# Non-commutative probability, joint distributions, conditioning, and the
associated polymorphisms

Palle E.T. Jorgensen  (Palle E.T. Jorgensen) Department of Mathematics, The
University of Iowa, Iowa City, IA 52242-1419, U.S.A.  palle-
<EMAIL_ADDRESS>and  James Tian  (James F. Tian) Mathematical Reviews, 416
4th Street Ann Arbor, MI 48103-4816, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We present a parallel between commutative and non-commutative polymorphisms.
Our emphasis is the applications to conditional distributions from stochastic
processes. In the classical case, both the measures and the positive definite
kernels are scalar valued. But the non-commutative framework (as motivated by
quantum theory) dictates a setting where instead now both the measures (in the
form of quantum states), and the positive definite kernels, are operator
valued. The non-commutative theory entails a systematic study of positive
operator valued measures, abbreviated POVMs. And quantum states (normal
states) are indexed by normalized positive trace-class operators. In the non-
commutative theory, the parallel to the commutative/scalar valued theory helps
us understand entanglement in quantum information. A further implication of
our study of the non-commutative framework will entail an interplay between
the two cases, scalar valued, vs operator valued.

###### Key words and phrases:

Polymorphisms, positive definite kernels, non-commutative Radon-Nikodym,
POVMs, quantum states, quantum information, operator-valued kernels.

###### 2020 Mathematics Subject Classification:

Primary 46L53. Secondary 46E22, 47A20, 60E05, 81P15, 81P47.

###### Contents

  1. 1 Introduction
  2. 2 Commutative polymorphisms
  3. 3 Positive operator valued measures (POVMs)
  4. 4 Quantum polymorphisms
  5. 5 Beyond the $Q$-polymorphisms

##  1\. Introduction

Motivated by questions in quantum information, we suggest here a non-
commutative setting for multi-variable measurement problems. With this we
extend both key notions and results from the framework of classical
polymorphism and Rokhlin disintegration theory to a particular non-commutative
setting. As is in the nature of quantum measurements [NGS+19, RB24, KMDS24]
such extensions will by necessity involve choices.

Our paper has two aims: we (1) present a non-commutative framework, and (2)
offer a noncommutative Rokhlin-type result that serves as a direct integral
disintegration.

To help motivate our results, we reference our earlier papers [JT24d, JT24b,
JT24a, JT24c], as well as the papers cited there [Arv10, SCC21, AA22, Gue22,
Eva80, JNP+11, BLW14, NGS+19, SW20].

For a direct link between Rokhlin/polymorphism on the one hand, and direct
integrals in Hilbert space on the other, see e.g., [HJW23]. While the use of
the rigorous Rokhlin factorizations/direct integrals, corresponding to classes
of polymorphisms, already serves as a powerful tool in mathematics, we note
that it further provides a precise (mathematical) formulation of conditioning
and Bayes’ rules in probability, i.e., describing the probability of an event,
based on prior knowledge of conditions influencing the event; see (2.1), and
(2.2). We show in Sections 3–5 of the paper that this approach in fact also
carries over to the non-commutative case, i.e., to quantum observables; see
especially (3.7), (4.2), (4.3), (4.7), and (4.8).

##  2\. Commutative polymorphisms

The notion, and the study of, polymorphisms was pioneered by Anatoly M.
Vershik, see [Ver05] and the papers cited below. The basic idea is that while
functions transform sets, we need multi-valued maps and other generalizations
to study different types of dynamics in measure theory. Polymorphisms,
systematically studied by Vershik, have many applications (see e.g., [HJW19,
JST20, JT20, JT21, BJ21, JT22, BJ22]). This includes Markov maps, transition
operators, two-dimensional distributions of Markov processes, and joinings of
measure spaces. Similar concepts in other fields include correspondences in
algebra, bifibrations in differential geometry, and Markov maps in probability
theory. In physics, polymorphism relates to the "coarse-graining" approach to
dynamics, where maps send a point to measures.

Recall that a given positive measure $\nu$ on $X\times X$
($\left(X,\mathscr{B}\right)$ being a measurable space) relative to the
product $\sigma$-algebra $\mathscr{B}_{2}$ is said to be a polymorphism for
$\mu_{1}$ and $\mu_{2}$ when $\mu_{i}$ are the respective marginal measures
$\mu_{i}:=\nu\circ\pi_{i}^{-1}$ relative to the projections
$\pi_{i}\left(\left(x_{1},x_{2}\right)\right)=x_{i}$, $i=1,2$.

Assume $\left(X,\mathscr{B}\right)$ is a standard measurable space. Let
$\mu_{1}\leftarrow\nu\rightarrow\mu_{2}$ be a polymorphism, then the marginal
measures $\mu_{i}$ yield conditional measures $\nu\left(\cdot\mid
x_{i}\right)$ defined as follows:

$\displaystyle\nu\left(dx_{1}dx_{2}\right)$
$\displaystyle=\mu_{1}\left(dx_{1}\right)\nu\left(dx_{2}\mid x_{1}\right)$
$\displaystyle=\mu_{2}\left(dx_{2}\right)\nu\left(dx_{1}\mid x_{2}\right),$
(2.1)

or equivalently, with $\nu$ specified on product sets $A\times B$,
$A,B\in\mathscr{B}$:

$\displaystyle\nu\left(A\times B\right)$
$\displaystyle=\int_{A}\mu_{1}\left(dx_{1}\right)\nu\left(B\mid x_{1}\right)$
$\displaystyle=\int_{B}\mu_{2}\left(dx_{2}\right)\nu\left(A\mid x_{2}\right).$
(2.2)

Formulas (2.1) and (2.2) are closely related to the Bayes rules in probability
and the slice decompositions of measures on product spaces in ergodic theory.
Bayes’ rules describe the probability of events based on prior knowledge of
conditions related to those events, hence conditional distributions.

In this paper, we consider a quantum version of polymorphisms within a non-
commutative setting. Extending results from the classical, commutative
framework to the non-commutative case allows us to explore new quantum
dynamics.

We will avoid certain technical details. However, for experts, we follow
Vershik in assuming our spaces are standard measure spaces. We recall that
Rokhlin’s papers on disintegrations cover this issue in depth, especially in
"separable" measure spaces. Our formulation Theorem 4.1 and following sections
assume certain properties for the measure spaces, specifically standard
measure spaces. This presentation is meant to be accessible to beginners. For
more details, readers can refer to [BJ21, BJ22, Ver05], and to the papers
cited there.

##  3\. Positive operator valued measures (POVMs)

In the below, we offer a non-commutative framework for polymorphisms. While it
is motivated by questions in quantum information, we believe that it is also
of independent interest. Recall (see above), that in the classical case, we
deal with a framework where both the measures and the positive definite
kernels are assumed scalar valued. But the non-commutative framework (as
motivated by quantum theory) dictates a setting where instead now both the
measures (in the form of quantum states), and the positive definite kernels,
are operator valued. Furthermore, the non-commutative theory entails a
systematic study of positive operator valued measures, abbreviated POVMs.

Our focus below will include the setting when quantum states are normal states
(in the sense of operator algebras), i.e., they are indexed by normalized
positive trace-class operators, see e.g., [Sak71, Sak68]. For the non-
commutative counterparts, the parallel commutative/scalar valued framework of
polymorphisms in Section 2 above helps us understand entanglement as it arises
in quantum information. A further implication of our study of the non-
commutative framework is that the study of important questions will entail an
interplay between the two cases, scalar valued, vs operator valued.

Let $\left(S,\mathscr{B}_{S}\right)$ be a measurable space, where $S$ is a
set, and $\mathscr{B}_{S}$ is a $\sigma$-algebra of subsets of $S$. Fix a
separable Hilbert space $\mathscr{H}$, and let
$\mathscr{L}\left(\mathscr{H}\right)$ denote the algebra of all bounded
operators in $\mathscr{H}$. By $\mathscr{H}$-valued (weakly) measurable
functions

$F:\left(S,\mathscr{B}_{S}\right)\rightarrow\mathscr{H},$

we mean that $s\mapsto\left\langle
h,F\left(s\right)\right\rangle_{\mathscr{H}}$ is measurable, for all
$h\in\mathscr{H}$. In particular, both
$s\mapsto\left\|F\left(s\right)\right\|_{\mathscr{H}}$ and
$s\mapsto\left\langle
F_{1}\left(s\right),F_{2}\left(s\right)\right\rangle_{\mathscr{H}}$ are
measurable.

######  Remark 3.1.

In what follows, we shall make use of weak (Pettis) integrals, a widely known
theory for which we omit details but refer to the following references:
[Kac22, Kal21, BD19]. These integrals play a crucial role in the subsequent
equations, particularly (3.1) and (3.2). For additional details regarding
operator valued integrals, readers are referred to [Cho23, NRP23, KMM23].

