# Lifshitz-scaling in CPT-even Lorentz-violating electrodynamics and GRB time
delay

1K. E. L. de Farias<EMAIL_ADDRESS>2T. A. M. Sampaio
<EMAIL_ADDRESS>3M. A. Anacleto<EMAIL_ADDRESS>1,3F.A. Brito<EMAIL_ADDRESS>3E. Passos<EMAIL_ADDRESS>1Departamento de Física, Universidade Federal da Paraíba,
Caixa Postal 5008, João Pessoa, Paraíba, Brazil.  2Instituto Federal de
Educação, Ciência e Tecnologia do Sertão Pernambucano - campus Salgueiro,
Caixa Postal 56000-000, Salgueiro, Pernambuco, Brazil.  3Departamento de
Física, Universidade Federal de Campina Grande,
Caixa Postal 10071, 58429-900, Campina Grande, Paraíba, Brazil.

###### Abstract

In this work, we consider the Hořava-Lifshitz scaling to rewrite higher-order
derivatives in Lorentz-violating CPT-even electrodynamics. The possibility of
the Lorentz invariance violation being originally suppressed by the quantum
gravity energy scale $M_{\rm QG}$, may be indeed circumvented in a scenario
with anisotropy between space and time above certain intermediate energy scale
$\Lambda_{\rm HL}\ll M_{\rm QG}$. To test the model, we have used
observational results of a cosmological time delay between high and low energy
photons from gamma-ray bursts (GRB090510) event. For specific values of the
Lifshitz critical exponent, we have found interesting lower bounds on the
values of $M_{\rm QG}$.

Keywords: Hořava-Lifshitz, Lorentz invariance violation, higher-order
derivatives, Time delay, Gamma-ray bursts.

###### pacs:

11.15.-q, 11.10.Kk

##  I Introduction

The Lorentz invariance is a fundamental ingredient of the standard model,
which accurately describes the properties of fundamental particles and
interactions up to TeV scale - energy scale achieved by the Large Hadron
Collider (LHC) experiment. However, the idea that this invariance can be
violated at high energies, presumably in some quantum gravity energy scale
$M_{QG}$ that has to be equal or around to the Planck scale ($\sim 10^{19}{\rm
GeV}$) has been previously considered in several areas of theoretical physics
(see, for example, Ref. Liberati (2013) and related references therein). In
the present work, the Lorentz invariance violation (LIV) is introduced through
higher-order derivative operators. These LIV models work as effective field
theories in the ultraviolet (UV) scale regime and thus lead to changes in the
dispersion relation for high energy particles Myers and Pospelov (2003) (see
also Ref. Bolokhov and Pospelov (2008)).

The study of non-renormalizable terms in the LIV scenario from an effective
field theory was initially proposed by Myers and Pospelov who used higher-
order derivative operators coupled with a non-dynamics four-vector, $n_{\mu}$,
that interact with scalar, fermionic and gauge fields Myers and Pospelov
(2003). For the electromagnetic field sector, we have, for example, the
introduction of a CPT-odd term with dimension-5 operator:

$\displaystyle S^{(5)}_{\rm MP}=-\frac{\xi}{M_{\rm QG}}\int
d^{4}x\,n^{\alpha}F_{\alpha\delta}n^{\gamma}\partial_{\gamma}n_{\beta}\tilde{F}^{\beta\delta}.$
(1)

Based on this idea, it is also possible to construct a CPT-even term with
dimension-6 operator following the same Myers-Pospelov criteria:

$S^{(6)}_{\rm MP}=-\frac{\zeta}{2M_{\rm QG}^{2}}\int
d^{4}x\,\tilde{F}^{\mu\nu}n_{\mu}n^{\gamma}\big{(}n\cdot\partial\big{)}^{2}\tilde{F}_{\gamma\nu},$
(2)

where $\xi$ and $\zeta$ are dimensionless parameters that control the LIV
intensity over $M_{QG}$ scale suppression,
$F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$,
$\tilde{F}^{\beta\delta}=\frac{1}{2}\epsilon^{\beta\delta\rho\omega}F_{\rho\omega}$,
$\tilde{F}^{0i}=\frac{1}{2}\epsilon^{ijk}F_{jk}$ and
$\tilde{F}_{0i}=\frac{1}{2}\epsilon_{ilm}F^{lm}$

Unlike electromagnetism modified with the original Myers-Pospelov term in Eq.
(1), which may violate causality due to birefringence effects that appear in
its dispersion relation, it was shown in the Ref. Sampaio and Passos (2019)
that the use of CPT-even LIV terms as in Eq.(2) modifying the electromagnetic
sector, can result in a consistent theory (preserving causality and
unitarity). Because the CPT-even terms do not produce birefringence effects in
the dispersion relation, it becomes possible to perform phenomenological tests
with the cosmological time delay between photons, since there is no impediment
due to the different polarization modes interfering with the time of flight of
the particles, as highlighted by Ref. Liberati and Maccione (2009).

For the future purpose of studying effects on the time of flight of the
particles, we can choose a simple scenario by which we have only LIV effects
on boosts transformations. The four-vector $n_{\mu}$ (which refers to the
background field of the model) can be restricted to a time-like case
${(n_{\mu}\equiv 1,\vec{0})}$ (this is known as isotropic models, better
discussed in Refs. Colladay and Kostelecký (1998); Reyes (2010)).

However, both the original Myers-Pospelov modification, Eq. (1), and the
dimension-6 modification in Eq. (2) can bring problems to electromagnetic
field theory that are associated with the non-renormalization. To deal with
these problems, we can use an intermediate energy scale $\Lambda_{HL}$ that
can still maintain the anisotropy between space and time, such as in the
Hořava-Lifshitz theory Horava (2011, 2009); Visser (2009); Anacleto  _et al._
(2018), that manages to renormalize theories that were previously non-
renormalizable. In this theory, there is an explicit asymmetry between the
temporal and spatial coordinates by which the Lorentz invariance is broken.
This asymmetry is controlled by the critical Lifshitz exponent $z$. In order
to make the coupling between the standard model and a phenomenologically
relevant Lorentz-violating Effective Field Theory (EFT) possible, the Ref.
Pospelov and Shang (2012) shows us an upper limit of $\Lambda_{\rm HL}\leq
10^{10}{\rm GeV}$.

