def transversity_amps_qcdf(q2, wc, par, B, V, **kwargs):
"""QCD factorization corrections to B->Vll transversity amplitudes."""
mB = par['m_' + B]
mV = par['m_' + V]
scale = config['renormalization scale']['bvll']
# using the b quark pole mass here!
mb = running.get_mb_pole(par)
N = flavio.physics.bdecays.bvll.amplitudes.prefactor(q2, par, B, V) / 4
ta = {}
T_perp_ = T_perp(q2, par, wc, B, V, scale, **kwargs)
if q2 == 0.:
T_para_ = 0.
ta['perp_L'] = N * sqrt(2) * 2 * (mB**2 - q2) * mb * T_perp_
else:
T_para_ = T_para(q2, par, wc, B, V, scale, **kwargs)
ta['perp_L'] = N * sqrt(2) * 2 * (mB**2 - q2) * mb / q2 * T_perp_
ta['perp_R'] = ta['perp_L']
ta['para_L'] = -ta['perp_L']
ta['para_R'] = ta['para_L']
if q2 == 0.:
ta['0_L'] = 0.
ta['0_R'] = 0.
else:
ta['0_L'] = (N * mb *
(mB**2 - q2)**2) / (mB**2 * mV * sqrt(q2)) * T_para_
ta['0_R'] = ta['0_L']
ta['t'] = 0
ta['S'] = 0
return ta
def lB_minus(q2, par, B):
flavio.citations.register("Beneke:2001at")
mB = par['m_' + B]
mb = running.get_mb_pole(par)
LambdaBar = mB - mb
w0 = 2 * LambdaBar / 3
return 1 / (exp(-q2 / mB / w0) / w0 * (-ei(q2 / mB / w0) + 1j * pi))
def get_input(par, B, V, scale):
mB = par['m_' + B]
mb = running.get_mb_pole(par)
mc = running.get_mc_pole(par)
alpha_s = running.get_alpha(par, scale)['alpha_s']
q = meson_spectator[(B, V)] # spectator quark flavour
qiqj = meson_quark[(B, V)]
eq = quark_charge[q] # charge of the spectator quark
ed = -1 / 3.
eu = 2 / 3.
xi_t = ckm.xi('t', qiqj)(par)
xi_u = ckm.xi('u', qiqj)(par)
eps_u = xi_u / xi_t
return mB, mb, mc, alpha_s, q, eq, ed, eu, eps_u, qiqj
def get_input(par, B, V, scale):
mB = par['m_'+B]
mb = running.get_mb_pole(par)
mc = running.get_mc_pole(par)
alpha_s = running.get_alpha(par, scale)['alpha_s']
q = meson_spectator[(B,V)] # spectator quark flavour
qiqj = meson_quark[(B,V)]
eq = quark_charge[q] # charge of the spectator quark
ed = -1/3.
eu = 2/3.
xi_t = ckm.xi('t', qiqj)(par)
xi_u = ckm.xi('u', qiqj)(par)
eps_u = xi_u/xi_t
return mB, mb, mc, alpha_s, q, eq, ed, eu, eps_u, qiqj
def get_hqet_parameters(par, scale):
p = {}
alphas = running.get_alpha(par, scale, nf_out=5)['alpha_s']
p['ash'] = alphas / pi
p['mb'] = running.get_mb_pole(par)
p['mc'] = p['mb'] - 3.4
p['mb1S'] = running.get_mb_1S(par)
mBbar = (par['m_B0'] + 3 * par['m_B*0']) / 4
# eq. (25); note the comment about the renormalon cancellation thereafter
p['Lambdabar'] = mBbar - p['mb1S'] + par['lambda_1'] / (2 * p['mb1S'])
p['epsc'] = p['Lambdabar'] / (2 * p['mc'])
p['epsb'] = p['Lambdabar'] / (2 * p['mb'])
p['zc'] = p['mc'] / p['mb']
return p
def get_hqet_parameters(par, scale):
p = {}
alphas = running.get_alpha(par, scale, nf_out=5)['alpha_s']
p['ash'] = alphas / pi
p['mb'] = running.get_mb_pole(par)
p['mc'] = p['mb'] - 3.4
p['mb1S'] = running.get_mb_1S(par, scale)
mBbar = (par['m_B0'] + 3 * par['m_B*0']) / 4
# eq. (25); note the comment about the renormalon cancellation thereafter
p['Lambdabar'] = mBbar - p['mb1S'] + par['lambda_1'] / (2 * p['mb1S'])
p['epsc'] = p['Lambdabar'] / (2 * p['mc'])
p['epsb'] = p['Lambdabar'] / (2 * p['mb'])
p['zc'] = p['mc'] / p['mb']
return p
def Y(q2, wc, par, scale, qiqj):
"""Function $Y$ that contains the contributions of the matrix
elements of four-quark operators to the effective Wilson coefficient
$C_9^{\mathrm{eff}}=C_9 + Y(q^2)$.
