def exponential_nonresonant_lineshape(m2,
m0,
alpha,
ma,
mb,
mc,
md,
lr,
ld,
barrierFactor=True):
"""
Exponential nonresonant amplitude with orbital barriers
"""
if barrierFactor:
m = atfi.sqrt(m2)
q = atfk.two_body_momentum(md, m, mc)
q0 = atfk.two_body_momentum(md, m0, mc)
p = atfk.two_body_momentum(m, ma, mb)
p0 = atfk.two_body_momentum(m0, ma, mb)
b1 = orbital_barrier_factor(p, p0, lr)
b2 = orbital_barrier_factor(q, q0, ld)
return atfi.complex(b1 * b2 * atfi.exp(-alpha * (m2 - m0**2)),
atfi.const(0.))
else:
return atfi.complex(atfi.exp(-alpha * (m2 - m0**2)), atfi.const(0.))
def helicity_couplings_from_ls(ja, jb, jc, lb, lc, bls):
"""Helicity couplings from a list of LS couplings.
ja : spin of A (decaying) particle
jb : spin of B (1st decay product)
jc : spin of C (2nd decay product)
lb : B helicity
lc : C helicity
bls : dictionary of LS couplings, where:
keys are tuples corresponding to (L,S) pairs
values are values of LS couplings
Note that ALL j,l,s should be doubled, e.g. S=1 for spin-1/2, L=2 for p-wave etc.
:param ja:
:param jb:
:param jc:
:param lb:
:param lc:
:param bls:
"""
a = 0.
# print("%d %d %d %d %d" % (ja, jb, jc, lb, lc))
for ls, b in bls.items():
l = ls[0]
s = ls[1]
coeff = math.sqrt((l+1)/(ja+1))*atfi.clebsch(jb, lb, jc, -lc, s, lb-lc)*\
atfi.clebsch( l, 0, s, lb-lc, ja, lb-lc)
if coeff:
a += atfi.complex(atfi.const(float(coeff)), atfi.const(0.)) * b
return a
def model(x) :
m2dp = phsp.m2ab(x)
m2ppi = phsp.m2bc(x)
p4d, p4p, p4pi = phsp.final_state_momenta(m2dp, m2ppi)
dp_theta_r, dp_phi_r, dp_theta_d, dp_phi_d = atfk.helicity_angles_3body(p4d, p4p, p4pi)
# List of intermediate resonances corresponds to arXiv link above.
resonances = [
(atfd.breit_wigner_lineshape(m2dp, mass_lcst, width_lcst, md, mp, mpi, mlb, dr, db, OrbitalMomentum(5, 1), 2), 5, 1,
couplings[0][0], couplings[0][1]
),
(atfd.breit_wigner_lineshape(m2dp, mass_lcx, width_lcx, md, mp, mpi, mlb, dr, db, OrbitalMomentum(3, 1), 1), 3, 1,
couplings[1][0], couplings[1][1]
),
(atfd.breit_wigner_lineshape(m2dp, mass_lcstst, width_lcstst, md, mp, mpi, mlb, dr, db, OrbitalMomentum(3, -1), 1), 3, -1,
couplings[2][0], couplings[2][1]
),
(atfd.exponential_nonresonant_lineshape(m2dp, mass0, alpha12p, md, mp, mpi, mlb, OrbitalMomentum(1, 1), 0), 1, 1,
couplings[3][0], couplings[3][1]
),
(atfd.exponential_nonresonant_lineshape(m2dp, mass0, alpha12m, md, mp, mpi, mlb, OrbitalMomentum(1, -1), 0), 1, -1,
couplings[4][0], couplings[4][1]
),
(atfd.exponential_nonresonant_lineshape(m2dp, mass0, alpha32p, md, mp, mpi, mlb, OrbitalMomentum(3, 1), 1), 3, 1,
couplings[5][0], couplings[5][1]
),
(atfd.exponential_nonresonant_lineshape(m2dp, mass0, alpha32m, md, mp, mpi, mlb, OrbitalMomentum(3, -1), 1), 3, -1,
couplings[6][0], couplings[6][1]
),
]
density = atfi.const(0.)
