Homework 9

Question 1:

Write the F₂ structure function of the neutron as a sum of the flavor singlet and nonsinglet contributions,
for 4 flavors (i.e. u, d, strangeness and charm).

Question 2:

The beta-function β(g) in QCD can be expanded as a series in the strong coupling constant, g, as
β(g) = − [β₀ / (4π)²] g³ − [β₁ / (4π)⁴] g⁵ + ... Solve the equation for the effective coupling constant,
dg—/ dt = β ( g—), where t = log Q² and g—(0) = g, to show that the running coupling can be written as:
α_s(Q²) ≡ [g—² (Q²) / 4π] = [ 4π / (β₀log(Q²/Λ²) ] [ 1 − (β₁/β₀²) log(log(Q²/Λ²)) / log(Q²/Λ²) ].

Question 3:

Using the Feynman rules for QCD and QED (to be given in the 11/27 lecture), calculate the amplitude
for the photon-gluon fusion process, γ g → q q— at tree level (i.e. with a single quark exchange, and no
internal loops).