††thanks: Published in Phys. Lett. B 836 (2023) 137613
https://doi.org/10.1016/j.physletb.2022.137613

# Speed Variations of Cosmic Photons and Neutrinos from Loop Quantum Gravity

Hao Li  School of Physics, Peking University, Beijing 100871, China  Bo-Qiang
Ma<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871,
China  Center for High Energy Physics, Peking University, Beijing 100871,
China  Collaborative Innovation Center of Quantum Matter, Beijing, China

###### Abstract

Recently a series of analyses on the flight time of cosmic photons and
neutrinos suggests that the speed of light _in vacuo_ takes the energy-
dependent form $v(E)\simeq 1-E/E_{\text{LIV}}^{\gamma}$ with
$E_{\text{LIV}}^{\gamma}\approx 3.6\times 10^{17}~{}\text{GeV}$, and meanwhile
the speed of neutrinos is proposed to be $v(E)\simeq 1\pm
E/E_{\text{LIV}}^{\nu}$ with $E_{\text{LIV}}^{\nu}\approx 6.5\times
10^{17}~{}\text{GeV}$ and $\pm{}$ representing the helicity dependence. This
novel picture immediately urges us to provide a satisfactory theoretical
explanation. Among all the attempts to predict the speed variations from
quantum gravity, we find that loop quantum gravity can serve as a good
candidate for explaining the aforementioned picture consistently.

Lorentz invariance, as a basic principle of relativity, is suspected to break
due to the quantum gravity (QG) effect around the Planck scale
$E_{\text{Planck}}\approx 1.22\times 10^{19}~{}\text{GeV}{}$ (or the inverse
of the length scale $\ell_{P}\approx 1.6\times 10^{-33}~{}\text{cm}$). As the
footprint of the underlying QG effect, Lorentz invariance violation (LIV) is
too minuscule to be directly observed in the laboratories, thus high energy
cosmic particles like photons and neutrinos might provide one of the most
promising opportunities to reveal the tiny LIV effects [1]. In general, one
expects that to the leading order of LIV, the modified speed of a massless
particle with energy $E$ can be expressed as $v(E)\simeq c(1-\xi E)$, where
$\xi{}$ is the LIV scale about the order of magnitude of the Planck scale.
Therefore given the precision accuracy of current experimental facilities, the
speed difference between two massless particles with different energies can
only be distinguishable when their energies are high enough and the travel
time differences are accumulated after a long propagating distance. As a
result, cosmic photons and neutrinos, especially those from gamma-ray bursts
(GRBs) and active galactic nuclei (AGNs) and simultaneously energetic, open a
unique window to the search of LIV from their flights in the Universe [1].

LIV researches utilizing high energy cosmic photons and neutrinos show many
interesting results, and we focus on one of them obtained by the analyses of
GRB photon travel time differences and possible GRB neutrino travel time lags.
The corresponding scenario to explain the data can be summarized as

  1. 1.

For photons, we have $v(E)\simeq c(1-E/E_{\text{LIV}}^{\gamma})$ and
$E_{\text{LIV}}^{\gamma}\approx 3.6\times 10^{17}~{}\text{GeV}{}$ without
helicity dependence. This means that photons are all subluminal and high
energy photons propagate slower than low energy photons. More details can be
found in Refs. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

  2. 2.

For neutrinos and anti-neutrinos (or vice versa), their speeds are
$v(E)\simeq(1\mp E/E_{\text{LIV}}^{\nu})$ and $E_{\text{LIV}}^{\nu}\approx
6.5\times 10^{17}~{}\text{GeV}{}$. Consequently neutrinos and anti-neutrinos
(or Majorana neutrinos with opposite helicities) could be subluminal and
superluminal respectively or visa versa, with the sign of the speed variations
depending on the helicities. For more details one can refer to Refs. [13, 14,
6, 7, 15, 16, 17].

While full clarity on the suggested phenomenological picture must wait for
more data to verify, it makes sense to explore possible theoretical frameworks
that would provide support for the phenomenological picture and establish
whether by sharpening the phenomenology one could get insight on some pure-
theory proposals. Although there are various theoretical explanations for only
the photon sector or the neutrino sector, a combined explanation of both
photon and neutrino sectors is still desirable. To fill the gap between
phenomenological studies and theoretical constructions, we suggest that loop
quantum gravity (LQG) is able to provide a viable approach to understanding
these phenomena consistently.

