Measurement of the CP observables in $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ and
first observation of $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$

Steven R. Blusk
Department of Physics
Syracuse University
Syracuse, NY 13244, USA
Proceedings of CKM 2012, the 7th International Workshop on the CKM Unitarity
Triangle, University of Cincinnati, USA, 28th September - 2 October 2012

##  1 Introduction

A central goal of flavor physics is to measure the angle ${\gamma\equiv{\rm
arg}\left(-{V_{\rm ub}^{*}V_{\rm ud}\over V_{\rm cb}^{*}V_{\rm cd}}\right)}$
in the Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] mixing matrix, which is
currently known to a precision of about 10-12o [3]. The theoretically cleanest
methods employ $B\rightarrow DK$ decays, where the sensitivity to $\gamma$
results from the interference between $b\rightarrow c$ and $b\rightarrow u$
transitions. Since both transitions are $\mathcal{O}(\lambda^{3}$) in the
Wolfenstein parameter [4], large CP violating asymmetries are expected. One
powerful class of methods utilize $B^{-}\rightarrow DK^{-}$ where the $D$ is
detected in either a CP eigenstate [7], a flavor-specific mode [6], or a
multi-body decay [8]. An advantage of these decays is that they do not require
knowledge of the $b$-hadron flavor at production (flavor tagging), and only
rely on measuring the time integrated rates. Another powerful method to
extract $\gamma$ is to perform a time-dependent analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ [9,
10, 11] and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$. Time-dependent analyses of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$$(\pi^{+}\pi^{-})$ are only possible at hadron colliders, and
are a unique capability of LHCb.

The time-dependent decay rates of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to a flavor-specific final
state, $f=D^{+}_{s}K^{-}$, is given by:

$\displaystyle\frac{d\Gamma_{B^{0}_{s}\rightarrow f}(t)}{dt}$
$\displaystyle=\frac{1}{2}|A_{f}|^{2}(1+|\lambda_{f}|^{2})e^{-\Gamma_{s}t}$
$\displaystyle\left[\cosh\left(\frac{\Delta\Gamma_{s}t}{2}\right)+D_{f}\sinh\left(\frac{\Delta\Gamma_{s}t}{2}\right)\right.$
(1) $\displaystyle\left.\phantom{\Big{(}}+C_{f}\cos\left(\Delta
m_{s}t\right)-S_{f}\sin\left(\Delta m_{s}t\right)\right.\bigg{]},$
$\displaystyle\frac{d\Gamma_{\overline{B}^{0}_{s}\rightarrow f}(t)}{dt}$
$\displaystyle=\frac{1}{2}|A_{f}|^{2}\left|\frac{p}{q}\right|^{2}(1+|\lambda_{f}|^{2})e^{-\Gamma_{s}t}$
$\displaystyle\left[\cosh\left(\frac{\Delta\Gamma_{s}t}{2}\right)+D_{f}\sinh\left(\frac{\Delta\Gamma_{s}t}{2}\right)\right.$
(2) $\displaystyle\left.\phantom{\Big{)}}-C_{f}\cos\left(\Delta
m_{s}t\right)+S_{f}\sin\left(\Delta m_{s}t\right)\right.\bigg{]},$

where $A_{f}$ is the decay amplitude $A(B^{0}_{s}\rightarrow f)$ and
$\lambda_{f}=(q/p)(\overline{A}_{f}/A_{f})=|\lambda_{f}|e^{i(\Delta-(\gamma-2\beta_{s}))}$.
Here, $|\lambda_{f}|$ and $\Delta$ are the relative magnitude and strong phase
difference between the $b\rightarrow u$ and $b\rightarrow c$ transitions, and
$2\beta_{s}$ is the phase of $B^{0}_{s}$ mixing. The complex coefficients $p$
and $q$ relate the $B^{0}_{s}$ meson mass eigenstates, $B_{H,L}$, to the
flavor eigenstates, $B^{0}_{s}$ and $\overline{B}^{0}_{s}$ via:

$\begin{array}[]{c}B_{L}=pB^{0}_{s}+q\overline{B}^{0}_{s}\\\
B_{H}=pB^{0}_{s}-q\overline{B}^{0}_{s}\end{array},\qquad|p|^{2}+|q|^{2}=1\,.$
(3)

Similar equations can be written for the $C\\!P$-conjugate decays, replacing
$A_{f}$ by $\overline{A}_{\overline{f}}=A(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow\overline{f})$,
$\lambda_{f}$ by
$\overline{\lambda}_{\overline{f}}=(p/q)(A_{\overline{f}}/\overline{A}_{\overline{f}})$,
$|p/q|^{2}$ by $|q/p|^{2}$, $C_{f}$ by $C_{\overline{f}}$, $S_{f}$ by
$S_{\overline{f}}$, and $D_{f}$ by $D_{\overline{f}}$. The $C\\!P$ asymmetry
observables $C_{f}$, $S_{f}$, $D_{f}$, $C_{\overline{f}}$, $S_{\overline{f}}$
and $D_{\overline{f}}$ are then given by

$\displaystyle
C_{f}=C_{\overline{f}}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\,,\quad\quad
S_{f}=\frac{2\mathcal{I}m(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,,\quad\quad
D_{f}=\frac{2\mathcal{R}e(\lambda_{f})}{1+|\lambda_{f}|^{2}},\quad$
$\displaystyle
S_{\overline{f}}=\frac{2\mathcal{I}m(\overline{\lambda}_{\overline{f}})}{1+|\overline{\lambda}_{\overline{f}}|^{2}}\,,\quad\quad
D_{\overline{f}}=\frac{2\mathcal{R}e(\overline{\lambda}_{\overline{f}})}{1+|\overline{\lambda}_{\overline{f}}|^{2}}\,.$
(4)

Since CP violation in mixing is expected to be below the percent level, it
follows that $|q/p|=1$, $|\lambda_{f}|=|\overline{\lambda}_{\overline{f}}|$,
and consequently $C_{f}=C_{\overline{f}}$. Thus there are five observables
that depend on the 3 physics parameters of interest: $|\lambda_{f}|$, $\Delta$
and $\gamma-2\beta_{s}$. Similar expressions are applicable to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, however, there is a potential dilution due to
the varying strong phase across the $D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ Dalitz
plane.