# Decay density is an incoherent sum over initial and final state polarisations
# (assumong no polarisation for Lambda_b^0), and for each polarisation combination
# it is a coherent sum over intermediate states (including two polarisations of
# the intermediate resonance).
for pol_lb in [-1, 1] :
for pol_p in [-1, 1] :
ampl = atfi.complex(atfi.const(0.), atfi.const(0.))
for r in resonances :
lineshape = r[0]
spin = r[1]
parity = r[2]
if pol_p == -1 :
sign = CouplingSign(spin, parity)
coupling1 = atfi.complex(r[3][0], r[3][1]) * sign
coupling2 = atfi.complex(r[4][0], r[4][1]) * sign
else :
coupling1 = atfi.complex(r[3][0], r[3][1])
coupling2 = atfi.complex(r[4][0], r[4][1])
ampl += coupling1*lineshape*\
atfk.helicity_amplitude_3body(dp_theta_r, dp_phi_r, dp_theta_d, dp_phi_d, 1, spin, pol_lb, 1, 0, pol_p, 0)
ampl += coupling2*lineshape*\
atfk.helicity_amplitude_3body(dp_theta_r, dp_phi_r, dp_theta_d, dp_phi_d, 1, spin, pol_lb, -1, 0, pol_p, 0)
density += atfi.density(ampl)
return density
def square_dalitz_plot_jacobian(self, sample):
"""
sample: [mAB^2, mBC^2]
Return the jacobian determinant (|J|) of tranformation from dmAB^2*dmBC^2 -> |J|*dMpr*dThpr where Mpr, Thpr are defined in (AC) frame.
"""
mPrime = self.m_prime_ac(sample)
thPrime = self.theta_prime_ac(sample)
diff_AC = tf.cast(
atfi.sqrt(self.maxac) - atfi.sqrt(self.minac), atfi.fptype())
mAC = atfi.const(0.5) * diff_AC * (
Const(1.0) + atfi.cos(atfi.pi() * mPrime)) + tf.cast(
atfi.sqrt(self.minac), atfi.fptype())
mACSq = mAC * mAC
eAcmsAC = (atfi.const(0.5) *
(mACSq - tf.cast(self.mc2, atfi.fptype()) +
tf.cast(self.ma2, atfi.fptype())) / mAC)
eBcmsAC = (atfi.const(0.5) *
(tf.cast(self.md, atfi.fptype())**2.0 - mACSq -
tf.cast(self.mb2, atfi.fptype())) / mAC)
pAcmsAC = atfi.sqrt(eAcmsAC**2.0 - tf.cast(self.ma2, atfi.fptype()))
pBcmsAC = atfi.sqrt(eBcmsAC**2.0 - tf.cast(self.mb2, atfi.fptype()))
deriv1 = Pi() * atfi.const(0.5) * diff_AC * atfi.sin(
atfi.pi() * mPrime)
deriv2 = Pi() * atfi.sin(atfi.pi() * thPrime)
return atfi.const(4.0) * pAcmsAC * pBcmsAC * mAC * deriv1 * deriv2
def zemach_tensor(m2ab, m2ac, m2bc, m2d, m2a, m2b, m2c, spin):
"""Zemach tensor for 3-body D->ABC decay
:param m2ab:
:param m2ac:
:param m2bc:
:param m2d:
:param m2a:
:param m2b:
:param m2c:
:param spin:
"""
z = None
if spin == 0:
z = atfi.complex(atfi.const(1.), atfi.const(0.))
if spin == 1:
z = atfi.complex(m2ac - m2bc + (m2d - m2c) * (m2b - m2a) / m2ab,
atfi.const(0.))
if spin == 2:
z = atfi.complex(
(m2bc - m2ac + (m2d - m2c) * (m2a - m2b) / m2ab)**2 - 1. / 3. *
(m2ab - 2. * (m2d + m2c) + (m2d - m2c)**2 / m2ab) *
(m2ab - 2. * (m2a + m2b) + (m2a - m2b)**2 / m2ab), atfi.const(0.))
