# Negative Probability111In memory of Nozer Singpurwalla

Nick<EMAIL_ADDRESS>
Booth School of Business
University of Chicago Vadim<EMAIL_ADDRESS>
Systems Engineering and Operations Research
George Mason University

###### Abstract

Negative probabilities arise primarily in quantum theory and computing.
Bartlett provides a definition based on characteristic functions and
extraordinary random variables. As Bartlett observes, negative probabilities
must always be combined with positive probabilities to yield a valid
probability distribution before any physical interpretation is admissible.
Negative probabilities arise as mixing distributions of unobserved latent
variables in Bayesian modeling. Our goal is to provide a link with dual
densities and the class of scale mixtures of normal distributions. We provide
an analysis of the classic half coin distribution and Feynman’s negative
probability examples. A number of examples of dual densities with negative
mixing measures including the linnik distribution, Wigner distribution and the
stable distribution are provided. Finally, we conclude with directions for
future research.

Keywords: Bayes, Negative Probability, Half Coin, Quantum computing, Dual
Densities, Wigner, Feynman.

##  1 Introduction

This paper was motivated by numerous conversations with Nozer Singpurwalla in
2022. Nozer had a keen interest in quantum probability, negative probabilities
and the foundations of statistical inference Lindley (1985). He had a great
sense of interesting problems that spanned many scientific fields and was
fearless in his pursuit of such ideas.

Negative probabilities are a useful tool for explaining physical phenomena in
quantum mechanics. Many authors including Heisenberg (1931); Wigner (1932);
Dirac (1942), Feynman (1987) use negative probabilities in such a manner. They
also arise in quantum computing, see Polson et al. (2023). As Dirac noted,
“negative energies and probabilities should not be considered as nonsense.
They are well-defined concepts mathematically, like a negative of money.”

To illustrate our methodology, we show that the classic negative probability
half coin distribution is related to the Polya Gamma mixing distribution
(Barndorff-Nielsen et al., 1982; Polson et al., 2013). The Linnik distribution
(Devroye, 1990) can be expressed as a Gaussian scale mixture but with negative
mixing weights (Chu, 1973; West, 1987; Smith, 1981; Kingman, 1972). Mixtures
of Exponential and Gaussian distributions have a long history in Bayesian MCMC
algorithms and hierarchical representations of distributions Polson and Scott
(2015) and lead to EM algorithms for posterior mode and maximum likelihood
inference. Our results build on this literature by extending the class of
distributions to those with negative mixing weights.

From another perspective, Gill (2014) provides a simple proof of the famous
Bell’s inequality wing two applications of Hoeffding’s inequality. Gill also
relates Bell’s theorem to statistical causality, See also Tian and Pearl
(2013) for probability bounds. Dirac (1942) “negative energies and
probabilities should not be considered as nonsense. They are well-defined
concepts mathematically, like a negative of money.” Hudson (1974) shows that
that for the Wigner quasi-probability density to be a true density is that the
corresponding Schrödinger state function is the exponential of a quadratic
polynomial (a 2-dim multivariate normal).

The rest of the paper is organized as follows. Section 2 revisits the
definition of negative probability and extraordinary random variables due to
Bartlett. We consider a archetypal example of half-coin distribution due to
Szekely. Section 3 provides our results on new characterizations of scale
mixture of normals using dual densities (Good, 1995; Gneiting, 1997).
Bernstein’s theorem fo completely monotone functions is used to determine when
the mixing weights are non-negative. Our work shows that many results in
quantum mechanics are also related to the notion of dual densities and scale
mixtures of normal. Section 4 considers a number of examples, including the
Linnik family, the stable family and Wigner distribution. Finally, Section 5
concludes with the directions for future research.

###  1.1 Motivating Example

Feynman (1987) provides the following simple example of negative
probabilities. Feynman discusses the case with a conditional table for
$p(\mathrm{state}=j\mid E)$ for $j=(1,2,3)$ and $E=\\{A,B\\}$ with a negative
entry

|  | given A | given B
---|---|---|---
| 1 | 0.3 | -0.4
State | 2 | 0.6 | 1.2
| 3 | 0.1 | 0.2
Table 1: Conditional Probabilities for Feynman’s Example.

Notice that $p(\mathrm{state}=2|B)=1.2>1$ in order to offset the negative
conditional probability $p(\mathrm{state}=1|A)=-0.4$. The sum total of
probabilities is still one, and we have a valid probability distribution over
the states.

Suppose that the underlying base rates are given by $p(A)=0.7$, $p(B)=0.3$.
Now we calculate the observable marginal distributions.

$p(\mathrm{state}=1)=p(\mathrm{state}=1\mid A)p(A)+p(\mathrm{state}=1\mid
B)p(B)=0.7\times 0.30.3\times 0.4=0.09.$

Although $p(\mathrm{state}=2\mid B)=1.2$, is allowed to be greater than one,

$p(\mathrm{state}=2)=0.7\times 0.6+0.3\times 1.2=0.78,$

which is still a valid probability. Finally, we have

$p(\mathrm{state}=3)=0.7\times 0.1+0.3.2=0.13.$

We can see that have ordinary probabilities. The key point is that the law of
total probability still holds even though the mixing weights, which are
unobserved (latent) are allowed to contain negative values.

##  2 Negative Probabilities

Negative probabilities arise as convolutions of probability measure. Imagine a
random variable represented as a convolution

$Y=X+Z.$

Here, $Z$ can have an extraordinary probability distribution. $X$ and $Z$ are
independent in the usual statistical sense. We can think of $Y$ as observed
and $Z$ as a hidden latent state and $X$ the state of nature.

In terms of generating functions, by independence,

$G_{Y}(s)=E(s^{Y})=E(s^{X})E(s^{Z})=G_{X}(s)G_{Z}(s).$

The following is a fundamental result in negative probability.

#### Fundamental theorem

For every generalized g.f. $f(z)$ of a signed probability distribution there
exist two probability generating functions $g$ and $h$ of ordinary non-
negative distributions such that

$f(z)g(z)=h(z).$

The sum of independent random variables leads to a product if their generating
functions. Let $L_{1}^{+}$ denote the space of integrable densities. Then,

$f\in L_{1}^{+},g\in L_{1},\exists f\;\;s.t.\;f\star g\in L_{1}^{+},$

where the convolution

$(f\star g)(z)=\int f(z-x)g(x)dx$

The law of total probability is convolution for random variables.

#### Extraordinary Random Varaibles.

Bartlett (1945) provides the following definition and intuition definition of
extraordinary random variables.

Let $\phi_{X}(t)=E(e^{itX})$ denote the characteristic function of a non-
negative variable $X$.

They are defined via their characteristic functions. In terms of Fourier
transforms (aka characteristic functions) we have

$\phi_{Y}(t)=\phi_{X}(t)\phi_{Z}(t).$

Given $\phi_{Y},\phi_{X}$, we would like to identify the mixing measure of the
hidden variable $Z$. Solving for $\phi_{Z}(t)$ we have,

$\phi_{Z}(t)=\frac{\phi_{Y}(t)}{\phi_{X}(t)}=\phi_{Y}(t)\phi_{X}^{-1}(t)$

This has the same form, as the convolution product above!

Therefore, we write the characteristic function of $Z$ in terms of a random
variable denoted by $W$.

$\phi_{Z}(t)=\phi_{Y}(t)\phi_{W}(t),\text{ with
}\phi_{W}(t)=\phi_{X}^{-1}(t),$

However, the following identity holds

$\phi_{X}^{-1}(t)\phi_{X}^{-1}(t)=1=E(e^{it0}).$

Hence, $X+W\stackrel{{\scriptstyle P}}{{=}}0$.

Therefore, $W$ will have an extra-ordinary probability distribution. Thats is
will take negative values in parts of its domain.

A related concept is that of a van Dantzig pair of functions $f(s)$ and
$g(s)=1/f(is)$ are both characteristic functions. For applications, see Lukacs
(1972) and Polson (2021).

###  2.1 Mixture Convolutions

One can view the law of total probability as a convolution theorem for random
variables. Hence, a natural generalization of Feynman’s examples are mixture
distributions with negative weights. In this section we give a number of
examples. The classic example is the half coin. We provide a connection with
the Polya-Gamma mixture model Polson et al. (2014); Polson and Scott (2015).

Wigner (1932) shows that in quantum theory the joint density function $P(x,p)$
of the location and momentum of a particle cannot be non-negative everywhere
as it is always real and yet its integral over the whole space is zero. Hence
written asa convolution (aka Bayesian mixture model) the mixing weights can be
negative. Feynman provides the following concrete example.

#### Half Coin

The probability generating function (probability-generating function) is
defined by the formula $f(z)=\sum_{n=1}^{\infty}p_{n}z^{n}$. The pdf of a fair
coin is

$f(z)={\scriptstyle\frac{1}{2}}+{\scriptstyle\frac{1}{2}}z.$

If we assume that $\sum_{n=1}^{\infty}p_{n}=1$ and $\sum_{n}|p_{n}|<\infty$
but drop the non-negativity of its probabilities, we can define the half coin
as having pdf

$f_{\scriptstyle\frac{1}{2}}(z)=\sqrt{{\scriptstyle\frac{1}{2}}+{\scriptstyle\frac{1}{2}}z}=\frac{1}{\sqrt{2}}(1+\frac{1}{2}s-\frac{1}{8}s^{2}...)$

According to the Binomial theorem

$\sqrt{{\scriptstyle\frac{1}{2}}+{\scriptstyle\frac{1}{2}}z}=\frac{1}{\sqrt{2}}\sum_{n=0}^{\infty}\binom{1/2}{n}z^{n},$

where the coefficients are, with $C_{n}$ the $n$-th Catalan number,

$\binom{1/2}{n}=(-1)^{n-1}\frac{2C_{n-1}}{4^{n}}\;\;{\rm
and}\;\;C_{n}=\frac{1}{n+1}\binom{2n}{n}.$

#### Pólya-Gamma

The probability-generating function of the half-coin is related to that of the
Pólya-Gamma distribution. Let $X\sim PG(b,0)$. By definition, the moment-
generating function is

$E\left\\{e^{-tX}\right\\}=\frac{1}{\cosh^{b}(\sqrt{t})}=\frac{1}{(e^{\sqrt{t}}+e^{-\sqrt{t}})^{b}}$

Letting $s=e^{-t}$, yields p.g.f.

