An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 17.6

Quantum Chromodynamics — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · qcd-small-x-parton-distributions

Problem

**17.6 Behavior of parton distribution functions at small

x

.** It is possible to solve the Altarelli-Parisi equations analytically for very small

x

, using some physically motivated approximations. This discussion is based on a paper of Ralston.

^\dagger

(a) Show that the

Q^2

dependence of the right-hand side of the A-P equations can be expressed by rewriting the equations as differential equations in

\xi = \log\log\left(\frac{Q^2}{\Lambda^2}\right),

where

\Lambda

is the value of

Q^2

at which

\alpha_s(Q^2)

, evolved with the leading-order

\beta

function, formally goes to infinity.

^\dagger

J. P. Ralston, *Phys. Lett.* 172B, 430 (1986).