If αs(Q^2)=c / logQ^2, show that (10.37) leads to (∫x^n-1 q(x, Q^2) d x)/(∫x^n-1 q(x, Q0^2) d x)

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If αs(Q^2)=c / logQ^2, show that (10.37) leads to (∫x^n-1...

If $\alpha_{s}\left(Q^{2}\right)=c / \log Q^{2}$, show that (10.37) leads to $$ \frac{\int x^{n-1} q\left(x, Q^{2}\right) d x}{\int x^{n-1} q\left(x, Q_{0}^{2}\right) d x}=\left(\frac{\log Q^{2}}{\log Q_{0}^{2}}\right)^{A_{n}} $$ where $$ \begin{aligned} A_{n} &=\frac{c}{2 \pi} \int_{0}^{1} x^{n-1} P_{q q}(x) d x \\ &=\frac{c}{2 \pi} \frac{4}{3}\left(-\frac{1}{2}+\frac{1}{n(n+1)}-2 \sum_{j=2}^{n} \frac{1}{j}\right) \end{aligned} $$ That is, in QCD, the moments $(n \geq 1)$ of the quark structure functions decrease as calculable powers of $\log Q^{2} \cdot c$ is given by (7.65). The observant reader may have noticed what appears to be a contradiction in our interpretation of the $P$ functions. For example, we regard the $P_{q q}$ term as the correction factor to the quark density that arises from allowing for gluon emission. However, there are two diagrams: one with the gluon emitted from the initial quark line and the other with the gluon radiated from the final quark line; see Fig. 10.2. Our picture is only valid if the first diagram dominates. Then, the emitted gluon can be considered as part of the proton structure. It is a "partonlike" diagram. It turns out that both diagrams are required to ensure gauge invariance of the amplitude, but that the second only plays the role of canceling the contributions from the unphysical polarization states of the gluon. Adopting a physical gauge, in which we sum only over transverse gluons, only the first diagram remains.

If $\alpha_{s}\left(Q^{2}\right)=c / \log Q^{2}$, show that (10.37) leads to
$$
\frac{\int x^{n-1} q\left(x, Q^{2}\right) d x}{\int x^{n-1} q\left(x, Q_{0}^{2}\right) d x}=\left(\frac{\log Q^{2}}{\log Q_{0}^{2}}\right)^{A_{n}}
$$
where
$$
\begin{aligned}
A_{n} &=\frac{c}{2 \pi} \int_{0}^{1} x^{n-1} P_{q q}(x) d x \\
&=\frac{c}{2 \pi} \frac{4}{3}\left(-\frac{1}{2}+\frac{1}{n(n+1)}-2 \sum_{j=2}^{n} \frac{1}{j}\right)
\end{aligned}
$$
That is, in QCD, the moments $(n \geq 1)$ of the quark structure functions decrease as calculable powers of $\log Q^{2} \cdot c$ is given by (7.65).
The observant reader may have noticed what appears to be a contradiction in our interpretation of the $P$ functions. For example, we regard the $P_{q q}$ term as the correction factor to the quark density that arises from allowing for gluon emission. However, there are two diagrams: one with the gluon emitted from the initial quark line and the other with the gluon radiated from the final quark line; see Fig. 10.2. Our picture is only valid if the first diagram dominates. Then, the emitted gluon can be considered as part of the proton structure. It is a "partonlike" diagram. It turns out that both diagrams are required to ensure gauge invariance of the amplitude, but that the second only plays the role of canceling the contributions from the unphysical polarization states of the gluon. Adopting a physical gauge, in which we sum only over transverse gluons, only the first diagram remains.

Key Concepts

Running Coupling Constant

In Quantum Chromodynamics, the coupling constant ?s is not fixed but depends on the energy scale Q². Its behavior, often approximated by a 1/ln(Q²) dependence at high energies, embodies the concept of asymptotic freedom, where quarks interact weakly at short distances. This scale dependence is crucial for understanding the evolution of observables and parton distribution functions with changing energy scales.

Moments of Structure Functions

Moments of parton distribution functions are integrals over the momentum fraction x, weighted by a power of x, which provide a compact way to summarize the information contained in the distributions. These moments are important in theoretical predictions and in comparing with experimental data, as their scale dependence reflects the underlying QCD dynamics depicted by evolution equations.

Altarelli-Parisi (DGLAP) Evolution Equations

The evolution of parton distributions with respect to the energy scale Q² is governed by the Altarelli-Parisi (or DGLAP) equations. These integro?differential equations describe how a parton inside a hadron splits into other partons as the probing scale increases, thereby modifying the distribution functions in a predictable way that can be analyzed through moments and logarithmic corrections.

Splitting Functions

Splitting functions, such as Pqq, quantify the probability for a parton to emit another parton, altering the momentum carried by the original parton. They are a central component in the evolution equations, incorporating the physics of parton branching processes. These functions carry the calculable corrections to the parton densities and help predict the logarithmic Q² dependence of the structure function moments.

Gauge Invariance in QCD

Gauge invariance is a fundamental principle in QCD that ensures physical observables are independent of the specific gauge chosen to perform calculations. In practice, this leads to cancellations between contributions from different diagrams, particularly those involving unphysical polarization states. Adopting a physical gauge, where only transverse gluon polarizations are summed, simplifies the analysis and confirms that the dominant contributions come from the physically relevant processes, thereby maintaining the consistency of the parton model interpretation.

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