# Geodesic complexity of a tetrahedron

Donald M. Davis  Department of Mathematics, Lehigh University
Bethlehem, PA 18015, USA<EMAIL_ADDRESS>
(Date: June 18, 2023)

###### Abstract.

We prove that the geodesic complexity of a regular tetrahedron exceeds its
topological complexity by 1 or 2. The proof involves a careful analysis of
minimal geodesics on the tetrahedron.

###### Key words and phrases:

Geodesic complexity, topological robotics, geodesics, tetrahedron

2000 Mathematics Subject Classification: 53C22, 52B10, 55M30.

##  1\. Introduction

In [3], Farber introduced the concept of the topological complexity,
$\operatorname{TC}(X)$, of a topological space $X$, which is the minimal
number $k$ such that there is a partition

$X\times X=E_{0}\sqcup E_{1}\sqcup\cdots\sqcup E_{k}$

with each $E_{i}$ being locally compact and admitting a continuous function
$\phi_{i}:E_{i}\to P(X)$ such that $\phi_{i}(x_{0},x_{1})$ is a path from
$x_{0}$ to $x_{1}$.111Farber’s original definition involved partitions into
$k$ sets rather than $k+1$, but for technical reasons the definition here has
become more common. Here $P(X)$ is the space of paths in $X$, and each
$\phi_{i}$ is called a motion-planning rule. If $X$ is the space of
configurations of one or more robots, this models the number of rules required
to program the robots to move between any two configurations.

In [4], Recio-Mitter suggested that if $X$ is a metric space, then we require
that the paths $\phi_{i}(x_{0},x_{1})$ be minimal geodesics from $x_{0}$ to
$x_{1}$, and defined the geodesic complexity, $\operatorname{GC}(X)$, to be
the smallest number $k$ such that there is a partition

$X\times X=E_{0}\sqcup E_{1}\sqcup\cdots\sqcup E_{k}$

with each $E_{i}$ being locally compact and admitting a continuous function
$\phi_{i}:E_{i}\to P(X)$ such that $\phi_{i}(x_{0},x_{1})$ is a minimal
geodesic from $x_{0}$ to $x_{1}$. Each function $\phi_{i}$ is called a
geodesic motion-planning rule (GMPR).

One example discussed in [4] was when $X$ is (the surface of) a cube. It is
well-known that here $\operatorname{TC}(X)=\operatorname{TC}(S^{2})=2$, and he
showed that $\operatorname{GC}(X)\geq 3$.

In this paper, we let $X$ be a regular tetrahedron $T$, and prove

######  Theorem 1.1.

$\operatorname{GC}(T)=3$ or $4$.

Again, for comparison, $\operatorname{TC}(T)=\operatorname{TC}(S^{2})=2$.

In Section 2, we introduce what we call the expanded cut locus in order to
study the geodesics on $T$. In Section 3, we prove $\operatorname{GC}(T)\leq
4$, and in Section 4, we prove $\operatorname{GC}(T)\geq 3$. Despite
considerable effort, we have been unable to establish the precise value of
$\operatorname{GC}(T)$.

##  2\. Expanded cut locus

The cut locus of a point $P$ on a convex polyhedron is the set of points $Q$
such that there is more than one shortest path from $P$ to $Q$. For the
regular tetrahedron $T$, this is conveniently sketched on a flat model of $T$.
For $P\in T$, we define the expanded cut locus of $P$ to be the set of
terminal points of equal shortest paths from $P$ to versions of cut-locus
points $Q$ in a flat model of $T$, expanded so that the same face may appear
more than once.

In Figure 2.1 we illustrate the expanded cut locus of a point $P$. The open
segments $a\,U_{0}$ and $a\,U_{-}$ depict the same set of points in the
tetrahedron, and the segments from $P$ to points on each at equal distance
from $a$ depict equal shortest segments from $P$ to a point $Q$ in $T$. A
similar situation holds for segments from $d$ to two $U$-points, from $c$ to
two $L$-points, and from $b$ to two $L$-points. Also the small open segments
$U_{-}\,L_{-}$ and $U_{+}\,L_{+}$ are part of the expanded cut locus of $P$,
as they represent the same points in $T$, and segments from $P$ to points at
equal height on the two lines are equal minimal geodesics. The three
$U$-points represent the same point in $T$; the paths from $P$ to them are
equal shortest paths in $T$. Similarly for the three $L$-points. Thus the
expanded cut locus of $P$ is the entire red polygon in Figure 2.1 minus the
points $a$, $b$, $c$, and $d$.

The actual cut locus for this point $P$ is shown in Figure 2.2, which is a
flat version of part of $T$, but does not contain multiple versions of points.

######  Figure 2.1.

An expanded cut locus.

$P$$a$$b$$c$$b$$c$$d$$d$$b$$c$$d$$C$$M$$L_{0}$$L_{-}$$U_{-}$$U_{0}$$L_{+}$$U_{+}$

######  Figure 2.2.

The corresponding cut locus.

$L$$U$$c$$d$$a$$b$

The expanded cut locus of any point $P$ in the interior of triangle $aCM$ in
Figure 2.1, where $C$ is the centroid and $M$ the midpoint of $ac$, has a form
similar to the one depicted there. We make this precise in Theorem 2.3.

######  Theorem 2.3.

Suppose that in Figure 2.1 the coordinates of $a$, $b$, and $c$ are,
respectively, $(0,\sqrt{3})$, $(-1,0)$, and $(1,0)$, and
$P=(x,\alpha\sqrt{3})$ with $0<x<\frac{1}{2}$ and
$\frac{1}{3}+\frac{1}{3}x<\alpha<1-x$. Then the expanded cut locus of $P$ is
as depicted in Figure 2.1 and described above with

$\displaystyle U_{\pm}$ $\displaystyle=$ $\displaystyle(\pm
2+x,\sqrt{3}(1-\frac{x(2-x)}{3(1-\alpha)}))$ $\displaystyle U_{0}$
$\displaystyle=$ $\displaystyle(2-x,\sqrt{3}(1+\frac{x(2-x)}{3(1-\alpha)}))$
$\displaystyle L_{\pm}$ $\displaystyle=$ $\displaystyle(\pm 2+x,\sqrt{3}\
\frac{1-x^{2}}{3\alpha})$ (2.4) $\displaystyle L_{0}$ $\displaystyle=$
$\displaystyle(-x,\sqrt{3}\ \frac{x^{2}-1}{3\alpha})$

###### Proof.

Since

$\langle x,\sqrt{3}(\alpha-1)\rangle\cdot\langle 2-x,\sqrt{3}\
\frac{x(2-x)}{3(1-\alpha)}\rangle=0,$

$\overrightarrow{aP}\perp\overrightarrow{aU_{0}}$. Similarly the red lines
through $b$, $c$, and $d$ are perpendicular to the segments from $P$ to those
points. Another easy verification is that $\frac{1}{2}(U_{0}+U_{-})=a$, and
similarly for $b$, $c$, and $d$. So $Pa$ is the perpendicular bisector of
$U_{0}U_{-}$. That $\frac{1}{2}(U_{0}+U_{+})=d$ shows that $U_{0}$ and $U_{+}$
lie in the same relative position in triangle $bcd$. One readily sees that the
region inside the red polygon in Figure 2.1 exactly covers the four triangles
that comprise the tetrahedron.

This slick verification hides the way in which the formulas (2.4) were
obtained. We initially used the method of star unfolding and Voronoi diagrams
developed in [1], and applied to the cube in [2], using perpendicular
bisectors.

The triangle $abc$ in Figure 2.1 is divided into six congruent subtriangles.
The formulas (2.4) only apply to points $P$ in the interior of the upper right
subtriangle $aCM$, but the expanded cut locus of points in the other five
subtriangles can be obtained by obvious rotations and reflections. We now
consider the form of the expanded cut locus for points on the boundary of
triangle $aCM$.