$E(s^{X})=\left(\sqrt{s}+\frac{1}{\sqrt{s}}\right)^{-b}.$

Barndorff-Nielsen et al. (1982) (section 3.6) gives the mixing density

$f(u)=\sum_{k=0}^{\infty}\binom{-2\delta}{k}\frac{(\delta+k)}{B(\delta,\delta)}e^{-\frac{1}{2}(\delta+k)^{2}u}\;\;\delta>0.$

Hence, we see the equivalence with the half-coin, where $=-1/2$ and
$\delta=2$!

#### Bayes Rule for Negative Probabilities

Suppose that the prior is a negative weight mixture

$p(\theta)=\int_{0}^{\infty}e^{-\theta s}g(s)ds$

Given a likelihood $f(y|\theta)$, then we have posterior

$\displaystyle p(\theta|y)$
$\displaystyle=\frac{f(y|\theta)p(\theta)}{m(y)}=\frac{f(y|\theta)}{m(y)}\int_{0}^{\infty}e^{-\theta
s}g(s)ds$ $\displaystyle=\frac{1}{m(y)}\int_{0}^{\infty}e^{-\theta
s}\frac{f(y|\theta)}{h(s)}g(s)h(s)ds$
$\displaystyle=\frac{1}{m(y)}\int_{0}^{\infty}f^{\star}(y|s,\theta)g^{\star}(s)ds$

where $g^{\star}(s)=g(s)h(s)$ is a signed measure, and $f^{\star}$ is the
exponentially tilted probability measure

$f^{\star}(y|s,\theta)=\frac{e^{-\theta s}f(y|\theta)}{h(s)}.$

Hence propagation by Bayes rule exists for negative probabilities.

##  3 Dual Densities

The appearance of negative probability arises mainly in quantum theory.
Bartlett (1945) provides a formal extension of Kolmogorov’s mathematical
probability as follows. He introduces extraordinary random variables through
their characteristic functions. We relate this approach with the notion of
dual densities. A number of illustrations within the class of scale mixtures
of normals are provided. In particular the Linnik and Cauchy distributions. As
Bartlett observes, negative probabilities must always be combined with
positive ones to give an ordinary probability distribution before a physical
interpretation is admissible. This can always be done as the fundamental
theorem shows.

#### Mixture of Exponentials

$\frac{1}{\pi}\int_{0}^{\infty}e^{-tu}\sin(u^{1/2})du=\frac{1}{2\pi^{1/2}}\frac{1}{t^{3/2}}e^{-1/4t}=\varphi_{1/2}(t)$
$e^{-|t|^{1/2}}=\int_{0}^{\infty}e^{-xt}\varphi_{1/2}(t)dt.$

Good (1995); Gneiting (1997) introduced the concept of a dual density. Given a
density $p(x)$, its dual density is given by its characteristic function

$\hat{p}(t)=\mathbb{E}(e^{itX})=\int_{-\infty}^{\infty}e^{itx}p(x)dx,$

appropriately normalized. The characteristic function is simply the Fourier
transform (with sign reversal) of the probability density function.

The pair of dual densities $(p,\hat{p})$ follow a Heisenberg principle
principle, when one learns something about $p$ one has information about the
other, but they both cannot be observed at the same time.

#### Scale Mixtures of Normals.

The class of scale mixtures of normal looks like

$p(X)=\int_{0}^{\infty}v^{-1}\phi(v^{-1}X)p(v)dv,$

see Andrews and Mallows (1974); Carlin et al. (1992); Carlin and Polson
(1991).

An interesting class arises when $p$ is a scale mixture of normals with
bounded density

$p(x)=\int_{0}^{\infty}\dfrac{1}{\sqrt{2\pi v}}e^{-x^{2}/2v}dF(v),$

where $F$ is a mixing measure defined on $(0,\infty)$.

Another way to write a scale mixture is $X=\sqrt{V}Z$, where $Z\sim N(0,1)$.
The Laplace (a.k.a. double exponential) is a scale mixture of Cauchy.

Consider the characteristic function

$\phi(t)=\int_{0}^{\infty}e^{-vt^{2}/2}dF(v).$

It is the characteristic function of a normal

$E(e^{itZ})=\int_{-\infty}^{\infty}e^{itx}\dfrac{1}{\sqrt{2\pi
v}}e^{-x^{2}/2v}dx=e^{-vt^{2}/2}.$

In the case when both $p$ and $\phi$ are bounded and integrable

$p(0)=\int_{0}^{\infty}\dfrac{1}{\sqrt{2\pi v}}dF(v)<\infty,$

the dual density $\hat{p}$ is proportional to $\phi$ and is given by

$\hat{p}(t)=\hat{p}(0)\phi(t).$

The dual density is then a scale mixture of normal as well, with density

$\hat{p}(x)=\int_{0}^{\infty}\dfrac{1}{\sqrt{2\pi
v}}e^{-x^{2}/2v}d\hat{F}(v),$

where

$\dfrac{1}{\sqrt{2\pi v}}d\hat{F}(v)=\dfrac{1}{2\pi p(0)}dF(1/v),~{}v>0.$

This follows for

$\hat{p}(x)=\hat{p}(0)\phi(x)=\dfrac{1}{2\pi
p(0)}\int_{0}^{\infty}e^{-vx^{2}/2}dF(1)=\dfrac{1}{2\pi
p(0)}\int_{0}^{\infty}e^{-x^{2}/2v}dF(1/v).$

Suppose that $p(x)=\int_{0}^{\infty}(2\pi
v)^{-\frac{1}{2}}\exp(-x^{2}/2v)dF(v)$ is a normal scale mixture. Then the
dual density is also a normal scale mixture.

Then its dual characteristic function is given by

$\hat{p}(t)=\int_{-\infty}^{\infty}\exp(itx)\int_{0}^{\infty}(2\pi
v)^{-\frac{1}{2}}\exp(-x^{2}/2v)dF(v)=\int_{0}^{\infty}\exp(-\frac{1}{2}u\omega^{2})f(u)du$

Therefore,

$\hat{p}(t)=\int_{0}^{\infty}(2\pi v)^{-\frac{1}{2}}\exp(-x^{2}/2v)dG(v)\;{\rm
where}\;g(v)=u^{-\frac{3}{2}}f(u^{-1})$

See also Andrews and Mallows (1974); West (1987); Carlin and Polson (1991);
Carlin et al. (1992).

Given $p(x)$, the mixing measure can then be obtained via an inversion of a
Laplace transform.

$p_{X}(x)=\int_{0}^{\infty}\frac{1}{(2\pi v)^{1/2}}e^{-x^{2}/2v}dF(v),$
$\varphi(t)=E(e^{itX})=\int_{0}^{\infty}e^{-vt^{2}/2}dF(v).$

Scale mixtures. $\hat{p}(t):=\hat{p}(0)\varphi(t)$, so $2\pi
p(0)\hat{p}(0)=1$.

Gneiting (1997); Good (1995) show that if $p$ and $\hat{p}$ are normal scale
mixtures,

$\sigma_{p}\sigma_{\hat{p}}\geq 1\Leftrightarrow p,\hat{p}\text{ are normal}$

Can prove these identities by flopping between Fourier and Laplace transforms.

Some functions are invariant under this transform. For example, the
characteristic function of normal is again normal. We call the two densities
dual if if each is proportional to the characteristic function of the other.
The normal is its own dual. The pair: Cauchy-Laplace is another example.

$\dfrac{1}{2}e^{-|x|}=\int_{-\infty}^{\infty}e^{itx}\dfrac{1}{\pi}\dfrac{1}{1+x^{2}}dx.$

This is a special case of the t-distribution being dual with the Bessel
function Wishart and Bartlett (1932); Good (1995).

A function $f(x)$ is completely monotone if and only if it can be represented
as a Laplace transform of some distribution function $F(s)$ as

$f(x)=\int_{0}^{\infty}e^{-sx}dF(s)$

The exponential power is a Gaussian mixture for $\alpha\in(0,2]$ given by

$\exp(-|t|^{\alpha})=\int_{0}^{\infty}e^{-st^{2}/2}f(s)ds$

where $f(s)$ can be identified as a positive $\alpha$-stable r.v. with index
$\alpha/2$. when $\alpha=2$ we get the Cauchy/Laplace dual density pair.

Normal Scale Mixtures can be derived using the link between completely
monotone functions. Here $p(\sqrt{x})$ is completely monotone if
$(-1)^{k}\frac{d^{k}}{dx}p(\sqrt{x})\geq 0\;\forall k=1,2,3,\ldots$.
Bernstein’s theorem states that $p(x)$ is completely monotonic if and only if
there is a unique measure $G$ on $[0,\infty)$ such that
$p(x)=\int_{0}^{\infty}e^{-x\lambda}dG(\lambda)$. and Bernstein functions
which include the class of scale mixtures of normals. We will see that there
are two representations

$p({\bf
x})=\exp\left(-\phi(x)\right)=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\sqrt{\lambda}\exp\left(-\lambda
x^{2}\right)dG(\lambda)\iff\;\;p(\sqrt{x})\;{\rm completely\;monotone}$

Bernstein-Widder-Schoenberg theorem applied to
$p(x)=\exp\left(-\phi(x)\right)$.

