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\begin{document} 
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\preprint{}

\title{Limit on Quark-Antiquark Mass Difference from the Neutral Kaon System}

\author{Michael J. Fromerth and Johann Rafelski}
\affiliation{Department of Physics, University of Arizona, Tucson, AZ 85721}
\date{November 22, 2002}

\begin{abstract}
We quantify the limits on quark-antiquark mass differences 
imposed by the neutral kaon mass system.
In particular, we find that an upper limit to the 
mass difference of $10^{-3}$\,eV exists if mass 
differences across quark flavors are uncorrelated.
We also find that such a small mass difference 
would be difficult to detect in upcoming antihydrogen 
experiments, both due to its small effect on the 
atomic spectrum of antihydrogen, and because of 
the dominant role that any electron-positron mass 
difference would play.
\end{abstract}

\pacs{12.15.Ff,12.38.Mh,12.40.Yx}
% 12.15.Ff Quark and lepton masses and mixing (see also 
%	  14.60.P Neutrino mass and mixing)
% 12.40.Yx Hadron mass models and calculations
% 12.38.Mh Quark gluon plasma
\maketitle


The origin of quark and lepton masses remains at present 
unknown. It is generally presumed that by virtue of CPT
symmetry matter and antimatter will always have same fundamental
mass parameters. The experimental limits are at present 
not very good, and in fact the   baryon-antibaryon 
mass difference~\cite{PDG02}:
$$|m_p-m_{\bar p}|< 6\,10^{-8}m_p\simeq 60\,\mbox{eV}.$$
is much grater than the small baryon-antibaryon
asymmetry seen in the early Universe, which  corresponds
at deconfinement boundary  to a 
chemical potential of the order of $0.33eV\pm30\%$~\cite{From02}.
Consequently, it would seem that our understanding 
of the early Universe depends
sensitively on the implicit assumption of mass symmetry for
quarks and antiquarks. 

Here we show how the properties of the neutral kaon system 
constrain the mass difference between quarks and antiquarks.
From a detailed study of the kaon decay rates, it is observed 
that the mass difference between the $K_L$ and $K_S$ states 
is $\Delta m \equiv m_{K_L} - m_{K_S} 
= 3.463 \pm 0.010 \times 10^{-6}$\,eV~\cite{Alavi02}.
Because this mass difference is understood within the standard 
model to arise from second-order weak interactions that mix 
the $K^0$ and $\bar{K^0}$ states~\cite{Gaillard75,Perkins00}, the 
magnitude of the CPT-violating contribution to
particle-antiparticle mass difference is severely restricted.

Recently, it has been demonstrated by Greenberg~\cite{Greenberg02} 
that CPT breaking implies the violation of coordinate Lorentz invariance. 
In an extension of the standard model, Lorentz- and CPT-violating 
operators yielding a satisfactory quantum field theory have been
considered \cite{Colladay97,Colladay98,Kostelecky01a}. 
Examples include spontaneous Lorentz and CPT violation in the 
context of string field theory (see, e.g., Refs.~\cite{Kostelecky89,
Kostelecky91}) and non-commutative field theories~\cite{Carroll01}.
These models offer a basis for numerous precision experiments 
placing extremely tight bounds on Lorentz and CPT breaking. 
In this context, the neutral kaon system has been analyzed both
experimentally~\cite{Carosi90,Schwingenheuer95} and 
theoretically~\cite{Kostelecky98,Kostelecky99,Kostelecky01b}, while 
direct CPT violation in the neutrino sector has been explored in 
Refs.~\cite{Skadhauge02,Barenboim02a,Barenboim02b}.
Other work discussing theoretical implications of CPT violation 
include~\cite{Barmin84,Dejardin00,Klinkhamer01,Urbanowski02}.

One may  ask if such a hypothesis for a mass difference
 makes good physical sense, considering the well 
established principles of quantum field theory.
We believe that  a search for tacitly assumed limits to accepted
physical principles  is a very important step in 
verification of the paradigm which govern our view on the 
laws of physics. This attitude is more generally shared, and
with formation of a large number of antihydrogen atoms, we
can look forward to further experimental  tests of CPT symmetry  to 
take place at CERN~\cite{CERN02}. A momentum-dependent difference 
between particles and antiparticles is 
expected~\cite{Kostelecky98,Kostelecky99}, should there be 
a violation of Lorentz invariance, appearing in association 
with  CPT breaking~\cite{Greenberg02}. 

