More on the Existence of a Physical Higgs Particle...
James B. Glattfelder
coming from *other* considerations than the perturbative quantum field
interpretation of a mathematical mechanism used to overcome the problem
of gauge-invariant mass terms in the standard model (i.e. the
implementation of spontaneous symmetry breaking)?
I ask this, because up to now I hardly ever have heard of people really
doubting its existence ñ in fact the great (financial) effort going into
detecting this last elusive ingredient of the standard model (SM)
speaks for itself.
And there are reasons to doubt...
A short summary:
-Spontaneous symmetry breaking (SSB) occurs when the Lagrangian
of a theory is invariant under some symmetry, but the vacuum
state is constructed to be non-invariant. (Technically the
operators representing the symmetry group act as creation
operators.)
-SSB is mathematically implemented by the introduction of a fundamental
complex scalar field with the property that its vacuum expectation value
(VEV) is not zero. This yields the required modification to the vacuum.
(Note that only scalars, in contrast to spinors, vectors and
tensor-particles, can exhibit non-zero VEVís without violating the
Lorentz invariance of a theory ñ thatís why the trick with scalars
works).
-If, as is in the case of the SM, we are considering local symmetries,
then the mechanism of spontaneously breaking them via the vacuum is
referred to as the Higgs mechanism.
Already here the rigorousness of the arguments becomes diluted. SSB in
gauge theories faces a problem: The proof of the Goldstone theorem
(where a massless boson appears as a result of the breaking of a global
symmetry) requires the validity of all of the usual field theory axioms.
However, there is no gauge-fixing condition one can impose for which
gauge theories obey all the axioms. So technically, the Goldstone
theorem is not applicable to the SM (a fact rarely mentioned in the
literature).
But there exists a loophole which turns this potential embarrassment
into a virtue. One only thinks about ìdegrees of freedomî associated
with the Goldstone theorem and automatically gets rid of them by curing
the invariant-mass problem of the SM at the same time: they cancel the
time-like component of a gauge field, leaving three space-like
components of a massive gauge boson with positive norm (in the
Gupta-Bleuler qunatization method massive fields have unphysical states
of negative norm ñ a fact related to the gauge invariance of massive
bosons). This then is the real Higgs mechanism. It is implemented by a
special choice of gauge in which the time-like components of the massive
gauge fields and the scalar degrees of freedom vanish.
So then, after a short detour and the fear of getting wet feet we are
finally back on dry land. But things only get worse...
To continue the story, we introduce new scalar terms into the SM
Lagrangian, namely something similar to a kinetic, a mass and a
self-interaction term (i.e. a Klein-Gordon Lagrangian with a kinetic and
a potential term). (We also couple our scalar field to fermions via the
Yukawa coupling.) General considerations concerning the energy (i.e.
the Hamiltonian associated with the scalar Lagrangian) lets one conclude
that there is a unique vacuum configuration which is associated with the
VEV of the scalar field being zero. However, making the ìmassî parameter
negative (a tachyon state), yields the Mexican hat potential and the
vacuum becomes degenerate (due to rotational symmetry). The new minimum
of the potential is now reached for non-zero VEVís. In selecting a
specific solution for the scalar field (i.e. selecting a direction in
isospin space) the rotational symmetry of the vacuum is broken and a
stable physical vacuum is realized.
And now, finally, my question raised at the beginning slowly re-emerges.
Conventional (perturbative quantum field theory) has it, that the scalar
field oscillates around the ground state in effect shifting the field
and creating a first excitation of the ground state ñ our prominent
physical Higgs field:
\phi = h + v, (1)
v being related to the scalar Lagrangian parameters (in [GeV]) and
proportional to the VEV of the scalar field. Inserting (1) into the
scalar and Yukawa Lagrangians yields fermion and gauge boson mass terms
next to mass and interaction terms for the physical Higgs boson.
My problem with this whole idea (next to the question of the validity of
perturbative quantum field theories raised by string/M-theory and loop
quantum gravity) is the following:
A.) We get *all* the required mass term for the SM if we insert the
unperturbed scalar term into the Lagrangians. No Higgs particle needed
here. (NB: The often quoted line of argument counting the degrees of
freedom can, in a natural way, be re-interpreted in the non-perturbative
picture.)
B.) The logical flow leading to the whole issue of the Higgs mechanism
is disrupted by the ad hoc postulation of (1). The VEV is *never*
explicitly realized, for if so, h would become zero. On the other hand,
the non-perturbative approach relies directly on the VEV being taken,
leading to its non-zero value, which, inserted into the Lagrangians,
yields the required mass terms. It seems a bit of a waist of time
setting up this whole elaborate web of reasoning based on symmetry and
the vacuum, if in the end we never take the VEV of a scalar field. So
how is SSB implemented then?
C.) In being pragmatic, the people in favor of the perturbative idea,
only choose to acknowledge those terms in the calculation that they can
physically interpret (mass and interaction terms). However, inserting
(1) into the Lagrangians leads to additional terms in h, namely linear
ones that just get swept under the carpet.
