Custodial Symmetry: Understanding the Transformation Law

[TriTertButoxy]

I have never really understood the approximate 'Custodial symmetry' in the Standard Model. I've seen it being described in many texts, but I can't seem to be able to put my finger on it.

Would somebody please write down the transformation law for the Higgs fields under a 'custodial SU(2) transformation?' It would really help if I got that!

 

[blechman]

custodial isospin in the electroweak interaction is defined many ways, depending on what you want to do with it (they're all effectively the same up to field redefinitions). but a good starting place is the following: posit a GLOBAL symmetry:

SU(2)_L x SU(2)_R

Now gauge the SU(2)_L and identify it with the standard model gauge group, but only gauge the U(1)_R subgroup of the SU(2)_R part, and identify that with hypercharge. The special treatment of the U(1)_R explicitly breaks the R part of the symmetry, but if we turn off that special treatment (let g'=0) then the R symmetry is restored (up to the Yukawa couplings). So one can do a spurion analysis with g' and the fermion yukawa couplings.

To the extent that g' is small, this describes the standard model. The SU(2)_R is the "custodial isospin" symmetry (sometimes it is the SU(2)_D, but I usually use the former in my research; as I said, they're the same up to field redefinitions).

If you want to make the custodial isospin symmetry manifest, you can let the Higgs transform as a bifundamental of the L-R symmetry:

$$H\rightarrow LHR^\dagger$$

This symmetry is realized in the standard model if you group the right-handed fermions into doublets of the SU(2)_R symmetry and let g'=0. If you have g' nonzero, this symmetry is only realized if R=1.

Custodial isospin symmetry is very important for EW precision measurements - it ensures that the W-Z mass ratio (called $\rho$) cannot get too large, for example. Its corrections must be proportional to g' and yukawas, especially the top quark yukawa coupling. This is a well known result.

The extent to which custodial isospin symmetry is broken is a very powerful test for physics beyond the standard model.

Hope that helps.