$\displaystyle=\quad\smashoperator[]{\lim_{n\to\infty}^{}}{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}\left(\bigwedge_{i=0}^{n}{\sf
X}^{i}\neg T\right)$ continuity of measures
$\displaystyle=\quad\smashoperator[]{\lim_{n\to\infty}^{}}{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}({\sf
X}^{n}\neg T)$ $T$ is a sink
$\displaystyle=\quad\smashoperator[]{\lim_{n\to\infty}^{}}{\mathcal{E}}([s_{n}\notin
T])$ definition of $[s_{n}\notin T]$
$\displaystyle\geq\quad\liminf_{n\to\infty}{\mathcal{E}}({\mathtt{val}(s_{n})})$
as $[s_{n}\notin T]\geq{\mathtt{val}(s_{n})}$
$\displaystyle\geq\quad{\mathtt{val}(s_{0})}-\varepsilon$ Equation 25.

##  Appendix F Missing Proofs from Section 6

We prove Lemma 6.2 from the main body:

See 6.2

######  Proof F.1.

We prove the equality in item 1 by induction on $n$. For $n=0$ it is trivial.
For the step, suppose the equality holds for some $n$. Let $s_{0}s_{1}\cdots
s_{n}\in s_{0}S_{*}^{*}$ be a partial run in ${\mathcal{M}}_{*}$ with
$s_{n}\in S$.

Let $s_{n}\in S_{\Box}$ and $s_{n+1}\in S_{*}\cap S$. We have:

$\displaystyle{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(s_{n},s_{n+1})s_{n+1}S_{*}^{\omega})$
$\displaystyle={\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}S_{*}^{\omega})\cdot\sigma(s_{0}s_{1}\ldots
s_{n})((s_{n},s_{n+1}))\cdot\frac{{\mathtt{val}_{{\mathcal{M}}}(s_{n+1})}}{{\mathtt{val}_{{\mathcal{M}}}(s_{n})}}$
def. of $P_{*}$
$\displaystyle={\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot\sigma(s_{0}s_{1}\ldots
s_{n})((s_{n},s_{n+1}))\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n+1})}$ ind. hyp.
$\displaystyle={\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot\sigma(\overline{s_{0}s_{1}\ldots
s_{n}})(s_{n+1})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n+1})}$ $\sigma$ in
${\mathcal{M}}$
$\displaystyle={\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}(s_{n}s_{n+1})s_{n+1}}S^{\omega})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n+1})}$

Let $s_{n}\in S_{\ocircle}$ and $s_{n+1}\in S_{*}\cap S$. We have:

$\displaystyle{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}s_{n+1}S_{*}^{\omega})$ $\displaystyle=\
{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}S_{*}^{\omega})\cdot P_{*}(s_{n})(s_{n+1})$ $\displaystyle=\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot
P_{*}(s_{n})(s_{n+1})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n})}$ ind. hyp.
$\displaystyle=\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot
P(s_{n})(s_{n+1})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n+1})}$ def. of $P_{*}$
$\displaystyle=\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}s_{n+1}}S^{\omega})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n+1})}$

This completes the inductive step, and we have proved item 1.

Towards item 2, define an MDP
${\mathcal{M}}_{*}^{\prime}=(S_{*}^{\prime},S_{*\Box}^{\prime},S_{*\ocircle}^{\prime},{\longrightarrow}_{*}^{\prime},P_{*}^{\prime})$
with “intermediate” states like $(s,t)$ in ${\mathcal{M}}_{*}$, but with
transition probabilities as in ${\mathcal{M}}$; more precisely:

$\displaystyle S_{*\Box}^{\prime}\ =\ $ $\displaystyle S_{\Box}$
$\displaystyle S_{*\ocircle}^{\prime}\ =\ $ $\displaystyle
S_{\ocircle}\cup\\{(s,t)\in\mathord{{\longrightarrow}}\mid s\in S_{\Box}\\}$
$\displaystyle{\longrightarrow}_{*}^{\prime}\ =\ $
$\displaystyle\\{(s,(s,t))\in(S_{\Box}\times\mathord{{\longrightarrow}})\mid
s{\longrightarrow}t\\}\cup(S_{\ocircle}\times S)\cup\mbox{}$
$\displaystyle\\{((s,t),t)\in(\mathord{{\longrightarrow}}\times S)\mid s\in
S_{\Box}\\}$ $\displaystyle P_{*}^{\prime}(s,t)\ =\ $ $\displaystyle P(s,t)$
$\displaystyle P_{*}^{\prime}((s,t),t)\ =\ $ $\displaystyle 1$

Then we have

${\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}({\mathfrak{R}})\ =\
{\mathcal{P}}_{{\mathcal{M}},s_{o},\sigma}(\overline{{\mathfrak{R}}})\qquad\text{for
all measurable ${\mathfrak{R}}\subseteq s_{0}(S_{*}^{\prime})^{\omega}$.}$
(26)

Let $s_{0}s_{1}\cdots s_{n}\in s_{0}(S_{*}^{\prime})^{*}$. If
$s_{0}s_{1}\cdots s_{n}$ is a partial run in ${\mathcal{M}}_{*}$, then we
have:

$\displaystyle{\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(S_{*}^{\prime})^{\omega})$ $\displaystyle\ =\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})$ $\displaystyle\ \geq\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n})}$ $\displaystyle\
=\
{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}S_{*}^{\omega})$ item 1 $\displaystyle\ =\
{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(S_{*}^{\prime})^{\omega}\cap S_{*}^{\omega})$

Otherwise (i.e., $s_{0}s_{1}\cdots s_{n}$ is not a partial run in
${\mathcal{M}}_{*}$), the same inequality holds trivially. Invoking Lemma F.2
below with $S:=S_{*}^{\prime}$ and $s:=s_{0}$ and
$\mu({\mathfrak{R}}):={\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}({\mathfrak{R}}\cap
S_{*}^{\omega})$ and
$\mu^{\prime}({\mathfrak{R}}):={\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}({\mathfrak{R}})$
and $x:={\mathtt{val}_{{\mathcal{M}}}(s_{0})}$ yields

${\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}({\mathfrak{R}})\
\geq\
{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}({\mathfrak{R}}\cap
S_{*}^{\omega})\quad\text{for all measurable }{\mathfrak{R}}\subseteq
s_{0}(S_{*}^{\prime})^{\omega}\,.$

By Equation 26, the first inequality of item 2 follows.

Towards the second inequality of item 2, define
$\llbracket{\varphi}\rrbracket^{s_{0}}_{-}\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}\\{\rho\in
s_{0}(S_{*}^{\prime})^{\omega}\mid\overline{\rho}\in\llbracket{\varphi}\rrbracket^{s_{0}}\\}$.
If ${\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(S_{*}^{\prime})^{\omega}\cap\llbracket{\varphi}\rrbracket^{s_{0}}_{-})>0$,
then $s_{0}s_{1}\cdots s_{n}$ is a partial run in ${\mathcal{M}}_{*}$ and we
have:

$\displaystyle{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(S_{*}^{\prime})^{\omega}\cap S_{*}^{\omega})$ $\displaystyle=\
{\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}S_{*}^{\omega})$ $\displaystyle=\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot{\mathtt{val}_{{\mathcal{M}}}(s_{n})}$ item 1
$\displaystyle\geq\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})\cdot{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\llbracket{\varphi}\rrbracket^{s_{0}}\mid\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega})$ ${\varphi}$ is tail $\displaystyle=\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{s_{0}s_{1}\cdots
s_{n}}S^{\omega}\cap\llbracket{\varphi}\rrbracket^{s_{0}})$ $\displaystyle=\
{\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(S_{*}^{\prime})^{\omega}\cap\llbracket{\varphi}\rrbracket^{s_{0}}_{-})$

Otherwise (i.e.,
${\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}(s_{0}s_{1}\cdots
s_{n}(S_{*}^{\prime})^{\omega}\cap\llbracket{\varphi}\rrbracket^{s_{0}}_{-})=0$),
the same inequality holds trivially. Invoking Lemma F.2 with
$S:=S_{*}^{\prime}$ and $s:=s_{0}$ and
$\mu({\mathfrak{R}}):={\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}({\mathfrak{R}}\cap\llbracket{\varphi}\rrbracket^{s_{0}}_{-})$
and
$\mu^{\prime}({\mathfrak{R}}):={\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}({\mathfrak{R}}\cap
S_{*}^{\omega})$ and $x:=1/{\mathtt{val}_{{\mathcal{M}}}(s_{0})}$ yields

${\mathtt{val}_{{\mathcal{M}}}(s_{0})}\cdot{\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}({\mathfrak{R}}\cap
S_{*}^{\omega})\ \geq\
{\mathcal{P}}_{{\mathcal{M}}_{*}^{\prime},s_{0},\sigma}({\mathfrak{R}}\cap\llbracket{\varphi}\rrbracket^{s_{0}}_{-})\quad\text{for
all measurable }{\mathfrak{R}}\subseteq s_{0}(S_{*}^{\prime})^{\omega}\,.$

By Equation 26, the second inequality of item 2 follows.

Item 3 follows from item 2, with
$\overline{{\mathfrak{R}}}=\llbracket{\varphi}\rrbracket^{s_{0}}$.

The following lemma was used in the preceding proof.