Negative convolution arise when $p(\sqrt{x})$ is not completely monotone.

#### Wigner Distribution.

Heisenberg (1931) uncertainty principle asserts a limit to the precision with
which position $x$ and momentum $p$ of a particle can be known simultaneously,
namely

$\sigma_{x}\sigma_{p}\leq h/2$

where $h$ is Planck’s constant. Wigner (1932), exhibited a joint distribution
function $f(x,p)$ for position and momentum however some of its values have to
be negative and he asserts that “this cannot really be interpreted as the
simultaneous probability for coordinates and momentum” but can be used in
calculations as an auxiliary mixture measure. For a unit vector, $\psi$, the
Wigner distribution is defined as

$f_{\psi}(x,p)=\frac{1}{2\pi}\int\psi\left(x+s\frac{h}{2}\right)\psi^{\star}\left(x-s\frac{h}{2}\right)e^{isp}ds.$

For a recent discussion on the Wigner distribution see Gurevich and Vovk
(2020), Wigner’s quasi-probability distribution, which can be used to make
predictions about quantum systems.

#### Feynman.

Feynman provides a number of examples of physical system that involve negative
probabilities. A particle diffusing in $1$-dim in a rod has probability
$P(x,t)$ of being at $x$ at time $t$ and satisfies

$\frac{\partial}{\partial t}P(x,t)=-\frac{\partial^{2}}{\partial x^{2}}P(x,t)$

Suppose that at $x=0$ and $x=\pi$ the rod has absorbers so that $P(x,t)=0$ and
let $P(x,o)=f(x)$. What is $P(x,t)$ thereafter? The solution is given by

$P(x,t)=\sum_{n-=1}^{\infty}p_{n}\sin(nx)e^{-n^{2}t}$

where

$f(x)=\sum_{n=1}^{\infty}p_{n}\sin(nx)\;\;{\rm and}\;\;p_{n}=\frac{2}{\pi}\int
f(x)\sin(nx)dx$

A mixture with negative weights. Linnik distribution will provide an example
of this.

#### Linnik Distribution.

If we start with the Laplace transform identity for a Cauchy random variables

$\frac{1}{1+x^{2}}=\int_{0}^{\infty}e^{-tx}t^{-{\scriptstyle\frac{1}{2}}}\sin(t)dt,$

and allow $x\rightarrow{\scriptstyle\frac{1}{2}}x^{2}$, under the
transformation, this becomes a scale mixture of normals representation for the
Linnik distribution

$\frac{1}{1+x^{4}}=\int_{0}^{\infty}\sqrt{t}e^{-{\scriptstyle\frac{1}{2}}tx^{2}}t^{-1}\sin(t/2)dt$

If $P(x)\propto\frac{1}{1+x^{4}}$,
$h(\sigma)\propto\sigma^{-2}\sin(\sigma^{-2})$, we have

$P(x)=\int_{0}^{\infty}\sigma^{-1}\varphi(\sigma^{-1}x)h(\sigma)d\sigma.$

This result follows from the fact

$\int_{0}^{\infty}\frac{1}{4+x^{4}}=\dfrac{\pi}{8},$

which can be calculated using identity

$\frac{1}{1+x^{4}}=\frac{1}{(x^{2}-2x+2)(x^{2}+2x+2)}.$

Similarly, when we have scale mixtures of Gaussians (Chu, 1973; West, 1987;
Carlin et al., 1992),

$\int_{0}^{\infty}N(0,t^{-1}C)W(t)dt=\frac{1}{1+(x^{T}C^{-1}x)^{2}}$

where the weights $W(t)=t^{-n/2}\sin(\frac{t}{2})$ can be negative,
$x^{T}=(x_{1},x_{2},...,x_{n})$.

Let $n=2$, $C=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\ \end{array}\right),$ and
we have the representation.

The Linnik family for $0<\alpha\leq 2$ has a scale mixture of normals is given
by

$\varphi(t)=E(e^{itX})=\frac{1}{1+|t|^{\alpha}},\qquad\alpha\in(0,2].$

The mixing measure is given by

$\int_{0}^{\infty}e^{-v\beta}\frac{e^{-|t|^{\alpha}v\beta}}{\Gamma(1+1/\beta)}dv=\frac{1}{(1+|t|^{\alpha})^{1/\beta}}.$

$\alpha=4$, $\beta=1$. mixture representation with positive probability.
Linnik distribution can have negative mixing weights (Devroye, 1990; Chu,
1973; West, 1987).

#### Stable Distribution.

The general positive stable and exponential power are related via

$\displaystyle P(y)=ke^{-|y|^{a}}$ $\displaystyle=$ $\displaystyle
k\int_{0}^{\infty}e^{iyt}f(t)dt\qquad\left(f(t)=\int_{0}^{\infty}v^{-1}\varphi(v^{-1}t)g(v)dv\right)$
$\displaystyle=$ $\displaystyle
k\int_{0}^{\infty}u^{-1}g(u^{-1})u^{-1}\varphi(u^{-1}y)du$

where $u^{-1}g(u^{-1})$ is the new mixing measure. See also West (1987) for
full details.

##  4 Discussion

Negative probabilities correspond to extraordinary random variables. They
arise in many physical systems and quantum computing (Polson et al., 2023).
They are related to the notion of dual densities and representing densities s
characteristic functions rather than Laplace transforms (a.k.a. mixtures of
exponentials). We provide a number of examples, including Linnik family of
distributions. There are many avenues for future research, in particular
developing optimization algorithms. In particular, EM and MM algorithms that
apply Laplace transform.

Our goal is twofold. First, to show that Feynman’s approach is consistent with
that of Ramsey (1926), Lindley (1985), and Shimony (1955) who use betting
rates to derive the axioms of probability such as Bayes rule. Second, to take
definition by Bartlett (1945) of extraordinary random variables given by their
generating functions and to provide mixing distribution representations
involving negative probabilities.

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TESTING LORENTZ SYMMETRY USING HIGH ENERGY ASTROPHYSICS OBSERVATIONS

Floyd W. Stecker
NASA Goddard Space Flight Center
Greenbelt, MD 20771 U.S.A.

## Abstract

We discuss some of the tests of Lorentz symmetry made possible by
astrophysical observations of ultrahigh energy cosmic rays, $\gamma$-rays, and
neutrinos. These are among the most sensitive tests of Lorentz invariance
violation because they are the highest energy phenomena known to man.

###### Contents

  1. 1 Introduction
  2. 2 The Coleman–Glashow Formalism
  3. 3 LIV Modified Kinematics for QED
  4. 4 Ultrahigh Energy Cosmic Rays
    1. 4.1 Extragalactic Origin
    2. 4.2 The Greisen–Zatsepin–Kuz’min Effect
  5. 5 Modification of the GZK Effect
    1. 5.1 Energy Threshold
    2. 5.2 Detailed GZK Kinematics with LIV
  6. 6 Comparison of LIV-Modified UHECR Spectra with Observations
  7. 7 Modified GZK Neutrino Spectrum from a Modified UHECR Spectrum
  8. 8 Generalizing the Coleman–Glashow Formalism
    1. 8.1 Time of Flight from $\gamma$-ray Bursts
    2. 8.2 Gamma-Ray Absorption with Planck Suppressed LIV
  9. 9 Vacuum Birefringence
  10. 10 LIV in the Neutrino Sector
    1. 10.1 Fermion LIV Operators with $[d]>4$ LIV with Rotational Symmetry in SME
    2. 10.2 LIV in the Neutrino Sector I: Lepton Pair Emission
      1. 10.2.1 Lepton Pair Emission in the [_d_] = 4 Case
      2. 10.2.2 Vacuum $e^{+}e^{-}$ Pair Emission in the $[d]>4$ Cases
  11. 11 LIV in the Neutrino Sector II: Neutrino Splitting
  12. 12 The Neutrinos Observed by IceCube
  13. 13 Extragalactic Superluminal Neutrino Propagation
  14. 14 The Theoretical Neutrino Energy Spectrum
    1. 14.1 [d] = 4 $\cal{CPT}$ Conserving Operator Dominance
    2. 14.2 [d] = 6 $\cal{CPT}$ Conserving Operator Dominance
    3. 14.3 [d] = 5 CPT Violating Operator Dominance
  15. 15 Summary: Results for Superluminal Neutrinos
  16. 16 Stable Pions from LIV
  17. 17 Conclusions

##  1 Introduction

One of the main motivations for testing Lorentz symmetry comes from the search
for an answer to one of the most fundamental problems of modern physics: how
to reconcile general relativity with quantum physics. This problem is
particularly acute on the length scale where the effect of quantum
fluctuations on spacetime cannot be ignored. This scale, known as the Planck
length, is given by $\lambda_{Pl}=\sqrt{G\hbar/c^{3}}\sim 10^{-35}$ m [1]. It
corresponds to the very large energy scale, $E_{Pl}=M_{Pl}c^{2}=1.2\times
10^{19}$ GeV, where $M_{Pl}=\sqrt{\hbar c/G}$. (Hereafter, we use the
conventions for the low energy speed of light $c=1$ and $\hbar=1$.) The open
question of constructing a framework for a quantum theory of gravity is
intimately related to the question of Lorentz invariance violation [2, 3].

