aainstitutetext: School of Physics, Peking University, Beijing 100871,
Chinabbinstitutetext: Center for High Energy Physics, Peking University,
Beijing 100871, Chinaccinstitutetext: Collaborative Innovation Center of
Quantum Matter, Beijing, China

# Lorentz and CPT breaking in gamma-ray burst neutrinos from string
theory†††Published as JHEP 03 (2023) 230,
https://doi.org/10.1007/JHEP03(2023)230

Chengyi Li<EMAIL_ADDRESS>a,b,c and Bo-Qiang Ma‡‡‡Author to whom any
correspondence should be addressed<EMAIL_ADDRESS>
###### Abstract

Previous studies on high-energy gamma-ray burst neutrinos from IceCube suggest
a neutrino speed variation at the Lorentz violation (LV) scale of $\sim
6.4\times 10^{17}$ GeV, with opposite velocity variances between neutrinos and
antineutrinos. Within a space-time foam model, inspired by string theory, we
develop an approach to describe the suggested neutrino/antineutrino
propagation properties with both Lorentz invariance and CPT symmetry breaking.
A threshold analysis on the bremsstrahlung of electron-positron pair
($\nu\rightarrow\nu ee^{+}$) for the superluminal (anti)neutrino is performed.
We find that, due to the energy violation caused by the quantum foam, such
reaction may be restricted to occur at sufficient high energies and could even
be kinematically forbidden. Constraints on neutrino LV from vacuum $ee^{+}$
pair emission are naturally avoided. Future experiments are appealed to test
further the CPT violation of cosmic neutrinos and/or neutrino superluminality.

###### Keywords:

Non-Standard Neutrino Properties, Space-Time Symmetries, String and Brane
Phenomenology, Violation of Lorentz and/or CPT Symmetry

††arxiv: 2303.04765

##  1 Introduction

Astrophysical neutrinos are ideal portals to reveal the tiny Lorentz
invariance violation (LV) as postulated by some quantum gravity (QG) theories
Jacob:2006gn ; Amelino-Camelia:2009imt ; Amelino-Camelia:2015nqa . The IceCube
collaboration has reported the discovery of ultrahigh-energy neutrinos of
extragalactic origin, including a couple of PeV events IceCube:2013cdg ;
IceCube:2014jkq ; IceCube:2016uab . Recent studies Amelino-Camelia:2016fuh ;
Amelino-Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ; Huang:2022xto of
events in the (near-)TeV–PeV range suggest a linearly energy dependent speed
variation of neutrinos through their associations with gamma-ray bursts
(GRBs). Analyses lead to a Lorentz violation scale of $\sim 6\times 10^{17}$
GeV, comparable with that determined from GRB photons Shao:2009bv ;
Zhang:2014wpb ; Xu:2016zxn ; Liu:2018qrg ; Zhu:2021pmw . More intriguingly, it
is proposed Huang:2018ham that either neutrinos or antineutrinos travel faster
than the constant light speed $c$,444Henceforth natural units in which
$c=\hslash=1$ are adopted. whereas the other ones go slower than unity. This
can be explained by the CPT-odd feature of the linear Lorentz violation
Huang:2018ham ; Huang:2019etr ; Huang:2022xto , and leads further to the
Charge–Parity–Time (CPT) reversal symmetry breaking between neutrinos and
antineutrinos, or a matter–antimatter asymmetry Zhang:2018otj . But it is also
found that the attempt to interpret such phenomenological picture with field-
theoretic models of LV faces challenges due to the constraints on the
superluminal neutrino velocity and the corresponding LV from the kinematically
allowed anomalous channels, e.g., vacuum pair emission ($\nu\rightarrow\nu
ee^{+}$) Zhang:2018otj .

The main objective of the study we are here performing is to indicate that the
experimental finding of LV for GRB neutrinos Amelino-Camelia:2016fuh ;
Amelino-Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ; Huang:2022xto may
coincide with the predictions from certain QG scheme that cannot be cast in an
effective field theory (EFT) description, i.e., the quantum (Liouville
inspired) space-time foam model from string/D-brane theories. In fact, the
main idea has been outlined in a letter Li:2022sgs , and in this paper we
provide a thorough account of the calculations and elaborate on more detailed
results through in depth discussions. This framework has also been used
previously in explaining light speed variation from analyzing flight times of
GRB photons Shao:2009bv ; Zhang:2014wpb ; Xu:2016zxn ; Liu:2018qrg ;
Zhu:2021pmw in a consistent way Li:2021stm .

The prototype idea of the quantum structure of space-time at a microscopic
level—“space-time foam” devised by Wheeler Wheeler:1955stf —arises from the
uncertainties of quanta. For string/brane theory, such nontrivial foamy
structures are provided by solitonic defects in some Liouville-string inspired
models Ellis:1999stm ; Ellis:2004stm , according to which our Universe lives
on a (compactified) D(irichlet)3-brane, roaming in a higher-dimensional bulk
space, punctured by a population of D0-branes in type I/IIA strings
Ellis:1999stm ; Ellis:2004stm ; Ellis:2008stm , as we will consider below (or
in IIB superstrings, of wrapped-up D-branes which are effectively pointlike
Li:2009tt ). The D-brane defects (“D-particles”) appear to a braneworld
observer as flashing-on-and-off vacuum structures when they cross the brane.
Their interaction with open-string Standard-Model (SM) excitations involving
capture/splitting process and subsequent recoil reduces local Lorentz
invariance. Such models, dubbed string/D-defect (space-time) foam, go beyond
the local EFT approach to QG with a variety of applications to study a number
of phenomena, such as the so-induced vacuum refraction for photons
Mavromatos:2010pk and fermions Ellis:2000sfd , origin of neutrino masses
Mavromatos:2010fmg and mixing Alexandre:2008sfo ; Mavromatos:2009sfm ;
Mavromatos:2005sfm , as well as string cosmologies associated with the dark
sector of the Universe Mavromatos:2010sco .

Our aim is to show that the suggested neutrino speed variation can be
explained by means of the CPT-breaking aspects of such stringy QG models with
linear Lorentz violations. Constraints implied by vacuum pair emission by the
superluminal neutrino are addressed and found to be consistent with the
findings of Refs. Amelino-Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ;
Huang:2022xto in such a string theoretic context. We also propose several
viable ways on testing the CPT violation in the neutrino sector with future
(astrophysical) observations.

The paper is organized as follows. In Sec. 2, we introduce within the
framework of stringy space-time foam a scenario that admits CPT-violating
neutrinos and compute the dispersion relation, velocity and traveling times.
In Sect. 3 we discuss the phenomenological implications of the results
obtained by associating IceCube observations with GRBs on stringy QG. Section
4 is devoted to elaborating on the plausible mechanism permitting a stable
propagation in vacuo for the neutrino species against superluminal decays in
the model as reported in Li:2022sgs . In particular, the ways to escape the
threshold constraints are given. To conclude, a summary and discussion of our
results is depicted in Sec. 5.

##  2 Stringy D(efect)-foam

Consider the isotropic D-foam framework, as portrayed by the seminal works
Ellis:1999stm ; Ellis:2004stm ; Ellis:2008stm in this area, the
capture/splitting of a neutral open string such as a neutrino by a D-particle
causes a recoil motion of the latter, described by a deformed stringy
$\sigma$-model operator:

$V\ni\int_{\partial{\cal W}}\textrm{d}\tau\,\epsilon\,{\cal
U}_{\ell}x^{0}\Theta_{\epsilon}(x^{0})\partial_{n}x^{\ell},$ (1)

where $\partial_{n}$ is the normal derivative on the boundary of the
worldsheet $\partial{\cal W}$, $\Theta_{\epsilon\rightarrow
0^{+}}(t)=\frac{1}{2\pi\textrm{i}}\int_{-\infty}^{\infty}\frac{\textrm{d}q}{q-\textrm{i}\epsilon}\textrm{e}^{\textrm{i}qt}$,
and ${\cal U}$ is the spatial part of the recoil 4-velocity of the D-defect,
${\cal U}_{\ell}={\cal V}_{\ell}(1-\mbox{\boldmath${\cal
V}$\unboldmath}^{2})^{-1/2}\equiv{\cal V}_{\ell}\gamma_{{\cal V}}$, which, for
heavy (nonrelativistic) D-particles, reduces to the ordinary 3-velocity that
can be identified as

$\mbox{\boldmath${\cal V}$\unboldmath}=M_{s}^{-1}g_{s}\Delta{\bf
k}~{}\rightarrow~{}{\cal V}_{\|}\simeq
M_{s}^{-1}g_{s}\lambda^{(\ell)}k_{\ell},$ (2)

following the (logarithmic) conformal field theory methods Kogan:1995lca ;
Ellis:1996lrt ; Mavromatos:1998nz . Above, $\Delta{\bf k}$ is the momentum
transfer during a collision, $g_{s}\ll 1$ is the string coupling, and $M_{s}$
is the string scale. Suffixes $\|$ denote components along brane longitudinal
dimensions, i.e., $\ell=1,2,3$, and $\lambda$ is the ratio of $\Delta
k_{\ell}$ with respect to the incoming neutrino momentum, that is,
$\lambda^{(\ell)}=\Delta k_{\ell}/k_{\ell}$, taken to be stochastic Gaussian
Mavromatos:2005sfm with the moments:

$\langle\\!\langle\lambda^{(\ell)}\rangle\\!\rangle=0,\quad\langle\\!\langle\lambda^{(\ell)}\lambda^{(m)}\rangle\\!\rangle=\mathfrak{d}^{2}\delta^{\ell
m}.$ (3)

The variances
$\mathfrak{d}^{(\ell)}=\sqrt{\langle\\!\langle(\lambda^{(\ell)})^{2}\rangle\\!\rangle}\neq
0$ may in general differ from each direction $\ell$, and
$\langle\\!\langle\cdot\\!\cdot\\!\cdot\rangle\\!\rangle$ denotes an average
over both (i) statistical collections of D-particles and (ii) quantum stringy
fluctuations Mavromatos:1998nz , treated by resummation over worldsheet
genera.