######  Lemma F.2.

Let $S$ be countable and $s\in S$. Call a set of the form $swS^{\omega}$ for
$w\in S^{*}$ a _cylinder_. Let $\mu,\mu^{\prime}$ be measures on $sS^{\omega}$
defined in the standard way, i.e., first on cylinders and then extended to all
measurable sets ${\mathfrak{R}}\subseteq sS^{\omega}$. Suppose there is $x\geq
0$ such that $x\cdot\mu(\mathfrak{C})\leq\mu^{\prime}(\mathfrak{C})$ for all
cylinders $\mathfrak{C}$. Then
$x\cdot\mu({\mathfrak{R}})\leq\mu^{\prime}({\mathfrak{R}})$ holds for all
measurable ${\mathfrak{R}}\subseteq sS^{\omega}$.

######  Proof F.3.

Let $\mathcal{C}=\\{\mathfrak{C}\subseteq sS^{\omega}\mid\mathfrak{C}\text{
cylinder}\\}$ denote the class of cylinders. This class generates an algebra
$\mathcal{C}_{*}\supseteq\mathcal{C}$, which is the closure of $\mathcal{C}$
under finite union and complement. The classes $\mathcal{C}$ and
$\mathcal{C}_{*}$ generate the same $\sigma$-algebra $\sigma(\mathcal{C})$.
The class $\mathcal{C}_{*}$ is a set of countable disjoint unions of cylinders
[4, Section 2]. Hence
$x\cdot\mu({\mathfrak{R}})\leq\mu^{\prime}({\mathfrak{R}})$ for all
${\mathfrak{R}}\in\mathcal{C}_{*}$.

Define

$\mathcal{Q}=\\{{\mathfrak{R}}\in\sigma(\mathcal{C})\mid
x\cdot\mu({\mathfrak{R}})\leq\mu^{\prime}({\mathfrak{R}})\\}\,.$

We have
$\mathcal{C}\subseteq\mathcal{C}_{*}\subseteq\mathcal{Q}\subseteq\sigma(\mathcal{C})$.
We show that $\mathcal{Q}$ is a _monotone_ class, i.e., if
${\mathfrak{R}}_{1},{\mathfrak{R}}_{2},\ldots\in\mathcal{Q}$, then
${\mathfrak{R}}_{1}\subseteq{\mathfrak{R}}_{2}\subseteq\cdots$ implies
$\bigcup_{i}{\mathfrak{R}}_{i}\in\mathcal{Q}$, and
${\mathfrak{R}}_{1}\supseteq{\mathfrak{R}}_{2}\supseteq\cdots$ implies
$\bigcap_{i}{\mathfrak{R}}_{i}\in\mathcal{Q}$. Suppose
${\mathfrak{R}}_{1},{\mathfrak{R}}_{2},\ldots\in\mathcal{Q}$ and
${\mathfrak{R}}_{1}\subseteq{\mathfrak{R}}_{2}\subseteq\cdots$. Then:

$\displaystyle x\cdot\mu\Big{(}\bigcup_{i}{\mathfrak{R}}_{i}\Big{)}$
$\displaystyle=\sup_{i}x\cdot\mu({\mathfrak{R}}_{i})$ measures are continuous
from below $\displaystyle\leq\sup_{i}\mu^{\prime}({\mathfrak{R}}_{i})$
definition of $\mathcal{Q}$
$\displaystyle=\mu^{\prime}\Big{(}\bigcup_{i}{\mathfrak{R}}_{i}\Big{)}$
measures are continuous from below

So $\bigcup_{i}{\mathfrak{R}}_{i}\in\mathcal{Q}$. Using the fact that measures
are continuous from above, one can similarly show that if
${\mathfrak{R}}_{1},{\mathfrak{R}}_{2},\ldots\in\mathcal{Q}$ and
${\mathfrak{R}}_{1}\supseteq{\mathfrak{R}}_{2}\supseteq\cdots$ then
$\bigcap_{i}{\mathfrak{R}}_{i}\in\mathcal{Q}$. Hence $\mathcal{Q}$ is a
monotone class.

Now the _monotone class theorem_ (see, e.g., [4, Theorem 3.4]) implies that
$\sigma(\mathcal{C})\subseteq\mathcal{Q}$, thus
$\mathcal{Q}=\sigma(\mathcal{C})$. Hence
$x\cdot\mu({\mathfrak{R}})\leq\mu^{\prime}({\mathfrak{R}})$ for all
${\mathfrak{R}}\in\sigma(\mathcal{C})$.

We prove Lemma 6.7 from the main body:

See 6.7

######  Proof F.4.

For any state $s_{0}\in S_{*}\cap S$, let

${\mathfrak{R}}_{s_{0}}\stackrel{{\scriptstyle\text{{\tiny{def}}}}}{{=}}\\{s_{0}s_{1}\cdots\in
s_{0}S_{*}^{\omega}\mid\exists\,i\geq 1:s_{0}=s_{i}\\}$

denote the event of returning to $s_{0}$. Suppose ${\mathcal{M}}_{*}$ is not
universally transient. By Lemma 3.7(3) there exists $s_{0}\in S_{*}\cap S$
such that
${\mathtt{val}_{{\mathcal{M}}_{*},{\mathfrak{R}}_{s_{0}}}(s_{0})}=1$. We show
that, in ${\mathcal{M}}$, for any $C>0$ there exists a strategy under which
the expected number of returns to $s_{0}$ is at least $C$. By Lemma 3.7(4)
this implies that ${\mathcal{M}}$ is not universally transient.

Let $C>0$. Let ${\mathfrak{R}}$ be the event, in ${\mathcal{M}}_{*}$, starting
in $s_{0}$, of returning to $s_{0}$ at least
$2C/{\mathtt{val}_{{\mathcal{M}},{\varphi}}(s_{0})}$ times, and denote by $X$
the random variable counting the number of returns to $s_{0}$. Since
${\mathtt{val}_{{\mathcal{M}}_{*},{\mathfrak{R}}_{s_{0}}}(s_{0})}=1$, we also
have ${\mathtt{val}_{{\mathcal{M}}_{*},{\mathfrak{R}}}(s_{0})}=1$, and so
there exists a strategy $\sigma$ with
${\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}({\mathfrak{R}})\geq\frac{1}{2}$.
By the first inequality of Lemma 6.2.2 we have
${\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{{\mathfrak{R}}})\geq{\mathtt{val}_{{\mathcal{M}},{\varphi}}(s_{0})}\cdot\frac{1}{2}$.
It follows:

${\mathcal{E}}{}_{{\mathcal{M}},s_{0},\sigma}(X)\ \geq\
{\mathcal{P}}_{{\mathcal{M}},s_{0},\sigma}(\overline{{\mathfrak{R}}})\cdot
2C/{\mathtt{val}_{{\mathcal{M}},{\varphi}}(s_{0})}\ \geq\ C$

In [17, Lemma 6] a variant, say ${\mathcal{M}}_{+}$, of the conditioned MDP
${\mathcal{M}}_{*}$ from Definition 6.1 was proposed. This variant
${\mathcal{M}}_{+}$ differs from ${\mathcal{M}}_{*}$ in that
${\mathcal{M}}_{+}$ has only those states $s$ from ${\mathcal{M}}$ that have
an optimal strategy, i.e., a strategy $\sigma$ with
${\mathcal{P}}_{{\mathcal{M}},s,\sigma}({\varphi})={\mathtt{val}_{{\mathcal{M}}}(s)}$.
Further, for any transition $s{\longrightarrow}t$ in ${\mathcal{M}}_{+}$ where
$s$ is a controlled state, we have
${\mathtt{val}_{{\mathcal{M}}}(s)}={\mathtt{val}_{{\mathcal{M}}}(t)}$, i.e.,
${\mathcal{M}}_{+}$ does not have value-decreasing transitions emanating from
controlled states.

As a consequence, in contrast to ${\mathcal{M}}_{*}$, in ${\mathcal{M}}_{+}$
there is no need for intermediate states of the form $(s,t)$: Since
${\mathtt{val}_{{\mathcal{M}}}(s)}={\mathtt{val}_{{\mathcal{M}}}(t)}$, an
intermediate state $(s,t)$ would transition to $t$ with probability $1$.
Therefore, such intermediate states do not appear in ${\mathcal{M}}_{+}$.
Instead, in ${\mathcal{M}}_{+}$ there is a direct transition from $s$ to $t$
like in the original MDP ${\mathcal{M}}$. As a further consequence, the state
$s_{\bot}$ does not appear in ${\mathcal{M}}_{+}$ (it would not be reachable).

Any strategy $\sigma$ in ${\mathcal{M}}_{+}$ can be naturally applied also in
${\mathcal{M}}_{*}$: whenever $\sigma$ moves from a controlled state $s$ to a
state $t$ (hence $s$ and $t$ have the same value), in ${\mathcal{M}}_{*}$
strategy $\sigma$ moves instead to the random state $(s,t)$ (from which
${\mathcal{M}}_{*}$ transitions to $t$ with probability $1$).