Loop quantum gravity (LQG), one of the most promising theories aiming at
reconciling general relativity and quantum mechanics, has attracted a lot of
interest with numerous novel results. A comprehensive introduction to LQG can
be found in some excellent books such as [18, 19, 20], and for our purpose, we
find it appropriate to only gather the essential results as well as necessary
details whenever they are needed, with more information can be found in Refs.
[21, 22]. To calculate LIV of photons and neutrinos from LQG, one considers a
special set of semiclassical states called the _would be semiclassical states_
(WBSCs). A WBSC $\ket{W}$ has a characteristic length $\mathcal{L}$ which
describes the discreteness of the spacetime represented by this state. When
the de Broglie wavelength $\lambda{}$ of a particle satisfies
$\ell_{P}\ll\lambda$ and the characteristic length $\mathcal{L}$ satisfies
$\ell_{P}\ll\mathcal{L}\leq\lambda{}$, one can compute the WBSC expectation of
the corresponding Hamiltonian of the particle and obtain an effective
Hamiltonian from which we can read off the modified dispersion relation and
accordingly, the speed of that particle.

For the electromagnetic fields, the effective Hamiltonian results in the
following modified Maxwell equations [21]:

$\displaystyle A_{\gamma}(\gradient\times\vec{B})$
$\displaystyle-\frac{\partial\vec{E}}{\partial
t}+2\ell_{P}^{2}\theta_{3}\gradient^{2}(\gradient\times\vec{B})-2\theta_{8}\ell_{P}\gradient^{2}\vec{B}$
$\displaystyle+4\theta_{4}\mathcal{L}^{2}{\left(\frac{\mathcal{L}}{\ell_{P}}\right)}^{2\Upsilon_{\gamma}}\ell_{P}^{2}\gradient\times({\vec{B}}^{2}\vec{B})=0,$
$\displaystyle A_{\gamma}(\gradient\times\vec{E})$
$\displaystyle+\frac{\partial\vec{B}}{\partial
t}+2\ell_{P}^{2}\theta_{3}\gradient^{2}(\gradient\times\vec{E})$ (1)
$\displaystyle-2\theta_{8}\ell_{P}\gradient^{2}\vec{E}=0,$

with

$A_{\gamma}=1+\theta_{7}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2+2\Upsilon_{\gamma}},$
(2)

and we supplement to Eq. (1) the source-free conditions [21]:

$\gradient\cdot\vec{E}=\gradient\cdot\vec{B}=0.$ (3)

We leave the parameters $\theta_{i}$ (with $i=3,4,7,8$) and
$\Upsilon_{\gamma}$ to be determined by analyzing the phenomenological picture
later. However it should be noted that naturally we can expect that all the
$\theta_{i}$s are either of the order of magnitude of one or are extremely
close to zero. We neglect the non-linear part in Eq. (1) and assume that the
solutions can be considered as superposition of plane waves. Then substituting

$\vec{E}\propto e^{i(\vec{k}\cdot\vec{x}-\omega t)},\qquad\vec{B}\propto
e^{i(\vec{k}\cdot\vec{x}-\omega t)},\qquad k:=|\vec{k}|$

into Eqs. (1) and (3), we can get the modified dispersion relation [21]:

$\omega=k\left[1+\theta_{7}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2+2\Upsilon_{\gamma}}-2\theta_{3}{(k\ell_{P})}^{2}\pm
2\theta_{8}(k\ell_{P})\right],$ (4)

which gives the speed of light from $v=\partial\omega/\partial k$ [21]:

$v_{\pm}^{\gamma}(k,\mathcal{L})\simeq 1\pm
4\theta_{8}(k\ell_{P})-6\theta_{3}{(k\ell_{P})}^{2}+\theta_{7}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2\Upsilon_{\gamma}+2}.$
(5)

This formula is valid only for small $\ell_{P}/\mathcal{L}$.