Liouville dressing to the vertex operator (1) Ellis:2004stm then induces an
off-diagonal distortion in target-space geometry $\hat{g}$, with $0\ell$
components Kogan:1995lca ; Ellis:1999stm :

$g_{0\ell}(x^{0},{\cal V}_{\|})\sim\epsilon^{2}{\cal
V}_{\ell}t\theta(t)\textrm{e}^{-2\epsilon t}\sim{\cal V}_{\ell}.$ (4)

This results in a local background of Finsler type:
$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$, $h_{0\ell}={\cal
V}_{\ell\|}^{A}\sigma_{A}$, where $\sigma_{A}$, $A=1,2,3$ are appropriate
flavor matrices. This metric deformation (4) then affects the dispersion
relation of neutrinos with mass $m_{\nu}$ via
$k^{\alpha}k^{\beta}g_{\alpha\beta}(k)=-m_{\nu}^{2}$, yielding

$\displaystyle E({\bf k})$ $\displaystyle={\bf k}\cdot{\mbox{\boldmath${\cal
V}$\unboldmath}}_{\|}\pm\lvert{\bf k}\rvert\biggl{(}1+\frac{m_{\nu}^{2}}{{\bf
k}^{2}}+\Bigl{(}{\mbox{\boldmath${\cal V}$\unboldmath}}_{\|}\cdot\frac{{\bf
k}}{\lvert{\bf k}\rvert}\Bigr{)}^{2}\biggr{)}^{1/2}$ $\displaystyle\simeq{\cal
E}_{M}+\frac{g_{s}}{M_{s}}\sum_{\ell}(k^{\ell})^{2}\lambda^{(\ell)}+\frac{{\cal
E}_{M}}{2}\frac{g_{s}^{2}}{M_{s}^{2}}\sum_{\ell}(k^{\ell})^{2}(\lambda^{(\ell)})^{2},$
(5)

where ${\cal E}_{M}=\pm\sqrt{k^{\ell}k_{\ell}+m_{\nu}^{2}}$ denotes the
Minkowski energy with indefinite signature. The flavor structures have been
omitted in the above expressions by taking account of Eq. (3), i.e.,
$\langle\\!\langle\lambda_{A}^{(\ell)}\lambda{}_{B}^{(m)}\rangle\\!\rangle=\mathfrak{d}^{2}\delta^{\ell
m}\delta_{AB}$, since one needs to average (2) over the D-particle
populations, that is

$\displaystyle\langle\\!\langle E\rangle\\!\rangle$
$\displaystyle\eqqcolon\overline{E}(k)\simeq\Bigl{\langle}\\!\\!\Bigl{\langle}\pm\,\bigl{\lvert}{\cal
E}_{M}\bigr{\rvert}\Bigl{(}1+\frac{(\mbox{\boldmath${\cal
V}$\unboldmath}\cdot{\bf k})^{2}}{2{\bf
k}^{2}}\Bigr{)}\Bigr{\rangle}\\!\\!\Bigr{\rangle}$
$\displaystyle\simeq\pm\Bigl{(}1+\frac{g_{s}^{2}}{2M_{s}^{2}}\sideset{}{{}^{\prime}}{\sum}_{\ell}\bigl{(}\mathfrak{d}^{(\ell)}\bigr{)}^{2}k_{\ell}^{2}\Bigr{)}\bigl{(}\overline{k}^{2}\\!+m_{\nu}^{2}\bigr{)}^{1/2},$
(6)

for $\overline{k}\gg m_{\nu}$, where
$\overline{k}\coloneqq\langle\\!\langle\lvert{\bf k}\rvert\rangle\\!\rangle$
is the averaged modulus of ${\bf k}$, and
$\sum^{\prime}(\mathfrak{d}^{2}k^{\ell}k_{\ell})$ represents
$\sim\langle\\!\langle\sum(\lambda k^{\ell}k_{\ell})\rangle\\!\rangle$.

On the other hand, the kinematics of the defect/neutrino scattering further
yields Mavromatos:2012ii ,

$\displaystyle E_{\textrm{i}}$
$\displaystyle=E_{\textrm{f}}+M_{D}(\gamma_{{\cal V}}-1)+\delta v,$ (7)
$\displaystyle{\bf k}_{\textrm{i}}$ $\displaystyle={\bf
k}_{\textrm{f}}+\Delta{\bf k}={\bf k}_{\textrm{f}}+M_{D}\gamma_{{\cal
V}}\mbox{\boldmath${\cal V}$\unboldmath}_{\|},$ (8)

where $(E,{\bf k})_{\textrm{i/f}}$ is the 4-momentum for initial/final state,
$\delta v$ denotes the fluctuation of brane vacuum energy during the
scattering, with D-particle mass being $M_{D}=M_{s}/g_{s}$. Then, the explicit
formula of $\overline{E}_{\textrm{f}}$ arises (on average) from Eqs. (7) and
(2), by noting $\overline{E}_{\textrm{i}}=\overline{E}$, as

$\displaystyle\overline{E}_{\textrm{f}}(\overline{{\bf k}}_{\textrm{f}})$
$\displaystyle=\overline{E}_{\textrm{i}}(\overline{{\bf
k}}_{\textrm{i}})-\Bigl{\langle}\\!\\!\Bigl{\langle}\Bigl{(}\frac{1}{2}M_{D}\mbox{\boldmath${\cal
V}$\unboldmath}_{\|}^{2}+{\cal O}({\cal
V}^{4})\Bigr{)}\Bigr{\rangle}\\!\\!\Bigr{\rangle}$
$\displaystyle\simeq\overline{E}_{\textrm{i}}(\overline{{\bf
k}}_{\textrm{f}})-\frac{g_{s}}{2M_{s}}\sideset{}{{}^{\prime}}{\sum}_{\ell}\bigl{(}\mathfrak{d}^{(\ell)}\bigr{)}^{2}(k_{\textrm{f}}^{\ell})^{2},$
(9)

where the momentum conservation [Eq. (8)] is used, i.e., $\overline{{\bf
k}}_{\textrm{f}}=\overline{{\bf k}}_{\textrm{i}}$ given that
$\langle\\!\langle\mbox{\boldmath${\cal
V}$\unboldmath}_{\|}\rangle\\!\rangle=0$, and $\langle\\!\langle\delta
v\rangle\\!\rangle=0$ is assumed. Note that Eq. (2) reflects total combined
effects of the D-foam on neutrino energy–momentum relations from both metric
distortion and capture/splitting.

Noticeably, antiparticles of spin-$\frac{1}{2}$ fermions can be regarded as
“holes” with negative energies, one arrives at the effectively CPT-violating
dispersion relations for neutrino ($\nu$) and antineutrino ($\bar{\nu}$) in
vacuo (or, for Majorana neutrino with different chirality):

$\begin{split}\overline{E}_{\nu}(\overline{{\bf
k}})&\coloneqq\lvert\overline{E}_{\textrm{f}}^{(+)}(\overline{{\bf
k}};M_{D},m_{\nu})\rvert,\\\ \overline{E}_{\bar{\nu}}(\overline{{\bf
k}})&\coloneqq\lvert\overline{E}_{\textrm{f}}^{(-)}(\overline{{\bf
k}};M_{D},m_{\nu})\rvert,\end{split}$ (10)

where $(+)/(-)$ denotes positive/negative part of Eq. (2) so that a physical
particle always has energy $\overline{E}>0$.

It is essential to understand that the difference between $\nu$’s and
$\bar{\nu}$’s in Eq. (10) follows from the Dirac’s proposal of “hole theory”.
This applies only to fermions such as neutrinos, but not to bosons, as it is
based upon the exclusion principle. In fact, as was elucidated in Refs.
Ellis:2004stm ; Ellis:2008stm ; Li:2009tt ; Mavromatos:2010pk ; Li:2021stm and
will be mentioned in the next section, neutral bosons like photons are subject
to propagate subluminally, independent of their helicities, in this “medium”
of (stringy) vacuum defects. The reason for that may be traced back to the
fact that the propagator of a fermion is similar to the square root of that of
a vector boson.

###  2.1 Propagation velocities

For the discussion of the GRB neutrinos here of interest, we consider an
isotropic foam, which further requires $\lambda^{(\ell)}=\lambda$ for all
$\ell=1,2,3$. In such a case, the asymmetric dispersion relation (10) for
high-energy (anti)neutrinos in this D-foam geometry reads

$\displaystyle\overline{E}_{\nu}(\overline{{\bf k}})$
$\displaystyle=\overline{k}-\frac{g_{s}}{2M_{s}}\mathfrak{d}_{\nu}^{2}\overline{k}^{2}+{\cal
O}(1/M_{D}^{2}),$ (11) $\displaystyle\overline{E}_{\bar{\nu}}(\overline{{\bf
k}})$
$\displaystyle=\overline{k}+\frac{g_{s}}{2M_{s}}\mathfrak{d}_{\nu}^{2}\overline{k}^{2}+{\cal
O}(1/M_{D}^{2}),$ (12)

which reduces to our result in Ref. Li:2022sgs once higher-order corrections
are negligible. Here,
$\langle\\!\langle\lambda^{2}\rangle\\!\rangle=(\mathfrak{d}^{(\ell)})^{2}|_{\ell=1,2,3}\equiv\mathfrak{d}_{\nu}^{2}>0$
can be naturally up to ${\cal O}(1)$. The relativistic limit is used to omit
the mass term, $m_{\nu}\simeq 0$.

To get group velocities, one can certainly employ the relation $v=\partial
E/\partial\lvert{\bf k}\rvert$; it is possible, though, that such velocity law
fails in QG, and that Ellis:2008stm the free propagation of particles may even
disentangle from their dispersion relations.555For that case, both neutrinos
and antineutrinos would be subluminal due to causality-respectful delays
experienced by them as interacting with the foam of Ellis:2008stm ; whereas
there could be superluminal propagation in a type IIB stringy D-foam Li:2009tt
. Nevertheless, from a conservative point of view, we shall still assume
Hamiltonian dynamics, and as such, the dispersion relation (11) then yields a
deformed subluminal neutrino velocity as

$\overline{v}_{\nu}\coloneqq\frac{\partial\overline{E}}{\partial\overline{k}}=1-g_{s}\frac{\mathfrak{d}_{\nu}^{2}\overline{k}}{M_{s}}\simeq
1-{\cal O}\Bigl{(}g_{s}\frac{n_{D}\overline{E}}{M_{s}}\Bigr{)},$ (13)

where we substituted the lowest order dispersion
$\overline{k}\simeq\overline{E}$. The parameter $\mathfrak{d}_{\nu}^{2}$
depends on the density of D-particles, $n_{D}$, which can be essentially
arbitrary in the model, as is the (stringy) quantum-gravity mass
$M_{\textrm{sQG}}\coloneqq M_{s}/(\mathfrak{d}^{2}g_{s})$. Though
$n_{D}=n_{D}(t)$ could in general vary with the cosmological epochs
Mavromatos:2010pk , say, it might evolve when the time elapses, we introduce a
hypothetical uniform foam situation, i.e.,
$n_{D}(t_{\textrm{late}})=n_{D}^{\star}\simeq\textrm{const.}$, at relatively
late eras of the Universe (for, e.g., redshifts $\lesssim 10$).

Similarly, from Eq. (12), the velocity defect, i.e.,
$\delta_{v}^{D}\coloneqq\overline{v}-1$, for an antineutrino propagating in a
quantum D0-brane foam becomes

$0<\delta_{v}^{D}=\frac{\overline{k}}{M_{\textrm{sQG}}}\propto\frac{n_{D}^{\star}}{M_{s}}\overline{E},$
(14)

which implies that antineutrinos are superluminal particles. That is an
important feature of our approach toward D-foam induced LV neutrino
propagation, and is crucial for generating desired phenomenologies which will
be discussed shortly. For our purpose, symmetric corrections of ${\cal
O}(\mathfrak{d}^{2})$ in particle and antiparticle sectors are assumed in the
discussion, such that the amounts of CPT violation are the same for both
$\nu$’s and $\bar{\nu}$’s. (An asymmetry between neutrino and antineutrino
sectors can be involved once there will be a need to reconcile with
phenomenological constraints.)