In the following we describe 
how the measured mass asymmetry of the  neutral kaons limits
the mass difference between quarks and antiquarks.
We define the $K_L$ and $K_S$ states in the standard formalism~\cite{Perkins00}:
\begin{eqnarray}
K_L & = & \frac{1}{\sqrt{2 + 2 \epsilon^2}} 
  \left[ (1+\epsilon) K^0 + (1-\epsilon) \bar{K^0} \right] \ , \label{K_L} \\
K_S & = & \frac{1}{\sqrt{2 + 2 \epsilon^2}} 
  \left[ (1-\epsilon) K^0 - (1+\epsilon) \bar{K^0} \right] \ ,
\end{eqnarray}
where $K^0 = |d\bar{s}\rangle$, $\bar{K^0} = |\bar{d}s\rangle$, and 
$\epsilon \approx 2.3 \times 10^{-3}$ is the CP violation parameter.

We express the (assumed) CPT-violating mass difference between quarks and antiquarks as:
\begin{eqnarray}
m_{s,\bar{s}} & = & m_s^0 \pm \frac{\delta m_s}{2} \label{delta_ms} \\
m_{d,\bar{d}} & = & m_d^0 \pm \frac{\delta m_d}{2} \label{delta_md} \ ,
\end{eqnarray}
where the signs of $\delta m_s$ and $\delta m_d$ are undetermined. 


The mass operator for the neutral kaon system, with the quark mass differences, becomes:
\begin{eqnarray}
\hat{M}\ =\ \hat{M}^0_{K}\ +\ \hat{M}_{w}\ +\ \frac{f}{2}\, 
 \left[\, (\delta m_d - \delta m_{\bar{s}})\, |d\bar{s}\rangle 
\langle d \bar{s}|\ +\ \right.\nonumber\\ \left.
+\ (-\delta m_{\bar{d}} + \delta m_s)\, |\bar{d} s \rangle 
\langle \bar{d} s|\, \right] ,\ \
\label{mass}
\end{eqnarray}
where $\hat{M}^0_{K}$ is the neutral kaon mass excluding weak interactions, 
$\hat{M}_{w}$ is the mass contribution due to weak interactions, and the 
third term is the effect that the change in the current quark masses of 
Eqs.~(\ref{delta_ms}) and (\ref{delta_md}) would have on the kaon mass.

The form of the third term arises because in a model of hadronic structure 
(e.g., the bag model), the response of the hadronic mass is linear with 
respect to the change in quark mass if expanded about a finite quark mass~\cite{Letessier02}.
Note that a similar effect arises in non-relativistic quark models.
Furthermore, the scaling factor $f$ is of order unity, as confirmed by 
the features of hadronic mass splittings.

  From Eqs.~(\ref{K_L})--(\ref{mass}), 
the mass difference between $K_L$ and $K_S$ becomes:
\begin{eqnarray}
\Delta m & = & \langle K_L | \hat{M} | K_L \rangle \ -\ \langle K_S | \hat{M} | K_S \rangle \nonumber \\
 & = & \Delta m_{w}\ +\ 2\, \epsilon f\, \left[\, (m_{\bar{s}} - m_s) - (m_{\bar{d}} - m_d)\, \right]
\end{eqnarray}
where $\Delta m_{w} \equiv\langle K_L | \hat{M}_w | K_L \rangle - \langle K_S | \hat{M}_w | K_S 
\rangle$ and terms of $\epsilon^2$ or higher have been neglected.
Since it is understood that $\Delta m \simeq \Delta m_{w}$, this immediately 
yields the result:
\begin{equation}
\left| (m_{\bar{s}} - m_s) - (m_{\bar{d}} - m_d) \right| \ \ll\ 
\frac{\Delta m}{2 \epsilon f}\ \approx\ 10^{-3}\, {\rm eV} \ .
\label{direct_cpt}
\end{equation}

