EWSMOneLoopEW Class Reference

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

#include <EWSMOneLoopEW.h>

Detailed Description

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

Author

HEPfit Collaboration

Copyright

GNU General Public License

This class handles one-loop EW contributions of \(O(\alpha)\) 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_f()),
• \(\delta\kappa_{\mathrm{rem}}^{f}\)   (with deltaKappa_rem_f()),
• \(\rho_{ij}^W\)   (with rho_GammaW()).

See also the description of EWSM class for their definitions.

The above quantities are computed with the help of the auxiliary member functions defined in the current class. It is noted that gauge-dependent quantities in the current class are given in the Unitary gauge. For example, the functions SigmabarWW_bos(), SigmabarWW_bos(), etc. represent the self-energies of the gauge bosons. The following table shows the comparisons of the definitions of the self-energies in the current class to those in [Bardin:1999ak] and [Hollik:1988ii] :

Current class	Bardin & Passarino [Bardin:1999ak]	Hollik [Hollik:1988ii]
\(\displaystyle\Sigma_{WW}(s)=\frac{\alpha}{4\pi s_W^2 } \overline{\Sigma}_{WW}(s)\)	\(\displaystyle\frac{\alpha}{4\pi s_W^2 } \Sigma_{WW}(s)\)	\(\displaystyle\Sigma^W(s)\)
\(\displaystyle\Sigma_{ZZ}(s)=\frac{\alpha}{4\pi s_W^2 c_W^2 } \overline{\Sigma}_{ZZ}(s)\)	\(\displaystyle\frac{\alpha}{4\pi s_W^2 c_W^2 } \Sigma_{ZZ}(s)\)	\(\displaystyle\Sigma^Z(s)\)
\(\displaystyle\Pi_{\gamma\gamma}(s)=\frac{\alpha}{4\pi}\,\overline{\Pi}_{\gamma\gamma}(s)\)	\(\displaystyle\frac{\alpha}{4\pi}\,\left[ -\Pi_{\gamma\gamma}(s) \right]\)	\(\displaystyle\Pi^\gamma(s)=\frac{\partial\Sigma^\gamma(s)}{\partial s}\)
\(\displaystyle\Pi_{Z\gamma}(s) = \frac{\alpha}{4\pi s_W c_W}\overline{\Pi}_{Z\gamma}(s)\)	\(\displaystyle\frac{\alpha}{4\pi s_W c_W}\left[ - \Pi_{ZA}(s)\right]\)
\(\displaystyle\overline{\Sigma}^{\prime}_{WW}(s) = \frac{\partial\,\overline{\Sigma}_{WW}(s)}{\partial s}\)	\(\displaystyle- \frac{\alpha}{4\pi s_W^2 } {\cal W}(s)\)
\(\displaystyle\overline{\Sigma}^{\prime}_{ZZ}(s) =\frac{\partial\,\overline{\Sigma}_{ZZ}(s)}{\partial s}\)	\(\displaystyle- \frac{\alpha}{4\pi s_W^2 c_W^2 } c_W^2 {\cal Z}(s)\)

These self-energies are decomposed into the sum of bosonic and fermionic contributions:

\begin{align*} \overline{\Sigma}_{WW}(s) = \overline{\Sigma}^{\rm bos}_{WW}(s) + \overline{\Sigma}^{\rm fer}_{WW}(s)\,, \\ \overline{\Sigma}_{ZZ}(s) = \overline{\Sigma}^{\rm bos}_{ZZ}(s) + \overline{\Sigma}^{\rm fer}_{ZZ}(s)\,, \\ \overline{\Pi}_{\gamma\gamma}(s) = \overline{\Pi}_{\gamma\gamma}^{\rm bos}(s) + \overline{\Pi}_{\gamma\gamma}^{\rm fer}(s)\,, \\ \overline{\Pi}_{Z\gamma}(s) = \overline{\Pi}_{Z\gamma}^{\rm bos}(s) + \overline{\Pi}_{Z\gamma}^{\rm fer}(s)\,. \end{align*}

Most of the formulae used in the current class have been copied or derived from from those in Bardin and Passarino's book [Bardin:1999ak] as well as from [Sirlin:1980nh], [Marciano:1980pb], [Bardin:1981sv], [Akhundov:1985fc], [Bardin:1986fi] and [Bardin:1989di].

Definition at line 105 of file EWSMOneLoopEW.h.

Public Member Functions
double	DeltaAlpha_l (const double s) const
	Leptonic contribution of \(O(\alpha)\) to the electromagnetic coupling \(\alpha(s)\), denoted as \(\Delta\alpha_{\mathrm{lept}}^{\alpha}\). More...
double	DeltaAlpha_t (const double s) const
	Top-quark contribution of \(O(\alpha)\) to the electromagnetic coupling \(\alpha(s)\), denoted as \(\Delta\alpha_{\mathrm{top}}^{\alpha}\). More...
gslpp::complex	deltaKappa_rem_f (const Particle f, const double Mw_i) const
	Remainder contribution of \(O(\alpha)\) to the effective couplings \(\kappa_Z^f\), denoted as \(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha}\). More...
gslpp::complex	deltaKappa_rem_tmp (const double deltaf, const gslpp::complex uf, const double Mw_i) const
	Remainder contribution of \(O(\alpha)\) to the effective couplings \(\kappa_Z^f\), denoted as \(\delta\kappa_{\rm rem}^{f,\,\alpha}\). More...
double	DeltaR_rem (const double Mw_i) const
	Remainder contribution of \(O(\alpha)\) to \(\Delta r\), denoted as \(\Delta r_{\rm rem}^{\alpha}\). More...
double	DeltaRbar_rem (const double Mw_i) const
	\(\Delta \bar{r}_{\rm rem}^{\alpha}\). More...
double	DeltaRho (const double Mw_i) const
	Leading one-loop contribution of \(O(\alpha)\) to \(\Delta\rho\), denoted as \(\Delta\rho^{\alpha}\). More...
gslpp::complex	deltaRho_rem_f (const Particle f, const double Mw_i) const
	Remainder contribution of \(O(\alpha)\) to the effective couplings \(\rho_Z^f\), denoted as \(\delta\rho_{\mathrm{rem}}^{f,\, \alpha}\). More...
gslpp::complex	deltaRho_rem_tmp (const gslpp::complex u_f, const double Mw_i) const
	Remainder contribution of \(O(\alpha)\) to the effective couplings \(\rho_Z^f\), denoted as \(\delta\rho_{\rm rem}^{f,\alpha}\). More...
double	DeltaRhobar (const double mu, const double Mw_i) const
	\(\Delta\overline{\rho}\). More...
double	DeltaRhobarW (const double mu, const double Mw_i) const
	\(\Delta\overline{\rho}_W\). More...
	EWSMOneLoopEW (const EWSMcache &cache_i)
	Constructor. More...
gslpp::complex	FbarWa_0 (const double s) const
	The form factor \(\overline{\mathcal{F}}_{Wa}^0\). More...
gslpp::complex	FbarWa_t (const double s, const double Mw_i) const
	The form factor \(\overline{\mathcal{F}}_{Wa}^t\). More...
gslpp::complex	FW (const double s, const Particle f, const double Mw_i) const
	The unified form factor \(\mathcal{F}_W\) for \(Z\to f\bar{f}\). More...
gslpp::complex	FWa_0 (const double s, const double Mw_i) const
	The form factor \(\mathcal{F}_{Wa}^0\). More...
gslpp::complex	FWa_t (const double s, const double Mw_i) const
	The form factor \(\mathcal{F}_{Wa}^t\). More...
gslpp::complex	FWn_0 (const double s, const double Mw_i) const
	The form factor \(\mathcal{F}_{Wn}^0\). More...
gslpp::complex	FWn_t (const double s, const double Mw_i) const
	The form factor \(\mathcal{F}_{Wn}^t\). More...
gslpp::complex	FZ (const double s, const double Mw_i) const
	The unified form factor \(\mathcal{F}_Z\). More...
gslpp::complex	FZa_0 (const double s, const double Mw_i) const
	The form factor \(\mathcal{F}_{Za}^0\). More...
gslpp::complex	PibarGammaGamma_bos (const double mu, const double s, const double Mw_i) const
	The bosonic contribution to the self-energy of the photon in the Unitary gauge, \(\overline{\Pi}^{\mathrm{bos}}_{\gamma\gamma}(s)\). More...
gslpp::complex	PibarGammaGamma_fer (const double mu, const double s) const
	The fermionic contribution to the self-energy of the photon in the Unitary gauge, \(\overline{\Pi}^{\mathrm{fer}}_{\gamma\gamma}(s)\). More...
gslpp::complex	PibarGammaGamma_fer (const double mu, const double s, const Particle f) const
	The fermionic contribution to the self-energy of the photon in the Unitary gauge, associated with loops of a lepton or quark. \(\overline{\Pi}^{\mathrm{fer},f}_{\gamma\gamma}(s)\). More...
gslpp::complex	PibarZgamma_bos (const double mu, const double s, const double Mw_i) const
	The bosonic contribution to the self-energy of the \(Z\)- \(\gamma\) mixing in the Unitary gauge, \(\overline{\Pi}^{\mathrm{bos}}_{Z\gamma}(s)\). More...
gslpp::complex	PibarZgamma_fer (const double mu, const double s, const double Mw_i) const
	The fermionic contribution to the self-energy of the \(Z\)- \(\gamma\) mixing in the Unitary gauge, \(\overline{\Pi}^{\mathrm{fer}}_{Z\gamma}(s)\). More...
double	rho_GammaW (const Particle fi, const Particle fj, const double Mw_i) const
	EW radiative corrections to the width of \(W \to f_i \bar{f}_j\), denoted as \(\rho^W_{ij}\). More...
double	rho_GammaW_tmp (const double Qi, const double Qj, const double Mw_i) const
	EW radiative corrections to the widths of \(W \to f_i \bar{f}_j\), denoted as \(\rho^W_{ij}\). More...
gslpp::complex	SigmabarPrime_WW_bos_Mw2 (const double mu, const double Mw_i) const
	The derivative of the bosonic contribution to the self-energy of the \(W\) boson for \(s=M_W^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{bos}}_{WW}(M_W^2)\). More...
gslpp::complex	SigmabarPrime_WW_fer_Mw2 (const double mu, const double Mw_i) const
	The derivative of the fermionic contribution to the self-energy of the \(W\) boson for \(s=M_W^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{fer}}_{WW}(M_W^2)\). More...
gslpp::complex	SigmabarPrime_ZZ_bos_Mz2 (const double mu, const double Mw_i) const
	The derivative of the bosonic contribution to the self-energy of the \(Z\) boson for \(s=M_Z^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{bos}}_{ZZ}(M_Z^2)\). More...
gslpp::complex	SigmabarPrime_ZZ_fer_Mz2 (const double mu, const double Mw_i) const
	The derivative of the fermionic contribution to the self-energy of the \(Z\) boson for \(s=M_Z^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{fer}}_{ZZ}(M_Z^2)\). More...
gslpp::complex	SigmabarWW_bos (const double mu, const double s, const double Mw_i) const
	The bosonic contribution to the self-energy of the \(W\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{bos}}_{WW}(s)\). More...
gslpp::complex	SigmabarWW_fer (const double mu, const double s, const double Mw_i) const
	The fermionic contribution to the self-energy of the \(W\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{fer}}_{WW}(s)\). More...
gslpp::complex	SigmabarZZ_bos (const double mu, const double s, const double Mw_i) const
	The bosonic contribution to the self-energy of the \(Z\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{bos}}_{ZZ}(s)\). More...
gslpp::complex	SigmabarZZ_fer (const double mu, const double s, const double Mw_i) const
	The fermionic contribution to the self-energy of the \(Z\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{fer}}_{ZZ}(s)\). More...
double	TEST_DeltaRhobar_bos (const double Mw_i) const
	A test function. More...
double	TEST_DeltaRhobarW_bos (const double Mw_i) const
	A test function. More...
gslpp::complex	TEST_FWn (const double s, const double mf, const double Mw_i) const
	A test function for \(\mathcal{F}_{Wn}\) with a finite fermion mass. More...

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

Constructor & Destructor Documentation

EWSMOneLoopEW()

EWSMOneLoopEW::EWSMOneLoopEW

(

const EWSMcache &

cache_i

)

Constructor.

Parameters

[in]

cache_i

a reference to an object of type EWSMcache

Definition at line 11 of file EWSMOneLoopEW.cpp.

: cache(cache_i)
{
}

Member Function Documentation

DeltaAlpha_l()

double EWSMOneLoopEW::DeltaAlpha_l

(

const double

s

)

const

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

This function uses the function PiGammaGamma_fer_l() for the fermionic contribution to the photon self-energy.

Parameters

[in]

s

invariant mass squared

Returns

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

Definition at line 19 of file EWSMOneLoopEW.cpp.

{
    double Mz = cache.getSM().getMz();
 
    double oneLoop[3];
    oneLoop[0] = -PibarGammaGamma_fer(Mz, s, cache.getSM().getLeptons(StandardModel::ELECTRON)).real()
            + PibarGammaGamma_fer(Mz, 0.0, cache.getSM().getLeptons(StandardModel::ELECTRON)).real();
    oneLoop[1] = -PibarGammaGamma_fer(Mz, s, cache.getSM().getLeptons(StandardModel::MU)).real()
            + PibarGammaGamma_fer(Mz, 0.0, cache.getSM().getLeptons(StandardModel::MU)).real();
    oneLoop[2] = -PibarGammaGamma_fer(Mz, s, cache.getSM().getLeptons(StandardModel::TAU)).real()
            + PibarGammaGamma_fer(Mz, 0.0, cache.getSM().getLeptons(StandardModel::TAU)).real();
 
    return ( cache.getSM().getAle() / 4.0 / M_PI
            * (oneLoop[0] + oneLoop[1] + oneLoop[2]));
}

DeltaAlpha_t()

double EWSMOneLoopEW::DeltaAlpha_t

(

const double

s

)

const

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

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

Parameters

[in]

s

invariant mass squared

Returns

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

Definition at line 35 of file EWSMOneLoopEW.cpp.