Suppose that $F$ and $G$ are measurable $\mathscr{H}$-valued functions
$\left(S,\mathscr{B}_{S}\right)\rightarrow\mathscr{H}$. Further, let $\mu$ be
a positive measure on $\left(S,\mathscr{B}_{S}\right)$. The induced covariance
operator $C=C_{F,G}$ is defined as follows: For $\varphi\in\mathscr{H}$,

$C\left(\varphi\right)\coloneqq\int_{S}\left\langle
F\left(s\right),\varphi\right\rangle_{\mathscr{H}}G\left(s\right)\mu\left(ds\right).$
(3.1)

######  Proposition 3.2.

The operator $C=C_{F,G}$ is of trace class from $\mathscr{H}$ to $\mathscr{H}$
if both $F$ and $G$ satisfy

$\int\left\|F\left(s\right)\right\|_{\mathscr{H}}^{2}\mu\left(ds\right)<\infty.$
(3.2)

In this case,

$\text{trace}_{\mathscr{H}\rightarrow\mathscr{H}}\left(C\right)=\int_{S}\left\langle
F\left(s\right),G\left(s\right)\right\rangle_{\mathscr{H}}\mu\left(ds\right).$

###### Proof.

The operators under the integral sign in (3.1) are rank-1 operators, written
in Dirac’s ket-bra notation as

$\left|G\left(\cdot\right)\left\rangle\right\langle
F\left(\cdot\right)\right|.$

Note that, for all $s\in S$, we have:

$\text{trace}\left|G\left(s\right)\left\rangle\right\langle
F\left(s\right)\right|=\left\langle
F\left(s\right),G\left(s\right)\right\rangle_{\mathscr{H}}.$

The assumption (3.2) then allows us to take the trace under the integral sign
in (3.1), and the desired result follows. ∎

######  Definition 3.3.

A projection-valued measure (PVM) on $\left(S,\mathscr{B}_{S}\right)$ is a
function

$Q\colon\mathscr{B}_{S}\rightarrow\mathscr{L}\left(\mathscr{H}\right)$

with the following properties:

  1. (1)

$Q\left(\emptyset\right)=0$, $Q\left(S\right)=I$, where $I$ is the identity
operator on $\mathscr{H}$.

  2. (2)

$Q\left(A\right)^{2}=Q\left(A\right)^{*}=Q\left(A\right)$, $\forall
A\in\mathscr{B}_{S}$.

  3. (3)

$Q\left(A\cap B\right)=Q\left(A\right)Q\left(B\right)$, $\forall
A,B\in\mathscr{B}_{S}$.

  4. (4)

$Q$ is $\sigma$-additive. That is, for all
$\left\\{A_{i}\right\\}_{i\in\mathbb{N}}$ in $\mathscr{B}_{S}$, with
$A_{i}\cap A_{j}=\emptyset$ for $i\neq j$, we have

$Q\left(\cup_{i}A_{i}\right)=\sum_{i}Q\left(A_{i}\right),$ (3.3)

where the limit holds in the strong operator topology. In particular,

$\left\langle
h,Q\left(\cup_{i}A_{i}\right)k\right\rangle_{\mathscr{H}}=\sum_{i}\left\langle
h,Q\left(A_{i}\right)k\right\rangle_{\mathscr{H}},\;\forall
h,k\in\mathscr{H}.$

Note that $\left\langle h,Q\left(A_{i}\right)k\right\rangle_{\mathscr{H}_{i}}$
are scalar-valued measures on $\left(S,\mathscr{B}_{S}\right)$.

######  Definition 3.4.

A positive operator-valued measure (POVM) on a measurable space
$\left(S,\mathscr{B}_{S}\right)$ is a function

$Q\colon\mathscr{B}_{S}\rightarrow\mathscr{L}\left(\mathscr{H}\right)$

satisfying the following conditions:

  1. (1)

$Q\left(\emptyset\right)=0$, $Q\left(S\right)=I$;

  2. (2)

$Q\left(E\right)\in\mathscr{L}\left(\mathscr{H}\right)_{+}$ for every
$E\in\mathscr{B}_{S}$, i.e., $Q\left(E\right)$ is a positive (self-adjoint)
operator on $\mathscr{H}$;

  3. (3)

Equation (3.3) holds true.

We point the reader to the following fundamental result, Theorem 3.5. A
detailed proof is omitted, which is widely available in the literature; see,
e.g., [BKS13, BFGP22, Ben21, Ben19, TV07, VTC87].

######  Theorem 3.5 (Dilation).

Suppose $Q\colon S\rightarrow\mathscr{L}\left(\mathscr{H}\right)$ is a POVM,
then there is a “bigger” Hilbert space $\mathscr{K}$, an isometry
$V\colon\mathscr{H}\rightarrow\mathscr{K}$ and a projection-valued measure
$P\colon\mathscr{B}_{S}\rightarrow\mathscr{L}\left(\mathscr{K}\right)$, such
that

$Q\left(A\right)=V^{*}P\left(A\right)V,\quad\forall A\in\mathscr{B}_{S}.$

In this context, we recall the following result on positive operator-valued
kernels, and an associated version of non-commutative Radon-Nikodym
derivatives.

######  Theorem 3.6 ([JT24a, JT24b]).

Let $K:S\times S\rightarrow\mathcal{L}\left(\mathscr{H}\right)$ be an
operator-valued positive kernel. Then there exists a reproducing kernel
Hilbert space (RKHS) $\mathcal{K}$, and a family of mappings
$V_{s}:\mathscr{H}\rightarrow\mathcal{K}$, $s\in S$, such that

$K\left(s,t\right)=V_{s}^{*}V_{t}.$ (3.4)

Furthermore, for p.d. kernels $K,L:S\times
S\rightarrow\mathcal{L}\left(\mathscr{H}\right)$, it holds that

$L\leq K$ (3.5)

if and only if there exists an operator $\Gamma$ in $\mathscr{K}$,
$0\leq\Gamma\leq I_{\mathscr{K}}$, such that

$L\left(s,t\right)=V_{s}^{*}\Gamma V_{t}.$ (3.6)

######  Remark 3.7.

(1) A kernel $K:S\times S\rightarrow\mathcal{L}\left(\mathscr{H}\right)$ is
said to be positive definite if

$\sum_{i,j=1}^{n}\left\langle
h_{i},K\left(s_{i},s_{j}\right)h_{j}\right\rangle_{\mathscr{H}}\geq 0$

for all $\left(s_{i}\right)_{1}^{n}$ in $S$, $\left(h_{i}\right)_{1}^{n}$ in
$\mathscr{H}$, and all $n\in\mathbb{N}$. It is straightforward to check that
if $K$ factors as in (3.4), then it is positive definite. Thus, (3.4) is a
characterization of $\mathcal{L}\left(\mathscr{H}\right)$-valued p.d. kernels.

(2) The ordering “$L\leq K$” in (3.5) means that

$\sum_{i,j=1}^{n}\left\langle
h_{i},L\left(s_{i},s_{j}\right)h_{j}\right\rangle_{\mathscr{H}}\leq\sum_{i,j=1}^{n}\left\langle
h_{i},K\left(s_{i},s_{j}\right)h_{j}\right\rangle_{\mathscr{H}}$

for all $\left(s_{i}\right)_{1}^{n}$ in $S$, $\left(h_{i}\right)_{1}^{n}$ in
$\mathscr{H}$, and all $n\in\mathbb{N}$.

(3) The operator $\Gamma$ in (3.6) is positive self-adjoint. It will be
referred to as the Radon-Nikodym derivative of $L$ with respect to $K$,
denoted

$\Gamma=\frac{dL}{dK}.$ (3.7)

(4) In fact, the space $\mathscr{kK}$ in Theorem 3.5 may be chosen to be a
RKHS, and the theorem follows from the general approach in Theorem 3.6 by
setting

$K\left(A,B\right)\coloneqq Q\left(A\cap B\right)$

for $A,B\in\mathscr{B}_{S}$.

##  4\. Quantum polymorphisms

The gist of the classical polymorphism in the Rokhlin setting is a
disintegration. In this section, we (1) present a non-commutative framework,
and (2) offer a non-commutative Rokhlin-type result (Theorem 4.1) that serves
as a direct integral disintegration. As in the nature of the subject, one must
make choices when extending from the commutative case to the non-commutative
case.

For $i=1,2$, let $\left(X_{i},\mathscr{B}_{X_{i}}\right)$ be a measurable
space with Borel $\sigma$-algebra $\mathscr{B}_{X_{i}}$, and
$\pi_{i}:X\rightarrow X_{i}$ the canonical projection. Let
$Q:\mathscr{B}_{X}\rightarrow\mathscr{H}$ be a POVM acting in a Hilbert space
$\mathscr{H}$, where $X=X_{1}\times X_{2}$ with the product $\sigma$-algebra
$\mathscr{B}_{X}$. Set

$Q_{i}=Q\circ\pi_{i}^{-1}$ (4.1)

as a POVM on $\mathscr{B}_{X_{i}}$.

