1990je].

Parameters

[in]

r

the ratio \(r=s/(4m_t^2)\)

Returns

\(A'_1(r)\)

Attention

This function is valid for \(0 \leq r < 1\).

Definition at line 309 of file EWSMTwoLoopQCD.cpp.

{
    if (r < 0.0 || r >= 1.0) throw std::runtime_error("r is out of range in EWSMTwoLoopQCD::A1prime");
 
    /* Zeta functions */
    double zeta_2 = cache.getZeta2();
    double zeta_3 = cache.getZeta3();
 
    if (r == 0.0) return (3.0 * (7.0 / 4.0 - zeta_2 - 2.0 * zeta_3));
 
    double Mz = cache.getSM().getMz();
    double Mt = cache.getSM().getMtpole();
 
    /* Logarithms etc */
    double Phi, gamma, h;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Phi = cache.Phi_QCD2();
        gamma = cache.gamma_QCD2();
        h = cache.h_QCD2();
    } else {
        Phi = asin(sqrt(r));
        gamma = log(2.0 * sqrt(r));
        h = log(2.0 * sqrt(1.0 - r));
    }
 
    /* Clausen functions */
    double Cl3_2Phi, Cl3_4Phi, Cl2_2Phi, Cl2_4Phi;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Cl3_2Phi = cache.Cl3_2Phi();
        Cl3_4Phi = cache.Cl3_4Phi();
        Cl2_2Phi = cache.Cl2_2Phi();
        Cl2_4Phi = cache.Cl2_4Phi();
    } else {
        ClausenFunctions* myClausen;
        myClausen = new ClausenFunctions();
        Cl3_2Phi = myClausen->Cl3(2.0 * Phi);
        Cl3_4Phi = myClausen->Cl3(4.0 * Phi);
        Cl2_2Phi = myClausen->Cl2(2.0 * Phi);
        Cl2_4Phi = myClausen->Cl2(4.0 * Phi);
        delete myClausen;
    }
 
    /* for Bprime and Dprime below */
    gsl_complex OneMinusE2Iphi = gsl_complex_rect(1.0 - cos(2.0 * Phi), -sin(2.0 * Phi));
    gsl_complex OneMinusE4Iphi = gsl_complex_rect(1.0 - cos(4.0 * Phi), -sin(4.0 * Phi));
    double log_real;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        log_real = cache.logV1primeAndA1prime();
    } else {
        log_real = GSL_REAL(gsl_complex_log(OneMinusE2Iphi))
                - 2.0 * GSL_REAL(gsl_complex_log(OneMinusE4Iphi));
    }
 
    double PhiPrime = 1.0 / 2.0 / sqrt(r * (1.0 - r));
    double Phi2Prime = Phi / sqrt(r * (1.0 - r));
    double gammaPrime = 1.0 / 2.0 / r;
    double hPrime = -1.0 / 2.0 / (1.0 - r);
 
    // A1(r) = 4.0*A*B + C*D + E
    double A = r - 3.0 / 2.0 + 1.0 / 2.0 / r;
    double Aprime = 1.0 - 1.0 / 2.0 / r / r;
    double B = 2.0 * Cl3_2Phi - Cl3_4Phi + 8.0 / 3.0 * Phi * (Cl2_2Phi - Cl2_4Phi)
            - 4.0 / 3.0 * Phi * Phi * (gamma + 2.0 * h);
    double Bprime = -2.0 / sqrt(r * (1.0 - r))*(Cl2_2Phi - Cl2_4Phi)
            + 8.0 / 3.0 * PhiPrime * (Cl2_2Phi - Cl2_4Phi)
            + 8.0 / 3.0 * Phi * (-log_real / sqrt(r * (1.0 - r)))
            - 4.0 / 3.0 * Phi2Prime * (gamma + 2.0 * h)
            - 4.0 / 3.0 * Phi * Phi * (gammaPrime + 2.0 * hPrime);
    double C = sqrt(1.0 / r - 1.0);
    double Cprime = -1.0 / 2.0 / r / sqrt(r * (1.0 - r));
    double D = 8.0 / 3.0 * (r - 1.0)*(-Cl2_2Phi + Cl2_4Phi + 2.0 * Phi * (gamma + 2.0 * h))
            - 2.0 * Phi * (r - 3.0 + 1.0 / 4.0 / r);
    double Dprime = 8.0 / 3.0 * (-Cl2_2Phi + Cl2_4Phi + 2.0 * Phi * (gamma + 2.0 * h))
            + 8.0 / 3.0 * (r - 1.0)*(log_real / sqrt(r * (1.0 - r))
            + 2.0 * PhiPrime * (gamma + 2.0 * h)
            + 2.0 * Phi * (gammaPrime + 2.0 * hPrime))
            - 2.0 * PhiPrime * (r - 3.0 + 1.0 / 4.0 / r)
            - 2.0 * Phi * (1.0 - 1.0 / 4.0 / r / r);
    double Eprime = 8.0 * Phi2Prime * (r - 11.0 / 12.0 + 5.0 / 48.0 / r + 1.0 / 32.0 / r / r)
            + 8.0 * Phi * Phi * (1.0 - 5.0 / 48.0 / r / r - 1.0 / 16.0 / r / r / r)
            - (-2.0 * zeta_3 + 1.0 / 4.0) / r / r;
    return (4.0 * Aprime * B + 4.0 * A * Bprime + Cprime * D + C * Dprime + Eprime);
 