Lorentz symmetry implies scale-free spacetime. The group of Lorentz
transformations is unbounded in energy (boost) space. Owing to the uncertainty
principle, the higher the particle energy attained, the smaller the scale of
physics that can be probed. Very high energy interactions probe physics at
ultra-short distance intervals, $\lambda\propto 1/E$. Thus, to probe physics
at small distance intervals, particularly the nature of space and time, we
need to go to ultrahigh energies. Cosmic $\gamma$-rays and neutrinos and
cosmic-ray nucleons provide the highest observable energies in the universe.

This paper reviews the topic of testing the exactness of Lorentz symmetry
using astrophysical observations of ultrahigh energy cosmic rays,
$\gamma$-rays and neutrinos. (For a review of tests of Lorentz symmetry in the
gravitational sector, see [4].) Cosmic $\gamma$-rays have been observed at
energies up to $\cal{O}$$(10^{3})$ GeV. Neutrinos of astrophysical origin have
been observed at energies up to $\cal{O}$$(10^{6})$ GeV. Even ultrahigh energy
cosmic ray nuclei have been observed only up to $\cal{O}$$(10^{11})$ GeV
energy. These energies are much lower than $M_{Pl}$. However, the effects of
violations of Lorentz invariance (LIV) can be manifested at energies much
lower than $M_{Pl}$. (By Noether’s theorem, Lorentz symmetry implies Lorentz
invariance; here we use these terms interchangeably.)

There are many ways that LIV terms in the free particle Lagrangian can affect
physics at astrophysically-accessible energies. In particular, the effect of
LIV can change the kinematics of particle interactions and decays and also the
threshold energies for these processes [5, 6]. LIV can also modify neutrino
oscillations [5, 7].

##  2 The Coleman–Glashow Formalism

As an initial framework for exploring the effects of LIV, we start with the
phenomenological approach of Coleman and Glashow [5]. This formalism has the
advantage of (1) simplicity, (2) preserving the $SU(3)\otimes SU(2)\otimes
U(1)$ standard model of strong and electroweak interactions, (3) having the
perturbative term in the Lagrangian consist of operators of mass dimension
four that thus preserves power counting renormalizability and (4) being
rotationally invariant in a preferred frame that can be taken to be the rest
frame of the CBR (Cosmic background radiation). Coleman and Glashow start with
a standard-model free-particle Lagrangian,

${\cal L}=\partial_{\mu}\Psi^{*}{\bf Z}\partial^{\mu}\Psi-\Psi^{*}{\bf
M}^{2}\Psi$ (1)

where the matrices are in boldface, $\Psi$ is a column vector of $n$ fields
with U(1) invariance, and the positive Hermitian matrices ${\bf Z}$, and M2
can be transformed so that ${\bf Z}$ is the identity and M2 is diagonalized to
produce the standard theory of $n$ decoupled free fields. (We adopt the
standard notation of denoting four-vector indexes by Greek letters and three-
vector spatial indexes by Latin letters).

They then add a leading order perturbative, Lorentz violating term constructed
from only spatial derivatives with rotational symmetry so that:

${{\cal L}\rightarrow{\cal L}+\partial_{i}\Psi\eta\partial^{i}\Psi},$ (2)

where $\eta$ is a small dimensionless Hermitian matrix that commutes with M2
so that the fields remain separable and the resulting single particle energy-
momentum eigenstates go from eigenstates of M2 at low energy to eigenstates of
$\eta$ at high energies.

To leading order, this term shifts the poles of the propagator, resulting in
the free particle dispersion relation:

$E^{2}=\vec{p}\ ^{2}+m^{2}+\eta\vec{p}\ ^{2}.$ (3)

This can be put in the standard form for the dispersion relation:

$E^{2}=\vec{p\ }{{}^{2}}c_{MAV}^{2}+m^{2}c_{MAV}^{4},$ (4)

by shifting the renormalized mass by the small amount $m\rightarrow
m/(1+\eta)$ and shifting the velocity from $c\ (=1)$ by the amount
$c_{MAV}=\sqrt{(1+\eta)}\simeq 1+\eta/2$. The group velocity is given by:

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

which goes to $c_{MAV}$ in the limit of large $|\vec{p}|$. Thus, Coleman and
Glashow identify $c_{MAV}$ to be the maximum attainable velocity of the free
particle. Using this formalism, it becomes apparent that, in principle,
different particles can have different maximum attainable velocities (MAVs),
which can be different from $c$. Hereafter, we denote the MAV of a particle of
type $i$ by $c_{i}$ and the difference:

$c_{i}-c_{j}~{}=~{}{{\eta_{i}-\eta_{j}}\over{2}}~{}\equiv~{}\delta_{ij}.$ (6)

##  3 LIV Modified Kinematics for QED

Using the formalism of [5], we denote maximum attainable velocity (MAV) of a
particle of type $i$ by $c_{i}$. We further define the difference
$c_{i}-c_{j}\equiv\delta_{ij}$, and specifically, here,
$c_{e\gamma}\equiv\delta\ll c$ where $c\equiv 1$ is the low energy photon
velocity. These definitions will be used to discuss the physics implications
of cosmic ray and cosmic $\gamma$-ray observations [6].

If $\delta<0$, the decay of a photon into an electron-positron pair is
kinematically allowed for photons with energies exceeding $E_{\rm
max}=m_{e}\,\sqrt{2/|\delta|}$. This decay would take place rapidly, so that
photons with energies exceeding $E_{\rm max}$ could not be observed either in
the laboratory or as cosmic rays. Since photons have been observed with
energies $E_{\gamma}\geq$ 50 TeV from the Crab Nebula [8], this implies that
$E_{\rm max}\geq 50\;$TeV, or that $|\delta|<2\times 10^{-16}$.

If, on the other hand, $\delta>0$, electrons become superluminal if their
energies exceed $E_{\rm max}/\sqrt{2}$. Electrons traveling faster than light
will emit light at all frequencies by a process of ‘vacuum Čerenkov
radiation’. The electrons then would rapidly lose energy until they become
subluminal. Because electrons have been seen in the cosmic radiation with
energies up to $\sim\,$2 TeV [9], it follows that $\delta<3\times 10^{-14}$. A
smaller, but more indirect, upper limit on $\delta$ for the $\delta>0$ case
can be obtained from theoretical considerations of $\gamma$-ray emission from
the Crab Nebula. Its emission above 0.1 GeV, extending into the TeV range, is
thought to be Compton emission of the same relativistic electrons that produce
its synchrotron radiation at lower energies [10]. The Compton component,
extends to 50 TeV and thus implies the existence of electrons having energies
at least this great in order to produce 50 TeV photons, even in the extreme
Klein-Nishina limit. This indirect argument, based on the reasonable
assumption that the 50-TeV $\gamma$-rays are from Compton interactions, leads
to a smaller upper limit on $\delta$, viz., $\delta<10^{-16}$. A further
constraint on $\delta$ for $\delta>0$ follows from a change in the threshold
energy for the pair production process $\gamma+\gamma\rightarrow e^{+}+e^{-}$.
This follows from the fact that the square of the four-momentum is changed to
give the threshold condition:

$2\epsilon
E_{\gamma}(1-cos\theta)~{}-~{}2E_{\gamma}^{2}\delta~{}\geq~{}4m_{e}^{2},$ (7)

where $\epsilon$ is the energy of the low energy photon and $\theta$ is the
angle between the two photons. The second term on the left-hand side comes
from the fact that $c_{\gamma}=\partial E_{\gamma}/\partial p_{\gamma}$. It
follows that the condition for a significant increase in the energy threshold
for pair production is $E_{\gamma}\delta/2$ $\geq$ $m_{e}^{2}/E_{\gamma}$, or
equivalently,

$\delta\geq{2m_{e}^{2}/E_{\gamma}^{2}}.$ (8)

The $\gamma$-ray spectrum of the active galaxy Mkn 501 while flaring extended
to $E_{\gamma}\geq 24$ TeV [10] and exhibited the high energy absorption
expected from $\gamma$-ray annihilation by extragalactic pair-production
interactions with extragalactic infrared photons [11, 12, 13]. This has led
Stecker and Glashow [6] to point out that the Mkn 501 spectrum presents
evidence for pair-production with no indication of Lorentz invariance
violation (LIV) up to a photon energy of $\sim$20 TeV and to thereby place a
quantitative constraint on LIV given by
$\delta<2m_{e}^{2}/E_{\gamma}^{2}\simeq 10^{-15}$. This constraint on positive
$\delta$ is more secure than the smaller, but indirect, limit given above. See
[14] for further considerations regarding the modification of $\gamma$-ray
spectra by LIV.

##  4 Ultrahigh Energy Cosmic Rays

###  4.1 Extragalactic Origin

Ultrahigh energy cosmic rays (UHECRs) appear to have an isotropic distribution
in arrival direction. Because of their ultrahigh energy, their propagation is
little affected by galactic magnetic fields. Thus, owing to their observed
isotropy, cosmic rays above 10 EeV ($10^{19}$ eV) are believed to be of
extragalactic origin.

###  4.2 The Greisen–Zatsepin–Kuz’min Effect

Shortly after the discovery of the 2.7 K CBR, Greisen [15] and Zatsepin and
Kuz’min [16] predicted that pion-producing interactions of UHECR protons with
the microwave photons of the CBR should produce a spectral cutoff at $E\sim$
50 EeV. The flux of UHECRs is thus expected to be attenuated by such meson-
producing interactions. This effect is generally known as the “GZK
(Greisen–Zatsepin–Kuz’min) effect”. Owing to these interactions, extragalactic
protons with energies above $\sim$100 EeV should be attenuated while
propagating from distances beyond $\sim$$100$ Mpc [17]. The cross-section for
this process peaks at the $\Delta$ baryon resonance,

$p+\gamma\rightarrow\Delta\rightarrow N\pi.$ (9)

The energy threshold for photomeson interactions of UHECR protons of initial
laboratory energy $E$ with low energy photons of the 2.7 blackbody CBR having
laboratory energy $\epsilon$ 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:

$4\epsilon E=m(2M+m)$ (10)

for the proton, and:

$4\epsilon E_{\pi}={{m^{2}(2M+m)}\over{M+m}}$ (11)

for the pion, where $M$ is the rest mass of the proton and $m$ is the rest
mass of the pion [17].

