def angularcoeffs_h_Gbasis_v(phi, H, Htilde, q2, mB, mV, mqh, mql, ml1, ml2):
qp = -cmath.exp(1j * phi) # here it is assumed that q/p is a pure phase, as appropriate for B and Bs mixing
laB = lambda_K(mB**2, mV**2, q2)
laGa = lambda_K(q2, ml1**2, ml2**2)
E1 = sqrt(ml1**2+laGa/(4 * q2))
E2 = sqrt(ml2**2+laGa/(4 * q2))
CH = {k: complex(v).conjugate() for k, v in H.items()}
CHtilde = {k: complex(v).conjugate() for k, v in Htilde.items()}
G = {}
G[0,0,0] = (
4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','V'] * CH['pl','V'])+2 * _Re(-qp * Htilde['mi','V'] * CH['mi','V'])+2 * _Re(-qp * Htilde['0','V'] * CH['0','V'])+2 * _Re(-qp * Htilde['pl','A'] * CH['pl','A'])+2 * _Re(-qp * Htilde['mi','A'] * CH['mi','A'])+2 * _Re(-qp * Htilde['0','A'] * CH['0','A']))
+4 * ml1 * ml2/3 * (2 * _Re(-qp * Htilde['pl','V'] * CH['pl','V'])+2 * _Re(-qp * Htilde['mi','V'] * CH['mi','V'])+2 * _Re(-qp * Htilde['0','V'] * CH['0','V'])-2 * _Re(-qp * Htilde['pl','A'] * CH['pl','A'])-2 * _Re(-qp * Htilde['mi','A'] * CH['mi','A'])-2 * _Re(-qp * Htilde['0','A'] * CH['0','A']))
+4/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['S'] * CH['S'])+4/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['P'] * CH['P'])
+16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','Tt'] * CH['pl','Tt'])+2 * _Re(-qp * Htilde['mi','Tt'] * CH['mi','Tt'])+2 * _Re(-qp * Htilde['0','Tt'] * CH['0','Tt']))
+8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','T'] * CH['pl','T'])+2 * _Re(-qp * Htilde['mi','T'] * CH['mi','T'])+2 * _Re(-qp * Htilde['0','T'] * CH['0','T']))
+16/3 * (ml1 * E2+ml2 * E1) * _Im((-qp * Htilde['pl','V'] * CH['pl','Tt'] + _Co(-qp) * H['pl','V'] * CHtilde['pl','Tt'])+(-qp * Htilde['mi','V'] * CH['mi','Tt'] + _Co(-qp) * H['mi','V'] * CHtilde['mi','Tt'])+(-qp * Htilde['0','V'] * CH['0','Tt'] + _Co(-qp) * H['0','V'] * CHtilde['0','Tt']))
+8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im((-qp * Htilde['pl','A'] * CH['pl','T'] + _Co(-qp) * H['pl','A'] * CHtilde['pl','T'])+(-qp * Htilde['mi','A'] * CH['mi','T'] + _Co(-qp) * H['mi','A'] * CHtilde['mi','T'])+(-qp * Htilde['0','A'] * CH['0','T'] + _Co(-qp) * H['0','A'] * CHtilde['0','T'])))
G[0,1,0] = (4 * sqrt(laGa)/3 * (
_Re((-qp * Htilde['pl','V'] * CH['pl','A'] + _Co(-qp) * H['pl','V'] * CHtilde['pl','A'])-(-qp * Htilde['mi','V'] * CH['mi','A'] + _Co(-qp) * H['mi','V'] * CHtilde['mi','A']))
+2 * sqrt(2)/q2 * (ml1**2-ml2**2) * _Re((-qp * Htilde['pl','T'] * CH['pl','Tt'] + _Co(-qp) * H['pl','T'] * CHtilde['pl','Tt'])-(-qp * Htilde['mi','T'] * CH['mi','Tt'] + _Co(-qp) * H['mi','T'] * CHtilde['mi','Tt']))
+2 * (ml1+ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','A'] * CH['pl','Tt'] + _Co(-qp) * H['pl','A'] * CHtilde['pl','Tt'])-(-qp * Htilde['mi','A'] * CH['mi','Tt'] + _Co(-qp) * H['mi','A'] * CHtilde['mi','Tt']))
+sqrt(2)*(ml1-ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','V'] * CH['pl','T'] + _Co(-qp) * H['pl','V'] * CHtilde['pl','T'])-(-qp * Htilde['mi','V'] * CH['mi','T'] + _Co(-qp) * H['mi','V'] * CHtilde['mi','T']))
-(ml1-ml2)/sqrt(q2) * _Re((-qp * Htilde['0','A'] * CH['P'] + _Co(-qp) * H['0','A'] * CHtilde['P']))-(ml1+ml2)/sqrt(q2) * _Re((-qp * Htilde['0','V'] * CH['S'] + _Co(-qp) * H['0','V'] * CHtilde['S']))
+_Im(sqrt(2) * (-qp * Htilde['0','T'] * CH['P'] + _Co(-qp) * H['0','T'] * CHtilde['P'])+2 * (-qp * Htilde['0','Tt'] * CH['S'] + _Co(-qp) * H['0','Tt'] * CHtilde['S']))
))
G[0,2,0] = -2/9 * laGa/q2 * (
-2 * _Re(-qp * Htilde['pl','V'] * CH['pl','V'])-2 * _Re(-qp * Htilde['mi','V'] * CH['mi','V'])+2 * 2 * _Re(-qp * Htilde['0','V'] * CH['0','V'])-2 * _Re(-qp * Htilde['pl','A'] * CH['pl','A'])-2 * _Re(-qp * Htilde['mi','A'] * CH['mi','A'])+2 * 2 * _Re(-qp * Htilde['0','A'] * CH['0','A'])
-2 * (-2 * _Re(-qp * Htilde['pl','T'] * CH['pl','T'])-2 * _Re(-qp * Htilde['mi','T'] * CH['mi','T'])+2 * 2 * _Re(-qp * Htilde['0','T'] * CH['0','T']))-4 * (-2 * _Re(-qp * Htilde['pl','Tt'] * CH['pl','Tt'])-2 * _Re(-qp * Htilde['mi','Tt'] * CH['mi','Tt'])+2 * 2 * _Re(-qp * Htilde['0','Tt'] * CH['0','Tt'])))
G[2,0,0] = (-4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','V'] * CH['pl','V'])+2 * _Re(-qp * Htilde['mi','V'] * CH['mi','V'])-2 * 2 * _Re(-qp * Htilde['0','V'] * CH['0','V'])+2 * _Re(-qp * Htilde['pl','A'] * CH['pl','A'])+2 * _Re(-qp * Htilde['mi','A'] * CH['mi','A'])
-2 * 2 * _Re(-qp * Htilde['0','A'] * CH['0','A']))-4 * ml1 * ml2/3 * (2 * _Re(-qp * Htilde['pl','V'] * CH['pl','V'])+2 * _Re(-qp * Htilde['mi','V'] * CH['mi','V'])-2 * 2 * _Re(-qp * Htilde['0','V'] * CH['0','V'])-2 * _Re(-qp * Htilde['pl','A'] * CH['pl','A'])
-2 * _Re(-qp * Htilde['mi','A'] * CH['mi','A'])+2 * 2 * _Re(-qp * Htilde['0','A'] * CH['0','A']))+8/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['S'] * CH['S'])
+8/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['P'] * CH['P'])
-16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','Tt'] * CH['pl','Tt'])+2 * _Re(-qp * Htilde['mi','Tt'] * CH['mi','Tt'])-2 * 2 * _Re(-qp * Htilde['0','Tt'] * CH['0','Tt']))
-8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','T'] * CH['pl','T'])+2 * _Re(-qp * Htilde['mi','T'] * CH['mi','T'])-2 * 2 * _Re(-qp * Htilde['0','T'] * CH['0','T']))
