Fast Evolution of Parton Distributions: partondistribution Class Reference

Fast Evolution of Parton Distributions

#include <partondistribution.h>

Inheritance diagram for partondistribution:

Public Member Functions
	partondistribution (void)

	partondistribution (int n, double Q0, double A[], double alpha[], double beta[], int eta=1)

	partondistribution (int n, double Q0, const std::valarray< double > &A, const std::valarray< double > &alpha, const std::valarray< double > &beta, int eta=1)

double	get_Q0 (void) const

int	get_eta (void) const

complex_d	get_Mellin_transform (complex_d z) const

double	get_F (double x, double z) const

double	get_dF (double x, double z) const

double	get_ddF (double x, double z) const

double	get_dddF (double x, double z) const

double	get_c0 (double x) const

virtual bool	with_qed (void) const

virtual double	get_electric_charge (void) const

Protected Attributes
int	n

double	Q0

std::valarray< double >	A

std::valarray< double >	alpha

std::valarray< double >	beta

int	eta

Private Member Functions
double	root_safeI (double a, double b, double x) const

Friends
partondistribution	operator+ (const partondistribution &f1, const partondistribution &f2)

Detailed Description

The class partondistribution represents the parameterization of a parton distribution in the form

$x f(x,Q_0^2) = \sum\limits_{i=0}^{n-1} A_i x^{\alpha_i} (1-x)^{\beta_i}$

In addition this class has a data member $\eta = \pm 1$ , which selects odd or even moments for the evolution. The default value for $\eta$ in the constructor is $+1$ , which is good for the singlet distributions and non-singlet non-valence distributions like

$(u+\bar{u}) - ( d + \bar{d} ),$

$(u+\bar{u}) + ( d + \bar{d} ) - 2 ( s + \bar{s} ), ...$

Valence non-singlet distributions like

$(u-\bar{u}), (d - \bar{d}), ...$

need $\eta=-1$ , which has to passed explicitly to the constructor.

Constructor & Destructor Documentation

§ partondistribution() [1/3]

partondistribution ( void )

Default constructor

§ partondistribution() [2/3]

partondistribution	(	int	nr,
		double	Q0r,
		double	Ar[],
		double	alphar[],
		double	betar[],
		int	etar = `1`
	)

Standard constructor

§ partondistribution() [3/3]

partondistribution	(	int	nr,
		double	Q0r,
		const std::valarray< double > &	Ar,
		const std::valarray< double > &	alphar,
		const std::valarray< double > &	betar,
		int	etar = `1`
	)

Standard constructor from valarray

Member Function Documentation

§ get_c0()

double get_c0 ( double x ) const

Returns for fixed $x$ the position of the minimum in the $z$ -plane of the function

$F(x,z) = x^{-z} \sum\limits_{i=0}^{n-1} A_i B(z+\alpha_i-1,1+\beta_i).$

on the real axis right to the rightmost pole.

§ get_dddF()

double get_dddF	(	double	x,
		double	z
	)		const

Evaluates

$\frac{d^3}{dz^3} F(x,z) = \sum\limits_{i=0}^{n-1} \left( G_i''' + 3 G_i'' G_i' + G_i'^3 \right) F_i$

where

$G_i''' = \Psi''(z+\alpha_i-1) - \Psi''(z+\alpha_i+\beta_i)$

§ get_ddF()

double get_ddF	(	double	x,
		double	z
	)		const

Evaluates

$\frac{d^2}{dz^2} F(x,z) = \sum\limits_{i=0}^{n-1} \left( G_i'' + G_i'^2 \right) F_i$

where

$G_i'' = \Psi'(z+\alpha_i-1) - \Psi'(z+\alpha_i+\beta_i)$

§ get_dF()

double get_dF	(	double	x,
		double	z
	)		const

Evaluates

$\frac{d}{dz} F(x,z) = \sum\limits_{i=0}^{n-1} G_i' F_i$

where

$F_i = x^{-z} A_i B(z+\alpha_i-1,1+\beta_i),$

$G_i = \ln \left( x^{-z} B(z+\alpha_i-1,1+\beta_i) \right)$

$G_i' = -\ln x + \Psi(z+\alpha_i-1) - \Psi(z+\alpha_i+\beta_i)$

§ get_electric_charge()

virtual double get_electric_charge ( void ) const

inlinevirtual

Return zero for a partondistribution. The method is overridden by the derived class partondistribution_with_qed.

Reimplemented in partondistribution_with_qed.

§ get_eta()

int get_eta ( void ) const

Returns $\eta$ .

§ get_F()

double get_F	(	double	x,
		double	z
	)		const

Evaluates the function

$F(x,z) = x^{-z} \sum\limits_{i=0}^{n-1} A_i B(z+\alpha_i-1,1+\beta_i).$

for the given pair $(x,z)$ .

§ get_Mellin_transform()

complex_d get_Mellin_transform ( complex_d z ) const

Evaluates the Mellin transform of the parton distribution

$f^z(Q_0^2) = \sum\limits_{i=0}^{n-1} A_i B(z+\alpha_i-1,1+\beta_i)$

at the point $z$ .

§ get_Q0()

double get_Q0 ( void ) const

Returns $Q_0$ .

§ root_safeI()

double root_safeI	(	double	a,
		double	b,
		double	x
	)		const

private

Solves numerically the equation

$\frac{d}{dz} F(x,z) = 0$

for $z$ .

§ with_qed()

virtual bool with_qed ( void ) const

inlinevirtual

Return false for a partondistribution. The method is overridden by the derived class partondistribution_with_qed.

Reimplemented in partondistribution_with_qed.

Friends And Related Function Documentation

§ operator+

partondistribution operator+	(	const partondistribution &	f1,
		const partondistribution &	f2
	)

friend

Addition of two parton distributions

The sum of two parton distributions is given by

$\sum\limits_{i=0}^{n-1} A_i x^{\alpha_i} (1-x)^{\beta_i} + \sum\limits_{i=0}^{n'-1} A_i' x^{\alpha_i'} (1-x)^{\beta_i'}$

The values of $Q_0$ and $\eta$ of the two summands have to agree, otherwise the function throws an exception.

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