{
    double xt = s / cache.getSM().getMtpole() / cache.getSM().getMtpole();
    double tmp = 1.0 + xt * 0.1071;
    tmp *= -4.0 / 45.0 * cache.getSM().getAle() / M_PI*xt;
    return tmp;
}

deltaKappa_rem_f()

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

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

This function handles the remainder contribution \(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha}\) for \(Z\to f\bar{f}\).

Parameters

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

Returns

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

See also

deltaKappa_rem_tmp()

Definition at line 143 of file EWSMOneLoopEW.cpp.

{
    if (f.is("TOP")) return ( gslpp::complex(0.0, 0.0, false));
 
    double Mz = cache.getSM().getMz();
    double Mw = Mw_i;
    gslpp::complex uf = (3.0 * cache.v_f(f, Mw) * cache.v_f(f, Mw) + cache.a_f(f) * cache.a_f(f))
            / 4.0 / cache.getSM().cW2(Mw) * FZ(Mz*Mz, Mw) + FW(Mz*Mz, f, Mw);
 
    return ( deltaKappa_rem_tmp(cache.delta_f(f, Mw), uf, Mw));
}

deltaKappa_rem_tmp()

gslpp::complex EWSMOneLoopEW::deltaKappa_rem_tmp	(	const double	deltaf,
		const gslpp::complex	uf,
		const double	Mw_i
	)		const

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

This function represents \(\delta\kappa_{\rm rem}^{f,\,\alpha}\), given in terms of the flavour-dependent quantities \(\delta_f\) and \(u_f\):

\[ \delta\kappa_{\rm rem}^{f,\alpha} = \frac{\alpha}{4\pi s_W^2} \left[ \overline{\Pi}_{Z\gamma}(M_Z^2)\big|_{\mu=M_Z} + \frac{\delta_f^2}{4 c_W^2}{\cal F}_Z(M_Z^2) - u_f + \left( \frac{1}{12 c_W^2} + \frac{4}{3} \right)\ln c_W^2 \right] \]

where \(\delta_f\) and \(u_f\) are defined as

\begin{align} \delta_f &= v_f - a_f = -2Q_f s_W^2\,, \\ u_f &= \frac{3v_f^2+a_f^2}{4 c_W^2}{\cal F}_Z(M_Z^2) + {\cal F}_W(M_Z^2) \end{align}

with the so-called unified form factors \({\cal F}_Z(M_Z^2)\) and \({\cal F}_W(M_Z^2)\).

Parameters

[in]	deltaf	the quantity \(\delta_f\)
[in]	uf	the quantity \(u_f\)
[in]	Mw_i	the \(W\)-boson mass

Returns

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

See also

deltaKappa_rem_l() and deltaKappa_rem_q() as well as FZ(), FW_l() and FW_q()

Definition at line 124 of file EWSMOneLoopEW.cpp.

{
    double Mz = cache.getSM().getMz();
    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 dKappa_rem(0.0, 0.0, false);
    dKappa_rem = (PibarZgamma_bos(Mz, Mz*Mz, Mw) + PibarZgamma_fer(Mz, Mz*Mz, Mw))
            + deltaf * deltaf / 4.0 / cW2 * FZ(Mz*Mz, Mw) - uf
            + (1.0 / 12.0 / cW2 + 4.0 / 3.0) * log_cW2;
    dKappa_rem *= cache.getSM().getAle() / 4.0 / M_PI / sW2;
    return dKappa_rem;
}

DeltaR_rem()

double EWSMOneLoopEW::DeltaR_rem

(

const double

Mw_i

)

const

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

The \(O(\alpha)\) remainder contribution \(\Delta r_{\mathrm{rem}}^{\alpha}\) is given by

\[ \Delta r_{\rm rem}^\alpha = \frac{\alpha}{4\pi s_W^2} \bigg[ - \frac{2}{3} s_W^2 + s_W^2\overline{\Pi}_{\gamma\gamma}^{t}(0)\big|_{\mu=M_Z} + s_W^2{\rm Re}\Big[\overline{\Pi}_{\gamma\gamma}^{\ell+5q}(M_Z^2)\big|_{\mu=M_Z}\Big] + \Delta\overline{\rho}_W\big|_{\mu=M_Z} + \left( 4 - \frac{25}{4}c_W^2 + \frac{3}{4} c_W^4 + \frac{9 c_W^2}{4 s_W^2} \right)\ln c_W^2 + \frac{11}{2} - \frac{5}{8} c_W^2 (1 + c_W^2) \bigg]. \]

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

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

Definition at line 49 of file EWSMOneLoopEW.cpp.

{
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    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 PiGammaGamma_t_0 = PibarGammaGamma_fer(Mz, 0.0, cache.getSM().getQuarks(QCD::TOP)).real();
    double PiGammaGamma_l5q_Mz2 = PibarGammaGamma_fer(Mz, Mz2).real()
            - PibarGammaGamma_fer(Mz, Mz2, cache.getSM().getQuarks(QCD::TOP)).real();
 
    double DR_rem = -2.0 / 3.0 * sW2 + sW2 * PiGammaGamma_t_0
            + sW2 * PiGammaGamma_l5q_Mz2 + DeltaRhobarW(Mz, Mw)
            + (4.0 - 25.0 / 4.0 * cW2 + 3.0 / 4.0 * cW2 * cW2 + 9.0 * cW2 / 4.0 / sW2)
            * log_cW2
            + 11.0 / 2.0 - 5.0 / 8.0 * cW2 * (1.0 + cW2);
    DR_rem *= cache.getSM().getAle() / 4.0 / M_PI / sW2;
    return DR_rem;
}

DeltaRbar_rem()

double EWSMOneLoopEW::DeltaRbar_rem

(

const double

Mw_i

)

const

\(\Delta \bar{r}_{\rm rem}^{\alpha}\).

The quantity \(\Delta \bar{r}_{\rm rem}^{\alpha}\), which contributes to \(\rho_Z^f\) and \(\kappa_Z^f\), is given by

\[ \Delta \bar{r}_{\rm rem}^{\alpha} = \bigg[ \Delta r_{\rm rem}^\alpha - \frac{\alpha}{4\pi} \overline{\Pi}_{\gamma\gamma}^{t}(0)\big|_{\mu=M_Z} - \frac{\alpha}{4\pi} {\rm Re}\left[ \overline{\Pi}_{\gamma\gamma}^{\ell+5q}(M_Z^2)\big|_{\mu=M_Z} \right] \bigg], \]

where the definition of \(\Delta r_{\rm rem}^\alpha\) is given in DeltaR_rem().

Parameters

[in]

Mw_i

the \(W\)-boson mass

Returns

\(\Delta \bar{r}_{\rm rem}^{\alpha}\)

Definition at line 73 of file EWSMOneLoopEW.cpp.

{
    double Mz = cache.getSM().getMz();
    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 DRbar_rem = -2.0 / 3.0 * sW2 + DeltaRhobarW(Mz, Mw)
            + (4.0 - 25.0 / 4.0 * cW2 + 3.0 / 4.0 * cW2 * cW2 + 9.0 * cW2 / 4.0 / sW2)
            * log_cW2
            + 11.0 / 2.0 - 5.0 / 8.0 * cW2 * (1.0 + cW2);
    DRbar_rem *= cache.getSM().getAle() / 4.0 / M_PI / sW2;
    return DRbar_rem;
}

DeltaRho()

double EWSMOneLoopEW::DeltaRho

(

const double

Mw_i

)

const

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

The leading one-loop contribution is written in terms of the quantity \(\Delta\overline{\rho}\):

\[ \Delta\rho^{\alpha} = - \frac{\alpha}{4\pi s_W^2}\Delta\overline{\rho}\big|_{\mu=M_Z}, \]

where \(\Delta\overline{\rho}\) is renormalized at the scale \(M_Z\), and computed with the function DeltaRhobar().

Parameters

[in]

Mw_i

the \(W\)-boson mass \(M_W\)

Returns

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

Definition at line 43 of file EWSMOneLoopEW.cpp.

{
 
    return ( -cache.getSM().getAle() / 4.0 / M_PI / cache.getSM().sW2(Mw_i) * DeltaRhobar(cache.getSM().getMz(), Mw_i));
}

deltaRho_rem_f()

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

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

This function handles the remainder contribution \(\delta\rho_{\mathrm{rem}}^{f,\, \alpha}\) for \(Z\to f\bar{f}\).

Parameters

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

Returns

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

See also

deltaRho_rem_tmp()

Definition at line 113 of file EWSMOneLoopEW.cpp.

{
    if (f.is("TOP")) return ( gslpp::complex(0.0, 0.0, false));
 
    double Mz = cache.getSM().getMz();
    double Mw = Mw_i;
    gslpp::complex uf = (3.0 * cache.v_f(f, Mw) * cache.v_f(f, Mw) + cache.a_f(f) * cache.a_f(f))
            / 4.0 / cache.getSM().cW2(Mw) * FZ(Mz*Mz, Mw) + FW(Mz*Mz, f, Mw);
    return ( deltaRho_rem_tmp(uf, Mw));
}

deltaRho_rem_tmp()

gslpp::complex EWSMOneLoopEW::deltaRho_rem_tmp	(	const gslpp::complex	u_f,
		const double	Mw_i
	)		const

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

This function represents \(\delta\rho_{\rm rem}^{f,\alpha}\), given in terms of the flavour-dependent quantity \(u_f\):

\[ \delta\rho_{\rm rem}^{f,\alpha} = \frac{\alpha}{4\pi s_W^2} \left\{ - \frac{1}{ c_W^2} {\rm Re}\big[\overline{\Sigma}^{\prime}_{ZZ}(M_Z^2)\big|_{\mu=M_Z} \big] - \Delta\overline{\rho}_W\big|_{\mu=M_Z} + 2u_f - \left[ \frac{1}{6 c_W^2} - \frac{1}{3} + \frac{3}{4} c_W^2 (1+ c_W^2) + \frac{9 c_W^2}{4 s_W^2} \right]\ln c_W^2 - \frac{11}{2} + \frac{5}{8} c_W^2(1+ c_W^2) \right\}, \]

where \(u_f\) is defined as

\[ u_f = \frac{3v_f^2+a_f^2}{4 c_W^2}{\cal F}_Z(M_Z^2) + {\cal F}_W(M_Z^2) \]

with the so-called unified form factors \({\cal F}_Z(M_Z^2)\) and \({\cal F}_W(M_Z^2)\).

Parameters

[in]	u_f	the quantity \(u_f\)
[in]	Mw_i	the \(W\)-boson mass

Returns

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

See also

deltaRho_rem_l() and deltaRho_rem_q() as well as FZ(), FW_l() and FW_q()

Definition at line 91 of file EWSMOneLoopEW.cpp.

{
    double Mz = cache.getSM().getMz();
    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 dRho_rem(0.0, 0.0, false);
    dRho_rem = -(SigmabarPrime_ZZ_bos_Mz2(Mz, Mw).real()
            + SigmabarPrime_ZZ_fer_Mz2(Mz, Mw).real()) / cW2
            - DeltaRhobarW(Mz, Mw) + 2.0 * uf
            - (1.0 / 6.0 / cW2 - 1.0 / 3.0 + 3.0 / 4.0 * cW2 * (1.0 + cW2) + 9.0 * cW2 / 4.0 / sW2)
            * log_cW2
            - 11.0 / 2.0 + 5.0 / 8.0 * cW2 * (1.0 + cW2);
    dRho_rem *= cache.getSM().getAle() / 4.0 / M_PI / sW2;
    return dRho_rem;
}

DeltaRhobar()

double EWSMOneLoopEW::DeltaRhobar	(	const double	mu,
		const double	Mw_i
	)		const

\(\Delta\overline{\rho}\).

The quantity \(\Delta\overline{\rho}\), which is associated with \(\Delta\rho\) as explained in the description of DeltaRho(), is defined as

\[ \Delta\overline{\rho}\big|_{\mu} = \frac{1}{M_W^2}\left[ {\rm Re}\,\overline{\Sigma}_{WW}(M_W^2)\big|_{\mu} - {\rm Re}\,\overline{\Sigma}_{ZZ}(M_Z^2)\big|_{\mu} \right], \]

where \(\mu\) denotes the renormalization scale.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\Delta\overline{\rho}\big|_{\mu}\)

Definition at line 803 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    return ( (SigmabarWW_bos(mu, Mw*Mw, Mw).real() + SigmabarWW_fer(mu, Mw*Mw, Mw).real()
            - SigmabarZZ_bos(mu, Mz*Mz, Mw).real() - SigmabarZZ_fer(mu, Mz*Mz, Mw).real())
            / Mw / Mw);
}

DeltaRhobarW()

double EWSMOneLoopEW::DeltaRhobarW	(	const double	mu,
		const double	Mw_i
	)		const

\(\Delta\overline{\rho}_W\).