-16/3 * (ml1 * E2+ml2 * E1) * _Im((-qp * Htilde['pl','V'] * CH['pl','Tt'] + _Co(-qp) * H['pl','V'] * CHtilde['pl','Tt'])+(-qp * Htilde['mi','V'] * CH['mi','Tt'] + _Co(-qp) * H['mi','V'] * CHtilde['mi','Tt'])-2 * (-qp * Htilde['0','V'] * CH['0','Tt'] + _Co(-qp) * H['0','V'] * CHtilde['0','Tt']))
-8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im((-qp * Htilde['pl','A'] * CH['pl','T'] + _Co(-qp) * H['pl','A'] * CHtilde['pl','T'])+(-qp * Htilde['mi','A'] * CH['mi','T'] + _Co(-qp) * H['mi','A'] * CHtilde['mi','T'])-2 * (-qp * Htilde['0','A'] * CH['0','T'] + _Co(-qp) * H['0','A'] * CHtilde['0','T'])))
G[2,1,0] = (-4 * sqrt(laGa)/3 * (_Re((-qp * Htilde['pl','V'] * CH['pl','A'] + _Co(-qp) * H['pl','V'] * CHtilde['pl','A'])-(-qp * Htilde['mi','V'] * CH['mi','A'] + _Co(-qp) * H['mi','V'] * CHtilde['mi','A']))
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * _Re((-qp * Htilde['pl','T'] * CH['pl','Tt'] + _Co(-qp) * H['pl','T'] * CHtilde['pl','Tt'])-(-qp * Htilde['mi','T'] * CH['mi','Tt'] + _Co(-qp) * H['mi','T'] * CHtilde['mi','Tt']))
+2 * (ml1+ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','A'] * CH['pl','Tt'] + _Co(-qp) * H['pl','A'] * CHtilde['pl','Tt'])-(-qp * Htilde['mi','A'] * CH['mi','Tt'] + _Co(-qp) * H['mi','A'] * CHtilde['mi','Tt']))
+sqrt(2) * (ml1-ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','V'] * CH['pl','T'] + _Co(-qp) * H['pl','V'] * CHtilde['pl','T'])-(-qp * Htilde['mi','V'] * CH['mi','T'] + _Co(-qp) * H['mi','V'] * CHtilde['mi','T']))
+2 * (ml1-ml2)/sqrt(q2) * _Re((-qp * Htilde['0','A'] * CH['P'] + _Co(-qp) * H['0','A'] * CHtilde['P']))+2 * (ml1+ml2)/sqrt(q2) * _Re((-qp * Htilde['0','V'] * CH['S'] + _Co(-qp) * H['0','V'] * CHtilde['S']))
-2 * _Im(sqrt(2) * (-qp * Htilde['0','T'] * CH['P'] + _Co(-qp) * H['0','T'] * CHtilde['P'])+2 * (-qp * Htilde['0','Tt'] * CH['S'] + _Co(-qp) * H['0','Tt'] * CHtilde['S']))))
G[2,2,0] = (-2/9 * laGa/q2 * (2 * _Re(-qp * Htilde['pl','V'] * CH['pl','V'])+2 * _Re(-qp * Htilde['mi','V'] * CH['mi','V'])+4 * 2 * _Re(-qp * Htilde['0','V'] * CH['0','V'])+2 * _Re(-qp * Htilde['pl','A'] * CH['pl','A'])+2 * _Re(-qp * Htilde['mi','A'] * CH['mi','A'])
+4 * 2 * _Re(-qp * Htilde['0','A'] * CH['0','A'])-2 * (2 * _Re(-qp * Htilde['pl','T'] * CH['pl','T'])+2 * _Re(-qp * Htilde['mi','T'] * CH['mi','T'])+4 * 2 * _Re(-qp * Htilde['0','T'] * CH['0','T']))-4 * (2 * _Re(-qp * Htilde['pl','Tt'] * CH['pl','Tt'])+2 * _Re(-qp * Htilde['mi','Tt'] * CH['mi','Tt'])+4 * 2 * _Re(-qp * Htilde['0','Tt'] * CH['0','Tt']))))
G[2,1,1] = (4/sqrt(3) * sqrt(laGa) * ((-qp * Htilde['pl','V'] * CH['0','A'] + _Co(-qp) * H['pl','V'] * CHtilde['0','A'])+(-qp * Htilde['pl','A'] * CH['0','V'] + _Co(-qp) * H['pl','A'] * CHtilde['0','V'])-(-qp * Htilde['0','V'] * CH['mi','A'] + _Co(-qp) * H['0','V'] * CHtilde['mi','A'])-(-qp * Htilde['0','A'] * CH['mi','V'] + _Co(-qp) * H['0','A'] * CHtilde['mi','V'])
+(ml1+ml2)/sqrt(q2) * ((-qp * Htilde['pl','V'] * CH['S'] + _Co(-qp) * H['pl','V'] * CHtilde['S'])+(-qp * Htilde['S'] * CH['mi','V'] + _Co(-qp) * H['S'] * CHtilde['mi','V']))-sqrt(2) * 1j * ((-qp * Htilde['P'] * CH['mi','T'] + _Co(-qp) * H['P'] * CHtilde['mi','T'])-(-qp * Htilde['pl','T'] * CH['P'] + _Co(-qp) * H['pl','T'] * CHtilde['P'])
+sqrt(2)*((-qp * Htilde['S'] * CH['mi','Tt'] + _Co(-qp) * H['S'] * CHtilde['mi','Tt'])-(-qp * Htilde['pl','Tt'] * CH['S'] + _Co(-qp) * H['pl','Tt'] * CHtilde['S'])))
+(ml1-ml2)/sqrt(q2) * ((-qp * Htilde['pl','A'] * CH['P'] + _Co(-qp) * H['pl','A'] * CHtilde['P'])+(-qp * Htilde['P'] * CH['mi','A'] + _Co(-qp) * H['P'] * CHtilde['mi','A']))
-2 * 1j * (ml1+ml2)/sqrt(q2) * ((-qp * Htilde['pl','A'] * CH['0','Tt'] + _Co(-qp) * H['pl','A'] * CHtilde['0','Tt'])+(-qp * Htilde['0','Tt'] * CH['mi','A'] + _Co(-qp) * H['0','Tt'] * CHtilde['mi','A'])-(-qp * Htilde['pl','Tt'] * CH['0','A'] + _Co(-qp) * H['pl','Tt'] * CHtilde['0','A'])-(-qp * Htilde['0','A'] * CH['mi','Tt'] + _Co(-qp) * H['0','A'] * CHtilde['mi','Tt']))
-sqrt(2) * 1j * (ml1-ml2)/sqrt(q2) * ((-qp * Htilde['pl','V'] * CH['0','T'] + _Co(-qp) * H['pl','V'] * CHtilde['0','T'])+(-qp * Htilde['0','T'] * CH['mi','V'] + _Co(-qp) * H['0','T'] * CHtilde['mi','V'])-(-qp * Htilde['pl','T'] * CH['0','V'] + _Co(-qp) * H['pl','T'] * CHtilde['0','V'])-(-qp * Htilde['0','V'] * CH['mi','T'] + _Co(-qp) * H['0','V'] * CHtilde['mi','T']))
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * ((-qp * Htilde['pl','T'] * CH['0','Tt'] + _Co(-qp) * H['pl','T'] * CHtilde['0','Tt'])+(-qp * Htilde['pl','Tt'] * CH['0','T'] + _Co(-qp) * H['pl','Tt'] * CHtilde['0','T'])-(-qp * Htilde['0','T'] * CH['mi','Tt'] + _Co(-qp) * H['0','T'] * CHtilde['mi','Tt'])-(-qp * Htilde['0','Tt'] * CH['mi','T'] + _Co(-qp) * H['0','Tt'] * CHtilde['mi','T']))))
G[2,2,1] = (4/3 * laGa/q2 * ((-qp * Htilde['pl','V'] * CH['0','V'] + _Co(-qp) * H['pl','V'] * CHtilde['0','V'])+(-qp * Htilde['0','V'] * CH['mi','V'] + _Co(-qp) * H['0','V'] * CHtilde['mi','V'])+(-qp * Htilde['pl','A'] * CH['0','A'] + _Co(-qp) * H['pl','A'] * CHtilde['0','A'])+(-qp * Htilde['0','A'] * CH['mi','A'] + _Co(-qp) * H['0','A'] * CHtilde['mi','A'])
-2 * ((-qp * Htilde['pl','T'] * CH['0','T'] + _Co(-qp) * H['pl','T'] * CHtilde['0','T'])+(-qp * Htilde['0','T'] * CH['mi','T'] + _Co(-qp) * H['0','T'] * CHtilde['mi','T'])+2 * ((-qp * Htilde['pl','Tt'] * CH['0','Tt'] + _Co(-qp) * H['pl','Tt'] * CHtilde['0','Tt'])+(-qp * Htilde['0','Tt'] * CH['mi','Tt'] + _Co(-qp) * H['0','Tt'] * CHtilde['mi','Tt'])))))
