# Calderón–Zygmund operators on RBMO

Evgueni Doubtsov  St. Petersburg Department of Steklov Mathematical Institute,
Fontanka 27, St. Petersburg 191023, Russia<EMAIL_ADDRESS>and  Andrei V.
Vasin  St. Petersburg Department of Steklov Mathematical Institute, Fontanka
27, St. Petersburg 191023, Russia<EMAIL_ADDRESS>
###### Abstract.

Let $\mu$ be an $n$-dimensional finite positive measure on $\mathbb{R}^{m}$.
We obtain a $T1$ condition sufficient for the boundedness of Calderón–Zygmund
operators on $\textrm{RBMO}(\mu)$, the regular BMO space of Tolsa.

###### Key words and phrases:

Calderón–Zygmund operator, non-doubling measure, regular BMO space

###### 2020 Mathematics Subject Classification:

Primary 42B20; Secondary 42B35

This research was supported by the Russian Science Foundation (grant No.
18-11-00053).

##  1\. Introduction

Given a positive Radon measure $\mu$ on $\mathbb{R}^{m}$, Tolsa [6] introduced
$\mathrm{RBMO}(\mu)$, the regular BMO space with respect to $\mu$. This space
is suitable for the non-doubling measures $\mu$ and it has genuine properties
of the classical space BMO such as the John–Nirenberg inequality. Moreover, it
is proved in [6] that a bounded on $L^{2}(\mu)$ Calderón–Zygmund operator maps
$L^{\infty}(\mu)$ into $\mathrm{RBMO}(\mu)$. Motivated by this result, we
consider Calderón–Zygmund operators on $\mathrm{RBMO}(\mu)$: we obtain a T1
condition sufficient for the boundedness of Calderón–Zygmund operators on
$\mathrm{RBMO}(\mu)$; see Section 1.4 for a precise formulation.

###  1.1. Cubes and $n$-dimensional measures

In what follows, a cube is a closed cube in $\mathbb{R}^{m}$ with sides
parallel to the axes and centered at a point of $\mathrm{supp\,}\mu$. For a
cube $Q$, let $\ell=\ell(Q)$ denote its side-length. Also the notation
$Q(x,\ell)$ is used to indicate explicitly the center $x$ of the cube under
consideration.

As in [6], we always assume that $\mu$ is an $n$-dimensional measure on
$\mathbb{R}^{m}$ for a real number $n$, $0<n\leq m$. By definition, it means
that

(1.1) $\mu(Q)\leq C\ell^{n}(Q)\ \textrm{for any cube\
}Q\subset\mathbb{R}^{m},\ \ell(Q)>0,$

with a universal constant $C>0$.

###  1.2. Calderón–Zygmund operators

Let $d(\cdot,\cdot)$ denote the standard distance between points of
$\mathbb{R}^{m}$. A Calderón–Zygmund kernel associated with an $n$-dimensional
measure $\mu$ on $\mathbb{R}^{m}$ is a measurable function $\mathcal{K}(x,y)$
on $\mathbb{R}^{m}\times\mathbb{R}^{m}\setminus\\{(x,x):x\in\mathbb{R}^{m}\\}$
satisfying the following conditions:

(1.2) $|\mathcal{K}(x,y)|\leq Cd^{-n}(x,y),$ (1.3)
$\displaystyle|\mathcal{K}(x_{1},y)-\mathcal{K}(x_{2},y)|+|\mathcal{K}(y,x_{1})$
$\displaystyle-\mathcal{K}(y,x_{2})|$ $\displaystyle\leq
C\frac{d^{\delta}(x_{1},x_{2})}{d^{n+\delta}(x_{1},y)},\quad
2d(x_{1},x_{2})\leq d(x_{1},y),$

and

(1.4) $\left|\int_{Q(x,R)\setminus Q(x,r)}\mathcal{K}(x,y)\,d\mu(x)\right|\leq
C,\quad 0<r<R,$

where $C>0$ is a universal constant and $\delta$, $0<\delta\leq 1$, is a
regularity constant specific to the kernel $\mathcal{K}$.

######  Remark 1.

Restrictions (1.2) and (1.3) are standard. Condition (1.4) is a more special
cancellation property.

The Calderón-Zygmund operator associated to the kernel $\mathcal{K}(x,y)$ and
the measure $\mu$ is defined as

$Tf(x)=\int_{\mathbb{R}^{m}}\mathcal{K}(x,y)f(y)\,d\mu(y)$

for $x\notin\textrm{supp}(f\mu)$. So, in the general setting, one introduces
the following truncated operators $T_{\varepsilon}$, $\varepsilon>0$:

$T_{\varepsilon}f(x)=\int_{\mathbb{R}^{m}\setminus
Q(x,\varepsilon)}\mathcal{K}(x,y)f(y)\,d\mu(y).$

The operator $T$ is said to be bounded on $L^{p}(\mu)$ if the operators
$T_{\varepsilon}$ are bounded on $L^{p}(\mu)$ uniformly in $\varepsilon>0$.

###  1.3. Regular BMO space

In this section, we give an equivalent definition of $\mathrm{RBMO}(\mu)$, the
regular BMO space introduced by Tolsa [6].

####  1.3.1. Coefficients $K(Q,R)$

Given two cubes $Q\subset R$ in $\mathbb{R}^{m}$, put

$K(Q,R)=1+\sum_{j=1}^{N_{Q,R}}\frac{\mu(2^{j}Q)}{\ell^{n}(2^{j}Q)},$

where $N_{Q,R}$ is the minimal integer $k$ such that
$\ell(2^{k}Q)\geq\ell(R)$. Clearly, $K(Q,R)\geq 1$. On the other hand,
$K(Q,R)$ is bounded above by $C\log(\ell(R)/\ell(Q))$ because $\mu$ is
$n$-dimensional.

####  1.3.2. Doubling cubes

######  Definition 1.

Let $\alpha>1$ and $\beta>\alpha^{n}$. A cube $Q$ is called
$(\alpha,\beta)$-doubling if

$\mu(\alpha Q)<\beta\mu(Q).$

Let $\mu$ be a Radon measure on $\mathbb{R}^{m}$ and $\alpha>1$. As indicated
in [6], it is known that for a sufficiently large $\beta=\beta(\alpha,n)$, for
$\mu$-almost all $x\in\mathbb{R}^{m}$ there is a sequence of
$(\alpha,\beta)$-doubling cubes $\\{Q_{k}\\}_{k=1}^{\infty}$ centered at $x$
and with $\ell(Q_{k})$ tending to $0$ as $k\to\infty$. Let $\beta_{n}$ denote
two times the infimum of the corresponding constants $\beta(4,n)$.

######  Definition 2.

A cube $Q\subset\mathbb{R}^{m}$ is called doubling if $Q$ is
$(4,\beta_{n})$-doubling.

######  Remark 2.

The original definition of a doubling cube and further results in [6] are
given for $\alpha=2$ and under assumption $1\leq n\leq m$. Nevertheless, it is
known that the results of Tolsa [6] are extendable to larger values of
$\alpha$ and for $0<n\leq m$; see, for example, [4] for a generalization of
this theory for a wide class of measures on appropriate metric spaces. So, in
what follows, we use the above definition with $\alpha=4$ and still refer to
original results of Tolsa [6].

####  1.3.3. Definition of $\mathrm{RBMO}$

######  Definition 3.

The space $\mathrm{RBMO}(\mu)$ consists of those $f\in L^{1}_{loc}(\mu)$ for
which there exists a constant $C_{\mathfrak{E}}>0$ and a collection of
constants $\\{f_{Q}\\}$ (one constant for each doubling cube
$Q\subset\mathbb{R}^{m}$) such that

(1.5) $\frac{1}{\mu(Q)}\int_{Q}|f-f_{Q}|\,d\mu\leq C_{\mathfrak{E}}$

and

(1.6) $|f_{Q}-f_{R}|\leq C_{\mathfrak{E}}K(Q,R)$

for all doubling cubes $Q$, $R$, $Q\subset R$. Let
$\|f\|=\|f\|_{\mathfrak{E}}$ denote the infimum of the corresponding constants
$C_{\mathfrak{E}}>0$.

Standard arguments guarantee that $\|\cdot\|$ is a norm on the space
$\mathrm{RBMO}(\mu)$ modulo constants.

###  1.4. Main theorem

Suppose that a Calderón-Zygmund operator $T$ is bounded on $L^{2}(\mu)$. Then,
as mentioned above, $T$ maps $L^{\infty}(\mu)$ boundedly into
$\mathrm{RBMO}(\mu)$. Moreover, in the classical situation of homogeneous
metric spaces and under additional assumption $T1=0$, the operator $T$ is
known to be bounded on $\mathrm{BMO}$ type spaces; see, for example, [5, Ch.
4, Sect. 4]. In the present paper, we obtain a $T1$ condition sufficient for
the boundedness of $T$ on $\mathrm{RBMO}(\mu)$.

Given a cube $Q\subset\mathbb{R}^{m}$, put

$K(Q)=K(Q,2^{k}Q),$

where $k$ is the smallest positive integer such that
$\mu(2^{k}Q)>\frac{1}{2}\mu(\mathbb{R}^{m})$.

######  Theorem 1.1.

Let $\mu$ be a finite positive $n$-dimensional measure on $\mathbb{R}^{m}$.
Let $T$ be a Calderón–Zygmund operator bounded on $L^{2}(\mu)$. Assume that
for each doubling cube $Q\subset\mathbb{R}^{m}$, there exists a constant
$b_{Q}$ such that

(1.7)
$\frac{1}{\mu(Q)}\int_{Q}|T1-b_{Q}|\,d\mu\leq\frac{C}{K(Q)}\quad\textrm{for
all doubling cubes}\ Q$

and

(1.8) $|b_{Q}-b_{R}|\leq C\frac{K(Q,R)}{K(Q)}\quad\textrm{for all doubling
cubes}\ Q,R,\ Q\subset R,$

where the constant $C>0$ does not depend on $Q$ and $R$. Then $T$ is bounded
on $\mathrm{RBMO}(\mu)$.

######  Remark 3.

We say that $T$ is bounded on $RBMO(\mu)$ if the operators $T_{\varepsilon}$,
$\varepsilon>0$, are uniformly bounded on $RBMO(\mu)$. Also, (1.7) and (1.8)
similarly mean that these estimates hold for $T_{\varepsilon}$ uniformly in
$\varepsilon>0$.

######  Remark 4.

We implicitly assume in Theorem 1.1 that $T1\in L^{\infty}(\mu)$. Indeed, this
property follows from (1.4); see Lemma 2.5. If $T1$ is a constant, then
Theorem 1.1 clearly guarantees that $T$ is bounded on $\mathrm{RBMO}(\mu)$.

######  Remark 5.

Property (1.7) is, in a sense, similar to the oscillation condition used in
[1], where a $T1$ theorem for $\textrm{BMO}_{H}$ in the Hermite-Calderón-
Zygmund setting is obtained.

The final step in the proof of Theorem 1.1 uses $\|\cdot\|$ only; however, the
following semi-norm on $\mathrm{RBMO}(\mu)$ is crucial for certain auxiliary
results.

######  Definition 4.

Let $f\in L^{1}_{loc}(\mu)$. Fix a constant $\rho>1$. Let
$\|f\|_{\mathfrak{A},\rho}$ denote the infimum of the constants
$C_{\mathfrak{A}}=C_{\mathfrak{A},\rho}>0$ with the following properties: for
each cube $Q$, there exists $f_{Q}\in\mathbb{R}$ such that

(1.9) $\displaystyle\sup_{Q}\frac{1}{\mu(\rho
Q)}\int_{Q}|f(x)-f_{Q}|\,d\mu(x)$ $\displaystyle\leq C_{\mathfrak{A}},$ (1.10)
$\displaystyle|f_{Q}-f_{R}|$ $\displaystyle\leq
C_{\mathfrak{A}}K(Q,R)\quad\text{for any two cubes\ }Q\subset R.$

######  Remark 6.

The original semi-norm on $\mathrm{RBMO}(\mu)$ from [6] is different from
those introduced in Definitions 3 and 4. Nevertheless, all these semi-norms
are equivalent; see Section 2 for further details.

###  1.5. Notation

As usual, the symbol $C$ denotes an absolute constant whose value can vary
from line to line. Notation $C_{\mathfrak{A}}$, $C_{\mathfrak{B}}$, etc. is
used in certain specific situations.

###  1.6. Organization of the paper

Auxiliary results are presented in Section 2. Section 3 is devoted to
estimates related to the main technical decomposition of functions from
$\mathrm{RBMO}$; equivalence of Definitions 3 and 4 is essential on this step.
The proof of Theorem 1.1 is given in Section 4.

##  2\. Auxiliary results

###  2.1. Equivalent definitions of $\mathrm{RBMO}$

As mentioned in the introduction, Definition 3 is not the original one for
$\mathrm{RBMO}(\mu)$ in [6]. In the present section, we show that Definition 3
and the definitions of the regular BMO from [6] are equivalent.

Firstly, recall several notions introduced by Tolsa [6]. Let $\rho>1$ and
$f\in L^{1}_{loc}(\mu)$. Given a cube $Q\subset\mathbb{R}^{m}$, let
$\left<f\right>_{Q}$ denote the standard $\mu$-average of $f$ over $Q$, that
is,

$\left<f\right>_{Q}=\frac{1}{\mu(Q)}\int_{Q}f\,d\mu.$

$\bullet$ Let $\|f\|_{\mathfrak{B},\rho}$ denote the infimum of the constants
$C_{\mathfrak{B}}=C_{\mathfrak{B},\rho}>0$ with the following properties:

$\displaystyle\frac{1}{\mu(\rho Q)}\int_{Q}|f-\langle
f\rangle_{\widetilde{Q}}|d\mu$ $\displaystyle\leq C_{\mathfrak{B}}\ \text{for
any cube $Q$ (centered at some point of $\textrm{supp}(\mu)$)},$
$\displaystyle|\langle f\rangle_{Q}-\langle f\rangle_{R}|$ $\displaystyle\leq
C_{\mathfrak{B}}K(Q,R)\ \text{for any two doubling cubes\ }Q\subset R,$

where $\widetilde{Q}$ denotes the smallest doubling cube in the sequence
$Q,4Q,4^{2}Q,\dots$.

######  Remark 7.

The original definition of $\widetilde{Q}$ in [6] uses the sequence
$Q,2Q,2^{2}Q,\dots$. The present definition of $\widetilde{Q}$ is based on
$\alpha=4$; related details are given in Definition 2 and Remark 2. See also
[4] for similar definitions with $\alpha>1$.

