# Deformation-induced CPT violation
in entangled pairs of neutral kaons

Andrea Bevilacqua<EMAIL_ADDRESS>National Centre for Nuclear
Research, ul. Pasteura 7, 02-093 Warsaw, Poland  Jerzy Kowalski-Glikman
<EMAIL_ADDRESS>University of Wrocław, Faculty of Physics
and Astronomy, pl. M. Borna 9, 50-204 Wrocław, Poland  National Centre for
Nuclear Research, ul. Pasteura 7, 02-093 Warsaw, Poland  Wojciech Wiślicki
<EMAIL_ADDRESS>corresponding author National Centre for
Nuclear Research, ul. Pasteura 7, 02-093 Warsaw, Poland

###### Abstract

In this paper we consider description of kaon – anti-kaon interference in the
context of a theory with deformed $\cal CPT$ symmetry. In the case of such
theoretical models, deviations from the standard $\cal CPT$ invariance is
related to the momentum carried by the particles; in particular the rest
masses of particles and antiparticles are equal. We find that the decay
intensity of kaon – anti-kaon pair has three contributing terms: the correct-
parity, the wrong-parity, and the interference between them, all of which are
affected by deformation. Using the fact that the presence of such terms were
not observed we estimate the magnitude of deformation parameter $\kappa\gtrsim
10^{18}$ GeV, very close to the expected Planck mass scale. This raises hopes
that the effect, if exists, could be detected in a near future accelerator
experiments. An uncertainty of the energy measurement is crucial for accuracy
of these predictions.

##  I Introduction

It was realized some time ago Amelino-Camelia:1999hpv  that, contrary to what
was believed at the time, quantum gravity phenomenology is not only feasible,
but that some effects of quantum gravitational origin can be observed using
current or near-future technologies. Today, with the advent of multi-messenger
astronomy, new observational opportunities using astrophysical sources of high
energy particles are opening up (see Amelino-Camelia:2008aez , Addazi:2021xuf
, AlvesBatista:2023wqm  for recent reviews). Unfortunately, there are still
obstacles to the implementation of this research program in the astrophysical
realm, such as low event statistics and poor understanding of the
astrophysical sources and the medium through which the cosmic messengers
travel. It is therefore tempting to look for opportunities to observe quantum
gravity-induced effects in the realm of better controlled and offering much
better statistics terrestrial high-energy experiments.

One of the most promising theoretical frameworks predicting possible
observable effects at energies much lower than the characteristic scale of
quantum gravity, the Planck energy $\kappa\simeq 10^{19}$ GeV, is the quantum
gravity induced Lorentz Invariance Violation (LIV)111This general term
describes both the quantum gravity induced manifest (spontaneous) Lorentz
invariance breaking, as in the Kostelecky model Kostelecky:1988zi , and
theories, in which Lorentz invariance is deformed, like in Doubly Special
Relativity Amelino-Camelia:2000stu  or Relative Locality Amelino-
Camelia:2011lvm . . For example, a large class of LIV models predict the
energy dependence of the velocity of massless particles, an effect which might
be observable in principle Amelino-Camelia:1997ieq , Addazi:2021xuf ,
AlvesBatista:2023wqm .

Another potentially observable effect arising as a consequence of quantum
gravity might be possible deviations from $\cal CPT$ symmetry. According to
Greenberg theorem Greenberg:2002uu  violation of $\cal CPT$ symmetry is a
consequence of Lorentz symmetry breaking and vice versa. Specifically, models
of violation of the $\cal CPT$ and Lorentz symmetry invariance were proposed
in the context of extensions of the Standard Model with electroweak
interactions coupled to gravity kostelecky_1 , or models of gravity with
decoherence bernabeu_1  at energy scale of the Planck mass $m_{P}\simeq
10^{19}$ GeV.

In a recent development, deformations of $\cal CPT$ symmetry have been studied
in the context of $\kappa$ deformations Arzano:2016egk , Arzano:2019toz ,
Arzano:2020rzu , Arzano:2020jro , bevilacqua_1 . In this case we have to do
with modification of both continuous (Poincaré) and discrete symmetries, but
all of the symmetries are still present and are not violated. The distinctive
feature of the models with deformed (as opposite to violated) $\cal CPT$
symmetry is that the deviations from the standard $\cal CPT$-invariance are
momentum dependent. In particular, the rest masses of particles and
antiparticles are equal. As a consequence the decay rates of particles and
antiparticles at rest are equal, but they start differ for the moving ones
Arzano:2019toz , Lobo:2020qoa . Although the size of this mismatch is tiny at
energies available so far in accelerators, and is still beyond reach of the
accuracy of detectors built at present, its quadratic dependence on particle
momenta, as well as rapidly developing acceleration and detection
technologies, motivate theoretical efforts in this direction.

A natural experimental framework where such a hypothesis can be examined is
provided by any particle-antiparticle system with well-controlled momenta of
both particles. When a particle-antiparticle pair is produced in an unstable
resonance decay, their total linear momentum vanishes in the resonance rest
frame. The decay products are entangled, and quantum interference provides a
sensitive tool to compare lifetimes by measuring their decay and interference
spectrum in their relative decay times. With this possibility in mind, the
authors of bernabeu_2  proposed a parametrization of the entangled state of
two neutral kaons in the case when $\cal CPT$-symmetry is violated. In the
present paper we reconsider the production of an entangled pair of neutral
kaons in the context of $\kappa$-$\cal CPT$ deformation proposed in
Arzano:2019toz , Arzano:2020jro , bevilacqua_1 .

In the following Sect. II we will recall briefly the theory of entangled pairs
of neutral kaons and its modification proposed in bernabeu_2 . Then in Sect.
III we discuss how the behavior of kaons is modified in deformed theory.
Finally, Sect. IV is devoted to a discussion of bounds on the deformation
parameter $\kappa$ that arise if no deformation effects are observed.

##  II Entangled pairs of neutral kaons

If one considers the $K^{0}\overline{K}^{0}$ pair in the non-deformed context,
created in a decay of the $\Phi^{0}(1020)$ resonance with $J^{PC}=1^{--}$ and
flying apart with momenta $\pm\mathbf{p}$ in the centre-of-mass frame of the
whole system, requirements of the Bose-Einstein statistics and conservation of
the total angular momentum provide their state to be

$\displaystyle|\psi\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|K^{0}(\mathbf{p})\rangle|\overline{K}^{0}(-\mathbf{p})\rangle-|\overline{K}^{0}(\mathbf{p})\rangle|K^{0}(-\mathbf{p})\rangle\right).$
(1)

Expressing the state $|\psi\rangle$ in the basis of the neutral kaon mass
eigenstates $|K_{S,L}\rangle$

$\displaystyle|K_{S,L}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2(1+|\varepsilon\pm\delta|^{2})}}\left[(1+\varepsilon\pm\delta)\big{|}K^{0}\rangle\pm(1-\varepsilon\mp\delta)|\overline{K}^{0}\rangle\right]$
(2)

one makes use of two experimentally measurable complex parameters quantifying
the $\cal CP$ and $\cal CPT$ violation222We use the notation of Ref. pdg . In
Eq. (2) the upper signs refer to $K_{S}$ and the lower ones to $K_{L}$. The
experimental value of $|\varepsilon|$ is of order $2\times 10^{-3}$ and
significantly differs from zero, whereas the value of $|\delta|$ is of order
$10^{-4}$ and is consistent with zero within one standard deviation pdg . We
neglect $\delta$ in further calculations. As shown in Sect. IV, corrections
due to a possible slight difference of coefficients multiplying
$|K_{S}\rangle$ and $|K_{L}\rangle$ in Eq. (2) with nonzero $\delta$ can
marginally affect only the non-deformed part of the decay intensity, and we
neglect it in what follows.

In the basis (2) one gets an expression for $|\psi\rangle$

$\displaystyle|\psi\rangle$ $\displaystyle=$ $\displaystyle
N\left(|K_{L}(\mathbf{p})\rangle|K_{S}(-\mathbf{p})\rangle-|K_{S}(\mathbf{p})\rangle|K_{L}(-\mathbf{p})\rangle\right),$
(3)

where the normalization factor $N$ depends on $\varepsilon$ and $\delta$.

The decay rate into final states $|f_{1(2)}\rangle$ at times $t_{1(2)}$ is
equal to the squared transition amplitude of $|\psi\rangle$ to the final state
of both kaons

$\displaystyle I(f_{1},t_{1};f_{2},t_{2})=|\langle
f_{1},t_{1};f_{2},t_{2}|H|\psi\rangle|^{2}$

Hereon we consider the case of identical final states $|\pi^{+}\pi^{-}\rangle$
for the $K_{S}$ and $K_{L}$ decays, which is advantageous for a possible
detection of novel, $\cal CPT$-violating effects, due to the $\cal
CP$-suppression of the $K_{L}\rightarrow\pi^{+}\pi^{-}$ decays. Since we
consider only decays to the specific final state $\pi^{+}\pi^{-}$, the decay
widths $\Gamma_{S,L}$ refer to decays of $K_{S,L}$ to this final state.

