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EWSMThreeLoopQCD Class Reference
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A class for \(O(\alpha\alpha_s^2)\) three-loop corrections to the EW precision observables. More...

#include <EWSMThreeLoopQCD.h>

Detailed Description

A class for \(O(\alpha\alpha_s^2)\) three-loop corrections to the EW precision observables.

Author
HEPfit Collaboration
Copyright
GNU General Public License

This class handles three-loop QCD contributions of \(O(\alpha\alpha_s^2)\) 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.

Definition at line 33 of file EWSMThreeLoopQCD.h.

Public Member Functions

double DeltaAlpha_l (const double s) const
 Leptonic contribution of \(O(\alpha\alpha_s^2)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{lept}}^{\alpha\alpha_s^2}(s)\). More...
 
double DeltaAlpha_t (const double s) const
 Top-quark contribution of \(O(\alpha\alpha_s^2)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{top}}^{\alpha\alpha_s^2}(s)\). More...
 
gslpp::complex deltaKappa_rem_f (const Particle f, const double Mw_i) const
 Remainder contribution of \(O(\alpha\alpha_s^2)\) to the effective couplings \(\kappa_Z^f\), denoted as \(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha\alpha_s^2}\). More...
 
double DeltaR_rem (const double Mw_i) const
 Remainder contribution of \(O(\alpha\alpha_s^2)\) to \(\Delta r\), denoted as \(\Delta r_{\mathrm{rem}}^{\alpha\alpha_s^2}\). More...
 
double DeltaRho (const double Mw_i) const
 Leading three-loop QCD contribution of \(O(\alpha\alpha_s^2)\) to \(\Delta\rho\), denoted as \(\Delta\rho^{\alpha\alpha_s^2}\). More...
 
gslpp::complex deltaRho_rem_f (const Particle f, const double Mw_i) const
 Remainder contribution of \(O(\alpha\alpha_s^2)\) to the effective couplings \(\rho_Z^f\), denoted as \(\delta\rho_{\mathrm{rem}}^{f,\, \alpha\alpha_s^2}\). More...
 
 EWSMThreeLoopQCD (const EWSMcache &cache_i)
 Constructor. More...
 

Private Member Functions

double deltaQCD_3 (const double Mw_i) const
 The function \(\delta^{\mathrm{QCD}}_3\). More...
 
gslpp::complex deltaQCD_kappa3 (const double Mw_i) const
 The function \(\delta^{\mathrm{QCD}}_{\kappa, 3}\). More...
 

Private Attributes

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

Constructor & Destructor Documentation

◆ EWSMThreeLoopQCD()

EWSMThreeLoopQCD::EWSMThreeLoopQCD ( const EWSMcache &  cache_i)

Constructor.

Parameters
[in]cache_ia reference to an object of type EWSMcache

Definition at line 10 of file EWSMThreeLoopQCD.cpp.

11: cache(cache_i)
12{
13}
EWSMThreeLoopQCD::cache
const EWSMcache & cache
A reference to an object of type EWSMcache.
Definition: EWSMThreeLoopQCD.h:153

Member Function Documentation

◆ DeltaAlpha_l()

double EWSMThreeLoopQCD::DeltaAlpha_l ( const double  s) const

Leptonic contribution of \(O(\alpha\alpha_s^2)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{lept}}^{\alpha\alpha_s^2}(s)\).

This contribution vanishes at \(O(\alpha\alpha_s^2)\).

Parameters
[in]sinvariant mass squared
Returns
\(\Delta\alpha_{\mathrm{lept}}^{\alpha\alpha_s^2}(s)=0\)

Definition at line 18 of file EWSMThreeLoopQCD.cpp.

19{
20 return (0.0);
21}

◆ DeltaAlpha_t()

double EWSMThreeLoopQCD::DeltaAlpha_t ( const double  s) const

Top-quark contribution of \(O(\alpha\alpha_s^2)\) to the electromagnetic coupling \(\alpha\), denoted as \(\Delta\alpha_{\mathrm{top}}^{\alpha\alpha_s^2}(s)\).