    /* TEST: Exact - Expansion */
    //std::cout << "A1:" 
    //          <<  (4.0*Aprime*B + 4.0*A*Bprime + Cprime*D + C*Dprime + Eprime)
    //             - (4.0*zeta_3 - 49.0/18.0 + 1378.0/405.0*r )
    //          << std::endl;
 
    /* Expansion for r << 1 */
    //return (4.0*zeta_3 - 49.0/18.0 + 1378.0/405.0*r);    
}

DeltaAlpha_l()

double EWSMTwoLoopQCD::DeltaAlpha_l

(

const double

s

)

const

Leptonic contribution of \(O(\alpha\alpha_s)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{lept}}^{\alpha\alpha_s}(s)\).

This contribution vanishes at \(O(\alpha\alpha_s)\).

Parameters

[in]

s

invariant mass squared

Returns

\(\Delta\alpha_{\mathrm{lept}}^{\alpha\alpha_s}(s)=0\)

Definition at line 20 of file EWSMTwoLoopQCD.cpp.

{
    return (0.0);
}

DeltaAlpha_t()

double EWSMTwoLoopQCD::DeltaAlpha_t

(

const double

s

)

const

Top-quark contribution of \(O(\alpha\alpha_s)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{top}}^{\alpha\alpha_s}(s)\).

A simple numerical formula presented in [Kuhn:1998ze] has been employed.

Parameters

[in]

s

invariant mass squared

Returns

\(\Delta\alpha_{\mathrm{top}}^{\alpha\alpha_s}(s)\)

Definition at line 25 of file EWSMTwoLoopQCD.cpp.

{
    double xt = s / cache.getSM().getMtpole() / cache.getSM().getMtpole();
    double als;
    if (s == cache.getSM().getMz() * cache.getSM().getMz())
        als = cache.getSM().getAlsMz();
    else
        als = cache.Als(sqrt(s), FULLNNLO);
    double tmp = (5.062 + xt * 0.8315) * als / M_PI;
    tmp *= -4.0 / 45.0 * cache.getSM().getAle() / M_PI*xt;
    return tmp;
}

deltaKappa_rem_f()

gslpp::complex EWSMTwoLoopQCD::deltaKappa_rem_f	(	const Particle	f,
		const double	Mw_i
	)		const

Remainder contribution of \(O(\alpha\alpha_s)\) to the effective couplings \(\kappa_Z^f\), denoted as \(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha\alpha_s}\).

The \(O(\alpha\alpha_s)\) remainder contribution to \(\kappa_{Z}^{f}\) is decomposed as

\[ \delta\kappa_{\mathrm{rem}}^{f,\,\alpha\alpha_s} = 2 \Delta\kappa^{ud} + \Delta\kappa^{tb} - \frac{c_W^2}{s_W^2}\Delta\rho^{\alpha\alpha_s}, \]

where \(\Delta\kappa^{ud}\) and \(\Delta\kappa^{tb}\) are associated with corrections to the self-energies of the gauge bosons with loops of a light-quark doublet and with those of the \(t\)- \(b\) doublet, respectively, and \((c_W^2/s_W^2)\Delta\rho^{\alpha\alpha_s}\) is the leading contribution of \(O(\alpha\alpha_s)\) to \(\kappa_{Z}^{f}\).

Parameters

[in]	f	a lepton or quark
[in]	Mw_i	the \(W\)-boson mass

Returns

\(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha\alpha_s}\)

Definition at line 58 of file EWSMTwoLoopQCD.cpp.

{
    if (f.is("TOP")) return ( gslpp::complex(0.0, 0.0, false));
    double Mw = Mw_i;
    return ( (2.0 * DeltaKappa_ud(Mw) + DeltaKappa_tb(Mw))
            - cache.getSM().cW2(Mw) / cache.getSM().sW2(Mw) * DeltaRho(Mw));
}

DeltaKappa_tb()

gslpp::complex EWSMTwoLoopQCD::DeltaKappa_tb

(

const double

Mw_i

)

const

Heavy-quark contribution to \(\kappa_Z^f\), denoted as \(\Delta\kappa^{tb}\).