Writing the kaons states in terms of plane waves, after changing variables
into $T=t_{1}+t_{2}$ and $\Delta t=|t_{1}-t_{2}|$, and integrating over T from
$\Delta t$ to infinity, one gets

$\displaystyle I(\Delta
t)=\frac{4|f_{L}|^{2}|f_{S}|^{2}}{\overline{\Gamma}}\left[e^{-\Gamma_{L}\Delta
t}+e^{-\Gamma_{S}\Delta t}-2e^{-\overline{\Gamma}\Delta t}\cos(\Delta E\Delta
t)\right],$ (4)

where

$\displaystyle f_{S,L}$ $\displaystyle=$
$\displaystyle\langle\pi^{+}\pi^{-}|H|K_{S,L}\rangle$ (5)

and $\Delta E=E_{L}-E_{S}$,
$\overline{\Gamma}=\frac{1}{2}(\Gamma_{L}+\Gamma_{S})$. The identity of the
final states of both kaon decays ensures that there is no additional phase in
the oscillation term in Eq. (4).

The ratio of the amplitudes in Eq. (5) is usually denoted by
$\eta_{+-}=f_{L}/f_{S}=|\eta_{+-}|e^{i\varphi}$ and is equal to the sum of two
parameters quantifying two mechanisms of $\cal CP$ violation. The indirect,
measured by $\varepsilon$, is due to mixing of $K^{0}$ and $\overline{K}^{0}$,
and the direct one, measured by $\varepsilon^{\prime}$, is due to decays of
$K_{S}$ and $K_{L}$ to states with $CP=-1$ and $CP=+1$, respectively, such
that $\eta_{+-}=\varepsilon+\varepsilon^{\prime}$. Direct $\cal CP$ violation
is much weaker, such that $\varepsilon^{\prime}$ is of order $10^{-6}$, but is
significantly different than zero pdg . In further considerations we omit
$\varepsilon^{\prime}$, for simplicity.

###  II.1 Entangled pairs of neutral kaons in $\omega$-model

As it was argued by authors of Ref. bernabeu_2 , Lorentz and $\cal CPT$
invariance violation symmetry of quantum gravitational origin may lead to
correction to (1). Such correction is parameterized by a complex parameter
$\omega$ and the modified state reads

$\displaystyle|\psi\rangle_{\omega}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|K^{0}(\mathbf{p})\rangle|\overline{K}^{0}(-\mathbf{p})\rangle-|\overline{K}^{0}(\mathbf{p})\rangle|K^{0}(-\mathbf{p})\rangle\right)$
(6) $\displaystyle+$
$\displaystyle\frac{\omega}{\sqrt{2}}\left(|K^{0}(\mathbf{p})\rangle|\overline{K}^{0}(-\mathbf{p})\rangle+|\overline{K}^{0}(\mathbf{p})\rangle|K^{0}(-\mathbf{p})\rangle\right).$

The complex parameter $\omega$ accounts for a possible $\cal CPT$ violation
and its magnitude may be of the order
$|\omega|\simeq\sqrt{(m_{K}^{2}/\kappa)/\Delta\Gamma}$, where
$\Delta\Gamma=\Gamma_{S}-\Gamma_{L}$ and $\kappa$ is the LIV parameter,
presumably of order of Planck mass. Recent experimental value of $|\omega|$ is
of order $10^{-4}$ and is consistent with zero kloe2 .

Expressing the state $|\psi\rangle$ in the basis of the neutral kaon mass
eigenstates $|K_{S}\rangle$ and $|K_{L}\rangle$ one gets

$\displaystyle|\psi\rangle_{\omega}$ $\displaystyle=$ $\displaystyle
N\Bigg{[}\left(|K_{L}(\mathbf{p})\rangle|K_{S}(-\mathbf{p})\rangle-|K_{S}(\mathbf{p})\rangle|K_{L}(-\mathbf{p})\rangle\right)$
(7) $\displaystyle+$
$\displaystyle\omega\left(|K_{S}(\mathbf{p})\rangle|K_{S}(-\mathbf{p})\rangle-|K_{L}(\mathbf{p})\rangle|K_{L}(-\mathbf{p})\rangle\right)\Bigg{]},$

Using the same procedure for calculating decay intensity as in the standard
case, modified intensity $I_{\omega}(\Delta t)$. After developing our
formalism, in Sec. IV.B we give a comparison of the two-kaon decay intensity
of Ref. bernabeu_2  and the one derived in the next Section below.

##  III Entangled pairs of neutral kaons in $\kappa$-deformed field theory

In this section we will consider theories with deformed $\cal CPT$ symmetry,
with deviations from the standard $\cal CPT$-invariance arising in the case of
moving particles. Specifically, in what follows we will use the theory with
$\kappa$-deformation derived and discussed in details in Arzano:2019toz  and
bevilacqua_1 , but the formalism we are using can be easily extended to more
general class of deformed theories, in which one has to do with non-linearly
modified momentum composition and the law of momentum conservation.

###  III.1 Formalism

A two-particle state consisting of a particle and antiparticle can be
constructed from the first principles in the framework of the
$\kappa$-deformed theory of scalar fields bevilacqua_1 .

If the particle’s four-momentum amounts to $p=(E,\mathbf{p})$ then for the
antiparticle it is given by its antipodal map
$-S(p)=(-S_{0}(p),-\mathbf{S}(p))$, where

$\displaystyle S_{0}(p)$ $\displaystyle=$ $\displaystyle-
p_{4}+\frac{\kappa^{2}}{E+p_{4}}$ $\displaystyle\simeq$
$\displaystyle-E+\frac{\mathbf{p}^{\,2}}{\kappa}+{\cal O}(1/\kappa^{2})$
$\displaystyle\mathbf{S}(p)$ $\displaystyle=$
$\displaystyle-\frac{\kappa\mathbf{p}}{E+p_{4}}$ (8) $\displaystyle\simeq$
$\displaystyle-\mathbf{p}+\frac{E\mathbf{p}}{\kappa}+{\cal O}(1/\kappa^{2}),$

where $p_{4}=\sqrt{\kappa^{2}+m^{2}}$.

Now let us consider two kaons produced in decay of a $\phi_{0}(1020)$
resonance, in its centre-of-mass frame. In the deformed theory instead of the
state $|\psi\rangle$ (1) we have its deformed counterpart

$\displaystyle|\psi\rangle_{\kappa}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|K^{0}(\mathbf{p})\rangle|\overline{K}^{0}(\mathbf{S}(p))\rangle-|\overline{K}^{0}(-\mathbf{S}(p))\rangle|K^{0}(-\mathbf{p})\rangle\right)$
(9) $\displaystyle=$ $\displaystyle
2^{-3/2}\Big{(}|K_{L}(-\mathbf{S}(p))\rangle|K_{L}(-\mathbf{p})\rangle-|K_{L}(\mathbf{p})\rangle|K_{L}(\mathbf{S}(p))\rangle$
$\displaystyle+|K_{L}(-\mathbf{S}(p))\rangle|K_{S}(-\mathbf{p})\rangle-|K_{S}(\mathbf{p})\rangle|K_{L}(\mathbf{S}(p))\rangle$
$\displaystyle-|K_{S}(-\mathbf{S}(p))\rangle
K_{L}(-\mathbf{p})\rangle+|K_{L}(\mathbf{p})\rangle|K_{S}(\mathbf{S}(p))\rangle$
$\displaystyle-|K_{S}(-\mathbf{S}(p))\rangle|K_{S}(-\mathbf{p})\rangle+|K_{S}(\mathbf{p})\rangle|K_{S}(\mathbf{S}(p)\Big{)}.$

Notice that, like in the non-deformed case, the total spatial momentum of the
two particles is zero since
$\mathbf{p}\oplus\mathbf{S}({p})=\mathbf{S}({p})\oplus\mathbf{p}=0$. The
energy is however always positive, so that particles with spatial momentum
$\mathbf{p}$ or $-\mathbf{p}$ have energy $E$, while particles with momentum
$\mathbf{S}({p})$ or $-\mathbf{S}(p)$ have energy $-S_{0}(p)$ (which is
positive on-shell). Notice also that in the mass-eigenstate representation the
terms proportional to $K_{L}\,K_{L}$ and $K_{S}\,K_{S}$ appear in the deformed
state (9).

We now boost the two-kaon state (9) with respect to the centre-of-mass frame
by large boost with parameter $\gamma\gg 1$. We boost the two-particle state
as described in Arzano:2022ewc , bevilacqua_2 , considering only terms up to
first order in the $1/\kappa$ expansion. For simplicity we consider the limit
of kaons produced at rest in the centre-of-mass frame. This is justified since
$\frac{|\mathbf{p}|}{m}\sim 0.01$. After the boost, both kaons are travelling
in the same direction (although with different modulus of the spatial
momentum).