##  5 Modification of the GZK Effect

We have previously discussed the observations of UHECRs and their expected
attenuation over extragalactic distances owing to the GZK effect. We now turn
our attention to how the GZK effect can be modified by a very small violation
of Lorentz symmetry.

###  5.1 Energy Threshold

If Lorentz symmetry is slightly broken so that $c_{\pi}~{}>~{}c_{p}$, it
follows from Equations (6), (11) and (22) and that the threshold energy for
photomeson is altered 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:

$4\epsilon E_{\pi}={{m^{2}(2M+m)}\over{M+m}}+2\delta_{\pi p}E_{\pi}^{2}$ (12)

Equation (12) is a quadratic equation with real roots only under the
condition:

$\delta_{\pi
p}\leq{{2\epsilon^{2}(M+m)}\over{m^{2}(2M+m)}}\simeq\epsilon^{2}/m^{2}.$ (13)

Defining $\epsilon_{0}\equiv kT_{CBR}=2.35\times 10^{-4}$ eV with
$T_{CBR}=2.725\pm 0.02$ K, Equation (13) can be rewritten:

$\delta_{\pi p}\leq 3.23\times 10^{-24}(\epsilon/\epsilon_{0})^{2}.$ (14)

Equation (14) implies that if Lorentz symmetry is violated and for
$\delta_{\pi p}>0$, photomeson interactions with CBR photons will not occur
for $\delta_{\pi p}\geq\sim 3\times 10^{-24}$. This condition, together with
Equation (12), 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 CBR blackbody spectrum, other
interactions involving higher energy protons and photons with smaller values
of $\eta$ 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.

###  5.2 Detailed GZK Kinematics with LIV

We have seen that kinematical relations governing photomeson interactions are
changed in the presence of even a small violation of Lorentz invariance. We
now consider this in more detail [18].

Following Equations (6) and (22), we denote:

$E^{2}=p^{2}+2\delta_{a}p^{2}+{m_{a}}^{2}$ (15)

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

In the cms (center-of-momentum system), the energy of particle $a$ is then
given by:

$\sqrt{s_{a}}=\sqrt{E^{2}-p^{2}}=\sqrt{2\delta_{a}p^{2}+m_{a}^{2}}\ \geq\ 0.$
(16)

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

$v={{\partial E}\over{\partial p}}\neq{{E}\over{p}}.$ (17)

The modified kinematical relations containing LIV have a strong effect on the
amount of energy transferred from an incoming proton to the pion produced in
the subsequent interaction, i.e., the inelasticity [18, 19]. The total
inelasticity, $K$, is an average of $K_{\theta}$, which depends on the angle
between the proton and photon momenta, $\theta$:

$K=\frac{1}{\pi}\int\limits_{0}^{\pi}{K_{\theta}d\theta}.$ (18)

The primary effect of LIV on photopion production is a reduction of phase
space allowed for the interaction. This results from the limits on the allowed
range of interaction angles integrated over in order to obtain the total
inelasticity from Equation (18). For real-root solutions for interactions
involving higher energy protons, the range of kinematically allowed angles in
Equation (18) becomes severely restricted. The modified inelasticity that
results is the key in determining the effects of LIV on photopion production.
The inelasticity rapidly drops for higher incident proton energies.

Figure 1 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 [18]. 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 1: 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
[18].

With this modified inelasticity, the proton energy loss rate by photomeson
production is given by:

${{1}\over{E}}{{dE}\over{dt}}=-{{\epsilon_{0}c}\over{2\pi^{2}\gamma^{2}}\hbar^{3}c^{3}}\int\limits_{\epsilon_{th}}^{\infty}d\epsilon^{\prime}~{}\epsilon^{\prime}~{}\sigma(\epsilon^{\prime})K(\epsilon^{\prime})\ln[1-e^{-\epsilon^{\prime}/2\gamma\epsilon_{0}^{\prime}}]$
(19)

where $\epsilon_{th}$ is the photon threshold energy for the interaction and
$\sigma(\epsilon^{\prime})$ is the total $\gamma$-p cross-section with
contributions from direct pion production, multipion production and the
$\Delta$ resonance.

The corresponding proton attenuation length is given by $\ell=cE/r(E)$, where
the energy loss rate $r(E)\equiv(dE/dt)$. This attenuation length is plotted
in Figure 2 for various values of $\delta_{\pi p}$ along with the unmodified
pair production attenuation length from pair production interactions,
$p+\gamma_{CBR}\rightarrow e^{+}+e^{-}$.

Figure 2: The calculated proton attenuation lengths as a function of proton
energy modified by LIV for various values of $\delta_{\pi p}$ (solid lines),
shown with the attenuation length for pair production unmodified by LIV
(dashed lines). From top to bottom, the curves are for $\delta_{\pi p}=1\times
10^{-22}$, $3~{}\times~{}10^{-23}$, $2~{}\times~{}10^{-23}$,
$1~{}\times~{}10^{-23}$, $3\times 10^{-24}$, 0 (no Lorentz violation) [18].

##  6 Comparison of LIV-Modified UHECR Spectra with Observations

An analytic calculation of LIV-modified UHECR proton spectra was obtained
using Equation (19) [18] (see also [19]). In order to take account of the
probable redshift evolution of UHECR production in astronomical sources, one
must take account of the following considerations [18]:

1\. The CBR photon number density increases as $(1+z)^{3}$, and the CBR photon
energies increase linearly with $(1+z)$. The corresponding energy loss for
protons at any redshift $z$ is thus given by:

$\displaystyle r_{\gamma p}(E,z)=(1+z)^{3}r[(1+z)E].$ (20)

2\. It is assumed that the average UHECR volume emissivity is of the form
given by $q(E_{i},z)=K(z)E_{i}^{-\Gamma}$ where $E_{i}$ is the initial energy
of the proton at the source and $\Gamma=2.55$. The source evolution is assumed
to be $K(z)\propto(1+z)^{3.6}$ with $z\leq 2.5$ so that $K(z)$ is roughly
proportional to the empirically-determined $z$-dependence of the star
formation rate. $K(z=0)$ and $\gamma$ are normalized to fit the data below the
GZK threshold.

The curves calculated in [18] assuming various values of $\delta_{\pi p}$ are
shown in Figure 3 along with the Auger data from [20]. They show that even a
very small amount of LIV that is consistent with both a GZK effect and with
the present UHECR data can lead to a “recovery” of the UHECR spectrum at
higher energies.

Figure 3: Comparison of Auger data [20] with calculated spectra for various
values of $\delta_{\pi p}$, taking $\delta_{p}=0$ (see the 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)
[18].

##  7 Modified GZK Neutrino Spectrum from a Modified UHECR Spectrum

Subsequent to the photopion interactions of Equation (9), the charged pions
that are produced decay to produce neutrinos. It follows from the modified
kinematics and inelasticity for photopion interactions as presented in Section
5 (see Figure 1) that there will be a modification of the spectrum of the “GZK
neutrinos” that result.

Figure 4 shows the corresponding total neutrino flux (all species) for the
same choices of $\delta_{\pi p}$ as the UHECR spectra presented in figure 3.
As expected, increasing $\delta_{\pi p}$ leads to a decreased flux of higher
energy photomeson neutrinos because the interactions involving higher energy
UHECRs are suppressed [21]. It is also evident that the peak energy of the
neutrino energy flux spectrum (EFS), $E\Phi(E)$, shifts to lower energies with
increasing $\delta_{\pi p}$.

Figure 4: Neutrino fluxes (of all species) corresponding to the UHECR models
considered in Figure 3. 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 [21].

This possible LIV effect on GZK neutrinos may be tested using radio arrays
built on Antarctic ice that are based on the Askaryan effect such as the
Askaryan Radio Array (ARA) [22] and the Antarctic Ross Ice-Shelf Antenna
Neutrino Array (ARIANNA) [23]. GZK effect neutrinos can also be observed by
space-based telescopes looking down at the atmosphere to study the horizontal
air showers they produce [24, 25].

##  8 Generalizing the Coleman–Glashow Formalism

The Coleman–Glashow formalism preserves a renormalizable Lagrangian structure,
with the Lagrangian density being of mass-dimension $[d]=4$. More generally,
it is possible to construct an extended standard model (SME) Lagrangian as an
effective field theory formalism that is testable and that holds at energies
$E\ll M_{Pl}$ [2]. This formalism admits LIV terms composed of operators of
dimension $[d]>4$ suppressed by values of $M_{Pl}^{(n-4)}$. It is motivated by
two ideas: (1) any LIV extant at accelerator energies must be extremely small
and (2) physics must be strongly modified at the Planck energy, $M_{Pl}$.
These considerations are reflected by adding small Lorentz-violating terms in
the free particle Lagrangian that are suppressed by powers of some quantum
gravity energy scale $E_{QG}$ $\sim$ ${\cal{O}}(M_{Pl})$. Sums of such terms
are also possible and, as such, admit energy-dependent values of $\delta$,

$\delta=\sum_{n=0,1,2}\lambda_{n}\left(\frac{E}{M_{Pl}}\right)^{n}.$ (21)

where we will consider only terms up to _n_ = 2. In the SME formalism, the _n_
= 1 term corresponds to the lowest order $\cal{CPT}$-odd operator, and the _n_
= 2 term corresponds to the lowest order $\cal{CPT}$-even operator. We now
consider two applications of this formalism.