This correspondence is exploited in the proof of the following lemma from the
main body:

See 6.8

######  Proof F.5.

Suppose ${\mathcal{M}}$ is universally transient. We show that
${\mathcal{M}}_{+}$ is universally transient. Indeed, let $s_{0}$ be any state
in ${\mathcal{M}}_{+}$, and let $\sigma$ be any strategy in
${\mathcal{M}}_{+}$. Write ${\mathfrak{R}}$ for the event of returning to
$s_{0}$ in ${\mathcal{M}}_{*}$, and $\overline{{\mathfrak{R}}}$ for the event
of returning to $s_{0}$ in ${\mathcal{M}}_{+}$. We have
${\mathcal{P}}_{{\mathcal{M}}_{+},s_{0},\sigma}(\overline{{\mathfrak{R}}})={\mathcal{P}}_{{\mathcal{M}}_{*},s_{0},\sigma}({\mathfrak{R}})$.
Since ${\mathcal{M}}_{*}$ is universally transient by Lemma 6.7, by Lemma
3.7(3) this probability is less than $1$. Applying Lemma 3.7(3) again, it
follows that ${\mathcal{M}}_{+}$ is universally transient.


RADEC_ERR |  E |  arcsec |  Combined positional error, uncorrected
RA_CORR |  D |  deg |  Right ascension (ICRS), corrected
DEC_CORR |  D |  deg |  Declination (ICRS), corrected
RADEC_ERR_CORR |  D |  arcsec |  Combined positional error, corrected
EXT |  E |  arcsec |  Source extent parameter
EXT_ERR |  E |  arcsec |  Extent error
EXT_LIKE |  E |  .. |  Extent likelihood
DET_LIKE∗ |  E |  .. |  Detection likelihood
ML_RATE∗ |  E |  cts/s |  Source count rate measured by PSF-fitting
ML_RATE_ERR∗ |  E |  cts/s |  1-$\sigma$ count rate error
ML_CTS∗ |  E |  cts |  Source net counts measured from count rate
ML_CTS_ERR∗ |  E |  cts |  1-$\sigma$ source counts error
ML_FLUX∗ |  E |  erg/cm2/s |  Source flux in the detection band
ML_FLUX_ERR∗ |  E |  erg/cm2/s |  1-$\sigma$ source flux error
ML_EXP∗ |  E |  s |  Vignetted exposure time at the source position
ML_BKG∗ |  E |  cts/arcmin2 |  Background at the source position
inArea90 |  L |  |  True if inside the inner region with 0.2-2.3 keV vignetted exposure above 500s, which comprises 90% of the total area
2\. Forced PSF-fitting results for seven energy Bands (Table. 23.1);
7$\times$15 columns
DET_LIKE_Band |  D |  .. |  Detection likelihood
ML_RATE_Band |  D |  cts/s |  Source count rate
ML_RATE_ERR_Band |  D |  cts/s |  1-$\sigma$ combined count rate error
ML_RATE_LOWERR_Band |  D |  cts/s |  1-$\sigma$ lower count rate error
ML_RATE_UPERR_Band |  D |  cts/s |  1-$\sigma$ upper count rate error
ML_CTS_Band |  D |  cts |  Source net counts
ML_CTS_ERR_Band |  D |  cts |  1-$\sigma$ combined counts error
ML_CTS_LOWERR_Band |  D |  cts |  1-$\sigma$ lower counts error
ML_CTS_UPERR_Band |  D |  cts |  1-$\sigma$ upper counts error
ML_FLUX_Band |  D |  erg/cm2/s |  Source flux
ML_FLUX_ERR_Band |  D |  erg/cm2/s |  1-$\sigma$ combined flux error
ML_FLUX_LOWERR_Band |  D |  erg/cm2/s |  1-$\sigma$ lower flux error
ML_FLUX_UPERR_Band |  D |  erg/cm2/s |  1-$\sigma$ upper flux error
ML_EXP_Band |  D |  s |  Vignetted exposure time at the source position
ML_BKG_Band |  D |  cts/arcmin2 |  Background at the source position
3\. Aperture photometry results for seven energy Bands (Table. 23.1);
7$\times$5 columns
APE_CTS_Band |  J |  cts |  Total counts extracted within the aperture
APE_EXP_Band |  D |  s |  Exposure map value at the given position
APE_BKG_Band |  D |  cts |  Background counts extracted within the aperture, excluding nearby sources using the source map
APE_RADIUS_Band |  D |  pixels |  Extraction radius
APE_POIS_Band |  D |  .. |  Poisson probability that the extracted counts (APE_CTS) are a background fluctuation

242424 The suffix in the column name of forced PSF fitting and aperture
photometry results (section 2 and 3) indicates the energy band (Table. 23).
The columns of the three-band detected hard catalogue are almost identical to
those of the single-band detected main and supplementary catalogues, except
that the PSF-fitting output columns (those marked with $a$ subscript asterisk)
have four sets of data with an energy-band suffix (0, 1, 2, and3) in the
column name rather than only one set of data without this energy-band suffix
(as listed in section 1). Moreover, the ID_main and ID_hard columns are only
for the hard and main catalogues, respectively.

The catalogues are available with this paper and on the eROSITA Early Data
Release
website252525https://erosita.mpe.mpg.de/edr/eROSITAObservations/Catalogues/.
Table 23 lists the energy bands and corresponding ECFs used in this work. The
ECFs are calculated assuming an absorbed power law with a slope of 2.0 and
with a Galactic absorbing column density of 3$\times 10^{20}$ cm-2.

The main and the supplementary catalogues from the 0.2–2.3 keV single-band
detection share the same columns as described in Table 24. The seven-band
forced photometry is done independently for the single-band and three-band
detected sources. In the case of the three-band detected hard catalogue, four
sets of output columns from the source detection PSF-fitting (not the forced
PSF-fitting) have the same names as listed in section 2 of Table 24 , but with
an energy-band suffix in the column name indicating the three source-detection
bands (1, 2, and 3) or the summary of the three bands (0). The only exception
is the ML_EXP_Band, which is only for the three individual bands (1, 2, and
3), but not for the summary case (0). The hard catalogue has an additional
column ID_main, which stores the source ID in the main catalogue for the
overlapping sources.

The current versions of the catalogues contain a small number of NULL values
in the PSF-fitting error measurement columns, for example, ML_RATE_ERR,
RADEC_ERR_ERR, and EXT_ERR, because in some cases the ermldet algorithm that
determines the parameter errors failed to converge. When the determination of
the positional error failed (490 sources in the main catalogue; 1.8% of the
total), a mean RADEC_ERR_CORR of $4\aas@@fstack{\prime\prime}0$ was assumed,
which is the median value of RADEC_ERR.

##  Appendix E eFEDS simulations

In order to inspect and optimise the source detection for eROSITA, we ran
extensive simulations as described in Liu et al. (2021c). The simulation of
eFEDS is briefly summarised here.

We used sixte-2.6.2262626https://www.sternwarte.uni-
erlangen.de/research/sixte/ to create mock eFEDS event files and implemented
18 realisations. To make them as representative as possible to the real data,
we input the real eROSITA calibration files and eFEDS attitude file, and AGN,
star, and cluster catalogues generated from cosmological simulations (Comparat
et al. 2020). We measured the mean X-ray and particle background spectra from
the eFEDS data and simulated the source signal, X-ray background, and particle
background separately to account for their different vignetting. We ran the
eFEDS source detection pipeline on the mock data and then associated the
detected sources with the input on the basis of the source-ID flag on each
photon. By checking the number of photons contributed by each input source in
the core region of a detected source, we identified its primary and secondary
(if existed) input counterparts. The point sources (AGN and stars) and
extended sources (clusters) were considered separately. In this way, we were
able to identify not only the cases where one input source was uniquely
matched to one detected source that was correctly classified (as point- or
extended source) or not, but also the cases where one input source was
detected as multiple sources and the cases where one detected source was due
to a blending of multiple input sources.

Based on the input-output association, we were able to measure the source
detection efficiency, that is, the completeness (detected fraction) and
contamination level (fraction of spurious sources). The final source detection
strategy was chosen according to the impact on source detection efficiency
caused by any adjustment of the source detection procedure and parameters.


# Efficiency corrections for mutually inclusive variables and particle
identification effect for mixed-cumulants in heavy-ion collisions

Arghya Chatterjee<EMAIL_ADDRESS>Key Laboratory of Quark & Lepton
Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China  Toshihiro Nonaka
<EMAIL_ADDRESS>Tomonaga Center for the History of the
Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  ShinIchi Esumi
<EMAIL_ADDRESS>Tomonaga Center for the History of the
Universe, University of Tsukuba, Tsukuba, Ibaraki 305, Japan  Xiaofeng Luo
<EMAIL_ADDRESS>Key Laboratory of Quark & Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China

###### Abstract

Mix-cumulants of conserved charge distributions are sensitive observables for
probing properties of the QCD medium and phase transition in heavy-ion
collisions. To perform precise measurements, efficiency correction is one of
the most important step. In this study, using the binomial efficiency model,
we derive efficiency correction formulas for mutually exclusive and inclusive
variables. The UrQMD model is applied to verify the validity of these formulas
for different types of correlations. Furthermore, we investigate the effect of
the multiplicity loss and contamination emerging from the particle
identifications. This study provides important step toward future measurements
of mixed-cumulants in relativistic heavy-ion collisions.