The quantity \(\Delta\kappa^{tb}\) is associated with \(O(\alpha\alpha_s)\) corrections to the self-energies of the gauge bosons with loops of the \(t\)- \(b\) doublet. The expression of \(\Delta\kappa^{tb}\) is given by

\[ \Delta\kappa^{tb} = \frac{\alpha\alpha_s(M_t^2)}{\pi^2} \frac{1}{4s_W^4} \Bigg\{ 4 c_W^2 w_t \Big[ v_t^2 V_1(r^Z_{4t}) + a_t^2A_1(r^Z_{4t}) - F_1(x^W_{t}) \Big] + 4s_W^2 \left( |Q_t| - 4s_W^2 Q_t^2 \right) z_t V_1(r^Z_{4t}) + \Big[ v_b^2 + a_b ^2 + s_W^2 \left( |Q_b| - 4s_W^2 Q_b^2 \right) \Big]\ln z_t \Bigg\} + i \frac{\alpha\alpha_s(M_Z^2)}{4\pi s_W^2} \left( \frac{1}{3} - \frac{4}{9}s_W^2 \right), \]

where the definitions of the symbols can be read from the codes below. See [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu] and Chapter 8 of [Bardin:1999ak].

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta\kappa^{tb}\)

Definition at line 507 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
    double Mt = cache.getSM().getMtpole();
    double wt = Mt * Mt / Mw / Mw;
    double zt = Mt * Mt / Mz / Mz;
    double rZ4t = Mz * Mz / 4.0 / Mt / Mt;
    double xWt = Mw * Mw / Mt / Mt;
 
    double vt = cache.v_f(cache.getSM().getQuarks(QCD::TOP), Mw);
    double at = cache.a_f(cache.getSM().getQuarks(QCD::TOP));
    double Qt = cache.Q_f(cache.getSM().getQuarks(QCD::TOP));
    double vb = cache.v_f(cache.getSM().getQuarks(QCD::BOTTOM), Mw);
    double ab = cache.a_f(cache.getSM().getQuarks(QCD::BOTTOM));
    double Qb = cache.Q_f(cache.getSM().getQuarks(QCD::BOTTOM));
 
    /* Logarithm */
    double log_zt = -2.0 * cache.logMZtoMTOP();
 
    gslpp::complex DeltaKappa(0.0, 0.0, false);
    DeltaKappa = 4.0 * cW2 * wt * (vt * vt * V1(rZ4t) + at * at * A1(rZ4t) - F1(xWt, Mw))
            + 4.0 * sW2 * (fabs(Qt) - 4.0 * sW2 * Qt * Qt) * zt * V1(rZ4t)
            + (vb * vb + ab * ab + sW2 * (fabs(Qb) - 4.0 * sW2 * Qb * Qb)) * log_zt;
    DeltaKappa += M_PI * sW2 * (1.0 / 3.0 - 4.0 / 9.0 * sW2)*(gslpp::complex::i());
    DeltaKappa /= 4.0 * sW2*sW2;
    DeltaKappa *= cache.getSM().getAle() * cache.alsMt() / M_PI / M_PI;
    return DeltaKappa;
}

DeltaKappa_ud()

gslpp::complex EWSMTwoLoopQCD::DeltaKappa_ud

(

const double

Mw_i

)

const

Light-quark contribution to \(\kappa_Z^f\), denoted as \(\Delta\kappa^{ud}\).

The quantity \(\Delta\kappa^{ud}\) is associated with \(O(\alpha\alpha_s)\) corrections to the self-energies of the gauge bosons with loops of the light-quark doublets. The expression of \(\Delta\kappa^{ud}\) is given by

\[ \Delta\kappa^{ud} = \frac{\alpha\alpha_s(M_Z^2)}{\pi^2} \frac{c_W^2}{4 s_W^4}\ln c_W^2 + i \frac{\alpha\alpha_s(M_Z^2)}{4\pi s_W^2} \left( 1 - \frac{20}{9}s_W^2 \right), \]

where the definitions of the symbols can be read from the codes below. See [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu] and Chapter 8 of [Bardin:1999ak].

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta\kappa^{ud}\)

Definition at line 491 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
 
    /* Logarithm */
    double log_cW2 = cache.log_cW2(Mw);
 
    gslpp::complex DeltaKappa(0.0, 0.0, false);
    DeltaKappa = cW2 / 4.0 / sW2 / sW2 * log_cW2
            + M_PI / 4.0 / sW2 * (1.0 - 20.0 / 9.0 * sW2)*(gslpp::complex::i());
    DeltaKappa *= cache.getSM().getAle() * cache.getSM().getAlsMz() / M_PI / M_PI;
    return DeltaKappa;
}

deltaQCD_2()

double EWSMTwoLoopQCD::deltaQCD_2

(

)

const

The function \(\delta^{\mathrm{QCD}}_2\).

This function is associated with the leading two-loop QCD contribution of \(O(\alpha\alpha_s m_t^2/M_Z^2)\) to \(\Delta\rho\), as explained in the description of DeltaRho(). See [Kniehl:1989yc], [Halzen:1990je] and Chapter 8 of [Bardin:1999ak].

