††institutetext: Department of Physics and Astronomy, Rutherford Building,
University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

# CP violatingTri-bimaximal-Cabibbo mixing

D. V. Ahluwalia<EMAIL_ADDRESS>
###### Abstract

In view of the new data from the Daya Bay and RENO collaborations, King has
presented a very natural deformation of tri-bimaximal mixing. Here we show
that $L/E$ flatness of the $e$-like event ratio in the atmospheric neutrino
data, when coupled with King’s observation that the smallest neutrino mixing
angle, $\theta_{13}$, seems to be related to the largest quark mixing angle
(the Cabibbo angle $\theta_{C}$), leads to a CP violating tri-bimaximal-
Cabibbo mixing. King’s tri-bimaximal-Cabibbo mixing follows as a leading order
approximation from our result.

###### Keywords:

Neutrino physics, CP violation

The precise form of the neutrino mixing matrix, $U$, that defines the
relationship between the flavour and mass eigenstates, $|\nu_{\ell}\rangle$
and $|\nu_{j}\rangle$ respectively Chau:1984fp ; Beringer:2012bj , reads

$|\nu_{\ell}\rangle=\sum_{j}U^{\ast}_{\ell
j}|\nu_{j}\rangle,\quad\ell=e,\mu,\tau,\quad j=1,2,3,$ (1)

and the knowledge of the masses for the underlying mass eigenstates, arise
from yet unknown physics. Nevertheless, once the parameters that determine the
mixing matrix and the mass-squared differences are deciphered from the data
one can derive their phenomenological consequences on supernova explosions
Ahluwalia:2004dv ; Lunardini:2007vn ; Duan:2006an ; Duan:2007sh , on the
synthesis of elements Yoshida:2006sk , on the cosmic microwave background and
the distribution of large-scale structure Lesgourgues:2006nd . In particular,
if the neutrino mixing angle $\theta_{13}\neq 0$ then one can obtain CP
violation in the neutrino sector with many interesting physical consequences
Khlopov:1981nq ; Frampton:2002qc ; Balantekin:2007es .

The T2K, MINOS, and Double CHOOZ indications that the smallest neutrino mixing
angle $\theta_{13}$ may be non-zero Abe:2011ph ; Adamson:2011qu ; Abe:2011fz
has now been confirmed by the results of the Daya Bay and RENO collaborations
An:2012eh ; Ahn:2012nd . King has made the observation King:2012vj  that the
smallest neutrino mixing angle $\theta_{13}$, seems to be related to the
largest quark mixing angle, the Cabibbo angle $\theta_{C}$ Cabibbo:1963yz , or
equivalently to the Wolfenstein parameter, $\lambda=0.2253\pm 0.0007$
Wolfenstein:1983yz ; Beringer:2012bj :111It is worth noting that Mohapatra and
Smirnov had earlier conjectured King’s observation (Mohapatra:2006gs, , Sec.
3.1).

$\theta_{13}~{}\mbox{(or,
}\theta_{reac}\mbox{)}=\arcsin\left(\frac{\sin\theta_{C}}{\sqrt{2}}\right)=\arcsin\left(\frac{\lambda}{\sqrt{2}}\right).$
(2)

To this observation we now add that the $L/E$ — where $L$ is the neutrino
source-detector distance and $E$ is the neutrino energy — flatness of the
$e$-like event ratio observed for atmospheric neutrinos Fukuda:1998mi
requires that

$\theta_{23}~{}\mbox{(or,
}\theta_{atm}\mbox{)}=\frac{\pi}{4},\quad\delta=\pm\frac{\pi}{2}.$ (3)

This observation was first made in reference Ahluwalia:2002tr . The $\delta$
obtained in Ahluwalia:2002tr  was also introduced recently as an Ansatz in
Ref. Zhang:2012ys .

Global analysis of neutrino oscillation data by two independent groups shows:
(a) $\delta$ to be $\left(0.83^{+0.54}_{-0.64}\right)\pi$ for the normal mass
hierarchy while allowing for the full $[0,2\pi]$ range for the inverted mass
hierarchy Tortola:2012te , (b) $\delta\approx\pi$ with no significant
difference between the normal and inverted mass hierarchies Fogli:2012ua . A
detailed study of these two papers reveals that there is no statistically
significant indication which disfavours $\delta=\pm\pi/2$. Regarding
$\theta_{23}$: (a) the first of the mentioned groups obtains
$\sin^{2}\theta_{23}=0.49^{+0.08}_{-0.05}$ for the normal mass hierarchy, and
$\sin^{2}\theta_{23}=0.53^{+0.05}_{-0.07}$ for the inverted mass hierarchy
(these values are consistent with $\theta_{23}=\pi/4$), while (b) the second
group finds a slight preference for $\theta_{23}<\pi/4$.

Both groups agree with the tri-bimaximal mixing value for the remaining angle
Tortola:2012te ; Fogli:2012ua

$\theta_{12}~{}\mbox{(or,
}\theta_{\odot}\mbox{)}=\arcsin\left(\frac{1}{\sqrt{3}}\right).$ (4)

With all the angles and phases thus fixed, the neutrino mixing matrix for the
choice $\delta=\pi/2$ in equation (3) takes the form

$U^{+}=\begin{pmatrix}\sqrt{\frac{2}{3}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}&\sqrt{\frac{1}{3}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}&i\frac{1}{\sqrt{2}}\lambda\\\
-\frac{1}{\sqrt{6}}\left(1-i\lambda\right)&\frac{1}{\sqrt{3}}\left(1+i\frac{1}{2}\lambda\right)&\frac{1}{\sqrt{2}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}\\\
\frac{1}{\sqrt{6}}\left(1+i\lambda\right)&-\frac{1}{\sqrt{3}}\left(1-i\frac{1}{2}\lambda\right)&\frac{1}{\sqrt{2}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}\end{pmatrix}.$
(5)

Its counterpart, $U^{-}$, for $\delta=-\pi/2$ is obtained by letting $i\to-i$
in $U^{+}$. As a measure of CP violation, following Beringer:2012bj , we
define the asymmetries

$A_{CP}^{(\ell^{\prime}\ell)}\colonequals
P(\nu_{\ell}\to\nu_{\ell^{\prime}})-P(\bar{\nu}_{\ell}\to\bar{\nu}_{\ell^{\prime}}),$
(6)

and find

$\displaystyle A_{CP}^{(\mu e)}=-A^{(\tau e)}_{CP}=A_{CP}^{(\tau\mu)}$
$\displaystyle=\mp\frac{1}{3}\lambda\left(2-\lambda^{2}\right)\left(\sin\frac{\Delta
m^{2}_{32}}{2p}L+\sin\frac{\Delta m^{2}_{21}}{2p}L+\sin\frac{\Delta
m^{2}_{13}}{2p}L\right)$ $\displaystyle\approx\mp 0.146\left(\sin\frac{\Delta
m^{2}_{32}}{2p}L+\sin\frac{\Delta m^{2}_{21}}{2p}L+\sin\frac{\Delta
m^{2}_{13}}{2p}L\right),$ (7)

where all symbols have their usual meaning. The $\mp$ sign holds for
$\delta=\pm\frac{\pi}{2}$. For $\lambda=0$, or equivalently $\theta_{13}=0$,
the $U^{\pm}$ reduce to the standard tri-bimaximal mixing matrix
Harrison:2002er .222This may be compared with (Stancu:1999ct, , Eq. (26)) that
gives an interpolating matrix with $\theta_{\odot}$ as a variable. In one
limit the interpolating matrix gives the bimaximal mixing Vissani:1997pa ;
Ahluwalia:1998xb ; Barger:1998ta  and in another it yields tri-bimaximal
mixing Harrison:2002er .

The result (7) is modified by matter effects Wolfenstein:1977ue ;
Mikheev:1986gs . Its general features are studied in detail by various authors
Gava:2008rp ; Balantekin:2007es ; Kneller:2009vd ; Kisslinger:2012se . In
gravitational environments the following argument suggests that one must
expect a significant modification to the result (7). Neutrino oscillations
provide us with a set of flavour oscillation clocks. These clocks must
redshift according to the general expectations of the theory of general
relativity. In gravitational environments of neutron stars the dimensionless
gravitational potential is $\Phi^{NS}_{grav}\approx 0.2$ (cf. for Earth,
$\Phi^{\oplus}_{grav}\approx 6.95\times 10^{-10}$). For a given source-
detector distance, and a given energy, the asymmetries $A_{CP}$ for supernovae
modeling must be accordingly modified Ahluwalia:1996ev ; Ahluwalia:1998jx ;
Konno:1998kq ; Wudka:2000rf ; Mukhopadhyay:2005gb ; Singh:2003sp  at the
$20\%$ level, or thereabouts.

An examination of the $U^{\pm}$ immediately shows that the expectation values
of the $\nu_{\mu}$ and $\nu_{\tau}$ masses are identical. To
$\mathcal{O}(\lambda^{2})$ the $U^{-}$ obtained above reproduces to King’s
result (King:2012vj, , Eq. (8)) for $\delta=\pi/2$. The presented $U^{\pm}$
not only accommodate the implications of the Daya Bay and RENO collaborations,
but also the L/E flatness of the $e$-like event ratio seen in the atmospheric
neutrino data while respecting all other known data on neutrino oscillations.

###### Acknowledgements.

The result presented here was obtained on 10 May 2012, and was presented the
next day at a MatScience Seminar. The author thanks Institute of Mathematical
Sciences (“MatScience”, Chennai, India) for its hospitality and for its
vibrant scholarly environment.

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# Acceleration as a circular motion along an imaginary circle: Kubo-Martin-
Schwinger condition for accelerating field theories in imaginary-time
formalism

Victor E. Ambrus<EMAIL_ADDRESS>Maxim N. Chernodub
<EMAIL_ADDRESS>Department of Physics, West University of
Timisoara,
Bd. Vasile Pârvan 4, Timisoara 300223, Romania  Institut Denis Poisson,
Université de Tours, Tours 37200, France

###### Abstract

We discuss the imaginary-time formalism for field theories in thermal
equilibrium in uniformly accelerating frames. We show that under a Wick
rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point
in a two-dimensional subspace tangential to the acceleration direction and the
imaginary time. We demonstrate that the accelerated version of the Kubo-
Martin-Schwinger (KMS) condition implies an identification of all spacetime
points related by integer-multiple rotations in the tangential subspace about
this Euclidean Rindler event-horizon point, with the rotational quanta defined
by the thermal acceleration, $\alpha=a/T$. In the Wick-rotated Rindler
hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic
boundary conditions in terms of the imaginary proper time (rapidity)
coordinate. Our findings pave the way to study, using first-principle lattice
simulations, the Hawking-Unruh radiation in geometries with event horizons,
phase transitions in accelerating Early Universe and early stages of quark-
gluon plasma created in relativistic heavy-ion collisions.

