# Neutrino Physics and Astrophysics

Floyd W<EMAIL_ADDRESS>NASA Goddard Space Flight Center
Greenbelt, MD 20771 USA
and University of California Los Angeles
Los Angeles, CA 90095

###### Abstract

The search for a theory that unifies general relativity and quantum theory has
focused attention on models of physics at the Planck scale. One possible
consequence of models such as string theory may be that Lorentz invariance is
not an exact symmetry of nature. We discuss here some possible experimental
and observational tests of Lorentz invariance involving neutrino physics and
astrophysics.

##  Chapter 0 Testing Lorentz Invariance with Neutrinos

Today we say that the law of relativity is supposed to be true at all
energies, but someday somebody may come along and say how stupid we were. We
do not know where we are stupid until we stick our neck out…And the only way
to find out that we are wrong is to find out what our predictions are. It is
absolutely necessary to make constructs.

\- Richard Feynman

###### Contents

  1. 0 Testing LIV with Neutrinos
    1. 1 Introduction
      1. 1 Neutrinos and Tests of Lorentz Invariance Violation
    2. 2 Effective Field Theories and Neutrino Physics
    3. 3 Free particle propagation and modified kinematics
      1. 1 Mass dimension [d] = 4 LIV with rotational symmetry
      2. 2 Fermion LIV operators with $[d]>4$ LIV with rotational symmetry in the ”Standard Model Extension” EFT
    4. 4 LIV in the neutrino sector I - Neutrino Oscillations
    5. 5 Velocity difference between neutrinos and photons from TXS 0506+056
    6. 6 LIV in the neutrino sector II - Lepton Pair Emission in vacuo
      1. 1 Lepton pair emission in the [d] = 4 case
      2. 2 Vacuum $e^{+}e^{-}$ Pair Emission in the $[d]>4$ cases
    7. 7 LIV in the neutrino sector III - decay by neutrino pair emission (neutrino splitting)
    8. 8 The IceCube Observations
    9. 9 The Energy Spectrum from Extragalactic Superluminal Neutrino Propagation
      1. 1 [d] = 4 $\cal{CPT}$ Conserving Operator Dominance
      2. 2 [d] = 6 $\cal{CPT}$ Conserving Operator Dominance
      3. 3 [d] = 5 $\cal{CPT}$ Violating Operator Dominance
    10. 10 Summary of the results from LIV kinematic effects for superluminal neutrinos
    11. 11 LIV in the Ultrahigh Energy Cosmic Ray Spectrum and the Subsequent Ultrahigh Energy Neutrino Spectrum
      1. 1 The GZK Effect
      2. 2 The Effect of LIV Kinematics on the GZK Process
      3. 3 The Photomeson Neutrino Spectrum
    12. 12 Stable Pions from LIV
    13. 13 Observational tests with new neutrino telescopes

###  1 Introduction

Modern theoretical physics stands on two strong legs. General relativity,
which sprang from the mind of Albert Einstein over a century ago, has been
crucial to a description of physics on the largest scales. Quantum field
theory, developed in the middle of the 20th century, has led to the modern
view of physics on extremely small scales, allowing a detailed description of
the interactions of the subatomic particles. It has also played a crucial role
in the macroscopic world, giving physicists a framework for the description of
emergent phenomena in condensed matter. However, a unification of these two
fundamental branches of modern physics has been heretofore illusive.

Even more problematic, the two theories appear to be incompatible,
particularly failing at the nexus point, known as the Planck scale of
$\lambda_{Pl}=\sqrt{G\hbar/c^{3}}\sim 10^{-35}$ m. [1]. It is at this scale
that the Compton wavelength and the Schwarzschild radius of a particle of mass
$m_{Planck}$ are equal. This implies that for intervals at or near the Planck
scale, the distortion of space-time geometry owing to quantum effects becomes
catastrophic. Thus, we say that at this scale space-time physics as we know it
”breaks down” [2]. More likely, it is a red flag of our ignorance. We may say
that the big-bang started at the Planck scale, but we really don’t know what
happened at that time. There has even been speculation of a pre-Plankian or
trans-Planckian era [3, 4] that influenced our present cosmology [5].

So what about physics at the Planck scale? In their efforts to provide a UV
(i.e., high energy) completion for a quantum theory of general relativity,
many theorists postulate drastic modifications to space-time at the Planck
scale (e.g., Ref. 6). Two examples of this are the introduction of ”extra
dimensions” and the postulation of a fundamental discreteness of space-time,
with the building blocks of nature being extended objects. At stake is nothing
less than our understanding of the nature of space and time on which all of
physics is fashioned.

One possible modification to space-time structure that has received quite a
bit of attention is the idea that the Lorentz space-time symmetry of
relativity is not an exact symmetry of nature. Such a proposal is rather
conservative as compared with other quantum gravity ideas. Historically the
symmetry groups that have been used to model physical phenomena have
inevitably evolved over time, with Galilean symmetry being replaced by Lorentz
symmetry and other particle physics symmetries being broken. Lorentz symmetry
violation has been explored within string theory, loop quantum gravity,
Hořava-Lifshitz gravity, causal dynamical triangulations, non-commutative
geometry, doubly special relativity, among others (See, e.g., Refs. 7 and 8
and references therein).

An example of the importance of neutrino physics is the long recognized
importance of the role of neutrino masses and oscillations in cosmology, in
particular, their role in determining the large scale structure of the
Universe [9, 10]. Neutrino studies and observations can also provide sensitive
tests of Lorentz invariance violation, which may be a the result of quantum
gravity physics (QG). Studies of neutrino oscillations as well as observations
of astrophysical neutrinos can be important in this regard.

####  1 Neutrinos and Tests of Lorentz Invariance Violation

Empirical studies of physical phenomena involving neutrinos, both in the
laboratory and from cosmic sources, can be a useful probe in searching for new
physics. In this chapter we will discuss the prospect of using observations of
high energy neutrinos produced in astrophysical sources in order to search for
traces of physics at the Planck-scale. Such a project is advantageous for
several reasons. Cosmic high energy neutrinos are unaffected by magnetic
fields and effectively unaffected by interactions with matter. They can
therefore reach us from cosmologically great distances, thus providing an
extremely long baseline for probing the smallest deviations from Lorentz
invariance and standard model physics. Because of the astronomically long
baselines and very high energies of such neutrinos, they are therefore ideal
probes for testing Lorentz and $\cal{CPT}$ symmetry as well as other new
physics. In addition, considerations based on dimensional analysis indicate
that LIV effects should generally increase with energy.

In the following sections, we will examine some of the observational
consequences of LIV in the neutrino sector within the context of effective
field theory (EFT), one that holds up to some limiting high energy scale that
we take to be the Planck scale.

###  2 Effective Field Theories and Neutrino Physics

While it is not possible to directly investigate space-time physics at the
Planck energy of $\sim 10^{19}$ GeV, many lower energy testable effects have
been predicted to arise from the violation of Lorentz invariance (LIV) at or
near the Planck scale. The subject of investigating LIV has therefore
generated much interest in the particle physics and astrophysics communities.
Here we choose to discuss LIV within the useful framework of effective field
theories (EFT).222There are non-EFT scenarios for either violating or
modifying Lorentz invariance (see, e.g., Refs. 7, 8, 11 and references
therein). However, these models do not easily lend themselves to particle
physics tests as the dynamics of particle interactions is less well
understood. We will not consider such scenarios here. An effective field
theory is a theory that is incomplete, but one that is an excellent
approximation below a certain limiting energy scale. An example of an EFT is
the Fermi four-fermion theory of weak interactions [12] that holds up to an
energy $\sim$ 100 GeV. This theory was replaced by the exact electroweak
theory of Glashow, Weinberg and Salam. (See, e.g., Ref. 13). Using EFT methods
allows one to pose questions involving LIV that can have well-defined
empirical answers without having the knowledge of an exact theory.

In the localized quantum field theory formalism the evolution of the state of
a quantum system is determined by a unitary Hermitian operator that is a
functional of a Lagrangian density, colloquially called the Lagrangian [14,
15]. The effect of incorporating possible new physics in such an EFT framework
is accomplished by employing new operators and incorporating new free
parameters into the Lagrangian of the theory with such operators that being
constructed by identifying the relevant fields and symmetries that determine
the possible new physics. Using such an effective field theory formalism,
Lorentz invariance violation can be incorporated by the addition of new terms
in the free particle Lagrangian that explicitly break Lorentz invariance.
Since it is well known that Lorentz invariance holds quite well at accelerator
energies, the extra Lorentz violating terms in the effective field theory
Lagrangian must be very small up to such energies.

