# Field renormalization of scalar Yang-Mills

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In most books, one can find the field renormalization $Z_3$ in Yang-Mills with fermionic matter in the fundamental. In the $\overline{MS}$ scheme, tt is given by $$Z_3 = 1 + \frac{g^2}{16\pi^2 \epsilon} \left[ \frac{10}{3} T_A - \frac{8}{3} n_F T_F \right] + {\cal O}(g^3)$$ One often writes $\delta_3 = Z_3 - 1$.

This formula can be found in

1. Eq. (73.33) of Srednicki.
2. Eq. (26.83) of Matt Schwartz' book.
3. Eq. (16.74) of P&S etc.

However, I can't find any reference that lists how this result is modified if there's scalar matter present. I've already computed it and I get $$Z_3 = 1 + \frac{g^2}{16\pi^2 \epsilon} \left[ \frac{10}{3} T_A - \frac{8}{3} n_F T_F - \frac{2}{3} n_S T_F \right] + {\cal O}(g^3)$$ Is this correct?

In particular, I'm really interested in computing ALL the counterterms of Yang-Mills in the presence of scalar matter. Is there any reference out there that already does that so I can match my results?

This post imported from StackExchange Physics at 2015-01-23 12:48 (UTC), posted by SE-user Prahar
retagged Jan 23, 2015
I don't think "most books" have the wavefunctions renormalization correction. Maybe "most introductory QFT books", haha.

This post imported from StackExchange Physics at 2015-01-23 12:48 (UTC), posted by SE-user JeffDror
@JeffDror - haha! of course. but since I mostly end up reading QFT books anyway, "most books" for me means exactly that

This post imported from StackExchange Physics at 2015-01-23 12:48 (UTC), posted by SE-user Prahar
Also to partially answer your question. Any book with a section on GUTs should provide the correct formula, though I've never seen a reference actually derive it in full.

This post imported from StackExchange Physics at 2015-01-23 12:48 (UTC), posted by SE-user JeffDror

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