Classes
struct	contribution_factors
	Multiplicative coefficients for the potentials and current matching coefficients. More...

struct	Et_Gammat_range
	Grid range in rescaled energy and width coordinates. More...

class	integrator
	Class for integrating a function of type Integrand. More...

struct	options
	Options for \(q\bar{q}\) functions. More...

struct	scheme
	Mass scheme. More...

struct	threshold
	Center-of-mass energy of a quark threshold. More...

Enumerations
enum	scheme_name { PS, PSshift, OneS, OneSshift, Pole, MSshift }
	Names of mass schemes. More...

enum	SM { none = 0, QED = 1 << 0, Higgs = 1 << 1 , all = (1 << 3) - 1 }
	Standard Model effects beyond QCD. More...

enum	production_channel { photon_only = 0, S_wave_only, all }
	Production channels. More...

Functions
double	alpha_s_top (double mu, double alpha_s_mZ=QQbar_threshold::alpha_s_mZ)
	The strong coupling used for top-related functions. More...

double	alpha_s_top (double mu, alpha_s_options alpha_s_mu0)
	The strong coupling used for top-related functions. More...

double	alpha_s_bottom (double mu, double alpha_s_mZ=QQbar_threshold::alpha_s_mZ)
	The strong coupling used for bottom-related functions. More...

double	alpha_s_bottom (double mu, alpha_s_options alpha_s_mu0)
	The strong coupling used for bottom-related functions. More...

double	axial_Z_coupling (double T3, double)
	Axial vector coupling of fermion to Z boson. More...

double	vector_Z_coupling (double T3, double e)
	Vector coupling of fermion to Z boson. More...

double	ttbar_energy_level (int n, double mu, double m, order o, options opt=top_options())
	Computes the binding energy \(E_n\) of an S-wave \(t\bar{t}\) bound state. More...

double	ttbar_energy_level (int n, double mu, top_properties M, order o, options opt=top_options())
	Computes the binding energy \(E_n\) of an S-wave \(t\bar{t}\) bound state. More...

double	bbbar_energy_level (int n, double mu, double m, order o, options opt=bottom_options())
	Computes the binding energy \(E_n\) of an S-wave \(b\bar{b}\) bound state. More...

template<class Integrand >
double	integrate (Integrand I, double x_min, double x_max)
	Compute the integral of a function. More...

template<class Integrand >
integrator< Integrand >	make_integrator (Integrand I)
	helper function to construct an integrator object More...

void	load_grid (char const *filename)
	Loads a precomputed grid. More...

void	load_grid (std::string const &filename)

void	load_grid (std::ifstream &&in)

void	load_nonresonant_grid (char const *filename)
	Loads a precomputed grid for the non-resonant contribution. More...

void	load_nonresonant_grid (std::string const &filename)

void	load_nonresonant_grid (std::ifstream &&in)

std::string	grid_directory ()
	Get directory with default grids. More...

Et_Gammat_range	grid_range ()
	Get grid range. More...

bool	is_shifted_mass (scheme_name s)
	Determines whether the given scheme is a shifted scheme. More...

scheme_name	to_shifted_mass (scheme_name s)
	Returns the corresponding shifted scheme. More...

bool	is_shifted_mass (scheme const &s)
	Determines whether the given scheme is a shifted scheme. More...

double	bottom_pole_mass (double mu, double m, order o, options opt=bottom_options())
	Computes the pole mass from a bottom quark mass in a different scheme. More...

double	top_pole_mass (double mu, double m, order o, options opt=top_options())
	Computes the pole mass from a top quark masses in a different scheme. More...

double	top_pole_mass (double mu, top_properties M, order o, options opt=top_options())
	Computes the pole mass from a top quark masses in a different scheme. More...

double	top_ISR_log (double sqrt_s)
	Computes the logarithm generated through initial state radiation. More...

double	bottom_ISR_log (double sqrt_s)
	Computes the logarithm generated through initial state radiation. More...

double	ISR_log (double sqrt_s, double alpha)
	Computes the logarithm generated through initial state radiation. More...

double	structure_function (double x, double beta)
	Computes the electron (or positron) structure function. More...

double	modified_structure_function (double x, double beta)
	Modified electron (or positron) structure function. More...

double	luminosity_function (double x, double beta)
	Computes the electron (or positron) luminosity function. More...

double	modified_luminosity_function (double t, double beta)
	Modified electron (or positron) luminosity function. More...

std::string	bottom_internal_settings (double mu, double m, order o, options opt)
	Describe internal settings for bottom-related functions. More...

std::string	bottom_internal_settings (double sqrt_s, double mu, double m, order o, options opt)
	Describe internal settings for bottom-related functions. More...

