# Two weight bump conditions for compactness of commutators

Adam Mair  Adam Mair
Department of Mathematics
University of Alabama
Tuscaloosa, AL 35487, USA<EMAIL_ADDRESS>and  Kabe Moen  Kabe Moen
Department of Mathematics
University of Alabama
Tuscaloosa, AL 35487, USA<EMAIL_ADDRESS>
###### Abstract.

We prove certain two weight bump conditions are sufficient for the compactness
of the commutator $[b,T]$ where $b\in CMO$ and $T$ is a Calderón-Zygmund
operator. This is the first result for compactness in the two weight setting
without additional assumptions on the individual weights.

###### 2000 Mathematics Subject Classification:

42B20, 42B25

The second author is supported by Simons Collaboration Grant for
Mathematicians, 160427. A version of this paper will serve as a chapter in the
second author’s Ph.D. dissertation.

##  1\. Introduction

Let $T$ be a Calderón-Zygmund operator (CZO) and $b$ be a $BMO$ function.
Define the commutator of $b$ and $T$ to be

$[b,T]f=bTf-T(bf).$

In this note we are interested in conditions on $b$ and $(u,v)$ that ensure
compactness of $[b,T]$ between two different weighted spaces $L^{p}(v)$ and
$L^{p}(u)$.

The boundedness of the commutator is well established. Coifmann, Rochberg, and
Weiss [6] showed that $[b,T]$ is bounded on $L^{p}(\mathbb{R}^{n})$ when $b\in
BMO$. Chung, Pereyra, and Pérez [4] showed that $[b,T]$ is bouneded on
$L^{p}(w)$ when $w$ belongs to the Muckenhoupt $A_{p}$ class and gave the
sharp quantitative control of the constants. The two weight boundedness of
$[b,T]$ from $L^{p}(v)$ to $L^{p}(u)$ is significantly more difficult. Cruz-
Uribe and the second author [8] showed that if one strengthens the two weight
$A_{p}$ condition with certain logarithmic bumps then $[b,T]$ is a bounded
linear operator from $L^{p}(v)$ to $L^{p}(u)$ (see below for relevant
definitions). We also refer readers to more recent results by Lerner, Ombrosi,
and Riveria-Ríos [18], who improved the results in [8]. Finally, we mention in
passing the work of Isralowitz, Pott, and Treil [14] and Cruz-Uribe, the
second author, and Tran [9] in which the two weight boundedness of $[b,T]$ and
higher order commutators is obtained assuming more general assumptions on $b$
and the weights $(u,v)$.

Recall that a linear operator between two Banach spaces $T:X\rightarrow Y$ is
compact if $T(B_{X})$ has compact closure in $Y$ (here $B_{X}$ is the unit
ball in $X$). The compactness of commutators was first investigated by
Uchiyama [21], who was motivated by applications to complex analysis and
Hankel operators. Uchiyama showed that if $b\in CMO$, the closure of
$C_{c}^{\infty}(\mathbb{R}^{n})$ in the $BMO$ norm, then $[b,T]$ is a compact
operator on $L^{p}(\mathbb{R}^{n})$. The original statement in [21] was that
$b\in VMO$, the space _vanishing mean oscillation_ , is sufficient for
compactness of $[b,T]$ on $L^{p}(\mathbb{R}^{n})$. However the spaces $CMO$
and $VMO$ do not coincide over $\mathbb{R}^{n}$ and the correct space for
compactness of commutators is $CMO$. This misnomer has appeared several times
in the literature (including [5]) as the article by Bourdaud [2] points out.
Wu and Yang [22] show that $CMO$ actually characterizes the compactness of
$[b,T]$ on $L^{p}(\mathbb{R}^{n})$ for specific operators $T$ like the Riesz
transforms. Iwaniec [15] proved a Fredholm alternative for Beltrami equations
as an application of the compactness of $[b,T]$. Cruz and Clop [5] were also
interested in applications to quasiregular mappings and showed that commutator
$[b,T]$ is compact on $L^{p}(w)$ when $b\in CMO$ and $w\in A_{p}$. Recently
Hytönen and Lappas [12] have shown an extrapolation theorem for compact
operators on weighted spaces. Finally we mention some very recent two weight
results which are related to the so-called Bloom $BMO$. Namely, under certain
weighted $BMO$ assumptions on $b$, it is shown that the operator $[b,T]$ is a
compact operator from $L^{p}(v)$ to $L^{p}(u)$ provided one assumes the side
condition $u,v\in A_{p}$. We refer the interested reader to Chen, Lacey, Li,
and Vempati [3] and Hytönen, Oikari, and Sinko [13] for more on compactness in
this setting. Our main result is the following.

######  Theorem 1.

Suppose $T$ is a Calderón-Zygmund operator and $b\in CMO$. If $u$ and $v$ are
weights that satisfy

(1.1) $\sup_{Q}\|u^{\frac{1}{p}}\|_{L^{p}(\log
L)^{2p-1+\delta},Q}\|v^{-\frac{1}{p}}\|_{L^{p^{\prime}}(\log
L)^{2p^{\prime}-1+\delta},Q}<\infty$

for some $\delta>0$, then the operator $[b,T]$ is a compact operator from
$L^{p}(v)$ to $L^{p}(u)$.

Condition (1.1) is a strengthening of the classic two weight $A_{p}$
condition. It does not require individual assumptions on $u$ and $v$. For
example, $u$ can be _any_ weight and $v=M^{k}u$ where $k=\lfloor 2p\rfloor+1$
and $M^{k}$ is the $k$-fold composition of the Hardy-Littlewood maximal
operator $M$ (see [20]). We also remark that conditions for compactness of the
higher order iterated commutators are more in-depth and will be addressed in a
forthcoming paper.

##  2\. Preliminaries

We begin with the space of functions of bounded mean oscillation. A measurable
function $b$ is said to belong to $BMO$ if

$\|b\|_{BMO}=\sup_{Q}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\Big{|}b(x)-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}b\Big{|}\,dx<\infty.$

The quantity $\|b\|_{BMO}$ is a semi-norm which can be made into a norm by
considering $BMO$ modulo constants. Henceforth, we will just refer to
$\|b\|_{BMO}$ as the norm of $b$ in $BMO$. The space $CMO$ is the subspace
obtained by taking the closure of $C^{\infty}_{c}(\mathbb{R}^{n})$ with
respect to the norm $\|\cdot\|_{BMO}$.

The Hardy-Littlewood maximal operator given by

$Mf(x)=\sup_{Q\ni
x}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|f(y)|\,dy.$

A Calderón-Zygmund operator is an $L^{2}(\mathbb{R}^{n})$ bounded operator
associated to a kernel in the sense

$Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)\,dy,\quad f\in
L^{\infty}_{c}(\mathbb{R}^{n}),x\notin\mathsf{supp}\,f.$

We will assume the kernel $K(x,y)$, defined on $\\{(x,y):x\not=y\\}$,
satisfies the size estimate

$|K(x,y)|\leq\frac{C}{|x-y|^{n}}$

and smoothness estimate

$|\nabla K(x,y)|\leq\frac{C}{|x-y|^{n+1}}.$

We will also need the maximal truncation operator which is given by

$T^{\sharp}f(x)=\sup_{\eta>0}\left|\int_{|x-y|>\eta}K(x,y)f(y)\,dy\right|.$

It is well-known that the Hardy-Littlewood maximal operator, CZOs, and
commutators of CZOs with $BMO$ functions are bounded on
$L^{p}(\mathbb{R}^{n})$ for all $1<p<\infty$. Moreover, these operators are
also bounded on $L^{p}(w)$ when the weight $w$ belongs to $A_{p}$:

$\sup_{Q}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}w\right)\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}w^{1-p^{\prime}}\right)^{p-1}<\infty.$

Following the book [7], we will work with pairs of weights $(u,v)$, that is,
non-negative $L^{1}_{\mathsf{loc}}(\mathbb{R}^{n})$ functions, such that $u>0$
on a set of positive measure and $v>0$ almost everywhere. We will work a
strengthened version of the two weight $A_{p}$ condition

$\sup_{Q}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}u\right)\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}v^{1-p^{\prime}}\right)^{p-1}<\infty.$

In order to understand the bump conditions we need some background on Young
functions and Orlicz spaces. A Young function,
$\Phi:[0,\infty)\rightarrow[0,\infty)$, is an increasing, convex function that
satisfies $\Phi(0)=0$ and $\Phi(t)/t\rightarrow\infty$ as
$t\rightarrow\infty$. Given a Young function $\Phi$ one may define the Orlicz
average of $f$ over a cube $Q$

$\|f\|_{\Phi,Q}=\inf\left\\{\lambda>0:\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\Phi\Big{(}\frac{|f(x)|}{\lambda}\Big{)}\,dx\leq
1\right\\}$

and this quantity is a norm. For our purposes we will only need Young
functions of the form $\Phi(t)=t^{p}\big{[}\log(e+t)\big{]}^{a}$, where $p>1$
and $a\in\mathbb{R}$. In this case we will use the notation

$\|f\|_{\Phi,Q}=\|f\|_{L^{p}(\log L)^{a},Q}$

and refer to such averages as logarithmic bumps.