######  Theorem 4.1.

Let $Q$, $Q_{i}$, $i=1,2$, be as specified above. Then there exist commuting
projection-valued measures $P_{1}:\mathscr{B}_{X_{1}}\rightarrow\mathscr{H}$
and $P_{2}:\mathscr{B}_{X_{2}}\rightarrow\mathscr{H}$, such that for all
$B\in\mathscr{B}_{X_{2}}$, the associated Radon-Nikodym derivative is given by

$\frac{dQ\left(\cdot\times B\right)}{dQ_{1}}=P_{2}\left(B\right).$ (4.2)

Similarly, for all $A\in\mathscr{B}_{X_{1}}$,

$\frac{dQ\left(A\times\cdot\right)}{dQ_{2}}=P_{1}\left(A\right).$ (4.3)

###### Proof.

Let $Q=V^{*}PV$ be the dilation of $Q$ from Theorem 3.5. Then,

$\displaystyle Q\left(A\times B\right)$ $\displaystyle=V^{*}P\left(A\times
B\right)V$ $\displaystyle=V^{*}P\left(\left(A\times
X_{2}\right)\cap\left(X_{1}\times B\right)\right)V$
$\displaystyle=V^{*}P\left(A\times X_{2}\right)P\left(X_{1}\times B\right)V$
$\displaystyle=V^{*}P_{1}\left(A\right)P_{2}\left(B\right)V$ (4.4)

where $P_{1}\left(A\right)=P\left(A\times X_{2}\right)$, and
$P_{2}\left(B\right)=P\left(X_{1}\times B\right)$, for all
$A\in\mathscr{B}_{X_{1}}$ and $B\in\mathscr{B}_{X_{2}}$. It is clear that
$P_{i}$’s are commuting PVMs.

Note that

$V^{*}P_{1}\left(\cdot\right)V=Q_{1}\left(\cdot\right),$

and

$V^{*}P_{1}\left(\cdot\right)P_{2}\left(B\right)V=Q\left(\cdot\times B\right)$

for every fixed $B\in\mathscr{B}_{X_{2}}$.

It then follows from Theorem 3.6 that $P_{2}\left(B\right)$ is the Radon-
Nikodym derivative of $Q\left(\cdot\times B\right)$ with respect to $Q_{1}$,
which is (4.2).

The proof of (4.3) is similar and is omitted. ∎

######  Corollary 4.2.

Fix $B\in\mathscr{B}_{X_{2}}$, consider $Q\left(\cdot\times B\right)$ and
$Q_{1}$ as p.d. kernels on $\mathscr{B}_{X_{1}}\times\mathscr{B}_{X_{1}}$.
Then

$Q\left(\cdot\times B\right)\leq Q_{1}\left(\cdot\right).$

Similarly, for fixed $A\in\mathscr{B}_{X_{1}}$,

$Q\left(A\times\cdot\right)\leq Q_{2}\left(\cdot\right).$

######  Corollary 4.3.

Let $Q$ and $Q_{i}=Q\circ\pi_{i}^{-1}$, $i=1,2$, be as specified in (4.1). Let
$\mathscr{H}_{Q}$, resp. $\mathscr{H}_{Q_{i}}$, be the corresponding Hilbert
spaces; see Theorem 4.1. Then $Q_{i}\rightarrow Q$ induces a contraction
$f_{i}\mapsto f_{i}\circ\pi_{i}$ from $\mathscr{H}_{Q_{i}}$ into
$\mathscr{H}_{Q}$, i.e., we have

$\left\|f_{i}\circ\pi_{i}\right\|_{\mathscr{H}_{Q}}\leq\left\|f_{i}\right\|_{\mathscr{H}_{Q_{i}}},$

for all $f_{i}\in\mathscr{H}_{Q_{i}}$, $i=1,2$.

######  Theorem 4.4.

Let $Q:\mathscr{B}_{X}\rightarrow\mathcal{L}\left(\mathscr{H}\right)$ be a
POVM, $Q_{i}=Q\circ\pi_{i}^{-1}$, where $\pi_{i}:X=X_{1}\times
X_{2}\rightarrow X_{i}$, $i=1,2$, and

$Q\left(A\times B\right)=V^{*}P_{1}\left(A\right)P_{2}\left(B\right)V$

as in (4.4). Then

$Q_{1}\otimes
Q_{2}:\mathscr{B}_{X}\rightarrow\mathcal{L}\left(\mathscr{H}\otimes\mathscr{H}\right)$
(4.5)

is a POVM in $\mathscr{H}\otimes\mathscr{H}$, and

$Q_{1}\otimes Q_{2}\left(A\times B\right)=\left(V\otimes
V\right)^{*}P_{1}\left(A\right)\otimes P_{2}\left(B\right)\left(V\otimes
V\right),$ (4.6)

for all $A\in\mathscr{B}_{X_{1}}$, $B\in\mathscr{B}_{X_{2}}$.

Moreover,

$\frac{dQ_{1}\otimes Q_{2}\left(\cdot\times B\right)}{dQ_{1}\otimes
Q_{2}\left(\cdot\times X_{2}\right)}=I\otimes P_{2}\left(B\right),$ (4.7)

and

$\frac{dQ_{1}\otimes Q_{2}\left(A\times\cdot\right)}{dQ_{1}\otimes
Q_{2}\left(X_{1}\times\cdot\right)}=P_{1}\left(A\right)\otimes I.$ (4.8)

###### Proof.

For (4.6), it suffices to check the following: Let $a\otimes
b\in\mathscr{H}\otimes\mathscr{H}$, then

$\displaystyle\left\langle a\otimes b,Q_{1}\left(A\right)\otimes
Q_{2}\left(B\right)\left(a\otimes
b\right)\right\rangle_{\mathscr{H}\otimes\mathscr{H}}$ $\displaystyle=$
$\displaystyle\left\langle
a,Q_{1}\left(A\right)a\right\rangle_{\mathscr{H}}\left\langle
b,Q_{2}\left(B\right)b\right\rangle_{\mathscr{H}}$ $\displaystyle=$
$\displaystyle\left\langle
a,V^{*}P_{1}\left(A\right)Va\right\rangle_{\mathscr{H}}\left\langle
b,V^{*}P_{2}\left(B\right)Vb\right\rangle_{\mathscr{H}}$ $\displaystyle=$
$\displaystyle\left\langle a\otimes b,\left(V\otimes
V\right)^{*}P_{1}\left(A\right)\otimes P_{2}\left(B\right)\left(V\otimes
V\right)\left(\left(a\otimes
b\right)\right)\right\rangle_{\mathscr{H}\otimes\mathscr{H}}.$

The right-side of (4.6) is a dilation of $Q_{1}\otimes Q_{2}$, and it can be
written as

$Q_{1}\otimes Q_{2}\left(A\times B\right)=\left(V\otimes
V\right)^{*}\left(P_{1}\left(A\right)\otimes I\right)\left(I\otimes
P_{2}\left(B\right)\right)\left(V\otimes V\right),$

where $P_{1}\left(A\right)\otimes I$ and $I\otimes P_{2}\left(B\right)$ are
commuting PVMs in $\mathscr{H}\otimes\mathscr{H}$, defined on
$\mathscr{B}_{X_{1}}$ and $\mathscr{B}_{X_{2}}$, respectively. Therefore,
(4.7)–(4.8) follow from a similar argument as in the proof of Theorem 4.1. ∎

##  5\. Beyond the $Q$-polymorphisms

Here, we aim to extend the $Q$-version of quantum polymorphisms (Section 4) in
two different settings. Recall that in the classical (commutative) setting, a
polymorphism $\mu$ for $\mu_{1}$ and $\mu_{2}$ is a joint probability
distribution having $\mu_{1},\mu_{2}$ as marginals.

One possible extension is based on the following questions:

  * •

What is the right notion of polymorphism $poly(s_{1},s_{2})$ for a fixed pair
of quantum states, $s_{i}$?

  * •

With a pair of quantum states, $s_{1}$ and $s_{2}$, what is the best quantum
notion for $poly(s_{1},s_{2})$? What of its properties are different from that
of the classical case in our paper?

  * •

Can we get quantum graphs analogous to the graphs from [JT24d], having the
sets $poly(s_{1},s_{2})$ as edges? What is the set of extreme-points in
$poly(s_{1},s_{2})$?

  * •

What is the role of entanglement in such an extension of the results in our
paper to quantum?

In particular, we note the following correspondence:

$\displaystyle\begin{matrix}\text{quantum states (trace operators)}\\\
\text{on $\mathscr{H}_{1}\otimes\mathscr{H}_{2}$}\end{matrix}$
$\displaystyle\Longleftrightarrow\text{joint measures/distributions}$ partial
traces $\displaystyle\Longleftrightarrow\text{marginal distributions}$

The second approach is through operator-valued positive definite (p.d.)
kernels. In terms of kernels, an analogous notion of polymorphism is as
follows:

Given two scalar-valued kernels $c_{1},c_{2}$ defined on $S\times S$, a
polymorphism is a $\mathcal{L}\left(\mathscr{H}\right)$-valued p.d. kernel on
$S\times S$, such that $c_{1}=tr\left(\rho_{1}K\right)$, and
$c_{2}=tr\left(\rho_{2}K\right)$.

Here, the states $\rho_{i}$ play the role of canonical coordinate projections.
The idea is that by slicing $K$ in various ways, one is able to recover
$c_{i}$ as certain “marginals”.

