import sympy
from equation_database.util.doc import bib, equation
from equation_database.util.parse import frac
[docs]
@equation()
def equation_2_1_30(
sigma_0=sympy.Symbol("sigma_0"),
alpha=sympy.Symbol("alpha"),
e_q=sympy.Symbol("e_q"),
Q=sympy.Symbol("Q"),
):
"""
Cross section for $\\gamma^* \\to q \\bar q$
Args:
sigma_0: norm cross section
alpha: fine structure constant
e_q: electric charge of the quark
Q: mass of the virtual photon
"""
return sympy.Eq(sigma_0, 3 * alpha * e_q**2 * Q)
[docs]
@equation()
def equation_2_3_32(
sigma=sympy.Symbol("sigma"),
sigma_0=sympy.Symbol("sigma_0"),
alpha_s=sympy.Symbol("alpha_s"),
x_1=sympy.Symbol("x_1"),
x_2=sympy.Symbol("x_2"),
):
"""
Differentiated cross section for $e^+e^- \\to q \\bar q g$
Args:
sigma: cross section
sigma_0: norm cross section
alpha_s: strong coupling constant
x_1: quark momentum fraction
x_2: antiquark momentum fraction
"""
return sympy.Eq(
sympy.Derivative(sigma, x_1, x_2) / sigma_0,
2 * alpha_s / 3 / sympy.pi * (x_1**2 + x_2**2) / ((1 - x_1) * (1 - x_2)),
)
[docs]
@equation()
def equation_4_3_20(
e=sympy.Symbol("e"),
e_q=sympy.Symbol("e_q"),
g_s=sympy.Symbol("g_s"),
Q=sympy.Symbol("Q"),
u=sympy.Symbol("u"),
t=sympy.Symbol("t"),
):
"""
$\\gamma^* g \\to q \\bar q$ scattering averaged matrix element
Args:
e: electric charge
e_q: electric charge of the quark
g_s: strong coupling constant
Q: mass of the virtual photon
u: Mandelstam variable u
t: Mandelstam variable t
"""
return (
e**2
* e_q**2
* g_s**2
* frac("4/8")
* frac("1/2")
* 8
* (u / t + t / u + 2 * Q**2 * (u + t + Q**2) / (t * u))
)
[docs]
@bib()
def bibtex():
# https://www.desy.de/~jung/qcd_and_mc_2009-2010/R.Field-Applications-of-pQCD.pdf
return r"""
@book{Field:1989uq,
author = "Field, R. D.",
title = "{Applications of Perturbative QCD}",
volume = "77",
year = "1989"
}
"""