###  8.1 Time of Flight from $\gamma$-ray Bursts

Energy-dependent MAVs constructed by assuming that LIV is related to physics
at the Planck scale have been used in time-of-flight observations of photon
arrival times from cosmologically-distant $\gamma$-ray bursts. For this
analysis, LIV modifications to the photon dispersion relation were taken to be
the sum of a series of terms in $E/E_{QG}$ [26],

$E^{2}\simeq
p^{2}\times\left[1-\overset{n=2}{\underset{n=1}{\sum}}\mathit{s_{\pm}}\left(\frac{E}{E_{QG}}\right)^{n}\right],$
(22)

where it was expected that $E_{QG}$ = ${\cal{O}}$$(M_{Pl})$. In Equation (22),
$\mathit{s_{\pm}}$ is the “sign of LIV”, taken equal to $+$1 ($-$1) for a
decrease (increase) in photon MAV with an increasing photon energy. In case
the $n=1$ term is suppressed, owing to conservation of $\cal{CPT}$, the $n=2$
is assumed to dominate.

Adopting the notation of [27], we define an“LIV parameter” $\tau_{n}$ as the
ratio of the time delay between photons of the highest observed energy,
$E_{h}$, and the lowest observed energy, $E_{l}$, and the quantity
$(E_{h}^{n}$ $-$ $E_{l}^{n})$. Then:

$\tau_{n}\equiv\frac{\Delta
t}{E_{h}^{n}-E_{l}^{n}}\simeq\mathit{s_{\pm}}\frac{(1+n)}{2H_{0}}\frac{1}{E_{QG}^{n}}\times\kappa_{n},$
(23)

where:

$\kappa_{n}\equiv\int\limits_{0}^{z}\frac{(1+z^{\prime})^{n}}{\sqrt{\Omega_{\Lambda}+\Omega_{\rm
M}(1+z^{\prime})^{3}}}dz^{\prime}$ (24)

where $\kappa_{n}$ is a cosmological distance parameter. (Values of $\Omega$
are conventionally defined as fractions of the critical density needed to
close the universe, $3H_{0}/8\pi G$.) Here, as usual, $z$ is redshift; $H_{0}$
is the Hubble constant; and values of $\Omega_{\lambda}$ and $\Omega_{m}$ are
designated for the dark energy density and total matter density, respectively.
The Hubble constant is taken to be $H_{0}=$ 67.8 km s-1 Mpc-1;
$\Omega_{\Lambda}$ = 0.7; and $\Omega_{\rm M}$ = 0.3.

Three different statistical analyses on Fermi observations of three
$\gamma$-ray bursts have been used to derive strong constraints on $\tau_{n}$
[27]. The most stringent limits for $s=+1$ at the 95% confidence level are
obtained from the $\gamma$-ray burst GRB 090510 and are $E_{QG}>7.6M_{Pl}$ and
$E_{QG}>1.3\times 10^{11}$ GeV for linear and quadratic leading order LIV-
induced vacuum dispersion, respectively.

An early suggestion for testing quantum gravity [28] employed the natural idea
of identifying $E_{QG}\equiv M_{Pl}$. A later “D-brane” model, inspired by
string theory concepts, suggested the prospect that only photons would exhibit
an energy-dependent velocity [29]. That model envisions a universe filled with
a gas of point-like D-branes that only interact with photons. It predicts that
vacuum has an energy-dependent index of refraction that causes only a
retardation, again with $E_{QG}\simeq M_{Pl}$. Such models are disfavored by
the analysis of the Fermi observations that, on its face, appears to suggest
$E_{QG}>7.6M_{Pl}$.

It is, of course, unphysical to invoke the interpretation $E_{QG}>M_{Pl}$, as
implied in the $n=1$ case. However, if we write the first-order dispersion
relation for the group velocity of a photon with energy $E$ as:

$c_{\gamma}(E)=\frac{\partial E}{\partial
p}\simeq\left[1-\lambda_{1}{{E}\over{M_{Pl}}}\right],$ (25)

in analogy with Equations (5) and (21), we find the limit $\lambda_{1}<0.13$,
in contrast with the prediction $\lambda_{1}\simeq 1$.

###  8.2 Gamma-Ray Absorption with Planck Suppressed LIV

Motivated by the time-of-flight result above, we drop the condition from
Equation (22) that $\lambda=\pm 1$ and consider $\lambda_{n}\neq 1$. Using
Equation (21), the energy threshold for relation (8) is modified slightly in
the Planck-suppressed LIV case. If we again take the condition for LIV to
significantly change the threshold energy for absorption as given by the
relation (8), with $\delta$ given by Equation (21), the criterion for an
absorption feature in the $\gamma$-ray energy spectrum of a source to be
affected by an LIV term with Planck suppression becomes:

$E_{\gamma}>\left({{2m_{e}^{2}M_{Pl}^{n}}\over{{\lambda_{n}}}}\right)^{1\over{n+2}}$
(26)

where $n$ is the lowest value for the LIV term that dominates in the series
(21). The $n=1$ case was first treated in [30]. This topic has been considered
further in [14].

##  9 Vacuum Birefringence

Important fundamental constraints on LIV come from searches for the vacuum
birefringence effect predicted within the framework of the effective field
theory (EFT) analysis of [31] (see also [32, 33])111Vacuum birefringence from
LIV differs from the identically-labeled effect that can occur in a strong
magnetic field from interactions with virtual electrons, causing the index of
refraction in vacuo to be different for the two polarization modes, depending
on their orientation to the magnetic field.. Within this framework, applying
the Bianchi identities to the leading order Maxwell equations in vacuo, a
$\cal{CPT}$-odd mass dimension five operator term is derived of the form:

${{\cal{L}}_{\gamma}={{\xi}\over{M_{Pl}}}{n^{a}F_{ad}n\cdot\partial(n_{b}{F}^{bd})}}$
(27)

It is shown in [31] that the expression given in Equation (27) is the only
dimension five modification of the free photon Lagrangian that preserves both
rotational symmetry and gauge invariance. This leads to a modification in the
dispersion relation proportional to $\xi(\omega/M_{Pl})=\xi(E/M_{Pl})$, with
the new dispersion relation given by:

$\omega^{2}~{}=~{}k^{2}\pm\xi\,k^{3}/M_{Pl}.$ (28)

with photons of opposite helicity having slightly different velocities. This
leads to a rotation in the angle of polarization,

$\theta(t)=\left[\omega_{+}(k)-\omega_{-}(k)\right]t_{P}/2~{}\simeq~{}\xi
k^{2}t_{P}/2M_{Pl}$ (29)

for a plane wave with wave-vector $k$, where $\xi k/M_{Pl}\ll 1$ and where
$t_{P}$ is the propagation time.

Observations of polarized radiation from distant sources can thus be used to
place an upper bound on $\xi$. The vacuum birefringence constraint arises from
the fact that if the angle of polarization rotation (29) were to differ by
more than $\pi/2$ over the energy range covered by the observation, the
instantaneous polarization at the detector would fluctuate sufficiently for
the net polarization of the signal to be suppressed well below any observed
value. The difference in rotation angles for wave-vectors $k_{1}$ and $k_{2}$
is:

$\Delta\theta=\xi(k_{2}^{2}-k_{1}^{2})L_{P}/2M_{Pl},$ (30)

where we have replaced the propagation time $t_{P}$ by the propagation
distance $L_{P}$ from the source to the detector.

If polarization is detected from a source at redshift $z$, this yields the
constraint:

$|\xi|<{{\pi
M_{Pl}}\over{\int\displaylimits_{0}^{z}dz^{\prime}{[k_{2}(z^{\prime})^{2}-k_{1}(z^{\prime})^{2}]|dL_{P}(z^{\prime})/dz^{\prime}|}}}$
(31)

where $k_{1,2}(z^{\prime})=(1+z^{\prime})\cdot k_{1,2}(z^{\prime}=0)$ and:

$\Bigl{|}{dL_{P}\over{dz^{\prime}}}\Bigr{|}={{c}\over{H_{0}}}{{1}\over{(1+z^{\prime})\sqrt{\Omega_{\Lambda}+(1+z^{\prime})^{3}\Omega_{m}}}}.$
(32)

Present observations of hard X-rays from Mpc-distant $\gamma$-ray bursts put
limits on $\xi$ of order $10^{-15}$–$10^{-16}$, e.g., [34, 35].

##  10 LIV in the Neutrino Sector

There are many ways that LIV terms in the free particle Lagrangian can affect
neutrino physics. We will here consider only one of these consequences, viz.,
the effect of resulting changes in the kinematics of particle interactions. (A
discussion of the effect of LIV on modifying neutrino oscillations may be
found in [7].) These changes can modify the threshold energies for particle
interactions, allowing or forbidding such interactions [5, 6]. Observational
evidence for this effect can be searched for by examining the energy spectrum
of high energy cosmic neutrinos obtained by the IceCube collaboration. We base
our discussion of this evidence on the well delineated framework of the
standard model extension (SME) formalism of effective field theory (EFT) [32],
as given in more detail in [36].

###  10.1 Fermion LIV Operators with $[d]>4$ LIV with Rotational Symmetry in
SME

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 [37]. Therefore, we only consider Dirac
neutrinos.