return z
def uniform_random(rnd, x1, x2):
"""
Uniform random numbers from x1 to x2
"""
if isinstance(x1, float): x1 = atfi.const(x1)
if isinstance(x2, float): x2 = atfi.const(x2)
return x1 + rnd * (x2 - x1)
def generate_combinatorial(cuts, rnd):
"""
Generate random combinations of three tracks
"""
meankpt = cuts[0]
meanpipt = cuts[1]
ptcut = cuts[2]
p4pi1 = generate_4momenta(rnd[:, 0:3], meanpipt, ptcut, atfi.const(mpi))
p4pi2 = generate_4momenta(rnd[:, 3:6], meanpipt, ptcut, atfi.const(mpi))
p4k = generate_4momenta(rnd[:, 6:9], meankpt, ptcut, atfi.const(mk))
return p4k, p4pi1, p4pi2
def toymc_model(x, switches=4 * [1]):
return model(
x,
switches=switches,
mrho=atfi.const(0.770),
wrho=atfi.const(0.150),
mkst=atfi.const(0.892),
wkst=atfi.const(0.050),
a1r=atfi.const(1.0),
a1i=atfi.const(0.0),
a2r=atfi.const(0.5),
a2i=atfi.const(0.0),
a3r=atfi.const(2.0),
a3i=atfi.const(0.0),
)
def generate_kstar(cuts, rnd):
"""
Generate random combinations of Kstar and a pion track
"""
meankstarpt = cuts[4]
meanpipt = cuts[1]
ptcut = cuts[2]
mkstargen = breit_wigner_random(rnd[:, 0], mkstar, wkstar)
ones = atfi.ones(rnd[:, 0])
p = atfk.two_body_momentum(mkstargen, mk * ones, mpi * ones)
zeros = atfi.zeros(p)
mom = [
atfk.lorentz_vector(atfk.vector(p, zeros, zeros),
atfi.sqrt(p**2 + mk**2)),
atfk.lorentz_vector(-atfk.vector(p, zeros, zeros),
atfi.sqrt(p**2 + mpi**2))
]
mom = generate_rotation_and_boost(mom, mkstargen, meankstarpt, ptcut,
rnd[:, 2:8])
p4k = mom[0]
p4pi1 = mom[1]
p4pi2 = generate_4momenta(rnd[:, 8:11], meanpipt, ptcut, atfi.const(mpi))
return p4k, p4pi1, p4pi2
def subthreshold_breit_wigner_lineshape(m2,
m0,
gamma0,
ma,
mb,
mc,
md,
dr,
dd,
lr,
ld,
barrier_factor=True):
"""
Breit-Wigner amplitude (with the mass under kinematic threshold)
with Blatt-Weisskopf formfactors, mass-dependent width and orbital barriers
"""
m = atfi.sqrt(m2)
mmin = ma + mb
mmax = md - mc
tanhterm = atfi.tanh((m0 - ((mmin + mmax) / 2.)) / (mmax - mmin))
m0eff = mmin + (mmax - mmin) * (1. + tanhterm) / 2.
q = atfk.two_body_momentum(md, m, mc)
q0 = atfk.two_body_momentum(md, m0eff, mc)
p = atfk.two_body_momentum(m, ma, mb)
p0 = atfk.two_body_momentum(m0eff, ma, mb)
ffr = blatt_weisskopf_ff(p, p0, dr, lr)
ffd = blatt_weisskopf_ff(q, q0, dd, ld)
width = mass_dependent_width(m, m0, gamma0, p, p0, ffr, lr)
bw = relativistic_breit_wigner(m2, m0, width)
ff = ffr * ffd
if barrier_factor:
b1 = orbital_barrier_factor(p, p0, lr)
b2 = orbital_barrier_factor(q, q0, ld)
ff *= b1 * b2
return bw * atfi.complex(ff, atfi.const(0.))