Returns

\(\delta^{\mathrm{QCD}}_2\)

Definition at line 69 of file EWSMTwoLoopQCD.cpp.

{
    return ( -2.0 / 3.0 * (1.0 + 2.0 * cache.getZeta2()));
}

DeltaR_rem()

double EWSMTwoLoopQCD::DeltaR_rem

(

const double

Mw_i

)

const

Remainder contribution of \(O(\alpha\alpha_s)\) to \(\Delta r\), denoted as \(\Delta r_{\mathrm{rem}}^{\alpha\alpha_s^2}\).

The \(O(\alpha\alpha_s)\) remainder contribution to \(\Delta r\) is decomposed as

\[ \Delta r_{\mathrm{rem}}^{\alpha\alpha_s} = 2 \Delta r^{ud} + \Delta r^{tb} + \frac{c_W^2}{s_W^2} \Delta\rho^{\alpha\alpha_s}, \]

where \(\Delta r^{ud}\) and \(\Delta r^{tb}\) are associated with corrections to the self-energies of the gauge bosons with loops of a light-quark doublet and with those of the \(t\)- \(b\) doublet, respectively, and \(\Delta\rho^{\alpha\alpha_s}\) is the leading contribution.

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta r_{\mathrm{rem}}^{\alpha\alpha_s^2}\)

Definition at line 44 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    return ( (2.0 * DeltaR_ud(Mw) + DeltaR_tb(Mw))
            + cache.getSM().cW2(Mw) / cache.getSM().sW2(Mw) * DeltaRho(Mw));
}

DeltaR_tb()

double EWSMTwoLoopQCD::DeltaR_tb

(

const double

Mw_i

)

const

Heavy-quark contribution to \(\Delta r\), denoted as \(\Delta r^{tb}\).

The quantity \(\Delta r^{tb}\) is associated with \(O(\alpha\alpha_s)\) corrections to the self-energies of the gauge bosons with loops of the \(t\)- \(b\) doublet. The expression of \(\Delta r^{tb}\) is given by

\[ \Delta r^{tb} = \frac{\alpha\alpha_s(M_t^2)}{\pi^2} \Bigg\{ Q_t^2 V_1'(0) + \frac{c_W^2}{s_W^4}\frac{w_t}{4}\left[ \zeta(2) + \frac{1}{2} \right] - \frac{z_t}{s_W^4} \Big[ v_t^2 V_1(r^Z_{4t}) + a_t^2\left[A_1(r^Z_{4t}) - A_1(0)\right] \Big] + \frac{c_W^2 - s_W^2}{s_W^4} w_t \Big[ F_1(x^W_{t}) - F_1(0) \Big] - \frac{v_ba_b}{2 s_W^4} \ln z_t \Bigg\}, \]

where the definitions of the symbols can be read from the codes below. See [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu] and Chapter 8 of [Bardin:1999ak].

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta r^{tb}\)

Definition at line 418 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
    double Mz = cache.getSM().getMz();
    double Mt = cache.getSM().getMtpole();
    double wt = Mt * Mt / Mw / Mw;
    double zt = Mt * Mt / Mz / Mz;
    double rZ4t = Mz * Mz / 4.0 / Mt / Mt;
    double xWt = Mw * Mw / Mt / Mt;
 
    double vt = cache.v_f(cache.getSM().getQuarks(QCD::TOP), Mw);
    double at = cache.a_f(cache.getSM().getQuarks(QCD::TOP));
    double vb = cache.v_f(cache.getSM().getQuarks(QCD::BOTTOM), Mw);
    double ab = cache.a_f(cache.getSM().getQuarks(QCD::BOTTOM));
 
    /* Zeta functions */
    double zeta_2 = cache.getZeta2();
 
    /* Logarithm */
    double log_zt = -2.0 * cache.logMZtoMTOP();
 
    double DeltaR;
    DeltaR = pow(cache.Q_f(cache.getSM().getQuarks(QCD::TOP)), 2.0) * V1prime(0.0)
            + cW2 / sW2 / sW2 * wt / 4.0 * (zeta_2 + 1.0 / 2.0)
            - zt / sW2 / sW2 * (vt * vt * V1(rZ4t) + at * at * (A1(rZ4t) - A1(0.0)))
            + (cW2 - sW2) / sW2 / sW2 * wt * (F1(xWt, Mw) - F1(0.0, Mw))
            - vb * ab / 2.0 / sW2 / sW2*log_zt;
    DeltaR *= cache.getSM().getAle() * cache.alsMt() / M_PI / M_PI;
    return DeltaR;
}

DeltaR_ud()

double EWSMTwoLoopQCD::DeltaR_ud

(

const double

Mw_i

)

const

Light-quark contribution to \(\Delta r\), not including \(\Delta\alpha^{l+5q}(M_Z^2)\), denoted as \(\Delta r^{ud}\).