Using Equation (22), one can define an effective mass, $\tilde{m}(E)$, that is
a useful parameter for analyzing LIV-modified kinematics. The effective mass
is constructed to include the effect of the LIV terms. We define an effective
mass $\tilde{m}_{I}(E)$ for a particle for type $I$ using the dispersion
relation (22) as [5, 36]:

$\tilde{m}_{I}^{2}(E)=m_{I}^{2}+2\delta_{I}E_{I}^{2},$ (33)

where the velocity parameters $\delta_{I}$ are now energy-dependent
dimensionless coefficients for each species, $I$, that are contained in the
Lagrangian. Furthermore, we define the parameter
$\delta_{IJ}\equiv\delta_{I}-\delta_{J}$ as the Lorentz-violating difference
between the MAVs of particles $I$ and $J$. In general, $\delta_{IJ}$ will
therefore be of the form given by Equation (21).

If we wish to assume the dominance of Planck-suppressed terms in the
Lagrangian as tracers of Planck-scale physics, it follows that that
$\lambda_{\nu e,0}\ll\lambda_{\nu e,1},\lambda_{\nu e,2}$. (Several mechanisms
have been proposed for the suppression of the LIV $[d]=4$ term in the
Lagrangian. See, e.g., the review in Reference [38].) Alternatively, we may
postulate the existence of only Planck-suppressed terms in the Lagrangian,
i.e., $\lambda_{\nu e,0}=0$. We can further simplify by noting the important
connection between LIV and $\cal{CPT}$ violation. Whereas a local interacting
theory that violates $\cal{CPT}$ invariance will also violate Lorentz
invariance [39], the converse does not follow; an interacting theory that
violates Lorentz invariance may, or may not, violate $\cal{CPT}$ invariance.
LIV terms of odd mass dimension $[d]=4+n$ are $\cal{CPT}$-odd and violate
$\cal{CPT}$, whereas terms of even mass dimension are $\cal{CPT}$-even and do
not violate $\cal{CPT}$ [40].

We can then specify a dominant term for $\delta_{IJ}$ in Equation (21)
depending on our choice of $\cal{CPT}$. Considering Planck-mass suppression,
the dominant term that admits $\cal{CPT}$ violation is the $n=1$ term in
Equation (21). On the other hand, if we require $\cal{CPT}$ conservation, the
$n=2$ term in Equation (21) is the dominant term. Thus, we can choose as a
good approximation to Equation (21), a single dominant term with one
particular power of $n$ by specifying whether we are considering
$\cal{CPT}$-even or odd LIV. As a result, $\delta_{IJ}$ reduces to Equation
(21) with $n=1$ or $n=2$ depending on the status of $\cal{CPT}$. We note that
in the SME formalism, since $[d]$-odd LIV operators are $\cal{CPT}$-odd, the
$\cal{CPT}$-conjugation property implies that neutrinos can be superluminal,
while antineutrinos are subluminal or vice versa [37]. This will have
consequences in interpreting our results, as we will discuss later.

In Equations (21) and (22), we have not designated a helicity index on the
$\lambda$ coefficients. The fundamental parameters in the Lagrangian are
generally helicity dependent. 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 [41]. Thus, the contribution to
$\lambda_{\nu e,1}$ 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 [36].

###  10.2 LIV in the Neutrino Sector I: Lepton Pair Emission

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 [5, 42] and can
thus cause significant observational effects. An example of such an
interaction is $\nu_{e}$ “splitting”, i.e.,
$\nu_{e}\rightarrow\nu_{e}+\nu_{i}+\bar{\nu_{i}}$ where $i$ is a flavor index.
Neutrino splitting can be represented as a rotation of the Feynman diagram for
neutrino-neutrino scattering, which is allowed by relativity. However, absent
a violation of Lorentz invariance, neutrino splitting is forbidden by
conservation of energy and momentum. In the the case of LIV-allowed
superluminal neutrinos, the dominant pair emission reactions are neutrino
splitting and its close cousin, vacuum electron-positron pair emission (VPE)
$\nu_{i}\rightarrow\nu_{i}+e^{+}+e^{-}$, as these are the reactions with the
lightest final state masses. We now set up a simplified formalism to calculate
the possible observational effect of these two specific anomalous interactions
on the interpretation of the neutrino spectrum observed by the IceCube
collaboration.

####  10.2.1 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 processes are
very similar (see Figure 5).

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

${\Gamma\ =\gamma_{\mu}^{-1}}{{G_{F}^{2}m_{\mu}^{5}}\over{192\pi^{3}}}$ (34)

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 (33) 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
$\tilde{m_{\nu}}^{2}(E)\simeq 2\delta_{\nu e}E_{\nu}^{2}$. (At 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:

$m_{\mu}^{2}\ \rightarrow\ \tilde{m_{\nu}}^{2}(E)\simeq 2\delta_{\nu
e}E_{\nu}^{2}$ (35)

from which it follows that:

${\gamma_{\mu}^{2}}\ \rightarrow\ {{E_{\nu}^{2}}\over{2\delta_{\nu
e}E_{\nu}^{2}}}=(2\delta_{\nu e})^{-1}.$ (36)

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

$\Gamma\ \propto\ (2\delta_{\nu e})^{1/2}G_{F}^{2}(2\delta_{\nu
e}E_{\nu}^{2})^{5/2}$ (37)

which gives the proportionality:

$\Gamma\ \propto\ G_{F}^{2}\ \delta_{\nu e}^{3}E_{\nu}^{5}$ (38)

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

Figure 5: Diagrams for muon decay (top), charged current-mediated vacuum
electron-positron pair emission (VPE) (bottom left) and neutral current-
mediated neutrino splitting and VPE (bottom right). Time runs from left to
right, and the flavor index $i$ represents $e,\mu$, or $\tau$ neutrinos.

The energy threshold for $e^{+}e^{-}$ pair production is given by [6]:

$E_{th}=m_{e}\sqrt{{{2}\over{{\delta_{\nu e}}}}}$ (39)

with $\delta\equiv\delta_{\nu e}$ given by Equation (21), the rate for the VPE
process, $\nu\to\nu\,e^{+}\,e^{-}$, via the neutral current $Z$-exchange
channel, has been calculated to be [42]:

$\Gamma=\frac{1}{14}\frac{G_{F}^{2}(2\delta)^{3}E_{\nu}^{5}}{192\,\pi^{3}}=1.31\times
10^{-14}\delta^{3}E_{\rm GeV}^{5}\ \ {\rm GeV}.$ (40)

with the mean fractional energy loss per interaction from VPE of 78% [42].

In general, the charged current (CC) $W$-exchange channels 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 (39) 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.

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

Using Equations (21) and (38) and the dynamical matrix element taken from the
simplest case [43], we can generalize Equation (40) for arbitrary values of
$n=([d]-4)$ [36].

$\Gamma=\frac{G_{F}^{2}}{192\,\pi^{3}}[(1-2s_{W}^{2})^{2}+(2s_{W}^{2})^{2}]\zeta_{n}\lambda_{n}^{3}\frac{E_{\nu}^{3n+5}}{M_{Pl}^{3n}}$
(41)

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

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

$\Gamma=1.72\times 10^{-14}\lambda_{1}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{3}\
{\rm GeV},$ (42)

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

$\Gamma=1.91\times 10^{-14}\lambda_{2}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{6}\
{\rm GeV}.$ (43)

##  11 LIV in the Neutrino Sector II: Neutrino Splitting

The process of neutrino splitting in the case of superluminal neutrinos, i.e.,
$\nu\rightarrow 3\nu$, is relatively unimportant in the $[d]=4,n=0$ case [42]
owing to the small velocity difference between neutrino flavors obtained from
neutrino oscillation data [44, 45, 46]. 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 [47]. 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 three 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 [36]:

$\Gamma=5.16\times 10^{-14}\lambda_{1}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{3}\
{\rm GeV},$ (44)

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

$\Gamma=5.73\times 10^{-14}\lambda_{2}^{3}E_{\rm GeV}^{5}\ (E/M_{Pl})^{6}\
{\rm GeV}.$ (45)

The threshold energy for neutrino splitting is proportional to the neutrino
mass so that it is negligible compared to that given by Equation (39).

##  12 The Neutrinos Observed by IceCube

As of this writing, the IceCube collaboration has identified $87_{-10}^{+14}$
events from neutrinos of astrophysical origin with energies above 10 TeV, with
the error in the number of astrophysical events determined by the modeled
subtraction of both conventional and prompt atmospheric neutrinos and also
penetrating atmospheric muons at energies below 60 TeV [48].

There are are four indications that 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 consistent with isotropy [50, 49, 48];
(2) at least one of the PeV neutrinos came from a direction off the galactic
plane [48]; (3) the diffuse galactic neutrino flux is expected to be well
below that observed by IceCube [51]; (4) 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 [52].

Above 60 TeV, the IceCube data are roughly consistent with a spectrum given by
$E_{\nu}^{2}(dN_{\nu}/dE_{\nu})\simeq\ 10^{-8}\ {\rm GeV}{\rm cm}^{-2}{\rm
s}^{-1}$ [50, 49, 48]. However, IceCube has not detected any neutrino-induced
events from the Glashow resonance effect at
$E_{\bar{\nu}_{e}}=M_{W}^{2}/2m_{e}=6.3$ PeV [53]. At the Glashow resonance
energy electrons in the IceCube volume provide enhanced target cross-sections
for $\bar{\nu}_{e}$’s through the $W^{-}$ resonance channel,
$\bar{\nu}_{e}+e^{-}\rightarrow W^{-}\rightarrow shower$. This enhancement
effect is expected to be about a factor of $\sim$$10$ [50]. Owing to
oscillations, it is expected that 1/3 of the potential 6.3 PeV neutrinos would
be ${\nu}_{e}$’s plus $\bar{\nu}_{e}$’s unless new physics is involved.