def breit_wigner_lineshape(m2,
m0,
gamma0,
ma,
mb,
mc,
md,
dr,
dd,
lr,
ld,
barrier_factor=True,
ma0=None,
md0=None):
"""
Breit-Wigner amplitude with Blatt-Weisskopf formfactors, mass-dependent width and orbital barriers
"""
m = atfi.sqrt(m2)
q = atfk.two_body_momentum(md, m, mc)
q0 = atfk.two_body_momentum(md if md0 is None else md0, m0, mc)
p = atfk.two_body_momentum(m, ma, mb)
p0 = atfk.two_body_momentum(m0, ma if ma0 is None else ma0, mb)
ffr = blatt_weisskopf_ff(p, p0, dr, lr)
ffd = blatt_weisskopf_ff(q, q0, dd, ld)
width = mass_dependent_width(m, m0, gamma0, p, p0, ffr, lr)
bw = relativistic_breit_wigner(m2, m0, width)
ff = ffr * ffd
if barrier_factor:
b1 = orbital_barrier_factor(p, p0, lr)
b2 = orbital_barrier_factor(q, q0, ld)
ff *= b1 * b2
return bw * atfi.complex(ff, atfi.const(0.))
def polynomial_nonresonant_lineshape(m2,
m0,
coeffs,
ma,
mb,
mc,
md,
lr,
ld,
barrierFactor=True):
"""
2nd order polynomial nonresonant amplitude with orbital barriers
coeffs: list of atfi.complex polynomial coefficients [a0, a1, a2]
"""
def poly(x, cs):
return cs[0] + cs[1] * atfi.complex(
x, atfi.const(0.)) + cs[2] * atfi.complex(x**2, atfi.const(0.))
if barrierFactor:
m = atfi.sqrt(m2)
q = atfk.two_body_momentum(md, m, mc)
q0 = atfk.two_body_momentum(md, m0, mc)
p = atfk.two_body_momentum(m, ma, mb)
p0 = atfk.two_body_momentum(m0, ma, mb)
b1 = orbital_barrier_factor(p, p0, lr)
b2 = orbital_barrier_factor(q, q0, ld)
return poly(m - m0, coeffs) * atfi.complex(b1 * b2, atfi.const(0.))
else:
return poly(m - m0, coeffs)
def spin_rotation_angle(pa, pb, pc, bachelor=2):
"""Calculate the angle between two spin-quantisation axes for the 3-body D->ABC decay
aligned along the particle B and particle A.
pa, pb, pc : 4-momenta of the final-state particles
bachelor : index of the "bachelor" particle (0=A, 1=B, or 2=C)
:param pa:
:param pb:
:param pc:
:param bachelor: (Default value = 2)
"""
if bachelor == 2:
return atfi.const(0.)
pboost = lorentz_vector(
-spatial_components(pb) / scalar(time_component(pb)),
time_component(pb))
if bachelor == 0:
pa1 = spatial_components(lorentz_boost(pa, pboost))
pc1 = spatial_components(lorentz_boost(pc, pboost))
return atfi.acos(scalar_product(pa1, pc1) / norm(pa1) / norm(pc1))
if bachelor == 1:
pac = pa + pc
pac1 = spatial_components(lorentz_boost(pac, pboost))
pa1 = spatial_components(lorentz_boost(pa, pboost))
return atfi.acos(scalar_product(pac1, pa1) / norm(pac1) / norm(pa1))
return None
def _with_form_factor(self, dataset: dict) -> tf.Tensor:
inv_mass_squared = dataset[self._dynamics_props.inv_mass_name]
inv_mass = atfi.sqrt(inv_mass_squared)
mass0 = self._dynamics_props.resonance_mass
gamma0 = self._dynamics_props.resonance_width
m_a = atfi.sqrt(dataset[self._dynamics_props.inv_mass_name_prod1])
m_b = atfi.sqrt(dataset[self._dynamics_props.inv_mass_name_prod2])
meson_radius = self._dynamics_props.meson_radius
l_orbit = self._dynamics_props.orbit_angular_momentum
q_squared = two_body_momentum_squared(inv_mass, m_a, m_b)
q0_squared = two_body_momentum_squared(mass0, m_a, m_b)
ff2 = blatt_weisskopf_ff_squared(q_squared, meson_radius, l_orbit)
ff02 = blatt_weisskopf_ff_squared(q0_squared, meson_radius, l_orbit)
width = gamma0 * (mass0 / inv_mass) * (ff2 / ff02)
# So far its all in float64,
# but for the sqrt operation it has to be converted to complex
width = atfi.complex(width, tf.constant(
0.0, dtype=tf.float64)) * atfi.sqrt(
