Question

Use momentum conservation to justify $$ \int_0^1 d z z\left(q_S\left(z, Q^2\right)+g\left(z, Q^2\right)\right)=1 . $$ Hence, determine the $\delta(1-z)$ term in $P_{g g}$ and verify $$ P_{g g}(z)=6\left(\frac{1-z}{z}+\frac{z}{(1-z)_{+}}+z(1-z)\right)+\left(\frac{11}{2}-\frac{n_f}{3}\right) \delta(1-z), $$ where $n_f$ is the number of quark flavors.

   Use momentum conservation to justify
$$
\int_0^1 d z z\left(q_S\left(z, Q^2\right)+g\left(z, Q^2\right)\right)=1 .
$$

Hence, determine the $\delta(1-z)$ term in $P_{g g}$ and verify
$$
P_{g g}(z)=6\left(\frac{1-z}{z}+\frac{z}{(1-z)_{+}}+z(1-z)\right)+\left(\frac{11}{2}-\frac{n_f}{3}\right) \delta(1-z),
$$
where $n_f$ is the number of quark flavors.

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Quarks and leptons: introductory course in modern particle physics

Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan… 1st Edition

Chapter 10, Problem 14

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Step 1

The integral $$ \int_0^1 d z z\left(q_S\left(z, Q^2\right)+g\left(z, Q^2\right)\right)=1 $$ represents the conservation of momentum in a proton, where $q_S(z, Q^2)$ is the momentum distribution function of the singlet quark combination and $g(z, Q^2)$ is the Show more…

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Use momentum conservation to justify $$ \int_0^1 d z z\left(q_S\left(z, Q^2\right)+g\left(z, Q^2\right)\right)=1 . $$ Hence, determine the $\delta(1-z)$ term in $P_{g g}$ and verify $$ P_{g g}(z)=6\left(\frac{1-z}{z}+\frac{z}{(1-z)_{+}}+z(1-z)\right)+\left(\frac{11}{2}-\frac{n_f}{3}\right) \delta(1-z), $$ where $n_f$ is the number of quark flavors.

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