EvolDC1Buras Class Reference

\( |\Delta F = 1 | \) Evolutor Class More...

#include <EvolDC1Buras.h>

Inheritance diagram for EvolDC1Buras:

Detailed Description

\( |\Delta F = 1 | \) Evolutor Class

\( |\Delta C = 1 | \) Evolutor Class

This evolutor is properly written to study \( |\Delta F = 1 | \) processes such as radiative and semileptonic weak decays of the B meson; it is implemented for the evolution of the 10 Wilson coefficients (related to the 2 current x current + 4 QCD penguins + em & chromo penguins + 2 semilptonic ones) in the Chetyrkin, Misiak and Munz basis at the NLO in \( \alpha_{strong} \); principal reference: hep-ph/0612329

This evolutor is properly written for Charm Physics studies; it is implemented for the evolution of the 6 Wilson coefficients (related to the 2 current x current + 4 QCD penguin operators) in the standard (Buras) basis at the NLO in \( \alpha_{strong} \); principal reference: hep-ph/9512380v1

Public Member Functions
gslpp::matrix< double >	AnomalousDimension_DC1_Buras (orders order, unsigned int n_u, unsigned int n_d) const
	a method returning the anomalous dimension matrix given in the standard basis More...
gslpp::matrix< double > &	DC1EvolBuras (double mu, double M, orders order, schemes scheme=NDR)
	a method returning the evolutor related to the high scale \( M \) and the low scale \( \mu \) More...
	EvolDC1Buras (unsigned int dim_i, schemes scheme, orders order, const StandardModel &model)
	EvolDC1Buras constructor. More...
gslpp::matrix< double >	StrongThresholds () const
	a method returning the matrix threshold for the QCD penguins at the NLO More...
virtual	~EvolDC1Buras ()
	EvolDC1Buras destructor. More...
Public Member Functions inherited from RGEvolutor
gslpp::matrix< double > *	Evol (orders order)
	Evolution matrix set at a fixed order of QCD coupling. More...
gslpp::matrix< double > *	Evol (orders_qed order_qed)
	Evolution matrix set at a fixed order of Electroweak coupling. More...
gslpp::matrix< double > **	getEvol () const
double	getM () const
	Retrieve the upper scale of the Wilson Coefficients. More...
	RGEvolutor (unsigned int dim, schemes scheme, orders order)
	constructor More...
	RGEvolutor (unsigned int dim, schemes scheme, orders order, orders_qed order_qed)
	constructor More...
void	setEvol (const gslpp::matrix< double > &m, orders order_i)
void	setEvol (const gslpp::matrix< double > &m, orders_qed order_qed_i)
void	setEvol (unsigned int i, unsigned int j, double x, orders order_i)
void	setEvol (unsigned int i, unsigned int j, double x, orders order_i, orders_qed order_qed)
void	setM (double M)
	Sets the upper scale for the running of the Wilson Coefficients. More...
void	setMu (double mu)
	Sets the lower scale for the running of the Wilson Coefficients. More...
void	setScales (double mu, double M)
	Sets the upper and lower scale for the running of the Wilson Coefficients. More...
virtual	~RGEvolutor ()
	destructor More...
Public Member Functions inherited from WilsonTemplate< gslpp::matrix< double > >
double	getMu () const
orders	getOrder () const
orders_qed	getOrder_qed () const
schemes	getScheme () const
unsigned int	getSize () const
virtual void	resetCoefficient ()
void	setScheme (schemes scheme)
	WilsonTemplate (const WilsonTemplate< gslpp::matrix< double > > &orig)
	WilsonTemplate (unsigned int dim, schemes scheme_i, orders order_i, orders_qed order_qed_i=NO_QED)
virtual	~WilsonTemplate ()

Private Member Functions
void	DC1EvolBuras (double mu, double M, double nf, schemes scheme)
	a void type method storing properly the magic numbers for the implementation of the evolutor More...
void	DC1PenguinThresholds (double M, orders order)
	a void type method for the implementation of the NLO threshold effects in the evolutor More...