In Pérez [19], it is shown that if the pair of weights satisfies the condition

(2.1)
$\sup_{Q}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}u\right)\|v^{-\frac{1}{p}}\|_{L^{p^{\prime}}(\log
L)^{p^{\prime}-1+\delta},Q}<\infty,$

for some $\delta>0$ then $M:L^{p}(v)\rightarrow L^{p}(u)$. For CZOs bump
conditions on both weights are needed. Lerner [17] showed that the condition

(2.2) $\sup_{Q}\|u^{\frac{1}{p}}\|_{L^{p}(\log
L)^{p-1+\delta},Q}\|v^{-\frac{1}{p}}\|_{L^{p^{\prime}}(\log
L)^{p^{\prime}-1+\delta},Q}<\infty$

for some $\delta>0$ is sufficient for the boundedness of $T$ and $T^{\sharp}$
from $L^{p}(v)$ to $L^{p}(u)$. Finally, for commutators of $BMO$ functions the
condition

(2.3) $\sup_{Q}\|u^{\frac{1}{p}}\|_{L^{p}(\log
L)^{2p-1+\delta},Q}\|v^{-\frac{1}{p}}\|_{L^{p^{\prime}}(\log
L)^{2p^{\prime}-1+\delta},Q}<\infty$

is sufficient for the boundedness of $[b,T]$ from $L^{p}(v)$ to $L^{p}(u)$
(see [8, Theorem 1.3]). We note that the condition (2.3) is stronger than both
(2.2) and (2.1). In particular, if $(u,v)$ satisfy (2.3), then all of the
operators $M,T,$ and $T^{\sharp},$ are bounded from $L^{p}(v)$ to $L^{p}(u)$
as well as the commutator $[b,T]$ when $b\in BMO$. We will use this fact in
our proof. We also point out that this is the benefit of the bump conditions,
which are universal conditions that are not operator specific.

To prove compactness of the operator $[b,T]$ we will need a characterization
of compact subsets of Lebesgue spaces, known as the Kolomogorov-Riesz theorem.
The survey paper by Hanch-Olsen and Holden [11] contains the basic statement
and history of the original theorem. A characterization of compact subsets of
weighted Lebesgue spaces has a more complicated development and was just
recently resolved. Early weighted versions of the Kolomogorv-Riesz theorem
required extra assumptions on the weight $u$, such as $u\in A_{p}$, $u\in
A_{\infty}$, or $u^{1-p_{0}^{\prime}}\in L^{1}_{\textsf{loc}}(\mathbb{R}^{n})$
for some $p_{0}$ (see [5] and [23]). Guo and Zhao [10] proved a general
Kolomogorov-Riesz theorem in Banach function spaces that generalize
$L^{p}(u)$. This generalization is paramount for proving two weight
compactness, because we are not assuming additional assumptions on the weight
$u$.

######  Theorem 2 ([10]).

Let $1\leq p<\infty$, $u$ be a weight, and $\mathcal{F}\subseteq L^{p}(u)$. If
the family $\mathcal{F}$ satisfies

  1. (a)

$\mathcal{F}$ is a bounded subset, i.e.,

$\sup_{f\in\mathcal{F}}\|f\|_{L^{p}(u)}\leq C;$

  2. (b)

$\mathcal{F}$ uniformly vanishes at infinity, that is,

$\lim_{N\rightarrow\infty}\sup_{f\in\mathcal{F}}\int_{|x|>N}|f(x)|^{p}u(x)\,dx=0;$

  3. (c)

$\mathcal{F}$ is equicontinuous, that is,

$\lim_{h\rightarrow
0}\sup_{f\in\mathcal{F}}\int_{\mathbb{R}^{n}}|f(x+h)-f(x)|^{p}u(x)\,dx<\epsilon,$

then the family $\mathcal{F}$ has compact closure in $L^{p}(u)$.

Finally, it will be useful to regularize our operators, an idea that goes back
to Krantz and Li [16] (see also [1]). Given $\eta>0$, we let $K_{\eta}(x,y)$
be a smooth truncation of the $K(x,y)$ such that

  1. (a)

$K_{\eta}(x,y)=0$ if $|x-y|\leq\eta$;

  2. (b)

$K_{\eta}(x,y)=K(x,y)$ if $|x-y|>2\eta$;

  3. (c)

$K_{\eta}$ satisfies the same size and regularity estimates as $K$, namely

(2.4) $|K_{\eta}(x,y)|\leq\frac{C}{|x-y|^{n}}\quad\text{and}\quad|\nabla
K_{\eta}(x,y)|\leq\frac{C}{|x-y|^{n+1}}.$

We let $T^{\eta}$ be the operator associated to the kernel $K_{\eta}$. We also
note that $T^{\eta}$ is a bounded operator from $L^{p}(v)$ to $L^{p}(u)$
whenever $M$ and $T^{\sharp}$ are also bounded, for example if $(u,v)$ satisfy
(2.2) or (2.3). Indeed,

$|T^{\eta}f(x)|\leq\left|\int_{\eta<|x-y|\leq
2\eta}K_{\eta}(x,y)f(y)\,dy\right|+\left|\int_{2\eta<|x-y|}K(x,y)f(y)\,dy\right|\\\
\leq\int_{\eta<|x-y|\leq
2\eta}\frac{|f(y)|}{|x-y|^{n}}\,dy+T^{\sharp}f(x)\leq\frac{1}{\eta^{n}}\int_{|x-y|\leq
2\eta}|f(y)|\,dy+T^{\sharp}f(x)\\\ \leq CMf(x)+T^{\sharp}f(x).$

##  3\. Proof of Theorem 1

We will make several reductions to prove the compactness of $[b,T]$ from
$L^{p}(v)$ to $L^{p}(u)$. First we use the well-known fact that the space of
compact operators, $\mathcal{K}(L^{p}(v),L^{p}(u))$, is a closed subset of the
space of bounded operators, $\mathcal{B}(L^{p}(v),L^{p}(u))$, in the operator
norm topology. The precise bound for $[b,T]$ from [8] is that if $b\in BMO$
and $(u,v)$ satisfy 2.3, then

$\|[b,T]f\|_{L^{p}(u)}\leq C\|b\|_{BMO}\|f\|_{L^{p}(v)}$

and in particular

$\|[b,T]\|_{\mathcal{B}(L^{p}(v),L^{p}(u))}\leq C\|b\|_{BMO}.$

The commutator is linear in $b$ which implies

$\|[b_{1},T]-[b_{2},T]\|_{\mathcal{B}(L^{p}(v),L^{p}(u))}=\|[b_{1}-b_{2},T]\|_{\mathcal{B}(L^{p}(v),L^{p}(u))}\leq
C\|b_{1}-b_{2}\|_{BMO}.$

Thus if $b\in CMO$ and $\\{b_{j}\\}$ is a sequence of smooth functions
converging to $b$ in $BMO$, then $[b_{j},T]\rightarrow[b,T]$ in the operator
topology. It is this estimate that breaks down for the higher order iterated
commutators. Next, notice that if $b\in C_{c}^{\infty}(\mathbb{R}^{n})$ then
by using the mean value theorem on $b$ we have

$\displaystyle|[b,T]f(x)-[b,T^{\eta}]f(x)|$ $\displaystyle\leq
C\int_{|x-y|\leq 2\eta}|b(x)-b(y)|\frac{|f(y)|}{|x-y|^{n}}\,dy$
$\displaystyle\leq C\|\nabla b\|_{\infty}\int_{|x-y|\leq
2\eta}\frac{|f(y)|}{|x-y|^{n-1}}\,dy$ $\displaystyle\leq C\|\nabla
b\|_{\infty}\sum_{k=0}^{\infty}\int_{2^{-k}\eta<|x-y|\leq
2^{-k+1}\eta}\frac{|f(y)|}{|x-y|^{n-1}}\,dy$ $\displaystyle\leq\eta C\|\nabla
b\|_{\infty}\sum_{k=0}^{\infty}2^{-k}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1}\eta)}{|f(y)|}\,dy$
$\displaystyle\leq\eta C\|\nabla b\|_{\infty}Mf(x).$

This implies that

$\|[b,T^{\eta}]-[b,T]\|_{\mathcal{B}(L^{p}(v),L^{p}(u))}\leq\eta
C_{b}\|M\|_{\mathcal{B}(L^{p}(v),L^{p}(u))}$

and hence $[b,T^{\eta}]$ converges to $[b,T]$ as $\eta\rightarrow 0$ in the
operator topology. It suffices to show that for a fixed $\eta>0$ and a fixed
$b\in C_{c}^{\infty}(\mathbb{R}^{n})$ that $[b,T^{\eta}]$ is a compact
operator from $L^{p}(v)$ to $L^{p}(u)$. Fix $b\in
C_{c}^{\infty}(\mathbb{R}^{n})$ and $\eta>0$ and note that our estimates may
depend on $b$ and $\eta$. Consider the unit ball in $L^{p}(v)$,

$B_{L^{p}(v)}=\\{f\in L^{p}(v):\|f\|_{L^{p}(v)}\leq 1\\}.$

To show that $[b,T^{\eta}]$ is a compact operator we will show that
$\mathcal{F}=[b,T^{\eta}](B_{L^{p}(v)})$ satisfies (a), (b), and (c) of
Theorem 2, hence proving that $\mathcal{F}$ has compact closure in $L^{p}(u)$.

