import sympy
from equation_database.util.doc import bib, equation
from equation_database.util.parse import frac
[docs]
@equation()
def table_7_1_qqp_qqp(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$qq' \\to qq'$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s: Mandelstam variable s
t: Mandelstam variable t
u: Mandelstam variable u
"""
return frac("4/9") * (s**2 + u**2) / (t**2)
[docs]
@equation()
def table_7_1_qqpb_qqpb(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$q\\bar{q}' \\to q\\bar{q}'$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s: Mandelstam variable s
t: Mandelstam variable t
u: Mandelstam variable u
"""
return frac("4/9") * (s**2 + u**2) / (t**2)
[docs]
@equation()
def table_7_1_qq_qq(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$qq \\to qq$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s: Mandelstam variable s
t: Mandelstam variable t
u: Mandelstam variable u
"""
return frac("4/9") * ((s**2 + u**2) / (t**2) + (s**2 + t**2) / (u**2)) - frac(
"8/27"
) * s**2 / (u * t)
[docs]
@equation()
def table_7_1_qqb_qpqpb(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$q\\bar{q} \\to q'\\bar{q}'$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
"""
return frac("4/9") * ((t**2 + u**2) / (s**2))
[docs]
@equation()
def table_7_1_qqb_qqb(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$q\\bar{q} \\to q\\bar{q}$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
"""
return frac("4/9") * ((s**2 + u**2) / (t**2) + (t**2 + u**2) / (s**2)) - frac(
"8/27"
) * u**2 / (s * t)
[docs]
@equation()
def table_7_1_qqb_gg(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$q\\bar{q} \\to gg$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
"""
return frac("32/27") * (t**2 + u**2) / (t * u) - frac("8/3") * (t**2 + u**2) / (
s**2
)
[docs]
@equation()
def table_7_1_gg_qqb(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$gg \\to q\\bar{q}$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
"""
return frac("1/6") * (t**2 + u**2) / (t * u) - frac("3/8") * (t**2 + u**2) / (s**2)
[docs]
@equation()
def table_7_1_gq_gq(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$gq \\to gq$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
"""
return frac("-4/9") * (s**2 + u**2) / (s * u) + (u**2 + s**2) / t**2
[docs]
@equation()
def table_7_1_gg_gg(s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u")):
"""
$gg \\to gg$
The invariant matrix elements squared for two-to-two parton subprocesses with massless partons. The colour and spin indices are averaged (summed) over initial (final) states.
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
"""
return frac("9/2") * (3 - t * u / s**2 - s * u / t**2 - s * t / u**2)
[docs]
@equation()
def table_7_2_qq_ag(
s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u"), N=sympy.Symbol("N")
):
"""$q\\bar q \\to \\gamma^* g$
Lowest order processes for virtual photon production. The colour and spin indices are averaged (summed) over initial (final) states. For a real photon (s + t + u) = 0 and for SU(3) we have N = 3
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
N : Number of colors
"""
return (N**2 - 1) / N**2 * (t**2 + u**2 + 2 * s * (s + t + u)) / (t * u)
[docs]
@equation()
def table_7_2_gq_aq(
s=sympy.Symbol("s"), t=sympy.Symbol("t"), u=sympy.Symbol("u"), N=sympy.Symbol("N")
):
"""$gq \\to \\gamma^* q$
Lowest order processes for virtual photon production. The colour and spin indices are averaged (summed) over initial (final) states. For a real photon (s + t + u) = 0 and for SU(3) we have N = 3
Args:
s : Mandelstam variable s
t : Mandelstam variable t
u : Mandelstam variable u
N : Number of colors
"""
return -1 / N * (s**2 + u**2 + s * t * (s + t + u) / (s * u))
[docs]
@bib()
def bibtex():
bibtex: str = r"""
@book{Ellis:1996mzs,
author = "Ellis, R. Keith and Stirling, W. James and Webber, B. R.",
title = "{QCD and collider physics}",
doi = "10.1017/CBO9780511628788",
isbn = "978-0-511-82328-2, 978-0-521-54589-1",
publisher = "Cambridge University Press",
volume = "8",
month = "2",
year = "2011"
}
"""
return bibtex