The quantity \(\Delta r^{ud}\) is associated with \(O(\alpha\alpha_s)\) corrections to the self-energies of the gauge bosons with loops of the light-quark doublets. The expression of \(\Delta r^{ud}\) is given by

\[ \Delta r^{ud} = - \frac{\alpha\alpha_s(M_Z^2)}{\pi^2}\frac{c_W^2 - s_W^2}{4s_W^4}\,\ln c_W^2. \]

See [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu] and Chapter 8 of [Bardin:1999ak].

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta r^{ud}\)

Definition at line 402 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
 
    /* Logarithm */
    double log_cW2 = cache.log_cW2(Mw);
 
    double DeltaR;
    DeltaR = -log_cW2;
    DeltaR *= (cW2 - sW2) / 4.0 / sW2 / sW2;
    DeltaR *= cache.getSM().getAle() * cache.getSM().getAlsMz() / M_PI / M_PI;
    return DeltaR;
}

DeltaRho()

double EWSMTwoLoopQCD::DeltaRho

(

const double

Mw_i

)

const

Leading two-loop QCD contribution of \(O(\alpha\alpha_s)\) to \(\Delta\rho\), denoted as \(\Delta\rho^{\alpha\alpha_s}\).

The formula used here is given by

\[ \Delta\rho^{\alpha\alpha_s} = 3\,X_t^\alpha\frac{\alpha_s(m_t^2)}{\pi} \delta^{\mathrm{QCD}}_2, \]

where \(X_t^\alpha = \alpha\, m_t^2/(16\pi s_W^2 M_W^2)\), and \(\delta^{\mathrm{QCD}}_2\) is computed via deltaQCD_2(). See, e.g., Chapter 8 of [Bardin:1999ak]. This quantity contributes to \(\Delta r\) and the \(Zf\bar{f}\) effective couplings \(\rho_Z^f\) and \(\kappa_Z^f\). See also the description of EWSM class.

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta\rho^{\alpha\alpha_s}\)

Definition at line 38 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    return ( 3.0 * cache.Xt_alpha(Mw) * cache.alsMt() / M_PI * deltaQCD_2());
}

deltaRho_rem_f()

gslpp::complex EWSMTwoLoopQCD::deltaRho_rem_f	(	const Particle	f,
		const double	Mw_i
	)		const

Remainder contribution of \(O(\alpha\alpha_s)\) to the effective couplings \(\rho_Z^f\), denoted as \(\delta\rho_{\mathrm{rem}}^{f,\, \alpha\alpha_s}\).

The \(O(\alpha\alpha_s)\) remainder contribution to \(\rho_{Z}^{f}\) is decomposed as

\[ \delta\rho_{\mathrm{rem}}^{f,\,\alpha\alpha_s} = 2 \Delta\rho^{ud} + \Delta\rho^{tb} - \Delta\rho^{\alpha\alpha_s}, \]

where \(\Delta\rho^{ud}\) and \(\Delta\rho^{tb}\) are associated with corrections to the self-energies of the gauge bosons with loops of a light-quark doublet and with those of the \(t\)- \(b\) doublet, respectively, and \(\Delta\rho^{\alpha\alpha_s}\) is the leading contribution of \(O(\alpha\alpha_s)\) to \(\rho_{Z}^{f}\).

Parameters

[in]	f	a lepton or quark
[in]	Mw_i	the \(W\)-boson mass

Returns

\(\delta\rho_{\mathrm{rem}}^{f,\, \alpha\alpha_s}\)

Definition at line 51 of file EWSMTwoLoopQCD.cpp.

{
    if (f.is("TOP")) return ( gslpp::complex(0.0, 0.0, false));
    double Mw = Mw_i;
    return ( (2.0 * DeltaRho_ud(Mw) + DeltaRho_tb(Mw)) - DeltaRho(Mw));
}

DeltaRho_tb()

double EWSMTwoLoopQCD::DeltaRho_tb

(

const double

Mw_i

)

const

Heavy-quark contribution to \(\rho_Z^f\), denoted as \(\Delta\rho^{tb}\).

The quantity \(\Delta\rho^{tb}\) is associated with \(O(\alpha\alpha_s)\) corrections to the self-energies of the gauge bosons with loops of the \(t\)- \(b\) doublet. The expression of \(\Delta\rho^{tb}\) is given by

\[ \Delta\rho^{tb} = \frac{\alpha\alpha_s(M_t^2)}{\pi^2} \frac{1}{4s_W^2 c_W^2} \Big\{ - \left[ v_t^2 V_1'(r^Z_{4t}) + a_t^2 A_1'(r^Z_{4t}) \right] + 4 z_t \left[ v_t^2 V_1(r^Z_{4t}) + a_t^2 A_1(r^Z_{4t}) \right] + v_b^2 + a_b^2 - 4 z_t\, F_1(0) \Big\}, \]

where the definitions of the symbols can be read from the codes below. See [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu] and Chapter 8 of [Bardin:1999ak].

