Decay of superluminal neutrinos in the collinear approximation J.M. Carmona1 2J.L. Cortés1 2yJ.J. Relancio3 2zand M.A. Reyes1 2x 1Departamento de Física Teórica Universidad de Zaragoza Zaragoza 50009 Spain

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Decay of superluminal neutrinos in the collinear approximation

J.M. Carmona,1, 2, ∗J.L. Cortés,1, 2, †J.J. Relancio,3, 2, ‡and M.A. Reyes1, 2, §

1Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain

2Centro de Astropartículas y Física de Altas Energías (CAPA),

Universidad de Zaragoza, Zaragoza 50009, Spain

3Departamento de Matemáticas y Computación, Universidad de Burgos, 09001 Burgos, Spain

The kinematics of the three body decay, with a modiﬁed energy-momentum relation of the par-

ticles due to a violation of Lorentz invariance, is presented in detail in the collinear approximation.

The results are applied to the decay of superluminal neutrinos producing an electron-positron or a

neutrino-antineutrino pair. Explicit expressions for the energy distributions, required for a study of

the cascade of neutrinos produced in the propagation of superluminal neutrinos, are derived.

I. INTRODUCTION

In a theory of quantum gravity (QG), our classical notion of spacetime will surely be modiﬁed. It is natural to

expect that the symmetries associated with the structure of spacetime (Poincaré invariance) will then also be modiﬁed.

There are in fact indications from some candidates of QG that the Lorentz symmetry could be violated or deformed

at very high energies [1–5]. Lacking an understanding of the origin of the departure from Lorentz invariance, it is

an open question how this departure will aﬀect diﬀerent particles. Neutrinos are very especial ingredients of the

Standard Model of particles physics (SM) due to their quantum numbers (behavior under the diﬀerent interactions).

This peculiarity may well be behind the origin of their extremely small masses and makes them very good candidates

to look for physics beyond the SM, including a possible violation of the Lorentz symmetry. A manifestation of this

violation is a modiﬁcation of the standard relativistic expression for the energy as a function of the momentum, which,

for Planckian (or another eﬀective high-energy scale) corrections, will be more important at higher energies. In the

case of modiﬁcations for which a higher value of the energy for a given momentum corresponds to a higher velocity

(derivative of the energy with respect to the momentum) than in special relativity, neutrinos can become superluminal

as a consequence of the Lorentz invariance violation (LIV).

There has been an impressive progress in the possibility of observing neutrinos of increasingly higher energies,

including the recent observations of high-energy astrophysical neutrinos up to energies of the order of PeV [6–8].

There are prospects to extend this energy range by three orders of magnitude (EeV) in the near future (see the last

section of Ref. [9]). These observations would be aﬀected by the modiﬁcations in the propagation of the neutrinos

from their sources to their detection in the presence of LIV. In particular, superluminal neutrinos are no longer stable

particles, being able to decay, producing an electron-positron pair or a neutrino-antineutrino pair, through the weak

interaction. These decays would lead to a suppression, stronger at higher energies, of the detected ﬂux of neutrinos.

After the initial claim on superluminal propagation by the OPERA experiment in 2011 [10], a number of theoretical

models involving superluminal neutrinos were discussed in the literature [11–15]. Superluminal neutrinos can decay

producing an electron-positron pair if the relation between the energy and momentum for electrons and positrons is

not modiﬁed or if the modiﬁcation is smaller than in the case of neutrinos. One can assume that this is the case,

since there are very stringent limits on such modiﬁcation of the energy-momentum relation for electrons ([16–21],

see additional references in [22]); in fact, the inferred sensitivities for LIV in neutrinos that one would get from

experiments involving the charged-lepton sector using gauge invariance arguments would exceed typical constraints

in the neutrino sector by several orders of magnitude [23].