A simple numerical formula presented in [Kuhn:1998ze] has been employed. See also [Chetyrkin:1995ii], [Chetyrkin:1996cf] and [Chetyrkin:1997mb].

Parameters
[in]sinvariant mass squared
Returns
\(\Delta\alpha_{\mathrm{top}}^{\alpha\alpha_s^2}(s)\)

Definition at line 23 of file EWSMThreeLoopQCD.cpp.

24{
25 double xt = s / cache.getSM().getMtpole() / cache.getSM().getMtpole();
26 double log_t, als;
27 if (s == cache.getSM().getMz() * cache.getSM().getMz()) {
28 log_t = 2.0 * cache.logMZtoMTOP();
29 als = cache.getSM().getAlsMz();
30 } else {
31 double mu = sqrt(s);
32 log_t = log(s / pow(cache.mf(cache.getSM().getQuarks(QCD::TOP), mu), 2.0));
33 als = cache.Als(sqrt(s), FULLNNLO);
34 }
35 double tmp = ((28.220 + 9.702 * log_t)
36 + xt * (6.924 + 1.594 * log_t))
37 * pow(als / M_PI, 2.0);
38 tmp *= -4.0 / 45.0 * cache.getSM().getAle() / M_PI*xt;
39 return tmp;
40}
FULLNNLO
@ FULLNNLO
Definition: OrderScheme.h:39
EWSMcache::Als
double Als(const double mu, const orders order) const
The strong coupling .
Definition: EWSMcache.h:366
EWSMcache::logMZtoMTOP
double logMZtoMTOP() const
A cache method.
Definition: EWSMcache.cpp:120
EWSMcache::mf
double mf(const Particle f, const double mu=0.0, const orders order=FULLNNLO) const
The mass of an SM fermion.
Definition: EWSMcache.cpp:49
EWSMcache::getSM
const StandardModel & getSM() const
Definition: EWSMcache.h:56
QCD::TOP
@ TOP
Definition: QCD.h:328
QCD::getMtpole
const double getMtpole() const
A get method to access the pole mass of the top quark.
Definition: QCD.h:600
QCD::getQuarks
const Particle & getQuarks(const QCD::quark q) const
A get method to access a quark as an object of the type Particle.
Definition: QCD.h:536
StandardModel::getMz
const double getMz() const
A get method to access the mass of the boson .
Definition: StandardModel.h:753
StandardModel::getAlsMz
const double getAlsMz() const
A get method to access the value of .
Definition: StandardModel.h:771
StandardModel::getAle
const double getAle() const
A get method to retrieve the fine-structure constant .
Definition: StandardModel.h:789
s
Test Observable.

◆ deltaKappa_rem_f()

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

Remainder contribution of \(O(\alpha\alpha_s^2)\) to the effective couplings \(\kappa_Z^f\), denoted as \(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha\alpha_s^2}\).

The formula used here is given by

\[ \delta\kappa_{\mathrm{rem}}^{f,\alpha\alpha_s^2} = - 3\,X_t^\alpha \frac{c_W^2}{s_W^2} \biggl(\frac{\alpha_s(m_t^2)}{\pi}\biggr)^2 \bigl( \delta^{\mathrm{QCD}}_3 + \mathrm{Re}\,[\delta^{\mathrm{QCD}}_{\kappa,\,3}]\bigr), \]

where \(\delta^{\mathrm{QCD}}_3\) and \(\delta^{\mathrm{QCD}}_3\) are computed via deltaQCD_3() and deltaQCD_kappa3(), respectively. See [Avdeev:1994db], [Chetyrkin:1995ix] and [Chetyrkin:1995js].

Parameters
[in]fa lepton or quark
[in]Mw_ithe \(W\)-boson mass
Returns
\(\delta\kappa_{\mathrm{rem}}^{f,\, \alpha\alpha_s^2}\)

Definition at line 72 of file EWSMThreeLoopQCD.cpp.

