# Efficient full state tomography of the orbital angular momentum states of
light using Helicity sorter

Joel M. Sunil<EMAIL_ADDRESS>Department of Physics, BITS Pilani K K
Birla Goa Campus, NH17B, Zuarinagar, Goa 403726, India  Sandeep K. Goyal
<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of
Science Education & Research (IISER) Mohali, Sector 81 SAS Nagar, Manauli PO
140306, Punjab, India.

###### Abstract

Orbital angular momentum (OAM) of light is a promising degree of freedom for
quantum communication and quantum computation. However, efficient methods to
transform and measure OAM modes are still not available readily. Here, we
propose a method for full quantum state tomography for arbitrary OAM states of
light. The key component in our scheme is the helicity sorter, a device that
can sort OAM modes of a light beam based on their Helicity. We present two
setups to implement the Helicity sorted, one requires $\mathcal{O}(N)$ number
of optical elements for $2^{N}$ OAM modes, and the other uses spatial light
modulators and is independent of the number of OAM modes. Ours is the first
method to provide the efficient and scalable full quantum state tomography of
the OAM modes and can be extremely important for quantum communication and
quantum computation.

##  I Introduction

Quantum information processing (QIP) offers a new avenue for efficient
computation and most secure communication. QIP tasks require quantum systems,
which can be efficiently manipulated, measured and are robust against
environmental effects. Orbital angular momentum (OAM) states of light is among
the most prominent candidates for QIP tasks [1, 2, 3, 4, 5, 6, 7]. One of the
biggest difficulties in working with OAM states of light is performing the
quantum measurements and state tomography on these states. There are several
schemes to perform measurements on the OAM states of light [8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 16, 19, 20, 21], but all of them have their
limitations. Here, we propose a method to perform full quantum state
tomography (QST) on an arbitrary state of a light beam.

QST estimates the density matrix of a system by performing a complete set of
measurements on an ensemble of identically prepared systems. Typically, QST
protocol require at least $N+1$ mutually independent measurement operators in
order to estimate the density operator of $N$-dimensional system [22, 23, 24].
The measurement operators whose eigenbases are mutually unbiased are the best
choice for QST protocol [25, 26, 27, 28]. However, for large dimensions $N$,
performing measurements in $N+1$ mutually independent operators can be a
difficult task.

QST on the OAM states of light is especially challenging [29, 30, 26]. Often
the researchers are content with measuring the population of each of the OAM
modes which amounts to partial state tomography. This can be achieved by
sorting the OAM modes in distinct paths and performing intensity measurement
in each path. This method requires cumbersome interferometric setups and are
nonscalable [8, 9]. OAM modes can also be measured by performing a log-polar
transformation that converts the helical phase into a linear phase [10, 11,
12].

QST on the OAM states is further made difficult by the fact that unitary
transformations that are required to perform measurements in the mutually
unbiased bases are very resource intensive. Methods to perform only SU(2)
transformations efficiently are known on specific OAM subspaces [31, 32].

There are alternate ways to perform measurements on the OAM modes that include
schemes by measuring the transverse intensity profile of the light beams [13,
14, 15, 16, 17, 18, 16, 19, 20, 21]. The Auxilliary Hilbert Space Tomography
(AHST) proposed by Liu et. al [21] is one such scheme that performs a partial
QST on the OAM states.

In this paper, we present a method to perform a full QST on the OAM states.
The key component of our scheme is a device named helicity sorter. As the name
suggests, the helicity sorter can sort the OAM states based on the helicity.
This allows us to perform operations on negative and positive subspaces of the
OAM modes independently and SU($2$) transformations between these two
subspaces. Here we propose two schemes to realize helicity sorter, one uses
only linear optical setup and requires $O(N)$ number of optical elements to
sort $2^{N}$ OAM modes, whereas other scheme uses SLMs and is independent of
the number $N$. The proposed QST scheme uses the AHST method of partial QST
along with the Helicity sort.

Ours is the first method to perform a full QST on OAM states of light. Another
highlight of our scheme is that it requires only three measurement settings
for any arbitrary number of OAM modes. The linear optical nature and the
finite number of measurement settings make our scheme efficient and scalable,
and can be a turning point in the OAM based quantum computation and
communication.

The paper is organised as follows: in Sec. II we present the relevant
background needed for our work. In Sec. III we present the scheme for the full
QST and the implementation schemes for the Helicity sorter. We conclude in
Sec. IV.

##  II Background

In this section, we first introduce two linear optical devices that are
essential to realizing a helicity-sorter: the OAM-sorter and the Gouy phase
radial mode sorter. We also summarize the AHST scheme for QST [21].

###  II.1 The OAM-sorter

An OAM sorter is a Mach-Zehnder interferometer that can sort OAM modes based
on their $\ell$ values [8, 9, 33]. For example, the device can be tuned in
such a way that the even and odd $\ell$ modes will be sorted in separate
paths. For an OAM sorter we require a linear optical device called Dove prism
(DP). The action $\mathcal{D}(\beta)$ of a DP rotated by an angle $\beta$
about the axis of propagation of light on OAM modes can be written

$\displaystyle\mathcal{D}(\beta)\ket{\ell}=e^{i2\ell\beta}\ket{-\ell}.$ (1)

In an OAM-sorter two DPs are introduced in one arm of a Mach-Zehnder
interferometer (Fig. 1). One of the DP is rotated at an angle $\beta$ and the
other is kept at zero angle. This setup of DPs will leave the OAM mode
unaffected and impart a phase $e^{i2\ell\beta}$ to the mode.

If a state $\ket{\psi}=\sum_{\ell}g_{\ell}\ket{\ell}$ enters the Mach-Zehnder
setup, then in path $a$ and path $b$ the states read

$\displaystyle\ket{\psi_{a}}$
$\displaystyle=\sum_{\ell}g_{\ell}e^{i2\ell\beta}\ket{\ell}$
$\displaystyle\ket{\psi_{b}}$ $\displaystyle=\sum_{\ell}g_{\ell}\ket{\ell}.$
(2)

At the output of the interferometer the states reads
$(\ket{\psi_{a}}\pm\ket{\psi_{b}})/\sqrt{2}$. Therefore, choosing
$\beta=\frac{\pi}{2}$ only either even or odd modes will survive in the output
modes. Hence, one can sort the even and odd modes. The same setup can be used
with different value of $\beta$ to sort special sets of OAM modes [8, 9, 33].

Figure 1: The OAM-sorter. Here we have a Mach-Zehnder interferometer with two
Balanced Beam Splitter (BBS). Two DPs are placed in the path $a$ of the
interferometer which attribute an $\ell$ dependent phase to the OAM modes.

###  II.2 Radial mode sorter

As the name suggests, a radial mode sorter sorts the radial modes instead of
$\ell$ modes [34]. Similar to the OAM sorter, the radial mode sorter also uses
a Mach-Zehnder interferometer, but instead of using the DPs, it uses a
combination of three lenses which introduce radial mode-dependent Gouy phase.

Figure 2: The Gouy-phase radial mode sorter. One arm of the Mach-Zehnder
interferometer contains three lenses of focal length $f_{1}$, $f_{2}$ and
$f_{3}$ which lead to a relative Gouy phase between the beams in the two arms.

The radial mode sorter consists of a Mach-Zehnder interferometric setup with a
lens configuration or Gouy Phase setup in one of the arms as shown in Fig. 2.
The lenses cause the state in that arm to accumulate a Gouy phase
$(|\ell|+2p+1)\alpha$ relative to the other arm. Here $p$ is the radial index
of the LG mode. The focal lengths of the lenses $f_{1},\>f_{2}$ and $f_{3}$,
and the distances between them $\>d_{1}$ and $d_{2}$ are the parameters that
determine the phase $\alpha$, and are chosen such that the beam waist of the
beams in the two arms remain the same. Following the analogy with the OAM
sorter one can see that this setup can sort the LG modes based on the radial
index $p$ for a fixed value of $\ell$.

###  II.3 Auxilliary Hilbert Space Tomography

A number of schemes to perform partial QST have been proposed over the years
[35, 36, 29]. Among them, one of the scheme which promises the most efficient
and scalable QST for OAM states is the AHST [21]. This is a single shot method
to perform the QST using the intensity distribution $I(r,\phi)$ of the light
beam containing the OAM states.

Let the state of the beam is $\rho$. The intensity profile measured for this
state is given by:

$\begin{split}I(r,\,\phi)&=\bra{r,\,\phi}\rho\ket{r,\,\phi}\\\
&=A\sum_{\ell_{1},\,\ell_{2}}f_{\ell_{1}}(r,\,\phi)f_{\ell_{2}}^{*}(r,\,\phi)\rho_{\ell_{1},\ell_{2}},\end{split}$
(3)

where $\rho_{\ell_{1},\ell_{2}}=\bra{\ell_{1}}\rho\ket{\ell_{2}}$ are the
matrix elements of $\rho$ in the OAM basis and
$f_{\ell}(r,\,\phi)=\braket{r,\,\phi}{\ell}$ is the OAM mode in the polar
coordinates.

AHST method exploits the fact that the Fourier transform of
$f_{\ell_{1}}(r,\,\phi)f_{\ell_{2}}^{*}(r,\,\phi)$, i.e.,
$P_{\ell_{1},\,\ell_{2}}(r_{f},\,\phi_{f})=\mathcal{F}[f_{\ell_{1}}(r,\,\phi)f_{\ell_{2}}^{*}(r,\,\phi)]$
satisfies the orthogonality relation

$\displaystyle\iint
P_{\ell_{1},\,\ell_{2}}P^{*}_{\ell_{1}^{\prime},\,\ell_{2}^{\prime}}e^{\frac{\pi^{2}r_{f}^{2}w_{o}^{2}}{2}}r_{f}dr_{f}d\phi_{f}=\frac{2}{\pi
w_{o}^{2}}\delta_{\ell_{1}\ell_{1}^{\prime}}\delta_{\ell_{2}\ell_{2}^{\prime}},$
(4)

for $\ell_{1},\ell_{2},\ell_{1}^{\prime},\ell_{2}^{\prime}\geq 0$. $w_{o}$ is
the beam waist at the plane of measurement. Using the orthogonality property
we can calculate all the elements of the density matrix
$\rho_{\ell_{1},\ell_{2}}$ as

$\rho_{\ell_{1},\ell_{2}}=\frac{\pi
wo^{2}}{2}\int_{0}^{\infty}\int_{0}^{2\pi}\mathcal{F}[I(r,\,\phi)]P^{*}_{\ell_{1},\,\ell_{2}}\\\
\times e^{\frac{\pi^{2}r_{f}^{2}w_{o}^{2}}{2}}r_{f}dr_{f}d\phi_{f}.$ (5)

We are thus able to perform QST on any state prepared in the subspace spanned
by $\\{\ket{\ell},\>\ell\geq 0\\}$.