We will assume that the EFT that breaks the symmetry of Lorentz invariance is
a good approximation to a ”true theory” and is one that holds up to the order
of the Planck scale. Within the context of such an EFT, one can postulate the
existence of additional terms in free particle Lagrangians that break Lorentz
invariance, some of which violate $\cal{CPT}$ invariance. Within this EFT
framework one can then describe effects arising from Planck-scale physics that
can be manifested at lower energies, however being suppressed at lower
energies by terms in the Lagrangians inversely proportional to powers of the
Planck mass. Such an EFT formalism was proposed with in the context of string
theory [16, 17]. In the rest of this chapter we will discuss the empirical
effects of an EFT that incorporates operators in the Lagrangian that violate
both Lorentz invariance and $\cal{CPT}$ invariance in the neutrino sector. In
particular, we will place limits on a class of non-renormalizable, mass-
dimension five and six Lorentz invariance violating operators that may be the
result of Planck-scale physics. (See Appendix A for a discussion of
Lagrangians and mass dimensions in EFT.)

###  3 Free particle propagation and modified kinematics

LIV modifications to the Lagrangian can affect neutrino physics in various
ways. We will consider here only two of these consequences, viz., (1) their
effect on modifying neutrino oscillations and (2) the effect of resulting
changes in the kinematics of particle interactions. Such modifications give
rise to an energy dependent effective neutrino mass, and so change the
patterns of neutrino oscillations. They also introduce corrections to the
matrix elements for existing interactions as well as create new interactions
between standard model fermions. However, for our purposes, the most important
effect these terms have is to change the kinematics of particle interactions,
leaving unchanged the matrix elements governed by existing standard model
physics.

Such changes can modify the threshold energies for particle interactions,
allowing or forbidding such interactions [18, 19, 20]. Since the Lorentz
violating operators change the free field behavior and dispersion relation,
interactions such as fermion-antifermion pair emission by neutrinos become
kinematically allowed [18, 19] and can cause significant observational effects
if the neutrinos are slightly superluminal. Absent a violation of Lorentz
invariance, such interactions are forbidden by conservation of energy and
momentum.

We now set up a simplified formalism to calculate the observational effect of
these two specific anomalous interactions on the neutrino spectrum seen in
IceCube.

####  1 Mass dimension [d] = 4 LIV with rotational symmetry

For an introduction demonstrating how LIV terms affect particle kinematics, we
consider the simple example of a free scalar particle Lagrangian with an
additional small dimension-4 Lorentz violating term, assuming rotational
symmetry [18].

$\Delta\mathcal{L}_{f}=\partial_{i}\Phi^{*}{\bf\epsilon}\partial^{i}\Phi.$ (1)

which is not Lorentz invariant because it only involves spacial derivatives.

This modified Lagrangian leads to a propagator for a particle of mass $m$

$-iD^{-1}~{}=~{}(p_{(4)}^{2}~{}-~{}m^{2})~{}+~{}\epsilon p^{2}.$ (2)

so that we obtain the dispersion relation

$p_{(4)}^{2}~{}=~{}E^{2}~{}-~{}p^{2}~{}\Rightarrow~{}m^{2}~{}+~{}\epsilon
p^{2}.$ (3)

In this example, the low energy ”speed of light” maximum attainable particle
velocity, here equal to 1 by convention, is replaced by a new maximum
attainable velocity (MAV) as $v_{MAV}\neq 1$, which is changed by

$\delta v\equiv\delta=\epsilon/2.$ (4)

Then,

${{\partial
E}\over{\partial|\vec{p}|}}={{|\vec{p}|}\over{\sqrt{|\vec{p}|^{2}+m^{2}v_{MAV}^{2}}}}v_{MAV},$
(5)

which goes to $v_{MAV}$ at relativistic energies, $|\vec{p}|^{2}\gg m^{2}$.

For the [d] = 4 case, the superluminal velocity of particle $I$ that is
produced by the existence of one or more LIV terms in the free particle
Lagrangian will be denoted by

$v_{I,MAV}\equiv 1+\delta_{I}$ (6)

We will always be in the relativistic limit $|\vec{p}|^{2}\gg m^{2}$ for both
neutrinos and electrons. Thus, the neutrino or electron velocity is just given
by equation (6) [18, 20].

####  2 Fermion LIV operators with $[d]>4$ LIV with rotational symmetry in the
”Standard Model Extension” EFT

In considering Lagrangians with $[d]>4$, we can employ the EFT framework known
as the ”Standard Model Extension” (SME) [16]. This formalism was inspired by
string theory and invokes small terms in the Lagrangian beyond the standard
model that violate Lorentz invariance and $\cal{CPT}$ invariance while keeping
the internal gauge symmetries of the standard model in the Lagrangian. (See
appendix B.) As in the [d] = 4 case above [18], we will consider here only the
isotropic terms in the SME formalism and relate our treatment to those SME
parameters in the conclusion. This ”spherical cow”, plain vanilla
approximation is justified by our present lack of empirical knowledge.333The
only additional LIV terms in the free particle Lagrangians that we consider
here are assumed to be rotationally invariant. The ”standard model extension”
(SME) EFT includes hundreds of possible anisotropic terms as well [16, 21, 22,
23]. Since there is presently no observational evidence for any LIV, we chose
the simplest isotropic approach here. We assume that the LIV terms are
isotropic in the rest system of the 2.73 K cosmic background radiation (CBR),
a preferred system picked out by the universe itself. In actuality, we are
moving with respect to this system by a velocity $\sim 10^{-3}$ of the speed
of light (hereafter taking $c=\hbar=1$).

In the cases where $[d]>4$ Planck-suppressed operators dominate, there will be
LIV terms that are proportional to $(E/M_{Pl})^{n}$, where $n=[d]-4$, leading
to values of $\delta_{I}$ that are energy dependent and are taken to be
suppressed by appropriate powers of the Planck mass (see appendix B).

If we again assume rotational invariance in the rest frame of the CBR and
consider only the effects of Lorentz violation on freely propagating cosmic
neutrinos. Thus, we only need to examine Lorentz violating modifications to
the neutrino kinetic terms. Majorana neutrino couplings are ruled out in SME
in the case of rotational symmetry [23]. Therefore we only consider Dirac
neutrinos.

A useful simplified formalism for analyzing Lorentz-modified kinematics, one
that highlights the physical processes, is to wrap the additional Lorentz
violating terms into an effective mass term, $\tilde{m}_{I}(E)$, which is the
right hand side of equation (3) labeled by a particle species index $I$. We
can further identify $\tilde{m}(E)$ using equation (3), yielding the relation

$\tilde{m}_{I}^{2}(E)=m_{I}^{2}+2\delta_{I}E^{2},$ (7)

where the velocity parameters $\delta_{I}$ are now energy dependent
dimensionless coefficients for each species that can be directly identified
from the fundamental parameters in the Lagrangian. We have implicitly assumed
that $\delta_{I}$ is defined relative the velocity of light, $c=1$. We extend
this definition by defining the parameter
$\delta_{IJ}\equiv\delta_{I}-\delta_{J}$ as the Lorentz violating difference
between the velocities of particles $I$ and $J$. In general $\delta_{IJ}$ will
therefore be of the form

$\delta_{IJ}=\sum_{n=0,1,2}\kappa_{IJ,n}\left(\frac{E}{M_{Pl}}\right)^{n}.$
(8)

Assuming the dominance of Planck-mass suppressed terms in the Lagrangian as
tracers of Planck scale physics, it follows that that $\kappa_{\nu e,0}=0$. We
will also assume that $\kappa_{\nu e,0}\ll\kappa_{\nu e,1},\kappa_{\nu e,2}$.
Constraints are therefore most directly expressed in terms of limits on
$\kappa_{\nu e,1}$ and $\kappa_{\nu e,2}$. For $[d]>4$ the superluminal
velocity excesses are given as integral multiples of $\kappa_{\nu e,1}$ and
$\kappa_{\nu e,2}$ through the group velocity relation given by equation (5).
The $\kappa_{\nu e,0}$ coefficient has been tightly constrained from
observations of extraterrestrial PeV neutrinos by the IceCube collaboration
[24, 25].444Several mechanisms have been proposed for the suppression of the
LIV $[d]=4$ term in the Lagrangian. See, e.g., the review in Ref. 8. 555In
equations (8) and (9) we have not designated a helicity index on the $\kappa$
coefficients. The fundamental parameters in the Lagrangian are generally
helicity dependent [26]. In the $n=1$ case a helicity dependence must be
generated in the electron sector due to the $\cal{CPT}$ odd nature of the LIV
term. However, the constraints on the electron coefficient are extremely tight
from observations of the Crab nebula (see above, see also Ref. 27). Thus, the
contribution to $\kappa_{\nu e,1}$ from the electron sector can be neglected.
In the $n=2$ case, which is $\cal{CPT}$ even, we can set the left and right
handed electron coefficients to be equal by imposing parity symmetry [28].