G[2,2,2] = -8/3 * laGa/q2 * ((-qp * Htilde['pl','V'] * CH['mi','V'] + _Co(-qp) * H['pl','V'] * CHtilde['mi','V'])+(-qp * Htilde['pl','A'] * CH['mi','A'] + _Co(-qp) * H['pl','A'] * CHtilde['mi','A'])-2 * ((-qp * Htilde['pl','T'] * CH['mi','T'] + _Co(-qp) * H['pl','T'] * CHtilde['mi','T'])+2 * (-qp * Htilde['pl','Tt'] * CH['mi','Tt'] + _Co(-qp) * H['pl','Tt'] * CHtilde['mi','Tt'])))
prefactor = sqrt(laB)*sqrt(laGa)/(2**9 * pi**3 * mB**3 * q2)
return {k: prefactor*v for k, v in G.items()}
def angularcoeffs_h_Gbasis_v(phi, H, Htilde, q2, mB, mV, mqh, mql, ml1, ml2):
qp = -cmath.exp(1j * phi) # here it is assumed that q/p is a pure phase, as appropriate for B and Bs mixing
laB = lambda_K(mB**2, mV**2, q2)
laGa = lambda_K(q2, ml1**2, ml2**2)
E1 = sqrt(ml1**2+laGa/(4 * q2))
E2 = sqrt(ml2**2+laGa/(4 * q2))
G = {}
G[0,0,0] = (
4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','V'] * _Co(H['pl','V']))+2 * _Re(-qp * Htilde['mi','V'] * _Co(H['mi','V']))+2 * _Re(-qp * Htilde['0','V'] * _Co(H['0','V']))+2 * _Re(-qp * Htilde['pl','A'] * _Co(H['pl','A']))+2 * _Re(-qp * Htilde['mi','A'] * _Co(H['mi','A']))+2 * _Re(-qp * Htilde['0','A'] * _Co(H['0','A'])))
+4 * ml1 * ml2/3 * (2 * _Re(-qp * Htilde['pl','V'] * _Co(H['pl','V']))+2 * _Re(-qp * Htilde['mi','V'] * _Co(H['mi','V']))+2 * _Re(-qp * Htilde['0','V'] * _Co(H['0','V']))-2 * _Re(-qp * Htilde['pl','A'] * _Co(H['pl','A']))-2 * _Re(-qp * Htilde['mi','A'] * _Co(H['mi','A']))-2 * _Re(-qp * Htilde['0','A'] * _Co(H['0','A'])))
+4/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['S'] * _Co(H['S']))+4/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['P'] * _Co(H['P']))
+16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','Tt'] * _Co(H['pl','Tt']))+2 * _Re(-qp * Htilde['mi','Tt'] * _Co(H['mi','Tt']))+2 * _Re(-qp * Htilde['0','Tt'] * _Co(H['0','Tt'])))
+8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','T'] * _Co(H['pl','T']))+2 * _Re(-qp * Htilde['mi','T'] * _Co(H['mi','T']))+2 * _Re(-qp * Htilde['0','T'] * _Co(H['0','T'])))
+16/3 * (ml1 * E2+ml2 * E1) * _Im((-qp * Htilde['pl','V'] * _Co(H['pl','Tt']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['pl','Tt']))+(-qp * Htilde['mi','V'] * _Co(H['mi','Tt']) + _Co(-qp) * H['mi','V'] * _Co(Htilde['mi','Tt']))+(-qp * Htilde['0','V'] * _Co(H['0','Tt']) + _Co(-qp) * H['0','V'] * _Co(Htilde['0','Tt'])))
+8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im((-qp * Htilde['pl','A'] * _Co(H['pl','T']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['pl','T']))+(-qp * Htilde['mi','A'] * _Co(H['mi','T']) + _Co(-qp) * H['mi','A'] * _Co(Htilde['mi','T']))+(-qp * Htilde['0','A'] * _Co(H['0','T']) + _Co(-qp) * H['0','A'] * _Co(Htilde['0','T']))))
G[0,1,0] = (4 * sqrt(laGa)/3 * (
_Re((-qp * Htilde['pl','V'] * _Co(H['pl','A']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['pl','A']))-(-qp * Htilde['mi','V'] * _Co(H['mi','A']) + _Co(-qp) * H['mi','V'] * _Co(Htilde['mi','A'])))
+2 * sqrt(2)/q2 * (ml1**2-ml2**2) * _Re((-qp * Htilde['pl','T'] * _Co(H['pl','Tt']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['pl','Tt']))-(-qp * Htilde['mi','T'] * _Co(H['mi','Tt']) + _Co(-qp) * H['mi','T'] * _Co(Htilde['mi','Tt'])))
+2 * (ml1+ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','A'] * _Co(H['pl','Tt']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['pl','Tt']))-(-qp * Htilde['mi','A'] * _Co(H['mi','Tt']) + _Co(-qp) * H['mi','A'] * _Co(Htilde['mi','Tt'])))
+sqrt(2)*(ml1-ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','V'] * _Co(H['pl','T']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['pl','T']))-(-qp * Htilde['mi','V'] * _Co(H['mi','T']) + _Co(-qp) * H['mi','V'] * _Co(Htilde['mi','T'])))
-(ml1-ml2)/sqrt(q2) * _Re((-qp * Htilde['0','A'] * _Co(H['P']) + _Co(-qp) * H['0','A'] * _Co(Htilde['P'])))-(ml1+ml2)/sqrt(q2) * _Re((-qp * Htilde['0','V'] * _Co(H['S']) + _Co(-qp) * H['0','V'] * _Co(Htilde['S'])))
+_Im(sqrt(2) * (-qp * Htilde['0','T'] * _Co(H['P']) + _Co(-qp) * H['0','T'] * _Co(Htilde['P']))+2 * (-qp * Htilde['0','Tt'] * _Co(H['S']) + _Co(-qp) * H['0','Tt'] * _Co(Htilde['S'])))
))
G[0,2,0] = -2/9 * laGa/q2 * (
-2 * _Re(-qp * Htilde['pl','V'] * _Co(H['pl','V']))-2 * _Re(-qp * Htilde['mi','V'] * _Co(H['mi','V']))+2 * 2 * _Re(-qp * Htilde['0','V'] * _Co(H['0','V']))-2 * _Re(-qp * Htilde['pl','A'] * _Co(H['pl','A']))-2 * _Re(-qp * Htilde['mi','A'] * _Co(H['mi','A']))+2 * 2 * _Re(-qp * Htilde['0','A'] * _Co(H['0','A']))
-2 * (-2 * _Re(-qp * Htilde['pl','T'] * _Co(H['pl','T']))-2 * _Re(-qp * Htilde['mi','T'] * _Co(H['mi','T']))+2 * 2 * _Re(-qp * Htilde['0','T'] * _Co(H['0','T'])))-4 * (-2 * _Re(-qp * Htilde['pl','Tt'] * _Co(H['pl','Tt']))-2 * _Re(-qp * Htilde['mi','Tt'] * _Co(H['mi','Tt']))+2 * 2 * _Re(-qp * Htilde['0','Tt'] * _Co(H['0','Tt']))))
G[2,0,0] = (-4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','V'] * _Co(H['pl','V']))+2 * _Re(-qp * Htilde['mi','V'] * _Co(H['mi','V']))-2 * 2 * _Re(-qp * Htilde['0','V'] * _Co(H['0','V']))+2 * _Re(-qp * Htilde['pl','A'] * _Co(H['pl','A']))+2 * _Re(-qp * Htilde['mi','A'] * _Co(H['mi','A']))
-2 * 2 * _Re(-qp * Htilde['0','A'] * _Co(H['0','A'])))-4 * ml1 * ml2/3 * (2 * _Re(-qp * Htilde['pl','V'] * _Co(H['pl','V']))+2 * _Re(-qp * Htilde['mi','V'] * _Co(H['mi','V']))-2 * 2 * _Re(-qp * Htilde['0','V'] * _Co(H['0','V']))-2 * _Re(-qp * Htilde['pl','A'] * _Co(H['pl','A']))