We can also perform QST on any state prepared in the basis
$\ket{\ell},\>\ell\leq 0$ using the exact same method because

$\braket{r,\,\phi}{\ell_{1}}\braket{\ell_{2}}{r,\,\phi}=\braket{r,\,\phi}{-\ell_{2}}\braket{-\ell_{1}}{r,\,\phi}$
(6) $\implies P_{\ell_{1},\,\ell_{2}}=P_{-\ell_{2},\,-\ell_{1}}.$ (7)

For any state prepared in the basis $\ket{\ell},\>\ell\leq 0$ we can obtain
the density matrix from the intensity profile using:

$\rho_{-\ell_{1},-\ell_{2}}=\frac{\pi
wo^{2}}{2}\int_{0}^{\infty}\int_{0}^{2\pi}\mathcal{F}[I(r,\,\phi)]P^{*}_{\ell_{2},\,\ell_{1}}\\\
\times e^{\frac{\pi^{2}r_{f}^{2}w_{o}^{2}}{2}}r_{f}dr_{f}d\phi_{f}.$ (8)

Unfortunately, AHST can not be used for states prepared in the full basis
$\ket{\ell}$ i.e., if the beam contains both positive and negative modes. It
is because from 6 the terms $\ket{\ell_{1}}\bra{\ell_{2}}$ and
$\ket{-\ell_{2}}\bra{-\ell_{1}}$ have the same intensity profile.

##  III Full state tomography in OAM space of light

Although, the AHST scheme for QST works perfectly, the only limitation of this
scheme is that it works only for a fixed helicity. Therefore, if the optical
beam under consideration has positive as well as negative modes then we can
not use this scheme. In this section, we present the scheme to perform full
QST for an arbitrary mixed state of OAM modes. In order to do so, we require a
helicity sorter setup which can separate the positive and negative $\ell$
modes. In the following, we show how combining the helicity sorter with AHST
method enables us to perform the full QST on the entire OAM space.

Similar to the OAM sorter and the radial mode sorter, the helicity sorter can
also be though of as a Mach-Zehnder interferometer with two input and two
output modes. Let $a$ and $b$ are the two arms of the interferometer. Let
$P_{\pm}$ are the projectors on the $\pm$ subspaces of the OAM such that
$P_{\pm}=\sum_{\ell}\ket{\pm\ell}\bra{\pm\ell}~{}\forall~{}\ell\geq 0$. Then
formally the action $S$ of the helicity sorter can be written as

$\displaystyle\begin{split}S=\ket{a}\bra{a}\otimes
P_{+}&+\ket{a}\bra{b}\otimes P_{-}\\\ &+\ket{b}\bra{a}\otimes
P_{-}+\ket{b}\bra{b}\otimes P_{+}.\end{split}$ (9)

If $\rho$ is the density matrix of the light beam prepared in the basis
$\\{\ket{\ell}\\}$ with $\ell\neq 0$, then we can write it as

$\displaystyle\rho=\begin{pmatrix}\rho_{+}&\sigma\\\
\sigma^{\dagger}&\rho_{-}\end{pmatrix},$ (10)

where $\rho_{-}$ and $\rho_{+}$ are the marginal density matrices in the
negative and positive subspace, respectively, and $\sigma$ and
$\sigma^{\dagger}$ are the cross terms between these two subspaces. Here, the
OAM basis is ordered in the following way
$\ket{1},\,\ket{2}...\ket{n},\,\ket{-1},\,\ket{-2}...\ket{-n}$ for the ease of
calculations. If initially, the beam was in the path $a$ of the helicity
sorter, then after passing through the heicity sorter the state reads:

$\displaystyle\rho_{f}$
$\displaystyle=S(\ket{a}\bra{a}\otimes\rho)S^{\dagger}$ (11)
$\displaystyle=\ket{a}\bra{a}\otimes\rho_{+}+\ket{a}\bra{b}\otimes\sigma+\ket{b}\bra{a}\otimes\sigma^{\dagger}+\ket{b}\bra{b}\otimes\rho_{-}.$
(12)

Performing AHST measurements in the paths $a$ and $b$ will yield the marginal
density matrices $\rho_{\pm}$. However, the cross terms $\sigma$ will be lost
in this procedure. In order to estimate the terms $\sigma$ we need to perform
$H_{x}$ and $H_{y}$ operations on the density matrix $\rho$, which are given
by

$\displaystyle H_{x}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathds{1}&\mathds{1}\\\
\mathds{1}&-\mathds{1}\end{pmatrix},\quad
H_{y}=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathds{1}&i\mathds{1}\\\
i\mathds{1}&\mathds{1}\end{pmatrix},$ (13)

where $\mathds{1}$ is the identity matrix. $H_{x}$ and $H_{y}$ perform SU($2$)
transformations on all pairs $\ket{\pm\ell},\>\ell>0$ simultaneously.

Figure 3: The scheme to perform full QST. Here the operator $U$ representing
the measurement setting can take three forms: $U=\mathds{1}$, (ii) $U=H_{x}$,
and (iii) $U=H_{y}$. The full helicity sorter (FHS) sorts the $\pm$ subspaces
of the OAM modes. AHST method is used to estimate the marginal density
operators for both subspaces. Combining the results for the three measurement
setting one can estimate the full density matrix of the OAM modes.

Once we have all the ingredients, we can use the setup shown in Fig. 3 to
perform the full QST. In order to perform full QST, we need to have three
different measurements settings corresponding to (i) $U=\mathds{1}$, (ii)
$U=H_{x}$, and (iii) $U=H_{y}$. Let marginal states measured by CCD1 are
$\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ for the three measurement settings,
respectively. Similarly, the marginal states measured at CCD2 are $\nu_{1}$,
$\nu_{2}$ and $\nu_{3}$. Therefore,

$\begin{gathered}\mu_{1}=\hat{P}_{+}\rho\hat{P}_{+},\quad\nu_{1}=\hat{P}_{-}\rho\hat{P}_{-},\\\
\mu_{2}=\hat{P}_{+}(H_{x}\rho
H_{x})\hat{P}_{+},\quad\nu_{2}=\hat{P}_{-}(H_{x}\rho H_{x})\hat{P}_{-},\\\
\mu_{3}=\hat{P}_{+}(H_{y}\rho
H_{y})\hat{P}_{+},\quad\nu_{3}=\hat{P}_{-}(H_{y}\rho
H_{y})\hat{P}_{-}.\end{gathered}$ (14)

From these six marginal density matrices we can obtain the full density matrix
$\rho$ as follows:

$\rho_{+}=\mu_{1},\quad\rho_{-}=\nu_{1},$ (15)
$\bra{\ell_{1}}\sigma\ket{-\ell_{2}}=\bra{\ell_{1}}\left(\mu_{2}-i\mu_{3}-\frac{1-i}{2}\mu_{1}\right)\ket{\ell_{2}}\\\
-\frac{1-i}{2}\bra{-\ell_{1}}\nu_{1}\ket{-\ell_{2}}.$ (16)

Here $\ell_{1}$ and $\ell_{2}$ are taken to be positive to avoid the
repetition. Hence, one can perform full QST on OAM states of light using a
helicity sorter and AHST method. In the following, we present two schemes to
implement the helicity-sorter on OAM states of light, and the implementation
schemes for $H_{x}$ and $H_{y}$ operations.

###  III.1 Helicity-sorter

Figure 4: The implementation of a helicity-sorter. BBS stands for Balanced
Beam Splitter. The Dove prisms are at a relative angle $\frac{\alpha}{2}$
which rotates the beam by $\alpha$. The parameters of the Gouy Phase setup are
chosen such that the beam accumulates a Gouy phase of $\alpha$ relative to the
other arm.

The helicity-sorter is a hybrid of the OAM sorter and the radial mode sorter
(see Fig. 4). This setup consists of a Mach-Zehnder interferometer with one
arm containing the OAM sorter and the radial mode sorter. Let the initial
state of the beam entering the port $a$ of the BBS is $\ket{\psi}$ which is
written as

$\displaystyle\ket{a}\otimes\ket{\psi}=\ket{a}\otimes\left(\sum_{\ell}c_{\ell}\ket{\ell}\right),$
(17)

where $c_{\ell}$ are the expansion coefficients and the sum is over all the
OAM modes from $-\infty$ to $\infty$. Here we are working with OAM modes with
$p=0$.

The action of the BBS is given by $\ket{a}\to(\ket{a}+\ket{b})/\sqrt{2}$ and
$\ket{b}\to(\ket{a}-\ket{b})/\sqrt{2}$. Upon passing through the beam splitter
the state of the beam transforms as

$\displaystyle\ket{a}\otimes\ket{\psi}\to\frac{1}{\sqrt{2}}\left(\ket{a}+\ket{b}\right)\otimes\ket{\psi}.$
(18)

In path $a$ we have the Gouy phase setup followed by the dove prism setup that
rotates the beam by an angle $\alpha$. The $\ket{\ell}$ mode in path $a$
transforms to $e^{i(|\ell|\alpha+\ell\alpha)}\ket{\ell}$. Therefore, the state
after passing through these two arrangements can be written as

$\displaystyle\ket{\psi_{1}}=\frac{1}{\sqrt{2}}\left(\ket{a}\otimes\sum_{\ell}e^{i(|\ell|\alpha+\ell\alpha)}c_{\ell}\ket{\ell}+\ket{b}\otimes\sum_{\ell}c_{\ell}\ket{\ell}\right).$
(19)

Upon passing through the second beamsplitter, the state transforms to
$\ket{\psi_{2}}$ which is given by

$\displaystyle\begin{split}\ket{\psi_{2}}=&\frac{1}{2}\left(\ket{a}\otimes\sum_{\ell}[1+e^{i(|\ell|\alpha+\ell\alpha)}]c_{\ell}\ket{\ell}\right.\\\
&+\left.\ket{b}\otimes\sum_{\ell}[1-e^{i(|\ell|\alpha+\ell\alpha)}]c_{\ell}\ket{\ell}\right).\end{split}$
(20)

One can immediately see that for the negative values of $\ell$ the coefficient
of $\ket{b}$ become zero, hence, path $b$ does not support negative OAM modes
irrespective of the choice of $\alpha$. However, the positive modes can be in
both, path $a$ as well as in path $b$.

We can see that by choosing different values of $\alpha$, different set of
positive OAM modes will be present in path $b$. For example, choosing
$\alpha=0$ will result in all positive and negative modes to be in path $a$.
Any other value of $\alpha$ will yield some positive OAM modes in path $b$ and
other remaining OAM modes in $a$ along with all the negative modes. Therefore,
we can call this setup as a partial helicity sorter and represent it by
$HS(\alpha)$. In order to realize a full helicity sorter (FHS) we treat even
and odd OAM modes separately.

The helicity sorter for odd modes: We have already seen from Eq. (20) that all
the OAM modes for $\ell<0$ are in path $a$. Now choosing $\alpha=\pi/2$ will
result in all the odd positive modes being sorted into path $b$. The helicity
sorter for odd modes ($HS_{o}$) can be simply implemented using a partial
helicity sorter with $\alpha=\pi/2$: $HS_{o}=HS(\pi/2)$.

