Title: A possible statistical mechanism of anomalous neutrino velocity in OPERA experiment?
ArXiv ID: 1109.5727
Date: 2011-10-18
Authors: Researchers from original ArXiv paper
📝 Abstract
The set of kinetic equations describing the process of conversion of a beam of protons into mesons and then to neutrinos is solved. The asymptotic evolution of the density profile of neutrinos is essentially the same as that obtained in the previous version of the note for a simple model of uniformly damped wave-packet. It shows again that the recently reported "superluminal neutrinos" could be considered as a purely statistical effect due to the fact that the detected neutrinos represent a biased sample of initial protons.
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The set of kinetic equations describing the process of conversion of a beam of protons into mesons and then to neutrinos is solved. The asymptotic evolution of the density profile of neutrinos is essentially the same as that obtained in the previous version of the note for a simple model of uniformly damped wave-packet. It shows again that the recently reported “superluminal neutrinos” could be considered as a purely statistical effect due to the fact that the detected neutrinos represent a biased sample of initial protons.
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The set of kinetic equations describing the process of conversion of a beam of protons into mesons and then to neutrinos is solved. The asymptotic evolution of the density profile of neutrinos is essentially the same as that obtained in the previous version of the note for a simple model of uniformly damped wave-packet. It shows again that the recently reported "superluminal neutrinos" [1] could be considered as a purely statistical effect due to the fact that the detected neutrinos represent a biased sample of initial protons.
Consider a one-dimensional model which describes propagation of a beam of particles with the speed of light. The beam consists of protons with the density p(x, t), mesons with the density m(x, t) and neutrinos with the density n(x, t). The kinetic equations read
where Γ p , Γ m , Γ n are damping rates for protons, mesons and neutrinos respectively, while γ p and γ m describe transformation rated of protons into mesons and mesons into neutrinos.
The system of equations ( 1) can be easily solved and for the initial conditions :
Under the physically justified condition Γ n « Γ p , Γ m and for t » 1/Γ n the expression (2) simplifies to
which essentially coincides with the result of the previous version of this note. For the initial Gaussian density profile (concentrated at the origin for t = 0) the neutrino density evolves asymptotically as
where d is the width of the density profile and N -the normalization constant. The time resolved density profile (4) at the point x = L » c/Γ n can be recast into the form
where the shift
can be misinterpreted as a consequence of a higher velocity v ≃ c(1 + δL/L). The origin of this shift is purely statistical, the neutrinos created at the longer distance from the proton source, and hence arriving earlier at the detector, are slightly less damped than those created at the shorter distance.
To apply this result to the OPERA experiment [1] one should notice that in the experiment one compares the density profile for the protons -the “grandparents of the neutrinos” with the profile for neutrinos. The former is measured under conditions where damping can be neglected and the time-resolved shape does not display the shift δL. On the other hand, the highly damped density profile of neutrinos is shifted. The damping rate Γ n describes an overall effect of all processes which reduce the fraction of neutrinos produced by the source in CERN which are detected in Gran Sasso Lab. Finally, one can compare the relevant numbers. The initial proton density profile is composed of five peaks which can be approximated by the Gaussians of the width d ≃ 600m. This value combined with the value of L ≃ 700km and the measured relative shift δL/L ≃ 2.5 × 10 -5 yield the overall damping factor (the prefactor γ p γ m /Γ p Γ m is neglected)
This value is consistent (within a reasonable accuracy) with the ratio N ν /N p ∼ 10 -16 where N ν ∼ 10 4 is the total number of neutrinos detected in OPERA and N p ∼ 10 20 is the number of the corresponding protons.