$\bullet$ Let $\|f\|_{\mathfrak{C},\rho}$ denote the infimum of the constants
$C_{\mathfrak{C}}=C_{\mathfrak{C},\rho}>0$ with the following properties: for
any cube Q

$\displaystyle\int_{Q}|f-\langle f\rangle_{Q}|\,d\mu$ $\displaystyle\leq
C_{\mathfrak{C}}\mu(\rho Q),$ $\displaystyle|\langle f\rangle_{Q}-\langle
f\rangle_{R}|$ $\displaystyle\leq C_{\mathfrak{C}}K(Q,R)\left(\frac{\mu(\rho
Q)}{\mu(Q)}+\frac{\mu(\rho R)}{\mu(R)}\right)\quad\textrm{for any two cubes\
}Q\subset R.$

$\bullet$ Let $\|f\|_{\mathfrak{D}}$ denote the infimum of the constants
$C_{\mathfrak{D}}>0$ with the following properties:

(2.1) $\int_{Q}|f-\langle f\rangle_{Q}|\,d\mu\leq
C_{\mathfrak{D}}\mu(Q)\quad\textrm{for any doubling cube\ }Q$

and

(2.2) $|\langle f\rangle_{Q}-\langle f\rangle_{R}|\leq
C_{\mathfrak{D}}K(Q,R)\quad\textrm{for any two doubling cubes\ }Q\subset R.$

The property $\|f\|_{\mathfrak{B},\rho}<\infty$ is used to define the regular
BMO space in [6]. By [6, Lemma 2.6], the norms $\|\cdot\|_{\mathfrak{A},\rho}$
are equivalent for different $\rho>1$; by [6, Lemma 2.8],
$\|\cdot\|_{\mathfrak{B},\rho}$ and $\|\cdot\|_{\mathfrak{A},\rho}$ are
equivalent. By [6, Lemma 2.10], $\|\cdot\|_{\mathfrak{C},\rho}$ and
$\|\cdot\|_{\mathfrak{D}}$ are equivalent to $\|\cdot\|_{\mathfrak{B},\rho}$.

Standard arguments show that Definition 3 and the property
$\|f\|_{\mathfrak{D}}<\infty$ define the same space, with equivalent norms.
Indeed, if $\|f\|_{\mathfrak{D}}<\infty$, then trivially
$f\in\mathrm{RBMO}(\mu)$ with $C_{\mathfrak{E}}=\|f\|_{\mathfrak{D}}$. Now,
assume that $f\in\mathrm{RBMO}(\mu)$. Property (1.5) guarantees that

$|\left<f\right>_{Q}-f_{Q}|\leq C_{\mathfrak{E}}$

for any doubling $Q$. Hence, (1.5) implies (2.1) with
$C_{\mathfrak{D}}=2C_{\mathfrak{E}}$; (1.6) implies (2.2) with
$C_{\mathfrak{D}}=3C_{\mathfrak{E}}$. Therefore, $\|\cdot\|_{\mathfrak{D}}$
and $\|\cdot\|_{\mathfrak{E}}$ are equivalent.

In the arguments related to the proof of Theorem 1.1, we will use
$\|\cdot\|_{\mathfrak{A},\rho}$. Thus, for further reference, we separately
formulate a particular conclusion from the above arguments as the following
lemma.

######  Lemma 2.1.

Given a constant $\rho>1$, $\|\cdot\|_{\mathfrak{A},\rho}$ is an equivalent
norm on the space $\mathrm{RBMO}(\mu)$ modulo constants.

###  2.2. John–Nirenberg inequality for $\mathrm{RBMO}$

Tolsa [6] proved the following version of the John–Nirenberg inequality for
$\mathrm{RBMO}$.

######  Theorem 2.2 (see [6, Theorem 3.1]).

Let $f\in\mathrm{RBMO}(\mu)$, $\rho>1$, and let $\\{b_{Q}\\}_{Q}$ be a
collection of numbers satisfying

$\displaystyle\sup_{Q}\frac{1}{\mu(\rho Q)}\int_{Q}|f(x)-b_{Q}|\,d\mu(x)$
$\displaystyle\leq C\|f\|_{\mathfrak{B},\rho}$ $\displaystyle|b_{Q}-b_{R}|$
$\displaystyle\leq CK(Q,R)\|f\|_{\mathfrak{B},\rho}$

for any two cubes $Q\subset R$, with an absolute constant $C>0$. Then for any
cube Q and any $\lambda>0$, we have

$\mu\\{x\in Q:|f(x)-b_{Q}|>\lambda\\}\leq C_{J\\!N}\mu(\rho
Q)\exp\left(\frac{-c_{J\\!N}\lambda}{\|f\|_{\mathfrak{B},\rho}}\right)$

with $C_{J\\!N},c_{J\\!N}>0$ depending on $\rho$.

We will need the following corollary related to Definition 4.

######  Lemma 2.3.

Let $1<p<\infty$ and $f\in\mathrm{RBMO}(\mu)$, $\rho>1$. Let $\\{f_{Q}\\}_{Q}$
be such numbers that (1.9) and (1.10) hold with
$C_{\mathfrak{A}}=2\|f\|_{\mathfrak{A},\rho}$. Then

$\left(\frac{1}{\mu(\rho Q)}\int_{Q}|f-f_{Q}|^{p}\right)^{\frac{1}{p}}\leq
C\|f\|$

for any cube $Q\subset\mathbb{R}^{m}$, where
$C=C(C_{J\\!N},c_{J\\!N},p,\rho)>0$.

###### Proof.

A standard argument is applicable. Indeed, by Theorem 2.2 and Lemma 2.1,

$\displaystyle\frac{1}{\mu(\rho Q)}\int_{Q}|f-f_{Q}|^{p}\,d\mu$
$\displaystyle=\frac{1}{\mu(\rho
Q)}\int_{0}^{\infty}p\lambda^{p-1}\mu\\{x:|f(x)-f_{Q}|>\lambda\\}\,d\lambda$
$\displaystyle\leq
C_{J\\!N}\int_{0}^{\infty}p\lambda^{p-1}\exp\left(\frac{-c_{J\\!N}\lambda}{\|f\|_{\mathfrak{A},\rho}}\right)\,d\lambda$
$\displaystyle\leq C\|f\|_{\mathfrak{A},\rho}^{p}$ $\displaystyle\leq
C\|f\|^{p},$

as required. ∎

###  2.3. Properties of $\mathrm{RBMO}$

######  Lemma 2.4.

Let $\mu$ be a finite $n$-dimensional measure, $f\in\mathrm{RBMO}(\mu)$,
$\rho>1$, and let $\\{f_{Q}\\}_{Q}$ be numbers such that (1.9) and (1.10) hold
with $C_{\mathfrak{A}}=2\|f\|_{\mathfrak{A},\rho}<\infty$. Then

$|f_{Q}|\leq C_{f}K(Q)\quad\textrm{for all cubes}\ Q\subset\mathbb{R}^{m},$

where the number $C_{f}>0$ does not depend on $Q$.

###### Proof.

Without loss of generality, we assume that $\mu(\mathbb{R}^{m})=1$. Fix a cube
$Q_{1}$ such that $\ell(Q_{1})\geq 1$ and $\mu(Q_{1})>0$. Put

$C_{1}(f)=|f_{Q_{1}}|.$

Firstly, let $Q_{2}$ be a cube such that $\mu(Q_{2})\geq\frac{1}{2}$. Select a
cube $Q_{3}$ such that $Q_{3}\supset Q_{1}\cup Q_{2}$. By the choice of
$f_{Q_{j}}$, $j=1,2,3$,

$\displaystyle|f_{Q_{2}}|$
$\displaystyle\leq|f_{Q_{2}}-f_{Q_{3}}|+|f_{Q_{3}}-f_{Q_{1}}|+|f_{Q_{1}}|$
$\displaystyle\leq
2K(Q_{2},Q_{3})\|f\|_{\mathfrak{A},\rho}+2K(Q_{1},Q_{3})\|f\|_{\mathfrak{A},\rho}+C_{1}(f).$

Since $\mu$ is $n$-dimensional, we have $\ell(Q_{2})\geq\varkappa>0$. Hence,

$K(Q_{m},Q_{3})\leq 1+\sum_{j\geq
1}\frac{\mu(2^{j}Q_{m})}{\ell^{n}(2^{j}Q_{m})}\leq 1+\sum_{j\geq
1}\frac{1}{\ell^{n}(2^{j}Q_{m})}\leq C$

for $m=1,2$. Therefore,

$|f_{Q_{2}}|\leq C\|f\|_{\mathfrak{A},\rho}+C_{1}(f):=C_{0}(f).$

Now, consider a cube $Q\subset\mathbb{R}^{m}$ such that $\mu(Q)<\frac{1}{2}$.
Let $k$ be the smallest positive integer such that
$\mu(2^{k}Q)\geq\frac{1}{2}$. Since the cube $2^{k}Q$ has the properties of
$Q_{2}$, we obtain

$\displaystyle|f_{Q}|$ $\displaystyle\leq|f_{Q}-f_{2^{k}Q}|+|f_{2^{k}Q}|$
$\displaystyle\leq 2K(Q,2^{k}Q)\|f\|_{\mathfrak{A},\rho}+C_{0}(f)$
$\displaystyle\leq C_{f}K(Q),$

as required. ∎

###  2.4. Estimates of $T1$

######  Lemma 2.5.

Let $\mu$ be a finite positive measure on $\mathbb{R}^{m}$. Let $T$ be a
Calderón–Zygmund operator. Then $T1\in L^{\infty}(\mu)$.

###### Proof.

Since $\mu$ is a finite measure, we have

$\displaystyle|T_{\varepsilon}1(x)|$
$\displaystyle\leq\int_{\mathbb{R}^{m}\setminus
Q(x,2)}\left|\mathcal{K}(x,y)\right|\,d\mu(y)+\left|\int_{Q(x,2)\setminus
Q(x,\varepsilon)}\mathcal{K}(x,y)\,d\mu(y)\right|$ $\displaystyle\leq
C\mu(\mathbb{R}^{m})+C\leq C$

by (1.2) and (1.4). ∎

######  Lemma 2.6.

Let the assumptions of Theorem 1.1(ii) hold. Then

(2.3)
$\frac{1}{\mu(Q)}\int_{Q}|T1-\left<T1\right>_{Q}|\,d\mu\leq\frac{C}{K(Q)}\quad\textrm{for
any doubling cube\ }Q$

and

(2.4) $|\left<T1\right>_{Q}-\left<T1\right>_{R}|\leq
C\frac{K(Q,R)}{K(R)}\quad\textrm{for any two doubling cubes\ }Q\subset R,$

where the constant $C>0$ does not depend on $Q$ and $R$.

###### Proof.

Put $h=T1$. Firstly, let $Q\subset\mathbb{R}^{m}$ be a doubling cube. The
following standard arguments guarantee that (1.7) implies (2.3):

$\displaystyle\int_{Q}|h-\left<h\right>_{Q}|\,d\mu$
$\displaystyle\leq\int_{Q}|h-b_{Q}|\,d\mu+\int_{Q}\left|b_{Q}-\frac{1}{\mu(Q)}\int_{Q}h\,d\mu\right|\,d\mu$
$\displaystyle\leq\frac{C\mu(Q)}{K(Q)}+\frac{1}{\mu(Q)}\int_{Q}\int_{Q}|h-b_{Q}|\,d\mu\,d\mu$
$\displaystyle\leq\frac{C\mu(Q)}{K(Q)}.$

Secondly, for any doubling cube $R\supset Q$, we have

$|b_{R}-\left<h\right>_{R}|\leq\frac{1}{\mu(R)}\int_{R}|h-b_{R}|\,d\mu\leq\frac{C}{K(R)}\leq\frac{CK(Q,R)}{K(R)}$

by (1.7). Clearly, $K(R)\leq CK(Q)$ for $Q\subset R$. Therefore,

$\displaystyle|\left<h\right>_{Q}-\left<h\right>_{R}|$
$\displaystyle\leq|\left<h\right>_{Q}-b_{Q}|+|b_{Q}-b_{R}|+|b_{R}-\left<h\right>_{R}|$
$\displaystyle\leq\frac{CK(Q,R)}{K(R)}$

by (1.8). The proof of the lemma is finished. ∎

##  3\. Main construction

Let $T$ be a Calderón-Zygmund operator bounded on $L^{2}(\mu)$. Let
$f\in\mathrm{RBMO}(\mu)$, $\rho=2$. Using Definition 4, for each cube
$Q\subset\mathbb{R}^{m}$, select a number $f_{2Q}$ such that (1.5) and (1.6)
hold with $C_{\mathfrak{A},\rho}=2\|f\|_{\mathfrak{A},\rho}$ and with $2Q$ in
the place of $Q$. In particular, the assumptions of Lemma 2.3 are satisfied.
Also, to explain further arguments and estimates, it is worth mentioning that
$2Q$ is not necessarily doubling even if $Q$ is a doubling cube.

In the present section, we give estimates related to the following functions:

$\displaystyle f_{1}=f_{1,Q}$ $\displaystyle=f_{2Q},$ $\displaystyle
f_{2}=f_{2,Q}$ $\displaystyle=(f-f_{2Q})\chi_{2Q},$ $\displaystyle
f_{3}=f_{3,Q}$ $\displaystyle=(f-f_{2Q})\chi_{\mathbb{R}^{m}\setminus 2Q}.$

Observe that

$f=f_{1}+f_{2}+f_{3}.$

This decomposition ascends to [3]; see also [2].

Let $b_{2,Q}=0$ and

$b_{3,Q}=\frac{1}{\mu(Q)}\int_{Q}Tf_{3,Q}(y)\,d\mu(y).$

In the following lemma, we assume that $Q$ is a doubling cube.

######  Lemma 3.1.

There exists a constant $C>0$ such that

$\frac{1}{\mu(Q)}\int_{Q}|Tf_{k}-b_{k,Q}|\,d\mu\leq C\|f\|,\quad k=2,3,$

for any doubling cube $Q$.

###### Proof.

Put

$I_{k}=\frac{1}{\mu(Q)}\int_{Q}|Tf_{k}-b_{k,Q}|\,d\mu,\quad k=2,3.$

For $k=2$, we have

$I_{2}\leq\left(\frac{1}{\mu(Q)}\int_{Q}|Tf_{2}|^{2}\,d\mu\right)^{\frac{1}{2}}$

by Hölder’s inequality. Since $T$ is bounded on $L^{2}(\mu)$, the definition
of $f_{2}$ guarantees that

$\displaystyle I_{2}\leq
C\left(\frac{1}{\mu(Q)}\int_{2Q}|f-f_{2Q}|^{2}\,d\mu\right)^{\frac{1}{2}}.$

Using the doubling property of $Q$ and Lemma 2.3 with $\rho=p=2$ and $2Q$ in
the place of $Q$, we obtain

$\displaystyle I_{2}$ $\displaystyle\leq
C\left(\frac{\mu(4Q)}{\mu(Q)}\right)^{\frac{1}{2}}\left(\frac{1}{\mu(4Q)}\int_{2Q}|f-f_{2Q}|^{2}\,d\mu\right)^{\frac{1}{2}}$
$\displaystyle\leq C\|f\|,$

as required for $k=2$.