To compare Eq. (5) with the aforementioned form of photon speed, we first drop
the third term which is of the second order of $\ell_{P}$ on the right hand
side. Then we adopt the mobile scale which relates the characteristic scale
$\mathcal{L}$ and the momentum $k$ by $\mathcal{L}=k^{-1}$. The photon speed
then can be written as

$v_{\pm}^{\gamma}(k)\simeq 1\pm
4\theta_{8}(\ell_{P}k)+\theta_{7}{(\ell_{P}k)}^{2\Upsilon_{\gamma}+2}.$ (6)

Obviously only the third term on the right hand side produces the helicity-
independent correction and we must set $\Upsilon_{\gamma}=-1/2$ such that our
picture can be reproduced qualitatively. Therefore we are led to consider the
following form:

$v_{\pm}^{\gamma}(k)\simeq 1+(\theta_{7}\pm 4\theta_{8})(\ell_{P}k).$ (7)

The $\theta_{8}$ term here still leads to birefringence effects and as a
result the total rotation angle between two oppositely polarized photons with
the same energy can be written as [23, 24, 25]

$\lvert\Delta\theta(E,z)\rvert\simeq\frac{2\theta_{8}\ell_{P}E^{2}}{H_{0}}\int_{0}^{z}\frac{(1+z^{\prime})\,\text{d}z^{\prime}}{\sqrt{\Omega_{m}{(1+z^{\prime})}^{3}+\Omega_{\Lambda}}},$
(8)

where $E$ is the observed energy, $z$ is the source redshift, and
$H_{0},\Omega_{m}$ and $\Omega_{\Lambda}{}$ are the cosmological constants
[26]. Therefore the best constraint on the rotation angle in the literature
[25] can be transformed into a constraint on $\theta_{8}$: $\theta_{8}\lesssim
10^{-16}$, for which we can simply let $\theta_{8}$ be zero since photons
usually have energies below $10^{6}~{}\text{GeV}{}$ such that
$\ell_{P}p\lesssim 10^{-13}$, and therefore
$\theta_{8}\ell_{P}p\sim{(\ell_{P}p)}^{2}$, and consequently we could drop the
term. As a result we have

$v_{\pm}^{\gamma}(k)\simeq 1+\theta_{7}\ell_{P}k,$ (9)

and

$\theta_{8}=0,\ \frac{1}{\theta_{7}\ell_{P}}\approx-3.6\times
10^{17}~{}\text{GeV}{},\ \text{and }\Upsilon_{\gamma}=-\frac{1}{2}.$ (10)

That is to say, we have $\lvert\theta_{7}\rvert\approx 33.9$, which is a
reasonable numerical result near the order $\mathcal{O}(1)$.

On the other hand, for massless Majorana fermions, we have [22]

$\displaystyle\left(i\frac{\partial}{\partial
t}-i\hat{A}\vec{\sigma}\cdot\gradient+\frac{\hat{C}}{2\mathcal{L}}\right)\xi(t,\vec{x})$
$\displaystyle=0,$ $\displaystyle\left(i\frac{\partial}{\partial
t}+i\hat{A}\vec{\sigma}\cdot\gradient-\frac{\hat{C}}{2\mathcal{L}}\right)\chi(t,\vec{x})$
$\displaystyle=0,$ (11)

where $\sigma^{i}$ with $i=1,2,3$ denote the standard Pauli matrices, $\xi{}$
is a Weyl spinor, $\chi(t,\vec{x})=i\sigma_{2}\xi^{*}(t,\vec{x})$ and the two
coefficients (indeed operators) are defined as [22]

$\displaystyle\hat{A}:=1+\kappa_{1}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}+1}+\kappa_{2}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2\Upsilon_{f}+2}+\frac{\kappa_{3}}{2}\ell_{P}^{2}\gradient^{2},$
$\displaystyle\hat{C}/\hbar:=\kappa_{4}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}}+\kappa_{5}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2\Upsilon_{f}+1}$
$\displaystyle{\ \ \ \ \ \ \ \ \
}+\kappa_{6}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{3\Upsilon_{f}+2}+\frac{\kappa_{7}}{2}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}}\ell_{P}^{2}\gradient^{2}.$
(12)

Similarly, a reasonable assumption is that all the $\kappa_{i}$s are either of
the order of magnitude of one or just can be set to zero. Meanwhile we assume
$\kappa_{i}=0$ for $i=4,5,6$ since these parameters merely contribute to the
renormalization of the mass which is of no interest in this work [22]. As a
result, we use the simplified definition of $\hat{C}$ in the following:

$\hat{C}=\frac{\hbar\kappa_{7}}{2}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}}\ell_{P}^{2}\gradient^{2}.$
(13)

According to Ref. [22], the dispersion relation can be expressed as

$E_{\pm}=\sqrt{A_{f}^{2}p^{2}+{\left(\frac{C_{f}}{2\mathcal{L}}\right)}^{2}\pm
B_{f}p},$ (14)

where $p:=\lvert\vec{p}\rvert{}$ and

$\displaystyle
A_{f}=1+\kappa_{1}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}+1}+\kappa_{2}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2\Upsilon_{f}+2}$
$\displaystyle{\ \ \ \ \ \ \ \ \ \ \ \
}-\frac{\kappa_{3}}{2}\ell_{P}^{2}p^{2},$ $\displaystyle
B_{f}=A_{f}\frac{C_{f}}{\mathcal{L}},$ $\displaystyle
C_{f}=-\frac{\hbar\kappa_{7}}{2}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}}\ell_{P}^{2}p^{2}.$
(15)

The subscript $\pm{}$ in Eq. (14) indicates the helicity-dependence of the
dispersion relation. Then the speed of massless fermions are [22]

$\displaystyle v_{\pm}^{f}(p,\mathcal{L})$ $\displaystyle\simeq
1-\frac{3}{2}\kappa_{3}{(\ell_{P}p)}^{2}+{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{\Upsilon_{f}+1}$
(16)
$\displaystyle{}\times\left[\kappa_{1}\mp\frac{\kappa_{7}}{2}(\ell_{P}p)\right]+\kappa_{2}{\left(\frac{\ell_{P}}{\mathcal{L}}\right)}^{2\Upsilon_{f}+2}.$

We should point out that Eq. (16) is obtained by expanding to the first
several orders of ${(\ell_{P}/\mathcal{L})}^{\Upsilon_{f}}$, so that this
expansion might not be valid for certain values of $\Upsilon_{f}$.
Nevertheless we leave this problem to be addressed after fixing the parameters
in Eq. (16) utilizing the aforementioned phenomenological picture.

In a very similar manner, we first drop the $\ell_{P}^{2}$ term in Eq. (16)
and adopt the mobile scale $\mathcal{L}=p^{-1}$ again, obtaining

$\displaystyle v_{\pm}^{f}(p)$ $\displaystyle\simeq
1+{\left(\ell_{P}p\right)}^{\Upsilon_{f}+1}$ (17)
$\displaystyle{}\times\left[\kappa_{1}\mp\frac{\kappa_{7}}{2}(\ell_{P}p)\right]+\kappa_{2}{\left(\ell_{P}p\right)}^{2\Upsilon_{f}+2},$

which is valid for Majorana neutrinos. Obviously if we want both the signs
$\pm{}$ can be taken for the linear corrections, we must set $\Upsilon_{f}=-1$
and obtain

$v_{\pm}^{f}(p)\simeq
1+(\kappa_{1}+\kappa_{2})\mp\frac{\kappa_{7}}{2}(\ell_{P}p).$ (18)

However if in the limit $p\to 0$ we also require $v^{f}(0)=1$,
$\kappa_{1}+\kappa_{2}$ has to be zero and eventually we have

$v_{\pm}^{f}(p)\simeq 1\mp\frac{\kappa_{7}}{2}(\ell_{P}p),$ (19)

with the parameters determined as

$\kappa_{1}+\kappa_{2}=0,\ \frac{2}{\kappa_{7}\ell_{P}}\approx 6.5\times
10^{17}~{}\text{GeV}{},\ \text{and }\Upsilon_{f}=-1,$ (20)

from which we obtain $\lvert\kappa_{7}\rvert\approx 37.5$, and it is
noteworthy that $\lvert\theta_{7}/\kappa_{7}\rvert\approx 1$.

