Re: Questions about Higgs scalars

On 2006-08-22, Jay R. Yablon <jyablon@xxxxxxxxxxxx> wrote:

I attach a few pages from Halzen and Martin's (H&M) Book "Quarks &
Leptons . . ." at the link below:

http://home.nycap.rr.com/jry/Papers/Halzen%20&%20Martin.pdf

H&M develop the Higgs field in the customary manner, using an SU(2)
doublet wherein the symmetry is broken so that the upper member is 0 and
the lower member is (1/sqrt(2))(v + h(x)). The h(x) of course is a
field and x = (t,x,y,z) is a label not a dynamical variable. As I
understand QFT, one can start with a collection of harmonic oscillators
separated by a distance l and with positions labeled as q_a (t) where a
is a label for each oscillator. In the continuum limit, where l is made
very small, q_a (t) is promoted to a field such as, in this case, h(x).
That, I believe, is Igor's forward arrow.

Now, look at Figure 15.1 in the H&M excerpt linked here. The electron
and the W^+ are both shown, in the Feynman diagrams, interacting with
h^0, which, if I read these Feynman diagrams correctly, I take to be a
single Higgs scalar particle, not a field.

You are reading too much into these Feynman diagrams. The lines in them
do not represent particles that are flying around, say, in this room.
They are a graphical representation for integrals over propagators and
vertex amplitudes that represent expectation values of products of
field operators between Fock states.

In ordinary quantum mechanics you can have a Hamiltonian that can be
written as H = H_0 + V, where the eigenstates of H_0 are known. To
first order in perturbation theory, the shift in the energy level of
the H_0 eigenstate |psi> is proportional to <psi|V|psi>.

Similarly, in QFT, you can have an interaction term V=eeh in the
Lagrangian, where e and h represent the electron and Higgs fields. But
there, it is the correction to the scattering amplitude that will be
proportional to the expectation value <out|V|in> the interaction term
between two Fock states. If the |in> state contains an electron quantum
and a Higgs quantum, while the |out> state contains only an electron
quantum, the afore mentioned expectation value can be represented as
the first diagram in H&M's Fig. 15.1, forgetting for the moment the
propagator factors. At the level of detail given in the exerpt you
provided, all we can say is that the leading order correction to the
electron-Higgs scattering amplitude will be proportional to g*m_e/M_W
(times some kinematic integrals over incoming/outgoig momenta and
angles). All this reasoning is done from the field theory perspective.
Only very rarely do modern authors turn back to the multi-particle
point of view.

So:
1) How do H&M, without ceremony, make the jump from the dynamical Higgs
scalar field h(x) to the h^0 particle shown in Figure 15.1? [...]

Like I said above, they don't. All are fields, everywhere.

2) H&M never define the ^0 superscript in h^0. How does one interpret
this? As a ground state? As a designator of a particle not a field?
Something else?

I think the h^0 label simply emphasizes that after spontaneous symmetry
breaking, only the phi^0 component of the Higgs doublet is taken as
non-zero, hence the similar label on h.

3) Would it be at all appropriate to represent the h^0 in Figure 15.1,
which, again, I read as representing a single particle, using a
single-particle scalar wavefunction of the form:

h^0 = N e^(-ip^u x_u)?

Wrong question to ask here. Think fields and field quanta, not
particles and wave functions.

Hope this helps.

Igor

.

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