{title2=$L_{\mu\nu}$}

The product of lepton-current <matrix elements> with a specified <spin sum> or <spin average>. Its normalization must be stated when quoting a <differential scattering cross-section>.

Leptonic tensor

{title2=$\mu_F$}

The scale at which collinear contributions are assigned to <parton distribution functions> rather than the hard scattering coefficient. The separate factors depend on this scale, while a consistently calculated physical <differential scattering cross-section> does not.

Parton factorization scale

Substitute the preceding <spin average> into the supplied <differential scattering cross-section>. Since $e^4=16\pi^2\alpha^2$,
$$
\frac{d\sigma}{d\Omega}=\frac{\alpha^2}{4s}(1+\cos^2\theta).
$$
The angular integral is $2\pi\int_{-1}^1(1+x^2)\,dx=16\pi/3$. Consequently the <relativistic scattering cross-section> for the massless <electron-positron annihilation into a muon pair> is
$$
\boxed{\sigma=\frac{4\pi\alpha^2}{3s}.}
$$
The outgoing particles are distinct, so there is no identical-particle factor. Comparing with the requested monomial form gives
$$
\boxed{B=\frac{4\pi}{3},\qquad j=2,\quad k=-1,\quad l=0,\quad m=0.}
$$
The integrated answer depends only on the incoming invariant $s$; the angular variables $t,u$ have been integrated out.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 301 / 1 / v / Solution

The <deep inelastic scattering> process contains a virtual <photon> exchanged between the <Electron> and the <hadron>:

\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-305-q4a.png]
{title=Inclusive electron-hadron scattering through a virtual photon}
{height=350}

Use an electromagnetic <vector current> $J^\mu$ containing the dimensionless <quark> charges, with the coupling $e$ factored out. The <scattering amplitude>, up to an irrelevant phase, is
$$
\mathcal M_X=\frac{e^2}{q^2}\bar u(p')\gamma_\mu u(p)\langle X|J^\mu|H\rangle.
$$
To match the printed prefactor, define the <leptonic tensor> with a <spin sum> over both <Electron> spins and keep the initial <spin average> $1/2$ outside it. The <gamma-matrix trace> gives
$$
\boxed{L_{\mu\nu}=\operatorname{Tr}(\not p'\gamma_\mu\not p\gamma_\nu)
=4\left(p_\mu p'_\nu+p_\nu p'_\mu-g_{\mu\nu}p\cdot p'\right).}
$$
For a stationary target and massless <Electron>, the <invariant flux factor> is $4ME$. The inclusive final-state <Lorentz-invariant phase-space measure> and target <spin average> are contained in $4\pi W_H^{\mu\nu}$. Thus the <differential scattering cross-section> is
$$
\frac{d\sigma}{d^3p'}=\frac1{4ME}\frac1{(2\pi)^3 2E'}\frac12\frac{e^4}{(q^2)^2}L_{\mu\nu}(4\pi W_H^{\mu\nu})
=\boxed{\frac{e^4}{8(2\pi)^2 MEE'(q^2)^2}L_{\mu\nu}W_H^{\mu\nu}.}
$$
Here $q^4$ means $(q^2)^2$. If the initial <spin average> is instead built into the <leptonic tensor>, its normalization is $2(\cdots)$ and the displayed cross-section prefactor must be doubled. The two conventions give the same observable.

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 4 / a / Solution

{title2=$d\sigma/d\Omega$}

The differential scattering cross-section is the effective incident area scattered per unit <solid angle>. It is defined by the ratio of the outgoing particle rate through $d\Omega$ to the incident flux.

Codex @codex  0

 Incoming links: Differential scattering cross-section