[Paper Review] On ultra-relativistic approximations, unobservable phases and other hand-waving in the derivation of the neutrino oscillation length

Jean-Michel Lévy|ArXiv.org|Jan 5, 2009

Neutrino Physics Research3 citations

TL;DR

This paper rigorously re-derives the neutrino oscillation length using Lorentz-invariant plane-wave formalism, exposing and correcting widespread hand-waving in standard derivations—particularly the flawed ultra-relativistic approximation that replaces time with distance. It proves that deviations from the standard oscillation length are negligible for ultrarelativistic neutrinos, confirming the robustness of the conventional formula under precise mathematical treatment.

ABSTRACT

A wrong derivation of the phase of a propagating massive particle which has repeatedly appeared during the last years has the advantage of leading once more to the important question of the phase difference between the mass eigenstates constitutive of an oscillating neutrino described in the plane wave formalism. Serious errors of principle are pointed at in a number of simple calculations and the oscillation length is derived in a way which allows to show that the standard result can suffer but minute variations in the ultrarelativistic case.

Motivation & Objective

• To correct widespread pedagogical and technical errors in the derivation of the neutrino oscillation length, especially in ultra-relativistic approximations.
• To identify and expose the misuse of the approximation $ T = L $, which incorrectly replaces time with distance in phase calculations.
• To demonstrate that the standard oscillation length formula is robust under precise derivation, with only minute deviations possible in the ultrarelativistic regime.
• To establish the Lorentz invariance of the oscillation phase by deriving it from first principles without ad hoc approximations.
• To clarify that the phase difference between mass eigenstates is physically observable and gauge-invariant in the context of wave packets and propagators.

Proposed method

• Derives the phase of a massive particle using the relativistic energy-momentum relation $ E^2 = p^2 + m^2 $, avoiding the flawed $ T = L $ substitution.
• Uses the exact phase expression $ \Phi = pL - ET $, with $ L = vT $, and applies the correct kinematic relations $ p = vE $ for any velocity $ v $.
• Applies Lorentz invariance by showing that the phase $ \Phi = -m\tau $, where $ \tau $ is proper time, is invariant across frames.
• Parametrizes deviations from the standard velocity using $ 1/v = 1/v_0 + \epsilon $, where $ v_0 $ is the energy-weighted average velocity of mass eigenstates.
• Derives the oscillation length as $ L = \frac{4\bar{p}\pi}{\delta m^2 + 2\bar{p}\delta E \epsilon} $, showing dependence on $ \epsilon $, and bounds $ \epsilon $ using kinematic constraints.
• Compares the derived result to the standard formula $ L_0 = \frac{4\pi\bar{p}}{\delta m^2} $, showing that $ L \approx L_0 (1 - \bar{p}\epsilon / \sqrt{s}) $, with small corrections.

Experimental results

Research questions

• RQ1 Why is the standard ultra-relativistic approximation $ T = L $ physically unjustified in deriving the neutrino oscillation phase?
• RQ2 How can the oscillation phase be derived without hand-waving, ensuring Lorentz invariance and consistency with the rest frame?
• RQ3 What are the quantitative limits on deviations from the standard oscillation length in the ultrarelativistic regime?
• RQ4 How does the choice of velocity for a mixed neutrino state affect the oscillation phase and length?
• RQ5 Why is the phase difference between mass eigenstates physically meaningful despite claims of gauge non-invariance or unobservability?

Key findings

• The standard derivation that replaces $ T $ with $ L $ in $ \Phi = pL - ET $ is incorrect and leads to a phase that is half the correct value in the rest frame.
• The correct phase is $ \Phi = -\frac{m^2 T}{E} = -\frac{m^2 L}{p} $, which reduces to $ -\frac{m^2 L}{2E} $ in the ultrarelativistic limit, matching the standard formula.
• Deviations from the standard oscillation length are bounded by $ \epsilon $, which is extremely small due to the small difference in velocities between mass eigenstates.
• In a $ \pi \to \mu\nu $ decay, $ \bar{p}/\sqrt{s} \approx 0.21 $, and $ \epsilon $ is suppressed by $ |v_1 - v_2| $, leading to corrections of order $ \bar{p}\epsilon / \sqrt{s} \ll 1 $.
• The oscillation length is $ L = \frac{L_0}{1 + \bar{p}\epsilon / \sqrt{s}} $, showing that deviations are negligible for light, ultrarelativistic neutrinos.
• The phase $ \delta\Phi = -x \frac{\delta m^2}{2\bar{p}} $ is Lorentz-invariant when $ x $ is interpreted as the distance traveled in the frame of momentum $ \bar{p} $, confirming its physical consistency.