std::string	top_internal_settings (double mu, double m, order o, options opt)
	Describe internal settings for top-related functions. More...

std::string	top_internal_settings (double sqrt_s, scales mu, top_properties M, order o, options opt)
	Describe internal settings for top-related functions. More...

options	bottom_options ()
	Default option settings for bottom-related functions. More...

options	top_options ()
	Default option settings for top-related functions. More...

double	ttbar_residue (int n, double mu, double m, order o, options opt=top_options())
	Computes the residue \(Z_n\) of an S-wave \(t\bar{t}\) bound state. More...

complex	ttbar_residue (int n, double mu, top_properties M, order o, options opt=top_options())
	Computes the residue \(Z_n\) of an S-wave \(t\bar{t}\) bound state. More...

double	bbbar_residue (int n, double mu, double m, order o, options opt=bottom_options())
	Computes the residue \(Z_n\) of an S-wave \(b\bar{b}\) bound state. More...

double	top_width (double mu, double m, order o, options opt=top_options())
	Computes the top quark decay width. More...

double	ttbar_width (int n, double mu, top_properties M, order o, options opt=top_options())
	Computes the decay width of a top-antitop bound state. More...

double	ttbar_xsection (double sqrt_s, scales mu, top_properties M, order o, options opt=top_options())
	Computes the cross section \(e^+e^- \to W^+W^-b\bar{b}\). More...

double	ttbar_xsection (double sqrt_s, double mu, top_properties M, order o, options opt=top_options())
	Computes the cross section \(e^+e^- \to W^+W^-b\bar{b}\). More...

double	bbbar_xsection (double sqrt_s, double mu, double m, order o, options opt=bottom_options())
	Computes the cross section \(e^+e^- \to b\bar{b}\). More...

double	ttbar_threshold (double mu, double m, order o, options opt=top_options())
	Computes the naive threshold for \(t\bar{t}\) production. More...

double	bbbar_threshold (double mu, double m, order o, options opt=bottom_options())
	Computes the naive threshold for \(b\bar{b}\) production. More...

double	bbbar_R_ratio (double sqrt_s, double mu, double m, order o, options opt=bottom_options())
	Bottom contribution to the hadronic R ratio. More...

double	ttbar_R_ratio (double sqrt_s, scales mu, top_properties M, order o, options opt=top_options())
	Top contribution to the hadronic R ratio. More...

Variables
constexpr double	alpha_s_mZ = 0.1184
	Default value for strong coupling at the scale mZ.

constexpr double	alpha_mZ = 1/128.944
	QED coupling at the scale mZ.

constexpr double	alpha_Y = 1/132.274
	QED coupling at the scale mu_alpha_Y.

constexpr double	mu_alpha_Y = 10.2
	Typical scale for Y resonances.

constexpr double	mZ = 91.1876
	Mass of the Z boson.

constexpr double	mW = 80.385
	Mass of the W boson.

constexpr double	me = 0.0005109989461
	Electron mass.

constexpr double	G_F = 0.000011663787
	Fermi constant - this is exclusively used for calculating the top width.

constexpr double	m_Higgs = 125.
	Default value for mass of the Higgs boson.

constexpr double	alpha_QED = 1/137.035999074
	Fine structure constant.

constexpr double	e_u = 2./3.
	Electric charge of the top quark in units of the positron charge.

constexpr double	e_d = -1./3.
	Electric charge of the bottom quark in units of the positron charge.

constexpr double	e = -1
	Electric charge of the electron in units of the positron charge.

constexpr double	cw2 = mWmW/(mZmZ)
	cosine of the weak mixing angle squared

constexpr double	sw2 = 1 - cw2
	sine of the weak mixing angle squared

constexpr double	T3_nu = 1./2.
	Weak isospin of neutrino.

constexpr double	T3_e = -1./2.
	Weak isospin of electron.

constexpr double	T3_u = 1./2.
	Weak isospin of top.

constexpr double	T3_d = -1./2.
	Weak isospin of bottom.

constexpr double	mb_SI = 4.203
	Reference scale-invariant mass for bottom quarks.

constexpr double	mu_thr = 2*mb_SI
	Decoupling threshold for bottom quarks.

constexpr int	nl_bottom = 4
	Number of light flavours for bottom-related functions.

constexpr int	nl_top = 5
	Number of light flavours for top-related functions.

constexpr double	mu_f_bottom = 2.
	Default PS scale for bottom.

constexpr double	mu_f_top = 20.
	Default PS scale for top.

constexpr double	invGeV2_to_pb = 389379300.
	Conversion factor from 1/GeV^2 to picobarn.

