Griffiths Introduction to Elementary Particles · Problem 8.4

Quantum Chromodynamics — David J. Griffiths

Physics — QFT · hadronic-tensor-symmetry

Problem

Prove equation (8.16). [Hint: First show that qμLμν=0q_\mu L^{\mu\nu} = 0. Then argue that we may as well take KμνK^{\mu\nu} such that qμKμν=0q_\mu K^{\mu\nu} = 0, in the sense that any term in KμνK^{\mu\nu} that does not obey qμKμν=0q_\mu K^{\mu\nu} = 0 will contribute nothing to LμνKμνL_{\mu\nu}K^{\mu\nu}.] Comment: Equation (8.16) actually follows more simply and generally from charge conservation at the proton vertex, but I have not developed the formalism here to make this argument (see Halzen and Martin, ref. 2, Sects. 8.2 and 8.3). One way to proceed is as follows. Take qμ=(0,0,0,q)q^\mu = (0,0,0,q); then qμLμν=0⇒Lμν=(□00000)q_\mu L^{\mu\nu} = 0 \Rightarrow L^{\mu\nu} = \begin{pmatrix} \square & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}. So LμνKμν=(□00000)(□×××)L^{\mu\nu}K_{\mu\nu} = \begin{pmatrix} \square & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} \square \\ \times & \times & \times \end{pmatrix}, and ×\times may as well be zero.

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