There is an important connection between LIV and $\cal{CPT}$ violation.
Whereas a local interacting theory that violates $\cal{CPT}$ invariance will
also violate Lorentz invariance, the converse does not follow; an interacting
theory that violates Lorentz invariance may, or may not, violate $\cal{CPT}$
invariance [29, 30]. $\cal{CPT}$ is a symmetry that is conserved in a field
theory formulation that satisfies the following three assumptions: (1)
locality, (2) Lorentz invariance, and (3) hermiticity of the Hamiltonian. As
an example, if strings, not points, are the fundamental elements of physics,
$\cal{CPT}$ invariance can be violated because the locality condition of the
$\cal{CPT}$ theorem [14] does not hold [17, 31].

In the framework of the standard model extension (SME) formalism [21] LIV
terms of even mass dimension $[d]=4+n$, are $\cal{CPT}$-even and do not
violate $\cal{CPT}$, whereas LIV terms of odd mass dimension are
$\cal{CPT}$-odd and violate $\cal{CPT}$ [22]. We can then specify a dominant
term for $\delta_{IJ}$ in equation (8) depending on our choice of $\cal{CPT}$.
Thus, in considering Planck-mass suppressed LIV terms, the dominant term that
admits $\cal{CPT}$ violation is the $n=1$ term in equation (8). On the other
hand, if we require $\cal{CPT}$ conservation, the $n=2$ term in equation (8)
is the dominant term. Therefore, we can choose as a good approximation to
equation (8), a single dominant term with one particular power of $n$ by
specifying whether we are considering $\cal{CPT}$ even or odd LIV. We can then
reduce the sum of the terms in the expansion given by equation (8) to the
leading terms only. As a result, $\delta_{IJ}$ reduces to

$\delta_{IJ}~{}\equiv\kappa_{IJ,n}\left({{E}\over{{M_{Pl}}}}\right)^{n}$ (9)

with $n=1$ or $n=2$ depending on the status of $\cal{CPT}$.

It will also be important later on to note that in the SME formalism, since
odd-[d] LIV operators are $\cal{CPT}$ odd, the $\cal{CPT}$-conjugation
property implies that neutrinos can be superluminal while antineutrinos are
subluminal or vice versa [23]. This will have consequences in interpreting the
results given in the later sections.

Figure 1: Alan Kostelecký and the author (left to right in the foreground) at
the CPT 2013 Symposium. In the background are Mike Snow (waving) and Rob
Potting (Courtesy of Neil Russell)

###  4 LIV in the neutrino sector I - Neutrino Oscillations

Let us now consider the effect on neutrino oscillations of Lorentz violating
terms in the Lagrangian that are suppressed by powers of the Planck mass. We
again note that the dominant term that admits $\cal{CPT}$ violation is the
$n=1$ term in equation (8); the dominant term that conserves $\cal{CPT}$ is
the $n=2$ term in equation (8). Given Planck mass suppression, we choose one
of these two terms to be the single dominant term with one particular power of
$n$, depending on whether $\cal{CPT}$ is conserved or not. As a result,
$\delta_{IJ}$ is given by equation (9). We take the effective mass,
$\tilde{m}$, to be given by equation (7) where the $m_{i}$ denotes one of the
three possible mass eigenstates of the neutrino.

We then consider a neutrino with flavor $I$ with momentum $p$ transitioning
into flavor $J$. The amplitude for this neutrino to be in a mass eigenstate
$i$ is then denoted by the matrix $U_{Ii}$ where ${\bf U}$ denotes the unitary
matrix, $\sum U^{{\dagger}}_{Ji}U_{Ii}=\partial_{IJ}$, where here the symbol
$\partial_{IJ}$ denotes the usual mathematical delta function to distinguish
it from to the velocity difference $\delta_{IJ}$ as defined in equation (8).

These considerations change the usual relations for neutrino oscillations. For
example, in the case of atmospheric $\nu_{\mu}$ oscillations, the survival
probability in the Lorentz invariant case is determined by the time evolution
of the mass eigenstates and is given by

$P_{\nu_{\mu}}\ \simeq\ 1-sin^{2}(2\vartheta_{23})sin^{2}\left({\Delta
m^{2}_{atm}L}\over{4E}\right)$ (10)

where $\Delta m^{2}\ \equiv m_{i}^{2}$ \- $m_{j}^{2}$, so that the oscillation
periods are determined by parameters involving the differences between the
squares of the neutrino mass eigenstates. Thus, the individual neutrino masses
themselves are not determined by the oscillations.

We can include the effects of LIV terms in the Lagrangian by making the
substitutions

$m^{2}_{atm}\ \rightarrow\ \ \tilde{m}^{2}_{atm}(E)\ =\
m^{2}_{atm}+2\delta_{ij}E^{2}$ (11)

where $\tilde{m}^{2}(E)$ is given by equation (7). In that case, equation (10)
is modified by an additional LIV term proportional to the difference in
neutrino velocities, $\delta_{IJ}$. It immediately follows that

$P_{\nu_{\mu}}\ \simeq\ 1-sin^{2}(2\vartheta_{23})sin^{2}\left({{\Delta
m^{2}_{atm}L}\over{4E}}+{{\delta_{IJ}}}{{EL\over{2}}}\right)$ (12)

where the square of the difference between the mass eigenvalues $\Delta
m^{2}_{atm}=\Delta m^{2}_{31}$ [18, 32, 33]. Recent results on atmospheric
neutrino oscillations [34, 35, 36] (also Gonzalez-Garcia and Maltoni, private
communication) give an upper limit for the difference in velocities between
the $\nu_{\mu}$ and $\nu_{\tau}$ neutrinos,
$\delta_{\nu_{\mu}\nu_{\tau}}<\cal{O}$ $(10^{-26})$.

It is interesting to note that if Lorentz invariance is violated, equation
(12) implies that neutrino oscillations would occur even if neutrinos are
massless or if the square of their mass differences is zero.

We note that the LIV term in equation (12) is linear in energy. Therefore, it
can dominate at very high energies and very large distances , as is the case
for many astrophysical applications. This dominance can be even more profound
for high mass dimensions where $\delta_{IJ}$ is given by equation (9).

###  5 Velocity difference between neutrinos and photons from TXS 0506+056

The detection of a neutrino event detected by the IceCube collaboration in
temporal and spacial coincidence with a $\gamma$-ray flare from the blazar TXS
0506+056 [37] has enabled an analysis of the limits on the velocity difference
between neutrinos and photons, $\delta_{\nu\gamma}$ [38, 39]. The result from
Ref. 38 gives

$\delta_{\nu\gamma}\leq 4.2\times 10^{-12}{{\Delta t}\over{{\rm 7\ days}}}$
(13)

assuming that the time difference is $\sim$7 days.

###  6 LIV in the neutrino sector II - Lepton Pair Emission in vacuo

Since the Lorentz violating operators change the free field behavior and
dispersion relation, interactions such as fermion-antifermion pair emission by
slightly superluminal neutrinos become kinematically allowed [18, 40] and can
thus cause significant observational effects.

The dominant pair emission reactions that become allowed if Lorentz invariance
is violated are vacuum electron-positron pair emission (VPE)
$\nu_{i}\rightarrow\nu_{i}+e^{+}+e^{-}$, and its close cousin, neutrino
splitting, $\nu_{e}\rightarrow\nu_{e}+\nu_{i}+\bar{\nu_{i}}$ (where $i$ is a
flavor index). These are the reactions involving the leptons with the lightest
final state masses. Neutrino splitting can be represented as a rotation of the
Feynman diagram for neutrino-neutrino scattering.

####  1 Lepton pair emission in the [d] = 4 case

In this section we consider the constraints on the LIV parameter $\delta_{\nu
e}$. We first relate the rates for superluminal neutrinos with that of a more
familiar tree level, weak force mediated standard model decay process: muon
decay, $\mu^{-}\rightarrow\nu_{\mu}+\bar{\nu}_{e}+e^{-}$, as the process are
very similar (see Figure 2).

Figure 2: Diagrams for muon decay (top), charged current mediated VPE (bottom
left), and neutral current mediated neutrino splitting/VPE (bottom right).
Time runs from left to right and the flavor index $i$ represents $e,\mu$, or
$\tau$ neutrinos.

For muons with a Lorentz factor $\gamma_{\mu}$ in the observer’s frame the
decay rate is found to be [12]

${\Gamma\ =\gamma_{\mu}^{-1}}{{G_{F}^{2}m_{\mu}^{5}}\over{192\pi^{3}}}$ (14)

where $G_{F}^{2}=g^{4}/(32M_{W}^{4})$, is the square of the Fermi constant
equal to $1.360\times 10^{-10}\ {\rm GeV}^{-4}$, with $g$ being the weak
coupling constant and $M_{W}$ being the $W$-boson mass in electroweak theory.

