QCD Library
Classes | Enumerations | Functions | Variables
QCD Namespace Reference

Classes

class  MSbar_mass
 Implements the running and decoupling of a quark mass in the $\overline{\rm MS}$ scheme. More...
 
struct  reference_coupling
 Value for strong coupling constant at a reference scale. More...
 
struct  reference_mass
 Value for $\overline{\rm MS}$ quark mass at a reference scale. More...
 
class  strong_coupling
 Implements the running and decoupling of the strong coupling constant. More...
 
struct  threshold
 Threshold for the decoupling of a quark flavour. More...
 

Enumerations

enum  scheme { ,
  pole = 0, OS = pole, MSbar, MS = MSbar,
  scale_invariant, SI = scale_invariant
}
 Mass schemes. More...
 

Functions

QCD_CONSTEXPR_AFTER_CXX14 double beta (unsigned order, int nf, double alpha_s)
 The QCD beta function. More...
 
QCD_CONSTEXPR_AFTER_CXX14 double beta (unsigned order, int nf)
 Perturbative coefficients of the QCD beta function. More...
 
QCD_CONSTEXPR_AFTER_CXX14 unsigned beta_max_nf_pow (unsigned order)
 Highest power of number of flavours in appearing in the beta function. More...
 
QCD_CONSTEXPR_AFTER_CXX14 double beta_nf_coeff (unsigned order, unsigned nfpow)
 Coefficient of some power of number of flavours in the beta function. More...
 
QCD_CONSTEXPR_AFTER_CXX14 double gamma_m (unsigned order, int nf, double alpha_s)
 The QCD mass anomalous dimension. More...
 
QCD_CONSTEXPR_AFTER_CXX14 double gamma_m (unsigned order, int nf)
 Perturbative coefficients of the QCD mass anomalous dimension. More...
 
QCD_CONSTEXPR_AFTER_CXX14 unsigned gamma_m_max_nf_pow (unsigned order)
 Highest power of number of flavours in appearing in the mass anomalous dimension. More...
 
QCD_CONSTEXPR_AFTER_CXX14 double gamma_m_nf_coeff (unsigned order, unsigned nfpow)
 Coefficient of some power of number of flavours in the mass anomalous dimension. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_alpha_s (unsigned order, unsigned nf, double lm, double alpha_s)
 Decoupling constant for the strong coupling constant. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_alpha_s_inv (unsigned order, unsigned nf, double lm, double alpha_s)
 Inverse decoupling constant for the strong coupling constant. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_alpha_s (unsigned order, unsigned nf, double lm)
 Coefficients of the decoupling constant. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_alpha_s_inv (unsigned order, unsigned nf, double lm)
 Coefficients of the inverse decoupling constant. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_m (unsigned order, unsigned nf, double lm, double alpha_s)
 Decoupling constant for the mass in the $\overline{\rm MS}$ scheme. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_m_inv (unsigned order, unsigned nf, double lm, double alpha_s)
 Inverse decoupling constant for the mass in the $\overline{\rm MS}$ scheme. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_m (unsigned order, unsigned nf, double lm)
 Coefficients of the decoupling constant. More...
 
template<scheme s>
QCD_CONSTEXPR_AFTER_CXX14 double zeta_m_inv (unsigned order, unsigned nf, double lm)
 Coefficients of the inverse decoupling constant. More...
 

Variables

constexpr double alpha_s_mz = 0.1184
 Default value for the strong coupling at the Z scale.
 
constexpr double mz = 91.18
 Default value for the Z boson mass.
 
constexpr unsigned beta_max_order = 4
 Highest implemented order of the beta function.
 
constexpr int n_c = 3
 Dimension of the fundamental representation.
 
constexpr int d_r = 3
 Dimension of the fundamental representation.
 
constexpr double c_A = 3.
 Quadratic Casimir in the adjoint representation.
 
constexpr double c_F = 4./3.
 Quadratic Casimir in the fundamental representation.
 
constexpr double T_f = 1./2.
 Trace normalisation.
 
constexpr double d_abcd_AA = 135.
 Product of two symmetrised traces in the adjoint representation. More...
 
constexpr double d_abcd_AF = 15./2.
 Product of two symmetrised traces in the adjoint and fundamental representation. More...
 
constexpr double d_abcd_FF = 5./12.
 Product of two symmetrised traces in the fundamental representation. More...
 
constexpr double N_A = 8.
 Dimension of the adjoint representation.
 
constexpr double mc_OS = 1.5
 charm quark mass
 
constexpr double mb_OS = 4.7
 bottom quark mass
 
constexpr double mt_OS = 175.
 top quark mass
 

Detailed Description

Enumeration Type Documentation

◆ scheme

enum scheme

Mass schemes.

Enumerator
pole 

Pole scheme.

OS 

On-shell scheme (alias for the pole scheme)

MSbar 

$\overline{\rm MS}$ scheme

MS 

Alias for the $\overline{\rm MS}$ scheme.

scale_invariant 

Scale-invariant scheme.

