Re: Lorentz violation of the Standard Model

J.C. Yoon wrote:

I meant to say that
"The SM is written as a Lorentz-invariant
action,but that of a massive fermion with
massless property."

The point is whether the fermion with one
chirality of SM Lagrangian satisfy the Dirac
equations or not.

According to the Dirac equation, there is no
such massive fermion field that has only one
chirality and it is always a combination of
two.

We just cannot represent a massive fermion
with one chirality unless there is a massive
fermion that has massless property, which is
inconsistent with the Dirac equations.

I suspect I am hindering, not helping, this
discussion. But I'll try one more time to
say something helpful...

Before spontaneous symmetry breaking (the Higgs
mechanism), the leptons are expressed masslessly
in the SM Lagrangian as a left-handed doublet like
(neutrino, electron)_L as shown in eq(20.75) of
Peskin & Schoeder. The right-handed leptons are
expressed as singlets, e.g: e_R. It is irrelevant
what equations of motion emerge from minimizing
the Action at this stage, because we have not yet
included the full details of the Higgs field in
the Lagrangian. The Higgs particle is a Lorentz
scalar, but an SU(2) doublet. The left-handed
leptons and right-handed leptons are postulated to
couple to the Higgs field, as shown in eq(20.98)
of P&S. Therefore, it is physically meaningless to
try and think of the original massless left-handed
and right-handed fields in isolation. They cannot
be meaningfully divorced from their couplings to
each other via the Higgs field.

Then, it is postulated that the Higgs field
has a non-zero vev (vacuum expectation value).
So even when there are no observable Higgs
particles flying around, there is still the
non-zero Higgs vev, which couples the left- and
right-handed fermion fields together in a term
which is indistinguishable from a Dirac mass term.

The SM is not saying that the original massless
fermion fields are "really" massive. It is saying
that fermion mass is an emergent phenomenon,
arising from the way the unavoidable non-zero
Higgs vev couples the massless left- and
right-handed fermion fields together, in such a
way that they appear physically to be one massive
Dirac fermion field.

.

Relevant Pages

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