The helicity sorter for even modes: The helicity sorter for even modes
requires additional consideration. One way of constructing the helicity sorter
for even modes ($HS_{e}$) is by using SLMs. An SLM is a device that can be
used to apply a phase mask on any OAM mode. Using SLM one can add or subtract
one unit of OAM which can cause the transformation
$\ket{\ell}\rightarrow\ket{\ell\pm 1}$ [37, 38]. Hence, an SLM can be used to
convert even modes to odd modes by adding one unit of OAM. The odd modes can
be easily sorted using the setup in Fig. 4. Once they are sorted, we can
subtract one unit of OAM using an SLM and retrieve the even modes back.

Another method to realize $HS_{e}$ is by using $N-1$ number of partial
helicity sorters where the largest value of topological charge in the given
OAM state is $\ell_{max}<2^{N}$. The FHS $HS_{e}$ can be implemented in two
steps:

  1. 1.

We first filter out all the positive modes. In order to do so, we pass the
light beam that contains only even OAM modes, through a partial helicity
sorter with $\alpha=\pi/4$, i.e., $HS(\pi/4$. Using Eq. (20) we can see that
$HS(\pi/4)$ filters out all the positive modes $\ell=4k+2$ to path $b_{1}$
(Fig. 5) and all the negative modes and all the positive modes of the form
$\ell=4k$ in path $a$ as shown in Fig. 5. We then pass the output of path $a$
through $HS(\alpha=\pi/8)$ to separate all the positive modes of the form
$8k+4$ from all the other modes. Continuing in this fashion, we need to pass
the beam through a total of $N-1$ partial helicity sorters with parameters
$\alpha=\frac{\pi}{4},\,\frac{\pi}{8},...,\,\frac{\pi}{2^{N}}$, to filter out
all the positive modes, for $\ell_{max}<2^{N}$. At the output of
$HS(\pi/2^{N})$ we are left with only the negative modes in path $a$.

  2. 2.

In order to recombine all the even positive modes scattered across $N-1$
paths: $b_{1}$, $b_{2}$,…$b_{N-1}$ (Fig. 5), we use $N-1$ OAM sorters with
$\beta=\frac{\pi}{4},\,\frac{\pi}{8},...,\,\frac{\pi}{2^{N}}$. The OAM sorter
with $\beta=\frac{\pi}{4}$ recombines modes of the form $4k+2$ with $4k$ (2)
i.e., the modes in path $b_{1}$ with the output of the OAM sorter with
$\beta=\frac{\pi}{8}$. The OAM sorter with $\beta=\frac{\pi}{8}$ recombines
modes of the form $8k+4$ with $8k$ i.e., the modes in path $b_{2}$ with the
output of the OAM sorter with $\beta=\frac{\pi}{16}$ and so on. We get all the
positive modes in the same path at the output port of the OAM sorter with
$\beta=\pi/4$.

This gives us the helicity sorter for even modes $HS_{e}$ Fig. 5. It is
composed of $N-1$ partial helicity-sorters and $N-1$ OAM sorters.

Figure 5:  The implementation of the helicity sorter for even modes $HS_{e}$
using $N-1$ partial helicity sorter and $N-1$ OAM sorters.

Hence, we can realize the helicity sorter for even and odd modes separately.
In order to achieve the FHS we first sort the even and odd modes followed by
separate helicity sorters on even and odd modes (Fig. 6). We then recombine
the even positive and odd positive modes together, and the even negative and
odd negative modes together using separate OAM sorters (Fig. 6). Next we
present the implementation scheme for $H_{x}$ and $H_{y}$ operators.

Figure 6:  Schematic diagram for FHS.

###  III.2 Implementation scheme for $H_{x}$ and $H_{y}$

The $H_{x}$ (13) is a Hadamard operator acting on all the pairs of states
$\ket{\pm\ell}$ simultaneously. From Eq. $\eqref{Hx}$, the action of $H_{x}$
on the OAM states is given by

$H_{x}\ket{\ell}=\begin{cases}\frac{1}{\sqrt{2}}(\ket{\ell}+\ket{-\ell})&\ell>0\\\
\frac{1}{\sqrt{2}}(\ket{\ell}-\ket{-\ell})&\ell<0\\\ \end{cases}\>.$ (21)

It can be implemented using the FHS as shown in Fig. 7a. Let the initial state
in path $a$ is

$\displaystyle\ket{a}\otimes\ket{\psi}=\ket{a}\otimes\left(\sum_{\ell}c_{\ell}\ket{\ell}\right).$
(22)

The action of FHS on this state yields

$\ket{\psi_{1}}=\ket{a}\otimes\sum_{\ell}c_{-\ell}\ket{-\ell}+\ket{b}\otimes\sum_{\ell}c_{\ell}\ket{\ell},\>\ell>0.$
(23)

The dove prism in path b converts the positive modes to negative modes giving:
$\ket{\psi_{1}^{\prime}}=\ket{a}\otimes\sum
c_{-\ell}\ket{-\ell}+\ket{b}\otimes\sum c_{\ell}\ket{-\ell}$, thus we have
negative modes in both paths. The BBS then mixes the modes in the two paths
and the second dove prism converts the modes in path $a$ to positive modes,
giving:

$\ket{\psi_{2}}=\frac{1}{\sqrt{2}}\left(\ket{a}\otimes\sum_{\ell}\left(c_{\ell}+c_{-\ell}\right)\ket{\ell}\right.\\\
\left.+\ket{b}\otimes\sum_{\ell}\left(c_{\ell}-c_{-\ell}\right)\ket{-\ell}\right).$
(24)

(a)  (b)

(c)

Figure 7: a) The schematics for $H_{x}$ operator. b) The implementation of the
phase gate $P(\theta)$. c) The schematics for $H_{y}$ operator.

When we recombine the two beams using the FHS we get the Hadamard transformed
state (for $\ell>0$):

$\ket{\psi_{3}}=\frac{1}{\sqrt{2}}\left(\sum_{\ell}\left(c_{\ell}+c_{-\ell}\right)\ket{\ell}+\sum_{\ell}\left(c_{\ell}-c_{-\ell}\right)\ket{-\ell}\right).$
(25)

The action of $H_{y}$ on the OAM states is given by (13):

$H_{y}\ket{\ell}=\begin{cases}\frac{1}{\sqrt{2}}(\ket{\ell}+i\ket{-\ell})&\ell>0\\\
\frac{1}{\sqrt{2}}(i\ket{\ell}+\ket{-\ell})&\ell<0\\\ \end{cases}.$ (26)

The transformation $H_{y}$ can be implemented by combining the transformation
$H_{x}$ with a phase gate $P(\theta)$ and can be written as

$H_{y}=P(\pi/2)H_{x}P(\pi/2).$ (27)

The phase gate $P(\theta)$ selectively phase shifts only the negative modes by
a phase $\theta$:

$P(\theta)=\begin{pmatrix}\mathds{1}&0\\\
0&e^{i\theta}\mathds{1}\end{pmatrix}.$ (28)

The phase gate can be implemented using the FHS as shown in Fig. 7b by
applying the phase shift of $\theta$ only on the negative modes. Combining the
$H_{x}$ setup with two of the phase gate setups will result in $H_{y}$ setup.
Since, two FHS applied in series result in identity transformation, the full
setup for the $H_{y}$ looks almost like the one for $H_{x}$ as can be seen
from Fig. 7c.

##  IV Conclusion

In conclusion, we have presented a method to perform full state tomography for
OAM states of a light beam. We have devised method to perform helicity sorter
and specific unitary transformations on all the $\pm\ell$ OAM modes
simultaneously, which along with AHST method of partial state tomography is
used to achieve the task. We have proposed two methods to perform the helicity
sorter, one using the SLMs and other using linear optical devices. The number
of optical components in scheme with linear optical devices scales as
$\mathcal{O}(N)$ for sorting $2^{N}$ OAM modes, whereas with SLMs, the same
setup can be used to sort any arbitrary number of OAM modes. Since helicity
sorter, unitary transformations, and the AHST method, all require only linear
optical components, the proposed scheme for full QST is scalable and
efficient. This is the first scheme to perform full QST on OAM modes and can
be important for quantum computation and quantum communication using OAM
modes.

###### Acknowledgements.

S.K.G. acknowledges the financial support from Inter-disciplinary Cyber
Physical Systems (ICPS) programme of the Department of Science and Technology,
India, (Grant No.:DST/ICPS/QuST/Theme-1/2019/12).

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# First determination of the $C\\!P$ content of
${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and updated determination of the $C\\!P$
contents of ${D}\to\pi^{+}\pi^{-}\pi^{0}$ and ${D}\to K^{+}K^{-}\pi^{0}$

S. Malde  C. Thomas  G. Wilkinson  P. Naik  C. Prouve  J. Rademacker  J. Libby
M. Nayak  T. Gershon  R. A. Briere  University of Oxford, Denys Wilkinson
Building, Keble Road, OX1 3RH, United Kingdom  European Organisation for
Nuclear Research (CERN), CH-1211, Geneva 23, Switzerland  University of
Bristol, Bristol, BS8 1TL, United Kingdom  Indian Institute of Technology
Madras, Chennai 600036, India  University of Warwick, Coventry, CV4 7AL,
United Kingdom  Carnegie Mellon University, Pittsburgh, Pennsylvania 15213,
USA

###### Abstract

Quantum-correlated $\psi(3770)\to{D}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}$ decays collected by the CLEO-c
experiment are used to perform a first measurement of $F_{+}^{4\pi}$, the
fractional $C\\!P$-even content of the self-conjugate decay
${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, obtaining a value of $0.737\pm 0.028$.
An important input to the measurement comes from the use of ${D}\to K^{0}_{\rm
S}\pi^{+}\pi^{-}$ and ${D}\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ decays to tag the
signal mode. This same technique is applied to the channels
${D}\to\pi^{+}\pi^{-}\pi^{0}$ and ${D}\to K^{+}K^{-}\pi^{0}$, yielding
$F_{+}^{\pi\pi\pi^{0}}=1.014\pm 0.045\pm 0.022$ and
$F_{+}^{KK\pi^{0}}=0.734\pm 0.106\pm 0.054$, where the first uncertainty is
statistical and the second systematic. These measurements are consistent with
those of an earlier analysis, based on $C\\!P$-eigenstate tags, and can be
combined to give values of $F_{+}^{\pi\pi\pi^{0}}=0.973\pm 0.017$ and
$F_{+}^{KK\pi^{0}}=0.732\pm 0.055$. The results will enable the three modes to
be included in a model-independent manner in measurements of the unitarity
triangle angle $\gamma$ using $B^{\mp}\to{D}K^{\mp}$ decays, and in time-
dependent studies of $C\\!P$ violation and mixing in the ${D}^{0}$ ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ system.