##  I Introduction

Heavy-ion collisions at relativistic energies produce matter at extreme energy
density and temperature conditions. This matter is likely to comprise
deconfined quarks and gluons and is called quark-gluon plasma (QGP). A primary
objective of heavy-ion collision experiments is to explore the phase structure
of the hot dense QCD matter. The QCD phase structure can be expressed as a
function of temperature ($T$) and baryon chemical potential ($\mu_{\rm B}$)
Rajagopal (1999). QCD based model calculations predict that at large $\mu_{\rm
B}$ values the transition from hadronic matter to QGP is of the first order
Stephanov (2006); Bowman and Kapusta (2009). The end point of the first order
phase transition boundary is known as the QCD critical point (CP), after which
there is no genuine phase transition, except a smooth crossover from hadronic
to quark-gluon degrees of freedom Aoki et al. (2006); Gupta et al. (2011). One
of the major approaches to exploring the QCD phase structure is via the
measurements of event-by-event fluctuations of conserved quantities, such as
net-baryon ($B$), net-charge ($Q$) and net-strangeness ($S$) Stephanov et al.
(1998); Bzdak et al. (2020); Luo and Xu (2017); Chatterjee and Mohan (2012);
Garg et al. (2013). In a thermodynamic system, $r$-th order fluctuations
(”cumulants”) of event-by-event net-multiplicity distributions are related to
the $r$-th order thermodynamic susceptibilities of the corresponding conserved
charges that diverge near the critical point Stephanov (2009); Cheng et al.
(2009); Asakawa et al. (2009); Stephanov (2011). Furthermore, these measured
cumulants have also been used to extract freeze-out parameters ($T$ and
$\mu_{\text{B}}$) by comparing them with model calculations from lattice QCD
and hadron resonance gas (HRG) Bazavov et al. (2012a); Borsanyi et al. (2013);
Alba et al. (2014); Flor et al. (2020); Gupta et al. (2020). Owing to the
experimental constraints on measuring neutral particle yields, net-proton and
net-kaons are adopted as experimental proxies of net-baryon and net-
strangeness respectively. In the last decade, the STAR and PHENIX experiments
at the relativistic heavy ion collider (RHIC) have measured the second, third
and forth order cumulants of net-proton Aggarwal et al. (2010); Luo (2016);
Adamczyk et al. (2014a); Adam et al. (2021); Abdallah et al. (2021), net-
charge Adamczyk et al. (2014b); Adare et al. (2016) and net-kaon Adamczyk et
al. (2018) multiplicity distributions over a wide range of collision energies
to determine non-monotonic energy dependence behaviours, as an indicator of
the CP presence. Within current statistical uncertainties, no distinctive
signatures of the CP have been inferred from the net-charge and net-kaon
measurements. However, recent measurements of forth order to second order
cumulant ratios of net-proton multiplicity distributions exhibit non-monotonic
energy dependence as a function of $\sqrt{{s_{\rm NN}}}~{}$ Adam et al.
(2021). Although, before drawing any substantial physics conclusion from
event-by-event fluctuation measurements, we need to carefully investigate the
different background contributions Luo et al. (2013); Chatterjee et al.
(2020a); Zhang et al. (2020); Westfall (2015); Zhou and Jia (2018); Chatterjee
et al. (2020b).

Recently, the STAR experiment reported the first measurement of second-order
mixed-cumulants between net-charge, net-proton, and net-kaon multiplicity
distributions in the first phase of the beam energy scan (BES-I) program at
RHIC Adam et al. (2019). These mixed-cumulants are related to the off-diagonal
thermodynamic susceptibilities that carry the correlation between different
conserved charges of QCD Koch et al. (2005); Gavai and Gupta (2006); Majumder
and Muller (2006); Bluhm and Kampfer (2008); Ding et al. (2015); Chatterjee et
al. (2016); Bellwied et al. (2020). The importance of the second-order mixed-
cumulants was first highlighted in the context of normalized baryon-
strangeness susceptibilities ($C_{B,S}=-3\chi^{1,1}_{B,S}/\chi^{2}_{S}$) in
Ref. Koch et al. (2005), which are expected to exhibit a rapid change with the
onset of deconfinement. These quantities can be investigated by measuring the
energy dependence ratios of off-diagonal over diagonal cumulant ratios between
net-baryon and net-strangeness ($C^{1,1}_{B,S}/C^{2}_{S}$). Another research
objective originates from the comparisons between the ideal HRG model and
lattice QCD calculations. The baryon-charge susceptibility
($\chi^{1,1}_{B,Q}$) exhibits a significant difference between the lattice and
ideal HRG calculations above the crossover transition temperature, even in at
the lowest order Bazavov et al. (2012b); Vovchenko et al. (2017). A similar
difference between the lattice and HRG calculations can also be observed in
higher-order baryon susceptibilities (($\chi^{4}_{B}$)), which is more
statistically challenged in the experimental measurement Karsch (2017).
Similar to diagonal cumulants, the mixed-cumulants are also limited by the
lack of neutral particle detection capability. The measurements of charge-
baryon or charge-strangeness mixed-cumulants are less affected by such an
experimental limitation, as the neutral particles do not contribute to such
charge correlations, and can be approximated by $C^{1,1}_{Q,B}\approx
C^{1,1}_{Q,p}$ and $C^{1,1}_{Q,S}\approx C^{1,1}_{Q,k}$ Chatterjee et al.
(2016). In contrast, baryon-strangeness mixed-cumulants cannot be approximated
by the proton-kaon off-diagonal. However, the relationship between these
cumulants have been studied in Ref. Chatterjee et al. (2016); Yang et al.
(2017).

Recent measurements of second-order mixed-cumulants at the RHIC energy range
($\sqrt{{s_{\rm NN}}}~{}$= 7.7-200 GeV) agree well with different model
predictions for the net proton-kaon off-diagonal correlator
($C_{p,k}=\sigma^{1,1}_{p,k}/\sigma^{2}_{k}$). However, the charge-proton
($C_{Q,p}=\sigma^{1,1}_{Q,p}/\sigma^{2}_{p}$) and charge-kaon
($C_{Q,k}=\sigma^{1,1}_{Q,k}/\sigma^{2}_{k}$) correlators significantly
deviate from the model predicted values. In this study, we demonstrate that
the deviations observed in charge-proton and charge-kaon correlators are owing
to the efficiency double counting. We argue that, to correct the effects of
efficiency for the charge-proton and charge-kaon mixed-cumulants, the particle
identification for charge needs to be performed with the estimation of
corresponding efficiencies. However, a significant number of charged tracks
are missed for particle identification as different detectors used Adam et al.
(2019). In this study, we focus on the 2nd-order mixed-cumulant for two
variables to elucidate and simplify several important points on the efficiency
correction. Although the points also apply for higher-order mixed-cumulants,
including more than two variables cases, these extensions should be
straightforward and are expected to be investigated in future studies.

This paper is organized as follows. In Sec II, cumulants, mixed-cumulants, and
their efficiency corrections are introduced. The formulas for the efficiency
correction of the 2nd-order mixed-cumulant is discussed for two types of
correlations. In Sec. III, we perform numerical analysis using the UrQMD model
to verify the importance of adopting appropriate formulas, depending on the
correlation type. Here, the effects of double-counting are discussed, and the
potential effects of the multiplicity loss owing to particle identification
are investigated. Finally, we summarize this study in Sec. IV.

##  II Efficiency correction

###  II.1 Cumulants and mixed-cumulants

In statistics, any distribution can be characterized by different order
moments or cumulants. The $r$th-order moment of variable $N$ is defined by the
$r$th order derivative of moment generating function $G(\theta)$:

$\displaystyle G(\theta)$ $\displaystyle=$
$\displaystyle\sum_{N}e^{N\theta}P(N)=\langle e^{N\theta}\rangle,$ (1)
$\displaystyle\langle N^{r}\rangle$ $\displaystyle=$
$\displaystyle\frac{d^{r}}{d\theta^{r}}G(\theta)\Bigl{|}_{\theta=0},$ (2)

where $P(N)$ is a probability distribution function, and $\langle\cdot\rangle$
represents an average over events. Cumulants are defined by the cumulant
generating function $T(\theta)$, which is the logarithm of moment generating
function:

$\displaystyle T(\theta)$ $\displaystyle=$ $\displaystyle{\rm ln}G(\theta),$
(3) $\displaystyle\langle N^{r}\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\frac{d^{r}}{d\theta^{r}}T(\theta)\Bigl{|}_{\theta=0},$ (4)

where $\langle\cdot\rangle_{\rm c}$ represents the cumulant of the variable
inside the bracket. From Eqs. 1–4, the 1st and 2nd-order cumulants are
expressed in terms of moments

$\displaystyle\langle N\rangle_{\rm c}$ $\displaystyle=$ $\displaystyle\langle
N\rangle,$ (5) $\displaystyle\langle N^{2}\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\langle N^{2}\rangle-\langle N\rangle^{2},$ (6)

Similarly, the multivariate moments and cumulants are defined by the
multivariate generating function. In this study, we focus on the two-variable
case, which we call ”mixed-” moments or cumualnts, where the moment generating
function is given by

$\displaystyle G(\theta_{1},\theta_{2})$ $\displaystyle=$
$\displaystyle\sum_{N_{1},N_{2}}e^{\theta_{1}N_{1}}e^{\theta_{2}N_{2}}P(N_{1},N_{2})=\langle
e^{\theta_{1}N_{1}}e^{\theta_{2}N_{2}}\rangle,$ (7) $\displaystyle\langle
N_{1}^{r_{1}}N_{2}^{r_{2}}\rangle$ $\displaystyle=$
$\displaystyle\frac{\partial^{r_{1}}}{\partial\theta_{1}^{r_{1}}}\frac{\partial^{r_{2}}}{\partial\theta_{2}^{r_{2}}}G(\theta_{1},\theta_{2})\Big{|}_{\theta_{1}=\theta_{2}=0}.$
(8)