Detailed Description

Enumeration Type Documentation

◆ scheme_name

enum scheme_name

Names of mass schemes.

If a shifted scheme is used in the evaluation of an expression the mass is first converted to the pole scheme at the given order and then inserted into the expression for the pole mass.

For insertion schemes the leading-order conversion to the pole scheme is used. Higher-order corrections in the relation to the pole mass are treated as insertions of counterterms [Beneke:2014].

See also: is_shifted_mass

Enumerator
PS	The potential-subtracted insertion scheme.
PSshift	The potential-subtracted shift scheme.
OneS	The 1S insertion scheme.
OneSshift	The 1S shift scheme.
Pole	The pole scheme.
MSshift	The \(\overline{\rm MS}\) scheme.

◆ SM

enum SM

strong

Standard Model effects beyond QCD.

Effects can be combined with the | operator e.g.

options opt;

opt.beyond_QCD = SM::QED | SM::Higgs;

activates both QED and Higgs effects.

Enumerator
none	No corrections beyond QCD.
QED	Corrections due to photons.
Higgs	Corrections due to the physical Higgs boson.
all	All Standard Model corrections.

◆ production_channel

enum production_channel

strong

Production channels.

Enumerator
photon_only	Production only via a virtual photon.
S_wave_only	Production only via S-wave photon or Z.
all	All possible production channels.

Function Documentation

◆ alpha_s_top() [1/2]

double QQbar_threshold::alpha_s_top	(	double	mu,
		double	alpha_s_mZ = `QQbar_threshold::alpha_s_mZ`
	)

The strong coupling used for top-related functions.

Parameters

mu	renormalisation scale
alpha_s_mZ	strong coupling at the scale mZ

Returns: The value of the strong coupling theory at the given scale

Both input and output value for the strong coupling are in the theory with five active flavours.

◆ alpha_s_top() [2/2]

double QQbar_threshold::alpha_s_top	(	double	mu,
		alpha_s_options	alpha_s_mu0
	)

The strong coupling used for top-related functions.

Parameters

mu	renormalisation scale
alpha_s_mu0	a pair {mu0, alpha_s} of a reference scale mu0 and the strong coupling at the scale mu0

Returns: The value of the strong coupling at the given renormalisation scale

Both input and output value for the strong coupling are in the theory with five active flavours.

◆ alpha_s_bottom() [1/2]

double QQbar_threshold::alpha_s_bottom	(	double	mu,
		double	alpha_s_mZ = `QQbar_threshold::alpha_s_mZ`
	)

The strong coupling used for bottom-related functions.

Parameters

mu	renormalisation scale
alpha_s_mZ	strong coupling at the scale mZ

Returns: The value of the strong coupling theory at the given scale

For the input value five active flavours are assumed. The output value is given in the four-flavour theory. Decoupling is performed at the scale mu_thr assuming a scale-invariant bottom mass mb_SI.

◆ alpha_s_bottom() [2/2]

double QQbar_threshold::alpha_s_bottom	(	double	mu,
		alpha_s_options	alpha_s_mu0
	)

The strong coupling used for bottom-related functions.

Parameters

mu	renormalisation scale
alpha_s_mu0	a pair {mu0, alpha_s} of a reference scale mu0 and the strong coupling at the scale mu0

Returns: The value of the strong coupling at the given renormalisation scale

For the input value, five active flavours are assumed if mu0 is above the bottom decoupling threshold mu_thr, and four active flavours if mu0 is below mu_thr. The output value is always given in the four-flavour theory. If necessary, decoupling is performed at the scale mu_thr assuming a scale-invariant bottom mass mb_SI.

◆ axial_Z_coupling()

double QQbar_threshold::axial_Z_coupling	(	double	T3,
		double
	)

inline

Axial vector coupling of fermion to Z boson.

Parameters

T3	Weak isospin

Returns: a_f, the axial vector coupling to the Z boson

◆ vector_Z_coupling()

double QQbar_threshold::vector_Z_coupling	(	double	T3,
		double	e
	)

inline

Vector coupling of fermion to Z boson.

Parameters

T3	Weak isospin
e	Electric charge in units of positron charge

Returns: v_f, the vector coupling to the Z boson

◆ ttbar_energy_level() [1/2]

double QQbar_threshold::ttbar_energy_level	(	int	n,
		double	mu,
		double	m,
		order	o,
		options	opt = `top_options()`
	)

Computes the binding energy \(E_n\) of an S-wave \(t\bar{t}\) bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
m	top quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The binding energy \(E_n\).