However, without further restrictions, any two scalar valued p.d. kernels can
be linked this way:

######  Lemma 5.1.

For $c_{1},c_{2}:S\times S\rightarrow\mathbb{C}$, p.d., there exists an
$\mathcal{L}\left(\mathscr{H}\right)$-valued p.d. kernel $K$ on $S\times S$,
and positive trance class operators $\rho_{1},\rho_{2}$, such that
$c_{i}=tr\left(\rho K\right)$.

###### Proof.

Given $c_{1},c_{2}$, let $\mathscr{H}$ be any Hilbert space, and set
$\rho_{1}=\left|a\left\rangle\right\langle a\right|$,
$\rho_{2}=\left|b\left\rangle\right\langle b\right|$, where $a,b$ are unit
vectors in $\mathscr{H}$ and $a\perp b$. Then the kernel
$K=c_{1}\otimes\rho_{1}+c_{2}\otimes\rho_{2}$ has the desired properties. ∎

Several key questions arise from this context:

  * •

What are the scalar valued covariance kernels $c$ which admit such
representations?

  * •

Fix $K$, find $\text{SVK}(K)\coloneqq$ {all scalar valued covariance kernels
$c$ with this representation for some density $\rho$}.

  * •

Fix $\rho$, find $\text{SVK}(\rho)\coloneqq$ {all scalar valued covariance
kernels $c$ with this representation for some $K$}.

  * •

Introduce an action on all scalar valued covariance kernels $c$ via pairs
$(K,\rho)$. Is that action transitive? What about the case when $K$ is the
same for two scalar valued covariance kernels?

Some observations related to our discussion of normal states, i.e., those from
normalized trace class operators $\rho$, and the corresponding scalar valued
p.d. kernels $trace\left(\rho K\left(\cdot,\cdot\right)\right)$ defined from
some operator valued $K$:

  1. (1)

Compact operators are precisely the norm limits of finite rank operators,
which means that trace class operators, being compact, fit this framework.

  2. (2)

There is a correspondence between $\mathcal{L}\left(\mathscr{H}\right)$-valued
p.d. kernels $K$ on $S\times S$, and scalar valued p.d. kernels $\tilde{K}$ on
$T\times T$ , where $T=\left(S,\mathscr{H}\right)$.

Questions:

  * •

Can we use the RKHS $\mathscr{H}(\tilde{K})$ to understand the $K$ induced
scalar valued states $trace\left(\rho K\left(\cdot,\cdot\right)\right)$ ?

  * •

How does the non-product form for $\rho$ (entanglement states) reflect itself
in properties of $trace\left(\rho K\left(\cdot,\cdot\right)\right)$ ?

  * •

Are polymorphisms the natural framework for addressing these questions?

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# IDEAL: An Open-Source Identifier Name Appraisal Tool

Anthony Peruma1, Venera Arnaoudova2, Christian D. Newman1<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Rochester Institute of
Technology, Rochester, NY, USA  2Washington State University, Pullman, WA, USA

###### Abstract

Developers must comprehend the code they will maintain, meaning that the code
must be legible and reasonably self-descriptive. Unfortunately, there is still
a lack of research and tooling that supports developers in understanding their
naming practices; whether the names they choose make sense, whether they are
consistent, and whether they convey the information required of them. In this
paper, we present IDEAL, a tool that will provide feedback to developers about
their identifier naming practices. Among its planned features, it will support
linguistic anti-pattern detection, which is what will be discussed in this
paper. IDEAL is designed to, and will, be extended to cover further anti-
patterns, naming structures, and practices in the near future. IDEAL is open-
source and publicly available, with a demo video available at:
https://youtu.be/fVoOYGe50zg

##  I Introduction

Program comprehension is a precursor to all software maintenance task [1]; it
is essential that a developer understands the code they will be modifying.
Therefore, maintaining the internal quality of the code over its lifetime is
of paramount importance. As fundamental elements in the source code,
identifier names account, on average, for almost 70% of the characters in a
software system’s codebase [2] and play a significant part in code
comprehension [3, 4]. Low quality identifiers can hinder developers’ ability
to understand the code [5, 6]; well-constructed names can improve
comprehension activities by an estimated 19% [7].

However, there is still very little support for developers in terms of helping
them craft high-quality identifier names. Research has examined the terms or
structure of names [7, 2, 8, 9, 10] and produced readability metrics and
models [11, 12, 13] to try and address this problem. However, they still fall
short of providing tangible advice for improving naming practices in
developers’ day-to-day activities. The work we present in this paper is
designed to operate within an IDE, or a CLI, setting and provide real-time
advice to developers about their naming practices.

###  I-A Goal

Our work aims to provide the research and developer community with an open-
source tool, IDEAL, that detects and reports violations in identifier names
for multiple programming languages using static analysis techniques. In
addition to identifying the identifier(s) exhibiting naming issues in the
source code, IDEAL also provides necessary information for each reported
violation so that appropriate action(s) can be taken to correct the issue. We
envision IDEAL utilized by developers in crafting and maintaining high-quality
identifier names in their projects and also by the research community to study
the distribution and effect that various poor naming practices have in the
field.

###  I-B Contribution

IDEAL is a multi-language platform for identifier name analysis. It is
context-aware; treating test and production names differently since they have
different characteristics [14, 15]. It allows for project-specific
configurations and is based on srcML [16], allowing it to support multiple
programming languages (specifically, Java and C#). IDEAL is publicly available
[17] as an open-source tool to facilitate extension and use within the
researcher and developer communities.

##  II Linguistic Anti-Patterns

While the idea behind IDEAL is to support a broad range of identifier naming
best practices based on research, we needed a strong place to start fleshing
the tool out. We chose to implement the linguistic anti-patterns, which were
first conceptualized by Arnaoudova et al. [18]. The primary reasons for this
are that the anti-patterns are well-researched and they represent real,
tangible identifier naming problems. Further, modern IDEs currently do not
support the semantics-aware naming problem detection embodied by the
Linguistic Anti-patterns and the current implementations of anti-patterns are:
Limited to singular languages, not open source, limited to a single IDE
environment, and/or do not provide enough information to the developer to help
ameliorate naming issues. Thus, they are a good place for IDEAL to begin
providing a direct, positive influence.

Linguistic anti-patterns represent deviations from well-established lexical
naming practices in source code and act as indicators of poor naming quality.
This degradation in quality results in inconsistencies in the source code,
leading to misinterpretations causing an increase in developer cognitive load
[19]. Detecting such naming violations in the source code is typically a
tedious and error-prone task for developers that requires an understanding of
the system and a manual analysis of the complete source code. Thus, tool
support is warranted.

To this extent, studies in linguistic anti-pattern detection investigate the
use of static analysis and artificial intelligence (AI) as detection
mechanisms. Two variants of LAPD (Linguistic Anti-Pattern Detector) by
Arnaoudova et al. [18, 20] utilize static analysis to detect these anti-
patterns in C++ and Java source code. The C++ version of the tool is available
as a standalone command-line executable (but is not open source and not
extendable), while the Java version is available as an Eclipse Checkstyle
plugin. The authors report an average precision of 72% for the C++ variant of
their tool. Fakhoury et al. [21] construct and compare AI-based linguistic
anti-pattern detection models for Java source code. The authors report
F1-Scores of 88.77% for traditional machine learning models and 74.53% for
deep neural network models. However, these models only report on the presence
or absence of a linguistic anti-pattern; details around the type of anti-
pattern present are not provided. In contrast, since IDEAL is built on srcML,
it supports multiple programming languages. IDEAL also provides finer-grain
feedback on the types of anti-patterns present and how to fix them; making it
easy to use for developers and researchers. It is also made to be extended
with further anti-patterns not supported by prior tools, that have been found
through prior research [15, 14].

Table I summarizes the linguistic anti-patterns currently detected by IDEAL.
Anti-Patterns A.* to F.* are the set of original anti-patterns defined by
Arnaoudova et al. [18], while the anti-patterns G.* are anti-patterns unique
to IDEAL. Our project website [17] provides code snippets from real-world
open-source systems that highlight examples of these anti-patterns. We should
also note that as an open-source tool IDEAL provides the necessary
infrastructure for the inclusion of additional anti-patterns.