We apply the effective energy-dependent mass-squared formalism given by
equation (7) to determine the scaling of the emission rate with the $\delta$
parameter and with energy. Noting that for any reasonable neutrino mass,
$m_{\nu}\ll 2\delta_{\nu e}E_{\nu}^{2}$, it follows that
$\tilde{m_{\nu}}^{2}(E)\simeq 2\delta_{\nu e}E_{\nu}^{2}$.666At relativistic
energies, assuming that the Lorentz violating terms yield small corrections to
$E$ and $p$, it follows that $E\simeq p$.

We therefore make the substitution

$m_{\mu}^{2}\ \rightarrow\ \tilde{m_{\nu}}^{2}(E)\simeq 2\delta_{\nu
e}E_{\nu}^{2}$ (15)

from which it follows that

${\gamma_{\mu}^{2}}\ \rightarrow\ {{E_{\nu}^{2}}\over{2\delta_{\nu
e}E_{\nu}^{2}}}=(2\delta_{\nu e})^{-1}.$ (16)

The rate for the vacuum pair emission processes (VPE) is then

$\Gamma\ \propto\ (2\delta_{\nu e})^{1/2}G_{F}^{2}(2\delta_{\nu
e}E_{\nu}^{2})^{5/2}$ (17)

which gives the proportionality

$\Gamma\ \propto\ G_{F}^{2}\ \delta_{\nu e}^{3}E_{\nu}^{5}$ (18)

showing the strong dependence of the decay rate on both $\delta_{\nu e}$ and
$E_{\nu}$.

The energy threshold for $e^{+}e^{-}$ pair production is given by[19]

$E_{th}=m_{e}\sqrt{{{2}\over{{\delta_{\nu e}}}}}$ (19)

We now define $\delta\equiv\delta_{\nu e}$ as given by equation (9). The rate
for the VPE process, $\nu\to\nu\,e^{+}\,e^{-}$ via the neutral current
$Z$-exchange channel, has been calculated to be [40]777For a different
treatment giving compatible results see [41]

$\Gamma=\frac{1}{14}\frac{G_{F}^{2}(2\delta)^{3}E_{\nu}^{5}}{192\,\pi^{3}}=1.31\times
10^{-14}\delta^{3}E_{\rm GeV}^{5}\ \ {\rm GeV},$ (20)

consistent with our derivation of the dependences on $\delta$ and $E_{\nu}$
given by equation (18). The mean fractional energy loss per interaction from
VPE is 78% [40]. Thus, the energy loss rate is given by ($v_{\nu}\simeq$ 1)

$\frac{dE}{dx}=-\frac{25}{448}\frac{G_{F}^{2}}{192\pi^{3}}E^{6}\delta^{3}$
(21)

It follows from equation (21) that for a superluminal neutrino propagating
from a distance $L$ and observed with a conservatively measured terminal
energy $E_{T}$, the upper limit on $\delta_{\nu}$ is given by [40]

$E_{T}^{-5}\equiv 0.256\ \delta_{\nu}^{3}\frac{G_{F}^{2}}{192\pi^{3}}L$ (22)

Using equation (22), the upper limit constraints on $\delta_{\nu}$ from
observations of various sources are shown in Table 1.

Table 1: Upper Limits on $\delta_{\nu}$ from equation (22)

Source | Distance | $E_{T}$ | $\delta_{\nu,max}$ | Reference
---|---|---|---|---
IceCube upgoing |  $>$ 500 km | 100 TeV | $1.7\times 10^{-11}$ |  40
TXS 0506+056— | 1.36 Gpc | 183 TeV | $1.3\times 10^{-18}$ |  42

In general, the charged current $W$-exchange channels (CC) contribute as well.
However, this channel is only kinematically relevant for $\nu_{e}$’s, as the
production of $\mu$ or $\tau$ leptons by $\nu_{\mu}$’s or $\nu_{\tau}$’s has a
much higher energy threshold due to the larger final state particle masses
(equation (19) with $m_{e}$ replaced by $m_{\mu}$ or $m_{\tau}$), with the
neutrino energy loss from VPE being highly threshold dependent. Owing to
neutrino oscillations, neutrinos propagating over large distances spend 1/3 of
their time in each flavor state. Thus, the flavor population of neutrinos from
astrophysical sources is expected to be [$\nu_{e}$:$\nu_{\mu}$:$\nu_{\tau}$] =
[1:1:1] so that CC interactions involving $\nu_{e}$’s will only be important
1/3 of the time.

The vacuum Čerenkov emission (VCE) process, $\nu\rightarrow\nu+\gamma$, is
also kinematically allowed for superluminal neutrinos. However, since the
neutrino has no charge, this process entails the neutral current channel
production of a loop consisting of a virtual electron-positron pair followed
by its annihilation into a photon. Thus, the rate for VCE is a factor of
$\alpha$ lower than that for VPE.

Neutrino pair emission, a.k.a. neutrino splitting, is unimportant for energy
loss of superluminal neutrinos in the $[d]=4$ case because the fractional
energy loss per interaction is very low [40] owing to the small velocity
difference between neutrino flavors obtained from neutrino oscillation data
(See Section 4). However, as we see in the next section, this is not true in
the cases with $[d]>4$.

####  2 Vacuum $e^{+}e^{-}$ Pair Emission in the $[d]>4$ cases

.

Using equations (9) and (9) and the dynamical matrix element taken from the
simplest case [43], we can generalize equation (20) for arbitrary values of
$n=[d]-4$

$\Gamma=\frac{G_{F}^{2}}{192\,\pi^{3}}[(1-2s_{W}^{2})^{2}+(2s_{W}^{2})^{2}]\zeta_{n}\kappa_{n}^{3}\frac{E_{\nu}^{3n+5}}{M_{Pl}^{3n}}$
(23)

where $s_{W}$ is the sine of the Weinberg angle ($s_{W}^{2}=0.231$) and the
$\zeta_{n}$’s are numbers of $\cal{O}$(1). [43]

For the $n=1$ case we obtain the VPE rate

$\Gamma=1.72\times 10^{-14}\kappa_{1}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{3}\ {\rm
GeV},$ (24)

and for the $n=2$ case we obtain the VPE rate

$\Gamma=1.91\times 10^{-14}\kappa_{2}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{6}\ {\rm
GeV}.$ (25)

###  7 LIV in the neutrino sector III - decay by neutrino pair emission
(neutrino splitting)

The process of neutrino splitting in the case of superluminal neutrinos, i.e.,
$\nu\rightarrow 3\nu$ is relatively unimportant in the $[d]=4,n=0$ owing to
the small velocity difference between neutrino flavors obtained from neutrino
oscillation data (see Section 4). However, this is not true in the cases with
$[d]>4$. In the presence of $[d]>4$ $(n>0)$ terms in a Planck-mass suppressed
EFT, the velocity differences between the neutrinos, being energy dependent,
become significant [33]. The daughter neutrinos travel with a smaller
velocity. The velocity dependent energy of the parent neutrino is therefore
greater than that of the daughter neutrinos. Thus, the neutrino splitting
becomes kinematically allowed. Let us then consider the $n=1$ and $n=2$
scenarios.

Neutrino splitting is a neutral current (NC) interaction that can occur for
all 3 neutrino flavors. The total neutrino splitting rate obtained is
therefore three times that of the NC mediated VPE process above threshold.
Assuming the three daughter neutrinos each carry off approximately 1/3 of the
energy of the incoming neutrino, then for the $n=1$ case one obtains the
neutrino splitting rate [28]

$\Gamma=5.16\times 10^{-14}\kappa_{1}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{3}\ {\rm
GeV},$ (26)

and for the $n=2$ case we obtain the neutrino splitting rate

$\Gamma=5.73\times 10^{-14}\kappa_{2}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{6}\ {\rm
GeV}.$ (27)

The threshold energy for neutrino splitting is proportional to the neutrino
mass so that it is negligible compared to that given by equation (19). (See
also the detailed treatment of neutrino splitting given in Ref. [44].)

###  8 The IceCube Observations

The IceCube collaboration has identified hundreds of events from neutrinos of
astrophysical origin with energies above 10 TeV (see Chapter 5 by Halzen and
Kheirandish).

There are are four indications that the the bulk of cosmic neutrinos observed
by IceCube with energies above 0.1 PeV are of extragalactic origin:

(1) The arrival distribution of the reported events with $E>0.1$ PeV observed
by IceCube above atmospheric background is roughly consistent with isotropy
[45].

(2) The diffuse galactic neutrino flux is expected to be well below that
observed by IceCube [46, 45, 47].

(3) Upper limits on diffuse galactic $\gamma$-rays in the TeV-PeV energy range
imply that galactic neutrinos cannot account for the neutrino flux observed by
IceCube [48].