The bound state mass is given by \(2m_t+E_n\), where \(m_t\) is the mass of the top quark in the given scheme.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

See also: options, bbbar_energy_level, ttbar_residue

◆ ttbar_energy_level() [2/2]

double QQbar_threshold::ttbar_energy_level	(	int	n,
		double	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Computes the binding energy \(E_n\) of an S-wave \(t\bar{t}\) bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The binding energy \(E_n\).

This function includes corrections due to a non-zero top-quark width.

The bound state mass is given by \(2m_t+E_n\), where \(m_t\) is the mass of the top quark in the given scheme.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

See also: options, bbbar_energy_level, ttbar_residue

◆ bbbar_energy_level()

double QQbar_threshold::bbbar_energy_level	(	int	n,
		double	mu,
		double	m,
		order	o,
		options	opt = `bottom_options()`
	)

Computes the binding energy \(E_n\) of an S-wave \(b\bar{b}\) bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
m	bottom quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The binding energy \(E_n\).

The bound state mass is given by \(M_{\Upsilon(nS)} = 2m_b+E_n\), where \(m_b\) is the mass of the bottom quark in the given scheme.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_bottom}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::QED
options::alpha = alpha_Y
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 1220.3 (for nl == 4)
options::ml = 0
options::double_light_insertion = false

See also: options, ttbar_energy_level, bbbar_residue

◆ integrate()

double integrate	(	Integrand	I,
		double	x_min,
		double	x_max
	)

inline

Compute the integral of a function.

Parameters

integrand	Integrand function
x_min	Lower bound of the integration.
x_max	Upper bound of the integration.

Returns: The value of \(\int_{\texttt{x\_min}}^{\texttt{x\_max}} dx {\texttt{integrand(x)}}\)

◆ make_integrator()

integrator<Integrand> QQbar_threshold::make_integrator ( Integrand I )

inline

helper function to construct an integrator object

Parameters

I	The integrand

Returns: An integrator object for I

◆ load_grid() [1/3]

void QQbar_threshold::load_grid ( char const * filename )

Loads a precomputed grid.

Parameters

filename Name of the grid to be loaded

For order > 1, the cross section functions require a precomputed grid. Some grids are already packaged with this library. New grids can be generated with the QQbarGridCalc.m Mathematica package

See also: bbbar_xsection, ttbar_xsection, QQbarGridCalc.m, QQbarThreshold

◆ load_grid() [2/3]

void QQbar_threshold::load_grid ( std::string const & filename )

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

◆ load_grid() [3/3]

void QQbar_threshold::load_grid ( std::ifstream && in )

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters

in	stream to the grid to be loaded

◆ load_nonresonant_grid() [1/3]

void QQbar_threshold::load_nonresonant_grid ( char const * filename )

Loads a precomputed grid for the non-resonant contribution.

Parameters

filename Name of the grid to be loaded

Explicitly loading a grid is only required for exotic values of the top mass or the Z mass. For these cases grids can be generated with the QQbarGridCalc.m Mathematica package

◆ load_nonresonant_grid() [2/3]

void QQbar_threshold::load_nonresonant_grid ( std::string const & filename )

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

◆ load_nonresonant_grid() [3/3]

void QQbar_threshold::load_nonresonant_grid ( std::ifstream && in )

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters

in	stream to the grid to be loaded

◆ grid_directory()

std::string QQbar_threshold::grid_directory ( )

Get directory with default grids.

Returns: The directory containing the default grids

◆ grid_range()

Et_Gammat_range QQbar_threshold::grid_range ( )

Get grid range.

Returns: The coordinate range of the currently loaded grid.

See also: Et_Gammat_range

◆ is_shifted_mass() [1/2]

bool QQbar_threshold::is_shifted_mass ( scheme_name s )

inline

Determines whether the given scheme is a shifted scheme.

Parameters

s	The name of the mass scheme

Returns: true for shifted mass schemes, false otherwise

◆ to_shifted_mass()

scheme_name QQbar_threshold::to_shifted_mass ( scheme_name s )

inline

Returns the corresponding shifted scheme.

Parameters

s	The name of the original mass scheme

Returns: The corresponding shifted scheme

For example, this returns PSshift if the input scheme is PS. If the input scheme is a shifted scheme or the pole scheme the input scheme is returned.

◆ is_shifted_mass() [2/2]

bool QQbar_threshold::is_shifted_mass ( scheme const & s )

inline

Determines whether the given scheme is a shifted scheme.

Parameters

s	The mass scheme

Returns: true for shifted mass schemes, false otherwise

◆ bottom_pole_mass()

double QQbar_threshold::bottom_pole_mass	(	double	mu,
		double	m,
		order	o,
		options	opt = `bottom_options()`
	)

Computes the pole mass from a bottom quark mass in a different scheme.