For $k=3$, we have

$\displaystyle I_{3}$
$\displaystyle=\frac{1}{\mu(Q)}\int_{Q}\left|\frac{1}{\mu(Q)}\int_{Q}(Tf_{3}(x)-Tf_{3}(y))\,d\mu(y)\right|d\mu(x)$
$\displaystyle\leq\frac{1}{\mu(Q)}\frac{1}{\mu(Q)}\int_{Q}\int_{Q}\int_{\mathbb{R}^{m}\setminus
2Q}|\mathcal{K}(x,u)-\mathcal{K}(y,u)||f(u)-f_{2Q}|\,d\mu(u)\,d\mu(y)\,d\mu(x).$

For $x,y\in Q$ and $u\in\mathbb{R}^{m}\setminus 2Q$, the defining property of
$\mathcal{K}(\cdot,\cdot)$ guarantees that

$|\mathcal{K}(x,u)-\mathcal{K}(y,u)|\leq
C\frac{d^{\delta}(x,y)}{d^{n+\delta}(x,u)}.$

Therefore,

$\displaystyle I_{3}$
$\displaystyle\leq\frac{C}{\mu(Q)}\int_{Q}\int_{\mathbb{R}^{m}\setminus
2Q}\frac{\ell^{\delta}}{d^{n+\delta}(x,u)}|f(u)-f_{2Q}|\,d\mu(u)\,d\mu(x)$
$\displaystyle\leq\frac{C}{\mu(Q)}\int_{Q}\sum_{k=1}^{\infty}\frac{\ell^{\delta}}{d^{n+\delta}(x,u)}\int_{2^{k+1}Q\setminus
2^{k}Q}|f(u)-f_{2Q}|\,d\mu(u)\,d\mu(x)$ $\displaystyle\leq
C\sum_{k=1}^{\infty}\frac{\ell^{\delta}}{(2^{k-1}\ell)^{n+\delta}}\int_{2^{k+1}Q\setminus
2^{k}Q}|f(u)-f_{2Q}|\,d\mu(u)$ $\displaystyle\leq
C\sum_{k=1}^{\infty}\frac{\ell^{\delta}}{(2^{k}\ell)^{n+\delta}}\left(\int_{2^{k+1}Q}|f(u)-f_{2^{k+1}Q}|\,d\mu(u)+\mu(2^{k+1}Q)|f_{2Q}-f_{2^{k+1}Q}|\right)$

by the triangle inequality. The choice of the constants $f_{2Q}$ and
$f_{2^{k+1}Q}$ guarantees that

$I_{3}\leq
C\sum_{k=1}^{\infty}\frac{\ell^{\delta}}{(2^{k}\ell)^{n+\delta}}\left(\mu(2^{k+2}Q)+\mu(2^{k+1}Q)K(2Q,2^{k+1}Q)\right)\|f\|_{\mathfrak{A},2}.$

Since $\mu$ is $n$-dimensional, we have

$\displaystyle I_{3}$ $\displaystyle\leq
C\|f\|_{\mathfrak{A},2}\sum_{k=1}^{\infty}\frac{\ell^{\delta}(2^{k+2}\ell)^{n}}{(2^{k}\ell)^{n+\delta}}K(2Q,2^{k+1}Q)$
$\displaystyle\leq
C\|f\|_{\mathfrak{A},2}\sum_{k=1}^{\infty}\frac{K(2Q,2^{k+1}Q)}{2^{k\delta}}$
$\displaystyle\leq
C\|f\|_{\mathfrak{A},2}\left(1+\sum_{k=1}^{\infty}2^{-k\delta}\sum_{j=2}^{k+1}\frac{\mu(2^{j}Q)}{(\ell
2^{j})^{n}}\right)$ $\displaystyle\leq C\|f\|$

by Lemma 2.1. The proof of the lemma is finished. ∎

In the following lemma, the cubes under consideration are not assumed to be
doubling. However, in the proof of Theorem 1.1, we apply this lemma in the
doubling setting.

######  Lemma 3.2.

There exists a constant $C>0$ such that

$|b_{k,Q}-b_{k,R}|\leq C\|f\|K(Q,R),\quad k=2,3.$

for any two cubes $Q\subset R$.

###### Proof.

For $k=2$, the required estimate is trivial; so assume that $k=3$. Given cubes
$Q\subset R$, we have

$|b_{3,Q}-b_{3,R}|\leq|b_{3,Q}-b_{3,Q_{0}}|+|b_{3,Q_{0}}-b_{3,R}|=I+J,$

where $Q_{0}=2^{k}Q$ and $k$ is the minimal integer such that $2^{k}Q\supset
R$.

Observe that

$\displaystyle I$
$\displaystyle=\left|\frac{1}{\mu(Q_{0})}\int_{Q_{0}}\frac{1}{\mu(Q)}\int_{Q}Tf_{3,Q}(y)\,d\mu(y)\,d\mu(z)\right.$
$\displaystyle\qquad\qquad\qquad\left.-\frac{1}{\mu(Q)}\int_{Q}\frac{1}{\mu(Q_{0})}\int_{Q_{0}}Tf_{3,Q_{0}}(z)\,d\mu(z)\,d\mu(y)\right|$
$\displaystyle=\left|\frac{1}{\mu(Q)}\frac{1}{\mu(Q_{0})}\int_{Q}\int_{Q_{0}}Tf_{3}(y)-Tf_{3}(z)\,d\mu(z)\,d\mu(y)\right|.$

By the definitions of $f_{3,Q}$ and $f_{3,Q_{0}}$,

$\displaystyle Tf_{3,Q}(y)$ $\displaystyle-
Tf_{3,Q_{0}}(z)=\int_{\mathbb{R}^{m}\setminus
2Q}K(y,\cdot)(f-f_{2Q})d\mu-\int_{\mathbb{R}^{m}\setminus
2Q_{0}}K(z,\cdot)(f-f_{2Q_{0}})d\mu$ $\displaystyle=\int_{2Q_{0}\setminus
2Q}K(y,\cdot)(f-f_{2Q})\,d\mu+\int_{\mathbb{R}^{m}\setminus
2Q_{0}}K(y,\cdot)(f_{2Q_{0}}-f_{2Q})\,d\mu$
$\displaystyle\qquad\qquad\qquad+\int_{\mathbb{R}^{m}\setminus
2Q_{0}}(K(y,\cdot)-K(z,\cdot))(f-f_{2Q_{0}})\,d\mu$ $\displaystyle=D+E+F.$

We split $D$ into dyadic sums as follows:

$\displaystyle D$ $\displaystyle=\sum_{j=1}^{k}\int_{2^{j+1}Q\setminus
2^{j}Q}K(y,\cdot)(f-f_{2Q})\,d\mu$
$\displaystyle=\sum_{j=1}^{k}\int_{2^{j+1}Q\setminus
2^{j}Q}K(y,\cdot)(f-f_{2^{j+1}Q})\,d\mu$
$\displaystyle\qquad\qquad\qquad+\sum_{j=1}^{k}(f_{2^{j+1}Q}-f_{2Q})\int_{2^{j+1}Q\setminus
2^{j}Q}K(y,\cdot)\,d\mu$ $\displaystyle=D_{1}+D_{2}.$

To estimate $D_{1}$, we apply (1.2) and obtain

$|K(y,x)|\leq\frac{C}{d^{n}(x,y)}\leq
C\frac{1}{(\ell(Q)2^{j+2})^{n}}\quad\textrm{for}\ y\in Q,\ x\in
2^{j+1}Q\setminus 2^{j}Q.$

Hence,

(3.1) $\displaystyle|D_{1}|$ $\displaystyle\leq
C\sum_{j=1}^{k}\frac{\mu(2^{j+2}Q)}{(\ell(Q)2^{j+2})^{n}}\frac{1}{\mu(2^{j+2}Q)}\int_{2^{j+1}Q\setminus
2^{j}Q}|f-f_{2^{j+1}Q}|\,d\mu$ $\displaystyle\leq
CK(Q,Q_{0})\|f\|_{\mathfrak{A},2}$ $\displaystyle\leq CK(Q,Q_{0})\|f\|$

by Lemma 2.1. Below we repeatedly use the equivalence of
$\|\cdot\|_{\mathfrak{A},2}$ and $\|\cdot\|$ without explicit reference to
Lemma 2.1.

Next, applying summation by parts, we obtain

$\displaystyle D_{2}=(f_{2^{k+1}Q}-f_{2Q})$
$\displaystyle\int_{2^{k+1}Q\setminus 2Q}K(y,\cdot)\,d\mu$
$\displaystyle-\sum_{j=1}^{k-1}(f_{2^{j+2}Q}-f_{2^{j+1}Q})\int_{2^{j+1}Q\setminus
2Q}K(y,\cdot)\,d\mu.$

By the cancellation property (1.4),

$\left|\int_{2^{j+1}Q\setminus 2Q}K(y,\cdot)\,d\mu\right|\leq C,\quad
j=1,2,\dots,k.$

Thus, the choice of the numbers $f_{2^{j+1}Q}$, $j=0,1,\dots,k$, guarantees
that

(3.2) $\displaystyle|D_{2}|$ $\displaystyle\leq
CK(2Q,2Q_{0})\|f\|_{\mathfrak{A},2}+C\sum_{j=1}^{k-1}K(2^{j+1}Q,2^{j+2}Q)\|f\|_{\mathfrak{A},2}$
$\displaystyle\leq
CK(Q,Q_{0})\|f\|_{\mathfrak{A},2}+CK(2Q,2Q_{0})\|f\|_{\mathfrak{A},2}$
$\displaystyle\leq CK(Q,Q_{0})\|f\|.$

To estimate $E$, we also use the cancellation property (1.4) and we obtain

(3.3) $\displaystyle|E|$
$\displaystyle=\left|(f_{2Q}-f_{2Q_{0}})\int_{\mathbb{R}^{m}\setminus
2Q_{0}}K(y,\cdot)\,d\mu\right|$ $\displaystyle\leq C|f_{2Q}-f_{2Q_{0}}|$
$\displaystyle\leq CK(Q,Q_{0})\|f\|_{\mathfrak{A},2}$ $\displaystyle\leq
CK(Q,Q_{0})\|f\|.$

Now, consider $F$. We have

$|K(y,x)-K(z,x)|\leq C\frac{\ell^{\delta}(Q_{0})}{d^{n+\delta}(x,Q_{0})}$

for $x\in\mathbb{R}^{m}\setminus 2Q_{0}$.

Therefore,

$\displaystyle|F|$ $\displaystyle\leq
C\ell^{\delta}(Q_{0})\int_{\mathbb{R}^{m}\setminus
2Q_{0}}\frac{|f-f_{2Q_{0}}|}{d^{n+\delta}(x,Q_{0})}\,d\mu(x)$
$\displaystyle\leq
C\sum_{j=1}^{\infty}\frac{\ell^{\delta}(Q_{0})}{(2^{j+2}\ell(Q_{0}))^{n+\delta}}\int_{2^{j+1}Q_{0}\setminus
2^{j}Q_{0}}|f-f_{2Q_{0}}|\,d\mu$ $\displaystyle\leq
C\sum_{j=1}^{\infty}\frac{\ell^{\delta}(Q_{0})}{(2^{j+2}\ell(Q_{0}))^{n+\delta}}\left(\int_{2^{j+1}Q_{0}\setminus
2^{j}Q_{0}}|f-f_{2^{j+1}Q_{0}}|\,d\mu\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+|f_{2^{j+1}Q_{0}}-f_{2Q_{0}}|\int_{2^{j+1}Q_{0}\setminus
2^{j}Q_{0}}\,d\mu\right)=F_{1}+F_{2}.$

Firstly,

(3.4) $\displaystyle F_{1}$
$\displaystyle\leq\sum_{j=1}^{\infty}\frac{\ell^{\delta}(Q_{0})\mu(2^{j+2}Q_{0})}{(2^{j+2}\ell(Q_{0}))^{n+\delta}}\|f\|_{\mathfrak{A},2}$
$\displaystyle\leq
C\sum_{j=1}^{\infty}\frac{\|f\|_{\mathfrak{A},2}}{(2^{j})^{\delta}}$
$\displaystyle\leq C\|f\|.$

Secondly,

$\displaystyle F_{2}$
$\displaystyle\leq\sum_{j=1}^{\infty}\frac{\ell^{\delta}(Q_{0})\mu(2^{j+2}Q_{0})}{(2^{j+2}\ell(Q_{0}))^{n+\delta}}\|f\|_{\mathfrak{A},2}K(2Q_{0},2^{j}Q_{0})$
$\displaystyle\leq
C\sum_{j=1}^{\infty}\frac{\|f\|_{\mathfrak{A},2}K(2Q_{0},2^{j}Q_{0})}{(2^{j})^{\delta}}.$

Since $\mu$ is $n$-dimensional, we have $K(2Q_{0},2^{j}Q_{0})\leq Cj$ with a
universal constant $C>0$. Thus,

(3.5) $F_{2}\leq
C\|f\|_{\mathfrak{A},2}\sum_{j=1}^{\infty}\frac{j}{(2^{j})^{\delta}}\leq
C\|f\|.$

Combining (3.1–3.5) and integrating with respect to $z$ and $y$, we obtain the
required estimate for $I=|b_{3,Q}-b_{3,Q_{0}}|$. Therefore, it remains to
estimate $J$.

