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 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_C9(par, wc, q2, scale, qiqj):
alpha_s = running.get_alpha(par, scale)['alpha_s']
mb = running.get_mb_pole(par)
mc = running.get_mc_pole(par)
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)
Ls = log(sh)
# computing this once to save time
delta_tmp = wc['C1_'+qiqj] * F_19(muh, z, sh) + wc['C2_'+qiqj] * F_29(muh, z, sh)
delta_t = wc['C8eff_'+qiqj] * F_89(Ls, sh) + delta_tmp
delta_u = delta_tmp + wc['C1_'+qiqj] * Fu_19(q2, mb, scale) + wc['C2_'+qiqj] * Fu_29(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 T_perp(q2, par, wc, B, V, scale,
include_WA=True, include_O8=True, include_QSS=True):
if not include_WA and not include_O8 and not include_QSS:
raise ValueError("At least one contribution to the QCDF corrections has to be switched on")
mB = par['m_'+B]
mV = par['m_'+V]
mc = running.get_mc_pole(par)
EV = En_V(mB, mV, q2)
fB = par['f_'+B]
fVperp = flavio.physics.running.running.get_f_perp(par, V, scale)
fVpara = par['f_'+V]
N = pi**2 / 3. * fB * fVperp / mB
a1_perp = par['a1_perp_'+V]
a2_perp = par['a2_perp_'+V]
def phiV_perp(u):
return phiV(u, a1_perp, a2_perp)
def T_minus(u):
return 0
def T_plus(u):
T = 0
if include_O8:
T += T_perp_plus_O8(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale)
if include_QSS:
T += T_perp_plus_QSS(q2, par, wc, B, V, u, scale)
return N / lB_plus(par=par, B=B) * phiV_perp(u) * T
def T_powercorr_1(u):
T=0
if include_WA:
T += T_perp_WA_PowC_1(q2, par, wc, B, V, scale)
ubar = 1 - u
# cf. (51) of hep-ph/0412400
return N * phiV_perp(u)/(ubar+u*q2/mB**2) * T
u_sing = (mB**2 - 4*mc**2)/(-q2 + mB**2)
if u_sing < 1:
points = [u_sing]
else:
points = None
T_tot = flavio.math.integrate.nintegrate_complex( lambda u: T_plus(u) + T_minus(u) + T_powercorr_1(u), 0, 1, points=points)
if include_WA:
# cf. (51) of hep-ph/0412400
T_tot += N / lB_plus(par=par, B=B) * fVpara/fVperp * mV/(1-q2/mB**2) * T_perp_WA_PowC_2(q2, par, wc, B, V, scale)
return T_tot
def T_perp(q2, par, wc, B, V, scale,
include_WA=True, include_O8=True, include_QSS=True):
if not include_WA and not include_O8 and not include_QSS:
raise ValueError("At least one contribution to the QCDF corrections has to be switched on")
mB = par['m_'+B]
mV = par['m_'+V]
mc = running.get_mc_pole(par)
EV = En_V(mB, mV, q2)
fB = par['f_'+B]
fVperp = par['f_perp_'+V]
fVpara = par['f_'+V]
N = pi**2 / 3. * fB * fVperp / mB
a1_perp = par['a1_perp_'+V]
a2_perp = par['a2_perp_'+V]
def phiV_perp(u):
return phiV(u, a1_perp, a2_perp)
def T_minus(u):
return 0
def T_plus(u):
T = 0
if include_O8:
T += T_perp_plus_O8(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale)
if include_QSS:
T += T_perp_plus_QSS(q2, par, wc, B, V, u, scale)
return N / lB_plus(par=par, B=B) * phiV_perp(u) * T
def T_powercorr_1(u):
T=0
if include_WA:
T += T_perp_WA_PowC_1(q2, par, wc, B, V, scale)
ubar = 1 - u
# cf. (51) of hep-ph/0412400
return N * phiV_perp(u)/(ubar+u*q2/mB**2) * T
u_sing = (mB**2 - 4*mc**2)/(-q2 + mB**2)
if u_sing < 1:
points = [u_sing]
else:
points = None