Parameters

mu	renormalisation scale
m	bottom quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: the bottom pole mass

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_bottom}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::QED
options::alpha = alpha_Y
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 1220.3 (for nl == 4)
options::ml = 0
options::double_light_insertion = false

Example

This computes the pole mass for a PS bottom quark mass of 4.5 GeV at a scale of 5 GeV at NNNLO:

options opt = bottom_options();
opt.mass_scheme = {PSshift, 2};
double mb_PS = 4.5;
double mb_pole = pole_bottom_mass(5, mb_PS, N3LO, opt);

See also: mass_schemes.hpp

◆ top_pole_mass() [1/2]

double QQbar_threshold::top_pole_mass	(	double	mu,
		double	m,
		order	o,
		options	opt = `top_options()`
	)

Computes the pole mass from a top quark masses in a different scheme.

Parameters

mu	renormalisation scale
m	top quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: the top pole mass

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

Example

This computes the pole mass for a \(\overline{\rm MS}\) top quark mass \(m_t(m_t) = 162\) GeV at N3LO:

options opt = top_options();
double mt_mt = 162;
opt.mass_scheme = {MSbar, mt_mt};
double mt_pole = pole_top_mass(mt_mt, mt_mt, N3LO, opt);

See also: mass_schemes.hpp

◆ top_pole_mass() [2/2]

double QQbar_threshold::top_pole_mass	(	double	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Computes the pole mass from a top quark masses in a different scheme.

Parameters

mu	renormalisation scale
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: the top pole mass

This function includes corrections from a non-zero width in the mass conversion. Currently, this is only relevant for the conversion from the 1S scheme.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

See also: mass_schemes.hpp

◆ top_ISR_log()

double QQbar_threshold::top_ISR_log ( double sqrt_s )

Computes the logarithm generated through initial state radiation.

Parameters

sqrt_s center of mass energy

Returns: \(\beta = -2*\alpha(m_Z)/\pi [\log(m_e^2/s) + 1]\)

See also: alpha_mZ, me, structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ bottom_ISR_log()

double QQbar_threshold::bottom_ISR_log ( double sqrt_s )

Computes the logarithm generated through initial state radiation.

Parameters

sqrt_s center of mass energy

Returns: \(\beta = -2*\alpha(m_\Upsilon)/\pi [\log(m_e^2/s) + 1]\)

See also: alpha_Y, me, structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ ISR_log()

double QQbar_threshold::ISR_log	(	double	sqrt_s,
		double	alpha
	)

Computes the logarithm generated through initial state radiation.

Parameters

sqrt_s	center of mass energy
alpha	electromagnetic coupling constant

Returns: \(\beta = -2*\alpha/\pi [\log(m_e^2/s) + 1]\)

See also: me, structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ structure_function()

double QQbar_threshold::structure_function	(	double	x,
		double	beta
	)

Computes the electron (or positron) structure function.

Parameters

x	fraction of center-of-mass energy squared between 0 and 1
beta	value of the logarithm to be resummed, see ISR_log

Returns: the value of the structure function

See also: me, structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ modified_structure_function()

double QQbar_threshold::modified_structure_function	(	double	x,
		double	beta
	)

Modified electron (or positron) structure function.

Parameters

t	convolution parameter between 0 and 1
beta	value of the logarithm to be resummed, see ISR_log

Returns: the value of the modified structure function

The modified structure function \(\tilde{\Gamma}_{ee}(t, \beta)\) is given by \(\tilde{\Gamma}_{ee}(t, \beta) = 2/\beta t^{2/\beta-1} \Gamma_{ee}(1-t^{2/\beta}, \beta)\), where \(\Gamma_{ee}(1-t^{2/\beta}, \beta)\) is the electron structure function.

See also: structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ luminosity_function()

double QQbar_threshold::luminosity_function	(	double	x,
		double	beta
	)

Computes the electron (or positron) luminosity function.

Parameters

x	fraction of center-of-mass energy squared between 0 and 1
beta	value of the logarithm to be resummed, see ISR_log

Returns: the value of the luminosity function

The luminosity function \(L\) is defined as \(L(x) = \int_x^1 \frac{dy}{y} f(y)f(x/y)\), where \(f\) is the electron structure function

See also: structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ modified_luminosity_function()

double QQbar_threshold::modified_luminosity_function	(	double	t,
		double	beta
	)

Modified electron (or positron) luminosity function.

