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

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

Verify that for $\gamma^{*} \mathrm{~g} \rightarrow \mathrm{q} \overline{\mathrm{q}}$, $$ \overline{|\mathscr{I}|^{2}}=32 \pi^{2}\left(e_{q}^{2} \alpha \alpha_{s}\right) \frac{1}{2}\left(\frac{\hat{u}}{\hat{t}}+\frac{\hat{t}}{\hat{u}}-\frac{2 \hat{s} Q^{2}}{\hat{t} \hat{u}}\right), $$ 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 contribution $(10.40)$ where $g(y)$ 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 $\bar{q}$ 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 $\gamma^{*} \mathrm{~g} \rightarrow \mathrm{q} \overline{\mathrm{q}}$,
$$
\overline{|\mathscr{I}|^{2}}=32 \pi^{2}\left(e_{q}^{2} \alpha \alpha_{s}\right) \frac{1}{2}\left(\frac{\hat{u}}{\hat{t}}+\frac{\hat{t}}{\hat{u}}-\frac{2 \hat{s} Q^{2}}{\hat{t} \hat{u}}\right),
$$
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 contribution
$(10.40)$
where $g(y)$ 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 $\bar{q}$ 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 Polarization Sum

In perturbative calculations of scattering processes involving photons, the polarization sum is used to account for all possible physical photon polarization states. The substitution of the sum of polarization vectors with the metric tensor (e.g. ? ?*? ?? = -g??) simplifies the computation of squared amplitudes and is a standard technique in quantum field theory.

Photon-Gluon Fusion

This is a fundamental process in Quantum Chromodynamics where a virtual photon interacts with a gluon within a hadron, producing a quark–antiquark pair. It is an important mechanism for probing the internal structure of the proton, as it directly connects the electromagnetic interactions (via the photon) to the strong interactions (via the gluon).

Gluon Density in the Proton

The gluon density represents the probability distribution for finding a gluon inside a proton with a given fraction of the proton's momentum. This concept is central to the parton model and the analysis of deep inelastic scattering, as it influences observable quantities such as the proton structure functions.

Splitting Functions

Splitting functions, such as Pqg(z) = ½(z² + (1-z)²), describe the probability of a parton (like a gluon) splitting into other partons (a quark–antiquark pair, in this case) with specific momentum fractions. They are a key element in the evolution equations that govern the scale dependence of parton distribution functions in perturbative QCD.

Proton Structure Functions and Scaling Violations

Proton structure functions, such as F2(x, Q²), encode information about the momentum distribution of quarks and gluons within the proton. Scaling violations, which are deviations from the simple scaling behavior predicted by the naive parton model, arise due to QCD radiative corrections and provide insights into the dynamics of parton splitting and the behavior of the strong force at different energy scales.

Transcript

00:01 Here we're going to investigate electric potential energy from the interaction of two point charges, and that potential energy is given as the coolum constant times the product of the two charges over their separation.

00:20 And we're going to use that potential energy of a configuration in order to figure out kinetic energy when the particles are.

00:32 Are allowed to move.

00:34 So we will be using conservation of energy.

00:39 So a reminder of how that works is that the sum of initial kinetic plus potential has to equal the final situation.

00:54 But first of all, we're going to figure out the potential energy in order to bring two protons together separated by three femtometers, which is 10 to the minus 50 meters.

01:09 So we'll call that potential energy proton proton interaction.

01:16 And i am going to use the version of the kulham constant with three significant figures, seeing that that's what they are doing with the numbers given.

01:33 So 8 .99 times 10 to the 9th is our kulam constant.

01:39 In si units, and then our two charges, we have two protons, each with the fundamental e of charge.

01:52 So we'll have to square that.

01:55 And then they are separated by a total of six femtometers.

02:08 And that unit, if everything's in si units, we should have joules.

02:14 So it's positive, which means that you will need to do some work to put those.

02:21 Particles into position.

02:28 Now the next part of the assembly of the configuration is we are going to bring in an alpha particle and put it at the top of a triangle.

02:39 So that alpha particle has basically a helium nucleus, but it has a charge of two times the electric fundamental positive and we can figure out its mass.

03:00 I think it's given to you.

03:05 Let's see.

03:08 The mass is given as 6 .64 times 10 to the minus 27.

03:20 So essentially four times the size of a proton because it is made of two neutrons and two protons.

03:29 But anyway, we want to figure out the energy needed to make this configuration.

03:38 And so we're first going to have to figure out the potential energy between the proton and the alpha particle.

03:50 And that alpha particle is interacting with two of them.

03:54 So we'll have to take that into account when we figure out the entire potential.

04:04 Energy.

04:07 And we'll see how to do that.

04:10 And their separation.

04:11 Now this is something i called up in the triangle d.

04:17 And they give us that the height of a little triangle in the geometry is three femtometers and its base is three femtometers.

04:30 And i'll draw that little right triangle.

04:33 And we can use the pythagorean theorem then to d .d .e.

04:37 Determine the hypotenuse d squared is equal to 3 squared plus 3 squared, femtomito squared, femtometer squared.

04:53 And working that out, it's the square root of 18.

05:04 How big is that 6 .4 .24, make your pardon.

05:18 Okay, so that goes into the r portion of our potential energy.

05:27 And since i put everything in si units, that should come out in jewels.

05:45 And let me check my numbers here.

05:48 That's always a good idea with such large exponents.

05:56 Well, not exactly large, but cumbersome that way.

06:12 And again, that is a positive potential energy.

06:18 Now, here's the thought about the total potential energy of assembling the entire configuration of three charges is first of all we put down our two protons, then we brought in the alpha particle, and it interacted with both protons.

06:38 So we have to multiply it by two.

06:43 And adding all that together, we get 2 .56 times 10 to the minus 13 joules for the entire.

06:58 Triangle with three positive charges.

07:02 Again, that's a positive quantity, meaning that those charges did not want to be brought together and stuck in place.

07:12 Okay, so now we're going to look at conservation of energy...

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