def ff_isgurwise(process, q2, par, scale, n=3, t0=None):
r"""Central value of $B\to P$ form factors in the standard convention
and BCL parametrization (arXiv:0807.2722) for $f_0$ and $f_+$, but using
an improved Isgur-Wise relation in the heavy quark limit for $f_T$.
"""
pd = process_dict[process]
mpl = par[process + ' BCL m+']
m0 = par[process + ' BCL m0']
mB = par['m_'+pd['B']]
mP = par['m_'+pd['P']]
ff = {}
a={}
a['f+'] = [ par[process + ' BCL' + ' a' + str(j) + '_f+'] for j in range(n) ]
# only the first n-1 parameters for f0 are taken from par
# the nth one is chosen to fulfill the kinematic constraint f+(0)=f0(0)
a['f0'] = [ par[process + ' BCL' + ' a' + str(j) + '_f0'] for j in range(n-1) ]
fplus_q20 = pole('f+', mpl, 0) * param_fplusT(mB, mP, a['f+'], 0, t0)
f0_q20 = pole('f0', m0, 0) * param_f0(mB, mP, a['f0'], 0, t0)
an_f0 = (fplus_q20-f0_q20)/z(mB, mP, 0, t0)**(n-1)
a['f0'].append(an_f0)
# evaluate FFs
ff['f+'] = pole('f+', mpl, q2) * param_fplusT(mB, mP, a['f+'], q2, t0)
ff['f0'] = pole('f0', m0, q2) * param_f0(mB, mP, a['f0'], q2, t0)
ff = isgur_wise(process, q2, ff, par, scale=scale)
return ff
def ff(process, q2, par, scale):
r"""Central value of $B\to P$ form factors in the lattice convention
CLN parametrization.
See arXiv:hep-ph/9712417 and arXiv:1703.05330.
"""
pd = process_dict[process]
mB = par['m_' + pd['B']]
mP = par['m_' + pd['P']]
w = max((mB**2 + mP**2 - q2) / (2 * mB * mP), 1)
phqet = hqet.get_hqet_parameters(par, scale)
ash = phqet['ash']
epsc = phqet['epsc']
epsb = phqet['epsb']
zc = phqet['zc']
CV1 = hqet.CV1(w, zc)
CV2 = hqet.CV2(w, zc)
CV3 = hqet.CV3(w, zc)
CT1 = hqet.CT1(w, zc)
CT2 = hqet.CT2(w, zc)
CT3 = hqet.CT3(w, zc)
L = hqet.L(par, w)
# leading, universal Isgur-Wise function
rho2 = par['CLN rho2_xi']
c = par['CLN c_xi']
z = common.z(mB, mP, q2, t0='tm')
xi = 1 - 8 * rho2 * z + (64 * c - 16 * rho2) * z**2
h = {}
h['h+'] = xi * (1 + ash * (CV1 + (w + 1) / 2 * (CV2 + CV3)) +
(epsc + epsb) * L[1] + epsc**2 * par['B->D CLN deltac_h+'])
h['h-'] = xi * (ash * (w + 1) / 2 * (CV2 - CV3) + (epsc - epsb) * L[4])
h['hT'] = xi * (1 + ash * (CT1 - CT2 + CT3) + (epsc + epsb) *
(L[1] - L[4]))
return h_to_f(mB, mP, h, q2)
def ff(process, q2, par, n=3, t0=None):
r"""Central value of $B\to P$ form factors in the standard convention
and BCL parametrization (arXiv:0807.2722).
The standard convention defines the form factors $f_+$, $f_0$, and $f_T$.