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta\rho^{tb}\)

Definition at line 467 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
    double Mt = cache.getSM().getMtpole();
    double zt = Mt * Mt / Mz / Mz;
    double rZ4t = Mz * Mz / 4.0 / Mt / Mt;
 
    double vt = cache.v_f(cache.getSM().getQuarks(QCD::TOP), Mw);
    double at = cache.a_f(cache.getSM().getQuarks(QCD::TOP));
    double vb = cache.v_f(cache.getSM().getQuarks(QCD::BOTTOM), Mw);
    double ab = cache.a_f(cache.getSM().getQuarks(QCD::BOTTOM));
 
    double DeltaRho;
    DeltaRho = -(vt * vt * V1prime(rZ4t) + at * at * A1prime(rZ4t))
            + 4.0 * zt * (vt * vt * V1(rZ4t) + at * at * A1(rZ4t))
            + vb * vb + ab * ab - 4.0 * zt * F1(0.0, Mw);
    DeltaRho /= 4.0 * sW2*cW2;
    DeltaRho *= cache.getSM().getAle() * cache.alsMt() / M_PI / M_PI;
    return DeltaRho;
}

DeltaRho_ud()

double EWSMTwoLoopQCD::DeltaRho_ud

(

const double

Mw_i

)

const

Light-quark contribution to \(\rho_Z^f\), denoted as \(\Delta\rho^{ud}\).

The quantity \(\Delta\rho^{ud}\) is associated with \(O(\alpha\alpha_s)\) corrections to the self-energies of the gauge bosons with loops of the light-quark doublets. The expression of \(\Delta\rho^{ud}\) is given by

\[ \Delta\rho^{ud} = \frac{\alpha\alpha_s(M_Z^2)}{\pi^2} \frac{1}{4s_W^2 c_W^2} \left( v_u^2 + v_d^2 + a_u^2 + a_d^2 \right), \]

where the definitions of the symbols can be read from the codes below. See [Kniehl:1989yc], [Halzen:1990je], [Kniehl:1991gu] and Chapter 8 of [Bardin:1999ak].

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta\rho^{ud}\)

Definition at line 451 of file EWSMTwoLoopQCD.cpp.

{
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
 
    double DeltaRho;
    DeltaRho = pow(cache.v_f(cache.getSM().getQuarks(QCD::UP), Mw), 2.0)
            + pow(cache.v_f(cache.getSM().getQuarks(QCD::DOWN), Mw), 2.0)
            + pow(cache.a_f(cache.getSM().getQuarks(QCD::UP)), 2.0)
            + pow(cache.a_f(cache.getSM().getQuarks(QCD::TOP)), 2.0);
    DeltaRho /= 4.0 * sW2*cW2;
    DeltaRho *= cache.getSM().getAle() * cache.getSM().getAlsMz() / M_PI / M_PI;
    return DeltaRho;
}

F1()

double EWSMTwoLoopQCD::F1	(	const double	x,
		const double	Mw_i
	)		const

The function \(F_1(x)\).

The expression for \(F_1(x)\) can be found in [Kniehl:1989yc] and [Halzen:1990je]. See also Chapter 8 of [Bardin:1999ak].

Parameters

[in]	x	the ratio \(x=s/m_t^2\)
[in]	Mw_i	the \(W\)-boson mass

Returns

\(F_1(x)\)

Attention

This function is valid for \(0 \leq x < 1\).

Definition at line 74 of file EWSMTwoLoopQCD.cpp.

{
    if (x < 0.0 || x >= 1.0) throw std::runtime_error("x is out of range in EWSMTwoLoopQCD::F1");
 
    /* Zeta functions */
    double zeta_2 = cache.getZeta2();
    double zeta_3 = cache.getZeta3();
 
    if (x == 0.0) return (23.0 / 16.0 - zeta_2 / 2.0 - 3.0 / 2.0 * zeta_3);
 
    /* Dilogarithm and Trilogarithm */
    double Li2_x, Li3_x, Li3_mx_1mx;
    double Mw = Mw_i;
    double Mt = cache.getSM().getMtpole();
    if (x == Mw * Mw / Mt / Mt) {
        Li2_x = cache.Li2_MW2toMTOP2(Mw);
        Li3_x = cache.Li3_MW2toMTOP2(Mw);
        Li3_mx_1mx = cache.Li3_for_F1(Mw);
    } else {
        Li2_x = cache.getPolyLog().Li2(x).real(); // x <= 1.0
        Li3_x = cache.getPolyLog().Li3(x);
        Li3_mx_1mx = cache.getPolyLog().Li3(-x / (1.0 - x));
    }
 
    double b = log(1.0 - x);
 
    double F1;
    F1 = (x - 3.0 / 2.0 + 1.0 / 2.0 / x / x)
            *(-Li3_x - Li3_mx_1mx + b / 3.0 * (2.0 * Li2_x - zeta_2) + b * b * b / 6.0)
            + 1.0 / 3.0 * (x + 1.0 / 2.0 - 1.0 / 2.0 / x) * Li2_x
            + b * b / 6.0 * (x - 3.0 / 4.0 - 3.0 / 2.0 / x + 5.0 / 4.0 / x / x)
            - b / 4.0 * (x - 5.0 / 2.0 + 2.0 / 3.0 / x + 5.0 / 6.0 / x / x)
            + zeta_3 * (x - 3.0 / 2.0) + zeta_2 / 3.0 * (x - 7.0 / 4.0 - 1.0 / 2.0 / x)
            + 13.0 / 12.0 - 5.0 / 24.0 / x;
    return F1;
}

