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\begin{document}

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In a seminal series of papers, Sen has shown how to obtain an exact conformal
field theory description of non-BPS D-branes (see \cite{sen} for two of the
main papers and two recent reviews).
In this setup, a non-BPS D($p-1$)-brane remains after
tachyon condensation on a D$p$--anti-D$p$-brane pair.
In type II theories the resulting object still
has a tachyonic mode on its worldvolume, but performing the orientifold to
type I stabilizes the non-BPS brane.
In a related development, D-brane charges have been shown to take values in
appropriate K-theory groups of space-time. A major result is that all
lower-dimensional D-branes can be considered in a unifying manner as
non-trivial excitations on the appropriate configuration of higher-dimensional
branes. For type IIB this was demonstrated by Witten in \cite{witten}, where
all branes are built from sufficiently many D9--anti-D9
pairs. As to type IIA, Horava outlined the construction of lower branes from
non-BPS D9-branes \cite{horava}.

As is well-known, type II BPS D-branes couple to Ramond-Ramond gauge fields
through the Wess-Zumino action \cite{GHM} (see \cite{anomalous} for
string computations checking the form of this action):
\begin{equation} \label{wzbps}
S_{\rm WZ}=\frac{T_p}{\kappa}\int_{p+1} C\wedge {\rm Tr}\, {\rm e}^{2\pi\a '\,F+B}\wedge
\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}~.
\end{equation}
Here $T_p/\kappa$ denotes the D$p$-brane tension, $C$ a formal sum of R-R
potentials, $F$ the gauge field on  the brane and $B$ the NS-NS two-form.
The trace is over the Chan-Paton indices.
Further, $R_T$ and $R_N$ are the curvatures of the tangent and normal
bundles of the D-brane world-volume, and $\hat{A}$ denotes the A-roof genus.
In the setup of Ref.~\cite{witten}, where one starts with an unstable configuration of
supersymmetric branes
and anti-branes, the coupling on the BPS-brane that remains after tachyon condensation
is inherited from the similar coupling of the parent branes.
In the scenario of Ref.~\cite{horava}, one starts from non-BPS branes, which
can decay into a lower BPS brane via tachyon condensation.
In this case, it has  not been clear how the resulting objects acquire
the desired couplings in \eq{wzbps}.
In this note we argue that all type II non-BPS branes couple
universally to Ramond-Ramond fields as given by
\begin{equation} \label{wznonbps}
S'_{\rm WZ}=a\int_{p+1} C\wedge d\,{\rm Tr}\, T \, {\rm e}^{2\pi\a '\,F+B}\wedge
\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}~,
\end{equation}
where $T$ is the real, adjoint tachyon field living on the non-BPS brane and
$a$ is a constant.\footnote{This constant will be fixed in the next paragraph by imposing that the
BPS D($p-1$)-brane we find there have the expected R-R charge. Then this action predicts the R-R charges
of the lower BPS D-branes that can be constructed from the non-BPS D$p$-brane.}
One term of this action (the one describing the coupling of a non-BPS D$p$-brane
to $C_p$) was discussed in Refs.~\cite{sen, horava}.
Below we will show how, upon tachyon condensation,
these non-BPS ``Wess-Zumino'' couplings induce the appropriate Wess-Zumino
action for the resulting BPS-branes. The cases D$p$ $\rightarrow$ D($p-1$)
and D$p$ $\rightarrow$ D($p-3$) will be treated in detail. It will turn out, for instance,
that the \mbox{R-R} charges of the D8-branes and D6-branes one constructs from unstable D9-branes
\cite{horava} have the expected ratio.
Moreover, we check the presence of these R-R couplings by performing
various disc amplitudes with an open string tachyon inserted at the boundary.
\paragraph{Relation to Wess-Zumino action} In Ref.~\cite{horava}
Horava described how to construct BPS D($p-2k-1$)-branes as
bound states of (sufficiently many) unstable D$p$-branes. The lower-dimensional BPS
branes arise as the result of the condensation of a tachyon field
into a vortex configuration, accompanied by non-trivial gauge fields.
We indicate now how the R-R couplings that we propose in \eq{wznonbps} account
for the R-R couplings (\ref{wzbps}) that the stable lower-dimensional brane
emerging from the condensation must possess.
\par
Consider first a single non-BPS D$p$-brane. There is a  real
tachyon field living on its worldvolume. The tachyon potential
is assumed to be such that the vacuum manifold consists of the two points
$\{T_0,-T_0\}$.\footnote{The symbol $T_0$ should not be confused with the tension of a D0-brane, which
will never explicitly appear in this paper.}
The tachyon
can condense to a non-trivial (anti)-kink configuration $T(x)$ depending on a
single coordinate.
The R-R coupling (\ref{wznonbps}) on the D$p$-brane reads in this case
\begin{equation}
\label{m1}
a\int_{p+1} C\wedge dT\wedge {\rm e}^{2\pi\alpha' F + B} \wedge
\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}~,
\end{equation}
the first term of which was suggested in Ref. \cite{horava} and shown to be present by a
disc computation (in an alternative formalism) in Ref.~\cite{sen}.
It involves the topological density  $\partial_x T(x)$, which is localized at
the core of the kink and is such that $\int dT(x) = \pm 2 T_0$.
In the limit of zero size we would have $dT(x) = 2 T_0\delta(x-x_0) dx$, and the above
action would take the form\footnote{%
Actually, trying to follow the reduction of a D$p$-brane
to a lower-dimensional one, there is a puzzle
concerning the gravitational part
$\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}$.
The directions along the parent brane transverse to the
smaller brane contribute originally to $\hat A(R_T)$. It is not clear to us how they are
reassigned to the normal bundle in the reduced action.
In fact, this problem seems also to be present for the reduction of brane-antibrane pairs
to lower-dimensional BPS branes as in Ref.~\cite{witten}, where only the standard
WZ actions (\ref{wzbps}) are involved.}
of the usual Wess-Zumino
effective action for a BPS D($p-1$)-brane, localized in the $x$-direction at
$x_0$:
\begin{equation}
\label{m2}
2T_0a\int_{p} C\wedge {\rm e}^{2\pi\alpha' F + B}
\wedge \sqrt{\hat{A}(R_T)/\hat{A}(R_N)}~.
\end{equation}
In reality, the D($p-1$)-brane will have a certain thickness in the direction of the kink.
\par
Note that the constant $a$ can be fixed in terms of $T_0$ by equating $2T_0a$ with the tension
$T_{p-1}/\kappa$ of a BPS D($p-1$)-brane. This being done, the remainder of this paragraph provides a
non-trivial check on our couplings in \eq{wznonbps}.
\par
As a less trivial example,
let us start from two unstable D$p$-branes. The tachyon field $T$,
transforming in the adjoint of the ${\rm U}(2)$ gauge group,
can form a non-trivial vortex configuration in co-dimension three.
The tachyon potential is assumed to be such that the minima of $T$ have
the eigenvalues $(T_0,-T_0)$, so that the vacuum manifold is
${\cal V} = {\rm U}(2)/({\rm U}(1)\times {\rm U}(1))=S^2$. The possible
stable vortex configurations  $T({\bf x})$, depending on $3$ coordinates
$x^i$ transverse to the ($p-2$)-dimensional core of the vortex, are classified by the
non-trivial embeddings of the ``sphere at infinity'' $S^{2}_\infty$ into the
vacuum manifold, namely by $\pi_{2}({\cal V}) = {\bf Z}$.

