[1]Giuseppe Gaetano Luciano

[1]Department of Chemistry, Physics and Environmental and Soil Sciences,
Escola Politècninca Superior, Universidad de Lleida, Av. Jaume II, 69, Lleida,
25001, Spain

# Kaniadakis entropy in extreme gravitational and cosmological environments: a
review on the state-of-the-art and future prospects

<EMAIL_ADDRESS>*

###### Abstract

Kaniadakis ($\kappa$-deformed) statistics is being widely used for describing
relativistic systems with non-extensive behavior and/or interactions. It is
built upon a one-parameter generalization of the classical Boltzmann-Gibbs-
Shannon entropy, possessing the latter as a particular sub-case. Recently,
Kaniadakis model has been adapted to accommodate the complexities of systems
under the influence of gravity. The ensuing framework exhibits a rich
phenomenology that allows for a deeper understanding of the most extreme
conditions of the Universe. Here we present the state-of-the-art of
$\kappa$-statistics, discussing its virtues and drawbacks at different energy
scales. Special focus is dedicated to gravitational and cosmological
implications, including effects on the expanding Universe in dark energy
scenarios. This review highlights the versatility of Kaniadakis paradigm,
demonstrating its broad application across various fields and setting the
stage for further advancements in statistical modeling and theoretical
physics.

###### keywords:

Kaniadakis entropy, Relativity, Statistics, Gravity, Cosmology

##  1 Introduction

Boltzmann-Gibbs (BG) theory forms the bedrock of classical statistical
mechanics, providing a robust framework to connect the microscopic properties
of multi-degree-of-freedom systems to their observable behaviors.
Nevertheless, its scope is confined when applied to systems in non-equilibrium
conditions, possessing long-range interactions or displaying non-extensive
features. These limitations have motivated the development of generalized
formalisms that extend beyond BG theory, while including the latter as a
special case.

Among the most common generalizations, Tsallis statistics [1] introduces a
deformed entropic functional that accommodates non-extensive behaviors typical
of large-scale systems. The resulting mechanics has attracted growing interest
in the last decades, particularly due to its capability to address the area-
scaling problem of black hole entropy [2]. A similar deformation occurs in
Barrow model [3], motivated by a quantum gravitational (fractal-like)
structure of the horizon geometry of black holes (see also [4, 5] for
applications to Cosmology), while a power-law corrected entropy appears in [6,
7] inspired by the idea that the origin of black-hole entropy may lie in the
entanglement of quantum fields between inside and outside of the horizon.

On the other hand, Kaniadakis ($\kappa$-deformed) statistics arises from the
effort to construct a self-consistent relativistic theory [8, 9, 10].
Concretely, its foundations are tightly connected with the observation that,
in special relativity, the mathematical expressions of physical observables
like the momentum, the energy etc., are one-parameter (light speed) continuous
deformations of the corresponding quantities in the classical physics. In
turn, Lorentz group symmetries impose a proper one-parameter deformation of
the classical BG entropy too. The ensuing entropic functional allows to
construct a coherent relativistic statistical theory, which maintains the main
properties (maximum entropy principle, thermodynamic stability and Lesche
stability, expansivity, etc.) of the classical statistical mechanics.
Deviations from the latter are quantified by the dimensionless parameter
$-1<\kappa<1$, which, in general, contains information on the relativistic
features of the system. In other terms, the physical interpretation of
$\kappa$ is connected with the complex (i.e. non-classical) nature of the
phenomenon under investigation. The special limit $\kappa\rightarrow 0$
recovers the classical entropy. In light of these considerations, it is licit
to regard Kaniadakis paradigm as an effective generalization of BG entropy
that naturally emerges in various high-energy contexts, such as astrophysics,
cosmology, gravitation and particle physics [11, 12]. In parallel, it is worth
mentioning that alternative studies have proposed $\kappa$-theory as an
approach to epidemiology [13], seismology [14], economics [15] and natural
sciences [16], posing it as a serious candidate to extend the ordinary
statistical mechanics into systems exhibiting non-trivial correlations [17].

Starting from the above premises, in the present work we review the impact of
Kaniadakis entropy on gravitational and cosmological systems. We shall not
deal with other modified entropies, e.g. Tsallis, Landsberg-Vedral,
Sharma–Mittal, Rény and Barrow entropies (see [18, 19, 20, 3, 21, 22, 23, 24,
25] for more details and applications). However, similar considerations could
be applied and expanded upon in those contexts as well. On one hand, we
explore how the $\kappa$-statistics affects established predictions and
concepts derived from the standard BG framework. On the other hand, we shed
the light on some emergent phenomena and novel behaviors that are not
accounted for by the conventional theory. These findings indicate that
Kaniadakis entropy could not only challenge the existing BG paradigm, but also
offer profound opportunities for advancing both theoretical and applied
aspects of our understanding of gravitational and cosmological environments in
extreme regimes.

The structure of the work is organized as follows: in the next Section, we
review the theoretical foundations and mathematical background of Kaniadakis
statistics. In Sec. 3, we focus on astrophysical applications, while Sec. 4 is
dedicated to modified theories of gravity and black hole
$\kappa$-thermodynamics. Generalized models of dark energy and effects on the
history of the Universe are discussed in Sec. 5. Finally, the last section
contains a summary of the work. Unless stated otherwise, we adopt Planck units
$\hslash=c=k_{b}=G=1$.

##  2 Mathematical background

Kaniadakis statistics naturally emerges as a one-parameter continuous
deformation of the BG mechanics within Einstein special relativity [8, 9]. It
preserves the mathematical structure of the classical framework, which is in
turn recovered as the deformation parameter tends to zero. The resulting
theory is suitable to describe the behavior of systems exhibiting power law
distributions and a large class of experimentally observed phenomena in high
energy physics.

The basic ingredient of $\kappa$-statistics is the modified entropy [8, 9]

$S_{\kappa}=-\sum_{i}n_{i}\ln_{\kappa}n_{i}\,,$ (1)

where the $\kappa$-logarithm is defined as

$\hskip
76.82234pt\ln_{\kappa}x\equiv\frac{x^{\kappa}-x^{-\kappa}}{2\kappa}\,,\qquad-1<\kappa<1\,.$
(2)

The generalized BG distribution for the $i$-th energy level $E_{i}$ is
obtained by maximizing, after properly constrained, the entropy (1). As a
result, one finds [8, 9]

$n_{i}\equiv\exp_{\kappa}\left[-\beta\left(E_{i}-\mu\right)\right],$ (3)

where $\mu$ is the chemical potential,
$\beta=\left(\sqrt{1-\kappa^{2}}\,\hskip 0.56905ptT\right)^{-1}$ the inverse
temperature of the system and the $\kappa$-exponential is defined by

$\ln_{\kappa}\left[\exp_{\kappa}\left(x\right)\right]=x\,\,\,\,\,\Longrightarrow\,\,\,\,\,\exp_{\kappa}(x)\equiv\left(\sqrt{1+\kappa^{2}x^{2}}+\kappa
x\right)^{1/\kappa}\,.$ (4)

The plot of $n_{i}$ versus $E_{i}/T$ is displayed in Fig. 1 for different
values of $\kappa$ (since $\exp_{\kappa}(x)$ is even with respect to
$\kappa\rightarrow-\kappa$, here and henceforth we restrict to the $\kappa>0$
domain). It can be noted that the higher the deformation parameter, the larger
the departure from the Maxwell-Boltzmann distribution (black solid curve). The
discrepancy becomes more evident for sufficiently large $E_{i}/T$, where
Kaniadakis distribution has a power-law tail, as opposed to the exponential
Maxwell-Boltzmann decay.

Figure 1: Plot of $n_{i}$ in Eq. (3) versus $E_{i}/T$, for fixed $\mu=0$ and
different values of $\kappa>0$.