Private Attributes
double	a [3][10]
double	alsMZ_cache
double	b [3][10][10][10]
double	c [3][10][10][10]
double	d [3][10][10][10]
unsigned int	dim
gslpp::vector< gslpp::complex >	e
gslpp::matrix< gslpp::complex >	gg
gslpp::matrix< gslpp::complex >	h
gslpp::matrix< gslpp::complex >	js
gslpp::matrix< gslpp::complex >	jss
gslpp::matrix< gslpp::complex >	jssv
gslpp::matrix< gslpp::complex >	jv
const StandardModel &	model
double	Mz_cache
int	nd
int	nu
gslpp::matrix< gslpp::complex >	s_s
gslpp::matrix< gslpp::complex >	v
gslpp::matrix< gslpp::complex >	vi
gslpp::matrix< gslpp::complex >	vij

Additional Inherited Members
Protected Member Functions inherited from WilsonTemplate< gslpp::matrix< double > >
gslpp::matrix< double > *	Elem (orders order) const
gslpp::matrix< double > *	Elem (orders_qed order_qed) const
void	setElem (const gslpp::matrix< double > &v, orders order_i)
void	setElem (const gslpp::matrix< double > &v, orders_qed order_qed_i)
Protected Attributes inherited from RGEvolutor
double	M
Protected Attributes inherited from WilsonTemplate< gslpp::matrix< double > >
gslpp::matrix< double > *	elem [MAXORDER_QED+1]
double	mu
orders	order
orders_qed	order_qed
schemes	scheme
unsigned int	size

Constructor & Destructor Documentation

EvolDC1Buras()

EvolDC1Buras::EvolDC1Buras	(	unsigned int	dim_i,
		schemes	scheme,
		orders	order,
		const StandardModel &	model
	)

EvolDC1Buras constructor.

Parameters

dim	an unsigned integer for the dimension of the evolutor
scheme	an enum "schemes" for the regularization scheme of the evolutor
order	an enum "orders" for the order of perturbation theory of the evolutor
model	an object of StandardModel class

Definition at line 14 of file EvolDC1Buras.cpp.

:           RGEvolutor(dim_i, scheme, order), model(model),
            v(dim_i,0.), vi(dim_i,0.), js(dim_i,0.), h(dim_i,0.), gg(dim_i,0.), s_s(dim_i,0.),
            jssv(dim_i,0.), jss(dim_i,0.), jv(dim_i,0.), vij(dim_i,0.), e(dim_i,0.), dim(dim_i)  
{
    
    /*magic numbers a & b */
    
    for(int L=2; L>-1; L--){
        
    /* L=2 --> u,d,s,c (nf=4)  L=1 --> u,d,s,c,b (nf=5) L=0 --> u,d,s,c,b,t (nf=6)*/
        
    nu = L;  nd = L;
    if(L == 1){nd = 3; nu = 2;} 
    if(L == 0){nd = 3; nu = 3;}
    
    AnomalousDimension_DC1_Buras(LO,nu,nd).transpose().eigensystem(v,e);
    vi = v.inverse();
    for(unsigned int i = 0; i < dim; i++){
       a[L][i] = e(i).real();
       for (unsigned int j = 0; j < dim; j++) {
           for (unsigned int k = 0; k < dim; k++)  {
                b[L][i][j][k] = v(i, k).real() * vi(k, j).real();
               }
           }
       }
    
    // LO evolutor in the standard basis
    
    gg = vi * AnomalousDimension_DC1_Buras(NLO,nu,nd).transpose() * v;
    double b0 = model.Beta0(6-L);
    double b1 = model.Beta1(6-L);
    for (unsigned int i = 0; i < dim; i++){
        for (unsigned int j = 0; j < dim; j++){
            s_s.assign( i, j, (b1 / b0) * (i==j) * e(i).real() - gg(i,j));    
            if(fabs(e(i).real() - e(j).real() + 2. * b0)>0.00000000001){
                h.assign( i, j, s_s(i,j) / (2. * b0 + e(i) - e(j)));
                }
            }
        }
    js = v * h * vi;
    jv = js * v;
    vij = vi * js;
    jss = v * s_s * vi;
    jssv = jss * v;        
    for (unsigned int i = 0; i < dim; i++){
        for (unsigned int j = 0; j < dim; j++){
            if(fabs(e(i).real() - e(j).real() + 2. * b0) > 0.00000000001){
                for(unsigned int k = 0; k < dim; k++){
                        c[L][i][j][k] = jv(i, k).real() * vi(k, j).real();
                        d[L][i][j][k] = -v(i, k).real() * vij(k, j).real();
                        }
                    }
            else{    
                for(unsigned int k = 0; k < dim; k++){
                   c[L][i][j][k] = (1./(2. * b0)) * jssv(i, k).real() * vi(k, j).real();
                   d[L][i][j][k] = 0.;
                   }   
                }
            }
        }
    }
}

~EvolDC1Buras()

EvolDC1Buras::~EvolDC1Buras ( )

virtual

EvolDC1Buras destructor.

Definition at line 78 of file EvolDC1Buras.cpp.