Apart from  the ``center of mass'' ${\rm U}(1)$
subgroup we are in the situation of the Georgi-Glashow model,
where the tachyon field $T({\bf x})= T^a({\bf x}) \sigma^a$
($\sigma^a$ being the Pauli matrices) sits in the adjoint of  ${\rm SU}(2)$,
and the vacuum manifold is described by $T^a T^a = T_0^2$.
The vortex configuration of winding number one, which is the 't Hooft-Polyakov
monopole, is of the form
\begin{equation}
\label{m3}
T({\bf x}) = f(r) \sigma_a x^a~,
\end{equation}
where $r$ is the radial distance in the three transverse directions,
and the prefactor $f(r)$ goes to a constant for $r\to 0$ and approaches
$T_0/r$ for $r\to\infty$.

The finite energy requirement implies that $D_i T^a$
vanishes sufficiently fast at infinity, from which it follows that a vortex is accompanied by a
non-trivial gauge field; for the case (\ref{m3}) above, the non-trivial
part of the SU(2) gauge field has the form
\begin{equation}
\label{m4}
{\cal A}^a_i({\bf x}) = h(r) \epsilon^a{}_{ij} x^j~,
\end{equation}
with $h(r)$ approaching a constant for $r\to 0$, while $h(r)\sim 1/r^2$
at infinity.
The field-strength in the unbroken ${\rm U}(1)$
direction,
\begin{equation}
\label{m5}
{\cal G}_{ij} = {T^a\over T_0} {\cal F}^a_{ij}~~,
\end{equation}
corresponds to a
non-trivial U(1) bundle on the sphere at infinity, {\it i.e.} the magnetic
charge $g = \int_{S^{2}_\infty} {\cal G}$ is non-zero (and in fact equals
the winding number of the vortex in appropriate units).
Thus there is a magnetic charge density
in the transverse directions, defined by $d{\cal G} = \rho({\bf x}) d^3x$,
which is concentrated at the core of the vortex
solution. In the zero size limit,  there would be a point-like magnetic charge
at the location of the core: $\rho({\bf x}) = g\, \delta^3({\bf x} - {\bf x}^0)$.
\par
The  WZ action (\ref{wznonbps}) for the D$p$-brane can be rewritten as
\begin{equation}
\label{m6}
a\int_{p+1} C\wedge d\,{\rm Tr}\{ T\, {\rm e}^{2\pi\alpha' {\cal F}}\}
\wedge{\rm e}^{2\pi\alpha' {\hat F} + B} \wedge
\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}~,
\end{equation}
where we have split the U(2) field-strength into its SU(2) part ${\cal F}$
and its U(1) part $\hat F$. Inserting the 't Hooft-Polyakov configuration
for the tachyon and the SU(2) gauge field, we see that Eq.~(\ref{m5}) involves
precisely the magnetic monopole field ${\cal G}= T^a {\cal F}^a$; we get indeed
\begin{eqnarray}
\label{m7}
& & 2\pi\alpha' a\int_{p+1}  C\wedge d{\rm Tr}\{ T\,{\cal F}\}\wedge
{\rm e}^{2\pi\alpha' {\hat F} + B} \wedge
\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}\nonumber\\
&=& 2\pi\alpha' a T_0
\int_{p+1}  C\wedge \rho({\bf x}) d^3x\wedge
{\rm e}^{2\pi\alpha' {\hat F} + B} \wedge
\sqrt{\hat{A}(R_T)/\hat{A}(R_N)}~.
\end{eqnarray}
Thus we have a distribution of D($p-3$)-brane charge localized at the
core of the vortex; in particular, in the limit of zero-size core we recover
the R-R couplings (\ref{wznonbps}) of a BPS D($p-3$)-brane that supports the
U(1) gauge field $\hat F$.
\par
Since the minimal magnetic charge $g$ is $4\pi$ in our units, Eq.~(\ref{m7})
and the remark after \eq{m2} lead to the expected ratio $4\pi^2\alpha'$ for the R-R charges of
D($p-3$)-~and D($p-1$)-branes.
\par
The mechanism described above generalizes
to the reduction of a non-BPS D$p$-brane to a D$(p-2k-1)$-brane via
tachyon condensation, described in \cite{horava}. In this case, it is
convenient to start with $2^k$ unstable D$p$-branes. The configuration of
vorticity one for the tachyon field, which sits in the adjoint of U$(2^k)$, is
of the form
\begin{equation}
\label{m8}
T({\bf x}) = f(r) \, \Gamma_i x^i~,
\end{equation}
where $r$ is the radius in the $2k+1$ transverse dimensions $x^i$, and the
$\Gamma$-matrices in these dimensions are viewed as
$U(2^k)$ elements. Eq.~(\ref{m8}) is a direct generalization of the 't Hooft-%
Polyakov case, Eq.~(\ref{m3}). Again, the finite energy requirement should imply
a non-trivial gauge field configuration, leading to a non-zero
generalized magnetic charge $\int_{S^{2k}_\infty}{\rm Tr}\{T ({\cal F})^k\}$.
In such a background, the WZ action (\ref{wznonbps}) contains the factor
$d\,{\rm Tr}\{T {\cal F}^k\}= \rho({\bf x}) d^{2k+1}x$; the (generalized) magnetic
charge density $\rho$ is concentrated at the core of the vortex, and in the
zero-size limit reduces to a delta-function in the transverse space. Thus we
are left with the WZ action for a D$(p-2k-1)$-brane.
\paragraph{String computation}
To compute the disc scattering amplitudes necessary to check \eq{wznonbps}, it is convenient to
conformally map the disc to the upper half plane and use the ``doubling trick'' as described, for
instance, in Ref.~\cite{hashimoto}. This trick consists in replacing, {\it e.g.}, $\bar X^{\mu}
(\bar z)$ by $S^\mu_{~\nu} X^\nu(\bar z)$, where $S^{\mu}_{~\nu}$ is diagonal, with entries $1$ in the
worldvolume and $-1$ in the transverse directions, and then treating the fields depending on
$\bar z$  as if $\bar z$ were a holomorphic variable living on the lower half plane. The fermionic
$\psi^\mu$ fields are treated in
the same way. As to the spin fields in the \mbox{R-R} sector, for BPS D$p$-branes in type IIA
$\bar S^{\dot\alpha}(\bar z)$ is replaced by
$(\gamma^0\gamma^1\cdots\gamma^p)^{\dot\alpha}{}_\beta\,S^\beta(\bar z)$ (where the chirality flips
because $p$ is even). For type IIB  $\bar S^{\alpha}(\bar z)$ is replaced by
$(\gamma^0\gamma^1\cdots\gamma^p)^{\alpha}{}_\beta\,S^\beta(\bar z)$, where now $p$ is odd. For the
non-BPS D$p$-branes we are studying here, $p$ is odd in IIA and even in IIB, so that there is a
chirality flip in IIB and not in IIA. The explicit computations below will be done for IIA, but the
story is, of course, completely analogous for IIB.