73{
74 if (f.is("TOP"))
75 return ( gslpp::complex(0.0, 0.0, false));
76 else {
77 double Mw = Mw_i;
78 return ( -3.0 * cache.Xt_alpha(Mw) * cache.getSM().cW2(Mw) / cache.getSM().sW2(Mw)
79 * pow(cache.alsMt() / M_PI, 2.0)
80 *(deltaQCD_3(Mw) + deltaQCD_kappa3(Mw).real()));
81 }
82}
EWSMThreeLoopQCD::deltaQCD_3
double deltaQCD_3(const double Mw_i) const
The function .
Definition: EWSMThreeLoopQCD.cpp:87
EWSMThreeLoopQCD::deltaQCD_kappa3
gslpp::complex deltaQCD_kappa3(const double Mw_i) const
The function .
Definition: EWSMThreeLoopQCD.cpp:114
EWSMcache::Xt_alpha
double Xt_alpha(const double Mw_i) const
The quantity with the coupling .
Definition: EWSMcache.h:355
EWSMcache::alsMt
double alsMt() const
The strong coupling at NNLO.
Definition: EWSMcache.h:378
Mw
An observable class for the -boson mass.
Definition: Mw.h:22
Particle::is
bool is(std::string name_i) const
Definition: Particle.cpp:23
StandardModel::cW2
virtual const double cW2(const double Mw_i) const
The square of the cosine of the weak mixing angle in the on-shell scheme, denoted as .
Definition: StandardModel/src/StandardModel.cpp:1076
StandardModel::sW2
virtual const double sW2(const double Mw_i) const
The square of the sine of the weak mixing angle in the on-shell scheme, denoted as .
Definition: StandardModel/src/StandardModel.cpp:1087

◆ deltaQCD_3()

double EWSMThreeLoopQCD::deltaQCD_3 ( const double  Mw_i) const
private

The function \(\delta^{\mathrm{QCD}}_3\).

This function is associated with the leading three-loop QCD contribution of \(O(\alpha\alpha_s^2(m_t^2/M_Z^2+1+M_Z^2/m_t^2))\) to \(\Delta\rho\), as explained in the description of DeltaRho(). See [Avdeev:1994db], [Chetyrkin:1995ix], [Chetyrkin:1995js] and Chapter 8 of [Bardin:1999ak].

Parameters
[in]Mw_ithe \(W\)-boson mass
Returns
\(\delta^{\mathrm{QCD}}_3\)

Definition at line 87 of file EWSMThreeLoopQCD.cpp.

88{
89 double dQCD3;
90 double lZ = 2.0 * cache.logMZtoMTOP();
91 double Mw = Mw_i;
92 double sW2 = cache.getSM().sW2(Mw);
93 double log2 = cache.getLog2();
94 double zeta2 = cache.getZeta2();
95 double zeta3 = cache.getZeta3();
96 double zeta4 = cache.getZeta4();
97 double S2 = cache.getS2(), D3 = cache.getD3(), B4 = cache.getB4();
98 double MZtoMT = cache.getSM().getMz() / cache.getSM().getMtpole();
99 double nf = 6.0;
100 dQCD3 = 157.0 / 648.0 - 3313.0 / 162.0 * zeta2 - 308.0 / 27.0 * zeta3
101 + 143.0 / 18.0 * zeta4 - 4.0 / 3.0 * zeta2 * log2
102 + 441.0 / 8.0 * S2 - B4 / 9.0 - D3 / 18.0
103 - (1.0 / 18.0 - 13.0 / 9.0 * zeta2 + 4.0 / 9.0 * zeta3) * nf
104 + pow(MZtoMT, 2.0)
105 *(-17.224 + 0.08829 * lZ + 0.4722 * lZ * lZ
106 + (22.6367 + 1.2527 * lZ - 0.8519 * lZ * lZ) * sW2)
107 + pow(MZtoMT, 4.0)
108 *(-7.7781 - 0.07226 * lZ + 0.004938 * lZ * lZ
109 + (21.497 + 0.05794 * lZ - 0.006584 * lZ * lZ) * sW2
110 - 21.0799 * sW2 * sW2);
111 return dQCD3;
112}
EWSMcache::getD3
double getD3() const
A get method to access the constant .
Definition: EWSMcache.h:197
EWSMcache::getZeta4
double getZeta4() const
A get method to access the value of the zeta function .
Definition: EWSMcache.h:155
EWSMcache::getB4
double getB4() const
A get method to access the constant .
Definition: EWSMcache.h:214
EWSMcache::getS2
double getS2() const
A get method to access the constant .
Definition: EWSMcache.h:180
EWSMcache::getZeta3
double getZeta3() const
A get method to access the value of the zeta function .
Definition: EWSMcache.h:146
EWSMcache::getLog2
double getLog2() const
A get method to access the constant .
Definition: EWSMcache.h:223
EWSMcache::getZeta2
double getZeta2() const
A get method to access the value of the zeta function .
Definition: EWSMcache.h:137