Mixed-cumulants are then defined as

$\displaystyle T(\theta_{1},\theta_{2})$ $\displaystyle=$ $\displaystyle{\rm
ln}G(\theta_{1},\theta_{2}),$ (9) $\displaystyle\langle
N_{1}^{r_{1}}N_{2}^{r_{2}}\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\frac{\partial^{r_{1}}}{\partial\theta_{1}^{r_{1}}}\frac{\partial^{r_{2}}}{\partial\theta_{2}^{r_{2}}}T(\theta_{1},\theta_{2})\Big{|}_{\theta_{1}=\theta_{2}=0}.$
(10)

From Eqs. 7–10, we obtain the 2nd-order mixed-cumulant in terms of moments and
mixed-moments:

$\displaystyle\langle N_{1}N_{2}\rangle_{\rm c}=\langle
N_{1}N_{2}\rangle-\langle N_{1}\rangle\langle N_{2}\rangle.$ (11)

###  II.2 Binomial model

The particle detection efficiency of each detector is always limited. The
event-by-event particle multiplicity distributions are convoluted owing to
this finite detector efficiency. The efficiency correction needs to be
performed to recover the true multiplicity distribution. For simplicity, we
assume that the detection efficiency can be approximated by the binomial
efficiency response function Bzdak and Koch (2012a, 2015a). The mean value
(first-order moment/cumulant) can be easily reconstructed by division with the
binomial efficiency response; however, its influence on higher-order cumulants
is complicated and depends on the probability distribution of efficiency Bzdak
et al. (2016); Nonaka et al. (2018); Esumi et al. (2021). Throughout this
paper, we focus on a simple assumption of the binomial distribution given by

$\displaystyle\tilde{P}(n)$ $\displaystyle=$
$\displaystyle\sum_{N}P(N)B_{\varepsilon,N}(n),$ (12) $\displaystyle
B_{\varepsilon,N}(n)$ $\displaystyle=$
$\displaystyle\frac{N!}{n!(N-n)!}\varepsilon^{n}(1-\varepsilon)^{N-n},$ (13)

where $\varepsilon$ represents the efficiency, while $N$ and $n$ are generated
and measured particles, respectively. In this case, the correction formulas
can be derived in a straightforward manner as discussed in the literature
Bialas and Peschanski (1986); Kitazawa and Asakawa (2012); Bzdak and Koch
(2012b, 2015b); Luo (2015); Kitazawa (2016); Nonaka et al. (2017); Luo and
Nonaka (2019). The efficiency correction for the 2nd-order mixed-cumulant is
given by Nonaka et al. (2017); Luo and Nonaka (2019)

$\displaystyle\langle\\!\langle K_{(x)}K_{(y)}\rangle\\!\rangle_{\rm c}$
$\displaystyle=$
$\displaystyle\langle\kappa_{(1,0,1)}\kappa_{(0,1,1)}\rangle_{\rm
c}+\langle\kappa_{(1,1,1)}\rangle_{\rm c}-\langle\kappa_{(1,1,2)}\rangle_{\rm
c},$ (14)

with

$\displaystyle K_{(x)}=\sum_{i}^{M}x_{i}n_{i},\;$ $\displaystyle
K_{(y)}=\sum_{i}^{M}y_{i}n_{i},$ (15) $\displaystyle\kappa{(r,s,t)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\frac{x_{i}^{r}y_{j}^{s}}{\varepsilon_{i}^{t}}n_{i}.$
(16)

where $\langle\\!\langle\cdot\rangle\\!\rangle$ represents the efficiency
correction, $M$ is the number of efficiency bins, $n_{i}$ indicates the number
of particles, $\varepsilon_{i}$ is the efficiency, $x_{i}$ and $y_{i}$ are the
electric charge of the particles at the $i$th efficiency bin. Notably Eqs. 15
and 16 can be rewritten in track-by-track notations as

$\displaystyle K_{(x)}=\sum_{j}^{n^{\rm tot}}x_{j},\;$ $\displaystyle
K_{(y)}=\sum_{i}^{n^{\rm tot}}y_{j},$ (17) $\displaystyle\kappa{(r,s,t)}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{n^{\rm
tot}}\frac{x_{j}^{r}y_{j}^{s}}{\varepsilon_{j}^{t}},$ (18)

where $n_{\rm tot}=\sum_{i}^{M}n_{i}$ is the number of measured particles in
one event, and the other variables are now track-wise with $j$ running over
the particles in the summation.

In the rest of this section, we consider two efficiency bins for simplicity.
Particles for each bin have the same efficiency values, $\varepsilon_{1}$ and
$\varepsilon_{2}$, respectively. The number of particles will be denoted by
$n_{1}$ and $n_{2}$.

###  II.3 Mutually exclusive variable

Let’s consider the correlation between two mutually exclusive variables. From
Eq. 10, the mixed-cumulant of generated particles are expanded in terms of
moments as

$\langle N_{1}N_{2}\rangle_{\rm c}=\langle N_{1}N_{2}\rangle-\langle
N_{1}\rangle\langle N_{2}\rangle.$ (19)

To perform the efficiency correction relative to measured particles, we
suppose

$\displaystyle x$ $\displaystyle=$ $\displaystyle(x_{1},x_{2})=(1,0),$ (20)
$\displaystyle y$ $\displaystyle=$ $\displaystyle(y_{1},y_{2})=(0,1),$ (21)

to consider $\langle\\!\langle K_{(x)}K_{(y)}\rangle\\!\rangle_{\rm c}$ with
$K_{(x)}=n_{1}$ and $K_{(y)}=n_{2}$. From Eq. 14, we get

$\displaystyle\langle\\!\langle K_{(1,0)}K_{(0,1)}\rangle\\!\rangle_{\rm c}$
$\displaystyle=$
$\displaystyle\langle\kappa{(1,0,1)}\kappa{(0,1,1)}\rangle_{\rm
c}+\langle\kappa{(1,1,1)}\rangle_{\rm c}-\langle\kappa{(1,1,2)}\rangle_{\rm
c}$ (22) $\displaystyle=$
$\displaystyle\Bigl{\langle}\frac{n_{1}}{\varepsilon_{1}}\frac{n_{2}}{\varepsilon_{2}}\Bigr{\rangle}_{\rm
c}$ (23) $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{1}\varepsilon_{2}}\langle
n_{1}n_{2}\rangle-\frac{1}{\varepsilon_{1}\varepsilon_{2}}\langle
n_{1}\rangle\langle n_{2}\rangle,$ (24)

which is the basic formula of the efficiency correction for the 2nd-order
mixed-cumulant of two variables.

###  II.4 Mutually inclusive variables

####  II.4.1 Problem

Next, we consider the correlation between $N_{1}$ and $N_{1}+N_{2}$, i.e, when
the two variables are not mutually exclusive. It is clear that we have the
self-correlation of $N_{1}$. The 2nd-order mixed-cumulant can be expanded as

$\displaystyle\langle N_{1}(N_{1}+N_{2})\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\langle N_{1}N_{2}\rangle-\langle N_{1}\rangle\langle
N_{2}\rangle+\langle N_{1}^{2}\rangle-\langle N_{1}\rangle^{2},$ (25)

where the last two terms represent the variance (2nd-order cumulant, $\langle
N_{1}^{2}\rangle_{\rm c}$). If we employ Eq. 24 for the efficiency correction,
we can just replace $n_{1}$ to $n_{1}+n_{2}$ as

$\displaystyle\langle\\!\langle K_{(1,0)}K_{(0,1)}\rangle\\!\rangle_{\rm c}$
$\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{1}\varepsilon_{2}^{\prime}}\langle
n_{1}(n_{1}+n_{2})\rangle-\frac{1}{\varepsilon_{1}\varepsilon_{2}^{\prime}}\langle
n_{1}\rangle\langle n_{1}+n_{2}\rangle,$ (26) $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{1}\varepsilon_{2}^{\prime}}\langle
n_{1}n_{2}\rangle-\frac{1}{\varepsilon_{1}\varepsilon_{2}^{\prime}}\langle
n_{1}\rangle\langle
n_{2}\rangle+\frac{1}{\varepsilon_{1}\varepsilon_{2}^{\prime}}\langle
n_{1}^{2}\rangle-\frac{1}{\varepsilon_{1}\varepsilon_{2}^{\prime}}\langle
n_{1}\rangle^{2},$ (27)

with $\varepsilon_{2}^{\prime}$ being the averaged efficiency for $N_{1}$ and
$N_{2}$ given by

