def g1_f1(wc_obj, par, lep, nu):
scale = flavio.config['renormalization scale']['kdecays']
ms = flavio.physics.running.running.get_ms(par, scale)
wc = get_wceff_fccc_std(wc_obj, par, 'su', lep, nu, ms, scale, nf=3)
f1 = par['Lambda->p f_1(0)'] * (wc['VL'] + wc['VR']).real
g1 = par['Lambda->p g_1(0)'] * (wc['VL'] - wc['VR']).real
return g1 / f1
def wc_eff(par, wc_obj, scale, nu):
r"""Lee-Yang effective couplings.
See eqS. (2), (9) of arXiv:1803.08732."""
# wilson coefficients
wc = get_wceff_fccc_std(wc_obj, par, 'du', 'e', nu, None, scale, nf=3)
# proton charges
g = proton_charges(par, scale)
gV = g['gV_u-d']
gA = g['gA_u-d']
gS = g['gS_u-d']
gP = g['gP_u-d']
gT = g['gT_u-d']
# radiative corrections
# Note: CVLSM is the universal Marciano-Sirlin result that needs to be
# divided out since it's already contained in the Deltas
CVLSM = get_CVLSM(par, scale, nf=3)
DeltaRV = par['DeltaRV']
DeltaRA = DeltaRV # not needed for superallowed, for neutron difference absorbed in lambda
rV = sqrt(1 + DeltaRV) / CVLSM
rA = sqrt(1 + DeltaRA) / CVLSM
# effective couplings
# note that C_i' = C_i
C = {}
C['V'] = gV * (wc['VL'] * rV + wc['VR'])
C['A'] = -gA * (wc['VL'] * rA - wc['VR'])
C['S'] = gS * (wc['SL'] + wc['SR'])
C['P'] = gP * (wc['SL'] - wc['SR'])
C['T'] = 4 * gT * (wc['T'])
return C
def _BR_BXclnu(par, wc_obj, lep, nu):
GF = par['GF']
scale = flavio.config['renormalization scale']['bxlnu']
mb_MSbar = flavio.physics.running.running.get_mb(par, scale)
wc = get_wceff_fccc_std(wc_obj, par, 'bc', lep, nu, mb_MSbar, scale, nf=5)
if lep != nu and all(C == 0 for C in wc.values()):
return 0 # if all WCs vanish, so does the BR!
kinetic_cutoff = 1. # cutoff related to the kinetic definition of mb in GeV
# mb in the kinetic scheme
mb = flavio.physics.running.running.get_mb_KS(par, kinetic_cutoff)
xl = par['m_' + lep]**2 / mb**2
# mc in MSbar at 3 GeV
mc = flavio.physics.running.running.get_mc(par, 3)
xc = mc**2 / mb**2
Vcb = flavio.physics.ckm.get_ckm(par)[1, 2]
alpha_s = flavio.physics.running.running.get_alpha(par, scale,
nf_out=5)['alpha_s']
# wc: NB this includes the EW correction already
# the b quark mass is MSbar here as it comes from the definition
# of the scalar operators
Gamma_LO = GF**2 * mb**5 / 192. / pi**3 * abs(Vcb)**2
r_WC = (
g(xc, xl) * (abs(wc['VL'])**2 + abs(wc['VR'])**2) - gLR(xc, xl) *
(wc['VL'] * wc['VR']).real + g(xc, xl) / 4. *
(abs(wc['SR'])**2 + abs(wc['SL'])**2) + gLR(xc, xl) / 2. *
(wc['SR'] * wc['SL']).real + 12 * g(xc, xl) * abs(wc['T'])**2
# the following terms vanish for vanishing lepton mass
+ gVS(xc, xl) * ((wc['VL'] * wc['SR']).real +
(wc['VR'] * wc['SL']).real) + gVSp(xc, xl) *
((wc['VL'] * wc['SL']).real + (wc['VR'] * wc['SR']).real) -
12 * gVSp(xc, xl) * (wc['VL'] * wc['T']).real + 12 * gVS(xc, xl) *
(wc['VR'] * wc['T']).real)
# eq. (26) of arXiv:1107.3100 + corrections (P. Gambino, private communication)
r_BLO = (
1
# NLO QCD
+ alpha_s / pi * pc1(xc, mb)
# NNLO QCD
+ alpha_s**2 / pi**2 * pc2(xc, mb)
# power correction
- par['mu_pi^2'] / (2 * mb**2) + (1 / 2. - 2 *
(1 - xc)**4 / g(xc, 0)) *
(par['mu_G^2'] - (par['rho_LS^3'] + par['rho_D^3']) / mb) / mb**2 +
d(xc) / g(xc, 0) * par['rho_D^3'] / mb**3
# O(alpha_s) power correction (only numerically)
+ alpha_s / pi * par['mu_pi^2'] * 0.071943 +
alpha_s / pi * par['mu_G^2'] * (-0.114774))
# average of B0 and B+ lifetimes
r_rem = (1 + par['delta_BXlnu']) # residual pert & non-pert uncertainty
return (par['tau_B0'] +
par['tau_B+']) / 2. * Gamma_LO * r_WC * r_BLO * r_rem
def _BR_BXclnu(par, wc_obj, lep, nu):
GF = par['GF']
scale = flavio.config['renormalization scale']['bxlnu']
mb_MSbar = flavio.physics.running.running.get_mb(par, scale)
wc = get_wceff_fccc_std(wc_obj, par, 'bc', lep, nu, mb_MSbar, scale, nf=5)
if lep != nu and all(C == 0 for C in wc.values()):
return 0 # if all WCs vanish, so does the BR!
