I'm trying to understand how the Higgs mass can influence EW precision tests. In order to do that I'm using the following document (section 4.3):


There are a couple of things that puzzle me... 1) Once I've computed S, T and U how can I compare them to experiments? What do they represent physically? 2) In the process of computing S and T he uses diagrams involving Goldstone bosons (Figure 2)a)b)). Could I just work in the unitary gauge where the Goldstone bosons are gone? If yes, how is it possible to get the same result (in some sense I would do the same computations but without the diagrams involving goldstone bosons)? I have the impression that S and T are observables and therefore gauge-invariant, which suggest that I can use the gauge I want.


First of all, the oblique parameters S,T and U are defined to be zero within the Standard Model (SM). This means, that the SM is a reference and therefore, these parameters are indications for physics beyond the Standard Model (BSM). They account for corrections in the vacuum polarizations of the EW gauge bosons and are chosen in a way, that different BSM phenomena only change one of these parameters.E.g. in a NHDM (n-Higgs Double Model) the parameters are calculated through higher order operators

$T\sim \frac{1}{\Lambda^2}|H^\dagger D_\mu H|^2$

$S\sim \frac{1}{\Lambda^2}H^\dagger W^{\mu\nu}B_{\mu\nu} H$

$U\sim \frac{1}{\Lambda^4}\left(H^\dagger W^{\mu\nu}H\right)\left(H^\dagger W_{\mu\nu} H\right)$

where $D_\mu$ is a covariant derivative and $W^{\mu\nu}$ and $B^{\mu\nu}$ the $SU(2)$ and $U(1)$ field strengths respectively. $H$ is a Higgs doublet.

S has something to say about 4th generation chiral fermions, T about isospin violation while U is usually not very useful in practice, since it is of higher order.

Now to your questions. The Higgs mass does influence the 'normalization' of the Peskin-Takeuchi parameters, since the SM is a reference point. Once you measure some quantity, you can compare to the theoretical predictions of your BSM theory (i.e. you see whether your predictions, which have a discrepancy to the SM predictions, fit the experimental data).

You are right about gauge invariance. However, unitary gauge is only worth choosing as long as you stay at tree level computations. Actually, I'm not quite sure, but I think in unitary gauge you still have to take $W^\pm$ and $Z$ bosons into account, which should give you the missing contributions from the Goldstone bosons.

  • $\begingroup$ Thank you for your explanation. I still don't understand everything of what you have explained (I'm new to the subject). Have you seen how the S T U parameters are defined in the document I have provided? How can they be zero in the SM? Doesn't $\Pi(q^2)$ contain corrections of the form " -O- " where the loop is given for instance by a top and bottom quark? $\endgroup$ – Worldsheep May 8 '15 at 17:17
  • $\begingroup$ Aren't S T and U rather parametrising the deviations from the SM at tree level? $\endgroup$ – Worldsheep May 9 '15 at 8:28
  • $\begingroup$ I took the definitions from a sheet of paper I found on my desk ;) In the paper you cited on page 18, (4.12)-(4.15) are the definitions. Not (all) the $\Pi$'s are zero, but $\hat{U}$ and $\hat{T}$ are defined as differences. These parameters are not restricted to tree level calculations. However, if you go to higher order, you also have to adjust the 'normalization', i.e. the SM computation. $\endgroup$ – Clever May 11 '15 at 10:41
  • $\begingroup$ Good answer, you can't use the unitary gauge for loop processes (you can see that by the propagator of the gauge boson), for example in the $B-\bar{B}$ mixing you have to take into account the Goldston bosons. $\endgroup$ – Karozo Sep 17 '15 at 22:31

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