###### keywords:

Acceleration , Unruh effect , KMS relation , Finite temperature field theory

††journal: Physics Letters B

##  1 Introduction

In the past decades, there has been a renewed interest in studying systems
with acceleration as toy models for understanding the dynamics of the quark-
gluon plasma fireball created in ultrarelativistic (non-central) heavy-ion
collisions [1]. Such systems exhibit large acceleration immediately after the
collision [2] until the central rapidity plateau develops as in the Björken
boost-invariant flow model [3], where the acceleration vanishes. A natural
question that arises for such a system is to what extent these extreme
kinematic regimes affect the thermodynamics of the plasma fireball, which sets
the stage for further evolution of the quark-gluon plasma. The environment of
the “Little Bangs” of high-energy heavy-ion collisions [4] sheds insights on
the properties of a primordial quark-gluon matter that once emerged at the
time of the Big Bang in the Early Universe [5].

Our knowledge of the non-perturbative properties of the quark-gluon plasma
originates from first-principle numerical simulations of lattice QCD, which is
formulated in Euclidean spacetime, by means of the imaginary-time formalism
[6]. Acceleration is closely related to rotation due to the resemblance of the
corresponding generators of Lorentz transformations of Minkowski spacetime. In
the case of non-central collisions, the angular velocity of the quark-gluon
fluid can reach values of the order of $\Omega\sim 10^{22}\,{\rm Hz}$ [7]
which translates to $\hbar\Omega\simeq 6\ {\rm MeV}\ll T_{c}$, where $T_{c}$
is the transition temperature to the quark-gluon plasma phase. The lattice
studies have so far been limited to the case of uniformly rotating systems in
Euclidean space-time, where the rotation parameter has to be analytically
continued to imaginary values [8] in order to avoid the sign problem that also
plagues lattice calculations at finite chemical potential [9]. Analytical
analyses of the effects of rotation on the phase diagram, performed in various
effective infrared models of QCD [10, 11, 12, 13, 14, 15, 16, 17], stay in
persistent contradiction with the first-principle numerical results [18, 19,
20, 21, 22, 23], presumably due to numerically-observed rotational instability
of quark-gluon plasma [21, 22, 23] (related to the thermal melting of the non-
perturbative gluon condensate [21]), splitting of chiral and deconfining
transitions [23, 24], or formation of a strongly inhomogeneous mixed
hadronic–quark-gluon-plasma phase induced by rotation [17, 25].

An earlier study of a Euclidean quantum field theory in an accelerating
spacetime with the Friedmann-Lemaître-Robertson-Walker metric has also
encountered the sign problem, which was avoided by considering a purely
imaginary Hubble constant [26]. On the contrary, our formulation of
acceleration in the imaginary-time formalism is free from the sign problem,
and thus, it can be formulated for physical, real-valued acceleration.
Throughout the paper, we use $\hbar=c=k_{B}=1$ units.

##  2 Global equilibrium in uniform acceleration

From a classical point of view, global equilibrium states in generic particle
systems are characterized by the inverse temperature four-vector
$\beta^{\mu}\equiv u^{\mu}(x)/T(x)$, associated with the local fluid velocity
$u^{\mu}$, with $\beta^{\mu}$ satisfying the Killing equation,
$\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu}=0$ [27, 28]. For an
accelerated system at equilibrium, one gets
$\beta^{\mu}\partial_{\mu}=\beta_{T}[(1+az)\partial_{t}+at\partial_{z}]$, with
$\beta_{T}=1/T$ where111Throughout our article, $T(x)$ denotes the local
temperature (1), while $T$ stands for the value of $T(x)$ at the origin
$t=z=0$. Also, for reasons that will become clear shortly later, we use the
notation $\beta_{T}$ instead of the conventional $\beta$ for the inverse
temperature at the coordinate origin. $T\equiv T({\boldsymbol{0}})$ represents
the temperature at the coordinate origin
${\boldsymbol{x}}_{\|}\equiv(t,z)={\boldsymbol{0}}$ in the longitudinal plane
spanned by the time coordinate $t$ and the acceleration direction $z$. The
local temperature $T(x)$, the local fluid velocity $u^{\mu}(x)$ and the local
proper acceleration $a^{\mu}(x)\equiv u^{\nu}\partial_{\nu}u^{\mu}$,
respectively,

$\displaystyle T(x)$
$\displaystyle\equiv(u_{\mu}\beta^{\mu})^{-1}=\frac{1}{\beta_{T}\sqrt{(1+az)^{2}-(at)^{2}}},$
(1) $\displaystyle u^{\mu}(x)\partial_{\mu}$
$\displaystyle=T(x)\beta_{T}\bigl{[}(1+az)\partial_{t}+at\partial_{z}\bigr{]}\,,$
(2) $\displaystyle a^{\mu}(x)\partial_{\mu}$
$\displaystyle=aT^{2}(x)\beta_{T}^{2}[at\partial_{t}+(1+az)\partial_{z}]\,,$
(3)

diverge at the Rindler horizon:

$\displaystyle(1+az)^{2}-(at)^{2}=0,\qquad\ z\geqslant-\frac{1}{a}\,.$ (4)

It is convenient to define the dimensionless quantity called the proper
thermal acceleration $\alpha=\sqrt{-\alpha^{\mu}\alpha_{\mu}}$ and the
corresponding four-vector
$\alpha^{\mu}=u^{\nu}\partial_{\nu}\beta^{\mu}=a^{\mu}/T(x)$, respectively:

$\displaystyle\alpha$ $\displaystyle=a\beta_{T}\,,$
$\displaystyle\alpha^{\mu}(x)\partial_{\mu}$
$\displaystyle=a\beta_{T}^{2}T(x)[at\partial_{t}+(1+az)\partial_{z}]\,.$ (5)

Note that, while the magnitude $\alpha$ of the thermal acceleration is a
space-time constant, the local acceleration
$a(x)=\sqrt{-a_{\mu}a^{\mu}}=\alpha T(x)$ depends on space and time
coordinates.

In classical theory, the energy-momentum tensor for an accelerating fluid in
thermal equilibrium reads

$T^{\mu\nu}=\mathcal{E}u^{\mu}u^{\nu}-\mathcal{P}\Delta^{\mu\nu},$ (6)

where $\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$. The local energy density
$\mathcal{E}$ and pressure $\mathcal{P}$ are characterized by the local
temperature (1). For a conformal system,

$\mathcal{E}=3\mathcal{P}=\frac{\nu_{\rm eff}\pi^{2}}{30}T^{4}(x),$ (7)

where $\nu_{\rm eff}$ is the effective bosonic degrees of freedom. In the case
of a massless, neutral scalar field, $\nu_{\rm eff}=1$, while for Dirac
fermions, $\nu_{\rm eff}=\frac{7}{8}\times 2\times 2=7/2$, taking into account
the difference between Bose-Einstein and Fermi-Dirac statistics ($7/8$), spin
degeneracy, as well as particle and anti-particle contributions.

##  3 Unruh and Hawking effects

Unruh has found that in a frame subjected to a uniform acceleration $a$, an
observer detects a thermal radiation with the temperature [29]:

$\displaystyle T_{U}\equiv\frac{1}{\beta_{U}}=\frac{a}{2\pi}\,,$ (8)

where we also defined the Unruh length $\beta_{U}$, which will be useful in
our discussions below.

The Unruh effect is closely related to the Hawking evaporation of black holes
[30, 31], which proceeds via the quantum production of particle pairs near the
event horizon of the black hole. The Hawking radiation has a thermal spectrum
with an effective temperature

$\displaystyle T_{H}=\frac{\kappa}{2\pi}\,,$ (9)

where $\kappa=1/(4M)$ is the acceleration due to gravity at the horizon of a
black hole of mass $M$. The similarity of both effects, suggested by the
equivalence of formulas for the Unruh temperature (9) and the Hawking
temperature (8), goes deeper as the thermal character of both phenomena
apparently originates from the presence of appropriate event horizons [32,
33]. In an accelerating frame, the event horizon separates causally
disconnected regions of spacetime, evident in the Rindler coordinates in which
the metric of the accelerating frame is conformally flat [34].

Quantum effects lead to acceleration-dependent corrections to Eq. (7) and may
also produce extra (anisotropic) contributions to the energy-momentum tensor
$T^{\mu\nu}$ of the system. Such corrections were already established using
the Zubarev approach [35, 36] or Wigner function formalism [37, 38], and one
remarkable conclusion is that the energy-momentum tensor $\Theta^{\mu\nu}$ in
an accelerating system exactly vanishes at the Unruh temperature (8), or,
equivalently, when the thermal acceleration (3) reaches the critical value
$\alpha=\alpha_{c}=2\pi$: $\Theta^{\mu\nu}(T=T_{U})=0$. A somewhat related
property is satisfied by thermal correlation functions in the background of a
Schwarzschild black hole, establishing the equivalence between Feynman and
thermal Green’s functions, with the latter one taken at the Hawking
temperature (9), cf. Ref. [33, 32].

As noted earlier, the energy density receives quantum corrections. For the
conformally-coupled massless real-valued Klein-Gordon scalar field and the
Dirac field, we have, respectively [36, 37, 38, 39, 40]:

$\displaystyle\mathcal{E}_{\rm scalar}$
$\displaystyle=\frac{\pi^{2}T^{4}(x)}{30}\biggl{[}1-\Bigr{(}\frac{\alpha}{2\pi}\Bigl{)}^{4}\biggr{]}\,,$
(10a) $\displaystyle\mathcal{E}_{\rm Dirac}$
$\displaystyle=\frac{7\pi^{2}T^{4}(x)}{60}\biggl{[}1-\Bigr{(}\frac{\alpha}{2\pi}\Bigl{)}^{2}\biggr{]}\biggl{[}1+\frac{17}{7}\Bigr{(}\frac{\alpha}{2\pi}\Bigl{)}^{2}\biggr{]}\,,$
(10b)

where we specially rearranged terms to make it evident that at the Unruh
temperature $T=T_{U}$ (or, equivalently, at $\alpha=2\pi$), the energy density
vanishes.