This work aims to study the effects of Hořava-Lifshitz scaling on Maxwell’s
electromagnetism modified by the presence of LIV term given by Eq. (2) to
construct an EFT, in which their respective energy scales are parameterized to
an intermediate energy scale governed by the critical exponent $z$.

Previous studies adopt certain corrections for the dispersion relations to
analyze, through gamma-ray bursts (GRB) measurements, changes in photon
polarization due to vacuum birefringence effects Gleiser and Kozameh (2001);
Coburn and Boggs (2003); Jacobson  _et al._ (2004); Götz  _et al._ (2011);
Laurent  _et al._ (2011) or to analyze the presence of time delay between the
arrival of two photons due to vacuum dispersive effects Ackermann  _et al._
(2009); Chang  _et al._ (2012); Rodriguez Martinez and Piran (2006); Nemiroff
_et al._ (2012); Wei  _et al._ (2017); Passos  _et al._ (2017) as a method of
obtaining boundaries on the LIV parameter or in the quantum gravity energy
scale. We will take advantage of UV-Lifshitz behaviour to find new bounds on
the LIV effects by using time delay measurements from the propagation of the
electromagnetic waves due to GRB events.

This paper is organized as follows: In section II, we have applied the Hořava-
Lifshitz scaling process for usual Maxwell electrodynamics, proving that it
preserves gauge invariance. In section III, we have applied the same process
in the presence of the dimension-6 LIV operator in the time-like case,
analyzing its gauge invariance and getting the modified dispersion relation.
In section IV, we have studied the phenomenological aspects for this model
using time-delay data in the propagation of photons seen in GRB090510 event,
obtaining lower bounds for the $M_{\rm QG}$ scale. Finally, in section V, we
have presented our conclusions and perspectives.

##  II Hořava-Lifshitz scaling in electrodynamics

The Hořava-Lifshitz theory has also been proposed as an alternative to
describing an ultraviolet (UV) behavior of general relativity Horava (2009).
It is known that general relativity is not UV renormalizable because its
coupling constant has a negative mass dimension, i.e., $[G]=[M]^{-2}$, with
$[M]$ being the mass unit. The main purpose of the Hořava-Lifshitz scaling is
to introduce a certain asymmetry between space and time parameterized by the
critical Lifshitz exponent $z$. Note that this can be approached in two ways:
$i)$ The time dimension is $[t]=[L]^{z}$, and the spatial coordinate
dimensions are $[\vec{r}]=[L]$; $ii)$ The time dimension is $[t]=[L]$, and the
spatial coordinate dimensions are $[\vec{r}]=[L]^{z}$. Here, $[L]=[M]^{-1}$ is
the length unit. Notice that in this scaling we have now $[G]=[M]^{z-3}$,
which for $z=3$ makes the coupling constant dimensionless. One of the
necessary conditions for renormalizability of a quantum field theory is that
the coupling constant should be dimensionless or dimensionful with positive
mass dimension ($z\geq 3$).

The effects of these asymmetries allow us an adjustment in the dimensionality
of the coupling constant, and then choose the value of the critical exponent
$z$. Thus, it is expected that we will deal only with renormalizable
interactions. Indeed this may happen, for example, in Ref. Horava (2009), in
which it was shown that $z\geq 3$ is sufficient to guarantee the
renormalizable character in UV scale of the general relativity. In this
section, we consider how this should affect electrodynamics.

###  II.1 Hořava-Lifshitz electrodynamics modifying the magnetic field sector

A possibility to insert the critical Lifshitz exponent in field theories is
found in Ref. Pospelov and Shang (2012). We can modify the magnetic sector of
the electrodynamics and rewrite Maxwell’s action as

$S_{M,HL}=-\frac{1}{2}\int
dtd^{3}\vec{x}\,\left[F_{0i}F^{0i}+\frac{1}{2}F_{ij}(-\Delta)^{z-1}F^{ij}\right],$
(3)

where $\Delta=-\partial_{i}\partial^{i}=\vec{\partial}\cdot\vec{\partial}$,
the metric used is ${\rm diag}(1,-1,-1,-1)$, and $z$, as we have
aforementioned, is the Lifshitz critical exponent. When $z>1$, this action has
an anisotropic behavior and for $z=1$, we recover the usual theory. In
principle one should include all the lower spatial derivative terms, but at
large $k$ (momentum of particles), which is the limit that we are mainly
interested in the highest spatial derivative term dominates and we will keep
only it. Note that the proposed model given by Eq. (3) must fit the first case
of anisotropy between space and time: $([t]=[L]^{z},\;[\vec{r}]=[L])$.
Consequently, we have the respective dimensionality rules:

$\displaystyle[\partial_{t}]=[L]^{-z},\;\;[\partial_{i}]=[L]^{-1}.$ (4)
$[A_{0}]=[L]^{-\frac{1}{2}(z+1)},\;\;[A_{i}]=[L]^{\frac{1}{2}(z-3)}.$ (5)

Note that, by assuming $z=1$, we recover Maxwell’s theory. From here, we can
define our terms with the new rescaling:

$\displaystyle t\rightarrow\Lambda_{\rm
HL}^{-z+1}t,\;\;A_{0}\rightarrow\Lambda^{\frac{1}{2}(z-1)}_{\rm HL}A_{0},$
$\displaystyle A_{i}\rightarrow\Lambda^{-\frac{1}{2}(z-1)}_{\rm
HL}A_{i},\;\;\partial_{t}\rightarrow\Lambda^{(z-1)}_{\rm HL}\partial_{t}.$ (6)

In this case, the dimension of Hořava-Lifshitz energy scale $\Lambda_{\rm HL}$
will follow the rule: $[\Lambda_{\rm HL}]=[L]^{-1}$, due to the rescaling
process. The other quantities will have the following dimensional structures:
$[t]=[L],[\partial_{t}]=[A_{0}]=[A_{i}]=[L]^{-1}$. Applying Eq.(II.1) to
Eq.(3), we obtain

$S_{M,HL}=-\frac{1}{2}\int
dtd^{3}\vec{x}\,\left[F_{0i}F^{0i}+\frac{1}{2\Lambda_{\rm
HL}^{2(z-1)}}F_{ij}(-\Delta)^{z-1}F^{ij}\right].$ (7)

As shown in Ref.Passos  _et al._ (2016), this theory preserves gauge
invariance.