"""
pd = process_dict[process]
mpl = par[process + ' BCL m+']
m0 = par[process + ' BCL m0']
mB = par['m_'+pd['B']]
mP = par['m_'+pd['P']]
ff = {}
a={}
for i in ['f+', 'fT']:
a[i] = [ par[process + ' BCL' + ' a' + str(j) + '_' + i] for j in range(n) ]
# only the first n-1 parameters for f0 are taken from par
# the nth one is chosen to fulfill the kinematic constraint f+(0)=f0(0)
a['f0'] = [ par[process + ' BCL' + ' a' + str(j) + '_f0'] for j in range(n-1) ]
fplus_q20 = pole('f+', mpl, 0) * param_fplusT(mB, mP, a['f+'], 0, t0)
f0_q20 = pole('f0', m0, 0) * param_f0(mB, mP, a['f0'], 0, t0)
an_f0 = (f0_q20-fplus_q20)/z(mB, mP, 0, t0)**(n-1)
a['f0'].append(an_f0)
# evaluate FFs
ff['f+'] = pole('f+', mpl, q2) * param_fplusT(mB, mP, a['f+'], q2, t0)
ff['fT'] = pole('fT', mpl, q2) * param_fplusT(mB, mP, a['fT'], q2, t0)
ff['f0'] = pole('f0', m0, q2) * param_f0(mB, mP, a['f0'], q2, t0)
return ff
def ff(process, q2, par, scale):
r"""Central value of $B\to V$ form factors in the lattice convention
CLN parametrization.
See arXiv:1703.05330.
"""
pd = process_dict[process]
mB = par['m_' + pd['B']]
mV = par['m_' + pd['V']]
w = max((mB**2 + mV**2 - q2) / (2 * mB * mV), 1)
phqet = hqet.get_hqet_parameters(par, scale)
ash = phqet['ash']
epsc = phqet['epsc']
epsb = phqet['epsb']
zc = phqet['zc']
# eq. (22) of arXiv:0809.0222
CV1 = hqet.CV1(w, zc)
CV2 = hqet.CV2(w, zc)
CV3 = hqet.CV3(w, zc)
CA1 = hqet.CA1(w, zc)
CA2 = hqet.CA2(w, zc)
CA3 = hqet.CA3(w, zc)
CT1 = hqet.CT1(w, zc)
CT2 = hqet.CT2(w, zc)
CT3 = hqet.CT3(w, zc)
L = hqet.L(par, w)
# leading, universal Isgur-Wise function
rho2 = par['CLN rho2_xi']
c = par['CLN c_xi']
z = common.z(mB, mV, q2, t0='tm')
xi = 1 - 8 * rho2 * z + (64 * c - 16 * rho2) * z**2
h = {}
h['V'] = xi * (1 + ash * CV1
+ epsc * (L[2] - L[5])
+ epsb * (L[1] - L[4]))
h['A1'] = xi * (1 + ash * CA1
+ epsc * (L[2] - L[5] * (w - 1)/(w + 1))
+ epsb * (L[1] - L[4] * (w - 1)/(w + 1))
+ epsc**2 * par['B->D* CLN deltac_hA1'])
h['A2'] = xi * (ash * CA2 + epsc * (L[3] + L[6]))
h['A3'] = xi * (1 + ash * (CA1 + CA3)
+ epsc * (L[2] - L[3] + L[6] - L[5])
+ epsb * (L[1] - L[4]))
h['T1'] = xi * (1 + ash * (CT1 + (w - 1)/2 * (CT2 - CT3))
+ epsc * L[2]
+ epsb * L[1]
+ epsc**2 * par['B->D* CLN deltac_hT1'])
h['T2'] = xi * (ash * (w + 1)/2 * (CT2 + CT3)
+ epsc * L[5]
- epsb * L[4])
h['T3'] = xi * (ash * CT2
+ epsc * (L[6] - L[3]))
return h_to_A(mB, mV, h, q2)
def ff(process, q2, par, scale, order_z=3, order_z_slp=2, order_z_sslp=1):
r"""Central value of $B\to V$ form factors in the lattice convention
CLN parametrization.
See arXiv:1703.05330.
