Superluminal Dark Neutrinos

Superluminal Dark Neutrinos arXiv:1110.0456v2 [hep-ph] 11 Oct 2011

Irina Ya. Aref’eva, Igor V. Volovich Steklov Mathematical Institute Russian Academy of Sciences Gubkin St. 8, 119991, Moscow, Russia e-mails:[email protected], [email protected]

Abstract The OPERA collaboration has claimed the discovery of supeluminal neutrino propagation. However the superluminal interpretation of the OPERA result was refuted by Cohen and Glashow because it was shown that such superluminal neutrinos would lose energy rapidly via the bremsstrahlung of electron-positron pairs (arXiv:1109.6562). We note that the superluminal interpretation is still possible if there exists a new (dark) neutrino which can propagate with a superluminal velocity and which couples with usual neutrinos only via the mass mixing leading to neutrino oscillations. It is supposed that the physical laws are invariant under rotations and translations in a preferred reference frame. Two possible pictures to parameterize departures from Lorentz invariance are discussed: a ”conventional” tachyonic dark neutrino and the modification of the Lagrangian by adding perturbations with the maximum attainable speed of the dark neutrino which is larger than the speed of light in vacuum. We analyze also the MINOS and SN1987a data and show that they are consistent with the conjecture that there exists the superluminal dark neutrino.

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The special theory of relativity is a cornerstone of modern fundamental physics having the upper limit of velocities which is the velocity of light in vacuum. The OPERA collaboration has recently announced the results about possible evidence for superluminal propagation of neutrinos [1]. Specifically, the CNGS beam of muon neutrinos with mean energy of 17 GeV produced at CERN, travels about 730 km to the OPERA detector in the Gran Sasso Laboratory in Italy. An early arrival time of the muon neutrinos with respect to the one computed assuming the speed of light in vacuum of 60 ns is reported. This anomaly corresponds to a relative difference of the muon neutrino velocity with respect to the speed of light δ ≡ (v − c)/c = 2.5 × 10−5 . Obviously such an astonishing claim requires of course extraordinary standards of proof including confirmation by independent experiments. The earlier MINOS experiment [2] reported a measurement of δ = 5 × 10−5 with lower neutrino energies peaking at 3 GeV. At the lower energy, in the 10 MeV range, a stringent limit of |δ| < 2 × 10−9 was set by the observation of (anti) neutrinos emitted by the SN1987a supernova [3]. There are various investigations of constraints on neutrino velocities and possible mechanisms for breaking the standard Lorentz invariance motivated by the OPERA claim [4]–[21]. It is known that in the case of superluminal propagation, certain otherwise forbidden processes are kinematically permitted, even in vacuum, being analogues to Cherenkov radiation. Cohen and Glashow [16] made an important observation that especially the process of pair bremsstrahlung νµ −→ νµ + e+ + e− .

(1)

places a severe constraint upon the superluminal velocities. From the neutral current weak interaction the authors of [16] have computed the rate of pair emission Γ by an energetic superluminal neutrino and dE/dx, the rate at which it loses energy in the high energy limit: ′

Γ = K G2F E 5 δ 3 , dE/dx = −KG2F E 6 δ 3 .

(2)

′

Here K and K are numerical constants and δ = vν2 − 1 while the speed of light is set to unity. Assuming δ is a constant one gets that neutrinos with initial energy E0 , after traveling a distance L, will have energy E as given by the formula E −5 − E0−5 = 5KG2F Lδ 3 ≡ ET−5 .

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Therefore neutrinos with initial energy greater than the terminal energy ET rapidly approach ET and the original beam would be strongly depleted and spectrally distorted upon its arrival at the Gran Sasso. Cohen and Glashow conclude [16] that the observation of neutrino with energies in excess of 12.5 GeV can not be reconciled with the claimed superluminal neutrino velocity measurement. We suggest that the superluminal interpretation of the OPERA data is still possible if there is a new (dark) neutrino which can propagate with a superluminal velocity and 2