The quantity \(\Delta\overline{\rho}_W\) is defined as

\[ \Delta\overline{\rho}_W\big|_{\mu} = \frac{1}{M_W^2}\left[ \overline{\Sigma}_{WW}(0)\big|_{\mu} - {\rm Re}\,\overline{\Sigma}_{WW}(M_W^2)\big|_{\mu} \right], \]

where \(\mu\) denotes the renormalization scale.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\Delta\overline{\rho}_W\big|_{\mu}\)

Definition at line 812 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    return ( (SigmabarWW_bos(mu, 0.0, Mw).real() + SigmabarWW_fer(mu, 0.0, Mw).real()
            - SigmabarWW_bos(mu, Mw*Mw, Mw).real() - SigmabarWW_fer(mu, Mw*Mw, Mw).real())
            / Mw / Mw);
}

FbarWa_0()

gslpp::complex EWSMOneLoopEW::FbarWa_0

(

const double

s

)

const

The form factor \(\overline{\mathcal{F}}_{Wa}^0\).

The form factor \(\overline{\mathcal{F}}_{Wa}^0\), associated with abelian-type diagrams of \(Z\to f\bar{f}\) with a virtual \(W\) boson, is given in the chiral limit by

\[ \overline{{\cal F}}_{Wa}^0(s) = 0\,. \]

See [Bardin:1999ak].

Parameters

[in]

s

momentum squared \(s\)

Returns

\(\overline{\mathcal{F}}_{Wa}^0\)

Definition at line 932 of file EWSMOneLoopEW.cpp.

{
    return ( gslpp::complex(0.0, 0.0, false));
}

FbarWa_t()

gslpp::complex EWSMOneLoopEW::FbarWa_t	(	const double	s,
		const double	Mw_i
	)		const

The form factor \(\overline{\mathcal{F}}_{Wa}^t\).

The form factor \(\overline{\mathcal{F}}_{Wa}^t\), associated with abelian-type diagrams of \(Z\to f\bar{f}\) with a virtual \(W\) boson and the heavy top quark, is given by

\[ \overline{{\cal F}}_{Wa}^t(s) = - w_t \Bigg\{ \left[ R_W + 2 - w_t(2 - w_t)R_W \right] M_W^2C_0(s;M_t,M_W,M_t) - \left( \frac{1}{2} - R_W + w_t R_W \right) \left[ - B_0(s;M_t,M_t)\big|_{\mu=M_W} + 1 \right] + w_t R_W \ln w_t \Bigg\}\,, \]

where the definitions of the symbols can be read from the codes below.

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\mathcal{F}}_{Wa}^t\)

Definition at line 1000 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Rw = Mw * Mw / s;
    double Mt = cache.getSM().getMtpole();
    double Mt2 = Mt*Mt;
    double wt = Mt2 / Mw2;
 
    /* Logarithm and two- and three-point one-loop functions */
    double log_wt = -2.0 * cache.logMZtoMTOP() - cache.log_cW2(Mw);
    gslpp::complex B0_Mw2_s_Mt2_Mt2;
    gslpp::complex C0_s_Mt2_Mw2_Mt2;
    if (s == Mz * Mz) {
        B0_Mw2_s_Mt2_Mt2 = cache.B0_Mw2_Mz2_Mt2_Mt2(Mw);
        C0_s_Mt2_Mw2_Mt2 = cache.C0_Mz2_Mt2_Mw2_Mt2(Mw);
    } else {
        B0_Mw2_s_Mt2_Mt2 = cache.getPV().B0(Mw2, s, Mt2, Mt2);
        C0_s_Mt2_Mw2_Mt2 = cache.getPV().C0(s, Mt2, Mw2, Mt2);
    }
 
    gslpp::complex FbarWa(0.0, 0.0, false);
    FbarWa = -wt * ((Rw + 2.0 - wt * (2.0 - wt) * Rw) * Mw * Mw * C0_s_Mt2_Mw2_Mt2
            - (1.0 / 2.0 - Rw + wt * Rw)*(-B0_Mw2_s_Mt2_Mt2 + 1.0)
            + wt * Rw * log_wt);
    return FbarWa;
}

FW()

gslpp::complex EWSMOneLoopEW::FW	(	const double	s,
		const Particle	f,
		const double	Mw_i
	)		const

The unified form factor \(\mathcal{F}_W\) for \(Z\to f\bar{f}\).

The so-called unified form factor \(\mathcal{F}_W\), associated with radiative corrections to the \(Z\to f\bar{f}\) vertex with a virtual \(W\) boson as well as with virtual \(W\) bosons, is given by

\[ {\cal F}_W(s) = c_W^2 {\cal F}_{Wn}^0(s) - \frac{1}{2}\sigma_{l'}^a {\cal F}_{Wa}^0(s) - \frac{1}{2}\overline{{\cal F}}_{Wa}^0(s)\,, \]

where the suprescripts "0" denote the chiral limit, \(\sigma_{f'}^a = |v_{f'} + a_{f'}| = 1 - 2|Q_{f'}|s_W^2 = 2c_W^2 - 1 + 2|Q_{f}| s_W^2\) with \(f'\) being the partner of \(f\) in the \(SU(2)_L\) doublet, and \({\cal F}_{Wn}^0(s)\), \({\cal F}_{Wa}^0(s)\) and \(\overline{{\cal F}}_{Wa}^0(s)\) correspond to the functions FWn_0(), FWa_0() and FbarWa_0(), respectively.

See [Bardin:1999ak].

Parameters

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

Returns

\(\mathcal{F}_W\)

Definition at line 1070 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double cW2 = cache.getSM().cW2(Mw);
 
    if (f.is("LEPTON")) {
        QCD::lepton lprime;
        if (f.is("NEUTRINO_1"))
            lprime = StandardModel::ELECTRON;
        else if (f.is("NEUTRINO_2"))
            lprime = StandardModel::MU;
        else if (f.is("NEUTRINO_3"))
            lprime = StandardModel::TAU;
        else if (f.is("ELECTRON"))
            lprime = StandardModel::NEUTRINO_1;
        else if (f.is("MU"))
            lprime = StandardModel::NEUTRINO_2;
        else if (f.is("TAU"))
            lprime = StandardModel::NEUTRINO_3;
        else
            throw std::runtime_error("EWSMOneLoopEW::FW(): " + f.getName() + " is not allowed");
        return ( cW2 * FWn_0(s, Mw) - cache.sigma_f(cache.getSM().getLeptons(lprime), Mw) / 2.0 * FWa_0(s, Mw)
                - FbarWa_0(s) / 2.0);
    } else if (f.is("QUARK")) {
        QCD::quark qprime;
        if (f.is("UP"))
            qprime = QCD::DOWN;
        else if (f.is("DOWN"))
            qprime = QCD::UP;
        else if (f.is("CHARM"))
            qprime = QCD::STRANGE;
        else if (f.is("STRANGE"))
            qprime = QCD::CHARM;
        else if (f.is("BOTTOM"))
            qprime = QCD::TOP;
        else
            throw std::runtime_error("EWSMOneLoopEW::FW(): TOP is not allowed");
        gslpp::complex FW(0.0, 0.0, false);
        FW = cW2 * FWn_0(s, Mw) - cache.sigma_f(cache.getSM().getQuarks(qprime), Mw) / 2.0 * FWa_0(s, Mw)
                - FbarWa_0(s) / 2.0;
        if (f.is("BOTTOM"))
            FW += cW2 * FWn_t(s, Mw) - cache.sigma_f(cache.getSM().getQuarks(qprime), Mw) / 2.0 * FWa_t(s, Mw)
            - FbarWa_t(s, Mw) / 2.0;
        return FW;
    } else
        throw std::runtime_error("EWSMOneLoopEW::FW_q(): Wrong argument");
}

FWa_0()

gslpp::complex EWSMOneLoopEW::FWa_0	(	const double	s,
		const double	Mw_i
	)		const

The form factor \(\mathcal{F}_{Wa}^0\).

The form factor \(\mathcal{F}_{Wa}^0\), associated with abelian-type diagrams of \(Z\to f\bar{f}\) with a virtual \(W\) boson, is given in the chiral limit by

\[ {\cal F}_{Wa}^0(s) = 2(R_W + 1)^2 s\, C_0(s;0,(\widetilde{M}_W^2)^{1/2},0) - (2R_W+3) \ln\left(-\frac{\widetilde{M}_W^2}{s}\right) - 2R_W - \frac{7}{2}\,, \]

where \(\widetilde{M}_W^2 \equiv M_W^2 - i\, M_W\Gamma_W\approx M_W^2-i\epsilon\), and the definitions of the other symbols can be read from the codes below.

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_{Wa}^0\)

Definition at line 909 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    double Rw = Mw * Mw / s;
 
    /* Logarithm and three-point one-loop functions */
    double log_Rw;
    gslpp::complex C0_s_0_Mw2_0;
    if (s == Mz * Mz) {
        log_Rw = cache.log_cW2(Mw);
        C0_s_0_Mw2_0 = cache.C0_Mz2_0_Mw2_0(Mw);
    } else {
        log_Rw = log(Rw);
        C0_s_0_Mw2_0 = cache.getPV().C0(s, 0.0, Mw*Mw, 0.0);
    }
 
    gslpp::complex FWa(0.0, 0.0, false);
    FWa = 2.0 * pow((Rw + 1.0), 2.0) * s * C0_s_0_Mw2_0
            - (2.0 * Rw + 3.0)*(log_Rw + M_PI * gslpp::complex::i()) - 2.0 * Rw - 7.0 / 2.0;
    return FWa;
}

FWa_t()

gslpp::complex EWSMOneLoopEW::FWa_t	(	const double	s,
		const double	Mw_i
	)		const

The form factor \(\mathcal{F}_{Wa}^t\).

The form factor \(\mathcal{F}_{Wa}^t\), associated with abelian-type diagrams of \(Z\to f\bar{f}\) with a virtual \(W\) boson and the heavy top quark, is given by

\begin{align} {\cal F}_{Wa}^t(s) &= 2(R_W +1)^2 s \left[ C_0(s;M_t,M_W,M_t) - C_0(s;0,(\widetilde{M}_W^2)^{1/2},0) \right] + (2R_W + 3) \left[ - B_0(s;M_t,M_t)\big|_{\mu=M_W} + \ln\left(-\frac{\widetilde{M}_W^2}{s}\right) + 2 \right] \\ &\quad - w_t \Bigg\{ \left( 3R_W +2-w_t - w_t^2 R_W \right) M_W^2 C_0(s;M_t,M_W,M_t) + \left( R_W + \frac{1}{2} + w_t R_W \right) \left[ 1 - B_0(s;M_t,M_t)\big|_{\mu=M_W} \right] \\ &\qquad\qquad - \left( 2 R_W + \frac{1}{2} - \frac{2}{w_t - 1} + \frac{3}{2}\frac{1}{(w_t - 1)^2} + w_t R_W \right) \ln w_t + \frac{3}{2}\frac{1}{w_t - 1} + \frac{3}{4} \Bigg\}\,, \end{align}

where \(\widetilde{M}_W^2 \equiv M_W^2 - i\, M_W\Gamma_W\approx M_W^2-i\epsilon\), and the definitions of the other symbols can be read from the codes below.

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_{Wa}^t\)

Definition at line 962 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Rw = Mw * Mw / s;
    double Mt = cache.getSM().getMtpole();
    double Mt2 = Mt*Mt;
    double wt = Mt2 / Mw2;
 
    /* Logarithm and two- and three-point one-loop functions */
    double log_wt = -2.0 * cache.logMZtoMTOP() - cache.log_cW2(Mw);
    double log_Rw;
    gslpp::complex B0_Mw2_s_Mt2_Mt2;
    gslpp::complex C0_s_Mt2_Mw2_Mt2, C0_s_0_Mw2_0;
    if (s == Mz * Mz) {
        log_Rw = cache.log_cW2(Mw);
        B0_Mw2_s_Mt2_Mt2 = cache.B0_Mw2_Mz2_Mt2_Mt2(Mw);
        C0_s_Mt2_Mw2_Mt2 = cache.C0_Mz2_Mt2_Mw2_Mt2(Mw);
        C0_s_0_Mw2_0 = cache.C0_Mz2_0_Mw2_0(Mw);
    } else {
        log_Rw = log(Rw);
        B0_Mw2_s_Mt2_Mt2 = cache.getPV().B0(Mw2, s, Mt2, Mt2);
        C0_s_Mt2_Mw2_Mt2 = cache.getPV().C0(s, Mt2, Mw2, Mt2);
        C0_s_0_Mw2_0 = cache.getPV().C0(s, 0.0, Mw2, 0.0);
    }
 
    gslpp::complex FWa(0.0, 0.0, false);
    FWa = 2.0 * (Rw + 1.0)*(Rw + 1.0) * s * (C0_s_Mt2_Mw2_Mt2 - C0_s_0_Mw2_0)
            + (2.0 * Rw + 3.0)*(-B0_Mw2_s_Mt2_Mt2 + log_Rw + M_PI * gslpp::complex::i() + 2.0)
            - wt * ((3.0 * Rw + 2.0 - wt - wt * wt * Rw) * Mw * Mw * C0_s_Mt2_Mw2_Mt2
            + (Rw + 1.0 / 2.0 + wt * Rw)*(1.0 - B0_Mw2_s_Mt2_Mt2)
            - (2.0 * Rw + 1.0 / 2.0 - 2.0 / (wt - 1.0)
            + 3.0 / 2.0 / (wt - 1.0) / (wt - 1.0) + wt * Rw) * log_wt
            + 3.0 / 2.0 / (wt - 1.0) + 3.0 / 4.0);
    return FWa;
}

FWn_0()

gslpp::complex EWSMOneLoopEW::FWn_0	(	const double	s,
		const double	Mw_i
	)		const

The form factor \(\mathcal{F}_{Wn}^0\).