The first amplitude we are going to compute is the two point function of one open string tachyon and
a R-R potential in the presence of a single non-BPS D$p$-brane in IIA.\footnote{This has been done before in a
formalism in which non-BPS D-branes are constructed in an alternative way \cite{sen}.}
This will establish the first term in the
expansion
of \eq{wznonbps}. We take the R-R vertex operator in the ($-1/2,-1/2$) picture (which
exhibits the R-R field strengths rather than the potentials):
\begin{equation}
V_{RR}=H_{\alpha\dot\beta}\, S^\alpha(z)\, \bar S^{\dot\beta}(\bar z)\,e^{{\rm i} k\cdot X(z,\bar z)}
\rightarrow H_{\alpha\dot\beta}\, S^\alpha(z)\, (\gamma^0\gamma^1\cdots\gamma^p)
^{\dot\beta}{}_{\dot\gamma}\,S^{\dot\gamma}(\bar z)\,e^{{\rm i} k\cdot X(z)}\,
e^{{\rm i} k\cdot S\cdot X(\bar z)}~,
\end{equation}
where $H_{\alpha\dot\beta}$
is the bispinor containing the R-R field strengths
and $k$ the momentum of the R-R
potential. We have omitted the superghost part and do not keep track of the overall normalization,
since we are not able to directly determine the constant $a$ in \eq{wznonbps} anyway.\footnote%
{However, we will be
interested in the relative normalization of this amplitude with respect to the ones with photons
inserted. The constant $a$ itself was fixed in the previous paragraph.}
The tachyon vertex operator is put in the $-1$ picture:
\begin{equation}
V_T=T(k')\,e^{{\rm i} k'\cdot X(y)}~~,
\end{equation}
where $T$ and $k'$ are the tachyon polarization and momentum and $y$ is a point on the real axis.
Again, the superghost part
is not displayed. The three insertion points $z$, $\bar z$ and $y$ can be fixed by introducing ghost
fields. Then the contributions of the ghost, superghost and $X$ sectors combine into $(z-\bar
z)^{5/4}$. The contraction of the two spin fields in the fermionic sector gives
\begin{equation}
<S^\alpha(z)\,S^{\dot\gamma}(\bar z)>=(z-\bar z)^{-5/4}\,C^{\alpha\dot\gamma}~~,
\end{equation}
with $C$ the charge conjugation matrix.
The amplitude becomes
\begin{equation}
T\,H_{\alpha\dot\beta}
(\gamma^0\gamma^1\cdots\gamma^p)^{\dot\beta}{}_{\dot\gamma}\,C^{\alpha\dot\gamma}\times K~~,
\end{equation}
where $K$ is a global factor.
Tracing over the spinor indices, only the part of $H_{\alpha\dot\beta}$
proportional to $H_{\mu_1\ldots\mu_{p+1}} (C\gamma^{\mu_1\ldots\mu_{p+1}})_{\alpha\dot\beta}$
contributes, making the amplitude proportional to
$T\, H_{\mu_1\ldots\mu_{p+1}}\epsilon^{\mu_1\ldots\mu_{p+1}}$. Upon integration by parts,
this confirms the first term of~\eq{wznonbps}.

There is a kinematical subtlety in this computation. String scattering amplitudes can only be
computed for on-shell external particles. It is easy to convince oneself that, since the tachyon
carries only momentum along the brane and the momentum along the brane is conserved, the tachyon and
the R-R potential cannot be both on-shell. As a way out, one could consider branes with Euclidean
signature, for which this kinematical problem does not occur, and then extrapolate the couplings one
finds there to their Minkowski cousins.

To check the second term of \eq{wznonbps}, depending linearly on $F$, we add to the previous
amplitude a vertex operator for a gauge field. This vertex operator is in the 0 picture:
\begin{equation}\label{photon}
V_A=A_\mu({\rm i}\dot X^\mu(w)+2\alpha'\,p\cdot\psi\psi^\mu(w))\,e^{{\rm i} p\cdot X(w)}~~,
\end{equation}
where this time we have kept track of all normalization factors. Here $A_\mu$ is the polarization of
the gauge field, $p$ is its momentum and $w$ is on the real axis. Only the fermionic part of the
photon vertex operator
can lead to terms of the type we are looking for (the photon should provide two gamma-matrices), so
we ignore possible contributions from the bosonic part. We will compute the amplitude to lowest order
in the photon momentum. This means that we will put $p$ equal to zero in the bosonic sector,
thus keeping only the explicit $p$ dotted with a $\psi$ in \eq{photon}. We follow the previous
computation as closely as possible by fixing again $z$, $\bar z$ and $y$, such that only $w$ needs
to be integrated over. In the limit of small photon momentum the ghost, superghost and $X$ sector
contributions are unchanged (they multiply to $(z-\bar z)^{5/4}$). The fermionic correlator is
\begin{equation}
2\alpha' p_\nu A_\mu <S^\alpha(z)\,\psi^\nu\psi^\mu(w)\,S^{\dot\gamma}(\bar z)>=
-{\rm i}\,\alpha' p_\nu A_\mu\,(\gamma^{\nu\mu})^{\alpha\dot\gamma}\, (w-z)^{-1}(w-\bar z)^{-1}
(z-\bar z)^{-1/4}.
\end{equation}
The resulting integral can be done by a contour integration:
\begin{equation}
(z-\bar z)\int_{-\infty}^{+\infty}\,dw\,(w-z)^{-1}(w-\bar z)^{-1}=2\pi{\rm i}~~,
\end{equation}
leading to
\begin{equation}
2\pi\alpha'\, p_\nu\,A_\mu\,T\,H_{\alpha\dot\beta}\,
(\gamma^{\nu\mu})^{\alpha\dot\gamma}(\gamma^0\gamma^1\cdots\gamma^p)^{\dot\beta}{}_{\dot\gamma}\,
\times K
\end{equation}
for the amplitude. This corresponds indeed to the term in \eq{wznonbps} linear in $F$. Note that the
factor $2\pi\alpha'$ multiplying $F$ in \eq{wznonbps} comes out correctly.

