Lorentz violating field theories and nonperturbative physics

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Lorentz violating field theories and
nonperturbative physics

• Brett Altschul
• Indiana University
• (Work with Alan Kostelecký)
• June 13, 2005

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Overview
Lorentz violation (LV) in nature is strongly
constrained by numerous experiments. Such
measurements have involved studies of trapped
charged particles, atomic spectra, meson
oscillations, and astrophysical photon
polarization measurements. There are numerous
candidate quantum gravity theories with LV, but
nobody knows whether these are the exception or
the rule.
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Outline

• Introduction
• The Standard Model Extension (SME)
• The Nonperturbative Renormalization Group
• Relevant Theories with LV
• Conclusion

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Introduction
In the last ten years, there has been growing
interest in the possibility that Lorentz symmetry
may not be exact. There are two broad reasons for
this interest Reason One Many theories that
have been put forward as candidates to explain
quantum gravity involve LV in some regime. (For
example, string theory, non-commutative geometry,
loop quantum gravity)
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Reason Two Lorentz symmetry is a basic building
block of both quantum field theory and the
General Theory of Relativity, which together
describe all observed phenomena. Anything this
fundamental should be tested. Much of the story
of modern theoretical physics is how important
symmetries do not hold exactly. There is no
excellent beauty that hath not somestrangeness
in the proportion. Francis Bacon
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Although many quantum gravity theories involve
LV, it is not clear how ubiquitous the Lorentz
violation really is. For example, the discovery
that in string theory the tachyon potential often
contains a minimum where Lorentz symmetry would
be spontaneously broken spurred a great deal of
interest in this subject. Kostelecký and
Samuels, PRD 39, 683 (1989) However, it now
seems that this minimum is probably NOT the true
vacuum.
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One way to answer the question of how common LV
really is is to look at the renormalization group
(RG). Within some relatively generic class of
theories, are the theories with LV more relevant
than those without? It turns out that the answer
is yes, at least for vector-like LV. There is a
large class of spontaneously Lorentz- and
CPT-violating theories, which are more relevant
than any Lorentz invariant theory.
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Standard Model Extension (SME)
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With those restrictions, the Lagrange density for
the free electron sector looks like
A separate set of coefficients will exist for
every elementary particle in the theory.
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One important effect of these Lorentz-violating
terms is to modify the velocity. For example,
with c present
From this expression, we can see when the
effective field theory breaks down. The velocity
may become superluminal when .
If , this is
. More generally, momentum eigenstates may not be
eigenstates of velocity.
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The coefficients need not be diagonal in flavor
space either. Like neutrino masses, they may mix
different species. In fact, a two-parameter
Lorentz-violating model can explain all observed
neutrino oscillations (except LSND). Kostelecký
and Mewes, PRD 69, 016005 (2004) However, many
possible parameters have not been probed. The
full neutrino sector has 102 Lorentz-violating
parameters.
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The photon sector contains more superficially
renormalizable couplings.
Most of these couplings are easy to constrain
with astrophysical polarimetry. However, some
will require more complicated measurements (e.g.
with Doppler shifts or electromagnetostatics).
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Nonperturbative RG
The renormalization group describes how a quantum
field theory changes when the cutoff (or
renormalization) scale is adjusted. For example,
we all know that QCD is asymptotically free, so
that the coupling goes logarithmically to zero at
high energies. For a long time, it was thought
that non-Abelian gauge theories were the only
asymptotically free field theories in four
dimensions.
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However, this conventional wisdom turns out to be
wrong! The RG equations for scalar field theory
have been known for a long time, and they can be
solved exactly in the linear approximation.
In this approximation, only one power of the
coupling can appear, so all the quantum
corrections come from tadpole loops.
Contributions to the four-point function.
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Solving the linearized RG equations, we can
demonstrate the well-known result that a
interaction (or one with any higher power) is
irrelevant. However, this only applies to
polynomial potentials. Nonpolynomial potentials
are perfectly allowed in canonical quantum
theory. In fact, when nonpolynomial potentials
are considered, there are interactions with every
possible anomalous dimension. Halpern and Huang,
PRL 74, 3526 (1995)
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Each loop contributes a regulated zero-separation
propagator in Euclidean space.
The factors of L will generate the RG
flow. Consider just a single-component scalar
field. A diagram with n loops has ,
divided by a symmetry factor of 2nn!, times a
combinatorial factor.
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This fact enables us to write each renormalized
vertex (and thus the whole renormalized
potential) as
Differentiating this and requiring
gives an ODE for Vr. The method
generalizes easily to N-component fields.
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For a N-component scalar field theory with SO(N)
invariance, the relevant potentials are
M is a confluent hypergeometric (Kummer) function.
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l is the anomalous dimension of the interaction.
If all the modes with Euclidean momenta in the
range are integrated out of
the theory and the fields rescaled accordingly,
then the renormalized g is shifted to
. The physics of these scalar potentials is
very complicated. Any calculations beyond
leading order are basically impossible, but
results for spontaneous symmetry breaking and
scattering have been worked out.
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This was all done in the linear approximation.
The full nonlinear RG is prohibitively complex,
except in the large N limit. Everything is
well-behaved in that limit, all the way down into
the IR. Gies, PRD 63, 065011 (2001) This
suggests that the nonpolynomial theory should be
well behaved even without the linearized
approximation.
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Relevant Theories with LV
The same nonperturbative RG methods can be
applied to theories with vector fields. After a
Wick rotation, minus the norm of a vector field
looks just like the magnitude squared
of a four-component scalar field,
. The only real complication is that with the
kinetic term , only three
components of the massive vector field are
propagating. The result is
.
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So there are relevant potentials for fields with
spacetime indices.
There is no gauge symmetry in the presence of
this potential. To study nonpolynomial potentials
with manifest gauge invariance would require a
completely new technique.
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What does this have to do with LV? Since the
field B has a vector index, if there is
spontaneous symmetry breaking, it will also be a
breaking of Lorentz invariance. In fact, all
viable potentials exhibit such breaking.
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If , the field acquires a spacelike
vev. If , the vev is timelike. In the
intermediate region, , the theories
are unstable, possessing no lowest-energy state.
(And of course, if , the interaction is
irrelevant.)
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The theory with the broken symmetry is very
attractive. There is a massless propagating
vector Goldstone mode, which looks like a photon
(in a temporal or axial gauge), plus
corrections. Couplings to the photon will also
couple to the vev, introducing Lorentz violation
in all the charged sectors of the
theory. Finally, the Lorentz-violating vev can
naturally be small, going as for large l.
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Conclusion
Many Lorentz-violating coefficients are strongly
constrained, but LV remains a strong candidate to
appear in a fundamental theory. There exists a
large class of relevant vector interactions, all
of which involve spontaneous violation of Lorentz
invariance. After the symmetry breaking, these
theories resemble QED, with emergent approximate
gauge symmetry, plus corrections.
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Thanks to V. A. Kostelecký, K. Huang, and E.
Altschul.
Thats all, folks!