The form factor \(\mathcal{F}_{Wn}^0\), associated with nonabelian-type diagrams of \(Z\to f\bar{f}\) with virtual \(W\) bosons, is given in the chiral limit by

\[ {\cal F}_{Wn}^0(s) = - 2 (R_W + 2) M_W^2 C_0(s;M_W,0,M_W) - \left( 2R_W + \frac{7}{3} - \frac{3}{2R_W} - \frac{1}{12R_W^2} \right) B_0(s; M_W, M_W)\big|_{\mu=M_W} + 2R_W + \frac{9}{2} - \frac{11}{18R_W} + \frac{1}{18R_W^2}\,, \]

where the definitions of the symbols can be read from the codes below.

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_{Wn}^0\)

Definition at line 937 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Rw = Mw * Mw / s;
 
    /* Two- and three-point one-loop functions */
    gslpp::complex B0_Mw_s_Mw2_Mw2;
    gslpp::complex C0_s_Mw2_0_Mw2;
    if (s == Mz * Mz) {
        B0_Mw_s_Mw2_Mw2 = cache.B0_Mw2_Mz2_Mw2_Mw2(Mw);
        C0_s_Mw2_0_Mw2 = cache.C0_Mz2_Mw2_0_Mw2(Mw);
    } else {
        B0_Mw_s_Mw2_Mw2 = cache.getPV().B0(Mw2, s, Mw2, Mw2);
        C0_s_Mw2_0_Mw2 = cache.getPV().C0(s, Mw2, 0.0, Mw2);
    }
 
    gslpp::complex FWn(0.0, 0.0, false);
    FWn = -2.0 * (Rw + 2.0) * Mw * Mw * C0_s_Mw2_0_Mw2
            - (2.0 * Rw + 7.0 / 3.0 - 3.0 / 2.0 / Rw - 1.0 / 12.0 / Rw / Rw) * B0_Mw_s_Mw2_Mw2
            + 2.0 * Rw + 9.0 / 2.0 - 11.0 / 18.0 / Rw + 1.0 / 18.0 / Rw / Rw;
    return FWn;
}

FWn_t()

gslpp::complex EWSMOneLoopEW::FWn_t	(	const double	s,
		const double	Mw_i
	)		const

The form factor \(\mathcal{F}_{Wn}^t\).

The form factor \(\mathcal{F}_{Wn}^t\), associated with nonabelian-type diagrams of \(Z\to f\bar{f}\) with virtual \(W\) bosons and the heavy top quark, is given by

\begin{align} {\cal F}_{Wn}^t(s) &= -2 (R_W +2) M_W^2 \left[ C_0(s;M_W,M_t,M_W) - C_0(s;M_W,0,M_W) \right] \\ &\quad + w_t \Bigg\{ \Bigg[ 3 R_W + \frac{5}{2} - \frac{2}{R_W } - w_t \left( 2 - \frac{1}{2R_W} \right) + w_t ^2 \left( \frac{1}{2} - R_W \right) \Bigg] M_W^2 C_0(s;M_W,M_t,M_W) \\ &\qquad\qquad +\left[ R_W + 1 - \frac{1}{4 R_W} - w_t \left( \frac{1}{2} - R_W \right) \right] \left[ B_0(s;M_W,M_W)\big|_{\mu=M_W} - 1 \right] \\ &\qquad\qquad + \left[ 2 R_W + \frac{1}{2} - \frac{2}{w_t - 1} + \frac{3}{2}\frac{1}{(w_t - 1)^2} - w_t \left( \frac{1}{2} - R_W \right) \right] \ln w_t - \frac{3}{2}\frac{1}{w_t -1} + \frac{1}{4} - \frac{1}{2 R_W} \Bigg\}\,, \end{align}

where the definitions of the symbols can be read from the codes below.

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_{Wn}^0\)

Definition at line 1029 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Rw = Mw * Mw / s;
    double Mt = cache.getSM().getMtpole();
    double Mt2 = Mt*Mt;
    double wt = Mt2 / Mw2;
 
    /* Logarithm and two- and three-point one-loop functions */
    double log_wt = -2.0 * cache.logMZtoMTOP() - cache.log_cW2(Mw);
    gslpp::complex B0_Mw2_s_Mw2_Mw2;
    gslpp::complex C0_s_Mw2_Mt2_Mw2, C0_s_Mw2_0_Mw2;
    if (s == Mz * Mz) {
        B0_Mw2_s_Mw2_Mw2 = cache.B0_Mw2_Mz2_Mw2_Mw2(Mw);
        C0_s_Mw2_Mt2_Mw2 = cache.C0_Mz2_Mw2_Mt2_Mw2(Mw);
        C0_s_Mw2_0_Mw2 = cache.C0_Mz2_Mw2_0_Mw2(Mw);
    } else {
        B0_Mw2_s_Mw2_Mw2 = cache.getPV().B0(Mw2, s, Mw2, Mw2);
        C0_s_Mw2_Mt2_Mw2 = cache.getPV().C0(s, Mw2, Mt2, Mw2);
        C0_s_Mw2_0_Mw2 = cache.getPV().C0(s, Mw2, 0.0, Mw2);
    }
 
    gslpp::complex FWn(0.0, 0.0, false);
    FWn = -2.0 * (Rw + 2.0) * Mw * Mw * (C0_s_Mw2_Mt2_Mw2 - C0_s_Mw2_0_Mw2)
            + wt * ((3.0 * Rw + 5.0 / 2.0 - 2.0 / Rw - wt * (2.0 - 1.0 / 2.0 / Rw)
            + wt * wt * (1.0 / 2.0 - Rw)) * Mw * Mw * C0_s_Mw2_Mt2_Mw2
            + (Rw + 1.0 - 1.0 / 4.0 / Rw - wt * (1.0 / 2.0 - Rw))*(B0_Mw2_s_Mw2_Mw2 - 1.0)
            + (2.0 * Rw + 1.0 / 2.0 - 2.0 / (wt - 1.0)
            + 3.0 / 2.0 / (wt - 1.0) / (wt - 1.0) - wt * (1.0 / 2.0 - Rw)) * log_wt
            - 3.0 / 2.0 / (wt - 1.0) + 1.0 / 4.0
            - 1.0 / 2.0 / Rw);
    return FWn;
}

FZ()

gslpp::complex EWSMOneLoopEW::FZ	(	const double	s,
		const double	Mw_i
	)		const

The unified form factor \(\mathcal{F}_Z\).

The so-called unified form factor \(\mathcal{F}_Z\), associated with radiative corrections to the \(Z\to f\bar{f}\) vertex with a virtual \(Z\) boson, is given by

\[ {\cal F}_Z(s) = {\cal F}_{Za}^0(s)\,, \]

where \({\cal F}_{Za}^0(s)\) corresponds to the function FZa_0().

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_Z\)

Definition at line 1065 of file EWSMOneLoopEW.cpp.

{
    return ( FZa_0(s, Mw_i));
}

FZa_0()

gslpp::complex EWSMOneLoopEW::FZa_0	(	const double	s,
		const double	Mw_i
	)		const

The form factor \(\mathcal{F}_{Za}^0\).

The form factor \(\mathcal{F}_{Za}^0\), associated with abelian-type diagrams of \(Z\to f\bar{f}\) with a virtual \(Z\) boson, is given in the chiral limit by

\[ {\cal F}_{Za}^0(s) = 2(R_Z + 1)^2 s\, C_0(s;0,(\widetilde{M}_Z^2)^{1/2},0) - (2R_Z+3) \ln\left(-\frac{\widetilde{M}_Z^2}{s}\right) - 2R_Z - \frac{7}{2}\,, \]

where \(\widetilde{M}_Z^2 \equiv M_Z^2 - i\, M_Z\Gamma_Z\approx M_Z^2-i\epsilon\), and the definitions of the other symbols can be read from the codes below.

See [Bardin:1999ak].

Parameters

[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_{Za}^0\)

Definition at line 883 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    double Rz = Mz * Mz / s;
 
    /* Logarithm and three-point one-loop functions */
    double log_Rz;
    gslpp::complex C0_s_0_Mz2_0;
    if (s == Mz * Mz) {
        log_Rz = 0.0;
        C0_s_0_Mz2_0 = cache.C0_Mz2_0_Mz2_0();
    } else if (s == Mw * Mw) {
        log_Rz = -cache.log_cW2(Mw);
        C0_s_0_Mz2_0 = cache.C0_Mw2_0_Mz2_0(Mw);
    } else {
        log_Rz = log(Rz);
        C0_s_0_Mz2_0 = cache.getPV().C0(s, 0.0, Mz*Mz, 0.0);
    }
 
    gslpp::complex FZa(0.0, 0.0, false);
    FZa = 2.0 * pow((Rz + 1.0), 2.0) * s * C0_s_0_Mz2_0
            - (2.0 * Rz + 3.0)*(log_Rz + M_PI * gslpp::complex::i()) - 2.0 * Rz - 7.0 / 2.0;
    return FZa;
}

PibarGammaGamma_bos()

gslpp::complex EWSMOneLoopEW::PibarGammaGamma_bos	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The bosonic contribution to the self-energy of the photon in the Unitary gauge, \(\overline{\Pi}^{\mathrm{bos}}_{\gamma\gamma}(s)\).

This function represents the \(O(\alpha)\) bosonic contribution to \(\overline{\Pi}_{\gamma\gamma}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Pi}^{\mathrm{bos}}_{\gamma\gamma}(s)\)

Definition at line 452 of file EWSMOneLoopEW.cpp.

{
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    double RW = pow(Mw, 2.0) / s;
    double RW2 = RW*RW;
    double RW3 = RW2*RW;
 
    /* Loop functions */
    double A0_Mw2;
    gslpp::complex B0_s_Mw2_Mw2;
    if (mu == Mz && s == Mz2) {
        A0_Mw2 = cache.A0_Mz2_Mw2(Mw);
        B0_s_Mw2_Mw2 = cache.B0_Mz2_Mz2_Mw2_Mw2(Mw);
    } else {
        A0_Mw2 = cache.getPV().A0(mu2, Mw2);
        B0_s_Mw2_Mw2 = cache.getPV().B0(mu2, s, Mw2, Mw2);
    }
 
    gslpp::complex Pi(0.0, 0.0, false);
    if (s == 0.0) {
        Pi = 7.0 * log(Mw2 / mu / mu) - 2.0 / 3.0;
    } else {
        Pi = -RW * ((4.0 + 17.0 / 3.0 / RW - 4.0 / 3.0 / RW2 - 1.0 / 12.0 / RW3) * B0_s_Mw2_Mw2
                + (4.0 - 4.0 / 3.0 / RW - 1.0 / 6.0 / RW2)*(A0_Mw2 / Mw2 + 1.0)
                - 1.0 / 18.0 / RW2 * (1.0 / RW - 13.0));
    }
    return Pi;
}

PibarGammaGamma_fer() [1/2]

gslpp::complex EWSMOneLoopEW::PibarGammaGamma_fer	(	const double	mu,
		const double	s
	)		const

The fermionic contribution to the self-energy of the photon in the Unitary gauge, \(\overline{\Pi}^{\mathrm{fer}}_{\gamma\gamma}(s)\).

This function represents the \(O(\alpha)\) fermionic contribution to \(\overline{\Pi}_{\gamma\gamma}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)

Returns

\(\overline{\Pi}^{\mathrm{fer}}_{\gamma\gamma}(s)\)

Definition at line 524 of file EWSMOneLoopEW.cpp.

{
    gslpp::complex Pi(0.0, 0.0, false);
    for (int i = 0; i < 6; i++) {
        Pi += PibarGammaGamma_fer(mu, s, cache.getSM().getLeptons((QCD::lepton) i));
        Pi += PibarGammaGamma_fer(mu, s, cache.getSM().getQuarks((QCD::quark) i));
    }
    return Pi;
}

PibarGammaGamma_fer() [2/2]

gslpp::complex EWSMOneLoopEW::PibarGammaGamma_fer	(	const double	mu,
		const double	s,
		const Particle	f
	)		const

The fermionic contribution to the self-energy of the photon in the Unitary gauge, associated with loops of a lepton or quark. \(\overline{\Pi}^{\mathrm{fer},f}_{\gamma\gamma}(s)\).

This function represents the \(O(\alpha)\) fermionic contribution to \(\overline{\Pi}_{\gamma\gamma}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	f	a lepton or quark

Returns

\(\overline{\Pi}^{\mathrm{fer},f}_{\gamma\gamma}(s)\)

Definition at line 485 of file EWSMOneLoopEW.cpp.

{
    // Neutrinos do not contribute, since Qf=0.
    if (f.is("NEUTRINO_1") || f.is("NEUTRINO_2")
            || f.is("NEUTRINO_3"))
        return 0.0;
 
    double mu2 = mu*mu;
    double mf2 = cache.mf2(f, mu);
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
 
    /* Loop functions */
    gslpp::complex Bf_s_mf2_mf2;
    if (mu == Mz && s == Mz2) {
        if (mf2 == 0.0) {
            Bf_s_mf2_mf2 = 0.0;
        } else {
            Bf_s_mf2_mf2 = cache.Bf_Mz2_Mz2_mf2_mf2(f);
        }
    } else if (mu == Mz && s == 0.0) {
        if (mf2 == 0.0) {
            Bf_s_mf2_mf2 = 0.0;
        } else {
            Bf_s_mf2_mf2 = cache.Bf_Mz2_0_mf2_mf2(f);
        }
    } else {
        if (mf2 == 0.0) {
            Bf_s_mf2_mf2 = 0.0;
        } else {
            Bf_s_mf2_mf2 = cache.getPV().Bf(mu2, s, mf2, mf2);
        }
    }
 
    double Qf = cache.Q_f(f);
    double colorfactor = (f.is("QUARK") ? 3. : 1.);
    return ( -4.0 * colorfactor * Qf * Qf * Bf_s_mf2_mf2);
}

PibarZgamma_bos()

gslpp::complex EWSMOneLoopEW::PibarZgamma_bos	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The bosonic contribution to the self-energy of the \(Z\)- \(\gamma\) mixing in the Unitary gauge, \(\overline{\Pi}^{\mathrm{bos}}_{Z\gamma}(s)\).