TABLE I: Summary of the linguistic anti-pattern detection rules IDEAL utilizes. Id | Pattern | Detection Strategy
---|---|---
A.1 |  “Get” more than accessor |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with ‘get’, the access specifier is public/protected, the name
contains the name of an attribute, the return type is the same as the
attribute type, and the body contains conditional statements
A.2 |  “Is” returns more than a Boolean |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with a predicate/affirmation related term and the return type
is not boolean
A.3 |  “Set” method returns |  |  Impacted Identifiers: Method Names
---
The name starts with ‘set’ and the return type is not void
A.4 |  Expecting but not getting single instance |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The last term in the name is singular and the name does not contain terms that
are a collection type and the return type is a collection
B.1 |  Not implemented condition |  |  Impacted Identifiers: Method Names
---
The name contains conditional related terms in the name or comment and body
does not conditional statements
B.2 |  Validation method does not confirm |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with a validation-related term, does not have a return type
and does not throw an exception
B.3 |  “Get” method does not return |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with a ‘get’ related term and the return type is void
B.4 |  Not answered question |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with a predicate/affirmation related term and the return type
is void
B.5 |  Transform method does not return |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with or an inner term constains a transformation term and the
return type is void
B.6 |  Expecting but not getting a collection |  |  Impacted Identifiers: Method Names (excludes test methods)
---
The name starts with a ‘get’ related term, the name contains a term that is
either plural or a collection type and the return type is not a collection-
based type
C.1 |  Method name and return type are opposite |  |  Impacted Identifiers: Method Names (excludes test methods)
---
An antonym relationship exists between terms in an identifiers name and data
type
C.2 |  Method signature and comment are opposite |  |  Impacted Identifiers: Method Names (excludes test methods)
---
An antonym relationship exists between either terms in an identifiers name or
data type and comments
D.1 |  Says one but contains many |  |  Impacted Identifiers: Attributes, Method Variables and Parameters
---
The last term in the name is singular and the data type is a collection
D.2 |  Name suggests Boolean but type does not |  |  Impacted Identifiers: Attributes, Method Variables and Parameters
---
The starting term should be predicate/affirmation related and the data type is
not boolean
E.1 |  Says many but contains one |  |  Impacted Identifiers: Attributes, Method Variables and Parameters
---
The last term in the name is plural and the data type is not a collection
F.1 |  Attribute name and type are opposite |  |  Impacted Identifiers: Attributes, Method Variables and Parameters
---
An antonym relationship exists between terms in an identifiers name and data
type
F.2 |  Attribute signature and comment are opposite |  |  Impacted Identifiers: Attributes, Method Variables and Parameters
---
An antonym relationship exists between either terms in an identifiers name or
data type and comments
G.1 |  Name contains only special characters |  |  Impacted Identifiers: Attributes, Method, Method Variables and Parameters
---
The name of the identifier is composed of only non-alphanumeric characters
G.2 |  Redundant use of “test” in method name |  |  Impacted Identifiers: Methods (excludes non-test methods)
---
The name starts with the term ‘test’

##  III IDEAL Architecture

Implemented as a command-line/console-based tool in Python, IDEAL integrates
with some well-known open-source libraries and tools in analyzing source code
to detect identifier name violations. Depicted in Figure 1 is a view of the
conceptual architecture of IDEAL. Broadly, IDEAL is composed of three layers–
Platform, Modules, and Interface. It utilizes well-known tools and libraries
used for natural language and static analysis, including Spiral [22], NLTK
[23], Wordnet [24], Stanford POS tagging [25], and srcML [16].

Figure 1: Conceptual architectural view of IDEAL.

##  IV Applicability

Practitioners. By integrating IDEAL into their development toolset and
workflow, developers are better equipped to maintain identifiers in their
source code. As a command-line/console application, the current version of
IDEAL supports integration with a build system. Hence, project teams can
analyze their entire project codebase, or just what was changed, during their
nightly build process and evaluate the report to determine violations that
need to be addressed.

Researchers. We envision the research community utilizing IDEAL in studies
around program comprehension. With the capability of batch-based analysis,
IDEAL supports researchers in conducting large-scale empirical studies.
Furthermore, by supporting Java and C#, IDEAL provides researchers to expand
their research to multiple programming languages and perform comparatison-
based studies. Finally, as an open-source tool, researchers are provided with
the opportunity to extend IDEAL by improving existing violation detection
strategies and introducing new anti-patterns.

Educators. IDEAL can be used in a classroom setting to teach students the
importance of constructing high-quality identifier names and their impact on
software maintenance and evolution. Through this, students will be better
prepared to write high-quality code when moving into the industry.

##  V Evaluation

To understand the effectiveness of IDEAL in correctly detecting identifier
naming violations, we subjected IDEAL to two types of evaluation activities.
First, we analyzed four popular open-source systems using IDEAL and manually
validated the detection results of a statistically significant sample. Our
next evaluation strategy involved assessing IDEAL on the sample dataset
utilized to evaluate LAPD by comparing the detection results. In the following
subsections, we provide details on these two evaluation activities, including
numbers around the correctness of IDEAL and qualitative findings based on our
manual analysis of source code.

###  V-A Evaluation on open-source systems

IDEAL can analyze systems implemented in any language supported by srcML.
However, currently, it has only been evaluated using Java and C#. Thus, we
selected two popular open-source systems for each of these programming
languages. To this extent, Retrofit [26] and Jenkins [27] were the two Java
systems, while Shadowsocks [28] and PowerShell [29] were the C# systems; Table
II summarizes the release of each system that was part of our evaluation
analysis. A breakdown of our validation results is available at [17].

For each of the four systems, we manually analyzed a random stratified
statistically significant (i.e., confidence level of 95% and confidence
interval of 10%) set of detected violations for each category. In total, we
manually verified 2,019 instances of naming violations spread across the four
systems. Table III provides a breakdown of the number of violation instances
for each category. As part of the manual analysis process and to mitigate
bias, the authors discussed specific violation instances that were subjective
and, at times, referenced literature (grey and reviewed) to aid in the
decision-making process. IDEAL reports an average precision of 75.27%, with 14
out of 19 violation types reporting a precision of over 50%. Though LAPD
reports an average precision of 72%, we manually validate 1,267 more instances
than LAPD. Furthermore, even though IDEAL supports customization per project
(e.g., specifying custom collection data types and terms), our evaluation
strategy did not utilize this feature in order to maintain consistency in
violation detection across the four systems.

TABLE II: Summary of the systems in our evaluation process. System | Language | Version |  | Release
---
Date
| Files
---
Analyzed
| Issues
---
Detected
Retrofit | Java | 2.9.0 | May-2020 | 282 | 192
Jenkins | Java | 2.293 | May-2021 | 1,688 | 4,818
Shadowsocks | C# | 4.4.0.0 | Dec-2020 | 88 | 275
PowerShell | C# | 7.1.3 | Mar-2021 | 1,290 | 8,455

From Table III, we observe that while IDEAL performs notably well in detecting
all A.*, D.*, and E.* violations (precision score of over 80%). These are
anti-patterns where the identifier either does or contains more than what is
required. In most instances, IDEAL can accurately process the return/data type
of the identifier to determine violations. However, there are also violations
that are challenging for IDEAL to analyze and hence result in a large volume
of false positives (e.g., C.2). Our manual analysis of these false-positive
instances shows patterns that, in most cases, are causing IDEAL to report them
as issues. First, since developers utilize custom data/return types for
identifiers in their code, IDEAL fails in identifying their intended purpose.
For instance, ‘EnvVars’ is a custom type created by a developer to hold a
collection of specific items. The developer returns this type in a method
called ‘getEnvironmentVariables2’. Since IDEAL is unaware that ‘EnvVars’ is a
collection-based type, it flags this as a violation since the method is
supposed to return a collection (i.e., this get method name contains a plural
term– ‘Variables’). We are confident that once developers configure IDEAL to
handle custom types, false positives, similar to this, will reduce. Our next
observation is on how IDEAL analyzes lexical relationships between words;
specifically, concerning antonyms (i.e., C.* and F.*). While IDEAL correctly
recognizes antonyms, the context around how these terms are used, either in
the identifier’s name or comment, is not considered, resulting in false
positives. Additionally, we also observe that naming habits/conventions also
cause the emergence of antonyms. For instance, consider the method
‘GetCompletionResult’ with a return type called ‘CompletionResult’. IDEAL
determines that ‘Get’ and ‘Result’ are antonyms, which are lexically valid,
but a false positive due to naming conventions. Similar to the last challenge,
context around the use of transformation terms (i.e., B.5) and conditional
terms (i.e., B.1) cause the reporting of a high volume of false positives.
While IDEAL correctly detects these terms in the source code, how the
developer utilizes the term in a name or comment is currently a challenge.

Finally, our manual review of the source code also allowed us to observe other
poor naming/coding practices, which can be future linguistic anti-patterns.
For example, the generic terms ‘data’ and ‘result’ are subjective. When used
as part of an identifier’s name, it is unknown if the identifier handles a
single item or collection of items. Likewise, the use of the type ‘var’ (in
C#) and ‘object’ also does not indicate the type of data the identifier
handles. Ideally, to convey the purpose/behavior of the identifier correctly,
developers need to be specific in naming identifiers and data types when
possible.