After analyzing Majorana neutrinos, we also study the Dirac case in the
following for completeness. Hereafter we further assume that $\kappa_{3}=0$
and the desired Dirac equations are

$\displaystyle\left(i\frac{\partial}{\partial
t}-i\vec{\sigma}\cdot\gradient+\frac{\hat{C}}{2\mathcal{L}}\right)\xi(t,\vec{x})=0,$
$\displaystyle\left(i\frac{\partial}{\partial
t}+i\vec{\sigma}\cdot\gradient-\frac{\hat{C}}{2\mathcal{L}}\right)\chi(t,\vec{x})=0,$
(21)

with $\chi=i\sigma_{2}\xi^{*}$. Let us assume that $\chi{}$ is independent of
$\xi{}$ temporarily, thus a general Dirac spinor can be written as

$\Psi=\begin{bmatrix}\xi\\\ \chi\end{bmatrix},$ (22)

and then it is not hard to verify that the following equation reproduces Eq.
(21):

$\left(i\gamma^{\mu}\partial_{\mu}-\frac{\hat{C}}{2\mathcal{L}}\gamma^{0}\gamma_{5}-m\right)\Psi=0,$
(23)

where the second term in the parenthesis explicitly violates Lorentz
invariance and we restore the mass term for later convenience. Of course Eq.
(23) may not be unique, but we only focus on it since we just attempt to show
that the results we just obtained are likely to apply to Dirac neutrinos as
well. To get the modified dispersion relation, we first multiply both sides of
this equation by
$(i\not{\partial}-\hat{C}\gamma^{0}\gamma_{5}/2\mathcal{L}+m)$, obtaining

$\left(-\partial^{2}-i\frac{\hat{C}}{2\mathcal{L}}\gamma_{5}\left(\gamma^{\mu}\gamma^{0}-\gamma^{0}\gamma^{\mu}\right)\partial_{\mu}-\frac{\hat{C}^{2}}{4\mathcal{L}^{2}}-m^{2}\right)\Psi=0,$
(24)

which, by straightforward calculation, can be simplified to be

$\left(-\partial^{2}+i\frac{\hat{C}}{\mathcal{L}}\vec{\Sigma}\cdot\gradient-\frac{\hat{C}^{2}}{4\mathcal{L}^{2}}-m^{2}\right)\Psi=0,$
(25)

with $\vec{\Sigma}:=\vec{\sigma}\otimes\text{diag}\\{1,1\\}$. Next we apply
$(-\partial^{2}-i\hat{C}\vec{\Sigma}\cdot\gradient/\mathcal{L}-\hat{C}^{2}/4\mathcal{L}^{2}-m^{2})$
to Eq. (25) and obtain the Klein-Gordon-type expression

$\left({\left(-\partial^{2}-\frac{\hat{C}^{2}}{4\mathcal{L}^{2}}-m^{2}\right)}^{2}+\frac{\hat{C}^{2}}{\mathcal{L}^{2}}\gradient^{2}\right)\Psi=0,$
(26)

which should be satisfied by all components of the Dirac spinor. Ignoring the
mass term again and substituting a plane wave solution into Eq. (26), we then
read off the modified dispersion relation for Dirac spinors:

$E=\sqrt{p^{2}\pm\frac{\ell_{P}^{2}\kappa_{7}}{2\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{f}}p^{3}-\frac{\ell_{P}^{4}\kappa_{7}^{2}}{16\mathcal{L}^{2}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{f}}p^{4}},$
(27)

or to the leading order:

$E\simeq
p\pm\frac{\ell_{P}^{2}\kappa_{7}}{4\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{f}}p^{2}.$
(28)

It is obvious that Eq. (28) also leads to Eq. (19) under the similar
assumptions:

$v^{f}=\frac{\partial E}{\partial p}\simeq
1\pm\frac{\kappa_{7}}{2}(\ell_{P}p).$ (29)

As a result, no matter whether neutrinos are Dirac particles or Majorana
particles, the phenomenological constraints from cosmic neutrino data analyses
[6, 15, 16, 17] can always be satisfied with the same parameters.