In what follows all energies, widths and decay times refer to this boosted
frame. Explicit formulas for boosted quantities are given at the end of this
subsection. The boosted state can be written as

$\displaystyle|\psi\rangle_{\kappa}$ $\displaystyle=$ $\displaystyle
2^{-3/2}\Big{(}|K_{L}(-\mathbf{S}(p))\rangle|K_{L}(\widehat{\mathbf{p}})\rangle-|K_{L}(\mathbf{p})\rangle|K_{L}(-\widehat{\mathbf{S}(p)})\rangle$
(10)
$\displaystyle+|K_{L}(-\mathbf{S}(p))\rangle|K_{S}(\widehat{\mathbf{p}})\rangle-|K_{S}(\mathbf{p})\rangle|K_{L}(-\widehat{\mathbf{S}(p)})\rangle$
$\displaystyle-|K_{S}(-\mathbf{S}(p))\rangle
K_{L}(\widehat{\mathbf{p}})\rangle+|K_{L}(\mathbf{p})\rangle|K_{S}(-\widehat{\mathbf{S}(p)})\rangle$
$\displaystyle-|K_{S}(-\mathbf{S}(p))\rangle|K_{S}(\widehat{\mathbf{p}})\rangle+|K_{S}(\mathbf{p})\rangle|K_{S}(-\widehat{\mathbf{S}(p)}\Big{)}.$

where the $\,\,\widehat{}\,\,$ over the spatial momenta on the rightmost
vector in each tensor product is just a notation to remind that the deformed
boost has been used Arzano:2022ewc , bevilacqua_2 . The spatial momenta in the
arguments are the physical momenta.

Let us now consider the decays of kaons to the final states $|f_{1}\rangle$
and $|f_{2}\rangle$ in times $t_{1}$ and $t_{2}$, respectively, and write down
all partial amplitudes contributing to the overall decay amplitude $\langle
f_{1},t_{1};f_{2},t_{2}|H|\psi\rangle_{\kappa}$

$\displaystyle\langle f_{1},t_{1};f_{2},t_{2}|H|\psi\rangle_{\kappa}$
$\displaystyle=$ $\displaystyle
Q_{1}-Q_{2}+Q_{3}-Q_{4}-Q_{5}+Q_{6}-Q_{7}+Q_{8},$ (11)

where

$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{L}(-\mathbf{S}(p)),K_{L}(\widehat{\mathbf{p}})\rangle=\langle
f_{1}|K_{L}\rangle\langle f_{2}|K_{L}\rangle
e^{-iE_{L}t_{1}-\Gamma_{L}t_{1}+i\zeta t_{1}}e^{-iE_{L}t_{2}-\Gamma_{L}t_{2}}$
$\displaystyle Q_{2}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{L}(\mathbf{p}),K_{L}(-\widehat{\mathbf{S}(p)})\rangle=\langle
f_{1}|K_{L}\rangle\langle f_{2}|K_{L}\rangle
e^{-iE_{L}t_{1}-\Gamma_{L}t_{1}}e^{-iE_{L}t_{2}-\Gamma_{L}t_{2}+i\zeta t_{2}}$
$\displaystyle Q_{3}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{L}(-\mathbf{S}(p)),K_{S}(\widehat{\mathbf{p}})\rangle=\langle
f_{1}|K_{L}\rangle\langle f_{2}|K_{S}\rangle
e^{-iE_{L}t_{1}-\Gamma_{L}t_{1}+i\zeta t_{1}}e^{-iE_{S}t_{2}-\Gamma_{S}t_{2}}$
$\displaystyle Q_{4}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{S}(\mathbf{p}),K_{L}(-\widehat{\mathbf{S}(p)})\rangle=\langle
f_{1}|K_{S}\rangle\langle f_{2}|K_{L}\rangle
e^{-iE_{S}t_{1}-\Gamma_{S}t_{1}}e^{-iE_{L}t_{2}-\Gamma_{L}t_{2}+i\zeta t_{2}}$
$\displaystyle Q_{5}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{S}(-\mathbf{S}(p)),K_{L}(\widehat{\mathbf{p}})\rangle=\langle
f_{1}|K_{S}\rangle\langle f_{2}|K_{L}\rangle
e^{-iE_{S}t_{1}-\Gamma_{S}t_{1}+i\zeta t_{1}}e^{-iE_{L}t_{2}-\Gamma_{L}t_{2}}$
$\displaystyle Q_{6}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{L}(\mathbf{p}),K_{S}(-\widehat{\mathbf{S}(p)})\rangle=\langle
f_{1}|K_{L}\rangle\langle f_{2}|K_{S}\rangle
e^{-iE_{L}t_{1}-\Gamma_{L}t_{1}}e^{-iE_{S}t_{2}-\Gamma_{S}t_{2}+i\zeta t_{2}}$
$\displaystyle Q_{7}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{S}(-\mathbf{S}(p)),K_{S}(\widehat{\mathbf{p}})\rangle=\langle
f_{1}|K_{S}\rangle\langle f_{2}|K_{S}\rangle
e^{-iE_{S}t_{1}-\Gamma_{S}t_{1}+i\zeta t_{1}}e^{-iE_{S}t_{2}-\Gamma_{S}t_{2}}$
$\displaystyle Q_{8}$ $\displaystyle=$ $\displaystyle\langle
f_{1},t_{1};f_{2},t_{2}|H|K_{S}(\mathbf{p}),K_{S}(-\widehat{\mathbf{S}(p)})\rangle=\langle
f_{1}|K_{S}\rangle\langle f_{2}|K_{S}\rangle
e^{-iE_{S}t_{1}-\Gamma_{S}t_{1}}e^{-iE_{S}t_{2}-\Gamma_{S}t_{2}+i\zeta
t_{2}}.$ (12)

In Eq. (12), $E_{L(S)}$ stand for boosted energies and $\Gamma_{L(S)}$ for
decay widths of the $K_{L(S)}$, respectively. The quantity
$\zeta=\mathbf{p}^{\,2}/\kappa$ accounts for the order $1/\kappa$ correction
to energy due to deformation. Boosting a two-particle system in the deformed
case is, in general, quite nontrivial Arzano:2022ewc , bevilacqua_2 , so that
different $\zeta_{i}$ would be needed, one for each $Q_{i}$. In this case,
however, we are only interested in the first order contribution in $1/\kappa$,
and one can show that the effect of deformation can be fully captured by the
single parameter $\zeta$. Indeed, the different $\zeta_{i}$ differ by
multiples of $m_{L}\Delta m/\kappa$, which can be safely ignored.

The two-kaon decay amplitude (11) contains terms with the correct $\cal CPT$
(we denote it by $C$) and wrong $\cal CPT$ parity (denoted by $W$). The first
consists of mixed terms $|K_{S}\rangle|K_{L}\rangle$ and
$|K_{L}\rangle|K_{S}\rangle$, and the second one of terms
$|K_{L}\rangle|K_{L}\rangle$ and $|K_{S}\rangle|K_{S}\rangle$:

$\displaystyle C$ $\displaystyle=$ $\displaystyle Q_{3}-Q_{4}-Q_{5}+Q_{6}$
$\displaystyle W$ $\displaystyle=$ $\displaystyle Q_{1}-Q_{2}-Q_{7}+Q_{8}$
(13)

The decay intensity $I=|\langle
f_{1},t_{1};f_{2},t_{2}|H|\psi\rangle_{\kappa}|^{2}$ can be then neatly
expressed as intensities of the correct-$\cal CPT$, wrong-$\cal CPT$, and
interference between the two. After changing variables into $T=t_{1}+t_{2}$
and $\Delta t=|t_{1}-t_{2}|$, and integrating over $T$ from $\Delta t$ to
infinity, one gets

$\displaystyle I_{\kappa}(\Delta t)=|C(\Delta t)|^{2}+|W(\Delta
t)|^{2}+2\Re[C(\Delta t)W(\Delta t)^{\ast}].$ (14)

In our further calculations we consider the case of the same final states
$|\pi^{+}\pi^{-}\rangle$ of the $K_{S}$ and $K_{L}$ decays. Formulas for other
combinations of final states can be derived in a similar way.

