Superluminal neutrino energy spectrum of OPERA and MINOS

and group velocity

equal to the speed of light c = 1 (in relativistic units). The modern theory expects, however, that neutrino has finite mass [1] m = m ν < 0.28 eV (3) that implies deviation from the energy spectrum (1) and velocity v = c. Recent experiments of the OPERA Collaboration [2] have revealed suprluminal motion of neutrino. The time between signals

measured at the baseline L = 730 km, was less than t 0 = L/c, implying that a small delay δt was negative and the velocity of neutrino

was definitely above the speed of light. The velocity shift (5) revealed almost no dependence on energy, and at the average energy

the time delay δt = -57.8 ± 7.8 [stat.] +8.3 -5.9 [sys.] ns (7) corresponded to

Superluminal neutrino was also observed by the MINOS Collaboration [3] as well as in supernova explosion SN1987a [4] . This fact is a serious puzzle to the researchers. There is no lack of hypotheses to explain it [5]. However, the value of superluminal velocity (8) imposes a severe constraint on the energy spectrum. The energy dependence of v was also explored by the OPERA collaboration [2], but it was not possible to warrant solid data because it was beyond the accuracy of the measurement on account of large errors. However, the energy dependence of (v -1) on E can contain very important information that may allow to understand the nature of neutrino. In the present paper we analyze the experimental data of the OPERA [2] and MINOS [3] and try to establish the range of possible energy spectrum of superluminal neutrino.

The most natural and plain idea is to treat neutrino as a massive tachyon whose energy spectrum

for an ultra-relativistic particle (E ≃ p ≫ m). It implies a tiny positive shift above the speed of light. However, according to (3) and (11), we cannot get estimation greater than v -1 ≃ 10 -22 (12) at E = 17 GeV. Otherwise, we have to expect very large tachyon mass m ≃ 120 MeV that contradicts to the expected upper bound (3). It is clear that superluminal neutrino cannot be a free tachyon with the energy spectrum (9), neither a free tardyon (ordinary particle) with the energy spectrum E = p 2 + m 2 . However, it is not a problem of the theory because there are many sophisticated arguments explaining the superluiminal velocity of neutrino [5]. Nevertheless, it is highly desirable to know the energy dependence of quantity

that is equivalent to dependence on the momentum f [p] for an ultra-relativistic particle. The knowledge of this dependence allows to restore the energy spectrum of neutrino

and test hypotheses of its nature.

The OPERA collaboration [2] has also obtained the following data at various energy that together with ( 6)-( 7) can be described by proportionality [6] δt ∼ E a-1 (18)

and, according to ( 5) and ( 18), we get [7,8] f

that is equivalent to

for an ultra-relativistic particle. Let us develop this interpretation in detail.

According to ( 7) and ( 15)-( 17

Substituting ( 20) in ( 14) we get the energy spectrum of ultra-relativistic neutrino

that satisfies both the OPERA and MINOS experiment. For example, choosing a = 1 in (28), we obtain velocity ( 8) and ( 23) within the accuracy of measurements when A = (2.2 ÷ 3.03) ×10 -5 .

Superluminal neutrino observed in experiments [2,3] cannot be a free tachyon because it mass (3) is not enough to correspond the observed velocity (8). The dependence (13) of the neutrino velocity v on the energy E may give much information about its energy spectrum (14). It can be taken as a power law (28) with parameters (26) and ( 27). It should be emphasized that parameter a in (28) must be positive, and the real energy spectrum lays somewhere between E = p + 1.5 × 10 -5 p 1.18 and E = p + 4.15 × 10 -4 p 0.4 . There is no possibility to establish a and A at high accuracy because of errors of experimental data. It is impossible even to clarify whether a > 1 or a < 1 and whether function v[E] is monotonically increasing. Of course, new measurements will reveal the exact energy spectrum of neutrino and clarify its physical nature. Now we can only state that superluminal neutrino is not a free tachyon and that the second term in the right side of (28) may give a hint to nonlinear self-interaction or external field acting on neutrino. It is the subject of further theoretical work.

The author is grateful to Erwin Schmidt for discussions.