◆ deltaQCD_kappa3()

gslpp::complex EWSMThreeLoopQCD::deltaQCD_kappa3 ( const double  Mw_i) const
private

The function \(\delta^{\mathrm{QCD}}_{\kappa, 3}\).

The sum \(\delta^{\mathrm{QCD}}_3 + \delta^{\mathrm{QCD}}_{\kappa, 3}\) corresponds to the \(O(\alpha\alpha_s^2(m_t^2/M_Z^2+1+M_Z^2/m_t^2))\) contribution to \(\delta\kappa_{\mathrm{rem}}^{f}\). See the arXiv version of [Chetyrkin:1995js] (and also [Avdeev:1994db], [Chetyrkin:1995ix] and Chapter 8 of [Bardin:1999ak]).

Parameters
[in]Mw_ithe \(W\)-boson mass
Returns
\(\delta^{\mathrm{QCD}}_{\kappa,3}\)

Definition at line 114 of file EWSMThreeLoopQCD.cpp.

115{
116 gslpp::complex dQCDk3;
117 double lZ = 2.0 * cache.logMZtoMTOP();
118 double Mw = Mw_i;
119 double sW2 = cache.getSM().sW2(Mw);
120 double MZtoMT = cache.getSM().getMz() / cache.getSM().getMtpole();
121 dQCDk3.real() = -deltaQCD_3(Mw)
122 + pow(MZtoMT, 2.0)
123 *((22.6367 + 1.2527 * lZ - 0.8519 * lZ * lZ) * sW2
124 + (-11.3184 - 0.6263 * lZ + 0.4259 * lZ * lZ) * sW2)
125 + pow(MZtoMT, 4.0)
126 *((21.497 + 0.05794 * lZ - 0.006584 * lZ * lZ) * sW2
127 + (-16.0186 - 0.02897 * lZ + 0.003292 * lZ * lZ) * sW2
128 - 21.0799 * sW2 * sW2 + 10.54 * sW2 * sW2);
129 dQCDk3.imag() = pow(MZtoMT, 2.0)
130 *((-1.968 + 2.676 * lZ) * sW2
131 + (2.6235 - 3.5682 * lZ) * sW2 * sW2)
132 + pow(MZtoMT, 4.0)
133 *((-0.09102 + 0.02069 * lZ) * sW2
134 + (0.1214 - 0.02758 * lZ) * sW2 * sW2);
135 return dQCDk3;
136}

◆ DeltaR_rem()

double EWSMThreeLoopQCD::DeltaR_rem ( const double  Mw_i) const

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

The three-loop remainder contribution of \(O(\alpha\alpha_s^2)\) is obtained from the \(O(\alpha)\) remainder contribution [Halzen:1990je] :

\[ \Delta r_{\mathrm{rem},ud}^{\alpha} \biggl[1+\frac{\alpha_s(M_Z^2)}{\pi} + 1.4097\, \biggl(\frac{\alpha_s(M_Z^2)}{\pi}\biggr)^2\biggr] = \Delta r_{\mathrm{rem},ud}^{\alpha} + \Delta r_{\mathrm{rem}}^{\alpha\alpha_s} + \Delta r_{\mathrm{rem}}^{\alpha\alpha_s^2}, \]

where \(\Delta r_{\mathrm{rem},ud}^{\alpha}\) is the one-loop light-quark contribution to \(\Delta r_{\mathrm{rem}}^{\alpha}\) and given by