Above 60 TeV, the IceCube data are crudely consistent with a spectrum given by
$E_{\nu}^{2}(dN_{\nu}/dE_{\nu})\simeq\ 10^{-8}$ ${\rm GeV}{\rm cm}^{-2}{\rm
s^{-1}}$ extending up to an energy $\sim$ 2.2 PeV, but dropping off above that
energy [49].

IceCube has also reported a candidate neutrino induced event at the Glashow
resonance of 6.3 PeV [50]. At this energy, electrons in the IceCube volume
provide enhanced target cross sections for $\bar{\nu}_{e}$’s through the
$W^{-}$ resonance channel, $\bar{\nu}_{e}+e^{-}\rightarrow
W^{-}\rightarrow~{}shower$ at the resonance energy
$E_{\bar{\nu}_{e}}=M_{W}^{2}/2m_{e}=6.3$ PeV [51].

###  9 The Energy Spectrum from Extragalactic Superluminal Neutrino
Propagation

Monte Carlo techniques can be used to determine the effect of neutrino
splitting and VPE on putative superluminal neutrinos propagating from
cosmological distances under the assumption of the dominance of Planck mass
suppressed LIV operators with $[d]>4$ [28]. The Monte Carlo codes used in Ref.
28 take account of energy losses by both neutrino splitting and VPE as well as
redshifting of neutrinos emitted from sources at cosmological distances. As in
Ref. 25 and Ref. 28, we here consider a scenario where the neutrino sources
have a redshift distribution that follows that of the star formation rate and
further assume a simple source spectrum roughly proportional to $E^{-2}$
between 100 TeV and 100 PeV. The redshift distribution describing star
formation appears to be roughly applicable for both active galactic nuclei and
$\gamma$-ray bursts.

The Monte Carlo events were generated using these two distributions. The final
results were normalized to an energy flux of
$E_{\nu}^{2}(dN_{\nu}/dE_{\nu})\simeq 10^{-8}\ {\rm GeV}{\rm cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}$, as is consistent with the IceCube data for both the
southern and northern hemisphere for energies between 60 TeV and 2 PeV [49].

The Monte Carlo runs in Ref. 28 were made using threshold energies between 10
PeV and 40 PeV for the VPE process, corresponding to values of $\delta_{\nu
e}$ between $5.2\times 10^{-21}\ {\rm and}~{}3.3\times 10^{-22}$. By
propagating the test neutrinos including energy losses from VPE, neutrino
splitting, and redshifting using this code, final neutrino spectra were
obtained and compared with the IceCube results.

Given that neutrinos detected by IceCube are extragalactic, cosmological
effects should be taken into account in deriving new LIV constraints. The
reasons are straightforward. As opposed to the extinction of high energy
extragalactic photons through electromagnetic interactions [52], neutrinos
survive from all redshifts because they only interact weakly. It follows that
since the universe is transparent to neutrinos, most of the cosmic PeV
neutrinos will come from sources at redshifts between $\sim$0.5 and $\sim$2
where most of the energy in the universe from astrophysical sources is
produced. Therefore, along with energy losses by VPE [40] and neutrino
splitting, energy losses by redshifting of neutrinos and the effect of the
cosmological $\Lambda$CDM redshift-distance relation

$D={{c}\over{H_{0}}}\int\limits_{0}^{z}\frac{dz^{\prime}}{(1+z^{\prime})\sqrt{\Omega_{\rm\Lambda}+\Omega_{\rm
M}(1+z^{\prime})^{3}}}$ (28)

need to be included in the determination of $\delta_{\nu}$.

As in Refs. 25 28, we assume here a flat $\Lambda$CDM universe with a Hubble
constant of $H_{0}=$ 67.8 km s-1 Mpc-1 along with $\Omega_{\rm\Lambda}$ = 0.7
and $\Omega_{\rm M}$ = 0.3. Thus the energy loss due to redshifting is given
by

$-(\partial\log E/\partial
t)_{redshift}=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}.$ (29)

The decay widths for the VPE process are given by equations (24) and (25) for
the cases $n=1$ and $n=2$ respectively while those for neutrino splitting are
given by equations (26) and (27).

####  1 [d] = 4 $\cal{CPT}$ Conserving Operator Dominance

In their seminal paper, using equation (20), Cohen and Glashow [40] showed how
the VPE process in the $[d]=4$ case implied powerful constraints on LIV. They
obtained an upper limit of $\delta<\cal{O}$ $(10^{-11})$ based on the initial
observation of high energy neutrinos by IceCube [53]. Based on later IceCube
observations, using the results of Ref. 40 and later IceCube results [54] and
assuming a distance to the neutrino source of 1 Gpc, an upper limit of
$\delta<\cal{O}$ $(10^{-18})$ was obtained in Ref. 55.

General predictions of limits on $\delta$ with cosmological factors taken into
account were then made, with the predicted spectra showing a pileup followed
by a cutoff [28]. Using the energy loss rate given by equation (20), a value
for $\delta_{\nu e}<10^{-20}$ was obtained based on a model of redshift
evolution of neutrino sources based on the and using Monte Carlo techniques to
take account of propagation effects as discussed in Section 9 [25, 28]. The
upper limit on $\delta_{e}$ is given by $\delta_{e}\leq 5\times 10^{-21}$.
Taking this into account, one gets the constraint
$\delta_{\nu}\leq(0.5-1)\times 10^{-20}$. The spectra derived therein for the
$[d]=4$ case also showed a pileup followed by a cutoff. The predicted cutoff
is determined by redshifting the threshold energy effect. Monte Carlo
techniques to take account of propagation effects as discussed in Section 9
[25, 28]. The upper limit on $\delta_{e}$ is found to be $\delta_{e}\leq
5\times 10^{-21}$. Taking this into account, one gets the constraint
$\delta_{\nu}\leq(0.5-1)\times 10^{-20}$. The spectra derived therein for the
$[d]=4$ case show a slight pileup followed by a cutoff. The predicted cutoff
is determined by redshifting the threshold energy effect. The blue curve in
Figure 4 holds for the [d] = 4 case because the energy losses are only from
VPE .

####  2 [d] = 6 $\cal{CPT}$ Conserving Operator Dominance

In both the $[d]=4$ and $[d]=6$ cases, the best fit matching the theoretical
propagated neutrino spectrum, normalized to an energy flux of
$E_{\nu}^{2}(dN_{\nu}/dE_{\nu})~{}\simeq 10^{-8}{\rm GeV}{\rm cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}$ below $\sim$ 0.3 PeV, is assumed in the calculations.

As found before [25, 28], the best fit match to the IceCube data corresponds
to a VPE rest-frame threshold energy $E_{\nu,\rm th}=10$ PeV, as shown in
Figure 3. Such a threshold energy corresponds to a value for $\delta_{\nu
e}\equiv\delta_{\nu}-\delta_{e}\leq\ 5.2\times 10^{-21}$. Given that
$\delta_{e}\leq\ 5\times 10^{-21}$, it is again found that
$\delta_{\nu}\leq(0.5-1)\times 10^{-20}$888We note that one can not assume
that $\delta_{\nu}$ and $\delta_{e}$ are equal. Models can be constructed
where $\delta_{\nu}$ and $\delta_{e}$ are independent and it has even been
suggested that LIV may occur only in the neutrino sector [43]., 999The $\nu$
is used here generically for all three neutrino flavors, $\nu_{e},\nu_{\mu}$,
and $\nu_{\tau}$ and $\delta_{\nu}$ is for assumed for all three flavors as
their velocity differences are very small (see Sect. 4).

As indicated in Figure 3, values of $E_{\nu,\rm th}$ less than 10 PeV are
inconsistent with the IceCube data. The result for a 10 PeV rest-frame
threshold energy is just consistent with the IceCube results, giving a cutoff
effect above 2 PeV. The apparent cutoff in the IceCube observations indicated
in Figure 3 led to the suggestion of LIV as a possible explanation for the
lack of observed neutrinos above 2 PeV [25, 28]. However, should confirmed
events be found corresponding to neutrino energies above 2 PeV, the values for
$\delta_{\nu}$ given above would be upper limits.

In the case of the $\cal{CPT}$ conserving $[d]=6$ operator (n = 2) dominance,
as in the $[d]=4$ case, the results shown in Figure 3 show a predicted high-
energy drop off in the propagated neutrino spectrum near the redshifted VPE
threshold energy and a pileup in the spectrum below that energy. This pileup
is caused by the propagation of higher energy neutrinos in energy space down
to energies within a factor of $\sim$ 5 below the VPE threshold.