Since the operator $[b,T^{\eta}]$ is bounded from $L^{p}(v)$ to $L^{p}(u)$, we
have that

$\|[b,T^{\eta}]f\|_{L^{p}(u)}\leq C\|f\|_{L^{p}(v)}\leq C$

for $f\in B_{L^{p}(v)}$ and thus $\mathcal{F}$ is a bounded subset of
$L^{p}(u)$, i.e., it satisfies (a). To see that $\mathcal{F}$ satisfies the
uniform vanishing at infinity (b), let $f\in B_{L^{p}(v)}$ and $Q$ be a cube
containing $\mathsf{supp}\,b$. Choose $N>1$ such that

$N>2\sup\\{|y|:y\in Q\\}.$

For such $N$ and $|x|>N$ we have

$\displaystyle||[b,T^{\eta}]f(x)|$
$\displaystyle=\left|\int_{\mathbb{R}^{n}}(b(x)-b(y))K_{\eta}(x,y)f(y)\,dy\right|$
$\displaystyle\leq
C\|b\|_{\infty}\int_{\mathsf{supp}\,b}\frac{|f(y)|}{|x-y|^{n}}\,dy$
$\displaystyle\leq\frac{C\|b\|_{\infty}}{|x|^{n}}\int_{Q}|f(y)|\,dy$
$\displaystyle\leq\frac{C\|b\|_{\infty}}{|x|^{n}}\|f\|_{L^{p}(v)}\left(\int_{Q}v^{-\frac{p^{\prime}}{p}}\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq\frac{C_{b,v}}{|x|^{n}}.$

We note that

$\left(\int_{Q}v^{-\frac{p^{\prime}}{p}}\right)^{\frac{1}{p^{\prime}}}\leq
C|Q|^{\frac{1}{p^{\prime}}}\|v^{-\frac{1}{p}}\|_{L^{p^{\prime}}(\log
L)^{2p^{\prime}-1+\delta},Q}<\infty$

and hence we have

$|[b,T^{\eta}]f(x)|\leq\frac{C_{b,v}}{|x|^{n}}$

with a finite constant $C_{b,v}$ that depends on $b$ and $v$, but not $f$.
Integrating, over $|x|>N$ yields

$\int_{|x|>N}|[b,T^{\eta}]f(x)|^{p}u(x)\,dx\leq
C_{b,v}\int_{|x|>N}\frac{u(x)}{|x|^{np}}\,dx.$

Since $M(\chi_{[-1,1]^{n}})(x)\geq C(1+|x|)^{-n}$ and $M$ is bounded from
$L^{p}(v)$ to $L^{p}(u)$ we have that

$\int_{\mathbb{R}^{n}}\frac{u(x)}{(1+|x|)^{np}}\,dx\leq
C\int_{\mathbb{R}^{n}}M(\chi_{[-1,1]^{n}})(x)^{p}u(x)\,dx\leq
Cv([-1,1]^{n})<\infty.$

This implies that

$\lim_{N\rightarrow\infty}\int_{|x|>N}\frac{u(x)}{|x|^{np}}\,dx=0$

and thus the family $\mathcal{F}$ satisfies (b) of Theorem 2.

We now prove the most difficult of the estimates, the equicontinuity estimate,
(c), from Theorem 2. Let $h\in\mathbb{R}^{n}$ be sufficiently small to be
chosen later and $f\in B_{L^{p}(v)}$. Consider

$\displaystyle T^{\eta}f(x+h)-T^{\eta}f(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}(b(x+h)-b(y))K_{\eta}(x+h,y)f(y)\,dy$
$\displaystyle\qquad-\int_{\mathbb{R}^{n}}(b(x)-b(y))K_{\eta}(x,y)f(y)\,dy$
$\displaystyle=(b(x+h)-b(x))\int_{\mathbb{R}^{n}}K_{\eta}(x,y)f(y)\,dy$
$\displaystyle\qquad+\int_{\mathbb{R}^{n}}(b(y)-b(x+h))(K_{\eta}(x,y)-K_{\eta}(x+h,y))f(y)\,dy$
$\displaystyle=Af(x)+Bf(x),$

where we have added and subtracted the term $b(x+h)T^{\eta}f(x)$. The first
term is easy to handle with the smoothness of $b$:

$|Af(x)|\leq|h|\|\nabla b\|_{\infty}|T^{\eta}f(x)|\leq C|h|\|\nabla
b\|_{\infty}(Mf(x)+T^{\sharp}f(x)).$

Hence

$\sup_{\|f\|_{L^{p}(v)}\leq 1}\|Af\|_{L^{p}(u)}\leq C|h|$

which will go to zero uniformly as $h\rightarrow 0$. For the other term,
notice that if $|h|<\frac{\eta}{4}$ and $|x-y|<\frac{\eta}{2}$ then
$|x+h-y|<|x-y|+|h|<\eta$. Hence $K_{\eta}(x+h,y)=K_{\eta}(x,y)=0$ for such
$|h|<\frac{\eta}{4}$, and we have

$\displaystyle|Bf(x)|$
$\displaystyle=\left|\int_{|x-y|\geq\frac{\eta}{2}}(b(y)-b(x+h))(K_{\eta}(x,y)-K_{\eta}(x+h,y))f(y)\,dy\right|$
$\displaystyle\leq
2\|b\|_{\infty}\int_{|x-y|\geq\frac{\eta}{2}}|K_{\eta}(x+h,y)-K_{\eta}(x,y)||f(y)|\,dy$
$\displaystyle\leq
2C|h|\|b\|_{\infty}\int_{|x-y|\geq\frac{\eta}{2}}\frac{|f(y)|}{|x-y|^{n+1}}\,dy$

where we have used the well-known fact that the gradient estimate (2.4)
implies the standard kernel estimate

$|K_{\eta}(x+h,y)-K_{\eta}(x,y)|\leq C\frac{|h|}{|x-y|^{n+1}}$

when $|x-y|\geq 2|h|$. Notice $|x-y|\geq 2|h|$ is satisfied since
$|x-y|\geq\frac{\eta}{2}\geq 2|h|$. We now have

$\displaystyle\int_{|x-y|\geq\frac{\eta}{2}}\frac{|f(y)|}{|x-y|^{n+1}}\,dy$
$\displaystyle\leq\sum_{k=0}^{\infty}\int_{2^{k-1}\eta\leq|x-y|<2^{k}\eta}\frac{|f(y)|}{|x-y|^{n+1}}\,dy$
$\displaystyle\leq\frac{C}{\eta}\sum_{k=0}^{\infty}2^{-k}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{k}\eta)}|f(y)|\,dy$
$\displaystyle\leq\frac{C}{\eta}Mf(x).$

The boundedness of the maximal function implies

$\sup_{\|f\|_{L^{p}(v)}\leq 1}\|Bf\|_{L^{p}(u)}\leq C|h|.$

These estimates do not depend on $f$ and go to zero uniformly as $h\rightarrow
0$ hence we have

$\lim_{h\rightarrow\infty}\sup_{\|f\|_{L^{p}(v)}\leq
1}\|[b,T^{\eta}]f(\cdot+h)-[b,T^{\eta}]f\|_{L^{p}(u)}=0.$

This concludes our proof thus showing that $[b,T]$ is a compact operator.

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# Strangeness-Correlations on the pseudo-critical line in (2+1)-flavor QCD

D. Bollweg  Computational Science Initiative, Brookhaven National Laboratory,
Upton, New York 11973, USA  H.-T. Ding  Key Laboratory of Quark & Lepton
Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China  J. Goswami  RIKEN Center for Computational
Science, Kobe 650-0047, Japan  F. Karsch  Fakultät für Physik, Universität
Bielefeld, D-33615 Bielefeld, Germany  Swagato Mukherjee  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973, USA  P. Petreczky
Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
C. Schmidt  Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld,
Germany

###### Abstract

We present some lattice QCD results on first ($\chi_{1}^{i}$) and second
($\chi_{2}^{i}$) cumulants of and correlations ($\chi_{11}^{ij}$) among net
baryon-number ($B$), strangeness ($S$) and electric charge ($Q$) along the
pseudo-critical line ($T_{pc}(\mu_{B})$) in the temperature ($T$)–baryon
chemical potential ($\mu_{B}$) phase diagram of (2+1)-flavor QCD. We point out
that violations of the isospin symmetric limit of vanishing electric charge
chemical potential are small along the $T_{pc}(\mu_{B})$ for the entire range
of $\mu_{B}$ covered in the RHIC beam energy scan. For the strangeness neutral
matter produced in heavy-ion collisions this leads to a close relation between
$\chi_{11}^{BS}$ and $\chi_{11}^{QS}$. We compare lattice QCD results for
$\chi_{11}^{BS}/\chi_{2}^{S}$ along the $T_{pc}(\mu_{B})$ line with
preliminary experimental measurements of $\chi_{11}^{BS}/\chi_{2}^{S}$ for
collision energies $7.7~{}{\rm GeV}\leq\sqrt{s_{{}_{NN}}}\leq 62.4~{}{\rm
GeV}$. While we find good agreements for $\sqrt{s_{{}_{NN}}}\geq 39$ GeV,
differences are sizeable at smaller values of $\sqrt{s_{{}_{NN}}}$. Moreover,
we compare lattice QCD results for the ratio of the strangeness ($\mu_{S}$) to
baryon ($\mu_{B}$) chemical potentials, which define a strangeness neutral
system with fixed electric charge to baryon number density, with experimental
results obtained by the STAR collaboration for $\mu_{S}/\mu_{B}$ using strange
baryon yields on the freeze-out line. Finally, we determine the baryon
chemical potential at the freeze-out ($\mu_{B}^{f}$) by comparing
$\chi_{1}^{B}/\chi_{2}^{B}$ along the $T_{pc}(\mu_{B})$ with the
experimentally measured net-proton cumulants $\chi_{1}^{p}/\chi_{2}^{p}$. We
find that $\\{\mu_{B}^{f},T_{pc}(\mu_{B}^{f})\\}$ are consistent with the
freeze-out parameters of the statistical-model fits to experimentally measured
hadron yields for $\sqrt{s_{{}_{NN}}}\geq 11.5$ GeV.