We have

$\displaystyle J$ $\displaystyle=|b_{3,Q_{0}}-b_{3,R}|$
$\displaystyle=\left|\frac{1}{\mu(R)}\int_{R}\frac{1}{\mu(Q_{0})}\int_{Q_{0}}Tf_{3,Q_{0}}(y)\,d\mu(y)\,d\mu(w)\right.$
$\displaystyle\qquad\qquad\qquad\left.-\frac{1}{\mu(Q_{0})}\int_{Q_{0}}\frac{1}{\mu(R)}\int_{R}Tf_{3,R}(w)\,d\mu(w)\,d\mu(y)\right|$
$\displaystyle=\left|\frac{1}{\mu(Q_{0})}\frac{1}{\mu(R)}\int_{Q_{0}}\int_{R}Tf_{3,Q_{0}}(y)-Tf_{3,R}(w)\,d\mu(w)\,d\mu(y)\right|.$

By the definitions of $f_{3,Q_{0}}$ and $f_{3,R}$,

$\displaystyle Tf_{3,Q_{0}}(z)$ $\displaystyle-
Tf_{3,R}(w)\\!=\int_{\mathbb{R}^{m}\setminus
2Q_{0}}K(z,\cdot)(f-f_{2Q_{0}})d\mu-\int_{\mathbb{R}^{m}\setminus
2R}K(w,\cdot)(f-f_{2R})d\mu$ $\displaystyle=-\int_{2Q_{0}\setminus
2R}K(w,\cdot)(f-f_{2R})\,d\mu+(f_{2Q_{0}}-f_{2R})\int_{\mathbb{R}^{m}\setminus
2Q_{0}}K(w,\cdot)\,d\mu$
$\displaystyle\qquad\qquad\qquad+\int_{\mathbb{R}^{m}\setminus
2Q_{0}}(K(z,\cdot)-K(w,\cdot))(f-f_{2Q_{0}})\,d\mu$
$\displaystyle=\mathcal{D}+\mathcal{E}+\mathcal{F}.$

Firstly, $\mathcal{D}$ is estimated similarly to $D$ and even simpler:

$\displaystyle\mathcal{D}$ $\displaystyle=-\int_{2Q_{0}\setminus
2R}K(w,\cdot)(f-f_{2Q_{0}})\,d\mu+(f_{2R}-f_{2Q_{0}})\int_{2Q_{0}\setminus
2R}K(w,\cdot)\,d\mu$ $\displaystyle=\mathcal{D}_{1}+\mathcal{D}_{2}.$

Since $\ell(R)$ and $\ell(Q_{0})$ are comparable and $\mu$ is $n$-dimensional,
we have

(3.6) $\displaystyle|\mathcal{D}_{1}|$ $\displaystyle\leq\int_{2Q_{0}\setminus
2R}\frac{C}{d^{n}(R,2Q_{0}\setminus 2R)}|f-f_{2Q_{0}}|\,d\mu$
$\displaystyle\leq\frac{C}{\ell^{n}(4R)}\frac{\mu(4Q_{0})}{\mu(4Q_{0})}\int_{2Q_{0}}|f-f_{2Q_{0}}|\,d\mu$
$\displaystyle\leq
C\frac{\mu(4Q_{0})}{\ell^{n}(4Q_{0})}\|f\|_{\mathfrak{A},2}$
$\displaystyle\leq C\|f\|.$

Using again that $\ell(R)$ and $\ell(Q_{0})$ are comparable and $\mu$ is
$n$-dimensional, we obtain

(3.7) $\displaystyle|\mathcal{D}_{2}|$
$\displaystyle\leq|f_{2Q_{0}}-f_{2R}|\frac{\mu(2Q_{0})}{d^{n}(R,2Q_{0}\setminus
2R)}$ $\displaystyle\leq C|f_{2Q_{0}}-f_{2R}|\frac{\mu(2Q_{0})}{\ell^{n}(R)}$
$\displaystyle\leq C|f_{2Q_{0}}-f_{2R}|$ $\displaystyle\leq
CK(2R,2Q_{0})\|f\|_{\mathfrak{A},2}$ $\displaystyle\leq C\|f\|.$

To estimate $\mathcal{E}$, we use the cancellation property (1.4) and obtain

(3.8) $\displaystyle|\mathcal{E}|$
$\displaystyle=\left|(f_{2R}-f_{2Q_{0}})\int_{\mathbb{R}^{m}\setminus
2Q_{0}}K(w,\cdot)\,d\mu\right|$ $\displaystyle\leq C|f_{2R}-f_{2Q_{0}}|$
$\displaystyle\leq CK(2R,2Q_{0})\|f\|_{\mathfrak{A},2}$ $\displaystyle\leq
C\|f\|.$

Now, we estimate $\mathcal{F}$. Property (1.3) guarantees that

$\displaystyle|\mathcal{F}|$
$\displaystyle=\left|\int_{\mathbb{R}^{m}\setminus
2Q_{0}}((K(z,\cdot)-K(w,\cdot))(f-f_{2Q_{0}})\,d\mu\right|$ $\displaystyle\leq
Cd^{\delta}(z,w)\int_{\mathbb{R}^{m}\setminus
2Q_{0}}\frac{|f(x)-f_{2Q_{0}}|}{d^{n+\delta}(x,Q_{0})}\,d\mu(x).$

We have

$d^{\delta}(z,w)\leq\ell^{\delta}(Q_{0}).$

Hence, using dyadic decompositions and summation by parts, we repeat the
arguments applied to estimate $|F|$ and we obtain the following analog of
(3.4) and (3.5):

(3.9) $|\mathcal{F}|\leq C\|f\|.$

Now, combining estimates (3.6–3.9), we conclude that

$J=|b_{3,Q}-b_{3,R}|\leq CK(Q,R)\|f\|$

for any two cubes $Q\subset R$. The proof of Lemma 3.2 is finished. ∎

##  4\. Proof of Theorem 1.1

Given an $f\in\mathrm{RBMO}(\mu)$, we have to prove that
$Tf\in\mathrm{RBMO}(\mu)$. Put $g=Tf$. In this section, for every doubling
cube $Q\subset\mathbb{R}^{m}$, we find a constant $g_{Q}\in\mathbb{R}$ such
that

(4.1) $\frac{1}{\mu(Q)}\int_{Q}|g-g_{Q}|\,d\mu\leq C$

and

(4.2) $|g_{Q}-g_{R}|\leq CK(Q,R)\quad\textrm{for any two doubling cubes\
}Q\subset R,$

where $C=C_{f}>0$.

So, given an $f\in\mathrm{RBMO}(\mu)$ and a doubling cube
$Q\subset\mathbb{R}^{m}$, we apply the construction described in Section 3 and
we obtain

$f=f_{1,Q}+f_{2,Q}+f_{3,Q}=f_{1}+f_{2}+f_{3}.$

Also, we have constants $b_{2,Q}$ and $b_{3,Q}$. By Lemma 2.6, properties
(2.3) and (2.4) hold for the constants $\left<T1\right>_{Q}$ and
$\left<T1\right>_{R}$. Put

$g_{Q}=f_{2Q}\left<T1\right>_{Q}+b_{2,Q}+b_{3,Q}.$

###  4.1. Oscillation condition (4.1)

We have

$\displaystyle\frac{1}{\mu(Q)}\int_{Q}|g-g_{Q}|\,d\mu$
$\displaystyle\leq\frac{1}{\mu(Q)}\left(\int_{Q}|Tf_{1}-f_{2Q}\left<T1\right>_{Q}|\,d\mu+\int_{Q}|Tf_{2}-b_{2,Q}|\,d\mu+\int_{Q}|Tf_{3}-b_{3,Q}|\,d\mu\right)$
$\displaystyle\leq\frac{1}{\mu(Q)}\int_{Q}|Tf_{1}-f_{2Q}\left<T1\right>_{Q}|\,d\mu+C\|f\|$

by Lemma 3.1. Recall that $Tf_{1}=Tf_{2Q}=f_{2Q}T1$. Therefore, applying (2.3)
and Lemma 2.4, we obtain

$\displaystyle\frac{1}{\mu(Q)}\int_{Q}|Tf_{1}-f_{2Q}\left<T1\right>_{Q}|\,d\mu$
$\displaystyle=\frac{|f_{2Q}|}{\mu(Q)}\int_{Q}|T1-\left<T1\right>_{Q}|\,d\mu$
$\displaystyle\leq\frac{C_{f}K(2Q)}{K(Q)}\leq C_{f}.$

Hence, the oscillation condition (4.1) is proved.

###  4.2. $K$-condition (4.2)

Let $Q\subset R$ be doubling cubes. Combining the triangle inequality and
Lemma 3.2, we obtain

$\displaystyle|g_{Q}-g_{R}|$
$\displaystyle\leq|f_{2Q}\left<T1\right>_{Q}-f_{2R}\left<T1\right>_{R}|+|b_{3,Q}-b_{3,R}|$
$\displaystyle\leq|f_{2Q}\left<T1\right>_{Q}-f_{2R}\left<T1\right>_{R}|+C\|f\|K(Q,R).$

Next, the choice of the constants $f_{2Q}$ and $f_{2R}$ guarantees that

$|f_{2Q}-f_{2R}|\leq C\|f\|_{\mathfrak{A},2}K(2Q,2R)\leq C\|f\|K(Q,R).$

Also, we have $T1\in L^{\infty}(\mu)$ by Lemma 2.5, hence
$|\left<T1\right>_{Q}|\leq C$. Therefore,

$\displaystyle|f_{2Q}\left<T1\right>_{Q}-f_{2R}\left<T1\right>_{2R}|$
$\displaystyle\leq|\left<T1\right>_{Q}||f_{2Q}-f_{2R}|+|f_{2R}|\left|\left<T1\right>_{Q}-\left<T1\right>_{R}\right|$
$\displaystyle\leq
C\|f\|K(Q,R)+|f_{2R}|\left|\left<T1\right>_{Q}-\left<T1\right>_{R}\right|.$

Applying Lemma 2.6 and Lemma 2.4, we obtain

$|f_{2R}|\left|\left<T1\right>_{Q}-\left<T1\right>_{R}\right|\leq
C_{f}K(2R)\frac{K(Q,R)}{K(R)}\leq C_{f}K(Q,R).$

Combining the above estimates, we conclude that (4.2) holds. This ends the
proof of the theorem.

## References

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aainstitutetext:  Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa,
Chiba 277-8583, Japanbbinstitutetext: Dipartimento di Fisica “E. Pancini",
Università di Napoli Federico II, via Cintia, 80126 Napoli,
Italyccinstitutetext: INFN sezione di Napoli, Complesso Universitario di Monte
S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italyddinstitutetext: Scuola
Superiore Meridionale, Largo S. Marcellino, 10, 80138 Napoli NA,
Italyeeinstitutetext: Quantum Theory Center ($\hbar$QTC) at IMADA and D-IAS,
Southern Denmark Univ., Campusvej 55, 5230 Odense M, Denmark

#  Scaling results
for
Charged sectors of near conformal QCD

Jahmall Bersini b, c, e Alessandra D’Alise d, e Clelia Gambardella b, c, d, e
Francesco Sannino<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We provide the leading near conformal corrections on a cylinder to the scaling
dimension $\Delta_{Q}^{\ast}$ of fixed isospin charge $Q$ operators defined at
the lower boundary of the Quantum Chromodynamics conformal window:

$\Delta_{Q}=\Delta_{Q}^{\ast}+\left(\frac{m_{\sigma}}{4\pi\nu}\right)^{2}\,Q^{\frac{\Delta}{3}}B_{1}+\left(\frac{m_{\pi}(\theta)}{4\pi\nu}\right)^{4}\
Q^{\frac{2}{3}(1-\gamma)}B_{2}+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\
.$

The results are expressed in powers of the dilaton and pion masses in units of
the chiral symmetry breaking scale $4\pi\nu$ with the theta-angle dependence
encoded directly in the pion mass. The characteristic $Q$-scaling is dictated
by the quark mass operator anomalous dimension $\gamma$ and the one
characterising the dilaton potential $\Delta$. The coefficients $B_{i}$ with
$i=1,2$ depend on the geometry of the cylinder and properties of the nearby
conformal field theory.

##  1 Introduction

Unveiling near-conformal properties of Quantum Chromodynamics (QCD) has
attracted much interest over the past several decades. This exploration was
spurred by the seminal work of Banks and Zaks Banks:1981nn that discovered the
existence of a perturbative infrared fixed (IR) point for massless QCD for a
number of flavours $N_{f}$ just below the loss of asymptotic freedom. As one
decreases $N_{f}$ relative to the fixed number of colours $N_{c}$ one expects
below a critical number of flavours $N_{f}^{c}$ the theory to undergo a
quantum phase transition. That this transition is bound to occur is clear from
the fact that for the observed number of light flavours the theory breaks
chiral symmetry dynamically generating a non-perturbative scale even in the
absence of explicit light quark masses. Estimates of the critical number of
flavours $N_{f}^{c}$ are all non-perturbative in nature and indicate that this
value is, for three colours QCD, around $N_{f}^{c}\sim 10-12$. The window in
the flavour-colour space where the theory displays IR conformality is termed
conformal window and its phase diagram for arbitrary matter representations is
presented in Sannino:2004qp ; Dietrich:2006cm . Ongoing lattice studies are
continuously searching to elucidate the confines and properties of the
conformal window as summarised in Cacciapaglia:2020kgq .

Several key questions are related to the dynamics near the lower boundary of
the conformal window. These range from a precise determination of its lower
edge to the characterisation of the quantum phase transition. One exciting
possibility is for the transition to lose conformality a la Berezinskii-
Kosterlitz-Thouless (BKT) Berezinsky:1970fr ; Kosterlitz:1973xp ;
Kosterlitz:1974nba . The latter occurs in two dimensions and was envisioned
for four dimensions in Miransky:1996pd ; Miransky:1984ef ; Appelquist:1996dq ;
Gies:2005as . An alternative is for the quantum transition to be a jumping one
Sannino:2012wy . The first possibility leads to an infrared non-conformal
physics displaying premonitory signs of near-conformality that can be observed
in power-law scaling of certain phenomenologically relevant operators
Holdom:1988gs ; Holdom:1988gr ; Cohen:1988sq . This dynamics is also known as
walking dynamics since the underlying gauge coupling has a region in the
renormalization time scale where the coupling is almost constant, and
therefore it walks rather than displaying a running behaviour. Such a walking
behaviour has recently been shown to mathematically describe the endemic state
of pandemics Complex:2021 via the epidemic renormalization group approach
DellaMorte:2020wlc . This methodology was also used to successfully predict
the second wave COVID-19 pandemic in Europe Secondwave:2021 providing
policymakers crucial epidemiological information. In terms of the spectrum of
the theory, in the confining and chiral symmetry broken phase occurring below
but near the lower end of the conformal, besides the ordinary Goldstons, it
has long been argued Leung:1985sn ; Bardeen:1985sm ; Yamawaki:1985zg ;
Sannino:1999qe that another precursor of a smooth quantum phase transition is
the occurrence of a dilaton in the theory. Its description at the effective
Lagrangian level goes back to the work of Coleman in coleman1988aspects and
for walking dynamics considered in Sannino:1999qe ; Hong:2004td ;
Dietrich:2006cm ; Dietrich:2005jn ; Appelquist:2010gy . Recent investigations
via effective approaches in different dynamical regimes have appeared in
Chacko:2012sy ; Matsuzaki:2013eva ; Hansen:2016fri ; Golterman:2016lsd ;
Cata:2019edh ; Golterman:2020tdq ; Appelquist:2020bqj ; Appelquist:2022mjb ;
Cata:2018wzl ; appelquist2020dilaton ; Orlando:2020yii ; Bersini:2022bnx ;
Bersini:2023uat . An explicit calculable example has been discussed in
Antipin:2011aa where it was possible to demonstrate the emergence of a dilaton
in a near-conformal field theory (CFT) alongside a precise study of all the
relevant scales emerging once conformality is lost. A complementary analysis
of the mass-induced confinement for gauge-fermion theories near the lower edge
of the conformal window was performed in Marcarelli:2022sbb . One can also use
first principle lattice simulations to disentangle the dilaton properties,
however, this is a difficult task since its quantum numbers are the ones of
the vacuum and therefore it is subject to large numerical noise. Nevertheless
there are attempts to fit effective approaches to lattice data as summarized
in Ingoldby:2023mtf ).