Some comments are in order here: first, it is an easy check that the laws of
the classical statistics are restored in the $\kappa\rightarrow 0$ limit. In
general, $\kappa$ is not fixed by the theory itself; rather, it needs to be
constrained or determined by aligning theoretical predictions with actual
experimental results. So far, the best agreement has been achieved by fitting
the spectrum of cosmic rays over several decades in energy, obtaining
$\kappa\sim\mathcal{O}(10^{-1})$ [8] (see Sec. 3 for details). Conversely,
cosmological studies yield much tighter constraints [17]. This suggests that,
although the behavior of the large-scale Universe might not deviate
significantly from the classical predictions, effects of $\kappa$-entropy
could still manifest through non-negligible imprints, especially in the early
Universe, which is not fully understood in the standard cosmological model.

For applications of $\kappa$-entropy to holographic gravity [26, 27, 28], it
proves convenient to recast Eq. (1) in the equivalent form [29, 30]

$S_{\kappa}=-\sum_{i=1}^{W}\frac{P_{i}^{1+\kappa}-P_{i}^{1-\kappa}}{2\kappa}\,,$
(5)

where $P_{i}$ gives the probability of the system being in the $i$-th state
and $W$ is total number of accessible states. For equiprobable configurations
$P_{i}=1/W$, Boltzmann’s entropy formula allows us to write Eq. (5) in terms
of the classical entropy $S=\log W$ as

$S_{\kappa}=\frac{1}{\kappa}\sinh\left(\kappa S\right)\,.$ (6)

Since Kaniadakis corrections are constrained around $\kappa=0$, it makes sense
to expand $S_{\kappa}$ to the leading order as

$S_{\kappa}=S+\frac{\kappa^{2}}{6}S^{3}+\mathcal{O}(\kappa^{4})\,.$ (7)

We shall see below that this relation serves as a starting point for many
applications of $\kappa$-entropy in the realm of black hole thermodynamics and
cosmological scenarios.

##  3 Astrophysical paradigms

Astrophysical systems often exhibit long-range interactions and non-trivial
dynamics not fitting the standard statistics. Also, they exist under extreme
relativistic conditions, representing the ideal “laboratory” to test the
correctness and predictability of $\kappa$-theory. Toward this end, we here
collect recent findings on astrophysical applications of $\kappa$-statistics.
The results are discussed in relation to challenging studies focused on
constraining the deformation parameter, highlighting implications for
understanding cosmic phenomena and shaping future astrophysical models.

###  3.1 Cosmic rays spectrum

As stated above, cosmic rays represent one of the most compelling systems to
explore Kaniadakis paradigm. These particles are essentially the normal nuclei
as in the standard cosmic abundances of matter. Therefore, they can be thought
of as an equivalent statistical ensemble of identical relativistic particles
with masses of order of the proton mass ($\sim 938\,\mathrm{MeV}$).

It is known that the spectrum of cosmic rays, which extends over 13 decades in
energy and spans 33 decades in particle flux, does not decay exponentially,
violating the BG equilibrium distribution. As a matter of fact, simple
calculations show that it has an approximately power-law behavior, with the
spectral index depending on the energy scale. Based on this observation, an
improved consistency between theoretical predictions and experimental data was
found in [8] by characterizing the cosmic rays spectrum through the Kaniadakis
distribution

$f(E)=\exp_{\kappa}\left(-\frac{1}{\kappa}\frac{E-\mu}{m}\right),$ (8)

where $m$ is the particle mass and the information on the temperature is
hidden in the parameter $\kappa$. One can check that the asymptotic regime of
$E\rightarrow\infty$ gives the approximated function
$f(E)\approx\left(\frac{m}{2E}\right)^{1/\kappa}$ [8], which indeed shows the
correct power-law decay.

The distribution (8) can be used to infer the cosmic rays flux $\Phi(E)\approx
p^{2}f(E)$, where $p^{2}=E^{2}-m^{2}$ is the relativistic particle momentum.
Following [8], computations are easily done by taking into account the
relativistic expression relating $E$ and $p$, yielding

$\Phi(E)=A\left[\left(\frac{E}{m}+1\right)^{2}-1\right]\exp_{\kappa}\left[-\beta\left(E-\mu\right)\right]\,,$
(9)

with $\beta$ being related to the inverse temperature. It is a direct check
that this expression, in agreement with observations, decays following the
power-law behavior $\Phi(E)\propto E^{-a}$, where
$a=\sqrt{1+\left(\frac{m}{T}\right)^{2}}-2$. In particular, comparison with BG
statistics reveals that Kaniadakis framework is better suited to fit the
observed cosmic rays flux for $A=10^{5}\,(\mathrm{m^{2}\,\hskip
1.13809ptsr\,\hskip 1.13809pts\,\hskip 1.13809ptGeV})^{-1}$,
$mc^{2}=938\,\mathrm{MeV}$, $\mu=-375\,\mathrm{MeV}$ and
$k_{B}T=208\,\mathrm{MeV}$, provided that $\kappa=0.2165$ [8].111We have here
restored full units for direct comparison with [8].

At this point, it is interesting to remark that the power law asymptotic
behavior of Eq. (9) is due to the presence of the $\kappa$-exponential, which,
in turn, can be naturally traced to the composition law of momenta in special
relativity through a suitable redefinition of the sum in the underlying
algebraic structure ($\kappa$ sum [8]). In other terms, such a power law
behavior of the cosmic flux can be regarded as a signature of the particle
relativistic nature. Such a result provides strong evidence that the intricate
nature of relativistic systems demands a reevaluation of the fundamentals of
the classical statistical mechanics. From this perspective, Kaniadakis model
stands out as a promising candidate to improve the current limits of BG theory
in the relativistic regime.

###  3.2 Open Stellar clusters

Stellar clusters are groups of stars held together by gravity. There are two
main types of such systems: globular clusters, which are large and spherical,
containing thousands to millions of stars, and open clusters, which are less
dense, with roughly hundreds to thousands of stars. Open clusters, often
located in the disk of galaxies like the Milky Way, are younger and less
tightly bound than globular clusters.

Because of their large number of constituents, open clusters obey statistical
laws that govern their collective properties and dynamics. There are three
dominant mechanisms underlying the long-term evolution of these systems:

  * $\bullet$

_collisional relaxation_ , which occurs when stars within a cluster undergo
close encounters or collisions, transferring energy and momentum among
themselves. Over time, such interactions lead to a uniform distribution of
velocities among stars, ultimately driving the cluster towards a more
dynamically relaxed state (see, for instance, [31] and references therein).

  * $\bullet$

_Lynden-Bell relaxation_ , which refers to the exchange of energy due to
gravitational interactions. This process plays a central role in adjusting the
velocities of stars within the cluster, leading to a more isotropic
distribution [32].

  * $\bullet$

_Relaxation associated with radial orbit instability_ , which pertains
specifically to the instability of stars due their gravitational interactions.
It involves the expansion of orbits and, in some cases, the escape of the most
energetic stars. This process contributes to the gradual dissolution of open
clusters over cosmic timescales [33].

If, on one hand, the first two mechanisms are adequately described by the
standard statistical mechanics, orbital relaxation has not yet been fully
understood. In this sense, a tentative solution was provided in [34],
introducing the Kaniadakis distribution

$\phi_{\kappa}(v_{r})\propto\exp_{\kappa}\left(-\frac{v_{r}^{2}}{\sigma^{2}}\right)$
(10)

for the stellar residual radial velocity $v_{r}$ in some baseline open
clusters. Application of the Kolmogorov-Smirnov statistical test [35] for each
observed cumulative distribution showed that Eq. (10) aligns with observations
more accurately than the Maxwell-Boltzmann exponential, assuming a non-trivial
dependence of $\kappa$ on stellar-cluster ages. In particular, for some
selected stellar clusters (e.g., Pleiades, NGC 1039, NGC 6475, NGC 2099, NGC
6633, Praesepe etc.), the entropic parameter is estimated around unity, while
a slightly lower value $\kappa\sim\mathcal{O}(10^{-1})$ is obtained for the
NGC 0752 system [34].