##  III Electrodynamics with the CPT-even dimension-six operator under
Lifshitz’s scaling

Based on subsec. II.1, where a Hořava-Lifshitz scaling was constructed for
Maxwell’s equation describing classical electrodynamics, let us now apply in
the same way the scaling for the dimension-6 operator, subject to the time-
like case given in Eq.(2). The incorporation of higher dimensional operators
present some challenges such as non-unitarity/ghosts due to the higher time
derivatives (see the Ref.Reyes (2010)). On the other hand, the purpose of HL
models is to keep the same number of time derivatives and yet to improve UV
behavior by including higher spatial derivatives characterizing a Lorentz
invariance violation, where this asymmetry is governed by a critical exponent
z. Hence the union of both theories becomes useful to phenomenological studies
in the UV regimes. Therefore, the dimension-6 Myers-Pospelov term, when
rescaled, becomes:

$S_{\rm MP,HL}=-\frac{\zeta}{2M_{\rm QG}^{2}}\int
dtd^{3}\vec{x}\frac{1}{\Lambda^{2(z-1)}_{\rm
HL}}\tilde{F}_{0i}\partial_{t}^{2}(-\Delta)^{z-1}\tilde{F}^{0i}.$ (8)

A difference that is quite evident here is the fact that for the usual
dimension-5 Myers-Pospelov term in Eq. (1) it is necessary to scale the
quantum gravity mass $M_{\rm QG}$ Passos  _et al._ (2016), while for the
dimension-6 operator, this is not the case.

So we can now write our complete model through our new rescaling. Adding the
modified Maxwell’s term in Eq. (7) to that of modified CPT-even Myers-Pospelov
Eq. (8), we arrive at

$S_{\rm MP,HL}=-\frac{1}{2}\int
dtd^{3}\vec{x}\,\left\\{\left[F_{0i}F^{0i}+\frac{1}{2\Lambda_{\rm
HL}^{2(z-1)}}F_{ij}(-\Delta)^{z-1}F^{ij}\right]+\frac{\zeta}{\Lambda^{2(z-1)}_{\rm
HL}M_{\rm
QG}^{2}}\tilde{F}_{0i}\partial_{t}^{2}(-\Delta)^{z-1}\tilde{F}^{0i}\right\\},$
(9)

in which both terms have the presence of LIV. Note that, for $z=1$ limit, we
recover the usual CPT-even dimension-6 in modified electrodynamics for the
isotropic case, as found in the non-renormalizable standard model extension
present in Ref.Kostelecky and Mewes (2009).

###  III.1 Gauge invariance

In order to analyze the gauge invariance of the model, we will show here that,
despite the Hořava-Lifshitz modification, the theory remains gauge invariant.
For this, we can make an analysis similar to Arnowitt-Deser-Misner’s formalism
in the Lifshitz gravity Pospelov and Shang (2012); Arnowitt  _et al._ (2008).
Hence, by considering the decomposition of the fields $A_{0}$ and
$A_{i}=A_{i}^{T}+\partial_{i}\varphi$, the Eq.(8) after some integrations by
part becomes

$\displaystyle\int
dtd^{3}\vec{x}\;\tilde{F}_{0i}\partial_{t}^{2}\tilde{F}^{0i}=\int
dtd^{3}\vec{x}\;A_{i}^{T}[\partial^{k}\partial_{t}^{2}(-\Delta)^{z-1}\partial^{j}-\eta^{jk}\partial_{t}^{2}(-\Delta)^{z}]A_{j}^{T}.$
(10)

The above result is then added to Eq. (7), since it is submitted by the field
decomposition and can be found in Passos  _et al._ (2016).

$\displaystyle S^{(2)}_{\rm MP,HL}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dtd^{3}\vec{x}\left\\{A^{T}_{j}\left[\partial_{t}^{2}+\frac{1}{\Lambda^{2(z-1)}_{\rm
HL}}(-\Delta)^{z}+\frac{1}{\Lambda^{2(z-1)}_{\rm HL}}\frac{\zeta}{M^{2}_{\rm
QG}}(\partial_{t})^{2}(-\Delta)^{z}\right]\eta^{jk}A^{T}_{k}\right.$ (11)
$\displaystyle\left.+\left(A_{0}+\dot{\varphi}\right)\Delta\left(A_{0}+\dot{\varphi}\right)\right\\}.$

This expression makes explicit the gauge symmetry

$A_{0}\rightarrow A_{0}^{\prime}=A_{0}+\dot{\omega}\ \ \ ;\ \ \
\varphi\rightarrow\varphi^{\prime}=\varphi-\omega,$ (12)

which is nothing but the original gauge symmetry: $A_{\mu}\rightarrow
A_{\mu}^{\prime}=A_{\mu}+\partial_{\mu}\omega$ (this is related to the fact
that the Eq.(9) is written in terms of the gauge invariant electric $F_{0i}$
and magnetic, $F_{ij}$ components).

###  III.2 Dispersion relation

Subjecting Eq. (11) to the gauge choice $A_{0}=0$, we obtain, after some
integrations by parts,

$\displaystyle S^{(2)}_{\rm MP,HL}=\frac{1}{2}\int
dtd^{3}\vec{x}A^{T}_{j}\left[\partial_{t}^{2}+\frac{1}{\Lambda^{2(z-1)}_{\rm
HL}}(-\Delta)^{z}+\frac{1}{\Lambda^{2(z-1)}_{\rm HL}}\frac{\zeta}{M^{2}_{\rm
QG}}(\partial_{t})^{2}(-\Delta)^{z}\right]\eta^{jk}A^{T}_{k}$ (13)

The equation of motion is

$\frac{\delta S^{(2)}_{\rm MP,HL}}{\delta
A^{T}_{j}}=\left(\partial_{t}^{2}+\frac{1}{\Lambda^{2(z-1)}_{\rm
HL}}(-\Delta)^{z}+\frac{1}{\Lambda^{2(z-1)}_{\rm HL}}\frac{\zeta}{M^{2}_{\rm
QG}}(\partial_{t})^{2}(-\Delta)^{z}\right)\eta^{km}A_{m}=0.$ (14)