As $P$ approaches the edge $aM$, $L_{\pm}$ approaches $U_{\pm}$. When $P$ is
on the edge, they coincide, and the two multiplicity-3 points in the cut locus
become a single multiplicity-4 point, which we will later call $B$, for
“both.” In Figure 2.5, we depict the two extreme cases, $P=a$ and $P=M$. The
continuum between them should be clear. We label the left one $P\approx a$,
because when $P=a$, the line passing through $a$ is not part of the expanded
cut locus, since the line connecting $P$ with points on the lines at equal
distance from $a$ in each direction are actually the same line in $T$. But for
points $P$ arbitrarily close to $a$, the lines from $P$ to points on the line
are not the same line in $T$.

######  Figure 2.5.

$P$ on an edge.

$P\approx a$$P=M$$a$$M$

As $P$ approaches the line $x=0$, $U_{+}$ and $U_{0}$ approach $d_{+}$ (the
version of $d$ on the positive side in Figure 2.1), and $U_{-}$ approaches
$d_{-}$. The diagram when $x=0$ is in Figure 2.6.

######  Figure 2.6.

$P$ on the line $x=0$.

$a$$d=U$$d=U$$c$$b$$d$$b$$c$$L$$L$$L$$C$

As $P$ moves from $a$ to $C$ along the line $x=0$, the point $L$ in Figure 2.6
moves from the centroid of $bcd$ to $d$. The limiting case $P=a$ has already
been discussed. However, if the $L$ in Figure 2.6 is moved to the centroid of
$bcd$, we obtain a picture which looks quite different from the left side of
Figure 2.5, which also depicts the case $P=a$. Even accounting for the fact
that when $P=a$, the line emanating from $a$ is not part of the expanded cut
locus, the diagrams still differ in that one has a vertical line on the left
side, whereas the other has a vertical line in the upper right. The
explanation is that paths from $a$ to corresponding points on those lines are
exactly the same path on $T$.

In Figure 2.7 we show the expanded cut locus when $P$ is at the centroid $C$
of $abc$, which is the case $L=d$ in Figure 2.6.

######  Figure 2.7.

$P$ at the centroid.

$U=L$$U=L$$U=L$

Finally, if $P$ is on the segment $CM$, $U_{\pm}=L_{\pm}(=B)$, and they lie on
edge $bd$. This is depicted in Figure 2.8. As $P$ moves from $C$ to $M$, $B$
moves from $d$ to the midpoint of $bd$.

######  Figure 2.8.

$P$ on the segment $CM$.

$d$$b$$c$$b$$d$$a$$d$$b$$B$$B$$B$$B$$C$$M$

##  3\. Upper bound

######  Theorem 3.1.

There is a decomposition

$T\times T=E_{1}\sqcup E_{2}\sqcup E_{3}\sqcup E_{4}\sqcup E_{5}$

with a GMPR $\phi_{i}$ on $E_{i}$.

###### Proof.

Let $G_{P}$ denote the polygon associated to the point $P$ sketched in red in
any of the figures of Section 2. More precisely, one must, of course, use the
formulas (2.4) to determine the vertices of the polygon, and if $P$ is
reflected across the line $x=0$ in Figure 2.1, then one must modify the
formulas to give the reflection of the polygon. If $P$ is at a vertex, there
are two choices for $G_{P}$, either as in Figure 2.5 or 2.6. It doesn’t
matter, but let’s choose 2.6.

The set $E_{1}$ is the complement of the total cut locus of $T$. It consists
of pairs $(P,Q)$ such that $Q$ is interior to the polygon $G_{P}$, together
with those for which $Q$ is a vertex of $T$, except for cases such as $(P,d)$
in Figure 2.6. (The only cases when a vertex $V$ is in the cut locus of a
point $P$ is when $P$ lies on a segment connecting the centroid of the face
opposite $V$ with one of the other vertices, including the centroid, but not
the vertices.) Here $\phi_{1}(P,Q)$ is the straight line from $P$ to $Q$ in
our expanded cut locus diagram.

The set $E_{2}$ consists of pairs $(P,Q)$ where $P$ is not a vertex and $Q$
lies in the interior of a cut-locus segment from a vertex $V$ to a $U$ or $L$
point, excluding cases in which $P$ lies on a segment from a vertex of face
$abc$ to the centroid $C$ of $abc$, and $V=d$. We choose $\phi_{2}(P,Q)$ to be
the path from $P$ to the appropriate point on the right side of the vector
from $P$ to $V$. For example, in Figures 2.1 and 2.2, $E_{2}$ contains $(P,Q)$
for all $Q$ in the open segments $aU$, $bL$, $cL$, and $dU$ in 2.2, and in 2.1
we choose the segments connecting $P$ with points on $aU_{0}$, $bL_{-}$,
$cL_{0}$, and $dU_{+}$. To maintain continuity of $\phi_{2}$, we had to
exclude points $(P,Q)$ with $P$ on the segment $aC$ and $Q$ on $dU$ because
shortest paths from the point $P$ in Figure 2.1 to $dU$ must pass through side
$ac$, whereas for points $P$ on the left side of $aC$ the diagram is reflected
and the shortest paths from $P$ to $dU$ will pass through side $ab$.

This requires some care because, for example, if $P$ is in face $abc$, the
cut-locus line out from vertex $d$ plays a different role than the others.
Because we have excluded points with $P$ on segments from a vertex to a
centroid, we can consider the domain of points $P$ for which $Q$ is on a cut-
locus line from vertex $d$ as three topologically disjoint sets $aCbd$,
$adcC$, and $bCcd$, as pictured in Figure 3.2.

######  Figure 3.2.

$P$-domains for lines through $d$.

$C$$d$$d$$d$$a$$c$$b$

The continuity of $\phi_{2}$ on each of these domains should be fairly clear,
but because of the different roles played by points in face $abc$ and the
other points, Figure 3.3 should make it clearer. What is pictured here is a
breakdown of the region $aCcd$ in Figure 3.2 into subregions together with,
for each subregion, the endpoints of the cut-locus segments out of vertex $d$
corresponding to points $P$ in the subregion. For example, output region 2 is
points $U_{+}$ in Figure 2.1 corresponding to points in input region 2, and
output region 6 is points $U_{0}$ in a rotated version of Figure 2.1
corresponding to points in input region 6. The entire segment between input
regions 5 and 6 maps to output point $b$. The dashed boundary of output
regions 5 and 6 are not in the image. We call the points $Q_{\text{max}}$ in
Figure 3.3 because they are the $Q$ farthest from $d$ for a point $P$.

######  Figure 3.3.

Largest $Q$ for varying $P$.

$1$$8$$7$$3$$6$$5$$4$$2$$2$$1$$3$$8$$4$$5$$7$$6$$b$$d$$c$$b$$a$Input $P$Output
$Q_{\max}$

The set $E_{3}$ consists of points $(P,Q)$ of two types. Type (1) has $P$ in
sets $\mathcal{I}$ defined as the interior of the set of points in a face
which are closer to one vertex than to the others. For example, in Figure 2.1,
one such region would be the interior of the quadrilateral in the upper third
of triangle $abc$. The points $Q$ associated to $P$ are the closed interval
$UL$. Type (2) has $P$ all points on segments connecting a vertex $V$ of a
face $abc$ with its centroid $C$, including $V$ but not $C$, and $Q$ in the
closed segment connecting the other vertex $d$ with the point $L$ associated
with $P$ as in Figure 2.6. Note that this can be considered as a $UL$ segment,
too.

For $P\in\mathcal{I}$ and $Q$ in the closed interval $UL$, we can choose
$\phi_{3}(P,Q)$ to be the appropriate point in $U_{+}L_{+}$, using rotations
of Figure 2.1. Then in Figure 2.6, we would choose as $\phi_{3}(P,Q)$ the path
that goes to the right from a point $P$ on $aC$ to the appropriate $Q$ on
$dL$.

The rest is easy. Let $E_{4}$ consist of pairs $(P,Q)$ such that $P$ is a
vertex and $Q$ the centroid of the opposite face, or $P$ is a centroid and $Q$
the opposite vertex. Since this is a discrete set, $\phi_{4}$ can be chosen
arbitrarily.

