Verify that for γ^* g →q q, |ϑR|^2=32 π^2(eq^2 α) (1)/(2)((û)/(t̂)+(t̂)/(û)-(2 ŝ Q^2)/(t̂ û)) ,

Step 1: **Understand the Process and Notation** The process $gamma^* g rightarrow q overline{q}$

Verify that for γ^* g →q q, |ϑR|^2=32 π^2(eq^2 α)...

Verify that for $\gamma^* \mathrm{~g} \rightarrow \mathrm{q} \overline{\mathrm{q}}$, $$ \overline{|\vartheta R|^2}=32 \pi^2\left(e_q^2 \alpha \alpha_s\right) \frac{1}{2}\left(\frac{\hat{u}}{\hat{t}}+\frac{\hat{t}}{\hat{u}}-\frac{2 \hat{s} Q^2}{\hat{t} \hat{u}}\right) \text {, } $$ using $\sum \varepsilon_\mu^* \varepsilon_\nu=-g_{\mu \nu}$ for the $\gamma^*$-polarization sum. Hence, show that (10.34) for the proton structure function contains the additional contributionwhere $g(y)$ is the gluon density in the proton and where $$ P_{q g}(z)=\frac{1}{2}\left(z^2+(1-z)^2\right) $$ represents the probability that a gluon annihilates into a $q \bar{q} \bar{q}$ pair such that the quark has a fraction $z$ of its momentum. Detailed measurements of the scaling violations of $F_2\left(x, Q^2\right)$ probe the gluon distribution inside the proton through $(10,40)$.

Verify that for $\gamma^* \mathrm{~g} \rightarrow \mathrm{q} \overline{\mathrm{q}}$,
$$
\overline{|\vartheta R|^2}=32 \pi^2\left(e_q^2 \alpha \alpha_s\right) \frac{1}{2}\left(\frac{\hat{u}}{\hat{t}}+\frac{\hat{t}}{\hat{u}}-\frac{2 \hat{s} Q^2}{\hat{t} \hat{u}}\right) \text {, }
$$
using $\sum \varepsilon_\mu^* \varepsilon_\nu=-g_{\mu \nu}$ for the $\gamma^*$-polarization sum. Hence, show that (10.34) for the proton structure function contains the additional contributionwhere $g(y)$ is the gluon density in the proton and where
$$
P_{q g}(z)=\frac{1}{2}\left(z^2+(1-z)^2\right)
$$
represents the probability that a gluon annihilates into a $q \bar{q} \bar{q}$ pair such that the quark has a fraction $z$ of its momentum. Detailed measurements of the scaling violations of $F_2\left(x, Q^2\right)$ probe the gluon distribution inside the proton through $(10,40)$.

Key Concepts

Photon-Gluon Fusion

In high-energy QCD processes, a virtual photon can interact with a gluon inside a proton resulting in the production of a quark–antiquark pair. This process is fundamental to the description of deep inelastic scattering (DIS) at next-to-leading order and is used to probe the internal gluonic structure of hadrons.

Polarization Sum in Gauge Theories

Summing over the polarizations of gauge bosons, such as photons or gluons, is a key technique in calculating scattering amplitudes. By using the completeness relation (for example, ? ?*? ?? = -g?? for a photon in covariant gauges), one systematically includes all physical polarization states and ensures gauge invariance in the theory.

Gluon Parton Distribution Functions (PDFs)

Gluon PDFs describe the probability of finding a gluon carrying a certain fraction of the proton’s momentum at a given scale. They are essential in QCD-based predictions for processes involving hadronic collisions, influencing cross sections and scaling violations in structure functions.

Splitting Functions

Splitting functions, like Pqg(z), quantify the probability that a parton (in this case, a gluon) splits into a pair of quark and antiquark with specific momentum fractions. These functions are central to the DGLAP evolution equations, which describe how parton distributions evolve with changes in momentum transfer Q².

Scaling Violations in Structure Functions

Scaling violations refer to the Q²-dependence of structure functions such as F?(x, Q²) in deep inelastic scattering. These deviations from Bjorken scaling arise due to QCD radiative corrections and are sensitive to the dynamical evolution of parton densities, thus providing insights into the gluonic content of the proton.

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