The generalization to multiple (low momentum) photon insertions is straightforward. The dependence
on the photon insertion points of the relevant part of the fermionic correlator factorizes, such
that each integration reduces to the one-dimensional integral described in the previous paragraph.
It is also easy to include Chan-Paton factors in the above computations, leading to the trace in
\eq{wznonbps}. Finally, one could check the presence of the gravitational terms in \eq{wznonbps}
explicitly. Since all graviton vertex operators can be inserted in the ($0,0$) picture, the various
contractions will be identical to the ones used in Ref.~\cite{anomalous}.

Note that, from a technical point of view, the only role of the tachyon in the above computations
is to provide its superghost part, allowing one to insert the R-R vertex operator in the $(-1/2,-1/2)$
picture, instead of the ($-3/2,-1/2$) one. Thus the inclusion of the tachyon proves wrong one's first
impression that non-BPS D-branes cannot couple to the closed string R-R sector because of the
GSO-projection. Apart from this, the above computations perfectly parallel their counterparts for
BPS D-branes.

\paragraph{Acknowledgement}
We would like to thank Pieter-Jan De Smet and Walter Troost for discussions.


text:
\section{Introduction}
Let $\randbin{n}$ be a uniformly distributed random binary string of length $n$. For any string $s$, let $\countuniq{m}(s)$ be the number of \emph{distinct} subsequences of $s$ of length $m$, and let $\countuniq{}(s)$ be the total number of distinct subsequences. We consider the random variables $\randcount{n}{m} = \countuniq{m}(\randbin{n})$ and $\randcount{n}{} = \countuniq{}(\randbin{n})$. Note that we are counting sub\emph{sequences}, not substrings (the latter would have to be contiguous portions of $s$). Questions about such random varaibles arise in the study of error-correcting codes for deletion channels \cite{MR2525669} -- communication channels in which each bit has some probability of being dropped, with the receiver getting no indication of where these deletions occurred. When a codeword $W$ is sent on such a channel, the receiver sees a subsequence of $W$.

We denote the expectation of a random variable $A$ by $\hat A$. Given a string $s = (s_0s_1\cdots s_{n-1})$, we write $s[i:j]$ for the substring $s = (s_is_{i+1}\cdots s_{j-1})$. A \emph{run} is a maximal constant substring. A sequence of length $m$ is called an $m$-sequence (similarly an $m$-subseqence, $m$-string et cetera).

\subsection{Subsequences of length $m$}
If a string $s$ begins with a run of length $k$, then we can assume with no loss of generality -- since we are only counting \emph{distinct} subsequences -- that  every nonempty subsequence starts at either $s_0$ or $s_k$. Now $\countuniq{m}(s)$ equals the number of distinct $m$-subsequences that start with zero, plus the number that start with 1, i.e.
\begin{equation}
\countuniq{m}(s) = \countuniq{m-1}(s[1:n]) + \countuniq{m-1}(s[k+1:n]).
\end{equation}
where the second term is zero when $k=n$.
Of course $\randbin{n}$ has probability $2^{-k}$ of starting with a run of length $k$ if $k<n$, and probability $2^{1-n}$ of starting with a run of length $n$ (i.e. being a constant sequence); thus (by linearity of expectation) $\hat\randcount{n}{m}$ satisfies the recurrence
\begin{equation}
  \label{eq:countuniqM}
\hat \randcount{n}{m} = \hat \randcount{n-1}{m-1} + \sum_{k \geq 1}^{n-1} 2^{-k} \hat\randcount{n-k-1}{m-1}
\end{equation}
Since $\hat\randcount{n}{m}=0$ when $m>n$ we can (for $m>1$) write this as
\begin{equation}
  \label{eq:countuniqM2}
\hat \randcount{n}{m}  = \sum_{k \geq 0}^{\infty} 2^{-k} \hat\randcount{n-k-1}{m-1}
\end{equation}
from which we obtain a recurrence relation similar to the recurrence for the binomial coefficents:
\begin{theorem}
\begin{equation}
\hat \randcount{n}{m} = \hat \randcount{n-1}{m-1} + \frac{1}{2}\hat\randcount{n-1}{m}
\end{equation}
with initial conditions $\hat\randcount{n}{n}=1$ and $\hat\randcount{n}{0}=1$.
\end{theorem}

Note in particular that $\hat\randcount{n}{n-1} = \frac{n+1}{2}$. This must be the expected number of runs in $X_n$, since an $(n-1)$-subsequence is determined entirely by the run from which one bit is deleted. Indeed the first bit of $\randbin{n}$ starts a new run with probability 1, while each subsequent bit starts a new run with probability $\frac{1}{2}$, again giving (by linearity of expectation) $\hat\randcount{n}{n-1}=\frac{n+1}{2}$.

