# Is there a w+(L or R)?

I was studying the interaction of Higgs field with the Right-Handed particles, and I don't know how to determine the quantum numbers before and after the boson W+- interaction with the component of Higgs scalar Field (h0).
I am following the Halzen's book.
The h0 has the following quantum numbers:
Isospin = 1/2 ;
Y= 1
Isospin3=-1/2;
The example on the book is related with er (electron right handed) that after interaction with h0 becomes el(electron left handed). But for w+ interaction with h0, there is no reference to w+ quantum number.

ChrisVer
Gold Member
No there are no Left/Right W bosons because they are not in the chiral representation of SU(2) .They rather come from the adjoint representation of it, and so they have polarizations (instead of helicities).
Which quantum numbers are you looking for and for which coupling?

I'm looking for the Isospin, isospin(third component) and the hypercharge of gauge boson W+- before and after the interaction with Higgs scalar field (h0).
Because is there no mention of it in Halzen's book, Griffths'book and Greiner's book of electroweak interactions.

vanhees71
Gold Member
2021 Award
You should start with the construction of the electroweak Lagrangian in the un-Higgsed form to understand the symmetry. It's a chiral SU(2)##\times##U(1). The SU(2) is weak isospin and the U(1) weak hypercharge symmetry. The corresponding gauge bosons transform under global gauge transformations according to the adjoint representation of this gauge group.

After Higgsing, it's not so simple anymore. In a "handwaving way", not to be taken literally. You can understand the Higgs mechanism as spontaneous symmetry breaking of the local gauge symmetry. In the electroweak standard model it's broken to the electromagnetic U(1). The Z and the photon fields are mixtures of weak-isospin and weak hypercharge fields. Thus it's not so easy to get the right transformation properties under the original gauge transformations in terms of the W, Z, and em. fields without expressing them in terms of the original gauge fields.

I already done that study. But my doubt remains. Maybe this picture illustrate my doubt. Which are the quantum numbers(T,T^3,Y) of the gauge boson W before and after interaction with h0?.

This picture is from Quarks and Leptons (Halzen & Martin) chapter 15, section 4.
PS: T -> weak isospin;
T^3-> third component of weak isospin;
Y-> hypercharge.;

ChrisVer
Gold Member
The W+- don't have definite isospin 3 neither hypercharge, they are combinations of $W_{1,2}^\mu$ fields before the SSB...So in any case you don't have to care about their isospin when coupling to the Higgs. The only thing that should and does conserve, is the electric charge Q (which itself is a combination of the hypercharge and isospin component)

Fabio Kopp
The W+- don't have definite isospin 3 neither hypercharge, they are combinations of $W_{1,2}^\mu$ fields before the SSB...So in any case you don't have to care about their isospin when coupling to the Higgs. The only thing that should and does conserve, is the electric charge Q (which itself is a combination of the hypercharge and isospin component)
Thanks for the help!