###  3.3 Jeans instability criterion

The Jeans instability serves as a criterion to ascertain the conditions for
gravitational instability and subsequent collapse in clouds of interstellar
gas. In its basic form, it states that a region of thermal gas will collapse
under its self-gravity if its size exceeds the critical length

$\lambda_{j}=\sqrt{\frac{\pi T}{\mu m_{H}\rho_{0}}}\,,$ (11)

where $T$ is the temperature of the gas, $\mu$ the mean molecular weight,
$m_{H}$ the mass of a hydrogen atom and $\rho_{0}$ the equilibrium mass
density of the system, respectively [36].

To conceptualize the Jeans length, the easiest way is to rewrite approximately
$\rho_{0}$ as the mass-to-volume ratio of the cloud, i.e. $\rho_{0}=M/r^{3}$.
Dropping irrelevant numerical factors, Eq. (11) becomes
$\lambda_{j}\propto\sqrt{\frac{Tr^{3}}{\mu M}}$. Hence, when $r=\lambda_{j}$,
we have $T=M\mu/r$, i.e. the thermal energy stemming from the agitation of
particles equals the self-gravitational energy and the cloud is in
equilibrium. Conversely, all scales larger than $\lambda_{j}$ are unstable to
gravitational collapse, whereas smaller scales are stable.

The critical length (11) is consistent with the classical equipartition
theorem. Nevertheless, due to the relativistic nature of interstellar gas
constituents, a derivation within Kaniadakis kinetic theory seems all the more
natural. This was investigated in [29], obtaining the modified expression

$\lambda_{c}^{\kappa}=\sqrt{\frac{\left(1+\frac{\kappa}{2}\right)}{\left(1+\frac{3}{2}\kappa\right)2\kappa}\frac{\Gamma\left(\frac{1}{2\kappa}-\frac{3}{4}\right)\Gamma\left(\frac{1}{2\kappa}+\frac{1}{4}\right)}{\Gamma\left(\frac{1}{2\kappa}+\frac{3}{4}\right)\Gamma\left(\frac{1}{2\kappa}-\frac{1}{4}\right)}}\,\lambda_{J}\,,$
(12)

where $\Gamma(x)$ denotes the Euler Gamma function. It is important to observe
that, for $\kappa=0$, this expression gives the correct limit
$\lambda_{c}^{\kappa}\rightarrow\lambda_{J}$, while considering
$0\leq\kappa<2/3$ results in $\lambda_{c}^{\kappa}>\lambda_{j}$. Intuitively,
relativistic statistical corrections contribute to increasing the thermal
kinetic energy of the cloud, thus making the system more stable against
gravitational perturbations. On the other hand, in the limit case
$\kappa=2/3$, the $\kappa$-generalized equipartition theorem suffers from a
divergence and the reasoning below the above derivation may need to be
reconsidered [29].

Further studies of Jeans instability appeared within the framework of gravity
beyond general relativity (see Sec. 4 for more discussion). Specifically, in
[37] a dispersion relation generalizing the Jeans modes was derived in
Eddington-inspired Born-Infeld model by deforming the Maxwell-Boltzmann
function to a family of power law distributions. Likewise, exploiting the
kinetic theory, the analysis of [38] explored the influence of Kaniadakis
statistics in the background of $f(R)$ gravity, which is a family of theories
that extend the Lagrangian of the Einstein–Hilbert action by including
suitable functions of the Ricci scalar $R$. The main outcome was that the
range of the unstable modes and the growth rates of density perturbations
decrease as the parameter $\kappa$ increases in both the high and low
frequency regime.

The behaviors of Jeans modes for gravitational systems composed of dark and
baryonic matters were the subject of investigation in [39]. The results showed
that the Jeans instability is suppressed due to relativistic corrections,
setting the conditions for new phenomenological models in cosmology.

##  4 Extended Gravity and Black hole thermodynamics

Einstein’s theory of General Relativity is the cornerstone of gravitational
physics. Its success in explaining a variety of phenomena stands as a
testament of its robustness. However, challenges arise when attempting to
unify this framework with quantum mechanics. Such discrepancies highlight the
need for a more comprehensive formalism, capable of describing gravity at both
macroscopic and microscopic scales.

In what follows we discuss the interplay between Kaniadakis theory and
extended approaches to gravity. These studies involve exploring to what extent
non-extensive statistical principles could shed light on unexplained
phenomena, especially in scenarios involving the extreme gravitational
conditions of black holes environments and the very early Universe.

###  4.1 Verlinde’s entropic gravity

Verlinde’s model is among the most engaging approaches that seek to connect
the concepts of entropy and gravity [40]. In a nutshell, it suggests that
gravity might not be a fundamental force, as described in Einstein’s theory,
but rather an outcome of the way microscopic information is encoded on the
surface of bodies. From this perspective, the gravitational force could emerge
from changes in the entropy associated with these information-bearing
boundaries.

Verlinde’s idea was used in [30] along with Kaniadakis statistical principles
to derive an effective Newton’s constant in the form

$G_{\kappa}=\frac{\left(1+\frac{3}{2}\kappa\right)2\kappa}{\left(1+\frac{\kappa}{2}\right)}\,\frac{\Gamma\left(\frac{1}{2\kappa}+\frac{3}{4}\right)\Gamma\left(\frac{1}{2\kappa}-\frac{1}{4}\right)}{\Gamma\left(\frac{1}{2\kappa}-\frac{3}{4}\right)\Gamma\left(\frac{1}{2\kappa}+\frac{1}{4}\right)}\,G\,,$
(13)

for $0\leq\kappa<2/3$, where we have restored the gravitational constant $G$
for the sake of clarity.

In Fig. 2 we plot the relative difference $\frac{\Delta G}{G}$ versus
$\kappa$, where $\Delta G\equiv G-G_{\kappa}$. It can be seen that
$G_{\kappa}\leq G$ for any allowed $\kappa$, with the equality being satisfied
as $\kappa\rightarrow 0$. Such a behavior entails that, for a given source,
the intensity of Verlinde’s gravity in Kaniadakis framework is reduced
comparing to the classical theory. Notice that this is an alternative way to
convey the result in Eq. (12).

The red dashed line in Fig. 2 marks the observational constraint
$|\frac{\Delta G}{G}|\lesssim 0.29$, which is obtained by cosmological
constraints on large-scales [41], and SNIa [42], PSR J0437-4715 [43] and
Planck [44] data in modified gravity frameworks. Hence, the constraint
$\kappa\lesssim 0.38$ is obtained on Kaniadakis parameter. Notice that this is
of the same order as the constraint from cosmic rays spectrum, although it is
much less stringent than other bounds of cosmological origin (see Sec. 5).

Also, it is interesting to observe that a smaller gravitational constant (of
course within the observational bounds) is known to be one of the mechanisms
that can alleviate the $\sigma_{8}$ tension, where $\sigma_{8}$ is the
standard deviation of the density fluctuation in an 8 $h^{-1}\,\mathrm{Mpc}$
radius sphere (see Sec. 5 for more discussion). Thus, Kaniadakis Cosmology may
provide an effective framework to explain this tension.

Figure 2: Plot of $\Delta G/G$ versus $\kappa$. The red dashed line marks the
observational bound $|\Delta G/G|\lesssim 0.29$ [41, 42, 43, 44, 45].

A related outcome is that self-gravitating systems will take longer to
collapse. This follows directly from the computation of the free fall time,
which in $\kappa$-statistics takes the generalized form [30]

$t_{FF}^{\kappa}=\sqrt{\frac{\left(1+\frac{\kappa}{2}\right)}{\left(1+\frac{3}{2}\kappa\right)2\kappa}\,\frac{\Gamma\left(\frac{1}{2\kappa}-\frac{3}{4}\right)\Gamma\left(\frac{1}{2\kappa}+\frac{1}{4}\right)}{\Gamma\left(\frac{1}{2\kappa}+\frac{3}{4}\right)\Gamma\left(\frac{1}{2\kappa}-\frac{1}{4}\right)}}\,t_{FF}\,,$
(14)

where $t_{FF}=\sqrt{\frac{3}{2\pi\rho_{0}}}$ is the standard expression. Since
the $\kappa$-dependent factor is the reciprocal square root of the correction
in Eq. (13), one has $t^{\kappa}_{FF}\geq t_{FF}$, where the inequality is
obviously saturated for $\kappa\rightarrow 0$.