See e.g. eq. (10) of 0811.1214v5."""
mb = running.get_mb_pole(par)
mc = running.get_mc_pole(par)
F_c = 4/3.*wc['C1_'+qiqj] + wc['C2_'+qiqj] + 6*wc['C3_'+qiqj] + 60*wc['C5_'+qiqj]
F_b = 7*wc['C3_'+qiqj] + 4/3.*wc['C4_'+qiqj] + 76*wc['C5_'+qiqj] + 64/3.*wc['C6_'+qiqj]
F_u = wc['C3_'+qiqj] + 4/3.*wc['C4_'+qiqj] + 16*wc['C5_'+qiqj] + 64/3.*wc['C6_'+qiqj]
F_4 = 4/3.*wc['C3_'+qiqj] + 64/9.*wc['C5_'+qiqj] + 64/27.*wc['C6_'+qiqj]
return ( h(s=q2, mq=mc, mu=scale) * F_c
- 1/2. * h(s=q2, mq=mb, mu=scale) * F_b
- 1/2. * h(s=q2, mq=0., mu=scale) * F_u
+ F_4 )
def delta_C7(par, wc, q2, scale, qiqj):
alpha_s = running.get_alpha(par, scale)['alpha_s']
mb = running.get_mb_pole(par)
mc = par['m_c BVgamma']
xi_t = ckm.xi('t', qiqj)(par)
xi_u = ckm.xi('u', qiqj)(par)
muh = scale/mb
sh = q2/mb**2
z = mc**2/mb**2
Lmu = log(scale/mb)
# computing this once to save time
delta_tmp = wc['C1_'+qiqj] * F_17(muh, z, sh) + wc['C2_'+qiqj] * F_27(muh, z, sh)
delta_t = wc['C8eff_'+qiqj] * F_87(Lmu, sh) + delta_tmp
delta_u = delta_tmp + wc['C1_'+qiqj] * Fu_17(q2, mb, scale) + wc['C2_'+qiqj] * Fu_27(q2, mb, scale)
# note the minus sign between delta_t and delta_u. This is because of a sign
# switch in the definition of the "Fu" functions between hep-ph/0403185
# (used here) and hep-ph/0412400, see footnote 5 of 0811.1214.
return -alpha_s/(4*pi) * (delta_t - xi_u/xi_t * delta_u)
def transversity_amps_qcdf(q2, wc, par, B, V, **kwargs):
"""QCD factorization corrections to B->Vll transversity amplitudes."""
mB = par['m_'+B]
mV = par['m_'+V]
scale = config['renormalization scale']['bvll']
# using the b quark pole mass here!
mb = running.get_mb_pole(par)
N = flavio.physics.bdecays.bvll.amplitudes.prefactor(q2, par, B, V)/4
T_perp_ = T_perp(q2, par, wc, B, V, scale, **kwargs)
T_para_ = T_para(q2, par, wc, B, V, scale, **kwargs)
ta = {}
ta['perp_L'] = N * sqrt(2)*2 * (mB**2-q2) * mb / q2 * T_perp_
ta['perp_R'] = ta['perp_L']
ta['para_L'] = -ta['perp_L']
ta['para_R'] = ta['para_L']
ta['0_L'] = ( N * mb * (mB**2 - q2)**2 )/(mB**2 * mV * sqrt(q2)) * T_para_
ta['0_R'] = ta['0_L']
ta['t'] = 0
ta['S'] = 0
return ta
def lB_plus(par, B):
mB = par['m_' + B]
mb = running.get_mb_pole(par)
LambdaBar = mB - mb
return 2 * LambdaBar / 3
def lB_minus(q2, par, B):
mB = par['m_' + B]
mb = running.get_mb_pole(par)
LambdaBar = mB - mb
w0 = 2 * LambdaBar / 3
return 1 / (exp(-q2 / mB / w0) / w0 * (-ei(q2 / mB / w0) + 1j * pi))
def lB_plus(par, B):
mB = par['m_'+B]
mb = running.get_mb_pole(par)
LambdaBar = mB - mb
return 2*LambdaBar/3
def lB_minus(q2, par, B):
mB = par['m_'+B]
mb = running.get_mb_pole(par)
LambdaBar = mB - mb
w0 = 2*LambdaBar/3
return 1/(exp(-q2/mB/w0)/w0 * (-ei(q2/mB/w0) + 1j*pi))
def lB_plus(par, B):
flavio.citations.register("Beneke:2001at")
mB = par['m_' + B]
mb = running.get_mb_pole(par)
LambdaBar = mB - mb
return 2 * LambdaBar / 3