Explicit expressions for all terms of Eq. (14) read:

$\displaystyle|C(\Delta t)|^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\overline{\Gamma}}|f_{L}|^{2}|f_{S}|^{2}\big{(}1+\cos(\zeta\Delta
t)\big{)}\left[e^{-\Gamma_{L}\Delta t}+e^{-\Gamma_{S}\Delta
t}-2e^{-\overline{\Gamma}\Delta t}\cos(\Delta E\Delta t)\right],$ (15)
$\displaystyle|W(\Delta t)|^{2}$ $\displaystyle=$ $\displaystyle
2\big{(}1-\cos(\zeta\Delta
t)\big{)}\Bigg{[}\frac{|f_{L}|^{4}}{\Gamma_{L}}e^{-\Gamma_{L}\Delta
t}+\frac{|f_{S}|^{4}}{\Gamma_{S}}e^{-\Gamma_{S}\Delta t}$ (16)
$\displaystyle-$ $\displaystyle
2\Re\,(f_{L}^{2}\bar{f}_{S}^{2})\frac{e^{-\overline{\Gamma}\Delta
t}}{\overline{\Gamma}^{2}+(\Delta E)^{2}}\left(\Delta E\sin(\Delta E\Delta
t)-\overline{\Gamma}\cos(\Delta E\Delta t)\right)\Bigg{]},$

and

$\displaystyle 2\Re\,[C(\Delta t)W^{\ast}(\Delta t)]=-16\sin(\zeta\Delta
t)\Bigg{[}|f_{L}|^{2}\Im\,(\bar{f}_{L}f_{S})\frac{\Delta
E}{(\Gamma_{L}+\overline{\Gamma})^{2}+(\Delta E)^{2}}e^{-\Gamma_{L}\Delta t}$
$\displaystyle+|f_{S}|^{2}\Im\,(f_{L}\bar{f}_{S})\frac{\Delta
E}{(\Gamma_{S}+\overline{\Gamma})^{2}+(\Delta E)^{2}}e^{-\Gamma_{S}\Delta t}$
$\displaystyle+|f_{L}|^{2}\Im\,(\bar{f}_{L}f_{S})\frac{e^{-\overline{\Gamma}\Delta
t}}{(\Gamma_{L}+\overline{\Gamma})^{2}+(\Delta
E)^{2}}\left[(\Gamma_{L}+\overline{\Gamma})\sin(\Delta E\Delta t)+\Delta
E\cos(\Delta E\Delta t)\right]$
$\displaystyle+|f_{S}|^{2}\Im\,(f_{L}\bar{f}_{S})\frac{e^{-\overline{\Gamma}\Delta
t}}{(\Gamma_{S}+\overline{\Gamma})^{2}+(\Delta
E)^{2}}\left[(\Gamma_{S}+\overline{\Gamma})\sin(\Delta E\Delta t)+\Delta
E\cos(\Delta E\Delta t)\right]\Bigg{]},$ (17)

Since the decay $K_{L}\rightarrow\pi^{+}\pi^{-}$ violates $\cal CP$, its
amplitude is suppressed with respect to $K_{S}\rightarrow\pi^{+}\pi^{-}$ by a
factor $|\varepsilon|=|f_{L}/f_{S}|\simeq 2\times 10^{-3}$.

###  III.2 Boosted quantities

From the phenomenological point of view, an important feature of the
$\kappa$-deformation is the dependence of the deformation-induced oscillation
frequency $\zeta$ on energy. Since $\zeta=\mathbf{p}^{\,2}/\kappa$, any
effects of deformation vanish in the particle’s rest frame. Let us thus write
all the boosted quantities explicitly in terms of $\gamma$. As shown in Ref.
bevilacqua_2 , for strong boosts and neglecting tiny effects of non-zero
transverse momenta, the arguments of the oscillating terms in Eqs. (15-17)
transform as

$\displaystyle(\Delta E\Delta t)^{\prime}$ $\displaystyle=$
$\displaystyle\gamma^{2}\,\Delta m\Delta t$ $\displaystyle(\zeta\Delta
t)^{\prime}$ $\displaystyle=$ $\displaystyle\gamma^{3}m^{2}\Delta t/\kappa.$
(18)

Decay widths contract as $\Gamma^{\prime}=\Gamma/\gamma$, decay times dilate
as $t^{\prime}=\gamma t$ and energies increase as $E^{\prime}=\gamma m$.
Therefore the arguments $\Gamma\Delta t$ of exponentials in Eqs. (15-17) are
Lorentz-invariant.

##  IV Observability of deformations

Having obtained explicit expressions for deformed decay intensity, let us now
turn to the question as to if the effect of deformation could be observable.

###  IV.1 Dominating terms and order of magnitude considerations

Let us start with Eqs. (15-17), which can be rewritten as follows:

$\displaystyle|C(\Delta t)|^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\overline{\Gamma}}|f_{S}|^{4}|\eta_{+-}|^{2}\big{(}1+\cos(\zeta\Delta
t)\big{)}\left[e^{-\Gamma_{L}\Delta t}+e^{-\Gamma_{S}\Delta
t}-2e^{-\overline{\Gamma}\Delta t}\cos(\Delta E\Delta t)\right],$ (19)
$\displaystyle|W(\Delta t)|^{2}$ $\displaystyle=$ $\displaystyle
2|f_{S}|^{4}\big{(}1-\cos(\zeta\Delta
t)\big{)}\Bigg{[}\frac{|\eta_{+-}|^{4}}{\Gamma_{L}}e^{-\Gamma_{L}\Delta
t}+\frac{1}{\Gamma_{S}}e^{-\Gamma_{S}\Delta t}$ (20) $\displaystyle-$
$\displaystyle 2|\eta_{+-}|^{2}\frac{e^{-\overline{\Gamma}\Delta
t}}{\overline{\Gamma}^{2}+(\Delta E)^{2}}\big{(}\Delta E\sin(\Delta E\Delta
t)-\overline{\Gamma}\cos(\Delta E\Delta t)\big{)}\Bigg{]},$

and

$\displaystyle 2\Re\,[C(\Delta t)W^{\ast}(\Delta t)]$ $\displaystyle=$
$\displaystyle-16|f_{S}|^{4}\sin(\zeta\Delta
t)\Bigg{[}|\eta_{+-}|^{3}\frac{\Delta
E}{(\Gamma_{L}+\overline{\Gamma})^{2}+(\Delta E)^{2}}e^{-\Gamma_{L}\Delta t}$
$\displaystyle+$ $\displaystyle|\eta_{+-}|\frac{\Delta
E}{(\Gamma_{S}+\overline{\Gamma})^{2}+(\Delta E)^{2}}e^{-\Gamma_{S}\Delta t}$
$\displaystyle+$
$\displaystyle|\eta_{+-}|^{3}\frac{e^{-\overline{\Gamma}\Delta
t}}{(\Gamma_{L}+\overline{\Gamma})^{2}+(\Delta
E)^{2}}\left[(\Gamma_{L}+\overline{\Gamma})\sin(\Delta E\Delta
t+\varphi)+\Delta E\cos(\Delta E\Delta t+\varphi)\right]$ $\displaystyle+$
$\displaystyle|\eta_{+-}|\frac{e^{-\overline{\Gamma}\Delta
t}}{(\Gamma_{S}+\overline{\Gamma})^{2}+(\Delta
E)^{2}}\left[(\Gamma_{S}+\overline{\Gamma})\sin(\Delta E\Delta
t+\varphi)+\Delta E\cos(\Delta E\Delta t+\varphi)\right]\Bigg{]}.$ (21)

In the correct-parity term (19) one finds a common factor $|\eta_{+-}|^{2}$
multiplying the whole expression, because in the initial state there is always
one short-lived $K_{S}$ and one long-lived $K_{L}$, both decaying into the
$\pi^{+}\pi^{-}$ final states, where the first is $\cal CP$-allowed and the
second one is $\cal CP$-suppressed. The wrong-parity term (20) consists of
either two $K_{S}$, whose decays are $\cal CP$-allowed, or two $K_{L}$
decaying in the $\cal CP$-suppressed mode. The latter ones thus incorporate
the $|\eta_{+-}|^{2}$ factors in the amplitudes ($|\eta_{+-}|^{4}$ in decay
widths) and the former ones are entirely favoured and thus have no
$\eta_{+-}$-dependent coefficients. The interference term (21) contains
coefficients proportional either to $|\eta_{+-}|$ for interference of
$|K_{L}\rangle|K_{L}\rangle$ with $|K_{L}\rangle|K_{S}\rangle$ states, or
$|\eta_{+-}|^{3}$ for interference of $|K_{S}\rangle|K_{S}\rangle$ with
$|K_{L}\rangle|K_{S}\rangle$.

Inspecting expressions (19 \- 21) one can see that in the kaon rest frames,
where $\zeta=0$, the terms of wrong parity $|W(\Delta t)|^{2}$, and
interference between the correct and wrong parity terms $\Re[C(\Delta
t)W^{\ast}(\Delta t)]$ disappear and reproduce the undeformed case.