In this article, we present the first measurements of these five CP
observables. First observations of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{s}\rightarrow D_{s1}(2536)^{+}\pi^{-}$
decays are also presented, along with measurements of their relative branching
fractions. All results are based on 1.0 fb-1 of integrated luminosity recorded
in 2011 by the LHCb experiment. More detailed documentation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ analyses can be found in Refs. [12] and [13],
respectively.

##  2 Event Selection

Signal $D^{+}_{s}$ candidates are formed by reconstructing
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$,
$D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{+}\pi^{-}\pi^{+}$. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates, only the $D^{+}_{s}\rightarrow
K^{+}K^{-}\pi^{+}$ decay is considered. The $D^{+}_{s}$ candidates are
required to form a good quality vertex, be spacially well separated from any
primary vertex (PV), and have an invariant mass consistent with the known
$D^{+}_{s}$ mass (within about 3 times the mass resolution). Multivariate
selection algorithms are employed to suppress the combinatorial background,
and typically have a signal efficiency of 80-90% while rejecting about 85% of
the combinatorial background. Invariant mass distributions for $D^{+}_{s}$
candidates are shown in Fig. 1 for the higher signal yield $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
decay, showing that clean signals are achievable even in the suppressed
$D^{+}_{s}$ decay modes.

Figure 1: Invariant mass distributions for $D^{+}_{s}$ candidates in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$ data sample, for (left) $K^{+}K^{-}\pi^{+}$, (middle)
$K^{+}\pi^{-}\pi^{+}$, and (right) $\pi^{+}\pi^{-}\pi^{+}$ final states.

Tighter particle identification requirements are applied to the $K^{-}$ or
$K^{-}\pi^{+}\pi^{-}$ recoiling from the $D^{+}_{s}$ to suppress cross-feed
from the favored $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decays. For the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ decays, the invariant mass of the
$\pi^{-}\pi^{+}\pi^{-}$ and $K^{-}\pi^{+}\pi^{-}$ systems are restricted to be
below 3000 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

##  3 Analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$

The invariant mass distributions for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$ are shown in Figs. 2 and 3. All three $D^{+}_{s}$ decay modes
have approximately equal $B^{0}_{s}$ mass resolutions, and are summed together
in these distributions. The signal shape is modeled as the sum of two Crystal
Ball [14] functions, with one exponential tail on each side of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal peak. A number of
specific backgrounds, due to either a missed particle (e.g. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\rho^{-}$,
with the $\pi^{0}$ undetected), a misidentified particle (e.g. $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
reconstructed as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$), or
both (e.g. $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\rho^{-}$ reconstructed as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$) are
accounted for using either data or simulation to model the shape of these
backgrounds. From an unbinned extended maximum likelihood fit, $27,965\pm 395$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$ and $1390\pm 98$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
signal events are selected.

Figure 2: Invariant mass distributions $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
candidates. The signal component is indicated by the dashed curve, and the
backgrounds are indicated by the various color-filled (shaded, in B/W) curves.
Figure 3: Invariant mass distributions $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
candidates. The signal component is indicated by the dashed curve, and the
backgrounds are indicated by the various color-filled (shaded, in B/W) curves.

The CP parameters are obtained by a fit to the decay time distribution of the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$ signal candidates. Two methods have been developed. The first,
referred to as sFit, uses sWeights [15] obtained from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ mass
fit to statistically subtract the background contribution. The second method,
referred to as cFit, is a conventional two-dimensional fit to the
reconstructed mass and decay time. The advantage of the first method is that
there is no need to model the time distribution of all the backgrounds, as
they are statistically removed via the sWeights. The statistical subtraction,
as presented here, uses events in the full mass fit region, and the
subtraction of this background leads to a larger statistical uncertainty than
if just a narrow signal region is used. For this reason, the second method is
expected to give a smaller statistical uncertainty; however it requires an
accurate model of the time distributions of the backgrounds that enter into
the signal region. For the analysis presented here, the sFit provides the
nominal result, and the cFit is used as a cross-check.

The measurement of the CP parameters in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
requires a fit to the time-dependent decay rates. The fit accounts for (i) the
acceptance versus reconstructed decay time, (ii) the decay time resolution,
and (iii) the effective tagging efficiency. The functional form of the
acceptance function is determined from simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$,
and its parameters are determined in a fit to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-}$
data, where the $B^{0}_{s}$ lifetime and mixing frequency, $\Delta m_{s}$, are
fixed to 1.51 ps and 17.69 ps-1 [17], respectively. The average decay time
resolution is about 50 fs, and is modeled by the sum of three Gaussian
functions, whose parameters are determined from simulation. The Gaussian width
parameters obtained from simulation are scaled up by 1.15 to account for
better resolution in the simulation than in data; this factor is determined by
comparing the width of the zero decay time component of prompt $D^{+}_{s}$
plus one random track in data and simulation. For the flavor tagging, only
opposite side (OS) taggers are currently used. These algorithms exploit the
correlation in flavor between the signal $b$ hadron at production, and the
other $b$ hadron in the event (referred to as the tag-$b$). In particular, the
charge of either an electron, a muon, or a kaon that does not come from any
$pp$ interaction vertex (or the signal $b$), or the charge of another
secondary vertex in the event, provide information on the flavor of the
tag-$b$ hadron. Because $b\overline{b}$ are produced in pairs, this translates
into a flavor determination of the signal $B^{0}_{s}$. The OS flavor tagging
algorithm was initially tuned using simulated decays, and then re-optimized
and calibrated to obtain the largest effective tagging efficiency using the
self-tagging $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{0}\rightarrow D^{*-}\mu^{+}\nu$ decays in data. In
general, the performance of the OS tagging algorithms are independent of the
signal $b$-hadron decay, and have a combined effective tagging efficiency of
$\epsilon D^{2}=1.90\%$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$.
Further details of the tagging algorithms can be found in Ref [16].