-2 * _Re(-qp * Htilde['mi','A'] * _Co(H['mi','A']))+2 * 2 * _Re(-qp * Htilde['0','A'] * _Co(H['0','A'])))+8/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['S'] * _Co(H['S']))
+8/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * 2 * _Re(-qp * Htilde['P'] * _Co(H['P']))
-16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','Tt'] * _Co(H['pl','Tt']))+2 * _Re(-qp * Htilde['mi','Tt'] * _Co(H['mi','Tt']))-2 * 2 * _Re(-qp * Htilde['0','Tt'] * _Co(H['0','Tt'])))
-8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (2 * _Re(-qp * Htilde['pl','T'] * _Co(H['pl','T']))+2 * _Re(-qp * Htilde['mi','T'] * _Co(H['mi','T']))-2 * 2 * _Re(-qp * Htilde['0','T'] * _Co(H['0','T'])))
-16/3 * (ml1 * E2+ml2 * E1) * _Im((-qp * Htilde['pl','V'] * _Co(H['pl','Tt']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['pl','Tt']))+(-qp * Htilde['mi','V'] * _Co(H['mi','Tt']) + _Co(-qp) * H['mi','V'] * _Co(Htilde['mi','Tt']))-2 * (-qp * Htilde['0','V'] * _Co(H['0','Tt']) + _Co(-qp) * H['0','V'] * _Co(Htilde['0','Tt'])))
-8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im((-qp * Htilde['pl','A'] * _Co(H['pl','T']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['pl','T']))+(-qp * Htilde['mi','A'] * _Co(H['mi','T']) + _Co(-qp) * H['mi','A'] * _Co(Htilde['mi','T']))-2 * (-qp * Htilde['0','A'] * _Co(H['0','T']) + _Co(-qp) * H['0','A'] * _Co(Htilde['0','T']))))
G[2,1,0] = (-4 * sqrt(laGa)/3 * (_Re((-qp * Htilde['pl','V'] * _Co(H['pl','A']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['pl','A']))-(-qp * Htilde['mi','V'] * _Co(H['mi','A']) + _Co(-qp) * H['mi','V'] * _Co(Htilde['mi','A'])))
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * _Re((-qp * Htilde['pl','T'] * _Co(H['pl','Tt']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['pl','Tt']))-(-qp * Htilde['mi','T'] * _Co(H['mi','Tt']) + _Co(-qp) * H['mi','T'] * _Co(Htilde['mi','Tt'])))
+2 * (ml1+ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','A'] * _Co(H['pl','Tt']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['pl','Tt']))-(-qp * Htilde['mi','A'] * _Co(H['mi','Tt']) + _Co(-qp) * H['mi','A'] * _Co(Htilde['mi','Tt'])))
+sqrt(2) * (ml1-ml2)/sqrt(q2) * _Im((-qp * Htilde['pl','V'] * _Co(H['pl','T']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['pl','T']))-(-qp * Htilde['mi','V'] * _Co(H['mi','T']) + _Co(-qp) * H['mi','V'] * _Co(Htilde['mi','T'])))
+2 * (ml1-ml2)/sqrt(q2) * _Re((-qp * Htilde['0','A'] * _Co(H['P']) + _Co(-qp) * H['0','A'] * _Co(Htilde['P'])))+2 * (ml1+ml2)/sqrt(q2) * _Re((-qp * Htilde['0','V'] * _Co(H['S']) + _Co(-qp) * H['0','V'] * _Co(Htilde['S'])))
-2 * _Im(sqrt(2) * (-qp * Htilde['0','T'] * _Co(H['P']) + _Co(-qp) * H['0','T'] * _Co(Htilde['P']))+2 * (-qp * Htilde['0','Tt'] * _Co(H['S']) + _Co(-qp) * H['0','Tt'] * _Co(Htilde['S'])))))
G[2,2,0] = (-2/9 * laGa/q2 * (2 * _Re(-qp * Htilde['pl','V'] * _Co(H['pl','V']))+2 * _Re(-qp * Htilde['mi','V'] * _Co(H['mi','V']))+4 * 2 * _Re(-qp * Htilde['0','V'] * _Co(H['0','V']))+2 * _Re(-qp * Htilde['pl','A'] * _Co(H['pl','A']))+2 * _Re(-qp * Htilde['mi','A'] * _Co(H['mi','A']))
+4 * 2 * _Re(-qp * Htilde['0','A'] * _Co(H['0','A']))-2 * (2 * _Re(-qp * Htilde['pl','T'] * _Co(H['pl','T']))+2 * _Re(-qp * Htilde['mi','T'] * _Co(H['mi','T']))+4 * 2 * _Re(-qp * Htilde['0','T'] * _Co(H['0','T'])))-4 * (2 * _Re(-qp * Htilde['pl','Tt'] * _Co(H['pl','Tt']))+2 * _Re(-qp * Htilde['mi','Tt'] * _Co(H['mi','Tt']))+4 * 2 * _Re(-qp * Htilde['0','Tt'] * _Co(H['0','Tt'])))))
G[2,1,1] = (4/sqrt(3) * sqrt(laGa) * ((-qp * Htilde['pl','V'] * _Co(H['0','A']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['0','A']))+(-qp * Htilde['pl','A'] * _Co(H['0','V']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['0','V']))-(-qp * Htilde['0','V'] * _Co(H['mi','A']) + _Co(-qp) * H['0','V'] * _Co(Htilde['mi','A']))-(-qp * Htilde['0','A'] * _Co(H['mi','V']) + _Co(-qp) * H['0','A'] * _Co(Htilde['mi','V']))
+(ml1+ml2)/sqrt(q2) * ((-qp * Htilde['pl','V'] * _Co(H['S']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['S']))+(-qp * Htilde['S'] * _Co(H['mi','V']) + _Co(-qp) * H['S'] * _Co(Htilde['mi','V'])))-sqrt(2) * 1j * ((-qp * Htilde['P'] * _Co(H['mi','T']) + _Co(-qp) * H['P'] * _Co(Htilde['mi','T']))-(-qp * Htilde['pl','T'] * _Co(H['P']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['P']))
+sqrt(2)*((-qp * Htilde['S'] * _Co(H['mi','Tt']) + _Co(-qp) * H['S'] * _Co(Htilde['mi','Tt']))-(-qp * Htilde['pl','Tt'] * _Co(H['S']) + _Co(-qp) * H['pl','Tt'] * _Co(Htilde['S']))))
+(ml1-ml2)/sqrt(q2) * ((-qp * Htilde['pl','A'] * _Co(H['P']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['P']))+(-qp * Htilde['P'] * _Co(H['mi','A']) + _Co(-qp) * H['P'] * _Co(Htilde['mi','A'])))
-2 * 1j * (ml1+ml2)/sqrt(q2) * ((-qp * Htilde['pl','A'] * _Co(H['0','Tt']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['0','Tt']))+(-qp * Htilde['0','Tt'] * _Co(H['mi','A']) + _Co(-qp) * H['0','Tt'] * _Co(Htilde['mi','A']))-(-qp * Htilde['pl','Tt'] * _Co(H['0','A']) + _Co(-qp) * H['pl','Tt'] * _Co(Htilde['0','A']))-(-qp * Htilde['0','A'] * _Co(H['mi','Tt']) + _Co(-qp) * H['0','A'] * _Co(Htilde['mi','Tt'])))