The above discussion focused on the free-field theory. In the interacting
case, a legitimate question is to what extent do the local kinematics
influence the phase structure of phenomenologically relevant field theories,
for example, to deconfinement and chiral thermal transitions of QCD. Central
to lattice finite-temperature studies is how to set the Euclidean-space
boundary conditions in the imaginary-time formalism. A static bosonic
(fermionic) system at finite temperature can be implemented by imposing
(anti-)periodicity in imaginary time $\tau=it$ with period given by the
inverse temperature, $\tau\rightarrow\tau+\beta_{T}$. These boundary
conditions are closely related to, and in fact, derived from the usual Kubo-
Martin-Schwinger (KMS) relation formulated for a finite-temperature state (at
vanishing acceleration), which translates into a condition written for the
scalar and fermionic thermal two-point functions [6, 41]:

$G_{F}(t)=G_{F}(t+i\beta_{T}),\quad S_{F}(t)=-S_{F}(t+i\beta_{T}),$ (11)

where we suppressed the dependence on the spatial coordinate $\boldsymbol{x}$
and the second four-point $x^{\prime}$. In the case of rotating states, the
KMS relation (11) gets modified to [17, 40, 42]

$\displaystyle G_{F}(t,\varphi)$
$\displaystyle=G_{F}(t+i\beta_{T},\varphi+i\beta_{T}\Omega),$ $\displaystyle
S_{F}(t,\varphi)$ $\displaystyle=-e^{-\beta_{T}\Omega
S^{z}}S_{F}(t+i\beta_{T},\varphi+i\beta_{T}\Omega),$ (12)

where $e^{-\beta_{T}\Omega S^{z}}$ is the spin part of the rotation with
imaginary angle $i\beta_{T}\Omega$ along the rotation ($z$) axis and
$S^{z}=\frac{i}{2}\gamma^{x}\gamma^{y}$ is the spin matrix. The purpose of the
present paper is to uncover the KMS relation and subsequent conditions for
fields and, consequently, for correlation functions in a uniformly accelerated
state.

##  4 Quantum field theory at constant acceleration

In Minkowski space, the most general solution of the Killing equation reads

$\beta^{\mu}=b^{\mu}+\varpi^{\mu}{}_{\nu}x^{\nu},$ (13)

where $b^{\mu}$ is a constant four-vector and $\varpi^{\mu\nu}$ is a constant,
anti-symmetric tensor. A quantum system in thermal equilibrium is
characterized by the density operator

$\hat{\rho}=e^{-b\cdot\hat{P}+\varpi:\hat{J}/2},$ (14)

where $\hat{P}^{\mu}$ and $\hat{J}^{\mu\nu}$ are the conserved four-momentum
and total angular momentum operator, representing the generators of
translations and of Lorentz transformations. In order to derive the KMS
relation, it is convenient to factorize $\hat{\rho}$ into a translation part
and a Lorentz transformation part, as pointed out in Ref. [37]:

$e^{-b\cdot\hat{P}+\varpi:\hat{J}/2}=e^{-\tilde{b}(\varpi)\cdot\hat{P}}e^{\varpi:\hat{J}/2},$
(15)

where $\tilde{b}$ is given by

$\tilde{b}(\varpi)^{\mu}=\sum_{k=0}^{\infty}\frac{i^{k}}{(k+1)!}(\varpi^{\mu}{}_{\nu_{1}}\varpi^{\nu_{1}}{}_{\nu_{2}}\cdots\varpi^{\nu_{k-1}}{}_{\nu_{k}})b^{\nu_{k}}.$
(16)

Focusing now on the accelerated system with reference inverse temperature
$\beta_{T}=1/T$, we have $b^{\mu}=\beta_{T}\delta^{\mu}_{0}$ and
$\varpi^{\mu}{}_{\nu}=\alpha(\delta^{\mu}_{3}g_{0\nu}-\delta^{\mu}_{0}g_{3\nu})$,
such that $\tilde{b}$ becomes

$\tilde{b}^{\mu}=B\delta^{\mu}_{0}+A\delta^{\mu}_{3},\quad
B=\frac{\sin\alpha}{a},\quad A=\frac{i}{a}(1-\cos\alpha),$ (17)

where $\alpha=a/T$ is the thermal acceleration (5). This observation allows
$\hat{\rho}=e^{-\beta_{T}\hat{H}+\alpha\hat{K}^{z}}$ to be factorized as

$\hat{\rho}=e^{-B\hat{H}+A\hat{P}^{z}}e^{\alpha\hat{K}^{z}}.$ (18)

A relativistic quantum field described by the field operator $\hat{\Phi}$
transforms under Poincaré transformations as

$\displaystyle
e^{i\tilde{b}\cdot\hat{P}}\hat{\Phi}(x)e^{-i\tilde{b}\cdot\hat{P}}$
$\displaystyle=\hat{\Phi}(x+\tilde{b}),$
$\displaystyle\hat{\Lambda}\hat{\Phi}(x)\hat{\Lambda}^{-1}$
$\displaystyle=D[\Lambda^{-1}]\hat{\Phi}(\Lambda x),$ (19)

where $\Lambda=e^{-\frac{i}{2}\varpi:\mathcal{J}}$ is written in terms of the
matrix generators
$(\mathcal{J}^{\mu\nu})_{\alpha\beta}=i(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}-\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha})$,
while $D[\Lambda]^{-1}=e^{\frac{i}{2}\varpi:S}$ is the spin part of the
inverse Lorentz transformation. Comparing Eq. (19) and (14), it can be seen
that the density operator $\hat{\rho}$ acts like a Poincaré transformation
with imaginary parameters [37]. Using now the factorization (18), it can be
seen that $\hat{\rho}$ acts on the field operator $\hat{\Phi}$ as follows:

$\hat{\rho}\hat{\Phi}(t,z)\hat{\rho}^{-1}=e^{-\alpha
S^{0z}}\hat{\Phi}({\tilde{t}},{\tilde{z}}),$ (20)

where

$\displaystyle{\tilde{t}}$
$\displaystyle=\cos(\alpha)t+i\sin(\alpha)z+\frac{i}{a}\sin(\alpha),$
$\displaystyle{\tilde{z}}$
$\displaystyle=i\sin(\alpha)t+\cos(\alpha)z-\frac{1}{a}[1-\cos(\alpha)].$ (21)

The spin term evaluates to $e^{-\alpha S^{0z}}=1$ in the scalar case (since
$S^{0z}=0$), while for the Dirac field,
$S^{0z}=\frac{i}{2}\gamma^{0}\gamma^{3}$ and

$e^{-\alpha
S^{0z}}=\cos\frac{\alpha}{2}-i\gamma^{0}\gamma^{3}\sin\frac{\alpha}{2}.$ (22)

##  5 KMS relation at constant uniform acceleration

Consider now the Wightman functions $G^{\pm}(x,x^{\prime})$ and
$S^{\pm}(x,x^{\prime})$ of the Klein-Gordon and Dirac theories, defined
respectively as

$\displaystyle G^{+}(x,x^{\prime})$
$\displaystyle=\langle\hat{\Phi}(x)\hat{\Phi}(x^{\prime})\rangle,$
$\displaystyle S^{+}(x,x^{\prime})$
$\displaystyle=\langle\hat{\Psi}(x)\hat{\overline{\Psi}}(x^{\prime})\rangle,$
$\displaystyle G^{-}(x,x^{\prime})$
$\displaystyle=\langle\hat{\Phi}(x^{\prime})\hat{\Phi}(x)\rangle,$
$\displaystyle S^{-}(x,x^{\prime})$
$\displaystyle=-\langle\hat{\overline{\Psi}}(x^{\prime})\hat{\Psi}(x)\rangle.$
(23)

When the expectation value $\langle\cdot\rangle$ is taken at finite
temperature and under acceleration, we derive the KMS relations:

$\displaystyle G^{+}(x,x^{\prime})$
$\displaystyle=G^{-}({\tilde{t}},{\tilde{z}};x^{\prime}),$ $\displaystyle
S^{+}(x,x^{\prime})$ $\displaystyle=-e^{-\alpha
S^{0z}}S^{-}({\tilde{t}},{\tilde{z}};x^{\prime}).$ (24)

The KMS relations also imply natural boundary conditions for the thermal
propagators:

$\displaystyle G_{F}({\tilde{t}},{\tilde{z}};x^{\prime})$
$\displaystyle=G_{F}(t,z;x^{\prime})\,,$ $\displaystyle
S_{F}({\tilde{t}},{\tilde{z}};x^{\prime})$ $\displaystyle=-e^{\alpha
S^{0z}}S_{F}(t,z;x^{\prime})\,,$ (25)

which are solved formally by [34, 40]

$\displaystyle G_{F}^{(\alpha)}(t,z;x^{\prime})$
$\displaystyle=\sum_{j=-\infty}^{\infty}G_{F}^{\rm
vac}(t_{(j)},z_{(j)};x^{\prime})\,,$ (26a) $\displaystyle
S_{F}^{(\alpha)}(t,z;x^{\prime})$
$\displaystyle=\sum_{j=-\infty}^{\infty}(-1)^{j}e^{-j\alpha S^{0z}}S_{F}^{\rm
vac}(t_{(j)},z_{(j)};x^{\prime})\,,$ (26b)

where $G^{\rm vac}_{F}(x,x^{\prime})$ and $S^{\rm vac}_{F}(x,x^{\prime})$ are
the vacuum propagators, while $t_{(j)}$ and $z_{(j)}$ are obtained by applying
the transformation in Eq. (21) $j\in{\mathbb{Z}}$ times:

$\displaystyle t_{(j)}$
$\displaystyle=t\cos(j\alpha)+\frac{i}{a}(1+az)\sin(j\alpha),$ $\displaystyle
z_{(j)}$
$\displaystyle=it\sin(j\alpha)+\frac{1}{a}(1+az)\cos(j\alpha)-\frac{1}{a}.$
(27)

In particular, $\tilde{t}=t_{(1)}$ and $\tilde{z}=z_{(1)}$. Due to the
periodicity of the trigonometric functions appearing above, in the case when
$\alpha/2\pi=p/q$ is a rational number represented as an irreducible fraction,
the sum over $j$ in Eqs. (26) contains only $q$ terms:

$\displaystyle G_{F}^{(p,q)}(t,z;x^{\prime})$
$\displaystyle=\sum_{j=0}^{q-1}G_{F}^{\rm vac}(t_{(j)},z_{(j)};x^{\prime}),$
(28a) $\displaystyle S_{F}^{(p,q)}(t,z;x^{\prime})$
$\displaystyle=\sum_{j=0}^{q-1}(-1)^{j}e^{-j\alpha S^{0z}}S_{F}^{\rm
vac}(t_{(j)},z_{(j)};x^{\prime}).$ (28b)

In particular, the case $\alpha=2\pi$ corresponds to $p=q=1$, while the
thermal propagators reduce trivially to the vacuum ones:
$G_{F}^{(1,1)}=G_{F}^{\rm vac}$ and $S_{F}^{(1,1)}=S_{F}^{\rm vac}$. Since
$e^{-q\alpha S^{0z}}=(-1)^{p}$ by virtue of Eq. (22), applying Eq. (25) $q$
times shows that
$S_{F}^{(p,q)}(t_{(q)},z_{(q)};x^{\prime})=(-1)^{p+q}S^{(p,q)}_{F}(t,z;x^{\prime})$
and thus $S_{F}^{(p,q)}$ cancels identically when $p+q$ is an odd integer.

