= Solution
Set $s=q^2$. The angular integral of the <electron-positron annihilation into a muon pair> differential cross-section is
$$
\int d\Omega(1+\cos^2\theta)=2\pi\int_{-1}^1(1+z^2)\,dz=\frac{16\pi}3.
$$
Therefore
$$
\boxed{\sigma_{\mu\mu}(s)=\frac{4\pi\alpha^2}{3s}.}
$$
The printed comparison of $\sqrt{s}$ with a quantity in $\mathrm{GeV}^2$ has inconsistent units: the high-energy condition means an energy large compared with particle masses, or equivalently an energy-squared large compared with their squared masses.
At leading order, <inclusive electron-positron annihilation into hadrons> proceeds through $e^+e^-\to\gamma^*\to q\overline q$, as in the first panel below. The <photon> couples to each <quark> with strength $eQ_f$; the corresponding contribution is $Q_f^2$ times the muon-pair rate per colour. Summing over the three orthogonal final colours and active flavours gives the <hadronic R ratio>
$$
\boxed{\sigma_{\mathrm{LO}}(s)=\frac{4\pi\alpha^2}{3s}
N_c\sum_{f\ \mathrm{active}}Q_f^2,\qquad
R_{\mathrm{LO}}=N_c\sum_fQ_f^2,\quad N_c=3.}
$$
For massless $u,d,s$ flavours this gives $R=2$, with charm included $R=10/3$, and with bottom as well $R=11/3$.
The approximation assumes unpolarized beams and negligible electron mass; active quark masses are small relative to $\sqrt{s}$, and heavy flavours below threshold are excluded. Photon exchange dominates, so $Z$ exchange and photon-$Z$ interference are neglected; this is appropriate sufficiently below the <Z boson> scale, not at arbitrary high energy. The coupling $\alpha$ must be evaluated consistently in numerator and reference rate. We also neglect higher-order QED radiation. Relating the inclusive partonic calculation to hadrons invokes <quark-hadron duality>: the energy should be in a perturbative continuum region, away from thresholds and narrow resonances, with power-suppressed nonperturbative effects neglected. An exclusive hadron channel cannot be obtained merely by multiplying the parton rate by the colour count.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-47-qcd-diagrams.png]
{title=Born hadronic production, real and virtual first-order QCD corrections, and their local counterterms}
At next order in <QCD>, the real-emission amplitudes are the two distinct diagrams for $e^+e^-\to\gamma^*\to q\overline qg$, with the <gluon> emitted from the quark or antiquark. They must be added before squaring, including their interference. Each amplitude is order $eg_s$ at the hadronic current vertex and gives a relative order-$g_s^2$ contribution to the rate.
The virtual order-$g_s^2$ amplitudes consist of the gluon correction to the photon-quark vertex and the quark and antiquark self-energy/external-state normalization contributions. The figure also shows the corresponding local vertex and field counterterms. In an amputated on-shell calculation the external-leg terms are represented by wave-function factors, rather than counted as extra final states. There is no gluon attachment to an electron, since it carries no colour. Gluon self-energy insertions and additional non-Abelian vertices enter at higher orders here. The rate correction is the Born-virtual interference $2\operatorname{Re}(\mathcal M_0^*\mathcal M_{\mathrm{virt}})$ plus the integrated squared real-emission amplitude, not the square of the one-loop amplitude.
Use <dimensional regularization>, $d=4-2\varepsilon$, to keep ultraviolet and infrared singularities in a common scheme. <Ultraviolet divergences> in the virtual quark and vertex contributions are removed by consistent <wave-function renormalization> and local counterterms; the conserved electromagnetic-current identity relates the vertex and field renormalizations. One uses a renormalized <QCD coupling> $g_s(\mu)$; its running changes this first correction only at the next perturbative order. In the massless on-shell scheme some external self-energy integrals are scaleless and vanish, but that statement combines ultraviolet and infrared poles and must not be used to discard the associated normalization terms inconsistently.
The remaining <infrared divergences> occur when a gluon is soft, and <collinear divergences> occur when it is parallel to a massless quark. They appear in both virtual diagrams and the real-emission phase-space integral. For the fully inclusive total rate, unresolved final states are summed and <real-virtual infrared cancellation in inclusive QCD> removes their poles. A resolved jet rate instead needs an <infrared-safe observable> and a resolution prescription. An isolated real or virtual correction is not a finite physical inclusive cross-section.
Define the dimensionless constant $A$ with the conventional loop factor extracted. Then the requested form is
$$
\boxed{\frac{\sigma_{\mathrm{LO+NLO}}}{\sigma_{\mathrm{LO}}}
=1+A\frac{g_s^2(\mu)}{16\pi^2}+O(g_s^4)
=1+A\frac{\alpha_s(\mu)}{4\pi}+O(\alpha_s^2).}
$$
Fixed numerical and colour factors are contained in $A$; choosing another extracted prefactor simply redefines that constant. No explicit loop coefficient is required to establish the form.
The Born and virtual processes give two energetic <particle jets>. Hard, well-separated real radiation gives a <three-jet event>; unresolved radiation changes jet widths and event-shape distributions, while the combined terms correct the inclusive rate. \b[Resolved three-jet events in electron-positron annihilation provided the direct gluon signature], observed at PETRA in 1979. Their interpretation is energetic quark, antiquark and gluon fragmentation, with approximately planar three-body momentum balance. The primary institutional account is https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=1643&printversion=1 .
Back to article page