The scale-invariant mass is the $\overline{\rm MS}$ mass at the scale of the $\overline{\rm MS}$ mass

SI 

Alias for the scale-invariant scheme.

Function Documentation

◆ beta() [1/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::beta ( unsigned  order,
int  nf,
double  alpha_s 
)
inline

The QCD beta function.

Parameters
orderPerturbative order of the beta function
nfNumber of active flavours
alpha_sValue of the strong coupling constant
Returns
The value of the beta function

The beta function is defined via the renormalisation group equation $\beta = \frac{d\ln\alpha_s(\mu)}{d\ln \mu^2} $, where $\alpha_s(\mu)$ is the strong coupling constant at the scale $\mu$.

◆ beta() [2/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::beta ( unsigned  order,
int  nf 
)
inline

Perturbative coefficients of the QCD beta function.

Parameters
orderPerturbative order of the coefficient
nfNumber of active flavours
Returns
The value of the coefficient of the beta function

The coefficients of the beta function are given by $\beta = - \sum_{i=0}^\infty \beta_i (\frac{\alpha_s}{\pi})^{i+1}$. For example, beta(0, 5) returns the value of $\beta_0$ for 5 active flavours.

◆ beta_max_nf_pow()

QCD_CONSTEXPR_AFTER_CXX14 unsigned QCD::beta_max_nf_pow ( unsigned  order)
inline

Highest power of number of flavours in appearing in the beta function.

Parameters
orderPerturbative order of the beta function
Returns
The highest power of the number of active flavours in the formula for the beta function up to the given order

◆ beta_nf_coeff()

QCD_CONSTEXPR_AFTER_CXX14 double QCD::beta_nf_coeff ( unsigned  order,
unsigned  nfpow 
)
inline

Coefficient of some power of number of flavours in the beta function.

Parameters
orderPerturbative order
nfpowPower of number of active flavour
Returns
The coefficient of $n_f^{\rm nfpow}$ in the formula for $\beta_{\rm order}$

◆ gamma_m() [1/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::gamma_m ( unsigned  order,
int  nf,
double  alpha_s 
)
inline

The QCD mass anomalous dimension.

Parameters
orderPerturbative order of the mass anomalous dimension
nfNumber of active flavours
alpha_sValue of the strong coupling constant
Returns
The value of the mass anomalous dimension

The mass anomalous dimension is defined via the renormalisation group equation $\gamma^m = \frac{d\ln m(\mu)}{d\ln \mu^2} $, where $m(\mu)$ is the running mass at the scale $\mu$.

◆ gamma_m() [2/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::gamma_m ( unsigned  order,
int  nf 
)
inline

Perturbative coefficients of the QCD mass anomalous dimension.

Parameters
orderPerturbative order of the coefficient
nfNumber of active flavours
Returns
The value of the coefficient of the mass anomalous dimension

The coefficients of the mass anomalous dimension are given by $\gamma^m = - \sum_{i=0}^\infty \gamma^m_i (\frac{\alpha_s}{\pi})^{i+1}$. For example, gamma_m(1, 5) returns the value of $\gamma^m_0$ for 5 active flavours.

◆ gamma_m_max_nf_pow()

QCD_CONSTEXPR_AFTER_CXX14 unsigned QCD::gamma_m_max_nf_pow ( unsigned  order)
inline

Highest power of number of flavours in appearing in the mass anomalous dimension.

Parameters
orderPerturbative order of the mass anomalous dimension
Returns
The highest power of the number of active flavours in the formula for the mass anomalous dimension up to the given order

◆ gamma_m_nf_coeff()

QCD_CONSTEXPR_AFTER_CXX14 double QCD::gamma_m_nf_coeff ( unsigned  order,
unsigned  nfpow 
)
inline

Coefficient of some power of number of flavours in the mass anomalous dimension.

Parameters
orderPerturbative order
nfpowPower of number of active flavour
Returns
The coefficient of $n_f^{\rm nfpow}$ in the formula for $\gamma^m_{\rm order}$

◆ zeta_alpha_s() [1/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_alpha_s ( unsigned  order,
unsigned  nf,
double  lm,
double  alpha_s 
)
inline

Decoupling constant for the strong coupling constant.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
alpha_sstrong coupling constant at the decoupling scale
Returns
The decoupling constant to compute the strong coupling in the theory with nf-1 flavours

The decoupling constant $\zeta_{\alpha_s}$ is defined by $\alpha_s^{(n_f-1)} = \zeta_{\alpha_s} \alpha_s^{(n_f)}$, where $\alpha_s^{(n_f)}$ is the coupling constant in the theory with $n_f$ active flavours.

All input parameters are defined in the theory with nf active flavours.