TABLE III: Summary of the detection correctness of IDEAL. Id. |  | Detected
---
Instances
| Validated
---
Samples
| True
---
Positives
| False
---
Positives
Precision
A.1 | 53 | 34 | 34 | 0 | 100.00%
A.2 | 45 | 37 | 37 | 0 | 100.00%
A.3 | 129 | 64 | 63 | 1 | 98.44%
A.4 | 341 | 127 | 102 | 25 | 80.31%
B.1 | 912 | 171 | 73 | 98 | 42.69%
B.2 | 446 | 166 | 165 | 1 | 99.40%
B.3 | 260 | 101 | 101 | 0 | 100.00%
B.4 | 18 | 16 | 5 | 11 | 31.25%
B.5 | 271 | 107 | 46 | 61 | 42.99%
B.6 | 827 | 159 | 128 | 31 | 80.50%
C.1 | 139 | 74 | 54 | 20 | 72.97%
C.2 | 294 | 112 | 13 | 99 | 11.61%
D.1 | 3,359 | 262 | 261 | 1 | 99.62%
D.2 | 83 | 53 | 53 | 0 | 100.00%
E.1 | 5,506 | 268 | 253 | 15 | 94.40%
F.1 | 38 | 32 | 19 | 13 | 59.38%
F.2 | 165 | 91 | 15 | 76 | 16.48%
G.1 | 1 | 1 | 1 | 0 | 100.00%
G.2 | 853 | 144 | 144 | 0 | 100.00%
Overall | 13,740 | 2,019 | 1,567 | 452 | 75.27%

###  V-B Comparison with LAPD

In this part of our evaluation we compare the correctness of IDEAL with LAPD.
To this extent, we analyze a sample of the source files that were utilized to
evaluate the effectiveness of LAPD and compare the results. Since IDEAL
implements the anti-patterns available in LAPD, it is essential to understand
the areas where IDEAL under- and overperforms. In total, we analyzed 209 Java
files and detected 294 violations. From this, both IDEAL and LAPD matched 199
true positive instances and 19 false positive instances. Furthermore, 47
instances identified as LAPD false positives were not detected by IDEAL,
highlighting where IDEAL outperforms LAPD. Most of these instances were
associated with C.2, D.1, and E.1. Finally, we also encounter instances where
IDEAL does not detect LAPD true positives. While some of these issues are due
to custom data types, we also encounter subjective instances, most of which
(10 instances) fall under D.2.

##  VI Conclusion and Future Work

This paper introduced IDEAL, an open-source configurable tool that detects 19
types of identifier naming violations in Java and C# code. A comprehensive
evaluation of IDEAL reports an average precision of 75.27%. Our future work
involves increasing support of additional anti-patterns and naming structures
(including naming structures derived in other research [14, 15]), utilizing a
source code specialized part-of-speech-tagger [30], and IDE integration. A
summary of the naming practices IDEAL will support is available in the
Identifier Name Structure Catalogue [31].

##  VII Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant No. 1850412.

## References

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# Power Counting to Saturation

Iain W. Stewart1 and Varun Vaidya2 1 Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  2 Department
of Physics, University of South Dakota, Vermillion, SD, 57069

###### Abstract

We present a description of saturation in small $x$ deep inelastic scattering
from power counting in a top-down effective theory derived from QCD. A
factorization formula isolates the universal physics of the nucleus at leading
power in $x$. The onset of saturation is then understood as a breakdown in the
expansion in an emergent power counting parameter, which is defined by the
matrix element of a gauge invariant operator. We demonstrate the presence of a
novel collinear-soft radiation mode, which clarifies the role played by the
medium size in deciding linear/non-linear evolution of the cross section in
$\ln x$.

††preprint:  MIT-CTP 5562

One of the primary objectives of a future Electron Ion Collider is an
investigation of the partonic structure of hadrons or large nuclei at high
energies. The partonic density at such high energies is enhanced due to
quantum radiative corrections leading to the phenomenon called saturation
Gribov  _et al._ (1983); Mueller (1990). The current understanding of
saturation is via the Color Glass Condensate (CGC) Iancu and Venugopalan
(2003); Gelis  _et al._ (2010); Kovchegov and Levin (2022) which treats the
hadron/nucleus as a classical source for small $x$ gluons. This is often
thought of as a bottom-up effective theory McLerran and Venugopalan (1999),
whose degrees of freedom consist of color sources $\rho$ at large $x$ with
momenta above a cutoff, $k^{+}>\Lambda^{+}$, and gauge fields $A^{\mu}$ at
small $x$ with $k^{+}<\Lambda^{+}$. The input is a gauge invariant
distribution $W_{\Lambda^{+}}[\rho]$ which determines the probability of a
configuration $\rho$, and at lowest order one solves classical Yang-Mills
equations for $A^{\mu}$ with $\rho$ as a source. This formalism incorporates
soft gluon radiative corrections via the non-linear B-JIMWLK or the BK (at
large $N_{C}$) Balitsky (1996); Jalilian-Marian  _et al._ (1997, 1998); Kovner
_et al._ (2000); Iancu  _et al._ (2001a, b); Ferreiro  _et al._ (2002);
Kovchegov (1999) equations which describe the evolution in the cutoff
$\Lambda^{+}$.

Our understanding of deep inelastic scattering (DIS) at large $x$ rests on the
notion of non-perturbative factorization which enables us to define operators
which separate the universal physics of the hadron from process dependent
short distance physics Sterman (1987); Catani and Trentadue (1989). This can
be thought of as a top-down effective field theory derived from QCD Bauer  _et
al._ (2002), where the power counting expansion parameter is $\Lambda_{\rm
QCD}/Q$. Here $Q^{2}=-q^{2}>0$ where $q^{\mu}$ is the momentum transfer of the
virtual photon in the electron-proton collision, $x=Q^{2}/(2P\cdot q)$ where
$P^{\mu}$ is the initial proton’s momentum, and we define $H=(P+q)^{2}$, the
photon-nucleus invariant mass.

The goal of this letter is to derive an effective field theory description of
small $x$ saturation physics from the top down, and connect it to the CGC
formalism. Starting from QCD we setup a factorization framework for DIS at
small $x$ at all orders in the strong coupling and leading power in $x$. This
leads to a definition for the universal physics of the nuclear medium as an
operator matrix element in the nuclear state. Next we show that we can
understand the onset of saturation in terms of a breakdown in an emergent
power counting parameter that appears due to multiple probe-medium
interactions. Finally we give a power counting argument to understand how the
form of the factorization changes as we enter the saturation regime and why
this will lead to a non-linear evolution in $\ln x$. Apart from the
kinematical scales that we will discuss in the next section, we foreshadow the
importance of three emergent length scales that are crucial for this
description. The first is $L_{D}$, the average distance beyond which color
sources in the medium are uncorrelated. The second is the time scale $t_{c}$
over which the partons in the probe maintain quantum coherence. Finally we
have the mean free path of the dipole probe $\lambda_{\text{mfp}}$ which is
the average distance between successive interactions of the probe with the
medium.

## Kinematics and Factorization:

We want to describe the small $x\ll 1$ region or equivalently
$\Lambda_{QCD}^{2}\ll Q^{2}\ll 2P\cdot q$, which is the high energy limit of
QCD where $H\approx 2P\cdot q$. Without loss of generality, we can orient our
axes so that $q^{\mu}$ has no transverse momentum and work in a frame with
$(q^{-},q^{+},q^{\perp})=(Q,-Q,0^{\perp})$, where we decompose
$q^{\mu}=q^{-}\frac{n^{\mu}}{2}+q^{+}\frac{\bar{n}^{\mu}}{2}+q^{\perp}$, with
$n^{\mu}\equiv(1,0,0,1)$, $\bar{n}^{\mu}\equiv(1,0,0,-1)$, and $\perp$ refers
to momenta in the $x$-$y$ plane. For a proton of mass $M_{p}$, the momentum
components in this frame are $(M_{p}^{2}Q/H,H/Q,0)$.

The dominant interaction is the photon fluctuating into a quark-anti quark
dipole which then scatters off the hadron/nucleus Gribov (1969). The medium
partons exchange a transverse momentum of order Q with the dipole and scatter
by an angle $\theta\sim q_{T}/E\sim Q^{2}/H=x\ll 1$ which is the power
counting parameter of the EFT.

In terms of this expansion parameter, the partons in the dipole scale as the
soft mode $p_{s}\sim\frac{H}{Q}\left(x,x,x\right)$ while the partons in the
medium scale as an $\bar{n}$ collinear mode
$p_{\bar{n}}\sim\frac{H}{Q}\left(x^{2},1,x\right)$. Both these modes are
propagating degrees of freedom with an invariant mass $p^{2}\sim Q^{2}$ . This
stage of the EFT is therefore designed to describe IR physics at the scale Q
while separating out the physics at the scale $H/Q$. The small angle
scattering between these modes is mediated by a Glauber mode whose scaling is
fixed by the constraint that both the dipole and medium partons must retain
their scaling in any interaction. Therefore, we can write
$p_{G}\sim\frac{H}{Q}\left(x^{2},x,x\right)$. These same EFT modes are used in
Neill  _et al._ (2023) to derive small-$x$ factorization for DIS with
applications to $\ln x$ resummation for DIS coefficient functions and
anomalous dimensions.