##  6 Imaginary-time formulation for acceleration

We now move to the Euclidean manifold by performing the Wick rotation to
imaginary time, $t\rightarrow\tau=it$. Then, Eq. (25) becomes

$\displaystyle G_{E}(\tau_{(1)},z_{(1)};x^{\prime})$
$\displaystyle=G_{E}(\tau,z;x^{\prime}),$ $\displaystyle
S_{E}(\tau_{(1)},z_{(1)};x^{\prime})$ $\displaystyle=-e^{\alpha
S^{0z}}S_{E}(\tau,z;x^{\prime}),$ (29)

and Eq. (26) reads, for the case when $\alpha/2\pi$ is an irrational number,

$\displaystyle G_{E}^{(\alpha)}(\tau,z;x^{\prime})$
$\displaystyle=\sum_{j=-\infty}^{\infty}G_{E}^{\rm
vac}(\tau_{(j)},z_{(j)};x^{\prime}),$ (30a) $\displaystyle
S_{E}^{(\alpha)}(\tau,z;x^{\prime})$
$\displaystyle=\sum_{j=-\infty}^{\infty}(-1)^{j}e^{-j\alpha S^{0z}}S_{E}^{\rm
vac}(\tau_{(j)},z_{(j)};x^{\prime}).$ (30b)

The case when $\alpha/2\pi=p/q$ must be treated along the lines summarized in
Eqs. (28) (see also discussion in Sec. 10). In the above, we considered
$j\in{\mathbb{Z}}$ and

$\displaystyle\tau_{(j)}$
$\displaystyle=\tau\cos(j\alpha)-\frac{1}{a}(1+az)\sin(j\alpha),$ (31a)
$\displaystyle z_{(j)}$
$\displaystyle=\tau\sin(j\alpha)+\frac{1}{a}(1+az)\cos(j\alpha)-\frac{1}{a}.$
(31b)

For the fields, the accelerated KMS conditions suggest the identification of
the fields at the points:

$\displaystyle\phi(\tau_{(j)},{\boldsymbol{x}}_{\|},z_{(j)})$
$\displaystyle=\phi(\tau,{\boldsymbol{x}}_{\|},z)\,,$ (32a)
$\displaystyle\psi(\tau_{(j)},{\boldsymbol{x}}_{\|},z_{(j)})$
$\displaystyle=(-1)^{j}e^{j\alpha
S^{0z}}\psi(\tau,{\boldsymbol{x}}_{\|},z)\,,$ (32b)

where the identified coordinates $(\tau_{(j)},z_{(j)})$ in the longitudinal
plane are given by Eq. (31) and ${\boldsymbol{x}}_{\|}=(x,y)$ are the
transverse coordinates which are unconstrained by acceleration. While the sums
of the form (26) may formally be divergent, the modified conditions (31) and
(32) give a finite solution to the accelerated KMS relations. The points
identified with the accelerated KMS condition (31) are illustrated in Fig. 1.

Figure 1:  The cyclic paths determined by the accelerating KMS boundary
condition (31) in the longitudinal plane spanned by the imaginary time $\tau$
and the acceleration direction $z$ of Wick-rotated Minkowski spacetime. Each
plot illustrates different accelerations $a$ encoded in the ratio
$\beta_{U}/\beta_{T}\equiv 2\pi T/a=3,4,5,10$ of the Unruh length $\beta_{U}$,
Eq. (8), to the thermal length $\beta_{T}=1/T$. The starting point of each
cyclic path, $(z,\tau)_{i}=(z_{i},0)$, with $z_{i}/\beta_{U}=-1,-1/2,\dots,1$,
is denoted by a hollow circle. The position of the Rindler horizon, collapsed
under the Wick rotation to a point (34), is denoted by the green star in each
plot.

##  7 Geometrical meaning of the accelerated KMS relation in imaginary-time
formalism

It is convenient, for a moment, to define a translationally shifted spatial
coordinate, ${\mathsf{z}}=z+1/a$, and rewrite Eq. (31) in the very simple and
suggestive form:

$\displaystyle\tau_{(j)}$
$\displaystyle=\tau\cos(j\alpha)-{\mathsf{z}}\sin(j\alpha),$
$\displaystyle{\mathsf{z}}_{(j)}$
$\displaystyle=\tau\sin(j\alpha)+{\mathsf{z}}\cos(j\alpha).$ (33)

In the shifted coordinates, the condition (4) for the Rindler horizon becomes
$a^{2}(\mathsf{z}^{2}+\tau^{2})=0$, which is solved by

$\displaystyle\tau={\mathsf{z}}=0\qquad\Leftrightarrow\qquad\tau=0,\quad
z=-\frac{1}{a}\,.$ (34)

Thus, we arrive at the following beautiful conclusion: in the Euclidean
spacetime of the imaginary-time formalism, the Rindler horizon (4) shrinks to
a single point (34). Thus, the accelerated KMS condition corresponds to the
identification of all points obtained by the discrete rotation of the space
around the Euclidean Rindler horizon point $(\tau,z)=(0,-1/a)$ with the unit
rotation angle defined by the reference thermal acceleration $\alpha=a/T$.

Our accelerated KMS condition, given in Eqs. (31) and (32), recovers the usual
finite-temperature KMS condition in the limit of vanishing acceleration.
Figure 2 demonstrates that in this limit,with $\alpha=a/T\to 0$, the proposed
KMS-type condition (27) for the acceleration is reduced to the standard
finite-temperature KMS-boundary condition [6] for which imaginary time $\tau$
is compactified to a circle of the length $\beta_{T}\equiv 1/T$ with the
points $(\tau,{\boldsymbol{x}})$ and $(\tau+\beta_{T}n,{\boldsymbol{x}})$,
$n\in{\mathbb{Z}}$, identified.

Figure 2:  The sets of points in the ($\tau,z$) plane which are identified by
our circular KMS condition (33) with the origin $(\tau,z)=(0,0)$ in a
thermally equilibrated system which experiences a uniform acceleration $a$
along the $z$ axis. The color distinguishes different acceleration strength
marked by different Unruh lengths $\beta_{U}=2\pi/|a|$. At vanishing
acceleration ($\beta_{U}/\beta_{T}\to\pm\infty$), condition (33) reduces to
the standard thermodynamic requirement of compactification of imaginary time
$\tau$ to a circle with the length $\beta_{T}=1/T$, while the Euclidean
Rindler horizon moves to (minus) spatial infinity. In the figure, each set of
points, corresponding to various ratios $\beta_{U}/\beta_{T}$, is connected by
a smooth line to guide the eye.

At the critical acceleration $\alpha=2\pi n$ (with $n\in{\mathbb{Z}}$), when
the background temperature $T$ equals to (an integer multiple of) the Unruh
temperature (8), the accelerated KMS conditions (31) do not constrain the
system anymore, $\tau_{(j)}=\tau$ and $z_{(j)}=z$, so that the system becomes
equivalent to a zero-temperature system in non-accelerated flat Minkowski
spacetime. This property, for $\alpha=2\pi$, has been observed in Refs. [35,
36, 37, 38].

In the situation where $2\pi/\alpha=\beta_{U}/\beta_{T}=n$ is an integer
number, the accelerated state at finite temperature can be implemented in
Euclidean space by imposing periodicity with respect to a specific set of
points that form a regular polygon with $n$ vertices located on the circle of
radius $\tau^{2}+z^{2}$. This is particularly convenient for lattice
simulations since the Euclidean action remains the standard one, allowing
accelerated systems to be modeled in the imaginary-time path integral
formalism without encountering the infamous sign problem.

##  8 KMS relations in Rindler coordinates

In the Minkowski Lorentz frame that we considered so far, the accelerating KMS
conditions (31) and (32) do not correspond to a boundary condition (as one
would naively expect from the KMS condition in thermal field theory) but
rather to a bulk condition: instead of relating the points at the boundary of
the imaginary-time Euclidean system, the accelerated KMS relations give us the
identification of the spacetime points in its interior.

While seemingly non-trivial in the form written in Eq. (27), the displacements
implied by the KMS relation correspond to the usual translation of the proper
time (rapidity) coordinate $\eta$ when employing the Rindler coordinates,

$at=e^{\zeta}\sinh(a\eta),\quad 1+az=e^{\zeta}\cosh(a\eta).$ (35)

It is easy to see that

$\displaystyle at_{(j)}$ $\displaystyle=e^{\zeta}\sinh(a\eta+ij\alpha),$ (36a)
$\displaystyle 1+az_{(j)}$ $\displaystyle=e^{\zeta}\cosh(a\eta+ij\alpha),$
(36b)

which implies that

$\eta_{(j)}=\eta+ij\beta_{T},\qquad\zeta_{(j)}=\zeta,$ (37)

in a seemingly perfect agreement with the usual KMS relation (11) for static
systems in Minkowski. However, there is also an unusual particularity of the
KMS conditions (37) in the Rindler coordinates (35).

The first relation in Eq. (37) suggests that the Wick rotation of the
Minkowski time $t=-i\tau$ should be supplemented with the Wick rotation of the
proper time in the accelerated frame $\eta=-i\theta/a$, where $\theta$ is the
imaginary rapidity.222Named in analogy with the rapidity coordinate
$\psi\equiv a\eta$. Then, the relation (35) in the imaginary (both Minkowski
and Rindler) time becomes as follows:

$a\tau=e^{\zeta}\sin\theta,\quad 1+az=e^{\zeta}\cos\theta,$ (38)

which shows that the imaginary rapidity becomes an imaginary coordinate with
the Euclidean Rindler KMS condition (37):

$\theta_{(j)}=\theta+j\alpha,\qquad\zeta_{(j)}=\zeta,\qquad
j\in{\mathbb{Z}}\,.$ (39)

Curiously, under the Wick transform, the rapidity becomes a cyclic compact
variable, $0\leqslant\theta<2\pi$, on which the imaginary-time condition (39)
imposes the additional periodicity with the period equal to the thermal
acceleration $\alpha$. Expectedly, at $\alpha=2\pi$ (or, equivalently, at
$T=T_{U}$), the boundary condition (39) becomes trivial.

The boundary conditions (39), characterized by the doubly-periodic imaginary
rapidity coordinate $\theta$, with periodicities $\theta\to\theta+2\pi$ and
$\theta\to\theta+\alpha$ (for $0\leqslant\alpha<2\pi$), can be easily
implemented in lattice simulations. Notice that this double periodicity has a
strong resemblance to the observation of Refs. [43, 44, 45] that the Euclidean
Rindler space can be identified with the space of the cosmic string which
possesses a conical singularity with the angular deficit
$\Delta\varphi=2\pi-\alpha$ [46, 47].

The KMS periodicity (39) of the compact imaginary rapidity $\theta$ is
formally sensitive to the rationality of the normalized thermal acceleration
$\alpha/(2\pi)$. Obviously, for $\alpha=2\pi p/q$, where $p<q$ are
nonvanishing irreducible integer numbers, the interplay of the two
periodicities will correspond to the single period $\theta\to\theta+2\pi/q$.

