In the center-of-mass frame of the parton process γ^* q1 →q2 g of Fig. 10.7, show that ŝ=2

step 1 Dear students here we have a given question question number 10 of chapter 46 according to the gi

In the center-of-mass frame of the parton process γ^* q1 →q2...

In the center-of-mass frame of the parton process $\gamma^{*} \mathrm{q}_{1} \rightarrow \mathrm{q}_{2} \mathrm{~g}$ of Fig. $10.7$, show that $$ \begin{aligned} &\hat{s}=2 k^{2}+2 k q_{0}-Q^{2}-4 k^{\prime 2}, \\ &\hat{t}=-Q^{2}-2 k^{\prime} q_{0}+2 k k^{\prime} \cos \theta=-2 k k^{\prime}(1-\cos \theta), \\ &\hat{u}=-2 k k^{\prime}(1+\cos \theta), \end{aligned} $$ where $k, k^{\prime}$ are the magnitudes of the center-of-mass momenta $\mathbf{k}, \mathbf{k}^{\prime}$. Note that for the virtual photon, $q_{0}^{2}-k^{2}-Q^{2}$. A useful result is $$ 4 k k^{\prime}=-\hat{t}-\hat{u}=\hat{s}+Q^{2} $$ The interesting quantity is the transverse momentum of the outgoing quark, $p_{T}=k^{\prime} \sin \theta$. Show that $$ p_{T}^{2}=\frac{\operatorname{sit} 2}{\left(s+Q^{2}\right)^{2}} $$ or, in the limit of small-angle scattering. $-\hat{t} \approx \hat{s}$, that $$ p_{T}^{2}=\frac{s(-\hat{\imath})}{s+Q^{2}} . $$ Further, show that for small scattering angles $(\cos \theta \simeq 1)$, $$ d \Omega=\frac{4 \pi}{s} d p_{T}^{2} $$ $\underset{\rightarrow q_{2} g .}{\text { Figy } 10.7 \text { Center-of-mass frame for } \gamma^{*} q_{1}}$ 10.4 The Gluon Emission Cross Section 213 It is clear from $(10.17)$ that at high energy ( $\vec{s}$ large), the $\gamma^{*} \mathrm{q} \rightarrow \mathrm{qg}$ cross section peaks as $-\hat{i} \rightarrow 0$. Referring back to Fig. $4.8$ and the related discussion, we see that this is due to quark exchange in the $t$ channel. We can therefore approximate the cross section by its forward peak. For forward scattering, we find, from (10.26) and (4.35), that $$ \frac{d \dot{\sigma}}{d p_{T}^{2}}=\frac{1}{16 \pi s^{2}} \overline{|S M|^{2}} $$

In the center-of-mass frame of the parton process $\gamma^{*} \mathrm{q}_{1} \rightarrow \mathrm{q}_{2} \mathrm{~g}$ of Fig. $10.7$, show that
$$
\begin{aligned}
&\hat{s}=2 k^{2}+2 k q_{0}-Q^{2}-4 k^{\prime 2}, \\
&\hat{t}=-Q^{2}-2 k^{\prime} q_{0}+2 k k^{\prime} \cos \theta=-2 k k^{\prime}(1-\cos \theta), \\
&\hat{u}=-2 k k^{\prime}(1+\cos \theta),
\end{aligned}
$$
where $k, k^{\prime}$ are the magnitudes of the center-of-mass momenta $\mathbf{k}, \mathbf{k}^{\prime}$. Note that for the virtual photon, $q_{0}^{2}-k^{2}-Q^{2}$. A useful result is
$$
4 k k^{\prime}=-\hat{t}-\hat{u}=\hat{s}+Q^{2}
$$
The interesting quantity is the transverse momentum of the outgoing quark, $p_{T}=k^{\prime} \sin \theta$. Show that
$$
p_{T}^{2}=\frac{\operatorname{sit} 2}{\left(s+Q^{2}\right)^{2}}
$$
or, in the limit of small-angle scattering. $-\hat{t} \approx \hat{s}$, that
$$
p_{T}^{2}=\frac{s(-\hat{\imath})}{s+Q^{2}} .
$$
Further, show that for small scattering angles $(\cos \theta \simeq 1)$,
$$
d \Omega=\frac{4 \pi}{s} d p_{T}^{2}
$$
$\underset{\rightarrow q_{2} g .}{\text { Figy } 10.7 \text { Center-of-mass frame for } \gamma^{*} q_{1}}$
10.4 The Gluon Emission Cross Section
213
It is clear from $(10.17)$ that at high energy ( $\vec{s}$ large), the $\gamma^{*} \mathrm{q} \rightarrow \mathrm{qg}$ cross section peaks as $-\hat{i} \rightarrow 0$. Referring back to Fig. $4.8$ and the related discussion, we see that this is due to quark exchange in the $t$ channel. We can therefore approximate the cross section by its forward peak. For forward scattering, we find, from (10.26) and (4.35), that
$$
\frac{d \dot{\sigma}}{d p_{T}^{2}}=\frac{1}{16 \pi s^{2}} \overline{|S M|^{2}}
$$

Key Concepts

Small-Angle/Forward Scattering Approximation

In high-energy scattering processes, particularly when interactions are dominated by forward scattering (i.e., small-angle deflections), the scattering amplitudes and differential cross sections can be approximated by assuming the scattering angle is very small. This approximation simplifies the angular dependencies and often leads to peaks in the cross section near zero momentum transfer. It is an essential tool in analytically evaluating cross sections and isolating the dominant contributions from, for instance, t-channel exchange processes.

Differential Cross Section

The differential cross section quantifies the probability of a scattering event occurring within a specific solid angle or momentum transfer interval. It is a foundational concept in collision theory and particle physics, providing detailed insights into the dynamics of individual scattering processes. Expressing the differential cross section in terms of invariant quantities like the Mandelstam variables allows for a general description that is independent of the observer's frame of reference.

Mandelstam Variables

These are Lorentz-invariant quantities (commonly denoted as s, t, and u) used to describe the energy and momentum transfer in a scattering process. They encapsulate the kinematic information of the interaction without reference to a specific frame, making them especially useful in high-energy particle physics. Their definitions and relationships (like s + t + u = sum of the masses squared, when applicable) allow for systematic analyses of scattering amplitudes and cross sections.

Center-of-Mass Frame

This is the inertial reference frame where the total momentum of the system is zero, which simplifies the analysis of collision problems. In this frame, symmetrical properties of the scattering process become evident, and kinematic relations (such as energy conservation and momentum magnitude relationships) can be expressed in a simpler form. It is particularly suited for evaluating invariant quantities like the Mandelstam variables and for isolating the effects of scattering angles.

Transverse Momentum

Transverse momentum (pT) refers to the component of momentum that is perpendicular to a designated reference direction (often the beam axis). It is a key observable in scattering processes, as it provides insight into the momentum transfer and dynamics of the interaction. The squares of the transverse momenta are often directly related to the Mandelstam variable t, and studying their distributions helps in understanding phenomena such as jet formation and the behavior of predicted peaks in scattering cross sections.

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