-sqrt(2) * 1j * (ml1-ml2)/sqrt(q2) * ((-qp * Htilde['pl','V'] * _Co(H['0','T']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['0','T']))+(-qp * Htilde['0','T'] * _Co(H['mi','V']) + _Co(-qp) * H['0','T'] * _Co(Htilde['mi','V']))-(-qp * Htilde['pl','T'] * _Co(H['0','V']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['0','V']))-(-qp * Htilde['0','V'] * _Co(H['mi','T']) + _Co(-qp) * H['0','V'] * _Co(Htilde['mi','T'])))
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * ((-qp * Htilde['pl','T'] * _Co(H['0','Tt']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['0','Tt']))+(-qp * Htilde['pl','Tt'] * _Co(H['0','T']) + _Co(-qp) * H['pl','Tt'] * _Co(Htilde['0','T']))-(-qp * Htilde['0','T'] * _Co(H['mi','Tt']) + _Co(-qp) * H['0','T'] * _Co(Htilde['mi','Tt']))-(-qp * Htilde['0','Tt'] * _Co(H['mi','T']) + _Co(-qp) * H['0','Tt'] * _Co(Htilde['mi','T'])))))
G[2,2,1] = (4/3 * laGa/q2 * ((-qp * Htilde['pl','V'] * _Co(H['0','V']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['0','V']))+(-qp * Htilde['0','V'] * _Co(H['mi','V']) + _Co(-qp) * H['0','V'] * _Co(Htilde['mi','V']))+(-qp * Htilde['pl','A'] * _Co(H['0','A']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['0','A']))+(-qp * Htilde['0','A'] * _Co(H['mi','A']) + _Co(-qp) * H['0','A'] * _Co(Htilde['mi','A']))
-2 * ((-qp * Htilde['pl','T'] * _Co(H['0','T']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['0','T']))+(-qp * Htilde['0','T'] * _Co(H['mi','T']) + _Co(-qp) * H['0','T'] * _Co(Htilde['mi','T']))+2 * ((-qp * Htilde['pl','Tt'] * _Co(H['0','Tt']) + _Co(-qp) * H['pl','Tt'] * _Co(Htilde['0','Tt']))+(-qp * Htilde['0','Tt'] * _Co(H['mi','Tt']) + _Co(-qp) * H['0','Tt'] * _Co(Htilde['mi','Tt']))))))
G[2,2,2] = -8/3 * laGa/q2 * ((-qp * Htilde['pl','V'] * _Co(H['mi','V']) + _Co(-qp) * H['pl','V'] * _Co(Htilde['mi','V']))+(-qp * Htilde['pl','A'] * _Co(H['mi','A']) + _Co(-qp) * H['pl','A'] * _Co(Htilde['mi','A']))-2 * ((-qp * Htilde['pl','T'] * _Co(H['mi','T']) + _Co(-qp) * H['pl','T'] * _Co(Htilde['mi','T']))+2 * (-qp * Htilde['pl','Tt'] * _Co(H['mi','Tt']) + _Co(-qp) * H['pl','Tt'] * _Co(Htilde['mi','Tt']))))
prefactor = sqrt(laB)*sqrt(laGa)/(2**9 * pi**3 * mB**3 * q2)
return {k: prefactor*v for k, v in G.items()}
def angularcoeffs_general_Gbasis_v(H, q2, mB, mV, mqh, mql, ml1, ml2):
laB = lambda_K(mB**2, mV**2, q2)
laGa = lambda_K(q2, ml1**2, ml2**2)
E1 = sqrt(ml1**2+laGa/(4 * q2))
E2 = sqrt(ml2**2+laGa/(4 * q2))
CH = {k: complex(v).conjugate() for k, v in H.items()}
G = {}
G[0,0,0] = (
4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (abs(H['pl','V'])**2+abs(H['mi','V'])**2+abs(H['0','V'])**2+abs(H['pl','A'])**2+abs(H['mi','A'])**2+abs(H['0','A'])**2)
+4 * ml1 * ml2/3 * (abs(H['pl','V'])**2+abs(H['mi','V'])**2+abs(H['0','V'])**2-abs(H['pl','A'])**2-abs(H['mi','A'])**2-abs(H['0','A'])**2)
+4/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * abs(H['S'])**2+4/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * abs(H['P'])**2
+16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','Tt'])**2+abs(H['mi','Tt'])**2+abs(H['0','Tt'])**2)
+8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','T'])**2+abs(H['mi','T'])**2+abs(H['0','T'])**2)
+16/3 * (ml1 * E2+ml2 * E1) * _Im(H['pl','V'] * CH['pl','Tt']+H['mi','V'] * CH['mi','Tt']+H['0','V'] * CH['0','Tt'])
+8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im(H['pl','A'] * CH['pl','T']+H['mi','A'] * CH['mi','T']+H['0','A'] * CH['0','T']))
G[0,1,0] = (4 * sqrt(laGa)/3 * (
_Re(H['pl','V'] * CH['pl','A']-H['mi','V'] * CH['mi','A'])
+2 * sqrt(2)/q2 * (ml1**2-ml2**2) * _Re(H['pl','T'] * CH['pl','Tt']-H['mi','T'] * CH['mi','Tt'])
+2 * (ml1+ml2)/sqrt(q2) * _Im(H['pl','A'] * CH['pl','Tt']-H['mi','A'] * CH['mi','Tt'])
+sqrt(2)*(ml1-ml2)/sqrt(q2) * _Im(H['pl','V'] * CH['pl','T']-H['mi','V'] * CH['mi','T'])
-(ml1-ml2)/sqrt(q2) * _Re(H['0','A'] * CH['P'])-(ml1+ml2)/sqrt(q2) * _Re(H['0','V'] * CH['S'])
+_Im(sqrt(2) * H['0','T'] * CH['P']+2 * H['0','Tt'] * CH['S'])
))
G[0,2,0] = -2/9 * laGa/q2 * (
-abs(H['pl','V'])**2-abs(H['mi','V'])**2+2 * abs(H['0','V'])**2-abs(H['pl','A'])**2-abs(H['mi','A'])**2+2 * abs(H['0','A'])**2
-2 * (-abs(H['pl','T'])**2-abs(H['mi','T'])**2+2 * abs(H['0','T'])**2)-4 * (-abs(H['pl','Tt'])**2-abs(H['mi','Tt'])**2+2 * abs(H['0','Tt'])**2))
G[2,0,0] = (-4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (abs(H['pl','V'])**2+abs(H['mi','V'])**2-2 * abs(H['0','V'])**2+abs(H['pl','A'])**2+abs(H['mi','A'])**2
-2 * abs(H['0','A'])**2)-4 * ml1 * ml2/3 * (abs(H['pl','V'])**2+abs(H['mi','V'])**2-2 * abs(H['0','V'])**2-abs(H['pl','A'])**2
-abs(H['mi','A'])**2+2 * abs(H['0','A'])**2)+8/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * abs(H['S'])**2
+8/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * abs(H['P'])**2
-16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','Tt'])**2+abs(H['mi','Tt'])**2-2 * abs(H['0','Tt'])**2)
-8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','T'])**2+abs(H['mi','T'])**2-2 * abs(H['0','T'])**2)
-16/3 * (ml1 * E2+ml2 * E1) * _Im(H['pl','V'] * CH['pl','Tt']+H['mi','V'] * CH['mi','Tt']-2 * H['0','V'] * CH['0','Tt'])
-8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im(H['pl','A'] * CH['pl','T']+H['mi','A'] * CH['mi','T']-2 * H['0','A'] * CH['0','T']))
G[2,1,0] = (-4 * sqrt(laGa)/3 * (_Re(H['pl','V'] * CH['pl','A']-H['mi','V'] * CH['mi','A'])
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * _Re(H['pl','T'] * CH['pl','Tt']-H['mi','T'] * CH['mi','Tt'])