◆ zeta_alpha_s_inv() [1/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_alpha_s_inv ( unsigned  order,
unsigned  nf,
double  lm,
double  alpha_s 
)
inline

Inverse decoupling constant for the strong coupling constant.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
alpha_sstrong coupling constant with nf active flavours at the decoupling scale
Returns
The decoupling constant to compute the strong coupling in the theory with nf+1 flavours

The inverse decoupling constant $\zeta_{\alpha_s}^{-1}$ is defined by $\alpha_s^{(n_f+1)} = \zeta_{\alpha_s}^{-1} \alpha_s^{(n_f)}$, where $\alpha_s^{(n_f)}$ is the coupling constant in the theory with $n_f$ active flavours.

◆ zeta_alpha_s() [2/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_alpha_s ( unsigned  order,
unsigned  nf,
double  lm 
)
inline

Coefficients of the decoupling constant.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
Returns
The perturbative coefficient of the decoupling constant at the given order

The coefficients $\zeta_{\alpha_s, i}$ are defined via $\zeta_{\alpha_s} = \sum_{i=0}^\infty \zeta_{\alpha_s, i} (\frac{\alpha_s}{\pi})^i$.

◆ zeta_alpha_s_inv() [2/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_alpha_s_inv ( unsigned  order,
unsigned  nf,
double  lm 
)
inline

Coefficients of the inverse decoupling constant.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
Returns
The perturbative coefficient of the inverse decoupling constant at the given order

The coefficients $\zeta_{\alpha_s, i}^{-1}$ are defined via $\zeta_{\alpha_s}^{-1} = \sum_{i=0}^\infty \zeta_{\alpha_s, i}^{-1} (\frac{\alpha_s}{\pi})^i$.

◆ zeta_m() [1/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_m ( unsigned  order,
unsigned  nf,
double  lm,
double  alpha_s 
)
inline

Decoupling constant for the mass in the $\overline{\rm MS}$ scheme.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
alpha_sstrong coupling constant at the decoupling scale
Returns
The decoupling constant to compute the mass in the theory with nf+1 flavours

The decoupling constant $\zeta_m$ is defined by $m^{(n_f-1)} = \zeta_m m^{(n_f)}$, where $m^{(n_f)}$ is the mass in the theory with $n_f$ active flavours.

All input parameters are defined in the theory with nf active flavours.

◆ zeta_m_inv() [1/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_m_inv ( unsigned  order,
unsigned  nf,
double  lm,
double  alpha_s 
)
inline

Inverse decoupling constant for the mass in the $\overline{\rm MS}$ scheme.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
alpha_sstrong coupling constant with nf active flavours at the decoupling scale
Returns
The decoupling constant to compute the mass in the theory with nf+1 flavours

The inverse decoupling constant $\zeta_m^{-1}$ is defined by $m^{(n_f+1)} = \zeta_m^{-1} m^{(n_f)}$, where $m^{(n_f)}$ is the mass in the theory with $n_f$ active flavours.

◆ zeta_m() [2/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_m ( unsigned  order,
unsigned  nf,
double  lm 
)
inline

Coefficients of the decoupling constant.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
Returns
The perturbative coefficient of the decoupling constant at the given order

The coefficients $\zeta_{m, i}$ are defined via $\zeta_m = \sum_{i=0}^\infty \zeta_{m, i} (\frac{\alpha_s}{\pi})^i$.

◆ zeta_m_inv() [2/2]

QCD_CONSTEXPR_AFTER_CXX14 double QCD::zeta_m_inv ( unsigned  order,
unsigned  nf,
double  lm 
)
inline

Coefficients of the inverse decoupling constant.

Template Parameters
sDecoupling scheme
Parameters
orderPerturbative order
nfNumber of active flavours before decoupling
lm$\ln(\mu^2/m^2)$, where $\mu$ is the decoupling scale and $m$ the mass of the decoupled quark
Returns
The perturbative coefficient of the inverse decoupling constant at the given order

The coefficients $\zeta_{m, i}^{-1}$ are defined via $\zeta_m^{-1} = \sum_{i=0}^\infty \zeta_{m, i}^{-1} (\frac{\alpha_s}{\pi})^i$.

Variable Documentation

◆ d_abcd_AA

constexpr double d_abcd_AA = 135.

Product of two symmetrised traces in the adjoint representation.

d_abcd_AA = (d^abcd_A)^2, where d^abcd_A is the trace over the symmetrised product of four generators in the adjoint representation

◆ d_abcd_AF

constexpr double d_abcd_AF = 15./2.

Product of two symmetrised traces in the adjoint and fundamental representation.

d_abcd_AF = (d^abcd_A) (d^abcd_F), where d^abcd_A/d^abcd_F is the trace over the symmetrised product of four generators in the adjoint/fundamental representation

◆ d_abcd_FF

constexpr double d_abcd_FF = 5./12.

Product of two symmetrised traces in the fundamental representation.

d_abcd_FF = (d^abcd_F)^2, where d^abcd_F is the trace over the symmetrised product of four generators in the fundamental representation