The QCD action can be systematically expanded in $\theta=x$ within the Soft
Collinear Effective Theory (SCET) Bauer  _et al._ (2003a, b, 2001); Bauer and
Stewart (2001); Bauer  _et al._ (2002), as shown in Rothstein and Stewart
(2016). This allows us to write an effective Hamiltonian at leading power in
$x$

$\displaystyle
H_{\text{eff}}=H_{s}+H_{\bar{n}}+H^{G}_{\bar{n}s}+e_{q}\bar{q}_{s}\gamma_{\mu}q_{s}L^{\mu}+\ldots$
(1)

This has an effective hard interaction mediated by a photon between the
electron current $L^{\mu}$ and the soft quark current, where $e_{q}$ is the
electric charge of the quark. Soft and collinear modes of SCET are decoupled
in the soft $H_{s}$ and collinear $H_{\bar{n}}$ Hamiltonians, but couple
through the Glauber Hamiltonian $H^{G}_{\bar{n}s}$ that mediates forward
sacttering and has the form

$\displaystyle\mathcal{H}_{G}^{\bar{n}s}$
$\displaystyle=C_{G}\\!\\!\sum_{i,j\in\\{q,g\\}}\mathcal{O}_{\bar{n}s}^{ij}\,,$
$\displaystyle\mathcal{O}_{\bar{n}s}^{ij}$
$\displaystyle=\mathcal{O}_{\bar{n}}^{iB}\frac{1}{\mathcal{P}_{\perp}^{2}}\mathcal{O}_{s}^{jB}\,,$
(2)

where $C_{G}(\mu)=8\pi\alpha_{s}(\mu)$ and ${\cal P}_{\perp}^{\mu}$ is a
derivative operator in the $\perp$ direction. The gauge invariant $\bar{n}$
collinear and soft operators are

$\displaystyle\mathcal{O}_{\bar{n}}^{qB}=\bar{\chi}_{\bar{n}}T^{B}\frac{\not{n}}{2}\chi_{\bar{n}},\
\ \
\mathcal{O}_{\bar{n}}^{gB}=\frac{i}{2}f^{BCD}\mathcal{B}_{\bar{n}\perp\mu}^{C}\frac{n}{2}\cdot(\mathcal{P}+\mathcal{P}^{\dagger})\mathcal{B}_{\bar{n}\perp}^{D\mu}$
$\displaystyle\mathcal{O}_{s}^{qB}=\bar{\chi}_{s}T^{B}\frac{\not{\bar{n}}}{2}\chi_{s},\
\ \
\mathcal{O}_{S}^{gB}=\frac{i}{2}f^{BCD}\mathcal{B}_{S\perp\mu}^{C}\frac{\bar{n}}{2}\cdot(\mathcal{P}+\mathcal{P}^{\dagger})\mathcal{B}_{S\perp}^{D\mu}\,,$
(3)

which are built out of the gauge invariant building blocks

$\displaystyle\chi_{\bar{n}}=W_{\bar{n}}^{\dagger}\xi_{\bar{n}},\ \ \ \
W_{\bar{n}}=\text{FT}\ {\bf P}\exp\Big{\\{}ig\int_{-\infty}^{0}dsn\cdot
A_{\bar{n}}(x+ns)\Big{\\}},$
$\displaystyle\chi_{S}=S_{\bar{n}}^{\dagger}\xi_{S},\ \ \ \
S_{\bar{n}}=\text{FT}\ {\bf P}\exp\Big{\\{}ig\int_{-\infty}^{0}ds\bar{n}\cdot
A_{S}(x+s\bar{n})\Big{\\}},$
$\displaystyle\mathcal{B}_{\bar{n}\perp}^{C\mu}T^{C}=\frac{1}{g}\Big{[}W_{\bar{n}}^{\dagger}iD_{\bar{n}\perp}^{\mu}W_{\bar{n}}\Big{]},\
\
\mathcal{B}_{S\perp}^{C\mu}T^{C}=\frac{1}{g}\Big{[}S_{\bar{n}}^{\dagger}iD_{S\perp}^{\mu}S_{\bar{n}}\Big{]}.$
(4)

FT stands for Fourier transform. These operators encode bare quark and gluons
dressed by Wilson lines. Since the dipole probe can interact multiple times
with the medium, we can think of it as an open quantum system evolving in a
nuclear environment. We therefore follow the evolution of the reduced density
matrix for our probe. The initial state density matrix can be written as

$\displaystyle\rho(0)=|e^{-}\rangle\langle e^{-}|\otimes\rho_{A}$ (5)

where the electron is disentangled from the nuclear medium ($\rho_{A}$). We
will evolve this with $H_{\text{eff}}$ and measure the properties of the
electron, while being inclusive over the hadronic final states. Typically in
DIS we write the cross section differential in $Q^{2}$ and $x$, which we
indicate with a measurement $\mathcal{M}$ which acts on the phase space of the
final state $e^{-}$. Since the Glauber mode couples the soft and collinear
sectors, we have to expand out the density matrix element order by order in
the Glauber Hamiltonian, $H_{\bar{n}s}^{G}$ in order to carry out soft-
collinear factorization,

$\displaystyle\Sigma$ $\displaystyle\equiv$
$\displaystyle\lim_{t\rightarrow\infty}\text{Tr}[e^{-iH_{\text{eff}}t}\rho(0)e^{iH_{\text{eff}}t}\mathcal{M}]=\sum_{i}\Sigma^{(i)}$
(6)

This is an expansion in the number of interactions of the probe with the
medium and is often referred to as the opacity expansion Gyulassy  _et al._
(2000). Taking the confinement length scale $\Lambda_{\rm QCD}^{-1}$ and
boosting to obtain a scale relevant for the size of a nucleon in the $y^{-}$
direction, $L_{D}\leavevmode\nobreak\ \sim M_{p}Q/(H\Lambda_{\rm QCD})$. In
reality $L_{D}$ might be substantially larger due to inter-nucleon
interactions, especially in larger nuclei. We also consider the mean free path
of the probe through the medium, $\lambda_{\text{mfp}}\sim
xQ/(\alpha_{s}(Q)^{2}\Gamma)$, which determines how often the probe interacts
with the medium, and depends on the interaction strength and medium density.
Here $\Gamma$ is the nucleon density per unit area, where
$\Gamma\sim\Lambda_{\rm QCD}^{2}$ for nucleons stacked next to one another. We
work in the regime $\lambda_{2}=L_{D}/\lambda_{\text{mfp}}\sim\Gamma/Q^{2}\ll
1$ where successive interactions of the probe with the medium happen with
color uncorrelated partons. The assumption $\lambda_{2}\ll 1$ dictates that
only even powers of $H^{G}_{\bar{n}s}$ contribute so as to have the same
number of Glauber exchanges on both sides of the cut. The factorization
formula at $O(2m)$ in this series is

$\displaystyle\Sigma^{(2m)}$ $\displaystyle=$
$\displaystyle\frac{|C_{G}(\mu)|^{2m}}{Q^{4}}\Bigg{[}\int
d^{+}p_{e}\mathcal{M}\Bigg{]}L_{\alpha\beta}\int_{\bar{y}\in\text{Nucleus}}d\bar{y}^{+}d^{2}\bar{y}_{\perp}$
(7)
$\displaystyle\text{Im}\Bigg{\\{}\Bigg{[}\Pi_{i=1}^{m}\int_{0}^{L_{N}^{-}}d\bar{y}_{i}^{-}\Theta(\bar{y}_{i}^{-}-\bar{y}_{i+1}^{-})\int\frac{d^{2}k_{i}^{\perp}}{(2\pi)^{2}}\mathcal{B}_{\bar{n}}(k_{i}^{\perp},\bar{y}_{i}^{-},\bar{y}^{+},\bar{y}^{\perp};\nu)\Bigg{]}S_{m}^{\alpha\beta}(k_{1}^{\perp},...k_{m}^{\perp};\bar{y}_{1}^{-},...\bar{y}_{m}^{-};\mu,\nu)\Bigg{\\}}.$

It is written in terms of $m$ copies of an $\bar{n}$-collinear medium
structure function $\mathcal{B}_{\bar{n}}$ and a soft function
$S_{m}^{\alpha\beta}$ describing the evolution of the Dipole probe. The
$\lambda_{2}\ll 1$ expansion leads to the product of ${\cal B}_{\bar{n}}$
functions, rather than a single multi-interaction $\bar{n}$ correlation
function. $\int d^{+}p_{e}$ is the phase space integral over the final state
electron and $L_{\alpha\beta}$ is the leptonic tensor. The coordinate
$\bar{y}$ indicates the position inside the nucleus. The physical picture is
that the successive interactions of the probe dipole occur with color
uncorrelated partons in the nucleus at a single value of
$\bar{y}^{+},\bar{y}^{\perp}$but at increasing values of $\bar{y}^{-}$ as
shown in Fig.1. The nucleus extends over a length $L_{N}^{-}$ in the $y^{-}$
direction. $k^{\perp}_{i}$ is the transverse momentum exchanged in the
$i^{th}$ interaction. The separation of collinear-soft physics leads to
rapidity divergences, for which we use the dimensional regularization like
rapidity regulator of Chiu  _et al._ (2012) with the corresponding
renormalization scale $\nu$. $\mathcal{B}_{\bar{n}}$ is a function of
$\bar{y}$ so that it can also describe an inhomogeneous medium. It is given by
a current-current correlator in the nuclear background

$\displaystyle\mathcal{B}_{\bar{n}}(k_{\perp},\bar{y};\nu)=\frac{1}{2}\frac{1}{k_{\perp}^{2}}(N_{C}^{2}-1)\int
d^{2}\hat{y}_{\perp}d\hat{y}^{-}e^{ik_{\perp}\cdot\hat{y}^{\perp}}$ (8)
$\displaystyle\times$
$\displaystyle\text{Tr}\Big{[}O^{A}_{\bar{n}}\left(\frac{1}{2}(-\hat{y}+\bar{y})\right)O^{A}_{\bar{n}}\left(\frac{1}{2}(\hat{y}+\bar{y})\right)\rho_{A}\Big{]}.$

This captures the $universal$ physics of the medium to all orders in
perturbation theory and is independent of any measurements that are made on
the probe. $\mathcal{O}_{\bar{n}}^{A}$ can be thought of as the operator
analog of the color source function $\rho$ in the CGC approach McLerran and
Venugopalan (1999). The medium structure function does not have any UV
divergences but obeys a renormalization group equation in rapidity which is
the BFKL equation at one loop.