Complementary ways to isolate the dilaton properties and more generally to
learn about the near-conformal dynamics of QCD are therefore vital to corner
the properties of the flavour-driven quantum phase transition. As we shall see
fixed charged sectors offer novel opportunities to disentangle near conformal
dynamics providing precious information on the sectors responsible for
breaking conformality. We start by recalling that central quantities in any
CFT are scaling dimensions of local operators. By the state-operator
correspondence Cardy:1984rp these are the energies of the corresponding states
on a non-trivial gravitational background. For example, the scaling dimension
of the lowest scaling operator of charge $Q$ denoted with $\Delta_{Q}^{\ast}$
is mapped into ground state energy $E_{Q}$ on the cylinder via the relation

$\Delta_{Q}^{\ast}=r\,E_{Q}\ ,$ (1)

with $r$ the radius of the cylinder. In the large charge limit we can compute
scaling dimensions of fixed charge operators by means of semiclassical
computations Hellerman:2015nra ; Banerjee:2017fcx ; Orlando:2019skh ;
Gaume:2020bmp ; Hellerman:2023myh ; Antipin:2020abu ; Antipin:2022naw ;
Antipin:2022hfe ; Antipin:2023tar ; Monin:2016jmo ; Badel:2019oxl ;
Cuomo:2020rgt ; Arias-Tamargo:2019xld . Following Orlando:2019skh ;
Orlando:2020yii ; Bersini:2022bnx it is natural to extend this relation to
near-conformal field theories by introducing the quantity:

$\Delta_{Q}\equiv r\,E_{Q}=\Delta_{Q}^{\ast}+{\rm near\leavevmode\nobreak\
CFT\leavevmode\nobreak\ terms}\ .$ (2)

The near CFT terms depend on the way the CFT is deformed. In the case at hand
we have two sources of conformal breaking: One stemming from an explicit quark
mass term and the other from the occurrence of an operator of dimension
$\Delta$ inducing a dilaton-potential. Therefore, in this work, we extend the
chiral Lagrangian to include a dilaton sector and use the large charge
expansion framework to determine the general expression for $\Delta_{Q}$
arriving at one of our central results:

$\Delta_{Q}=\Delta_{Q}^{\ast}+\left(\frac{m_{\sigma}}{4\pi\nu}\right)^{2}\,Q^{\frac{\Delta}{3}}B_{1}+\left(\frac{m_{\pi}(\theta)}{4\pi\nu}\right)^{4}\
Q^{\frac{2}{3}(1-\gamma)}B_{2}+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\
.$ (3)

The latter is expressed in powers of the dilaton and pion masses given in
units of the chiral symmetry breaking scale $4\pi\nu$. The theta-angle
dependence is explicitly encoded in the pion mass. The novel scalings in the
charge corrections are expressed in terms of the quark mass operator anomalous
dimension $\gamma$ and the one characterising the dilaton potential $\Delta$.
The coefficients $B_{i}$ with $i=1,2$ depend on the geometry of the cylinder
and the properties of the nearby CFT. The above result for $\Delta_{Q}$ in (3)
is obtained as the large charge limit of the general expression given in (42)
at leading order in the semiclassical expansion. The next-to-leading order, to
be computed in the future, requires the knowledge of the spectrum of
fluctuations that we have also determined here. Additionally, we have also
provided the phase diagram of QCD at non-zero isospin chemical potential in
the presence of the CP-violating topological term.

The paper is organised as follows. In Section 2 we introduce the QCD chiral
Lagrangian for generic $N_{f}$ including the $\theta$-angle and isospin
chemical potential $\mu$. The $\mu-\theta$ phase diagram is presented in Sec.
3. The dilaton potential and setup for the large charge expansion are
discussed in Sec. 4. Section 5 is devoted to the determination of $\Delta_{Q}$
while in Sec. 6 we first discuss the patterns of symmetry breaking and then
compute the spectrum of fluctuations. Section 7 is dedicated to the universal
contributions in the conformal limit. We offer our conclusions in Sec. 8.

##  2 Chiral Lagrangian at finite isospin and $\theta$-angle: Notation and
conventions

The low-energy dynamics of QCD at finite isospin density is described by the
chiral Lagrangian below

$\displaystyle\begin{split}\mathcal{L}&=\nu^{2}Tr\\{\partial_{\mu}\Sigma\partial^{\mu}\Sigma^{\dagger}\\}+m^{2}_{\pi}\nu^{2}Tr\\{M\Sigma+M^{\dagger}\Sigma^{\dagger}\\}+2i\mu\nu^{2}Tr\\{I\partial_{0}\Sigma\Sigma^{\dagger}-I\Sigma^{\dagger}\partial_{0}\Sigma\\}\\\
&+2\mu^{2}\nu^{2}Tr\\{II-\Sigma^{\dagger}I\Sigma I\\}\ ,\end{split}$ (4)

where $4\pi\nu$ is the energy scale of the chiral symmetry breaking, $\mu$ is
the isospin chemical potential, and

$\Sigma=e^{i\varphi/\nu}\,,\qquad\varphi=\Pi^{a}T^{a}+\frac{S}{\sqrt{N_{f}}}\
,$ (5)

with $T^{a}$ the $SU(N_{f})$ generators normalized as ${\rm
Tr}[T^{a}T^{b}]=\delta^{ab}$. The mass matrix and the isospin generator read

$M=\mathbbm{1}_{N_{f}}\,,\qquad
I=\frac{1}{2}\left(\begin{array}[]{cc}\mathbbm{1}_{N_{f}/2}&0\\\
0&-\mathbbm{1}_{N_{f}/2}\end{array}\right)\ ,$ (6)

where we assumed degenerate Goldstone bosons of mass $m_{\pi}$. The matrix $I$
generalizes the concept of isospin in the multi-flavor theory with the two
flavour case being the conventional QCD isospin.

Finally, as fully detailed in Witten:1980sp ; DiVecchia:1980yfw , the CP-
violating topological sector is included through the $\theta$-angle term

$\Delta\mathcal{L}_{\theta}=-a\nu^{2}\left(\theta-\frac{i}{2}Tr\\{\log\Sigma-\log\Sigma^{\dagger}\\}\right)^{2}\
,$ (7)

where $a$ is the energy scale associated with the axial anomaly.

##  3 Phase diagram in the $\mu$-$\theta$ plane

In the absence of the $\theta$ angle, the ground state of our theory
generalizes the $N_{f}=2$ case examined in detail in Son:2000xc and takes the
following form

$\displaystyle\Sigma_{c}$
$\displaystyle=\mathbbm{1}_{N_{f}}\cos\varphi+i\Sigma_{I}\sin\varphi\ ,$ (8)

with

$\displaystyle\Sigma_{I}$
$\displaystyle=\left(\begin{array}[]{cc}0&\mathbbm{1}_{N_{f}/2}\\\
\mathbbm{1}_{N_{f}/2}&0\\\ \end{array}\right)\ \cos\eta\
+i\left(\begin{array}[]{cc}0&-\mathbbm{1}_{N_{f}/2}\\\
\mathbbm{1}_{N_{f}/2}&0\end{array}\right)\ \sin\eta\,.$ (13)

The two terms in $\Sigma_{c}$ minimize, respectively, the mass and isospin
terms in the potential. In fact, it can be easily shown that the latter is
minimized when $\Sigma_{I}$ satisfies $I\Sigma_{I}=-\Sigma_{I}I$. To take into
account the effect of the $\theta$-angle on the vacuum state we introduce the
Witten variables $\alpha_{i}$ Witten:1980sp and arrive at our ansatz for the
ground state

$\Sigma_{0}=U(\alpha_{i})\Sigma_{c}\ ,\,\,\text{with}\
\,\,U(\alpha_{i})=\text{diag}\\{e^{-i\alpha_{1}}\,,\dots\,,e^{-i\alpha_{N_{f}}}\\}\
.$ (14)

It is useful to define the following quantities

$\bar{\theta}=\theta-\sum_{i}^{N_{f}}\alpha_{i}\
,\,\,\,X=\sum_{i=1}^{N_{f}}\cos\alpha_{i}\ ,$ (15)

in terms of which the Lagrangian of the theory evaluated on the ground state
ansatz reads

$\mathcal{L}[\Sigma_{0}]=2m_{\pi}^{2}\nu^{2}X\cos\varphi+N_{f}\mu^{2}\nu^{2}\sin^{2}\varphi-a\nu^{2}\bar{\theta}^{2}\,.$
(16)

The angle $\varphi$ and the Witten variables $\alpha_{i}$ are determined by
the equations of motion (EOMs) as

$\displaystyle\sin\varphi\left(N_{f}\cos\varphi-\frac{m_{\pi}^{2}X}{\mu^{2}}\right)$
$\displaystyle=0\ ,$ (17) $\displaystyle m_{\pi}^{2}\sin\alpha_{i}\cos\varphi$
$\displaystyle=a\bar{\theta}\ ,\,\,\,i=1,\dots,N_{f}\ ,$ (18)

while the angle $\eta$ does not appear in the equation of motion meaning that
there is a residual unbroken $U(1)$ isospin vector-symmetry stemming from the
original diagonal $SU(2)$ isospin. However, once we choose a specific value of
$\eta$ this amounts to the further spontaneous breaking of this leftover
$U(1)$. The first EOM has two solutions, namely $\varphi=0$ and
$\cos\varphi=\frac{m_{\pi}^{2}X}{N_{f}\mu^{2}}$. When the latter is realized
the theory is in a superfluid phase characterized by pion condensation. The
energy of the system in the two phases reads

$\displaystyle E(\theta)$
$\displaystyle=-2m_{\pi}^{2}\nu^{2}X+a\nu^{2}\bar{\theta}^{2}\ \,\,\,\,\,$
normal phase $\displaystyle\ \ (\varphi=0)$ $\displaystyle E(\theta)$
$\displaystyle=-\frac{m_{\pi}^{4}\nu^{2}}{N_{f}\mu^{2}}X^{2}-N_{f}\nu^{2}\mu^{2}+a\nu^{2}\bar{\theta}^{2}\
\,\,\,\,\,$ superfluid phase $\displaystyle\
\left(\cos\varphi=\frac{m_{\pi}^{2}X}{N_{f}\mu^{2}}\right)\ .$ (19)

We first observe that when $a\gg m^{2}_{\pi}$ all the $\theta$-dependence can
be read off the effective pion mass $m^{2}_{\pi}(\theta)=m^{2}_{\pi}X/N_{f}$.
The EOMs and the expression for the vacuum energy of the system are very
similar to the ones found in Bersini:2022jhs for two-color QCD with finite
baryon density. At $\theta=0$ the normal to superfluid phase transition occurs
at a critical value of the chemical potential $\mu_{c}=m_{\pi}$. Since the
$\theta$-vacuum may differ in the two phases, to study the superfluid
transition at non-vanishing values of $\theta$, we first need to determine the
$\theta$ dependence. We then solve eq.(18) by expanding in powers of
$m^{2}_{\pi}/a$ that we take to be small. Specifically, at the leading order
in $m^{2}_{\pi}/a$, we have

$\displaystyle\alpha_{i}=\begin{cases}\pi-\alpha(\theta)\,,\qquad&i=1,\dots,n\\\
\alpha(\theta)\,,\qquad&i=n+1,\dots,N_{f}\,,\end{cases}$ (20)

where

$\alpha(\theta)=\frac{\theta+(2k-n)\pi}{(N_{f}-2n)},\quad
k=0,\dots,N_{f}-2n-1\,,\quad n=0,...,\left[\frac{N_{f}-1}{2}\right]\,.$ (21)

The interval of values for $k$ is constrained because at fixed $n$ the
solutions are periodic in $k$ of period $N_{f}-2n$. In the normal phase the
energy is minimized when $X$ is maximized. Hence, it can be shown that the
$\theta$-vacuum corresponds to $n=0$ and the following values of
$\alpha(\theta)$

$\alpha(\theta)=\begin{cases}\frac{\theta}{N_{f}}\,,\qquad\theta\in[0,\pi]\\\
\frac{\theta-2\pi}{N_{f}}\,,\ \quad\theta\in[\pi,2\pi]\,.\end{cases}$ (22)

The physics at $\theta=\pi$ deserves further discussion In fact, the
Lagrangian possesses $CP$ symmetry when $\theta=\pi$ but the latter is
spontaneously broken by the vacuum leading to the known Dashen’s phenomenon
Dashen:1970et . In fact, at $\theta=\pi$ the two solutions for
$\alpha(\theta)$ in eq.(22) have the same energy leading to degenerate vacua
connected by a CP transformation. In Fig.1 we visualize the ground state
energy as a function of $\theta$ for the template case $N_{f}=2$ as well as a
plot of the $CP$ order parameter $\langle\widetilde{F}F\rangle$ associated
with the pseudo-scalar glueball condensate.

(a) $\theta$-dependence of the energy in the normal phase for $N_{f}=2$. (b)
$CP$ order parameter for $N_{f}=2$.

Figure 1: $\theta$-dependence of the normalized ground state energy and $CP$
order parameter $\langle F\widetilde{F}\rangle$ as a function of $\theta$ for
$N_{f}=2$ in the normal phase. Here $X_{i}=2\cos\left(\theta/2\right)$ and
$X_{ii}=2\cos\left(\theta/2-\pi\right)$ refers to the two solutions for
$\alpha(\theta)$ in eq.(22).

We now move to the $\theta$-dependence in the superfluid phase. The EOM
becomes

$\displaystyle\frac{m^{4}_{\pi}}{N_{f}\mu^{2}}X\sin\alpha_{i}$
$\displaystyle=a\bar{\theta}\ ,\,\,\,i=1,\dots,N_{f}\,,$ (23)

and admits the same solution (21) at the leading order in the natural
expansion parameter $\frac{m^{4}_{\pi}}{\mu^{2}}$. The crucial difference with
respect to the normal phase is that the energy now depends quadratically
instead of linearly on $X$. The situation is analogous to the case of two-
color QCD at finite baryon charge investigated in detail in Bersini:2022jhs
for even $N_{f}$. As has been found there, the ground state solution is the
same as the normal phase being given by eq.(22). Accordingly, for $N_{f}>2$,
at the crossing point $\theta=\pi$ spontaneous breaking of $CP$ occurs. On the
other hand, for $N_{f}=2$ the two solutions $\alpha(\theta)=\theta/N_{f}$ and
$\alpha(\theta)=\frac{\theta-2\pi}{N_{f}}$ have degenerate energy for all
values of $\theta$. As a consequence, the physics is analytic at $\theta=\pi$,
and Dashen’s phenomenon does not occur Bersini:2022jhs ; Metlitski:2005db .
The $\theta$ vacuum in the superfluid phase in the two flavour case is
illustrated in Fig.2.