atfi.complex(
(q_squared / q0_squared),
tf.constant(0.0, dtype=tf.float64),
))
return relativistic_breit_wigner(
inv_mass_squared, mass0, width) * atfi.complex(
mass0 * gamma0 * atfi.sqrt(ff2), atfi.const(0.0))
def _create_normalized_intensity(builder: IntensityBuilder, recipe: dict,
**_: dict) -> Callable:
model = builder.create_element(recipe["Intensity"])
dataset, volume = builder.get_normalization_data()
# its important to convert the dataset to tf tensors (memory footprint)
dataset = {x: tf.constant(y) for x, y in dataset.items()}
return _NormalizedIntensity(model, dataset, atfi.const(volume))
def invariant_mass_jacobian(self, sample):
"""
sample: [mAB^2, mBC^2]
Return the jacobian determinant (|J|) of tranformation from dmAB^2*dmBC^2 -> |J|*dmAB*dmBC. |J| = 4*mAB*mBC
"""
return (atfi.const(4.0) * atfi.sqrt(self.m2ab(sample)) *
atfi.sqrt(self.m2bc(sample)))
def generate_exp(rnd, x1, x2, alpha=None):
"""
Exponential random distribution with constant "alpha",
limited to the range x1 to x2
"""
if isinstance(x1, float): x1 = atfi.const(x1)
if isinstance(x2, float): x2 = atfi.const(x2)
if alpha is None or alpha == 0:
return uniform_random(rnd, x1, x2)
else:
if isinstance(alpha, float): alpha = atfi.const(alpha)
xi1 = atfi.exp(-x1 / alpha)
xi2 = atfi.exp(-x2 / alpha)
ximin = atfi.min(xi1, xi2)
ximax = atfi.max(xi1, xi2)
return atfi.abs(alpha * atfi.log(uniform_random(rnd, ximin, ximax)))
def generate_rho(cuts, rnd):
"""
Generate random combinations of rho -> pi pi and a kaon track
"""
meanrhopt = cuts[5]
meankpt = cuts[0]
ptcut = cuts[2]
mrhogen = breit_wigner_random(rnd[:, 0], mrho, wrho)
ones = atfi.ones(rnd[:, 0])
p = atfk.two_body_momentum(mrhogen, mpi * ones, mpi * ones)
zeros = atfi.zeros(p)
mom = [
atfk.lorentz_vector(atfk.vector(p, zeros, zeros),
atfi.sqrt(p**2 + mpi**2)),
atfk.lorentz_vector(-atfk.vector(p, zeros, zeros),
atfi.sqrt(p**2 + mpi**2))
]
mom = generate_rotation_and_boost(mom, mrhogen, meanrhopt, ptcut, rnd[:,
2:8])
p4k = generate_4momenta(rnd[:, 8:11], meankpt, ptcut, atfi.const(mk))
p4pi1 = mom[0]
p4pi2 = mom[1]
return p4k, p4pi1, p4pi2
def model(x, mrho, wrho, mkst, wkst, a1r, a1i, a2r, a2i, a3r, a3i, switches):
a1 = atfi.complex(a1r, a1i)
a2 = atfi.complex(a2r, a2i)
a3 = atfi.complex(a3r, a3i)
m2ab = phsp.m2ab(x)
m2bc = phsp.m2bc(x)
m2ac = phsp.m2ac(x)
hel_ab = phsp.cos_helicity_ab(x)
hel_bc = phsp.cos_helicity_bc(x)
hel_ac = phsp.cos_helicity_ac(x)
ampl = atfi.complex(atfi.const(0.0), atfi.const(0.0))
if switches[0]:
ampl += (
a1
* atfd.breit_wigner_lineshape(
m2ab, mkst, wkst, mpi, mk, mpi, md, rd, rr, 1, 1
)
* atfd.helicity_amplitude(hel_ab, 1)
)
if switches[1]:
ampl += (
a2
* atfd.breit_wigner_lineshape(
m2bc, mkst, wkst, mpi, mk, mpi, md, rd, rr, 1, 1
)
* atfd.helicity_amplitude(hel_bc, 1)
)
if switches[2]:
ampl += (
a3
* atfd.breit_wigner_lineshape(
m2ac, mrho, wrho, mpi, mpi, mk, md, rd, rr, 1, 1
)
* atfd.helicity_amplitude(hel_ac, 1)
)
if switches[3]:
ampl += atfi.cast_complex(atfi.ones(m2ab)) * atfi.complex(
atfi.const(5.0), atfi.const(0.0)
)
return atfd.density(ampl)
def momentum_scale(dm, moms):
"""
Function to calculate the momentum scale factor for kinematic fit
dm : invariant mass shift from the desired value
moms : list of 4-momenta of the final state particles
"""
psum = sum(moms) # sum of 4-momenta
pvecsum = atfk.spatial_components(psum)
esum = atfk.time_component(psum)
dedd = atfi.const(0.)