###  2.2 Lag in travel times of neutrinos

Before closing this section, we remark on the relation of the induced phase-
space dependent metrics (4) with Finsler geometries, which, over past few
years, are known to play a role in new physics as well as quantum gravity. The
Finsler structure depends on both coordinates and momenta, as is precisely the
situation encountered in (4). In some sense, the D-particle recoil may be
viewed as an example of Finsler geometry in string theory. One may define
appropriately the Finsler norm from the metric $\hat{g}({\cal V})$:
$F=[g_{\alpha\beta}(y({\cal V}))y^{\alpha}y^{\beta}]^{1/2}$ (here $y$’s denote
“velocities” in Finslerian frameworks thereof Zhu:2022blp ), and discuss
geodesics in such space-times as in Zhu:2022blp .

However as we have seen above, the effects of D-foam go beyond those encoded
in a Finsler-like metric. Since one need to take account of the statistical
effects of the quantum-fluctuating D-defects, this leads to a more general
structure: stochastic Finsler geometry Mavromatos:2010sco . In fact, for the
isotropic foam, the stochasticity, $\langle\\!\langle{\cal
V}_{\ell}\rangle\\!\rangle=0$ of (3), implies the restoration of Lorentz
symmetry on average but with nontrivial (higher-order) fluctuations expressed
as correlators by $\langle\\!\langle{\cal V}_{\ell}{\cal
V}_{m}\rangle\\!\rangle\propto\mathfrak{d}^{2}\delta_{\ell m}$,
$\mathfrak{d}^{2}\neq 0$. The effects of recoil distortions are washed out
statistically and the metric will be like a Riemannian metric. So one may deal
with particle propagation in an expanding space-time of the
Friedmann–Robertson–Walker (FRW) type as usual. The kinematical aspects of the
recoil-brane scattering on matter also enter in the dispersion relation and
velocity. The latter cannot be cast by a Finsler treatment.

Based upon the above considerations, we estimate the flight time differences
of neutrinos of astrophysical origin in a FRW Universe, where the momentum of
the neutrino is redshifted by cosmological expansion. Given that the scale
factor $a$ is related to the cosmic redshift $z^{\prime}$ by
$a=\frac{1}{1+z^{\prime}}$, the velocity at the time of $z^{\prime}$ becomes

$\overline{v}(\overline{k},z^{\prime})\simeq 1\mp
g_{s}\mathfrak{d}_{\nu}^{2}\Bigl{(}\frac{\overline{k}/a}{M_{s}}\Bigr{)}=1\mp\mathfrak{d}_{\nu}^{2}(1+z^{\prime})\Bigl{(}\frac{\overline{k}}{M_{D}}\Bigr{)}.$
(15)

The comoving distance $x_{c}$ between the source of the neutrinos, for example
a GRB, and the Earth is

$x_{c}=\int_{t_{0}}^{t_{\textrm{h/l}}}\overline{v}(\overline{k})\frac{\textrm{d}t}{a},$
(16)

where $t_{\textrm{h/l}}$ is the time when the high-/low-energy neutrinos
arrive. If they are emitted at the same time $t_{0}$ at the GRB, the time lag
can be obtained,

$\delta t\coloneqq
t_{\textrm{h}}-t_{\textrm{l}}\simeq\pm\,\mathfrak{d}_{\nu}^{2}\frac{\delta\overline{k}}{M_{D}}\int_{0}^{z}\frac{(1+z^{\prime})}{H(z^{\prime})}\textrm{d}z^{\prime},$
(17)

where
$\delta\overline{k}=\overline{k}_{\textrm{h}}-\overline{k}_{\textrm{l}}$, and
the Hubble parameter $H(z^{\prime})$ depends on the cosmological model. Since
for late eras, at which we assume an approximately constant density of
defects, the Hubble expansion on the D3-brane world is not affected
Mavromatos:2010sco , the standard Cold-Dark Matter model with a positive
cosmological constant ($\Lambda>0$) would be a good approximation of reality,
as current observation suggests, then we have
$H(z^{\prime})=H_{0}\sqrt{\Omega_{\textrm{M}}^{0}(1+z^{\prime})^{3}+\Omega_{\Lambda}^{0}}$,
where $H_{0}$ is the Hubble constant. Hence,

$\delta
t\simeq\pm\,\mathfrak{d}_{\nu}^{2}H_{0}^{-1}\frac{\delta\overline{E}}{M_{D}}\int_{0}^{z}\textrm{d}z^{\prime}\frac{(1+z^{\prime})}{\sqrt{\Omega_{\textrm{M}}^{0}(1+z^{\prime})^{3}+\Omega_{\Lambda}^{0}}},$
(18)

here $\overline{E}\simeq\overline{k}$ is identified as the neutrino
(antineutrino) energy observed on the Earth and $\delta\overline{E}\simeq
E_{\textrm{h}}$. For neutrinos, there is a “delay” in their arrival times
($\delta t_{\nu}>0$), i.e., high-energy neutrinos arrive later compared to
lower-energy ones; whereas for antineutrinos, $\delta t_{\bar{\nu}}<0$,
corresponding to the arrival time “advance”.

##  3 CPT-violating neutrino and phenomenological aspects

It has been realized, on the observational side, mainly over the last decade
that high-energetic neutrinos are one of the most promising portals to LV
physics Jacob:2006gn ; Amelino-Camelia:2009imt ; Amelino-Camelia:2015nqa . In
this respect, Amelino-Camelia et al. developed a strategy of analysis on
IceCube data to detect LV-modified laws of propagation for neutrinos in a
quantum space-time Amelino-Camelia:2016fuh , and recently a Lorentz-violating
picture is suggested by a series of model-independent studies on time-of-
flight lags of IceCube neutrino events with respect to the purportedly
associated light signals arriving from GRBs Amelino-Camelia:2016ohi ;
Huang:2018ham ; Huang:2019etr ; Huang:2022xto , which we proceed to review
briefly. In the residue of this section we will show that this may serve as a
support to the space-time foam model just presented.

###  3.1 Speed variation in IceCube GRB neutrinos

In Refs. Amelino-Camelia:2016fuh ; Amelino-Camelia:2016ohi ; Huang:2018ham ;
Huang:2019etr ; Huang:2022xto , adopting a LV-deformed dispersion relation
with a generic form, the authors get the modified propagation velocity for
neutrinos as

$v(E)=1-s_{n}\frac{1+n}{2}\Bigl{(}\frac{E}{E_{\textrm{LV},n}}\Bigr{)}^{n}$
(19)

where $n=1,2$ corresponds to linear or quadratic dependence of the energy,
$s_{n}=\pm 1$ is a sign factor of LV correction, and $E_{\textrm{LV},n}$ is
the $n$th-order LV scale. Neutrino mass, which has been constrained within 1
eV Huang:2014qwa , can be safely neglected in the velocity formula for
neutrinos with energies around or beyond TeV scale. Supposing the linear term
(i.e., $n=1$) in (19) dominates, a regularity fitting well with TeV and PeV
GRB neutrinos from IceCube (including also near-TeV events Huang:2019etr ) is
observed, indicating a neutrino speed variation
$v(E)=1-sE/E_{\textrm{LV}}^{\nu}$ with $s=\pm 1$, at a scale of

$E_{\textrm{LV}}^{\nu}=(6.4\pm 1.5)\times 10^{17}~{}\textrm{GeV},$ (20)

which is close to the Planck scale $E_{\textrm{Pl}}\simeq 1.22\times 10^{19}$
GeV. Such an energy scale is consistent with various time-of-flight
constraints, available today, from MeV neutrinos of supernova 1987A
Ellis:2008fc and that from higher-energy neutrinos registered at IceCube,
e.g., Wang:2016lne ; it is also compatible with the constraints Ellis:2018ogq
; Laha:2018hsh ; Wei:2018ajw from recent multimessenger observations of blazar
TXS 0506+056 coincident with $\sim 290$ TeV track-like neutrino event
IceCube:2018dna . We mention in passing that a light speed variation of
$v(E)=1-E/E_{\textrm{LV}}^{\gamma}$ with $E_{\textrm{LV}}^{\gamma}\gtrsim
3.60\times 10^{17}$ GeV has also been suggested from time-of-flight studies on
the energetic radiation from GRBs Shao:2009bv ; Zhang:2014wpb ; Xu:2016zxn ;
Amelino-Camelia:2016ohi ; Liu:2018qrg ; Zhu:2021pmw and from flares of active
galactic nuclei Li:2020uef . This latter LV scale is not the same as the LV
scale of neutrinos (20) but of the same order of magnitude.

Intriguingly, it is found that there exist both time “delay” ($s=+1$) and
“advance” ($s=-1$) events, which can be interpreted with different propagation
properties between neutrinos and antineutrinos Huang:2018ham ; Huang:2019etr ;
Huang:2022xto , implying that neutrinos are superluminal and antineutrinos are
subluminal, or vice versa, due to opposite signs of LV sign factor $s$. This
proposal is at present only a hypothesis, but a reasonable one, because the
IceCube detector cannot distinguish between neutrinos and antineutrinos
(except using the Glashow resonance, to be mentioned later). It is thus
necessary to verify this hypothesis with further experimental tests, and to
check then whether the revealed regularity Amelino-Camelia:2016ohi ;
Huang:2018ham ; Huang:2019etr ; Huang:2022xto can still persist or not with
more neutrino events by IceCube (or by other facilities) available in the
future.

###  3.2 D-foam as an explanation for neutrino in-vacuo dispersion feature

Notice that the phenomenological LV picture revisited in the previous
subsection is supported by a number of IceCube events among the gap over 4
orders of magnitude in neutrino energy scale, ranging from a few hundred GeV
up to about 2 PeV, with all these GRB neutrino events falling on a pair of
inclined lines surprisingly, see Refs. Amelino-Camelia:2016ohi ; Huang:2018ham
; Huang:2019etr ; Huang:2022xto and also Amelino-Camelia:2016fuh . Such result
yields a strong indication of a linear energy dependence in the propagation
speed of cosmic neutrinos, $\lvert v-1\rvert\sim{\cal
O}(E/E_{\textrm{LV}}^{\nu})$, with $E_{\textrm{LV}}^{\nu}$, characterizing
such a linear suppression of (leading) LV effect, at about $10^{17}$ GeV,
i.e., Eq. (20). On the other hand, the presence of the opposite sign factors
between neutrinos with their antiparticle counterparts Huang:2018ham clearly
indicates a CPT-violating propagation of neutrinos in cosmic space. Hence, it
is clear that the above two distinct LV properties, as exposed from current
time-of-flight data Huang:2018ham ; Huang:2019etr ; Huang:2022xto , are
consistent with the expected behavior for (anti)neutrinos propagating in a
gravitational “medium” of D-brane defects.