###### pacs:

11.10.Wx, 11.15.Ha, 12.38.Gc, 12.38.Mh

##  I Introduction

Higher order cumulants of fluctuations of conserved charges and correlations
among different conserved charges of Quantum Chromodynamics (QCD) are
sensitive observables for the occurrence of phase transitions in strongly
interacting matter and reflect the change of relevant degrees of freedom
responsible for fluctuations and correlations in hot and dense strong-
interaction matter. In particular, correlations between conserved, net baryon
number and net strangeness number have been suggested as sensitive probes Koch
_et al._ (2005) for the change of degrees of freedom carrying strangeness at
low temperature (hadrons) and high temperature (quarks), respectively. The
temperature and density dependence of conserved charge fluctuations and their
higher order cumulants is studied in lattice QCD calculations using Taylor
series expansions as well as in numerical simulations with imaginary chemical
potentials Gavai and Gupta (2001); Allton  _et al._ (2002); D’Elia and
Lombardo (2003); Borsanyi  _et al._ (2012).

Experimentally the cumulants of conserved charge densities are not directly
accessible. E.g. net proton Adam  _et al._ (2021) or kaon number Afanasiev
_et al._ (2002); Abelev  _et al._ (2013); Adam  _et al._ (2020) fluctuations
and correlations among them are used as proxies for baryon or strangeness
number fluctuations. Incorporating strange baryon contributions also allows to
construct proxies for strangeness and baryon number as well as strangeness and
electric charge correlations Braun-Munzinger  _et al._ (2015); Adam  _et al._
(2020); Bellwied  _et al._ (2020).

The experimentally observed charge fluctuations and correlations are expected
to be generated at the so-called chemical freeze-out temperature $T_{f}$. Also
this temperature and the baryon as well as strangeness chemical potentials
($\mu_{B}^{f},\mu_{S}^{f})$ that control the thermal conditions at the time of
freeze-out are not directly accessible in heavy ion collisions. Eventually
they could be deduced through a comparison of experimental data with QCD
thermodynamics, if the medium is in equilibrium at the time of freeze-out. In
practice the experimental determination of freeze-out parameters involves a
model dependent step. Experimentally determined particle yields are compared
to hadronization models Andronic  _et al._ (2018); Adamczyk  _et al._ (2017);
Acharya  _et al._ (2023). The thus determined freeze-out temperature at
various values of the beam energy, $T_{f}(\sqrt{s_{{}_{NN}}})$, is found to be
in good agreement with the pseudo-critical temperature,
$T_{pc}(\hat{\mu}_{B})$, that is determined in lattice QCD calculations
Bazavov  _et al._ (2019); Borsanyi  _et al._ (2020). Generally this is taken
as support for the assumption that hadron resonance gas (HRG) models provide a
good approximation to the thermodynamics of strong-interaction matter at the
time of freeze-out.

Aside from studies of higher order cumulants of proxies for net baryon-number
fluctuations, the calculation of correlations between low order cumulants of
conserved charge densities, e.g. correlation of net strangeness and net
baryon-number, is being performed in the beam energy scan at RHIC. We
previously presented comparisons of higher order cumulants of net baryon-
number fluctuations with results obtained by the STAR collaboration Bazavov
_et al._ (2020, 2017a) for the related proton number cumulants. We also
compared results of the STAR collaboration on strange hadron yields at several
values of the beam energy Adam  _et al._ (2020) with lattice QCD results
Bazavov  _et al._ (2014); Bollweg  _et al._ (2021a) on correlations between
net baryon-number and strangeness. This provided evidence for the influence of
additional strange hadrons, not listed by the Particle Data Group as
established strange hadron resonances Workman  _et al._ (2022).

We previously also calculated low order cumulants of net baryon-number
fluctuations at non-zero values of the baryon chemical potential, using high
statistics datasets generated in (2+1)-flavor QCD calculations at finite
temperature Bollweg  _et al._ (2022, 2021b). In these calculations we focused
on an analysis of second order cumulants at vanishing values of the chemical
potentials as well as an analysis of the convergence of the Taylor series and
the comparison with Padé resummed results obtained from such series
expansions. We concluded that expansions of second order cumulants, in
particular the second order cumulant for net baryon-number fluctuations, are
well controlled for baryon chemical potentials $\mu_{B}/T\lesssim 1.5$.

We extend this analysis here by calculating correlations between conserved
charge densities as function of the baryon chemical potential on the pseudo-
critical line in (2+1)-flavor QCD using Taylor series for second order
cumulants up to NNLO order, i.e. up to ${\cal O}(\mu_{B}^{4})$. We will use
these results to compare with recent preliminary results of the STAR
collaboration obtained for correlations between net baryon number and
strangeness densities Feng .

The data sets used for our calculation and further details relevant for our
data analysis, have been described previously Bazavov  _et al._ (2020). In
particular, as discussed in Bazavov  _et al._ (2020), using fits of lattice
QCD results obtained at different values of the lattice spacing to rational
polynomials, we obtain continuum extrapolated estimates for the
$\hat{\mu}_{B}$-dependence of various second order cumulants on the pseudo-
critical line, $T_{pc}(\hat{\mu}_{B})$, as well as extrapolations performed
directly for ratios of second order cumulants.

This paper is organized as follows. In Section II we introduce basic formulae
used for the calculation of Taylor expansions of second order cumulants. In
Section III we present results for various second order cumulants obtained in
lattice QCD calculations as functions of $T$ and $\hat{\mu}_{B}$. In Section
IV we present results for these cumulants evaluated on the pseudo-critical
line $T_{pc}(\mu_{B})$ and compare them with recent results of the STAR
collaboration obtained during the beam energy runs BES-II at RHIC. We give our
conclusions in Section V.

##  II Cumulants of conserved charge densities

Cumulants of conserved charge fluctuations at vanishing chemical potentials
for the conserved charges of $(2+1)$-flavor QCD, i.e. net baryon-number ($B$),
electric charge ($Q$) and strangeness ($S$), are obtained from the QCD
partition function, $\mathcal{Z}(T,V,\vec{\mu})$, as derivatives with respect
to the associated chemical potentials $\vec{\mu}=(\mu_{B},\mu_{Q},\mu_{S})$,

$\chi_{ijk}^{BQS}(T,V)=\left.\frac{1}{VT^{3}}\frac{\partial\ln\mathcal{Z}(T,V,\vec{\mu})}{\partial\hat{\mu}_{B}^{i}\partial\hat{\mu}_{Q}^{j}\partial\hat{\mu}_{S}^{k}}\right|_{\vec{\mu}=0}\;,$
(1)

with $\hat{\mu}_{X}\equiv\mu_{X}/T$. These cumulants can be used to set up
Taylor series for cumulants at non-zero values of the chemical potentials.
Starting with the Taylor series for the pressure,

$\displaystyle\frac{P}{T^{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{VT^{3}}\ln\mathcal{Z}(T,V,\vec{\mu})=\sum_{i,j,k=0}^{\infty}\frac{\chi_{ijk}^{BQS}}{i!j!\,k!}\hat{\mu}_{B}^{i}\hat{\mu}_{Q}^{j}\hat{\mu}_{S}^{k}\;,$
(2)

we easily obtain series for second order cumulants by taking appropriate
derivatives with respect to the chemical potentials. In these Taylor series we
replace the chemical potentials $\hat{\mu}_{Q}$ and $\hat{\mu}_{S}$ by series
in terms of $\hat{\mu}_{B}$ only,

$\displaystyle\mu_{S}/\mu_{B}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}s_{2i+1}(T)\ \hat{\mu}_{B}^{2i+1}\;,$ (3)
$\displaystyle\mu_{Q}/\mu_{B}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}q_{2i+1}(T)\ \hat{\mu}_{B}^{2i+1}\;,$ (4)

where the coefficients $s_{i}$, $q_{i}$ are fixed through constraints on the
net baryon-number density111We use here and in the following dimensionless
number densities. I.e. densities $n_{X}$, $X=B,\ Q,\ S$, are given in units of
$T^{3}$., $n_{B}$, and the ratio of electric charge and strangeness densities,
$n_{Q}/n_{S}$. We demand strangeness neutrality, $n_{S}=0$, and a fixed ratio
of net electric charge and baryon number densities, $n_{Q}/n_{B}=0.4$, which
is consistent with experimental conditions realized in heavy ion collisions of
lead-lead or gold-gold nuclei. The resultant coefficients $s_{i}$, $q_{i}$ are
given in Bazavov  _et al._ (2017b) for $i=1,3,5$. With this we obtain for the
next-to-leading order (NLO) expansion of second order cumulants