This function represents the \(O(\alpha)\) bosonic contribution to \(\overline{\Pi}_{Z\gamma}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Pi}^{\mathrm{bos}}_{Z\gamma}(s)\)

Definition at line 534 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double cW2 = cache.getSM().cW2(Mw);
    return ( PibarGammaGamma_bos(mu, s, Mw) * cW2);
}

PibarZgamma_fer()

gslpp::complex EWSMOneLoopEW::PibarZgamma_fer	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The fermionic contribution to the self-energy of the \(Z\)- \(\gamma\) mixing in the Unitary gauge, \(\overline{\Pi}^{\mathrm{fer}}_{Z\gamma}(s)\).

This function represents the \(O(\alpha)\) fermionic contribution to \(\overline{\Pi}_{Z\gamma}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Pi}^{\mathrm{fer}}_{Z\gamma}(s)\)

Definition at line 541 of file EWSMOneLoopEW.cpp.

{
    double ml2[6], mq2[6];
    for (int i = 0; i < 6; i++) {
        ml2[i] = cache.mf2(cache.getSM().getLeptons((QCD::lepton) i));
        mq2[i] = cache.mf2(cache.getSM().getQuarks((QCD::quark) i), mu);
    }
    double mu2 = mu*mu;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
 
    /* Loop functions */
    gslpp::complex Bf_s_ml2_ml2[6], Bf_s_mq2_mq2[6];
    if (mu == Mz && s == Mz2) {
        for (int i = 0; i < 6; i++) {
            if (i == 0 || i == 2 || i == 4)
                Bf_s_ml2_ml2[i] = 0.0; // Neutrinos do not contribute, since Ql=0.
            else
                Bf_s_ml2_ml2[i] = cache.Bf_Mz2_Mz2_mf2_mf2(cache.getSM().getLeptons((QCD::lepton) i));
            Bf_s_mq2_mq2[i] = cache.Bf_Mz2_Mz2_mf2_mf2(cache.getSM().getQuarks((QCD::quark) i));
        }
    } else {
        for (int i = 0; i < 6; i++) {
            if (i == 0 || i == 2 || i == 4)
                Bf_s_ml2_ml2[i] = 0.0; // Neutrinos do not contribute, since Ql=0.
            else
                Bf_s_ml2_ml2[i] = cache.getPV().Bf(mu2, s, ml2[i], ml2[i]);
            Bf_s_mq2_mq2[i] = cache.getPV().Bf(mu2, s, mq2[i], mq2[i]);
        }
    }
 
    gslpp::complex Pi(0.0, 0.0, false);
    double Ql, Qq;
    for (int i = 0; i < 6; i++) {
        Ql = cache.Q_f(cache.getSM().getLeptons((QCD::lepton) i));
        Pi += -(fabs(Ql) - 4.0 * sW2 * Ql * Ql) * Bf_s_ml2_ml2[i];
        //
        Qq = cache.Q_f(cache.getSM().getQuarks((QCD::quark) i));
        Pi += -3.0 * (fabs(Qq) - 4.0 * sW2 * Qq * Qq) * Bf_s_mq2_mq2[i];
    }
    return Pi;
}

rho_GammaW()

double EWSMOneLoopEW::rho_GammaW	(	const Particle	fi,
		const Particle	fj,
		const double	Mw_i
	)		const

EW radiative corrections to the width of \(W \to f_i \bar{f}_j\), denoted as \(\rho^W_{ij}\).

Parameters

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

Returns

\(\rho^W_{ij}\)

See also

rho_GammaW_tmp()

Definition at line 190 of file EWSMOneLoopEW.cpp.

{
    if (!((fi.is("NEUTRINO_1") && fj.is("ELECTRON")) || (fi.is("NEUTRINO_2") && fj.is("MU"))
            || (fi.is("NEUTRINO_3") && fj.is("TAU")) || (fi.is("UP") && fj.is("DOWN"))
            || (fi.is("CHARM") && fj.is("STRANGE")) || (fi.is("TOP") && fj.is("BOTTOM"))))
        throw std::runtime_error("EWSMOneLoopEW::rho_GammaW(): Wrong arguments");
    double Mw = Mw_i;
    return ( rho_GammaW_tmp(cache.Q_f(fi), cache.Q_f(fj), Mw));
}

rho_GammaW_tmp()

double EWSMOneLoopEW::rho_GammaW_tmp	(	const double	Qi,
		const double	Qj,
		const double	Mw_i
	)		const

EW radiative corrections to the widths of \(W \to f_i \bar{f}_j\), denoted as \(\rho^W_{ij}\).

The factor \(\rho^W_{ij}\) is decomposed as [Bardin:1986fi], [Bardin:1999ak]

\[ \rho^W_{ij} = 1 + \delta f^W_{ij} + \delta f^{\rm QED}_{ij}, \]

where \(\delta f^W_{ij}\) and \(\delta f^{\rm QED}_{ij}\) are given by

\begin{align} \delta f^W_{ij} &= \frac{\alpha}{4\pi s_W^2} \Bigg[ - \Delta\overline{\rho}_W \big|_{\mu=M_W} - {\rm Re}\big[\overline{\Sigma}^{\prime}_{WW}(M_W^2)\big]\big|_{\mu=M_W} + \frac{5}{8} c_W^2 (1 + c_W^2) - \frac{11}{2} - \frac{9}{4}\frac{c_W^2}{s_W^2}\ln c_W^2 \\ &\qquad\qquad\qquad + \left( -1 + \frac{1}{2\,c_W^2} + \frac{2 s_W^4}{c_W^2} Q_i Q_j \right) \left( V_1(M_W^2,M_Z^2) + \frac{3}{2} \right) + 2\,c_W^2 \left( V_2(M_W^2,M_W^2,M_Z^2) + \frac{3}{2} \right) \Bigg]\,, \\ \delta f^{\rm QED}_{ij} &= \frac{\alpha}{\pi} \left[ \frac{85}{18} - \frac{\pi^2}{3} + \frac{3}{4}\,Q_iQ_j \right]. \end{align}

The functions \(V_1\) and \(V_2\) are given in [Bardin:1981sv] :

\begin{align} V_1(M_W^2,M_Z^2) &= {\rm Re}\left[{\cal F}_{Za}^0(M_W^2)\right] - \frac{3}{2}\,, \\ V_2(M_W^2,M_W^2,M_Z^2) &= - 2(2 + c_W^2)M_Z^2\,{\rm Re}\left[C_0(M_W^2;\,M_W,0,M_Z)\right] - \left(\frac{1}{12 c_W^4} + \frac{5}{3 c_W^2} + 1 \right) {\rm Re}\left[B_0^F(M_W^2;\,M_Z,M_W)\Big|_{\mu=M_W}\right] \\ &\quad + \left( \frac{1}{12 c_W^4} + \frac{1}{c_W^2} + 1 \right)\log c_W^2 + \frac{1}{12 c_W^4} + \frac{13}{12 c_W^2} + \frac{59}{18}\,, \end{align}

where the imaginary parts have been neglected.

Parameters

[in]	Qi	the electric charge of f_i
[in]	Qj	the electric charge of f_j
[in]	Mw_i	the \(W\)-boson mass

Returns

\(\rho^W_{ij}\)

Attention

The masses of virtual fermions are neglected.

Definition at line 155 of file EWSMOneLoopEW.cpp.

{
    double QiQj = Qi*Qj;
    double Mz = cache.getSM().getMz();
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
 
    /* Logarithm and one-loop functions */
    double log_cW2 = cache.log_cW2(Mw);
    gslpp::complex B0_Mw2_Mw2_Mz2_Mw2 = cache.B0_Mw2_Mw2_Mz2_Mw2(Mw);
    gslpp::complex C0_Mw2_Mw2_0_Mz2 = cache.C0_Mw2_Mw2_0_Mz2(Mw);
 
    double V1 = FZa_0(Mw*Mw, Mw).real() - 3.0 / 2.0;
    double V2 = -2.0 * (2.0 + cW2) * Mz * Mz * C0_Mw2_Mw2_0_Mz2.real()
            - (1.0 / 12.0 / cW2 / cW2 + 5.0 / 3.0 / cW2 + 1.0) * B0_Mw2_Mw2_Mz2_Mw2.real()
            + (1.0 / 12.0 / cW2 / cW2 + 1.0 / cW2 + 1.0) * log_cW2
            + 1.0 / 12.0 / cW2 / cW2 + 13.0 / 12.0 / cW2 + 59.0 / 18.0;
 
    double deltafij_W, deltafij_QED;
    deltafij_W = -DeltaRhobarW(Mw, Mw)
            - SigmabarPrime_WW_bos_Mw2(Mw, Mw).real()
            - SigmabarPrime_WW_fer_Mw2(Mw, Mw).real()
            + 5.0 / 8.0 * cW2 * (1.0 + cW2) - 11.0 / 2.0 - 9.0 * cW2 / 4.0 / sW2 * log_cW2
            + (-1.0 + 1.0 / 2.0 / cW2 + 2.0 * sW2 * sW2 / cW2 * QiQj)*(V1 + 3.0 / 2.0)
            + 2.0 * cW2 * (V2 + 3.0 / 2.0);
    deltafij_W *= cache.getSM().getAle() / 4.0 / M_PI / sW2;
 
    deltafij_QED = 85.0 / 18.0 - M_PI * M_PI / 3.0 + 3.0 / 4.0 * QiQj;
    deltafij_QED *= cache.getSM().getAle() / M_PI;
 
    return ( 1.0 + deltafij_W + deltafij_QED);
}

SigmabarPrime_WW_bos_Mw2()

gslpp::complex EWSMOneLoopEW::SigmabarPrime_WW_bos_Mw2	(	const double	mu,
		const double	Mw_i
	)		const

The derivative of the bosonic contribution to the self-energy of the \(W\) boson for \(s=M_W^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{bos}}_{WW}(M_W^2)\).

See also the definition of the self-energy given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\prime,\mathrm{bos}}_{WW}(M_W^2)\)

Definition at line 589 of file EWSMOneLoopEW.cpp.

{
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    double mh = cache.getSM().getMHl();
    double mh2 = mh*mh;
    double sW2 = cache.getSM().sW2(Mw_i);
    double cW2 = cache.getSM().cW2(Mw_i);
    double cW4 = cW2*cW2;
    double rw = mh2 / Mw2;
    //double rz = mh2/Mz2;
 
    /* Loop functions */
    double A0_Mw2, A0_Mz2, A0_mh2;
    gslpp::complex B0_Mw2_Mz2_Mw2, B0_Mw2_0_Mw2, B0_Mw2_mh2_Mw2;
    gslpp::complex B0p_Mw2_Mz2_Mw2, B0p_Mw2_0_Mw2, B0p_Mw2_mh2_Mw2;
    if (mu == Mw) {
        A0_Mw2 = cache.A0_Mw2_Mw2(Mw);
        A0_Mz2 = cache.A0_Mw2_Mz2(Mw);
        A0_mh2 = cache.A0_Mw2_mh2(Mw);
        B0_Mw2_Mz2_Mw2 = cache.B0_Mw2_Mw2_Mz2_Mw2(Mw);
        B0_Mw2_0_Mw2 = cache.B0_Mw2_Mw2_0_Mw2(Mw);
        B0_Mw2_mh2_Mw2 = cache.B0_Mw2_Mw2_mh2_Mw2(Mw);
        B0p_Mw2_Mz2_Mw2 = cache.B0p_Mw2_Mw2_Mz2_Mw2(Mw);
        B0p_Mw2_0_Mw2 = cache.B0p_Mw2_Mw2_0_Mw2(Mw);
        B0p_Mw2_mh2_Mw2 = cache.B0p_Mw2_Mw2_mh2_Mw2(Mw);
    } else {
        A0_Mw2 = cache.getPV().A0(mu2, Mw2);
        A0_Mz2 = cache.getPV().A0(mu2, Mz2);
        A0_mh2 = cache.getPV().A0(mu2, mh2);
        B0_Mw2_Mz2_Mw2 = cache.getPV().B0(mu2, Mw2, Mz2, Mw2);
        B0_Mw2_0_Mw2 = cache.getPV().B0(mu2, Mw2, 0.0, Mw2);
        B0_Mw2_mh2_Mw2 = cache.getPV().B0(mu2, Mw2, mh, Mw2);
        B0p_Mw2_Mz2_Mw2 = cache.getPV().B0p(mu2, Mw2, Mz2, Mw2);
        B0p_Mw2_0_Mw2 = cache.getPV().B0p(mu2, Mw2, 0.0, Mw2);
        B0p_Mw2_mh2_Mw2 = cache.getPV().B0p(mu2, Mw2, mh2, Mw2);
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    Sigma = -(1.0 / 12.0 / cW4 + 2.0 / 3.0 / cW2 + 2.0 * cW2) * B0_Mw2_Mz2_Mw2
            + (1.0 / 12.0 / cW4 + 4.0 / 3.0 / cW2 - 17.0 / 3.0 - 4.0 * cW2) * Mw2 * B0p_Mw2_Mz2_Mw2
            - 2.0 * sW2 * B0_Mw2_0_Mw2 - 4.0 * sW2 * Mw2 * B0p_Mw2_0_Mw2
            + rw / 6.0 * (1.0 - rw / 2.0) * B0_Mw2_mh2_Mw2
            + (1.0 - rw / 3.0 + rw * rw / 12.0) * Mw2 * B0p_Mw2_mh2_Mw2
            + (1.0 / cW2 + 8.0 + rw) / 12.0 * A0_Mw2 / Mw2
            - (1.0 / cW2 + 9.0 - 8.0 * cW2 - 12.0 * cW4) / 12.0 * A0_Mz2 / Mw2
            - (rw - 1.0) / 12.0 * A0_mh2 / Mw2 + 4.0 / 9.0;
    return Sigma;
}

SigmabarPrime_WW_fer_Mw2()

gslpp::complex EWSMOneLoopEW::SigmabarPrime_WW_fer_Mw2	(	const double	mu,
		const double	Mw_i
	)		const

The derivative of the fermionic contribution to the self-energy of the \(W\) boson for \(s=M_W^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{fer}}_{WW}(M_W^2)\).