Thus, the enhancement in the overall effective area expected is a factor of
$\sim$3\. Taking account of the increased effective area between 2 and 6 PeV
and a decrease from an assumed neutrino energy spectrum of $E_{\nu}^{-2}$, we
would expect about three events at the Glashow resonance energy, provided that
the number of $\bar{\nu}_{e}$’s is equal to the number of ${\nu}_{e}$’s. Even
without considering the resonance effect, several neutrino events above 2 PeV
would be expected if the $E_{\nu}^{-2}$ spectrum extended to higher energies.
Thus, the lack of this flux of neutrinos above $\sim$$2$ PeV energy and at the
6.3-PeV resonance may be indications of a cutoff in the neutrino
spectrum222Since [36] was published, an event with a deposited energy of 2.6
PeV was reported by the IceCube collaboration. The energy of the neutrino that
produced this event may have been as high as 6 PeV, although this is uncertain
[54]..

##  13 Extragalactic Superluminal Neutrino Propagation

Monte Carlo techniques have been employed to determine the effect of neutrino
splitting and VPE on putative superluminal neutrinos [36, 41]. Extragalactic
neutrinos were propagated from cosmological distances taking account of the
resulting energy loss effects by VPE and redshifting in the $[d]=4$ case. In
the $[d]>4$ cases, energy loses include those from both neutrino splitting and
VPE, as well as redshifting. It was assumed that the neutrino sources have a
redshift distribution similar to that of the star formation rate [55]. Such a
redshift distribution appears to be roughly applicable for both active
galactic nuclei and $\gamma$-ray bursts. A simple neutrino source spectrum
proportional to $\sim$$E^{-2}$ was assumed between 100 TeV and 100 PeV, as is
the case for cosmic neutrinos observed by IceCube with energies above 60 TeV
[49]. The final results on the propagated spectrum were normalized to an
energy flux of $E_{\nu}^{2}(dN_{\nu}/dE_{\nu})\simeq 10^{-8}~{}{\rm
GeV}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}~{}{\rm sr}^{-1}$, consistent with the
IceCube data for both the Southern and Northern Hemisphere. Results were
obtained for VPE threshold energies between 10 PeV and 40 PeV as given by
Equation (39), corresponding to values of $\delta_{\nu e}$ between $5.2\times
10^{-21}\ {\rm and}~{}3.3\times 10^{-22}$.

Since the neutrinos are extragalactic and survive propagation from all
redshifts, cosmological effects must be taken into account in deriving new LIV
constraints. Most of the cosmic PeV neutrinos will come from sources at
redshifts between $\sim$0.5 and $\sim$2 [55]. The effect of the cosmological
$\Lambda$CDM redshift-distance relation is given by:

$D(z)={{c}\over{H_{0}}}\int\limits_{0}^{z}\frac{dz^{\prime}}{(1+z^{\prime})\sqrt{\Omega_{\Lambda}+\Omega_{\rm
M}(1+z^{\prime})^{3}}}.$ (46)

The energy loss due to redshifting is given by:

$-(\partial\log E/\partial
t)_{redshift}=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}.$ (47)

The decay widths for the VPE process are given by Equations (42) and (43) for
the cases $n=1$ and $n=2$, respectively, while those for neutrino splitting
are given by Equations (44) and (45).

##  14 The Theoretical Neutrino Energy Spectrum

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

In their seminal paper, using Equation (40), Cohen and Glashow [42] 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. Further 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 [56]. An upper
limit of $\delta<\cal{O}$ $(10^{-18})$ was obtained [57] based on later
IceCube observations [50].

Using the energy loss rate given by Equation (40), a value for $\delta_{\nu
e}<10^{-20}$ was obtained based on a model of the redshift evolution of
neutrino sources and using Monte Carlo techniques to take account of
propagation effects as discussed in Section 13 [41, 36]. The upper limit on
$\delta_{e}$ is given by $\delta_{e}\leq 5\times 10^{-21}$ [41]. 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.

###  14.2 [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 0.3 PeV, with the IceCube data corresponds to a
VPE rest-frame threshold energy $E_{\nu,\rm th}=10$ PeV, as shown in Figure 6
[41, 36]. This corresponds to $\delta_{\nu
e}\equiv\delta_{\nu}-\delta_{e}\leq\ 5.2\times 10^{-21}$. Given that
$\delta_{e}\leq\ 5\times 10^{-21}$, it is again found that
$\delta_{\nu}\leq(0.5-1)\times 10^{-20}$. As shown in Figure 7, 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.

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 6 show a 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
predicted drop off may be a possible explanation for the lack of observed
neutrinos above 2 PeV [41, 36]. This pileup is 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.

The pileup effect caused by the neutrino-splitting process is more pronounced
than that caused by the VPE process because neutrino splitting produces two
new lower energy neutrinos per interaction. This 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%
[42], 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 [58].

Figure 6: Propagated neutrino spectra including energy losses as described in
the text [36]. 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. The neutrino
spectra are normalized to the IceCube data both with (gray) and without
(black) an estimated flux of prompt atmospheric neutrinos subtracted [49].

Figure 7: Calculated _n_ = 2 spectra taking into account all three processes
(redshifting, neutrino splitting and VPE) occurring simultaneously for rest
frame VPE threshold energies of 10 PeV (black, as in Figure 6), 20 PeV (green)
and 40 PeV (blue). The IceCube data are as in Figure 6 [49].

###  14.3 [d] = 5 CPT Violating Operator Dominance

In the _n_ = 1 case, the dominant $[d]=5$ operator violates $\cal{CPT}$. Thus,
if the $\nu$ is superluminal, the $\bar{\nu}$ will be subluminal, and vice
versa. 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 $\bar{\nu_{e}}$. We note that $\nu-\bar{\nu}$
oscillation measurements would give the strongest constraints on the
difference in $\delta$’s between $\nu$’s and $\bar{\nu}$’s [45].

Since both VPE and neutrino-splitting interactions generate a particle-
antiparticle lepton pair, one of the pair particles will be superluminal
($\delta>0$), whereas the other particle will be subluminal ($\delta<0$) [33].
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 clear spectral cutoff occurs [36].

##  15 Summary: Results for Superluminal Neutrinos

In the SME EFT formalism [2, 32, 33], if the apparent cutoff above $\sim 2$
PeV in the neutrino spectrum shown in Figure 7 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 $\mathaccent 23{c}^{(6)}=-\lambda_{2}/M_{Pl}^{2}\geq-5.2\times 10^{-35}$
GeV-2 [36]. Such a cutoff would not occur if the dominant LIV term is a
$\cal{CPT}$-violating the $[d]=5$ operator (However, see Footnote 2.) An event
with energy significantly above 3 PeV would increase our constraints by as
much as an order of magnitude. However, there is still the lack of expected
Glashow resonance events to consider.

If the lack of neutrinos at the Glashow resonance is the result of LIV effects
as shown in Figures 6 and 7, this would imply that there will be no cosmogenic
[59, 60] ultrahigh energy neutrinos. A less drastic effect in the cosmogenic
neutrino spectrum can be caused by a violation of LIV in the hadronic sector
at the level of $10^{-22}$ [21].

A cutoff can naturally occur if it is produced by a maximum acceleration
energy in the sources. In that case, the parameters given above would be
reduced to upper limits. However, the detection of a pronounced pileup just
below the cutoff would be prima facie evidence of a $\cal{CPT}$-even LIV
effect, possibly related to Planck-scale physics. In fact, $\cal{CPT}$-even
LIV in the gravitational sector at energies below the Planck energy has been
considered in the context of Hořava–Lifshitz gravity [61, 62].

##  16 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 [63, 64]. In order for the pion to be
stable above a critical energy $E_{c}$, we require that its effective mass as
given by Equation (33) 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 $\delta_{\pi}<\delta_{\mu}$ [5]. Then, neglecting
the neutrino mass, this critical energy energy is given by:

$E_{c}={{\sqrt{m_{\pi}^{2}-m_{\mu}^{2}}}\over{\delta_{\mu}-\delta_{\pi}}}$
(48)

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 (21) or
Equation (21). For example, if we set $E_{c}$ = 6 PeV, in order to just avoid
the Glashow resonance, we get the requirement, $\delta_{\mu\pi}\simeq
1.5\times 10^{-8}$.

If a lack of multi-PeV neutrinos is caused by this effect, there will be no
pileup below the cutoff energy as opposed to the superluminal cases previously
discussed. This would make the stable pion case difficult to distinguish from
a natural cutoff caused by maximum cosmic-ray acceleration energies in the
neutrino sources.

##  17 Conclusions

One of the most profound problems of modern physics is how to reconcile
general relativity with quantum physics. A unification of these fundamental
principles must somehow occur at the Planck scale. While direct observational
evidence on the physics at the Planck scale itself is unattainable, within the
EFT framework, clues to “quantum gravity” phenomena can be searched for at
energies much lower than $M_{Pl}$. One of these clues may be a tiny violation
of Lorentz symmetry that may increase with energy. In this regard, it has been
recognized that since cosmic $\gamma$-rays and neutrinos and cosmic-ray
nucleons provide the highest observable energies in the universe, high energy
astrophysics observations can play an important role in searching for clues to
Planck-scale physics.

As discussed in this paper, such astrophysical searches have only thus far
yielded strong constraints on LIV, verifying the strength of Lorentz symmetry.
However, there may be a hint at possible Lorentz symmetry breaking at the
level of $10^{-20}$ in the neutrino sector at PeV energies. Better statistics
are needed to explore this possibility further.

Acknowledgments: I thank my collaborators in much of the work discussed here:
Stefano Liberati, David Mattingly and Sean Scully.

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