$\displaystyle\varepsilon_{2}^{\prime}=\frac{\langle
N_{1}\rangle\varepsilon_{1}+\langle N_{2}\rangle\varepsilon_{2}}{\langle
N_{1}\rangle+\langle N_{2}\rangle},$ (28)

which is not an appropriate efficiency corrected expression for mutually
inclusive variables. To confirm this, we suppose that two independent
variables $N_{1}$ and $N_{2}$ follow the Poisson distribution with $\langle
N_{1}\rangle=4$ and $\langle N_{2}\rangle=6$. It is known that the relation
$\langle N^{r}\rangle_{\rm c}=\langle N\rangle_{\rm c}$ holds for Poisson
distributions; thus, $\langle N^{2}\rangle_{\rm c}=\langle
N^{2}\rangle-\langle N\rangle^{2}$, which leads to $\langle
N_{1}(N_{1}+N_{2})\rangle_{\rm c}=4$ , from Eq. 25. The relationship $\langle
N_{1}N_{2}\rangle=\langle N_{1}\rangle\langle N_{2}\rangle$ was adopted for
the independent variables. We then consider the efficiency correction for
$\varepsilon_{1}=0.5$ and $\varepsilon_{2}=0.4$. Accordingly,

$\displaystyle\varepsilon_{2}^{\prime}=0.44,\;\langle n_{1}\rangle=2,\;\langle
n_{2}\rangle=2.4,$ (29) $\displaystyle\langle n_{1}^{2}\rangle_{\rm c}=\langle
n_{1}^{2}\rangle-\langle n_{1}\rangle^{2}=2,$ (30) $\displaystyle\langle
n_{1}n_{2}\rangle=\langle n_{1}\rangle\langle n_{2}\rangle,$ (31)

which leads to the efficiency corrected mixed-cumulant value

$\displaystyle\langle\\!\langle K_{(1,0)}K_{(0,1)}\rangle\\!\rangle_{\rm
c}=4.55.$ (32)

Hence, Eq. 24 is not valid for the self correlated or mutually inclusive case.

####  II.4.2 Solution

The solution is to adopt the appropriate indices for $x$ and $y$ in Eqs. 16
and 15. To consider $\langle\\!\langle K_{(x)}K_{(y)}\rangle\\!\rangle_{\rm
c}$ with $K_{(x)}=n_{1}$ and $K_{(y)}=n_{1}+n_{2}$, the indices should have
been

$\displaystyle x$ $\displaystyle=$ $\displaystyle(x_{1},x_{2})=(1,0),$ (33)
$\displaystyle y$ $\displaystyle=$ $\displaystyle(y_{1},y_{2})=(1,1),$ (34)

thus

$\displaystyle\langle\\!\langle K_{(1,0)}K_{(1,1)}\rangle\\!\rangle_{\rm c}$
$\displaystyle=$
$\displaystyle\langle\kappa{(1,0,1)}\kappa{(0,1,1)}\rangle_{\rm
c}+\langle\kappa{(1,1,1)}\rangle_{\rm c}-\langle\kappa{(1,1,2)}\rangle_{\rm
c}$ (35) $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{1}\varepsilon_{2}}\langle
n_{1}n_{2}\rangle-\frac{1}{\varepsilon_{1}\varepsilon_{2}}\langle
n_{1}\rangle\langle n_{2}\rangle+\frac{1}{\varepsilon_{1}^{2}}\langle
n_{1}^{2}\rangle-\frac{1}{\varepsilon_{1}^{2}}\langle
n_{1}\rangle^{2}+\frac{1}{\varepsilon_{1}}\langle
n_{1}\rangle-\frac{1}{\varepsilon_{1}^{2}}\langle n_{1}\rangle,$ (36)

where we determine two additional terms compared with Eq. 27. It is inferred
that the last four terms in Eq. 36 represent the efficiency correction of the
variance (2nd-order cumulant), $\langle\\!\langle
K_{(x)}^{2}\rangle\\!\rangle$, which corresponds to the last two terms in Eq.
25 111This can be confirmed by substituting $(x_{1},x_{2})=(1,0)$ and
$(y_{1},y_{2})=(1,0)$ into Eqs. 14–16. This indicates that the variance has to
be correctly considered for the mutually inclusive variable case, which cannot
be handled by Eq. 24.

We summarize this section as follows. The efficiency correction formula for
the 2nd-order mixed-cumulant was fully expanded for two cases: one is for two
mutually exclusive variables, and the other case assumes that one variable is
a subset of the other, to consider the self-correlation, as expressed in Eqs.
24 and 36. Both cases were determined to be incompatible with each other. The
proper correction formulas needs to be obtained by substituting appropriate
indices into Eqs. 14–16. This implies that the efficiency values have to be
handled properly for each variable, without averaging them, especially when
considering the self-correlation. It should be noted that the risk of using
the averaged efficiency has already been pointed out in Ref. Nonaka et al.
(2017) for higher-order cumulants of single-variables. The efficiency bins
always need to be carefully handled. The track-by-track efficiency via the
identified particle approach expressed in Eqs. 17 and 18 would be a better way
to handle all possible variations of efficiencies Luo and Nonaka (2019).
However, the particle identification needs to be applied to determine the
efficiencies for different particle species, which does not discard a small
amount of particles, depending on the overlapping area of the variables for
the particle identification. This effect will be studied by numerical
simulations in the next section Shao et al. (2006); Adamczyk et al. (2014b).

##  III Numerical analysis in UrQMD model

###  III.1 Closure test using UrQMD model

To validate the discussion from the previous section, we have analyzed the
second-order mixed cumulants from the UrQMD event generator at $\sqrt{{s_{\rm
NN}}}~{}$= 200 GeV. The UrQMD is a microscopic transport model, where the
space-time evolution of the fireball is considered in terms of the excitation
of color strings that fragment further into hadrons, the co-variant
propagation of hadrons and resonances that undergo scatterings, and finally
the decay of all the resonances Bass et al. (1998); Bleicher et al. (1999).
The collision energy dependence of the baryon stopping phenomenon is
dynamically incorporated in the UrQMD model. The UrQMD model has been
relatively successful and widely used in the heavy-ion phenomenology Bleicher
et al. (1999, 1998). Previously, this model was adopted to study several
observable fluctuation and cumulants Bleicher et al. (2000); Haussler et al.
(2006); Luo et al. (2013); Chatterjee et al. (2016); Xu et al. (2016); He and
Luo (2017); Mukherjee et al. (2018). More information on the UrQMD model can
be found in Ref. Bass et al. (1998); Bleicher et al. (1999). In this study, we
have adopted approximately one million events for Au+Au collisions at
$\sqrt{{s_{\rm NN}}}~{}$= 200 GeV to probe the efficiency correction effect on
mixed cumulants. The obtained results are presented for 9 different centrality
bins represented by the average number of participant nucleons ($\langle
N_{part}\rangle$). In this study, we have applied the same kinematic
acceptance $|\eta|<0.5$ and $0.4<p_{T}<1.6$ GeV/$c$ as STAR data Adam et al.
(2019). The collision centrality is defined using RefMult2 (charged particle
multiplicity within the pseudorapidity range $0.5<|\eta|<1.0$) to reduce the
centrality auto-correlation effect Luo et al. (2013); Chatterjee et al.
(2020a). Figure 1 illustrates the centrality dependence of second-order mixed-
cumulants of net-charge, net-proton, and net-kaon multiplicity distributions
from the UrQMD model. The gray solid points represent the ’true’ mixed-
cumulants values. To introduce the detector efficiency effect, we passed the
binomial filter to the counted particles number in each event. We adopted two
$p_{T}$-bin and positive-negative separate efficiencies similar to real data
analysis. We took this approach because different detector subsystems are used
for particle identification in a high or low momentum region Adam et al.
(2019). These subsystems have different efficiencies, and it is always better
to use their proper efficiency values over their average efficiencies Nonaka
et al. (2017). Subsequently, we estimated the mixed-cumulants with filtered
particle numbers represented by red square points, which are analogous to
efficiency uncorrected values.

Figure 1: (Color online) Centrality dependence of second-order mixed-cumulants
of net-charge ($Q=N_{Q^{+}}-N_{Q^{-}}$), net-proton ($p=N_{p}-N_{\bar{p}}$)
and net-kaon ($k=N_{k^{+}}-N_{k^{-}}$) multiplicities for Au+Au collisions at
200 GeV, using the UrQMD model. The efficiency corrections are performed
assuming variables are mutually exclusive (Eq. 24). Figure 2: (Color online)
Centrality dependence of second-order off-diagonal to diagonal cumulant ratios
for Au+Au collisions at 200 GeV, using the UrQMD model. The efficiency
correction are performed assuming the variables are mutually exclusive (Eq.
24).