"""
pd = process_dict[process]
mB = par['m_' + pd['B']]
mV = par['m_' + pd['V']]
w = max((mB**2 + mV**2 - q2) / (2 * mB * mV), 1)
phqet = hqet.get_hqet_parameters(par)
ash = phqet['ash']
epsc = phqet['epsc']
epsb = phqet['epsb']
zc = phqet['zc']
# eq. (22) of arXiv:0809.0222
CV1 = hqet.CV1(w, zc)
CV2 = hqet.CV2(w, zc)
CV3 = hqet.CV3(w, zc)
CA1 = hqet.CA1(w, zc)
CA2 = hqet.CA2(w, zc)
CA3 = hqet.CA3(w, zc)
CT1 = hqet.CT1(w, zc)
CT2 = hqet.CT2(w, zc)
CT3 = hqet.CT3(w, zc)
z = common.z(mB, mV, q2, t0='tm')
rho2 = par['CLN rho2_xi']
c = par['CLN c_xi']
xi3 = par['CLN xi3']
xi = hqet.xi(z, rho2, c, xi3, order_z=order_z)
L = hqet.Lz(par, w, z, order_z=order_z_slp)
ell = hqet.ell(par, z, order_z=order_z_sslp)
h = {}
h['V'] = xi * (1 + ash * CV1 + epsc * (L[2] - L[5]) + epsb *
(L[1] - L[4]) + epsc**2 * (ell[2] - ell[5]))
h['A1'] = xi * (1 + ash * CA1 + epsc * (L[2] - L[5] * (w - 1) /
(w + 1)) + epsb *
(L[1] - L[4] * (w - 1) / (w + 1)) + epsc**2 *
(ell[2] - ell[5] * (w - 1) / (w + 1)))
h['A2'] = xi * (ash * CA2 + epsc * (L[3] + L[6]))
h['A3'] = xi * (1 + ash * (CA1 + CA3) + epsc *
(L[2] - L[3] + L[6] - L[5]) + epsb *
(L[1] - L[4]) + epsc**2 *
(ell[2] - ell[3] + ell[6] - ell[5]))
h['T1'] = xi * (1 + ash * (CT1 + (w - 1) / 2 * (CT2 - CT3)) + epsc * L[2] +
epsb * L[1] + epsc**2 * ell[2])
h['T2'] = xi * (ash * (w + 1) / 2 *
(CT2 + CT3) + epsc * L[5] - epsb * L[4] + epsc * ell[5])
h['T3'] = xi * (ash * CT2 + epsc * (L[6] - L[3]) + epsc *
(ell[6] - ell[3]))
return h_to_A(mB, mV, h, q2)
def ff(process, q2, par, scale):
r"""Central value of $B\to V$ form factors in the lattice convention
CLN parametrization.
See arXiv:1703.05330.
"""
pd = process_dict[process]
mB = par['m_' + pd['B']]
mV = par['m_' + pd['V']]
w = max((mB**2 + mV**2 - q2) / (2 * mB * mV), 1)
phqet = hqet.get_hqet_parameters(par, scale)
ash = phqet['ash']
epsc = phqet['epsc']
epsb = phqet['epsb']
zc = phqet['zc']
# eq. (22) of arXiv:0809.0222
CV1 = hqet.CV1(w, zc)
CV2 = hqet.CV2(w, zc)
CV3 = hqet.CV3(w, zc)
CA1 = hqet.CA1(w, zc)
CA2 = hqet.CA2(w, zc)
CA3 = hqet.CA3(w, zc)
CT1 = hqet.CT1(w, zc)
CT2 = hqet.CT2(w, zc)
CT3 = hqet.CT3(w, zc)
L = hqet.L(par, w)
# leading, universal Isgur-Wise function
rho2 = par['CLN rho2_xi']
c = par['CLN c_xi']
z = common.z(mB, mV, q2, t0='tm')
xi = 1 - 8 * rho2 * z + (64 * c - 16 * rho2) * z**2
h = {}
h['V'] = xi * (1 + ash * CV1 + epsc * (L[2] - L[5]) + epsb * (L[1] - L[4]))
h['A1'] = xi * (1 + ash * CA1 + epsc * (L[2] - L[5] * (w - 1) /
(w + 1)) + epsb *
(L[1] - L[4] * (w - 1) /
(w + 1)) + epsc**2 * par['B->D* CLN deltac_hA1'])
h['A2'] = xi * (ash * CA2 + epsc * (L[3] + L[6]))
h['A3'] = xi * (1 + ash * (CA1 + CA3) + epsc *
(L[2] - L[3] + L[6] - L[5]) + epsb * (L[1] - L[4]))
h['T1'] = xi * (1 + ash * (CT1 + (w - 1) / 2 * (CT2 - CT3)) + epsc * L[2] +
epsb * L[1] + epsc**2 * par['B->D* CLN deltac_hT1'])
h['T2'] = xi * (ash * (w + 1) / 2 *
(CT2 + CT3) + epsc * L[5] - epsb * L[4])
h['T3'] = xi * (ash * CT2 + epsc * (L[6] - L[3]))
return h_to_A(mB, mV, h, q2)
def ff(process, q2, par, scale, order_z=3, order_z_slp=2, order_z_sslp=1):
r"""Central value of $B\to P$ form factors in the lattice convention
CLN parametrization.