Interestingly, the sensitivity of an effect to the denominator $q$ (and not to
the numerator $p$) of a relevant parameter is a signature of the fractal
nature of the effect. Such fractality is noted, for example, in particle
systems subjected to imaginary rotation implemented via rotwisted boundary
conditions [17, 48, 49], which leads, in turn, to the appearance of “ninionic”
deformation of particle statistics [50]. The suggested fractality of
acceleration in imaginary formalism is not surprising given the conceptual
similarity of acceleration and rotation with imaginary angular frequency [37,
38]. Below, we will show that, despite the fractal property of the system, the
KMS boundary condition (39) in Euclidean Rindler space correctly reproduces
results for accelerated particle systems.

##  9 Energy-momentum tensor with the accelerated KMS conditions

Now let us come back to the Wick-rotated Minkowski spacetime and verify how
the modified KMS conditions for the fields, Eqs. (31) and (32), and related
solutions for their two-point functions (30), can recover the known results in
field theories under acceleration. To this end, we start from a non-minimally
coupled scalar field theory with the Lagrangian [51, 52, 53]

$\displaystyle{\mathcal{L}}_{\xi}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-2\xi\partial_{\mu}\left(\phi\partial^{\mu}\phi\right),$
(40)

possessing the following energy-momentum tensor:

$\displaystyle\Theta^{\xi}_{\mu\nu}=(1-2\xi)\partial_{\mu}\phi\partial_{\nu}\phi$
$\displaystyle-2\xi\phi\partial_{\mu}\partial_{\nu}\phi$
$\displaystyle-\frac{1}{2}(1-4\xi)\delta_{\mu\nu}\partial_{\lambda}\phi\partial_{\lambda}\phi,$
(41)

where the values $\xi=0$ and $\xi=1/6$ of the coupling parameter correspond to
the canonical and conformal energy-momentum tensors, respectively. In terms of
the Euclidean Green’s function, $\Theta^{\xi}_{\mu\nu}$ can be written as

$\Theta^{\xi}_{\mu\nu}=\lim_{x^{\prime}\rightarrow
x}\left[(1-2\xi)\partial_{(\mu}\partial_{\nu^{\prime})}-\tfrac{1}{2}(1-4\xi)\delta_{\mu\nu}\partial_{\lambda}\partial_{\lambda^{\prime}}\right.\\\
\left.-\xi(\partial_{\mu}\partial_{\nu}+\partial_{\mu^{\prime}}\partial_{\nu^{\prime}})\right]\Delta
G^{(\alpha)}_{E}(x,x^{\prime}),$ (42)

where $\Delta
G^{(\alpha)}_{E}(x,x^{\prime})=G^{(\alpha)}_{E}(x,x^{\prime})-G_{E}^{\rm
vac}(x,x^{\prime})$ represents the thermal part of the Green’s function. For
the Dirac field,
$\Theta_{\mu\nu}=\frac{1}{2}\bar{\psi}\gamma^{E}_{\mu}\overleftrightarrow{\partial_{\nu}}\psi$
can be computed from the Euclidean two-point function
$S^{(\alpha)}_{E}(x,x^{\prime})$ via

$\Theta_{\mu\nu}=-\frac{1}{2}\lim_{x^{\prime}\rightarrow x}{\rm
tr}[\gamma^{E}_{\mu}(\partial_{\nu}-\partial_{\nu^{\prime}})\Delta
S^{(\alpha)}_{E}].$ (43)

The vacuum propagators satisfying $\Box G^{\rm
vac}_{E}(x,x^{\prime})=\gamma^{E}_{\mu}\partial_{\mu}S^{\rm
vac}_{E}(x,x^{\prime})=\delta^{4}(x-x^{\prime})$ are given by

$\displaystyle G^{\rm vac}_{E}(\Delta x)$
$\displaystyle=\frac{1}{4\pi^{2}\Delta X^{2}},$ (44) $\displaystyle S_{E}^{\rm
vac}(\Delta x)$ $\displaystyle=\gamma^{E}_{\mu}\partial_{\mu}G^{\rm
vac}_{E}(\Delta x)=-\frac{\gamma^{E}_{\mu}\partial_{\mu}}{2\pi^{2}\Delta
X^{4}},$ (45)

with $\Delta X^{2}=(\Delta\tau)^{2}+(\Delta{\boldsymbol{x}})^{2}$. Using Eq.
(30), the thermal expectation values of the normal-ordered energy-momentum
operator can be obtained in the case of the Klein-Gordon field as:

$\Theta^{\mu\nu}_{\xi}(x)=\sum_{j\neq 0}^{\infty}\frac{1}{4\pi^{2}\Delta
X_{(j)}^{4}}\left[(1-2\xi)(R^{(j)}_{\mu\nu}+R^{(j)}_{\nu\mu})\right.\\\
\left.-\delta_{\mu\nu}(1-4\xi)R^{(j)}_{\lambda\lambda}+2\xi(R^{(j)}_{\nu\lambda}R^{(j)}_{\mu\lambda}+\delta_{\mu\nu})\right]\\\
-\sum_{j\neq 0}\frac{\Delta x^{(j)}_{\lambda}\Delta
x^{(j)}_{\kappa}}{\pi^{2}\Delta
X^{6}_{(j)}}\left[(1-2\xi)(\delta_{\mu\lambda}R^{(j)}_{\nu\kappa}+\delta_{\nu\lambda}R^{(j)}_{\mu\kappa})\right.\\\
\left.-\delta_{\mu\nu}(1-4\xi)R^{(j)}_{\lambda\kappa}+2\xi(R^{(j)}_{\mu\lambda}R^{(j)}_{\nu\kappa}+\delta_{\mu\lambda}\delta_{\nu\kappa})\right],$
(46)

where $\Delta
X^{2}_{(j)}=\frac{4}{a^{2}}\sin^{2}\frac{j\alpha}{2}[(a\tau)^{2}+(1+az)^{2}]$
and $R^{(j)}_{\mu\nu}\equiv\partial_{\mu}\Delta x^{(j)}_{\nu}$ is given by

$R^{(j)}_{\mu\nu}=\begin{pmatrix}\cos(j\alpha)&0&0&\sin(j\alpha)\\\ 0&1&0&0\\\
0&0&1&0\\\ -\sin(j\alpha)&0&0&\cos(j\alpha)\end{pmatrix},$ (47)

such that $R^{(j)}_{\mu\lambda}R^{(j)}_{\nu\lambda}=\delta_{\mu\nu}$. For the
Dirac field, we find

$\Theta_{\mu\nu}=\sum_{j\neq
0}\frac{(-1)^{j}}{\pi^{2}}\left[\delta_{\mu\lambda}\cos\tfrac{j\alpha}{2}+\left(\delta_{\mu
0}\delta_{\lambda 3}-\delta_{\mu 3}\delta_{\lambda
0}\right)\sin\tfrac{j\alpha}{2}\right]\\\
\times\left[\frac{R_{\nu\lambda}^{(j)}+\delta_{\nu\lambda}}{\Delta
X^{4}_{(j)}}-\frac{4\Delta X^{(j)}_{\lambda}}{\Delta
X_{(j)}^{6}}(R^{(j)}_{\nu\kappa}+\delta_{\nu\kappa})\Delta
X^{(j)}_{\kappa}\right].$ (48)

Taking advantage of the relation
$(R^{(j)}_{\nu\kappa}+\delta_{\nu\kappa})\Delta
x^{(j)}_{\kappa}=-\frac{2}{a}\sin(j\alpha)[(1+az)\delta_{\nu
0}-a\tau\delta_{\nu 3}]$ and after switching back to the real time $t$, we
find

$\Theta^{\mu\nu}=\mathcal{E}u^{\mu}u^{\nu}-\mathcal{P}\Delta^{\mu\nu}+\pi^{\mu\nu},$
(49)

with $\mathcal{E}$, $\mathcal{P}$, and $u^{\mu}$ being the energy density,
isotropic pressure, and the fluid four-velocity (2), respectively. The shear-
stress tensor $\pi^{\mu\nu}$ is by construction traceless, symmetric and
orthogonal to $u^{\mu}$, discriminating between the energy-momentum tensors in
classical (6) and quantum (49) fluids. Due to the symmetries of the problem,
its tensor structure is fixed as

$\displaystyle\pi^{\mu\nu}=\frac{\pi_{s}}{2}\left(\Delta^{\mu\nu}-\frac{3\alpha^{\mu}\alpha^{\nu}}{\alpha^{\lambda}\alpha_{\lambda}}\right)\,,$
(50)

with $\alpha^{\mu}(x)$ being the local thermal acceleration (3), such that the
shear coefficient $\pi_{s}$ is the only degree of freedom of $\pi^{\mu\nu}$ in
Eq. (50). In the scalar case, we find for the components of (49):

$\displaystyle\mathcal{E}_{\xi}$ $\displaystyle=\frac{3[\alpha
T(x)]^{4}}{16\pi^{2}}\left[G_{4}(\alpha)+4\xi G_{2}(\alpha)\right],$
$\displaystyle\mathcal{P}_{\xi}$ $\displaystyle=\frac{[\alpha
T(x)^{4}]}{16\pi^{2}}\left[G_{4}(\alpha)+\frac{4}{3}\left(1-3\xi\right)G_{2}(\alpha)\right],$
$\displaystyle\pi_{s}^{\xi}$ $\displaystyle=-\frac{[\alpha
T(x)]^{4}}{12\pi^{2}}(1-6\xi)G_{2}(\alpha),$ (51)

with $G_{n}(\alpha)=\sum_{j=1}^{\infty}[\sin(j\alpha/2)]^{-n}$, in complete
agreement with the results in Ref. [37]. Formally, $G_{n}$ diverges, however
its value can be obtained from its analytical continuation to imaginary
acceleration $a=i\phi$,
$\widetilde{G}_{n}(\beta_{T}\phi)=i^{n}G_{n}(i\beta_{T}\phi)$. The sum can be
evaluated, in a certain domain around $\beta_{T}\phi>0$ [37], to:

$\displaystyle\widetilde{G}_{2}(\beta_{T}\phi)$
$\displaystyle=\frac{2\pi^{2}}{3\beta_{T}^{2}\phi^{2}}-\frac{2}{\beta_{T}\phi}+\frac{1}{6},$
$\displaystyle\widetilde{G}_{4}(\beta_{T}\phi)$
$\displaystyle=\frac{8\pi^{4}}{45\beta_{T}^{4}\phi^{4}}-\frac{4\pi^{2}}{9\beta_{T}^{2}\phi^{2}}+\frac{4}{3\beta_{T}\phi}-\frac{11}{90}.$
(52)

Substituting now $G_{n}(\alpha)={\rm
Re}[i^{-n}\widetilde{G}_{n}(i\beta_{T}\phi)\rfloor_{\phi\rightarrow-ia}]$ into
Eq. (51) gives Eq. (10) for the conformal coupling $\xi=1/6$. For minimal
coupling $\xi=0$ or a generic non-conformal coupling $\xi\neq 1/6$, we recover
the results of Refs. [37, 54].