Intending to find an equation that describes the dispersion relation for our
model, we consider the following Ansatz

$A^{k}(x)=\epsilon^{k}(p)\exp(-ik_{\mu}x^{\mu})\ \ \ \ ;\ \ \ \
k_{\mu}=(\omega,\vec{k}),$

where $k=|\vec{k}|$. Thus, we obtain

$-\omega^{2}+\frac{k^{2z}}{\Lambda^{2(z-1)}_{\rm HL}}-\frac{\zeta}{M^{2}_{\rm
QG}\Lambda^{2(z-1)}_{\rm HL}}\omega^{2}k^{2z}=0.$ (15)

It is worth mentioning that in the case where $z=1$, the Lifshitz modification
is absent in Eq. (15), as expected. However, we still have the presence of
$\mathcal{O}(\omega^{4})$ modification. It occurs due to the presence of the
dimension-6 operator. Finally, Eq. (15) leads to

$\omega=\pm\frac{k^{z}}{\Lambda^{z-1}_{\rm HL}}\left(1+\frac{\zeta}{M^{2}_{\rm
QG}\Lambda^{2(z-1)}_{\rm HL}}k^{2z}\right)^{-1/2}.$ (16)

It is immediately to note that in the case of $z=1$ and $\zeta=0$ we recover
the relativistic dispersion relation. We see that the intensity of dispersive
effects depends on the critical exponent. A preliminary observation of the Eq.
(16) is that there is no presence of birefringence sign (that always appears
in CPT-odd EFT theories). Hence it makes it possible to use this model to
calculate the temporal delay in cosmological events since the time of flight
of the photons does not change due to different polarization modes. Taking the
first approximation order in Eq. (16), we get the following dispersion
relation

$\omega=\pm\left(\frac{k^{z}}{\Lambda^{z-1}_{\rm
HL}}-\frac{1}{2}\frac{\zeta}{M^{2}_{\rm
QG}}\frac{{k^{3z}}}{\Lambda^{3(z-1)}_{\rm HL}}\right).$ (17)

We now follow the above expression to address the phenomenological issues. The
choice of $z=1$ recover the order $\zeta(M^{-2}_{\rm QG})k^{3}$, where is
connected with dimension-six operator. To $z=4/3$ the order of correction
obtained is $\zeta(M_{\rm QG}\Lambda_{\rm HL})^{-2}k^{4}$ associated to
operators of dimension-seven. However, by taking $z=2/3$ we found
$\zeta(M^{-2}_{\rm QG}\Lambda_{\rm HL})k^{2}$, in which the correction order
is associated with dimension-five. In the case of $z=1/3$ the correction order
obtained is $\zeta(M^{-1}_{\rm QG}\Lambda_{\rm HL})^{2}k$, in which is related
to operators of dimension-four. Moreover, by taking $\zeta=0$ we recover the
usual dispersion relation. Before ending this section some comments are in
order. The anisotropic scaling develops a fundamental role in improving the UV
behavior of non-renormalizable theories such as Einstein gravity that has
fundamental coupling of the theory given by the Newtonian constant with
negative mass dimension, i.e., $[G_{N}]=M^{-2}$. The Horava-Lifshitz gravity
takes advantage of this scaling to get $[G_{N}]=M^{z-3}$, which for $z\geq 3$
eliminates the problem of negative mass dimension. This is done in an
alternative theory of gravity, with dimensions now competing in the HL action
in a new fashion as $[K_{ij}]^{2}[d^{3}x][dt]/[G_{N}]$, where
$[K_{ij}]=M^{z}$. However, suppose one considers a deformed HL gravity by
introducing higher operators into the Lagrangian, say $\Delta^{z-1}$, this
would produce a Newtonian constant with dimension $[G_{N}]=M^{3z-5}$, which
now $z\geq 5/3$ would also render an improved UV behavior with smaller
critical exponent $z$. Although, in our setup there is no negative mass
dimension coupling as in gravity theories since we are working with gauge
fields, same reduction in the magnitude of $z$ occurs in a precise manner.

##  IV Phenomenological aspects: time delay between emitted photons

The temporal delay occurs when emitted photons from the same source are
detected at different times, being detected a more energetic photon after the
arrival of a less energetic photon, resulting in a small temporal delay.
According to Quantum Gravity Theory, this happens because, at high energies
next to the Planck scale, space stops being continuous and becomes a discrete
space, as if it would have spatial granulations.

These energetic photons may be generated from transient sources, that is,
sources that give a great peak of energy in a short time. When a highly
energetic photon goes through space from the source to us, it probes the
discreteness of space due to its high energy, suffering a temporal delay,
concerning the low energetic photon, in its detection.

Considering the equation (16), the photon velocity in our model becomes

$v=\frac{\partial\omega_{+}}{\partial k}=\frac{zk^{z-1}}{\Lambda_{\rm
HL}^{z-1}}-\frac{3z\zeta}{2}\frac{\Lambda^{2}_{\rm HL}}{M^{2}_{\rm
QG}}\frac{{k^{3z-1}}}{\Lambda^{(3z-1)}_{\rm HL}}$ (18)

which is a sub-luminal group velocity for photons that are under the LIV
effect. In order to study cosmological aspects, the expression (18) needs to
consider the universe expansion. Hence, in terms of the cosmological redshift
$\mathcal{Z}^{\prime}$111Please, note the difference between two notations:
$z$ will continue associated to the Lifshitz critical exponent, while
$\mathcal{Z}^{\prime}$ represents the cosmological redshift., the equation
(18) becomes

$v(\mathcal{Z}^{\prime})=\frac{z[k(1+\mathcal{Z^{\prime}})]^{z-1}}{\Lambda_{\rm
HL}^{z-1}}-\frac{3z\zeta}{2}\frac{\Lambda^{2}_{\rm HL}}{M^{2}_{\rm
QG}}\frac{{[k(1+\mathcal{Z^{\prime}})]^{3z-1}}}{\Lambda^{(3z-1)}_{\rm HL}},$
(19)

where $k(1+\mathcal{Z}^{\prime})$ is the photon momentum when it was emitted
from the source located at redshift $\mathcal{Z}^{\prime}$. The light-travel
distance, considering the universe expansion in a $\Lambda\mbox{CDM}$
Universe, is given by Rodriguez Martinez and Piran (2006):

$x(t)=\frac{1}{H_{0}}\int^{t}_{0}v(t^{\prime})dt^{\prime}=\frac{1}{H_{0}}\int_{0}^{\mathcal{Z}}v(\mathcal{Z}^{\prime})\frac{d\mathcal{Z}^{\prime}}{(1+\mathcal{Z}^{\prime})\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+\mathcal{Z}^{\prime})^{3}}},$
(20)

where $H_{0}\simeq 70\mbox{km/s/Mpc}=1.505\times 10^{-42}\rm GeV$ is the
Hubble constant, while $\Omega_{\Lambda}\simeq 0.685$ and $\Omega_{m}\simeq
0.315$ are the mass density and the dark energy density parameters,
respectively.