{}

Member Function Documentation

AnomalousDimension_DC1_Buras()

gslpp::matrix< double > EvolDC1Buras::AnomalousDimension_DC1_Buras	(	orders	order,
		unsigned int	n_u,
		unsigned int	n_d
	)		const

a method returning the anomalous dimension matrix given in the standard basis

Parameters

order	an enum "orders" for the order of perturbation theory of the ADM
n_u	an unsigned integer for the up-type number of d.o.f.
n_d	an unsigned integer for the down-type number of d.o.f.

Returns

the ADM at the order LO/NLO in the standard basis

Definition at line 81 of file EvolDC1Buras.cpp.

{
   
    /* anomalous dimension related to Delta F = 1 operators in Buras basis, hep-ph/9512380v1 */
    
    /* gamma(row, column) at the LO */
    
    unsigned int nf = n_u + n_d; /*n_u/d = active type up/down flavor d.o.f.*/
    gslpp::matrix<double> gammaDF1(dim, 0.);
    
    switch(order){
        
    case LO:
    
    gammaDF1(0,0) = -2.;
    gammaDF1(0,1) = 6. ;    
    
    gammaDF1(1,0) = 6.;
    gammaDF1(1,1) = -2.;
    
    if(nf<5){ 
    gammaDF1(1,2) = -2./9.;
    gammaDF1(1,3) = 2./3.;
    gammaDF1(1,4) = -2./9.;
    gammaDF1(1,5) = 2./3.;
    
    }
    
    gammaDF1(2,2) = -22./9.;
    gammaDF1(2,3) = 22./3.;
    gammaDF1(2,4) = -4./9.;
    gammaDF1(2,5) = 4./3.;
    
    gammaDF1(3,2) = 6.-2./9.*nf;
    gammaDF1(3,3) = -2.+2./3.*nf;
    gammaDF1(3,4) = -2./9.*nf;
    gammaDF1(3,5) = 2./3.*nf;
    
    gammaDF1(4,4) = 2.;
    gammaDF1(4,5) = -6.;
    
    gammaDF1(5,2) = -2./9.*nf;
    gammaDF1(5,3) = 2./3.*nf;
    gammaDF1(5,4) = -2./9.*nf;
    gammaDF1(5,5) = -16.+2./3.*nf;
   
    break;    
    
    case NLO:
        
    if (!(nf == 3 || nf == 4 || nf == 5 || nf == 6)){ 
                throw std::runtime_error("EvolDF1nlep::AnomalousDimension_B(): wrong number of flavours"); 
        }
    
    /* gamma(row, column) at the NLO */
    
    gammaDF1(0,0) = -21./2.-2./9.*nf;
    gammaDF1(0,1) = 7./2.+2./3.*nf;
    
    gammaDF1(1,0) = 7./2.+2./3.*nf;
    gammaDF1(1,1) = -21./2.-2./9.*nf;
    
    if(nf<5){    
        
    gammaDF1(0,2) = 79./9.;
    gammaDF1(0,3) = -7./3.;
    gammaDF1(0,4) = -65./9.;
    gammaDF1(0,5) = -7./3.;
    
    gammaDF1(1,2) = -202./243.;
    gammaDF1(1,3) = 1354./81.;
    gammaDF1(1,4) = -1192./243.;
    gammaDF1(1,5) = 904./81.;
    
    }
    
    gammaDF1(2,2) = -5911./486.+71./9.*nf;
    gammaDF1(2,3) = 5983./162.+1./3.*nf;
    gammaDF1(2,4) = -2384./243.-71./9.*nf;
    gammaDF1(2,5) = 1808./81.-1./3.*nf;
    
    
    gammaDF1(3,2) = 379./18.+56./243.*nf;
    gammaDF1(3,3) = -91./6.+808./81.*nf;
    gammaDF1(3,4) = -130./9.-502./243.*nf;
    gammaDF1(3,5) = -14./3.+646./81.*nf;
    
    gammaDF1(4,2) = -61./9.*nf;
    gammaDF1(4,3) = -11./3.*nf;
    gammaDF1(4,4) = 71./3.+61./9.*nf;
    gammaDF1(4,5) = -99.+11./3.*nf;
    
    gammaDF1(5,2) = -682./243.*nf;
    gammaDF1(5,3) = 106./81.*nf;
    gammaDF1(5,4) = -225./2.+1676./243.*nf;
    gammaDF1(5,5) = -1343./6.+1348./81.*nf;
        
    break;
    
    default:
            throw std::runtime_error("EvolDF1nlep::AnomalousDimensio_B_S(): order not implemented"); 
    }
   
    return (gammaDF1);
    
  }

DC1EvolBuras() [1/2]

void EvolDC1Buras::DC1EvolBuras ( double  mu, double  M, double  nf, schemes  scheme  )

private

a void type method storing properly the magic numbers for the implementation of the evolutor

Parameters

mu	a double for the low scale of the evolution
M	a double for the high scale of the evolution
nf	a double for the active number of flavors
scheme	an enum "schemes" for the regularization scheme of the evolutor

Definition at line 233 of file EvolDC1Buras.cpp.

