ql::Tools< TOutput, TMass, TScale > Class Template Reference

Standard Tools for the library. More...

#include <tools.h>

Inheritance diagram for ql::Tools< TOutput, TMass, TScale >:

Public Member Functions
	Tools ()
	The Constructor.
	~Tools ()
	The Destructor.
bool	iszero (TMass const &psq) const
	Check for zeros. More...
TOutput	cLn (TOutput const &z, TScale const &isig) const
	Log of complex argument with explicit sign for imag part. More...
TOutput	cLn (TScale const &x, TScale const &isig) const
TOutput	fndd (int const &n, TOutput const &x, TScale const &iep) const
	The fndd function. More...
TOutput	Lnrat (TOutput const &x, TOutput const &y) const
	Computes the ratio of logs. More...
TOutput	Lnrat (TScale const &x, TScale const &y) const
TMass	ddilog (TMass const &x) const
	Computes the dilog function for real argument. More...
TOutput	denspence (TOutput const &z, TScale const &isig) const
	Spence's function. More...
TOutput	spencer (TOutput const &zrat1, TOutput const &zrat2, TScale const &ieps1, TScale const &ieps2) const
	Spence's function for ratio arguments. More...
TOutput	xspence (TOutput const (&z1)[2], TScale const (&im1)[2], TOutput const &z2, TScale const &im2) const
	Difference of complex Spence's function. More...
TOutput	cspence (TOutput const &z1, TScale const &im1, TOutput const &z2, TScale const &im2) const
	Complex Spence's function. More...
TOutput	Li2omrat (TScale const &x, TScale const &y) const
	Expression for dilog(1-(x-i*ep)/(y-i*ep)). More...
TOutput	Li2omrat (TOutput const &x, TOutput const &y, TScale const &ieps1=-1, TScale const &ieps2=-1) const
TOutput	Li2omx2 (TScale const &v, TScale const &w, TScale const &x, TScale const &y) const
	Expression for dilog(1-(v-i*ep)*(w-i*ep)/(x-i*ep)/(y-i*ep)). More...
TOutput	Li2omx2 (TOutput const &v, TOutput const &w, TOutput const &x, TOutput const &y, TScale const &ieps1=-1, TScale const &ieps2=-1) const
TOutput	cLi2omx2 (TOutput const &z1, TOutput const &z2, TScale const &ieps1=-1, TScale const &ieps2=-1) const
TOutput	Li2omx (TMass const &x1, TMass const &x2, TScale const &ieps1, TScale const &ieps2) const
	Calculate Li[2](1-(x1+ieps1)*(x2+ieps2)) More...
TOutput	cLi2omx3 (TOutput const &z1, TOutput const &z2, TOutput const &z3, TScale const &ieps1, TScale const &ieps2, TScale const &ieps3) const
	Calculate Li[2](1-(z1+ieps1)*(z2+ieps2)*(z3+ieps3)) More...
TOutput	L0 (TMass const &x, TMass const &y) const
TOutput	L1 (TMass const &x, TMass const &y) const
TOutput	R3int (TOutput const &p, TOutput const &s1, TOutput const &s2, TOutput const &t1) const
TOutput	R3int (TOutput const &p, TOutput const &s1, TOutput const &s2, TOutput const &t1, TOutput const &t2, TOutput const &t3, TOutput const &t4) const
TOutput	R2int (TOutput const &a, TOutput const &b, TOutput const &y0) const
TOutput	Rint (TOutput const &y, TOutput const &z, TScale const &ieps) const
void	R (TOutput &r, TOutput &d, TOutput const &q) const
	Tools<TOutput, TMass, TScale>::R. More...
TOutput	Zlogint (TOutput const &z, TScale const &ieps) const
	Tools<TOutput, TMass, TScale>::qlZlogint. More...
TOutput	ltspence (int const &i_in, TOutput const &z_in, TScale const &s) const
TOutput	li2series (TOutput const &z, TScale const &isig) const
TOutput	ltli2series (TOutput const &z1, TScale const &s) const
TOutput	eta2 (TOutput const &a, TOutput const &b) const
TOutput	eta3 (TOutput const &a, TOutput const &b, TOutput const &c) const
TOutput	eta5 (TOutput const &a, TOutput const &b, TOutput const &c, TOutput const &d, TOutput const &e) const
TOutput	xetatilde (TOutput const (&z1)[2], TScale const (&im1)[2], TOutput const &z2, TScale const &im2, TOutput const (&l1)[2]) const
TOutput	xeta (TOutput const (&z1)[2], TScale const (&im1)[2], TOutput const &z2, TScale const &im2, TScale const &im12, TOutput const (&l1)[2]) const
int	eta (TOutput const &z1, TScale const &s1, TOutput const &z2, TScale const &s2, TScale const &s12) const
int	etatilde (TOutput const &c1, TScale const &im1x, TOutput const &c2, TScale const &im2x) const
void	kfn (TOutput(&res)[3], TScale &ieps, TMass const &xpi, TMass const &xm, TMass const &xmp) const
	The K-function. More...
void	solveabc (TMass const &a, TMass const &b, TMass const &c, TOutput(&z)[2]) const
	Solution of the quadratic equation. More...
void	solveabcd (TOutput const &a, TOutput const &b, TOutput const &c, TOutput const &d, TOutput(&z)[2]) const
	Solution of the quadratic equation passing the discriminant. More...
void	solveabcd (TOutput const &a, TOutput const &b, TOutput const &c, TOutput(&z)[2]) const
void	ratgam (TOutput &ratp, TOutput &ratm, TScale &ieps, TMass const &p3sq, TMass const &m3sq, TMass const &m4sq) const
	Ratio function. More...
void	ratreal (TMass const &si, TMass const &ta, TMass &rat, TScale &ieps) const
	Ratio function. More...