By using the Eq. (19) into Eq. (20) we find

$x(\mathcal{Z})=\frac{1}{H_{0}}\int_{0}^{\mathcal{Z}}\left\\{\frac{z[k(1+\mathcal{Z^{\prime}})]^{z-1}}{\Lambda_{\rm
HL}^{z-1}}-\frac{3z\zeta}{2}\frac{\Lambda^{2}_{\rm HL}}{M^{2}_{\rm
QG}}\frac{{[k(1+\mathcal{Z^{\prime}})]^{3z-1}}}{\Lambda^{(3z-1)}_{\rm
HL}}\right\\}\frac{d\mathcal{Z}^{\prime}}{(1+\mathcal{Z}^{\prime})\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+\mathcal{Z}^{\prime})^{3})}}.$
(21)

The Eq. (21) corresponds as well to the time of flight of the photons under
the effects of LIV induced by the considered model. In the low/intermediary
energy: $\Lambda_{\rm HL}\sim{\rm IR\,regime}$, the second term becomes
negligible. Thus, the photon travel time recovers the usual one in the
standard relativistic cosmology when we assume $z=1$ (returning to Maxwell’s
general results too). The first term of Eq. (21) is associated with the photon
time of flight under a not so high energy scale (many orders lower than the
quantum gravity scale), can be motivated by a suppression under the Lifshitz
energy scale $\Lambda_{HL}\ll M_{QG}$ where would induce a weak anisotropy at
space-time. Consequently, the time difference between a high-energy and a low-
energy photon in a Hořava-Lifshitz LIV scenario is222For more details of
cosmological time delay formula derivation due to LIV effects, see Ref Jacob
and Piran (2007).

$|\Delta t|\approx\frac{3z}{2}\zeta\frac{\Lambda^{3(1-z)}_{\rm HL}}{M^{2}_{\rm
QG}}H_{0}^{-1}(k_{h}^{3z-1}-k_{l}^{3z-1})\int_{0}^{\mathcal{Z}}\frac{(1+\mathcal{Z}^{\prime})^{3z-2}d\mathcal{Z}^{\prime}}{\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+\mathcal{Z}^{\prime})^{3}}},$
(22)

where $k_{h}$ and $k_{l}$ represents the higher energy and lower energy of the
photons, respectively. Starting from the fact that LIV effects would occur
just under $M_{QG}$ scale, we can consider $\zeta=1$ to estimate the quantum
gravity energy. Using upper bounds for phenomenological values of $\Delta
t_{\rm LIV}$, we can obtain lower bounds for $M_{QG}$, that is

$M_{\rm QG}=\sqrt{\frac{3z}{2}\zeta\frac{\Lambda^{3(1-z)}_{\rm HL}}{\Delta
t}H_{0}^{-1}(k_{h}^{3z-1}-k_{l}^{3z-1})\mathcal{I}(\mathcal{Z})},$ (23)

where $\mathcal{I}(\mathcal{Z})$ is the integral in Eq. (22).

To estimate this bound, we will use data from GRB090510, which was studied in
Ref. Ackermann  _et al._ (2009). This GRB proved to be a promising event for
the analysis of dispersive effects due to the highly energetic photons were
detected on the GeV scale. The source was located at a redshift
$\mathcal{Z}=0.90$. Approximately $859\mbox{ms}$ after the initial emission
around $100$ keV, a peak of $31\mbox{GeV}$ was detected (this corresponds to
the most conservative LIV time delay choice). With these data, we can
estimate, for different values of the critical exponent $z$, lower bounds for
$M_{QG}$, as can be seen in the table below.

Table IV A: Lower bounds on $M_{QG}$ in terms of particular values of Lifshitz critical exponent $z$. | z=1/2 | z=2/3 | z=1
---|---|---|---
$M_{\rm QG}(\mbox{GeV})$ | $5.93\times 10^{16}$ | $4.72\times 10^{14}$ | $2.71\times 10^{10}$

The Lifshitz energy scale used to arrive at that result is
$\Lambda_{HL}=10^{10}\mbox{GeV}$ assuming that is a lower bound, according to
Ref.Pospelov and Shang (2012). When we choose $z=1$, we eliminate the
contribution arising from the Hořava-Lifshitz modification. Therefore, we end
up with the pure LIV contribution of a dimension-6 CPT-even term, which brings
quadratic dispersive effects. The lower bound for $M_{\rm QG}$ in this case,
which is of the order of $10^{10}\mbox{GeV}$, is the same found in the
literature for this type of dimension-6 EFT.

For any value of $z\neq 1$, the Lifshitz modification will contribute to the
result, as it will count the presence of the intermediate energy scale
$\Lambda_{\rm HL}$. For $z=2/3$, for example, we have an analogous case to
what would be a linear order dispersive effects (from a pure Myers-Pospelov
term of dimension-5, for example). It is interesting to note that the lower
limit of $M_{QG}$ does not exceed Planck’s mass, unlike the results found in
the literature that consider a usual dispersion at a linear level.

From Eq. (23), we have plotted a graph to observe the behavior of constraints
on $M_{QG}$ as function of $z$ as it varies from $0$ to $3$. The line on the
top of graph 1 corresponds to the Planck mass, i.e., $M_{\rm P}=1.22\times
10^{19}\mbox{GeV}$.

Figure 1: The behavior of constraints on $M_{QG}$ in terms of Lifshitz
critical exponent $z$.