The investigation of phenomenological consequences of the decay of superluminal neutrinos involve the computation

of decay rates in a LIV framework. This has mainly been studied in the case of new dimension 4 (d= 4) operators

in the free Lagrangian for the neutrino ﬁeld, which gives a velocity of propagation for the neutrino which is diﬀerent

from cand is energy-independent. Such a modiﬁcation was used in the context of the former OPERA anomaly

to give a ﬁrst estimate of the production of an electron-positron pair by a superluminal neutrino by Cohen and

Glashow [24]. Detailed calculations trying to reproduce the Cohen and Glashow result in diﬀerent frameworks were

given in Refs. [25–27]. In particular, Ref. [25] showed that the result for the decay rate that is derived from a

Lagrangian containing LIV terms is diﬀerent from the Cohen and Glashow result, and depends on the explicit form

∗jcarmona@unizar.es

†cortes@unizar.es

‡jjrelancio@ubu.es

§mkreyes@unizar.es

arXiv:2210.02222v1 [hep-ph] 5 Oct 2022

of the interaction terms in the Lagrangian, computing the decay width for two diﬀerent models (‘model I’ and ‘model

II’ [25]). These two models corresponded to the ‘second example’ and ‘fourth example’ of Ref. [27] (while the ‘ﬁrst

example’ corresponded to the Cohen and Glashow result), where the dependence of decay rates on the choice of the

dynamical matrix elements was also examined. Moreover, Ref. [27] considered diﬀerent choices of modiﬁed dispersion

relation for neutrinos, going beyond the case of an energy-independent velocity.

The choice of the ‘model II’ in Ref. [25] and of the ‘fourth example’ in Ref. [27] was motivated by a gauge invariance

argument, which, as we will explain, is not satisfactory. There are in fact some concerns about the theoretical

consistency of an scenario where all the eﬀects of LIV are restricted to the neutrino sector. In the extension of the SM

within the eﬀective ﬁeld theory framework [1], one considers all possible terms involving the SM ﬁelds compatible with

the gauge symmetry of the SM. This would lead to gauge covariant derivatives, instead of usual derivatives, acting

on the SU (2) doublet of left handed ﬁelds, including the neutrino and charged lepton ﬁelds. Then, one would have,

in principle, together with the LIV corrections on neutrinos, a similar correction on the charged leptons, which may

be incompatible with the above mentioned constraints in the charged lepton sector. It is a technical curiosity that, as

explained in Ref. [28], one can have diﬀerent LIV parameters for a charged lepton and its neutrino in gauge-invariant

models under a restricted set of gauge transformations, within the SU (2)Lgauge group, if the models only involve the

interaction with the Z0(so that the interaction Lagrangian is diagonal in SU(2)Lspace). The ‘model II’ in Ref. [25]

and the ‘fourth example’ in Ref. [27] are precisely examples of this situation. Indeed, a limitation of all the above

mentioned calculations of decay rates of superluminal neutrinos is that they were made considering only the neutral

weak current. The complete model, however, contains the charged weak current, which means that the introduction

of a LIV term at the level of the covariant derivatives is not a way to reconcile gauge invariance with a LIV aﬀecting

diﬀerently neutrinos and charged leptons.

There remain two possibilities to escape the argument that the LIV corrections should aﬀect equally neutrinos as

charged leptons. The ﬁrst one is to assume that LIV corrections involve the lepton ﬁelds only through the gauge

invariant product of the Higgs doublet and the lepton doublet [27]. Then, one can use derivatives of the invariant

product and, when one replaces the Higgs doublet by its vacuum expectation value, the invariant product reduces to

the neutrino ﬁeld multiplied by a constant and one can obtain a LIV term involving only the neutrino ﬁeld. This is

one way to generate LIV eﬀects aﬀecting only the neutrino sector consistently with the gauge invariance of the SM.

A more speculative alternative is the possibility that, together with the loss of Lorentz invariance, one had also to

consider a departure from the gauge symmetries deﬁning the Lorentz invariant SM. Indeed, as argued in [29], LIV

violates gauge invariance within general relativity. Lacking a well deﬁned origin of the (possible) corrections to the

SM, and also taking into account the very special role that neutrinos play within the SM, one should keep an open

mind on the possibility of a relation between the violation of the Lorentz and the gauge symmetries of the SM, as

previously pointed out in [29, 30]. Any of these two possibilities leads to the introduction of LIV terms at the level

of the free Lagrangian for the neutrino ﬁelds, and exclude these terms at the level of the interaction with the gauge

ﬁelds.