Parameters

t	convolution parameter between 0 and 1
beta	value of the logarithm to be resummed, see ISR_log

Returns: the value of the modified luminosity function

The modified luminosity function \(\tilde{L}(t, \beta)\) is defined as \(\tilde{L}(t, \beta) = t^{1/\beta-1}/\beta L(1-t^{1/\beta}, \beta)\), where \(L\) is the luminosity function

See also: structure_function, luminosity_function, modified_structure_function, modified_luminosity_function

◆ bottom_internal_settings() [1/2]

std::string QQbar_threshold::bottom_internal_settings	(	double	mu,
		double	m,
		order	o,
		options	opt
	)

Describe internal settings for bottom-related functions.

Parameters

mu	renormalisation scale
m	quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: A string containing the names and values of internal variables

This function is intended for debugging purposes.

◆ bottom_internal_settings() [2/2]

std::string QQbar_threshold::bottom_internal_settings	(	double	sqrt_s,
		double	mu,
		double	m,
		order	o,
		options	opt
	)

Describe internal settings for bottom-related functions.

Parameters

sqrt_s	center-of-mass energy
mu	renormalisation scale
m	bottom quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: A string containing the names and values of internal variables

This function is intended for debugging purposes.

◆ top_internal_settings() [1/2]

std::string QQbar_threshold::top_internal_settings	(	double	mu,
		double	m,
		order	o,
		options	opt
	)

Describe internal settings for top-related functions.

Parameters

mu	renormalisation scale
m	quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: A string containing the names and values of internal variables

This function is intended for debugging purposes.

◆ top_internal_settings() [2/2]

std::string QQbar_threshold::top_internal_settings	(	double	sqrt_s,
		scales	mu,
		top_properties	M,
		order	o,
		options	opt
	)

Describe internal settings for top-related functions.

Parameters

sqrt_s	center-of-mass energy
mu	renormalisation scales
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: A string containing the names and values of internal variables

This function is intended for debugging purposes.

◆ bottom_options()

options QQbar_threshold::bottom_options ( )

Default option settings for bottom-related functions.

The default settings are:

options::contributions = (all set to 1)
options::alpha_s_mZ = alpha_s_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 0
options::invariant_mass_cut = 0
options::ml = 0
options::r4 = 1220.3 (for four active flavours)
options::alpha = alpha_Y;
options::mu_alpha = mu_alpha_Y;
options::resum_poles = 6;
options::beyond_QCD = SM::QED;
options::mass_scheme = {PSshift, mu_f_bottom};
options::production = production_channel::all;
options::expand_s = false;
options::double_light_insertion = false;

◆ top_options()

options QQbar_threshold::top_options ( )

Default option settings for top-related functions.

The default settings are:

options::contributions = (all set to 1)
options::alpha_s_mZ = alpha_s_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::invariant_mass_cut = m -mW
options::ml = 0
options::r4 = 824.12 (for five active flavours)
options::alpha = alpha_mZ;
options::mu_alpha = mZ;
options::resum_poles = 6;
options::beyond_QCD = SM::all;
options::mass_scheme = {PSshift, mu_f_bottom};
options::production = production_channel::all;
options::expand_s = true;
options::double_light_insertion = false;

◆ ttbar_residue() [1/2]

double QQbar_threshold::ttbar_residue	(	int	n,
		double	mu,
		double	m,
		order	o,
		options	opt = `top_options()`
	)

Computes the residue \(Z_n\) of an S-wave \(t\bar{t}\) bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
m	top quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The residue \(Z_n\).

The residue in the pole scheme is defined as \(Z_N = \frac{4m^2}{(2m+E_N)^2}c_v \bigg[c_v - \frac{E_N}{m}\frac{d_v}{3}\bigg]|\psi_N(0)|^2\,.\) \(E_N\) is the binding energy, \(\psi_N(0)\) is the wave function at the origin, and \(c_v\) and \(d_v\) are the hard current matching coefficients. Deviating from the general conversion rules, the residue \(Z_N^{\rm RS}\) in an insertion scheme is given by \(Z_N^{\rm RS} = m^{\rm RS} \times [Z_N/m^2]^{\rm RS}\), where the usual prescription for scheme conversion is applied to the term in square brackets.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::expand_s = false
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

See also: options, ttbar_energy_level, bbbar_residue

◆ ttbar_residue() [2/2]

complex QQbar_threshold::ttbar_residue	(	int	n,
		double	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Computes the residue \(Z_n\) of an S-wave \(t\bar{t}\) bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The residue \(Z_n\).

This function includes corrections due to a non-zero top quark width.