Let $E_{5}$ be the set of $(P,Q)$ such that $P$ lies in one of six
topologically disjoint sets, each of which is the union of lines from the
midpoint $M$ of an edge of $T$ to the adjacent vertices and centroids,
including $M$ but not the vertices or centroids. See Figure 3.4. A unique
point $Q=B$ is associated to each point $P$. Recall that when $U=L$, we call
it $B$. These are points of multiplicity 4, as in Figures 2.5 and 2.8. As long
as one chooses $\phi_{5}$ continuously on a set such as Figure 3.4, it can be
chosen arbitrarily.

######  Figure 3.4.

Typical set for $E_{5}$.

$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$a$$c$$d$$b$$B$$B$$B$$B$$M$

##  4\. Lower bound

######  Theorem 4.1.

The space $T\times T$ cannot be partitioned as $E_{1}\sqcup E_{2}\sqcup E_{3}$
with a GMPR on each $E_{1}$.

###### Proof.

Let $M$ be the midpoint of $ac$ in Figure 2.1, and $P^{\prime}$ a point on the
segment connecting $M$ and $P$ in that figure. The expanded cut locus for
$P^{\prime}$ is as in the figure, and as $P^{\prime}$ approaches $M$,
$L_{\pm}$ approaches $U_{\pm}$, and they and $U_{0}$ and $L_{0}$ approach the
midpoint of $bd$. We call this point $B$.

Suppose $(M,B)\in E_{1}$, and $\phi_{1}(M,B)$ is the path which goes down
(toward the limit of $L_{0}$ in Figure 2.1). (Going up is handled similarly,
reversing the roles of $U$ and $L$. We will consider later how to handle it
when $\phi_{1}(M,B)$ goes left or right.) We cannot have a sequence of
$P^{\prime}$ as in the figure with $P^{\prime}\to M$ and
$(P^{\prime},U_{P^{\prime}})\in E_{1}$ because that would imply
$\phi_{1}(P^{\prime},U_{P^{\prime}})\to\phi_{1}(M,B)$, which is impossible
since $\phi(P^{\prime},U_{P^{\prime}})$ must go either left, right, or up.
There is a sequence of such $P_{n}^{\prime}$ all in the same $E_{i}$, which we
call $E_{2}$, and, restricting more, all
$\phi_{2}(P_{n}^{\prime},U_{P_{n}^{\prime}})$ going in the same direction,
which we will suppose is left. We will consider later the minor modifications
required if $\phi_{2}(P_{n}^{\prime},U_{P_{n}^{\prime}})$ goes right or up.

For each such $P_{n}^{\prime}$, there is an interval of $Q$’s in the cut locus
of $P_{n}^{\prime}$ abutting $U_{P_{n}^{\prime}}$ along the segment from $d$
to $U_{P_{n}^{\prime}}$. (It is close to $U_{0}$ and $U_{+}$ in Figure 2.1.)
There cannot be infinitely many of these with $(P_{n}^{\prime},Q)\in E_{2}$
since $\phi(P_{n}^{\prime},Q)$ must go right or up, but
$\phi_{2}(P_{n}^{\prime},U_{P_{n}^{\prime}})$ goes left. If there were, for
infinitely many $n$, a sequence $Q_{n,m}$ approaching $U_{P_{n}^{\prime}}$
with $(P_{n},Q_{n,m})\in E_{1}$, then the sequence $(P_{n},Q_{n,n})$ would
approach $(M,B)$, but $\phi_{1}(P_{n},Q_{n,n})$ cannot approach
$\phi_{1}(M,B)$, since the possible directions differ. Thus there exist
sequences $Q_{n,m}\to U_{P_{n}^{\prime}}$ with $(P_{n}^{\prime},Q_{n,m})$ in a
new set $E_{3}$, and we may assume that $\phi_{3}(P_{n}^{\prime},Q_{n,m})$ all
have the same direction, which we may assume to be “up,” i.e., toward the
vicinity of $U_{0}$.

For each $(n,m)$, there exists a sequence $Q_{n,m,\ell}\to Q_{n,m}$ such that
the unique minimal geodesic from $P_{n}^{\prime}$ to $Q_{n,m,\ell}$ goes to
the right, i.e., in the vicinity of $U_{+}$. For each $(n,m)$, there cannot be
infinitely many $\ell$ with $(P_{n}^{\prime},Q_{n,m,\ell})\in E_{3}$, since
$\phi_{3}(P_{n}^{\prime},Q_{n,m})$ and $\phi(P_{n}^{\prime},Q_{n,m,\ell})$
have different directions. We restrict now to, for each $(n,m)$, an infinite
sequence of $\ell$ such that $(P_{n}^{\prime},Q_{n,m,\ell})\not\in E_{3}$.
Taking a diagonal limit on $m$ and $\ell$,
$(P_{n}^{\prime},Q_{n,m,\ell})\to(P_{n}^{\prime},U_{P_{n}^{\prime}})$; since
$\phi_{2}(P_{n}^{\prime},U_{P_{n}^{\prime}})$ and
$\phi(P_{n}^{\prime},Q_{n,m,\ell})$ have opposite directions,
$(P_{n}^{\prime},Q_{n,m,\ell})\not\in E_{2}$ for an infinite sequence of $m$’s
and all $\ell\geq L_{m}$ for an increasing sequence of integers $L_{m}$. Now
taking a diagonal limit over $n$, $m$, and $\ell$, we approach $(M,B)$. Since
the directions of $\phi_{1}(M,B)$ and $\phi(P_{n}^{\prime},Q_{n,m,\ell})$
differ, there must be an infinite sequence of $(P_{n}^{\prime},Q_{n,m,\ell})$
not in $E_{1}$. So it requires a fourth set $E_{4}$.

Now we discuss the minor changes for other cases to which we alluded above. If
$\phi_{2}(P_{n}^{\prime},U_{P_{n}^{\prime}})$ went right, instead of left,
then the $Q$’s will be chosen on the segment from vertex $a$ to
$U_{P_{n}^{\prime}}$, close to $U_{0}$ and $U_{-}$, and the rest of the
argument proceeds similarly.

If instead of going down or up, $\phi_{1}(M,B)$ goes left, then we consider
$P^{\prime}$ on a little segment going sharply down and left from $M$. The
expanded cut locus will be similar to that in Figure 2.1, but with
$U_{+}L_{+}$ and $U_{0}$ interchanged (and moved slightly to the other side of
line $bdb$), and similarly for $U_{-}L_{-}$ and $L_{0}$. These $P^{\prime}$
have $\phi(P^{\prime},U_{P^{\prime}})$ going up, down, or right, and an
argument like the one above works.

## References

  * [1] P. Agarwal, B. Arnov, J. O’Rourke, and C. Schevon, Star Unfolding of a Polytope with Applications, SIAM J. Comput., 26 (1997) 1689–1713.
  * [2] D.M.Davis and M.Guo, Isomorphism classes of cut loci on a cube, arXiv:2301.11366.
  * [3] M.Farber, Topological complexity of motion planning, Discr Comp Geom 29 (2003) 211–221.
  * [4] D. Recio-Mitter, Geodesic complexity of motion planning, J. Appl. Comput. Topol, 5 (2021) 141–178.


