Study the origin of the logQ^2 term. Recall that the γ^* q →qg cross section, d σ̇ / d pT^2, is

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Study the origin of the logQ^2 term. Recall that the γ^* q...

Study the origin of the $\log Q^{2}$ term. Recall that the $\gamma^{*} \mathrm{q} \rightarrow \mathrm{qg}$ cross section, $d \dot{\sigma} / d p_{T}^{2}$, is dominated by the forward peak. The $t$-channel quark propagator leads to a factor $1 / p_{T}^{4}$. Show that helicity conservation at the gluon vertex weakens this singularity by introducing a factor $p_{T}^{2}$ in the numerator.

Key Concepts

t-channel Propagator

In scattering amplitudes, the t-channel propagator represents the exchange of a particle between initial and final states, and its contribution to the amplitude is typically characterized by an inverse power of the momentum transfer. In the context of the process being considered, the t-channel quark propagator gives rise to a factor that scales as 1/p??, which signals a strong enhancement in the forward (small transverse momentum) region and leads to an apparent divergence upon integration over p?.

Helicity Conservation in Gauge Theories

Helicity conservation is an important feature of interactions in gauge theories, particularly in processes involving massless or nearly massless particles. In the specific vertex where a gluon is emitted, helicity conservation can restrict the allowed configurations of the particles, effectively introducing additional momentum factors in the numerator of the amplitude. This additional p?² factor moderates the otherwise severe singular behavior from the propagator, reducing the divergence encountered in the cross section.

Transverse Momentum Integration in Scattering Processes

The integration over transverse momentum in scattering processes is crucial in determining the overall behavior and divergence structure of the cross section. Although the propagator by itself leads to a strong, 1/p?? singularity, the interplay with vertex factors (such as those due to helicity conservation) alters the effective power of p? in the integrand. This modified behavior typically produces logarithmic dependencies, as integrals of the form ?dp?²/(p?²) yield logarithms, which in this context contribute to the log Q² term in the cross section.

Logarithmic Divergences in Cross Sections

Logarithmic divergences arise in quantum field theories when the integration over certain momentum regions (in this case, the forward, low-p? region) leads to integrals that behave as the logarithm of a momentum cutoff or scale (such as Q²). The cancellation of one power of the singular behavior by helicity conservation results in an integral proportional to ?dp?²/p?², which gives rise to a logarithm. This logarithmic dependence is a common occurrence in processes where soft or collinear emissions dominate the dynamics.

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