T_tot = flavio.math.integrate.nintegrate_complex( lambda u: T_plus(u) + T_minus(u) + T_powercorr_1(u), 0, 1, points=points)
if include_WA:
# cf. (51) of hep-ph/0412400
T_tot += N / lB_plus(par=par, B=B) * fVpara/fVperp * mV/(1-q2/mB**2) * T_perp_WA_PowC_2(q2, par, wc, B, V, scale)
return T_tot
def T_para(q2, par, wc, B, V, scale,
include_WA=True, include_O8=True, include_QSS=True):
if not include_WA and not include_O8 and not include_QSS:
raise ValueError("At least one contribution to the QCDF corrections has to be switched on")
mB = par['m_'+B]
mV = par['m_'+V]
mc = running.get_mc_pole(par)
fB = par['f_'+B]
fVpara = par['f_' + V]
EV = En_V(mB, mV, q2)
N = pi**2 / 3. * fB * fVpara / mB * (mV/EV)
a1_para = par['a1_para_'+V]
a2_para = par['a2_para_'+V]
def phiV_para(u):
return phiV(u, a1_para, a2_para)
def T_minus(u):
T = 0
if include_WA:
T += T_para_minus_WA(q2=q2, par=par, wc=wc, B=B, V=V, scale=scale)
if include_O8:
T += T_para_minus_O8(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale)
if include_QSS:
T += T_para_minus_QSS(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale)
return N / lB_minus(q2=q2, par=par, B=B) * phiV_para(u) * T
def T_plus(u):
if include_QSS:
return N / lB_plus(par=par, B=B) * phiV_para(u) * (
T_para_plus_QSS(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale))
else:
return 0
u_sing = (mB**2 - 4*mc**2)/(-q2 + mB**2)
if u_sing < 1:
points = [u_sing]
else:
points = None
T_tot = flavio.math.integrate.nintegrate_complex( lambda u: T_plus(u) + T_minus(u), 0, 1, points=points)
return T_tot
def T_para(q2, par, wc, B, V, scale,
include_WA=True, include_O8=True, include_QSS=True):
if not include_WA and not include_O8 and not include_QSS:
raise ValueError("At least one contribution to the QCDF corrections has to be switched on")
mB = par['m_'+B]
mV = par['m_'+V]
mc = running.get_mc_pole(par)
fB = par['f_'+B]
fVpara = par['f_' + V]
EV = En_V(mB, mV, q2)
N = pi**2 / 3. * fB * fVpara / mB * (mV/EV)
a1_para = par['a1_para_'+V]
a2_para = par['a2_para_'+V]
def phiV_para(u):
return phiV(u, a1_para, a2_para)
def T_minus(u):
T = 0
if include_WA:
T += T_para_minus_WA(q2=q2, par=par, wc=wc, B=B, V=V, scale=scale)
if include_O8:
T += T_para_minus_O8(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale)
if include_QSS:
T += T_para_minus_QSS(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale)
return N / lB_minus(q2=q2, par=par, B=B) * phiV_para(u) * T
def T_plus(u):
if include_QSS:
return N / lB_plus(par=par, B=B) * phiV_para(u) * (
T_para_plus_QSS(q2=q2, par=par, wc=wc, B=B, V=V, u=u, scale=scale))
else:
return 0
u_sing = (mB**2 - 4*mc**2)/(-q2 + mB**2)
if u_sing < 1:
points = [u_sing]
else:
points = None
T_tot = flavio.math.integrate.nintegrate_complex( lambda u: T_plus(u) + T_minus(u), 0, 1, points=points)
return T_tot
def Yu(q2, wc, par, scale, qiqj):
mc = running.get_mc_pole(par)
return ( (4/3.*wc['C1_'+qiqj] + wc['C2_'+qiqj])
* ( h(s=q2, mq=mc, mu=scale) - h(s=q2, mq=0, mu=scale) ))
def Yu(q2, wc, par, scale, qiqj):
flavio.citations.register("Beneke:2004dp")
mc = running.get_mc_pole(par)
return ((4 / 3. * wc['C1_' + qiqj] + wc['C2_' + qiqj]) *
(h(s=q2, mq=mc, mu=scale) - h(s=q2, mq=0, mu=scale)))