More generally,  $\hat\randcount{n}{n-m}$ is, for fixed $m$, a polynomial of degree $m$.
Let $\hat\randcount{n}{n-m} = p_m(n) = \sum_{i=0}^m \alpha_{m,i} n^i$. Now
\[
p_m(n) = p_m(n-1) + \frac{1}{2}p_{m-1}(n-1)
\]
thus equating coefficients on $n^{m-1}$ gives
\[
\alpha_{m,m-1} = -m\alpha_{m,m} + \alpha_{m,m-1} + \frac{1}{2}\alpha_{m-1,m-1}
\]
so
\[
\alpha_{m,m} = \frac{1}{2m}\alpha_{m-1,m-1} \ .
\]
Thus (since $\alpha_{0,0} = 1$) we obtain $\alpha_m = 1/(2^mm!)$ and we have the approximation
\begin{theorem}
\begin{equation}
  \label{eq:binomialApprox}
  \hat\randcount{n}{n-m} = 2^{-m} {n \choose m} + O(n^{m-1})
\end{equation}
\end{theorem}

\subsection{Total Number of Subsequences}
By the same reasoning as in (\ref{eq:countuniqM}) we have
\begin{equation}
\countuniq{}(s) = 1+ \countuniq{}(s[1:n]) + \countuniq{}(s[k+1:n]).
\end{equation}
when $s$ begins with a $k$-run (the initial $1$ counts the empty subsequence), and thus
\begin{equation}
  \label{eq:countuniqAll}
  \hat\randcount{n}{} = 1 + \hat\randcount{n-1}{} + \sum_{k \geq 1}2^{-k}\hat\randcount{n-k-1}{}
  = 1 + \sum_{k \geq 0}2^{-k}\hat\randcount{n-k-1}{} \ .
\end{equation}
Noting that
\begin{equation}
  \sum_{k \geq 1}2^{-k}\hat\randcount{n-k-1}{} = \frac{\hat\randcount{n-1}{}-1}{2}
\end{equation}
we have
\begin{equation}
  \hat\randcount{n}{} = \frac{1}{2} + \frac{3}{2}\hat\randcount{n-1}{}
\end{equation}
and $\hat\randcount{0}{}=1$, thus
\begin{theorem}
\begin{equation}
  \label{eq:countuniqAllSltn}
  \hat\randcount{n}{} = 2\left( \frac{3}{2} \right)^n - 1 \ .
\end{equation}
\end{theorem}
This improves an earlier result \cite{FlaxmanHS04} that $\hat\randcount{n}{} = O((3/2)^n)$.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
$CP$ violation (CPV) is one of the most important features in particle physics.
In the standard model (SM),
it arises from a unique physical phase of the Cabibbo-Kobayashi-Maskawa (CKM)
quark mixing matrix \cite{ckm}.
However, the origin of CPV remains unclear.
On the other hand, it is unknown whether CPV also occurs in the lepton sector.

In general, there are two possible sources of CPV: complex  couplings of the Yukawa interactions and
complex vacuum expectation values (VEVs) of the scalar fields,
which explicitly and spontaneously break $CP$ symmetry, such as the cases of the SM and
multi-Higgs doublet models~ \cite{CPV},
respectively.

Recently, a new spontaneous $CP$ violation mechanism has been
proposed by Friedberg and Lee \cite{timeon}.
They introduced a  gauge singlet pseudoscalar which couples to quarks
to yield a new type of Yukawa interactions.
As the Yukawa couplings
 are assumed to be real, the system is $CP$ and $T$ conserving.
Once the pseudoscalar acquires a VEV,
both $CP$ and $T$ are spontaneously broken.
The excitation of the pseudoscalar from its vacuum predicts a new neutral
 scalar field named as ``timeon''.
One of the most interesting features of the proposal is that the pseudoscalar is also
 responsible for the masses of up and down quarks,
 but not for the heavy quarks.
As a result, the timeon mass and the VEV can be much lower than electroweak scale.

The timeon model, however, has a potential problem of flavor changing neutral
 currents (FCNCs) mediated by the timeon.
As we shall explain later, the timeon mass is strongly constrained by the mass
 mixings of the neutral mesons and must be much larger than its VEV.
This fact implies the strongly interacting quartic timeon coupling and
the broken down of the perturbation.
In this paper, in order to save the model,
we introduce a
 small parameter $\epsilon$ and
 re-define the VEV as $\tau_0 \rightarrow \epsilon \tau_0$.
We will demonstrate that the small $\epsilon$ can relax the
phenomenological constraints, which permit
a much smaller mass of  the timeon.
We would also extend the timeon model to the lepton sector and
explore leptonic FCNC processes, such as the lepton number violating decays and
the muon anomalous magnetic moment.

The paper is organized as follows.
In Sec. 2, we give a brief review of the timeon model and calculate the
 mixing parameters in the neutral meson systems.
In Sec. 3, we extend the timeon model to the lepton sector.
We present the summary in Sec. 4.

\section{Quark sector}
\subsection{Model}
In the theory of timeon, proposed by
Friedberg and Lee~\cite{timeon},
the quark mass Hamiltonian is written as
\begin{eqnarray}
{\cal H}_q
 &=&\bar{u}_i[G_u + i\gamma_5  \tau_0 F]_{ij}u_j
        +\bar{d}_i[G_d + i\gamma_5  \tau_0 F]_{ij}d_j \nonumber\\
 &=&\bar{u}_{Li}[G_u + i \tau_0 F]_{ij}u_{Rj}
        +\bar{d}_{Li}[G_d + i \tau_0 F]_{ij}d_{Rj}
        +h.c.\ ,\label{eq:quark}
\end{eqnarray}
where the constant $\tau_0$ is the VEV of
the new $CP$ odd and $T$ odd,
gauge singlet pseudoscalar field $\tau(x)$, $i,j=1,\cdots, 3$ stand for family indices and
$G_{u,d}$ and $F$ are $3 \times 3$ matrices, given by
\begin{eqnarray}
G_{u,d}=
 \left(\begin{array}{ccc}
 b_{u,d} \eta_{u,d}^2 (1+\xi_{u,d}^2) & -b_{u,d} \eta_{u,d} &
 -b_{u,d} \xi_{u,d} \eta_ {u,d}\\
 -b_{u,d} \eta_{u,d} & b_ {u,d}+ a_{u,d} \xi_{u,d}^2 & -a_{u,d} \xi_{u,d} \\
 -b_{u,d} \xi_{u,d} \eta_{u,d} & -a_{u,d} \xi_{u,d} & a_{u,d} + b_{u,d}
 \end{array}\right)\label{eq:Gq}\
\end{eqnarray}
and
\begin{eqnarray}
F=
 \left(\begin{array}{ccc}
 \cos^2\alpha_q & \sin\alpha_q \cos\alpha_q \cos\beta_q &
  \sin\alpha_q \cos\alpha_q \sin\beta_q \\
 \sin\alpha_q \cos\alpha_q \cos\beta_q & \sin^2\alpha_q \cos^2\beta_q &
  \sin^2\alpha_q \sin\beta_q \cos\beta_q \\
 \sin\alpha_q \cos\alpha_q \sin\beta_q & \sin^2\alpha_q \sin\beta_q \cos\beta_q &
  \sin^2\alpha_q \sin^2\beta_q
 \end{array}\right)\ \label{eq:Fmatrix}
\end{eqnarray}
with the real parameters of $a_{u,d} , b_{u,d} , \eta_{u,d}$ and $\xi_{u,d}$ and two angles of
$\alpha_q$ and $\beta_q$, respectively.
Furthermore,  $\eta_{u,d}$ and $\xi_{u,d}$ in Eq. (\ref{eq:Gq}) can be also defined by four angles as
\begin{eqnarray}
&&\xi_u = \tan\phi_u,\ \eta_u = \tan\varphi_u \cos\phi_u\ , \\
&&\xi_d = \tan\phi_d,\ \eta_d = -\tan\varphi_d \cos\phi_d\ .
\end{eqnarray}
The potential of the pseudoscalar field is given by~\cite{timeon}
\begin{eqnarray}
V(\tau)&=& -{1\over 2}\lambda\tau^2\left(\tau_0^2-{1\over 2}\tau^2\right)\ ,
\end{eqnarray}
which leads to the mass of the new quantum, $i.e.$, timeon, to be
\begin{eqnarray}
M_T&=&(2\lambda)^{1\over 2}\tau_0\,.
\label{eq:MT}
\end{eqnarray}