In the fit to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$, the following parameters are fixed: $\Delta m_{s}=17.69$
ps-1, $\tau_{B_{s}}=1.51$ ps and
$\Delta\Gamma_{s}\equiv\Gamma_{s,L}-\Gamma_{s,H}=0.105$ ps-1 [17]. About 60%
of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$ candidates have no flavor tag; the time-dependent decay rates
for these untagged decays is given by the sum of the two expressions in Eq.
LABEL:eq:decay_rates_2, and the sensitivity to $\gamma$ enters through the
hyperbolic sine term. The decay time distribution of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
signal decays and projections of the fitted are shown in Fig. 4. The
projections show the four possible tagged decays, $B^{0}_{s}\rightarrow
D_{s}^{\pm}K^{\mp}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{\pm}K^{\mp}$,
as well as the untagged time-dependent decay rates $(B^{0}_{s},\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})\rightarrow D^{-}_{s}K^{+}$ and
$(B^{0}_{s},\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})\rightarrow
D^{+}_{s}K^{-}$.

Figure 4: Distribution of reconstruct decay time for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
signal decays (points with error bars), along with the results of the fit.
Projections of the decay rates versus the decay time for the four possible
flavor tagged decays, and the two untagged decays.

The fitted values for the CP parameters are

$\displaystyle C$ $\displaystyle=\phantom{-}1.01\pm 0.50\pm 0.23\,,$
$\displaystyle S_{f}$ $\displaystyle=-1.25\pm 0.56\pm 0.24\,,$ $\displaystyle
S_{\overline{f}}$ $\displaystyle=\phantom{-}0.08\pm 0.68\pm 0.28\,,$
$\displaystyle D_{f}$ $\displaystyle=-1.33\pm 0.60\pm 0.26\,,$ $\displaystyle
D_{\overline{f}}$ $\displaystyle=-0.81\pm 0.56\pm 0.26\,,$

where the first uncertainties are statistical and the second are systematic.
Several sources of systematic uncertainty have been considered. The dominant
sources are due to the precision on the effective flavor tagging efficiency
(0.16$\sigma_{\rm stat}$-0.23$\sigma_{\rm stat}$), variations in the
parameters that are fixed in the default fits (0.15$\sigma_{\rm
stat}$-0.42$\sigma_{\rm stat}$), and the correlation between the mass of
specific backgrounds and their reconstructed decay time (0.08$\sigma_{\rm
stat}$-0.27$\sigma_{\rm stat}$), where these uncertainties are expressed as a
fraction of the statistical error. These are the first measurements of the CP
parameters in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}$. With additional data and analysis refinements, reduction in
both the statistical and systematic uncertainties are expected.

##  4 First observation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$

The decay $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ can be analyzed in a similar way to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ to
measure the weak phase $\gamma$. While this decay has not yet been observed,
if one uses $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$
decays as a guide, it would naively be expected that its branching fraction is
1.5-2.0 times larger than $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$,
making this a potentially attractive decay mode to explore. The first step in
such an analysis is to firmly establish an observation of this decay and
measure its branching fraction (here, relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$). While searching for this decay, the decay
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ is also observed and its branching fraction is
measured relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$.

With the previously defined selections, Fig. 5 shows the invariant mass
distributions for (left) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates and (right) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. Significant $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signals are seen in both
spectra, and a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ signal is
seen in the $D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ mass distribution. The main sources
of background are $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$ (to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$), and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{*+}_{s}\pi^{-}\pi^{+}\pi^{-}$, and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{*+}_{s}K^{-}\pi^{+}\pi^{-}$ (to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$). Their shapes are taken from simulation, with
parameters that are allowed to vary within their uncertainties. Yields of
$5683\pm 83$ $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$, $216\pm 21$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $402\pm 33$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ are observed. After correcting for the relative
efficiencies, the ratio of branching fractions are measured to be

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.2\pm 0.5\pm 0.3)\times
10^{-2}$ $\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})}$ $\displaystyle=0.54\pm 0.07\pm 0.07,$

where the uncertainties are statistical and systematic, respectively.

Figure 5: Invariant mass distribution for (left) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates and (right) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ candidates. The fitted signal (dashed lines) and
background shapes (shaded/hatched regions) are shown, as described in the
text.

These are the first observations of these decays. Since $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ has a branching fraction that is about twice
as large as $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s}^{+}\pi^{-}$, and ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-})\sim
0.09\times{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-})$
[18], it follows that ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})$ is at least as large as ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-})$,
or as much as 50% larger. The ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-})$ is also sizeable, and is likely dominated by
contributions where an extra $s\overline{s}$ pair is produced in addition to
the weak decay (see Ref. [13] for more details).