{
 
    gslpp::matrix<double> resLO(dim, 0.), resNLO(dim, 0.), resNNLO(dim, 0.);
 
    int L = 6 - (int) nf;
    double alsM = model.Als(M) / 4. / M_PI;
    double alsmu = model.Als(mu) / 4. / M_PI;
    
    double eta = alsM / alsmu;
    
    for (unsigned int k = 0; k < dim; k++) {
        double etap = pow(eta, a[L][k] / 2. / model.Beta0(nf));
        for (unsigned int i = 0; i < dim; i++){
            for (unsigned int j = 0; j < dim; j++) {
                resNNLO(i, j) += 0.;
                
                if(fabs(e(i).real() - e(j).real() + 2. * model.Beta0(nf))>0.000000000001)  {
                    resNLO(i, j) += c[L][i][j][k] * etap * alsmu;
                    resNLO(i, j) += d[L][i][j][k] * etap * alsM;
                }
                else{
                    resNLO(i, j) += - c[L][i][j][k] * etap * alsmu * log(eta);    
                }        
                resLO(i, j) += b[L][i][j][k] * etap;
            }
        }
    }
    switch(order) {
        case NNLO:
            *elem[NNLO] = 0.;
        case NLO:
            *elem[NLO] = (*elem[LO]) * resNLO + (*elem[NLO]) * resLO;
        case LO:
            *elem[LO] = (*elem[LO]) * resLO;
            break;
        case FULLNNLO:
        case FULLNLO:
        default:
            throw std::runtime_error("Error in EvolDC1Buras::DC1EvolBuras()");
    } 
    
  }

DC1EvolBuras() [2/2]

gslpp::matrix< double > & EvolDC1Buras::DC1EvolBuras	(	double	mu,
		double	M,
		orders	order,
		schemes	scheme = NDR
	)

a method returning the evolutor related to the high scale \( M \) and the low scale \( \mu \)

Parameters

mu	a double for the low scale of the evolution
M	a double for the high scale of the evolution
order	order an enum "orders" for the order of perturbation theory of the evolutor
scheme	an enum "schemes" for the regularization scheme of the evolutor

Returns

the evolutor \( U (\mu , M) \)

Definition at line 188 of file EvolDC1Buras.cpp.

{
    switch (scheme) {
        case NDR:
            break;
        case LRI:
        case HV:
        default:
            std::stringstream out;
            out << scheme;
            throw std::runtime_error("EvolDC1::Df1EvolDC1(): scheme " + out.str() + " not implemented "); 
    }
 
    double alsMZ = model.getAlsMz();
    double Mz = model.getMz();
    if(alsMZ == alsMZ_cache && Mz == Mz_cache) {
        if (mu == this-> mu && M == this->M && scheme == this->scheme)
        return (*Evol(order));        
    }
    alsMZ_cache = alsMZ;
    Mz_cache = Mz;
 
    if (M < mu) {
        std::stringstream out;
        out << "M = " << M << " < mu = " << mu;
        throw out.str();
    }
 
    setScales(mu, M); // also assign evol to identity
 
    double m_down = mu;
    double m_up = model.AboveTh(m_down);
    double nf = model.Nf(m_down);
    while (m_up < M) {
        DC1EvolBuras(m_down, m_up, nf, scheme);
        DC1PenguinThresholds(m_up, order);
        m_down = m_up;
        m_up = model.AboveTh(m_down);
        nf += 1.;
    }
    DC1EvolBuras(m_down, M, nf, scheme);
    return (*Evol(order));
    
    }

DC1PenguinThresholds()

void EvolDC1Buras::DC1PenguinThresholds ( double  M, orders  order  )

private

a void type method for the implementation of the NLO threshold effects in the evolutor

Parameters

M	a double for the intermidiate scale of the threshold
order	an enum "orders" for the order of perturbation theory of the threshold (LO is trivial)

Definition at line 297 of file EvolDC1Buras.cpp.

