T -violation tests for relativity principles

Yvonne Y. Y. Wong §

Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

Abstract. We consider the implications of a violation of the equivalence principle or of Lorentz invariance in the neutrino sector for the T -asymmetry ΔP_{T} ≡ P (ν_{α} → ν_{β}) − P (ν_{β} → ν_{α}) in a three-flavour framework. We find that additional mixing due to these mechanisms, while obeying all present bounds, can lead to a substantial enhancement, suppression, and/or sign change in ΔP_{T} for the preferred energies and baselines of a neutrino factory. This in turn allows for the possibility of improving existing constraints by several orders of magnitude.

1. Introduction

The phenomenon of flavour oscillations follows from non-degenerate neutrino states and mixing. The “standard” mechanism stipulates that the degeneracy be broken by means of small neutrino masses. An alternative scenario that does not require neutrino masses is provided by a violation of the equivalence principle (VEP) in the neutrino sector [1, 2].

Phenomenologically, the standard and the VEP mechanisms differ in their energy dependences. In the two-flavour case, a formal transformation from the former to the latter scenario is accomplished by replacing in the oscillation probability δm^{2}/2E → 2E|ϕ|δγ, where δγ ≡ γ_{2} − γ_{1} is the difference in the neutrino–gravity couplings, and ϕ is the gravitational potential. Note that a violation of Lorentz invariance also contributes to lifting the neutrino degeneracy [3]. This scenario, however, has the same phenomenology as VEP [4].

The purpose of this work is not to promote VEP as a solution to the solar, atmospheric, and LSND neutrino puzzles—this has been shown by many to be futile (see [5] for a review), and the VEP breaking scale is severely constrained by null oscillation experiments, |ϕ|δγ <10^{−22} [6]. Rather, we examine the possibility of further constraining |ϕ|δγ with a new generation of experiments, namely, the measurement of the T -asymmetry ΔP_{T} ≡ P (ν_{α} → ν_{β}) − P (ν_{β} → ν_{α}) at a neutrino factory.

2. T -violation

Consider three-flavour oscillations governed by the Hamiltonian (in flavour basis)

H̃ = \frac{1}{2E} U_{M}MU_{M}^{†} + 2E U_{G}GU_{G}^{†} + V ≡ H^{M} + H^{G} + V, (1)

where M =Diag(−δm_{⊙}^{2}, 0, δm_{⊕}^{2}) is the mass term, G=|ϕ|×Diag(γ_{1}, γ_{2}, γ_{3}) comes from VEP, V = \sqrt{2} G_{F} N_{e} ×Diag(1, 0, 0) represents matter effects, U_{M} is the MNS matrix

§ To whom correspondence should be addressed (ywong@physics.udel.edu)

T -violation tests for relativity principles 2

with three angles and a CP phase δ, and U_{G} describes mixing in the VEP sector. We assume the solar and atmospheric neutrino anomalies to be completely explained by the square mass differences δm_{⊙}^{2} and δm_{⊕}^{2}, together with the angles θ_{⊙} and θ_{⊕} in U_{M}.

The asymmetry between the T -conjugate oscillation probabilities is given by

ΔP̃_{T} = 16J̃ sin \frac{Δ̃_{12}L}{2} sin \frac{Δ̃_{23}L}{2} sin \frac{Δ̃_{31}L}{2}, (2)

with Δ̃_{ij} = λ̃_{i} − λ̃_{j}, where λ̃_{1,2,3} are the eigenvalues of H̃, and J̃ is the effective Jarlskog factor. We wish to compare ΔP̃_{T} with the intrinsic T -asymmetry ΔP_{T} due to δ. Using the identity Δ_{12}Δ_{23}Δ_{31}J =Im(H_{eμ}H_{μτ} H_{τe}) [7, 8], and by restricting the neutrino energy and the experimental baseline to O(10)→O(30) GeV and O(3000)→ O(6000) km respectively, we find the ratio ΔP̃_{T} /ΔP_{T} to be well approximated by

\frac{ΔP̃_{T}}{ΔP_{T}} ≃ \frac{Im(H̃_{eμ}H̃_{μτ} H̃_{τe})}{Im(H_{eμ}H_{μτ} H_{τe})}, (3)

where H ≡ H^{M} + V is the Hamiltonian (1) without VEP contribution, assuming that |ϕ|δγ satisfies the condition 2E|ϕδγ| < |δm_{⊕}^{2}|/2E.

3. An example

The exact consequence of VEP on the T -asymmetry depends on the nature of H^{G}. We provide here the simplest example of pure ν_{e} ↔ ν_{τ} mixing in the VEP sector, with VEP breaking scale |ϕ|δγ and mixing angle φ. Equation (3) can now be written as

\frac{ΔP̃_{T}}{ΔP_{T}} ≃ 1 − \frac{4E^{2}}{δm_{⊙}^{2}} \frac{sin 2θ_{⊕}}{sin 2θ_{⊙}} |ϕ|δγ sin 2φ. (4)

Using the best fit values δm_{⊙}^{2} ≃ 5 × 10^{−5} eV^{2}, sin 2θ_{⊙} ≃ 0.8, and sin 2θ_{⊕} ≃ 1, we see that at E = 13 GeV, the VEP parameters |ϕ|δγ sin 2φ = ±10^{−25} can offset the T -asymmetry by ∓100 %, i.e., ΔP_{T} is enhanced or suppressed depending on the sign of |ϕ|δγ sin 2φ. Furthermore, if the offset exceeds 100 %—because of a large VEP breaking scale and/or a high neutrino energy, ΔP_{T} flips sign.

4. Conclusion

Extra neutrino mixing from VEP can enhance, suppress, and/or flip the sign of the intrinsic asymmetry between two T -conjugate oscillation processes, depending on the VEP breaking scale and the neutrino energy. Thus T -violation experiments run at two or more energies will allow us to either establish, or further constrain VEP and improve present bounds on |ϕ|δγ by at least two orders of magnitude.

References

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[4] Glashow S L, Halprin A, Krastev P I, Leung C N, and Pantaleone J 1997 Phys. Rev. D 56 2433 [5] Leung C N and Wong Y Y Y 2002 In preparation.

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[8] Harrison P F and Scott W G 2000 Phys. Lett. B 476 249