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T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-
Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7,
V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, E. Hicks48, W. Hofmann10, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, S. Kandybei40, M.
Karacson37, T.M. Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38,
B. Khanji6, Y.M. Kim46, M. Knecht38, S. Koblitz37, P. Koppenburg23, A.
Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25,
R. Kumar14,37, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J.
Luisier38, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37,
J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N.
Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A.
Martens7, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, D. Martins
Tostes1, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E.
Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R. McNulty12, C.
Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K.
Müller39, R. Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P.
Naik42, T. Nakada38, R. Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9,
M. Needham46, N. Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5,
N. Nikitin31, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47,
O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen49, B. Pal52, J. Palacios39, M. Palutan18, J. Panman37, A. Papanestis45,
M. Pappagallo13,b, C. Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D.
Patel48, M. Patel49, S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,37, A. Petrolini19,i, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D.
Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A.
Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A.
Powell51, T. du Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W.
Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40,
G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K.
Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47, F. Rodrigues2, P.
Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V. Romanovsky34, J. Rouvinet38,
T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P.
Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P.
Schaack49, M. Schiller11, S. Schleich9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y.
Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A.
Shires49, R. Silva Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37,
N.A. Smith48, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro49, B. Souza
De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U.
Straumann39, N. Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P.
Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C.
Thomas51,45, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S.
Topp-Joergensen51, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda
Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A. Vorobyev29, H.
Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25,37, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Yang3, R.
Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam

###### Abstract

A model-independent search for direct $C\\!P$ violation in the Cabibbo
suppressed decay $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ in a sample of
approximately 370,000 decays is carried out. The data were collected by the
LHCb experiment in 2010 and correspond to an integrated luminosity of 35 pb-1.
The normalized Dalitz plot distributions for $D^{+}$ and $D^{-}$ are compared
using four different binning schemes that are sensitive to different
manifestations of $C\\!P$ violation. No evidence for $C\\!P$ asymmetry is
found.

###### pacs:

13.25.Ft, 11.30.Er, 14.40.Lb

††preprint: APS/123-QED

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 |  |
---|---|---
|  | LHCb-PAPER-2011-017
|  | CERN-PH-EP-2011-163

##  I Introduction

To date $C\\!P$ violation (CPV) has been observed only in decays of neutral
$K$ and $B$ mesons. All observations are consistent with CPV being generated
by the phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ;
Kobayashi:1973fv  of the Standard Model (SM). In the charm sector, CKM
dynamics can produce direct $C\\!P$ asymmetries in Cabibbo suppressed
$D^{\pm}$ decays of the order of 10-3 or less Bianco:2003vb . Asymmetries of
up to around 1% can be generated by new physics (NP) Artuso:2008vf ;
Grossman:2006jg . In most extensions of the SM, asymmetries arise in processes
with loop diagrams. However, in some cases CPV could occur even at tree level,
for example in models with charged Higgs exchange.

In decays of hadrons, CPV can be observed when two different amplitudes with
non-zero relative weak and strong phases contribute coherently to a final
state. Three-body decays are dominated by intermediate resonant states, and
the requirement of a non-zero relative strong phase is fulfilled by the phases
of the resonances. In two-body decays, CPV leads to an asymmetry in the
partial widths. In three-body decays, the interference between resonances in
the two-dimensional phase space can lead to observable asymmetries which vary
across the Dalitz plot.

$C\\!P$-violating phase differences of $10^{\circ}$ or less do not, in
general, lead to large asymmetries in integrated decay rates, but they could
have clear signatures in the Dalitz plot, as we will show in Sect. III. This
means that a two-dimensional search should have higher sensitivity than an
integrated measurement. In addition, the distribution of an asymmetry across
phase space could hint at the underlying dynamics.

At present, no theoretical tools for computing decay fractions and relative
phases of resonant modes in $D$ decays have been applied to multibody $D^{+}$
decay modes, and no predictions have been made for how asymmetries might vary
across their Dalitz plots. A full Dalitz plot analysis of large data samples
could, in principle, measure small phase differences. However, rigorous
control of the much larger strong phases would be required. For this to be
achieved, better understanding of the amplitudes, especially in the scalar
sector, will be needed, and effects like three-body final state interactions
should be taken into account.

This paper describes a model-independent search for direct CPV in the Cabibbo
suppressed decay $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ in a binned Dalitz
plot.111Throughout this paper charge conjugation is implied, unless otherwise
stated. A direct comparison between the $D^{+}$ and the $D^{-}$ Dalitz plots
is made on a bin-by-bin basis. The data sample used is approximately 35 pb-1
collected in 2010 by the LHCb experiment at a centre of mass energy of
$\sqrt{s}=7$ TeV. This data set corresponds to nearly 10 and 20 times more
signal events than used in previous studies of this channel performed by the
BABAR Aubert:2005gj  and CLEO-c :2008zi  collaborations, respectively. It is
comparable to the dataset used in a more recent search for CPV in
$D^{+}\rightarrow\phi\pi^{+}$ decays at BELLE :2011en .

The strategy is as follows. For each bin in the Dalitz plot, a local $C\\!P$
asymmetry variable is defined Bediaga:2009tr ; Aubert:2008yd ,

$\mathcal{S}_{CP}^{i}=\frac{N^{i}(D^{+})-\alpha
N^{i}(D^{-})}{\sqrt{N^{i}(D^{+})+\alpha^{2}N^{i}(D^{-})}}\ ,\hskip
14.22636pt\alpha=\frac{N_{\mathrm{tot}}(D^{+})}{N_{\mathrm{tot}}(D^{-})},$ (1)

where $N^{i}(D^{+})$ and $N^{i}(D^{-})$ are the numbers of $D^{\pm}$
candidates in the $i^{\mathrm{th}}$ bin and $\alpha$ is the ratio between the
total $D^{+}$ and $D^{-}$ yields. The parameter $\alpha$ accounts for global
asymmetries, i.e. those that are constant across the Dalitz plot.

In the absence of Dalitz plot dependent asymmetries, the
$\mathcal{S}_{CP}^{i}$ values are distributed according to a Gaussian
distribution with zero mean and unit width. CPV signals are, therefore,
deviations from this behaviour. The numerical comparison between the $D^{+}$
and the $D^{-}$ Dalitz plots is made with a $\chi^{2}$ test
($\chi^{2}=\sum(\mathcal{S}_{CP}^{i})^{2}$). The number of degrees of freedom
is the number of bins minus one (due to the constraint of the overall
$D^{+}/D^{-}$ normalization). The $p$-value that results from this test is
defined as the probability of obtaining, for a given number of degrees of
freedom and under the assumption of no CPV, a $\chi^{2}$ that is at least as
high as the value observed lyons1989statistics . It measures the degree to
which we are confident that the differences between the $D^{+}$ and $D^{-}$
Dalitz plots are driven only by statistical fluctuations.

If CPV is observed, the $p$-value from this test could be converted into a
significance for a signal using Gaussian statistics. However, in the event
that no CPV is found, there is no model-independent mechanism for setting
limits on CPV within this procedure. In this case, the results can be compared
to simulation studies in which an artificial $C\\!P$ asymmetry is introduced
into an assumed amplitude model for the decay. Since such simulations are
clearly model-dependent, they are only used as a guide to the sensitivity of
the method, and not in the determination of the $p$-values that constitute the
results of the analysis.

The technique relies on careful accounting for local asymmetries that could be
induced by sources such as, the difference in the $K$–nucleon inelastic cross-
section, differences in the reconstruction or trigger efficiencies, left-right
detector asymmetries, etc. These effects are investigated in the two control
channels $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$.

The optimum sensitivity is obtained with bins of nearly the same size as the
area over which the asymmetry extends in the Dalitz plot. Since this is a
search for new and therefore unknown phenomena, it is necessary to be
sensitive to effects restricted to small areas as well as those that extend
over a large region of the Dalitz plot. Therefore two types of binning scheme
are employed. The first type is simply a uniform grid of equally sized bins.
The second type takes into account the fact that the $D^{+}\rightarrow
K^{-}K^{+}\pi^{+}$ Dalitz plot is dominated by the $\phi\pi^{+}$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}K^{+}$ resonances, so the
event distribution is highly non-uniform. This “adaptive binning” scheme uses
smaller bins where the density of events is high, aiming for a uniform bin
population. In each scheme, different numbers of bins are used in our search
for localized asymmetries.

The paper is organized as follows. In Sect. II, a description of the LHCb
experiment and of the data selection is presented. In Sect. III, the methods
and the binnings are discussed in detail. The study of the control channels
and of possible asymmetries generated by detector effects or backgrounds is
presented in Sect. IV. The results of our search are given in Sect. V, and the
conclusions in Sect. VI.