###  4.2 Loop quantum gravity

Loop Quantum Gravity (LQG) provides a theoretical attempt to unify quantum
mechanics and Einstein’s theory of gravity. A distinctive feature of this
approach is that geometrical quantities like area and volume are described by
quantum operators with discrete spectra. This reflects the expected granular
structure in the fabric of quantum spacetime, which is common to many models
of quantum gravity [46, 47]. Concretely, the size of the quantum of area in
Planck units is measured by the Immirzi parameter $\gamma=\log
2/\left(\pi\sqrt{3}\right)$, which is estimated by counting the number of
microstates of a given system classically [48] (see also [49] for more
physical insights on the meaning of $\gamma$).

In [50] Abreu _et al._ worked on a generalization of the Immirzi parameter to
non-extensive statistics. For a system of boundary surface $A$ in the
microcanonical ensemble, they obtained a new relation between the Immirzi
parameter, the $\kappa$ parameter and the surface area in the form

$\gamma_{\kappa}=\gamma\,\frac{A}{4\log\left[\exp_{\kappa}\left(A/4\right)\right]}\,,$
(15)

which implies $\gamma_{\kappa}>\gamma$, and $\gamma_{\kappa}\rightarrow\gamma$
for vanishing $\kappa$, as expected. Therefore, Kaniadakis corrections
contribute to modify, and in particular to increase the size of the quantum of
area, revealing a potential role of $\kappa$-statistics in the quest for a
quantum gravity formalism.

###  4.3 Modified gravity and expanding Universe

The success of the $\Lambda$-CDM model arises from its remarkable match with
the available cosmological observations. However, a number of issues challenge
its ability to describe the evolution of the Universe at both infrared and
ultraviolet scale. In this gap, modified cosmology based on extended gravity
provides one of the most fruitful paradigms to improve our current knowledge
of the Universe.

Based on a generalized model of dark energy with $\kappa$-entropy, the
analysis in [51] explored the cosmic evolution within the frameworks of $f(G)$
and $\tilde{f}(T)$ gravity, where $G$ and $T$ are the Gauss–Bonnet invariant
and torsion scalar, respectively. Using these models, the behavior of various
cosmic parameters was examined to discuss the accelerated expansion of the
Universe. Similar investigation was conducted in [52, 53] in Brans–Dicke
gravity, which extends Einstein’s theory by introduction of a scalar field
interacting with gravity. Such studies fit into a very active research trend
that aims to scrutinize the statistical properties behind the concept of dark
energy (see Sec. 5 for more discussion).

###  4.4 Black hole thermodynamics

Thermodynamics of black holes provides a unique arena for testing the
interplay between gravity, relativity and quantum theory. This domain emerges
by conceptualizing black holes as thermodynamic systems, revealing intriguing
parallels between their properties and classical thermodynamics [54, 55, 56,
57]. In this framework, an important concept is black hole entropy. Arguments
from different perspectives, including the holographic principle222The
holographic principle is a key concept in quantum gravity. It posits that the
information within a defined $d$-dim. space can be fully encoded on a
$(d-1)$-dim. surface surrounding that space. [26, 27, 28], suggest that the
entropy of a black hole should obey the Bekenstein-Hawking area law
$S_{BH}=A/4$, where $A$ is the horizon surface area [54]. However, it is clear
that, if we consider black holes as genuinely $3$-dim. systems, $S_{BH}$
cannot be identified as thermodynamic entropy, since it violates extensivity
[2].

To understand this puzzle, the thermodynamics of black holes was generalized
to accommodate non-extensive definitions of entropy [58, 59, 60, 61].
Interesting features were found in the context of thermodynamics based on Eq.
(6), showing that relativistic $\kappa$-entropy has a non-trivial impact onto
the phase structure and transitions of charged anti-de Sitter black holes
[58]. For concreteness, we display in Fig. 3 the behavior of the heat capacity
at constant pressure versus the entropy. Since divergences of $C_{p}$ are
associated to (first-order) small-to-large phase transitions of black holes,
it is evident that Kaniadakis corrections affect the black hole size at which
such critical phenomena occur. Further study of the $P-V$ diagram and Gibbs
free energy highlighted a phenomenology similar to that of van der Waals fluid
[58], with the critical exponents being now dependent on $\kappa$.

Figure 3: Plot of the heat capacity at constant pressure versus entropy. SBH
and LBH denote the “small” and “large” stable (i.e. $C_{p}>0$) domains,
respectively, while IBH is the “intermediate” unstable (i.e. $C_{p}<0$) region
(from [59]).

Along the same line, Cimidiker _et al._ analyzed the effects of a quantum
gravitational deformation of the Heisenberg Principle on non-extensive black
hole thermodynamics [61]. As a result, it was shown that the final state of a
black hole at the end of the evaporation turns out to be a remnant with finite
temperature and entropy. Apart from its own interest, this achievement could
be exploited to tackle the black hole information paradox in the future [62].

##  5 Cosmological insights

The observation that the Universe is currently accelerating presents a
significant conundrum in cosmology. To account for this phenomenon, the most
plausible mechanism is the existence of an unknown form of energy - the _dark
energy_ (DE) - which would permeate all of space. Because of its negative
pressure, DE would exert a repulsive gravitational force, providing the fuel
to drive the accelerated cosmic expansion. Although DE is thought to make up
about 70% of the energy budget of the Universe, its properties remain poorly
understood.

Since its theorization in the late 1990s, DE has been extensively studied
[63]. Among the various models proposed so far, holographic dark energy (HDE)
has a central role in quantum gravity, as it is founded on the application of
the holographic principle to the horizon degrees of the freedom of the
Universe [64]. In this model, the energy density of DE takes the form

$\rho_{de}=3C^{2}M_{p}^{2}L^{-2}\,,$ (16)

where $C$ is a constant of order unity and $L$ a characteristic length scale
that acts as IR cutoff (we have here restored the Planck mass $M_{p}$ for
clarity). The choice of the future event horizon
$L=R_{h}=a\int_{t}^{\infty}\frac{dt}{a}$ was proven to be in line with
phenomenology [64], though improvements and further analysis could be needed
[65].

###  5.1 Kaniadakis modification of Friedmann cosmology

Let us elaborate on implications of $\kappa$-entropy for cosmology. We focus
on the generalization of Eq. (16) and the predictions that emerge from the
modified Friedmann equations governing the Universe’s evolution.

####  5.1.1 Kaniadakis holographic dark energy

The relation (16) is framed in the semiclassical theory, where the Universe’s
entropy is described by the usual Bekenstein-Hawking area law $S_{BH}$ (see
Sec. 4.4). On the other hand, a generalized HDE model was proposed in [66],
using the entropy (7) and the future horizon as IR cutoff333Other descriptions
of KHDE set the apparent horizon as IR cutoff [67, 68, 69]. Besides not
getting back the standard HDE and thermodynamics as a sub-case, the
disadvantage of these models is that they need unacceptably large values of
$\kappa$ to fit observational results.. In the ensuing framework, the energy
density of Kaniadakis Holographic Dark Energy (KHDE) reads

$\rho_{de}^{\kappa}=3C^{2}M_{p}^{2}R_{h}^{-2}+\kappa^{2}M_{p}^{6}R_{h}^{2}\,.$
(17)