See also the definition of the self-energy given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\prime,\mathrm{fer}}_{WW}(M_W^2)\)

Definition at line 642 of file EWSMOneLoopEW.cpp.

{
    double ml2[6], mq2[6];
    for (int i = 0; i < 6; i++) {
        ml2[i] = cache.mf2(cache.getSM().getLeptons((QCD::lepton) i));
        mq2[i] = cache.mf2(cache.getSM().getQuarks((QCD::quark) i), mu);
    }
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
 
    /* Loop functions */
    gslpp::complex Bf_Mw2_mlprime2_ml2[3], Bf_Mw2_mqprime2_mq2[3];
    gslpp::complex Bfp_Mw2_mlprime2_ml2[3], Bfp_Mw2_mqprime2_mq2[3];
    gslpp::complex B1p_Mw2_mlprime2_ml2[3], B1p_Mw2_mqprime2_mq2[3];
    gslpp::complex B1p_Mw2_ml2_mlprime2[3], B1p_Mw2_mq2_mqprime2[3];
    if (mu == Mw) {
        for (int gen = 0; gen < 3; gen++) {
            Bf_Mw2_mlprime2_ml2[gen] = cache.Bf_Mw2_Mw2_mfprime2_mf2(gen, Mw);
            Bf_Mw2_mqprime2_mq2[gen] = cache.Bf_Mw2_Mw2_mfprime2_mf2(gen + 3, Mw);
            Bfp_Mw2_mlprime2_ml2[gen] = cache.Bfp_Mw2_Mw2_mfprime2_mf2(gen, Mw);
            Bfp_Mw2_mqprime2_mq2[gen] = cache.Bfp_Mw2_Mw2_mfprime2_mf2(gen + 3, Mw);
            B1p_Mw2_ml2_mlprime2[gen] = cache.B1p_Mw2_Mw2_mf2_mfprime2(gen, Mw);
            B1p_Mw2_mq2_mqprime2[gen] = cache.B1p_Mw2_Mw2_mf2_mfprime2(gen + 3, Mw);
            B1p_Mw2_mlprime2_ml2[gen] = cache.B1p_Mw2_Mw2_mfprime2_mf2(gen, Mw);
            B1p_Mw2_mqprime2_mq2[gen] = cache.B1p_Mw2_Mw2_mfprime2_mf2(gen + 3, Mw);
        }
    } else {
        for (int gen = 0; gen < 3; gen++) {
            Bf_Mw2_mlprime2_ml2[gen] = cache.getPV().Bf(mu2, Mw2, ml2[2 * gen + 1], ml2[2 * gen]);
            Bf_Mw2_mqprime2_mq2[gen] = cache.getPV().Bf(mu2, Mw2, mq2[2 * gen + 1], mq2[2 * gen]);
            Bfp_Mw2_mlprime2_ml2[gen] = cache.getPV().Bfp(mu2, Mw2, ml2[2 * gen + 1], ml2[2 * gen]);
            Bfp_Mw2_mqprime2_mq2[gen] = cache.getPV().Bfp(mu2, Mw2, mq2[2 * gen + 1], mq2[2 * gen]);
            B1p_Mw2_ml2_mlprime2[gen] = cache.getPV().B1p(mu2, Mw2, ml2[2 * gen], ml2[2 * gen + 1]);
            B1p_Mw2_mq2_mqprime2[gen] = cache.getPV().B1p(mu2, Mw2, mq2[2 * gen], mq2[2 * gen + 1]);
            B1p_Mw2_mlprime2_ml2[gen] = cache.getPV().B1p(mu2, Mw2, ml2[2 * gen + 1], ml2[2 * gen]);
            B1p_Mw2_mqprime2_mq2[gen] = cache.getPV().B1p(mu2, Mw2, mq2[2 * gen + 1], mq2[2 * gen]);
        }
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    double mf2, mfprime2;
    for (int gen = 0; gen < 3; gen++) {
        mf2 = ml2[2 * gen];
        mfprime2 = ml2[2 * gen + 1];
        Sigma += -(Bf_Mw2_mlprime2_ml2[gen] + Mw2 * Bfp_Mw2_mlprime2_ml2[gen]);
        Sigma += mfprime2 * B1p_Mw2_ml2_mlprime2[gen] + mf2 * B1p_Mw2_mlprime2_ml2[gen];
        //
        mf2 = mq2[2 * gen];
        mfprime2 = mq2[2 * gen + 1];
        Sigma += -3.0 * (Bf_Mw2_mqprime2_mq2[gen] + Mw2 * Bfp_Mw2_mqprime2_mq2[gen]);
        Sigma += 3.0 * (mfprime2 * B1p_Mw2_mq2_mqprime2[gen] + mf2 * B1p_Mw2_mqprime2_mq2[gen]);
    }
    return Sigma;
}

SigmabarPrime_ZZ_bos_Mz2()

gslpp::complex EWSMOneLoopEW::SigmabarPrime_ZZ_bos_Mz2	(	const double	mu,
		const double	Mw_i
	)		const

The derivative of the bosonic contribution to the self-energy of the \(Z\) boson for \(s=M_Z^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{bos}}_{ZZ}(M_Z^2)\).

See also the definition of the self-energy given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\prime,\mathrm{bos}}_{ZZ}(M_Z^2)\)

Definition at line 698 of file EWSMOneLoopEW.cpp.

{
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    double mh = cache.getSM().getMHl();
    double mh2 = mh*mh;
    double cW2 = cache.getSM().cW2(Mw);
    double cW4 = cW2*cW2;
    double rw = mh2 / Mw2;
    double rz = mh2 / Mz2;
 
    /* Loop functions */
    double A0_Mw2, A0_Mz2, A0_mh2;
    gslpp::complex B0_Mz2_Mw2_Mw2, B0_Mz2_mh2_Mz2;
    gslpp::complex B0p_Mz2_Mw2_Mw2, B0p_Mz2_mh2_Mz2;
    if (mu == Mz) {
        A0_Mw2 = cache.A0_Mz2_Mw2(Mw);
        A0_Mz2 = cache.A0_Mz2_Mz2();
        A0_mh2 = cache.A0_Mz2_mh2();
        B0_Mz2_Mw2_Mw2 = cache.B0_Mz2_Mz2_Mw2_Mw2(Mw);
        B0_Mz2_mh2_Mz2 = cache.B0_Mz2_Mz2_mh2_Mz2();
        B0p_Mz2_Mw2_Mw2 = cache.B0p_Mz2_Mz2_Mw2_Mw2(Mw);
        B0p_Mz2_mh2_Mz2 = cache.B0p_Mz2_Mz2_mh2_Mz2();
    } else {
        A0_Mw2 = cache.getPV().A0(mu2, Mw2);
        A0_Mz2 = cache.getPV().A0(mu2, Mz2);
        A0_mh2 = cache.getPV().A0(mu2, mh2);
        B0_Mz2_Mw2_Mw2 = cache.getPV().B0(mu2, Mz2, Mw2, Mw2);
        B0_Mz2_mh2_Mz2 = cache.getPV().B0(mu2, Mz2, mh2, Mz2);
        B0p_Mz2_Mw2_Mw2 = cache.getPV().B0p(mu2, Mz2, Mw2, Mw2);
        B0p_Mz2_mh2_Mz2 = cache.getPV().B0p(mu2, Mz2, mh2, Mz2);
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    Sigma = (1.0 / 4.0 / cW2 + 8.0 / 3.0 - 17.0 / 3.0 * cW2) * B0_Mz2_Mw2_Mw2
            + (1.0 / 12.0 / cW2 + 4.0 / 3.0 - 17.0 / 3.0 * cW2 - 4.0 * cW4) * Mz2 * B0p_Mz2_Mw2_Mw2
            + rw / 6.0 * (1.0 - rz / 2.0) * B0_Mz2_mh2_Mz2
            + (1.0 - rz / 3.0 + rz * rz / 12.0) * Mz2 / cW2 * B0p_Mz2_mh2_Mz2
            + (1.0 + 4.0 * cW2) / 3.0 / cW2 * A0_Mw2 / Mz2
            - (1.0 - rz) / 12.0 / cW2 * (A0_Mz2 - A0_mh2) / Mz2
            + 2.0 / 9.0 / cW2 - 10.0 / 9.0 + 4.0 / 3.0 * cW2;
    Sigma *= cW2;
    return Sigma;
}

SigmabarPrime_ZZ_fer_Mz2()

gslpp::complex EWSMOneLoopEW::SigmabarPrime_ZZ_fer_Mz2	(	const double	mu,
		const double	Mw_i
	)		const

The derivative of the fermionic contribution to the self-energy of the \(Z\) boson for \(s=M_Z^2\) in the Unitary gauge, \(\overline{\Sigma}^{\prime,\mathrm{fer}}_{ZZ}(M_Z^2)\).

See also the definition of the self-energy given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\prime,\mathrm{fer}}_{ZZ}(M_Z^2)\)

Definition at line 746 of file EWSMOneLoopEW.cpp.

{
    double ml2[6], mq2[6];
    for (int i = 0; i < 6; i++) {
        ml2[i] = cache.mf2(cache.getSM().getLeptons((QCD::lepton) i));
        mq2[i] = cache.mf2(cache.getSM().getQuarks((QCD::quark) i), mu);
    }
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
 
    /* Loop functions */
    gslpp::complex Bf_Mz2_ml2_ml2[6], Bf_Mz2_mq2_mq2[6];
    gslpp::complex Bfp_Mz2_ml2_ml2[6], Bfp_Mz2_mq2_mq2[6];
    gslpp::complex B0p_Mz2_ml2_ml2[6], B0p_Mz2_mq2_mq2[6];
    if (mu == Mz) {
        for (int i = 0; i < 6; i++) {
            Bf_Mz2_ml2_ml2[i] = cache.Bf_Mz2_Mz2_mf2_mf2(cache.getSM().getLeptons((QCD::lepton) i));
            Bf_Mz2_mq2_mq2[i] = cache.Bf_Mz2_Mz2_mf2_mf2(cache.getSM().getQuarks((QCD::quark) i));
            Bfp_Mz2_ml2_ml2[i] = cache.Bfp_Mz2_Mz2_mf2_mf2(cache.getSM().getLeptons((QCD::lepton) i));
            Bfp_Mz2_mq2_mq2[i] = cache.Bfp_Mz2_Mz2_mf2_mf2(cache.getSM().getQuarks((QCD::quark) i));
            B0p_Mz2_ml2_ml2[i] = cache.B0p_Mz2_Mz2_mf2_mf2(cache.getSM().getLeptons((QCD::lepton) i));
            B0p_Mz2_mq2_mq2[i] = cache.B0p_Mz2_Mz2_mf2_mf2(cache.getSM().getQuarks((QCD::quark) i));
        }
    } else {
        for (int i = 0; i < 6; i++) {
            Bf_Mz2_ml2_ml2[i] = cache.getPV().Bf(mu2, Mz2, ml2[i], ml2[i]);
            Bf_Mz2_mq2_mq2[i] = cache.getPV().Bf(mu2, Mz2, mq2[i], mq2[i]);
            Bfp_Mz2_ml2_ml2[i] = cache.getPV().Bfp(mu2, Mz2, ml2[i], ml2[i]);
            Bfp_Mz2_mq2_mq2[i] = cache.getPV().Bfp(mu2, Mz2, mq2[i], mq2[i]);
            B0p_Mz2_ml2_ml2[i] = cache.getPV().B0p(mu2, Mz2, ml2[i], ml2[i]);
            B0p_Mz2_mq2_mq2[i] = cache.getPV().B0p(mu2, Mz2, mq2[i], mq2[i]);
        }
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    double mf2, vf2, af2;
    for (int i = 0; i < 6; i++) {
        mf2 = ml2[i];
        vf2 = pow(cache.v_f(cache.getSM().getLeptons((QCD::lepton) i), Mw), 2.0);
        af2 = pow(cache.a_f(cache.getSM().getLeptons((QCD::lepton) i)), 2.0);
        Sigma += -(vf2 + af2)*(Bf_Mz2_ml2_ml2[i] + Mz2 * Bfp_Mz2_ml2_ml2[i])
                - 2.0 * af2 * mf2 * B0p_Mz2_ml2_ml2[i];
        //
        mf2 = mq2[i];
        vf2 = pow(cache.v_f(cache.getSM().getQuarks((QCD::quark) i), Mw), 2.0);
        af2 = pow(cache.a_f(cache.getSM().getQuarks((QCD::quark) i)), 2.0);
        Sigma += -3.0 * (vf2 + af2)*(Bf_Mz2_mq2_mq2[i] + Mz2 * Bfp_Mz2_mq2_mq2[i])
                - 6.0 * af2 * mf2 * B0p_Mz2_mq2_mq2[i];
    }
    return Sigma;
}

SigmabarWW_bos()

gslpp::complex EWSMOneLoopEW::SigmabarWW_bos	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The bosonic contribution to the self-energy of the \(W\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{bos}}_{WW}(s)\).

