Newsgroups: sci.physics
Subject: Fermion mass terms (was Re: Defining Photons)
From: mcirvin@husc8.harvard.edu (Mcirvin)
Date: 27 Jul 92 21:23:59 GMT
 <9976@sun13.scri.fsu.edu> <26JUL199218561022@zeus.tamu.edu> <24926@dog.ee.lbl.gov>
Distribution: na
Keywords: fermions Higgs

sichase@csa3.lbl.gov (SCOTT I CHASE) writes:

>Even in the Higgs model the mass of the electron is not entirely from 
>the Higgs-electron interaction, is it?  Up to a few percent of the mass
>still comes from radiative effects (depending on what cutoff mass you use
>when you renormalize). 

True, though the amount is indeed cutoff-dependent.  But I don't
think you get a radiative mass term unless the Higgs coupling or
some other kind of mass term is there in the first place.

>And (as Matt Austern is always pointing out every time
>I mention Higgses, the total body of evidence for the Higgs is so slim and 
>indirect that, IMHO, it would not be fair to take potshots at an outsider 
>for not knowing about them, yet.

Quite right; and many think that the truth is probably
more complicated than a fundamental Higgs scalar.  I'd bet on
some kind of field coupling being responsible for the masses,
though, if only because I don't like the idea of the gauge
symmetry being explicitly broken.

Also, it's a little misleading to call the Higgs field the
origin of mass in the world, since, though it does supply the
lepton and quark masses, the nucleon masses which make up
most of what you see on the bathroom scale are (according to
the standard model) the result of QCD chiral symmetry breaking,
and would be about the same if the up and down quarks were
massless.

Here's something I've been wondering about:  Just how
important are Dirac mass terms?  The Weinberg angle is related
to the electroweak coupling constants in such a way that 
electromagnetism couples to the vector current; it is a 
parity-conserving theory.  It seems to me that the only
thing enforcing this, at least at tree level, is the existence
of Dirac mass terms for fermions, coming from Higgs
Yukawa couplings.  Since, by the existence of these couplings,
it becomes necessary that axial rotations be spontaneously
broken, the massless (electromagnetic) part of the SU(2) x U(1)
quartet ends up coupled entirely to the vector current, and
electromagnetism conserves parity.  

Are the Yukawa couplings necessary?  In a world with the
standard model's gauge symmetries and a Higgs scalar, but
with no massive fermions (we can put in some massless
ones to define handedness), would the Weinberg angle be
an extra free parameter, independent of the electroweak
couplings?  

It seems to me that it would, just by consideration of
an Abelian toy model where the gauge symmetry
is U(1) right x U(1) left.  This makes the analysis
simpler by removing the doublets.  If you imagine these two
degrees of freedom as superpositions of a photon and a
kind of "Z," there's nothing preventing you from
constructing the Higgs potential so that the massive
"Z" is any combination you like of the right- and the
left-handed degrees of freedom, *unless* a fermion gets
a Dirac mass from the Higgs field.  Then the part of
U(1)R x U(1)L that is spontaneously broken is necessarily
axial U(1), and the photon couples to the vector current.
Unless I'm mistaken, the presence of the Dirac particle
is what fixes the "Weinberg angle" of this toy theory.

I don't understand anomalies sufficiently to know whether
they play any part in fixing the angle, but that's the
only other thing I can think of.

-- 
Matt McIrvin  mcirvin@husc.harvard.edu