The residue in the pole scheme is defined as \(Z_N = \frac{4m^2}{(2m+E_N)^2}c_v \bigg[c_v - \frac{E_N}{m}\frac{d_v}{3}\bigg]|\psi_N(0)|^2\,.\) \(E_N\) is the binding energy, \(\psi_N(0)\) is the wave function at the origin, and \(c_v\) and \(d_v\) are the hard current matching coefficients. Deviating from the general conversion rules, the residue \(Z_N^{\rm RS}\) in an insertion scheme is given by \(Z_N^{\rm RS} = m^{\rm RS} \times [Z_N/m^2]^{\rm RS}\), where the usual prescription for scheme conversion is applied to the term in square brackets.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::expand_s = false
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

See also: options, ttbar_energy_level, bbbar_residue

◆ bbbar_residue()

double QQbar_threshold::bbbar_residue	(	int	n,
		double	mu,
		double	m,
		order	o,
		options	opt = `bottom_options()`
	)

Computes the residue \(Z_n\) of an S-wave \(b\bar{b}\) bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
m	bottom quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The binding energy \(E_n\).

The residue in the pole scheme is defined as \(Z_N = \frac{4m^2}{(2m+E_N)^2}c_v \bigg[c_v - \frac{E_N}{m}\frac{d_v}{3}\bigg]|\psi_N(0)|^2\,.\) \(E_N\) is the binding energy, \(\psi_N(0)\) is the wave function at the origin, and \(c_v\) and \(d_v\) are the hard current matching coefficients. Deviating from the general conversion rules, the residue \(Z_N^{\rm RS}\) in an insertion scheme is given by \(Z_N^{\rm RS} = m^{\rm RS} \times [Z_N/m^2]^{\rm RS}\), where the usual prescription for scheme conversion is applied to the term in square brackets.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_bottom}
options::expand_s = false
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::QED
options::alpha = alpha_Y
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 1220.3 (for nl == 4)
options::ml = 0
options::double_light_insertion = false

See also: options, bbbar_energy_level, ttbar_residue

◆ top_width()

double QQbar_threshold::top_width	(	double	mu,
		double	m,
		order	o,
		options	opt = `top_options()`
	)

Computes the top quark decay width.

Parameters

mu	renormalisation scale
m	top quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The decay width

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::r4 = 824.12 (for nl == 5)

See also: options

◆ ttbar_width()

double QQbar_threshold::ttbar_width	(	int	n,
		double	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Computes the decay width of a top-antitop bound state.

Parameters

n	principal quantum number
mu	renormalisation scale
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The decay width

The supported options and their default values are:

options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)

See also: options

◆ ttbar_xsection() [1/2]

double QQbar_threshold::ttbar_xsection	(	double	sqrt_s,
		scales	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Computes the cross section \(e^+e^- \to W^+W^-b\bar{b}\).

Parameters

sqrt_s	center-of-mass energy
mu	renormalisation scales {soft, weak}
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The cross section in picobarn

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::production = production_channels::all
options::resum_poles = 6
options::expand_s = false
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0

See also: options, ttbar_threshold, threshold

◆ ttbar_xsection() [2/2]

double QQbar_threshold::ttbar_xsection	(	double	sqrt_s,
		double	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Computes the cross section \(e^+e^- \to W^+W^-b\bar{b}\).

Parameters

sqrt_s	center-of-mass energy
mu	overall value for all renormalisation scales
M	top quark mass and width
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The cross section in picobarn

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::production = production_channels::all
options::resum_poles = 6
options::expand_s = false
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0

See also: options, ttbar_threshold, threshold

◆ bbbar_xsection()

double QQbar_threshold::bbbar_xsection	(	double	sqrt_s,
		double	mu,
		double	m,
		order	o,
		options	opt = `bottom_options()`
	)

Computes the cross section \(e^+e^- \to b\bar{b}\).

Parameters

sqrt_s	center-of-mass energy
mu	soft renormalisation scale
m	bottom quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The cross section in picobarn

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_bottom}
options::production = production_channels::all
options::resum_poles = 6
options::expand_s = false
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::QED
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 1220.3 (for nl == 4)
options::ml = 0

See also: options, bbbar_threshold, threshold

◆ ttbar_threshold()

double QQbar_threshold::ttbar_threshold	(	double	mu,
		double	m,
		order	o,
		options	opt = `top_options()`
	)

Computes the naive threshold for \(t\bar{t}\) production.

Parameters

mu	renormalisation scale
m	top quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The center-of-mass energy of the \(t\bar{t}\) production threshold

The naive production threshold is twice the pole quark mass. If the mass is given in a different mass scheme the corresponding pole mass is used.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

◆ bbbar_threshold()

double QQbar_threshold::bbbar_threshold	(	double	mu,
		double	m,
		order	o,
		options	opt = `bottom_options()`
	)

Computes the naive threshold for \(b\bar{b}\) production.