Further simplification, suitable for subsequent quantitative assessments, can
be done by neglecting terms proportional to $1/\gamma$ and keeping only those
of order $\sim\gamma$. One thus gets an approximation valid for large boosts
$\gamma$

$\displaystyle|C(\Delta t)|^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{|\eta_{+-}|^{2}|f_{S}|^{4}}{\overline{\Gamma}}2\big{(}1+\cos(\zeta\Delta
t)\big{)}\left[e^{-\Gamma_{L}\Delta t}+e^{-\Gamma_{S}\Delta
t}-2e^{-\overline{\Gamma}\Delta t}\cos(\Delta E\Delta t)\right],$ (22)
$\displaystyle|W(\Delta t)|^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{|\eta_{+-}|^{2}|f_{S}|^{4}}{\overline{\Gamma}}\big{(}1-\cos(\zeta\Delta
t)\big{)}\left[|\eta_{+-}|^{2}\,e^{-\Gamma_{L}\Delta
t}+\frac{1}{|\eta_{+-}|^{2}}\,e^{-\Gamma_{S}\Delta t}\right].$ (23)

The interference term is negligible in this approximation.

In Eqs. (22,23) a common factor $|\eta_{+-}|^{2}|f_{S}|^{4}/\overline{\Gamma}$
is put in front of the time-dependent expressions, in order to make both terms
directly comparable. It can be seen that $|W(\Delta t)|^{2}$ is strongly
dominated by term proportional to $e^{-\Gamma_{S}\Delta t}$, for reasons
explained above, thus making it orders of magnitude larger than this for
$|C(\Delta t)|^{2}$.

###  IV.2 Relation to the $\omega$-model

The decay intensity in Eq. (8) of Ref. bernabeu_2  contains three terms

$\displaystyle I_{\omega}(\Delta t)=\left|\left\langle\pi^{+}\pi^{-}\mid
K_{S}\right\rangle\right|^{4}|C|^{2}\left|\tilde{\eta}_{+-}\right|^{2}\left[I_{1}+I_{2}+I_{12}\right],$
(24)

where

$\displaystyle I_{1}(\Delta t)$ $\displaystyle=\frac{e^{-\Gamma_{S}\Delta
t}+e^{-\Gamma_{L}\Delta t}-2e^{-\left(\Gamma_{S}+\Gamma_{L}\right)\Delta
t/2}\cos(\Delta M\Delta t)}{\Gamma_{L}+\Gamma_{S}}$ $\displaystyle
I_{2}(\Delta t)$
$\displaystyle=\frac{|\omega|^{2}}{\left|\tilde{\eta}_{+-}\right|^{2}}\frac{e^{-\Gamma_{S}\Delta
t}}{2\Gamma_{S}}$ (25)

and

$\displaystyle I_{12}(\Delta t)$ $\displaystyle=-\frac{4}{4(\Delta
M)^{2}+\left(3\Gamma_{S}+\Gamma_{L}\right)^{2}}\frac{|\omega|}{\left|\tilde{\eta}_{+-}\right|}$
$\displaystyle\times\left\\{2\Delta M\left[e^{-\Gamma_{S}\Delta
t}\sin\left(\phi_{+-}-\Omega\right)-e^{-\left(\Gamma_{S}+\Gamma_{L}\right)\Delta
t/2}\sin\left(\phi_{+-}-\Omega+\Delta M\Delta t\right)\right]\right.$
$\displaystyle\left.-\left(3\Gamma_{S}+\Gamma_{L}\right)\left[e^{-\Gamma_{S}\Delta
t}\cos\left(\phi_{+-}-\Omega\right)-e^{-\left(\Gamma_{S}+\Gamma_{L}\right)\Delta
t/2}\cos\left(\phi_{+-}-\Omega+\Delta M\Delta t\right)\right]\right\\}.$ (26)

In Eqs. (24-IV.2) the original notation is kept where
$\tilde{\eta}_{+-}=1/\eta_{+-}$ and $\Delta M=-\Delta m$, $\Omega$ stands for
the phase of $\omega$, and $\phi_{+-}=-\varphi$ for phase of
$\tilde{\eta}_{+-}$.

In Eq. (IV.2, $I_{1}(\Delta t)$ corresponds to the correct-parity term
$|C(\Delta t)|^{2}$ (19) but does not contain the deforming factor
$(1+\cos(\zeta\Delta t))$ which makes an important difference between these
two models. The $I_{2}(\Delta t)$ consists of a single term proportional to
$e^{-\Gamma_{S}}$ whereas in the wrong-parity term $|W(\Delta t)|^{2}$ (20)
there are two additional terms where the one proportional to $e^{-\Gamma_{L}}$
is strongly suppressed by $|\eta_{+-}|^{2}$ factor. The deforming term
$1-\cos(\zeta\Delta t)$ (20) modifies the expression analogously to
$\omega^{2}$ (IV.2). Finally, the interference $I_{12}(\Delta t)$ (IV.2)
corresponds to $2\Re[C(\Delta t)W^{\ast}(\Delta t)]$ (21). Their structures
are similar but the first depends linearly on $\omega$ and the second one on
$\sin(\zeta\Delta t)$.

Apart from all formal similarities, the fundamental physical difference is
that in the model presented here the $\cal CPT$ violation is present only in
the moving frame and it disappears in the kaon rest frame where
$|\psi\rangle_{\kappa}\rightarrow|\psi(\mathbf{p}=\mathbf{0})\rangle$ for
$\mathbf{p}\rightarrow\mathbf{0}$, cf. Eqs. (1, 9). The $\omega$-model of Ref.
bernabeu_2  is formulated for kaons at rest where the violation of $\cal CPT$
is allowed.

###  IV.3 Numerical estimates of deformation

Deformed oscillations in the decay-intensity spectrum provide new
opportunities to estimate the deformation scale $\kappa$. In particular, both
intensities (22) and (23) hint at what an experimentalist should look at to
find signatures of deformation in the decay spectra of interfering kaons. In
the following, let us discuss two semi-quantitative approaches to the problem,
bearing in mind that complete qualitative analyses can only be done using
simulations for specific experimental setups.

####  IV.3.1 Frequency of the deformed oscillations

The deformation-dependent oscillations, governed by $1\pm\cos(\zeta\Delta t)$
factors in $|C(\Delta t)|^{2}$ and $|W(\Delta t)|^{2}$ terms, have not yet
been attempted to be observed experimentally. We are not aware of any
observations made with boosts high enough to make such effects potentially
visible. Let us formulate the necessary, order-of-magnitude conditions for
such an observation without any sophisticated analysis of the shape of the
decay time intensity.

Considering an experiment with a large boost one may expect to see an effect,
provided that the deformed oscillations with frequency $\zeta$ are not too
slow compared to the undeformed ones with the frequency $\Delta E$. According
to Eq. (18), boosted periods of normal oscillations, amounting to $1/\Delta
E$, extend from $10^{16}$ GeV-1 down to $10^{10}$ GeV-1 for the corresponding
Lorentz $\gamma$ in the range from $10^{2}$ to $10^{5}$. For the same range of
$\gamma$, the boosted deformed oscillation periods $1/\zeta$ extend from
$10^{13}$ GeV-1 down to $10^{10}$ GeV-1. It is thus seen that the two
oscillation periods, $1/\Delta E$ and $1/\zeta$, become comparable only for
large $\gamma\sim 10^{5}$, corresponding to protons of an energy of $100$ TeV,
yet unattainable in accelerators.

The unobservability of deformed oscillations can provide us an upper limit for
$\kappa$. If one requires that $\cos(\zeta\Delta t)$ varies very slowly
compared to $\cos(\Delta E\Delta t)$, which means $\zeta\ll\Delta E$, by using
Eq. (18) one translates this inequality into condition

$\displaystyle\kappa\gg\gamma\times 0.5\times 10^{14}\;\mbox{GeV}.$ (27)

For example, using energy of $10$ TeV one gets a lower limit of at most
$10^{17}$ GeV, still located below the Planck energy scale $10^{19}$ GeV. This
provides us with an exciting possibility to limit $\kappa$ experimentally by
pushing further up future accelerating technologies.

####  IV.3.2 Intensity of the wrong-parity oscillations

Another way of limiting the value of deformation parameter $\kappa$ may be to
exploit strong amplification of the wrong-parity term $|W(\Delta t)|^{2}$ due
to the double $\cal CP$-favoured decay of the $|K_{S}\rangle|K_{S}\rangle$
state where each $K_{S}$ decays into the $\pi^{+}\pi^{-}$ pair. Since
$|\eta_{+-}|^{2}=0.4\times 10^{-5}$, one may expect that unbservability of the
wrong-parity term requires $1-\cos(\zeta\Delta t)$ to be limited from above by
$\sigma_{W}<10^{-6}$, depending on available experimental accuracy. Here we
assume that the suppression factor $\sigma_{W}$ is an order of magnitude
smaller than the amplification $|\eta_{+-}|^{2}$ in the wrong-parity term
(23). Realistic values of $\sigma_{W}$ depend on many factors, such as the
statistical errors and overall shape of the intensity spectrum due to the
apparatus acceptance.