In the next step, we correct the efficiency using input efficiency values. In
this case, we adopted unidentified charged particles for the net-charge ($Q$)
and applied Eq. 24, similar to the STAR measurement. The $p$-$k$ mixed
cumulant “true” value can be reproduced via this method, as they are mutually
exclusive variables. However, the efficiency correction for $Q$-$p$ and
$Q$-$k$ fails to reproduce the “true” values as we discussed in Sec. II. We
end up with a higher value for unidentified charge correlators. This leads to
a higher value in cumulant ratios $C_{Q,k}=\sigma_{Q,k}/\sigma^{2}_{k}$ and
$C_{Q,p}=\sigma_{Q,p}/\sigma^{2}_{p}$ obtained from “true” values, as
presented in Fig. 2. This shows that Eq. 24 is not valid for overlap or
mutually inclusive variables. The observation is qualitatively consistent with
the fact that $C_{p,k}$ values in Ref. Adam et al. (2019) agree well with the
model calculations, while $C_{Q,k}$ and $C_{Q,p}$ are significantly above the
model calculations. However, for mutually exclusive variables (like protons-
kaons), there is no issue. As we discussed before, to correct the mixed
cumulant for mutually inclusive variables, Eq. 36 is required. To apply Eq. 36
for $Q$-$k$ and $Q$-$p$ mixed cumulants, it is necessary to identify the
charged particles with their efficiencies. In Figs. 3 and 4, we solely
consider identified charged particles ($Q=\pi+k+p$). Then Eq. 36 becomes

$\displaystyle\langle\\!\langle N_{Q}N_{k}\rangle\\!\rangle_{\rm c}$
$\displaystyle=$
$\displaystyle\langle\\!\langle(N_{\pi}+N_{p}+N_{k})N_{k}\rangle\\!\rangle_{\rm
c}$ (37) $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{1}\varepsilon_{3}}\langle
N_{\pi}N_{k}\rangle-\frac{1}{\varepsilon_{1}\varepsilon_{3}}\langle
N_{\pi}\rangle\langle
N_{k}\rangle+\frac{1}{\varepsilon_{2}\varepsilon_{3}}\langle
N_{p}N_{k}\rangle-\frac{1}{\varepsilon_{2}\varepsilon_{3}}\langle
N_{p}\rangle\langle N_{k}\rangle+\frac{1}{\varepsilon_{3}^{2}}\langle
N_{k}^{2}\rangle-\frac{1}{\varepsilon_{3}^{2}}\langle N_{k}\rangle^{2}$
$\displaystyle+\frac{1}{\varepsilon_{3}}\langle
N_{k}\rangle-\frac{1}{\varepsilon_{3}^{2}}\langle N_{k}\rangle,$

where $\varepsilon_{1}$, $\varepsilon_{2}$, and $\varepsilon_{3}$ are the
efficiencies for pions, kaons, and protons, respectively. Now, we can
reproduce the “true” values, as presented in Figs. 3 and 4.

Figure 3: (Color online) Centrality dependence of second-order mixed-cumulants
of identified net-charge
($Q=(N_{\pi^{+}}+N_{k^{+}}+N_{p})-(N_{\pi^{-}}+N_{k^{-}}+N_{\bar{p}})$), net-
proton ($p=N_{p}-N_{\bar{p}}$), and net-kaon ($k=N_{k^{+}}-N_{k^{-}}$)
multiplicity for Au+Au collisions at 200 GeV, using the UrQMD model. The
efficiency corrections are done assuming the variables are mutually inclusive
(Eq. 36). Figure 4: (Color online) Centrality dependence of second-order off-
diagonal to diagonal cumulant ratios of identified charged particles for Au+Au
collisions at 200 GeV, using the UrQMD model. The efficiency correction are
performed assuming the variables are mutually inclusive (Eq. 36).

However, owing to the particle identification with optimal purity we may lose
a few pions, protons, and kaons. We also studied the effects of those tracks
for mixed cumulants via UrQMD simulations. Charged particle identification is
performed using the ionization energy loss inside the time projection chamber
(TPC) detector subsystem. We mimic the ionization energy loss curve in UrQMD
simulation using the STAR TPC resolution. Figure 5(a) presents the measured
$dE/dx$ distribution after passing the UrQMD input through the TPC simulation.
The measured values of $dE/dx$ are compared to the expected theoretical values
which is an extension of the Bethe-Bloch formula Abelev et al. (2009) (shown
as dashed lines in Fig. 5(a)). To identify particles X, a quantity
$n\sigma_{X}$ is defined as

$\displaystyle
n\sigma_{X}=\frac{1}{R}\ln\frac{[dE/dx]_{obs}}{[dE/dx]_{th,X}},$ (38)

where $[dE/dx]_{obs}$ represents the energy loss in the UrQMD simulation and
$[dE/dx]_{th,X}$ is the corresponding theoretical value for particle species
X. R represents the $dE/dx$ resolution, and we use R = 7.5% within our
analysis range. Figure 5(b) presents the $n\sigma$ distributions of
$p(\bar{p})$, $\pi^{\pm}$, and $k^{\pm}$. Typically, $n\sigma_{p}<2$ are
adopted for the $p(\bar{p})$ selection. Similarly, $n\sigma_{\pi}<2$ and
$n\sigma_{k}<2$ are adopted for pion and kaon selections, respectively.
However, from Fig. 5(a) we can deduce that at high momenta, the $dE/dx$ bands
for different particles are overlapped. We have also adopted the
$2\sigma$-rejection cut to improve the purity.

(a)

(b)

Figure 5: (Color online) (a) $dE/dx$ from UrQMD simulations plotted against charge$\times$momentum of individual particles for Au+Au collisions at $\sqrt{{s_{\rm NN}}}~{}$= 200 GeV. (b) $n\sigma$ distributions of protons, pions, and kaons. Figure 6: (Color online) $n\sigma$ acceptance dependence mean multiplicity of net-charge (Q), net-proton (p), and net-kaon (k) for Au+Au collisions at 200 GeV, using the UrQMD model. Figure 7: (Color online) $n\sigma$ acceptance dependence second-order off-diagonal and diagonal cumulants for Au+Au collisions at 200 GeV, using the UrQMD model. For | Legend | Particle Code | $n\sigma$ |  $2\sigma$-rejection | Baseline
---|---|---|---|---|---
Signal | S1 | Used | Applied | N/A | B1
S2 | Used | Applied | Applied | B2
S3 | N/A | Applied | Applied | B2
Baseline | B1 | Used | N/A | N/A | N/A
B2 | Used | N/A | Applied | N/A
Table 1: This table describes the information that is used to select the
particles for mixed-cumulant calculations in Fig. 7, in terms of the particle
species code given by UrQMD, $n\sigma$ and $2\sigma$-rejection cuts.
Corresponding legends in Fig. 6, 7, and 8 are presented in the second left
row. Figure 8: (Color online) $n\sigma$ acceptance dependence second-order
off-diagonal over diagonal cumulants ratios for Au+Au collisions at 200 GeV
using UrQMD model.

Figure 6 presents the event-by-event average of net-charge, net-kaon, and net-
proton multiplicities as a function of $n\sigma$. Similarly, figure 7 shows
mixed-cumulants as a function of the $n\sigma$ cut for three cases as follows
(refer to Tab. 1):

S1

: Information on the particle species is provided by UrQMD. The $n\sigma$ cut
is applied.

S2

: Information on the particle species is provided by UrQMD. Both $n\sigma$ and
$2\sigma$-rejection cuts are applied.

S3

: Particles are identified by both $n\sigma$ and $2\sigma$-rejection cuts.
This is the only possible cut in the experimental data analysis.

We note that “$Q$” is defined as the summation of the identified $\pi$, $K$,
and $p$. The “2$\sigma$-rejection cut” indicates the requirement of
$n\sigma_{K}>2.0$ and $n\sigma_{\pi}>2.0$ for proton identification,
$n\sigma_{p}<-2.0$ and $n\sigma_{\pi}>2.0$ for kaon identification, and
$n\sigma_{K}<-2.0$ and $n\sigma_{p}<-2.0$ for pion identification.
Furthermore, two baselines are calculated with the following conditions, which
are independent of the $n\sigma$ cut:

B1

: Information on the particle species is provided by UrQMD.

B2

: Information on the particle species is provided by UrQMD. The
$2\sigma$-rejection cut is applied.

The difference between two baselines is owing to the particle multiplicity.
There are more particles for B1 than B2 because the $2\sigma$-rejection cut is
applied to the latter case. Depending on whether the $2\sigma$-rejection cut
is applied, S1 can compared to B1, while S2 and S3 need to be compared to B2.
The descriptions of Fig. 7 are summarized in Tab. 1.

It is determined that the values of mixed-cumulants are close to the
corresponding baselines with a loose $n\sigma$ cut, and decreases with the
tightening the $n\sigma$ cut. As can be observed in Eq. 19, the mixed-cumulant
for the mutually inclusive case comprises second-order cumulants, as well as
second-order mixed-cumulants. It is well known that the former has a trivial
volume dependence Asakawa and Kitazawa (2016), which can also be confirmed
from the lower panels in Fig. 7 for the second-order cumulants. In Fig. 7,
therefore, only $\sigma^{11}_{p,k}$ is independent of the $n\sigma$ cut. The
difference between S2 and S3 would indicate the effect of contamination. This
can be confirmed from the fact that the difference becomes small with the
tightening of the $n\sigma$ cut, in addition to the 2$\sigma$-rejection cut.
Consequently we observed larger values of $\sigma^{11}_{Q,k}$ and
$\sigma^{11}_{Q,p}$ for S3, compared to those for S2.

To cancel the trivial volume dependence, the normalized mixed-cumulants are
also calculated in Fig. 8 as a function of the $n\sigma$ cut. All the results
for S1, S2, and S3 are inferred to be consistent with corresponding baselines
within uncertainties. This is because the probability of the contamination has
been significantly suppressed by the $2\sigma$-rejection cut. To confirm this,
normalized mixed-cumulants with only the $n\sigma$ cuts are calculated, where
the significant deviations are observed from the corresponding baselines,
owing to contamination. It should be noted that the $n\sigma$ dependence of
mixed-cumulants and normalized mixed-cumulants also depends on how the
intrinsic correlations between two variables change relative to the $n\sigma$
cut. In the current simulation, the energy losses of particles are randomly
smeared to implement the resolution of the detectors. Therefore, the
correlation terms (the first term on the right hand side of Eqs. 19 and 25)
are considered to be unaffected by the $n\sigma$ cut. However, experimentally,
this effect needs to be carefully studied by changing the criteria for
particle identifications.