As indicated above, LIV eﬀects motivated by quantum gravity are expected to become more relevant as the energy

increases, which means a velocity of superluminal neutrinos which depends on the energy, or, more precisely, on

(E/Λ)n, where Λis the quantum-gravity-motivated LIV scale, and nthe order of the correction. The linear case,

n= 1, corresponds to d= 5 operators in the Lagrangian, and the quadratic case, n= 2, to d= 6 operators. Besides

this motivation, a correction due to LIV in the neutrino energy-momentum relation increasing with the energy provides

a natural mechanism for the suppression of LIV eﬀects at low energies, where one has the more precise tests of Lorentz

invariance.

In [31], a ﬁrst attempt to consider n= 2 Planck-scale suppressed LIV eﬀects on the cosmogenic neutrino spectrum

was presented. An estimate of the decay width of a superluminal neutrino into three neutrinos (neutrino splitting),

based on a rough approximation of the integral over the phase-space volume of the three particles in the ﬁnal state,

led to the prediction that one would have a cutoﬀ at an energy in the interval (1018 eV,1019 eV), preceded by a bump

in the cosmogenic neutrino spectrum. Motivated by a hint of a suppression in the ﬁnal part of the ﬂux of astrophysical

neutrinos detected by IceCube [32], a study of the possible eﬀects of n= 1 and n= 2 LIV in the neutrino astrophysical

spectrum at energies around the PeV scale was pursued in [33, 34], using the expressions of the decay width which

had been obtained by the explicit calculations of Ref. [27]. The numerical results in Ref. [33] contained, however,

some uncertainties, because of two facts: the computation of Ref. [27] only included the pair-creation process, and

only through the Z0exchange. Neutrino splitting had been the subject of a detailed calculation in Ref. [35], but only

in the energy-independent (but ﬂavor-dependent) velocity case.

The aim of this work is to go beyond the previous limitations and present a calculation that allows one to include

both the neutrino splitting process and the charged weak bosons exchange in the computation of the decay width for

a generic n > 0 (n∈N)neutrino superluminal case. We will do that by considering systematically the three body

decay of a superluminal particle and will use this approach to determine the energy distribution of neutrinos in the

decay of a superluminal neutrino. This may be useful for more detailed studies of the possible eﬀects of this kind

of LIV in the propagation of very high-energy superluminal neutrinos. The near future prospects to have a more

precise determination of the neutrino astrophysical spectrum at energies above the PeV going up to EeV are a good

motivation for such studies.

We will begin by brieﬂy reviewing the introduction of superluminal neutrinos in the ﬁeld theory framework in Sec. II.

As mentioned above, this will be done by including a LIV term of dimension 4+nin the free Lagrangian of the neutrino

ﬁelds. In Sec. III, we present in detail the collinear approximation to the three body decay of a superluminal particle,

which is the main novelty of this work. This approximation is relevant when studying LIV corrections in a variety

of situations in high-energy astrophysics, including the decay of a highly energetic particle. Indeed, in the following

section we will apply the results of the collinear approximation to the decay of a superluminal neutrino producing

an electron-positron pair (Sec. IV A) and a neutrino-antineutrino pair (Sec. IV B) for a LIV correction relevant to

quantum gravity phenomenology (n > 0). We present a summary of the results in Sec. V.

II. MODIFIED DISPERSION RELATION FOR SUPERLUMINAL NEUTRINOS

We are going to consider the eﬀects of LIV on the neutrino sector of the SM by adding to the SM Lagrangian a LIV

correction, compatible with rotational invariance, involving only the neutrino ﬁelds. In order to make this correction

compatible with the very stringent limits on LIV, we assume that the LIV correction in the Lagrangian is a quadratic

term in the neutrino ﬁelds with (n+ 1) derivatives, so that its coeﬃcient is the n-th power of the inverse of a new

energy scale (Λ) parametrizing the LIV. Neutrino masses are irrelevant in the decays of superluminal neutrinos, which

is the eﬀect of the LIV correction we are going to study in this work, so we will treat them as massless particles.