(a) $\theta$-dependence of the energy in the superfluid phase for $N_{f}=2$.
(b) $CP$ order parameter for $N_{f}=2$.

Figure 2: $\theta$-dependence of the normalized ground state energy and $CP$
order parameter $\langle F\widetilde{F}\rangle$ as a function of $\theta$ for
$N_{f}=2$ in the superfluid phase. Here
$X_{i}^{2}=4\cos^{2}\left(\theta/2\right)$ and
$X_{ii}^{2}=4\cos^{2}\left(\theta/2-\pi\right)$ refers to the two solutions
for $\alpha(\theta)$ in eq.(22) while $\mu$ is measured in units of $m_{\pi}$.

Note that the pairs of completely degenerate solutions remain such to all
orders in $\frac{m_{\pi}^{2}}{a}$. In fact, given the EOM (23) for a certain
$\alpha(\theta)$

$\frac{m_{\pi}^{4}}{2a\mu^{2}}\sin(2\alpha(\theta))=\theta-
N_{f}\alpha(\theta)\,,$ (24)

we have the same EOM for $\alpha(\theta)+\pi$, upon shifting the
$\theta$-angle as $\theta\to\theta+N_{f}\pi$. Being $N_{f}$ even, this
corresponds to a shift by an integer multiple of $2\pi$ which leaves the
physics unaltered. Finally, we investigate the critical value of the isospin
chemical potential where the superfluid phase transition occurs. For
$m_{\pi}^{2}\ll a$, the latter occurs at a critical value of the chemical
potential given by

$\mu_{c}=m_{\pi}(\theta)=m_{\pi}\left[\sqrt{\absolutevalue{\cos\frac{\theta}{N_{f}}}}+\mathcal{O}\left(\frac{m_{\pi}^{2}}{a}\right)\right]\,,$
(25)

which is of particular interest when $N_{f}=2$ at $\theta=\pi$. In fact,
$\mu_{c}$ is almost vanishing since the effective pion mass
$m^{2}_{\pi}(\theta)\sim m^{2}_{\pi}\absolutevalue{\cos(\theta/2)}$
identically vanishes. Concretely, at $\theta\sim\pi$ we have

$\mu_{c}\sim
m_{\pi}\sqrt{\frac{m_{\pi}^{2}}{a}+\frac{\absolutevalue{\phi}}{2}}\,,\qquad\phi\equiv\theta-\pi\,.$
(26)

On the other hand, a vanishing pion mass at the effective Lagrangian level
signals incongruities since it would imply no explicit breaking of chiral
symmetry. However, there is no chiral symmetry restoration in the fundamental
QCD Lagrangian. This apparent paradox is solved by pointing out that the
global flavour symmetry is still broken when including higher-order mass terms
in the effective Lagrangian Smilga:1998dh ; Tytgat:1999yx .

##  4 Dilaton augmented chiral Lagrangian and the large charge expansion

In this section, following Orlando:2019skh we consider the dynamics near the
lower edge of the conformal window to determine the ground-state energy on
non-trivial metrical backgrounds that can be associated with scaling
dimensions of QCD operators carrying (generalized) isospin charge. The first
step is to upgrade the chiral Lagrangian to a conformally invariant theory via
the introduction of a scalar degree of freedom $\sigma$, the _dilaton_ , which
under scale dilations $x\mapsto e^{\lambda}x$ transforms as

$\sigma\mapsto\sigma-\frac{\lambda}{f}\ .$ (27)

Scale invariance can then be enforced at the effective action level by
coupling $\sigma$ to each operator $\mathcal{O}_{k}$ of dimension $k$
appearing in the Lagrangian as coleman1988aspects ; goldberger2007light

$\mathcal{O}_{k}\mapsto e^{(k-4)\sigma f}\ \mathcal{O}_{k}\ .$ (28)

The resulting theory features non-linearly realized dilation invariance with
$f$ and $\sigma$ being the length scale and the Goldstone boson associated
with the spontaneous breaking of conformal symmetry, respectively. Explicit
breaking of the latter can be taken into account introducing a potential term
for $\sigma$. Consider perturbing a CFT with an operator $\mathcal{O}$ with
conformal dimension $\Delta$, i.e.

$\mathcal{L}_{\text{CFT}}\rightarrow\mathcal{L}_{\text{CFT}}+\lambda_{\mathcal{O}}\mathcal{O}\,,$
(29)

with $\lambda_{\mathcal{O}}$ the corresponding coupling. For
$\lambda_{\mathcal{O}}\ll 1$ the perturbation generates the following dilaton
potential rattazzi2001comments ; goldberger2007light ; Chacko:2012sy

$V(\sigma)=\frac{m_{\sigma}^{2}e^{-4f\sigma}}{4(4-\Delta)f^{2}}-\frac{m_{\sigma}^{2}e^{-\Delta
f\sigma}}{\Delta(4-\Delta)f^{2}}+\mathcal{O}(\lambda_{\mathcal{O}}^{2})\ .$
(30)

This potential has been considered in various recent studies of QCD-like
theories in the near-conformal regime appelquist2020dilaton ;
Appelquist:2022mjb ; Bersini:2022bnx as well as in the description of dense
skyrmion matter Li:2016uzn ; Ma:2021nuf ; Bersini:2023uat . For $\Delta=2$ it
reduces to the well-known Higgs-like quartic potential. Note that the
potential (30) cannot be trusted for $\Delta\ll 1$ and its singularity at
$\Delta=0$ is unphysical Bersini:2023uat . When the perturbation is nearly
marginal, i.e. $\Delta\to 4$, for arbitrary values of $\lambda_{\mathcal{O}}$
the potential reads goldberger2007light

$V(\sigma)=-\frac{m_{\sigma}^{2}e^{-4f\sigma}}{16f^{2}}(1+4f\sigma)+\mathcal{O}\left((\Delta-4)^{2}\right)\
.$ (31)

The above agrees with the potential derived according to the counting scheme
proposed in Golterman:2016lsd ; Golterman:2020tdq . Since eq.(31) is the
$\Delta\to 4$ limit of eq.(30), in what follows we will assume the generic
form of the dilaton potential eq.(30).

Finally, the mass term operator has dimension $y=3-\gamma$, with $\gamma$
being the anomalous dimension of the chiral condensate. Unitarity dictates
$0<\gamma<2$ whereas the underlying four fermion operator becomes near-
marginal for $\gamma\simeq 1$. For these reasons, this work focuses on the
range $0<\gamma<1$.

Our second step is to employ the large charge expansion framework
Hellerman:2015nra ; Badel:2019oxl ; Gaume:2020bmp to determine the scaling
dimension $\Delta_{Q}$ of the lowest-lying operator with charge $Q$. To this
end, we exploit the approximate Weyl invariance of the near-conformal theory
to map the latter onto the cylinder $\mathbb{R}\times S^{3}$. We denote the
volume, the radius, and the Ricci scalar of $S^{3}$ as $V$, $r$, and
$R=\frac{6}{r^{2}}$ respectively. The advantage is that we can now consider an
approximate state-operator correspondence which links $\Delta_{Q}$ to the
ground state energy $E_{Q}$ at fixed $Q$ of the theory on the cylinder as

$\Delta_{Q}=rE_{Q}\,,\qquad E_{Q}=\mu Q-\mathcal{L}\ ,$ (32)

The dilaton-pion effective Lagrangian on $\mathbb{R}\times S^{3}$ reads

$\begin{split}\mathcal{L}_{\sigma}&=\nu^{2}Tr\\{\partial_{\mu}\Sigma\partial^{\mu}\Sigma^{\dagger}\\}\
e^{-2\sigma f}+m^{2}_{\pi}\nu^{2}Tr\\{M\Sigma+M^{\dagger}\Sigma^{\dagger}\\}\
e^{-y\sigma f}+2\mu^{2}\nu^{2}Tr\\{II-\Sigma^{\dagger}I\Sigma I\\}e^{-2\sigma
f}\\\
&+2i\mu\nu^{2}Tr\\{I\partial_{0}\Sigma\Sigma^{\dagger}-I\Sigma^{\dagger}\partial_{0}\Sigma\\}e^{-2\sigma
f}-a\nu^{2}\left(\theta-\frac{i}{2}Tr\\{\log\Sigma-\log\Sigma^{\dagger}\\}\right)^{2}\
e^{-4\sigma f}-\Lambda^{4}_{0}\ e^{-4\sigma f}\\\
&+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{R}{6f^{2}}\right)\
e^{-2\sigma f}-V(\sigma)\,,\end{split}$ (33)

where for later convenience we included the bare cosmological constant
$\Lambda_{0}$. In the conformal limit $m_{\pi}=m_{\sigma}=0$, the ground state
energy $E_{Q}$ can then be computed via a semiclassical expansion in the
double scaling limit

$\Lambda_{0}f\to 0\ ,\quad Q\to\infty\ ,\quad
Q(\Lambda_{0}f)^{4}=\text{fixed.}$ (34)

This can be seen by considering the expectation value of the evolution
operator $U=e^{-HT}$ in an arbitrary state $\ket{Q}$ with charge $Q$

$\braket{U}_{Q}\equiv\braket{Q}{e^{-HT}}{Q}\underset{T\to\infty}{\to}\mathcal{N}e^{-E_{Q}T}\,,$
(35)

with $H$ the Hamiltonian, $T$ the time interval, and $\mathcal{N}$ an
unimportant normalization factor. Then one can rescale the fields as
$\Sigma\to\nu f\Sigma$ and $e^{-f\sigma}\to\sqrt{Q}e^{-f\sigma}$ to exhibit
$Q$ as a new counting parameter in the path integral expression for
$\braket{U}_{Q}$. Accordingly, the scaling dimension of the lowest-lying
operator assumes the following form

$E_{Q}R=\Delta_{Q}=\sum_{j=-1}\frac{1}{Q^{j}}\Delta_{j}\left(Q(\Lambda_{0}f)^{4}\right)\
.$ (36)

The leading order $\Delta_{-1}$ is given by the classical ground state energy
on $\mathbb{R}\times S^{3}$ while the next-to-leading order $\Delta_{0}$ is
determined by the fluctuations around the classical trajectory. We refer the
reader interested in the details of the approach to Badel:2019oxl ;
Antipin:2020abu ; Antipin:2022naw . In the next section, we will determine the
classical ground state energy in the large $Q\gg(\Lambda_{0}f)^{4}$ limit. As
we shall see, deviations from conformality will be encoded in a set of
contributions that depend on the spacetime geometry due to the lack of Weyl
invariance.

##  5 Large charge expansion: leading order

As anticipated in the previous section, the state-operator correspondence
enables us to deduce the scaling dimension for the lowest-lying operator with
(generalised) isospin charge $Q$. This is achieved by determining the energy
associated with the vacuum structure inducing the superfluid phase transition.
We therefore evaluate the Lagrangian (33) on the ansatz (14), obtaining

$\begin{split}\small\mathcal{L}_{\sigma}\left[\Sigma_{0},\sigma_{0}\right]&=-e^{-4f\sigma_{0}}\Lambda_{0}^{4}-V(\sigma_{0})-\frac{R\
e^{-2f\sigma_{0}}}{12f^{2}}+2m^{2}_{\pi}\nu^{2}X\cos\varphi\
e^{-f\sigma_{0}y}\\\
&+N_{f}\mu^{2}\nu^{2}e^{-2f\sigma_{0}}\sin^{2}\varphi-a\nu^{2}e^{-4f\sigma_{0}}{\bar{\theta}}^{2}\,.\end{split}$
(37)

It is worth noticing that, replacing $m_{\pi}\rightarrow\sqrt{2}\,m_{\pi}$ and
$\mu\rightarrow\sqrt{2}\,\mu$, the resulting expression is identical to that
found for 2-color QCD at finite baryon density, as explored in Bersini:2022bnx
. Nevertheless we further the results obtained in the previous work
Bersini:2022bnx by being able to solve for generic dilaton potentials (30) and
mass deformations rather than concentrating on specific values of $\Delta$ and
$y$.