As the pileup effect shown in Figure 4, that is caused by the neutrino
splitting process is more pronounced than that caused by the VPE process. This
is because neutrino splitting produces two new lower energy neutrinos per
interaction. This difference would be a potential way of distinguishing a
dominance of $[d]>4$ Planck-mass suppressed interactions from $[d]=4$
interactions. Thus, with better statistics in the energy range above 100 TeV,a
significant pileup effect would be a signal of Planck-scale physics. Pileup
features are indicative of the fact that fractional energy loss from the last
allowed neutrino decay before the VPE process ceases is 78% [40] and that for
neutrino splitting is taken to be 1/3. The pileup effect is similar to that of
energy propagation for ultrahigh energy protons near the GZK threshold [56].

Values of $E_{\nu,\rm th}$ less than 10 PeV are inconsistent with the IceCube
data. The result for a 10 PeV rest-frame threshold energy, corresponding to
$\delta_{\nu e}=\ 5.2\times 10^{-21}$, is just consistent with the IceCube
results, giving a cutoff effect above $\sim$2 PeV. Thus for the conservative
case of no-LIV effect, e.g., if one assumes a cutoff in the intrinsic neutrino
spectrum of the sources, or one assumes a slightly steeper PeV-range neutrino
spectrum proportional to $E_{\nu}^{-2.3}$, we previously obtained the
constraint on superluminal neutrino velocity, $\delta_{\nu}=\delta_{\nu
e}+\delta_{e}\leq\ 1.0\times 10^{-20}$ [25].

In the case of the $\cal{CPT}$ conserving [d] = 6 operator (n = 2) dominance,
the results in Figure 3 appear to show a high-energy drop off in the
propagated neutrino spectrum near the redshifted VPE threshold energy and may
be consistent with a pileup in the spectrum below that energy, given the
statistics of small numbers. This predicted drop off may be a possible
explanation for the lack of observed neutrinos above 2 PeV [25, 28].101010The
report of a possible Glashow resonance from a 6.3 PeV $\bar{\nu_{e}}$ may
tighten the above constraints on $\delta_{\nu e}$. A pileup would be caused by
the propagation of the higher energy neutrinos in energy space down to
energies within a factor of $\sim$5 below the VPE threshold. This is
indicative of the fact that fractional energy loss from the last allowed
neutrino decay before the VPE process ceases is 0.78 [40] and that for
neutrino splitting is taken to be 1/3. The pileup effect is similar to that of
energy propagation for ultrahigh energy protons near the GZK threshold [56].
As is shown in Figure 4, including neutrino splitting in the calculation
increases the pileup effect.

Figure 3: Calculated n = 2 spectra taking into account of all three processes
(redshifting, neutrino splitting, and VPE) occurring simultaneously for rest
frame VPE threshold energies of 10 PeV (black), 20 PeV (green), and 40 PeV
(blue) [28]. The six year IceCube data are also shown (see Chapter 5 by Halzen
and Kheirandish.) Figure 4: Separately calculated n = 2 neutrino spectra with
the VPE case shown in blue and the neutrino splitting case shown in green. The
black spectrum takes account of all three processes (redshifting, neutrino
splitting, and VPE) occurring simultaneously. The rates for all cases are
fixed by setting the rest frame threshold energy for VPE at 10 PeV [28]. The
neutrino spectra are normalized to the six year IceCube data. Figure 5:
Calculated n = 0 (red) and n = 2 (black) neutrino spectrum obtained for the
VPE process only (no neutrino splitting) simultaneously with redshifting. The
rates for all cases are fixed by setting the threshold energy for VPE at 10
PeV [28]. Figure 6: Mean decay times for neutrino splitting process in the n =
2 case obtained by setting the threshold energy for VPE at 10 PeV (black), 20
PeV (green), and 40 PeV (blue) [28].

In order to determine threshold effects for the VPE process, a Monte Carlo
routine was employed in Ref. 28 to find the opening up of phase space,
assuming the same LIV parameters for every particle but with an electron mass
for two of the outgoing states. It was found that the entirety of phase space
is available when the energy reaches about 1.6 times that of threshold.
Threshold effects should therefore have little impact on the results of the
Monte Carlo calculation, as above 1.6$E_{th}$ the full phase space is open. A
Monte Carlo exploration of phase space for neutrino splitting yields similar
results. However, the threshold for this reaction is in the GeV range meaning
that full rates apply throughout our calculation.

In practice, neutrinos near threshold rarely pair produce before redshifting
below their threshold energy since their mean decay times increase with their
decreasing energy as they propagate. This also justifies the assumption that
the neutrino splitting and VPE rates are similar per decay channel.

Throughout the above calculation it was assumed that a neutrino loses 0.78 of
its initial energy per VPE interaction. Equation (23) shows that the VPE rates
do not differ by more than 45% between the $n=0$ and $n=2$ cases. This
reflects the difference in the phase space factors, since the dynamical matrix
elements are the same, indicating that this is also the maximum deviation in
the fraction of energy carried off by the neutrino in VPE. It is likely that
the deviation would be at most a third of that in a three-body decay, viz.,
15% meaning that the resulting energy fraction for the $n=2$ case could be as
high as 0.25. This effect was found to produce no discernible difference in
the spectra. In Ref. 28 we also tested an energy fraction of 0.5 and found
that even this extreme case would generate no observational consequences on
the pileup effect.

In Figure 5, plots the VPE process alone (along with redshifting) for the
$\cal{CPT}$\- conserving cases $n=0$ and $n=2$. It can be seen that the
resulting spectra are indistinguishable below threshold. Events above the
redshifted threshold pair-produce in relatively short times compared to
cosmological timescales regardless of the energy dependence, making the
spectra for $n=0$ and $n=2$ below the redshifted threshold indistinguishable.
One can only see the expected differences in the steepening of the spectra for
energies above threshold owing to the rate differences between $n=0$ and $n=2$
given by equation (23).

As can be seen in Figure 6, the mean decay times increase for the neutrino
splitting process with increasing choice of VPE threshold. The increased mean
decay times have the effect of reducing the pileup for increased choice of
threshold as fewer neutrino splitting events will occur. Thus the pileup
becomes a somewhat less sensitive test of Planck-scale effects with increasing
threshold energies. Figure 3 shows the effects of choosing different threshold
energies. The dominant process continues to be that of neutrino splitting but
with decreasing importance.

####  3 [d] = 5 $\cal{CPT}$ Violating Operator Dominance

In the n=1 case, the dominant [d] = 5 operator violates $\cal{CPT}$.

Both VPE and neutrino splitting generate a particle-antiparticle lepton pair.
The particles of this pair will have opposite helicities. This holds true for
both Dirac neutrinos and Majorana neutrinos, with the later being their own
antiparticles, In this case, one of the pair particles will be superluminal
($\delta>0$) whereas the other particle will be subluminal ($\delta<0$) [57].
Thus, of the daughter particles, one will be superluminal and interact, while
the other will only redshift. The overall result in the $[d]=5$ case is that
no strong spectral cutoff occurs [25].

However, the IceCube detector cannot distinguish neutrinos from antineutrinos.
The incoming $\nu(\bar{\nu}$) generates a shower in the detector, allowing a
measurement of its energy and direction. Even in cases where there is a muon
track, the charge of the muon is not determined.

There would be an exception for electron antineutrinos at 6.3 PeV, given an
expected enhancement in the event rate at the $W^{-}$ Glashow resonance, since
this resonance only occurs with $\bar{\nu_{e}}$. candidate Glashow resonance
event has now been reported by the IceCube collaboration [51]. A confirmation
of such resonance events might favor a $\cal{CPT}$-odd interpretation,
particularly if they are rarer than expected.

Figure 7: Calculated n = 1 neutrino spectra assuming 100% (black), 50% (blue)
and 0% (red) initial superluminal neutrinos (antineutrinos). The neutrino
spectra are normalized to the IceCube data.

Figure 7 shows the results in the $\cal{CPT}$-violating $n=1$ case, assuming
100%, 50% and 0% initial superluminal neutrinos (antineutrinos) and
propagating the spectrum using a Monte Carlo program and taking account of the
fact that in all cases, one of the daughter leptons is subluminal and
therefore does not undergo further interactions. As a sanity check, it was
found that in the 0% case only redshifting occurs, preserving the initial
$E^{-2}$ spectrum. The other cases show the effect of VPE and neutrino
splitting by both the initial fraction of superluminal neutrinos and the
superluminal daughter neutrinos.

Thus, as opposed to the $\cal{CPT}$-conserving $n=2$ case, no clearly
observable cut off is produced, with the possible unrealistic exception of
postulating that only superluminal $\nu$’s (or superluminal $\bar{\nu}$’s) are
produced in cosmic sources. That case, shown in black in Fig. 7, as well as
the other case of postulating no initial superluminal neutrinos, shown in red,
are shown for illustrative purposes. The 50/50 case, shown in blue, is more
realistic.

