# How to show the oblique parameters S, T, and U are coefficients of d=6 operators

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?

-

1
– Mistake Ink Aug 19 '12 at 12:13
Interested to know if you were able to work it out from there? – Mistake Ink Aug 19 '12 at 22:53
Thanks for the reference; I haven't looked at it carefully enough to see if I could work it out. In any case, the paper seems to be important enough to read through. – QuantumDot Aug 20 '12 at 0:32
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.915779709815979, "perplexity_flag": "middle"}