Although it is not known from the analyses on IceCube data whether neutrinos,
or antineutrinos, are superluminal, the D-particle foam scenario predicts that
the latter would propagate slightly faster than the constant speed of light,
so considering also the constraint that neutrinos and antineutrinos have the
same amounts of speed variation, we can relate the D-foam induced velocity
defect $\delta_{v}^{D}$ to the corresponding variation implied by the generic
dispersion (19). This yields,

$\begin{split}&\delta_{v}^{D}\simeq-\frac{E}{E_{\textrm{LV}}^{\nu}},\quad\textrm{for
subluminal}~{}~{}\nu\textrm{'s};\\\
&\delta_{v}^{D}\simeq\frac{E}{E_{\textrm{LV}}^{\nu}},\quad\textrm{for
superluminal}~{}~{}\bar{\nu}\textrm{'s}.\end{split}$ (21)

With the established correspondence (21) for the neutrino velocities, we can
then identify a combination of the foam parameters,
$M_{D}/\mathfrak{d}^{2}\simeq M_{\textrm{sQG}}$, i.e., stringy QG mass scale,
as the linear-order Lorentz violation scale $E_{\textrm{LV}}$ determined by
IceCube neutrinos:

$\frac{\mathfrak{d}_{\nu}^{2}g_{s}}{M_{s}}\simeq\frac{1}{E_{\textrm{LV}}^{\nu}}\simeq
1.6\times 10^{-18}~{}\textrm{GeV}^{-1},$ (22)

or, explicitly one has,

$M_{\textrm{sQG}}^{(\nu)}\simeq E_{\textrm{LV}}^{\nu}\simeq 6.4\times
10^{17}~{}\textrm{GeV}.$ (23)

Such result gives an estimation of the neutrino QG scale $\sim 10^{17}$ GeV
for the string/D-particle foam scenario. This is very much in line with one’s
intuitive expectation that the scale $M_{\textrm{sQG}}$ would be comparable to
the Planck mass: $M_{\textrm{sQG}}\sim M_{\textrm{Pl}}~{}(\simeq 10^{19}$
GeV).

Moreover, we note that the string model from the last section exhibits
likewise nontrivial optical ($\gamma$) properties of the vacuum in terms of a
velocity deformation for photon with different energy $\overline{\omega}$:

$\delta_{v}^{D}=-\frac{\overline{\omega}}{M_{\textrm{sQG}}^{(\gamma)}}<0.$
(24)

This arises from the capture/splitting process of photon open strings by
D-defects, in much the same way described in the previous section for the
neutrino-D-particle interactions. The main difference is that photons are
necessarily subluminal, thus, following our proposal Li:2021stm of
interpreting the light speed variation from GRB photons with (anisotropic)
D-foam models, one can do the same thing in the context of isotropic foam
scenario considered here. The arguments are similar to that of Ref. Li:2021stm
(see also Li:2021tcv ), and will not be repeated here. It is essential to
notice that, in that case, an estimate about the photon’s QG scale can be got
as $M_{\textrm{sQG}}^{(\gamma)}\simeq E_{\textrm{LV}}^{\gamma}\gtrsim
3.6\times 10^{17}$ GeV. Hence, we immediately observe

$M_{\textrm{sQG}}^{(\nu)}\gtrsim M_{\textrm{sQG}}^{(\gamma)},$ (25)

i.e., $M_{\textrm{sQG}}$ for neutrinos may slightly be higher than that of
photons but of the same order of magnitude. It should be noticed that this is
exactly what the D-foam model expects since, from a theoretical viewpoint, the
value of $M_{\textrm{sQG}}$ could depend upon quantities such as the string
coupling $g_{s}$, the density of defects $n_{D}$ and, importantly, on the
strength of particle interactions with foam defects, which may not be
universal among particle species Ellis:2003sfd .

Indeed, the neutrino interaction with the space-time D-defects could be very
slightly suppressed compared with those of photons, because, in such a
framework Li:2022ugz , only species that transform as trivial representations
of the SM gauge group, i.e., neutral particles, are susceptible to the
D-particle foam, in which case the fact that neutrinos are fermions with
nontrivial $\textrm{SU}(2)_{L}$ properties renders such foam effects somewhat
weakened. Nevertheless, neutrinos are at least electrically neutral
excitations, so that the space-time foam appears not completely transparent to
them, by invoking a possible background induced breaking of SU(2) gauge
symmetry of the Standard Model. Hence the velocities of neutrinos would
deviate from the low-energy velocity of light naturally less than photons, so
that, $\lvert\delta_{v(\nu)}^{D}\rvert\lesssim\delta_{v(\gamma)}^{D}$, which
just corresponds to the observation (25) inferred from the findings of speed
variation of high-energy cosmic neutrinos Amelino-Camelia:2016ohi ;
Huang:2018ham ; Huang:2019etr ; Huang:2022xto and cosmic photons Shao:2009bv ;
Zhang:2014wpb ; Xu:2016zxn ; Liu:2018qrg ; Zhu:2021pmw ; Li:2020uef .

We should emphasize, in passing, that this avoids also the conflict, as
indicated elsewhere Crivellin:2020oov for models where $\textrm{SU}(2)_{L}$
gauge invariance is still kept, with existing tight bounds on LV for the
charged leptons (see, e.g., Ref. Li:2022ugz ; He:2022jdl and those quoted in
Crivellin:2020oov ). It is essential to notice that, in the D-foam case,
Lorentz violation in neutrinos does not imply LV in the charged lepton (which
is actually Lorentz invariant for specifically stringy reasons to be mentioned
later), or vice versa. Therefore the result (22) would not lead to
unacceptably large Lorentz-violating effects in the charged-lepton (e.g.,
electron/positron) sectors, and the difficulty encountered by
Crivellin:2020oov for the field-theoretic LV-scenario explanation does not
arise when exploring a D-foam interpretation.

From Eq. (22), we can further deduce,

$\mathfrak{d}_{\nu}^{2}\sim
10^{-18}\,\frac{M_{s}}{g_{s}}~{}\textrm{GeV}^{-1}.$ (26)

This means that the dimensionless variance $\mathfrak{d}^{2}$, expressing
stochastic fluctuations of the D-particle recoil velocity, relates to the
value of the mass of the foam defect $M_{s}/g_{s}$, which in the modern
version of string theory is essentially a free parameter. So for
$\mathfrak{d}^{2}\sim{\cal O}(1)$ for instance, a sub-Planckian D-particle
mass $M_{s}/g_{s}\sim 10^{18}$ GeV is required so that the D-foam generates
the phenomenologically suggested neutrino speed variation, by means of a
Lorentz-/CPT-violating term $(g_{s}/M_{s})\mathfrak{d}^{2}k^{2}$ between $\nu$
and $\bar{\nu}$ sectors.

We should note that this type of CPT symmetry breaking may also lead to the
observed baryon asymmetry in a natural way through gravitational leptogenesis
of the early Universe Mavromatos:2012ii , which requires $M_{s}/g_{s}\sim
10^{25}\mathfrak{d}^{2}~{}\textrm{GeV}$ so as to provide the physically
observed lepton and, thus, baryon asymmetry. This seems to go beyond our
result in Eq. (26). However, as mentioned above, in space-time foam
situations, the density of D-particles $n_{D}$, which essentially the CPT-
violating parameter, $\mathfrak{d}^{2}$, as a (Gaussian) variance of the
fraction variable $\lambda$, depends upon, may vary with the cosmological time
scale $t$, in the sense that their bulk distribution may not be constant all
the time. We may expect, in general, a time dependent variance,
$\langle\\!\langle\lambda^{2}\rangle\\!\rangle(t)\equiv\mathfrak{d}^{2}(t)\propto
n_{D}(t)$. Thus, it could be possible that at very early eras of the Universe
there is a relatively dilute, but still statistically significant, population
of D-particles, by which a neutrino field can be encountered, thereby
resulting in a somewhat small variance $\mathfrak{d}$, upon averaging (2) over
the D-particle populations. As the time elapses, the brane Universe, which
roams in the bulk space, may however move into a region densely populated with
D-particles, so that for late epochs, say, redshifts $z\lesssim{\cal O}(10)$,
neutrinos coming from cosmologically remote sources, either GRBs or active
galaxies, interact with D-particles more frequently as they propagate over a
distance in the foamy space-time, yielding stronger stochastic effects ${\cal
O}(\mathfrak{d}^{2})$. For a Grand-Unified-Theory scale $\sim{\cal
O}(10^{15})$ GeV D-particle mass for instance, one acquires

$\mathfrak{d}_{\nu}(t_{\textrm{early}})\simeq
10^{-5},~{}~{}\mathfrak{d}_{\nu}(t_{\textrm{late}})\simeq 0.03,$ (27)

with corresponding distribution for the stochastic fluctuating D-particle
recoil velocities:666We stress here that the modeling of recoil operator
${\cal V}_{\ell}$ (or variable $\lambda$) by a Gaussian process is not based
on analysis of the underlying string theory, but a reasonable assumption for
characterizing stochasticity in recoiling space-time (D-)defects
Mavromatos:2005sfm .

${\cal
D}(\lambda;\mathfrak{d}^{2})\sim\exp\biggl{[}-\frac{1}{2}\biggl{(}\frac{\lambda}{\mathfrak{d}(t)}\biggr{)}^{2}\biggr{]},$
(28)

which then differs with time scale $t$, particularly between early
cosmological epochs, ${\cal D}_{\textrm{early}}$, and late eras, ${\cal
D}_{\textrm{late}}$, for reasons explained above. Hence, in such sense, there
is no conflict between our result (26) (or (23)) for explaining neutrino speed
variation from GRBs with the constraint from CPT-violating effects in the
early Universe Mavromatos:2012ii , and further, our estimate (27) indicates a
much denser, but uniform to a large extent, bulk D-particle foam from late- to
current-era Universe ($z\leq 10$).

In view of the above discussion, we conclude here that the finding of the
neutrino speed variation of the form of $v(E)=1\mp E/E_{\textrm{LV}}^{\nu}$
from IceCube observations is compatible with the string/D-foam motivated
deformation of velocity of neutrinos (13) (and of antineutrinos (14)) with
$M_{\textrm{sQG}}\simeq 6\times 10^{17}$ GeV, approaching the Planck mass,
through relating the stringy QG scale to the linear-order LV scale
$E_{\textrm{LV}}^{\nu}$ from GRBs. It is crucial that the propagation
properties suggested as differing between neutrinos and antineutrinos are
explained by means of CPT-violating aspects of such D-brane foam model of
Lorentz violation. This fact can thus in turn serve as a support to this type
of models inspired by string theory.

##  4 Inhibited superluminal neutrino in-vacuo decay

It was claimed however that assuming the usual conservation laws of energy and
momentum, as widely postulated in field-theoretic LV models, superluminal
neutrinos would dissipate much of their energies through kinematically allowed
anomalous processes in a vacuum Cohen:2011hx , such as, Cherenkov radiation
($\nu\rightarrow\nu\gamma$), neutrino splitting
($\nu\rightarrow\nu\nu\bar{\nu}$), and bremsstrahlung of electron-positron
pairs ($\nu\rightarrow\nu ee^{+}$), among which the last one, also known as
vacuum $ee^{+}$ pair emission (VPE), a neutral-current process, dominates the
neutrino energy losing. Such processes would result in significant depletion
of cosmic neutrino fluxes at high energies beyond which no neutrino should
arrive at Earth. This has been used to bound superluminal velocities of high-
energy neutrinos from IceCube IceCube:2013cdg ; IceCube:2014jkq ;
IceCube:2016uab ; IceCube:2018dna ; in particular, powerful constraints on LV
energy scale of $\gtrsim(10^{3}-10^{5})\times E_{\textrm{Pl}}$ are reported
Borriello:2013ala ; Stecker:2013jfa ; Diaz:2013wia ; Wang:2020tej . The same
constraint applies to antineutrinos, if they are slightly superluminal, for
the CP-conjugated processes as one can understand.

