# Why does the Higgs field only couple to opposite-chiralities fermions?

According to the Wikipedia page on the Standard Model, the Higgs field interact with fermions through a Yukawa interaction coupling only left to right chiralities. What is the reason for that? Is that the case with all Yukawa couplings?

-
Incidentally, Wikipedia's articles on the Standard Model are in serious need of attention, if anybody would care to volunteer. –  Harry Johnston Nov 23 '11 at 4:05

It is because once the Higgs couples two Weyl fermions together, they become the two chiralities of a massive charged fermion. The standard 4-component spinor formalism disguises how natural this coupling is because it makes every fermion into a Dirac fermion, and projects out the unphysical states at every vertex. If you don't do that, if you only include the 2 spinor corresponding to each physical field, the Higgs coupling is extremely natural: it is the most general renormalizable gauge-invariant coupling of an SU(2) doublet Higgs with hypercharge 1/2 (in one usual normalization) to chiral Weyl 2-spinors.

You can think of a Yukawa coupling as a mass term with a scalar field taking the role of the mass. The mass term for a charged fermion has to be of Dirac type, because it must be invariant under phase rotations, and only the term $\bar\psi\psi$ is invariant. This means that Higgs fields will always couple opposite chiralities of charged massive fermions, i.e. everything in the standard model that couples to the Higgs.

To see a situation where the coupling is for one chirality only, consider the nonrenormalizable two-Higgs two lepton interaction:

$$H H L L$$

Where L is the SU(2) doublet left-handed lepton field, H is the Higgs field, also an SU(2) doublet, and each L's SU(2) index is contracted with one of the H's SU(2) index using the SU(2) epsilon tensor, and the L's Lorentz indices are contracted with each other using the space-time epsilon tensor (in 2 index formalism). This nonrenormalizable term becomes a neutrino Majorana mass, and it is suppressed by a large scale, but it is the right order of magnitude to explain the majorana masses.

In this case, the two-Higgs field is coupled to a single chirality. The allowed couplings are always determined by matching all the SU(2),SU(3) and Lorentz indices together in a 2-spinor formalism, and it becomes the chirality restriction only by coincidence in the standard model.

If you had a fundamental SU(2) symmetric tensor Higgs field T (it would be the SU(2) "spin 1" representation, not spin 1/2) with twice the hypercharge of the standard model Higgs, it could give neutrinos a Majorana mass with renormalizable Yukawa couplings, just by replacing $HH$ with $T$ above, and contracting the indices the same way.

-