##  II Detector, dataset and selection

The LHCb detector Alves:2008zz  is a single-arm forward spectrometer with the
main purpose of measuring CPV and rare decays of hadrons containing $b$ and
$c$ quarks. A vertex locator (VELO) determines with high precision the
positions of the vertices of primary $pp$ collisions (PVs) and the decay
vertices of long-lived particles. The tracking system also includes a large
area silicon strip detector located in front of a dipole magnet with an
integrated field of around 4 Tm, and a combination of silicon strip detectors
and straw drift chambers placed behind the magnet. Charged hadron
identification is achieved with two ring-imaging Cherenkov (RICH) detectors.
The calorimeter system consists of a preshower, a scintillator pad detector,
an electromagnetic calorimeter and a hadronic calorimeter. It identifies high
transverse energy ($E_{\rm T}$) hadron, electron and photon candidates and
provides information for the trigger. Five muon stations composed of multi-
wire proportional chambers and triple-GEMs (gas electron multipliers) provide
fast information for the trigger and muon identification capability.

The LHCb trigger consists of two levels. The first, hardware-based level
selects leptonic and hadronic final states with high tranverse momentum, using
the subset of the detectors that are able to reduce the rate at which the
whole detector is read out to a maximum of 1 MHz. The second level, the High
Level Trigger (HLT), is subdivided into two software stages that can use the
information from all parts of the detector. The first stage, HLT1, performs a
partial reconstruction of the event, reducing the rate further and allowing
the next stage, HLT2, to fully reconstruct the individual channels. At each
stage, several selections designed for specific types of decay exist. As
luminosity increased throughout 2010 several changes in the trigger were
required. To match these, the datasets for signal and control modes are
divided into three parts according to the trigger, samples 1, 2 and 3, which
correspond to integrated luminosities of approximately 3, 5 and 28 pb-1
respectively. The magnet polarity was changed several times during data
taking.

The majority of the signal decays come via the hadronic hardware trigger,
which has an $E_{\rm T}$ threshold that varied between 2.6 and 3.6 GeV in
2010. In the HLT1, most candidates also come from the hadronic selections
which retain events with at least one high transverse momentum ($p_{\rm T}$)
track that is displaced from the PV. In the HLT2, dedicated charm triggers
select most of the signal. However, the signal yield for these channels can be
increased by using other trigger selections, such as those for decays of the
form $B\rightarrow DX$. To maintain the necessary control of Dalitz plot-
dependent asymmetries, only events from selections which have been measured
not to introduce charge asymmetries into the Dalitz plot of the
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ control mode are accepted.

The signal ( $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$) and control
($D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$) mode candidates are selected using the same criteria,
which are chosen to maximize the statistical significance of the signal.
Moreover, care is taken to use selection cuts that do not have a low
efficiency in any part of the Dalitz plot, as this would reduce the
sensitivity in these areas. The selection criteria are the same regardless of
the trigger conditions.

The event selection starts by requiring at least one PV with a minimum of five
charged tracks to exist. To control CPU consumption each event must also have
fewer than 350 reconstructed tracks. The particle identification system
constructs a relative log-likelihood for pion and kaon hypotheses,
$\mathrm{DLL}_{K\pi}$, and we require $\mathrm{DLL}_{K\pi}$ $>$ 7 for kaons
and $<$ 3 for pions. Three particles with appropriate charges are combined to
form the $D^{+}_{(s)}$ candidates. The corresponding tracks are required to
have a good fit quality ($\chi^{2}/{\rm ndf}<5$), $\mbox{$p_{\rm T}$}>$ 250
MeV$/c$, momentum $p>$ 2000 MeV$/c$ and the scalar sum of their $p_{\rm T}$
above 2800 MeV$/c$. Because a typical $D^{+}$ travels for around 8 mm before
decaying, the final state tracks should not point to the PV. The smallest
displacement from each track to the PV is computed, and a $\chi^{2}$
($\chi^{2}_{\mathrm{IP}}$), formed by using the hypothesis that this distance
is equal to zero, is required to be greater than 4 for each track. The three
daughters should be produced at a common origin, the charm decay vertex, with
vertex fit $\chi^{2}/{\rm ndf}<$ 10.

This ‘secondary’ vertex must be well separated from any PV, thus a flight
distance variable ($\chi^{2}_{\mathrm{FD}}$) is constructed. The secondary
vertex is required to have $\chi^{2}_{\mathrm{FD}}>100$, and to be downstream
of the PV. The $p_{\rm T}$ of the $D^{+}_{(s)}$ candidate must be greater than
1000 MeV$/c$, and its reconstructed trajectory is required to originate from
the PV ($\chi^{2}_{\mathrm{IP}}<12$).

Figure 1: Fitted mass spectra of (a) $K^{-}\pi^{+}\pi^{+}$ and (b)
$K^{-}K^{+}\pi^{+}$ candidates from samples 1 and 3, $D^{+}$ and $D^{-}$
combined. The signal mass windows and sidebands defined in the text are
labelled.  Figure 2: Dalitz plot of the $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$
decay for selected candidates in the signal window. The vertical $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ and horizontal
$\phi(1020)$ contributions are clearly visible in the data.

In order to quantify the signal yields ($S$), a simultaneous fit to the
invariant mass distribution of the $D^{+}$ and $D^{-}$ samples is performed. A
double Gaussian is used for the $K^{-}K^{+}\pi^{+}$ signal, whilst the
background ($B$) is described by a quadratic component and a single Gaussian
for the small contamination from $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$
above the $D^{+}_{s}$ peak. The fitted mass spectrum for samples 1 and 3
combined is shown in Fig. 1, giving the yields shown in Table 1. A weighted
mean of the widths of the two Gaussian contributions to the mass peaks is used
to determine the overall widths, $\sigma$, as 6.35 MeV/$c^{2}$ for
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$, 7.05 MeV/$c^{2}$ for
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$, and 8.0 MeV/$c^{2}$ for
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$. These values are used to define signal
mass windows of approximately 2$\sigma$ in which the Dalitz plots are
constructed. The purities, defined as $S/(B+S)$ within these mass regions, are
also shown in Table 1 for samples 1 and 3 in the different decay modes.

For sample 2, the yield cannot be taken directly from the fit, because there
is a mass cut in the HLT2 line that accepts the majority of the signal,
selecting events in a $\pm 25$ MeV$/c^{2}$ window around the nominal value.
However, another HLT2 line with a looser mass cut that is otherwise identical
to the main HLT2 line exists, although only one event in 100 is retained. In
this line the purity is found to be the same in sample 2 as in sample 3. The
yield in sample 2 is then inferred as the total $(S+B)$ in all allowed
triggers in the mass window times the purity in sample 3. Thus the overall
yield of signal $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ candidates in the three
samples within the mass window is approximately 370,000. The total number of
candidates ($S+B$) in each decay mode used in the analysis are given in Table
2. The Dalitz plot of data in the $D^{+}$ window is shown in Fig. 2.

