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.

 

[sir-pinski]

Custodial symmetry is a concept in particle physics that relates to the properties of the Higgs field and how it interacts with other particles. It is a symmetry that is present in the Standard Model of particle physics, which is the current best theory we have for explaining the fundamental particles and their interactions.

To understand custodial symmetry, it is important to first understand the concept of symmetry in physics. Symmetry refers to the idea that certain physical properties or laws remain the same even when certain transformations are applied. For example, the laws of physics should be the same regardless of whether you are standing on Earth or on Mars.

In the case of custodial symmetry, the transformation being considered is a rotation in the space of the Higgs field. This rotation is part of a larger symmetry group known as the SU(2) group. The Higgs field is a complex scalar field, meaning that it has both a magnitude and a direction in space. A rotation in this space would change the direction of the field, but not its magnitude.

The transformation law for the Higgs field under a custodial SU(2) transformation can be written as follows:

Φ' = UΦ

Where Φ is the original Higgs field, Φ' is the transformed Higgs field, and U is the transformation matrix. This matrix belongs to the SU(2) group and is a 2x2 unitary matrix, meaning that its inverse is equal to its conjugate transpose.

This transformation law tells us that the Higgs field is invariant under a rotation in its space, as long as the rotation is described by a matrix that belongs to the SU(2) group. This is what we mean by custodial symmetry - the Higgs field remains the same even though we have rotated it in its space.

Custodial symmetry is important because it helps to explain why certain particles, such as the W and Z bosons, have the same mass even though they have different electric charges. This is because the Higgs field is responsible for giving particles their mass, and custodial symmetry ensures that this is done in a balanced way for particles with different properties.

I hope this explanation has helped you understand custodial symmetry and its transformation law for the Higgs field. It is a complex concept, but an important one in our understanding of the fundamental particles and their interactions.