$\displaystyle\chi_{11}^{BS}(T,\vec{\mu})$ $\displaystyle=$
$\displaystyle\chi_{11}^{BS}+\frac{\hat{\mu}_{B}^{2}}{2!}\big{[}\chi_{31}^{BS}+2s_{1}\chi_{22}^{BS}+s_{1}^{2}\chi_{13}^{BS}$
$\displaystyle+2q_{1}\chi_{211}^{BQS}+2q_{1}s_{1}\chi_{112}^{BQS}+q_{1}^{2}\chi_{121}^{BQS}\big{]}+{\cal
O}(\hat{\mu}_{B}^{4})\;,$ $\displaystyle\chi_{11}^{QS}(T,\vec{\mu})$
$\displaystyle=$
$\displaystyle\chi_{11}^{QS}+\frac{\hat{\mu}_{B}^{2}}{2!}\big{[}\chi_{211}^{BQS}+2s_{1}\chi_{112}^{BQS}+s_{1}^{2}\chi_{13}^{QS}$
$\displaystyle+2q_{1}\chi_{121}^{BQS}+2q_{1}s_{1}\chi_{22}^{QS}+q_{1}^{2}\chi_{31}^{QS}\big{]}+{\cal
O}(\hat{\mu}_{B}^{4})\;,$ $\displaystyle\chi_{2}^{S}(T,\vec{\mu})$
$\displaystyle=$
$\displaystyle\chi_{2}^{S}+\frac{\hat{\mu}_{B}^{2}}{2!}\big{[}\chi_{22}^{BS}+s_{1}^{2}\chi_{4}^{S}+2s_{1}\chi_{13}^{BS}$
$\displaystyle+2q_{1}\chi_{112}^{BQS}+2q_{1}s_{1}\chi_{13}^{QS}+q_{1}^{2}\chi_{22}^{QS}\big{]}+{\cal
O}(\hat{\mu}_{B}^{4})\;,$ $\displaystyle\chi_{2}^{B}(T,\vec{\mu})$
$\displaystyle=$
$\displaystyle\chi_{2}^{B}+\frac{\hat{\mu}_{B}^{2}}{2!}\big{[}\chi_{4}^{B}+s_{1}^{2}\chi_{22}^{BS}+2s_{1}\chi_{31}^{BS}$
$\displaystyle+2q_{1}\chi_{31}^{BQ}+2q_{1}s_{1}\chi_{211}^{BQS}+q_{1}^{2}\chi_{22}^{BQ}\big{]}+{\cal
O}(\hat{\mu}_{B}^{4})\;.$

Explicit expressions for the order ${\cal O}(\hat{\mu}_{B}^{4})$ (NNLO)
expansion coefficients, which we will use in the following, can be obtained
easily. The leading order $\mu_{S}/\mu_{B}$ is related with the
$\chi_{11}^{BS}/\chi_{2}^{S}$ in the following way,

$\frac{\mu_{S}}{\mu_{B}}=-\frac{\chi_{11}^{BS}(T,0)}{\chi_{2}^{S}(T,0)}-\frac{\chi_{11}^{QS}(T,0)}{\chi_{2}^{S}(T,0)}q_{1}+\mathcal{O}(\mu_{B}^{2})\,.$
(9)

Figure 1: Top: Correlation between net baryon-number and strangeness number
densities normalized to the second order cumulant of strangeness fluctuations
versus $T$ for several values of $\hat{\mu}_{B}$. Shown are results obtained
on lattices with temporal extent $N_{\tau}=8$ (filled symbols) and $12$ (open
symbols) from ${\cal O}(\hat{\mu}_{B}^{2})$ (left) and ${\cal
O}(\hat{\mu}_{B}^{4})$ (right) Taylor series for $\chi_{11}^{BS}$ and
$\chi_{2}^{S}$, respectively. The lattice QCD results are compared to HRG
model calculations using the QMHRG2020 list (colored lines). For
$\hat{\mu}_{B}=0$ also a result based on the PDG-HRG list is shown (black
line). The grey band shows the continuum extrapolated results for the second
order cumulant ratio at vanishing chemical potential Bollweg  _et al._
(2021b). The yellow band shows the location of the pseudo-critical line
$T_{pc}(\hat{\mu}_{B})$. Bottom: same as figures on the top, but for
correlation between net electric charge and strangeness densities normalized
to the second order cumulant of strangeness fluctuations.

In the case of degenerate up and down quark masses, which generally is used in
lattice QCD calculations, the cumulants $\chi_{11}^{BS}$, $\chi_{11}^{QS}$,
and $\chi_{2}^{S}$ are directly related to each other,

$2\frac{\chi_{11}^{QS}(T,\vec{\mu})}{\chi_{2}^{S}(T,\vec{\mu})}-\frac{\chi_{11}^{BS}(T,\vec{\mu})}{\chi_{2}^{S}(T,\vec{\mu})}=1+\frac{\Delta^{BQS}(T,\vec{\mu})}{\chi_{2}^{S}(T,\vec{\mu})}\;.$
(10)

For $\mu_{Q}\neq 0$, one finds,

$\displaystyle\Delta^{BQS}(T,\vec{\mu})$
$\displaystyle=q_{1}\frac{\hat{\mu}_{B}^{2}}{2}\Bigg{(}q_{1}(2\chi^{BQS}_{031}-\chi^{BQS}_{022}-\chi^{BQS}_{121})$
(11) $\displaystyle-
s_{1}(2\chi^{BQS}_{013}-4\chi^{BQS}_{022}+2\chi^{BQS}_{112})$
$\displaystyle-\chi^{BQS}_{103}+4\chi^{BQS}_{121}-2\chi^{BQS}_{202}-\chi^{BQS}_{301}\Bigg{)}$
$\displaystyle+{\cal O}(\hat{\mu}_{B}^{4})\;.$

At vanishing electric charge chemical potential $\mu_{Q}$, i.e. in the isospin
symmetric case with $n_{Q}/n_{B}=0.5$, one thus obtains the sum rule,

$\displaystyle\Delta^{BQS}(T,\vec{\mu})$ $\displaystyle=$ $\displaystyle
0\;\;,\;{\rm for}\;\vec{\mu}=(\mu_{B},0,\mu_{S})\;.$ (12)

We will compare results for the second order cumulants with HRG model
calculations based on the list of hadrons published in the tables of the
Particle Data group Workman  _et al._ (2022) (PDG-HRG) as well as the
QMHRG2020 list compiled by the HotQCD CollaborationBollweg  _et al._ (2021b,
c). In order to probe the validity of Eq. 12 and the isospin symmetry
violations induced by a non-vanishing electric charge chemical potential in
HRG model calculations, we introduced an isospin symmetric version of these
hadron lists. For this purpose we set masses for charged strange hadrons equal
to the masses of their neutral partners. This concerns 7 strange hadrons, for
which mass differences are listed in the PDG tables:
$K,K^{*},K_{2}^{*},\Sigma,\Sigma(1385),\Xi,\Xi(1530)$. In all these cases
isospin violations, arising from the mass differences between charged and
neutral strange hadrons, are below 1%. We note that setting the masses equal
leads only to small changes in the Boltzmann weights entering HRG model
calculations. In the vicinity of the pseudo-critical temperature
$T_{pc,0}=156.5(1.5)$ MeV Bazavov  _et al._ (2019) this amounts to about
$0.6$%. In general this small difference is negligible. However, when
comparing QCD and HRG model results for $\Delta^{BQS}$ as given in Eq. 11,
this small change leads to differences which are of the same magnitude as the
modifications induced by a non-vanishing electric charge chemical potential
itself.

##  III Second order cumulants as a function of temperature and baryon
chemical potential

In Bollweg  _et al._ (2021b) we presented continuum extrapolated results for
all second order cumulants at vanishing values of the chemical potentials. For
non-vanishing values of the chemical potentials Taylor series for the diagonal
second order net baryon-number cumulant, $\chi_{2}^{B}(T,\hat{\mu}_{B})$, have
been presented in Bazavov  _et al._ (2020). Some preliminary results on
strangeness correlations and fluctuations at non-zero $\hat{\mu}_{B}$ have
also been shown in Bollweg  _et al._ (2021a); Karsch (2017).

Here we focus on the behavior of off-diagonal, second order cumulants. We
concentrate on a discussion of correlations between net strangeness number
($n_{S}$) and net baryon-number ($n_{B}$) densities as well as $n_{S}$ and net
electric charge ($n_{Q}$) density. We shall consider thermal conditions that
are of relevance for comparisons with experimental conditions realized in
heavy ion collisions, i.e. we consider a thermal medium with vanishing net
strangeness, $n_{S}=0$, and $n_{Q}/n_{B}=0.4$.

In Fig. 1 we show the ratios
$-\chi_{11}^{BS}(T,\hat{\mu}_{B})/\chi_{2}^{S}(T,\hat{\mu}_{B})$ (top) and
$\chi_{11}^{QS}(T,\hat{\mu}_{B})/\chi_{2}^{S}(T,\hat{\mu}_{B})$ (bottom),
obtained from ${\cal O}(\hat{\mu}_{B}^{2})$ (left) and ${\cal
O}(\hat{\mu}_{B}^{4})$ (right) Taylor series, respectively. As can be seen
these expansions agree well for small $\hat{\mu}_{B}$. The second and fourth
order expansions start to differ at temperatures below the pseudo-critical
line for the chiral transition (yellow bands) at $\hat{\mu}_{B}\simeq 1.5$.