Table 1:  Yield ($S$) and purity for samples 1 and 3 after the final selection. The purity is estimated in the 2$\sigma$ mass window.  Decay | Yield | Purity
---|---|---
| Sample 1+3 | Sample 1 | Sample 3
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ | $(3.284\pm 0.006)\times 10^{5}$ | 88% | 92%
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ | $(4.615\pm 0.012)\times 10^{5}$ | 89% | 92%
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ | $(3.3777\pm 0.0037)\times 10^{6}$ | 98% | 98%
Table 2: Number of candidates $(S+B)$ in the signal windows shown in Fig. 1 after the final selection, for use in the subsequent analysis.  | sample 1 | sample 2 | sample 3 | Total
---|---|---|---|---
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ | 84,667 | 65,781 | 253,446 | 403,894
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ | 126,206 | 91,664 | 346,068 | 563,938
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ | 858,356 | 687,197 | 2,294,315 | 3,839,868
Table 3:  The CLEO-c amplitude model “B” :2008zi  used in the simulation studies. The uncertainties are statistical, experimental systematic and model systematic respectively.  Resonance |  Amplitude |  Relative phase |  Fit fraction
---|---|---|---
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ |  1 (fixed) |  0 (fixed) | $25.7\pm 0.5^{+0.4+0.1}_{-0.3-1.2}$
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ | $\phantom{0}4.56\pm 0.13^{+0.10+0.42}_{-0.01-0.39}$ | $\phantom{-}\phantom{0}70\pm 6^{+1+16}_{-6-23}$ | $18.8\pm 1.2^{+0.6+3.2}_{-0.1-3.4}$
$\kappa(800)$ | $\phantom{0}2.30\pm 0.13^{+0.01+0.52}_{-0.11-0.29}$ | $\phantom{}-87\pm 6^{+2+15}_{-3-10}$ | $\phantom{0}7.0\pm 0.8^{+0.0+3.5}_{-0.6-1.9}$
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{2}(1430)^{0}$ | $\phantom{00}7.6\pm 0.8^{+0.5+2.4}_{-0.6-4.8}$ | $\phantom{-}171\pm 4^{+0+24}_{-2-11}$ | $\phantom{0}1.7\pm 0.4^{+0.3+1.2}_{-0.2-0.7}$
$\phi(1020)$ | $1.166\pm 0.015^{+0.001+0.025}_{-0.009-0.009}$ | $-163\pm 3^{+1+14}_{-1-5}$ | $27.8\pm 0.4^{+0.1+0.2}_{-0.3-0.4}$
$a_{0}(1450)^{0}$ | $\phantom{0}1.50\pm 0.10^{+0.09+0.92}_{-0.06-0.33}$ | $\phantom{-}116\pm 2^{+1+7}_{-1-14}$ | $\phantom{0}4.6\pm 0.6^{+0.5+7.2}_{-0.3-1.8}$
$\phi(1680)$ | $\phantom{0}1.86\pm 0.20^{+0.02+0.62}_{-0.08-0.77}$ | $-112\pm 6^{+3+19}_{-4-12}$ | $0.51\pm 0.11^{+0.01+0.37}_{-0.04-0.15}$

Within the $2\sigma$ $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ mass window, about
8.6% of events are background. Apart from random three-body track
combinations, charm backgrounds and two-body resonances plus one track are
expected. Charm reflections appear when a particle is wrongly identified in a
true charm three-body decay and/or a track in a four-body charm decay is lost.
The main three-body reflection in the $K^{-}K^{+}\pi^{+}$ spectrum is the
Cabibbo-favoured $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, where the incorrect
assignment of the kaon mass to the pion leads to a distribution that partially
overlaps with the $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ signal region, but
not with $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$. The four body, Cabibbo-favoured
mode $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ where a $\pi^{+}$ is lost
and the $\pi^{-}$ is misidentified as a $K^{-}$ will appear broadly
distributed in $K^{-}K^{+}\pi^{+}$ mass, but its resonances could create
structures in the Dalitz plot. Similarly, $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ and $\phi$ resonances
from the PV misreconstructed with a random track forming a three-body vertex
will also appear.

##  III Methods and binnings

Figure 3: $\mathcal{S}_{CP}$ across the Dalitz plot in a Monte Carlo pseudo-
experiment with a large number of events with (a) no CPV and (b) a 4∘ CPV in
the $\phi\pi$ phase. Note the difference in colour scale between (a) and (b).

Figure 4:  Layout of the (a) “Adaptive I” and (b) “Adaptive II” binnings on
the Dalitz plot of data.

Monte Carlo pseudo-experiments are used to verify that we can detect CPV with
the strategy outlined in Sect. I without producing fake signals, and to devise
and test suitable binning schemes for the Dalitz plot. They are also used to
quantify our sensitivity to possible manifestations of CPV, where we define
the sensitivity to a given level of CPV as the probability of observing it
with $3\sigma$ significance.

For the $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ Dalitz plot model, the result of
the CLEO-c analysis (fit B) :2008zi  is used. The amplitudes and phases of the
resonances used in this model are reproduced in Table 3. For simplicity, only
resonant modes with fit fractions greater than $2\%$ are included in the
pseudo-experiments. The fit fraction for a resonance is defined as the
integral of its squared amplitude over the Dalitz plot divided by the integral
of the square of the overall complex amplitude. An efficiency function is
determined from a two-dimensional second order polynomial fit to the Dalitz
plot distribution of triggered events that survive the selection cuts in the
GEANT-based Agostinelli:2002hh  LHCb Monte Carlo simulation for nonresonant
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$. A simple model for the background is
inferred from the Dalitz plots of the sidebands of the $D^{+}\rightarrow
K^{-}K^{+}\pi^{+}$ signal. It is composed of random combinations of $K^{-}$,
$K^{+}$, and $\pi^{+}$ tracks, $\phi$ resonances with $\pi^{+}$ tracks, and
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ resonances with
$K^{+}$ tracks. The CLEO-c Dalitz plot analysis has large uncertainties, as do
the background and efficiency simulations (due to limited numbers of MC
events), so the method is tested on a range of different Dalitz plot models.

Pseudo-experiments with large numbers of events are used to investigate how
CPV would be observed in the Dalitz plot. These experiments are simple “toy”
simulations that produce points in the Dalitz plot according to the
probability density function determined from the CLEO-c amplitude model with
no representation of the proton-proton collision, detector, or trigger. Figure
3(a) illustrates the values of $\mathcal{S}_{CP}^{i}$ observed with $8\times
10^{7}$ events and no CPV. This dataset is approximately 50 times larger than
the data sample under study. The resulting $\chi^{2}/{\rm ndf}$ is
$253.4/218$, giving a $p$-value for consistency with no CPV of 5.0%. This test
shows that the method by itself is very unlikely to yield false positive
results. Figure 3(b) shows an example test using $5\times 10^{7}$ events with
a $C\\!P$ violating phase difference of $4^{\circ}$ between the amplitudes for
the $\phi(1020)\pi^{+}$ component in $D^{+}$ and $D^{-}$ decays. The $p$-value
in this case is less than $10^{-100}$. The CPV effect is clearly visible, and
is spread over a broad area of the plot, changing sign from left to right.
This sign change means the CPV causes only a 0.1% difference in the total
decay rate between $D^{+}$ and $D^{-}$. This illustrates the strength of our
method, as the asymmetry would be much more difficult to detect in a
measurement that was integrated over the Dalitz plot. Even with no systematic
uncertainties, to see a 0.1% asymmetry at the $3\sigma$ level would require
$2.25\times 10^{6}$ events. With the method and much smaller dataset used here
we would observe this signal at the $3\sigma$ level with 76% probability, as
shown in Table 4 below.

The sensitivity to a particular manifestation of CPV depends on the choice of
binning. The fact that the $C\\!P$-violating region in most of the pseudo-
experiments covers a broad area of the Dalitz plot suggests that the optimal
number of bins for this type of asymmetry is low. Each bin adds a degree of
freedom without changing the $\chi^{2}$ value for consistency with no CPV.
However, if $C\\!P$ asymmetries change sign within a bin, they will not be
seen. Similarly, the sensitivity is reduced if only a small part of a large
bin has any CPV in it. To avoid effects due to excessive fluctuations, bins
that contain fewer than 50 candidates are not used anywhere in the analysis.
Such bins are very rare.

The binnings are chosen to reflect the highly non-uniform structure of the
Dalitz plot. A simple adaptive binning algorithm was devised to define
binnings of approximately equal population without separating $D^{+}$ and
$D^{-}$. Two binnings that are found to have good sensitivity to the simulated
asymmetries contain 25 bins (“Adaptive I”) arranged as shown in Fig. 4(a), and
106 bins (“Adaptive II”) arranged as shown in Fig. 4(b). For Adaptive I, a
simulation of the relative value of the strong phase across the Dalitz plot in
the CLEO-c amplitude model is used to refine the results of the algorithm: if
the strong phase varies significantly across a bin, $C\\!P$ asymmetries are
more likely to change sign. Therefore the bin boundaries are adjusted to
minimise changes in the strong phase within bins. The model-dependence of this
simulation could, in principle, influence the binning and therefore the
sensitivity to CPV, but it cannot introduce model-dependence into the final
results as no artificial signal could result purely from the choice of
binning. Two further binning schemes, “Uniform I” and “Uniform II”, are
defined. These use regular arrays of rectangular bins of equal size.

