The xT distribution (11.30) can be translated into an...

The $x_T$ distribution (11.30) can be translated into an "acollinearity" distribution $d \sigma / d \theta$, where $\theta$ is the angle between the $\mathrm{q}$ and $\overline{\mathrm{q}}$ jet directions defined in (11.20) and Fig. 11.7. Show that for $\theta$ not too large, $$ \frac{1}{\sigma} \frac{d \sigma}{d \theta}=\frac{8 \alpha_s}{3 \pi} \frac{1}{\theta} \log \left(\frac{1}{\theta^2}\right) . $$ An exact result can be obtained using (11.41) instead of (11.30).

The $x_T$ distribution (11.30) can be translated into an "acollinearity" distribution $d \sigma / d \theta$, where $\theta$ is the angle between the $\mathrm{q}$ and $\overline{\mathrm{q}}$ jet directions defined in (11.20) and Fig. 11.7. Show that for $\theta$ not too large,
$$
\frac{1}{\sigma} \frac{d \sigma}{d \theta}=\frac{8 \alpha_s}{3 \pi} \frac{1}{\theta} \log \left(\frac{1}{\theta^2}\right) .
$$

An exact result can be obtained using (11.41) instead of (11.30).

Key Concepts

Acollinearity Distribution

This concept refers to the angular deviation of the produced jets from being perfectly back?to?back. In high energy collisions, jets formed by quark and antiquark pairs are expected to emerge nearly in opposite directions, but radiative processes such as gluon emission introduce small angular deviations. Studying the distribution of these angular separations provides insights into the underlying QCD dynamics and the pattern of soft and collinear emissions.

Transverse Momentum Distribution

The transverse momentum (x_T) distribution measures the momentum component perpendicular to the beam direction. This distribution is closely related to the angular spread of the jets in the final state. By translating the x_T distribution into an angular (acollinearity) distribution, one links momentum space behavior with the angular correlation between the jets, allowing for the study of how QCD radiation affects jet kinematics.

Collinear Approximation

In many QCD processes, particularly at high energies, particles are emitted nearly collinearly. The collinear approximation simplifies the analysis by assuming small angles between the emitted partons. This approximation enables the derivation of analytic forms for distributions, such as the 1/? behavior observed in the differential cross section, and highlights the singular nature of QCD splitting in the collinear limit.

Logarithmic Enhancements

The appearance of logarithmic factors like log(1/?²) in the cross section is a hallmark of soft and collinear divergences in QCD. These logarithms, which result from the integration over regions of phase space where the emitted gluon is nearly collinear with the emitting quark, significantly enhance the probability of small-angle emissions. In perturbative QCD, summing these leading logarithmic contributions is crucial for accurate predictions.

Perturbative QCD and Gluon Emission

Perturbative QCD, involving calculations order-by-order in the strong coupling constant ?_s, provides the framework for understanding jet formation and radiation. The expression provided, with its dependence on ?_s and the angular variable ?, directly results from the underlying dynamics of gluon radiation from quarks. This framework explains how gluon bremsstrahlung leads to deviations from perfect collinearity, as encoded in the angular distribution of the jets.

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