The lattice QCD results are compared to HRG model calculations that utilize
the PDG-HRG and QMHRG2020 lists. As noted already in the analysis of second
order cumulants at vanishing chemical potential Bollweg  _et al._ (2021b) HRG
model calculations based on the QMHRG2020 list describe the lattice QCD
results well at low temperatures up to $T_{pc}(\hat{\mu}_{B})$. HRG model
calculations based on the PDG-HRG list Workman  _et al._ (2022), however, are
insufficient already for $T\raise
1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}130$ MeV. While
QMHRG2020 and lattice QCD calculations agree well at $T_{pc,0}\equiv
T_{pc}(0)$, differences become larger with increasing $\hat{\mu}_{B}$ also on
the pseudo-critical line.

The pseudo-critical line has been determined in lattice QCD calculations as a
Taylor series up to ${\cal O}(\hat{\mu}_{B}^{4})$ Bonati  _et al._ (2015);
Bazavov  _et al._ (2019); Borsanyi  _et al._ (2020),

$T_{pc}(\mu_{B})=T_{pc,0}\left[1-\kappa_{2}\hat{\mu}_{B}^{2}+\kappa_{4}\hat{\mu}_{B}^{4}\right]\;.$
(13)

We use results from Bazavov  _et al._ (2019), i.e. $T_{pc,0}=(156.5\pm 1.5)$
MeV and $\kappa_{2}=0.012(4)$. Within errors this is in agreement with results
obtained in Bonati  _et al._ (2015); Borsanyi  _et al._ (2020). The ${\cal
O}(\hat{\mu}_{B}^{4})$ correction in Eq. 13 has been found to vanish within
current statistical errors Bazavov  _et al._ (2019); Borsanyi  _et al._
(2020). We leave out the contribution of the ${\cal O}(\hat{\mu}_{B}^{4})$
correction in the following. The pseudo-critical temperature
$T_{pc}(\hat{\mu}_{B})$ is shown in Fig. 1 as a yellow band.

As can be seen in Fig. 1 the magnitude of the ratio
$\chi_{11}^{BS}(T,\hat{\mu}_{B})/\chi_{2}^{S}(T,\hat{\mu}_{B})$ increases with
increasing $\hat{\mu}_{B}$. On the pseudo-critical line this amounts to only a
moderate change of about (15-20)% between $\hat{\mu}_{B}=0$ and 1.5.

In the previous section we pointed out that the sum rule,
$2\chi_{11}^{QS}-\chi_{11}^{BS}=\chi_{2}^{S}$, holds in QCD at any value of
the temperature for isospin symmetric matter, i.e. for $\mu_{Q}=0$. The
difference of the cumulant ratios $\chi_{11}^{BS}/\chi_{2}^{S}$ and
$2\chi_{11}^{QS}/\chi_{2}^{S}$ thus equals unity for all $\hat{\mu}_{Q}=0$,

$2\frac{\chi_{11}^{QS}(T,\vec{\mu})}{\chi_{2}^{S}(T,\vec{\mu})}-\frac{\chi_{11}^{BS}(T,\vec{\mu})}{\chi_{2}^{S}(T,\vec{\mu})}=1\;\;{\rm
for}\;\;\vec{\mu}=(\mu_{B},0,\mu_{S})\\\ \;.$ (14)

For $\hat{\mu}_{Q}\neq 0$, e.g. for $n_{Q}/n_{B}=0.4$, we find that deviations
from unity increase with increasing $\hat{\mu}_{B}$. Deviations are largest in
the vicinity of $T_{pc}(\hat{\mu}_{B})$. However, they stay below222Note that
isospin violations in QCD at non-zero temperature are indeed expected to be
negligible Pisarski and Wilczek (1984). Nonetheless large violations of
isospin symmetry have been reported recently to occur in kaon production in
high-energy nucleus-nucleus collisions Brylinski  _et al._ (2023); Adhikary
_et al._ (2023). 0.5% even for $\hat{\mu}_{B}\simeq 2$. This is evident from
the fourth order Taylor series results shown in Fig. 2. At $\hat{\mu}_{B}=2$
the pseudo-critical temperature is about $T_{pc}(\hat{\mu}_{B}=2)\simeq 150$
MeV. The corresponding baryon chemical potential thus is $\mu_{B}\simeq 300$
MeV, which corresponds to conditions reached in heavy ion collisions at beam
energies below $11.5$ GeV. We thus expect Eq. 14 to hold for correlations of
net baryon-number or electric charge with net strangeness number at the LHC
and also at RHIC for almost the entire range of beam energies covered in the
beam energy scan.

Figure 2: Deviations from the sum rule, Eq. 14, valid in isospin symmetric
matter for the relation between strangeness and baryon-number correlation on
the one hand, and strangeness and electric charge correlations on the other
hand. Shown are lattice QCD results using $4^{\rm th}$ order Taylor series
results for the second order cumulants. Also shown are HRG model calculations
using the QMHRG2020 list with additional isospin symmetrization for 7 strange
hadrons as discussed in Sec. II.

We finally also note that the sum of $\chi_{11}^{BS}$ and $\chi_{11}^{QS}$
gives the correlation between light and strange quark densities for any value
of the chemical potentials $\vec{\mu}$,

$\chi_{11}^{us}(T,\vec{\mu})=-\left(\chi_{11}^{BS}(T,\vec{\mu})+\chi_{11}^{QS}(T,\vec{\mu})\right)\;.$
(15)

Making use of the very well satisfied sum rule between the second order
strange cumulants (cf. Eq. 14) we thus find that $\chi_{11}^{BS}$ also
provides a good approximation for the correlation between light and strange
quark densities,

$\displaystyle\frac{\chi_{11}^{us}(T,\hat{\mu}_{B})}{\chi_{2}^{S}(T,\hat{\mu}_{B})}$
$\displaystyle=$
$\displaystyle-\left(\frac{\chi_{11}^{BS}(T,\hat{\mu}_{B})}{\chi_{2}^{S}(T,\hat{\mu}_{B})}+\frac{\chi_{11}^{QS}(T,\hat{\mu}_{B})}{\chi_{2}^{S}(T,\hat{\mu}_{B})}\right)$
(16) $\displaystyle\simeq$
$\displaystyle-\frac{1}{2}\Bigg{(}1+3\frac{\chi_{11}^{BS}(T,\hat{\mu}_{B})}{\chi_{2}^{S}(T,\hat{\mu}_{B})}\Bigg{)}\;.$
(17)

We show QCD results for the correlation between light and strange quark
densities on the pseudo-critical line in Fig. 3. We note that the magnitude of
the light and strange quark correlations decrease with increasing
$\hat{\mu}_{B}$ in HRG model calculations as well as in the QCD calculation.
This arises from the increasing importance of multiple strange baryon
contributions on the pseudo-critical line.

Figure 3: Correlation between light and strange quark-number densities on the
pseudo-critical line. Shown are continuum estimates for second and fourth
order Taylor expansion results for $-\chi_{11}^{us}/\chi_{2}^{S}$ on the
pseudo-critical line that have been obtained from fits to data obtained in
simulations on lattices with temporal extent $N_{\tau}=8$ and 12. These
results are compared with HRG model results based on the PDG-HRG and QMHRG2020
hadron lists, respectively.

##  IV Second order cumulants on the pseudo-critical line

###  IV.1 Pseudo-critical line and experimentally determined freeze-out
conditions

On the pseudo-critical line, $T_{pc}(\hat{\mu}_{B})$, the ratio of strangeness
and baryon chemical potentials obtained by demanding $n_{S}=0$ varies little
with increasing $\hat{\mu}_{B}$. Taylor expansion results for
$\mu_{S}/\mu_{B}$ are shown in Fig. 4 and are compared with experimental
results obtained by the STAR Collaboration by either fitting particle yields
for a large set of strange and non-strange hadrons to a HRG motivated
hadronization model Adamczyk  _et al._ (2017) or by fitting only yields of
strange baryons Adamczyk  _et al._ (2017); Adam  _et al._ (2020). Also shown
in this figure are values for $\mu_{S}/\mu_{B}$ obtained in HRG model
calculations based on the PDG-HRG and QMHRG2020 lists for hadron resonances.
It is apparent that the former does not describe the QCD results well, whereas
the QMHRG2020 provides a good description of $\mu_{S}/\mu_{B}$ on the pseudo-
critical line at least for $\hat{\mu}_{B}\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}1$. Beyond this
small deviations from the QCD results also show up. HRG model results based on
PDG-HRG and QMHRG2020 differ by about 20%. This is similar to the differences
found for the ratio $\chi_{11}^{BS}/\chi_{2}^{S}$. The good agreement between
QCD and HRG model calculations based on QMHRG2020 at $T_{pc}(\hat{\mu}_{B})$
in calculations of $\chi_{11}^{BS}/\chi_{2}^{S}$, along with the agreement
between determinations of the experimentally determined $\mu_{S}/\mu_{B}$ with
QCD results, thus suggests that the experimental results for $\mu_{S}/\mu_{B}$
are indeed sensitive to strange particle content at the time of freeze-out.

Figure 4: The ratio $\mu_{S}/\mu_{B}$ on the pseudo-critical line in
(2+1)-flavor QCD versus $\hat{\mu}_{B}$ obtained for strangeness neutral
matter with $n_{Q}/n_{B}=0.4$. The bands show continuum estimates for second
and fourth order Taylor expansion results of $\mu_{S}/\mu_{B}$ on the pseudo-
critical line that have been obtained from fits to data obtained in
simulations on lattices with temporal extent $N_{\tau}=8$ and 12. Also shown
are results obtained by the STAR Collaboration Aggarwal  _et al._ (2011);
Adamczyk  _et al._ (2017); Adam  _et al._ (2020) and HRG model calculations
using the PDG-HRG and QMHRG2020 particle lists, respectively.