Based on this relation, modified Friedmann equations were derived that
reproduce the observed history of the Universe [66] (see also [70] for an
alternative derivation). This model also allowed to compute characteristic
quantities like the equation of state parameter of KHDE, the deceleration
parameter, the squared sound speed and the Hubble parameter. Experimental
implications of KHDE were further addressed in [71, 72, 73, 74]. Notably, the
analysis of [72] revealed that Kaniadakis horizon-entropy model may alleviate
the Hubble tension problem, representing a potentially successful candidate
for the description of the cosmic evolution. Further study was conducted in
[70] by investigating the validity of the generalized second law of
thermodynamics for the Universe enclosed by the apparent horizon. The obtained
equations provide a background to explore a new cosmological model built on
the $\kappa$-entropy, which might be used to study the evolution of the
Universe from early deceleration to the late-time acceleration and constrain
the $\kappa$-parameter accordingly. In this context, the analysis of [72]
reveals that
$|\beta|\equiv|\frac{\kappa\pi}{GH_{0}^{2}}|\lesssim\mathcal{O}(10^{-3}\div
10^{-2})$ for consistency with the current observed value of the deceleration
parameter $q_{0}$ in the presence of a non-vanishing cosmological constant
$\Lambda$. In turn, this entails the very stringent constraint
$|\kappa|\lesssim\mathcal{O}(10^{-126}\div 10^{-125})$, where we have set the
present Hubble rate $H_{0}\simeq 10^{-42}\,\mathrm{GeV}$. Also, the dynamical
system analysis of [72] shows that, in a Kaniadakis-entropy based evolutionary
scenario, the Universe past attractor is the matter-dominated epoch, while at
late times, the Universe is dominated by dark energy and undergoes an
accelerated expansion in both vanishing and non-vanishing $\Lambda$ cases. The
requested deviation from the standard Bekenstein-Hawking-like horizon scaling
is still quantified by $|\kappa|\simeq 10^{-125}$ for $\Lambda\neq 0$ and
$|\kappa|\simeq 10^{-123}$ for $\Lambda=0$. The latter less stringent
constraint can be understood by observing that, in the absence of an explicit
cosmological constant term, one needs a more significant deviation from
standard Cosmology to describe the late-time Universe acceleration.

While being more stringent than the astrophysical bounds discussed in Sec. 3,
the above cosmological constraints align with those inferred from other set of
cosmic measurements, e.g. Observational Hubble data, Pantheon supernova Type
Ia sample, H II galaxies, Strong lensing systems and Baryon acoustic
oscillations (see [72] for more details). Therefore, it is expected that
allowing even small deviations from the classical horizon entropy might lead
to a very interesting cosmological phenomenology, which is in good agreement
with observational behaviour and can solve some of the current tensions of the
$\Lambda$CDM model.

####  5.1.2 Baryogenesis

In [74] $\kappa$-modified Friedmann equations provided a possible solution to
baryogenesis, which is the observed imbalance between matter and antimatter in
the Universe. In order for this asymmetry to occur, the following three
Sakharov’s conditions must be satisfied: _i_) violation of the baryon number;
_ii_) loss of thermal equilibrium; _iii_) violation of C and CP symmetries
[75]. While the first and last requirements can be fulfilled by introducing a
suitable coupling between the spacetime and baryon current, there is no
mechanism to break the thermal equilibrium in the radiation-dominated era in
the standard cosmology. Interestingly enough, in [74] Kaniadakis effects were
found to generate fluctuations in the energy and pressure content of the
Universe, providing the right conditions to satisfy all three Sakharov’s
criteria. Observational consistency set $\kappa\simeq 10^{-10}$. Despite being
much smaller than the constraint from the cosmic rays spectrum, the non-
vanishing value of $\kappa$ signals the need of a Kaniadakis-like modification
of Friedmann cosmology to trigger baryogenesis.

In [76] a relation was established between the quantities related with the
energy content of the Universe and the cosmographic parameters that express
its geometry. Reconstructing the evolution of the deceleration parameter
versus the redshift, the model showed that Kaniadakis corrections could drive
the current cosmic acceleration without introducing any ad hoc component of DE
or cosmological constant.

####  5.1.3 IceCube PeV neutrinos

The IceCube Neutrino experiment was designed to detect neutrino sources in the
TeV regime [77]. Unexpectedly, the discovery of high-energy astrophysical
neutrino flux with energies of the order of PeV opened new scenarios in
astroparticle physics, due to the difficulties to reconcile the candidate
sources of these particles with the standard cosmological model. This tension
was alleviated in [78], assuming that the Universe’s evolution obeys
$\kappa$-modified Friedmann equations with the parameter $\kappa\simeq
10^{-37}$. The discrepancy with constraints of different origin served as
inspiration for an effective dynamical model, where relativistic statistical
corrections might become relevant in the very early Universe, while they
gradually turn off approaching the present time.

###  5.2 Inflation and perturbation growth

Inflationary cosmology proposes that the Universe underwent a rapid and
exponential expansion immediately after the Big Bang [79, 80, 81]. This model
was introduced to solve several key problems (e.g., flatness, horizon,
structure formation) in standard cosmology and explain the uniformity of the
cosmic microwave background radiation. Although inflation is largely accepted,
the underlying mechanism and its complete implications remain areas of ongoing
research and debate.

In the framework of $\kappa$-modified cosmology, inflation driven by a scalar
field was analyzed in [82] from the slow-roll condition perspective. The model
was successfully tested by comparing the predicted scalar spectral index and
tensor-to-scalar ratio to the latest BICEP and Planck data. Furthermore, it
was proven that Kaniadakis corrections affect the growth of primordial
fluctuations, which are the seeds of all structures in the Universe. To show
this, the relevant quantity to consider is the density contrast
$\delta\equiv\delta\rho/\rho$, where $\rho$ and $\delta\rho$ are the energy
density and fluctuations of the fluid that permeates the Universe,
respectively. Assuming $\delta\ll 1$ and the approximation (7), the following
modified evolution equation was obtained

$\delta^{\prime\prime}+\frac{3}{2a}\left(1+\frac{9\kappa^{2}}{64\rho^{2}}\right)\delta^{\prime}-\frac{3}{2a^{2}}\left(1+\frac{9\kappa^{2}}{32\rho^{2}}\right)\delta=0\,,$
(18)

where the prime denotes derivative with respect to the scale factor $a$. The
above relation was solved numerically. The solution is plotted in Fig. 4,
which shows that higher values of $\kappa$ correspond to a faster perturbation
growth comparing to the standard case (red solid curve). Therefore, even very
tiny deviations from classical statistics at microscopic level may result in
appreciable effects at very large scales in the Universe.

Figure 4: Plot of $\delta$ versus the redshift $z$ (from [82]).

Recently, the Universe’s evolution from inflation to reheating was studied in
[83], using a more generalized entropy that includes Kaniadakis functional as
a limit case. In that context, the inflaton field driving the accelerated
expansion was represented by the entropic energy density. For suitable ranges
of the entropic parameters, the Universe was found to enter to a reheating
stage after the end of inflation, when the entropic energy decays to
relativistic particles with a certain decay rate. On top of that, consistency
was achieved between predicted values of the observable inflation indices and
the recent Planck data.

It is worth noting that the study of the evolution equation of matter
overdensity $\delta$ is intimately connected with the $\sigma_{8}$ tension,
which is indeed related to the matter clustering and the fact that the Cosmic
Microwave Background (CMB) estimation [84] differs from the SDSS/BOSS direct
measurements [85, 86]. In [87, 88] it was argued that an increased friction
term in the matter-perturbation evolution (the second one in Eq. (18)) may
lead to smaller $\sigma_{8}$, providing a potential effective mechanism to
alleviate the $\sigma_{8}$ tension. Clearly, a more detailed analysis in the
present framework requires investigating the behavior of
$f\sigma_{8}(z)=f(z)\sigma(z)\equiv-\frac{d\log\delta(z)}{d\log
z}\,\sigma_{8}\frac{\delta(z)}{\delta(0)}$ against the $\Lambda$CDM scenario.
More work in this direction is reserved for the future. In passing, we mention
that the possibility of alleviating both the $H_{0}$ and $\sigma_{8}$ tension
in modified entropic scenarios was recently explored in [89] in the context of
Tsallis Cosmology.

###  5.3 Cosmological phase transitions

Inspired by similarities with the critical phenomena of anti-de Sitter black
holes, the efforts in [90] were focused on the thermodynamic phase structure
of a Friedmann–Lemaître–Robertson–Walker Universe in Kaniadakis statistics. In
particular, the study of the equation of state pointed out the existence of
non-trivial critical points, where a van der Waals-like phase transition
occurs. Although results were obtained in the leading order approximation (7),
they provide the basis for a reliable thermodynamic picture of the early
Universe, at least in the stable regions.

