An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 6.1

Radiative Corrections: Introduction — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · qed-form-factors-rosenbluth

Problem

Rosenbluth formula. As discussed Section 6.2, the exact electromagnetic interaction vertex for a Dirac fermion can be written quite generally in terms of two form factors

F_1(q^2)

and

F_2(q^2)

= \bar{u}(p')\left[\gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu}q_\nu}{2m}F_2(q^2)\right]u(p),

where

q = p' - p

and

\sigma^{\mu\nu} = \frac{1}{2}i[\gamma^\mu, \gamma^\nu]

. If the fermion is a strongly interacting particle such as the proton, the form factors reflect the structure that results from the strong interactions and so are not easy to compute from first principles. However, these form factors can be determined experimentally. Consider the scattering of an electron with energy

E \gg m_e

from a proton initially at rest. Show that the above expression for the vertex leads to the following expression (the Rosenbluth formula) for the elastic scattering cross section, computed to leading order in

\alpha

but to all orders in the strong interactions:

\frac{d\sigma}{d\cos\theta} = \frac{\pi\alpha^2\left[(F_1^2 - \frac{q^2}{4m^2}F_2^2)\cos^2\frac{\theta}{2} - \frac{q^2}{2m^2}(F_1+F_2)^2\sin^2\frac{\theta}{2}\right]}{2E^2[1+\frac{2E}{m}\sin^2\frac{\theta}{2}]\sin^4\frac{\theta}{2}},

where

\theta

is the lab-frame scattering angle and

F_1

and

F_2

are to be evaluated at the

q^2

associated with elastic scattering at this angle. By measuring

(d\sigma/d\cos\theta)

as a function of angle, it is thus possible to extract

F_1

and

F_2

. Note that when

F_1 = 1

and

F_2 = 0

, the Rosenbluth formula reduces to the Mott formula (in the massless limit) for scattering off a point particle (see Problem 5.1).

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