We start our discussion of cumulant ratios at finite temperature and non-zero
values of the chemical potentials by revisiting the analysis of the ratio of
mean, $\chi_{1}^{B}(T,\hat{\mu}_{B})$, and variance,
$\chi_{2}^{B}(T,\hat{\mu}_{B})$, of net baryon-number density distributions

$\displaystyle R_{12}^{B}$ $\displaystyle=$
$\displaystyle\frac{\chi_{1}^{B}(T,\hat{\mu}_{B})}{\chi_{2}^{B}(T,\hat{\mu}_{B})}$
(18) $\displaystyle=$
$\displaystyle\hat{\mu}_{B}\bigg{(}1+s_{1}\frac{\chi_{11}^{BS}}{\chi_{2}^{B}}+q_{1}\frac{\chi_{11}^{QS}}{\chi_{2}^{B}}\bigg{)}+{\cal
O}(\hat{\mu}_{B}^{3})\;,$

where we again choose $n_{S}=0$, and $n_{Q}/n_{B}=0.4$.

The ratio $R_{12}^{B}(T,\hat{\mu}_{B})$ has been analyzed by us previously and
was shown in Bazavov  _et al._ (2020) in a $\hat{\mu}_{B}$-range relevant for
the analysis of higher order cumulants. In Fig. 5 we show
$R_{12}^{B}(T,\hat{\mu}_{B})$ in a larger range of $\hat{\mu}_{B}$ values. The
continuum estimates based on up to ${\cal O}(\hat{\mu}_{B}^{5})$ Taylor series
for the net baryon-number density, $\chi_{1}^{B}$, and a ${\cal
O}(\hat{\mu}_{B}^{4})$ Taylor series for the net baryon number fluctuations,
$\chi_{2}^{B}$ (NNLO [5,4]) shown in this figure are statistically well
controlled for $\hat{\mu}_{B}\leq 1.5$. For large values of the chemical
potential, $\hat{\mu}_{B}\leq 2$, we only show the result (NLO [3,2]) based on
${\cal O}(\hat{\mu}_{B}^{3})$ and ${\cal O}(\hat{\mu}_{B}^{2})$ Taylor series
for $\chi_{1}^{B}$ and $\chi_{2}^{B}$, respectively. These continuum estimates
are obtained from fits to data taken on lattices with temporal extent
$N_{\tau}=8$ and $12$. As can be seen, the ratio $R_{12}^{B}$ starts to
deviate from HRG model calculations at about $\hat{\mu}_{B}\simeq 1$. The
Taylor series, however, is still well controlled at least up to
$\hat{\mu}_{B}\simeq 1.5$. We compare $R_{12}^{B}(T,\hat{\mu}_{B})$,
calculated in (2+1)-flavor QCD, with the corresponding ratio of net proton-
number and its variance, $R_{12}^{p}(\sqrt{s_{{}_{NN}}})$, Abdallah  _et al._
(2021) as follows. In order to convert the experimental control parameter,
$\sqrt{s_{{}_{NN}}}$, to a chemical potential value on the freeze-out line we
use the set of freeze-out temperatures ($T_{f}$) and baryon chemical
potentials ($\mu_{B}^{f}$) determined from the analysis of particle yields
using fits based on the thermodynamics of a hadron gas in the Grand Canonical
Ensemble Adamczyk  _et al._ (2017); Abdallah  _et al._ (2021).

Figure 5: The ratio $R_{12}^{B}(T(\mu_{B}),\mu_{B})$ on the pseudo-critical
line in (2+1)-flavor QCD versus $\hat{\mu}_{B}$ obtained for strangeness
neutral matter with $n_{Q}/n_{B}=0.4$. The bands show continuum estimates for
next-to-leading (NLO [3,2]) and next-to-next-to-leading (NNLO [5,4]) order
Taylor expansion results on the pseudo-critical line that have been obtained
from fits to data obtained in simulations on lattices with temporal extent
$N_{\tau}=8$ and 12 (see also discussion in the text). Also shown are results
obtained by the STAR Collaboration for the corresponding proton cumulant
ratio, $R_{12}^{p}$ for various beam energies $\sqrt{s_{{}_{NN}}}$Abdallah
_et al._ (2021). Results from Adamczyk  _et al._ (2017) have been used to
convert $\sqrt{s_{{}_{NN}}}$ to thermal parameters
$(T_{f},\hat{\mu}_{B}^{f})$. Furthermore, we show recent results for
$R_{12}^{p}$ obtained by the STAR Collaboration Pandav . For data taken at two
new beam energies, $\sqrt{s_{{}_{NN}}}=9.2$ GeV and 17.3 GeV we used the
parametrization of interpolating curves for freeze-out parameters given in
Andronic  _et al._ (2018). Solid curves show results from HRG model
calculations using the PDG-HRG and QMHRG2020 particle lists, respectively.

As can be seen in the figure, the experimental results for
$R_{12}^{p}(T_{f},\mu_{B}^{f})$ are close to the line of values for
$R_{12}^{B}(T,\mu_{B})$ on the pseudo-critical line. We thus determined a set
of freeze-out parameters obtained by demanding
$R_{12}^{p}(\sqrt{s_{{}_{NN}}})=R_{12}^{B}(T_{pc}(\mu_{B}^{f}),\mu_{B}^{f})$.
In Table 1 we compare the thus determined set of freeze-out parameters,
$\\{\mu_{B}^{f},T_{pc}(\mu_{B}^{f})\\}$, with the experimental set of freeze-
out parameters, $\\{\mu_{B}^{f},T_{ch}\\}$, obtained by comparing measured
particle yields to hadronization models Andronic  _et al._ (2018); Adamczyk
_et al._ (2017); Acharya  _et al._ (2023).

As can be seen in Fig. 5, NLO and NNLO QCD results agree well with each other
up to $\hat{\mu}_{B}\simeq 1.5$, where the errors of the NNLO expansion start
getting large. In Tab. 1, we tabulate the freeze-out parameters determined
from $R_{12}^{p}$ using NLO QCD results for $R_{12}^{B}(T,\hat{\mu}_{B})$ and
they agree well with the freeze-out parameters $\\{T_{f},\hat{\mu}_{B}^{f}\\}$
obtained from particle yields down to $\sqrt{s_{{}_{NN}}}\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}17.3$ GeV, which
also corresponds to $\hat{\mu}_{B}\simeq 1.5$. For smaller
$\sqrt{s_{{}_{NN}}}$ or $\hat{\mu}_{B}\simeq 2.0$ the NNLO lattice QCD results
have too large errors for a detailed quantitative comparison. Results from the
NLO expansion are, however, still in good agreement with the STAR data down to
$\sqrt{s_{{}_{NN}}}\simeq 11.5$ GeV. In order to compare QCD results with STAR
data at even lower beam energies statistically well controlled higher order
Taylor series will be necessary.

It also should be noted that the experimentally determined $R_{12}^{p}$
becomes larger than unity for $\sqrt{s_{{}_{NN}}}\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}17.3$ GeV. This is
consistent with lattice QCD results for $R_{12}^{B}$, which become larger than
unity for $\hat{\mu}_{B}\raise
1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}1.3$ or
$\mu_{B}\raise 1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}200$
MeV. On the other hand, HRG model calculations based on non-interacting,
point-like hadrons will always lead to $R_{12}^{B}(T,\hat{\mu}_{B})<1$. The
experimental data thus seem to reflect interactions in strong-interaction
matter at freeze-out (on the pseudo-critical line) that go beyond those taken
care of in HRG models through the presence of a tower of excited states and
resonances.

We give a comparison of the chemical potentials and pseudo-critical
temperatures on the pseudo-critical line,
$\\{T_{pc}(\hat{\mu}_{B}),\mu_{B}\\}$, that are obtained by demanding
$R_{12}^{B}=R_{12}^{p}$ and the freeze-out parameters,
$\\{T_{f},\mu_{B}^{f}\\}$, obtained from HRG model fits to measured hadron
yields in Table 1