##  6 Discussion and Conclusions

Kaniadakis theory is a one-parameter extension of the traditional statistics
that incorporates relativistic effects. It is based on the idea that in
extreme conditions, such as high energies or densities, standard statistical
models might not accurately describe the behavior of systems, requiring
suitable modifications. The effort of this work was to review recent advances
of Kaniadakis theory in extreme gravitational and cosmological environments.
The advantage of applying $\kappa$-framework to such systems is that they may
somehow amplify modified entropy effects, creating suitable conditions for
their direct or indirect detection.

In the first part of the work, we discussed astrophysical implications on open
stellar clusters, Jeans instability and gravitational collapse. We showed that
Kaniadakis statistics has a non-trivial impact on the classical predictions of
Jeans length and collapsing time, as it contributes to making gravitational
systems more stable. Afterwards, the focus was shifted onto the interplay
between Kaniadakis theory and modified gravity, with special attention to
Verlinde’s formalism and Loop Quantum Gravity. We also explored black hole
$\kappa$-thermodynamics, which exhibits a richer phenomenology compared to the
ordinary description. As a final study, we investigated Kaniadakis
modification of Friedmann cosmological equations. Remarkably, this framework
allows for alleviating tensions and explaining phenomena that cannot be
understood in the classical theory. Hence, it stands out as a potential
candidate to improve over the current limits of the standard cosmological
model. Clearly, the class of phenomena and predictions to explore is too vast
to cover in this review. More applications and insights of $\kappa$-theory are
discussed in [17].

Despite many virtues listed above, some aspects are yet to be clarified or
fully understood. As emphasized in Sec. 2, Kaniadakis theory is framed in the
context of special relativity. The question naturally arises of how it can be
adapted to accommodate the symmetries of general relativity. The issue of
generality of $\kappa$-statistics is also tackled in [91] from the condensed
matter perspective. Regrettably, Kaniadakis paradigm fails to provide a non-
extensive version of Ising model, unlike Tsallis’s distribution, which instead
stands out as a potentially valid generalization (see [92, 91, 93] for further
discussion on the subject). This drawback apparently challenges the role of
$\kappa$-theory as generalized statistical mechanics, at least in low-energy
regimes. Toward developing a more comprehensive framework, valuable insights
could be gained by working with a multiparameter functional that gathers all
the deformed entropies introduced so far under a single umbrella. Prospective
analysis appears in [94, 95, 60] in cosmology and black holes thermodynamics,
but further applications are under active consideration in gravity and
information theory. For instance, a challenging task is to look for footprints
in the spectrum of primordial gravitational waves, which carry unique
knowledge on the Universe’s initial conditions. On the other hand, based on
[96], the use of non-extensive entropy as a candidate for a generalized
quantum information theory could be explored for the purpose of solving some
paradoxes in quantum gravity [97]. Work is in progress along these directions
and will be presented elsewhere [62].

Acknowledgments The author would like to acknowledge the anonymous Referees
for their comments and recommendations, which contributed to improve the
quality of the manuscript. He is also grateful to the Spanish “Ministerio de
Universidades” for the awarded Maria Zambrano fellowship and funding received
from the European Union - NextGenerationEU.

## Declarations

The author has no conflicts of interest to declare. Data sharing is not
applicable to this article as no new datasets were generated or analyzed
during the current study.

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#  12th Workshop on the CKM Unitarity Triangle
Santiago de Compostela, 18-22 September 2023
Charm lifetime measurements at Belle II

Alan Schwartz1 on behalf of the Belle II Collaboration
1Physics Department  University of Cincinnati  Cincinnati  Ohio 45221

###### Abstract

We report precision measurements of lifetimes of charmed mesons and baryons
performed by the Belle II experiment. Specifically, we measure $D_{s}^{+}$,
$D^{+}$, $D^{0}$, $\Lambda_{c}^{+}$, and $\Omega_{c}^{0}$ lifetimes. Our
results for $\tau(D_{s}^{+})$, $\tau(D^{+})$, $\tau(D^{0})$, and
$\tau(\Lambda_{c}^{+})$ are the world’s most precise; our result for
$\tau(\Omega_{c}^{0})$ confirms that the $\Omega_{c}^{0}$ lifetime is longer
than that of the $\Lambda_{c}^{+}$ and the $\Xi_{c}^{0}$.

##  1 Introduction

The lifetime of a particle, like its mass, is one of its fundamental
properties. The lifetime is the reciprocal of the sum of all partial widths,
and thus all final states contribute to it. In this manner, the lifetime can
provide information about final states that are difficult to measure or
calculate.

Charm lifetimes are calculated within the framework of the Heavy Quark
Expansion (HQE) [1]. Using the optical theorem, the expansion takes the form
(for a $D$ meson) [2]:

$\displaystyle\Gamma({D})$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{{D}}}\sum_{X}\int\limits_{\rm
PS}(2\pi)^{4}\delta^{(4)}(p_{D}-p_{X})\,|\langle X(p_{X})|{\cal H}_{\rm
eff}|{D}(p_{D})\rangle|^{2}$ (1.1) $\displaystyle\rightarrow$
$\displaystyle\Gamma_{3}+\Gamma_{5}\frac{\langle{\cal
O}_{5}\rangle}{m_{c}^{2}}+\Gamma_{6}\frac{\langle{\cal
O}_{6}\rangle}{m_{c}^{3}}+...+16\pi^{2}\left(\tilde{\Gamma}_{6}\frac{\langle\tilde{\mathcal{O}}_{6}\rangle}{m_{c}^{3}}+\tilde{\Gamma}_{7}\frac{\langle\tilde{\mathcal{O}}_{7}\rangle}{m_{c}^{4}}+\ldots\right),$
(1.2)

where the summation in Eq. (1.1) is over all final states, and the
$\Gamma_{n}$ terms in Eq. (1.2) are Wilson coefficients that are expanded in
powers of $\alpha_{s}$ and calculated perturbatively. The
$\langle\mathcal{O}_{n}\rangle$ and $\langle\tilde{\mathcal{O}}_{n}\rangle$
terms are matrix elements of dimension-$n$ local operators and must be
calculated using non-perturbative methods.111No tilde denotes a two-quark
operator; a tilde denotes a four-quark operator [2]. Comparing HQE
calculations with experimental measurements tests our understanding of QCD.
Here we present several lifetime measurements of charmed hadrons performed by
the Belle II experiment [3].

Belle II runs at the SuperKEKB $e^{+}e^{-}$ collider [4] based at the KEK
laboratory in Tsukuba, Japan. The experiment studies weak decays of $B$ and
$D$ mesons, and $\tau$ leptons, with the goal of uncovering new physics. The
Belle II detector is described in Ref. [5]. The measurements reported here
were performed with early data sets, before the second layer of the silicon
pixel detector was fully installed. The lifetime is determined from a fit to
the decay time distribution. For a particle of mass $m$ and momentum
$\vec{p}$, the decay time is calculated as
$t=m\,(\vec{d}\cdot\hat{p})/|\vec{p}\,|$, where $\vec{d}$ is the displacement
vector from the $e^{+}e^{-}$ interaction point (IP), where the particle is
produced, to its decay vertex position. The IP in Belle II is measured every
30 minutes using $e^{+}e^{-}\to\mu^{+}\mu^{-}$ events. All charm lifetime
analyses impose a momentum requirement on the decaying meson or baryon to
eliminate those originating from $B$ decays, which are displaced from the IP
and would corrupt the lifetime measurement.

The uncertainty on $t$, denoted $\sigma_{t}$, is calculated event-by-event by
propagating the uncertainties on $\vec{d}$ and $\vec{p}$, taking into account
their correlations. The decay time resolution ($\langle\sigma_{t}\rangle$) is
approximately twice as precise as that achieved at Belle and Babar:
$80\\!-\\!90$ fs versus $\sim\\!200$ fs. For the analyses presented here, the
uncertainty $\sigma_{t}$ multiplied by a scaling factor is taken as the width
of a Gaussian resolution function used to fit the decay time distributions.