D.) On the horizon, using the non-perturbative interpretation yields
terms in v which (also usually ignored by pragmatic Higgs supporters)
could be interpreted as an energy density, thus opening up a window of
opportunity to relate SSB and the Higgs mechanism to the cosmological
constant and the early evolution of the universe, namely inflation. All
of this with simply using a scalar potential in which the mass parameter
dynamically varies.
My question: Is the Higgs mechanism more than just an algebraic
manipulation of the Lagrangian based on group theoretic considerations
required by phenomenology? Or are there other requirements?
The only hint of an answer I came across concerns unitarity for WW
scattering amplitudes at high energies. The consistent formulation of
the weak interactions as a theory of fields interacting weakly up to
high energies leads supposedly to a vector boson theory complemented by
a scalar Higgs field. However, an alternative to this scenario exists
where the W boson is then required to interact *strongly* at high
energies. Another hint at new physics? Note that the problem addressed
by unitarity constraints is associated with the perturbative treatment
of the SM.
See hep-ph/0011255; hep-ph/0004103; Gunion, Haber, Kane, Dawson, ìThe
Higgís Hunters Guideî, Addison Wesley, 1990, for details.
PS.
General references and technical details are readily available to anyone
who is interested.
Daniel Doro Ferrante
> Is there any real inevitability for the physical Higgs particle to exist
> coming from *other* considerations than the perturbative quantum field
> interpretation of a mathematical mechanism used to overcome the problem
> of gauge-invariant mass terms in the standard model (i.e. the
> implementation of spontaneous symmetry breaking)?
>
Another way to "implement" SSB is by the choice of boundary
conditions in the Schwinger-Dyson eqs. In this sense, all that you
described below for SSB is slightly changed. The new "framework" is to
write down Schwinger-Dyson's eqs for the theory you got and, in order to
solve those, you'll have to choose boundary conditions. That's exactly
how the phase structure shows up: different boundaries => different
phases. For further reference, take a look at: hep-th/9612079.
> My question: Is the Higgs mechanism more than just an algebraic
> manipulation of the Lagrangian based on group theoretic
> considerations required by phenomenology? Or are there other
> requirements?
>
See above. Given the above understanding, i'd say so.
--
Daniel
,-----------------------------------------------------------------------------.
> | olympus.het.brown.edu www.fma.if.usp.br <
> Daniel Doro Ferrante | <
> dani...@het.brown.edu | Thirteen at a table is unlucky only when <
> Linux Counter #34445 | the hostess has only twelve chops. <
> | -- Groucho Marx <
`-----------------------------------------------------------------------------'
Thomas Larsson
> Is there any real inevitability for the physical Higgs particle to exist
> coming from *other* considerations than the perturbative quantum field
> interpretation of a mathematical mechanism used to overcome the problem
> of gauge-invariant mass terms in the standard model (i.e. the
> implementation of spontaneous symmetry breaking)?
hints are right, its existence is pretty inevitable.
There are theoretical alternatives to a fundamental Higgs. In technicolor,
something Higgs-like appears as a bound state of two fermions, similar
to Cooper pairs in superconductivity. However, the simplest technicolor
models have been ruled out, and extended technicolor schemes seem too
arbitrary to yield any solid predictions.
> I ask this, because up to now I hardly ever have heard of people really
> doubting its existence ñ in fact the great (financial) effort going into
> detecting this last elusive ingredient of the standard model (SM)
> speaks for itself.
Something that people often miss is that the LHC is in fact a quite cheap
machine. In 1993, a total machine cost for LHC of 2230 MCHF was estimated.
(http://press.web.cern.ch/Press/Releases93/PR12.93Ecouncil.html)
Even with the recent overdraft, it should be well below 3 bn euros, which
is about the amount spent on the now-defunct SSC. And that machine only
resulted in half-finished tunnels in the Texan desert and a lot of
frustrated people, whereas the LHC will produce unique information not
attainable otherwise (not without even greater financial efforts, anyway).
But in the standard argumentation for the LHC, Higgs is only a warm-up
exercise. String people hope to see supersymmetry, large extra
dimensions, and a zoo of new particles. Personally I neither hope nor
expect that such things will show up. If they don't, people might finally
start to look for deep mathematics in quantum general relativity or in
the standard model, rather than speculating wildly without experimental
support.
But independent of your perspective, new experimental input is needed to
remove the present dead-lock situation in theoretical physics.
Aaron Bergman
thomas....@hdd.se (Thomas Larsson) wrote:
> "James B. Glattfelder" <j...@bluewin.ch> wrote in message
> news:<3CB5FBD3...@bluewin.ch>...
> > Is there any real inevitability for the physical Higgs particle to exist
> > coming from *other* considerations than the perturbative quantum field
> > interpretation of a mathematical mechanism used to overcome the problem
> > of gauge-invariant mass terms in the standard model (i.e. the
> > implementation of spontaneous symmetry breaking)?
>
> There are hints that the Higgs has already been seen at 115 GeV. If these
> hints are right, its existence is pretty inevitable.
