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Lorentz violation in Quantum Field Theory and an Experimental limit.
Ben Rybolt
12/7/2012
Introduction
In the Standard Model Lorentz violating terms are not allowed due to the assumption that space is
homogenous. The assumption implies that Laws of physics are invariant regardless of transformations in
space time. In the Standard Model Extension [1] we allow for terms to be added to the Standard Model
that are not Lorentz Invariant. In this paper I will discuss how Lorentz Violating terms can be measured
using neutrino oscillations.
The Standard Model Lagrangian for leptons is given by:
̅ 𝛾 𝜇 𝐷𝜇 Ψ
ℒ = 𝑖Ψ
In the standard model extension the lagrangian becomes [2]:
1
1
̅ 𝑎 𝛾 𝜇 𝐷𝜇 Ψ𝑎 − (𝑎𝐿 )𝜇𝑎𝑏 Ψ
̅ 𝑎 𝛾 𝜇 Ψ𝑏 + 𝑖(𝑐𝐿 )𝜇𝜈𝑎𝑏 Ψ
̅ 𝑎 𝛾 𝜇 𝐷𝜈 Ψ𝑏
ℒ = 𝑖Ψ
2
2
𝑎𝐿 represents all terms that contain a Lorentz violation and CPT violation while 𝑐𝐿 is only Lorentz
violating.
These additional terms designate a possible preferred direction in the universe. If these terms exist we
would like to devise a way to measure them or show that they are zero.
In the Standard Model Extension the effective Hamiltonian of neutrino oscilations can be written as
(ℎ𝑒𝑓𝑓 )𝑎𝑏 ~
(𝑚2 )𝑎𝑏
2𝐸
1
+ 𝐸 [(𝑎𝐿 )𝜇 𝜌𝜇 − (𝑐𝐿 )𝜇𝜈 𝜌𝜇 𝜌𝜈 ]𝑎𝑏
where E and 𝜌𝜇 are energy and momentum of the neutrino and(𝑎𝐿 )𝜇 , (𝑐𝐿 )𝜇𝜈 are Lorentz violating
terms.
If Lorentz violating terms exist than we know they must be very small because no previous experiment
has observed them. It has been proposed that neutrino oscillation experiments might be able to see
small violations because they act as interferometers. These test have already been done at accelerator
based detectors and this paper describes a way to investigate Lorentz violating terms at the Double
Chooz reactor based detector [3].
𝜈
𝜈
The probability of oscillation of a 𝜈𝑒 can be written as 𝑃𝜈𝑒→𝜈𝑒 = 1 − |(ℎ𝑒𝑓𝑓
) | 2 𝐿2 − |(ℎ𝑒𝑓𝑓
) | 2 𝐿2 for
𝑒𝜇
simplicity we will neglect the 𝑒 → 𝜏 transition so the probability becomes𝑃𝜈𝑒→𝜈𝑒 = 1 −
𝑒𝜏
𝜈
|(ℎ𝑒𝑓𝑓
) | 2 𝐿2.
𝑒𝜇
Further simplification can occur by neglecting the mass difference term. This assumption means that
neutrino oscillations are due to Lorentz violating terms. We do believe this to be the case but it will
allow us to put an upper limit on what the Violating terms could be. Previous experiments can further
simplify our expressing by requiring that all (𝑐𝐿 ) are zero. Taking all this into account the Hamiltonian
can be written:
1
(ℎ𝑒𝑓𝑓 )𝑎𝑏 ~ [(𝑎𝐿 )𝜇 𝜌𝜇 ]𝑎𝑏
𝐸
Notice that the Hamiltonian and thus the probability of oscillation depends on the direction of
propagation of the neutrinos. Since the laboratory is on earth we need to transform into a sun centered
reference frame. This can be done with a frame transformation that splits 𝑣 into a standard time
dependent piece and a sidereal time dependent piece [4]. Transforming from an earth based frame to a
sun centered frame yields
sin 𝛺0 𝑇
− sin 𝜔𝑠 𝑇𝑠
𝑣̅ = 𝑣𝑜 (− cos 𝜂 cos 𝛺0 𝑇) + 𝑣𝑟 ( cos 𝜔𝑠 𝑇𝑠 )
− sin 𝜂 cos 𝛺0 𝑇
0
Where 𝜂 is the angle between the earths Z axis and its rotational XY plane around the sun 𝜂 ≈ 23° . 𝛺0 is
the earths orbital angular frequency and 𝑣𝑜 is the earths orbital velocity. 𝑣𝑟 is the velocity of the lab due
to the Earth’s rotation and 𝜔𝑠 is the Earth’s sidereal frequency and 𝑇𝑠 is sidereal time. Observe that 𝑣̅𝑥
depends on − sin 𝜔𝑠 𝑇𝑠 , 𝑣̅𝑦 depends on cos 𝜔𝑠 𝑇𝑠 and 𝑣̅𝑧 has no sidereal time dependence. We can use
these dependencies to separate (𝑎𝐿 )𝜇 𝜌𝜇 into sidereal time independent parts[(𝑎𝐿 )𝑇 , (𝑎𝐿 )𝑍 ] and
dependent parts with [(𝑎𝐿 )𝑋 , (𝑎𝐿 )𝑌 ].