The classical ground state energy is computed by solving the following EOMs

$\displaystyle\sin\varphi(N_{f}\mu^{2}e^{-2f\sigma_{0}}\cos\varphi-
m_{\pi}^{2}Xe^{-f\sigma_{0}y})$ $\displaystyle=0\,,$ (38) $\displaystyle
ae^{-4f\sigma_{0}}\bar{\theta}-m_{\pi}^{2}\sin\alpha_{i}\cos\varphi
e^{-f\sigma_{0}y}$ $\displaystyle=0\,,\qquad i=1,..,N_{f}\,,$ (39)
$\displaystyle\frac{Re^{-2f\sigma_{0}}}{6f}+4af\nu^{2}e^{-4f\sigma_{0}}\bar{\theta}^{2}+4f\Lambda_{0}^{4}e^{-4f\sigma_{0}}-\frac{\partial
V(\sigma)}{\partial\sigma}{\bigg{\rvert}}_{\sigma=\sigma_{0}}+$
$\displaystyle-2fN_{f}\mu^{2}\nu^{2}e^{-2f\sigma_{0}}\sin^{2}\varphi-2fym_{\pi}^{2}\nu^{2}X\cos\varphi
e^{-f\sigma_{0}y}$ $\displaystyle=0\,,$ (40) $\displaystyle
2N_{f}\mu\nu^{2}e^{-2f\sigma_{0}}\sin^{2}\varphi$
$\displaystyle=\frac{Q}{V}\,,$ (41)

where the last equation corresponds to fixing the isospin charge density. By
solving the EOM via an expansion in the dilaton and pion masses, at the
leading order in the double scaling limit (34), we obtain

$\begin{split}\Delta_{Q}=rE_{Q}&=\frac{\pi^{2}}{8f^{2}}\left(6N_{f}(f\mu\nu
r)^{2}+1\right)\left(\frac{2N_{f}(f\mu\nu r)^{2}-1}{f^{2}\Lambda^{4}}\right)\
+m_{\sigma}^{2}\ \frac{\pi^{2}2^{1-\Delta}r^{4-\Delta}}{(\Delta-4)\Delta
f^{2}}\left(\frac{2N_{f}(f\mu\nu
r)^{2}-1}{f^{2}\Lambda^{4}}\right)^{\Delta/2}\ \\\
&-m_{\pi}^{4}N_{f}\cos^{2}(\alpha(\theta))\ 2^{2\gamma-3}\left(\frac{\pi\nu
r^{\gamma+1}}{r\mu}\right)^{2}\left(\frac{2N_{f}(f\mu\nu
r)^{2}-1}{f^{2}\Lambda^{4}}\right)^{2-\gamma}\
+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\,,\end{split}$
(42)

where $\mu$ is related to $Q$ as

$r\mu=\frac{\left(6\pi^{4}\nu^{2}N_{f}\right)^{1/3}+\left(\sqrt{81f^{6}\Lambda^{8}Q^{2}-6\pi^{4}\nu^{2}N_{f}}+9f^{3}\Lambda^{4}Q\right)^{2/3}}{f\left(6\pi\nu^{2}N_{f}\right)^{2/3}\left(\sqrt{81f^{6}\Lambda^{8}Q^{2}-6\pi^{4}\nu^{2}N_{f}}+9f^{3}\Lambda^{4}Q\right)^{1/3}}\
.$ (43)

The first term in (42) represents the scaling dimension in the conformal limit
$m_{\pi}=m_{\sigma}=0$ which depends only on the dimensionless combination
$\mu r$ (43) and, therefore, is insensitive to the spacetime geometry.
Noticeably, the leading correction in the pion mass is of order $m_{\pi}^{4}$
and its dependence on the geometry is tied to the anomalous dimension of the
chiral condensate through the universal factor $r^{2(\gamma+1)}$. Remarkably,
the whole $\theta$-dependence is encoded in the factor
$\cos^{2}(\alpha(\theta))$ with $\alpha(\theta)$ given in eq.(22). The first
term in the dilaton potential (30) redefines the cosmological constant as

$\Lambda^{4}\equiv\Lambda^{4}_{0}+\frac{m^{2}_{\sigma}}{4f^{2}(4-\Delta)}\,.$
(44)

The contribution stemming from the second term in the dilaton potential (30)
has been expanded in powers of $m_{\sigma}$ with the leading order quadratic
in the dilaton mass. The latter exhibits a universal dependence on the radius
of the sphere via the factor $r^{4-\Delta}$. It is interesting to further
expand our results (42) in the large charge limit $Q(\Lambda_{0}f)^{4}\gg 1$
where we can make contact with the universal predictions of the large charge
effective field theory (EFT) Hellerman:2015nra ; Gaume:2020bmp . We have

$\Delta_{Q}=rE_{Q}=\Delta_{Q}^{\ast}+\left(\frac{m_{\sigma}}{4\pi\nu}\right)^{2}\,Q^{\frac{\Delta}{3}}B_{1}+\left(\frac{m_{\pi}}{4\pi\nu}\right)^{4}\
\cos^{2}(\alpha(\theta))\
Q^{\frac{2}{3}(1-\gamma)}B_{2}+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\,,$
(45)

where $B_{1}$ and $B_{2}$ read

$\displaystyle B_{1}$
$\displaystyle=\frac{c_{2/3}2^{9-2\Delta}3^{\frac{\Delta}{2}-1}(\pi\nu
r)^{4-\Delta}\left(c_{4/3}N_{f}\right){}^{1-\frac{\Delta}{2}}}{(\Delta-4)\Delta}\left(1-\frac{\Delta\
c_{2/3}}{4c_{4/3}}Q^{-2/3}+{\cal O}\left(Q^{-4/3}\right)\right)\,,$ (46)
$\displaystyle B_{2}$ $\displaystyle=-\
3^{4-\gamma}2^{4\gamma-3}\pi^{2\gamma+2}c_{4/3}^{\gamma-4}N_{f}^{\gamma-1}(\nu
r)^{2(\gamma+1)}\left(1+\frac{(\gamma-4)c_{2/3}}{2c_{4/3}}Q^{-2/3}+{\cal
O}\left(Q^{-4/3}\right)\right)\,,$ (47)

while

$\Delta_{Q}^{\ast}=rE_{Q}=c_{4/3}Q^{4/3}+c_{2/3}Q^{2/3}+{\cal
O}\left(Q^{0}\right)\,,$ (48)

is the scaling dimension in the conformal limit which depends only on the
dimensionless coefficients defined below

$c_{4/3}=\frac{3}{8}\left(\frac{2\Lambda^{2}}{\pi
N_{f}\nu^{2}}\right)^{2/3}\,,\quad
c_{2/3}=\frac{1}{4f^{2}}\left(\frac{2\pi^{2}}{N_{f}\nu^{2}\Lambda^{4}}\right)^{1/3}\,.$
(49)

The conformal dimension $\Delta_{Q}^{\ast}$ exhibits the general structure
predicted by the large charge EFT. The non-conformal corrections feature a
characteristic $Q$-scaling which has been made manifest in eq.(45) and depends
on the parameters $\gamma$ and $\Delta$ encoding the explicit breaking of
scale invariance.

##  6 SSB and spectrum of fluctuations

We now move to determine the spectrum of the theory including the symmetry
breaking pattern starting from the latter. In fact, fixing the generalized
isospin charge results in the following symmetry breaking pattern

$\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times
U(1)_{V}\overset{N_{f}^{2}-1}{\rightsquigarrow}\ $ $\displaystyle
SU(N_{f})_{V}\times U(1)_{V}\longrightarrow
SU\left(\frac{N_{f}}{2}\right)_{u}\times
SU\left(\frac{N_{f}}{2}\right)_{d}\times U(1)_{I}\times U(1)_{V}$
$\displaystyle\overset{\frac{N_{f}^{2}}{4}}{\rightsquigarrow}SU\left(\frac{N_{f}}{2}\right)_{ud}\times
U(1)_{V}\,,$ (50)

where $\rightsquigarrow$ and $\longrightarrow$ denote, respectively,
spontaneous and explicit breaking. The first stage is the usual chiral
symmetry breaking with the ABJ anomaly already taken into account in the
breaking of the axial symmetry. The further explicit breaking is owed to the
introduction of the isospin charge while the last spontaneous breaking is
associated with pion condensation and the superfluid phase transition. In the
absence of the dilaton, the spectrum of light modes is composed of
$N_{f}^{2}/4$ massless Goldstone bosons with speed $v_{G}=1$ that parameterize
the coset $G/H=\displaystyle{\frac{SU(N_{f}/2)_{u}\times SU(N_{f}/2)_{d}\times
U(1)_{I}\times U(1)_{V}}{SU(N_{f}/2)_{ud}\times U(1)_{V}}}$. These modes
arrange themselves in the adjoint representation of the stability group
$SU(N_{f}/2)_{udV}\times U(1)_{V}$ plus a singlet which we denote as $\pi_{3}$
since it is associated with the spontaneous breaking of $U(1)_{I}$ i.e. to the
third Pauli matrix in the $N_{f}=2$ case. In addition, a pseudo-Goldstone mode
stems from the would-be spontaneous breaking of $U(1)_{A}$ which we call the
$S$ (singlet) mode and it is related to the $\eta^{\prime}$-meson
DiVecchia:2013swa . As mentioned above, the $U(1)_{A}$ symmetry is quantum
mechanically anomalous, and therefore the latter mode acquires a mass
proportional to the scale of the anomaly factor $\sqrt{a}$.

In what follows, we shall focus on the spectrum of Goldstone bosons since they
control the large charge dynamics. Specifically, we are interested in
analyzing how the spectrum of light modes changes when (near)conformal
dynamics is realized through the dilaton dressing. Precisely, conformal
invariance dictates the existence of a massless mode with speed
$v_{G}=\frac{1}{\sqrt{d-1}}=\frac{1}{\sqrt{3}}$ Hellerman:2015nra ; Son:2002zn
. As we shall see, the latter arises from the mixing between the singlet
$\pi_{3}$ with the dilaton that acts as its "radial mode" and changes its
speed from $v_{G}=1$ to $v_{G}=\frac{1}{\sqrt{3}}$. We consider the hierarchy
of scales $m_{\pi},m_{\sigma}\ll\mu\ll 4\pi\nu$ which ensures the validity of
chiral perturbation theory and assumes small deviation from conformality. To
determine the fluctuations’ spectrum, we expand around the vacuum solution as

$\displaystyle\Sigma=e^{\frac{2i}{\sqrt{N_{f}}}S\mathbbm{1}_{N_{f}}}e^{i\Omega}\Sigma_{0}e^{-i\Omega^{\dagger}}\,,$
(51)

where $\Sigma_{0}$ is the classical solution (8) while the fluctuations are
organized in the matrix $\Omega$ as

$\displaystyle\Omega=\left(\begin{array}[]{cc}\pi&0\\\ 0&-\pi^{t}\\\
\end{array}\right)\,.$ (54)

Here $\pi=\sum_{a=0}^{\text{dim G/H}}\pi^{a}T_{a}$ belongs to the algebra of
the coset space and we normalized the generators as
$Tr\left\\{T_{a}T_{b}\right\\}=\frac{\delta_{ab}}{2}$. After some
manipulations, we obtain

$\displaystyle
Tr\left\\{\partial_{\mu}\Sigma\partial^{\mu}\Sigma^{\dagger}\right\\}$
$\displaystyle=4\sin^{2}\varphi\
\partial_{\mu}\pi^{a}\partial^{\mu}\pi^{a}+4\partial_{\mu}S\partial^{\mu}S\,,$
(55) $\displaystyle
Tr\left\\{I\partial_{0}\Sigma\Sigma^{\dagger}-I\Sigma^{\dagger}\partial_{0}\Sigma\right\\}$
$\displaystyle=4iTr\left\\{I\partial_{0}\nu\right\\}\sin^{2}\varphi=2i\sqrt{N_{f}}\,\partial_{0}\pi^{3}\sin^{2}\varphi\,,$
(56) $\displaystyle Tr\left\\{M\Sigma+M^{\dagger}\Sigma^{\dagger}\right\\}$
$\displaystyle=2\cos\varphi\left[X\cos\left(\frac{2}{\sqrt{N_{f}}}S\right)+Z\sin\left(\frac{2}{\sqrt{N_{f}}}S\right)\right]\,,$
(57) $\displaystyle Tr\left\\{\log\Sigma-\log\Sigma^{\dagger}\right\\}$
$\displaystyle=4i\sqrt{N_{f}}S-2i\sum_{i}^{N_{f}}\alpha_{i}\ ,$ (58)

where we defined $Z\equiv\sum_{i=1}^{N_{f}}\sin\alpha_{i}$. Finally, by
expanding the dilaton field around its background solution as
$\sigma\rightarrow\sigma_{0}+\hat{\sigma}(t,\mathbf{x})$, we arrive at the
following quadratic Lagrangian

$\displaystyle\frac{\mathcal{L}_{\sigma}^{(2)}}{4\nu^{2}\sin^{2}\varphi\
e^{-2\sigma_{0}f}}=\left(\begin{array}[]{lll}\pi^{3}&\hat{\sigma}&S\end{array}\right)D^{-1}\left(\begin{array}[]{l}\pi^{3}\\\
\hat{\sigma}\\\ S\end{array}\right)+\sum_{a=1,a\neq
3}^{\text{dim}[{\tiny\yng(2,1)}]}\partial_{\mu}\pi^{a}\partial^{\mu}\pi^{a}\,,$
(63)

where the inverse propagator $D^{-1}$ is given by

$\displaystyle D^{-1}$
$\displaystyle=\left(\begin{array}[]{ccc}\omega^{2}-k^{2}&i\omega\mu
f\sqrt{N_{f}}&0\\\ -i\omega\mu
f\sqrt{N_{f}}&\frac{\omega^{2}-k^{2}}{8\nu^{2}\sin^{2}\varphi}-M_{\sigma}^{2}&\frac{1}{2}I_{\hat{\sigma}s}\\\
0&\frac{1}{2}I_{\hat{\sigma}s}&\frac{\omega^{2}-k^{2}}{\sin^{2}\varphi}-M_{s}^{2}\end{array}\right)\
,$ (67)

with

$\displaystyle I_{\hat{\sigma}S}$
$\displaystyle=\frac{f\mu^{2}\sqrt{N_{f}}\left(4a\mu^{2}\bar{\theta}N_{f}^{2}e^{-2f\sigma_{0}\gamma}-m_{\pi}^{4}XZ(3-\gamma)\right)}{\left(\mu^{4}N_{f}^{2}e^{2f\sigma_{0}(1-\gamma)}-m_{\pi}^{4}X^{2}\right)}\,,$
(68) $\displaystyle M_{\sigma}^{2}$
$\displaystyle=\frac{\mu^{2}N_{f}}{8\nu^{2}}\Bigg{[}\frac{\Delta\mu^{2}N_{f}m_{\sigma}^{2}e^{-f(\Delta-2)\sigma}}{(\Delta-4)\left(\mu^{4}N_{f}^{2}-m_{\pi}^{4}X^{2}e^{2(\gamma-1)f\sigma}\right)}$
$\displaystyle+\left.\frac{2f^{2}\left(2\mu^{2}N_{f}\left(\mu^{2}\nu^{2}N_{f}e^{2f\sigma}-4\left(a\nu^{2}\bar{\theta}^{2}+\Lambda^{4}\right)\right)+((\gamma-6)\gamma+7)\nu^{2}m_{\pi}^{4}X^{2}e^{2\gamma
f\sigma}\right)}{m_{\pi}^{4}X^{2}e^{2\gamma
f\sigma}-\mu^{4}N_{f}^{2}e^{2f\sigma}}\right)\Bigg{]}\,,$ (69) $\displaystyle
M_{S}^{2}$
$\displaystyle=\frac{a\mu^{4}N_{f}^{3}+\mu^{2}m_{\pi}^{4}X^{2}e^{2\gamma
f\sigma_{0}}}{\mu^{4}N_{f}^{2}e^{2f\sigma_{0}}-m_{\pi}^{4}X^{2}e^{2\gamma
f\sigma_{0}}}\ .$ (70)

The $\pi_{a}$ modes with $a\neq 3$ appearing in eq.(63) denote the Goldstone
modes transforming under the adjoint representation of $SU(N_{f}/2)$ and have
trivial dispersion relations $\omega=k$. On the other hand, $\pi_{3}$
transforms as a singlet of $SU(N_{f}/2)$, and due to this property, it mixes
with the dilaton and the $S$, as it is manifest in eq.(67). We pause to note
that eq.(70) is consistent with the Witten-Veneziano relation Witten:1979vv ;
Veneziano:1979ec . In fact, in the $m_{\pi}\to 0$ limit,$M_{S}^{2}$ reduces to

$\lim_{\sigma_{0}\to 0,\ m_{\pi}\to 0}\eqref{massofS}\implies
M^{2}_{S}=aN_{f}\ .$ (71)

