Testing Relativity TheoryWith Neutrinos

Brett Altschul

University of South Carolina

May 15, 2008

OverviewLorentz 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

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…)

Introduction

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

Standard Model Extension (SME)

Idea: Look for all operators that can contribute to Lorentz violation.

Then one usually adds restrictions:• locality• superficial renormalizability• gauge invariance• etc...

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SU(3)C × SU(2)L ×U(1)Y

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L =ψ iΓ μ∂μ − M( )ψ

M = m + / a − / b γ 5 +1

2H μν σ μν

Γ μ = γ μ + cνμγν − dνμγν γ 5

With those restrictions, the Lagrange density for a free fermion looks like:

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:

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vk = pk − ckj p j − c jk p j + c j kc jl pl( )E−1 − c0k

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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E ≈ m c

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c ≈ m MP

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E ≈ mMP

Most Lorentz-violating effects at high relativistic energies depend on a particle’s maximum achievable velocity (MAV).

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vmax =1+ δ( ˆ p ) =1− c jkˆ p j ˆ p k − c0 j

ˆ p j

The corresponding energy-momentum relation is

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E = m2 + 1+ 2δ( ˆ p )[ ]p2

The photon sector contains more superficially renormalizable couplings.

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L =−1

4F μν Fμν −

1

4kF( )μνυσ

F μν F ρσ +1

2kAF( )

μεμνρσ Aν F μν

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 E773Kostelecký

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 SuperKKostelecký, Mewes

birefringence photon kAF (CPT odd) —42 Carroll, Field, Jackiw

kF (CPT even) —32 to —37 Kostelecký, Mewes

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.

clock comparison various b, c, d (e, p, n) —22 to —30 Kostelecký, Lane

He-Xe b, d (n) —31 Bear et al.Cane et al.

torsion pend. spin-polarized solid

b, d (e) —29 Heckel et al.Hou et al.

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 Lorentz-violating parameters.

Neutrino Tests of Relativity

Since neutrinos are always relativistic, they are an interesting laboratory for looking for changes to special relativity.

Constraints on can be set in two ways:• time of flight measurements, and

• energy-momentum measurements.€

δ( ˆ p )

It’s well known that SN1987A neutrinos traveled to Earth with a speed that differed from by a fraction .

However, this boundapplies only to electronneutrinos moving in onedirection.

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< 2 ×10−9

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c =1

We can get better bounds by looking at energetic constraints.

We now feel con-fident that ultra-high-energy cosmicrays are primarilyprotons, with en-ergies up toGeV.

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6 ×109

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 would emit vacuum Cerenkov radiation.

The primary cosmic rays must also be immune to -decay, .

This is where the neutrinos come in.

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p+ → n + e+ + ν e

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β

This decay is disallowed only if

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δν ( ˆ p ) > −mn

ECR

≈ −2 ×10−11

This is only a one-sided bound if the neutrino MAV is isotropic.

However, an anistropic MAV is bounded on both sides at the level.

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10−10

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, themass makes a significant contribution.

Most other particles that a proton could decay into are also subject to similar or better bounds.

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ν μ

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μ+

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ντ

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τ +

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τ +

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.

That’s all, folks!

Thanks to V. A. Kostelecký and E. Altschul.