In Morii, Lim, Mukherjee, The Physics of the Standard Model and Beyond. 2004, ch. 8, they claim that the Peskin–Takeuchi oblique parameters S, T and U are in fact Wilson coefficients of certain dimension-6 operators. On page 212, they claim that the T parameter is described by $$O_T=(\phi^\dagger D_\mu \phi)\phi^\dagger D^\mu \phi)-\frac{1}{3}(\phi^\dagger D_\mu D^\mu\phi)(\phi^\dagger\phi)\,,$$ and the S parameter by $$O_S=[\phi^\dagger(F_{\mu\nu}^i\sigma^i)\phi]B^{\mu\nu}\,,$$ where $\phi$ is the Higgs doublet, $F_{\mu\nu}^i$ and $\sigma^i$ are the SU(2) weak isospin field strength and sigma matrices respectively, and $B^{\mu\nu}$ is the U(1) weak hypercharge field strength.
On the next page (p. 213), problem 8.6 asks us to show that these are the operators.
How do I precisely show that these higher-dimension operators give precisely the Peskin-Takeuchi parameters?