When treating the LIV as a ﬁrst order correction to the Lorentz invariant SM Lagrangian, one can use the SM ﬁeld

equations to reduce the number of LIV terms. This means that in the terms quadratic in the neutrino ﬁelds one can

replace any space derivative by a time derivative. As a consequence of the previous argument, one has a single LIV

term in the Lagrangian

L(ν)

LIV =−1

ΛnνlLγ0(i∂0)n+1 νlL ,(1)

where the subindex Lrefers to the left-handed chirality of the neutrino ﬁelds, lrefers to the three types of neutrinos

(e,µ,τ), and we assume that there is no ﬂavor dependence in the LIV terms, avoiding constrains from neutrino

oscillations. The choice of the sign in front of (1) is arbitrary and will be discussed later.

The LIV term in the Lagrangian modiﬁes the free theory of the neutrino ﬁeld. When we introduce a plane-wave

expansion for the neutrino ﬁeld

νL(t, ~x) = Zd3~

kh˜

k˜u(~

k)e−i˜

E−t+i~

k·~x +˜

d†

k˜v(~

k)ei˜

E+t−i~

k·~xi,(2)

we ﬁnd that the spinors ˜u,˜vhave to satisfy the equations

"γ0˜

E−−~γ ·~

k−γ0˜

En+1

−

Λn#˜u(~

k) = 0 ,"γ0˜

E+−~γ ·~

k+ (−1)n+1γ0˜

En+1

Λn#˜v(~

k) = 0 .(3)

In the chiral representation for the Dirac matrices, the spinors ˜u,˜vcan be written as ˜u= (χ, 0)T,˜v= (η, 0)T, and

the bi-spinors χ,ηsatisfy the equations

(~σ ·~

k)χ(~

k) = − ˜

E−−˜

En+1

−

Λn!χ(~

k),(~σ ·~

k)η(~

k) = − ˜

E++ (−1)n+1 ˜

En+1

Λn!η(~

k).(4)

The matrix (~σ ·~

k)has two eigenvalues ±|~

k|. Then we conclude that the relation between the momentum (~

k) and the

energy (E−) for a neutrino is

k|=˜

E−−˜

En+1

−

Λn,(5)

and the relation between the momentum (~

k) and the energy (E+) for an antineutrino is

k|=˜

E++ (−1)n+1 ˜

En+1

Λn.(6)

We see then that, for the choice of minus sign in (1), when nis even, both the neutrino and the antineutrino

are superluminal, while in the case of nodd the neutrino is superluminal and the antineutrino is subluminal. If we

considered a positive coeﬃcient in (1) instead, any superluminal state would become subluminal and vice versa.

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DecayofsuperluminalneutrinosinthecollinearapproximationJ.M.Carmona,1,2,J.L.Cortés,1,2,yJ.J.Relancio,3,2,zandM.A.Reyes1,2,x1DepartamentodeFísicaTeórica,UniversidaddeZaragoza,Zaragoza50009,Spain2CentrodeAstropartículasyFísicadeAltasEnergías(CAPA),UniversidaddeZaragoza,Zaragoza50009,Spain3Departamento...

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Decay of superluminal neutrinos in the collinear approximation J.M. Carmona1 2J.L. Cortés1 2yJ.J. Relancio3 2zand M.A. Reyes1 2x 1Departamento de Física Teórica Universidad de Zaragoza Zaragoza 50009 Spain.pdf

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Decay of superluminal neutrinos in the collinear approximation J.M. Carmona1 2J.L. Cortés1 2yJ.J. Relancio3 2zand M.A. Reyes1 2x 1Departamento de Física Teórica Universidad de Zaragoza Zaragoza 50009 Spain

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