Inclusive Annihilation and the Emergence of Jets

Inclusive $e^+e^-$ annihilation gives the cleanest first QCD prediction because the incoming state has no hadronic structure. The ratio to muon-pair production exposes the number of colors and quark charges; its first QCD correction shows how real and virtual partons combine into an inclusive hadronic observable, while resolved radiation gives the first three-jet events.

Required background. QCD fields, scales, and the perturbative domain fixes $C_F$, $N_c$, $\alpha_s$, and the relevant scale test. Bloch–Nordsieck and KLN cancellation supplies the inclusive real–virtual cancellation.

Helpful background. Infrared and collinear safety supplies the measurement-function test used for jets.

The inclusive annihilation ratio

Away from quark thresholds and narrow resonances, and at energies where photon exchange is the declared approximation, define

$$R(s)=\frac{\sigma(e^+e^-\to\text{hadrons};s)} {\sigma(e^+e^-\to\mu^+\mu^-;s)}.$$

For massless fermions at tree level,

$$\sigma(e^+e^-\to\mu^+\mu^-)=\frac{4\pi\alpha^2}{3s}.$$

Replacing the muon by a quark of electric charge $Q_f$ contributes $Q_f^2$, and summing its three colors gives

$$R_0(s)=N_c\sum_{f\,\mathrm{active}}Q_f^2, \qquad N_c=3.$$

The sum is over flavors that can be treated in the stated mass approximation. Near a threshold, the massless step-function picture fails; close to a narrow resonance, bound-state dynamics invalidates the smooth fixed-order description. At sufficiently high energy, $Z$ exchange and photon–$Z$ interference must also be included. The tree-level color and charge counting is derived in Schwartz 2014, §26.3.1, pp. 513–16.

The first QCD correction

At order $\alpha_s$, two classes contribute to the same inclusive final-state question:

• virtual gluon corrections to $e^+e^-\to q\bar q$; and
• real emission $e^+e^-\to q\bar qg$ integrated over unresolved as well as resolved configurations.

Each class is infrared divergent in isolation. For an inclusive measurement, the soft and collinear singularities cancel after ultraviolet renormalization and after summing the degenerate states. In the massless photon-exchange limit,

$$R(s)=R_0(s) \left[1+\frac{\alpha_s(\mu_R)}{\pi} +O(\!\left(\alpha_s^2\right)) +O\!\left(\frac{m_f^2}{s}\right) +O\!\left(\frac{\Lambda_{\mathrm{QCD}}^p}{s^{p/2}}\right) \right].$$

The coefficient is independent of an arbitrary separation between “two-parton” and “three-parton” regions; only their sum is the inclusive prediction. At finite order, the running of $\alpha_s$ leaves residual $\mu_R$ dependence of the first omitted order. The explicit one-loop calculation and cancellation are given in Schwartz 2014, §§26.3.2–26.3.3, pp. 516–17.

This result answers a precise question: an inclusive, electroweakly specified, sufficiently short-distance rate. It does not predict an event-by-event integer number of hadrons.

From partons to jets

At Born level the energetic quark and antiquark recoil back to back. Confinement replaces each colored parton by a collimated spray of hadrons, but energy flow still retains the underlying short-distance directions. A hard, wide-angle gluon in $q\bar qg$ produces a third energetic spray, so three-jet rates begin one power of $\alpha_s$ above the two-jet Born process.

To turn that statement into a calculation, introduce a measurement function $\mathcal M_n(p_1,\ldots,p_n)$. Infrared and collinear safety requires

$$\mathcal M_{n+1}(p_1,\ldots,p_n,k\to0) \longrightarrow \mathcal M_n(p_1,\ldots,p_n),$$

and

$$\mathcal M_{n+1}(\ldots,zp,(1-z)p,\ldots) \longrightarrow \mathcal M_n(\ldots,p,\ldots)$$

in unresolved soft and collinear limits. These conditions allow the real singular region to cancel the corresponding virtual singularity bin by bin. The classic two-cone energy-fraction definition is an explicit infrared-safe separation of two-jet from multijet events Sterman and Weinberg 1977, pp. 1436–39.

The physical sequence is therefore

$$\text{hard current} \longrightarrow q\bar q(+g+\cdots) \longrightarrow \text{parton shower} \longrightarrow \text{hadrons} \longrightarrow \text{IRC-safe jets or event shapes}.$$

Only the first radiation stages are perturbatively calculable without nonperturbative input. Hadronization corrections are suppressed for sufficiently inclusive high-scale observables but can be enhanced near endpoints or by narrow jet definitions.

Checks, domains, and failure modes

Charge and color check. Turning off QCD must recover $N_c\sum_fQ_f^2$. Dropping the color multiplicity or summing charges before squaring gives the wrong observable.

Cancellation check. Regulate real and virtual pieces in the same scheme and verify that infrared poles cancel in the inclusive or IRC-safe sum. A leftover pole usually signals an incomplete state sum or an unsafe measurement.

Scale check. Expanding a higher-order or resummed result to $O(\alpha_s)$ must reproduce the coefficient above in its common domain. Renormalization-scale variation should be performed coherently, not separately on real and virtual terms whose cancellation is required.

Boundary check. Add quark-mass effects near thresholds, electroweak exchanges where relevant, resummation when a jet-resolution variable is small, and hadronization corrections when the induced scale approaches $\Lambda_{\mathrm{QCD}}$.

Common pitfalls

Calling a parton count a jet count. Jets are outputs of a specified clustering or event-shape definition. A soft or collinear splitting must not change an IRC-safe classification.

Explaining finiteness by confinement. The perturbative inclusive rate is finite because degenerate real and virtual contributions cancel. Confinement explains why the detector sees hadrons, not the cancellation of perturbative poles.

Using the massless result at a resonance. Threshold velocities, Coulombic effects, finite widths, and bound states introduce different scales. They must be treated before interpreting $R(s)$ as a smooth short-distance series.

Handoff

The reusable output is

$$\left\{J_\mu^{\mathrm{EW}},\ s,\ \{Q_f,m_f\},\ n_f, \mathcal M,\ \mu_R,\ \text{perturbative order},\ \text{power and electroweak corrections}\right\}.$$

For identified jet shapes and resummation, continue to QCD radiation, jets, and event shapes. For an incoming hadron and its structure functions, continue to deep-inelastic scattering and the parton model.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §26.3, pp. 513–17. DOI.
• Sterman, George, and Steven Weinberg. “Jets from Quantum Chromodynamics.” Physical Review Letters 39, no. 23 (1977): 1436–39. DOI.