IIRC, they retracted the claim after further data analysis.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
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From: "Ralph E. Frost" <ref...@dcwi.com>
Newsgroups: sci.physics.research
Subject: Re: Intuition for the Lagrangian?
Date: Tue, 16 Apr 2002 09:02:13 -0500
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Rancid Moth <ranci...@yahoo.com> wrote in message
news:a9dr36$nga$1...@perki.connect.com.au...
> perhaps one way of looking at it asthetically would be....
>
> [N.B I have used terms that have very specific meansings, in a very loose
> way...and have put them in quotes]
>
> the Lagrangian L is defined as the difference between the kinetic(T) and
> potential energy (U) of the system in question. now restricting ourselves
> to conservative systems, we know that the total energy H of the system is
> conserved, and thus when either T or U vary, the other must also vary to
> compensate for the conservation of H. Now we think about the way they
vary
> over time from initial time t[i] to final time t[f]. The principle of
least
> action, basicaly states that the path taken is the one such that the
changes
> in T and U over the given time period, result in the "least" amount of
> difference between the potential and kinetic energy of the system
"averaged"
> over t[f]-t[i]. what does this mean? its all up to interpretation to a
> degree, but i like to think of it as...ok you have a particle, say, thats
> moving thus giving it kinetic energy. its being influenced by a force
> described by U. it seems only natural to me that the particles kinetic
> energy will vary such that the particle will take the path of "least
> resistance" _and_ exert as little "effort" to do so.
How about if you describe the situation as, ok, you are part of a system
that currently has a total energy H, and a portion of this energy is split
into activity (kinetic) and the remainder is present in a passive -
potential state. If you look at some hypotheical particle as doing an
economic transaction in order to move long, negotiating - trading bits of
action/energy with the passive surroundings, then what the path of least
resistance also is is the path of maximum motion or longest travel. Since
the particle can only buy it's way in along in whole pennies, I mean,
Planck-like units, then the trade automaticatically defines the path taken.
That is, let's say we have a region of where the potential energy is
arranged into mostly packed sphere except for a tubular, snake-like
channel. The particle comes along and finds an even trade in all
'directions' except for that where the passive energy is arranged in the
shape of a tube.
In that 'direction', the particle gets a "buy signal" and, like brokers on
the futures trading room floor, he starts buying wildly along the tube.
Mr. Particle is still only going to trade a unit for a unit, but the
direction of travel is set at the start of each transaction and,
length-distance-wise, according to that which seems important to us, he will
have traveled farther for that unit of energy expenditure.
>
> Its change in kinetic energy will be influenced by the changes that are
> going on in the potential force applied to it...but the changes to T are
> only going to be as small as they have to be relative to the changes in U
> and _such_ _that_ momentum and energy etc are all conserved. now taking
the
> difference of the two is a way to measure that change. and finding the
> extremum of that change for every instant of time t for the period
> t[f]-t[i], is the way to "minimize" the overal change of T for the
duration
> of the particles flight, with respect to the variations occuring in U.
>
>
> maybe that helps? maybe not.
It links to the abstract math expression. That's always helpful. But where
you suggest minimizing t, in the alternate inutitive model, allowing that
units of potential energy have shape allows one to see how the particle can
automatically maximize the longest path for EVERY unit of energy that is
traded in the transaction. Thus, distance isn't really so much distance as
it is a unit of energy with a particular shape.
Maybe that helps?
Thomas Larsson
> >
> > There are hints that the Higgs has already been seen at 115 GeV. If these
> > hints are right, its existence is pretty inevitable.
>
> IIRC, they retracted the claim after further data analysis.
>
Do you have any references?
Exactly what did they retract? As far as I remember, they had found the Higgs
with some 2.5 sigma certainty. Has this confidence level melted down, or is
it just that 2.5 sigma is not enough (which I think that they said in the first
place)?
Aaron Bergman
Thomas....@hdd.se (Thomas Larsson) wrote:
> Aaron Bergman <aber...@princeton.edu> wrote in message
> news:<abergman-EF4928...@news.bellatlantic.net>...
> > >
> > > There are hints that the Higgs has already been seen at 115 GeV. If these
> > > hints are right, its existence is pretty inevitable.
> >
> > IIRC, they retracted the claim after further data analysis.
> >
>
> Do you have any references?
Not off the top of my head.
> Exactly what did they retract? As far as I remember, they had found the Higgs
> with some 2.5 sigma certainty. Has this confidence level melted down, or is
> it just that 2.5 sigma is not enough (which I think that they said in the
> first place)?
I think they did further analysis, and the peak evaporated. Or something
like that.
James B. Glattfelder
Is it really beyond doubt, that the mathematical manipulation called
the "Higgs mechanism" should be interpreted within the context of
quantum field theory giving rise to the existence of a physical Higgs
particle?
I.e., are we really that sure, that in expanding the fundamental
scalar field around its ground state we are indeed doing the right
thing and are giving the "Higgs mechanism" *real* physical meaning?
And thus could it be, that the question concerning the origin of mass
within the context of the standard model is going to require a
different physical picture to interpret the subtleties of this
mathematical model of reality?