##  2 $D_{s}^{+}$ lifetime

The most recent measurement is that of the $D_{s}^{+}$ lifetime, which used
207 fb-1 of data [6]. The Cabibbo-favored (CF) decay $D_{s}^{+}\to\phi(\to
K^{+}K^{-})\pi^{+}$ is reconstructed, and candidates with an invariant mass
satisfying $M(\phi\pi^{+})\in[1.960,1.976]$ GeV/$c^{2}$ are used to measure
the lifetime. Fitting the $M(\phi\pi^{+})$ distribution, we obtain a signal
yield in this range of $116\times 10^{3}$ events with a purity of 92%.

The lifetime is determined from an unbinned maximum likelihood fit to the
decay time ($t$) distribution. As the uncertainty $\sigma_{t}$ is used in the
resolution function, and its distribution differs for signal and background
events, we include probability density functions (PDFs) for $\sigma_{t}$ in
the likelihood function to avoid biasing the fit results [7]. The likelihood
function for event $i$ is given by

$\displaystyle{\cal L}(\tau|t^{i},\sigma_{t}\\!\\!{}^{i})$ $\displaystyle=$
$\displaystyle f_{\rm
sig}\,P_{\rm{sig}}(t^{i}|\tau,\sigma_{t}\\!\\!{}^{i})\,P_{\rm{sig}}(\sigma_{t}\\!\\!{}^{i})\
+\
(1-f_{\rm{sig}})\,P_{\rm{bkg}}(t^{i}|\sigma_{t}\\!\\!{}^{i})\,P_{\rm{bkg}}(\sigma_{t}\\!\\!{}^{i})\,,$
(2.1)

where $f_{\rm sig}$ is the fraction of events that are signal decays, and
$P_{\rm{sig}}(t^{i}|\tau,\sigma_{t}^{i})$ is the signal PDF for measuring
$t_{i}$ given a lifetime $\tau$ and an uncertainty $\sigma_{t}^{i}$. The
background PDF $P_{\rm bkg}(t^{i}|\sigma_{t}\\!\\!{}^{i})$ is an exponental
function determined from events in the sideband
$M(\phi\pi^{+})\in[1.990,2.020]$ GeV/$c^{2}$. The PDF
$P_{\rm{bkg}}(\sigma_{t}\\!\\!{}^{i})$ is a histogram also determined from the
$M(\phi\pi^{+})$ sideband, and $P_{\rm{sig}}(\sigma_{t}\\!\\!{}^{i})$ is a
histogram determined from events in the $M(\phi\pi^{+})$ signal region after
subtracting off the $P_{\rm{bkg}}(\sigma_{t}\\!\\!{}^{i})$ contribution. The
fraction $f_{\rm sig}$ is determined from the fit to the $M(\phi\pi^{+})$
distribution.

The PDF for signal decays is

$\displaystyle P_{\rm sig}(t^{i}|\tau,\sigma_{t}\\!\\!{}^{i})\ =\
\frac{1}{\tau}\int
e^{-t^{\prime}/\tau}\,R(t^{i}-t^{\prime};\mu,s,\sigma_{t}\\!\\!{}^{i})\,dt^{\prime}\,,$
(2.2)

where $R$ is a Gaussian function with mean $\mu$ and a per-candidate standard
deviation $s\cdot\sigma_{t}^{i}$, with $s$ being a scaling factor. The
lifetime $\tau$ is determined by maximizing the total likelihood
$\Sigma_{i}{\cal L}_{i}$, where the summation runs over all signal candidates.
The floated parameters are $\tau$, $\mu$, and $s$. The result of the fit is
$\tau(D^{+}_{s})\ =\ (499.5\pm 1.7\pm 0.9)$ fs, where the first uncertainty is
statistical and the second is systematic. The projection of the fit result is
shown in Fig. 1. The sources of systematic uncertainty are listed in Table 1.

Figure 1: $D_{s}^{+}\to\phi\pi^{+}$ decay time distribution with the fit
result superimposed (left), and $\sigma_{t}$ distributions for background and
signal + background (right) [6].

.

Table 1: Systematic uncertainties for the $D_{s}^{+}$ lifetime measurement [6]. Source | Uncertainty (fs)
---|---
Resolution function | $+0.43$
Background $(t,\sigma_{t})$ distribution | $\pm 0.40$
Binning of $\sigma_{t}$ histogram PDF | $\pm 0.10$
Imperfect detector alignment | $\pm 0.56$
Sample purity | $\pm 0.09$
Momentum scale factor | $\pm 0.28$
$D^{+}_{s}$ mass | $\pm 0.02$
Total | $\pm 0.87$

##  3 $D^{+}$ and $D^{0}$ lifetimes

The measurements of the $D^{0}$ and $D^{+}$ lifetimes [8] are similar to that
for the $D_{s}^{+}$ but use a smaller data set: 78 fb-1. To reduce
backgrounds, $D^{0}$ and $D^{+}$ candidates are required to originate from
either $D^{*+}\to D^{0}\pi^{+}_{s}$ or $D^{*+}\to D^{+}\pi^{0}_{s}$ decays.
The pion daughter in these decays has very low momentum in the laboratory
frame and thus is labeled the “slow” pion ($\pi_{s}$). The $D^{0}$ and $D^{+}$
are reconstructed in the CF final states $K^{-}\pi^{+}$ and
$K^{-}\pi^{+}\pi^{+}$, respectively. To eliminate signal candidates
originating from $B\to DX$, we require that the $D^{*}$ momentum in the
$e^{+}e^{-}$ center-of-mass (CM) frame be greater than 2.5 GeV/$c$ for $D^{0}$
candidates and $>\\!2.6$ GeV/$c$ for $D^{+}$ candidates. For the lifetime
fits, we select $D^{0}$ candidates satisfying
$M(K^{-}\pi^{+})\in[1.851,1.878]$ GeV/$c^{2}$ and
$M(K^{-}\pi^{+}\pi^{+}_{s})\\!-\\!M(K^{-}\pi^{+})\in[144.94,145.90]$
MeV/$c^{2}$, and $D^{+}$ candidates satisfying
$M(K^{-}\pi^{+}\pi^{+})\in[1.855,1.883]$ GeV/$c^{2}$ and
$M(K^{-}\pi^{+}\pi^{+}\pi^{0}_{s})\\!-\\!M(K^{-}\pi^{+}\pi^{+})\in[138.0,143.0]$
MeV/$c^{2}$. These samples contain $171\times 10^{3}$ $D^{0}\to K^{-}\pi^{+}$
candidates (99.8% purity) and $59\times 10^{3}$ $D^{0}\to K^{-}\pi^{+}\pi^{+}$
candidates (91% purity).

As in the $D_{s}^{+}$ analysis, the lifetimes are determined from unbinned
maximum likelihood fits to the decay time, and the per-candidate uncertainty
$\sigma_{t}$ multiplied by a scaling factor is used as the width(s) of a
double Gaussian ($D^{0}$) or single Gaussian ($D^{+}$) resolution function.
The decay time PDF for $D^{+}$ background is obtained from the sideband
$M(K^{-}\pi^{+}\pi^{+})\in[1.758,1.814]\cup[1.936,1.992]$ GeV/$c^{2}$, whereas
$D^{0}$ background is negligible. The results of the fits are
$\tau(D^{0})=(410.5\pm 1.1\,\pm 0.8)$ fs and $\tau(D^{+})=(1030.4\pm 4.7\,\pm
3.1)$ fs, where the first uncertainty is statistical and the second is
systematic. The decay time distributions are shown in Fig. 2 along with
projections of the fit result. The sources of systematic uncertainty are
listed in Table 2.

Figure 2: Decay time distribution with the fit result superimposed for
$D^{0}\to K^{-}\pi^{+}$ (top) and $D^{+}\to K^{-}\pi^{+}\pi^{+}$ (bottom) [8].

.