See arXiv:hep-ph/9712417 and arXiv:1703.05330.
"""
pd = process_dict[process]
mB = par['m_' + pd['B']]
mP = par['m_' + pd['P']]
w = max((mB**2 + mP**2 - q2) / (2 * mB * mP), 1)
phqet = hqet.get_hqet_parameters(par)
ash = phqet['ash']
epsc = phqet['epsc']
epsb = phqet['epsb']
zc = phqet['zc']
CV1 = hqet.CV1(w, zc)
CV2 = hqet.CV2(w, zc)
CV3 = hqet.CV3(w, zc)
CT1 = hqet.CT1(w, zc)
CT2 = hqet.CT2(w, zc)
CT3 = hqet.CT3(w, zc)
z = common.z(mB, mP, q2, t0='tm')
rho2 = par['CLN rho2_xi']
c = par['CLN c_xi']
xi3 = par['CLN xi3']
xi = hqet.xi(z, rho2, c, xi3, order_z=order_z)
L = hqet.Lz(par, w, z, order_z=order_z_slp)
ell = hqet.ell(par, z, order_z=order_z_sslp)
h = {}
h['h+'] = xi * (1 + ash * (CV1 + (w + 1) / 2 * (CV2 + CV3))
+ (epsc + epsb) * L[1]
+ epsc**2 * ell[1])
h['h-'] = xi * (ash * (w + 1) / 2 * (CV2 - CV3)
+ (epsc - epsb) * L[4]
+ epsc**2 * ell[4])
h['hT'] = xi * (1 + ash * (CT1 - CT2 + CT3)
+ (epsc + epsb) * (L[1] - L[4])
+ epsc**2 * (ell[1] - ell[4]))
return h_to_f(mB, mP, h, q2)
def zs(mX, mP, par, q2):
mLb = par['m_Lambdab']
mB = par['m_B+']
zq2 = z(mB, mP, q2, t0=(mLb - mX)**2) # cf. eq. (34) of arXiv:1602.01399
return np.array([1, zq2, zq2**2])
def zs(mB, mV, q2, t0):
zq2 = z(mB, mV, q2, t0)
z0 = z(mB, mV, 0, t0)
return np.array([1, zq2-z0, (zq2-z0)**2])
def zs(mB, mV, q2, t0):
zq2 = z(mB, mV, q2, t0)
return np.array([1, zq2, zq2**2])
def param_f0(mB, mP, a_i, q2, t0=None):
Z = z(mB, mP, q2, t0)
k = np.arange(len(a_i))
return ( a_i * Z**k ).sum()
def param_fplusT(mB, mP, a_i, q2, t0=None):
Z = z(mB, mP, q2, t0)
n = len(a_i)
k = np.arange(n)
return ( a_i * (Z**k - (-1.)**(k - n) * k/n * Z**n) ).sum()
def zs(mB, mV, q2, t0):
zq2 = z(mB, mV, q2, t0)
z0 = z(mB, mV, 0, t0)
return np.array([1, zq2 - z0, (zq2 - z0)**2])
def zs(mB, mV, q2, t0):
zq2 = z(mB, mV, q2, t0)
return np.array([1, zq2, zq2 ** 2])
def zs(mX, mP, par, q2):
mLb = par['m_Lambdab']
mB = par['m_B+']
zq2 = z(mB, mP, q2, t0=(mLb - mX)**2) # cf. eq. (34) of arXiv:1602.01399
return np.array([1, zq2, zq2**2])