Protected Attributes
TScale	_pi
TScale	_pi2
TScale	_pio3
TScale	_pio6
TScale	_pi2o3
TScale	_pi2o6
TScale	_pi2o12
TScale	_zero
TScale	_half
TScale	_one
TScale	_two
TScale	_three
TScale	_four
TScale	_five
TScale	_six
TScale	_ten
TScale	_eps
TScale	_eps4
TScale	_eps7
TScale	_eps10
TScale	_eps14
TScale	_eps15
TScale	_xloss
TScale	_neglig
TScale	_reps
TOutput	_2ipi
TOutput	_ipio2
TOutput	_ipi
TOutput	_czero
TOutput	_chalf
TOutput	_cone
TOutput	_ctwo
TOutput	_cthree
TOutput	_cfour
TOutput	_ieps
TOutput	_ieps2
TOutput	_ieps50

Detailed Description

template<typename TOutput = complex, typename TMass = double, typename TScale = double>
class ql::Tools< TOutput, TMass, TScale >

Standard Tools for the library.

The container of common functions used by Topology.

Member Function Documentation

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::cLi2omx2	(	TOutput const &	z1,
		TOutput const &	z2,
		TScale const &	ieps1 = -1,
		TScale const &	ieps2 = -1
	)		const

Calculates Li[2](1-(z1+ieps1)*(z2+ieps2)) for complex z1,z2 Using +Li2(1-z1*z2) for z1*z2<1 and -Li2(1-1/(z1*z2))-1/2*(ln(z1)+ln(z2))^2 for z1*z2>1

Parameters

z1	input argument
z2	input argument
ieps1	sign of z1
ieps2	sign of z2

Returns

Li2 of the 1-product

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::cLi2omx3	(	TOutput const &	z1,
		TOutput const &	z2,
		TOutput const &	z3,
		TScale const &	ieps1,
		TScale const &	ieps2,
		TScale const &	ieps3
	)		const

Calculate Li[2](1-(z1+ieps1)*(z2+ieps2)*(z3+ieps3))

Generalization of cLi2omx2 for 3 arguments.

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::cLn	(	TOutput const &	z,
		TScale const &	isig
	)		const

Log of complex argument with explicit sign for imag part.

Computes the log of a complex number z. If the imag(z)=0 and real(z)<0 and extra ipi term is included.

Parameters

z	the complex argument of the logarithm
isig	the sign of the imaginary part

Returns

the complex log(z)

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::cLn	(	TScale const &	x,
		TScale const &	isig
	)		const

Computes the log of a real number x. If the x<0 and extra ipi term is included.

Parameters

x	the real argument of the logarithm
isig	the sign of the imaginary part

Returns

the complex log(x)

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::cspence	(	TOutput const &	z1,
		TScale const &	im1,
		TOutput const &	z2,
		TScale const &	im2
	)		const

Complex Spence's function.

Parameters

z1	input argument
im1	sign of z1
z2	input argument
im2	sign of z2

Returns

the complex Spence's function

template<typename TOutput , typename TMass , typename TScale >

TMass ql::Tools< TOutput, TMass, TScale >::ddilog

(

TMass const &

x

)

const

Computes the dilog function for real argument.

The dilog function for real argument

Parameters

x	the argument of the dilog

Returns

the dilog

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::denspence	(	TOutput const &	z,
		TScale const &	isig
	)		const

Spence's function.