However, these constraints become inapplicable, as we shall illustrate in the
following, in the sense that the dominant energy-losing channels may be
inhibited/forbidden in D-particle backgrounds, owing to so-induced deformation
of energy–momentum conservation.

To see this clearly, we perform here a threshold analysis on pair
bremsstrahlung as an illustration. In the case of space-time D-foam, high-
energy antineutrinos that are superluminal particles are expected to undergo
such processes and to lose energy until they are at or below the threshold.
Caution, however, is needed here. Due to the presence of recoiling D-defects
near the braneworld, the energy conservation law in the common sense may be
violated Ellis:2000sf , though, during reactions like
$\bar{\nu}\rightarrow\bar{\nu}ee^{+}$, the total energy remains conserved once
the kinetic energy of a heavy D-particle,
$E_{D\textrm{-kin}}=M_{D}(\gamma_{{\cal
V}}-1)\simeq\frac{1}{2}M_{D}\lvert\mbox{\boldmath${\cal
V}$\unboldmath}\rvert^{2}$, is taken into account. Indeed, this fact has
already been used in Eq. (7) as deriving the modified relativistic dispersion;
in general, one has Ellis:2000sf ,

$\displaystyle\sum_{i}^{(N)}(\pm)E_{i}$
$\displaystyle\simeq\frac{1}{2}\frac{M_{s}}{g_{s}}\mbox{\boldmath${\cal
V}$\unboldmath}_{(N)}^{2},$ (29) $\displaystyle\sum_{i}^{(N)}(\pm){\bf p}_{i}$
$\displaystyle\simeq M_{D}\mbox{\boldmath${\cal V}$\unboldmath}_{(N)},$ (30)

for a multi-($N$-)particle reaction. Here, $\mbox{\boldmath${\cal
V}$\unboldmath}_{(N)}$ is a proper generalization of (2), i.e.,
$\mbox{\boldmath${\cal V}$\unboldmath}=\sum_{i}(\pm)({\bf p}_{i}/M_{D})$, and
the notation $\sum(\pm)$ represents, for example in the case of
$\bar{\nu}\rightarrow\bar{\nu}ee^{+}$ as we discuss now:

$\displaystyle\sum_{i}(\pm)E_{i}$ $\displaystyle=E_{\textrm{in}}-\sum
E_{\textrm{out}}$ $\displaystyle=E-(E^{\prime}+E_{e^{-}}+E_{e^{+}}),$ (31)

where $E$, $E^{\prime}$ denote the energies for the incoming $\bar{\nu}$ and
outgoing $\bar{\nu}$, respectively. Hence, the sum of the energies of all
observable particles is not conserved due to a recoil-induced loss upon
averaging over the foam:

$\sum_{i}^{(N)}(\pm)\overline{E}_{i}\eqqcolon\delta\overline{E}_{D}^{(N)}=\frac{M_{D}}{2}\langle\\!\langle\mbox{\boldmath${\cal
V}$\unboldmath}_{(N)}^{2}\rangle\\!\rangle,$ (32)

which gives nonzero value due to the nontrivial stochastic fluctuations
$\langle\\!\langle{\cal V}^{\ell}{\cal V}_{\ell}\rangle\\!\rangle\neq 0$,
despite vanishing $\langle\\!\langle{\cal V}_{\ell}\rangle\\!\rangle$.

We should mention however that the energy lost $\delta\overline{E}_{D}$ due to
stochastic interactions with the D-defects is relevant mainly for reactions
with neutral elementary particles in the initial state, for which there is no
obstacle to interact with the D-particles. In particular, the D-foam is
predominantly transparent to (electrically) charged matter on account of
charge conservation Ellis:2003sfd , as justified by the strength of Cherenkov
constraints for the electron/positron sectors Li:2022ugz ; He:2022jdl .777The
Crab Nebula severely limits electron LV effect with the tightest constraint:
$E_{\textrm{LV}}^{e}\gtrsim 10^{26}~{}\textrm{GeV}\gg E_{\textrm{Pl}}$ for
$v_{e}-1\simeq E/E_{\textrm{LV}}^{e}$ Li:2022ugz ; He:2022jdl , by using PeV
$\gamma$-ray observed recently by LHAASO. Hence, mainly the anomalous decay
processes of the neutrino species, as discussed below, are affected dominantly
by the foam effect.

Consider now the superluminal $\bar{\nu}$ decay via
$\bar{\nu}\rightarrow\bar{\nu}ee^{+}$, the averaged energy–momentum
(non)conservation that follows from Eqs. (4) and (32) reads

$\displaystyle\overline{E}$
$\displaystyle=\overline{E}{}^{\prime}+\overline{E}_{e^{-}}+\overline{E}_{e^{+}}+\delta\overline{E}_{D}^{(4)},$
(33) $\displaystyle\overline{k}$
$\displaystyle=\overline{k}{}^{\prime}+\overline{p}_{e^{-}}+\overline{p}_{e^{+}},$
(34)

where we denote the 4-momenta of the emitted $\bar{\nu}$ and $ee^{+}$ pairs
as, $k^{\prime\alpha}=(\overline{E}{}^{\prime},\overline{{\bf
k}}{}^{\prime})$, $p_{e/e^{+}}^{\alpha}=(\overline{E},\overline{{\bf
p}})_{e/e^{+}}$, respectively. The spatial 3-momentum is conserved on the
average (34) due to the zero mean (3) of ${\cal V}$ arising from isotropy. The
Lorentz-violating energy defect, $\delta E$, is

$\displaystyle\overline{E}-\overline{k}\eqqcolon\delta E(\overline{{\bf
k}})\geq$ $\displaystyle~{}(\overline{E}_{e^{-}}-\overline{p}_{e^{-}})$
$\displaystyle\,+(\overline{E}_{e^{+}}-\overline{p}_{e^{+}})+\delta\overline{E}_{D}^{(4)}.$
(35)

In D-particle foam models, Lorentz violation is absent for charged-lepton
sectors, as explained:

$\overline{E}_{e^{\mp}}=\sqrt{m_{e}^{2}+\overline{p}_{e^{\mp}}^{2}}\simeq\overline{p}_{e^{\mp}}\Bigl{(}1+\frac{m_{e}^{2}}{2\overline{p}_{e^{\mp}}^{2}}\Bigr{)}.$
(36)

Inserting the modified dispersion for (anti)neutrinos (12) in Eq. (4), and
using (36) yields,

$\delta\overline{E}_{D}^{(4)}-\frac{g_{s}}{2M_{s}}\mathfrak{d}_{\nu}^{2}\overline{k}^{2}\leq-\frac{m_{e}^{2}}{2\overline{p}_{e^{-}}}-\frac{m_{e}^{2}}{2\overline{p}_{e^{+}}},$
(37)

from which the threshold condition can then be read off, by plugging in
$E_{\textrm{thr}}\lesssim\overline{E}\simeq\overline{k}=2\overline{p}_{e/e^{+}}$:

$\Bigl{(}\delta\overline{E}_{D}^{(4)}-\frac{g_{s}}{2M_{s}}\mathfrak{d}_{\nu}^{2}\overline{E}^{2}\Bigr{)}\overline{E}(\overline{{\bf
k}})\simeq-2m_{e}^{2},$ (38)

where we used the fact that the $ee^{+}$ pair takes most of the total
momentum, so that $E^{\prime}\simeq 0$ at threshold. Above, the energy
violation, $\delta\overline{E}_{D}$, in a 4-particle interaction follows from
Eq. (32) for $N=4$. The amount of such losses, during $\bar{\nu}$ Cherenkov-
like decays in the D-foam backgrounds, can be estimated, to leading order, as

$\delta\overline{E}_{D}^{(4)}\simeq\frac{g_{s}}{M_{s}}2\varsigma_{I}^{\nu}\overline{E}^{2}+\ldots,$
(39)

where the dimensionless factor $\varsigma_{I}$ that controls the intensity of
energy violation during reactions in which neutrinos (antineutrinos) get
involved is a priori distinct from the QG parameter $\mathfrak{d}^{2}$ that
appears in the modified dispersions (for details on (39), see Appendix A or
Ellis:2000sf ) but in principle of the same order of magnitude. The fact that
$\delta\overline{E}_{D}\geq 0$ (or $\varsigma_{I}>0$), i.e., energy is lost
during particle interactions in the D-foam, follows from the underlying
stringy treatment Mavromatos:1998nz of the recoil D-brane deformation. The
condition that observed neutrinos are near or below the threshold energy then
yields,

$-\frac{g_{s}}{2M_{s}}(4\varsigma_{I}^{\nu}-\mathfrak{d}_{\nu}^{2})\overline{k}^{3}\leq
2m_{e}^{2}.$ (40)

This inequality means that the stability of high-energy IceCube events can
merely provide bounds on a combined quantity of the fundamental parameters of
the foam. For example, given the most energetic event, #35 with energy of 2
PeV IceCube:2013cdg ,888A higher-energy $2.6$ PeV track event ATel #7856
IceCube:2016uab is deduced to be a “delay” event Huang:2018ham which
corresponds to a subluminal neutrino in our case, hence cannot be used to cast
bounds from (40). of the time “advance” type Huang:2018ham , and hence
probably a superluminal antineutrino event, we then infer a limit from (40):

$\lvert\mathfrak{d}_{\nu}^{2}-4\varsigma_{I}^{\nu}\rvert
g_{s}\sqrt{\alpha^{\prime}}\lesssim 1.3\times 10^{-25}~{}\textrm{GeV}^{-1}.$
(41)

Here $\alpha^{\prime}$ is the string’s Regge slope, related to the string mass
scale via $M_{s}=(\alpha^{\prime})^{-1/2}$.