Table 2: Systematic uncertainties for the $D^{0}$ and $D^{+}$ lifetime measurements [8]. Source |  $\tau(D^{0})$ (fs) |  $\tau(D^{+})$ (fs)
---|---|---
Resolution model | 0.16 | 0.39
Backgrounds | 0.24 | 2.52
Detector alignment | 0.72 | 1.70
Momentum scale | 0.19 | 0.48
Total | 0.80 | 3.10

##  4 $\Lambda_{c}^{+}$ lifetime

Measurement of the $\Lambda_{c}^{+}$ lifetime [9] is similar to that for the
$D_{(s)}^{+}$ and $D^{0}$ lifetimes. The reconstructed decay is
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$. However, some $\Lambda_{c}^{+}$ candidates
originate from $\Xi_{c}^{0}\to\Lambda_{c}^{+}\pi^{-}$ and
$\Xi_{c}^{+}\to\Lambda_{c}^{+}\pi^{0}$ decays; since $\tau(\Xi_{c}^{+})=152$
fs and $\tau(\Xi_{c}^{0})=453$ fs [10], such $\Lambda_{c}^{+}$ candidates
corrupt the $\tau(\Lambda_{c}^{+})$ measurement. Fortunately, they can be
identified from the displacement of the $\Lambda_{c}^{+}$ decay vertex from
the IP in the plane transverse to the beam line: $\Lambda_{c}^{+}$ candidates
from $\Xi_{c}$ decays typically have larger values. Fitting this distribution
yields $374\pm 88$ $\Xi_{c}\to\Lambda_{c}^{+}X$ decays in the entire
$\Lambda_{c}^{+}$ sample. These events are reduced by 40% by combining the
$\Lambda_{c}^{+}$ candidate with a $\pi^{-}$ or $\pi^{0}$ candidate and
vetoing those that satisfy
$M(pK^{-}\pi^{+}\pi^{-})-M(pK^{-}\pi^{+})\in[183.4,186.4]$ MeV/$c^{2}$ or
$M(pK^{-}\pi^{+}\pi^{0})-M(pK^{-}\pi^{+})\in[175.3,187.3]$ MeV/$c^{2}$. The
effect of the remaining $\Lambda_{c}^{+}$ candidates is evaluated using MC
simulation; the resulting bias of $+0.34$ fs is subtracted from the fitted
$\Lambda_{c}^{+}$ lifetime and included as a systematic uncertainty (see Table
3). The final fitted sample consists of $116\times 10^{3}$ events with 92.5%
purity. The fit result after the bias correction is
$\tau(\Lambda_{c}^{+})=(203.20\pm 0.89\pm 0.77)$ fs.

Table 3: Systematic uncertainties for the $\Lambda_{c}^{+}$ lifetime measurement [9]. Source | Uncertainty [fs]
---|---
Resolution model | 0.46
$\Xi_{c}$ contamination | 0.34
Non-$\Xi_{c}$ backgrounds | 0.20
Detector alignment | 0.46
Momentum scale | 0.09
Total | 0.77

##  5 $\Omega_{c}^{0}$ lifetime

Measurement of the $\Omega_{c}^{0}$ lifetime [11] follows the same procedure
as described above for the $\Lambda_{c}^{+},D^{+}_{(s)}$, and $D^{0}$
lifetimes. The reconstructed decay is $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$
followed by $\Omega^{-}\to\Lambda(\to p\pi^{-})K^{-}$. To eliminate
$\Omega_{c}^{0}$ candidates originating from $B\to\Omega_{c}^{0}X$, we require
$p_{\Omega_{c}^{0}}/p_{\rm max}>0.60$, where $p_{\rm max}=\sqrt{(E_{\rm
beam})^{2}-m(\Omega^{-}\pi^{+})^{2}}$, and the momentum $p_{\Omega^{0}_{c}}$
and beam energy $E_{\rm beam}$ are evaluated in the $e^{+}e^{-}$ CM frame. The
final candidate sample is much smaller than those of the other lifetime
measurements: 132 signal candidates with 67% purity – see Fig. 3 (left). The
final result is $\tau(\Omega_{c}^{0})=(243\pm 48\pm 11)$ fs, where the first
uncertainty is statistical and the second is systematic. The systematic
uncertainty is dominated by the resolution function ($\pm 6.2$ fs) and
background modeling ($\pm 8.3$) fs. The fitted decay time distribution is
shown in Fig. 3 (right) along with projections of the fit result. The measured
lifetime is significantly longer than the lifetime measured by Fermilab E687
[12], CERN WA89 [13], and FOCUS [14], but it agrees with more recent
measurements by LHCb [15]. This lifetime is unexpectedly longer than that of
lighter charmed baryons, i.e., the hierarchy of charm baryon lifetimes becomes
$\tau(\Xi_{c}^{+})>\tau(\Omega_{c}^{0})>\tau(\Lambda_{c}^{+})>\tau(\Xi_{c}^{0})$.

.

Figure 3: Invariant mass distribution for
$\Omega_{c}^{0}\to\Omega^{-}(\to\Lambda K^{-})\pi^{+}$ candidates (left);
decay time distribution for $\Omega_{c}^{0}$ candidates in the signal region
$M(\Omega^{-}\pi^{+})\in[2.68,2.71]$ GeV/$c^{2}$ (right top); decay time
distribution for $\Omega_{c}^{0}$ candidates in the sideband
$M(\Omega^{-}\pi^{+})\in[2.55,2.65]\cup[2.75,2.85]$ GeV/$c^{2}$ (right
bottom). The latter events are fitted to obtain the decay time PDF used for
background in the signal region [11].

##  6 Summary

Except for the $\Omega_{c}^{0}$ charm baryon, the lifetime measurements
presented here are the world’s most precise. For the $\Omega_{c}^{0}$, the
Belle II measurement confirms the unexpectedly long lifetime measured by LHCb
[15]. All measurements are compared to theory predictions [2, 16] in Table 4.
Currently, the theory uncertainties are large and preclude a precision test of
QCD. While all measurements agree with theory predictions within $2\sigma$,
the measured values of $\tau(D^{0})$, $\tau(D^{+})$, and
$\tau(\Lambda_{c}^{+})$ differ from the predictions by more than $1\sigma$.
Such differences can help discriminate among different calculational
approaches and ultimately improve our understanding of QCD.

Table 4: Comparison between measurements and HQE theory predictions for charm lifetimes. The limits listed correspond to 90% C.L. The HQE calculations for $\tau(D_{s}^{+})$ do not include the partial width for $D_{s}^{+}\to\tau^{+}\nu$, as $m_{\tau}>m_{c}$. | Belle II | King et al. [2] | Gratrex et al. [16]
---|---|---|---
|  |  | (MSR scheme)
$\tau(D^{0})$ [fs] |  $410.5\pm 1.1\pm 0.8$ [8] | $629\,^{+296}_{-167}$ | $595\,^{+344}_{-166}$
$\tau(D^{+})$ [fs] |  $1030.4\pm 4.7\pm 3.1$ [8] | $>1193$ | $>1690$
$\tau(D^{+}_{s})$ [fs] |  $499.5\pm 1.7\pm 0.9$ [6] | $637\,^{+381}_{-190}$ | $599\,^{+459}_{-180}$
$\tau(D^{+})/\tau(D^{0}))$ | $2.510\pm 0.016$ | $2.80\,^{+0.86}_{-0.90}$ | $2.89\,^{+0.78}_{-0.85}$
$\tau(D^{+}_{s})/\tau(D^{0})$ | $1.217\pm 0.006$ | $1.01\pm 0.15$ | $1.00\,^{+0.23}_{-0.21}$
$\tau(\Lambda_{c}^{+})$ [fs] |  $203.20\pm 0.89\pm 0.77$ [9] |  | $312\,^{+128}_{-96}$
$\tau(\Omega_{c}^{0})$ [fs] |  $243\pm 48\pm 11$ [11] |  | $237\,^{+111}_{-75}$
$\tau(\Omega_{c}^{0})/\tau(\Lambda_{c}^{+})$ | $1.20\pm 0.24$ |  | $0.83\,^{+0.30}_{-0.18}$

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