{
 
    double alsM = model.Als(M) / 4. / M_PI;
    gslpp::matrix<double> drsT(dim,0.);
    drsT = alsM * StrongThresholds();
    *elem[NLO] = (*elem[NLO]) + (*elem[LO]) * drsT;  
  }

StrongThresholds()

gslpp::matrix< double > EvolDC1Buras::StrongThresholds

(

)

const

a method returning the matrix threshold for the QCD penguins at the NLO

Returns

matrix threshold for QCD penguin operators

Definition at line 277 of file EvolDC1Buras.cpp.

{
 
// entries of the threshold matrix for the evolution at the NLO
    
gslpp::matrix<double> deltarsT(dim,0.);
 
deltarsT(2,3) = 5./27.;
deltarsT(2,5) = 5./27.;
deltarsT(3,3) = -5./9.;
deltarsT(3,5) = -5./9.;
deltarsT(4,3) = 5./27.;
deltarsT(4,5) = 5./27.;
deltarsT(5,3) = -5./9.;
deltarsT(5,5) = -5./9.;
 
return(deltarsT);
 
}

Member Data Documentation

a

double EvolDC1Buras::a[3][10]

private

Parameters

a	array of double for the magic numbers of the evolutor ( LO evolution )
b	array of double for the magic numbers of the evolutor ( LO evolution )
c	array of double for the magic numbers of the evolutor ( NLO evolution, associated to \( \alpha_{strong}(\mu) \) )
d	array of double for the magic numbers of the evolutor ( NLO evolution, associated to \( \alpha_{strong}(M) \) )

Definition at line 74 of file EvolDC1Buras.h.

alsMZ_cache

double EvolDC1Buras::alsMZ_cache

private

Definition at line 93 of file EvolDC1Buras.h.

b

double EvolDC1Buras::b[3][10][10][10]

private

Definition at line 74 of file EvolDC1Buras.h.

c

double EvolDC1Buras::c[3][10][10][10]

private

Definition at line 74 of file EvolDC1Buras.h.

d

double EvolDC1Buras::d[3][10][10][10]

private

Definition at line 74 of file EvolDC1Buras.h.

dim

unsigned int EvolDC1Buras::dim

private

Definition at line 92 of file EvolDC1Buras.h.

e

gslpp::vector<gslpp::complex> EvolDC1Buras::e

private

Definition at line 91 of file EvolDC1Buras.h.

gg

gslpp::matrix<gslpp::complex> EvolDC1Buras::gg

private

Definition at line 90 of file EvolDC1Buras.h.

h

gslpp::matrix<gslpp::complex> EvolDC1Buras::h

private

Definition at line 90 of file EvolDC1Buras.h.

js

gslpp::matrix<gslpp::complex> EvolDC1Buras::js

private

Definition at line 90 of file EvolDC1Buras.h.

jss

gslpp::matrix<gslpp::complex> EvolDC1Buras::jss

private

Definition at line 90 of file EvolDC1Buras.h.

jssv

gslpp::matrix<gslpp::complex> EvolDC1Buras::jssv

private

Definition at line 90 of file EvolDC1Buras.h.

jv

gslpp::matrix<gslpp::complex> EvolDC1Buras::jv

private

Definition at line 90 of file EvolDC1Buras.h.

model

const StandardModel& EvolDC1Buras::model

private

Definition at line 75 of file EvolDC1Buras.h.

Mz_cache

double EvolDC1Buras::Mz_cache

private

Definition at line 94 of file EvolDC1Buras.h.

nd

int EvolDC1Buras::nd

private

Definition at line 67 of file EvolDC1Buras.h.

nu

int EvolDC1Buras::nu

private

Parameters

nu	an unsigned integer for the up-type number of d.o.f.
nu	an unsigned integer for the down-type number of d.o.f.

Definition at line 67 of file EvolDC1Buras.h.

s_s

gslpp::matrix<gslpp::complex> EvolDC1Buras::s_s

private

Definition at line 90 of file EvolDC1Buras.h.

v

gslpp::matrix<gslpp::complex> EvolDC1Buras::v

private

Definition at line 90 of file EvolDC1Buras.h.

vi

gslpp::matrix<gslpp::complex> EvolDC1Buras::vi

private

Definition at line 90 of file EvolDC1Buras.h.

vij

gslpp::matrix<gslpp::complex> EvolDC1Buras::vij

private

Definition at line 90 of file EvolDC1Buras.h.

---

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

• EvolDC1Buras.h
• EvolDB1bsg.h
• EvolDB1Mll.h
• EvolDF1.h
• EvolDC1Buras.cpp