In the case of the Dirac field, one can easily check that
$\mathcal{E}_{D}=3\mathcal{P}_{D}$ and $\pi_{D}^{s}=0$, while

$\mathcal{P}_{D}=\frac{[\alpha T(x)]^{4}}{4\pi^{2}}S_{4}(\alpha),$ (53)

with
$S_{n}(\alpha)=-\sum_{j=1}^{\infty}(-1)^{j}\cos(j\alpha/2)/[\sin(j\alpha/2)]^{n}\rightarrow\widetilde{S}_{n}(\beta_{T}\phi)\equiv
i^{n}S_{n}(i\beta_{T}\phi)=-\sum_{j=1}^{\infty}(-1)^{j}\cosh(j\beta_{T}\phi/2)/[\sinh(j\beta_{T}\phi/2)]^{n}$,
which agrees with the results obtained in Ref. [38].

Finally, let us also illustrate the practical functionality of the
accelerating KMS boundary conditions (39) formulated in the imaginary-rapidity
Rindler space (38). For simplicity, we calculate the fluctuations of the
scalar field $\langle\phi^{2}\rangle$ using point-splitting and noticing that
the same method can be used to calculate also other quantities.

When expressed with respect to Rindler coordinates
$X=(\theta/a,\mathbf{x}_{\perp},\zeta)$, the Euclidean vacuum two-point
function $G_{E,R}^{\rm vac}(X,X^{\prime})$ given in Eq. (44) reads as follows:

$G_{\rm E,R}^{\rm
vac}=\frac{1}{4\pi^{2}}\left[\frac{2}{a^{2}}e^{\zeta+\zeta^{\prime}}(\cosh\Delta\zeta-\cos\Delta\theta)+\Delta{\boldsymbol{x}}_{\perp}^{2}\right]^{-1}.$
(54)

The KMS condition (39) implies that the Euclidean two-point function under
acceleration satisfies $G^{(\alpha)}_{\rm E,R}=\sum_{j\in{\mathbb{Z}}}G^{\rm
vac}_{\rm E,R}(\Delta\theta+j\alpha)$, where we consider vanishing spatial
distance between the points: $\zeta^{\prime}\to\zeta$ and
$\mathbf{x}_{\perp}^{\prime}\to\mathbf{x}_{\perp}$. Subtracting the vacuum
($j=0$) term that diverges in the $\Delta X\to 0$ limit, we get for the scalar
fluctuations:

$\displaystyle\langle\phi^{2}\rangle$ $\displaystyle=\lim_{\Delta\theta\to
0}\bigl{[}G^{(\alpha)}_{\rm E,R}(\Delta\theta)-G^{\rm vac}_{\rm
E,R}(\Delta\theta)\bigr{]}$ (55)
$\displaystyle=\frac{a^{2}e^{-2\zeta}}{8\pi^{2}}G_{2}(\alpha)=\frac{T^{2}(x)}{12}-\frac{a^{2}(x)}{48\pi^{2}}\,,\quad
0\leqslant a\leqslant 2\pi T\,,$

which agrees with the known result [37, 55].

##  10 Fractalization of thermodynamics

Let us consider the case when $\alpha/2\pi$ is a rational number, represented
as the irreducible fraction $p/q$. Then, the functions
$G_{n}(\alpha)\rightarrow
G_{n}^{(p,q)}(\alpha)=\frac{1}{2}\sum_{j=1}^{q-1}[\sin(\pi jp/q)]^{-n}$ are
regular and evaluate in the relevant $n=2$ and $n=4$ cases to

$G_{2}^{(p,q)}=\frac{q^{2}-1}{6},\quad
G_{4}^{(p,q)}=\frac{q^{4}+10q^{2}-11}{90}.$ (56)

The above results are independent of the numerator $p$ of the irreducible
fraction. The quadratic field fluctuations, shear stress coefficient
$\pi_{s}$, energy density, and pressure reduce to

$\displaystyle\langle\phi^{2}\rangle^{(p,q)}$ $\displaystyle=\frac{[\alpha
T(x)]^{2}}{96\pi^{2}}(q^{2}-1),$ (57a)
$\displaystyle\mathcal{E}_{\xi}^{(p,q)}$ $\displaystyle=\frac{[\alpha
T(x)]^{4}}{480\pi^{2}}(q^{2}-1)(q^{2}+11+60\xi),$ (57b)
$\displaystyle\mathcal{P}_{\xi}^{(p,q)}$ $\displaystyle=\frac{[\alpha
T(x)]^{4}}{1440\pi^{2}}(q^{2}-1)(q^{2}+31-60\xi),$ (57c)
$\displaystyle\pi_{s;\xi}^{(p,q)}$ $\displaystyle=-\frac{[\alpha
T(x)]^{4}}{72\pi^{2}}(1-6\xi)(q^{2}-1),$ (57d)

manifestly vanishing when $q^{2}=1$, i.e. for $\alpha=2\pi$.

In the case of the Dirac field, we have $S_{n}(\alpha)\rightarrow
S_{n}^{(p,q)}=-\frac{1}{2}\sum_{j=1}^{q-1}(-1)^{j}\cos(\pi jp/q)/[\sin(\pi
jp/q)]^{n}$. For the case $n=4$, the relation
$(-1)^{q-j}\cos[\pi(q-j)p/q]=(-1)^{j+p+q}\cos(\pi jp/q)$ implies that
$S_{4}^{(p,q)}$ vanishes when $p+q$ is an odd number. This happens whenever
$q$ is an even number in order to maintain the fraction $p/q$ irreducible.
When $q$ is odd, $S_{4}^{(p,q)}$ vanishes for all even values of $p$. When
both $p$ and $q$ are odd, $S_{4}^{(p,q)}$ can be computed analytically and the
final result can be summarized as

$S_{4}^{(p,q)}=\frac{7q^{2}+17}{720}(q^{2}-1)\times\frac{1+(-1)^{p+q}}{2}.$
(58)

The fermion pressure becomes

$\mathcal{P}^{(p,q)}_{D}=\frac{[\alpha
T(x)]^{4}}{2880\pi^{2}}(q^{2}-1)(7q^{2}+17)\frac{1+(-1)^{p+q}}{2}.$ (59)

##  11 Conclusions

In this paper, we derived the KMS relation for bosonic and fermionic quantum
systems at finite temperature under uniform acceleration. In Wick-rotated
Minkowski spacetime, the uniform acceleration requires the identification (31)
of the points in the bulk of the system along the discrete points lying on
circular orbits (32) about the Rindler horizon, which shrinks to a point (34)
under the Wick rotation. In the Wick-rotated Rindler coordinates, the KMS
relations reduce to standard (anti-)periodic boundary conditions in terms of
the imaginary rapidity coordinates. To illustrate the effectiveness of the
method, we considered the quantum thermal distributions of massless scalar and
Dirac particles under acceleration and found perfect agreement with results
previously derived in the literature.

Our work paves the way to systematic explorations of the influence of the
kinematic state of a system on its global equilibrium thermodynamic
properties. Our paper equips us with a rigorously formulated method in
imaginary-time formalism which allows us to construct the ground state of a
field theory in thermal equilibrium in a uniformly accelerating frame,
opening, in particular, a way for first-principle lattice simulations of
accelerated systems.

## Acknowledgements

This work is supported by the European Union - NextGenerationEU through the
grant No. 760079/23.05.2023, funded by the Romanian ministry of research,
innovation and digitalization through Romania’s National Recovery and
Resilience Plan, call no. PNRR-III-C9-2022-I8.

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# Modified particle lifetimes as a signature of deformed relativity

Pedro H. Morais  ID<EMAIL_ADDRESS>Physics Department, Federal
University of Paraíba, Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil.
Iarley P. Lobo  ID<EMAIL_ADDRESS>Department of Chemistry and Physics,
Federal University of Paraíba, Rodovia BR 079 - Km 12, 58397-000 Areia-PB,
Brazil.  Physics Department, Federal University of Lavras, Caixa Postal 3037,
37200-000 Lavras-MG, Brazil.  Christian Pfeifer  ID<EMAIL_ADDRESS>ZARM, University of Bremen, 28359 Bremen, Germany.  Rafael Alves Batista  ID
<EMAIL_ADDRESS>Instituto de Física Teórica UAM-CSIC, C/ Nicolás
Cabrera 13-15, 28049 Madrid, Spain.  Valdir B. Bezerra  ID
<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa
Postal 5008, 58059-900, João Pessoa, PB, Brazil.

###### Abstract

We demonstrate a compatibility between the relativity principle and the clock
postulate in deformed special relativity, by identifying the relevant deformed
Lorentz transformations in position space between arbitrary frames. This
result leads to a first-principles correction to the dilated lifetime of
fundamental particles. It turns out that these modified time dilations offer a
way to scrutinize Lorentz invariance (or deviations thereof) to high
precision.

Introduction. The characterization of what should be a quantum spacetime is
one of the routes that may lead us to an appropriate description of quantum
gravity. We expect that such a challenging task should pass through
intermediary steps before its full realization from the theoretical and
experimental points of view. For this reason, it is plausible to expect that
corrections to the Riemannian and general relativistic descriptions of gravity
should become manifest once we advance towards the Quantum Gravity (QG) scale
.In this sense, quantum gravity phenomenology plays a fundamental role by
translating the intuition of this area to observables, recognizable in our
current treatment of spacetime physics (for reviews on the subject we refer
the reader to Amelino-Camelia (2013); Addazi  _et al._ (2022)).

It is known that there exist formulations of such effective quantum gravity
spacetime geometries, in which some of the cornerstones of modern physics,
like the physical equivalence between local inertial observers, are preserved
Amelino-Camelia (2002). Those formulations that incorporate an invariant
length/energy scale, which is expected to exist from various fundamental
quantum gravity models, in a relativistic way, are known as Deformed Special
Relativity (DSR) models Magueijo and Smolin (2003); Majid and Ruegg (1994);
Bruno  _et al._ (2001); Barcaroli  _et al._ (2015); Girelli  _et al._ (2007).
They extend Einstein’s first postulate and their impact on observables is
intensively studied. Among the possibilities for the realization of this idea,
Finsler geometry stands out when one searches for a continuous description of
the kinematics of a particle in a curved spacetime with a fundamental length
Girelli  _et al._ (2007); Lobo and Pfeifer (2021); Lobo  _et al._ (2017); Zhu
and Ma (2023a). It describes the spacetime geometry fully in terms of an arc-
length functional (for the usefulness of different commutative geometries for
the description of phase and configuration spaces, please refer to the reviews
Pfeifer (2019); Albuquerque  _et al._ (2023); Zhu and Ma (2023b)). Finslerian
geodesics turn out to be the deformed trajectories of massless particles that
are analyzed in the rich phenomenology of time delays from gamma-ray bursts
Zhu and Ma (2022) and isometries of the Finsler metric are connected to local
deformations of the Lorentz symmetry Amelino-Camelia  _et al._ (2014). This
means for example, that if we call $E_{\text{Pl}}$ the Planck energy, then the
kinematics of a particle of mass $m$, energy $E$ and momentum $|\vec{p}|=|p|$,
subject to a modified dispersion relation (MDR) of the form ($\eta^{(n)}$ is a
dimensionless parameter that controls the perturbative approach that we are
going to follow)111We are using natural coordinates such that $c=\hbar=1$.