It is obvious that the constraint (41) set by the threshold effect of
$\bar{\nu}\rightarrow\bar{\nu}ee^{+}$ for superluminal antineutrino with
measured energy at $\sim 2$ PeV has nothing to do with the value we get for
the stringy QG scale in Eq. (22) (or (23)). To understand this situation from
another viewpoint, we invoke the explicit form of VPE threshold in D-foam
cases from the equation (40):

$E_{\textrm{thr}}^{\textrm{VPE}}\lesssim\biggl{[}\frac{M_{D}m_{e}^{2}}{(\mathfrak{d}_{\nu}/2)^{2}-\varsigma_{I}^{\nu}}\biggr{]}^{1/3},\quad\textrm{for}~{}~{}\bar{\nu}\textrm{'s}.$
(42)

It now becomes clear that the above kinematical threshold is finite only if
$4\varsigma_{I}<\mathfrak{d}^{2}$; otherwise, it would be either infinite (for
$\mathfrak{d}^{2}=4\varsigma_{I}$) or imaginary
($\mathfrak{d}^{2}<4\varsigma_{I}$), implying that the pair emission is not
permitted to happen in a vacuum. Even in the case of
$\mathfrak{d}^{2}>4\varsigma_{I}$, the process could be effectively inhibited
if one has $\varsigma_{I}\approx(\mathfrak{d}/2)^{2}$, in which case the
energy threshold would be bumped up to a very high energy scale, for instance
of order PeV, so that PeV-scale (anti)neutrinos will not be depleted by
$\bar{\nu}\rightarrow\bar{\nu}ee^{+}$. We are then able to observe
superluminal events of such energy as detected by IceCube in spite of implying
a very small difference between $\varsigma_{I}$ and
$\frac{1}{4}\mathfrak{d}^{2}$, namely, one may expect that (in late-era
Universe),

$\displaystyle\varsigma_{I}^{\nu}\sim 3\times
10^{-19}\,\frac{M_{s}}{g_{s}}~{}\textrm{GeV}^{-1},$ (43)

and that,

$\displaystyle(\mathfrak{d}_{\nu}^{2}-4\varsigma_{I}^{\nu})\lesssim 8.4\times
10^{-8},\ \ \textrm{for}~{}~{}\mathfrak{d}_{\nu}^{2}>4\varsigma_{I}^{\nu},$
(44)

which follows from (41) and the fact that, for $\mathfrak{d}^{2}\lesssim{\cal
O}(1)$, we have $M_{s}/g_{s}<6.4\times 10^{17}~{}\textrm{GeV}$ from Eq. (22).
The constraint (44), corresponding to a threshold energy around 2 PeV, which
is at the desired order for observing PeV IceCube events, as explained, is not
unnaturally small in the context of D-particle foam models, where, as we have
mentioned, both the parameters of the foam, $\mathfrak{d}^{2}$ and
$\varsigma_{I}$, are in general free to be adjusted.

We emphasize the fact that there is energy loss in particle reactions, as a
result of the nontrivial recoil of defects in the string/D-particle foam,
distinguishes our approach from those field-theoretic models, where energy
remains conserved in the usual sense though Lorentz invariance is broken. It
is the energy losses caused by the quantum D-brane foam in this approach that
raise the thresholds for superluminal neutrino decay, so as to permit a stable
propagation. Hence, in such sense, very tight constraints so far cast by means
of pair production threshold analyses Borriello:2013ala ; Stecker:2013jfa ;
Diaz:2013wia ; Wang:2020tej , which aim at limiting superluminal velocities
within LV frameworks that entail a usual conservation of energy and momentum,
are not applicable to our case. This can be understood more clearly if
inspecting again the foam-modified VPE threshold (42) but adopting the part of
$M_{D}/\mathfrak{d}^{2}$ from Eq. (22):

$E_{\textrm{thr}}^{\textrm{VPE}}\leq(4m_{e}^{2}{\cal E}_{*}^{\nu})^{1/3},$
(45)

or approximately, we may write $E_{\textrm{thr}}\simeq(4m_{e}^{2}{\cal
E}_{*}^{\nu})^{1/3}$, corresponding to the emission of a zero energy
antineutrino. Here a new energy scale ${\cal E}_{*}$ is defined as

${\cal
E}_{*}^{\nu}\coloneqq\frac{E_{\textrm{LV}}^{\nu}}{1-4\varsigma_{I}^{\nu}E_{\textrm{LV}}^{\nu}/M_{D}}.$
(46)

This indicates in a clear manner that the threshold limits as established in,
e.g., Refs. Borriello:2013ala ; Stecker:2013jfa ; Diaz:2013wia ; Wang:2020tej
, are actually imposed on the scale ${\cal E}_{*}$, though can be evaded if
${\cal E}_{*}\leq 0$ or easily satisfied in case of ${\cal E}_{*}>0$ by proper
assignment for the value of $\varsigma_{I}$, but not imposed upon the actual
neutrino Lorentz violation scale $E_{\textrm{LV}}^{\nu}$, which corresponds to
$M_{D}/\mathfrak{d}^{2}$, as in Eq. (22). The latter, as stressed, may only be
limited by in-vacuo velocity dispersion effect via time-of-flight analyses,
such as those performed in Refs. Amelino-Camelia:2016fuh ; Amelino-
Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ; Huang:2022xto , wherein a
neutrino speed variation at about $10^{17}$ GeV is suggested from IceCube
neutrinos when associated with GRB candidates, or studies thereof Wang:2016lne
; Ellis:2018ogq ; Laha:2018hsh ; Wei:2018ajw , where comparable flight-time
bounds are obtained. Here, we end up with a consistent description for the
findings of Refs. Huang:2018ham ; Huang:2019etr ; Huang:2022xto with the $\nu$
decay threshold constraint in the (stringy) QG framework adopted here.

We comment briefly here, for completeness, on a previous argument
Stecker:2014oxa of a possible $\sim 2$ PeV cutoff in the $\nu(\bar{\nu})$
spectrum inferred from initial IceCube data set IceCube:2013cdg , which could
be due to novel LV processes, particularly the neutrino splitting (NS), which,
as argued in Stecker:2014oxa , might play a more important role than pair
emission by the neutrinos. Though the claim of this drop off and its
interpretation with neutrino splittings are outdated, and actually disfavored
by more recent data (see below for details),999We are grateful to an anonymous
referee who reminded us that the cutoff at about 2 PeV has proven an outdated
proposal, and who encouraged us to comment on the role played by the newly
observed Glashow resonance at IceCube IceCube:2021rpz . it may still be useful
to provide an additional refutation with our approach, by invoking the
kinematical threshold for that LV splitting channel:101010We postpone the
discussion of superluminal $\bar{\nu}$-splitting, which is similar to that of
vacuum pair emission, to Appendix B.

$E_{\textrm{thr}}^{\textrm{NS}}\simeq\biggl{[}9m_{\nu}^{2}\frac{M_{s}}{g_{s}(\mathfrak{d}_{\nu}^{2}-2\varsigma_{I}^{\nu})}\biggr{]}^{1/3}.$
(47)

By adopting again the case in which
$\mathfrak{d}^{2}>4\varsigma_{I}(>2\varsigma_{I})$, and tuning their values,
as was implemented in the VPE case, the anomalous decay through splitting into
3 neutrinos would then lead to a drop in $\bar{\nu}$ (not $\nu$) fluxes above
2 PeV. However, we find that this requires
$(\mathfrak{d}^{2}-2\varsigma_{I})<{\cal O}(10^{-18}\\!-\\!10^{-19})$, which
is inconsistent with (44). This implies that there exists some (unnatural?)
cancellation in $\mathfrak{d}^{2}-2\varsigma_{I}$, indicating again, but from
the point of view of the string model that the cutoff claimed elsewhere
Stecker:2014oxa seems very suspicious.

##  5 Discussion and conclusion

It is worthwhile to stress again that the finding of neutrino speed variation,
and the consequent Lorentz/CPT violation for cosmic neutrinos as in Refs.
Amelino-Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ; Huang:2022xto and
Amelino-Camelia:2016fuh , which we have devoted ourself to understand in Ref.
Li:2022sgs , and in this paper, with a string theory model, still needs
further comprehensive examination with data. As these results are of
fundamental importance, it is necessary to check whether additional IceCube
neutrino events can still support the revealed regularity Amelino-
Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ; Huang:2022xto or not, as
well as checking the suggested subluminal and superluminal events, which, if
indeed correspond, respectively, to neutrinos and antineutrinos, would be a
strong sign for the type of CPT breaking as indicated by the theory.

At present, as we mentioned previously, IceCube cannot distinguish between
neutrinos from antineutrinos, so the intriguing correspondence uncovered by
Huang:2018ham ; Huang:2019etr ; Huang:2022xto is only a conjecture from
phenomenological point of view, while coincides with the prediction from a
D-foam scenario, as explained thoroughly. There is an exception for electron
antineutrinos at 6.3 PeV, at which the induced event rate is enhanced via the
$W^{-}$ Glashow resonance channel Glashow:1960zz , as this phenomenon only
occurs with $\bar{\nu}_{e}$’s. A candidate Glashow resonance event has lately
been reported by IceCube IceCube:2021rpz , indicating the existence of
$\gtrsim 6$ PeV cosmic antineutrinos. This may provide rare opportunities to
test the result of Huang:2018ham ; Huang:2019etr ; Huang:2022xto and the
D-foam models. Indeed, if one can infer through analysis according to the
strategy of Amelino-Camelia:2016fuh ; Amelino-Camelia:2016ohi ; Huang:2018ham
that such events correspond to “advance” (i.e., superluminal) events, it would
largely favor the interpretation explored here, and would also allow one to
tighten (44) by a factor of at least $(6/2)^{3}\approx 27$.

Together with the track-like event #7856 IceCube:2016uab with deposited energy
of $\sim 2.6$ PeV (which implies that the true neutrino energy can be of
${\cal O}(10)$ PeV), there appears to be no cutoff in the neutrino spectrum up
to $\sim 10$ PeV—the highest energy for which current data are available.
Still, this event (ATel #7856) is inferred to be of the “delay” type according
to prior analysis Huang:2018ham and hence probably a subluminal neutrino,
which is stable in the quantum D-brane foam. It is then natural to observe
such neutrino events from cosmologically remote sources, providing no extra
constraint for the string model, as opposed to the case of antineutrinos.

Thus, no clearly observable cutoff is produced once the effect of VPE and
similar process is strongly inhibited for superluminal neutrino species with
some pertinent choice of relevant QG parameters, as what current observation
actually indicates. That fact might suggest stringy space-time foam as
providing a possible realistic interpretation for all of these LV
phenomenologies.

We propose to search for higher-energy (anti)neutrino events, which, if
observed by IceCube or other neutrino telescopes in the future, would imply a
much smaller difference between $\mathfrak{d}^{2}$ and $4\varsigma_{I}$ (or,
$2\varsigma_{I}$) for the superluminal picture. Of course, one is also
encouraged to further examine the $\nu$\- and $\bar{\nu}$-spectra, which, if
both extend continuously (or fall smoothly), without any cutoff, may indicate
a more natural possibility that no splitting process or pair emission should
ever happen for the antineutrino even if traveling in vacuo slightly
superluminally—a case can be naturally realized by
$\varsigma_{I}\geq(\nicefrac{{1}}{{2}})\mathfrak{d}^{2}~{}(\Leftrightarrow\widetilde{\cal
E}_{*}\leq 0)$ in the model. For details on our this argument, we refer the
reader to Appendix B.

Before closing we mention that the stringent constraint provided by Lorentz
violation deformed patterns in neutrino oscillations IceCube:2014noc can also
be accounted for in D-foam situations, by recalling that the recoil and its
resulting effects discussed above is essentially geometrical and kinematical,
depending only upon neutrino energy (and the status of CPT). As such, the
D-brane foam is flavor blind, with $M_{\textrm{sQG}}$ being the same for all
neutrino species, hence with no effects on oscillations.111111Note also that
existing neutrino experiments Lisi:2000ndc are still far from achieving the
required sensitivity Alexandre:2008sfo to detect QG decoherence effects on
neutrino oscillations in D-foam backgrounds. Thus the exotic contribution
(additive to that $L_{m}$ from the mass difference) to the flavor oscillation
length, $L=2\pi/\lvert\Delta E\rvert$, like those arising in some generic
quantum gravity models vanishes (using
$\mathfrak{d}_{A}^{2}-\mathfrak{d}_{B}^{2}=\Delta\mathfrak{d}^{2}\sim 0$) in
that case:

$E\Delta\mathfrak{d}_{\nu}^{2}L_{\textrm{sQG}}\sim 2\pi\frac{M_{D}}{E}\
\Rightarrow\ L_{\textrm{sQG}}\gg L_{m},$ (48)

and $L-L_{m}={\cal O}(L_{m}/L_{\textrm{sQG}})$, from which it follows that
$1/L\approx 1/L_{m}$, $L_{m}=4\pi E/\Delta m_{\nu}^{2}$. So, the present data
IceCube:2014noc on neutrino oscillations do not prescribe the theory
contemplated in Ref. Li:2022sgs and in this work with a universal scale
$M_{\textrm{sQG}}\sim 10^{17}$ GeV.

