#### Transcript Lorentz violating field theories and nonperturbative physics

```Testing Relativity Theory
With Neutrinos
Brett Altschul
University of South Carolina
May 15, 2008
Overview
Lorentz invariance is extremely well tested.
Yet many candidate theories of quantum
gravity “predict” Lorentz violation in certain
regimes, especially at very high speeds.
Neutrino physics offers interesting ways to test
whether relativity still holds very close to the
speed of light.
Outline
•
•
•
•
Introduction
The Standard Model Extension (SME)
Tests of Relativity with Neutrinos
Conclusion
Introduction
In the last fifteen 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…)
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 some
strangeness in the proportion. — Francis Bacon
Standard Model Extension (SME)
Idea: Look for all operators that can
contribute to Lorentz violation.
• locality
• superficial renormalizability
•SU(3)C  SU(2) L  U(1)Y gauge invariance
• etc...
With those restrictions, the Lagrange density
for a free fermion looks like:
L  i    M 
1 
M  m  a b 5  H  
2
      c     d     5
A separate set of coefficients will exist for every
elementary
particle in the theory.

One important effect of these Lorentz-violating
terms is to modify the velocity. For example,
with c present:


1
vk  pk  ckj p j  c jk p j  c j kc jl pl E  c0k
From this expression, we can see when the
effective field theory breaks down. The
velocity may become superluminal when
E  m c . If c  m M P , this is E  mMP .
More generally, momentum eigenstates may
not be eigenstates of velocity.

Most Lorentz-violating effects at high
relativistic energies depend on a particle’s
maximum achievable velocity (MAV).
v max 1 ( pˆ ) 1 c jk pˆ j pˆ k  c 0 j pˆ j
The corresponding energy-momentum relation
is
E  m  1 2( pˆ )p
2
2
The photon sector contains more superficially
renormalizable couplings.
1 
1
1

 
L   F F  kF  F F  kAF   A F 
4
4
2
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).
Measurement Type
System
Coefficients
log Sensitivity
Source
oscillations
K (averaged)
a (d, s)
—20
E773
Kostelecký
K (sidereal)
a (d, s)
—21
KTeV
D (averaged)
a (u, c)
—16
FOCUS
D (sidereal)
a (u, c)
—16
FOCUS
B (averaged)
a (d, b)
—16
BaBar, BELLE,
DELPHI, OPAL
neutrinos
a, b, c, d
—19 to —26
SuperK
Kostelecký, Mewes
photon
kAF (CPT odd)
—42
Carroll, Field, Jackiw
kF (CPT even)
—32 to —37
Kostelecký, Mewes
birefringence
resonant cavity
photon
kF (CPT even)
—7 to —16
Lipa et al.
Muller et al.
Schiller et al.
Wolf et al.
anomaly frequency
e-/e+
b (e)
—23
Dehmelt et al.
e- (sidereal)
b, c, d (e)
—23
Mittleman et al.
mu/anti-mu
b (mu)
—22
Bluhm, Kostelecký, Lane
cyclotron frequency
H-/anti-p
c (e, p)
—26
Gabrielse et al.
hyperfine structure
H (sidereal)
b, d (e, p)
—27
Walsworth et al.
muonium (sid.)
b, d (mu)
—23
Hughes et al.
various
b, c, d (e, p, n)
—22 to —30
Kostelecký, Lane
He-Xe
b, d (n)
—31
Bear et al.
Cane et al.
spin-polarized
solid
b, d (e)
—29
Heckel et al.
Hou et al.
clock comparison
torsion pend.
The coefficients need not be diagonal in flavor
space either. Like neutrino masses, they may
mix different species.
In fact, three-parameter Lorentz-violating
models can explain all observed neutrino
oscillations (including LSND).
However, many possible parameters have not
been probed.
The “full” neutrino sector has 102 Lorentzviolating parameters.
Neutrino Tests of Relativity
Since neutrinos are always relativistic, they
are an interesting laboratory for looking for
changes to special relativity.
Constraints on  ( pˆ ) can be set in two ways:
• time of flight measurements, and
• energy-momentum measurements.


It’s well known that SN1987A neutrinos
traveled to Earth with a speed that differed
9

2

10
from c 1by a fraction
.

However, this bound
applies only to electron
neutrinos moving in one
direction.
We can get better bounds by looking at
energetic constraints.
We now feel confident that ultrahigh-energy cosmic
rays are primarily
protons, with en9
ergies up to 6 10
GeV.

The protons have to live long enough to travel
tens of Mpc to reach Earth.
Normally, that would be no problem, but
relativity violations might cause fast-moving
protons to decay, even if they’re stable at rest.
If the protons has speeds greater than 1, they
The primary cosmic rays must also be immune
to  -decay, p  n  e    e .
This is where the neutrinos come in.
This decay is disallowed only if
mn
 ( pˆ )  
 2 1011
ECR
This is only a one-sided bound if the neutrino

MAV is isotropic.
However, an anistropic MAV is bounded on
both sides at the 10 10 level.
The bounds are the same for the   . Just

swap the positron for a  .
The constraints on the MAV for   are worse by
a factor of 3, since in a  decay, the  
 contribution.
mass makes a significant




Most other particles that a proton could decay
into are also subject to similar or better
bounds.
Conclusion
Lorentz violation is an interesting possibility to
be part of the “Theory of Everything.”
Lorentz tests for ultrarelativistic particles like
neutrinos are parameterized by the MAV.
The fact that primary cosmic ray protons don’t
decay into neutrons sets stronger limits on the
neutrino MAV than time-of-flight
measurements.
Thanks to V. A. Kostelecký and E. Altschul.
That’s all, folks!
```