alpha_strong Class Reference

#include <coupling.h>

Public Types
enum	{ exact, truncate_in_one_over_L }
Static Public Member Functions
static double	beta_0 (int Nf)
static double	beta_1 (int Nf)
static int	get_active_flavours (double Q)
static double	get_lambda_QCD (double Q)
static double	get_lambda_QCD (int Nf)
static double	get_value (double Q, int Nf)
Static Public Attributes
static double	lambda_3 = 0.374
static double	lambda_4 = 0.327
static double	lambda_5 = 0.226
static double	charm_threshold = 1.3
static double	bottom_threshold = 4.7
static int	method = alpha_strong::truncate_in_one_over_L
Static Private Member Functions
static double	get_F (double Q, double a_s, int Nf)
static double	get_dF (double Q, double a_s, int Nf)
static double	root_safeI (double a, double b, double Q, int Nf)

Detailed Description

The strong coupling constant.

The class works with the quantity

$a_s = \frac{\alpha_s}{4\pi}$

.

If alpha_strong::method == alpha_strong::exact, the value of $a_s$ is obtained by solving numerically the equation

$\frac{1}{a_s} \mbox{} - \frac{\beta_1}{\beta_0} \ln \left( \beta_1 + \frac{\beta_0}{a_s} \right) \mbox{} - \beta_0 L = 0$

where $L = \ln \frac{Q^2}{\Lambda^2_{QCD}}$ .

If alpha_strong::method == alpha_strong::truncate_in_one_over_L, the value of $a_s$ is given by the formula

$a_s = \frac{1}{\beta_0 L} \left( 1 - \frac{\beta_1}{\beta_0^2} \frac{\ln L}{L} \right).$

Member Function Documentation

double beta_0 ( int Nf ) [static]

The first coefficient $\beta_0$ of the beta function

$\mu^2 \frac{d}{d\mu^2} a_s = - \varepsilon a_s \mbox{} - \beta_0 a_s^2 - \beta_1 a_s^3 - \beta_2 a_s^4 - ...$

with $a_s = \alpha_s / ( 4 \pi)$ and

$\beta_0 = \frac{11}{3} C_A - \frac{4}{3} T_R N_f$

double beta_1 ( int Nf ) [static]

The second coefficient $\beta_1$ of the beta function

$\mu^2 \frac{d}{d\mu^2} a_s = - \varepsilon a_s \mbox{} - \beta_0 a_s^2 - \beta_1 a_s^3 - \beta_2 a_s^4 - ...$

with $a_s = \alpha_s / ( 4 \pi)$ and

$\beta_1 = \frac{34}{3} C_A^2 - 4 \left( \frac{5}{3} C_A + C_F \right) T_R N_f$

int get_active_flavours ( double Q ) [static]

Returns the number of active flavours at the scale $Q$ .

double get_dF	(	double	Q,
		double	a_s,
		int	Nf
	)			`[static, private]`

Evaluates

$\frac{d}{da_s} F(Q,a_s) = - \frac{\beta_0}{a_s^2} \frac{1}{(\beta_0+\beta_1 a_s)}.$

double get_F	(	double	Q,
		double	a_s,
		int	Nf
	)			`[static, private]`

Evaluates

$F(Q,a_s) = \frac{1}{a_s} \mbox{} - \frac{\beta_1}{\beta_0} \ln \left( \beta_1 + \frac{\beta_0}{a_s} \right) \mbox{} - \beta_0 L$

with $L = \ln \frac{Q^2}{\Lambda^2_{QCD}}$ .

double get_lambda_QCD ( int Nf ) [static]

Returns the appropriate $\Lambda_{QCD}$ corresponding to $N_f$ active flavours.

double get_lambda_QCD ( double Q ) [static]

Returns the appropriate $\Lambda_{QCD}$ corresponding to the scale $Q$ .

double get_value	(	double	Q,
		int	Nf
	)			`[static]`

Returns the value of $a_s = \frac{\alpha_s}{4\pi}$ at the scale $Q$ with $N_f$ active flavours.

The parameter $N_f$ ensures that the correct formula is used close to flavour thresholds.

double root_safeI	(	double	a,
		double	b,
		double	Q,
		int	Nf
	)			`[static, private]`

Numerical solution of the equation

$\frac{1}{a_s} \mbox{} - \frac{\beta_1}{\beta_0} \ln \left( \beta_1 + \frac{\beta_0}{a_s} \right) \mbox{} - \beta_0 L = 0$

for $a_s$ .

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

alpha_strong Class Reference

Public Types

Static Public Member Functions

Static Public Attributes

Static Private Member Functions

Detailed Description

Member Function Documentation