V1()

double EWSMTwoLoopQCD::V1

(

const double

r

)

const

The function \(V_1(r)\).

The expression for \(V_1(r)\) can be found in [Kniehl:1989yc] and [Halzen:1990je]. See also Chapter 8 of [Bardin:1999ak].

Parameters

[in]

r

the ratio \(r=s/(4m_t^2)\)

Returns

\(V_1(r)\)

Attention

This function is valid for \(0 \leq r < 1\).

Definition at line 111 of file EWSMTwoLoopQCD.cpp.

{
    if (r < 0.0 || r >= 1.0) throw std::runtime_error("r is out of range in EWSMTwoLoopQCD::V1");
 
    /* Zeta functions */
    double zeta_3 = cache.getZeta3();
 
    if (r == 0.0) return (0.0);
 
    double Mz = cache.getSM().getMz();
    double Mt = cache.getSM().getMtpole();
 
    /* Logarithms etc */
    double Phi, gamma, h;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Phi = cache.Phi_QCD2();
        gamma = cache.gamma_QCD2();
        h = cache.h_QCD2();
    } else {
        Phi = asin(sqrt(r));
        gamma = log(2.0 * sqrt(r));
        h = log(2.0 * sqrt(1.0 - r));
    }
 
    /* Clausen functions */
    double Cl3_2Phi, Cl3_4Phi, Cl2_2Phi, Cl2_4Phi;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Cl3_2Phi = cache.Cl3_2Phi();
        Cl3_4Phi = cache.Cl3_4Phi();
        Cl2_2Phi = cache.Cl2_2Phi();
        Cl2_4Phi = cache.Cl2_4Phi();
    } else {
        ClausenFunctions* myClausen;
        myClausen = new ClausenFunctions();
        Cl3_2Phi = myClausen->Cl3(2.0 * Phi);
        Cl3_4Phi = myClausen->Cl3(4.0 * Phi);
        Cl2_2Phi = myClausen->Cl2(2.0 * Phi);
        Cl2_4Phi = myClausen->Cl2(4.0 * Phi);
        delete myClausen;
    }
 
    double V1;
    V1 = 4.0 * (r - 1.0 / 4.0 / r)
            *(2.0 * Cl3_2Phi - Cl3_4Phi + 8.0 / 3.0 * Phi * (Cl2_2Phi - Cl2_4Phi)
            - 4.0 / 3.0 * Phi * Phi * (gamma + 2.0 * h))
            + sqrt(1.0 / r - 1.0)*(8.0 / 3.0 * (r + 1.0 / 2.0)
            *(-Cl2_2Phi + Cl2_4Phi + 2.0 * Phi * (gamma + 2.0 * h))
            - 2.0 * Phi * (r + 3.0 / 2.0))
            + 8.0 * Phi * Phi * (r - 1.0 / 6.0 - 7.0 / 48.0 / r) + 13.0 / 6.0 + zeta_3 / r;
    return V1;
}

V1prime()

double EWSMTwoLoopQCD::V1prime

(

const double

r

)

const

The derivative of the function \(V_1(r)\).

The expression for \(V'_1(r)\) has been derived from \(V_1(r)\) in [Kniehl:1989yc] and [Halzen:1990je].

Parameters

[in]

r

the ratio \(r=s/(4m_t^2)\)

Returns

\(V'_1(r)\)

Attention

This function is valid for \(0 \leq r < 1\).

Definition at line 216 of file EWSMTwoLoopQCD.cpp.

{
    if (r < 0.0 || r >= 1.0) throw std::runtime_error("r is out of range in EWSMTwoLoopQCD::V1prime");
 
    /* Zeta functions */
    double zeta_3 = cache.getZeta3();
 
    if (r == 0.0) return (4.0 * zeta_3 - 5.0 / 6.0);
 
    double Mz = cache.getSM().getMz();
    double Mt = cache.getSM().getMtpole();
 
    /* Logarithms etc */
    double Phi, gamma, h;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Phi = cache.Phi_QCD2();
        gamma = cache.gamma_QCD2();
        h = cache.h_QCD2();
    } else {
        Phi = asin(sqrt(r));
        gamma = log(2.0 * sqrt(r));
        h = log(2.0 * sqrt(1.0 - r));
    }
 