+2 * (ml1+ml2)/sqrt(q2) * _Im(H['pl','A'] * CH['pl','Tt']-H['mi','A'] * CH['mi','Tt'])
+sqrt(2) * (ml1-ml2)/sqrt(q2) * _Im(H['pl','V'] * CH['pl','T']-H['mi','V'] * CH['mi','T'])
+2 * (ml1-ml2)/sqrt(q2) * _Re(H['0','A'] * CH['P'])+2 * (ml1+ml2)/sqrt(q2) * _Re(H['0','V'] * CH['S'])
-2 * _Im(sqrt(2) * H['0','T'] * CH['P']+2 * H['0','Tt'] * CH['S'])))
G[2,2,0] = (-2/9 * laGa/q2 * (abs(H['pl','V'])**2+abs(H['mi','V'])**2+4 * abs(H['0','V'])**2+abs(H['pl','A'])**2+abs(H['mi','A'])**2
+4 * abs(H['0','A'])**2-2 * (abs(H['pl','T'])**2+abs(H['mi','T'])**2+4 * abs(H['0','T'])**2)-4 * (abs(H['pl','Tt'])**2+abs(H['mi','Tt'])**2+4 * abs(H['0','Tt'])**2)))
G[2,1,1] = (4/sqrt(3) * sqrt(laGa) * (H['pl','V'] * CH['0','A']+H['pl','A'] * CH['0','V']-H['0','V'] * CH['mi','A']-H['0','A'] * CH['mi','V']
+(ml1+ml2)/sqrt(q2) * (H['pl','V'] * CH['S']+H['S'] * CH['mi','V'])-sqrt(2) * 1j * (H['P'] * CH['mi','T']-H['pl','T'] * CH['P']
+sqrt(2)*(H['S'] * CH['mi','Tt']-H['pl','Tt'] * CH['S']))
+(ml1-ml2)/sqrt(q2) * (H['pl','A'] * CH['P']+H['P'] * CH['mi','A'])
-2 * 1j * (ml1+ml2)/sqrt(q2) * (H['pl','A'] * CH['0','Tt']+H['0','Tt'] * CH['mi','A']-H['pl','Tt'] * CH['0','A']-H['0','A'] * CH['mi','Tt'])
-sqrt(2) * 1j * (ml1-ml2)/sqrt(q2) * (H['pl','V'] * CH['0','T']+H['0','T'] * CH['mi','V']-H['pl','T'] * CH['0','V']-H['0','V'] * CH['mi','T'])
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * (H['pl','T'] * CH['0','Tt']+H['pl','Tt'] * CH['0','T']-H['0','T'] * CH['mi','Tt']-H['0','Tt'] * CH['mi','T'])))
G[2,2,1] = (4/3 * laGa/q2 * (H['pl','V'] * CH['0','V']+H['0','V'] * CH['mi','V']+H['pl','A'] * CH['0','A']+H['0','A'] * CH['mi','A']
-2 * (H['pl','T'] * CH['0','T']+H['0','T'] * CH['mi','T']+2 * (H['pl','Tt'] * CH['0','Tt']+H['0','Tt'] * CH['mi','Tt']))))
G[2,2,2] = -8/3 * laGa/q2 * (H['pl','V'] * CH['mi','V']+H['pl','A'] * CH['mi','A']-2 * (H['pl','T'] * CH['mi','T']+2 * H['pl','Tt'] * CH['mi','Tt']))
prefactor = sqrt(laB)*sqrt(laGa)/(2**9 * pi**3 * mB**3 * q2)
return {k: prefactor*v for k, v in G.items()}
def angularcoeffs_general_Gbasis_v(H, q2, mB, mV, mqh, mql, ml1, ml2):
laB = lambda_K(mB**2, mV**2, q2)
laGa = lambda_K(q2, ml1**2, ml2**2)
E1 = sqrt(ml1**2+laGa/(4 * q2))
E2 = sqrt(ml2**2+laGa/(4 * q2))
G = {}
G[0,0,0] = (
4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (abs(H['pl','V'])**2+abs(H['mi','V'])**2+abs(H['0','V'])**2+abs(H['pl','A'])**2+abs(H['mi','A'])**2+abs(H['0','A'])**2)
+4 * ml1 * ml2/3 * (abs(H['pl','V'])**2+abs(H['mi','V'])**2+abs(H['0','V'])**2-abs(H['pl','A'])**2-abs(H['mi','A'])**2-abs(H['0','A'])**2)
+4/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * abs(H['S'])**2+4/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * abs(H['P'])**2
+16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','Tt'])**2+abs(H['mi','Tt'])**2+abs(H['0','Tt'])**2)
+8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','T'])**2+abs(H['mi','T'])**2+abs(H['0','T'])**2)
+16/3 * (ml1 * E2+ml2 * E1) * _Im(H['pl','V'] * _Co(H['pl','Tt'])+H['mi','V'] * _Co(H['mi','Tt'])+H['0','V'] * _Co(H['0','Tt']))
+8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im(H['pl','A'] * _Co(H['pl','T'])+H['mi','A'] * _Co(H['mi','T'])+H['0','A'] * _Co(H['0','T'])))
G[0,1,0] = (4 * sqrt(laGa)/3 * (
_Re(H['pl','V'] * _Co(H['pl','A'])-H['mi','V'] * _Co(H['mi','A']))
+2 * sqrt(2)/q2 * (ml1**2-ml2**2) * _Re(H['pl','T'] * _Co(H['pl','Tt'])-H['mi','T'] * _Co(H['mi','Tt']))
+2 * (ml1+ml2)/sqrt(q2) * _Im(H['pl','A'] * _Co(H['pl','Tt'])-H['mi','A'] * _Co(H['mi','Tt']))
+sqrt(2)*(ml1-ml2)/sqrt(q2) * _Im(H['pl','V'] * _Co(H['pl','T'])-H['mi','V'] * _Co(H['mi','T']))
-(ml1-ml2)/sqrt(q2) * _Re(H['0','A'] * _Co(H['P']))-(ml1+ml2)/sqrt(q2) * _Re(H['0','V'] * _Co(H['S']))
+_Im(sqrt(2) * H['0','T'] * _Co(H['P'])+2 * H['0','Tt'] * _Co(H['S']))
))
G[0,2,0] = -2/9 * laGa/q2 * (
-abs(H['pl','V'])**2-abs(H['mi','V'])**2+2 * abs(H['0','V'])**2-abs(H['pl','A'])**2-abs(H['mi','A'])**2+2 * abs(H['0','A'])**2
-2 * (-abs(H['pl','T'])**2-abs(H['mi','T'])**2+2 * abs(H['0','T'])**2)-4 * (-abs(H['pl','Tt'])**2-abs(H['mi','Tt'])**2+2 * abs(H['0','Tt'])**2))
G[2,0,0] = (-4/9 * (3 * E1 * E2+laGa/(4 * q2)) * (abs(H['pl','V'])**2+abs(H['mi','V'])**2-2 * abs(H['0','V'])**2+abs(H['pl','A'])**2+abs(H['mi','A'])**2
-2 * abs(H['0','A'])**2)-4 * ml1 * ml2/3 * (abs(H['pl','V'])**2+abs(H['mi','V'])**2-2 * abs(H['0','V'])**2-abs(H['pl','A'])**2
-abs(H['mi','A'])**2+2 * abs(H['0','A'])**2)+8/3 * (E1 * E2-ml1 * ml2+laGa/(4 * q2)) * abs(H['S'])**2
+8/3 * (E1 * E2+ml1 * ml2+laGa/(4 * q2)) * abs(H['P'])**2
-16/9 * (3 * (E1 * E2+ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','Tt'])**2+abs(H['mi','Tt'])**2-2 * abs(H['0','Tt'])**2)
-8/9 * (3 * (E1 * E2-ml1 * ml2)-laGa/(4 * q2)) * (abs(H['pl','T'])**2+abs(H['mi','T'])**2-2 * abs(H['0','T'])**2)
-16/3 * (ml1 * E2+ml2 * E1) * _Im(H['pl','V'] * _Co(H['pl','Tt'])+H['mi','V'] * _Co(H['mi','Tt'])-2 * H['0','V'] * _Co(H['0','Tt']))
-8 * sqrt(2)/3 * (ml1 * E2-ml2 * E1) * _Im(H['pl','A'] * _Co(H['pl','T'])+H['mi','A'] * _Co(H['mi','T'])-2 * H['0','A'] * _Co(H['0','T'])))
G[2,1,0] = (-4 * sqrt(laGa)/3 * (_Re(H['pl','V'] * _Co(H['pl','A'])-H['mi','V'] * _Co(H['mi','A']))
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * _Re(H['pl','T'] * _Co(H['pl','Tt'])-H['mi','T'] * _Co(H['mi','Tt']))