###### keywords:

charm decay, quantum correlations, $C\\!P$ violation

††journal: Physics Letters B

##  1 Introduction

Studies of the process $B^{\mp}\to{D}K^{\mp}$, where $D$ indicates a neutral
charmed meson reconstructed in a state accessible to both ${D}^{0}$ and
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays, give
sensitivity to the unitarity triangle angle $\gamma\equiv\arg({-V_{\rm
ud}V^{*}_{\rm ub}/V_{\rm cd}V_{\rm cb}^{*}})$ (also denoted $\phi_{3}$).
Improved knowledge of $\gamma$ is necessary for testing the Standard Model
description of $C\\!P$ violation. In a recent publication [1] it was shown how
inclusive three-body self-conjugate $D$ meson decays can be used for this
purpose, provided their fractional $C\\!P$-even content is known, a quantity
denoted $F_{+}$ (or $F_{+}^{f}$ when it is necessary to designate the specific
decay $f$). Measurements of $F_{+}$ for the decays
${D}\to\pi^{+}\pi^{-}\pi^{0}$ and ${D}\to K^{+}K^{-}\pi^{0}$ were performed,
making use of quantum-correlated ${D}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}$ decays coherently produced at the
$\psi(3770)$ resonance and collected by the CLEO-c detector. In this Letter a
first measurement is presented of the $C\\!P$ content of the four-body mode
${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, again exploiting CLEO-c $\psi(3770)$
data. This fully-charged and relatively abundant final state [2] can be
reconstructed with good efficiency by the LHCb detector and hence is a
promising mode for improving the determinaton of $\gamma$ at that experiment,
as well as at Belle II.

The three-body analysis reported in Ref. [1] exploited events in which one $D$
meson is reconstructed in the signal mode and the other ‘tagging’ meson in its
decay to a $C\\!P$ eigenstate. The measurement of $F_{+}^{4\pi}$ presented in
this Letter follows the same method, but augments it with other approaches, in
particular a complementary strategy in which the tagging modes are ${D}\to
K^{0}_{\rm S,L}\pi^{+}\pi^{-}$, and attention is paid to where on the Dalitz
plot this tag decay occurs. In order to benefit from this strategy for the
previously studied decays, this Letter also presents measurements of
$F_{+}^{\pi\pi\pi^{0}}$ and $F_{+}^{KK\pi^{0}}$ using ${D}\to K^{0}_{\rm
S,L}\pi^{+}\pi^{-}$ tags. Throughout the effects of $C\\!P$ violation in the
charm system are neglected, which is a good assumption given theoretical
expectations and current experimental limits [2, 3, 4]. However, as discussed
in Ref. [5], knowledge of $F_{+}$ also allows such $D$ decays to be used to
study $C\\!P$-violating observables and mixing parameters through time-
dependent measurements at facilities where the mesons are produced
incoherently.

The remainder of the Letter is structured as follows. Section 2 introduces the
$C\\!P$-even fraction $F_{+}$, derives the relations that are used to measure
its value at the $\psi(3770)$ resonance, and reviews how knowledge of $F_{+}$
allows non-$C\\!P$ eigenstates to be cleanly employed in the measurement of
$\gamma$ with $B^{\mp}\to{D}K^{\mp}$ decays. Section 3 describes the data set
and event selection. Sections 4, 5 and 6 presents the determination of $F_{+}$
using $C\\!P$ tags, ${D}\to K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ tags and other
tags, respectively. In Sect. 7 combinations of the individual sets of results
are performed for each signal mode; for ${D}\to\pi^{+}\pi^{-}\pi^{0}$ and
${D}\to K^{+}K^{-}\pi^{0}$ these combinations include the results from Ref.
[1]. Section 8 gives the conclusions.

##  2 Measuring the $C\\!P$ content of a self-conjugate $D$-meson decay and
the consequences for the $\gamma$ determination with $B^{\mp}\to{D}K^{\mp}$

Let the amplitude of a ${D}^{0}$ meson decaying to a self-conjugate final
state $f$ be written as $\mathcal{A}({{D}^{0}}\to f(\mathbf{x}))\equiv
a_{x}e^{i\theta_{x}}$, where $\mathbf{x}$ indicates a particular point in the
decay phase space and $\theta_{x}$ is a $C\\!P$-conserving strong phase. The
amplitude is normalised such that

$\displaystyle\int_{\mathbf{x}\in\mathcal{D}}\\!\\!|\mathcal{A}({{D}^{0}}\to
f(\mathbf{x}))|^{2}\,\mathrm{d}\mathbf{x}=\mathcal{B}(f),$ (1)

where $\mathcal{B}(f)$ is the branching fraction of the ${D}^{0}$ decay and
$\mathcal{D}$ indicates the entire phase space. The ${D}^{0}$ decay amplitude
at $\mathbf{\bar{x}}$ is denoted $a_{\bar{x}}e^{i\theta_{\bar{x}}}$, where
$\mathbf{\bar{x}}$ indicates the point in phase space reached by applying a
$C\\!P$ transformation to the final-state system at $\mathbf{x}$. $C\\!P$
violation in the charm system is neglected, which implies that the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decay amplitude at
$\mathbf{\bar{x}}$ is equal to the ${D}^{0}$ amplitude at $\mathbf{x}$. It is
useful to define the strong phase difference
$\Delta\theta_{x}\equiv\theta_{x}-\theta_{\bar{x}}$.

It is possible to express the $C\\!P$-even fraction in terms of the decay
amplitudes introduced above. Let the $C\\!P$ eigenstates be
$|{{D}_{CP\pm}}\rangle\equiv(|{{D}^{0}}\rangle\pm|{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rangle)/\sqrt{2}$ and consider
the decay ${{D}^{0}}\to f$ in terms of these states. The total $C\\!P$-even
fraction of the inclusive decay is defined as

$\displaystyle F_{+}^{f}$
$\displaystyle\equiv\frac{\int_{\mathbf{x}\in\mathcal{D}}|\langle
f(\mathbf{x})|{{D}_{CP+}}\rangle|^{2}\,\mathrm{d}\mathbf{x}}{\int_{\mathbf{x}\in\mathcal{D}}|\langle
f(\mathbf{x})|{{D}_{CP+}}\rangle|^{2}+|\langle
f(\mathbf{x})|{{D}_{CP-}}\rangle|^{2}\,\mathrm{d}\mathbf{x}},$ (2)

and so

$\displaystyle F_{+}^{f}$
$\displaystyle=\frac{\int_{\mathbf{x}\in\mathcal{D}}a_{x}^{2}+a_{\bar{x}}^{2}+2a_{x}a_{\bar{x}}\cos\Delta\theta_{x}\,\mathrm{d}\mathbf{x}}{\int_{\mathbf{x}\in\mathcal{D}}2(a_{x}^{2}+a_{\bar{x}}^{2})\,\mathrm{d}\mathbf{x}}$
$\displaystyle=\frac{1}{2}\left[1+\frac{1}{\mathcal{B}(f)}\int_{\mathbf{x}\in\mathcal{D}}\\!\\!a_{x}a_{\bar{x}}\cos\Delta\theta_{x}\,\mathrm{d}\mathbf{x}\right].$
(3)

Note also that the following relation is always true in the absence of $C\\!P$
violation:

$\displaystyle\int_{\mathbf{x}\in\mathcal{D}}\\!\\!a_{x}a_{\bar{x}}\sin\Delta\theta_{x}\,\mathrm{d}\mathbf{x}=0.$
(4)

Now consider a quantum-correlated ${D}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}$ system produced in the decay of a
$\psi(3770)$ meson. One of the $D$ mesons in the system decays to $f$ at the
point $\mathbf{x}$, the other to $g$ at $\mathbf{y}$, where in general the
phase space of the two decays is different. The amplitude of the latter decay
is denoted $b_{y}e^{i\phi_{y}}$ in analogy with the terminology used above.

The amplitude of the $\psi(3770)\to{D}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}\to fg$ correlated wavefunction can be
written [6]

$\displaystyle\mathcal{A}(f(\mathbf{x})|g(\mathbf{y}))=\frac{1}{\sqrt{2}}\left[a_{x}e^{i\theta_{x}}b_{\bar{y}}e^{i\phi_{\bar{y}}}-a_{\bar{x}}e^{i\theta_{\bar{x}}}b_{y}e^{i\phi_{y}}\right].$
(5)

The resulting decay probability is then

$\displaystyle\mathcal{P}(f(\mathbf{x})|g(\mathbf{y}))\propto$
$\displaystyle\Big{[}a_{x}^{2}b_{\bar{y}}^{2}+a_{\bar{x}}^{2}b_{y}^{2}-2a_{x}b_{\bar{y}}a_{\bar{x}}b_{y}\Big{(}\cos\Delta\theta_{x}\cos\Delta\phi_{y}+\sin\Delta\theta_{x}\sin\Delta\phi_{y}\Big{)}\Big{]}.$
(6)

If both $D$ mesons decay to the same final state the probability is divided by
two to avoid double counting. This formula can be used to determine the
population of quantum-correlated decays either integrated over all phase space
or after dividing the phase space into bins.

The number of ‘double-tagged’ candidates in which one $D$ meson decays to $f$
and the other to $g$, integrating over the phase space of each decay, is

$\displaystyle
M(f|g)=\mathcal{Z}\mathcal{B}(f)\mathcal{B}(g)\Big{[}1-\left(2F_{+}^{f}-1\right)\Big{(}2F_{+}^{g}-1\Big{)}\Big{]},$
(7)

where $\mathcal{Z}$ is a normalisation constant common to all decay modes. An
important special case, considered in Sect. 4, is where the tagging-mode $g$
is a $C\\!P$ eigenstate, and $(2F_{+}^{g}-1)$ reduces to $\pm 1$. Section 6
describes an analysis of classes of double-tags where this is not the case.

Alternatively, when the tagging-mode $g$ is a multibody decay, its phase space
may be divided into bins. Integrating over the phase space of $f$ results in
the following decay probability in bin $i$ of the phase space of $g$:

$\displaystyle\mathcal{P}(f|g_{i})\propto\int_{\mathbf{y}\in\mathcal{D}_{i}}b_{y}^{2}+b_{\bar{y}}^{2}-\left(2F_{+}^{f}-1\right)b_{y}b_{\bar{y}}\cos\Delta\phi_{y}\,\mathrm{d}\mathbf{y},$
(8)

where $\mathcal{D}_{i}$ indicates the phase space encompassed by bin $i$. In
Sect. 5 this relation is exploited for the tags $D\to K^{0}_{\rm
S,L}\pi^{+}\pi^{-}$.

