Deep inelastic scattering from a photon. Consider the problem of deep-inelastic scattering of an electron from a photon. This process can actually be measured by analyzing the reaction e+e−→e+e−+X in the regime where the positron goes forward, with emission of a collinear photon, which then has a hard reaction with the electron. Let us analyze this process to leading order in QED and to leading-log order in QCD. To predict the photon structure functions, it is reasonable to integrate the renormalization group equations with the initial condition that the parton distribution for photons in the photon is δ(x−1) at Q2=(21 GeV)2. Take Λ=150 MeV. Assume for simplicity that there are four flavors of quarks, u, d, c, and s, with charges 2/3, −1/3, 2/3, −1/3, respectively, and that it is always possible to ignore the masses of these quarks.
(a) Use the Altarelli-Parisi equations to compute the parton distributions for quarks and antiquarks in the photon, to leading order in QED and to zeroth order in QCD. Compute also the probability that the photon remains as a photon as a function of Q2.
(b) Formulate the problem of computing the moments of W2 for the photon as a problem in operator mixing. Compute the relevant anomalous dimension matrix γ. You should be able to assemble this matrix from familiar ingredients without doing further Feynman diagram computations.
(c) Compute the n=2 moments of the photon structure functions as a function of Q2.
(d) Describe qualitatively the evolution of the photon structure function as a function of x and Q2.