This function represents the \(O(\alpha)\) bosonic contribution to \(\overline{\Sigma}_{WW}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\mathrm{bos}}_{WW}(s)\)

Definition at line 203 of file EWSMOneLoopEW.cpp.

{
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    double mh = cache.getSM().getMHl();
    double mh2 = mh*mh;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
    double cW4 = cW2*cW2;
    double RW = pow(Mw, 2.0) / s;
    double RW2 = RW*RW;
    double RW3 = RW2*RW;
    double rw = pow(mh / Mw, 2.0);
 
    /* Loop functions */
    double A0_Mw2, A0_Mz2, A0_mh2;
    gslpp::complex B0_s_Mz2_Mw2, B0_s_0_Mw2, B0_s_mh2_Mw2;
    gslpp::complex B0p_s_Mz2_Mw2, B0p_s_mh2_Mw2; /* for s==0.0 */
    if (mu == Mz && s == 0.0) {
        A0_Mw2 = cache.A0_Mz2_Mw2(Mw);
        A0_Mz2 = cache.A0_Mz2_Mz2();
        A0_mh2 = cache.A0_Mz2_mh2();
        B0_s_Mz2_Mw2 = cache.B0_Mz2_0_Mz2_Mw2(Mw);
        B0_s_0_Mw2 = cache.B0_Mz2_0_0_Mw2(Mw);
        B0_s_mh2_Mw2 = cache.B0_Mz2_0_mh2_Mw2(Mw);
        B0p_s_Mz2_Mw2 = cache.B0p_Mz2_0_Mz2_Mw2(Mw);
        B0p_s_mh2_Mw2 = cache.B0p_Mz2_0_mh2_Mw2(Mw);
    } else if (mu == Mz && s == Mw2) {
        A0_Mw2 = cache.A0_Mz2_Mw2(Mw);
        A0_Mz2 = cache.A0_Mz2_Mz2();
        A0_mh2 = cache.A0_Mz2_mh2();
        B0_s_Mz2_Mw2 = cache.B0_Mz2_Mw2_Mz2_Mw2(Mw);
        B0_s_0_Mw2 = cache.B0_Mz2_Mw2_0_Mw2(Mw);
        B0_s_mh2_Mw2 = cache.B0_Mz2_Mw2_mh2_Mw2(Mw);
    } else {
        A0_Mw2 = cache.getPV().A0(mu2, Mw2);
        A0_Mz2 = cache.getPV().A0(mu2, Mz2);
        A0_mh2 = cache.getPV().A0(mu2, mh2);
        B0_s_Mz2_Mw2 = cache.getPV().B0(mu2, s, Mz2, Mw2);
        B0_s_0_Mw2 = cache.getPV().B0(mu2, s, 0.0, Mw2);
        B0_s_mh2_Mw2 = cache.getPV().B0(mu2, s, mh2, Mw2);
        B0p_s_Mz2_Mw2 = cache.getPV().B0p(mu2, s, Mz2, Mw2);
        B0p_s_mh2_Mw2 = cache.getPV().B0p(mu2, s, mh2, Mw2);
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    if (s == 0.0) {
        Sigma = Mw2 * (2.0 / 3.0 * (1.0 / cW2 - 4.0 - 4.0 * cW2 + cW4) * B0_s_Mz2_Mw2
                + (1.0 / 12.0 / cW4 + 2.0 / 3.0 / cW2 - 3.0 / 2.0
                + 2.0 / 3.0 * cW2 + 1.0 / 12.0 * cW4) * Mw2 * B0p_s_Mz2_Mw2
                - 17.0 * sW2 / 6.0 * B0_s_0_Mw2 + 5.0 / 12.0 * sW2
                - 1.0 / 12.0 * (-10.0 + 2.0 * rw) * B0_s_mh2_Mw2
                + 1.0 / 12.0 * pow(1.0 - rw, 2.0) * Mw2 * B0p_s_mh2_Mw2
                - 1.0 / 12.0 * (24.0 - 2.0 * cW2 + cW4) * A0_Mw2 / Mw2
                - 1.0 / 6.0 * A0_mh2 / Mw2
                - 1.0 / 12.0 * (1.0 + 14.0 * cW2 + 9.0 * cW4) * A0_Mz2 / Mw2
                - 1.0 / 6.0 * (1.0 / cW2 + 22.0 + cW2 + cW4 + rw));
    } else {
        Sigma = Mw2 * (((1.0 / 12.0 / cW4 + 2.0 / 3.0 / cW2 - 3.0 / 2.0 + 2.0 / 3.0 * cW2
                + 1.0 / 12.0 * cW4) * RW
                + 2.0 / 3.0 * (1.0 / cW2 - 4.0 - 4.0 * cW2 + cW4)
                - (3.0 / 2.0 + 8.0 / 3.0 * cW2 + 3.0 / 2.0 * cW4) / RW
                + 2.0 / 3.0 * cW2 * (1.0 + cW2) / RW2 + 1.0 / 12.0 * cW4 / RW3) * B0_s_Mz2_Mw2
                - sW2 / 6.0 * (-5.0 * RW + 17.0 + 17.0 / RW - 5.0 / RW2) * B0_s_0_Mw2
                - 1.0 / 12.0 * (-pow(1.0 - rw, 2.0) * RW - 10.0 + 2.0 * rw - 1.0 / RW)
                * B0_s_mh2_Mw2
                - 1.0 / 12.0 * ((1.0 / cW2 - 2.0 + cW2 - cW4 + rw) * RW
                + 24.0 - 2.0 * cW2 + cW4
                + (-10.0 + cW2 + cW4) / RW - cW4 / RW2) * A0_Mw2 / Mw2
                - 1.0 / 12.0 * (-(1.0 / cW2 + 9.0 - 9.0 * cW2 - cW4) * RW
                + 1.0 + 14.0 * cW2 + 9.0 * cW4
                + cW2 / RW * (1.0 - 9.0 * cW2) - cW4 / RW2) * A0_Mz2 / Mw2
                + 1.0 / 12.0 * ((mh2 - Mw2) / s - 2.0) * A0_mh2 / Mw2
                - 1.0 / 6.0 * (1.0 / cW2 + 22.0 + cW2 + cW4 + rw)
                + 1.0 / 9.0 * ((6.0 + 3.0 * cW2 + 7.0 / 2.0 * cW4) / RW
                - (1.0 + 3.0 / 2.0 * cW2 + 5.0 / 2.0 * cW4) / RW2
                + cW4 / 2.0 / RW3));
    }
    return Sigma;
}

SigmabarWW_fer()

gslpp::complex EWSMOneLoopEW::SigmabarWW_fer	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The fermionic contribution to the self-energy of the \(W\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{fer}}_{WW}(s)\).

This function represents the \(O(\alpha)\) fermionic contribution to \(\overline{\Sigma}_{WW}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\mathrm{fer}}_{WW}(s)\)

Definition at line 287 of file EWSMOneLoopEW.cpp.

{
    double ml2[6], mq2[6];
    for (int i = 0; i < 6; i++) {
        ml2[i] = cache.mf2(cache.getSM().getLeptons((QCD::lepton) i));
        mq2[i] = cache.mf2(cache.getSM().getQuarks((QCD::quark) i), mu);
    }
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
 
    /* Loop functions */
    gslpp::complex B1_s_ml2_mlprime2[3], B1_s_mq2_mqprime2[3];
    gslpp::complex B1_s_mlprime2_ml2[3], B1_s_mqprime2_mq2[3];
    gslpp::complex Bf_s_mlprime2_ml2[3], Bf_s_mqprime2_mq2[3];
    if (mu == Mz && s == 0.0) {
        for (int gen = 0; gen < 3; gen++) {
            B1_s_ml2_mlprime2[gen] = cache.B1_Mz2_0_mf2_mfprime2(gen);
            B1_s_mq2_mqprime2[gen] = cache.B1_Mz2_0_mf2_mfprime2(gen + 3);
            B1_s_mlprime2_ml2[gen] = cache.B1_Mz2_0_mfprime2_mf2(gen);
            B1_s_mqprime2_mq2[gen] = cache.B1_Mz2_0_mfprime2_mf2(gen + 3);
            Bf_s_mlprime2_ml2[gen] = cache.Bf_Mz2_0_mfprime2_mf2(gen);
            Bf_s_mqprime2_mq2[gen] = cache.Bf_Mz2_0_mfprime2_mf2(gen + 3);
        }
    } else if (mu == Mz && s == Mw2) {
        for (int gen = 0; gen < 3; gen++) {
            B1_s_ml2_mlprime2[gen] = cache.B1_Mz2_Mw2_mf2_mfprime2(gen, Mw);
            B1_s_mq2_mqprime2[gen] = cache.B1_Mz2_Mw2_mf2_mfprime2(gen + 3, Mw);
            B1_s_mlprime2_ml2[gen] = cache.B1_Mz2_Mw2_mfprime2_mf2(gen, Mw);
            B1_s_mqprime2_mq2[gen] = cache.B1_Mz2_Mw2_mfprime2_mf2(gen + 3, Mw);
            Bf_s_mlprime2_ml2[gen] = cache.Bf_Mz2_Mw2_mfprime2_mf2(gen, Mw);
            Bf_s_mqprime2_mq2[gen] = cache.Bf_Mz2_Mw2_mfprime2_mf2(gen + 3, Mw);
        }
    } else {
        for (int gen = 0; gen < 3; gen++) {
            B1_s_ml2_mlprime2[gen] = cache.getPV().B1(mu2, s, ml2[2 * gen], ml2[2 * gen + 1]);
            B1_s_mq2_mqprime2[gen] = cache.getPV().B1(mu2, s, mq2[2 * gen], mq2[2 * gen + 1]);
            B1_s_mlprime2_ml2[gen] = cache.getPV().B1(mu2, s, ml2[2 * gen + 1], ml2[2 * gen]);
            B1_s_mqprime2_mq2[gen] = cache.getPV().B1(mu2, s, mq2[2 * gen + 1], mq2[2 * gen]);
            Bf_s_mlprime2_ml2[gen] = cache.getPV().Bf(mu2, s, ml2[2 * gen + 1], ml2[2 * gen]);
            Bf_s_mqprime2_mq2[gen] = cache.getPV().Bf(mu2, s, mq2[2 * gen + 1], mq2[2 * gen]);
        }
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    double mf2, mfprime2;
    for (int gen = 0; gen < 3; gen++) {
        mf2 = ml2[2 * gen];
        mfprime2 = ml2[2 * gen + 1];
        if (s != 0.0) Sigma += -s * Bf_s_mlprime2_ml2[gen];
        Sigma += mfprime2 * B1_s_ml2_mlprime2[gen] + mf2 * B1_s_mlprime2_ml2[gen];
        //
        mf2 = mq2[2 * gen];
        mfprime2 = mq2[2 * gen + 1];
        if (s != 0.0) Sigma += 3.0 * (-s * Bf_s_mqprime2_mq2[gen]);
        Sigma += 3.0 * (mfprime2 * B1_s_mq2_mqprime2[gen] + mf2 * B1_s_mqprime2_mq2[gen]);
    }
    return Sigma;
}

SigmabarZZ_bos()

gslpp::complex EWSMOneLoopEW::SigmabarZZ_bos	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The bosonic contribution to the self-energy of the \(Z\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{bos}}_{ZZ}(s)\).

This function represents the \(O(\alpha)\) bosonic contribution to \(\overline{\Sigma}_{ZZ}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\mathrm{bos}}_{ZZ}(s)\)

Definition at line 348 of file EWSMOneLoopEW.cpp.

{
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
    double mh = cache.getSM().getMHl();
    double mh2 = mh*mh;
    double cW2 = cache.getSM().cW2(Mw);
    double cW4 = cW2*cW2;
    double RW = pow(Mw, 2.0) / s;
    double RW2 = RW*RW;
    double RW3 = RW2*RW;
    double rw = pow(mh / Mw, 2.0);
 
    /* Loop functions */
    double A0_Mw2, A0_Mz2, A0_mh2;
    gslpp::complex B0_s_Mw2_Mw2, B0_s_mh2_Mz2;
    if (mu == Mz && s == Mz2) {
        A0_Mw2 = cache.A0_Mz2_Mw2(Mw);
        A0_Mz2 = cache.A0_Mz2_Mz2();
        A0_mh2 = cache.A0_Mz2_mh2();
        B0_s_Mw2_Mw2 = cache.B0_Mz2_Mz2_Mw2_Mw2(Mw);
        B0_s_mh2_Mz2 = cache.B0_Mz2_Mz2_mh2_Mz2();
    } else {
        A0_Mw2 = cache.getPV().A0(mu2, Mw2);
        A0_Mz2 = cache.getPV().A0(mu2, Mz2);
        A0_mh2 = cache.getPV().A0(mu2, mh2);
        B0_s_Mw2_Mw2 = cache.getPV().B0(mu2, s, Mw2, Mw2);
        B0_s_mh2_Mz2 = cache.getPV().B0(mu2, s, mh2, Mz2);
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    if (s == 0.0) {
        throw std::runtime_error("EWSMOneLoopEW::SigmabarZZ_bos(): Undefined for s=0.0");
    } else {
        Sigma = Mw2 * (-cW4 * (4.0 + 17.0 / 3.0 / RW - 4.0 / 3.0 / RW2 - 1.0 / 12.0 / RW3)
                * B0_s_Mw2_Mw2
                + 1.0 / 12.0 * ((1.0 / cW4 - 2.0 / cW2 * rw + rw * rw) * RW
                + 10.0 / cW2 - 2.0 * rw + 1.0 / RW) * B0_s_mh2_Mz2
                - cW2 * (4.0 - 4.0 / 3.0 / RW - 1.0 / 6.0 / RW2) * A0_Mw2 / Mz2
                + 1.0 / 12.0 * ((Mz2 - mh2) / s + 1.0)*(A0_Mz2 - A0_mh2) / cW2 / Mz2
                - 1.0 / 12.0 * A0_mh2 / cW2 / Mz2
                - (1.0 / 6.0 / cW2 + 4.0 * cW4 + 1.0 / 6.0 * rw
                - (1.0 / 18.0 + 4.0 / 3.0 * cW4) / RW
                + 1.0 / 9.0 * cW4 * (5.0 - 1.0 / 2.0 / RW) / RW2));
    }
    return Sigma;
}

SigmabarZZ_fer()

gslpp::complex EWSMOneLoopEW::SigmabarZZ_fer	(	const double	mu,
		const double	s,
		const double	Mw_i
	)		const

The fermionic contribution to the self-energy of the \(Z\) boson in the Unitary gauge, \(\overline{\Sigma}^{\mathrm{fer}}_{ZZ}(s)\).