Parameters

z	the argument
isig	the sign of z

template<typename TOutput , typename TMass , typename TScale >

int ql::Tools< TOutput, TMass, TScale >::eta	(	TOutput const &	z1,
		TScale const &	s1,
		TOutput const &	z2,
		TScale const &	s2,
		TScale const &	s12
	)		const

Parameters

z1
s1
z2
s2
s12

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::eta2	(	TOutput const &	a,
		TOutput const &	b
	)		const

Parameters

a
b

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::eta3	(	TOutput const &	a,
		TOutput const &	b,
		TOutput const &	c
	)		const

Parameters

a
b
c

Returns

template<typename TOutput , typename TMass , typename TScale >

int ql::Tools< TOutput, TMass, TScale >::etatilde	(	TOutput const &	c1,
		TScale const &	im1x,
		TOutput const &	c2,
		TScale const &	im2x
	)		const

Parameters

c1
im1x
c2
im2x

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::fndd	(	int const &	n,
		TOutput const &	x,
		TScale const &	iep
	)		const

The fndd function.

Implementation of the formulae of Denner and Dittmaier [5].

\[ f_{n}(x) = \ln \left( 1 - \frac{1}{x} \right) + \sum_{l=n+1}^{\infty} \frac{x^{n-l}}{l+1} \]

Parameters

n	the lower index of
x	the argument of the function
iep	the epsilon value

Returns

function DD from Eq. 4.11 of [5].

template<typename TOutput , typename TMass , typename TScale >

bool ql::Tools< TOutput, TMass, TScale >::iszero

(

TMass const &

psq

)

const

Check for zeros.

Compares the Abs(psq) to the cutoff

Parameters

msq	the input to be tested

Returns

true if |psq| < onshell cut-off

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::kfn	(	TOutput(&)	res[3],
		TScale &	ieps,
		TMass const &	xpi,
		TMass const &	xm,
		TMass const &	xmp
	)		const

The K-function.

Calculate the K-function give in Eq. 2.7 of [1]

\[ K(p^2,m,m_p) = \frac{1-\sqrt{1-4m m_p / (z-(m-m_p)^2)}}{1+\sqrt{1-4m m_p / (z-(m-m_p)^2)}} \]

and fill x[0] = -K, x[1] = 1+K, x[2] = 1-K, the roots are allowed to be imaginary. ieps gives the sign of the imaginary part of -K: 1 -> +i*eps

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::L0	(	TMass const &	x,
		TMass const &	y
	)		const

Parameters

x	input argument
y	input argument

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::L1	(	TMass const &	x,
		TMass const &	y
	)		const

Parameters

x	input argument
y	input argument

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Li2omrat	(	TScale const &	x,
		TScale const &	y
	)		const

Expression for dilog(1-(x-i*ep)/(y-i*ep)).

Parameters

x	numerator
y	denominator

Returns

the Li2 ratio

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Li2omx	(	TMass const &	x1,
		TMass const &	x2,
		TScale const &	ieps1,
		TScale const &	ieps2
	)		const

Calculate Li[2](1-(x1+ieps1)*(x2+ieps2))

Calculate Li[2](1-(x1+ieps1)*(x2+ieps2)) for real x1,x2 Using +Li2(1-x1*x2) for x1*x2<1 and -Li2(1-1/(x1*x2))-1/2*(ln(x1)+ln(x2))^2 for x1*x2>1

Parameters

x1	numerator
x2	denominator
ieps1	sign of x1
ieps2	sign of x2

Returns

the ratio Li2

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Li2omx2	(	TScale const &	v,
		TScale const &	w,
		TScale const &	x,
		TScale const &	y
	)		const

Expression for dilog(1-(v-i*ep)*(w-i*ep)/(x-i*ep)/(y-i*ep)).

expression for dilog(1-(v-i*ep)*(w-i*ep)/(x-i*ep)/(y-i*ep)) for real v,w,x and y

Parameters

v	numerator
w	numerator
x	denominator
y	denominator

Returns

the dilog ratio

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Li2omx2	(	TOutput const &	v,
		TOutput const &	w,
		TOutput const &	x,
		TOutput const &	y,
		TScale const &	ieps1 = -1,
		TScale const &	ieps2 = -1
	)		const

Calculates Li[2](1-(z1+ieps1)*(z2+ieps2)) for complex z1,z2 Using +Li2(1-z1*z2) for z1*z2<1 and -Li2(1-1/(z1*z2))-1/2*(ln(z1)+ln(z2))^2 for z1*z2>1

Parameters

z1	input argument
z2	input argument
ieps1	sign of z1
ieps2	sign of z2

Returns

Li2 of the 1-product

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::li2series	(	TOutput const &	z,
		TScale const &	isig
	)		const

Parameters

z
isig

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Lnrat	(	TOutput const &	x,
		TOutput const &	y
	)		const

Computes the ratio of logs.

Computes \({\rm Lnrat}(x,y) = \log(x-i \epsilon)-\log(y-i \epsilon)\)

Parameters

x	TMass object for the numerator
y	TMass object for the denumerator

Returns

returns the ratio of logs

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::ltli2series	(	TOutput const &	z1,
		TScale const &	s
	)		const

Parameters

z1
s

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::ltspence	(	int const &	i_in,
		TOutput const &	z_in,
		TScale const &	s
	)		const

Parameters

i_in
z_in
s

Returns

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::R	(	TOutput &	r,
		TOutput &	d,
		TOutput const &	q
	)		const

Tools<TOutput, TMass, TScale>::R.

