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: ̅ Ψ ℒ = Ψ In the standard model extension the lagrangian becomes [2]: 1 1 ̅ Ψ − ( ) Ψ ̅ Ψ + ( ) Ψ ̅ Ψ ℒ = Ψ 2 2 represents all terms that contain a Lorentz violation and CPT violation while 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 (ℎ ) ~ (2 ) 2 1 + [( ) − ( ) ] where E and are energy and momentum of the neutrino and( ) , ( ) 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]. The probability of oscillation of a can be written as → = 1 − |(ℎ ) | 2 2 − |(ℎ ) | 2 2 for simplicity we will neglect the → transition so the probability becomes→ = 1 − |(ℎ ) | 2 2. 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 ( ) are zero. Taking all this into account the Hamiltonian can be written: 1 (ℎ ) ~ [( ) ] 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 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 0 − sin ̅ = (− cos cos 0 ) + ( cos ) − sin cos 0 0 Where is the angle between the earths Z axis and its rotational XY plane around the sun ≈ 23° . 0 is the earths orbital angular frequency and is the earths orbital velocity. is the velocity of the lab due to the Earth’s rotation and is the Earth’s sidereal frequency and is sidereal time. Observe that ̅ depends on − sin , ̅ depends on cos and ̅ has no sidereal time dependence. We can use these dependencies to separate ( ) into sidereal time independent parts[( ) , ( ) ] and dependent parts with [( ) , ( ) ]. (ℎ ) = [(( ) + ( ) ̂ ̂ ̂ ) + ( ) + ( ) ] In the beam’s frame this can be written as ̂ ( ) + [ ̂ sin − ̂ cos ]( ) (ℎ ) = [( ) − ̂ sin − ̂ cos ]( ) ] + [− ̂, ̂ and ̂ hold directional information depending on direction and location of the neutrino Where signal. Grouping terms with like sidereal time dependence we obtain [5]. 2 → = 1 − 2 [|() + ( ) sin + ( ) | ] () = [−( ) − 0.29 ( ) ] ( ) = [−0.91( ) + 0.29( ) ] ( ) = [0.29( ) + 0.91( ) ] 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 − probability is dominated by the constant () term and the sidereal time dependent terms ( ) and ( ) are small compared to the uncertainties and this effect is consistent with no sidereal time dependence. According to the fit all ( ) terms (if they do exist) must be on the order of ~10−20 GeV. Although in this calculation a non-zero () 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 [( ) , ( ) ] 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

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