In the n = 1, $\cal{CPT}$-odd case, the details of the kinematics are
different from the n = 0 and n = 2, $\cal{CPT}$-even cases because in the
$\cal{CPT}$-odd case the signs of $\delta$ are opposite for $\nu$’s and
$\bar{\nu}$’s. If we assume that they are equal and opposite, then the rate
given in equation (23) would maximally be altered by replacing the $\delta$
with 2$\delta$. Since the source kinematics dominate as the daughter energies
are comparable, doubling delta should overestimate their contribution to the
overall rate. Applying Monte Carlo techniques to explore the phase space [28],
one finds that the subliminal particle will carry away a slightly higher
fraction of the energy after the split ($\sim 40$%) in the $\cal{CPT}$-odd
case. By making these modifications to the Monte Carlo code one finds that
there is little observational difference between the modified results and
those obtained assuming the same rate as given by equation (23). An exact
treatment of the kinematics for $\cal{CPT}$-odd, which are complex, are
therefore unnecessary and our spectral results in the $\cal{CPT}$-odd case
given in Figure 7 are a good approximation to an exact treatment.

###  10 Summary of the results from LIV kinematic effects for superluminal
neutrinos

In the previous sections we have explored and summarized the effects of
$[d]>4$ Planck-mass suppressed operators on the propagation and resulting
energy spectrum of superluminal neutrinos of extragalactic origin. We have
expressed these Lorentz violating perturbations as a modifications of the
energy-momentum dispersion relation in the form $\delta_{\nu
e}\simeq\delta_{\nu}\equiv\delta_{n}$ (see discussion in IIIA) for Planck mass
suppressed energy dependent values of $\delta_{n}$ as defined in equation (9).
These terms can arise from higher dimension operators in the EFT formalism
[21].

In the SME EFT formalism [16, 21, 22], if the apparent cutoff above $\sim 2$
PeV in the neutrino spectrum shown in Figure 3 is caused by LIV, this would
result from an EFT with either a dominant $[d]=4$ term with $\mathaccent
23{c}^{(4)}=-\delta_{\nu e}=5.2\times 10^{-21}$, or by a dominant $[d]=6$ term
with $\mathaccent 23{c}^{(6)}=-\kappa_{2}/M_{Pl}^{2}\geq-5.2\times 10^{-35}$
GeV-2 (Refs. 23, 28, see appendix B) 111111The IceCube collaboration has
looked for deviations in $\nu_{\mu}$ oscillations between horizontal and
vertical muon fluxes of atmospheric origin in the detector [58] (See Section
4). The horizontal path length is much shorter than the vertical path length
and is used for normalization. They obtain constraints of order $10^{-36}$
GeV-2 on the $[d]=6$ operator involved. Such a cutoff would not occur if the
dominant LIV term is a $\cal{CPT}$-violating the $[d]=5$ operator. Future
detections of astrophysical neutrinos with energy above $\sim$2 PeV would
indicate that the above numbers should be considered to be upper limits on
these parameters.

Furthermore, we note that, should the $\cal{CPT}$-violating [d] = 5 operator
dominate, we would find an absence of a clear cutoff in the propagated
neutrino spectrum.

###  11 LIV in the Ultrahigh Energy Cosmic Ray Spectrum and the Subsequent
Ultrahigh Energy Neutrino Spectrum

####  1 The GZK Effect

Shortly after the discovery of the 3K cosmic background radiation (CBR),
Greisen [59] and Zatsepin and Kuz’min [60] predicted that pion-producing
interactions of such cosmic ray protons with the CBR should produce a spectral
cutoff at $E\sim$ 50 EeV. The flux of ultrahigh energy cosmic rays (UHECR) is
expected to be attenuated by such photomeson producing interactions. This
effect is generally known as the ”GZK effect”. Owing to this effect, protons
with energies above $\sim$100 EeV should be attenuated from distances beyond
$\sim 100$ Mpc because they interact with the CBR photons with a resonant
photoproduction of pions [61]. The GZK effect is not a true cutoff, but a
suppression of the ultrahigh energy cosmic ray flux owing to an energy
dependent propagation time against energy losses by such interactions, a time
which is only $\sim$300 Myr for 100 EeV protons [61]. At high redshifts, $z$,
the target photon density increases by $(1+z)^{3}$ and both the photon and
initial cosmic ray energies increase by $(1+z)$. If the source spectrum is
hard enough, there could also be a relative enhancement just below the “GZK
energy” owing to a “pileup” of cosmic rays starting out at higher energies and
crowding up in energy space at or below the predicted GZK cutoff energy [56]
(See also Section 2).

####  2 The Effect of LIV Kinematics on the GZK Process

We now consider the kinematics of the photomeson production process leading to
the GZK effect. Near threshold, where single pion production dominates,

$p+\gamma\rightarrow p+\pi.$ (30)

Using the normal Lorentz invariant kinematics, the energy threshold for
photomeson interactions of UHECR protons of initial laboratory energy $E$ with
low energy photons of the CBR with laboratory energy $\omega$, is determined
by the relativistic invariance of the square of the total four-momentum of the
proton-photon system. This relation, together with the threshold inelasticity
relation $E_{\pi}=m/(M+m)E$ for single pion production, yields the threshold
conditions for head on collisions in the laboratory frame

$4\omega E=m(2M+m)$ (31)

for the proton, and

$4\omega E_{\pi}={{m^{2}(2M+m)}\over{M+m}}$ (32)

in terms of the pion energy, where M is the rest mass of the proton and m is
the rest mass of the pion.

If LI is broken so that $\delta_{\pi}~{}>~{}\delta_{p}$ ($\delta_{\pi p}>0)$,
it follows from equations (3) and (5) that the threshold energy for photomeson
production is altered [18] because the square of the four-momentum is shifted
from its LI form so that the threshold condition in terms of the pion energy
becomes

$4\omega E_{\pi}={{m^{2}(2M+m)}\over{M+m}}+2\delta_{\pi p}E_{\pi}^{2}$ (33)

Equation (33) is a quadratic equation with real roots only under the condition

$\delta_{\pi
p}\leq{{2\omega^{2}(M+m)}\over{m^{2}(2M+m)}}\simeq\omega^{2}/m^{2}.$ (34)

Defining $\omega_{0}\equiv kT_{CBR}=2.35\times 10^{-4}$ eV with
$T_{CBR}=2.725\pm 0.02$ K, equation (34) can be rewritten

$\delta_{\pi p}\leq 3.23\times 10^{-24}(\omega/\omega_{0})^{2}.$ (35)

If LIV occurs and $\delta_{\pi p}>0$, photomeson production can only take
place for interactions of CBR photons with energies large enough to satisfy
equation (35). This condition, together with equation (33), implies that while
photomeson interactions leading to GZK suppression can occur for “lower
energy” UHE protons interacting with higher energy CBR photons on the Wien
tail of the spectrum, other interactions involving higher energy protons and
photons with smaller values of $\omega$ will be forbidden. Thus, the observed
UHECR spectrum may exhibit the characteristics of GZK suppression near the
normal GZK threshold, but the UHECR spectrum can ”recover” at higher energies
owing to the possibility that photomeson interactions at higher proton
energies may be forbidden.

The kinematical relations governing photomeson interactions are changed in the
presence of even a small violation of Lorentz invariance. Following equations
(3) and (4), we denote

$E^{2}=p^{2}+2\delta_{a}p^{2}+{m_{a}}^{2}$ (36)

where $\delta_{a}$ is the difference between the MAV for the particle a and
the speed of light in the low momentum limit ($c=1$).

The square of the cms energy of particle $a$ is then given by

$\sqrt{s_{a}}=\sqrt{E^{2}-p^{2}}=\sqrt{2\delta_{a}p^{2}+m_{a}^{2}}\geq 0.$
(37)

Owing to LIV, in the cms the particle will not generally be at rest when $p=0$
because

$v={{\partial E}\over{\partial p}}\neq{{p}\over{E}}.$ (38)

The modified kinematical relations containing LIV have a strong effect on the
amount of energy transferred from a incoming proton to the pion produced in
the subsequent interaction, i.e., the inelasticity [62, 63].

Figure 8 shows the calculated proton inelasticity modified by LIV for a value
of $\delta_{\pi p}=3\times 10^{-23}$ as a function of both CBR photon energy
and proton energy [63]. Other choices for $\delta_{\pi p}$ yield similar
plots. The principal result of changing the value of $\delta_{\pi p}$ is to
change the energy at which LIV effects become significant. For a choice of
$\delta_{\pi p}=3\times 10^{-23}$, there is no observable effect from LIV for
$E_{p}$ less than $\sim 200$ EeV. Above this energy, the inelasticity
precipitously drops as the LIV term in the pion rest energy approaches
$m_{\pi}$.