+2 * (ml1+ml2)/sqrt(q2) * _Im(H['pl','A'] * _Co(H['pl','Tt'])-H['mi','A'] * _Co(H['mi','Tt']))
+sqrt(2) * (ml1-ml2)/sqrt(q2) * _Im(H['pl','V'] * _Co(H['pl','T'])-H['mi','V'] * _Co(H['mi','T']))
+2 * (ml1-ml2)/sqrt(q2) * _Re(H['0','A'] * _Co(H['P']))+2 * (ml1+ml2)/sqrt(q2) * _Re(H['0','V'] * _Co(H['S']))
-2 * _Im(sqrt(2) * H['0','T'] * _Co(H['P'])+2 * H['0','Tt'] * _Co(H['S']))))
G[2,2,0] = (-2/9 * laGa/q2 * (abs(H['pl','V'])**2+abs(H['mi','V'])**2+4 * abs(H['0','V'])**2+abs(H['pl','A'])**2+abs(H['mi','A'])**2
+4 * abs(H['0','A'])**2-2 * (abs(H['pl','T'])**2+abs(H['mi','T'])**2+4 * abs(H['0','T'])**2)-4 * (abs(H['pl','Tt'])**2+abs(H['mi','Tt'])**2+4 * abs(H['0','Tt'])**2)))
G[2,1,1] = (4/sqrt(3) * sqrt(laGa) * (H['pl','V'] * _Co(H['0','A'])+H['pl','A'] * _Co(H['0','V'])-H['0','V'] * _Co(H['mi','A'])-H['0','A'] * _Co(H['mi','V'])
+(ml1+ml2)/sqrt(q2) * (H['pl','V'] * _Co(H['S'])+H['S'] * _Co(H['mi','V']))-sqrt(2) * 1j * (H['P'] * _Co(H['mi','T'])-H['pl','T'] * _Co(H['P'])
+sqrt(2)*(H['S'] * _Co(H['mi','Tt'])-H['pl','Tt'] * _Co(H['S'])))
+(ml1-ml2)/sqrt(q2) * (H['pl','A'] * _Co(H['P'])+H['P'] * _Co(H['mi','A']))
-2 * 1j * (ml1+ml2)/sqrt(q2) * (H['pl','A'] * _Co(H['0','Tt'])+H['0','Tt'] * _Co(H['mi','A'])-H['pl','Tt'] * _Co(H['0','A'])-H['0','A'] * _Co(H['mi','Tt']))
-sqrt(2) * 1j * (ml1-ml2)/sqrt(q2) * (H['pl','V'] * _Co(H['0','T'])+H['0','T'] * _Co(H['mi','V'])-H['pl','T'] * _Co(H['0','V'])-H['0','V'] * _Co(H['mi','T']))
+2 * sqrt(2) * (ml1**2-ml2**2)/q2 * (H['pl','T'] * _Co(H['0','Tt'])+H['pl','Tt'] * _Co(H['0','T'])-H['0','T'] * _Co(H['mi','Tt'])-H['0','Tt'] * _Co(H['mi','T']))))
G[2,2,1] = (4/3 * laGa/q2 * (H['pl','V'] * _Co(H['0','V'])+H['0','V'] * _Co(H['mi','V'])+H['pl','A'] * _Co(H['0','A'])+H['0','A'] * _Co(H['mi','A'])
-2 * (H['pl','T'] * _Co(H['0','T'])+H['0','T'] * _Co(H['mi','T'])+2 * (H['pl','Tt'] * _Co(H['0','Tt'])+H['0','Tt'] * _Co(H['mi','Tt'])))))
G[2,2,2] = -8/3 * laGa/q2 * (H['pl','V'] * _Co(H['mi','V'])+H['pl','A'] * _Co(H['mi','A'])-2 * (H['pl','T'] * _Co(H['mi','T'])+2 * H['pl','Tt'] * _Co(H['mi','Tt'])))
prefactor = sqrt(laB)*sqrt(laGa)/(2**9 * pi**3 * mB**3 * q2)
return {k: prefactor*v for k, v in G.items()}
def br_inst(par, wc, B, l1, l2):
r"""Branching ratio of $B_q\to\ell_1^+\ell_2^-$ in the absence of mixing.
Parameters
----------
- `par`: parameter dictionary
- `B`: should be `'Bs'` or `'B0'`
- `lep`: should be `'e'`, `'mu'`, or `'tau'`
"""
# paramaeters
GF = par['GF']
alphaem = running.get_alpha(par, 4.8)['alpha_e']
ml1 = par['m_' + l1]
ml2 = par['m_' + l2]
mB = par['m_' + B]
tauB = par['tau_' + B]
fB = par['f_' + B]
# appropriate CKM elements
if B == 'Bs':
xi_t = ckm.xi('t', 'bs')(par)
elif B == 'B0':
xi_t = ckm.xi('t', 'bd')(par)
N = xi_t * 4 * GF / sqrt(2) * alphaem / (4 * pi)
beta = sqrt(lambda_K(mB**2, ml1**2, ml2**2)) / mB**2
prefactor = abs(N)**2 / 32. / pi * mB**3 * tauB * beta * fB**2
P, S = amplitudes(par, wc, B, l1, l2)
return prefactor * (abs(P)**2 + abs(S)**2)
def br_inst(par, wc, B, l1, l2):
r"""Branching ratio of $B_q\to\ell_1^+\ell_2^-$ in the absence of mixing.
Parameters
----------
- `par`: parameter dictionary
- `B`: should be `'Bs'` or `'B0'`
- `lep`: should be `'e'`, `'mu'`, or `'tau'`
"""
# paramaeters
GF = par['GF']
alphaem = running.get_alpha(par, 4.8)['alpha_e']
ml1 = par['m_'+l1]
ml2 = par['m_'+l2]
mB = par['m_'+B]
tauB = par['tau_'+B]
fB = par['f_'+B]
# appropriate CKM elements
if B == 'Bs':
xi_t = ckm.xi('t','bs')(par)
elif B == 'B0':
xi_t = ckm.xi('t','bd')(par)
N = xi_t * 4*GF/sqrt(2) * alphaem/(4*pi)
beta = sqrt(lambda_K(mB**2,ml1**2,ml2**2))/mB**2
beta_p = sqrt(1 - (ml1 + ml2)**2/mB**2)
beta_m = sqrt(1 - (ml1 - ml2)**2/mB**2)
prefactor = abs(N)**2 / 32. / pi * mB**3 * tauB * beta * fB**2
P, S = amplitudes(par, wc, B, l1, l2)
return prefactor * ( beta_m**2 * abs(P)**2 + beta_p**2 * abs(S)**2 )
def prefactor(q2, par, B, V, lep):
GF = par['GF']
scale = config['renormalization scale']['bvll']
ml = par['m_' + lep]
mB = par['m_' + B]
mV = par['m_' + V]
tauB = par['tau_' + B]
laB = lambda_K(mB**2, mV**2, q2)
laGa = lambda_K(q2, ml**2, 0.)
qi_qj = meson_quark[(B, V)]
if qi_qj == 'bu':
Vij = ckm.get_ckm(par)[0, 2] # V_{ub} for b->u transitions
if qi_qj == 'bc':
Vij = ckm.get_ckm(par)[1, 2] # V_{cb} for b->c transitions
if q2 <= ml**2:
return 0
return 4 * GF / sqrt(2) * Vij
def prefactor(q2, par, B, V, lep):
GF = par['GF']
scale = config['renormalization scale']['bvll']
ml = par['m_'+lep]
mB = par['m_'+B]
mV = par['m_'+V]
tauB = par['tau_'+B]
laB = lambda_K(mB**2, mV**2, q2)
laGa = lambda_K(q2, ml**2, 0.)
qi_qj = meson_quark[(B, V)]
if qi_qj == 'bu':
Vij = ckm.get_ckm(par)[0,2] # V_{ub} for b->u transitions
if qi_qj == 'bc':
Vij = ckm.get_ckm(par)[1,2] # V_{cb} for b->c transitions
if q2 <= ml**2:
return 0
return 4*GF/sqrt(2)*Vij
def helicity_amps_qcdf(q2, wc, par, B, V, **kwargs):
if q2 > 6:
warnings.warn("The QCDF corrections should not be trusted for q2 above 6 GeV^2")
mB = par['m_'+B]
mV = par['m_'+V]
X = sqrt(lambda_K(mB**2,q2,mV**2))/2.
ta = transversity_amps_qcdf(q2, wc, par, B, V, **kwargs)
h = flavio.physics.bdecays.angular.transversity_to_helicity(ta)
return h
def helicity_amps_qcdf(q2, wc, par, B, V, **kwargs):
if q2 > 6:
warnings.warn("The QCDF corrections should not be trusted for q2 above 6 GeV^2")
mB = par['m_'+B]
mV = par['m_'+V]
X = sqrt(lambda_K(mB**2,q2,mV**2))/2.