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ decay has also been analyzed to search for
intermediate excited $D_{sj}$ states. For $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates within 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal peak, the mass
difference, $\Delta M\equiv M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s})$ is
computed for both $\pi^{-}\pi^{+}$ mass combinations. The resulting mass
difference spectrum is shown in Fig. 6. The signal is fit with a Breit-Wigner
convolved with a Gaussian resolution function whose width is fixed to the
expected $\Delta M$ resolution. A signal of $20.0\pm 5.1$ events is observed
with a $\Delta M$ value and width consistent with the $D_{s1}(2536)^{+}$
state. Applying corrections for the relative efficiency, the ratio of
branching fractions is measured to be

$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-},~{}D_{s1}^{+}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-})}=(4.0\pm 1.0\pm 0.4)\times 10^{-3}.$

The excess of events is 5.9 standard deviations over the expected background,
thus establishing the first observation of this decay.

Figure 6: Distribution of the difference in invariant mass,
$M(D^{+}_{s}\pi^{-}\pi^{+})-M(D^{+}_{s})$, using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ candidates within 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}_{s}$ mass
(points) and in the upper $B^{0}_{s}$ mass sidebands (filled histogram). The
fit to the distribution is shown, as described in the text.

##  5 Summary

First measurements of the CP observables in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$
decay have been reported. With the larger data sample recorded in 2012, and
the larger data set anticipated in the future, this decay will contribute
significantly to the determination of the weak phase $\gamma$. First
observations of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ are also reported. The former can be used in a
similar way to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$ to
extract $\gamma$. After including $D^{+}_{s}\rightarrow\pi^{+}\pi^{-}\pi^{+}$
and $D^{+}_{s}\rightarrow K^{-}\pi^{+}\pi^{-}$ decays, and reoptimizing the
selection for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}K^{-}\pi^{+}\pi^{-}$ only, the yield in this mode more than doubles
with a comparable signal-to-background. The yield in this mode is therefore
expected to have about 35-40% of that obtained in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s}^{+}K^{-}$. The
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D_{s1}(2536)^{+}\pi^{-}$ decay is also observed for the first time, and its
branching fraction relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{+}_{s}\pi^{-}\pi^{+}\pi^{-}$ is presented.

## Acknowledgements

I gratefully acknowledge support from the National Science Foundation, which
makes this research possible.

## References

  * [1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
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  * [10] R. Fleischer, Nucl. Phys. B671, 459 (2003).
  * [11] K. De Bruyn, R. Fleischer, R. Knegjens, M. Merk, M. Schiller and N. Tuning, Nucl. Phys. B868, 351 (2012).
  * [12] LHCb collaboration, LHCb-CONF-2012-029.
  * [13] R. Aaij (LHCb collaboration), LHCb-PAPER-2012-033, arXiv:1211.1541, submitted to Phys. Rev. D.
  * [14] T. Skwarnicki, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02.
  * [15] M. Pivk and F. R. Le Diberder, Nucl. Instrum. Meth. A555, 356 (2005).
  * [16] R. Aaij et. al. (LHCb Collaboration), Eur. Phys. J. C72, 2022 (2012); LHCb-CONF-2012-026; Also see contribution by J. Wishahi in these proceedings.
  * [17] Heavy Flavor Averaging Group, D. Asner et al., Averages of b-hadron, chadron, and tau-lepton Properties, arXiv:1010.1589, Online updates available at http://www.slac.stanford.edu/xorg/hfag/.
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11institutetext:  School of Physics, Peking University, Beijing 100871, China
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China

# Kink scattering in a Lorentz-violating $\phi^{6}$ model

Haobo Yan111E-mail<EMAIL_ADDRESS>(corresponding author)

###### Abstract

The role played by a Lorentz-violating term on the outcomes of kink scattering
in the $\phi^{6}$ model is investigated by using the Fourier spectral method.
Impacts of the Lorentz-violating term on the critical velocities, the location
of two-bounce windows, and the maximal values of various types of energy
densities are analyzed. Some novel features of kink-antikink collisions are
discussed. The interactions between three and four kinks are also considered.

##  1 Introduction

In $(1+1)$-dimensional nonlinear scalar field models with more than one
degenerated vacua, there may exist a type of classical solution called kink,
which smoothly connects different vacua and has localized energy density. In
the past decades, kink and its higher-dimensional extension, domain wall, have
been extensively studied in both condensed matter physics and high energy
physics [1].

A related topic that has been repeatedly discussed recently, is the scattering
of non-integrable kinks. Unlike the collision of integrable kinks where the
kinks simply pass through each other, a pair of incoming non-integrable kinks
(for instance, $\phi^{4}$ kinks) can be bounced back after a finite time of
the collision or form a bound state called bion/oscillon, depending on their
initial velocities, say $v_{0}$ and $-v_{0}$ for symmetric collisions [2, 3,
4, 5]. The velocity intervals within which kinks are bounced back after $n$
collisions are called $n$-bounce windows ($n$BWs). For the $\phi^{4}$ model,
the $n$-bounce windows form an interesting fractal structure [6].

Bounce windows have also been found in many other non-integrable kink models
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. A
well-accepted explanation for the appearance of bounce windows is the resonant
energy exchange between the translational and vibrational modes of the kink
[5, 9]. This idea is shown recently using a collective coordinate model [26].
In addition to double kink collisions, the simultaneous collisions of multi-
kink at the same point have also been investigated in many works [27, 28, 29,
30, 31, 32]. Such collisions can produce a very large energy density around
the collision point, and the properties of the high energy region are
important for predicting the maximal number of kink-antikink pairs formed in
wave train collisions [33]. In some models [27, 34], one can also pinpoint the
location of bounce windows by observing the behaviors of maximal energy
densities, which vary smoothly within the bounce windows but become chaos
elsewhere. Other interesting topics include deformed $\phi^{4}$ theory [17],
deformed sine-Gordon model[35], etc.