Figure 8: The calculated proton inelasticity modified by LIV for $\delta_{\pi
p}=3\times 10^{-23}$ as a function of CBR photon energy and proton energy [63,
64]. Figure 9: Comparison of the latest Auger data with calculated spectra for
various values of $\delta_{\pi p}$, taking $\delta_{p}=0$ (see text). From top
to bottom, the curves give the predicted spectra for $\delta_{\pi p}=1\times
10^{-22},6\times 10^{-23},4.5\times 10^{-23},3\times 10^{-23},2\times
10^{-23},1\times 10^{-23},3\times 10^{-24},0$ (no Lorentz violation) [63, 64].

####  3 The Photomeson Neutrino Spectrum

We now turn our attention to calculating the photomeson neutrino spectrum that
would result from the UHECR calculation detailed in the previous section. To
determine this, we use the data on the cross section for pion production
compiled in reference [65] and summarized in reference [66]. Near threshold
the the total photomeson cross section is dominated by the emission of single
pions. The most significant channel to consider involves the intermediate
production of the $\Delta$ resonance [61]:

$p+\gamma\rightarrow\Delta\rightarrow N+\pi$ (39)

This channel strongly dominates the photomeson production process near
threshold. Since the UHECR flux falls steeply with energy, it follows that the
bulk of the pions leading to the production of neutrinos will be produced
close to the threshold.

For a proton interacting with the CBR, a pion and a nucleon are produced. The
outgoing nucleon has probability of 2/3 to be a proton and 1/3 probability to
be a neutron from isospin considerations. Should the resulting nucleon be a
neutron, then the resulting pion is a $\pi^{+}$. Thus approximately twice the
number of neutral pions are produced relative to charged pions from resonant
pion production. However direct pion production, which accounts for about 20%
of the total cross section, produces charged pions almost exclusively meaning
that all told, approximately equal numbers of neutral and charged pions are
produced around threshold. Neutral pions decay into photons so one only need
only consider the charged pions for neutrino production. Three neutrinos of
roughly equal energy result from the decay chain of the
$\pi^{+}\rightarrow\mu^{+}\nu_{\mu}\rightarrow
e^{+}\bar{\nu{{}_{\mu}}}\nu_{e}$.

It is straightforward to determine the neutrinos produced and their energies
from the ultrahigh energy cosmic rays. We follow closely the calculation of
the neutrino flux as described in reference [46] and references therein. The
key is to determine the amount of energy that is carried away by the pion.
This follows directly from the inelasticity and the incident proton energy
calculated using equation (32). For simplity, we assume here that all of the
sources have the same primary injection spectrum and distribution. One can
then calculate the total neutrino flux by integrating over proton energy,
photon energy, and redshift, assuming the standard $\Lambda$CDM cosmology.

The effect of LIV on the photomeson neutrino production is manifested through
the modification of the inelasticity of the interaction, since this determines
the amount of energy that is carried away by the pion and therefore the
resultant neutrino energy. The biggest impact of including LIV is to suppress
the production of the higher energy photopions and therefore the resulting
higher energy neutrinos.

Figure 10 shows the corresponding total neutrino flux (all species) for the
same choices of $\delta_{\pi p}$ as the UHECR spectra presented in figure 9.
As expected, increasing $\delta_{\pi p}$ leads to a decreased flux of higher
energy photomeson neutrinos as the interactions involving higher energy UHECRs
are suppressed [63].

Figure 10: Neutrino fluxes (of all species) corresponding to the UHECR models
considered in Figure 9. From left to right, the curves give the predicted
fluxes for $\delta_{\pi p}=1\times 10^{-22},6\times 10^{-23},3\times
10^{-23},1\times 10^{-23},0$. [67]

###  12 Stable Pions from LIV

Almost all neutrinos are produced by pion decay. It has been suggested that if
LIV effects can prevent the decay $\pi\rightarrow\mu+\nu$ of charged pions
above a threshold energy, thereby eliminating higher energy neutrinos at the
Glashow resonance energy and above [68, 69]. In order for the pion to be
stable above a critical energy $E_{c}$, we require that its effective mass as
given by equation (7) is less than the effective mass of the muon that it
would decay to, i.e., $\tilde{m}_{\pi}<\tilde{m}_{\mu}$. This situation
requires the condition that $\delta_{\pi}<\delta_{\mu}$ [18]. Then, neglecting
the neutrino mass, this critical energy energy is given by

$E_{c}={{\sqrt{m_{\pi}^{2}-m_{\mu}^{2}}}\over{\delta_{\mu}-\delta_{\pi}}}$
(40)

noting that if, as before, we write
$\delta_{\mu\pi}\equiv\delta_{\mu}-\delta_{\pi}$, then for small
$\delta_{\mu\pi}$, it follows that
$\sqrt{2\delta_{\mu\pi}}\simeq\delta_{\mu\pi}$. In terms of SME formalism with
Planck-mass suppressed terms, $\delta_{\mu\pi}$ is given by equation (8) or
equation (9). For example, if we set $E_{c}$ = 6 PeV, in order to just avoid
the Glashow resonance, we get the requirement, $\delta_{\mu\pi}\simeq
1.5\times 10^{-8}$. However, it should be noted that a Glashow resonance event
has been recently reported in Ref. 50.

###  13 Observational tests with new neutrino telescopes

Future neutrino detectors are being planned or constructed (see chapters 6 and
7): IceCube-Gen 2 [70], the Askaryan effect detectors ARA [71] and ARIANNA
[72], and space-based telescopes such as OWL [73], EUSO [74], and a more
advanced OWL-type instrument called POEMMA [75]. They will provide more
sensitive tests of LIV.

### Acknowledgments

I would like to acknowledge my collaborators: Stefano Liberati, David
Mattingly and Sean Scully. I thank Francis Halzen for helpful comments on the
IceCube neutrino telescope. I also thank Maria Gonzalez-Garcia and Michele
Maltoni for helpful comments.

[Appendix A: Dimensional Analysis with Mass Dimensions] It is routine in
particle theory to use ”natural” units, setting $\hbar$ = $c=1$. Thus, in
natural units, the dimensions of $\hbar$ and $c$ are then $[\hbar]=[c]=0$. One
defines the dimension of mass $[m]=+1$. It then follows that $[E]=[p]=+1$, and
$[t]=[x]=-1$. The derivative operator $\partial_{\mu}$ then has dimension
$[\partial_{\mu}]$ = +1.

The time evolution of a quantum system is described by a unitary
transformation, $e^{iS}$ where $S$ is the dimensionless action (remembering
$[\hbar]=0$), given by the 4-integral of the Lagrangian density (or
”Lagrangian”), $\cal{L}$,

$\\\ \\\ \\\ \\\ \\\ \\\ S=\int{d^{4}}x\ \cal{L}\\\ \\\ \\\ \\\ \\\ \\\ \ $

so that, since $[S]=0$, $[\cal{L}]$ = +4. Scalar fields, $\phi$, scale
inversely with $x$ so that [$\phi$] = +1. Fermion fields have $[\psi]$ = +3/2.

Since $[\cal{L}]$ = +4, any operator consisting of fields adding up to
$[\cal{O}]$ = +(4+n) must be balanced by a mass term $M^{-n}$, where $M$ is
the scale of the EFT. Thus, for the Fermi EFT with the 4-fermion operator
$G_{F}[\psi][\psi][\psi][\psi]$, the Fermi coupling constant $G_{F}$ must have
mass dimension $[G_{F}]$ = -2. In fact, its value is 1.166 $\times 10^{-5}$
GeV-2 and is related to the weak mass scale, $M_{W}^{-2}$.

[Appendix B: Standard Model Extension Isotropic Diagonalizable Terms]

The standard model extension (SME) formalism restricted to diagonalizable and
isotropic terms yields a very simple class of models [23] These models have
simultaneously diagonalizable Lorentz-violating operators and they also are
manifestly rotationally invariant in a preferred frame. Thus, they can be
related to the formalism used in Section 3.

For these models, it is convenient to work in the diagonal basis and in the
preferred frame, however, see footnotes in Section 2. In the diagonal basis
and preferred frame, the energy of a neutrino specific neutrino species,
assuming the dominance of operators of even dimension $d$ with $\cal{CPT}$
even, can be written in the form

${{\mathaccent 23{E}}}_{\nu}={|{\bf p}|}+\frac{m_{\nu}^{2}}{2|{\bf
p}|}+\sum_{d}|{\bf p}|^{d-3}\,{{\mathaccent 23{c}}}^{(d)}_{.}$ (41)

In general, three coefficients for Lorentz violation appear for each $d$, one
for each neutrino species. For relativistic neutrinos, $|{\bf p}|\gg m_{\nu}$
and the second term in equation (41) can be neglected. See reference 23 for a
general treatment including $\cal{CPT}$-odd terms and anisotropic terms.

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