$E^{2}-p^{2}=m^{2}+\eta^{(n)}\frac{|p|^{n+2}}{E_{\text{Pl}}^{n}}\,$ (1)

are determined by the arc-length functional,

$s=\int F(\dot{t},\dot{x})d\lambda\,,$ (2)

where

$F(\dot{t},\dot{x})=\sqrt{\dot{t}^{2}-\dot{x}^{2}}+\frac{\eta^{(n)}}{2}\left(\frac{m}{E_{\text{Pl}}}\right)^{n}\frac{|\dot{x}|^{n+2}}{(\dot{t}^{2}-\dot{x}^{2})^{\frac{n+1}{2}}}\,.$
(3)

Here, “dot” means derivative with respect to the parameter $\lambda$ and
$|\dot{x}|=|\dot{\vec{x}}|$. For simplicity we shall assume a treatment in
$1+1$ dimensions in the following. The function $F$ is called Finsler function
and this result has been derived in more details and generality in Lobo and
Pfeifer (2021). This equivalence of treatments is due to the fact that a
Finsler function above is the Lagrangian derived from a Legendre
transformation Rodrigues and Lobo (2022) of the nonquadratic Hamiltonian
defined by the MDR (1), as was originally discussed in Girelli  _et al._
(2007).

Time dilation from the Clock Postulate. Recently, a novel aspect of the
Finslerian description of quantum spacetime was found in Lobo and Pfeifer
(2021) . By applying the Clock Postulate (CP) (which states that the proper
time an observer measures between two events is given by the arc-length of its
worldline in spacetime between the two events), we found that the dilated
laboratory lifetime $t$ of a particle with velocity $v=dx/dt$ relative to the
laboratory (lab) and proper rest frame lifetime $\tau$ is

$t=\gamma\tau\left[1-\frac{\eta^{(n)}}{2}\left(\frac{m}{E_{\text{Pl}}}\right)^{n}(\gamma^{2}-1)^{\frac{n+2}{2}}\right]\,,$
(4)

where $\gamma^{-1}=\sqrt{1-v^{2}}$. We clearly see the Planck-scale
corrections beyond Special Relativity (SR) induced by the MDR (1). In order to
connect this expression with observations, it is necessary to express the
velocity $v$ in terms of the particle’s energy in Finsler geometry, which
simply reads

$\displaystyle\begin{split}E&=m\frac{\partial
F}{\partial\dot{t}}\Bigg{|}_{\lambda=t}\\\
&=m\gamma\left[1-\frac{\eta^{(n)}}{2}(n+1)\left(\frac{m}{E_{\text{Pl}}}\right)^{n}(\gamma^{2}-1)^{\frac{n+2}{2}}\right].\end{split}$
(5)

Curiously, as we discussed in Lobo  _et al._ (2022), this expression is
actually a deformed Lorentz transformation from the rest frame to the
laboratory. Expression (5) can be inverted, giving

$\gamma=\frac{E}{m}\left[1+\frac{\eta^{(n)}}{2}(n+1)\left(\frac{m}{E_{\text{Pl}}}\right)^{n}\left[\left(\frac{E}{m}\right)^{2}-1\right]^{\frac{n+2}{2}}\right]\,.$
(6)

From this expression we easily find from the CP the Finslerian description of
the dilated lifetime of a particle. Let $\tau$ be the particle’s lifetime at
rest and $m$ and $E$ be it’s mass and energy which obey a MDR of the form (1).
Then, the particle lifetime $t$ in the laboratory frame is

$\displaystyle
t=\gamma_{\text{CP}}\tau=\frac{E}{m}\left[1+\frac{n\eta^{(n)}}{2}\left(\frac{|p|}{m}\right)^{2}\left(\frac{|p|}{E_{\text{Pl}}}\right)^{n}\right]\tau\,.$
(7)

We call the modified Lorentz factor that dilates the lifetime
$\gamma=\gamma_{\text{CP}}$, since it was calculated from an extension of the
clock postulate to an effective quantum spacetime described in terms of
Finsler geometry that is used for describing the kinematics of a particle
subject to a MDR.

A similar effect was described preliminarly in Trimarelli (2022), in which a
corrected Lorentz factor is suggested to be
$\gamma_{\text{LIV}}=E/m_{\text{LIV}}$, where
$m_{\text{LIV}}^{2}=m^{2}+\eta^{(n)}|p|^{n+2}/E_{\text{Pl}}^{n}$ is the right-
hand side of Eq.(1) (LIV stands for Lorentz Invariance Violation). The first
order (and dominant) correction of this expression gives

$\gamma_{\text{LIV}}=E/m_{\text{LIV}}\approx\frac{E}{m}\left[1-\frac{\eta^{(n)}}{2}\left(\frac{|p|}{m}\right)^{2}\left(\frac{|p|}{E_{\text{Pl}}}\right)^{n}\right]\,.$
(8)

The expressions for $\gamma_{\text{LIV}}$ and $\gamma_{\text{CP}}$ look
similar at first order (one simply translates superluminal effects from
$\gamma_{\text{LIV}}$ to be subluminal in $\gamma_{\text{CP}}$). However, only
in the CP case the concept of time emerges in a natural way due to the
Finslerian approach employed. In the LIV case one could say that there seems
to be no deeper reason to suppose that a Lorentz factor that dilates lifetimes
should be modified the way it is proposed.

Interestingly, in both cases, this kind of correction presents an amplifying
factor given by $(|p|/m)^{2}$, which can furnish large values for ultra-high-
energy cosmic rays (UHECRs). This is the reason why dilated lifetimes have
been considered as potential observables in the search for quantum gravity and
deviations from Lorentz invariance Trimarelli (2022), in addition to other
effects such as modified interaction thresholds Abreu  _et al._ (2022).

Despite the clear physical interpretation and mathematical formulation of the
CP approach, one could be tempted to state that an actual comparison between
times in different frames must be derived from an actual map between
observers. In order to be coherent with the DSR roots of the Finsler relation
with quantum gravity phenomenology, such an effect must come from a Deformed
Lorentz Transformation (DLT) involving spacetime coordinates – a step that is
missing so far in this approach.

This is precisely the goal of this letter. We seek Therefore to show that in
this DSR scenario, just like in SR, the result concerning the CP is actually
an isometry of the Finsler measure or a DLT between frames that move relative
to each other with velocity $v$.

Compatibility between the Clock Postulate and a Deformed Lorentz
Transformation. To prove that Eq.(7) is indeed a DLT, we use the geodesic
equation that determines the relation $x(t)$. To find it, we use a conserved
quantity given by the spatial momentum $p=m\partial F/d\dot{x}$, as this
expression is parametrization-invariant, we use the laboratory time as
parameter and solve this equation for $dx/dt$. Finally, we use Eq.(7) to
express this solution as a function of $\tau$, $E$, $p$ and $m$ as222In fact,
the relation between a propagation distance $L_{xy}=|x|$, the transverse
momentum $|p|=p_{\text{T}}$, the mass of a particle from the PDG
$m=M_{\text{PDG}}$ and the proper time $\tau$ is the basis for the measurement
of particles’ proper lifetimes in accelerators ALICE collaboration (2023).
Therefore, our result describes discrepancies that could emerge for
measurements done with future experiments (with higher energies than those
attainable today) and with better precision Lobo and Pfeifer (2023).

$\displaystyle
x=-\frac{p\,\tau}{m}\left[1+\frac{\eta^{(n)}}{2}\frac{(2m^{2}+nE^{2})}{m^{2}}\left(\frac{|p|}{E_{\text{Pl}}}\right)^{n}\right]\,.$
(9)

This is basically the geodesic solution in the proper time parametrization. If
we use the above expression along with (7) to calculate the Finsler function
(3), a direct calculation shows that for on-shell particles

$\displaystyle F(\dot{t},\dot{x})$
$\displaystyle=\sqrt{\dot{t}^{2}-\dot{x}^{2}}+\frac{\eta^{(n)}}{2}\left(\frac{m}{E_{\text{Pl}}}\right)^{n}\frac{|\dot{x}|^{n+2}}{(\dot{t}^{2}-\dot{x}^{2})^{\frac{n+1}{2}}}$
$\displaystyle=\dot{\tau}=F(\dot{\tau},0)\,.$ (10)

This proves that the set of transformations given by Eqs.(7) and (9)
corresponds to an isometry in Finsler geometry, therefore, they constitute a
Deformed Lorentz Transformation.

An alternative expression for this transformation can be found by expressing
$E$ and $p$ as a function of the velocity $v$ from $p_{\mu}=m\partial
F/\partial\dot{x}^{\mu}$ in the lab time parametrization. In this case, the
transformations for $t$ and $x$ are simply

$\displaystyle t$
$\displaystyle=\gamma\tau\left[1-\frac{\eta^{(n)}}{2}\left(\frac{m}{E_{\text{Pl}}}\right)^{n}(\gamma^{2}-1)^{\frac{n+2}{2}}\right]=\gamma_{\text{CP}}\tau\,,$
(11) $\displaystyle x$
$\displaystyle=v\gamma\tau\left[1-\frac{\eta^{(n)}}{2}\left(\frac{m}{E_{\text{Pl}}}\right)^{n}(\gamma^{2}-1)^{\frac{n+2}{2}}\right]=v\gamma_{\text{CP}}\tau\,.$
(12)

It is straightforward to verify that indeed $v$ is the velocity of the
particle in the lab frame, since we check that $dx/dt=v$.

