Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 3.6

Classical field theory — Schwartz M.D. z-lib.org

Physics — QFT · yukawa-potential

Problem

Yukawa potential. (a) Calculate the equations of motion for a massive vector AμA_\mu from the Lagrangian L=−14Fμν2+12m2Aμ2−AμJμ,(3.87)\mathcal{L} = -\frac{1}{4}F^2_{\mu\nu} + \frac{1}{2}m^2 A^2_\mu - A_\mu J^\mu, \tag{3.87} where Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. Assuming ∂μJμ=0\partial_\mu J^\mu = 0, use the equations to find a constraint on AμA_\mu. (b) For JμJ_\mu the current of a point charge, show that the equation of motion for A0A_0 reduces to A0(r)=e4π2ir∫−∞∞k dkk2+m2eikr.(3.88)A_0(r) = \frac{e}{4\pi^2 ir} \int_{-\infty}^{\infty} \frac{k\, dk}{k^2 + m^2} e^{ikr}. \tag{3.88} (c) Evaluate this integral with contour integration to get an explicit form for A0(r)A_0(r). (d) Show that as m→0m \to 0 you reproduce the Coulomb potential. (e) In 1935 Yukawa speculated that this potential might explain what holds protons together in the nucleus. What qualitative features does this Yukawa potential have, compared to a Coulomb potential, that make it a good candidate for the force between protons? What value for mm might be appropriate (in MeV)? (f) Plug the constraint on AμA_\mu that you found in part (a) back into the Lagrangian, simplify, then rederive the equations of motion. Can you still find the constraint? What is acting as a Lagrange multiplier in Eq. (3.87)?

Related theory

Browse the problem library