As  seen from Eq. (\ref{eq:Gq}), $G_{u,d}^* =G_{u,d}$ and $\det G_{u,d}=0$, and thus
 up and down quarks of the first generation  are massless before $\tau(x)$ receives the VEV.
After $\tau(x)$ gets the VEV, $CP$ and $T$ are spontaneously broken
and the first generation quarks acquire small masses.
Note that since the timeon is not responsible for the masses of heavy quarks,
its VEV can be much lower than the electroweak scale.

\subsection{Flavor Changing Neutral Currents}
In Ref. \cite{timeon}, it was estimated that
 $\tau_0 \simeq 33\ {\rm MeV}$, which implies that
 \begin{eqnarray}
 M_T&\leq& 47\ {\rm MeV}
 \label{Eq:MT0}
 \end{eqnarray}
 from Eq. (\ref{eq:MT}) if
 $\lambda\leq 1$.
 Therefore, the mass of the timeon is much lower than
the electroweak scale.
However, a light timeon could give rise to dangerous
 FCNC processes because in general $G_{u(d)}$ and $F$
 cannot be diagonalized simultaneously.
In fact, in the mass eigenstate basis of quarks, flavor changing timeon
 couplings are given by
 %
\begin{eqnarray}
 &&F^u \equiv U_{u}^{T}FU_{u}=
  \left(\begin{array}{ccc}
   0.08 & -0.14 & 0.23 \\
   -0.14 & 0.25 & -0.41 \\
   0.23 & -0.41 & 0.67
  \end{array}\right)\ , \nonumber\\
 &&F^d \equiv U_{d}^{T}FU_{d}=
    \left(\begin{array}{ccc}
   0.15 & -0.18 & 0.31 \\
   -0.18 & 0.21 & -0.36 \\
   0.31 & -0.36 & 0.64
  \end{array}\right)\ ,
  \label{Eq:FU}
\end{eqnarray}
where $U_u$ and $U_d$ are the unitary matrices to diagonalize  up and down-type
 quarks, respectively.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c||c|c|}\hline
 parameter & input & parameter &input \\ \hline
 $m_u(2\ {\rm GeV})$ & $2.5\times 10^{-3}\ {\rm GeV}$ & $f_K$ & $0.16\ {\rm GeV}$ \\ \hline
 $m_d(2\ {\rm GeV})$ & $5\times 10^{-3}\ {\rm GeV}$ & $f_{B_s}$ & $0.24\ {\rm GeV}$ \\ \hline
 $m_s(2\ {\rm GeV})$ & $0.095\ {\rm GeV}$ & $f_{B_d}$ & $0.198\ {\rm GeV}$ \\ \hline
 $m_c(m_c)$ & $1.25\ {\rm GeV}$ & $f_D$ & $0.223\ {\rm GeV}$ \\ \hline
 $m_b(m_b)$ & $4.2\ {\rm GeV}$ & $M_K$ & $0.497\ {\rm GeV}$ \\ \hline
 $M_{B_s}$ & $5.366\ {\rm GeV}$ & $M_{B_d}$ & $5.280\ {\rm GeV}$ \\ \hline
 %
 $M_D$ & $1.865\ {\rm GeV}$ &  &  \\ \hline
\end{tabular}
\end{center}
\caption{Values of parameters used in the calculation.
Here, we have used the same quark masses as those in Ref. \cite{timeon},
while the decay constants and the meson masses  are the center values  in
 Ref. \cite{kifune}.}
\label{tab:para}
\end{table}
From the values in Eq. (\ref{Eq:FU})  and inputs in Table \ref{tab:para},
the mass mixing parameter  in the neutral $K$ meson system due to the timeon contribution at tree level is
estimated to be
\begin{eqnarray}
 \Delta M_K^{timeon}
 &=&2\left| \frac{2F_{ds}^d (F_{sd}^d)^{*} }{M_T^2}
 <K^0|\bar{d}_L^\alpha s_R^\alpha \bar{d}_R^\beta s_L^\beta |\bar{K^0}>
 \right.\nonumber\\
 &&\hspace{1cm}\left. +\frac{(F_{ds}^d)^2 + (F_{sd}^d)^{*2} }{M_T^2}
 <K^0|\bar{d}_L^\alpha s_R^\alpha \bar{d}_L^\beta s_R^\beta |\bar{K^0}>
 \right| \nonumber\\
 &\sim & \frac{1.76\times 10^{-3}\ {\rm GeV}^3}{M_T^2}\,.

 \label{eq:mk}
\end{eqnarray}
From the experimental data of
$\Delta M_K^{exp}\simeq 3.5\times 10^{-15}\ {\rm GeV}$,
without loss of generality we find the lower limit of the timeon mass as
\begin{eqnarray}
M_T \ge 7\times 10^{5}\ {\rm GeV}\,,
 \label{Eq:MT1}
 \end{eqnarray}
which is much higher than that in Eq. (\ref{Eq:MT0}).
Here, we have ignored the renormalization group effects as well as some hadronic uncertainties.
We note that the mixing parameters  in other neutral meson systems also lead to similar constraints on $M_T$.
Therefore,
in the timeon model of Ref. \cite{timeon}, $M_T\gg \tau_0$,
which requires the strongly interacting quartic timeon coupling, $i.e.$, $\lambda\gg 1$.