The case $z=1/3$ and does not represent any physical relevance since the Eq.
(23) there is no energy dependence, canceling dispersive effects. For $z<1/3$,
we get imaginary results. Then we should consider the constraints to $z>1/3$.
Furthermore, the values next to $z=1$ produce weak lower bounds of $M_{\rm
QG}$. However, it is interesting to see that the function does not exceed the
Planck energy scale $M_{\rm P}$ under any circumstances.

For $z\rightarrow 1/3$, we get some interesting behavior. Normally the low
energy $k_{l}$ is discarded in some models for being many orders smaller than
the high energy value $k_{h}$, but in our model, it has an interesting role.
This happens because $k_{l}^{3z-1}$ assumes values close to $k_{h}^{3z-1}$ in
the Eq. (23), resulting in a reduction in the graph growth and bringing up the
$M_{\rm QG}$ to values near to $M_{\rm P}$. There is a maximum $M_{\rm QG}$
value close to $z=1/3$ before the graph starts to decay. This particular value
depends on the energy difference between the photons. It is worth to highlight
that the acceptable physical values lie in the interval $1/3<z\leq 1$,
however, as can be seen in the graphic 1 the best results are the values in
the interval of $z=1/3$ and $z=1/2$, which characterizes a presence of
intermediate energy between operators of dimension four and five. This result
is possible due to the introduction of critical exponent $z$, where we can get
control of the energy parameter. The peak in this graph is obtained when
$z=0.347$, i.e., $M_{\rm QG}\geq 3.02\times 10^{18}\mbox{GeV}$, which is a
strong lower bound, since it is relatively close to $M_{\rm P}$.

##  V Conclusions

In this work, we have analyzed the LIV of a Lifshitz-scaling to a CPT-even
Maxwell-Myers-Pospelov model, where the Lifshitz-type gauge-invariant of usual
electrodynamics can be recovered. The dispersion relation of the theory was
computed, showing a vacuum dispersive behavior with a dependence on the
Lifshitz critical exponent $z$, which induces some anisotropy between space
and time near to an intermediate energy scale $\Lambda_{\rm HL}$. There is no
appearance of vacuum birefringence, which allowed us to test this model by
analyzing the time delay between photons considering cosmological distances.

Finally, we have obtained lower bounds for the $M_{\rm QG}$ for any physical
relevant Lifshitz critical exponent related to our model. These limits were
determined using the observational results of the gamma-ray burst GRB090510.
For $z=1$, which corresponds to the usual dimensional-6 operator, we found the
same results already obtained in the literature for quadratic order dispersion
($\geq 10^{10}\mbox{GeV}$). For $z=2/3$, that is related to a linear
dispersion, was shown that the bound for $M_{\rm QG}$ recovers a realistic
scenario where LIV may be possible ($\geq 10^{14}\mbox{GeV}$, i.e., below
Planck’s mass). This result is an improvement from the outcome found in the
literature where the Hořava-Lifshitz scaling is absent, which predicts bounds
beyond Planck’s mass. Finally, the greater difference in energies of the two
considered photons was found for values of $z\rightarrow 1/3$, that is where
the bounds of $M_{\rm QG}$ tend to approximate to the Planck’s mass ($\geq
10^{18}\mbox{GeV}$). Therefore, this can be relevant for further
phenomenological investigations and opening a window for future researches in
order to understand the role of LIV in the standard model considering a
quantum gravity theory, such as Hořava-Lifshitz gravity. We also emphasize
that this behavior of Planck energy as a function of $z$, differs from what
was observed in Ref.Passos  _et al._ (2016) which show that the upper bounds
for the time delay derived from a Lifshitz-scaling to a CPT-odd Maxwell-Myers-
Pospelov model is not much sensitive with respect to the critical exponent
$z$.

###### Acknowledgements.

We would like to thank CNPq, CAPES and PRONEX/CNPq/FAPESQ-PB (Grant no.
165/2018), for partial financial support. MAA, FAB and EP acknowledge support
from CNPq (Grant nos. 306962/2018-7, 312104/2018-9, 304852/2017-1).

