ABSTRACT
Using the property of the delta function given by equation 2.32, the delta function can be re-expressed:
δ[(p1 + p2 − p23)2 −m24] = δ(E3 − E˜3)
2 ∣∣∣∣W − E˜3√E˜23−m23 |p1 + p2| cos θ
∣∣∣∣ , (B.5)
where W = E1 + E2, and E˜3 is the solution to
s + m23 −m24 − 2WE˜3 + 2|p1 + p2| cos θ √
E˜23 −m23 = 0. (B.6) Integrating over E3, the Lorentz-invariant phase-space element in a very general, but impractical, form is
dLips(s; p3, p4) = N3N4 16π2
√ E˜23 −m23∣∣∣∣W − E˜3√E˜23−m23 |p1 + p2| cos θ
∣∣∣∣ dΩ. (B.7)
The result is general and includes no approximations. There are two common reference frames used in practical calculations: the center-of-mass frame, and the laboratory frame in which one of the initial particles is at rest. In the center-of-mass reference frame (CMS), |p1 + p2| = 0 and
dLips(s; p3, p4) = N3N4 16π2
|p3|√ s dΩ center-of-mass frame , (B.8)
where
|p3| = √
E˜23 −m23 and E˜3 = s + m23 −m24
2 √ s
. (B.9)
In quantum electrodynamics, the result is often not this general. For two photons in the initial state
dLips(s; p3, p4) = N3N4 32π2
√ 1− 4m
s dΩ CMS: γ + γ → 3 + 4, (B.10)
where m ≡ m3 = m4. For two photons in the final state
dLips(s; p3, p4) = N3N4 32π2
dΩ CMS: 1 + 2→ γ + γ. (B.11) For processes with no photons, the expression does not simplify unless at least E3 ≈ |p3|:
dLips(s; p3, p4) = N3N4 32π2
s + m23 −m24 s
dΩ CMS: E3 ≈ |p3|. (B.12)
Obvious additional simplifications follow, if some of the masses can be ignored. Equation B.8 can be easily adapted to two-body decays of a particle at rest:
dLips(m1; p2, p3) = N2N3 16π2
|p | m1
dΩ decay 1→ 2 + 3 . (B.13)
The other reference frame of interest is the laboratory rest frame. In the rest frame of particle 1, p1 = 0 and
dLips(s; p3, p4) = N3N4 16π2
|p2| cos θ dΩ laboratory frame .