The requirement

$\displaystyle 1-\cos(\zeta\Delta t)<\sigma_{W}$ (28)

translates into a condition

$\displaystyle\kappa>\frac{\gamma^{3}m^{2}\Delta t_{r}}{2\sqrt{\sigma_{W}}},$
(29)

where $t_{r}$ refers to decay time in the kaon rest frame. For $\gamma\sim
10^{3}$, $\sigma_{W}=10^{-6}$ and $\Delta t_{r}=\tau_{S}=1.4\times 10^{14}$
GeV-1, the condition (29) provides with a limitation $\kappa>0.2\times
10^{25}\;\mbox{GeV}$, well above the Planck scale. Interestingly, this limit
may be much tightened from below by using smaller boosts. It can be also
lowered by using smaller decay times, depending on the experimental time
resolution. For currently available time resolutions of $\sigma_{t}=4.5$ fs
lhcb_res  the lower limit on $\kappa$ can be calculated from Eq. (29) by
taking $\gamma\Delta t_{r}=\sigma_{t}$ which gives

$\displaystyle\kappa>0.2\times 10^{18}\;\mbox{GeV}.$ (30)

Strong amplification of $|W(\Delta t)|^{2}$ (23) makes this method potentially
more efficient in detecting the wrong $\cal CPT$ than the comparison of
oscillation frequencies. The $|\eta_{+-}|^{-2}$ factor multiplies the short-
time component $e^{-\Gamma_{S}\Delta t}$ in Eq. (23) where it decreases fast
with $\Delta t$. This may be also advantageous due to higher sensitivity to
shape compared to slowly varying component $e^{-\Gamma_{L}\Delta t}$.

####  IV.3.3 Resonant oscillations

Potentially the most stringent estimation of $\kappa$ is provided at high
energy when the deformed frequency $\zeta$ becomes equal to the normal
oscillation frequency $\Delta E$. According to Eq. (18), this kind of
resonance happens at $\gamma^{\ast}=(\Delta m/m^{2})\kappa$. For instance, if
the deformation $\kappa$ is of the order of the Planck’s mass, $\kappa\simeq
10^{19}$ GeV, then $\gamma^{\ast}\simeq 0.2\times 10^{5}$, corresponding to
energies $10^{2}-10^{3}$ TeV. Using condition $\zeta=\Delta E$ as a prediction
for $\kappa$ one gets

$\displaystyle\kappa=\gamma\times 0.45\times 10^{14}\;\mbox{GeV}$ (31)

which represents a point estimate of $\kappa$, and not a lower limit, thus
demonstrating the potential power of the method.

Doing predictions for future experiments it has to be realized, however, that
the precision of the estimates (27), (30) and (31) predominantly depends on
the uncertainty of determination of the energy $E$ which directly propagates
to the uncertainty of the Lorentz boost $\gamma=E/m$. In current high-energy
experiments, the resolution of energy or momentum is typically at the percent
level. Resolutions in $\Delta m$ and $m$ are $2\times 10^{-3}$ and $3\times
10^{-5}$ pdg , respectively, and can be neglected in these predictions. The
uncertainty of energy measurement thus critically affects the precision of
$\kappa$ and for the energy’s uncertainty $10^{-2}$ one only determines our
estimates with an error of $\pm\gamma\times 10^{12}$. To make this method
efficient, one has to significantly improve the accuracy of the energy
measurement.

##  V Conclusions

In this paper, we investigated kaon interferometry in the quantum gravity-
inspired model with deformed (as opposed to manifestly violated) continuous
(Poincaré) and discrete ($\cal CPT$) symmetries. We found that in this case
the decay intensity of kaon – anti-kaon pair has three contributing terms: the
right-parity, the wrong-parity, and the interference between them, all of
which are affected by deformation. The magnitude of deviations from the
predictions of the standard, undeformed, $\cal CPT$-invariant theory is
proportional to the momenta carried by kaons, and, if real, can be in
principle observed.

A distinctive feature of the approach of the present paper is its derivation
from the concepts of non-commutative geometry and deformed conservation rules.
Our prediction for the boost-dependent violation of the $\cal CPT$ symmetry
was derived from the first principles. Its dependence on energy and decay
times provides us with a clear prescription on how to examine this scenario on
the fast-moving technology frontiers of energy and precision.

The system of kaons offers to possible observational opportunities: one
associated with an additional momentum-dependent oscillation frequency and
another being a consequence of the emergence of a wrong parity term, that is
not allowed by the standard $\cal CPT$-invariance. We estimate that the first
method provides the bound $\kappa\gtrsim 10^{14}$ GeV in the case of the boost
$\gamma\sim 10^{5}$ (an order of magnitude higher than that attainable with
the present accelerator techniques). The second method $\kappa\gtrsim 10^{18}$
GeV, is already close to the expected value of $\kappa$, of the order of
Planck mass. The third method, potentially the most powerful, can predict the
value of $\kappa$ and not its lower limit. For $\gamma\sim 10^{5}$ one gets
$\kappa\sim 10^{19}$ GeV, so the Planck’s energy scale. This suggests that it
is feasible that the theory with deformed $\cal CPT$-invariance might be ruled
out, or confirmed in the not-too-distant accelerator experiments fcc .
However, to make these measurements conclusive, one has to significantly
improve the accuracy of the experimental determination of particles’ boost.

Finally, let us compare our model with the one proposed in Ref. bernabeu_2 .
In the parametrization of that paper, the $\omega$ parameter, quantifying an
admixture of states with an ill-defined $\cal CPT$, is found only in terms
originating from the $|K_{L}\rangle|K_{L}\rangle$ and
$|K_{S}\rangle|K_{S}\rangle$ states and thus, apart from the overall
normalization, modifies only the wrong-parity and interference terms. In our
case, the $\kappa$-dependent factors influence oscillations of all terms
(19)-(21), each in a different manner. If identified directly with the factor
multiplying the wrong-parity term of the intensity (20), the $|\omega|^{2}\sim
1-\cos(\zeta\Delta t)$ and is energy- and time-dependent.

## Acknowledgment

For AB and JKG this work was supported by funds provided by the National
Science Center, project no. 2019/33/B/ST2/00050.

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#  On Data Sampling Strategies for Training Neural Network Speech Separation
Models
††thanks: This work was supported by the Centre for Doctoral Training in
Speech and Language Technologies (SLT) and their Applications funded by UK
Research and Innovation [grant number EP/S023062/1]. This work was also funded
in part by 3M Health Information Systems, Inc.

William Ravenscroft, Stefan Goetze, and Thomas Hain  0000-0002-0780-3303
0000-0003-1044-7343 0000-0003-0939-3464 Department of Computer Science, The
University of Sheffield, Sheffield, United Kingdom
{jwravenscroft1, s.goetze<EMAIL_ADDRESS>
###### Abstract

Speech separation remains an important area of multi-speaker signal
processing. DNN models have attained the best performance on many speech
separation benchmarks. Some of these models can take significant time to train
and have high memory requirements. Previous work has proposed shortening
training examples to address these issues but the impact of this on model
performance is not yet well understood. In this work, the impact of applying
these training signal length (TSL) limits is analysed for two speech
separation models: SepFormer, a transformer model, and Conv-TasNet, a
convolutional model. The WJS0-2Mix, WHAMR and Libri2Mix datasets are analysed
in terms of signal length distribution and its impact on training efficiency.
It is demonstrated that, for specific distributions, applying specific TSL
limits results in better performance. This is shown to be mainly due to
randomly sampling the start index of the waveforms resulting in more unique
examples for training. A SepFormer model trained using a TSL limit of 4.42s
and dynamic mixing (DM) is shown to match the best-performing SepFormer model
trained with DM and unlimited signal lengths. Furthermore, the 4.42s TSL limit
results in a 44% reduction in training time with WHAMR.

###### Index Terms:

speech separation, context modelling, data sampling, speech enhancement,
transformer

##  I Introduction

Speech separation models are used in a number of downstream speech processing
tasks, from meeting transcription to assistive hearing [1, 2, 3, 4]. Often,
speakers are at a far-field distance from the microphone, which creates
additional challenges for speech separation due to interference from noise and
reverberation in the signal [5, 6, 7].

DNN separation models have led to significant improvements on anechoic data
but there is a performance gap when these models are used for more distorted
speech data [8, 9, 10]. Many of these models use Transformer or bidirectional
long short term memory (BLSTM) layers [11, 9, 10] which can consume large
amounts of memory and have quadratic time-complexity, i.e. for $L$ input
frames of data the model performs at least $L^{2}$ operations [12]. This is a
particular concern in training when memory requirements are higher due to
storing gradients for each operation required in the back-propagation stage
[13]. This increased computational load also means longer training times. One
way to compensate for the memory requirements is to use a batch size of 1 [9]
which leads to even longer training times as more parameter updates are
performed. Another approach that reduces memory requirements and allows for
larger batch sizes is to reduce the mixture signal length [14]. This reduces
the training time but potentially at the expense of performance.

In this work, the first aim is to address if there are TSL limits at which no
additional performance gain can be attained by deep neural network (DNN)
speech separation models. It is shown that, depending on model and dataset
selection, there is a TSL limit at which not only no additional performance
gain is attained but actually limiting the TSL to a specific value can lead to
notably improved performance. This effect is demonstrated to be due to a
random sampling of the start index when using TSL limits. Further evaluations
show the benefit of having more unique training examples than using the full
signal lengths. Finally, the application of TSL limits used with DM [15] is
evaluated.