pdpdd = atfi.const(0.)
for mom in moms:
pm = atfk.p(mom) # Absolute momentum
em = atfk.time_component(mom) # Energy
pvec = atfk.spatial_components(mom) # 3-momentum
s = momentum_resolution(pm)
dedd += s * pm**2 / em
pdpdd += atfk.scalar_product(pvecsum, pvec) * s
return -dm / (2. * esum * dedd - 2. * pdpdd)
def wigner_capital_d(phi, theta, psi, j, m1, m2):
"""Calculate Wigner capital-D function.
phi,
theta,
psi : Rotation angles
j : spin (in units of 1/2, e.g. 1 for spin=1/2)
m1 and m2 : spin projections (in units of 1/2, e.g. 1 for projection 1/2)
:param phi:
:param theta:
:param psi:
:param j2:
:param m2_1:
:param m2_2:
"""
i = atfi.complex(atfi.const(0), atfi.const(1))
return Exp(-i*atfi.cast_complex(m1/2.*phi)) * \
atfi.cast_complex(wigner_small_d(theta, j, m1, m2)) * \
Exp(-i * atfi.cast_complex(m2 / 2. * psi))
def _model(a1r, a1i, a2r, a2i, a3r, a3i, switches=4 * [1]):
a1 = atfi.complex(a1r, a1i)
a2 = atfi.complex(a2r, a2i)
a3 = atfi.complex(a3r, a3i)
ampl = atfi.cast_complex(atfi.ones(m2ab)) * atfi.complex(
atfi.const(0.0), atfi.const(0.0))
if switches[0]:
ampl += a1 * bw1 * hel_ab
if switches[1]:
ampl += a2 * bw2 * hel_bc
if switches[2]:
ampl += a3 * bw3 * hel_ac
if switches[3]:
ampl += atfi.cast_complex(atfi.ones(m2ab)) * atfi.complex(
atfi.const(5.0), atfi.const(0.0))
return atfd.density(ampl)
def relativistic_breit_wigner(m2, mres, wres):
"""
Relativistic Breit-Wigner
"""
if wres.dtype is atfi.ctype():
return tf.math.reciprocal(
atfi.cast_complex(mres * mres - m2) -
atfi.complex(atfi.const(0.), mres) * wres)
if wres.dtype is atfi.fptype():
return tf.math.reciprocal(atfi.complex(mres * mres - m2, -mres * wres))
return None
def complex_two_body_momentum(md, ma, mb):
"""Momentum of two-body decay products D->AB in the D rest frame.
Output value is a complex number, analytic continuation for the
region below threshold.