Table 4:  Results from sets of 100 pseudo-experiments with different $C\\!P$ asymmetries and Adaptive I and II binnings. $p(3\sigma)$ is the probability of a 3$\sigma$ observation of CPV. $\langle S\rangle$ is the mean significance with which CPV is observed.  CPV | Adaptive I | Adaptive II
---|---|---
| $p(3\sigma)$ | $\langle S\rangle$ | $p(3\sigma)$ | $\langle S\rangle$
no CPV | 0 | 0.84$\sigma$ | 1% | 0.84$\sigma$
$6^{\circ}$ in $\phi(1020)$ phase | 99% | 7.0$\sigma$ | 98% | 5.2$\sigma$
$5^{\circ}$ in $\phi(1020)$ phase | 97% | 5.5$\sigma$ | 79% | 3.8$\sigma$
$4^{\circ}$ in $\phi(1020)$ phase | 76% | 3.8$\sigma$ | 41% | 2.7$\sigma$
$3^{\circ}$ in $\phi(1020)$ phase | 38% | 2.8$\sigma$ | 12% | 1.9$\sigma$
$2^{\circ}$ in $\phi(1020)$ phase | 5% | 1.6$\sigma$ | 2% | 1.2$\sigma$
$6.3\%$ in $\kappa(800)$ magnitude | 16% | 1.9$\sigma$ | 24% | 2.2$\sigma$
$11\%$ in $\kappa(800)$ magnitude | 83% | 4.2$\sigma$ | 95% | 5.6$\sigma$
Table 5:  Results from sets of 100 pseudo-experiments with $4^{\circ}$ CPV in the $\phi(1020)$ phase and different Dalitz plot models. $p(3\sigma)$ is the probability of a 3$\sigma$ observation of CPV. $\langle S\rangle$ is the mean significance with which CPV is observed. The sample size is comparable to that seen in data.  Model | Adaptive I | Adaptive II
---|---|---
| $p(3\sigma)$ | $\langle S\rangle$ | $p(3\sigma)$ | $\langle S\rangle$
B (baseline) | 76% | 3.8$\sigma$ | 41% | 2.7$\sigma$
A | 84% | 4.3$\sigma$ | 47% | 2.9$\sigma$
B2 (add $f_{0}(980)$) | 53% | 3.2$\sigma$ | 24% | 2.2$\sigma$
B3 (vary $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ magn.) | 82% | 4.0$\sigma$ | 41% | 2.8$\sigma$
B4 (vary $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$ phase) | 73% | 3.7$\sigma$ | 38% | 2.7$\sigma$

The adaptive binnings are used to determine the sensitivity to several
manifestations of CPV. With 200 test experiments of approximately the same
size as the signal sample in data, including no asymmetries, no
$C\\!P$-violating signals are observed at the 3$\sigma$ level with Adaptive I
or Adaptive II. The expectation is 0.3.

With the chosen binnings, a number of sets of 100 pseudo-experiments with
different $C\\!P$-violating asymmetries are produced. The probability of
observing a given signal in either the $\phi(1020)$ or $\kappa(800)$
resonances with 3$\sigma$ significance is calculated in samples of the same
size as the dataset. The results are given in Table 4. The CPV shows up both
in the $\chi^{2}/{\rm ndf}$ and in the width of the fitted $\mathcal{S}_{CP}$
distribution.

For comparison, the asymmetries in the $\phi$ phase and $\kappa$ magnitude
measured by the CLEO collaboration using the same amplitude model were $(6\pm
6^{+0+6}_{-2-2})^{\circ}$ and $(-12\pm 12^{+6+2}_{-1-10})\%$,222The
conventions used in the CLEO paper to define asymmetry are different, so the
asymmetries in Table II of :2008zi  have been multiplied by two in order to be
comparable with those given above. where the uncertainties are statistical,
systematic and model-dependent, respectively. Table 4 suggests that, assuming
their model, we would be at least 95% confident of detecting the central
values of these asymmetries.

The sensitivity of the results to variations in the Dalitz plot model and the
background is investigated, and example results for the $C\\!P$ asymmetry in
the $\phi(1020)$ phase are shown in Table 5. In this table, models A and B are
taken from the CLEO paper, model B2 includes an $f_{0}(980)$ contribution that
accounts for approximately 8% of events, and models B3 and B4 are variations
of the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{0}(1430)^{0}$
amplitude and phase within their uncertainties. As expected, the sensitivity
to CPV in the resonances of an amplitude model depends quite strongly on the
details of the model. This provides further justification for our model-
independent approach. However, a reasonable level of sensitivity is retained
in all the cases we tested. Thus, when taken together, the studies show that
the method works well. It does not yield fake signals, and should be sensitive
to any large CPV that varies significantly across the Dalitz plot even if it
does not occur precisely in the way investigated here.

##  IV Control modes

It is possible that asymmetries exist in the data that do not result from CPV,
for example due to production, backgrounds, instrumental effects such as left-
right differences in detection efficiency, or momentum-dependent differences
in the interaction cross-sections of the daughter particles with detector
material. Our sensitivity to such asymmetries is investigated in the two
Cabibbo favoured control channels, where there is no large CPV predicted. The
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ control mode has an order of magnitude
more candidates than the Cabibbo-suppressed signal mode, and is more sensitive
to detector effects since there is no cancellation between $K^{+}$ and $K^{-}$
reconstruction efficiencies. Conversely, the $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$ control mode is very similar to our signal mode in terms of
resonant structure, number of candidates, kinematics, detector effects, and
backgrounds.

Figure 5: (a) Distribution of $\mathcal{S}_{CP}$ values from $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ from a test with 900 uniform bins. The mean of the fitted
Gaussian distribution is $0.015\pm 0.034$ and the width is $0.996\pm 0.023$.
(b) Distribution of $\mathcal{S}_{CP}$ values from $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$ with 129 bins. The fitted mean is $-0.011\pm 0.084$ and the
width is $0.958\pm 0.060$.

Figure 6:  Dalitz plots of (a) $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, showing
the 25-bin adaptive scheme with the $\mathcal{S}_{CP}$ values, and (b)
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$, showing the three regions referred
to in the text. The higher and lower $K^{-}\pi^{+}$ invariant mass
combinations are plotted in (a) as there are identical pions in the final
state.

The control modes and their mass sidebands defined in Fig. 1 are tested for
asymmetries using the method described in the previous section. Adaptive and
uniform binning schemes are defined for $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$
and $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$. They are applied to samples 1–3
and each magnet polarity separately. In the final results, the asymmetries
measured in data taken with positive and negative magnet polarity are combined
in order to cancel left-right detector asymmetries. The precise number of bins
chosen is arbitrary, but care is taken to use a wide range of tests with
binnings that reflect the size of the dataset for the decay mode under study.

For $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, five different sets of bins in
each scheme are used. A very low $p$-value would indicate a local asymmetry.
One test with 25 adaptive bins in one of the subsamples (with negative magnet
polarity) has a $p$-value of 0.1%, but when combined with the positive
polarity sample the $p$-value increases to 1.7%. All other tests yield
$p$-values ranging from 1–98%. Some example results are given in Table 6. A
typical distribution of the $\mathcal{S}_{CP}$ values with a Gaussian fit is
shown in Fig. 5(a) for a test with 900 uniform bins. The fitted values of the
mean and width are consistent with one and zero respectively, suggesting that
the differences between the $D^{+}$ and the $D^{-}$ Dalitz plots are driven
only by statistical fluctuations.