The remainder of this paper is structured as follows. In Section II, the
signal model is introduced. The separation networks and datasets used are
described in Section III, and Section IV, respectively. Section V presents
evaluations for varying TSL limit for each separation network and dataset.
Section VI explores splitting signals to generate more train examples and
whether DM mitigates the effects gains found using TSL limits. Final
conclusions are provided in Section VII.

##  II Signal Model

The noisy reverberant speech separation problem is defined as aiming to
estimate $C$ speech signals $\hat{s}_{c}[i]$ for sample index $i$ and speaker
number $c\in\\{1,\ldots,C\\}$ from the discrete time-domain mixture

$x[i]=\sum_{c=1}^{C}s_{c}[i]\ast h_{c}[i]+\nu[i]$ (1)

of length $L_{x}$. The $\ast$ operator denotes convolution, $h_{c}[i]$ is a
room impulse response (RIR) corresponding to speaker $c$ and $\nu[i]$ denotes
additive background noise.

##  III Separation Models

The SepFormer [9] and Conv-TasNet [14] models are both widely researched
TasNets. The structure for these TasNets is to have a time-domain neural
encoder which encodes a mixture signal block $\mathbf{x}_{\ell}$ of size
$L_{\mathrm{BL}}$ to $\mathbf{w}_{\ell}$ followed by a mask estimation network
which estimates a series of masks $\mathbf{m}_{\ell}$ for each of $C$
speakers. These masks are used to separate the encoded features which are then
decoded back in to a time domain signal $\mathbf{s}_{\ell,c}$ using a neural
decoder. An example of the architecture for $C=2$ speakers can be seen in
Figure 1.

Figure 1: Architecture of the SepFormer and Conv-TasNet models, exemplary for
$C=2$ speakers. The $\odot$ symbol denotes the Hadamard product.

Both Conv-TasNet and SepFormer are trained using the utterance-level
permutation-invariant scale-invariant signal-to-distortion ratio (SISDR)
objective function [16, 14, 9, 17]. Models are trained according to the best
performing models in each of their original papers [14, 9] unless stated
otherwise. Batch sizes of $2$ and $4$ are used for SepFormer and Conv-TasNet
respectively except where otherwise stated.

###  III-A SepFormer Network

The SepFormer model is briefly introduced in this section. SepFormer is chosen
because it is a large transformer model that is among the state-of-the-art
models on several speech separation benchmarks [9, 18]. The SepFormer uses a
1D convolutional layer for encoding the signal proceeded by a ReLU activation
function. The decoder is a single transposed 1D convolutional layer. The mask
estimation network uses a dual-path structure [11] whereby a series of
Transformer layers are stacked such that each alternating layer computes a
multihead attention layer on either the local or global context of the
sequence. The local processing is achieved by first splitting the input signal
into overlapping chunks of a predetermined size $K$ turning a batched 3D
tensor into a 4D tensor. The output of the local Transformer layer is a 4D
tensor which is then reshaped by swapping the axes of encoded chunks and the
number of chunks before being fed into the global Transformer layer. The final
stage is to reconstruct a 3D tensor using the overlap-add method to produce
$C$ sequences of masks. The encoded features are then masked and decoded back
into the time domain.

###  III-B Conv-TasNet

The Conv-TasNet model contrasts the SepFormer in that it is a much smaller
model (25.8M vs. 3.5M parameters in the implementations used here) and the
only global information it processes is the overall signal energy, whereas the
SepFormer model has global access to all information in the input signal due
to the transformer layers used. Conv-TasNet uses a temporal convolutional
network (TCN) sequence model for the mask estimator in Figure 1. The encoder
and decoder of the network are the same as those used for the SepFormer model
but with a different number of output channels in the configuration used in
this paper. The mask estimation network is composed of a 1D pointwise
convolution (P-Conv) bottleneck layer, a TCN and a 1D P-Conv projection layer
with a ReLU activation function to produce the sequence of masks. The TCN is
composed of a series of convolutional blocks consisting of P-Conv and
depthwise-separable convolution (DS-Conv) layers with kernel size $P$. The
convolutional blocks are configured in stacks of $X$ blocks with increasing
dilation factors $f\in\\{2^{0},2^{1},\ldots,2^{X-1}\\}$. Stack are repeated
$R$ times with the dilation factor reset at the start of each stack [14].

##  IV Datasets

Three corpora of 2-speaker mixtures are analysed in this work. The trends
demonstrated later in Sections V and VI for the 2-speaker scenario are assumed
to generalize to higher $C$ values. For all corpora, the $8$kHz min
configuration is used. The min configuration refers to mixtures being
truncated to the shortest utterance in a mixture as opposed to padding shorter
utterances to the longest utterances.

###  IV-A WSJ0-2Mix and WHAMR

WSJ0-2Mix and WHAMR are both simulated 2-speaker corpora derived from the WSJ0
corpus [19, 20]. WSJ0-2Mix takes speech samples from WSJ0 and overlaps them at
speech-to-speech ratios between $0$ and $5$ dB. WHAMR is a noisy reverberant
extension of WSJ0-2Mix with noise from the WHAM [21] dataset mixed with the
loudest speaker at signal-to-noise ratios between $-6$ and $3$ dB.

###  IV-B Libri2Mix

Libri2Mix is a simulated 2-speaker mixture corpus derived from the LibriSpeech
and WHAM corpora [22]. Speech samples come from the LibriSpeech corpus [23]
and noise samples come from the WHAM corpus. Instead of SSRs, LibriMix uses
loudness units relative to full scale (LUFS) measured in dB to set the
loudness of speakers and noise in the mixtures. Speakers have a loudness
between -25 and -33 LUFS and noise has a loudness between $-38$ and $-30$
LUFS. For training, the train-100 dataset was chosen as it has a very similar
TSL distribution to the alternate train-360 dataset but with less examples,
meaning shorter training times.

###  IV-C Signal Length Distributions

The distribution, density estimation (DE) and the mean and standard deviation
of the mixture signal length in the WHAMR train tr and test tt sets can be
seen in Figure 2.

Figure 2: Distributions of mixture signal lengths in WSJ0-2Mix/WHAMR (left)
and Libri2Mix (right) for both train (top) and test (bottom) sets. DE is shown
by solid green lines, mean values are indicated by dashed red lines and
standard deviation values by dash-dotted blue lines.

WSJ0-2Mix and WHAMR have identical signal distributions as WHAMR is derived
from the former. These distributions are shown in the left panel. The train
and test sets in WHAMR have similar distributions of signal length with mean
values within 0.3s of one another and standard deviations within 0.1s of one
another. This contrasts the distributions of the Libri2Mix dataset [22] also
shown in Figure 2 where the train-100 and test sets have a difference in mean
value of $6.2$s and difference in standard deviation of $1.79$s.

##  V Training Signal Length Analysis

Evaluations of varying the TSL limit are presented in this section. For all
evaluations, the improvement in SISDR over the input mixture signal, denoted
by $\Delta$ SISDR, is used as the evaluation metric. SISDR measures the energy
of distortions in the estimated speech signals and is one of the most common
metrics used in recent monaural speech separation literature [11, 9, 10].

###  V-A Initial TSL Limit Evaluations

As a first experiment, twelve SepFormer models are trained and evaluated on
WSJ0-2Mix, WHAMR and Libri2Mix, each with a different TSL limit. Twelve
logarithmically spaced signal limits $T_{\mathrm{lim}}$ are selected between
$0.5$s and $10$s:

$\displaystyle T_{\mathrm{lim}}\in\\{$ $\displaystyle
0.5,0.66,0.86,1.13,1.49,1.95,$ $\displaystyle
2.56,3.36,4.42,5.8,7.62,10\\}\text{s}.$ (2)

The notation $L_{\mathrm{lim}}$ is used for the respective discrete sample
index, i.e. $L_{\mathrm{lim}}=T_{\mathrm{lim}}\mathrm{f}_{s}$ for sampling
rate $\mathrm{f}_{s}$. When cutting the training signal lengths such that
$L_{x}\leq L_{\mathrm{lim}}$ the starting sample index of the signal is
randomly selected from the uniform distribution
$\mathcal{U}\left(0,1+max\left(0,L_{x}-L_{\mathrm{lim}}\right)\right)$.
Performance for SepFormer models trained and evaluated on all three datasets
is compared in Figure 3.

Figure 3: SepFormer results for varying the TSL limit for the anechoic
WSJ0-2Mix (top), Libri2Mix (middle) and WHAMR (bottom) test sets.