## References

  * Liberati (2013) S. Liberati, Classical and Quantum Gravity 30, 133001 (2013).
  * Myers and Pospelov (2003) R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003), arXiv:hep-ph/0301124 [hep-ph] .
  * Bolokhov and Pospelov (2008) P. A. Bolokhov and M. Pospelov, Phys. Rev. D77, 025022 (2008), arXiv:hep-ph/0703291 [HEP-PH] .
  * Sampaio and Passos (2019) T. A. M. Sampaio and E. Passos, Int. J. Mod. Phys. A34, 1950192 (2019), arXiv:1911.00048 [hep-th] .
  * Liberati and Maccione (2009) S. Liberati and L. Maccione, Ann. Rev. Nucl. Part. Sci. 59, 245 (2009), arXiv:0906.0681 [astro-ph.HE] .
  * Colladay and Kostelecký (1998) D. Colladay and V. A. Kostelecký, Phys. Rev. D 58, 116002 (1998).
  * Reyes (2010) C. M. Reyes, Phys. Rev. D82, 125036 (2010), arXiv:1011.2971 [hep-ph] .
  * Horava (2011) P. Horava, Phys. Lett. B694, 172 (2011), arXiv:0811.2217 [hep-th] .
  * Horava (2009) P. Horava, Phys. Rev. D79, 084008 (2009), arXiv:0901.3775 [hep-th] .
  * Visser (2009) M. Visser, Physical Review D 80, 025011 (2009).
  * Anacleto  _et al._ (2018) M. Anacleto, F. Brito, E. Maciel, A. Mohammadi, E. Passos, W. Santos, and J. Santos, Phys. Lett. B 785, 191 (2018), arXiv:1806.08273 [hep-th] .
  * Pospelov and Shang (2012) M. Pospelov and Y. Shang, Phys. Rev. D 85, 105001 (2012).
  * Gleiser and Kozameh (2001) R. J. Gleiser and C. N. Kozameh, Phys. Rev. D64, 083007 (2001), arXiv:gr-qc/0102093 [gr-qc] .
  * Coburn and Boggs (2003) W. Coburn and S. E. Boggs, Nature 423, 415 (2003), arXiv:astro-ph/0305377 [astro-ph] .
  * Jacobson  _et al._ (2004) T. A. Jacobson, S. Liberati, D. Mattingly, and F. W. Stecker, Phys. Rev. Lett. 93, 021101 (2004), arXiv:astro-ph/0309681 [astro-ph] .
  * Götz  _et al._ (2011) D. Götz, S. Covino, R. Hascoët, A. Fernandez-Soto, F. Daigne, R. Mochkovitch, and P. Esposito, Monthly Notices of the Royal Astronomical Society 413, 2173 (2011).
  * Laurent  _et al._ (2011) P. Laurent, D. Gotz, P. Binetruy, S. Covino, and A. Fernandez-Soto, Phys. Rev. D83, 121301 (2011), arXiv:1106.1068 [astro-ph.HE] .
  * Ackermann  _et al._ (2009) M. Ackermann _et al._ (Fermi GBM/LAT), Nature 462, 331 (2009), arXiv:0908.1832 [astro-ph.HE] .
  * Chang  _et al._ (2012) Z. Chang, Y. Jiang, and H.-N. Lin, Astroparticle Physics 36, 47 (2012).
  * Rodriguez Martinez and Piran (2006) M. Rodriguez Martinez and T. Piran, JCAP 0604, 006 (2006), arXiv:astro-ph/0601219 [astro-ph] .
  * Nemiroff  _et al._ (2012) R. J. Nemiroff, J. Holmes, and R. Connolly, Phys. Rev. Lett. 108, 231103 (2012), arXiv:1109.5191 [astro-ph.CO] .
  * Wei  _et al._ (2017) J.-J. Wei, B.-B. Zhang, L. Shao, X.-F. Wu, and P. Mészáros, Astrophys. J. 834, L13 (2017), arXiv:1612.09425 [astro-ph.HE] .
  * Passos  _et al._ (2017) E. Passos, M. Anacleto, F. Brito, O. Holanda, G. Souza, and C. Zarro, Phys. Lett. B 772, 870 (2017), arXiv:1612.02769 [hep-th] .
  * Passos  _et al._ (2016) E. Passos, E. M. C. Abreu, M. A. Anacleto, F. A. Brito, C. Wotzasek, and C. A. D. Zarro, Phys. Rev. D93, 085022 (2016), arXiv:1603.01558 [hep-th] .
  * Arnowitt  _et al._ (2008) R. L. Arnowitt, S. Deser, and C. W. Misner, Gen. Rel. Grav. 40, 1997 (2008), arXiv:gr-qc/0405109 [gr-qc] .
  * Kostelecky and Mewes (2009) V. A. Kostelecky and M. Mewes, Phys. Rev. D80, 015020 (2009), arXiv:0905.0031 [hep-ph] .
  * Jacob and Piran (2007) U. Jacob and T. Piran, Nature Phys. 3, 87 (2007), arXiv:hep-ph/0607145 [hep-ph] .


11institutetext: 1 ELI Beamlines, Institute of Physics ASCR, Prague, Czech
Republic
2 FMP, Charles University, Prague, Czech Republic
3 FNSPE, Czech Technical University in Prague, Prague, Czech Republic

# Development of LPA based hard X-ray sources at ELI Beamlines

U. Chaulagain1 M. Lamač1,2 M. Raclavsky1,3 K. H. Rao1 J. Nejdl1,3 S. A Weber1
and S. V. Bulanov1

##  1 Introduction

ELI-Beamlines is a part of the ELI (Extreme Light Infrastructure) project and
will soon become one of the most powerful laser facilities in the world [1].
One of the main objectives of ELI Beamlines center, is to provide beams of
laser-driven femtoseconds pulses of hard X-ray radiation to the users from
various fields of research [2]. Based on radiation from electrons accelerated
in plasma, the Betatron and Inverse Compton X-ray sources have the common
properties to be compact and to deliver collimated, incoherent, and
femtosecond radiation. The Betatron X-ray source is laser-driven broadband
hard X-ray source, generated from the transverse oscillations of electrons
during their acceleration and propagated along the direction of electron
velocity. It is characterized with a source size of the order of a micrometer,
a pulse duration of the order of few femtoseconds. These unique features of
the source will be open to various applications in atomic physics to
biomolecular sciences as well as in the industry. Here we report progress on
development of these noval X-ray sources at ELI Beamlines facility. We also
report on an advanced interferometry method for characterization of gas jets
that are routinely used in laser-plasma accleratror (LPA) as a target.

##  2 LPA based X-ray sources at ELI beamlines

ELI Gammatron beamline [3] is multidisciplinary, user-oriented hard X-ray
sources of radiation based on the Laser Wakefield acceleration [4]. It
provides X-ray pulses with a photon flux beyond $10^{11}$ photons per pulse
with optimized laser and plasma conditions [5, 6, 7]. The X-ray pulses can be
either broadband or quasi-monoenergetic with photon energies ranging from a
few keV up to a few MeV based on Betatron radiation or Inverse Compton
radiation. These X-ray sources have pulse duration of few femtoseconds as they
inherit the temporal properties of the accelerated electron bunch.

Figure 1: ELI Gammatron beamline located in the E2 experimental hall of ELI-
Beamlines.

Fig. 1 shows the Gammatron beamline located at the experimnetal hall E2. The
driving laser (Green beam) is the state-of-the-art diode-pumped Ti:sapphire
HAPLS laser (30J, 10 Hz). The beam is guided from the laser compressor to the
interaction chamber using six turning box chambers equipped with high laser
damage thresold ion beam sputtered dilectric coated mirrors [11]. The laser
beam focus to supersonic gas jets drives a LPA in which electrons accelerate
to GeV level and produce ultrashort pulse of hard X-ray radiation. The gas jet
target allows to operate the source at high-repetition rate, typical gases
used are pure He, mixture of He with Nitrogen, or even simply dry air[8]. The
all-optical ultrafast X-ray pulse from the Gammatron beamline, with jitter-
free synchronization with the pump IR beam, due to their way of generation
from the high-power optical laser pulses, provides an extensive platform to
perform time-resolved ultrafast studies on atomic time scales. A dedicated
user station has been designed to enable various time-resolved experiments. We
have designed a novel multi-lane KB X-ray focusing optics covering a wide
spectral range up to 25 keV which aims to exploit the broadband nature of the
betatron X-ray source for different applications ranging from phase contrast
imaging [9] to time-resolved X-ray spectroscopy and diffraction of bulk
samples.