To understand the relevance of the $C\\!P$-even fraction in the measurement of
the unitarity-triangle angle $\gamma$ consider the decay of a $B^{-}$ meson to
${D}K^{-}$, following which the $D$ meson decays to a self-conjugate final
state $f$ consisting of three or more particles. The amplitude of the $B^{-}$
decay is a superposition of two decay paths:

$\displaystyle\mathcal{A}(B^{-})=\mathcal{A}(B^{-}\to{{D}^{0}}K^{-})\mathcal{A}({{D}^{0}}\to
f)+\mathcal{A}(B^{-}\to{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-})\mathcal{A}({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\to f).$ (9)

Following the formalism developed above, the decay amplitude of the ${D}^{0}$
meson at the point $\mathbf{x}$ in the phase space is denoted
$a_{x}e^{i\theta_{x}}$. The decay amplitude of the $B^{-}$ meson at this point
in phase space is therefore

$\displaystyle\mathcal{A}(B^{-}(\mathbf{x}))=\mathcal{A}(B^{-}\to{{D}^{0}}K^{-})\Big{[}a_{x}e^{i\theta_{x}}+r_{B}e^{i(\delta_{B}-\gamma)}a_{\bar{x}}e^{i\theta_{\bar{x}}}\Big{]},$
(10)

where $r_{B}$ and $\delta_{B}$ are respectively the ratio of moduli and the
strong phase difference between the suppressed and favoured $B^{-}$ decay
amplitudes. The resulting decay probability is

$\displaystyle\mathcal{P}(B^{-}(\mathbf{x}))\propto
a_{x}^{2}+r_{B}^{2}a_{\bar{x}}^{2}+2r_{B}a_{x}a_{\bar{x}}\cos\left(\delta_{B}-\gamma+\theta_{x}-\theta_{\bar{x}}\right)$
(11)
$\displaystyle=a_{x}^{2}+r_{B}^{2}a_{\bar{x}}^{2}+2r_{B}a_{x}a_{\bar{x}}\Big{[}\cos(\delta_{B}-\gamma)\cos\Delta\theta_{x}-\sin(\delta_{B}-\gamma)\sin\Delta\theta_{x}\Big{]}.$

The expression for $B^{+}(\mathbf{x})$ is identical except that the sign in
front of $\gamma$ is reversed and $\mathbf{x}\leftrightarrow\mathbf{\bar{x}}$.
The total yield of $B^{\mp}$ decays is determined by integrating over the
entire $D$ phase space:

$\displaystyle Y^{\mp}$
$\displaystyle=h^{\mp}\int_{\mathbf{x}\in\mathcal{D}}\\!\\!\mathcal{P}(B^{\mp}(\mathbf{x}))\,\mathrm{d}\mathbf{x}$
$\displaystyle=h^{\mp}\left[1+r_{B}^{2}+\left(2F_{+}^{f}-1\right)2r_{B}\cos(\delta_{B}\mp\gamma)\right],$
(12)

where $h^{\mp}$ is a normalisation constant and Eqs. 2 and 4 have been
employed. This expression is very similar to that derived in Ref. [7] for the
case when the $D$ meson decays to a $C\\!P$ eigenstate and is indeed identical
in the event $F_{+}^{f}=0$ or 1. Hence measurements of $Y^{\mp}$, and
observables built from these yields [1], can be used to obtain information on
the angle $\gamma$ and the other parameters of the $B^{\mp}$ decay, provided
that $F^{f}_{+}$ is known. In Ref. [1] it is demonstrated how the effects of
${D}^{0}$ ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing,
neglected in Eq. 12, may also be accommodated.

##  3 Data set and event selection

The data set analysed consists of $e^{+}e^{-}$ collisions produced by the
Cornell Electron Storage Ring (CESR) at $\sqrt{s}=3.77$ GeV corresponding to
an integrated luminosity of 818 $\rm pb^{-1}$ and collected with the CLEO-c
detector. The CLEO-c detector is described in detail elsewhere [8]. Monte
Carlo simulated samples of signal decays are used to estimate selection
efficiencies. Possible background contributions are determined from a generic
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ simulated
sample corresponding to approximately fifteen times the integrated luminosity
of the data set. The EVTGEN generator [9] is used to simulate the decays. The
detector response is modelled using the GEANT software package [10].

Table 1 lists the $D$-meson final states considered in the analysis. Double-
tag candidates are reconstructed in which one $D$ meson decays into
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and the other into a $C\\!P$ eigenstate, or
where one $D$ meson decays into $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$,
$\pi^{+}\pi^{-}\pi^{0}$ or $K^{+}K^{-}\pi^{0}$ and the other into one of the
mixed-$C\\!P$ modes $K^{0}_{\rm S}\pi^{+}\pi^{-}$ or $K^{0}_{\rm
L}\pi^{+}\pi^{-}$. The combinations $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ vs.
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ vs.
$\pi^{+}\pi^{-}\pi^{0}$ are also reconstructed.

Table 1: $D$-meson final states reconstructed in this analysis. Type | Final states
---|---
Mixed $C\\!P$ |  $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}\pi^{0}$, $K^{+}K^{-}\pi^{0}$, $K^{0}_{\rm S,L}\pi^{+}\pi^{-}$
$C\\!P$-even |  $K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $K^{0}_{\rm S}\pi^{0}\pi^{0}$, $K^{0}_{\rm L}\pi^{0}$, $K^{0}_{\rm L}\omega$
$C\\!P$-odd |  $K^{0}_{\rm S}\pi^{0}$, $K^{0}_{\rm S}\omega$, $K^{0}_{\rm S}\eta$, $K^{0}_{\rm S}\eta^{\prime}$

The unstable final state particles are reconstructed in the following decay
modes: $\pi^{0}\to\gamma\gamma$, $K^{0}_{\rm S}\to\pi^{+}\pi^{-}$,
$\omega\to\pi^{+}\pi^{-}\pi^{0}$, $\eta\to\gamma\gamma$,
$\eta\to\pi^{+}\pi^{-}\pi^{0}$ and
$\eta^{\prime}\to\eta(\gamma\gamma)\pi^{+}\pi^{-}$. The $\pi^{0}$, $K^{0}_{\rm
S}$, $\omega$, $\eta$ and $\eta^{\prime}$ reconstruction procedure is
identical to that used in Ref. [11].

Final states that do not contain a $K^{0}_{\rm L}$ are fully reconstructed via
the beam-constrained candidate mass,
$m_{\text{bc}}\equiv\sqrt{s/(4c^{4})-\mathbf{p}_{D}^{2}/c^{2}}$, where
$\mathbf{p}_{D}$ is the $D$-candidate momentum, and $\Delta E\equiv
E_{D}-\sqrt{s}/2$, where $E_{D}$ is the $D$-candidate energy. The
$m_{\text{bc}}$ and $\Delta E$ distributions of correctly reconstructed
$D$-meson candidates peak at the nominal ${D}^{0}$ mass and zero,
respectively. Neither $\Delta E$ nor $m_{\text{bc}}$ distributions exhibit any
peaking structure for combinatoric background. The double-tag yield is
determined from counting events in signal and sideband regions of
$m_{\text{bc}}$ after placing requirements on $\Delta E$ [1, 12, 11, 13]. The
selection criteria of candidates involving the modes ${D}\to K^{+}K^{-}$ and
${D}\to\pi^{+}\pi^{-}$ do not include the cosmic ray muon and radiative Bhabha
vetoes that are described in Ref. [1]. This is because these sources of
background do not contaminate the double-tag sample, and the vetoes are found
to perturb the selection efficiency of the other $D$ meson in the event. When
selecting ${D}\to K^{0}_{\rm S}\pi^{+}\pi^{-}$ candidates it is demanded that
the $K^{0}_{\rm S}$ decay products form a vertex that is significantly
displaced from the $e^{+}e^{-}$ collision point; in contrast, for
$D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and ${D}\to\pi^{+}\pi^{-}\pi^{0}$
candidates the $\pi^{+}\pi^{-}$ vertex must be consistent with originating
from the collision point in order to suppress contamination from $D\to
K^{0}_{\rm S}\pi^{+}\pi^{-}$ and ${D}\to K^{0}_{\rm S}\pi^{0}$ decays,
respectively.

The double-tag yield determination procedure is identical to that presented in
Refs. [12, 11] except for the selections where the signal decay is
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and the tag decay is $K^{+}K^{-}$,
$\pi^{+}\pi^{-}$, $\pi^{+}\pi^{-}\pi^{0}$ or $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$,
which are all dominated by a background from continuum production of light
quark-antiquark pairs. For these modes an unbinned maximum likelihood fit is
performed to the distribution of the average $m_{\text{bc}}$ of the two
$D$-meson candidates. The background is modelled with an ARGUS function [14]
and the signal is modelled with the sum of two Crystal Ball functions [15]
with power-law tails on opposite sides. The parameters of the Crystal Ball
functions are fixed from fits to large samples of simulated data.

Figures 1 (a) and (b) show the average $m_{\text{bc}}$ distributions for
$C\\!P$-tagged ${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ candidates, summed over
all tag modes that are $C\\!P$-even and $C\\!P$-odd eigenstates, respectively,
where the $C\\!P$-tag final state does not contain a $K^{0}_{\rm L}$ meson.
Figure 2 shows the average $m_{\text{bc}}$ distributions for
${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, ${D}\to\pi^{+}\pi^{-}\pi^{0}$ and
${D}\to K^{+}K^{-}\pi^{0}$ candidates tagged with ${D}\to K^{0}_{\rm
S}\pi^{+}\pi^{-}$ decays, while Figs. 3 (a)–(c) show the Dalitz-plot
distributions of the tag decay for these three signal modes.

|  |
---|---|---
Figure 1: Distributions of ${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ candidates tagged by $C\\!P$-eigenstates. Sub-figures (a) and (b) show average $m_{\text{bc}}$ distributions for $C\\!P$-even tags and $C\\!P$-odd tags not involving $K^{0}_{\rm L}$ mesons, respectively. Sub-figure (c) shows the $m^{2}_{\text{miss}}$ distribution for candidates tagged by $C\\!P$ eigenstates that contain a $K^{0}_{\rm L}$ meson. The shaded histogram is the estimated peaking background and the vertical dotted lines indicate the signal region. |  |
---|---|---
Figure 2: Average $m_{\text{bc}}$ distributions for (a) ${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, (b) ${D}\to\pi^{+}\pi^{-}\pi^{0}$ and (c) ${D}\to K^{+}K^{-}\pi^{0}$ candidates tagged by a ${D}\to K^{0}_{\rm S}\pi^{+}\pi^{-}$ decay. The shaded histogram is the estimated peaking background and the vertical dotted lines indicate the signal region. |  |
---|---|---
|  |
Figure 3: Dalitz-plot distributions for $D\to K^{0}_{\rm S}\pi^{+}\pi^{-}$ reconstructed against (a) $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, (b) $D\to\pi^{+}\pi^{-}\pi^{0}$ and (c) $D\to K^{+}K^{-}\pi^{0}$, and $D\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ reconstructed against (d) $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, (e) $D\to\pi^{+}\pi^{-}\pi^{0}$ and (f) $D\to K^{+}K^{-}\pi^{0}$. The axis labels $m_{\pm}^{2}$ are the invariant-mass squared of the $\pi^{\pm}K^{0}_{\rm S,L}$ pair.  |  |
---|---|---
Figure 4: $m^{2}_{\text{miss}}$ distributions for (a)
${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, (b) ${D}\to\pi^{+}\pi^{-}\pi^{0}$ and
(c) ${D}\to K^{+}K^{-}\pi^{0}$ candidates tagged by a ${D}\to K^{0}_{\rm
L}\pi^{+}\pi^{-}$ decay. The shaded histogram is the estimated peaking
background and the vertical dotted lines indicate the signal region.