Parameters

q

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::R2int	(	TOutput const &	a,
		TOutput const &	b,
		TOutput const &	y0
	)		const

Finite Triangle Li2 sum.

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::R3int	(	TOutput const &	a,
		TOutput const &	s1,
		TOutput const &	s2,
		TOutput const &	t1
	)		const

Finite Triangle Li2 sum.

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::R3int	(	TOutput const &	a,
		TOutput const &	s1,
		TOutput const &	s2,
		TOutput const &	t1,
		TOutput const &	t2,
		TOutput const &	t3,
		TOutput const &	t4
	)		const

Finite Triangle Li2 sum.

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::ratgam	(	TOutput &	ratp,
		TOutput &	ratm,
		TScale &	ieps,
		TMass const &	p3sq,
		TMass const &	m3sq,
		TMass const &	m4sq
	)		const

Ratio function.

Calculate the function

\[ R(p^2,m,m_p) = \frac{p_3^2+m_4^2-m_3^2+\sqrt{(p_3^2+m_4^2-m_3^2)^2-4 p_3^2 m_4^2}}{-p_3^2+m_4^2-m_3^2+\sqrt{(p_3^2+m_4^2-m_3^2)^2-4 p_3^2 m_4^2}} \]

where the roots are allowed to be imaginary.

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::ratreal	(	TMass const &	si,
		TMass const &	ta,
		TMass &	rat,
		TScale &	ieps
	)		const

Ratio function.

Calculate the function

\[ R = \frac{\sigma-i \epsilon}{\tau-i \epsilon} \]

where sigma and tau are real and ieps give the sign if i*pi.

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Rint	(	TOutput const &	y,
		TOutput const &	z,
		TScale const &	ieps
	)		const

Parameters

y
z
ieps

Returns

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::solveabc	(	TMass const &	a,
		TMass const &	b,
		TMass const &	c,
		TOutput(&)	z[2]
	)		const

Solution of the quadratic equation.

Solution of a quadratic equation a*z^2+b*z+c=0.

Parameters

a	coefficient
b	coefficient
c	coefficient
l	result [-im,+im]

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::solveabcd	(	TOutput const &	a,
		TOutput const &	b,
		TOutput const &	c,
		TOutput const &	d,
		TOutput(&)	z[2]
	)		const

Solution of the quadratic equation passing the discriminant.

Solution of a quadratic equation a*z^2+b*z+c=0, with a give input discriminant.

Parameters

a	coefficient
b	coefficient
c	coefficient
d	discriminant
l	result [-im,+im]

template<typename TOutput , typename TMass , typename TScale >

void ql::Tools< TOutput, TMass, TScale >::solveabcd	(	TOutput const &	a,
		TOutput const &	b,
		TOutput const &	c,
		TOutput(&)	z[2]
	)		const

Solution of a quadratic equation a*z^2+b*z+c=0, with a give input discriminant.

Parameters

a	coefficient
b	coefficient
c	coefficient
d	discriminant
l	result [-im,+im]

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::spencer	(	TOutput const &	zrat1,
		TOutput const &	zrat2,
		TScale const &	ieps1,
		TScale const &	ieps2
	)		const

Spence's function for ratio arguments.

Parameters

zrat1	first argument
zrat2	second argument
ieps1	sign for zrat1
ieps2	sign for zrat2

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::xeta	(	TOutput const (&)	z1[2],
		TScale const (&)	im1[2],
		TOutput const &	z2,
		TScale const &	im2,
		TScale const &	im12,
		TOutput const (&)	l1[2]
	)		const

Parameters

z2
im2
im12

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::xetatilde	(	TOutput const (&)	z1[2],
		TScale const (&)	im1[2],
		TOutput const &	z2,
		TScale const &	im2,
		TOutput const (&)	l1[2]
	)		const

Parameters

z2
im2

Returns

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::xspence	(	TOutput const (&)	z1[2],
		TScale const (&)	im1[2],
		TOutput const &	z2,
		TScale const &	im2
	)		const

Difference of complex Spence's function.

Parameters

z2	input arguments
im2	input signs.

Returns

the difference of cspence functions

template<typename TOutput , typename TMass , typename TScale >

TOutput ql::Tools< TOutput, TMass, TScale >::Zlogint	(	TOutput const &	z,
		TScale const &	ieps
	)		const

Tools<TOutput, TMass, TScale>::qlZlogint.

Parameters

z
ieps

Returns

---

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

• /home/carrazza/repo/github/qcdloop/src/qcdloop/tools.h
• /home/carrazza/repo/github/qcdloop/src/tools.cc