from __future__ import annotations
import numpy as np
import tensorflow as tf
import zfit
from zfit import z
from zfit.core.space import ANY_LOWER, ANY_UPPER, Space
from zfit.util import ztyping
@z.function(wraps="tensor")
def relbw_pdf_func(x, m, gamma):
"""Calculate the relativistic Breit-Wigner PDF.
Args:
x: value(s) for which the CDF will be calculated.
m: Mean value
gamma: width
Returns:
`tf.Tensor`: The calculated PDF values.
Notes:
Based on code from this [github gist](https://gist.github.com/andrewfowlie/cd0ed7e6c96f7c9e88f85eb3b9665b97#file-bw-py-L87-L110)
"""
alpha = gamma / m
gamma2 = m**2 * (1.0 + alpha**2) ** 0.5
k = 2.0 ** (3.0 / 2.0) * m**2 * alpha * gamma2 / (np.pi * (m**2 + gamma2) ** 0.5)
return k / ((x**2 - m**2) ** 2 + m**4 * alpha**2)
[docs]
class RelativisticBreitWigner(zfit.pdf.BasePDF):
_N_OBS = 1
def __init__(
self,
m: ztyping.ParamTypeInput,
gamma: ztyping.ParamTypeInput,
obs: ztyping.ObsTypeInput,
*,
extended: ztyping.ParamTypeInput | None = None,
norm: ztyping.NormTypeInput | None = None,
name: str = "RelativisticBreitWigner",
label: str | None = None,
):
"""Relativistic Breit-Wigner distribution.
Formula for PDF and CDF are based on https://gist.github.com/andrewfowlie/cd0ed7e6c96f7c9e88f85eb3b9665b97
Args:
m: the average value
gamma: the width of the distribution
obs: |@doc:pdf.init.obs| Observables of the
model. This will be used as the default space of the PDF and,
if not given explicitly, as the normalization range.
The default space is used for example in the sample method: if no
sampling limits are given, the default space is used.
The observables are not equal to the domain as it does not restrict or
truncate the model outside this range. |@docend:pdf.init.obs|
extended: |@doc:pdf.init.extended| The overall yield of the PDF.
If this is parameter-like, it will be used as the yield,
the expected number of events, and the PDF will be extended.
An extended PDF has additional functionality, such as the
``ext_*`` methods and the ``counts`` (for binned PDFs). |@docend:pdf.init.extended|
norm: |@doc:pdf.init.norm| Normalization of the PDF.
By default, this is the same as the default space of the PDF. |@docend:pdf.init.norm|
name: |@doc:pdf.init.name| Name of the PDF.
Maybe has implications on the serialization and deserialization of the PDF.
For a human-readable name, use the label. |@docend:pdf.init.name|
label: |@doc:pdf.init.label| Human-readable name
or label of
the PDF for a better description, to be used with plots etc.
Has no programmatical functional purpose as identification. |@docend:pdf.init.label|
"""
params = {"m": m, "gamma": gamma}
super().__init__(obs=obs, params=params, name=name, extended=extended, norm=norm, label=label)
@zfit.supports()
def _unnormalized_pdf(self, x: tf.Tensor, params) -> tf.Tensor:
"""Calculate the PDF at value(s) x.
Args:
x : Either one value or an array
Returns:
`tf.Tensor`: The value(s) of the unnormalized PDF at x.
"""
m = params["m"]
gamma = params["gamma"]
x = x[0]
return relbw_pdf_func(x, m=m, gamma=gamma)
@z.function(wraps="tensor")
def arctan_complex(x):
r"""Function that evaluates arctan(x) using tensorflow but also supports complex numbers. It is defined as.
.. math::
\mathrm{arctan}(x) = \frac{i}{2} \left(\ln(1-ix) - \ln(1+ix)\right)
Args:
x: tf.Tensor
Returns:
.. math:: \mathrm{arctan}(x)
Notes:
Formula is taken from https://www.wolframalpha.com/input/?i=arctan%28a%2Bb*i%29
TODO: move somewhere?
"""
return 1 / 2 * 1j * (tf.math.log(1 - 1j * x) - tf.math.log(1 + 1j * x))
@z.function(wraps="tensor")
def relbw_cdf_func(x, m, gamma):
"""Analytical function for the CDF of the relativistic Breit-Wigner distribution.
Args:
x: value(s) for which the CDF will be calculated.
m: Mean value
gamma: width
Returns:
`tf.Tensor`: The calculated CDF values.
Notes:
Based on code from this [github gist](https://gist.github.com/andrewfowlie/cd0ed7e6c96f7c9e88f85eb3b9665b97#file-bw-py-L112-L154)
"""
gamma = z.to_complex(gamma)
m = z.to_complex(m)
x = z.to_complex(z.unstack_x(x))
alpha = gamma / m
gamma2 = m**2 * (1.0 + alpha**2) ** 0.5
k = 2.0 ** (3.0 / 2.0) * m**2 * alpha * gamma2 / (np.pi * (m**2 + gamma2) ** 0.5)
arg_1 = z.to_complex(-1) ** (1.0 / 4.0) / (-1j + alpha) ** 0.5 * x / m
arg_2 = z.to_complex(-1) ** (3.0 / 4.0) / (1j + alpha) ** 0.5 * x / m
shape = -1j * arctan_complex(arg_1) / (-1j + alpha) ** 0.5 - arctan_complex(arg_2) / (1j + alpha) ** 0.5
norm = z.to_complex(-1) ** (1.0 / 4.0) * k / (2.0 * alpha * m**3)
cdf_ = shape * norm
return z.to_real(cdf_)
def relbw_integral(limits: ztyping.SpaceType, params: dict, model) -> tf.Tensor:
"""Calculates the analytic integral of the relativistic Breit-Wigner PDF.
Args:
limits: An object with attribute rect_limits.
params: A hashmap from which the parameters that defines the PDF will be extracted.
model: Will be ignored.
Returns:
The calculated integral.
"""
del model
lower, upper = limits.rect_limits
lower_cdf = relbw_cdf_func(x=lower, m=params["m"], gamma=params["gamma"])
upper_cdf = relbw_cdf_func(x=upper, m=params["m"], gamma=params["gamma"])
return upper_cdf - lower_cdf
# These lines of code adds the analytic integral function to RelativisticBreitWigner PDF.
relbw_integral_limits = Space(axes=(0,), limits=(((ANY_LOWER,),), ((ANY_UPPER,),)))
RelativisticBreitWigner.register_analytic_integral(func=relbw_integral, limits=relbw_integral_limits)