    /* Clausen functions */
    double Cl3_2Phi, Cl3_4Phi, Cl2_2Phi, Cl2_4Phi;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        Cl3_2Phi = cache.Cl3_2Phi();
        Cl3_4Phi = cache.Cl3_4Phi();
        Cl2_2Phi = cache.Cl2_2Phi();
        Cl2_4Phi = cache.Cl2_4Phi();
    } else {
        ClausenFunctions* myClausen;
        myClausen = new ClausenFunctions();
        Cl3_2Phi = myClausen->Cl3(2.0 * Phi);
        Cl3_4Phi = myClausen->Cl3(4.0 * Phi);
        Cl2_2Phi = myClausen->Cl2(2.0 * Phi);
        Cl2_4Phi = myClausen->Cl2(4.0 * Phi);
        delete myClausen;
    }
 
    /* for Bprime and Dprime below */
    gsl_complex OneMinusE2Iphi = gsl_complex_rect(1.0 - cos(2.0 * Phi), -sin(2.0 * Phi));
    gsl_complex OneMinusE4Iphi = gsl_complex_rect(1.0 - cos(4.0 * Phi), -sin(4.0 * Phi));
    double log_real;
    if (r == Mz * Mz / 4.0 / Mt / Mt) {
        log_real = cache.logV1primeAndA1prime();
    } else {
        log_real = GSL_REAL(gsl_complex_log(OneMinusE2Iphi))
                - 2.0 * GSL_REAL(gsl_complex_log(OneMinusE4Iphi));
    }
 
    double PhiPrime = 1.0 / 2.0 / sqrt(r * (1.0 - r));
    double Phi2Prime = Phi / sqrt(r * (1.0 - r));
    double gammaPrime = 1.0 / 2.0 / r;
    double hPrime = -1.0 / 2.0 / (1.0 - r);
 
    // V1(r) = 4.0*A*B + C*D + E
    double A = r - 1.0 / 4.0 / r;
    double Aprime = 1.0 + 1.0 / 4.0 / r / r;
    double B = 2.0 * Cl3_2Phi - Cl3_4Phi + 8.0 / 3.0 * Phi * (Cl2_2Phi - Cl2_4Phi)
            - 4.0 / 3.0 * Phi * Phi * (gamma + 2.0 * h);
    double Bprime = -2.0 / sqrt(r * (1.0 - r))*(Cl2_2Phi - Cl2_4Phi)
            + 8.0 / 3.0 * PhiPrime * (Cl2_2Phi - Cl2_4Phi)
            + 8.0 / 3.0 * Phi * (-log_real / sqrt(r * (1.0 - r)))
            - 4.0 / 3.0 * Phi2Prime * (gamma + 2.0 * h)
            - 4.0 / 3.0 * Phi * Phi * (gammaPrime + 2.0 * hPrime);
    double C = sqrt(1.0 / r - 1.0);
    double Cprime = -1.0 / 2.0 / r / sqrt(r * (1.0 - r));
    double D = 8.0 / 3.0 * (r + 1.0 / 2.0)
            *(-Cl2_2Phi + Cl2_4Phi + 2.0 * Phi * (gamma + 2.0 * h))
            - 2.0 * Phi * (r + 3.0 / 2.0);
    double Dprime = 8.0 / 3.0 * (-Cl2_2Phi + Cl2_4Phi + 2.0 * Phi * (gamma + 2.0 * h))
            + 8.0 / 3.0 * (r + 1.0 / 2.0)
            *(log_real / sqrt(r * (1.0 - r))
            + 2.0 * PhiPrime * (gamma + 2.0 * h)
            + 2.0 * Phi * (gammaPrime + 2.0 * hPrime))
            - 2.0 * PhiPrime * (r + 3.0 / 2.0) - 2.0 * Phi;
    double Eprime = 8.0 * Phi2Prime * (r - 1.0 / 6.0 - 7.0 / 48.0 / r)
            + 8.0 * Phi * Phi * (1.0 + 7.0 / 48.0 / r / r) - zeta_3 / r / r;
    return (4.0 * Aprime * B + 4.0 * A * Bprime + Cprime * D + C * Dprime + Eprime);
 
 
    /* TEST: Exact - Expansion */
    //std::cout << "V1: " 
    //          << (4.0*Aprime*B + 4.0*A*Bprime + Cprime*D + C*Dprime + Eprime)
    //              - (4.0*zeta_3 - 5.0/6.0 + 656.0/81.0*r)
    //          << std::endl;
 
    /* Expansion for r << 1 */
    //return (4.0*zeta_3 - 5.0/6.0 + 656.0/81.0*r);
}

Member Data Documentation

cache

const EWSMcache& EWSMTwoLoopQCD::cache

private

A reference to an object of type EWSMcache.

Definition at line 376 of file EWSMTwoLoopQCD.h.

---

The documentation for this class was generated from the following files:

• EWSMTwoLoopQCD.h
• EWSMTwoLoopQCD.cpp