RHIC beam energy | QCD, pseudo-critical line: using $R_{12}^{p}$ | HIC, freeze-out line: using yields
---|---|---
$\sqrt{s_{{}_{NN}}}$ [GeV] |  $\mu_{B}$ [MeV] |  $\mu_{S}$ [MeV] |  $T_{pc}$ [MeV] |  $\mu_{B}$ [MeV] |  $\mu_{S}$ [MeV] |  $T_{ch}$ [MeV]
7.7 | — | — | — | 398.2(16.4) | 89.5(6.0) | 144.3(4.8)
9.2 | — | — | — | 358.3 | — | 144.4
11.5 | 297.9(12.4) | 73.5(4.8) | 148.9(2.9) | 287.3(12.5) | 64.5(4.7) | 149.4(5.2)
14.5 | 274.8(9.1) | 68.2(3.7) | 150.1(2.6) | 264 | — | 151.6
17.3 | 245.3(6.5) | 61.3(2.8) | 151.6(2.2) | 218.6 | — | 154.7
19.6 | 222.3(4.7) | 55.7(2.2) | 152.5(2.0) | 187.9(8.6) | 43.2(3.8) | 153.9(5.2)
27.0 | 170.5(2.1) | 42.9(1.3) | 154.2(1.6) | 144.4(7.2) | 33.5(3.6) | 155.0(5.1)
39.0 | 120.2(1.3) | 30.3(1.2) | 155.4(1.5) | 103.2(7.4) | 24.5(3.8) | 156.4(5.7)
54.4 | 89.3(0.9) | 22.5(0.6) | 155.9(1.5) | 83 | — | 160.0
62.4 | 81.5(0.8) | 20.5(0.5) | 156.0(1.5) | 69.2(5.6) | 16.7(3.3) | 160.3(4.9)
200 | 28.3(0.3) | 7.1(0.2) | 156.4(1.5) | 28.4(5.8) | 5.6(3.9) | 164.3(5.3)
Table 1: The left hand column lists the various RHIC beam energies,
$\sqrt{s_{{}_{NN}}}$, used by the STAR collaboration Abdallah  _et al._
(2021); Pandav  for their measurements of net proton-number cumulants and
$B-S$ correlations. The three columns in the middle give the chemical
potentials and the temperature values on the pseudo-critical line that
correspond to $R_{12}^{B}=R_{12}^{p}$. The last three columns show
corresponding freeze-out parameters obtained by fitting particle yields to a
hadronization model based on the grand canonical ensemble Adamczyk  _et al._
(2017). Results for $\sqrt{s_{{}_{NN}}}=14.5$ GeV and 54.4 GeV are taken from
Abdallah  _et al._ (2021). For $\sqrt{s_{{}_{NN}}}=9.2$ GeV and 17.3 GeV we
used the parametrization of interpolating curves for freeze-out parameters
given in Andronic  _et al._ (2018).

###  IV.2 Strangeness correlations on the pseudo-critical line

Figure 6: Left: Correlation between net strangeness and net baryon-number
densities normalized to the second order cumulant of strangeness fluctuations
on the pseudo-critical line. The bands show continuum estimates for second and
fourth order Taylor expansion results on the pseudo-critical line that have
been obtained from fits to data obtained in simulations on lattices with
temporal extent $N_{\tau}=8$ and 12. In the case of fourth order expansion we
show results only up to $\mu_{B}/T=1.5$. Right: same as left hand figure but
for correlations between net strangeness and net electric charge densities.
The data shown in the left hand figure use the preliminary results for
$\chi_{11}^{BS}/\chi_{2}^{S}$ obtained by the STAR Collaboration Feng . Data
points shown in the right hand figure are obtained from the STAR data making
use of the relation given in Eq. 10. Also shown in this figure is the result
obtained in Braun-Munzinger  _et al._ (2015) from an analysis of ALICE data on
strange particle yields (see also discussions in the text).

In Fig. 6 (left) we show results for the ratio $\chi_{11}^{BS}/\chi_{2}^{S}$
and compare with preliminary results for this ratio obtained by the STAR
Collaboration Feng . As can be seen the Taylor series for the ratio
$\chi_{11}^{BS}/\chi_{2}^{S}$ converges well at least for $\hat{\mu}_{B}\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}1.5$. Similar to
the observation made for other observables results in (2+1)-flavor QCD and HRG
model calculations based on the QMHRG2020 particle list also agree well in
this $\hat{\mu}_{B}$-range. The preliminary STAR results, however, agree well
with the QCD result only for $\sqrt{s_{{}_{NN}}}=62.4$ GeV and 39 GeV. For
smaller $\sqrt{s_{{}_{NN}}}$ differences are significant. We note that this
difference does not arise from a possible ambiguity in relating values of
$\sqrt{s_{{}_{NN}}}$ to values for $\hat{\mu}_{B}$ derived from experimental
data.

As discussed above correlations between electric charge and strangeness are
closely related to $\chi_{11}^{BS}$. On the pseudo-critical line and for
conditions met in heavy ion collisions ($n_{S}=0,n_{Q}/n_{B}=0.4$) violations
of the sum rule, Eq. 14, are smaller than 0.5% for $\hat{\mu}_{B}\leq 2$,
which covers almost the entire range of the RHIC BES-II energy range in
collider mode. In Fig. 6 (right) we show the ratio
$\chi_{11}^{QS}/\chi_{2}^{S}$ on the pseudo-critical line and also compare
with preliminary STAR results for $\chi_{11}^{BS}/\chi_{2}^{S}$ Feng  that
have been converted to $\chi_{11}^{QS}/\chi_{2}^{S}$ using Eq. 14.

Also shown in this figure is a result for $\chi_{11}^{QS}/\chi_{2}^{S}$
obtained in Braun-Munzinger  _et al._ (2015) from particle yields measured by
the ALICE collaboration. This result has been obtained using data on strange
particle yields, obtained by the ALICE Collaboration at the LHC Abelev  _et
al._ (2013, 2015). The second order cumulants $\chi_{2}^{S},\ \chi_{11}^{BS}$
and $\chi_{11}^{QS}$ have been constructed from measured particle yields
taking into account feed down corrections from $\phi$-mesons and neutral kaons
Braun-Munzinger  _et al._ (2015). The second order cumulants, constructed in
this way, obey the QCD sum rule, Eq. 14, except for contributions arising from
the feed down corrections. As pointed out in Braun-Munzinger  _et al._ (2015)
certain decay channels, contributing in particular to $\chi_{11}^{BS}$, are
not known experimentally. Moreover, it is difficult to take care of feed-down
corrections arising from decays of experimentally not well controlled higher
kaon resonances and additional strange hadrons. This suggests that an accurate
experimental determination of $\chi_{11}^{BS}$ will remain to be difficult
without achieving better control over contributions arising from additional
resonances and their decay channels.

##  V Conclusions

We pointed out that the determination of freeze-out parameters
$(T_{f},\mu_{B}^{f},\mu_{S}^{f})$ from particle yields in heavy ion collisions
is consistent with QCD parameters describing thermal conditions realized on
the pseudo-critical line for strangeness neutral matter with
$n_{Q}/n_{B}=0.4$. In particular, we find that at
($T_{f},\mu_{B}^{f},\mu_{S}^{f})$, corresponding to $\sqrt{s_{{}_{NN}}}\geq
11.5$ GeV, the new results of the STAR collaboration obtained for the ratio of
mean net proton-number and the net proton number fluctuations, $R_{12}^{p}$,
are in good agreement with $R_{12}^{B}$ calculated in lattice QCD on the
pseudo-critical line. Similarly we find that the ratio of strangeness and
baryon chemical potentials, obtained from strange baryon yields and QCD
calculations, respectively, are in good agreement. This suggests that the
values of thermal control parameters $(T,\mu_{B},\mu_{S})$, characterizing
strong-interaction matter created at a given $\sqrt{s_{{}_{NN}}}$, are well
understood at least for $\sqrt{s_{{}_{NN}}}\geq 11.5$ GeV.

Furthermore, we presented QCD results for the normalized baryon-number
strangeness correlation $\chi_{11}^{BS}/\chi_{2}^{S}$ as function of the
baryon chemical potential $\hat{\mu}_{B}$ on the pseudo-critical line
$T_{pc}(\hat{\mu}_{B})$. The results, based on fourth order Taylor series of
second order cumulants, are shown to be well controlled for $\hat{\mu}_{B}\leq
1.5$ or equivalently $\mu_{B}\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}200$ MeV. For
conditions realized in heavy ion collisions at the time of freeze-out this
corresponds to beam energies $\sqrt{s_{{}_{NN}}}\geq 17.3$ GeV. Although
results for $\chi_{11}^{BS}/\chi_{2}^{S}$ obtained in QCD calculations agree
well with experimental data for $\sqrt{s_{{}_{NN}}}=62.4$ GeV and 39 GeV, we
observe significant differences for smaller beam energies. We thus conclude
that there is some tension between current experimental results on strangeness
and baryon number correlations and those appearing in strong-interaction
matter on the pseudo-critical line as described by equilibrium QCD
thermodynamics. To verify or falsify that QCD thermodynamics can provide a
good description of second order cumulant ratios of fluctuations and
correlations of conserved charges, as measured in heavy ion collisions using
various proxies, the origin of these differences clearly needs to be analyzed
further.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) Proj. No. 315477589-TRR 211; and the PUNCH4NFDI
consortium supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) with project number 460248186 (PUNCH4NFDI). This material
is based upon work supported by The U.S. Department of Energy, Office of
Science, Office of Nuclear Physics through Contract Nos. DE-SC0012704, and
within the frameworks of Scientific Discovery through Advanced Computing
(SciDAC) award Fundamental Nuclear Physics at the Exascale and Beyond, and the
NSFC under grant No. 12325508 and the National Key Research and Development
Program of China under Grant No. 2022YFA1604900.

This research used awards of computer time provided by the U.S. Department of
Energy’s INCITE and ALCC programs at the Argonne and the Oak Ridge Leadership
Computing Facilities. The Argonne Leadership Computing Facility at Argonne
National Laboratory is supported by the Office of Science of the U.S. DOE
under Contract No. DE-AC02-06CH11357. The Oak Ridge Leadership Computing
Facility at the Oak Ridge National Laboratory is supported by the Office of
Science of the U.S. DOE under Contract No. DE-AC05-00OR22725. Computations for
this work were carried out in part on facilities of the USQCD Collaboration,
funded by the Office of Science of the U.S. Department of Energy.

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