In order to solve the problem, in this paper, we introduce a small dimensionless
 parameter $\epsilon$ for the terms related to $F$ in Eq. (\ref{eq:quark}), which is equivalent
to a  re-defined  VEV  of
\begin{eqnarray}
 \tau_0 \rightarrow \epsilon\tau_0 \simeq 33\ {\rm MeV} \,.\label{eq:vev2}
\end{eqnarray}
If $\epsilon$ is very small,
the constraints from the FCNC processes can be relaxed
becouse the left hand side of Eq. (\ref{Eq:MT1}) is replaced by
$M_T/\epsilon$.
In this case,
the constraint from $\Delta M_K$ becomes
\begin{eqnarray}
 M_T \ge 151\ {\rm GeV}\ .
\end{eqnarray}
By assuming that $\tau_0\simeq M_T\sim 151\ {\rm GeV}$, $i.e.$, $\lambda\simeq 1/2$,
we  obtain that $\epsilon \sim 0.22\times 10^{-3}$ from
 Eq. (\ref{eq:vev2}).

However, If we use a  smaller $\tau_0$ or a larger $\epsilon$,
$M_T$ has to be larger
 than $151\ {\rm GeV}$.
 %
In this paper,
we take $151\ {\rm GeV}$ as the lowest value of the timeon mass.
Consequently, we can also estimate the mixings
in $B_s$, $B_d$ and $D$ meson systems from the timeon contributions.
By using  $M_T \sim 151\ {\rm GeV}$, we obtain
\begin{eqnarray}
 &&\Delta M_{B_s}^{timeon} \sim 0.365\times 10^{-13}\ {\rm GeV}
   <\Delta M_{B_s}^{exp}\simeq 1.17\times 10^{-11}\ {\rm GeV}\ ,\\
 &&\Delta M_{B_d}^{timeon} \sim 0.178\times 10^{-13}\ {\rm GeV}
   <\Delta M_{B_d}^{exp}\simeq 3.34\times 10^{-13}\ {\rm GeV}\ ,\\
 &&\Delta M_{D}^{timeon} \sim 0.205\times 10^{-14}\ {\rm GeV}
   <\Delta M_{D}^{exp}\simeq 1.4\times 10^{-14}\ {\rm GeV}\ .
\end{eqnarray}
It is clear that all the FCNC processes are suppressed if $\epsilon\le 0.22\times 10^{-3}$
and $M_T \ge 151\ {\rm GeV}$.

Now we would like to briefly comment on the possible origin of the small parameter
 $\epsilon$.
Since $\tau(x)$ is a gauge singlet scalar field,
it cannot be
 incorporated into the renormalizable $SU(2)_L \times U(1)_Y$ model.
One way out is to consider
a non-renormalizable interacting term, such as,
\begin{eqnarray}
 \frac{F_{ij}}{\Lambda}\bar{\psi}_{Li} \Phi \psi_{Rj} \tau\ ,
\end{eqnarray}
where $\Phi$ is a $SU(2)_L$ doublet scalar and
 $\Lambda$ is a typical energy scale of the model.
In this case, $<\Phi>/\Lambda$ might be able to explain the existence of
the small parameter $\epsilon$.
As the construction of realistic models goes beyond our purpose of
this paper, we do not discuss this point further.

\section{Lepton sector}
\subsection{Model}
In this section, we extend the timeon model to the lepton sector.
The lepton mass Hamiltonian is similar to Eq. (\ref{eq:quark}),
 given by
\begin{eqnarray}
{\cal H}_l
 &=&\bar{\ell}_{i}[G_\ell + i\gamma_5 \epsilon \tau_0 F]_{ij}\ell_{j}
        +\bar{\nu}_i^{(c)}[G_\nu + i\gamma_5 \epsilon_\nu \tau_0 F]_{ij}\nu_j\ ,
         \label{eq:lepton}
\end{eqnarray}
where $\ell_i$ and $\nu_i$ represent charged leptons and neutrinos, respectively.
Note that in the charged lepton sector, we use the same parameter $\epsilon$ as that of
the quark sector, whereas in the
neutrino sector we assume a different small parameter $\epsilon_\nu$ as
 neutrino masses would be due to some new physics such as the
See-Saw mechanism, which may have a different origin from that of the charged fermions
\footnote{We do not restrict ourselves to the case of Majorana neutrinos.}.
The matrix $F$ takes the same form as Eq. (\ref{eq:Fmatrix}) but described
by different angles $\alpha_l$ and $\beta_l$.
For  $G_{\ell,\nu}$, we assume the following simple textures of
\begin{eqnarray}
G_\ell=
 \left(\begin{array}{ccc}
 0 & 0 & 0 \\
 0 & b_\ell + a_\ell & a_\ell \\
 0 & a_\ell & a_\ell + b_\ell
 \end{array}\right)\label{eq:Gell}\ ,
\end{eqnarray}
\begin{eqnarray}
G_\nu=
 \left(\begin{array}{ccc}
 \frac{1}{2}b_\nu & \sqrt{\frac{1}{2}}b_\nu & 0 \\
 \sqrt{\frac{1}{2}}b_\nu &  b_\nu & 0 \\
 0 & 0 & a_\nu + b_\nu
 \end{array}\right)\label{eq:Gnu}\,,
\end{eqnarray}
respectively.
Note that this corresponds to
$\eta_\ell = \xi_\nu = 0,\xi_\ell=-1$ and $\eta_\nu = -\sqrt{\frac{1}{2}}$
\cite{FL}.

Here, we would like to comment on several features of the model.
As shown in Ref. \cite{FL}, Eqs. (\ref{eq:Gell}) and (\ref{eq:Gnu})
imply the exact tri-bimaximal lepton mixing matrix \cite{TB}.
That is,
\begin{eqnarray}
 \sin^2 \theta_{12}= \frac{1}{3}\ ,\ \sin^2 \theta_{23} = \frac{1}{2}\ ,\
 \sin^2 \theta_{13} =0\ .
\end{eqnarray}
After $\tau(x)$ receives the VEV,  the mixing matrix deviates from exact tri-bimaximal
mixing.
However, if $\alpha_l =0$, there is no flavor changing timeon coupling in the charged
 lepton sector, while $\sin^2 \theta_{13}=0$ and
$\sin^2 \theta_{23}=1/2$ remain.
On the other hand,
if $\alpha_l\neq 0$, $\sin^2 \theta_{13}$ and $\sin^2 \theta_{23}$ will be different from
$0$ and $1/2$, respectively, and flavor changing timeon couplings in the charged
 lepton sector are induced.
As in the case of the quark sector, $CP$ is spontaneously broken.
We note that if $\alpha_l =0$, the $CP$ violating Dirac phase in the neutrino sector will not show up
due to $\sin^2 \theta_{13}=0$.