\[ \Delta r_{\mathrm{rem},ud}^{\alpha} = - \frac{\alpha}{\pi} \frac{c_W^2 - s_W^2}{4s_W^4}\,\ln c_W^2, \]

and the QCD corrections are associated with the \(R\) ratio (see, e.g., [Baikov:2008jh]):

\[ R = \frac{11}{3}\biggl[ 1 + \frac{\alpha_s(M_Z^2)}{\pi} + 1.4097\, \biggl(\frac{\alpha_s(M_Z^2)}{\pi}\biggr)^2 + \cdots \biggr]. \]

Parameters
[in]Mw_ithe \(W\)-boson mass
Returns
\(\Delta r_{\mathrm{rem}}^{\alpha\alpha_s^2}\)

Definition at line 48 of file EWSMThreeLoopQCD.cpp.

49{
50 double Mw = Mw_i;
51 double sW2 = cache.getSM().sW2(Mw);
52 double cW2 = cache.getSM().cW2(Mw);
53
54 /* Logarithm */
55 double log_cW2 = cache.log_cW2(Mw);
56
57 // O(alpha_s) correction to Delta r^{ud} of O(alpha alpha_s).
58 double DeltaR;
59 DeltaR = -log_cW2;
60 DeltaR *= (cW2 - sW2) / 4.0 / sW2 / sW2;
61 DeltaR *= cache.getSM().getAle() * cache.getSM().getAlsMz() / M_PI / M_PI;
62 DeltaR *= 1.4097 * cache.getSM().getAlsMz() / M_PI;
63 return DeltaR;
64}
EWSMcache::log_cW2
double log_cW2(const double Mw_i) const
A cache method.
Definition: EWSMcache.cpp:148

◆ DeltaRho()

double EWSMThreeLoopQCD::DeltaRho ( const double  Mw_i) const

Leading three-loop QCD contribution of \(O(\alpha\alpha_s^2)\) to \(\Delta\rho\), denoted as \(\Delta\rho^{\alpha\alpha_s^2}\).

The formula used here is given by

\[ \Delta\rho^{\alpha\alpha_s^2} = 3\,X_t^\alpha\biggl(\frac{\alpha_s(m_t^2)}{\pi}\biggr)^2 \delta^{\mathrm{QCD}}_3, \]

where \(X_t^\alpha = \alpha\, m_t^2/(16\pi s_W^2 M_W^2)\), and \(\delta^{\mathrm{QCD}}_3\) is computed via deltaQCD_3(). See [Avdeev:1994db], [Chetyrkin:1995ix], [Chetyrkin:1995js] and 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_ithe \(W\)-boson mass
Returns
\(\Delta\rho^{\alpha\alpha_s^2}\)

Definition at line 42 of file EWSMThreeLoopQCD.cpp.

43{
44 double Mw = Mw_i;
45 return ( 3.0 * cache.Xt_alpha(Mw) * pow(cache.alsMt() / M_PI, 2.0) * deltaQCD_3(Mw));
46}

◆ deltaRho_rem_f()

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

Remainder contribution of \(O(\alpha\alpha_s^2)\) to the effective couplings \(\rho_Z^f\), denoted as \(\delta\rho_{\mathrm{rem}}^{f,\, \alpha\alpha_s^2}\).

This contribution is not implemented, since it is tiny and negligible.

Parameters
[in]fa lepton or quark
[in]Mw_ithe \(W\)-boson mass
Returns
\(\delta\rho_{\mathrm{rem}}^{f,\, \alpha\alpha_s^2}=0\)

Definition at line 66 of file EWSMThreeLoopQCD.cpp.

67{
68 if (f.is("TOP")) return ( gslpp::complex(0.0, 0.0, false));
69 return ( gslpp::complex(0.0, 0.0, false));
70}

Member Data Documentation

◆ cache

const EWSMcache& EWSMThreeLoopQCD::cache
private

A reference to an object of type EWSMcache.

Definition at line 153 of file EWSMThreeLoopQCD.h.

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