In addition, a high peak brightness hard X-ray Betatron source is being
commissioned within the plasma physics platform (P3) in the E3 experimental
hall [12, 13]. The P3 aims to carry wide range of advanced plasma physics
experiments including HED physics [14], laboratory astrophysics [15, 16] and
high field interactions. These experiments are very important, in particular,
to benchmark the models and simulation codes that are often used to model the
astrophysical process [17, 18]. The P3 platform is designed to perform
advanced plasma physics reserach using multiple lasers beams; with both the
high energy long pulse and high power short pulse lasers [19]. Presence of a
hard X-ray backlighter adds the capability of P3 in diagnostics of highly
ionized dense plasmas genreated during high-intensity laser matter
interaction.

##  3 Advanced interferometry target characterization station for LPA targets

Figure 2: a) Schematics of four pass set up for interferrometric gas jet
characterization. b) Comparison of spatial phase distributions obtained at one
pass (left), and four passes (right) of the probe beam through the gas jet
generated by a 1-mm cylindrical nozzle (P= 3 bar, Ar gas).

We have developed a novel optical probing technique with significantly
increased sensitivity and high spatial resolution suitable for
characterization of low-density LPA gas targets. These features are a result
of unique optical configuration employing multiple passes of the probe beam
through the object with two relay-imaging arms and polarization switching
using passive optical elements. With multiple passes of the probe beam through
the sample to enable an increase of the accumulated phase and improve the
sensitivity. The scheme of four pass interferrometry is shown in 2a. In a two-
pass configuration a two-fold increase in the sensitivity compared to the
standard single-pass configuration [20, 21] and with four-fold in four pass
was found. Fig. 2b shows the comaprision of the measurment at single pass to
four pass configuartion, we observed a four-fold increase in the phase
sensitivity. We used this advanced interferometric technique to perform a
tomographic characterization of slit nozzle with complex spatial distribution
of the gas density [22].

##  4 Acknowledgments

Supported by the project Advanced research using high intensity laser produced
photons and particles (ADONIS) (CZ.02.1.01/0.0/0.0/16_019/0000789) from
European Regional Development Fund and Project LM2018141 of the Ministry of
Education, Youth and Sports within targeted support of Large infrastructures.

## References

  * [1] B. Rus, et al. “Outline of the ELI-Beamlines facility", Proc. SPIE 8080, 808010 (2011).
  * [2] J. Nejdl, et al. “Update on laser-driven X-ray sources at ELI Beamlines", Proc. X-ray Lasers, Vol. vol. 11886, p. 1188608, SPIE (2020).
  * [3] U. Chaulagain, et al. “ELI Gammatron Beamline: Dawn of ultrafast hard X-ray science" (to be published).
  * [4] T. Tajima, and J. M. Dawson. “Laser electron accelerator" PRL 43, no. 4:267 (1979).
  * [5] M. Kozlova, et al. “Hard X Rays from Laser-Wakefield Accelerators in Density Tailored Plasmas" Phys. Rev. X. 10.1: 011061 (2020).
  * [6] S. Fourmaux, et al. “Laser-based synchrotron X-ray radiation experimental scaling" Opt. Express 28.3: 3147-3158 (2020).
  * [7] M. Lamač,et al. “Two-color non linear resonances in betatron oscillations of laser accelerated relativistic electrons“, PRR, 3(3), p.033088 (2021).
  * [8] K. Bohacek, et al. “Stable electron beams from laser wakefield acceleration with few-terawatt driver using a supersonic air jet" Nucl. Instrum. Methods Phys. Res. A: Accel. Spectrom. Detect. Assoc. Equip. 883 24-28 (2018).
  * [9] U. Chaulagain, et al., “X-ray phase contrast imaging of biological samples using a betatron X-ray source generated in a laser wakefield accelerator“, Proc. X-ray Lasers (2018).
  * [10] M. Raclavsky et al., “Multi- Lane Mirror for Broadband Applications of Betatron X-ray Source“, Photonics, 8, 579 (2021).
  * [11] T. Willemsen, et al. “Large area ion beam sputtered dielectric ultrafast mirrors for petawatt laser beamlines" Opt. Express, 30(4), pp.6129-6141 (2022).
  * [12] S. Weber, et al. “P3: An installation for high-energy density plasma physics and ultra-high intensity laser–matter interaction at ELI-Beamlines” MRE, 2(4), pp.149-176 (2017).
  * [13] U. Chaulagain, et al. “LWFA-driven betatron source for Plasma Physics Platform at ELI Beamlines" International Conf. on X-ray Lasers. Springer, Cham, (2018).
  * [14] F. Suzuki-Vidal, et al. “Counterpropagating radiative shock experiments on the Orion laser" PRL 119, no.5:055001 (2017).
  * [15] U. Chaulagain, et al. "Structure of a laser-driven radiative shock" HEDP, 17, pp.106-113 (2015).
  * [16] R. L. Singh, et al. “Experimental study of the interaction of two laser-driven radiative shocks at the PALS laser" HEDP 23, 20-30 (2017).
  * [17] M. Cotelo, et al. "Simulation of radiative shock waves in Xe of last PALS experiments" HEDP 17:68-73 (2015).
  * [18] F. Suzuki-Vidal, et al. "First radiative shock experiments on the SG-II laser" HPLSE 9 (2021).
  * [19] N. Jourdain, et al. “The L4n laser beamline of the P3-installation: Towards high-repetition rate high-energy density physics at ELI-Beamlines" MRE 6.1:015401 (2021).
  * [20] J. Nejdl, et al. “Imaging Michelson interferometer for a low-density gas jet characterization” RSI, 90(6), 065107 (2019).
  * [21] U. Chaulagain, et al. “Tomographic characterization of gas jets for laser-plasma acceleration with increased sensitivity”, Proc. X-ray Lasers Vol. 11886, p. 118860A, SPIE (2020).
  * [22] S. Karatodorov, et al. “Multi-pass probing for high-sensitivity tomographic interferometry” Sci. Rep., 11(1),1- 10 (2021).