Many $K^{0}_{\rm L}$ mesons do not deposit any reconstructible signal in the
detector. However, double-tag candidates can be fully reconstructed using a
missing-mass squared ($m^{2}_{\text{miss}}$) technique [16] for tags
containing a single $K^{0}_{\rm L}$ meson. Yields are determined from the
signal and sideband regions of the $m^{2}_{\text{miss}}$ distribution. Figure
1 (c) shows the $m^{2}_{\text{miss}}$ distributions for
${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ candidates tagged with either a
$K^{0}_{\rm L}\pi^{0}$ or $K^{0}_{\rm L}\omega$ decay. Figure 4 shows the
$m^{2}_{\text{miss}}$ distributions for ${D}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$,
${D}\to\pi^{+}\pi^{-}\pi^{0}$ and ${D}\to K^{+}K^{-}\pi^{0}$ candidates tagged
with ${D}\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ decays, and Figs. 3 (d)–(f) show the
corresponding tag-side Dalitz-plot distributions.

In events where more than one pair of decays is reconstructed an algorithm is
applied to select a single double-tag candidate based on the information
provided by the $m_{\text{bc}}$ and $\Delta E$ variables. The particular
choice of metric varies depending on the category of double tag and is
optimised through simulation studies.

The peaking background estimates are determined from the generic Monte Carlo
sample of ${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$
events. For double tags involving a $C\\!P$ mode without a $K^{0}_{\rm L}$
meson the peaking backgrounds are found to constitute 5-10% of the selected
events, and are predominantly from residual $D\to K^{0}_{\rm S}\pi^{+}\pi^{-}$
contamination. The peaking backgrounds for final states with a $K^{0}_{\rm L}$
are generally larger; for $K^{0}_{\rm L}\pi^{0}$ and $K^{0}_{\rm L}\omega$
this contamination amounts to 15–20% of the signal yield, whereas for
$K^{0}_{\rm L}\pi^{+}\pi^{-}$ it is $\sim 10\%$ of the signal yield. The
dominant source of peaking background in each case is the equivalent decay
containing a $K^{0}_{\rm S}$ instead of a $K^{0}_{\rm L}$ meson. The
contamination from specific modes in the other categories of double tags is
typically 10% or less. The statistical uncertainties on these background
estimates arising from the finite size of the simulated samples are included
in the total statistical uncertainties on the signal yields.

The measured double-tag event yields after background subtraction are given in
Table 2.

Table 2: Double-tagged signal yields after background subtraction. Information on the entries marked ‘$\dagger$’, not studied in the current analysis, can be found in Ref. [1]. | $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | $\pi^{+}\pi^{-}\pi^{0}$ | $K^{+}K^{-}\pi^{0}$
---|---|---|---
$K^{+}K^{-}$ |  $19.3$ $\pm$ | $6.3$ | $\dagger$ | $\dagger$
$\pi^{+}\pi^{-}$ |  $3.3$ $\pm$ | $8.2$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\pi^{0}\pi^{0}$ |  $18.6$ $\pm$ | $5.2$ | $\dagger$ | $\dagger$
$K^{0}_{\rm L}\pi^{0}$ |  $49.2$ $\pm$ | $10.9$ | $\dagger$ | $\dagger$
$K^{0}_{\rm L}\omega$ |  $22.0$ $\pm$ | $6.5$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\pi^{0}$ |  $112.8$ $\pm$ | $11.0$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\omega$ |  $41.0$ $\pm$ | $6.8$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\eta(\gamma\gamma)$ |  $18.8$ $\pm$ | $4.5$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\eta(\pi^{+}\pi^{-}\pi^{0})$ |  $3.1$ $\pm$ | $2.7$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\eta^{\prime}$ |  $9.3$ $\pm$ | $3.3$ | $\dagger$ | $\dagger$
$K^{0}_{\rm S}\pi^{+}\pi^{-}$ |  $217.9$ $\pm$ | $16.8$ |  $289.2$ $\pm$ | $17.6$ |  $52.5$ $\pm$ | $7.8$
$K^{0}_{\rm L}\pi^{+}\pi^{-}$ |  $485.0$ $\pm$ | $26.3$ |  $291.1$ $\pm$ | $19.2$ |  $78.1$ $\pm$ | $11.1$
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ |  $41.0$ $\pm$ | $16.3$ |  $75.5$ $\pm$ | $15.7$ | —

Knowledge of the single-tag yields of the $C\\!P$-eigenstate modes is required
for normalisation purposes. Since the single-tag reconstruction criteria
applied are identical to those employed in Ref. [1], all information on these
yields is taken from the earlier publication. It is also necessary to know the
single-tag yield for the decay $D\to\pi^{+}\pi^{-}\pi^{0}$. A fit to the
$m_{\text{bc}}$ distribution returns a result of $29998\pm 320$ signal
candidates, after the subtraction of small peaking-background contributions.

##  4 Analysis with the $C\\!P$ tags

The yields of the single and double tags containing a $C\\!P$ eigenstate are
used as inputs to determine the $C\\!P$-even fraction, $F_{+}^{4\pi}$.
Following on from Eq. 7, the expected number of observed events, $M$, where
one $D$ meson decays to the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ final state, and
the other decays to $X$, a $C\\!P$ eigenstate with eigenvalue $\eta_{C\\!P}$,
is given by

$M(4\pi|X)=2N_{{D}{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}}}\,\mathcal{B}(4\pi)\mathcal{B}(X)\varepsilon(4\pi|X)\left[1-\eta_{C\\!P}\Big{(}2F_{+}^{4\pi}-1\Big{)}\right],$
(13)

where $N_{{D}{{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}}}$ is the number
of $D$ ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}$ pairs,
$\mathcal{B}(4\pi)$ and $\mathcal{B}(X)$ are the branching fractions for the
two reconstructed final states and $\varepsilon(4\pi|X)$ is the efficiency of
reconstructing such a double tag. The double tag yield is denoted by $M^{-}$
($M^{+}$) for $C\\!P$-even ($C\\!P$-odd) tags. Experimentally it is
advantageous to eliminate dependence on $N_{{D}{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}}}$, the branching fractions and the
reconstruction efficiency, which can be achieved by normalising by the single-
tag yields. The yield of single tags, $S^{+}$ ($S^{-}$) decaying to a
$C\\!P$-odd ($C\\!P$-even) eigenstate $X$, is given by

$S(X)=2N_{{D}{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}}}\,\mathcal{B}(X)\varepsilon(X),$ (14)

where $\varepsilon(X)$ is the reconstruction efficiency of the single tag. The
small effects of ${D}^{0}$ ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing are eliminated from the
measurement by correcting the measured single-tag yields $S^{\pm}_{\rm meas}$
such that $S^{\pm}=S^{\pm}_{\rm meas}/(1-\eta_{C\\!P}y_{D})$ where
$y_{D}=(0.62\pm 0.08)\%$ is the well-known $D$-mixing parameter [17]. A
further correction is applied in the case of the tags $K^{+}K^{-}$ and
$\pi^{+}\pi^{-}$ because of the differing selection requirements for the
single and double-tag case, as described in Sect. 3. This correction factor is
determined by taking the ratio of the selection efficiency of the single tag
from simulation with the two differing selections. It is determined to be 1.15
and 1.10 for the $D\to K^{+}K^{-}$ and $D\to\pi^{+}\pi^{-}$ modes,
respectively, with an uncertainty of $\pm 0.05$. The other uncertainties on
these single-tag yields are assigned following the same procedure described in
Ref. [1].

For the case of the two $C\\!P$ tags involving a $K^{0}_{\rm L}$ meson a
different treatment is required, since it is not possible to measure the
single-tag yield directly for these modes. Following the procedure described
in Ref. [1], the effective single-tag yield is evaluated using Eq. 14, where
the effective single-tag efficiency $\varepsilon(K_{\rm L}^{0}X)$ is
calculated from the ratio of $\varepsilon(4\pi|K_{\rm
L}^{0}X)/\varepsilon(4\pi)$, and the leading systematic uncertainties are
associated with the branching fractions and the value used for the effective
single-tag efficiency. The effective single-tag yields are determined to be
21726 $\pm$ 3497 and 9124 $\pm$ 4105 for $K_{\rm L}^{0}\pi^{0}$ and $K_{\rm
L}^{0}\omega$, respectively.

Assuming that the reconstruction efficiencies of each $D$ meson are
independent, then the ratio of the double-tagged and single-tagged yields are
independent of the branching fraction and reconstruction efficiency of the
$C\\!P$ tag and $N_{{D}{{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}}}$.
This ratio is defined as $N^{+}\equiv M^{+}/S^{+}$, with an analogous
expression for $N^{-}$. The $C\\!P$-even fraction $F_{+}^{4\pi}$ is then given
by

$F_{+}^{4\pi}=\frac{N^{+}}{N^{+}+N^{-}}.$ (15)

The measured values for $N^{+}$ and $N^{-}$ for each $C\\!P$ tag are displayed
in Fig. 5. It can be seen that there is consistency between the individual
tags for each measurement. The mean value $<N^{+}>=(5.54\pm 0.46)\times
10^{-3}$ is significantly larger than $<N^{-}>=(1.80\pm 0.32)\times 10^{-3}$,
indicating that the $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ final state is
predominantly $C\\!P$ even.

|
---|---
Figure 5: $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ results for (a) $N^{+}$ and (b)
$N^{-}$. In each plot the vertical yellow band indicates the value obtained
from the combination of all tags. The black portion of the uncertainty
represents the statistical uncertainty only while the red represents the
total.

If the acceptance across the phase space of the
$D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decay is not uniform it has the potential
to bias the measurement of $F_{+}^{4\pi}$. Using simulated data the selection
efficiency of individual pions in $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decays is
determined in bins of momentum and polar angle with respect to the beam
direction. The candidates in data are then weighted by the normalised
efficiency. Each pion is treated independently and an overall weight,
typically lying within 5–10% of unity, is found by multiplying the individual
weights. The scaled signal yields are used to re-determine $F_{+}^{4\pi}$ and
the difference between this and the value found without efficiency correction
is 0.008, which is taken as the systematic uncertainty due to non-uniform
acceptance.

Using the $C\\!P$ tags only, and accounting for the correlations between the
systematic uncertainties, yields $F_{+}^{4\pi}=0.754\pm 0.031\pm 0.021$, where
the first uncertainty is statistical and the second systematic.