kinetic_cutoff = 1. # cutoff related to the kinetic definition of mb in GeV
# mb in the kinetic scheme
mb = flavio.physics.running.running.get_mb_KS(par, kinetic_cutoff)
xl = par['m_'+lep]**2/mb**2
# mc in MSbar at 3 GeV
mc = flavio.physics.running.running.get_mc(par, 3)
xc = mc**2/mb**2
Vcb = flavio.physics.ckm.get_ckm(par)[1, 2]
alpha_s = flavio.physics.running.running.get_alpha(par, scale, nf_out=5)['alpha_s']
# wc: NB this includes the EW correction already
# the b quark mass is MSbar here as it comes from the definition
# of the scalar operators
Gamma_LO = GF**2 * mb**5 / 192. / pi**3 * abs(Vcb)**2
r_WC = ( g(xc, xl) * (abs(wc['VL'])**2 + abs(wc['VR'])**2)
- gLR(xc, xl) * (wc['VL']*wc['VR']).real
+ g(xc, xl)/4. * (abs(wc['SR'])**2 + abs(wc['SL'])**2)
+ gLR(xc, xl)/2. * (wc['SR']*wc['SL']).real
+ 12*g(xc, xl) * abs(wc['T'])**2
# the following terms vanish for vanishing lepton mass
+ gVS(xc, xl) * ((wc['VL']*wc['SR']).real
+ (wc['VR']*wc['SL']).real)
+ gVSp(xc, xl) * ((wc['VL']*wc['SL']).real
+ (wc['VR']*wc['SR']).real)
- 12*gVSp(xc, xl)* (wc['VL']*wc['T']).real
+ 12*gVS(xc, xl) * (wc['VR']*wc['T']).real
)
# eq. (26) of arXiv:1107.3100 + corrections (P. Gambino, private communication)
r_BLO = ( 1
# NLO QCD
+ alpha_s/pi * pc1(xc, mb)
# NNLO QCD
+ alpha_s**2/pi**2 * pc2(xc, mb)
# power correction
- par['mu_pi^2']/(2*mb**2)
+ (1/2. - 2*(1-xc)**4/g(xc, 0))*(par['mu_G^2'] - (par['rho_LS^3'] + par['rho_D^3'])/mb)/mb**2
+ d(xc)/g(xc, 0) * par['rho_D^3']/mb**3
# O(alpha_s) power correction (only numerically)
+ alpha_s/pi * par['mu_pi^2'] * 0.071943
+ alpha_s/pi * par['mu_G^2'] * (-0.114774)
)
# average of B0 and B+ lifetimes
r_rem = (1 + par['delta_BXlnu']) # residual pert & non-pert uncertainty
return (par['tau_B0']+par['tau_B+'])/2. * Gamma_LO * r_WC * r_BLO * r_rem
def BR_BXclnu(par, wc_obj, lep):
r"""Total branching ratio of $\bar B^0\to X_c \ell^- \bar\nu_\ell$"""
GF = par['GF']
scale = flavio.config['renormalization scale']['bxlnu']
kinetic_cutoff = 1. # cutoff related to the kinetic definition of mb in GeV
# mb in the kinetic scheme
mb = flavio.physics.running.running.get_mb_KS(par, kinetic_cutoff)
# mc in MSbar at 3 GeV
mc = flavio.physics.running.running.get_mc(par, 3)
mb_MSbar = flavio.physics.running.running.get_mb(par, scale)
rho = mc**2 / mb**2
Vcb = flavio.physics.ckm.get_ckm(par)[1, 2]
alpha_s = flavio.physics.running.running.get_alpha(par, scale,
nf_out=5)['alpha_s']
# wc: NB this includes the EW correction already
# the b quark mass is MSbar here as it comes from the definition
# of the scalar operators
wc = get_wceff_fccc_std(wc_obj, par, 'bc', lep, mb_MSbar, scale, nf=5)
Gamma_LO = GF**2 * mb**5 / 192. / pi**3 * abs(Vcb)**2 * g(rho)
r_WC = (abs(wc['V'])**2 + abs(wc['Vp'])**2 - gLR(rho) / g(rho) *
(wc['V'] * wc['Vp']).real + mb_MSbar**2 / 4. *
(abs(wc['S'])**2 + abs(wc['Sp'])**2) +
mb_MSbar**2 / 2. * gLR(rho) / g(rho) * (wc['S'] * wc['Sp']).real +
12 * abs(wc['T'])**2)
# eq. (26) of arXiv:1107.3100 + corrections (P. Gambino, private communication)
r_BLO = (
1
# NLO QCD
+ alpha_s / pi * pc1(rho, mb)
# NNLO QCD
+ alpha_s**2 / pi**2 * pc2(rho, mb)
# power correction
- par['mu_pi^2'] / (2 * mb**2) + (1 / 2. - 2 * (1 - rho)**4 / g(rho)) *
(par['mu_G^2'] - (par['rho_LS^3'] + par['rho_D^3']) / mb) / mb**2 +
d(rho) / g(rho) * par['rho_D^3'] / mb**3
# O(alpha_s) power correction (only numerically)
+ alpha_s / pi * par['mu_pi^2'] * 0.071943 +
alpha_s / pi * par['mu_G^2'] * (-0.114774))
# average of B0 and B+ lifetimes
return (par['tau_B0'] + par['tau_B+']) / 2. * Gamma_LO * r_WC * r_BLO