Since our model is a special case of Ref. \cite{timeon},
the analytic study
is the same as that in Ref. \cite{timeon}.
Because the values of $\epsilon$ and $\tau_0$ are already determined
as $\epsilon=0.22\times 10^{-3}$ and $\tau_0 = 151\ {\rm GeV}$ in the quark sector,
 the model is described by seven
 parameters: $a_\ell , b_\ell , a_\nu , b_\nu , \alpha_l , \beta_l$ and $\epsilon_\nu$,
which can be  fixed by  seven physical quantities.
In our numerical calculation, we take~\cite{PDG}
\begin{eqnarray}
 m_e = 0.511\ {\rm MeV} ,\ m_\mu = 105.658\ {\rm MeV} ,\
 m_\tau = 1776.84 \pm 0.17\ {\rm MeV} ,
 \label{Eq:D1}
\end{eqnarray}
and~\cite{osi}
\begin{eqnarray}
 \Delta m_{21}&=&(7.45- 7.88)\times 10^{-5}\ {\rm eV}^2 \,,\
 \Delta m_{31}=(2.29-2.52)\times 10^{-3}\ {\rm eV}^2 ,
 \nonumber\\
 \sin^2 \theta_{12}&=& 0.288- 0.326 \,,\
 \sin^2 \theta_{23}= 0.44- 0.57 \,.
 \label{Eq:D2}
\end{eqnarray}
One may think that if we also use $\sin^2 \theta_{13}$, we can determine the
 value of $\epsilon\tau_0$ without the result in
 the quark sector.
However, the experimental upper limit of $\sin^2 \theta_{13}$ is too mild to constrain
 the model.

Nevertheless, the value of $\sin^2 \theta_{13}$ is obtained as a prediction
 of the model.
 From the  quantities in Eqs. (\ref{Eq:D1}) and  (\ref{Eq:D2}), we get
$\alpha_l \simeq 82.89^\circ$.
Hence, our model gives a small  $\sin^2 \theta_{13}$ as well as FCNCs
 mediated by the timeon.
In Figure \ref{fig:m1-rct}, we  show the allowed values of
$m_1^\nu$ and $  \sin^2 \theta_{13}$.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.8\textwidth]{m1-rct.eps}
\caption{\footnotesize
The predictions of the model in $m_1^\nu - \sin^2 \theta_{13}$ plane.
}\label{fig:m1-rct}
\end{center}
\end{figure}

\subsection{Leptonic Processes}
As  mentioned in the previous subsection, $\alpha_l \neq 0$ results in the
existence of flavor changing timeon couplings.
The branching ratios of the lepton flavor violating (LFV)
 decays due to these coupling are given by~\cite{lfv}
\begin{eqnarray}
&&Br(\mu\rightarrow e\gamma)
=\frac{\alpha_{em}\tau_\mu}{2^{10}\pi^4}
  \frac{m_\mu^3 m_\tau^2}{M_T^4}
  \left(\epsilon F_{\mu\tau}\ \epsilon F_{e\tau}\right)^2
  \left| \ln\frac{m_\tau^2}{M_T^2} + \frac{3}{2} \right|^2\ ,
  \label{eq:meg}\\
  &&Br(\ell^- \rightarrow \ell_3^- \ell_2^+ \ell_1^-)
=\frac{5}{3}\frac{\tau_\ell}{2^{11}\pi^3}\frac{m_\ell^5}{M_T^4}
  \left(\epsilon F_{\ell_3\ell}\ \epsilon F_{\ell_2\ell_1}\right)^2\,,
\label{eq:t3m}
\end{eqnarray}
where $\alpha_{em}$ is the fine structure constant, $\ell=\mu$ or $\tau$, $\ell_i=e$ or $\mu$, and $\tau_{\ell}$
is the life time of the $\ell$ lepton.
From Eqs. (\ref{eq:meg}) and (\ref{eq:t3m}), we obtain
\begin{eqnarray}
&&Br(\mu\rightarrow e\gamma)
\simeq 10^{-16} - 10^{-21}\,,
\nonumber\\
&&Br(\mu^- \rightarrow e^- e^+ e^-)
\simeq 10^{-21} - 10^{-26}\,,
\nonumber\\
&&Br(\tau^- \rightarrow e^- e^+ e^-)
\simeq 3\times 10^{-21}\,,
\nonumber\\
&&Br(\tau^- \rightarrow \mu^- \mu^+ \mu^-)
\simeq 10^{-19} - 10^{-35}\ .
\end{eqnarray}
Similarly, we find that the timeon contribution to
the muon anomalous magnetic moment is
\begin{eqnarray}
 \Delta a_\mu &=&\frac{1}{8\pi^2}
 \frac{m_\mu^2 m_\tau^2}{M_T^2}
 \left(\epsilon F_{\mu\tau}\right)^2
  \left( \ln\frac{m_\tau^2}{M_T^2} + \frac{3}{2} \right)\nonumber\\
 &\simeq& 10^{-16} - 10^{-21}\ .
\end{eqnarray}
It is clear that all contributions to the leptonic processes from the timeon are too small to be measured in the
 near future.

\section{Summary}
We have investigated the timeon model, which is a new kind of
 spontaneous $CP$ violation mechanism, recently proposed by Friedberg and Lee~\cite{timeon}.
However, the original model has a potential problem of FCNCs mediated by the timeon
 if the timeon mass $M_T$ is smaller than $7\times 10^5\ {\rm GeV}$.
In order to avoid  the problem, we have introduced a small dimensionless parameter
to suppress FCNCs and thus allow a smaller $M_T$.
 Due to the constraints from the mass mixing parameters in the $K^0-\bar{K}^0$ and other neutral meson systems,
 we have shown that  $M_T \ge 151\ {\rm GeV}$.

We have also extended the timeon model to the lepton sector.
We have found  that our simple model predicts a small but non-zero $\sin^2 \theta_{13}$ as well as
 non-zero lepton flavor violating timeon couplings.
However, all contributions to the LFV decays and muon anomalous magnetic moment due to the timeon
  are not measurable. \\

 \noindent {\bf Acknowledgments}

This work is supported in part by
the National Science Council of ROC under Grant No:
 NSC-95-2112-M-007-059-MY3.