Table 6: Results ($p$-values, in %) from tests with the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ control channel using the uniform and adaptive binning schemes. The values correspond to tests performed on the whole dataset in the mass windows defined in Sect. II.  | 1300 bins | 900 bins | 400 bins | 100 bins | 25 bins
---|---|---|---|---|---
Uniform | 73.8 | 17.7 | 72.6 | 54.6 | 1.7
Adaptive | 81.7 | 57.4 | 65.8 | 30.0 | 11.8

For the $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ mode a different procedure is
followed due to the smaller sample size and to the high density of events
along the $\phi$ and the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ bands. The Dalitz plot is
divided into three zones, as shown in Fig. 6. Each zone is further divided
into 300, 100 and 30 bins of same size. The results are given in Table 7. In
addition, a test is performed on the whole Dalitz plot using 129 bins chosen
by the adaptive algorithm, and a version of the 25-bin scheme outlined in
Sect. III scaled by the ratio of the available phase space in the two modes.
These tests yield $p$-values of 71.5% and 34.3% respectively.

Table 7: Results ($p$-values, in %) from tests with the $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ control channel using the uniform binning scheme. The values correspond to tests performed separately on Zones A-C, with samples 1-3 and both magnet polarities combined.  bins | Zone A | Zone B | Zone C
---|---|---|---
300 | 20.1 | 25.3 | 14.5
100 | 41.7 | 84.6 | 89.5
30 | 66.0 | 62.5 | 24.6

Other possible sources of local charge asymmetry in the signal region are the
charm contamination of the background, and asymmetries from CPV in
misreconstructed $B$ decays. In order to investigate the first possibility,
similar tests are carried out in the mass sidebands of the
$D^{+}_{(s)}\rightarrow K^{-}K^{+}\pi^{+}$ signal (illustrated in Fig. 1).
There is no evidence for asymmetries in the background.

From a simulation of the decay $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ the
level of secondary charm ($B\rightarrow DX$) in our selected sample is found
to be 4.5%. The main discriminating variable to distinguish between prompt and
secondary charm is the impact parameter (IP) of the $D$ with respect to the
primary vertex. Given the long $B$ lifetime, the IP distribution of secondary
charm candidates is shifted towards larger values compared to that of prompt
$D^{+}$ mesons.

The effect of secondary charm is investigated by dividing the data set
according to the value of the candidate IP significance ($\chi^{2}_{IP}$). The
subsample with events having larger $\chi^{2}_{IP}$ are likely to be richer in
secondary charm. The results are shown in Table 8. No anomalous effects are
seen in the high $\chi^{2}_{IP}$ sample, so contamination from secondary charm
with CPV does not affect our results for studies with our current level of
sensitivity.

Table 8: Results ($p$-values, in %) from tests with the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ samples divided according to the impact parameter with respect to the primary vertex. The tests are performed using the adaptive binning scheme with 25 bins.  | $\chi^{2}_{IP}<6$ | $\chi^{2}_{IP}>6$
---|---|---
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ | 8.5 | 88.9
$D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ | 52.0 | 30.6

The analysis on the two control modes and on the sidebands in the final states
$K^{-}K^{+}\pi^{+}$ and $K^{-}\pi^{+}\pi^{+}$ gives results from all tests
that are fully consistent with no asymmetry. Therefore, any asymmetry observed
in $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ is likely to be a real physics effect.

Table 9: Fitted means and widths, $\chi^{2}/{\rm ndf}$ and $p$-values for consistency with no CPV for the $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ decay mode with four different binnings.  Binning | Fitted mean | Fitted width | $\chi^{2}/{\rm ndf}$ |  $p$-value (%)
---|---|---|---|---
Adaptive I | $\phantom{-}0.01\pm 0.23$ | $1.13\pm 0.16$ | 32.0/24 | 12.7
Adaptive II | $-0.024\pm 0.010$ | $1.078\pm 0.074$ | 123.4/105 | 10.6
Uniform I | $-0.043\pm 0.073$ | $0.929\pm 0.051$ | 191.3/198 | 82.1
Uniform II | $-0.039\pm 0.045$ | $1.011\pm 0.034$ | 519.5/529 | 60.5

Figure 7: Distribution of ${\mathcal{S}}^{i}_{\it CP}$ in the Dalitz plot for
(a) “Adaptive I”, (b) “Adaptive II”, (c) “Uniform I” and (d) “Uniform II”. In
(c) and (d) bins at the edges are not shown if the number of entries is not
above a threshold of 50 (see Sect. III).

Figure 8: Distribution of ${\mathcal{S}}^{i}_{\it CP}$ fitted to Gaussian
functions, for (a) “Adaptive I”, (b) “Adaptive II”, (c) “Uniform I” and (d)
“Uniform II”. The fit results are given in Table 9.

##  V Results

The signal sample with which we search for $C\\!P$ violation consists of
403,894 candidates selected within the $K^{-}K^{+}\pi^{+}$ mass window from
1856.7 to 1882.1 MeV$/c^{2}$, as described in Sect. II. There are 200,336 and
203,558 $D^{+}$ and $D^{-}$ candidates respectively. This implies a
normalization factor $\alpha=N_{\rm tot}(D^{+})/N_{\rm tot}(D^{-})=0.984\pm
0.003$, to be used in Eq. 1.

The strategy for looking for signs of localized CPV is discussed in the
previous sections. In the absence of local asymmetries in the control channels
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\rightarrow
K^{-}K^{+}\pi^{+}$ and in the sidebands of the $K^{-}K^{+}\pi^{+}$ mass
spectrum, we investigate the signal sample under different binning choices.

First, the adaptive binning is used with 25 and 106 bins in the Dalitz plot as
illustrated in Fig. 4. Then CPV is investigated with uniform binnings, using
200 and 530 bins of equal size. For each of these binning choices, the
significance ${\cal S}^{i}_{\it CP}$ of the difference in $D^{+}$ and $D^{-}$
population is computed for each bin $i$, as defined in Eq. 1. The
$\chi^{2}/{\rm ndf}=\sum_{i}({\cal S}^{i}_{\it CP})^{2}/{\rm ndf}$ is
calculated and the $p$-value is obtained. The distributions of
${\mathcal{S}}^{i}_{\it CP}$ are fitted to Gaussian functions.

The $p$-values are shown in Table 9. The Dalitz plot distributions of
${\mathcal{S}}^{i}_{\it CP}$ are shown in Fig. 7. In Fig. 8 the distributions
of ${\mathcal{S}}^{i}_{\it CP}$ and the corresponding Gaussian fits for the
different binnings are shown. The $p$-values obtained indicate no evidence for
CPV. This is corroborated by the good fits of the ${\mathcal{S}}^{i}_{\it CP}$
distributions to Gaussians, with means and widths consistent with 0 and 1,
respectively.

As further checks, many other binnings are tested. The number of bins in the
adaptive and uniform binning schemes is varied from 28 to 106 and from 21 to
530 respectively. The samples are separated according to the magnet polarity
and the same studies are repeated. In all cases the $p$-values are consistent
with no CPV, with values ranging from 4% to 99%. We conclude that there is no
evidence for CPV in our data sample of $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$.

##  VI Conclusion

Due to the rich structure of their Dalitz plots, three body charm decays are
sensitive to $C\\!P$ violating phases within and beyond the Standard Model.
Here, a model-independent search for direct $C\\!P$ violation is performed in
the Cabibbo suppressed decay $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ with 35 pb-1
of data collected by the LHCb experiment, and no evidence for CPV is found.
Several binnings are used to compare normalised $D^{+}$ and $D^{-}$ Dalitz
plot distributions. This technique is validated with large numbers of
simulated pseudo-experiments and with Cabibbo favoured control channels from
the data: no false positive signals are seen. To our knowledge this is the
first time a search for CPV is performed using adaptive bins which reflect the
structure of the Dalitz plot.

Monte Carlo simulations illustrate that large localised asymmetries can occur
without causing detectable differences in integrated decay rates. The
technique used here is shown to be sensitive to such asymmetries. Assuming the
decay model, efficiency parameterisation and background model described in
Sect. III we would be 90% confident of seeing a $C\\!P$ violating difference
of either $5^{\circ}$ in the phase of the $\phi\pi^{+}$ or 11% in the
magnitude of the $\kappa(800)K^{+}$ with $3\sigma$ significance. Since we find
no evidence of CPV, effects of this size are unlikely to exist.

##  VII Acknowledgments

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.

## References

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