##  5 Analysis with the ${D}\to K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ tags

For each of the signal samples that are tagged by $D\to K^{0}_{\rm
S}\pi^{+}\pi^{-}$ or $D\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ decays the Dalitz plot
of the tag mode is divided into eight pairs of symmetric bins by the line
$m_{+}^{2}=m_{-}^{2}$, where $m_{\pm}^{2}$ is the invariant-mass squared of
the $\pi^{\pm}K^{0}_{\rm S,L}$ pair. The bins lying on one side of this line
($m_{+}^{2}>m_{-}^{2}$) are labelled $-1\to-8$, and those on the other side
$1\to 8$. The binning definition follows the ‘Equal $\Delta\delta_{D}$ BABAR
2008’ scheme of Ref. [18], in which the boundaries are chosen according to the
strong-phase prediction of a model developed by the BaBar collaboration [19].
The expected distribution of entries is symmetric and so the analysis
considers the absolute bin number $|i|$, which contains the contents of the
pair of bins $-i$ and $i$.

Following Eq. 8, the expected population of bin $|i|$ for signal decays with
$K^{0}_{\rm S}\pi^{+}\pi^{-}$ tags is

$M_{|i|}=h\left[K_{i}+K_{-i}-\Big{(}2F_{+}-1\Big{)}2c_{i}\sqrt{K_{i}K_{-i}}\right],$
(16)

where $h$ is a normalisation factor specific to the signal category, $K_{i}$
is the flavour-tagged fraction, being the proportion of $K^{0}_{\rm
S}\pi^{+}\pi^{-}$ decays to fall in bin $i$ in the case that the mother
particle is known to be a $D^{0}$ meson, and $c_{i}$ is the cosine of the
strong-phase difference between $D^{0}$ and $\bar{D}^{0}$ decays averaged in
bin $i$ and weighted by the absolute decay rate (a precise definition may be
found in Ref. [6]). The only difference between the form of this expression
and the case when the signal decays into a pure $CP$-even eigenstate [6] is
the prefactor of $(2F_{+}-1)$ in the final term.

Similarly, when the tagging meson decays to $K^{0}_{\rm L}\pi^{+}\pi^{-}$ then
the number of double-tag decays produced in bin $|i|$ is

$M^{\prime}_{|i|}=h^{\prime}\left[K_{i}^{\prime}+K_{-i}^{\prime}+\Big{(}2F_{+}-1\Big{)}2c^{\prime}_{i}\sqrt{K^{\prime}_{i}K^{\prime}_{-i}}\,\right],$
(17)

where the primed quantities are now specific to this case. The reversed sign
in front of the final term reflects the fact that the $K^{0}_{\rm L}$ meson is
almost entirely a $C\\!P$-odd eigenstate.

The values of $c_{i}$ and $c^{\prime}_{i}$ within these bins have been
measured by the CLEO collaboration [18]. The values of the $K_{i}$ parameters
are taken from an analysis of the predictions of various $B$-factory models
[20, 21, 22, 19] presented in Ref. [13], and those of the $K^{\prime}_{i}$
parameters from measurements performed with CLEO-c data [23].

The double-tagged samples are analysed to determine the background-subtracted
signal yield in each Dalitz-plot bin. The distribution of background between
the different bins is assigned according to its category. Flat background is
assumed to contribute proportionally to the bin area. Peaking backgrounds that
occur on the signal side affect the distribution of tag decays in $K^{0}_{\rm
S,L}\pi^{+}\pi^{-}$ phase space according to their nature. For example, in the
case of $D\to K^{0}_{\rm S}\pi^{0}$ decays that are wrongly reconstructed as
$D\to\pi^{+}\pi^{-}\pi^{0}$, the tag decay will be in a $C\\!P$-even state and
distributed accordingly. Similarly, the distribution of $K^{0}_{\rm
S}(\pi^{0}\pi^{0})\pi^{+}\pi^{-}$ decays that are misreconstructed as
$K^{0}_{\rm L}\pi^{+}\pi^{-}$ tags is well understood and modelled
appropriately. The distribution of the residual $K^{0}_{\rm S}\pi^{+}\pi^{-}$
vs. $K^{0}_{\rm S}\pi^{+}\pi^{-}$ events that contaminate the
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ vs. $K^{0}_{\rm S}\pi^{+}\pi^{-}$ selection is
determined from data by inverting the $K^{0}_{\rm S}$ veto on the signal
decay.

It is also necessary to account for relative bin-to-bin efficiency variations
in the background-subtracted signal yields. The correction factors are
determined from simulation and typically differ $\lesssim 5\%$ from unity. The
signal yields in each bin after background subtraction and relative efficiency
correction are shown in Table 3 for $K^{0}_{\rm S}\pi^{+}\pi^{-}$ tags and in
Table 4 for $K^{0}_{\rm L}\pi^{+}\pi^{-}$ tags.

Table 3: Double-tagged signal yields vs. $K^{0}_{\rm S}\pi^{+}\pi^{-}$ after background subtraction in absolute bin numbers of the ${D}\to K^{0}_{\rm S}\pi^{+}\pi^{-}$ Dalitz plot. The yields are corrected for relative bin-to-bin efficiency variations and then scaled so that the totals match the values in Table 2. $|i|$ | $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | $\pi^{+}\pi^{-}\pi^{0}$ | $K^{+}K^{-}\pi^{0}$
---|---|---|---
1 |  $30.8$ $\pm$ | $7.0$ |  $29.9$ $\pm$ | $6.3$ |  $12.6$ $\pm$ | $4.1$
2 |  $19.8$ $\pm$ | $5.3$ |  $19.1$ $\pm$ | $4.8$ |  $4.6$ $\pm$ | $2.6$
3 |  $16.4$ $\pm$ | $4.5$ |  $27.2$ $\pm$ | $5.2$ |  $6.9$ $\pm$ | $2.5$
4 |  $10.1$ $\pm$ | $3.4$ |  $18.5$ $\pm$ | $4.4$ |  $1.6$ $\pm$ | $1.5$
5 |  $55.1$ $\pm$ | $8.1$ |  $96.9$ $\pm$ | $10.0$ |  $8.4$ $\pm$ | $3.1$
6 |  $21.1$ $\pm$ | $5.1$ |  $31.2$ $\pm$ | $5.8$ |  $4.4$ $\pm$ | $2.4$
7 |  $27.7$ $\pm$ | $6.0$ |  $34.6$ $\pm$ | $6.3$ |  $7.7$ $\pm$ | $2.8$
8 |  $36.9$ $\pm$ | $6.8$ |  $31.8$ $\pm$ | $6.0$ |  $6.2$ $\pm$ | $2.5$
Table 4: Double-tagged signal yields vs. $K^{0}_{\rm L}\pi^{+}\pi^{-}$ after background subtraction in absolute bin numbers of the ${{D}^{0}}\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ Dalitz plot. The yields are corrected for relative bin-to-bin efficiency variations. $|i|$ | $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ | $\pi^{+}\pi^{-}\pi^{0}$ | $K^{+}K^{-}\pi^{0}$
---|---|---|---
1 |  $134.1$ $\pm$ | $13.9$ |  $89.2$ $\pm$ | $11.1$ |  $17.3$ $\pm$ | $6.1$
2 |  $59.2$ $\pm$ | $8.9$ |  $32.9$ $\pm$ | $6.9$ |  $8.8$ $\pm$ | $4.0$
3 |  $55.4$ $\pm$ | $8.7$ |  $31.0$ $\pm$ | $6.3$ |  $4.1$ $\pm$ | $3.1$
4 |  $20.3$ $\pm$ | $5.8$ |  $7.0$ $\pm$ | $3.7$ |  $0.1$ $\pm$ | $1.9$
5 |  $46.0$ $\pm$ | $8.7$ |  $6.7$ $\pm$ | $4.8$ |  $2.1$ $\pm$ | $3.1$
6 |  $24.6$ $\pm$ | $6.2$ |  $14.7$ $\pm$ | $5.0$ |  $10.0$ $\pm$ | $3.7$
7 |  $61.2$ $\pm$ | $9.0$ |  $46.7$ $\pm$ | $7.8$ |  $17.6$ $\pm$ | $4.7$
8 |  $84.1$ $\pm$ | $10.8$ |  $62.9$ $\pm$ | $8.9$ |  $18.1$ $\pm$ | $5.1$

A log-likelihood fit is performed to the efficiency-corrected signal yields of
each sample, assuming the expected distributions given by Eqs. 16 and 17. The
fit parameters are the $C\\!P$-even fraction and the overall normalisation.
The values of $K_{i}$, $K_{i}^{\prime}$, $c_{i}$ and $c_{i}^{\prime}$ are also
fitted, but with their measurement uncertainties and correlations imposed with
Gaussian constraints. Separate fits are performed for the $D\to K^{0}_{\rm
S}\pi^{+}\pi^{-}$ tags, the $D\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ tags, and for
both samples combined. Fits to large ensembles of simulated experiments
demonstrate that the returned uncertainties are reliable and that there is no
significant bias in the procedure. All data fits are found to be of good
quality. The results are plotted in Fig. 6 for the $D\to K^{0}_{\rm
S}\pi^{+}\pi^{-}$ tags and in Fig. 7 for the $D\to K^{0}_{\rm
L}\pi^{+}\pi^{-}$ tags. The numerical results for the $C\\!P$-even fraction
are given in Table 5 for $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and in Table 6 for
$D\to\pi^{+}\pi^{-}\pi^{0}$ and $D\to K^{+}K^{-}\pi^{0}$.

|  |
---|---|---
Figure 6: Data (points) and fit results (solid line) in absolute bin numbers for $K^{0}_{\rm S}\pi^{+}\pi^{-}$ tags vs. (a) $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, (b) $D\to\pi^{+}\pi^{-}\pi^{0}$ and (c) $D\to K^{+}K^{-}\pi^{0}$. Also shown in each case is the expectation if $F_{+}=0$ (dotted line) or $F_{+}=1$ (dashed line).  |  |
---|---|---
Figure 7: Data (points) and fit results (solid line) in absolute bin numbers for $K^{0}_{\rm L}\pi^{+}\pi^{-}$ tags vs. (a) $D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, (b) $D\to\pi^{+}\pi^{-}\pi^{0}$ and (c) $D\to K^{+}K^{-}\pi^{0}$. Also shown in each case is the expectation if $F_{+}=0$ (dotted line) or $F_{+}=1$ (dashed line).  Table 5: The $F_{+}^{4\pi}$ fit results for the $D\to K^{0}\pi^{+}\pi^{-}$ tags, where the first uncertainty is statistical and the second systematic. The row $K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ indicates the configuration where the $C\\!P$-even fraction is a common fit parameter shared between the $D\to K^{0}_{\rm S}\pi^{+}\pi^{-}$ and $D\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ samples. Tag | $F_{+}^{4\pi}$
---|---
$K^{0}_{\rm S}\pi^{+}\pi^{-}$ | 0.828 $\pm$ | 0.074 $\pm$ | 0.014
$K^{0}_{\rm L}\pi^{+}\pi^{-}$ | 0.670 $\pm$ | 0.057 $\pm$ | 0.039
$K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ | 0.737 $\pm$ | 0.049 $\pm$ | 0.024
Table 6: The $F_{+}^{\pi\pi\pi^{0}}$ and $F_{+}^{KK\pi^{0}}$ fit results for the $D\to K^{0}\pi^{+}\pi^{-}$ tags, where the first uncertainty is statistical and the second systematic. The row $K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ indicates the configuration where the $C\\!P$-even fraction is a common fit parameter shared between the $D\to K^{0}_{\rm S}\pi^{+}\pi^{-}$ and $D\to K^{0}_{\rm L}\pi^{+}\pi^{-}$ samples.  Tag | $F_{+}^{\pi\pi\pi^{0}}$ | $F_{+}^{KK\pi^{0}}$
---|---|---
$K^{0}_{\rm S}\pi^{+}\pi^{-}$ | 1.034 $\pm$ | 0.054 $\pm$ | 0.023 | 0.573 $\pm$ | 0.152 $\pm$ | 0.046
$K^{0}_{\rm L}\pi^{+}\pi^{-}$ | 0.971 $\pm$ | 0.075 $\pm$ | 0.033 | 0.916 $\pm$ | 0.181 $\pm$ | 0.066
$K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ | 1.014 $\pm$ | 0.045 $\pm$ | 0.022 | 0.734 $\pm$ | 0.106 $\pm$ | 0.054