This function represents the \(O(\alpha)\) fermionic contribution to \(\overline{\Sigma}_{ZZ}(s)\), whose definition is given in the detailed description of the current class.

Parameters

[in]	mu	renormalization scale \(\mu\)
[in]	s	momentum squared \(s\)
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\overline{\Sigma}^{\mathrm{fer}}_{ZZ}(s)\)

Definition at line 399 of file EWSMOneLoopEW.cpp.

{
    double ml2[6], mq2[6];
    for (int i = 0; i < 6; i++) {
        ml2[i] = cache.mf2(cache.getSM().getLeptons((QCD::lepton) i));
        mq2[i] = cache.mf2(cache.getSM().getQuarks((QCD::quark) i), mu);
    }
    double mu2 = mu*mu;
    double Mw = Mw_i;
    double Mz = cache.getSM().getMz();
    double Mz2 = Mz*Mz;
 
    /* Loop functions */
    gslpp::complex Bf_s_ml2_ml2[6], Bf_s_mq2_mq2[6];
    gslpp::complex B0_s_ml2_ml2[6], B0_s_mq2_mq2[6];
    if (mu == Mz && s == Mz2) {
        for (int i = 0; i < 6; i++) {
            Bf_s_ml2_ml2[i] = cache.Bf_Mz2_Mz2_mf2_mf2(cache.getSM().getLeptons((QCD::lepton) i));
            Bf_s_mq2_mq2[i] = cache.Bf_Mz2_Mz2_mf2_mf2(cache.getSM().getQuarks((QCD::quark) i));
            B0_s_ml2_ml2[i] = cache.B0_Mz2_Mz2_mf2_mf2(cache.getSM().getLeptons((QCD::lepton) i));
            B0_s_mq2_mq2[i] = cache.B0_Mz2_Mz2_mf2_mf2(cache.getSM().getQuarks((QCD::quark) i));
        }
    } else {
        for (int i = 0; i < 6; i++) {
            Bf_s_ml2_ml2[i] = cache.getPV().Bf(mu2, s, ml2[i], ml2[i]);
            Bf_s_mq2_mq2[i] = cache.getPV().Bf(mu2, s, mq2[i], mq2[i]);
            B0_s_ml2_ml2[i] = cache.getPV().B0(mu2, s, ml2[i], ml2[i]);
            B0_s_mq2_mq2[i] = cache.getPV().B0(mu2, s, mq2[i], mq2[i]);
        }
    }
 
    gslpp::complex Sigma(0.0, 0.0, false);
    if (s == 0.0) {
        throw std::runtime_error("EWSMOneLoopEW::SigmaZZ_fer(): Undefined for s=0.0");
    } else {
        double mf2, vf2, af2;
        for (int i = 0; i < 6; i++) {
            mf2 = ml2[i];
            vf2 = pow(cache.v_f(cache.getSM().getLeptons((QCD::lepton) i), Mw), 2.0);
            af2 = pow(cache.a_f(cache.getSM().getLeptons((QCD::lepton) i)), 2.0);
            if (s != 0.0) Sigma += -(vf2 + af2) * s * Bf_s_ml2_ml2[i];
            Sigma += -2.0 * af2 * mf2 * B0_s_ml2_ml2[i];
            //
            mf2 = mq2[i];
            vf2 = pow(cache.v_f(cache.getSM().getQuarks((QCD::quark) i), Mw), 2.0);
            af2 = pow(cache.a_f(cache.getSM().getQuarks((QCD::quark) i)), 2.0);
            if (s != 0.0) Sigma += -3.0 * (vf2 + af2) * s * Bf_s_mq2_mq2[i];
            Sigma += -3.0 * 2.0 * af2 * mf2 * B0_s_mq2_mq2[i];
        }
    }
    return Sigma;
}

TEST_DeltaRhobar_bos()

double EWSMOneLoopEW::TEST_DeltaRhobar_bos

(

const double

Mw_i

)

const

A test function.

\(\Delta\overline{\rho}^{\mathrm{bos}}\) is given without the use of the self-energies of the gauge bosons.

Parameters

[in]

Mw_i

the \(W\)-boson mass \(M_W\)

Returns

\(\Delta\overline{\rho}^{\mathrm{bos}}\)

Attention

The renormalization scale is fixed to be \(\mu=M_W\).

Definition at line 823 of file EWSMOneLoopEW.cpp.

{
    double Mz = cache.getSM().getMz();
    double mh = cache.getSM().getMHl();
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
    double cW4 = cW2*cW2;
    double rw = pow(mh / Mw, 2.0);
    double rz = pow(mh / Mz, 2.0);
 
    /* Logarithm and one-loop functions */
    double log_cW2 = cache.log_cW2(Mw);
 
    /* B0 functions for mu=Mw */
    gslpp::complex B0_Mw2_Mz2_Mw2_Mw2 = cache.B0_Mz2_Mz2_Mw2_Mw2(Mw) + log_cW2;
    gslpp::complex B0_Mw2_Mz2_mh2_Mz2 = cache.B0_Mz2_Mz2_mh2_Mz2() + log_cW2;
    gslpp::complex B0_Mw2_Mw2_Mz2_Mw2 = cache.B0_Mz2_Mw2_Mz2_Mw2(Mw) + log_cW2;
    gslpp::complex B0_Mw2_Mw2_mh2_Mw2 = cache.B0_Mz2_Mw2_mh2_Mw2(Mw) + log_cW2;
 
    double DRhobar;
    DRhobar = -(1.0 / 12.0 / cW2 + 4.0 / 3.0 - 17.0 / 3.0 * cW2 - 4.0 * cW4)
            *(B0_Mw2_Mz2_Mw2_Mw2.real() - 1.0 / cW2 * B0_Mw2_Mw2_Mz2_Mw2.real())
            + (1.0 - 1.0 / 3.0 * rw + 1.0 / 12.0 * rw * rw) * B0_Mw2_Mw2_mh2_Mw2.real()
            - (1.0 - 1.0 / 3.0 * rz + 1.0 / 12.0 * rz * rz) / cW2 * B0_Mw2_Mz2_mh2_Mz2.real()
            + 1.0 / 12.0 * sW2 * rw * rw * (log(rw) - 1.0)
            - (1.0 / 12.0 / cW4 + 1.0 / 2.0 / cW2 - 2.0 + 1.0 / 12.0 * rw) * log_cW2
            - 1.0 / 12.0 / cW4 - 19.0 / 36.0 / cW2 - 133.0 / 18.0 + 8.0 * cW2;
    return DRhobar;
}

TEST_DeltaRhobarW_bos()

double EWSMOneLoopEW::TEST_DeltaRhobarW_bos

(

const double

Mw_i

)

const

A test function.

\(\Delta\overline{\rho}_W^{\mathrm{bos}}\) is given without the use of the self-energies of the gauge bosons.

Parameters

[in]

Mw_i

the \(W\)-boson mass \(M_W\)

Returns

\(\Delta\overline{\rho}_W^{\mathrm{bos}}\)

Attention

The renormalization scale is fixed to be \(\mu=M_W\).

Definition at line 854 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double sW2 = cache.getSM().sW2(Mw);
    double cW2 = cache.getSM().cW2(Mw);
    double cW4 = cW2*cW2;
    double mh = cache.getSM().getMHl();
    double rw = pow(mh / Mw, 2.0);
 
    /* Logarithm and one-loop functions */
    double log_cW2 = cache.log_cW2(Mw);
 
    /* B0 functions for mu=Mw */
    gslpp::complex B0_Mw2_Mz2_Mw2 = cache.B0_Mz2_Mw2_Mz2_Mw2(Mw) + log_cW2;
    gslpp::complex B0_Mw2_mh2_Mw2 = cache.B0_Mz2_Mw2_mh2_Mw2(Mw) + log_cW2;
 
    double DRhobarW;
    DRhobarW = -(1.0 / 12.0 / cW4 + 4.0 / 3.0 / cW2 - 17.0 / 3.0 - 4.0 * cW2) * B0_Mw2_Mz2_Mw2.real()
            - (1.0 - 1.0 / 3.0 * rw + 1.0 / 12.0 * rw * rw) * B0_Mw2_mh2_Mw2.real()
            + (3.0 / 4.0 / (1.0 - rw) + 1.0 / 4.0 - 1.0 / 12.0 * rw) * rw * log(rw)
            + (1.0 / 12.0 / cW4 + 17.0 / 12.0 / cW2 - 3.0 / sW2 + 1.0 / 4.0) * log_cW2
            + 1.0 / 12.0 / cW4 + 11.0 / 8.0 / cW2 + 139.0 / 36.0 - 177.0 / 24.0 * cW2
            + 5.0 / 8.0 * cW4 - 1.0 / 12.0 * rw * (7.0 / 2.0 - rw);
    return DRhobarW;
}

TEST_FWn()

gslpp::complex EWSMOneLoopEW::TEST_FWn	(	const double	s,
		const double	mf,
		const double	Mw_i
	)		const

A test function for \(\mathcal{F}_{Wn}\) with a finite fermion mass.

Parameters

[in]	s	momentum squared \(s\)
[in]	mf	the mass of the fermion in the loop
[in]	Mw_i	the \(W\)-boson mass \(M_W\)

Returns

\(\mathcal{F}_{Wn}\)

Definition at line 1121 of file EWSMOneLoopEW.cpp.

{
    double Mw = Mw_i;
    double Mw2 = Mw*Mw;
    double Rw = Mw2 / s;
    double mf2 = mf*mf;
    double wf = mf2 / Mw2;
 
    /* Logarithm and two- and three-point one-loop functions */
    double log_wf = log(wf);
    double A0_Mw2 = cache.getPV().A0(Mw2, Mw2);
    double A0_mf2 = cache.getPV().A0(Mw2, mf2);
    gslpp::complex B0_Mw2_s_Mw2_Mw2 = cache.getPV().B0(Mw2, s, Mw2, Mw2);
    gslpp::complex B0_Mw2_0_mf2_Mw2 = cache.getPV().B0(Mw2, 0.0, mf2, Mw2);
    gslpp::complex C0_s_Mw2_mf2_Mw2 = cache.getPV().C0(s, Mw2, mf2, Mw2);
    //gslpp::complex C0_s_Mw2_0_Mw2 = cache.getPV().C0(s, Mw2, 0.0, Mw2);
 
    gslpp::complex FWn(0.0, 0.0, false);
    /* Eq.(5.586) in Bardin and Passarino's book */
    FWn = ((2.0 + wf)*(1.0 - wf)*(1.0 - wf) * Rw + 4.0 - 5.0 / 2.0 * wf + 2.0 * wf * wf
            - wf * wf * wf / 2.0 + wf * (2.0 - wf / 2.0) / Rw) * Mw * Mw * C0_s_Mw2_mf2_Mw2
            - (-(2.0 + wf)*(1.0 - wf) * Rw - 3.0 + 3.0 / 2.0 * wf - wf * wf / 2.0)
            *(B0_Mw2_s_Mw2_Mw2 - B0_Mw2_0_mf2_Mw2)
            - (2.0 / 3.0 - wf / 2.0 + (3.0 / 2.0 - wf / 4.0) / Rw + 1.0 / 12.0 / Rw / Rw) * B0_Mw2_s_Mw2_Mw2
            - A0_mf2 / Mw / Mw - (2.0 / 3.0 + 1.0 / 6.0 / Rw) * A0_Mw2 / Mw / Mw
            - 2.0 / 3.0 - wf / 2.0 + (4.0 / 9.0 + wf / 4.0) / Rw - 1.0 / 18.0 / Rw / Rw;
 
    /* Eq.(5.500) in Bardin and Passarino's book */
    double wW;
    if (mf == 0.0) {
        wW = 3.0 / 2.0;
    } else {
        wW = 5.0 / 4.0 * wf + 3.0 * (1.0 - wf / 2.0) * wf * wf / (wf - 1.0) / (wf - 1.0) * log_wf
                - 3.0 / 2.0 / (wf - 1.0);
    }
 
    return ( -FWn + wW);
}

Member Data Documentation

cache

const EWSMcache& EWSMOneLoopEW::cache

private

A reference to an object of type EWSMcache.

Definition at line 846 of file EWSMOneLoopEW.h.

---

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

• EWSMOneLoopEW.h
• EWSMOneLoopEW.cpp