Despite many interesting discoveries, non-integrable kink collisions are
rarely considered in Lorentz-violating models. Considering the fact that
Lorentz symmetry breaking naturally arises in many fundamental theories, such
as canonical and loop quantum gravity [36, 37, 38], noncommutative quantum
field theory [39], and might have observational implications [40, 41] (see
[42] for more references), it would be interesting to consider kink collisions
in Lorentz-violating non-integrable scalar field models. However, the breaking
of Lorentz symmetry hinders one from deriving moving kink solutions by simply
boosting static ones, which are necessary for the construction of the initial
configuration. Therefore, the study of kink collisions was limited to the
models that preserve the Lorentz invariance for a long time, until Bazeia and
others proposed a model in which there is a simple transformation relationship
between the static and dynamic kink solutions [43].

In this paper, we study the kink interactions in a Lorentz-violating
$\phi^{6}$ model using the traveling kink solution reported in Ref. [43]. In
the following section, the Lorentz-violating $\phi^{6}$ model is considered
and the kink solutions in the Lorentz-violating $\phi^{6}$ model are derived.
Then we analyze the linear perturbation modes of both the single kink and
double kink solutions. The numerical simulation of the kink collisions, as
well as the dependencies of several physical quantities on the Lorentz-
violating parameter, is conducted subsequently.

##  2 Model and solution

The $(1+1)$-dimensional Lorentz-violating $\phi^{6}$ theory is defined by the
Lagrangian density:

$\displaystyle\mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}\kappa^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\phi^{2}(1-\phi^{2})^{2},$
(1)

where we introduced the conventional Minkowski metric
$\eta^{\mu\nu}\equiv\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ and a tensor
$\kappa^{\mu\nu}\equiv\begin{pmatrix}0&\alpha\\\ \alpha&0\end{pmatrix}$,
representing the Lorentz violation.

Since the $\phi^{6}$ model contains three degenerate vacua ${\\{-1,0,1\\}}$,
kinks and antikinks take two types, each of which is connected to neighboring
vacua and has to be studied separately.

The Lagrangian (Eq. 1) yields the EoM for the field

$\frac{\partial^{2}\phi}{\partial t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}+2\alpha\frac{\partial^{2}\phi}{\partial x\partial
t}+3\phi^{5}-4\phi^{3}+\phi=0$ (2)

The static kink solution interpolating $(0,1)$ is found to be the same as the
ordinary $\phi^{6}$ model solution

$\phi_{K}(x)=\phi_{s(0,1)}(x)=\sqrt{\frac{1+\tanh(x)}{2}}.$ (3)

The corresponding antikink is

$\phi_{\bar{K}}(x)=\phi_{(1,0)}(x)=\phi_{K}(-x),$ (4)

and kinks belong to the other topological sector are

$\phi_{(0,-1)}(x)=-\phi_{K}(x),\phi_{(-1,0)}(x)=-\phi_{K}(-x).$ (5)

To obtain a moving kink solution, one has to resort to a more sophisticated
method, for the ordinary Lorentz boost simply does not hold in this Lorentz-
violating model. However, one can still carry out a boost-like approach by
deforming the Lorentz factor $\gamma\equiv 1/\sqrt{1-v^{2}+2\alpha v}$ [43].
The traveling kink solution can then be expressed as

$\phi_{(0,1)}(x_{0},t)=\sqrt{\frac{1+\tanh[\gamma(x-x_{0}-v_{0}t)]}{2}},$ (6)

where $x_{0}$ and $v_{0}$ are the initial position and velocity of the kink.

As is known, the breaking of the Lorentz invariance breaks the symmetry (the
width, the energy density, etc) between moving kink and antikink, and kink
collisions would be asymmetrical. The spatial configurations of kink-antikink
($K\bar{K}$) and antikink-kink ($\bar{K}K$) pairs at $\alpha=0,1,2$ are shown
in Fig. 1. As $\alpha$ increases, the asymmetry between the kink and the
antikink increases. Note that the energy density distribution of the same kink
is also different if it is moving oppositely, as indicated by the dashed
lines.

Figure 1: Configurations of $K\bar{K}$ and $\bar{K}K$ topology for
$v_{0}=0.2$, $x_{0}=10$, and $\alpha=0,1,2$. The antikinks are always more
energetic than kinks, and this asymmetry grows as $\alpha$ increases. The
dashed lines show the difference of maximal energy densities between right-
moving and left-moving (anti)kinks.

##  3 Linear stability and the vibrational mode

To prove the stability of the kink solutions and get a better understanding of
the bounce windows, the linear perturbative states of the static kink
solutions should be examined. Consider a small perturbation $\delta\phi(x,t)$
around a background (a static kink or a two-kink configuration), by taking the
variation of the action with respect to $\delta\phi(x,t)$ and Fourier expand
the field $\phi(x,t)$, we immediately arrive at a Schrödinger-like equation:

$\mathscr{H}f_{n}=\tilde{\omega}_{n}^{2}f_{n},$ (7)

where a factorizable Hamiltonian
$\mathscr{H}\equiv-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}x^{2}}+\frac{1}{\theta}\frac{{\mathrm{d}}^{2}\theta}{{\mathrm{d}}x^{2}}=\left(\frac{{\mathrm{d}}}{{\mathrm{d}}x}+\frac{1}{\theta}\frac{{\mathrm{d}}\theta}{{\mathrm{d}}x}\right)\left(-\frac{{\mathrm{d}}}{{\mathrm{d}}x}+\frac{1}{\theta}\frac{{\mathrm{d}}\theta}{{\mathrm{d}}x}\right)$
(with $\theta=\partial_{x}\phi_{s}$) is introduced, and
$\tilde{\omega}_{n}^{2}\equiv\left(1+\alpha^{2}\right)\omega_{n}^{2}$ is then
always non-negative according to the supersymmetric quantum mechanics, thus
proving our claim that the static solutions are stable against small
perturbations. Note that $f_{n}(x)$ is only the expansion coefficient of
perturbation, and $g_{n}(x)\equiv f_{n}(x)\mathrm{e}^{{i}\alpha\omega_{n}x}$
is the complete eigenfunction.