:param md:
:param ma:
:param mb:
"""
return atfi.sqrt(
atfi.complex(
(md**2 - (ma + mb)**2) * (md**2 - (ma - mb)**2) / (4 * md**2),
atfi.const(0.)))
def _with_form_factor(self, dataset: dict) -> tf.Tensor:
inv_mass = atfi.sqrt(dataset[self._dynamics_props.inv_mass_name])
m_a = atfi.sqrt(dataset[self._dynamics_props.inv_mass_name_prod1])
m_b = atfi.sqrt(dataset[self._dynamics_props.inv_mass_name_prod2])
meson_radius = self._dynamics_props.meson_radius
l_orbit = self._dynamics_props.orbit_angular_momentum
q_squared = two_body_momentum_squared(inv_mass, m_a, m_b)
return atfi.complex(
atfi.sqrt(
blatt_weisskopf_ff_squared(q_squared, meson_radius, l_orbit)),
atfi.const(0.0),
)
def _bw_ff_squared(x):
if l_orbit == 0:
return atfi.const(1.0)
if l_orbit == 1:
return (2 * x) / (x + 1)
if l_orbit == 2:
return (13 * x * x) / ((x - 3) * (x - 3) + 9 * x)
if l_orbit == 3:
return (277 * x * x * x) / (x * (x - 15) * (x - 15) + 9 *
(2 * x - 5) * (2 * x - 5))
if l_orbit == 4:
return (12746 * x * x * x * x) / ((x * x - 45 * x + 105) *
(x * x - 45 * x + 105) + 25 * x *
(2 * x - 21) * (2 * x - 21))
def breit_wigner_decay_lineshape(m2, m0, gamma0, ma, mb, meson_radius,
l_orbit):
"""
Breit-Wigner amplitude with Blatt-Weisskopf form factor for the decay products,
mass-dependent width and orbital barriers
Note: This function does not include the production form factor.
"""
inv_mass = atfi.sqrt(m2)
q_squared = atfk.two_body_momentum_squared(inv_mass, ma, mb)
q0_squared = atfk.two_body_momentum_squared(m0, ma, mb)
ff2 = blatt_weisskopf_ff_squared(q_squared, meson_radius, l_orbit)
ff02 = blatt_weisskopf_ff_squared(q0_squared, meson_radius, l_orbit)
width = gamma0 * (m0 / inv_mass) * (ff2 / ff02)
# So far its all in float64,
# but for the sqrt operation it has to be converted to complex
width = atfi.complex(width, atfi.const(0.0)) * atfi.sqrt(
atfi.complex(
(q_squared / q0_squared),
atfi.const(0.0),
))
return relativistic_breit_wigner(m2, m0, width) * atfi.complex(
m0 * gamma0 * atfi.sqrt(ff2), atfi.const(0.0))
def hankel1(x):
if l == 0:
return atfi.const(1.0)
if l == 1:
return 1 + x * x
if l == 2:
x2 = x * x
return 9 + x2 * (3.0 + x2)
if l == 3:
x2 = x * x
return 225 + x2 * (45 + x2 * (6 + x2))
if l == 4:
x2 = x * x
return 11025.0 + x2 * (1575.0 + x2 * (135.0 + x2 * (10.0 + x2)))
def generate_4momenta(rnd, meanpt, ptcut, m):
"""
Generate random 4-momenta according to specified mean Pt, minimum Pt, and mass of the particle, flat in eta and phi
"""
pt = generate_pt(rnd[:, 0], meanpt, ptcut, atfi.const(200.)) # Pt in GeV
eta = generate_eta(rnd[:, 1]) # Eta
phi = generate_phi(rnd[:, 2]) # Phi
theta = 2. * atfi.atan(atfi.exp(-eta))
p = pt / atfi.sin(theta) # Full momentum
e = atfi.sqrt(p**2 + m**2) # Energy
px = p * atfi.sin(theta) * atfi.sin(phi)
py = p * atfi.sin(theta) * atfi.cos(phi)
pz = p * atfi.cos(theta)
return atfk.lorentz_vector(atfk.vector(px, py, pz), e)
def toymc_model(x, switches=4 * [1]):
return model(x)(
switches=switches,
a1r=atfi.const(1.0),
a1i=atfi.const(0.0),
a2r=atfi.const(0.5),
a2i=atfi.const(0.0),
a3r=atfi.const(2.0),
a3i=atfi.const(0.0),
)