Equation~(\ref{direct_cpt}) places rather stringent limits 
on direct CPT violation in $d$ and $s$ quarks.
If the size of the CPT violation across quark flavors is 
uncorrelated, then an upper limit to the mass difference 
between quarks and antiquarks of {\it each} flavor is much 
less than $10^{-3}$\,eV.
Otherwise, the size of the CPT violation across $s$ and $d$ 
flavors must be highly correlated, such that 
$(m_{\bar{s}} - m_s) \simeq (m_{\bar{d}} - m_d)$. 
This would imply that a CPT violating force does not
distinguish between the first and second particle generation.

In the former case, such a small mass difference between $d$ and $\bar{d}$ 
quarks would be challenging to measure in the antihydrogen experiments.
The wavelengths of atomic transitions in hydrogen scale with the inverse 
of the reduced mass, $\lambda \propto (m_p + m_e)/m_e m_p$.
As a result, the relative shift in wavelength due to a mass difference 
in hydrogen and antihydrogen atoms is:
\begin{eqnarray}
\left| \frac{\delta \lambda}{\lambda} \right| 
& = & \frac{m_p m_e}{m_p + m_e} 
\left(\frac{\delta m_e}{m_e^2} + \frac{\delta m_p}{m_p^2} \right) \nonumber \\
 & \simeq & \left[\,\frac{1}{m_e}\, \delta m_e \ +\ 
\frac{m_e}{m_p^2}\, f\, \left(2 \delta m_u + \delta m_d \right)\, \right] \ . 
\label{waveshift}
\end{eqnarray}
If the CPT violation is exclusively in quarks and antiquarks, 
and the mass difference between $u$ and $\bar{u}$ is also 
$\ll 10^{-3}$\,eV, then the resulting shift in wavelength becomes:
\begin{equation}
\left| \frac{\delta \lambda}{\lambda} \right| \ll  2 \times 10^{-15} \ .
\end{equation}
Therefore, a technique at least this sensitive to 
wavelength would have to be used to detect direct 
CPT violation in the quark sector, in order to be competitive with 
the experimental limits set by the neutral kaon system.
Note that Eq.~(\ref{waveshift}) also predicts that 
a mass difference between electrons and positrons 
equal in magnitude to the quark-antiquark mass difference 
will dominate the wavelength shift by a factor 
$m_p^2/3 m_e^2 \simeq 10^6$, thus primarily addressing the 
CPT violation in the lepton (electron) sector. 

We find that the current upper limit to the mass difference 
between quarks and antiquarks in the $d$ and $s$ flavors is 
$\ll 10^{-3}\, {\rm eV}$ if the magnitude of the CPT 
violation is uncorrelated across flavors. In this case, the relative precision 
with which the strange quark mass difference is determined appears to be 
by far the most precise such value presently known:
\[
\left|\frac{m_s-m_{\bar s}}{m_s+m_{\bar s}}\right|\ll 10^{-11}\,,
\]
providing a strong constraint for any CPT model considered, and assuring
that a possible quark mass asymmetry is not relevant in the determination of
the physical conditions in the early Universe.

The $d$-quark mass difference would have the effect 
of shifting the wavelengths of the antihydrogen atomic 
spectrum by $\ll  2 \times 10^{-15}$ relative to the hydrogen spectrum.
Furthermore, the effect of a mass difference in the leptonic 
sector is likely to dominate over any quark effect.
It therefore seems unlikely that the upcoming 
antihydrogen experiments will be able to improve 
on the quark-antiquark mass difference limits already 
imposed by the neutral kaon system, except perhaps if the
CPT violating force were to cause a practically equal
mass effect for both $d,\,\bar d$ and $s,\,\bar s$ quarks.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\vspace{0.5cm}\noindent\it Acknowledgments:\\}
%\acknowledgments
The authors wish to thank Elliott Cheu and Ralf Lehnert for their helpful comments.
Supported  by a grant from the U.S. Department of Energy,  DE-FG03-95ER40937\,. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bibliographystyle{unsrt}
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\end{document}