which couples with the usual neutrino only via the mass mixing leading to neutrino oscillations. We don‘t assume the Yukawa coupling for the dark neutrino and the seesaw mechanism, so the dark neutrino is different from the sterile neutrino which could suffer from the difficulty with pair bremsstrahlung. For the dark neutrino there is no process of pair bremsstrahlung (1) since the dark neutrino sector couples with the standard model sector only by means the mass mixing. To investigate possible violations of Lorentz symmetry it is supposed that the physical laws are invariant under rotations and translations in a preferred reference frame. Two possible pictures to parameterize departures from Lorentz invariance are discussed: a ”conventional” tachyonic dark neutrino and the modification of the Lagrangian by adding perturbations with the maximum attainable speed of the dark neutrino which is larger than the speed of light in vacuum. The last picture was considered by Coleman and Glashow for the ”usual” neutrino [22, 23]. They considered the case of spacetime translations along with exact rotational symmetry in the rest frame of the cosmic background radiation, but allow small departures from boost invariance in this frame. Perturbative departures from Lorentz invariance are then parametrized in terms of a fixed time-like 4-vector. We analyze also the MINOS and SN1987a data and show that they are consistent with the conjecture that there exists the superluminal dark neutrino. Neutrino oscillations were predicted by Pontecorvo [24]. It arises from a mixture between the mass and lepton flavor (electron, muon or tau) eigenstates of neutrinos. A neutrino created with a specific flavor can later be measured to have a different flavor. The transformation from the flavor to the mass eigenstates is performed by means of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. Let us discuss first a toy model of particle oscillations when there are only two scalar fields: the field φ with the usual mass m and the superluminal (dark) tachyon field χ. The Lagrangian is 1 1 1 1 L = (∂µ φ)2 − m2 φ2 + (∂µ χ)2 + M 2 χ2 + λφχ + Lint (φ) 2 2 2 2

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where Lint (φ) depends only on the field φ. The superluminal dark sector χ couples with the usual field φ only via the quadratic term λφχ. To quantize the tachyon field χ we have to restrict ourself integration in the momentum space over momenta greater than M, so we have violation of Lorentz invariance. There are particle oscillations described by this Lagrangian. If we neglect the interaction term Lint (φ) then one can go from the flavor basis (φ, χ) to the mass eigenstates basis (ϕ1 , ϕ2 ) with the mass squares (κ21 , −κ22 ) by using the PMNS matrix: φ = cos θϕ1 + sin θϕ2 , χ = − sin θϕ1 + cos θϕ2 .

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For the creation and annihilation operators it will be just the Bogoliubov transformation. The mean value of energy for the φ-particles will be 2

E = cos θ

q

p2

+

κ21 3

2

q

+ sin θ p2 − κ22

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and the mean velocity is p sin2 θ p cos2 θ +q v=q p2 + κ21 p2 − κ22

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with a rather complicated dependence from energy E. To describe propagation of the φ particle we consider the amplitude < f |e−itH |f > where the state |f > is a tensor product of the one particle state in the physical sector of the field φ and (bare) vacuum in the dark sector of the field χ. The role of the wave packets in the theory of particle oscillations is considered in [25]. This amplitude describes particle oscillations and the particle of the physical field φ propagates with the superluminal velocity due to the coupling with the dark sector of the χ-field. In this system there is entanglement between two sectors and one could disentangle φ-particles by taking trace over the dark sector. Vacuum decay in the presence of tachyons was discussed in [26]. Gravity with Lorentz violation and superluminal propagation in various models with extra dimensions have been considered in [27]-[31]. Note also the string models with right neutrinos on the branes [32]. Instead of coupling with the tachyon field one could use the Coleman-Glashow parametrization of the violation of Lorentz invariance. In this case the field χ has a normal mass and one adds the term − 2ǫ (∂i χ)2 which violates the Lorentz invariance: 1 1 ǫ 1 1 L = (∂µ φ)2 − m2 φ2 + (∂µ χ)2 − (∂i χ)2 − M 2 χ2 + λφχ + Lint (φ) 2 2 2 2 2

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It is tempting to speculate that the Lagrangians of this type could be obtained by the dimensional reduction in the Kaluza-Klein or superstring theories when instead of Minkowsky we reduce to a spacetime with less symmetries. One can assume that true fundamental constants live in ten dimensions and the four-dimensional fundamental constants, including not only the Newton constant but also the speed of light and even the Planck constant, can be obtained as approximated values depending on the compactification. The minimal extension of the standard model Lagrangian LSM along of this discussion with a Majorana mass term looks as follows L = LSM +

mαβ c ν¯ νLβ + h.c. + LLor (χ) 2 Lα

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Here α, β = e, µ, τ, χ. The field χ now denotes the superluminal dark Majorana spinor and the quadratic term LLor (χ) involves only the field χ and violates Lorentz invariance. Similarly one can build couplings with the Dirac superluminal dark fields. Whether one can construct a gauge invariant formulation of such theories it has to be seen.

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Acknowledgments The work is partially supported by grants RFFI 11-01-00894, NS 8265.2010.1 (IA) and RFFI 11-01-00828-a and NS 7675.2010.1 (I.V.).

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