##  IV Summary

In this study, we discussed the efficiency correction problem for mixed-
cumulants. This study provided a comprehensive extension of the binomial
efficiency correction formula for second-order mixed accumulators in two
different cases: one case is for mutually exclusive variables and the other is
for mutually inclusive variables. We infer that different efficiency
correction formulas need to be applied to mixed-cumulants, depending on the
type of variable pairs. To apply the binomial efficiency correction for
$Q$-$k$ and $Q$-$p$ mixed cumulants, it is necessary to identify the charged
particles with their corresponding efficiencies.

It should be noted that the efficiency correction for mixed-cumulants in the
case of mutually inclusive variables has already been discussed in Ref.
Vovchenko and Koch (2021). In the proposed formulas, two different levels of
efficiencies were implemented for each variable, such as $N_{Q}$ and $N_{p}$
in the case of $\sigma^{11}_{Q,p}$. The tracking efficiency was applied to
$N_{Q}$, while the proton identification efficiency was applied on top of the
tracking efficiency for $N_{p}$. The method has the advantage that we can keep
charged particles as much as possible without identifying each particle
species contained in $N_{Q}$. This implies that the averaged efficiencies for
pions, kaons, and protons were used for $N_{Q}$. However, we must remember
that using the averaged efficiency does not provide the true solution, which
depends on underlying probability distributions of the number of particles and
the difference in efficiency between particle species Nonaka et al. (2017). It
is also important to note that the identity method would be useful for the
measurements of mixed-cumulants Gazdzicki et al. (2020), which enables us to
measure fluctuations without the multiplicity loss, owing to particle
identification. It would be desirable for novel ideas to address the two-step
efficiency based on the identity method Gazdzicki et al. (2020).

At this stage, the identification of each particle species and subsequent
implemention of the appropriate efficiency is the simplest approach.
Therefore, we further investigated the effect of the loss in multiplicity
owing to particle identifications, using numerical simulations. In the case of
mutually inclusive variables, the mixed-cumulants exhibited a monotonic
decrease as the cut value of particle identification tightened. This can be
explained by a trivial volume dependence. In contrast, the normalized mixed-
cumulants were found to be independent of the cut value for the particle
identification. This is because the intrinsic correlations between different
particle species were assumed to be independent of the variables of the
particle identification, which could not be the case in real experiments.
Therefore, it is recommended to verify these effects by changing the criteria
for the particle identifications. This work provides an important reference
for future measurements of mixed-cumulants in relativistic heavy-ion
collisions.

##  V Acknowledgement

We thank Volker Koch, Masakiyo Kitazawa, Nihar Rajan Sahoo, Prithwish Tribedy,
Volodymyr Vovchenko, Tapan Nayak, and Nu Xu for stimulating discussions. This
work is supported by the National Key Research and Development Program of
China (Grant No. 2020YFE0202002 and 2018YFE0205201), the National Natural
Science Foundation of China (Grant No. 11828501, 11890711 and 11861131009),
Ito Science Foundation and JSPS KAKENHI Grant No. 25105504, 19H05598.

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##  Appendix A Efficiency correction for higher-order mixed-cumulants

The efficiency correction formulas for higher-order mixed-cumulants are
provided in Ref. Nonaka et al. (2017) as

$\displaystyle\langle\\!\langle K_{(x)}^{2}K_{(y)}\rangle\\!\rangle_{\rm c}$
$\displaystyle=$
$\displaystyle\langle\kappa{(1,0,1)}^{2}\kappa{(0,1,1)}\rangle_{\rm
c}+2\langle\kappa{(1,0,1)}\kappa{(1,1,1)}\rangle_{\rm
c}-2\langle\kappa{(1,0,1)}\kappa{(1,1,2)}\rangle_{\rm c}$ (39)
$\displaystyle+\langle\kappa{(0,1,1)}\kappa{(2,0,1)}\rangle_{\rm
c}-\langle\kappa{(0,1,1)}\kappa{(2,0,2)}\rangle_{\rm
c}+\langle\kappa{(2,1,1)}\rangle_{\rm c}-3\langle\kappa{(2,1,2)}\rangle_{\rm
c}+2\langle\kappa{(2,1,3)}\rangle_{\rm c},$ $\displaystyle\langle\\!\langle
K_{(x)}^{2}K_{(y)}^{2}\rangle\\!\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\langle\kappa{(1,0,1)}^{2}\kappa{(0,1,1)}^{2}\rangle_{\rm c}$
(40) $\displaystyle+\langle\kappa{(1,0,1)}^{2}\kappa{(0,2,1)}\rangle_{\rm
c}-\langle\kappa{(1,0,1)}^{2}\kappa{(0,2,2)}\rangle_{\rm
c}+\langle\kappa{(0,1,1)}^{2}\kappa{(2,0,1)}\rangle_{\rm
c}-\langle\kappa{(0,1,1)}^{2}\kappa{(2,0,2)}\rangle_{\rm c}$
$\displaystyle+4\langle\kappa{(1,0,1)}\kappa{(0,1,1)}\kappa{(1,1,1)}\rangle_{\rm
c}-4\langle\kappa{(1,0,1)}\kappa{(0,1,1)}\kappa{(1,1,2)}\rangle_{\rm c}$
$\displaystyle+2\langle\kappa{(1,0,1)}\kappa{(1,2,1)}\rangle_{\rm
c}-6\langle\kappa{(1,0,1)}\kappa{(1,2,2)}\rangle_{\rm
c}+4\langle\kappa{(1,0,1)}\kappa{(1,2,3)}\rangle_{\rm c}$
$\displaystyle+2\langle\kappa{(0,1,1)}\kappa{(2,1,1)}\rangle_{\rm
c}-6\langle\kappa{(0,1,1)}\kappa{(2,1,2)}\rangle_{\rm
c}+4\langle\kappa{(0,1,1)}\kappa{(2,1,3)}\rangle_{\rm c}$
$\displaystyle-4\langle\kappa{(1,1,1)}\kappa{(1,1,2)}\rangle_{\rm
c}+2\langle\kappa{(1,1,1)}^{2}\rangle_{\rm
c}+2\langle\kappa{(1,1,2)}^{2}\rangle_{\rm c}$
$\displaystyle+\langle\kappa{(2,0,1)}\kappa{(0,2,1)}\rangle_{\rm
c}-\langle\kappa{(2,0,1)}\kappa{(0,2,2)}\rangle_{\rm
c}-\langle\kappa{(2,0,2)}\kappa{(0,2,1)}\rangle_{\rm
c}+\langle\kappa{(2,0,2)}\kappa{(0,2,2)}\rangle_{\rm c}$
$\displaystyle+\langle\kappa{(2,2,1)}\rangle_{\rm
c}-7\langle\kappa{(2,2,2)}\rangle_{\rm c}+12\langle\kappa{(2,2,3)}\rangle_{\rm
c}-6\langle\kappa{(2,2,4)}\rangle_{\rm c},$ $\displaystyle\langle\\!\langle
K_{(x)}^{3}K_{(y)}\rangle\\!\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\langle\kappa{(1,0,1)}^{3}\kappa{(0,1,1)}\rangle_{\rm c}$ (41)
$\displaystyle+3\langle\kappa{(1,0,1)}^{2}\kappa{(1,1,1)}\rangle_{\rm
c}-3\langle\kappa{(1,0,1)}^{2}\kappa{(1,1,2)}\rangle_{\rm
c}+3\langle\kappa{(2,0,1)}\kappa{(1,0,1)}\kappa{(0,1,1)}\rangle_{\rm c}$
$\displaystyle-3\langle\kappa{(2,0,2)}\kappa{(1,0,1)}\kappa{(0,1,1)}\rangle_{\rm
c}+3\langle\kappa{(1,0,1)}\kappa{(2,1,1)}\rangle_{\rm
c}-9\langle\kappa{(1,0,1)}\kappa{(2,1,2)}\rangle_{\rm c}$
$\displaystyle+6\langle\kappa{(1,0,1)}\kappa{(2,1,3)}\rangle_{\rm
c}+3\langle\kappa{(2,0,1)}\kappa{(1,1,1)}\rangle_{\rm
c}-3\langle\kappa{(2,0,1)}\kappa{(1,1,2)}\rangle_{\rm
c}-3\langle\kappa{(2,0,2)}\kappa{(1,1,1)}\rangle_{\rm c}$
$\displaystyle+3\langle\kappa{(2,0,2)}\kappa{(1,1,2)}\rangle_{\rm
c}+\langle\kappa{(3,0,1)}\kappa{(0,1,1)}\rangle_{\rm
c}-3\langle\kappa{(3,0,2)}\kappa{(0,1,1)}\rangle_{\rm
c}+2\langle\kappa{(3,0,3)}\kappa{(0,1,1)}\rangle_{\rm c}$
$\displaystyle+\langle\kappa{(3,1,1)}\rangle_{\rm
c}-7\langle\kappa{(3,1,2)}\rangle_{\rm c}+12\langle\kappa{(3,1,3)}\rangle_{\rm
c}-6\langle\kappa{(3,1,4)}\rangle_{\rm c}.$

The substitution of appropriate indices for $x$ and $y$ in Eq. 15 is required,
as discussed in Sec. II.