This is a remarkable result. For the first time we have simultaneously a DLT
involving space and time coordinates and the boost parameter is undoubtedly
identified as the velocity of a particle $dx/dt$. Besides that, this result is
compatible with the CP (just like in SR), which allows us to indeed describe
what would be a Planck-scale correction to the twin paradox. For this reason,
we are confident to state that Eq.(7) actually defines a DSR Lorentz factor

$\displaystyle\gamma_{\text{CP}}=\gamma_{\text{DSR}}\,.$ (13)

Let us compare our findings here with the ones made in Trimarelli (2022) for
the LIV case, which are currently being analyzed using UHECRs. One may be
tempted to translate the results found in that paper using
$\eta^{(n)}\mapsto-n\eta^{(n)}$ (since only very-high energy relativistic
particles would effectively contribute to the effect). But, we should notice
that besides the modification in the particle’s lifetime, also a modified
velocity is used as input

$v_{\text{LIV}}=\beta_{\text{LIV}}=\frac{|p|}{m\gamma_{\text{LIV}}}\approx\frac{|p|}{E}\left[1+\frac{\eta^{(n)}}{2}\left(\frac{|p|}{m}\right)^{2}\left(\frac{|p|}{E_{\text{Pl}}}\right)^{n}\right]\,.$
(14)

In our case, the relation between the velocity $v$ and the momenta is
naturally given by the definition of the spatial physical momentum

$p=m\frac{\partial
F}{\partial\dot{x}}\Bigg{|}_{\lambda=t}=-mv\gamma\left[1+\frac{\eta^{(n)}}{2}\frac{(mv)^{n}(v^{2}-2-n)}{E_{\text{Pl}}^{n}(1-v^{2})^{\frac{n+2}{2}}}\right]\,,$
(15)

from which we can calculate its absolute value $|p|$ and, using the relation
between the Lorentz factor and the energy (5), we derive the following

$\displaystyle
v_{\text{DSR}}=\beta_{\text{DSR}}=\frac{|p|}{E}\left[1+\frac{(2+n)\eta^{(n)}}{2}\left(\frac{|p|}{E_{\text{Pl}}}\right)^{n}\right]\,.$
(16)

As expected, this result is actually the velocity of the particle defined from
the MDR (1), since

$v=\Bigg{|}\frac{\partial E}{\partial
p}\Bigg{|}\stackrel{{\scriptstyle\text{MDR}}}{{=\mathrel{\mkern-3.0mu}=\mathrel{\mkern-3.0mu}=}}v_{\text{DSR}}\,.$
(17)

For this reason, even in a LIV scenario, one should use Eq.(16) instead of
(14). So, we can actually drop the symbol DSR out of the velocity in (16),
since it is simply the velocity of the particle read from the MDR. With this
observation, we complete the analysis connecting the rest and the lab frames.
The next natural step consists in connecting general spacetime rectangular
coordinates of different frames that move relative to each other with velocity
$v$, i.e., $(t,x)\mapsto(t^{\prime},x^{\prime})$, which reduces to the
previous case when the target frame is $(\tau,0)\mapsto(t,x)$.

General Deformed Lorentz Transformation. In order to generalize the previous
isometry such that we connect two arbitrary rectangular coordinates, it is
sufficient to propose a transformation involving the boost parameter $v$, the
spacetime coordinates $(t,x)$ and the velocities $(\dot{t},\dot{x})$ that
reduces to the previous case when the target frame obeys $x=0=\dot{x}$. Since
this is a transformation that shall leave the Finsler function invariant (and
consequently the metric), it should not depend on the parametrization
$\lambda$ of the velocities $(\dot{t},\dot{x})$.

The functions that naturally satisfy this requirement are the momentum
components

$\displaystyle E(\dot{t},\dot{x})=m\frac{\partial
F(\dot{t},\dot{x})}{\partial\dot{t}},\quad p(\dot{t},\dot{x})=m\frac{\partial
F(\dot{t},\dot{x})}{\partial\dot{x}}\,,$ (18)

which satisfy $E=m$ in the particle rest frame, i.e. for $x=0=v$, see (5) and
(15).

A generalisation of the transformations (11) and (12) for transformations
between arbitrary frames $(t,x)\mapsto(t^{\prime},x^{\prime})$ is constructed
from combinations of factors of the type $E^{n-r}|p|^{r}/E_{\text{Pl}}^{n}$,
where $E$ and $p$ are treated as functions of $(\dot{t},\dot{x})$. A general
ansatz for such transformations to first order in $\eta^{(n)}$ is of the
following form:

$\displaystyle t^{\prime}=$
$\displaystyle\,\,\,(t+xv)\gamma+\frac{\eta^{(n)}}{2}t\gamma\left(\frac{E}{E_{\text{Pl}}}\right)^{n}\left(\gamma^{2}-1\right)^{\frac{n+2}{2}}$
(19)
$\displaystyle+\frac{\eta^{(n)}}{2E_{\text{Pl}}}\left(t\sum_{r=0}^{n}\alpha_{r}E^{n-r}|p|^{r}+x\sum_{r=0}^{n}\beta_{r}E^{n-r}|p|^{r}\right)\,,$
$\displaystyle x^{\prime}=$
$\displaystyle\,\,\,(tv+x)\gamma+\frac{\eta^{(n)}}{2}tv\gamma\left(\frac{E}{E_{\text{Pl}}}\right)^{n}\left(\gamma^{2}-1\right)^{\frac{n+2}{2}}$
(20)
$\displaystyle+\frac{\eta^{(n)}}{2E_{\text{Pl}}}\left(x\sum_{r=0}^{n}\delta_{r}E^{n-r}|p|^{r}+t\sum_{r=0}^{n}\lambda_{r}E^{n-r}|p|^{r}\right)\,,$

where we isolated the terms from the sum which are non-vanishing in the rest
frame, i,e, $p=0=x$, $E=m$, for clarity.

Imposing the isometry condition
$F^{2}(\dot{t}^{\prime},\dot{x}^{\prime})=F^{2}(\dot{t},\dot{x})$ on this
transformation and noticing that
$E=m\dot{t}/\sqrt{\dot{t}^{2}-\dot{x}^{2}}+{\cal O}(m/E_{\text{Pl}})$ and
$p=-m\dot{x}/\sqrt{\dot{t}^{2}-\dot{x}^{2}}+{\cal O}(m/E_{\text{Pl}})$ are
conserved functions of velocities, we derive an expression involving powers
and factors of $\dot{t}$ and $|\dot{x}|$, besides terms like
$(|\dot{x}|+v\dot{t})^{n}$, for which we can use the binomial theorem to
express it in terms of combinations of $\dot{t}^{n-r}|\dot{x}|^{r}$.
Furthermore, imposing that this transformation should reduce to that of
Eqs.(11), (12) when $p=0=x$ and $E=m$, we find the following conditions

$\displaystyle\alpha_{0}$ $\displaystyle=0=\lambda_{0},\,$
$\displaystyle\beta_{0}$ $\displaystyle=-v^{1+n}\gamma^{3+n},\,$ (21)
$\displaystyle\delta_{0}$ $\displaystyle=-v^{n}\gamma^{1+n}(\gamma^{2}-1),$
(22) $\displaystyle\alpha_{n}$
$\displaystyle=v\beta_{n}+\gamma-\gamma^{-1},\,$ $\displaystyle\lambda_{n}$
$\displaystyle=\beta_{n}+v\gamma(1+\gamma^{n})\,,$ (23)
$\displaystyle\delta_{n}$
$\displaystyle=v\beta_{n}+\gamma^{-1}+\gamma^{1+n}\,,$ (24)

and for $1\leq r\leq n-1$,

$\displaystyle\alpha_{r}$
$\displaystyle=v\beta_{r},\,\qquad\lambda_{r}=\beta_{r}+v^{1+n-r}\gamma^{1+n}\binom{n}{r}\,,$
(25) $\displaystyle\delta_{r}$
$\displaystyle=v\beta_{r}+v^{n-r}\gamma^{1+n}\binom{n}{r}\,,$ (26)

where $\binom{n}{r}$ is the binomial coefficient. We see that we have the
freedom to choose $n$ arbitrary functions of $v$, namely $\beta_{r}\,(1\leq
r\leq n)$ that should be null when $v\rightarrow 0$ in order to guarantee that
we recover the identity transformation when the velocity is zero. As this
transformation preserves the Finsler function, it also preserves the metric
and consequently the MDR (1) that is calculated from the norm of the momenta
Girelli  _et al._ (2007); Barcaroli  _et al._ (2015); Lobo  _et al._ (2017).

A simple choice for the $\beta_{r}$ is setting all $\beta_{r}=0$ for $1\leq
r\leq n$. Another possible choice is fixing $\beta_{r}$ is the comparison of
this transformation with one arising from the action of boosts in a quantum
algebraic approach. For example, similar ambiguities are found at the
bicrossproduct basis of $\kappa$-Poincaré-inspired Finsler isometries, which
could be fixed by comparing the generators of the transformations found from
the geometric and algebraic approaches Amelino-Camelia  _et al._ (2014).

Conclusion. Phenomenological models which break Lorentz invariance lead to a
modified Lorentz factor which encodes the time dilation between different
observer frames. This prediction triggered the search for such phenomenology
using the mass content of EAS from UHECR data Trimarelli (2022), thus using
deformations of particle lifetimes as a window to Planck scale physics.

In this letter, we have used Finsler geometry to show that actually a similar
correction (with the opposite sign) emerges naturally from an approach that
deforms, rather than breaks, Lorentz symmetry, such that the modified
dispersion relation’s form is kept invariant when transforming between frames
in a way that not only preserves the relativity principle, but also the so-
called clock postulate (the observer’s proper time is the line element of its
trajectory) from special relativity to a deformed special relativity. This can
be seen in the discussion that leads to Eqs.(7) and (13). Therefore,
unintentionally, what has been considered in previous analysis using UHECR
data would be a deformation instead of a violation of Lorentz symmetry with
basically opposite signs on the correction. The phenomenological consequences
of these two scenarios are manifestly different. For instance, some processes
that are forbidden in the Lorentz-invariance case would be allowed in the LIV
scenario but not in DSR. These additional observables would ultimately allow
us to distinguish between these scenarios.

Besides that, we have shown that one must not use a velocity given by
$\beta_{\text{LIV}}=|p|/m\,\gamma_{\text{LIV}}$ (Eq.(14)), as done in
Trimarelli (2022) since it is incompatible with the actual velocity of a
particle in the lab frame, which must be read from the dispersion relation,
whose expression can also be naturally derived from our analysis, as can be
seen in the discussion that surrounds Eqs.(16) and (17).

We also generalized this result to a transformation between two lab frames
that move relative to each other with velocity $v$, given by Eqs.(19)-(26),
and that reduces to the previous case in the comoving limit $x=0=p$ and $E=m$.
We believe that the search for quantum gravity effects from cosmic-ray data
could benefit from the findings of this letter within the scope of the
deformation of Lorentz symmetry and a next natural step would be to consider
these findings in future analyses. As a final remark, as UHECR observatories
improve their detection techniques and capabilities of reconstruction of air
showers, the prospects for detecting the effects discussed here will become
even better, especially with future facilities Coleman  _et al._ (2023).

Acknowledgments. P. H. M. thanks Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brazil (CAPES) - Finance Code 001 for financial support. I.
P. L. was supported by the National Council for Scientific and Technological
Development - CNPq grant 306414/2020-1 and by the grant 3197/2021, Paraíba
State Research Foundation (FAPESQ). C. P. is funded by the excellence cluster
QuantumFrontiers funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC-2123
QuantumFrontiers – 390837967. R. A. B. is funded by the “la Caixa” Foundation
(ID 100010434) and the European Union’s Horizon 2020 research and innovation
program under the Marie Marie Sklodowska-Curie-Curie grant agreement No
847648, fellowship code LCF/BQ/PI21/11830030. V. B. B. was supported by the
National Council for Scientific and Technological Development - CNPq grant
307211/2020-7. The authors like to acknowledge networking support by the COST
Action CA18108.

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