The dominant systematic uncertainty is associated with the distribution of the
continuum and combinatoric backgrounds. This is assessed by repeating the
measurement with these contributions switched to those found in the sidebands
of the signal distributions. Shifts of $0.020$, $0.019$ and $0.037$ are
observed for $F_{+}^{4\pi}$, $F_{+}^{\pi\pi\pi^{0}}$ and $F_{+}^{KK\pi^{0}}$,
respectively. The uncertainty associated with the measurement errors on
$K_{i}$, $K_{i}^{\prime}$, $c_{i}$ and $c_{i}^{\prime}$ is estimated by re-
running the fit with these quantities set as fixed parameters and subtracting
in quadrature the new fit uncertainty from that obtained with the original
procedure. This component is found to be $0.013$ for
$D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, $0.010$ for $D\to\pi^{+}\pi^{-}\pi^{0}$
and $0.025$ for $D\to K^{+}K^{-}\pi^{0}$, and is accounted as a systematic
uncertainty in the final results. An uncertainty is evaluated to account for
non-uniformities in acceptance across phase space. For
$D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ this contribution is calculated with the
same procedure as in Sect. 4, and found to be $0.002$ for the joint
$K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ fit. For $D\to\pi^{+}\pi^{-}\pi^{0}$ and $D\to
K^{+}K^{-}\pi^{0}$ the acceptance uncertainties are taken to be 0.001 and
$0.010$, respectively, as determined in Ref. [1]. Other sources of bias are
evaluated to be small. The total systematic uncertainties are given in Tables
5 and 6 and are in all cases significantly smaller than the corresponding
statistical uncertainties.

##  6 Other tags

The double-tagged yield of $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ vs.
$\pi^{+}\pi^{-}\pi^{0}$ can be used to determine $F_{+}^{4\pi}$, benefiting
from the well-measured value of $F_{+}^{\pi\pi\pi^{0}}$. The ratio of double-
tag and $D\to\pi^{+}\pi^{-}\pi^{0}$ single-tag yields is defined as
$N^{\pi\pi\pi^{0}}\equiv M(4\pi|\pi\pi\pi^{0})/S({\pi\pi\pi^{0}})$, where a
very small correction is applied to the measured single-tag yield to account
for mixing effects. Following from Eq. 7, the ratio $N^{\pi\pi\pi^{0}}/N^{+}$
removes dependence on the signal branching fraction and reconstruction
efficiency and is given by

$\frac{N^{\pi\pi\pi^{0}}}{N^{+}}=\frac{\left[1-\Big{(}2F_{+}^{4\pi}-1\Big{)}\left(2F_{+}^{\pi\pi\pi^{0}}-1\right)\right]}{2F_{+}^{4\pi}},$
(18)

which can be rearranged to yield

$F_{+}^{4\pi}=\frac{N^{+}F_{+}^{\pi\pi\pi^{0}}}{N^{\pi\pi\pi^{0}}-N^{+}+2N^{+}F_{+}^{\pi\pi\pi^{0}}}.$
(19)

The choice of $N^{+}$ in the denominator of Eq. 18 is preferred to $N^{-}$ as
it is measured with better relative precision.

Taking as input the yields given in Sect. 3, the value of $N^{+}$ reported in
Sect. 4 and the final result for $F_{+}^{\pi\pi\pi^{0}}$ presented in Sect. 7
implies $F_{+}^{4\pi}=0.695\pm 0.050\pm 0.021$, where the uncertainties are
statistical and systematic, respectively. The main contributions to the
systematic uncertainty arise from: the measurement of the $D\to\pi\pi\pi^{0}$
single-tag yield and small violations of the efficiency-factorisation ansatz
assumed in Eq. 18; the understanding of the peaking background component in
the sample; and the possible effects of non-uniform acceptance.

The self-tagged yield of $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ vs.
$\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ also carries information on the value of
$F_{+}^{4\pi}$. This sample is however only used for a consistency check, as
there are large backgrounds from both the continuum and from misidentification
of $D\to K_{\rm S}^{0}\pi\pi$ decays that are a potential source of
significant systematic bias. Furthermore, the predicted yield and measurement
uncertainty means that the result from analysis of these double tags would
have low weight in the combined measurement of $F_{+}^{4\pi}$. Using Eq. 7 the
number of observed self-tagged events is given by

$M(4\pi|4\pi)=4\mathcal{R}F_{+}^{4\pi}\Big{(}1-F_{+}^{4\pi}\Big{)},$ (20)

where $\mathcal{R}=N_{{D}{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}}}\mathcal{B}(4\pi)^{2}\varepsilon(4\pi|4\pi)$.
The predicted double-tagged yield using the value of $F_{+}^{4\pi}$ obtained
from the $C\\!P$ tags is 17$\pm$2, which is consistent with the measured yield
reported in Table 2.

##  7 Combination of results

The results for $F_{+}^{4\pi}$ from the $C\\!P$ tags, the $K^{0}_{\rm
S,L}\pi^{+}\pi^{-}$ tags and the $\pi^{+}\pi^{-}\pi^{0}$ tag are summarised in
Table 7. They are compatible and are therefore combined, taking account of
correlated uncertainties. Correlations arise from the non-flat Dalitz plot
acceptance between all three measurements and the use of $N^{+}$ as an input
to both the $C\\!P$ tags and $\pi^{+}\pi^{-}\pi^{0}$ tag measurements. There
is a further small correlation between the results obtained with the $C\\!P$
and $\pi^{+}\pi^{-}\pi^{0}$ tags, associated with the uncertainty on the value
of the mixing parameter $y_{D}$. The final result is $F_{+}^{4\pi}=0.737\pm
0.028$.

Table 7: Results for $F_{+}^{4\pi}$ for each tag category, and combined. When two uncertainties are shown, the first is statistical and the second systematic. For the combined result the total uncertainty is given.  Tag | $F_{+}^{4\pi}$
---|---
$C\\!P$ eigenstates | 0.754 $\pm$ | 0.031 $\pm$ | 0.021
$K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ | 0.737 $\pm$ | 0.049 $\pm$ | 0.024
$\pi^{+}\pi^{-}\pi^{0}$ | 0.695 $\pm$ | 0.050 $\pm$ | 0.021
Combined | 0.737 $\pm$ 0.028

Table 8 summarises the results on $F_{+}^{\pi\pi\pi^{0}}$ and
$F_{+}^{KK\pi^{0}}$ obtained with $K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ tags,
together with those determined from $C\\!P$ tags. The $K^{0}_{\rm
S,L}\pi^{+}\pi^{-}$ measurements confirm the results of the earlier analysis.
A combination is performed, accounting for the sole source of correlated
uncertainties, which is that arising from the non-uniform acceptance over the
Dalitz plots. Results of $F_{+}^{\pi\pi\pi^{0}}=0.973\pm 0.017$ and
$F_{+}^{KK\pi^{0}}=0.732\pm 0.055$ are obtained. The $K^{0}_{\rm
S,L}\pi^{+}\pi^{-}$ tags improve the relative precision on
$F_{+}^{\pi\pi\pi^{0}}$ by 6% and on $F_{+}^{KK\pi^{0}}$ by 10%.

Table 8: Results for $F_{+}^{\pi\pi\pi^{0}}$ and $F_{+}^{KK\pi^{0}}$ for each tag category, and combined. The $C\\!P$-eigenstate tag results are from Ref. [1]. When two uncertainties are shown, the first is statistical and the second systematic. For the combined result the total uncertainty is given.  Tag | $F_{+}^{\pi\pi\pi^{0}}$ | $F_{+}^{KK\pi^{0}}$
---|---|---
$C\\!P$ eigenstates | 0.968 $\pm$ | 0.017 $\pm$ | 0.006 | 0.731 $\pm$ | 0.058 $\pm$ | 0.021
$K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ | 1.014 $\pm$ | 0.045 $\pm$ | 0.022 | 0.734 $\pm$ | 0.106 $\pm$ | 0.054
Combined | 0.973 $\pm$ 0.017 | 0.732 $\pm$ 0.055

##  8 Conclusions

A first measurement has been made of the $C\\!P$-even fraction of the decay
$D\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, exploiting quantum-correlated double-tags
involving $C\\!P$-eigenstates, a binned Dalitz-plot analysis of the modes
$D\to K^{0}_{\rm S,L}\pi^{+}\pi^{-}$, and $D\to\pi^{+}\pi^{-}\pi^{0}$ decays.
The result, $F_{+}^{4\pi}=0.737\pm 0.028$, when considered alongside the
relatively high branching fraction, indicates that this channel is a valuable
addition to the suite of $D$ decays that can be harnessed for the measurement
of the unitarity-triangle angle $\gamma$ through the process $B^{\mp}\to
DK^{\pm}$. The decays $D\to K^{0}_{\rm S,L}\pi^{+}\pi^{-}$ have also been
employed as a tag to measure the $C\\!P$ contents of the modes
$D\to\pi^{+}\pi^{-}\pi^{0}$ and $D\to K^{+}K^{-}\pi^{0}$. The results confirm
the conclusion of a previous analysis [1], based on $C\\!P$-eigenstate tags,
and also suggested by earlier amplitude-model studies [24, 25, 26], that the
$C\\!P$-even content of the $\pi^{+}\pi^{-}\pi^{0}$ final state is very high,
and therefore this decay too is a powerful mode for the measurement of
$\gamma$. Combining the two sets of measurements yields
$F_{+}^{\pi\pi\pi^{0}}=0.973\pm 0.017$ and $F_{+}^{KK\pi^{0}}=0.732\pm 0.055$.
Now that their $C\\!P$-even fractions have been measured, all three decay
modes may also be used for studies of indirect $C\\!P$ violation and mixing in
the ${D}^{0}$ ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ system
[5].

## Acknowledgments

This analysis was performed using CLEO-c data, and as members of the former
CLEO collaboration we thank it for this privilege. We are grateful for support
from the UK Science and Technology Facilities Council, the UK-India Education
and Research Initiative, and the European Research Council under FP7.

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