Parameters

mu	renormalisation scale
m	bottom quark mass
o	perturbative order (LO, NLO, N2LO, or N3LO)
opt	further options

Returns: The center-of-mass energy of the \(b\bar{b}\) production threshold

The naive production threshold is twice the pole quark mass. If the mass is given in a different mass scheme the corresponding pole mass is used.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_bottom}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::QED
options::alpha = alpha_Y
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 1220.3 (for nl == 4)
options::ml = 0
options::double_light_insertion = false

◆ bbbar_R_ratio()

double QQbar_threshold::bbbar_R_ratio	(	double	sqrt_s,
		double	mu,
		double	m,
		order	o,
		options	opt = `bottom_options()`
	)

Bottom contribution to the hadronic R ratio.

The bottom contribution to the hadronic R ratio is defined by bbbar_R_ratio = \(\sigma_b/\sigma_{pt} \) with \(\sigma_{pt} = \frac{4 \pi \alpha(\mu_\alpha)^2}{3s}\), where \(\sigma_b\) is the bottom production cross section bbbar_xsection converted to natural units. \(\alpha(\mu_\alpha)\) is the running on-shell QED coupling at the scale \(\mu_\alpha\) set by the options.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_bottom}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::QED
options::alpha = alpha_Y
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 1220.3 (for nl == 4)
options::ml = 0
options::double_light_insertion = false

◆ ttbar_R_ratio()

double QQbar_threshold::ttbar_R_ratio	(	double	sqrt_s,
		scales	mu,
		top_properties	M,
		order	o,
		options	opt = `top_options()`
	)

Top contribution to the hadronic R ratio.

The bottom contribution to the hadronic R ratio is defined by ttbar_R_ratio = \(\sigma_t/\sigma_{pt} \) with \(\sigma_{pt} = \frac{4 \pi \alpha(\mu_\alpha)^2}{3s}\), where \(\sigma_t\) is the top production cross section bbbar_xsection converted to natural units. \(\alpha(\mu_\alpha)\) is the running on-shell QED coupling at the scale \(\mu_\alpha\) set by the options.

The supported options and their default values are:

options::mass_scheme = {PSshift, mu_f_top}
options::alpha_s_mZ = alpha_s_mZ
options::contributions = (all set to 1)
options::beyond_QCD = SM::all
options::alpha = alpha_mZ
options::m_Higgs = m_Higgs
options::Yukawa_factor = 1
options::r4 = 824.12 (for nl == 5)
options::ml = 0
options::double_light_insertion = false

Classes

Enumerations

Functions

Variables

Detailed Description

Enumeration Type Documentation

◆ scheme_name

◆ SM

◆ production_channel

Function Documentation

◆ alpha_s_top() [1/2]

◆ alpha_s_top() [2/2]

◆ alpha_s_bottom() [1/2]

◆ alpha_s_bottom() [2/2]

◆ axial_Z_coupling()

◆ vector_Z_coupling()

◆ ttbar_energy_level() [1/2]

◆ ttbar_energy_level() [2/2]

◆ bbbar_energy_level()

◆ integrate()

◆ make_integrator()

◆ load_grid() [1/3]

◆ load_grid() [2/3]

◆ load_grid() [3/3]

◆ load_nonresonant_grid() [1/3]

◆ load_nonresonant_grid() [2/3]

◆ load_nonresonant_grid() [3/3]

◆ grid_directory()

◆ grid_range()

◆ is_shifted_mass() [1/2]

◆ to_shifted_mass()

◆ is_shifted_mass() [2/2]

◆ bottom_pole_mass()

Example

◆ top_pole_mass() [1/2]

Example

◆ top_pole_mass() [2/2]

◆ top_ISR_log()

◆ bottom_ISR_log()

◆ ISR_log()

◆ structure_function()

◆ modified_structure_function()

◆ luminosity_function()

◆ modified_luminosity_function()

◆ bottom_internal_settings() [1/2]

◆ bottom_internal_settings() [2/2]

◆ top_internal_settings() [1/2]

◆ top_internal_settings() [2/2]

◆ bottom_options()

◆ top_options()

◆ ttbar_residue() [1/2]

◆ ttbar_residue() [2/2]

◆ bbbar_residue()

◆ top_width()

◆ ttbar_width()

◆ ttbar_xsection() [1/2]

◆ ttbar_xsection() [2/2]

◆ bbbar_xsection()

◆ ttbar_threshold()

◆ bbbar_threshold()

◆ bbbar_R_ratio()

◆ ttbar_R_ratio()