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 CS(wc, par, scale, Z, N):
"""Scalar electron-nucleon coupling."""
me = proton_charges(par, scale)
CSu = -(wc['CSRR_eeuu'].imag + wc['CSRL_eeuu'].imag)
CSd = -(wc['CSRR_eedd'].imag + wc['CSRL_eedd'].imag)
CS0 = me['gS_u+d'] * (CSu + CSd)
CS1 = me['gS_u-d'] * (CSu - CSd)
return CS0 + (Z - N) / (Z + N) * CS1
def CR_mue(wc_obj, par, nucl):
r"""Conversion rate independent of the target nucleus"""
mm = par['m_mu']
scale = flavio.config['renormalization scale']['mudecays']
#####overlap integrals and other parameters#####
pc = proton_charges(par, scale)
GuSp = (pc['gS_u+d'] + pc['gS_u-d']) / 2
GdSp = (pc['gS_u+d'] - pc['gS_u-d']) / 2
GsSp = pc['gS_s']
GuSn = GdSp
GdSn = GuSp
GsSn = GsSp
D = par['D ' + nucl] * mm**(5 / 2)
Sp = par['Sp ' + nucl] * mm**(5 / 2)
Vp = par['Vp ' + nucl] * mm**(5 / 2)
Sn = par['Sn ' + nucl] * mm**(5 / 2)
Vn = par['Vn ' + nucl] * mm**(5 / 2)
omega_capt = par['GammaCapture ' + nucl]
#####Wilson Coefficients######
#####Conversion Rate obtained from hep-ph/0203110#####
flavio.citations.register("Kitano:2002mt")
wc = wc_obj.get_wc('mue', scale, par, nf_out=3)
prefac = -np.sqrt(2) / par['GF']
AL = prefac / (4 * mm) * wc['Cgamma_emu']
AR = prefac / (4 * mm) * wc['Cgamma_mue'].conjugate()
gRV = {
'u': prefac * (wc['CVRR_mueuu'] + wc['CVLR_uumue']) / 2,
'd': prefac * (wc['CVRR_muedd'] + wc['CVLR_ddmue']) / 2,
's': prefac * (wc['CVRR_muess'] + wc['CVLR_ssmue']) / 2,
}
gLV = {
'u': prefac * (wc['CVLR_mueuu'] + wc['CVLL_mueuu']) / 2,
'd': prefac * (wc['CVLR_muedd'] + wc['CVLL_muedd']) / 2,
's': prefac * (wc['CVLR_muess'] + wc['CVLL_muess']) / 2,
}
gRS = {
'u': prefac * (wc['CSRR_emuuu'] + wc['CSRL_emuuu']).conjugate() / 2,
'd': prefac * (wc['CSRR_emudd'] + wc['CSRL_emudd']).conjugate() / 2,
's': prefac * (wc['CSRR_emuss'] + wc['CSRL_emuss']).conjugate() / 2,
}
gLS = {
'u': prefac * (wc['CSRR_mueuu'] + wc['CSRL_mueuu']) / 2,
'd': prefac * (wc['CSRR_muedd'] + wc['CSRL_muedd']) / 2,
's': prefac * (wc['CSRR_muess'] + wc['CSRL_muess']) / 2,
}
lhc = (AR.conjugate() * D + (2 * gLV['u'] + gLV['d']) * Vp +
(gLV['u'] + 2 * gLV['d']) * Vn +
(GuSp * gLS['u'] + GdSp * gLS['d'] + GsSp * gLS['s']) * Sp +
(GuSn * gLS['u'] + GdSn * gLS['d'] + GsSn * gLS['s']) * Sn)
rhc = (AL.conjugate() * D + (2 * gRV['u'] + gRV['d']) * Vp +
(gRV['u'] + 2 * gRV['d']) * Vn +
(GuSp * gRS['u'] + GdSp * gRS['d'] + GsSp * gRS['s']) * Sp +
(GuSn * gRS['u'] + GdSn * gRS['d'] + GsSn * gRS['s']) * Sn)
omega_conv = 2 * par['GF']**2 * (abs(lhc)**2 + abs(rhc)**2)
return omega_conv / omega_capt
def CR_mue(wc_obj, par, nucl):
r"""Conversion rate independent of the target nucleus"""
mm = par['m_mu']
scale = flavio.config['renormalization scale']['mudecays']
#####overlap integrals and other parameters#####
# GuV = par['GuV']
# GdV = par['GdV']
# GsV = par['GsV']
pc = proton_charges(par, scale)
GuS = (pc['gS_u+d'] + pc['gS_u-d']) / 2
GdS = (pc['gS_u+d'] - pc['gS_u-d']) / 2
GsS = pc['gS_s']
D = par['D ' +nucl]*mm**(5/2)
Sp = par['Sp '+nucl]*mm**(5/2)
Vp = par['Vp '+nucl]*mm**(5/2)
Sn = par['Sn '+nucl]*mm**(5/2)
Vn = par['Vn '+nucl]*mm**(5/2)
GC = par['GammaCapture '+nucl]
#####Wilson Coefficients######
#####Conversion Rate obtained from hep-ph/0203110#####
wc = wc_obj.get_wc('mue', scale, par, nf_out=3)
AL = np.sqrt(2)/(4*par['GF']*mm)*wc['Cgamma_emu'].conjugate()
AR = np.sqrt(2)/(4*par['GF']*mm)*wc['Cgamma_mue']
gRV = {'u': 4*(wc['CVRR_mueuu'] + wc['CVLR_mueuu']),
'd': 4*(wc['CVRR_muedd'] + wc['CVLR_muedd']),
's': 4*(wc['CVRR_muess'] + wc['CVLR_muess'])}
gLV = {'u': 4*(wc['CVLR_uumue'] + wc['CVLL_mueuu']),
'd': 4*(wc['CVLR_ddmue'] + wc['CVLL_muedd']),
's': 4*(wc['CVLR_ssmue'] + wc['CVLL_muess'])}
gRS = {'u': 4*(wc['CSRR_emuuu'].conjugate() + wc['CSRL_mueuu']),
'd': 4*(wc['CSRR_emudd'].conjugate() + wc['CSRL_muedd']),
's': 4*(wc['CSRR_emuss'].conjugate() + wc['CSRL_muess'])}
gLS = {'u': 4*(wc['CSRL_emuuu'].conjugate() + wc['CSRR_mueuu']),
'd': 4*(wc['CSRL_emudd'].conjugate() + wc['CSRR_muedd']),
's': 4*(wc['CSRL_emuss'].conjugate() + wc['CSRR_muess'])}
lhc = (AR.conjugate()*D + (2*gLV['u'] + gLV['d'])*Vp
+(gLV['u'] + 2*gLV['d'])*Vn
+ (GuS* gLS['u'] + GdS*gLS['d'] + GsS*gLS['s'])*Sp
+ (GuS*gLS['u'] + GdS*gLS['d'] + GsS*gLS['s'])*Sn)
rhc = (AL.conjugate()*D + (2*gRV['u'] + gRV['d'])*Vp
+(gRV['u'] + 2*gRV['d'])*Vn
+ (GuS* gRS['u'] + GdS*gRS['d'] + GsS*gRS['s'])*Sp
+ (GuS*gRS['u'] + GdS*gRS['d'] + GsS*gRS['s'])*Sn)
return 2*(par['GF']**2)*(abs(lhc)**2 + abs(rhc)**2)/GC