Most importantly, we further analyze the two parameters $\Upsilon_{\gamma}$
and $\Upsilon_{f}$ which play a crucial role in producing the desired energy
dependence and helicity dependence. Let $A^{i}_{a}$ be the real connection
variables, then these two parameters are determined by [21, 22]

$\displaystyle\bra{W,\uline{\vec{E}},\uline{\vec{B}}}\dots
A_{ia}\dots\ket{W,\uline{\vec{E}},\uline{\vec{B}}}\approx\dots\frac{1}{\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{\gamma}}\dots,$
$\displaystyle\bra{W,\xi}\dots
A_{ia}\dots\ket{W,\xi}\approx\dots\frac{1}{\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{f}}\delta_{ia}\dots$
(30)

respectively, where $\ket{W,\uline{\vec{E}},\uline{\vec{B}}}$ and
$\ket{W,\xi}$ are the corresponding WBSCs. A straightforward and satisfactory
choice for $\Upsilon_{\gamma}$ and $\Upsilon_{f}$ may be
$\Upsilon_{\gamma}=\Upsilon_{f}=1$ [21, 22] since it makes both the
expectation values in Eq. (30) vanish in the limit $\hbar\to 0$ and a
vanishing expectation value of this type means that the spacetime discreteness
disappears once the quantum effects are turned off. Thus it seems that
$\Upsilon_{\gamma}=-1/2$ and $\Upsilon_{f}=-1$ are both unacceptable. However
if we speculate that $\Upsilon_{\bullet}$ (the slot $(\bullet)$ represents
$\gamma{}$ or $f$) is a function of the spin of the corresponding
particles,111It may also depend on other quantum numbers, but here we have few
clues what the form the dependence could be, so we take them into
consideration by introducing some parameters. this problem might be addressed
as follows. For convenience, we assume that the limit $\hbar\to 0$ can be
taken continuously and one permissible way to take this limit is
$\hbar(t)=\hbar_{0}(1-t)$ where $0\leq t\leq 1$ and $\hbar_{0}$ is the Planck
constant usually measured in laboratories. We further rewrite the spins of
photons and fermions as $s(t;\gamma)=\hbar(t)/\hbar_{0}$ and
$s(t;f)=\hbar(t)/2\hbar_{0}$ respectively such that the spins vanish when
$\hbar{}$ is zero and take their ordinary values when $\hbar{}$ is the real-
world one. There are still many possibilities of how $\Upsilon_{\bullet}{}$
depends on the spin $s$, while here as an example we choose the simplest one
that $\Upsilon_{\bullet}{}$ depends on $s$ linearly:
$\Upsilon_{\bullet}(s)=s-3/2$. Then the expectation values behave like

$\displaystyle\frac{1}{\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{\gamma}(s)}\propto\sqrt{1-t}^{-t-\frac{1}{2}},$
$\displaystyle\frac{1}{\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{f}(s)}\propto\sqrt{1-t}^{-\frac{t}{2}-1},$
(31)

which diverge as $t\to 1$ so the linear form is not valid. In addition there
is no free parameter describing the effects of other quantum numbers and this
also makes the linear form disfavored. We then try a quadratic function:
$\Upsilon_{\bullet}(s)=a_{1}s^{2}+a_{2}s+a_{3}$ where $a_{i}$ with $i=1,2,3$
are parameters that may be dependent on other quantum numbers. A direct
calculation suggests that

$\Upsilon_{\bullet}(s)=(3+2a_{3})s^{2}-(\frac{7}{2}+3a_{3})s+a_{3}.$ (32)

We do not have any knowledge of the exact value of $a_{3}$, but let us
consider the expectation values:

$\displaystyle\frac{1}{\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{\gamma}(s)}\propto\sqrt{1-t}^{(3+2a_{3})t^{2}-(\frac{5}{2}+a_{3})t-\frac{1}{2}},$
$\displaystyle\frac{1}{\mathcal{L}}\left(\frac{\ell_{P}}{\mathcal{L}}\right)^{\Upsilon_{f}(s)}\propto\sqrt{1-t}^{(\frac{3}{4}+\frac{a_{3}}{2})t^{2}+(\frac{1}{4}+\frac{a_{3}}{2})t-1},$
(33)