For the WHAMR corpus, an increase in overall $\Delta$ SISDR performance from
the $0.5$s to $1.95$s limit can be observed. The optimal TSL is at $3.36$s.
Between $3.36$s and $10$s performance decreases again by $-1.4$dB. This may
seem surprising as the general convention with training DNNs is that more data
normally results in improved overall performance. A similar trend is observed
for WSJ0-2Mix where there is a notable increase between 0.5s and $3.36$s and
then a drop in performance of $0.8$dB between $4.4$s and $10$s. For Libri2Mix,
the performance saturates before a TSL limit of $4.42$s. There is no drop in
performance as the TSL limit approaches $10$s which is likely due to the
Libri2Mix training set having a more uniform distribution below signal lengths
of $10s$ than the WHAMR or WSJ0-2Mix datasets, cf. Figure 2.

The results for the WHAMR evaluation set are separated into quartiles of
mixture signal length for the following experiment. $\Delta$ SISDR results for
each quartile are shown in Figure 4.

Figure 4: TSL analysis of the $1$st to $4$th signal length quartiles in the
WHAMR evaluation set.

Comparing Q1 to Q4 shows that, with a sufficiently large TSL ($\geq 1.95$s),
the best separation performance in SISDR is found on the longest signal
lengths, regardless of TSL. A loss in SISDR performance is still observed from
$3.36$s to 10s regardless of which quartile is evaluated.

###  V-B Training Time Evaluation

The average training epoch duration (ED) for the SepFormer model on WHAMR and
Libri2Mix training sets are shown in Figure 5. Note the ED for WSJ0-2Mix is
omitted for brevity but is similar to WHAMR due to having the same TSL
distribution (cf. Figure 2). All models were trained on the same hardware to
control any impact this has on speed.

Figure 5: Comparison of average epoch duration (in mins) for the SepFormer
model on the Libri2Mix and WHAMR training sets.

The average EDs for the WHAMR dataset have a sigmoidal shape due to the
majority of the signal lengths being concentrated around the mean signal
length of the training set ($5.6$s, cf. Figure 2). Libri2Mix has a more linear
relationship between TSL limit and ED due to the more uniform shape of the
signal length distribution below $10$s in the train-100 set, cf. Figure 2.
Reducing the TSL limit has more benefit in terms of ED for the Libri2Mix
dataset when iterating over all training examples.

###  V-C Fixed vs. Random Start Index

In Section V-A, the start index of each shortened signal was randomly sampled
from a uniform distribution. In this section, this is compared to using a
fixed start sample. A start sample of 1999 ($=0.25$s at 8kHz) was used for
signals where the original mixture signal length was larger than
$L_{\mathrm{lim}}$, else the entire signal length was used. The motivation for
this was that many training examples contain silence at the beginning of
clips. It was considered desirable to omit as much silence to make for a
fairer comparison with the randomly sampled clips which are assumed to have a
lower likelihood of beginning with silence. Results in Figure 6 confirm that
the loss in performance from a TSL limit of $3.36$s to $10$s with WHAMR is due
to randomly sampling the start index.

Figure 6: Comparison of TSL variation for the SepFormer model trained and
evaluated on the WHAMR datasets on a subset of TSL limits in the range
$[1.95,7.62]$s.

The performance saturates at a TSL limit of $5.8$s when using a fixed start
index. This is similar to the performance saturation point of Libri2Mix in
Figure 3 demonstrating the performance drop in higher TSLs seen before on
WHAMR (cf. Figure 3) is related to both a non-uniform TSL distribution and the
random sampling used.

###  V-D Transformer vs. Convolutional Model

Results comparing the Conv-TasNet model (cf. Section III-B) to the SepFormer
model are shown in Figure 7. The loss in performance above $3.36$s is not
observed for the Conv-TasNet model. All SISDR results above
$T_{\mathrm{lim}}=1.95$s are within in $0.5$dB of each other suggesting Conv-
TasNet is more invariant to the TSL limit if the limit is sufficiently large.
This is possibly due to the $1.53$s receptive field of the Conv-TasNet models
being smaller than these particular TSLs limits [24].

Figure 7: Comparison of SepFormer and Conv-TasNet across TSL limits
$T_{\mathrm{lim}}\in[4.42,7.62]$ using the WHAMR corpus.

##  VI Signal Splitting and Dynamic Mixing

Next, two sampling strategies are evaluated to (i) investigate whether the
performance gained by TSL with random sampling on shorter sequences still
holds when the same quantity of audio data in terms of length in seconds is
used and (ii) whether using TSL limits still result in performance gains with
DM, i.e. simulating new speech mixtures for each epoch [15].

###  VI-A Signal Splitting

A signal splitting strategy was designed such that a batch of inputs
$\boldsymbol{\mathrm{X}}\in\mathbb{R}^{M\times L_{x}}$ was reshaped to
$\boldsymbol{\mathrm{X}}^{\prime}\in\mathbb{R}^{MD\times\frac{L_{x}}{D}}$ for
batch size $M$. Signal length $L_{x}$ is still limited such that $L_{x}\leq
L_{\mathrm{lim}}$. The motivation of this method is to evaluate the importance
of training on the entire sequence length compared to the raw data quantity
used in seconds. Computational complexity in training is also reduced. TSLs
for $T_{\mathrm{lim}}\in[4.42,10]$s are analysed for $D=2$. Figure 8 shows
that $D=2$ improves performance for shorter TSL limits ($T_{\mathrm{lim}}\leq
5.8$s) compared to $D=1$ (the original shape). However for
$T_{\mathrm{lim}}\geq 7.62$s the performance is similar to $D=1$. As in Figure
3 this is likely due to the TSL distribution of WHAMR.

Figure 8: Comparison of split signal and batch reshape training $D=2$ against
full signal training $D=1$ for the SepFormer model.

###  VI-B Dynamic Mixing

DM was proposed to improve performance of various speech separation models
[15]. DM often results in a $1.0$ to $1.5$dB SISDR performance improvement
dependant upon the model and dataset [15, 25]. The random start index sampling
used in Section V is similar to DM in that it provides the model with unique
training examples each epoch but without simulating new mixtures. In this
section using DM and TSL limits is compared against just using TSL limits to
see if further performance gains can be attained using both approaches. The DM
results for the WHAMR corpus are shown in Figure 9. It can be seen that with
DM the drop in performance is less (at $7.62$s) than without DM.

Figure 9: Comparison of SepFormer model training using TSL limits with and
without dynamic mixing being used as well on WHAMR evaluation set.

The best performing model $T_{\mathrm{lim}}=4.4$s is compared to the Sepformer
model with no TSL limit in Table I. The batch size reported for the model with
no TSL is the largest it was found possible to train on a $32$GB Nvidia V$100$
GPU. It can be seen that using the TSL limited model is able to match its
performance with an average ED reduction of $44$%, highlighting the benefit of
this approach.

Table I: Comparison of best performing SepFormer models on WHAMR with and without TSL limits. $M$ denotes batch size and the average epoch duration (ED) is reported in minutes. Model | $M$ |  $T_{\mathrm{lim}}$ (s) |  $\Delta$ SISDR (dB) | ED (mins)
---|---|---|---|---
SepFormer + DM | 1 | $-$ | 14.0 | 85
SepFormer + DM | 2 | 7.62 | 13.6 | 74
SepFormer + DM | 2 | 4.42 | 14.0 | 59

##  VII Conclusion

In this paper it was shown that TSL limits can affect the overall performance
for speech separation models in a number of ways. For WSJ0 derived speech
separation benchmarks, i.e. WSJ0-2Mix and WHAMR, it is optimal to use
shortened training examples randomly sampled from the original examples due to
the signal length distribution of these corpora. For the Libri2Mix dataset,
the same method led to shorter training times with no notable loss in
performance. The SepFormer model was compared to the Conv-TasNet model and it
was shown that the Conv-TasNet performance has less variation. Using dynamic
mixing and TSL limits with random sampling was shown to be able to match the
performance of the SepFormer model trained DM using full sequence lengths on
WHAMR with a 44% reduction in training time. With some previous literature
opting to limit TSLs [14, 11] and others not [10, 9] on the same benchmarks,
the results in this paper suggest that this is not a fair comparison and that
TSL limiting is important to factor in when analysing results, particularly
for the WSJ0-2Mix and WHAMR benchmarks.

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  *[TSL]: training signal length
  *[DM]: dynamic mixing
  *[BLSTM]: bidirectional long short term memory
  *[DNN]: deep neural network
  *[RIR]: room impulse response
  *[TasNets]: time-domain audio separation network
  *[SISDR]: scale-invariant signal-to-distortion ratio
  *[ReLU]: rectified linear unit
  *[TCN]: temporal convolutional network
  *[P-Conv]: pointwise convolution
  *[DS-Conv]: depthwise-separable convolution
  *[SSRs]: speech-to-speech ratio
  *[LUFS]: loudness units relative to full scale
  *[DE]: density estimation
  *[DNNs]: deep neural network
  *[ED]: epoch duration
  *[EDs]: epoch duration
  *[TSLs]: training signal length