Homework 7

Question 1:
Using the explicit form for the leptonic tensor in eN elastic scattering, verify the charge conservation
condition, q^μ L_{μ ν} = q^ν L_{μ ν} = 0.

Question 2:
(a)
Show that the Lorentz-invariant phase space, dLips(p',k'), for unpolarized scattering can be written
in the laboratory (or target rest) frame as dΩ d|k'| (|k'|/p'₀) δ (p'₀ + |k'| - |k| - M) / (4π)² .
(b)
Show further that this reduces to dLips(p',k') = (|k'|²/M |k|) dΩ / (4π)² .
(c)
Finally, changing variables from dΩ to dQ², show that d|Q²| = 2 |k'|² dcos(θ) .
[Note that the quantities in bold face, e.g. k or k', denote 3-momentum vectors.]

Question 3:
In the "pion pole model" the induced pseudoscalar form factor of the nucleon, G_P(Q²), is given by
4M f_π g_πNN(Q²) / (Q² + m_π²), where g_πNN(Q²) is the form factor at the pion-nucleon-nucleon vertex.
Using conservation of the axial vector current in the chiral limit, derive the Goldberger-Treiman
relation, M G_A(Q²) = f_π g_πNN(Q²).

How well does it hold experimentally?