The remaining dispersion relations are obtained by solving the equation
det$(D^{-1})$. The results describe two gapped modes $\omega_{1,2}$ with mass

$\displaystyle M^{2}_{1,2}$
$\displaystyle=-\frac{1}{2}\sin^{2}\varphi\left[M_{S}^{2}+8\nu^{2}f^{2}\mu^{2}N_{f}+8\nu^{2}M_{\sigma}^{2}\pm\sqrt{\left(M_{S}^{2}-8\nu^{2}\left(f^{2}\mu^{2}N_{f}+M_{\sigma}^{2}\right)\right){}^{2}+8\nu^{2}I_{\hat{\sigma}S}^{2}}\right]\,,$
(72)

and a massless mode $\omega_{3}$ with speed given by

$\displaystyle v_{3}^{2}$
$\displaystyle=\frac{I_{\hat{\sigma}s}^{2}-4M^{2}_{\sigma}M^{2}_{S}}{I_{\hat{\sigma}s}^{2}-4M^{2}_{S}\left(M^{2}_{\sigma}+f^{2}N_{f}\mu^{2}\right)}\
.$ (73)

In parallel with the previous section, we determine the dispersion relations
considering corrections in both $m_{\sigma}^{2}$ and $m_{\pi}^{2}$. According
to this expansion, we obtain

$\displaystyle\omega^{2}_{i}$
$\displaystyle=\omega^{2\ast}_{i}+\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{\frac{\Delta-2}{2}}m^{2}_{\sigma}D_{1}(\omega_{i}^{2\ast},\Delta)+\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{-\gamma-3}m^{4}_{\pi}\cos^{2}(\alpha(\theta))D_{2}(\omega_{i}^{2\ast},\gamma)$
$\displaystyle+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\,,\qquad\text{for}\
i=1,2,3\,.$ (74)

The coefficients $D_{1}$ and $D_{2}$ read

$\displaystyle D_{1}(\omega_{i}^{2\ast},\Delta)$
$\displaystyle=\frac{1}{12(\Delta-4)f^{2}\left(\mu^{2}\nu^{2}N_{f}\left(aN_{f}\left(6f^{2}\mu^{2}\nu^{2}N_{f}+k^{2}-\omega_{i}^{2\ast}\right)+8f^{2}\Lambda^{4}\left(2k^{2}-3\omega_{i}^{2\ast}\right)\right)+3\Lambda^{4}\left(k^{2}-\omega_{i}^{2\ast}\right)^{2}\right)}$
$\displaystyle\times\left[aN_{f}\left(2f^{2}\mu^{2}\nu^{2}N_{f}\left((3\Delta-10)k^{2}-3(\Delta-2)\omega_{i}^{2\ast}\right)-\left(k^{2}-\omega_{i}^{2\ast}\right)^{2}\right)\right.$
$\displaystyle\left.+4f^{2}\Lambda^{4}\left(k^{2}-\omega_{i}^{2\ast}\right)\left((3\Delta-8)k^{2}-3\Delta\omega_{i}^{2\ast}\right)\right]\,,$
(75) $\displaystyle D_{2}(\omega_{i}^{2\ast},\gamma)$
$\displaystyle=\frac{\mu^{4}\nu^{8}\Lambda^{-4(\gamma+4)}(\mu\nu)^{2\gamma}N_{f}^{\gamma+4}}{96\left(\mu^{2}\nu^{2}N_{f}\left(aN_{f}\left(k^{2}-\omega_{i}^{2\ast}\right)+8f^{2}\Lambda^{4}\left(2k^{2}-3\omega_{i}^{2\ast}\right)\right)+6af^{2}\mu^{4}\nu^{4}N_{f}^{3}+3\Lambda^{4}\left(k^{2}-\omega_{i}^{2\ast}\right)^{2}\right)}$
$\displaystyle\times\left[-2(\gamma(3\gamma-10)-5)f^{2}\Lambda^{4\gamma}\mu^{2-2\gamma}\left(k^{2}-\omega_{i}^{2\ast}\right)\left(\nu^{2}N_{f}\right)^{1-\gamma}\left(a\mu^{2}\nu^{2}N_{f}^{2}+2\Lambda^{4}\left(k^{2}-\omega_{i}^{2\ast}\right)\right)\right.$
$\displaystyle\left.+3\left(k^{2}-\omega_{i}^{2\ast}\right)^{2}\left(\frac{\mu^{2}\nu^{2}N_{f}}{\Lambda^{4}}\right)^{-\gamma}\left(a\mu^{2}\nu^{2}N_{f}^{2}+2\Lambda^{4}\left(k^{2}-\omega_{i}^{2\ast}\right)\right)-16(\gamma-1)f^{2}\omega_{i}^{2\ast}\Lambda^{4\gamma}\mu^{2-2\gamma}\left(\nu^{2}N_{f}\right)^{1-\gamma}\right.$
$\displaystyle\left.\times\left(a\mu^{2}\nu^{2}N_{f}^{2}+2\Lambda^{4}\left(k^{2}-\omega_{i}^{2\ast}\right)\right)+\Lambda^{4\gamma}(\mu\nu)^{-2\gamma}N_{f}^{-\gamma}\left(4f^{2}\mu^{2}\nu^{2}N_{f}\left(k^{2}-3\omega_{i}^{2\ast}\right)+\left(k^{2}-\omega_{i}^{2\ast}\right)^{2}\right)\right.$
$\displaystyle\times\left.\left(6\Lambda^{4}\left(k^{2}+\mu^{2}-\omega_{i}^{2\ast}\right)-a(\gamma-7)\mu^{2}\nu^{2}N_{f}^{2}\right)\right]$
(76)

while the dispersion relations in the conformal limit $m_{\pi}=m_{\sigma}=0$
have the following simple form

$\displaystyle\omega^{2\ast}_{1}$
$\displaystyle=k^{2}+6f^{2}\mu^{2}\nu^{2}N_{f}+2f\mu\nu\sqrt{N_{f}\left(9f^{2}\mu^{2}\nu^{2}N_{f}+2k^{2}\right)}\,,$
(77) $\displaystyle\omega^{2\ast}_{2}$
$\displaystyle=k^{2}+\frac{a\mu^{2}\nu^{2}N_{f}^{2}}{2\Lambda^{4}}\,,$ (78)
$\displaystyle\omega^{2\ast}_{3}$
$\displaystyle=k^{2}+6f^{2}\mu^{2}\nu^{2}N_{f}-2f\mu\nu\sqrt{N_{f}\left(9f^{2}\mu^{2}\nu^{2}N_{f}+2k^{2}\right)}\,,$
(79)

where we can recognize the expected mode with a square mass of order $a$
stemming from the axial anomaly as well as a gapped mode with mass
$12N_{f}f^{2}\mu^{2}\nu^{2}$ and a Goldstone boson with speed
$v_{3}^{2}=\frac{1}{3}$ as dictated by scale invariance. We conclude the
section by providing explicit expression for $M_{1,2}^{2}$ and $v_{3}^{2}$

$\displaystyle v^{2}_{3}$ $\displaystyle=\frac{1}{3}+m^{2}_{\sigma}\
\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{\Delta/2}\frac{\Lambda^{4}}{9f^{2}N_{f}^{2}\mu^{4}\nu^{4}}$
$\displaystyle-m^{4}_{\pi}\cos^{2}(\alpha(\theta))\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{-\gamma-3}\frac{\mu^{4}\nu^{8}N_{f}^{2}}{144\Lambda^{16}}(\gamma-3)(\gamma+1)+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\,,$
(80) $\displaystyle M^{2}_{1}$
$\displaystyle=12N_{f}f^{2}\mu^{2}\nu^{2}+m^{2}_{\sigma}\
\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{\Delta/2}\frac{\Delta}{\Delta-4}\left(\frac{2\Lambda^{4}}{N_{f}\mu^{2}\nu^{2}}\right)$
$\displaystyle-m^{4}_{\pi}\cos^{2}(\alpha(\theta))\
\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{-\gamma-3}\frac{f^{2}\mu^{6}\nu^{10}N_{f}^{3}}{8\Lambda^{16}}(\gamma^{2}-6\gamma+7)+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\,,$
(81) $\displaystyle M^{2}_{2}$
$\displaystyle=\frac{a\mu^{2}\nu^{2}N_{f}^{2}}{2\Lambda^{4}}-m^{2}_{\sigma}\
\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{\Delta/2}\frac{a}{6(\Delta-4)f^{2}\mu^{2}\nu^{2}}$
$\displaystyle-m^{4}_{\pi}\cos^{2}(\alpha(\theta))\
\left(\frac{N_{f}\mu^{2}\nu^{2}}{2\Lambda^{4}}\right)^{-\gamma-3}\frac{\mu^{6}\nu^{8}N_{f}^{2}}{96\Lambda^{20}}\left(a(\gamma-4)\nu^{2}N_{f}^{2}-6\Lambda^{4}\right)\
+\mathcal{O}\left(m_{\sigma}^{4},m_{\pi}^{8},m_{\sigma}^{2}m_{\pi}^{4}\right)\,.$
(82)

##  7 Casimir energy contribution to $\Delta_{Q}$ in the conformal limit

As discussed in Sec.4, in the conformal limit $m_{\pi}=m_{\sigma}=0$ our
results (45) correspond to the leading contribution to the conformal dimension
$\Delta_{Q}$ of the lowest-lying charge $Q$ operator in the double scaling
limit (34). However, as has been first shown in Hellerman:2015nra ,
$\Delta_{Q}$ can also be computed in the strongly coupled regime by
constructing an EFT for the relativistic Goldstone modes stemming from the SSB
induced by fixing the charge. In $d=4$ dimensions, the large charge EFT
predicts Hellerman:2015nra ; Cuomo:2020rgt

$\Delta_{Q}=k_{4/3}Q^{4/3}+k_{2/3}Q^{2/3}+k_{0}\log Q+{\cal
O}\left(Q^{0}\right)\,,$ (83)

where the coefficients $k_{4/3}$ and $k_{2/3}$ are related to the Wilson
coefficients of the EFT and cannot, therefore, be computed within the EFT
approach. The calculated coefficients $c_{4/3}$ and $c_{2/3}$ given in eq.(49)
can be seen as the leading contribution to $k_{4/3}$ and $k_{2/3}$ in a
perturbative expansion of the latter around $(\Lambda_{0}f)^{4}=0$
Badel:2019oxl . On the other hand, the coefficient $k_{0}$ is a purely quantum
contribution related to the Casimir energy of the relativistic Goldstone
bosons. Importantly, its value is universal being entirely determined by
symmetry and the number of spacetime dimensions. In particular, it can be
computed exactly from the knowledge of the low energy spectrum. In fact,
consider the low-energy action for a Goldstone mode $\chi$

$S_{G}=\int_{\mathcal{M}}\ \differential t\ \differential\mathbf{x}\
\left(\frac{1}{2}(\partial_{t}\chi)^{2}+\frac{v^{2}_{G}}{2}\big{(}\gradient{\chi}\big{)}^{2}\right)\
.$ (84)

The corresponding Casimir energy is given by

$E_{\rm
Casimir}=\frac{1}{2}Tr\\{\log\big(-\partial^{2}_{t}-v^{2}_{G}\laplacian\big)\\}=\frac{1}{4\pi}\int_{-\infty}^{\infty}\
\differential\omega\sum_{\mathbf{p}}\
\log\big(\omega^{2}+v^{2}_{G}E^{2}(\mathbf{p})\big{missing})=\frac{v_{G}}{2}\sum_{\mathbf{p}}E(\mathbf{p})\,.$
(85)

Here $E(\mathbf{p})^{2}$ denotes the eigenvalues of the Laplacian operator on
$S^{3}$. The above contribution scales as $Q^{0}$ and exhibits a pole for
$d\to 4$ in dimensional regularization. The latter is related to a $\log Q$
term with a universal coefficient $-\frac{v_{G}}{48}$ stemming from the
renormalization of the vacuum energy Cuomo:2020rgt . Hence, to calculate
$k_{0}$ we simply need to sum the contributions of the various Goldstone
bosons. The symmetry breaking pattern in the chiral limit reads

$\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}\times
U(1)_{A}$ $\displaystyle\rightarrow SU\left(\frac{N_{f}}{2}\right)_{uL}\times
SU\left(\frac{N_{f}}{2}\right)_{dL}\times
SU\left(\frac{N_{f}}{2}\right)_{uR}\times
SU\left(\frac{N_{f}}{2}\right)_{dR}\times U(1)_{I}\times U(1)_{V}$
$\displaystyle\overset{\frac{3}{4}N_{f}^{2}-2}{\leadsto}SU\left(\frac{N_{f}}{2}\right)_{ud}\times
U(1)_{V}\,.$ (86)

The resulting $\frac{3}{4}N_{f}^{2}-2$ Goldstone bosons have speed $v_{G}=1$
except the $\pi^{3}$ mode which has $v_{G}=\frac{1}{\sqrt{3}}$. We therefore
obtain

$k_{0}=-\frac{1}{48}\left(\frac{1}{\sqrt{3}}+\frac{3}{4}N_{f}^{2}-3\right).$
(87)

##  8 Conclusions

We uncovered near-conformal properties of finite isospin density QCD on a non-
trivial gravitational background. Specifically, we determined the ground state
energy $\Delta_{Q}/r$ with $r$ radius of $\mathbb{R}\times S^{3}$ via the
semiclassical large charge expansion. In the conformal limit this energy maps,
via state-operator correspondence, into the scaling dimension
$\Delta_{Q}^{\ast}$ of the lowest lying operator of fixed isospin charge $Q$.

One of our main results given in (3) is the determination of the leading near
conformal corrections to $\Delta_{Q}^{\ast}$ at the lower boundary of the QCD
conformal window. We showed that the characteristic $Q$-scalings due to the
near conformal corrections are induced by the quark mass operator anomalous
dimension $\gamma$ as well as the conformal dimension $\Delta$ of the operator
responsible for dynamically deforming QCD away from the conformal window. Our
results and methodology work as a template to obtain similar results for QCD-
like theories such as two-color QCD at nonzero baryon density.

Additionally, we determined the pattern of symmetry breaking and the
associated physical spectrum and their dispersion relations. These latter
results will also help determine the next-to-leading order large charge
contributions. Last but not least we discussed the $\mu-\theta$ QCD phase
diagram.

## Acknowledgments

The work of J.B. was supported by the World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan; and also supported by the
JSPS KAKENHI Grant Number JP23K19047. The work of F.S. is partially supported
by the Carlsberg Foundation, grant CF22-0922. A.D.A. expresses sincere
appreciation to the University of Southern Denmark and D-IAS for their
hospitality during the crucial final stages of the work.

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