which are zero at $t\to 1$ so long as $a_{3}>0$ and recover the desired
behavior at $t\to 0$. The parameter $a_{3}$ is likely to encode the potential
effects of other quantum numbers as we expect and as a result the problem of
turning off quantum effects is likely to be solved. Another associated problem
is whether the perturbative expressions Eq. (5) and Eq. (16) are still valid
with the present parameter choice. It is clear that for photons, since
$2+2\Upsilon_{\gamma}=1>0$, the expansion with respect to small
$\ell_{P}/\mathcal{L}$ (or $\ell_{P}k$) does not bring any new difficulty. In
the fermion case, instead of perturbing with respect to
$\ell_{P}/\mathcal{L}$, one can alternatively expand with respect to
$\ell_{P}$. With this fact we can easily see that the dispersion relation (14)
again yields the speed (16). As a result, although the choices of parameters
may violate the perturbative expansions (5) and (16), the phenomenologically
determined ones are still consistent with these perturbative expressions and
the associated analyses are valid as well.

Lastly, we should point out that to obtain the desired dispersion relations
Eq. (4) and Eq. (14), we need to introduce the description of LQG based on
weave states (WBSCs in this work) [22, 21], and consequently the results
depend on the assumption of weave states and also the assumed properties of
weave states. LQG still is an incomplete theory (in the literal sense that its
“Hamiltonian constraint” has not been solved) and therefore all the
predictions obtained from LQG require some assumptions. It is of notice that
the much-studied description of LQG based on weave states [22, 21] has the
sort of phenomenological implications described in this manuscript. Therefore
the results from LQG with assumed weave states make sense because they can be
directly related to observable phenomena and conversely the phenomenological
analyses may be used to examine the description of LQG based on weave states,
leading towards better understanding of the LQG framework and further progress
of LQG theories.

In summary, we find that loop quantum gravity (LQG) provides a viable way to
understanding the phenomenological suggestion of the speed variations of
cosmic photons and neutrinos. Comparing the theoretical calculation and
phenomenological picture we determine several parameters of LQG induced LIV,
which conversely may be helpful for the understanding of LQG itself. For
example, we determine the exact values of $\Upsilon_{\gamma}$ and
$\Upsilon_{f}$, which reflect some properties of the WBSCs. Besides, we find
that $\theta_{7}$ and $\kappa_{7}$ are of the desired order of magnitude and a
simple assumption of $\theta_{7}\approx\kappa_{7}$ reproduces the
phenomenological result $E_{\text{LIV}}^{\gamma}/E_{\text{LIV}}^{\nu}\approx
1:2$, which is an interesting observation. We thus conclude that LQG seems to
be a promising theory for explaining the novel picture suggested from
phenomenological analyses on the time of flight of cosmic photons [2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12] and neutrinos [13, 14, 6, 7, 15, 16, 17]. With the
suggested LQG parameters, we expect that more observable signals could testify
the predictions of loop quantum gravity in the future.

###### Acknowledgements.

This work is supported by National Nature Science Foundation of China (Grant
No. 12075003).

##  Appendix A Conventions

The signature is chosen to be $\eta^{\mu\nu}=(+1,-1,-1,-1)$. The Pauli
matrices $\sigma^{\mu}{}$ are

$\displaystyle\sigma^{0}=\left(\begin{array}[]{c@{\quad}c}1&0\\\
0&1\end{array}\right),\ \sigma^{1}=\left(\begin{array}[]{c@{\quad}c}0&1\\\
1&0\end{array}\right),{}$ (38)
$\displaystyle\sigma^{2}=\left(\begin{array}[]{c@{\quad}c}0&-i\\\
i&0\end{array}\right),\ \sigma^{3}=\left(\begin{array}[]{c@{\quad}c}1&0\\\
0&-1\end{array}\right),$ (43)

and $\bar{\sigma}^{\mu}:=(\sigma^{0},-\vec{\sigma})$. We adopt the same
convention for the $\gamma{}$ matrices as that in the book of Peskin and
Schroeder [27], and hence the $\gamma{}$ matrices are expressed as

$\gamma^{\mu}=\left(\begin{array}[]{c@{\quad}c}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{array}\right).$ (44)

Correspondingly, $\gamma_{5}:=i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$,
which takes the simple form: $\gamma_{5}=\text{diag}\\{-1,-1,+1,+1\\}{}$, and
the helicity projection operators are

$P_{L}=\frac{1-\gamma_{5}}{2},\quad P_{R}=\frac{1+\gamma_{5}}{2}.$ (45)

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