To summarize, a CPT-violating propagation of neutrinos (antineutrinos), whose
velocities scale linearly with their energies, can emerge from a
string/D-brane theory model for space-time foam, being the type of isotropic,
Lorentz-invariant on average, but Gaussian stochastically fluctuating. Given
that high-energy astrophysical neutrinos have gained a prominent role in
probing such novel quantum-gravitational effect, we argue that the finding of
the neutrino speed variation $v=1\mp E/E_{\textrm{LV}}^{\nu}$ at about
$10^{17}$ GeV from IceCube GRB neutrino events Huang:2018ham ; Huang:2019etr ;
Huang:2022xto can serve as a support for such theory. The existences of both
time “delay” and “advance” events with equal amounts of speed variances are
well accounted for by means of CPT-breaking aspects of such scenario,
indicating that neutrinos are subluminal while antineutrinos travel at speeds
in excess of the light speed $c$. For superluminal antineutrino, we show that
the novel energy (non)conservation implied by the recoiling D-particle foam
may offer a viable mechanism in coping with the challenge due to the anomalous
decay channels like electron-positron pair production. In this respect, those
threshold constraints upon the superluminal velocities can be naturally
avoided, and accordingly, being consistent with the findings of Amelino-
Camelia:2016ohi ; Huang:2018ham ; Huang:2019etr ; Huang:2022xto in this
string-theory-inspired context. We appeal to further test both theoretically
and experimentally the superluminality, Lorentz invariance violation and CPT
violation for cosmic neutrinos. One may anticipate that more evidence will be
observed as more neutrino data is accrued, enabling the stringy foam
interpretation endorsed here to be either verified or falsified. Such efforts
would contribute to fundamental physics as well as resolving important issues
concerning the nature of space-time.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant No. 12075003).

##  Appendix A Energy violation in reactions

We briefly review the issues of energy nonconservation during interactions in
a recoiling D-foam. The treatment essentially follows Ellis:2000sf , of which
the result is then applied to $\bar{\nu}$-decays by emitting $ee^{+}$ pairs
for our purposes.

Following the theorem highlighted in Ref. Ellis:2000sf , which states that
during particle reactions the energy is in general violated, but being
conserved for the complete system (i.e., matter plus recoiling D-particle
inaccessible by a low-energy braneworld observer), i.e., Eq. (29), the energy
violation in a simplest case, namely a $1\rightarrow 2$-body decay (or a
3-particle vertex), is

$\overline{E}_{1}-\overline{E}_{2}-\overline{E}_{3}=\frac{1}{2}\frac{M_{s}}{g_{s}}\langle\\!\langle\mbox{\boldmath${\cal
V}$\unboldmath}_{(3)}^{2}\rangle\\!\rangle,$ (49)

where $\mbox{\boldmath${\cal V}$\unboldmath}_{(3)}=[{\bf p}_{1}-({\bf
p}_{2}+{\bf p}_{3})]/M_{D}$ is the recoil velocity of a D-particle. We expect
the average $\langle\\!\langle\cdot\\!\cdot\\!\cdot\rangle\\!\rangle$ over
effects of the foam medium to be proportional to:

$\langle\\!\langle{\cal V}^{2}\rangle\\!\rangle\sim{\cal
O}(p^{2})(\varsigma_{I}/M_{D}^{2}).$ (50)

Here $\varsigma_{I}$, as is made clear below, parametrizes energy loss during
reactions, and may in general differ from the variance $\mathfrak{d}^{2}$
appeared in particle dispersion relations.

To estimate the above quantity, one may use the conservation for the (average)
3-momenta, $\langle\\!\langle M_{D}\mbox{\boldmath${\cal
V}$\unboldmath}\rangle\\!\rangle=0$, incorporating also the recoil of the
background D-particle. This has been performed in Ellis:2000sf , where we
refer the reader for details. Eq. (49) then becomes

$\delta\overline{E}_{D}^{(3)}=\frac{\varsigma_{I}}{M_{D}}\overline{p}_{2}^{2}+\frac{\varsigma_{I}}{2M_{D}}\sum_{i}\delta
p_{i}^{2}+\ldots,$ (51)

where $\overline{p}_{2}$ is the momentum of a particle emitted during decay,
say, particle 2, while particle 1 is converted into particle 3; $\delta
p^{2}\equiv\langle\\!\langle{\bf p}^{2}\rangle\\!\rangle-\overline{p}^{2}$ is
minute variance of $\overline{p}^{2}$, and the $\ldots$ denote stochastic
quantum uncertainties. The latter two are purportedly negligible compared to
$\overline{p}_{2}^{2}/M_{D}$, hence can never cancel the leading term.

We stress that, to leading order in $1/M_{D}\sim M_{\textrm{Pl}}^{-1}$, the
matrix element for any particle reaction should in principle agree with that
in low-energy relativistic field theory, while the “recoil” gravitational
background that induces a violation of energy conservation would modify the
usual kinematics of quantum field-theoretic result for (threshold) reactions.
Multi-particle processes with $N\geq 4$ may be factorized as products of
3-point vertices and, $\delta\overline{E}_{D}$ is thus determined via summing
up the corresponding violations in every 3-body interaction (51), irrelevant
to the particle species considered. In this case, $\overline{p}_{2}$ may
denote a typical momentum of a mediator that may be exchanged in, e.g., a
4-particle interaction.

For the pair emission, i.e.,
$\bar{\nu}\rightarrow\bar{\nu}Z^{*}\rightarrow\bar{\nu}ee^{+}$, a 4-point
amplitude which can be viewed as a product of two fundamental leptonic
vertices mediated by the exchange of an off-shell $Z^{0}$ boson with momentum
$\overline{p}_{2}$, as in standard electroweak theory, the outgoing
$\bar{\nu}$ with energy $\simeq 0$ at threshold,
$\overline{k}\simeq\overline{p}_{2}=2\overline{p}_{e}$, then yields,

$\delta\overline{E}_{D}^{(4)}\simeq\frac{2\varsigma_{I}^{\nu}}{M_{D}}\overline{k}^{2}+\ldots\gtrsim\frac{g_{s}\varsigma_{I}^{\nu}}{M_{s}}2E_{\textrm{thr}}^{2},$
(52)

where $E_{\textrm{thr}}$ is the threshold for the vacuum pair emission, as
given by (45), and the foam parameter $\varsigma_{I}$ characterizes energy
violation during processes involving the neutrino sector (indicated by the
superfix $\nu$).

##  Appendix B Neutrino splitting in D-foam

To understand how a superluminal antineutrino splits into three
($\bar{\nu}\rightarrow\bar{\nu}\nu\bar{\nu}$) in D-foam situations, it
suffices to calculate its threshold energy. Let $\overline{E},\overline{k}$ be
the (average) energy and momentum of the incoming antineutrino, and
$(\overline{E}{}^{\prime},\overline{{\bf
k}}{}^{\prime}),(\overline{E},\overline{{\bf k}})_{\nu/\bar{\nu}}$ be the
4-momenta carried respectively by the outgoing $\bar{\nu}$, and by emitted
$\nu\bar{\nu}$ pairs. The D-foam modified conservation law reads

$\displaystyle\overline{E}$
$\displaystyle=\overline{E}{}^{\prime}+\overline{E}_{\nu}+\overline{E}_{\bar{\nu}}+\delta\overline{E}_{D}^{(4)},$
$\displaystyle\overline{k}$
$\displaystyle=\overline{k}{}^{\prime}+\overline{k}_{\nu}+\overline{k}_{\bar{\nu}},$
(53)

where the fact that all momenta are collinear at threshold is taken into
account. For deformed dispersion relations with QG term
$(\mp\nicefrac{{1}}{{2}})(\mathfrak{d}\overline{k})^{2}/M_{D}$, i.e., Eqs.
(11) and (12) but with mass-related term $\langle\\!\langle
m_{\nu}^{2}/(2\lvert{\bf k}\rvert)\rangle\\!\rangle\simeq
m_{\nu}^{2}/(2\overline{k})$, the threshold equation can be derived as

$\displaystyle\frac{m_{\nu}^{2}}{\overline{k}}$
$\displaystyle-\frac{m_{\nu}^{2}}{\overline{k}{}^{\prime}}-\frac{m_{\nu}^{2}}{\overline{k}_{\nu}}-\frac{m_{\nu}^{2}}{\overline{k}_{\bar{\nu}}}$
$\displaystyle=2\times\delta\overline{E}_{D}^{(4)}+\frac{g_{s}\mathfrak{d}_{\nu}^{2}}{M_{s}}\bigl{(}(\overline{k}{}^{\prime})^{2}-\overline{k}^{2}-\overline{k}_{\nu}^{2}+\overline{k}_{\bar{\nu}}^{2}\bigr{)}.$
(54)

Note that the minute neutrino mass is not negligible here, as this reaction
exclusively involves the neutrino sector. By putting $\overline{k}\simeq
E_{\textrm{thr}}\simeq 3\overline{k}{}^{\prime}=3\overline{k}_{\nu/\bar{\nu}}$
therefore

$4m_{\nu}^{2}=-\overline{k}\Bigl{(}\delta\overline{E}_{D}^{(4)}-\frac{g_{s}}{9M_{s}}4\mathfrak{d}_{\nu}^{2}\overline{k}^{2}\Bigr{)}.$
(55)

The amount of energy violation, $\delta\overline{E}_{D}^{(4)}$, in this
process, is again given by the sum of the violations in each of the two
3-point interactions (51), assuming factorization of relevant scattering
amplitudes, via virtual $Z$ exchange, as $\bar{\nu}$-splitting can be viewed
as a “rotation” of the neutrino-neutrino scattering process. Hence,

$\delta\overline{E}_{D}^{(4)}\simeq\frac{2\varsigma_{I}^{\nu}}{M_{D}}\Bigl{(}\frac{2\overline{k}}{3}\Bigr{)}^{2}=\frac{8\varsigma_{I}^{\nu}}{9M_{D}}\overline{k}^{2},$
(56)

where we assume that the three outgoing $\nu$ (or $\bar{\nu}$) each carry off
approximately $1/3$ of the energy of the incoming $\bar{\nu}$, i.e.,
$\overline{k}=\overline{p}_{2}+\overline{k}{}^{\prime}(=\overline{k}/3)$, as
used above.

