Superluminal Neutrinos in a Pseudoscalar Potential

The OPERA collaboration recently reported the evidence of superluminal propagation of muon neutrinos [1]. An earlier report by the MINOS collaboration [2] reached the similar conclusion with a lower statistical confidence. If the result is not due to an unknown instrumental effect, this fact inevitably points towards new physics. Since the release of the OPERA data, many ideas have been disposed to interpret the data, e.g. [3][4][5][6][7][8]. The difficulties of various interpretations have been discussed [9][10][11][12][13]. One of the difficulties is to account for the energy dependence of neutrino speed anomaly ∆ = (v -c)/c. While OPERA reports ∆ = (2.48 ± 0.28(stat.) ± 0.30(sys.) × 10 -5 , Ēν = 17 GeV

(1) ∆ 1 = (2.18 ± 0.77(stat.) ± 0.30(sys.) × 10 -5 , Ēν,1 = 13.9 GeV

(2)

The MINOS data, with less significance, suggests ∆ = (5.1 ± 2.9) × 10 -5 , Ēν = 3 GeV .

(

This further sharpens the ∆ contrast in the 10 MeV to 10 GeV range.

Here we propose a superluminal neutrino theory without introducing Lorentz Invariance Violation (LIV) or particles with imaginary masses (tachyons). We envisage a pseudoscalar potential under the influence of which the neutrino propagates from CERN to OPERA detector.

We introduce a pseudoscalar potential φ > 0, which can be energy-dependent, but is constant in space (or can be approximated as a constant for the spatial range in consideration). The Dirac equation for a neutrino with mass m ν in such a pseudoscalar potential can be written as

• Email: sarira@nucleares.unam.mx, zhang@physics.unlv.edu where α, β and γ 5 are the Dirac matrices. The natural units with c = 1 and = 1 are adopted here. The neutrino wave function ψ can be expressed in terms of two-component spinors u and v as

where N is the normalization constant. These two spinors u and v satisfy the coupled equations

Solving these two equations, we obtain the neutrino energy

For

The group velocity of the neutrino is then

where φ = dφ/dE ν , or

The condition of superluminal neutrinos (∆ > 0) can be written as

III. CONSTRAINING THE FORM OF φ WITH DATA

The condition ( 13) is satisfied for a wide range of φ forms. Below we attempt to construct an analytical form of φ that gives superluminal motion and satisfies the observational constraints.

An energy-independent potential ( φ = 0) naturally satisfies the condition (13), so that

Such a form is similar to many theories invoking LIV or tachyon particles e.g. [8,9,13], which is ruled out by the OPERA differential speed data. The ratio of velocity difference in this model is

According to equations ( 2) and ( 3), the ratio between two average neutrino energies is about Ēν,2 / Ēν,1 ∼ 3.1. This corresponds to a predicted ∆ 1 /∆ 2 ∼ 9.5, while the observed ratio is ∆ 1 /∆ 2 = 0.79 +1.11 -0.50 . So a more complicated form of φ with a much shallower E ν dependence on ∆ is needed.

For a pseudoscalar potential of the form

the superluminal condition ( 13) can be translated to

The ratio of velocity difference in this model is ∆ 1 /∆ 2 = ( Ēν,1 / Ēν,2 ) 2α-2 . Solving for α using the OPERA data, one gets 0.72 < |α| < 1.55 (17) with the typical value |α| ∼ 1.1. This range is inconsistent with the condition (16), and is in the subluminal regime. Therefore the power law model cannot interpret the OPERA data.

The pseudoscalar potential of the form

gives

The superluminal condition is α < 1/2 + E ν /E 0 . However, when E ν ≪ E 0 and E ν ≫ E 0 , the velocity difference has the energy-dependence in the form of ∆ ∝ E -2 ν and ∆ ∝ E -1 ν , respectively. Both dependences are too steep to account for the OPERA data in equations ( 2) and ( 3). This form is therefore not favored.

We consider the potential of the form

The superluminal condition ( 13) can be translated to

for α < 1/2 (the branch α > 1/2 is not favored since the E ν -dependence of ∆ is very steep). The velocity difference takes the form

(22) For a not very high E 0 , the superluminal condition ( 21) is easily satisfied. The steep ∝ E -2 ν dependence is compensated by other factors in equation ( 22) so that a shallow E ν -dependence can be achieved in the superluminal regime. By properly adjusting E 0 and normalization C, the OPERA data can be interpreted.

We have shown that if neutrinos travel in a pseudoscalar potential, superluminal propagation is possible under the condition given in equation ( 13), without violating special relativity. In order to interpret the OPERA data, a shallow energy-dependence is required in the superluminal regime. A potential form similar to equation (20) with α < 1/2 can meet such a requirement.

The very small |∆| derived for SN 1987A of 10 MeV neutrinos is difficult to account for with such a potential. If one adjusts E 0 to interpret OPERA data, the 10 MeV neutrinos would be in the subluminal regime with a large |∆| violating the data constraint. If one instead adjusts 10 MeV to the transition point from superluminal to subluminal, i.e. E ν ∼ E ν,c = E 0 exp( 21-2α ) ∼ 10 MeV,

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