ta = transversity_amps_qcdf(q2, wc, par, B, V, **kwargs)
h = flavio.physics.bdecays.angular.transversity_to_helicity(ta)
return h
def helicity_amps_p(q2, mB, mP, mqh, mql, ml1, ml2, ff, wc, prefactor):
laB = lambda_K(mB**2, mP**2, q2)
h = {}
h['V'] = sqrt(laB)/(2*sqrt(q2)) * (
2*mqh/(mB+mP)*(wc['7']+wc['7p'])*ff['fT']+(wc['v']+wc['vp'])*ff['f+'] )
h['A'] = sqrt(laB)/(2*sqrt(q2)) * (wc['a']+wc['ap'])*ff['f+']
h['S'] = (mB**2-mP**2)/2. * ff['f0'] * (
(wc['s']+wc['sp'])/(mqh-mql) + (ml1-ml2)/q2*(wc['v']+wc['vp']) )
h['P'] = (mB**2-mP**2)/2. * ff['f0'] * (
(wc['p']+wc['pp'])/(mqh-mql) + (ml1+ml2)/q2*(wc['a']+wc['ap']) )
h['T'] = -1j*sqrt(laB)/(2*(mB+mP)) * (wc['t']-wc['tp']) * ff['fT']
h['Tt'] = -1j*sqrt(laB)/(2*(mB+mP)) * (wc['t']+wc['tp']) * ff['fT']
return {k: prefactor*v for k, v in h.items()}
def angularcoeffs_general_Gbasis_p(h, q2, mB, mP, mqh, mql, ml1, ml2):
laB = lambda_K(mB**2, mP**2, q2)
laGa = lambda_K(q2, ml1**2, ml2**2)
E1 = sqrt(ml1**2+laGa/(4 * q2))
E2 = sqrt(ml2**2+laGa/(4 * q2))
G = {}
G[0] = (
( 4*(E1*E2 + ml1*ml2) + laGa/(3*q2) ) * abs(h['V'])**2
+ ( 4*(E1*E2 - ml1*ml2) + laGa/(3*q2) ) * abs(h['A'])**2
+ ( 4*(E1*E2 - ml1*ml2) + laGa/( q2) ) * abs(h['S'])**2
+ ( 4*(E1*E2 + ml1*ml2) + laGa/( q2) ) * abs(h['P'])**2
+ 16*(E1*E2 + ml1*ml2 - laGa/(12*q2)) * abs(h['Tt'])**2
+ 8*(E1*E2 - ml1*ml2 - laGa/(12*q2)) * abs(h['T'])**2
+ 16 * (ml1*E2 + ml2*E1) * _Im( h['V'] * _Co(h['Tt']) )
+ 8*sqrt(2)*(ml1*E2 - ml2*E1) * _Im( h['A'] * _Co(h['T']) ) )
G[1] = -4*sqrt(laGa) * (
_Re( (ml1+ml2)/sqrt(q2) * h['V'] * _Co(h['S'])
+ (ml1-ml2)/sqrt(q2) * h['A'] * _Co(h['P']) )
- _Im( 2 * h['Tt'] * _Co(h['S']) + sqrt(2) * h['T'] * _Co(h['P'])) )
G[2] = -4*laGa/(3*q2) * (
abs(h['V'])**2 + abs(h['A'])**2 - 2*abs(h['T'])**2 - 4*abs(h['Tt'])**2 )
prefactor = sqrt(laB)*sqrt(laGa)/(2**9 * pi**3 * mB**3 * q2)
return {k: prefactor*v for k, v in G.items()}
def helicity_amps_v(q2, mB, mV, mqh, mql, ml1, ml2, ff, wc, prefactor):
laB = lambda_K(mB**2, mV**2, q2)
H = {}
H['0','V'] = (4 * 1j * mB * mV)/(sqrt(q2) * (mB+mV)) * ((wc['v']-wc['vp']) * (mB+mV) * ff['A12']+mqh * (wc['7']-wc['7p']) * ff['T23'])
H['0','A'] = 4 * 1j * mB * mV/sqrt(q2) * (wc['a']-wc['ap']) * ff['A12']
H['pl','V'] = 1j/(2 * (mB+mV)) * (+(wc['v']+wc['vp']) * sqrt(laB) * ff['V']-(mB+mV)**2 * (wc['v']-wc['vp']) * ff['A1'])+1j * mqh/q2 * (+(wc['7']+wc['7p']) * sqrt(laB) * ff['T1']-(wc['7']-wc['7p']) * (mB**2-mV**2) * ff['T2'])
H['mi','V'] = 1j/(2 * (mB+mV)) * (-(wc['v']+wc['vp']) * sqrt(laB) * ff['V']-(mB+mV)**2 * (wc['v']-wc['vp']) * ff['A1'])+1j * mqh/q2 * (-(wc['7']+wc['7p']) * sqrt(laB) * ff['T1']-(wc['7']-wc['7p']) * (mB**2-mV**2) * ff['T2'])
H['pl','A'] = 1j/(2 * (mB+mV)) * (+(wc['a']+wc['ap']) * sqrt(laB) * ff['V']-(mB+mV)**2 * (wc['a']-wc['ap']) * ff['A1'])
H['mi','A'] = 1j/(2 * (mB+mV)) * (-(wc['a']+wc['ap']) * sqrt(laB) * ff['V']-(mB+mV)**2 * (wc['a']-wc['ap']) * ff['A1'])
H['P'] = 1j * sqrt(laB)/2 * ((wc['p']-wc['pp'])/(mqh+mql)+(ml1+ml2)/q2 * (wc['a']-wc['ap'])) * ff['A0']
H['S'] = 1j * sqrt(laB)/2 * ((wc['s']-wc['sp'])/(mqh+mql)+(ml1-ml2)/q2 * (wc['v']-wc['vp'])) * ff['A0']
H['0','T'] = 2 * sqrt(2) * mB * mV/(mB+mV) * (wc['t']+wc['tp']) * ff['T23']
H['0','Tt'] = 2 * mB * mV/(mB+mV) * (wc['t']-wc['tp']) * ff['T23']
H['pl','T'] = 1/(sqrt(2) * sqrt(q2)) * (+(wc['t']-wc['tp']) * sqrt(laB) * ff['T1']-(wc['t']+wc['tp']) * (mB**2-mV**2) * ff['T2'])
H['mi','T'] = 1/(sqrt(2) * sqrt(q2)) * (-(wc['t']-wc['tp']) * sqrt(laB) * ff['T1']-(wc['t']+wc['tp']) * (mB**2-mV**2) * ff['T2'])
H['pl','Tt'] = 1/(2 * sqrt(q2)) * (+(wc['t']+wc['tp']) * sqrt(laB) * ff['T1']-(wc['t']-wc['tp']) * (mB**2-mV**2) * ff['T2'])
H['mi','Tt'] = 1/(2 * sqrt(q2)) * (-(wc['t']+wc['tp']) * sqrt(laB) * ff['T1']-(wc['t']-wc['tp']) * (mB**2-mV**2) * ff['T2'])
return {k: prefactor*v for k, v in H.items()}
def br_kll(par, wc_obj, K, l1, l2, ld=True):
r"""Branching ratio of $K\to\ell_1^+\ell_2^-$"""
# parameters
wc = get_wc(wc_obj, par, l1, l2)
GF = par['GF']
alphaem = par['alpha_e']
ml1 = par['m_'+l1]
ml2 = par['m_'+l2]
mK = par['m_K0']
tauK = par['tau_'+K]
fK = par['f_K0']
# CKM part of the eff. operator prefactor N is included in Peff and Seff
N = 4 * GF / sqrt(2) * alphaem / (4 * pi)
beta = sqrt(lambda_K(mK**2, ml1**2, ml2**2)) / mK**2
beta_p = sqrt(1 - (ml1 + ml2)**2 / mK**2)
beta_m = sqrt(1 - (ml1 - ml2)**2 / mK**2)
prefactor = 2 * abs(N)**2 / 32. / pi * mK**3 * tauK * beta * fK**2
Peff, Seff = amplitudes_eff(par, wc, K, l1, l2, ld=ld)
return prefactor * (beta_m**2 * abs(Peff)**2 + beta_p**2 * abs(Seff)**2)
def br_kll(par, wc_obj, K, l1, l2, ld=True):
r"""Branching ratio of $K\to\ell_1^+\ell_2^-$"""
# parameters
wc = get_wc(wc_obj, par, l1, l2)
GF = par['GF']
alphaem = par['alpha_e']
ml1 = par['m_'+l1]
ml2 = par['m_'+l2]
mK = par['m_K0']
tauK = par['tau_'+K]
fK = par['f_K0']
# appropriate CKM elements
N = 4 * GF / sqrt(2) * alphaem / (4 * pi)
beta = sqrt(lambda_K(mK**2, ml1**2, ml2**2)) / mK**2
beta_p = sqrt(1 - (ml1 + ml2)**2 / mK**2)
beta_m = sqrt(1 - (ml1 - ml2)**2 / mK**2)
prefactor = 2 * abs(N)**2 / 32. / pi * mK**3 * tauK * beta * fK**2
Peff, Seff = amplitudes_eff(par, wc, K, l1, l2, ld=ld)
return prefactor * (beta_m**2 * abs(Peff)**2 + beta_p**2 * abs(Seff)**2)