For the single kink solution $\phi_{s(0,1)}(x)=\sqrt{\frac{1+\tanh(x)}{2}}$,
the effective potential in the Schrödinger-like equation is
$V_{\mathrm{eff}}=\frac{\partial_{x}^{3}\phi_{s}}{\partial_{x}\phi_{s}}=\frac{15}{4}(\tanh(x)+1)^{2}-6(\tanh(x)+1)+1$
and only translational mode survives.

For the two-kink static configurations,

$\begin{cases}\phi_{K\bar{K}}(x,0)=\phi_{\bar{K}}(x-x_{0})+\phi_{K}(x+x_{0})-1,\\\
\phi_{\bar{K}K}(x,0)=\phi_{K}(x-x_{0})+\phi_{\bar{K}}(x+x_{0}),\end{cases}$
(8)

the effective potential becomes
$V_{\mathrm{eff}}^{2K}=15\phi_{2K}^{4}-12\phi_{2K}^{2}+1$, where $\phi_{2K}$
denotes either $K\bar{K}$ or $\bar{K}K$ pair. It turns out that the $K\bar{K}$
pair still supports only translational mode, but $\bar{K}K$ configuration
supports many of the vibrational modes, just as the case in the ordinary
$\phi^{6}$ model [9]. The eigenvalues $\tilde{\omega}_{n}^{2}$ depend both on
the half-separation $x_{0}$ and the Lorentz-violating parameter $\alpha$. The
eigenfunctions $g_{n}(x)$ for the first five modes of $\bar{K}K$ pair with
half-separation $x_{0}=10$ ($14$ modes in total) are shown in Fig. 2. The
$\alpha-$dependence of the vibrational modes indicates that the kink collision
results will be affected by the Lorentz violation. Various novel kink-
collision phenomena will be discussed in the following sections.

(a) $\alpha=0$

(b) $\alpha=1$

(c) $\alpha=2$

Figure 2: Eigenfunctions of modes of $\bar{K}K$ static solution for $\alpha=0$
(ordinary $\phi^{6}$ model) and $\alpha=1,2$. $0\leq n\leq 4$ stands for the
zero mode and the $n\textsuperscript{th}$ excited state, and $x_{0}=10$ is the
half-separation distance. The spatial wave function $g_{n}(x)\equiv
f_{n}(x)e^{i\alpha\omega_{n}x}$ gets more nodes under Lorentz violation.

##  4 Simulation and results

In this section, we study the asymmetrical kink collisions. Due to the absence
of exact kink scattering solutions of the non-integrable model, one has to
resort to numerical simulation. The initial conditions of kink scattering are
set by superpositions of individual kinks and antikinks, which is Eq. 8 and

$\begin{cases}\dot{\phi}_{K\bar{K}}(x,0)=\dot{\phi}_{\bar{K}}(x-x_{0})+\dot{\phi}_{K}(x+x_{0}),\\\
\dot{\phi}_{\bar{K}K}(x,0)=\dot{\phi}_{K}(x-x_{0})+\dot{\phi}_{\bar{K}}(x+x_{0}).\end{cases}$
(9)

Since a single kink has an exponentially decaying tail, the overlap in the
initial conditions can be safely ignored at a large distance. Initial
conditions for more kinks can be obtained in the same way.

The Fourier spectrum method [44] with periodic boundary conditions is used to
solve the EoM numerically, and one may control the validity of the simulation
by insisting on energy conservation. The total energy of the configuration is
calculated by

$E(t)=\frac{1}{2}\int_{-\infty}^{\infty}\left[\left(\frac{\partial\phi}{\partial
t}\right)^{2}+\left(\frac{\partial\phi}{\partial
x}\right)^{2}+\frac{1}{2}\phi^{2}\left(1-\phi^{2}\right)^{2}\right]\mathrm{d}x,$
(10)

and can be approximated by

$E_{\mathrm{th}}\approx E\left(v_{0}\right)+E\left(-v_{0}\right),$ (11)

where $E\left(v_{0}\right)=\gamma\left(1+\alpha v_{0}\right)M_{K}$ is the
energy of a moving kink and the mass of the kink is
$M_{K}=\int[(\partial_{x}\phi_{(0,1)})^{2}/2+V]dx=\frac{1}{4}$. The relative
error between $E_{\mathrm{th}}$ and the calculated energy $E_{\mathrm{num}}$
will be checked to guarantee the validity of simulation results. The EoM is
discretized, with the time step chosen automatically by MATLAB ode45 solver,
typically around $0.002$, to ensure the relative error of energy to be less
than $10^{-9}$. The space step is chosen to be $0.1$ and $0.05$ to guarantee
the results are convergent and no significant change happens with further
precision improvement.

