An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 17.1

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

Physics — QFT · qcd-renormalization-group

Problem

17.1 Two-loop renormalization group relations. (a) In higher orders of perturbation theory, the expression for the QCD

\beta

function will be a series

\beta(g) = -\frac{b_0}{(4\pi)^2}g^3 - \frac{b_1}{(4\pi)^4}g^5 - \frac{b_2}{(4\pi)^6}g^7 + \cdots.

Integrate the renormalization group equation and show that the running coupling constant is now given by

\alpha_s(Q^2) = \frac{4\pi}{b_0}\left[\frac{1}{\log(Q^2/\Lambda^2)} - \frac{b_1}{b_0^2}\frac{\log\log(Q^2/\Lambda^2)}{(\log(Q^2/\Lambda^2))^2} + \cdots\right],

where the omitted terms decrease as

(\log(Q^2/\Lambda^2))^{-2}

. (b) Combine this formula with the perturbation series for the

e^+e^-

annihilation cross section:

\sigma(e^+e^- \to \text{hadrons}) = \sigma_0 \cdot \left(3\sum_f Q_f^2\right) \cdot \left[1 + \frac{\alpha_s}{\pi} + a_2\left(\frac{\alpha_s}{\pi}\right)^2 + \mathcal{O}(\alpha_s^3)\right].

The coefficient

a_2

depends on the details of the renormalization conditions defining

\alpha_s

. Show that the leading two terms in the asymptotic behavior of

\sigma(s)

for large

s

depend only on

b_0

and

b_1

and are independent of

a_2

and

b_2

. Thus the first *two* coefficients of the QCD

\beta

function are independent of the renormalization prescription.