We end up with some remarks below based on the following kinematical
threshold:

$E_{\textrm{thr}}^{\textrm{NS}}\simeq\sqrt[3]{\frac{9M_{D}m_{\nu}^{2}}{(\mathfrak{d}_{\nu}^{2}-2\varsigma_{I}^{\nu})}}=(9m_{\nu}^{2}\widetilde{\cal
E}_{*}^{\nu})^{1/3},$ (57)

with

$\widetilde{\cal
E}_{*}^{\nu}\coloneqq\frac{E_{\textrm{LV}}^{\nu}}{1-2\varsigma_{I}^{\nu}E_{\textrm{LV}}^{\nu}/M_{D}}.$
(58)

The splitting process opens only if $\mathfrak{d}^{2}>2\varsigma_{I}$,
otherwise is forbidden, with the very case of
$(\mathfrak{d},\varsigma_{I})=(0,0)$, yielding an infinite $E_{\textrm{thr}}$,
as in the SM. If $\sqrt{2\varsigma_{I}}<\mathfrak{d}\leq
2\sqrt{\varsigma_{I}}$, the decay channel through emitting $ee^{+}$ pairs is
not allowed, though $\bar{\nu}$-splitting can still happen. This latter
reaction can be suppressed at the kinematical level in case of
$\varsigma_{I}\approx\frac{1}{2}\mathfrak{d}^{2}$, thereby making the
antineutrino practically stable in vacuo, against any splitting effect at high
energies. The threshold (57) is proportional to the mass of order
$m_{\nu}\lesssim 1$ eV Huang:2014qwa . To push $E_{\textrm{thr}}$ to a
sufficiently high scale, e.g., greater than 2 PeV, such that we are always
below threshold when comparing with the IceCube data, an exceedingly small
value of $(\mathfrak{d}^{2}-2\varsigma_{I})$ would be required. A more natural
solution to that could be
$\varsigma_{I}\geq(\nicefrac{{1}}{{2}})\mathfrak{d}^{2}$, so that
antineutrinos would never decay in D-foam geometries, despite traveling faster
than light, and there is thus no need to fine-tune the model parameter.

## References

  * (1) U. Jacob and T. Piran, Nat. Phys. 3, 87 (2007).
  * (2) G. Amelino-Camelia and L. Smolin, Phys. Rev. D 80, 084017 (2009).
  * (3) G. Amelino-Camelia, D. Guetta and T. Piran, ApJ 806, 269 (2015).
  * (4) IceCube Collaboration: M. G. Aartsen et al., Phys. Rev. Lett. 111, 021103 (2013); Science 342, 1242856 (2013); Phys. Rev. Lett. 113, 101101 (2014).
  * (5) IceCube Collaboration: M. G. Aartsen et al., Phys. Rev. Lett. 117, 241101 (2016) [Erratum 119, 259902 (2017)].
  * (6) IceCube Collaboration: M. G. Aartsen et al., ApJ Lett. 805, L5 (2015); ApJ 824, 115 (2016); 843, 112 (2017).
  * (7) G. Amelino-Camelia, L. Barcaroli, G. D’Amico, N. Loret and G. Rosati, Phys. Lett. B 761, 318 (2016).
  * (8) G. Amelino-Camelia, G. D’Amico, G. Rosati and N. Loret, Nat. Astron. 1, 0139 (2017).
  * (9) Y. Huang and B.-Q. Ma, Commun. Phys. 1, 62 (2018) doi:10.1038/s42005-018-0061-0 [arXiv:1810.01652].
  * (10) Y. Huang, H. Li and B.-Q. Ma, Phys. Rev. D 99, 123018 (2019).
  * (11) Y. Huang and B.-Q. Ma, Fundamental Research (2022) doi:10.1016/j.fmre.2022.05.022.
  * (12) L. Shao, Z. Xiao and B.-Q. Ma, Astropart. Phys. 33, 312 (2010).
  * (13) S. Zhang and B.-Q. Ma, Astropart. Phys. 61, 108 (2015).
  * (14) H. Xu and B.-Q. Ma, Astropart. Phys. 82, 72 (2016); Phys. Lett. B 760, 602 (2016); JCAP 01 (2018) 050.
  * (15) Y. Liu and B.-Q. Ma, Eur. Phys. J. C 78, 825 (2018).
  * (16) J. Zhu and B.-Q. Ma, Phys. Lett. B 820, 136518 (2021); 820, 136546 (2021).
  * (17) X. Zhang and B.-Q. Ma, Phys. Rev. D 99, 043013 (2019).
  * (18) C. Li and B.-Q. Ma, Phys. Lett. B 835, 137543 (2022).
  * (19) C. Li and B.-Q. Ma, Phys. Lett. B 819, 136443 (2021); Results Phys. 26, 104380 (2021).
  * (20) J. A. Wheeler, Phys. Rev. 97, 511 (1955); J. A. Wheeler, K. Ford and M. Goldhaber, Phys. Today 52, 63 (1998).
  * (21) J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Gen. Rel. Grav. 32, 127 (2000); Phys. Rev. D 61, 027503 (1999); 62, 084019 (2000).
  * (22) J. Ellis, N. E. Mavromatos and M. Westmuckett, Phys. Rev. D 70, 044036 (2004); 71, 106006 (2005); J. Ellis, N. E. Mavromatos, D. V. Nanopoulos and M. Westmuckett, Int. J. Mod. Phys. A 21, 1379 (2006).
  * (23) J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Lett. B 665, 412 (2008); 674, 83 (2009); 694, 61 (2010); Int. J. Mod. Phys. A 26, 2243 (2011).
  * (24) T. Li, N. E. Mavromatos, D. V. Nanopoulos and D. Xie, Phys. Lett. B 679, 407 (2009); T. Li and D. V. Nanopoulos, Eur. Phys. J. C 72, 2044 (2012).
  * (25) N. E. Mavromatos, Int. J. Mod. Phys. A 25, 5409 (2010), and references therein.
  * (26) J. Ellis, N. E. Mavromatos, D. V. Nanopoulos and G. Volkov, Gen. Rel. Grav. 32, 1777 (2000); J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Rev. D 65, 064007 (2002).
  * (27) N. E. Mavromatos, Phys. Rev. D 83, 025018 (2011); J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Rev. D 96, 086012 (2017).
  * (28) J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Rev. D 63, 024024 (2000); J. Alexandre, K. Farakos, N. E. Mavromatos and P. Pasipoularides, Phys. Rev. D 77, 105001 (2008); 79, 107701 (2009).
  * (29) N. E. Mavromatos, S. Sarkar and W. Tarantino, Phys. Rev. D 80, 084046 (2009); Mod. Phys. Lett. A 28, 1350045 (2013).
  * (30) N. E. Mavromatos and S. Sarkar, Phys. Rev. D 72, 065016 (2005); J. Bernabeu, N. E. Mavromatos and S. Sarkar, Phys. Rev. D 74, 045014 (2006).
  * (31) N. E. Mavromatos, S. Sarkar and A. Vergou, Phys. Lett. B 696, 300 (2011); N. E. Mavromatos, V. A. Mitsou, S. Sarkar and A. Vergou, Eur. Phys. J. C 72, 1956 (2012); T. Elghozi, N. E. Mavromatos, M. Sakellariadou and M. F. Yusaf, JCAP 02 (2016) 060.
  * (32) I. I. Kogan and N. E. Mavromatos, Phys. Lett. B 375, 111 (1996); I. I. Kogan, N. E. Mavromatos and J. F. Wheater, Phys. Lett. B 387, 483 (1996).
  * (33) J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Int. J. Mod. Phys. A 12, 2639 (1997); 13, 1059 (1998).
  * (34) N. E. Mavromatos and R. J. Szabo, Phys. Rev. D 59, 104018 (1999).
  * (35) N. E. Mavromatos and S. Sarkar, Eur. Phys. J. C 73, 2359 (2013).
  * (36) See, e.g., J. Zhu and B.-Q. Ma, Phys. Rev. D 105, 124069 (2022), and references therein.
  * (37) Y. Huang and B.-Q. Ma, The Universe 2, 65–71 (2014) [arXiv:1407.4357].
  * (38) J. Ellis, N. Harries, A. Meregaglia, A. Rubbia and A. S. Sakharov, Phys. Rev. D 78, 033013 (2008).
  * (39) Z.-Y. Wang, R.-Y. Liu and X.-Y. Wang, Phys. Rev. Lett. 116, 151101 (2016).
  * (40) J. Ellis, N. E. Mavromatos, A. S. Sakharov and E. K. Sarkisyan-Grinbaum, Phys. Lett. B 789, 352 (2019).
  * (41) R. Laha, Phys. Rev. D 100, 103002 (2019).
  * (42) J.-J. Wei et al., JHEAp 22 (2019) 1.
  * (43) The IceCube Collaboration, Fermi-LAT, MAGIC et al., Science 361, eaat1378 (2018); IceCube Collaboration: M. G. Aartsen et al., Science 361, 147 (2018).
  * (44) H. Li and B.-Q. Ma, Sci. Bull. 65, 262 (2020).
  * (45) C. Li and B.-Q. Ma, Phys. Rev. D 104, 063012 (2021); Sci. Bull. 66, 2254 (2021).
  * (46) J. R. Ellis, N. E. Mavromatos and A. S. Sakharov, Astropart. Phys. 20, 669 (2004); J. Ellis, N. E. Mavromatos, D. V. Nanopoulos and A. S. Sakharov, Nature (London) 428, 386 (2004); Int. J. Mod. Phys. A 19, 4413 (2004).
  * (47) C. Li and B.-Q. Ma, Phys. Lett. B 829, 137034 (2022).
  * (48) P. He and B.-Q. Ma, Phys. Lett. B 835, 137536 (2022).
  * (49) A. Crivellin, F. Kirk and M. Schreck, JHEP 04 (2021) 082.
  * (50) A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. 107, 181803 (2011).
  * (51) E. Borriello, S. Chakraborty, A. Mirizzi and P. D. Serpico, Phys. Rev. D 87, 116009 (2013).
  * (52) F. W. Stecker, Astropart. Phys. 56, 16 (2014).
  * (53) J. S. Diaz, V. A. Kostelecký and M. Mewes, Phys. Rev. D 89, 043005 (2014).
  * (54) K. Wang, S.-Q. Xi, L. Shao, R.-Y. Liu, Z. Li and Z.-K. Zhang, Phys. Rev. D 102, 063027 (2020).
  * (55) J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Rev. D 63, 124025 (2001).
  * (56) F. W. Stecker, S. T. Scully, S. Liberati and D. Mattingly, Phys. Rev. D 91, 045009 (2015).
  * (57) IceCube Collaboration: M. G. Aartsen et al., Nature (London) 591, 220 (2021) [Erratum 592, E11 (2021)].
  * (58) S. L. Glashow, Phys. Rev. 118, 316 (1960).
  * (59) See, e.g., Super-Kamiokande Collaboration: K. Abe et al., Phys. Rev. D 91, 052003 (2015); IceCube Collaboration: M. G. Aartsen et al., Nat. Phys. 14, 961 (2018); R. Abbasi et al., Nat. Phys. 18, 1287 (2022).
  * (60) See, for instance, E. Lisi, A. Marrone and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000); G. L. Fogli, E. Lisi, A. Marrone, D. Montanino and A. Palazzo, Phys. Rev. D 76, 033006 (2007); IceCube Collaboration: R. Abbasi et al., Phys. Rev. D 79, 102005 (2009).