###  4.1 Impacts on the collision structure

It is important to look at how the Lorentz-violating term affects the global
distribution of BW and $v_{c}$. The $K\bar{K}$ and $\bar{K}K$-type collision
structures for $\alpha=0,0.5,1$ are shown in Fig. 3. As $\alpha$ increases,
the critical velocity decreases. The critical difference between this
structure and that of the $\phi^{4}$ model is no bounce window is found in
$K\bar{K}$ collisions, and no bounce window higher than the
$2\textsuperscript{nd}$ order in $\bar{K}K$ collisions is found, rendering the
structure of the Lorentz-violating $\phi^{6}$ model terminated after the
second level, which is consistent with the ordinary $\phi^{6}$ model [9].

One can make another two observations regarding the oscillations in the
$K\bar{K}$ structure, as shown in Fig. 3(b). First, the moments of
$n\textsuperscript{th}$ oscillations of the bions plot a series of curves as
functions of $v_{0}$. As $v_{0}$ increases, the curves decrease in general,
but there exist certain windows where all the curves start to rise, which is
called the rising windows (RW). The existence of RWs is also confirmed in
Lorentz-violating collisions. Furthermore, by zooming in on the windows one
would find the RWs decompose into smaller increasing and decreasing segments.
The first two RWs are highlighted, and the first is enlarged on the right.
Second, in the ordinary $\phi^{6}$ model, the amplitudes of the oscillations
are relatively preserved, while the presence of Lorentz violation damps the
oscillation. This damping effect is most evident near the critical velocity,
where the amplitudes of oscillations quickly decay to near zero. The field
values at the origin of $v_{0}=0.15$ for $\alpha=0,0.5,1$ are plotted in Fig.
4 to illustrate the damping effect. We emphasize that this is due to the
asymmetry of the Lorentz-violating $K\bar{K}$ collisions. The bions formed
from the collisions deviate from $x=0$ as time evolves, so the field values at
$x=0$ come mainly from radiations instead of the bions themselves. However,
the bions formed in the $\bar{K}K$ collisions are not found to deviate from
the collision points.

(a) $\bar{K}K$

(b) $K\bar{K}$

Figure 3: Field values of collisions of (a) $\bar{K}K$ and (b) $K\bar{K}$ at
the central spatial point as functions of velocities for $\alpha=0,0.5$ and
$1$. As $\alpha$ increases, the critical velocity decreases. There is no 3BW
and higher-order bounce window in this figure. Collision points at initial
velocities $v_{0}=0.03315$ have been colored white in plot (a). In plot (b),
there is no BW, but there exist windows (RW) where the $n\textsuperscript{th}$
oscillation moments increases as $v_{0}$ increases, with the first two RWs
highlighted and the first RW enlarged on the right. Figure 4: The field values
at the origin of $v_{0}=0.15$ for $\alpha=0,0.5,1$ of $K\bar{K}$ collisions.
The amplitudes of oscillations decay if $\alpha\neq 0$.

The typical numerical results with $v_{0}=0.03315$ for antikink-kink
collisions have been selected in Fig. 3(a) and shown in Fig. 5. The first row
of figures shows all 3 types of $\phi(x,t)$ configurations, namely bion, two-
bounce, and inelastic scattering. The second row shows the corresponding
energy density configurations, where the asymmetrical nature of collisions is
clearly shown by the behavior of radiation. Note that albeit the asymmetry,
the scattering still happens at $x=0$, which can be seen in this row of energy
density plots and in Fig. 3 where the amplitudes of the first oscillation at
$x=0$ are almost the same for $\alpha=0,0.5$, and $1$. The field values at the
origin of the first three 2BWs (with the second 2BW being a false window) at
$\alpha=1$ are presented in Fig. 6. The number $n$ of oscillations of
$\phi(0,t)$ between two collisions is the same as that of the ordinary
$\phi^{6}$ model [9], which together with Fig. 3 means that the Lorentz
violation may change the positions of 2BWs and $v_{c}$, but preserve certain
invariant quantities, i.e. the number of collective mode oscillations and the
number of 2BWs.

(a) $\alpha=0$, bion.

(b) $\alpha=0.5$, two-bounce.

(c) $\alpha=1$, inelastic scattering.

Figure 5: The numerical results for antikink-kink collision at
$v_{0}=0.03315$. The first row shows the field configuration $\phi(x,t)$, and
the second row shows the energy density $\rho(x,t)$. The left, middle and
right columns are results of $\alpha=0$, $\alpha=0.5$, and $\alpha=1$, and
correspond to the bound state, two-bounce window, and inelastic scattering
solutions, respectively. Figure 6: The field values at the origin of the first
three 2BWs (including a false window) of $\bar{K}K$ collisions at $\alpha=1$.
The numbers of oscillations are the same as those at $\alpha=0$.

The critical velocity of $K\bar{K}$ and $\bar{K}K$ collisions in the Lorentz-
violating $\phi^{6}$ model are both monotonically decreasing functions of the
parameter $\alpha$, as shown in the data points of Fig. 7. This is just the
case in the $\phi^{4}$ model. And it is of great interest if these results are
compared with those of the Lorentz-violating $\phi^{4}$ model. Despite being
calculated from different scalar models or different topologies, the data of
the critical velocities present similar varying trends. The following
empirical formula is introduced to fit the data:

$v(\alpha)=v(0)\frac{\mathrm{e}^{-\tau\alpha}}{\sqrt{1+\alpha^{2}}},$ (12)

and the corresponding fitted curves are also shown in Fig. 7.

To explain their similar behavior under Lorentz invariance breaking, notice
that the energies of the kink and the antikink are no longer the same (see
Fig. 1). The excessive energy of the kink (antikink) is then wasted, as a
result, higher $\alpha$ means lower effective colliding velocity. Eq. 12 then
provides a prediction for the critical velocities under Lorentz violation.