(ℎ𝑒𝑓𝑓 )𝑒𝜇 = [((𝑎𝐿 )𝑇 + (𝑎𝐿 ) 𝑧
𝑝̂𝑦
𝑝̂𝑧
𝑝̂𝑥
) + (𝑎𝐿 )𝑥 + (𝑎𝐿 )𝑦 ]𝑒𝜇
𝐸
𝐸
𝐸
In the beam’s frame this can be written as
̂ 𝑧 (𝑎𝐿 ) 𝑧 + [𝑁
̂ 𝑦 sin 𝜔𝑠 𝑇𝑠 − 𝑁
̂ 𝑥 cos 𝜔𝑠 𝑇𝑠 ](𝑎𝐿 )𝑥
(ℎ𝑒𝑓𝑓 )𝑒𝜇 = [(𝑎𝐿 )𝑇 − 𝑁
̂ 𝑥 sin 𝜔𝑠 𝑇𝑠 −𝑁
̂ 𝑦 cos 𝜔𝑠 𝑇𝑠 ](𝑎𝐿 )𝑦 ]𝑒𝜇
+ [−𝑁
̂𝑥, 𝑁
̂ 𝑦 and 𝑁
̂ 𝑧 hold directional information depending on direction and location of the neutrino
Where 𝑁
signal. Grouping terms with like sidereal time dependence we obtain [5].
2
𝑃𝜈𝑒→𝜈𝑒 = 1 − 𝐿2 [|(𝐶)𝑒𝜇 + (𝐴𝑠 )𝑒𝜇 sin 𝑤𝑠 𝑇𝑠 + (𝐴𝑐 )𝑒𝜇 𝑐𝑜𝑠𝑤𝑠 𝑇𝑠 | ]
(𝐶)𝑒𝜇 = [−(𝑎𝐿 )𝑇 − 0.29 (𝑎𝐿 )𝑍 ]
(𝐴𝑠 )𝑒𝜇 = [−0.91(𝑎𝐿 )𝑋 + 0.29(𝑎𝐿 )𝑌 ]
(𝐴𝑐 )𝑒𝜇 = [0.29(𝑎𝐿 ) 𝑋 + 0.91(𝑎𝐿 )𝑌 ]
Here the particular geometry of the Double Chooz experiment has been applied.
Experimental Results
The Double Chooz experiment measures Neutrinos from two nuclear reactors ~1 km away. The detector
measures ~ 40 neutrinos per day. In order to look for sidereal time variations neutrino events where
divided in to 24 bins representing one sidereal day (23.934 hours). The data set analyzed represented
~230 days of live time. With this long of a data set we can be sure that daily fluctuations will not affect
the sidereal time dependence.
You can see that the 𝑒 − 𝜇 probability is dominated by the constant (𝐶)𝑒𝜇 term and the sidereal time
dependent terms (𝐴𝑠 )𝑒𝜇 and (𝐴𝑐 )𝑒𝜇 are small compared to the uncertainties and this effect is
consistent with no sidereal time dependence. According to the fit all (𝑎𝐿 )𝜇 terms (if they do exist) must
be on the order of ~10−20 GeV. Although in this calculation a non-zero (𝐶)𝑒𝜇 term was obtained, this is
not a sure signal for Lorentz violation. In the beginning we made an assumption that the mass term in
the effective Hamiltonian was zero. Experiments show that there is a mass term and its affect is causes a
non-sidereal time dependent oscillation consistent with the 90% deficit we observed in the neutrino
signal.
Conclusion
Lorentz Violating terms in the Standard Model Extension predict a sidereal time dependent oscillation in
neutrinos in earth based experiments. For a simplified model where we assume only two flavors of
neutrino we have observed no sidereal time dependent oscillation and put an upper limit on the
strength of two parameters [(𝑎𝐿 ) 𝑋 , (𝑎𝐿 )𝑌 ] to <10−20 GeV.
[1]D. Colladay and A. Kostelecky, CPT Violation and the Standard Model, Phys. Rev. D 55, 6760 (1997).
[2]V.A. Kostelecky and M. Mewes, Lorentz and CPT violation in the neutrino sector, Phys Rev D 70,
031902
[3]Y. Abe et al., First Test of Lorentz Violation with a Reactor-Based Antineutrino Experiment,
arXiv:1209.5810[hep-ex]
[4]V.A. Kostelecky and M. Mewes Phys. Rev D. 66, 056005, Appendix C
[5] V.A. Kostelecky and M. Mewes Phys., Lorentz Violation and Short-baseline Neutrino Experiments,
Phys Rev D 70, 076002