Figure 7: The critical velocity $v_{c}$ of $\phi^{4}$ and $\phi^{6}$ model
collisions and reconstructed fitting curves as functions of the Lorentz-
violating parameter $\alpha\in[0,2]$. The fitted decaying parameters $\tau$
are $0.046,0.290,0.260$ for $\bar{K}K$ collision, $K\bar{K}$ collision, and
$\phi^{4}$ collision, respectively.

The dependence of the maximal energy densities on $\alpha$ provides rich
information about the kink-scattering behavior under Lorentz breaking. The
total energy density can be written as

$\rho(x,t)=k(x,t)+u(x,t)+p(x,t),$ (13)

where $k(x,t)=\frac{1}{2}\left(\frac{\partial\phi}{\partial t}\right)^{2}$,
$u(x,t)=\frac{1}{2}\left(\frac{\partial\phi}{\partial x}\right)^{2}$, and
$p(x,t)=V(\phi)$ are the kinetic energy density, gradient energy density, and
potential energy density, respectively.

From the experience of Ref. [27, 34], one would expect this energy density
structure to be chaotic unless the velocity falls in the inelastic scattering
region or the bounce windows, showing fractal information about the scattering
structure. However, this is not the case anymore for the $\phi^{6}$ model. In
Fig. 8, maximal values of $\rho(x,t)$ and its components for kink scattering,
with initial conditions $x_{0}=10$ and $v_{0}=0.03315$, are presented. The
curve is smooth and no chaotic zone is observed, regardless of the final state
of the configuration. The lacking of the single vibrational mode forbids the
violent energy exchange between different energy levels, the maximal energy
densities then lost their fractal chaotic behavior.

Figure 8: Maximal values of $\rho(x,t)$ and its components for $v_{0}=0.03315$
kink scattering as functions of $\alpha$.

###  4.2 Impacts on multi-kink collision

The authors in Ref. [28] have studied the multi-kink collision of the ordinary
$\phi^{6}$ model in great detail. And it would be of interest to study how the
results depend on the Lorentz violation. We follow the initial value settings
in Ref.[28] to guarantee the collisions occur at the same point when
$\alpha=0$ (the upper row in Fig. 9). The parameter $\alpha$ is then slowly
increased to study the impact of small Lorentz violation (the lower row in
Fig. 9 shows the results of $\alpha=0$).

(a) $(0,1,0,1)$

(b) $(1,0,1,0)$

(c) $(0,1,0,1,0)$

Figure 9: The spacetime configuration for three- and four-kink collisions. The
captions indicate the topology of the multi-kinks. The first row and the
second row are fields for $\alpha=0$ and $\alpha=0.1$, respectively.

The three-kink collisions of topology $K\bar{K}K$ are presented in Fig. 9(a)
and Fig. 9(b). In the ordinary $\phi^{6}$ model, the field ends up with a
right-moving kink and a small-amplitude left-moving bion. When $\alpha$ is
turned on, the velocity of the kink and bion is decreased, whilst the
amplitude of the bion is increased. In this case, part of the kinetic energy
is transferred to the potential energy of the bion. The $\bar{K}K\bar{K}$
collisions are similar to those of $K\bar{K}K$, in the sense of forming bions.
The four-kink collisions of $K\bar{K}K\bar{K}$ are shown in Fig. 9(c). When
$\alpha=0$, the four kinks collide and end up with two kinks and radiation in
between. When $\alpha=0.1$, the collision creates two bions, and the bions
travel slower than the kinks at $\alpha=0$. This tendency of forming bions is
valid for small Lorentz violation parameters ($\alpha<0.1$). In this range,
the kinks give more energy to the bions at higher $\alpha$.

It is now clear that a small $\alpha$ could lead to a very different final
state, especially for the multi-kink collisions. In the example presented in
Fig. 9, a small Lorentz violation tends to suppress the speed of kinks and
bions and uses the energy to build bions or make them stronger.

##  5 Conclusions

In this study, we were concerned mainly with the kink scattering of a Lorentz-
violating $\phi^{6}$ model. The static and boosted kink solutions were
obtained in the spirit of Ref. [43]. It was found that with Lorentz invariance
broken, traveling kinks and antikinks became asymmetrical. The following
analysis proved the stability of kink solutions. The eigenfunctions of the
$\bar{K}K$ linear perturbation for $\alpha=0,1,2$, $0\leq n\leq 4$ and
separation $x_{0}=10$ were calculated, as shown in 2.

After the general remarks, the asymmetrical kink collisions are studied for
both the $K\bar{K}$ and $(\bar{K}K)$ configurations. The Lorentz-violating
$\phi^{6}$ model has no BW for $K\bar{K}$ collisions and no 3BW or higher
bounce windows for $\bar{K}K$ collisions. The structure in Fig. 3 shows how
$v_{c}$ decreases due to the presence of $\alpha$. We observed rising windows
in the structure of $K\bar{K}$ scatterings, and the damping of the field value
at the origin, or equivalently, the deviation of bions from the origin under
Lorentz violation. To the author’s knowledge, these two effects have not been
reported yet.

We also discovered an empirical formula Eq. 12 of $v_{c}$ dependencies on
$\alpha$ for the two topologies in the $\phi^{6}$ model and that of the
$\phi^{4}$ model. One can study higher-order Lorentz-violating models to
generalize the formula. We then studied the impacts of $\alpha$ on maximal
energy densities. Due to the lacking of different modes of single kinks, the
dependencies have no intricate structure as that of the $\phi^{4}$ model.

The impact of Lorentz violation on the multi-kink collisions was also studied.
We found that the presence of small $\alpha$ would change the collision
process. In those examples, the kinetic energy is transferred to forming bions
or enlarging their amplitude.

##  6 Acknowledgements

The author is grateful to Yuan Zhong for helpful discussions.

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