$\displaystyle\frac{d\mathcal{B}_{\bar{n}}(k_{\perp},\bar{y};\nu)}{d\ln\nu}=-\int
d^{2}u_{\perp}K_{\text{BFKL}}(\vec{k}_{\perp},\vec{u}_{\perp})\mathcal{B}_{\bar{n}}(u_{\perp},\bar{y};\nu)$
$\displaystyle=-\frac{\alpha_{s}N_{C}}{\pi^{2}}\int
d^{2}u_{\perp}\Bigg{[}\frac{\mathcal{B}_{\bar{n}}(u_{\perp},\bar{y};\nu)}{(\vec{u}_{\perp}-\vec{k}_{\perp})^{2}}-\frac{k_{\perp}^{2}\mathcal{B}_{\bar{n}}(k_{\perp},\bar{y};\nu)}{2u_{\perp}^{2}(\vec{u}_{\perp}-\vec{k}_{\perp})^{2}}\Bigg{]}.$
(9)

Its natural rapidity scale corresponds to $\nu\sim H/Q$ for which logarithms
in this function are minimized. By RG consistency, we have

$\displaystyle\frac{dS^{\alpha\beta}_{m}(k_{1\perp},...k_{m\perp},\\{\bar{y}_{i}^{-}\\};\mu,\nu)}{d\ln\nu}=\sum_{i=1}^{n}\int
d^{2}u_{\perp}K_{\text{BFKL}}(u_{\perp},k_{i\perp})$ $\displaystyle\times
S^{\alpha\beta}_{m}(k_{1\perp},..k_{i-1\perp},u_{\perp},k_{i+1\perp},..k_{m\perp},\\{\bar{y}_{i}^{-}\\};\mu,\nu).$
(10)

The natural rapidity scale for this function is $\nu\sim Q$. We can use the
rapidity RG equation to resum the large logarithms in $x$ by solving the BFKL
equation, running the function $\mathcal{B}_{\bar{n}}$ from $\nu\simeq s/Q$ to
$\nu\simeq Q$ while setting $\nu\simeq Q$ in the Dipole function, leading to
an RG improved function
$\mathcal{B}_{\bar{n}}^{\text{R}}(k_{\perp},\bar{y},x)$. At leading log, we
can resum the series in Eq.(6) to all orders in the Glauber interactions

$\displaystyle\Sigma=\int\\!\\!d^{2}\bar{y}^{\perp}d\bar{y}^{+}\\!\\!\int\\!\\!d^{2}b\>\mathcal{F}(\vec{b},Q)\Bigg{\\{}1-{\bf
P}e^{-\mbox{$\int_{0}^{L^{-}_{N}}$}\frac{d\bar{y}^{-}}{\lambda_{\text{mfp}}\left(\vec{b},Q,\bar{y}\right)}}\Bigg{\\}},$
(11)

which is a path ordered exponential with $\lambda_{\text{mfp}}$ given by a RG
improved medium structure function $\mathcal{B}_{\bar{n}}^{\text{R}}$,

$\displaystyle\lambda^{-1}_{\text{mfp}}(\vec{b},Q,\bar{y})$
$\displaystyle=\\!\\!\int\\!\\!\\!\frac{d^{2}k_{\perp}}{(2\pi)^{2}k_{\perp}^{2}}|C_{G}(k_{\perp})|^{2}\mathcal{B}_{\bar{n}}^{\text{R}}(\vec{k}_{\perp},\bar{y},x)$
$\displaystyle\qquad\times\Big{[}e^{i\vec{b}\cdot k_{\perp}}-1\Big{]}.$ (12)

Here $\lambda_{\text{mfp}}$ is an emergent length scale, which can be
interpreted as the mean free path of the probe in the medium in the $y^{-}$
direction.

Figure 1: Interaction of the probe with the medium.

It depends on the local properties of the medium at $\bar{y}$. The function
$\mathcal{F}(\vec{b},Q)$ includes the transverse component of the dipole
function $S_{n}^{\alpha\beta}$ at tree level,

$\displaystyle\mathcal{F}(\vec{b},Q)=\frac{1}{Q^{3}}\Bigg{[}\int
d^{+}p_{e}\mathcal{M}\Bigg{]}L_{\alpha\beta}g_{\perp}^{\alpha\beta}\int
d^{2}p_{\perp}d^{2}\bar{p}_{\perp}e^{i\vec{b}\cdot(\vec{p}_{\perp}+\vec{\bar{p}}_{\perp})}$
$\displaystyle\int_{0}^{1}\\!\\!dz\left(z^{2}\\!+\\!(1\\!-\\!z)^{2}\right)\Bigg{[}\frac{\vec{p}_{\perp}}{Q^{2}z(1\\!-\\!z)\\!+\vec{p}_{\perp}^{2}}+\frac{\vec{\bar{p}}_{\perp}}{Q^{2}z(1\\!-\\!z)\\!+\vec{\bar{p}}_{\perp}^{2}}\Bigg{]}^{2}$

with the transverse size of the dipole $\vec{b}\sim 1/Q$, as this is the only
scale in this function. The emergence of the mean free path automatically
defines a new power counting parameter in the EFT

$\displaystyle\lambda_{1}=\int_{0}^{L^{-}_{N}}\frac{d\bar{y}^{-}}{\lambda_{\text{mfp}}\left(\vec{b},Q,\bar{y}\right)}\,.$
(13)

The onset of saturation is just the breakdown of the power expansion in this
parameter, when $\lambda_{1}\sim 1$. Given the fact that the transverse size
of the dipole $\vec{b}$ is conjugate to the scale Q, we can define the
saturation momentum scale $Q_{s}(\bar{y})$ via

$\displaystyle\int_{0}^{L^{-}_{N}}\frac{d\bar{y}^{-}}{\lambda_{\text{mfp}}\left(|\vec{b}|=1/Q_{s},Q_{s},\bar{y}\right)}=1\,.$
(14)

This corresponds to the regime where the dipole is likely to have multiple
interactions with the medium.

Eq.(13) will be equivalent to the GGM formula Mueller (1990)

$\displaystyle\lambda_{1}=\Gamma(\bar{y}_{\perp})\frac{\alpha_{s}\pi^{2}}{N_{C}}b_{\perp}^{2}xG_{N}(x,1/b_{\perp})\,,$
(15)

if we model the nucleus as a bunch of non-interacting nucleons with some
effective Glauber gluon density $xG_{N}(x,1/b_{\perp})$ and a density of
nucleons per unit area $\Gamma(\bar{y})$. Rescaling
$k_{\perp}\to\hat{k}_{\perp}/|b_{\perp}|$ in Eq.(Kinematics and
Factorization:) using Eq.(8), we find an overall factor $\lambda_{1}\propto
b_{\perp}^{2}$. We can then equate the $\hat{k}_{\perp}$ and $\bar{y}^{-}$
integral of the rescaled $\mathcal{B}$ with
$\Gamma(\bar{y}_{\perp})xG_{N}(x,1/b_{\perp})$; the number density of small
$x$ gluons per unit area. $G_{N}$ absorbs an $\alpha_{s}N_{c}$, leaving
$\alpha_{s}\pi^{2}/N_{c}$ as the prefactor. This yields Eq.(15). In the CGC
formalism $xG_{N}(x,1/b_{\perp})$ is obtained by treating the nucleus as a
source for small x gluons with radiative corrections from the BFKL/BK
equations. The resummation modifies the saturation scale from its tree level
value $Q_{s_{0}}$ to have a power law scaling with $x$;
$Q_{s}(x)=Q_{s_{0}}x^{-2.44\alpha_{s}N_{C}/\pi^{2}}$. The boundary condition
for this resummation, in our case $\mathcal{B}(k_{\perp})$, is described in
the CGC by the MV model McLerran and Venugopalan (1994a, b, c).

## Role of the medium size:

From Eq.(7), we observe potential sensitivity to another length scale
$L^{-}_{N}$; the extent of the medium in the $y^{-}$ direction. The relevant
radiation sensitive to this scale will have momenta $p^{+}\sim 1/L^{-}_{N}$,
and coherence time $t_{c}\sim L_{N}^{-}$. If $N$ is the typical number of
nucleons seen by the probe along its path, then the rest frame size of the
nucleus is $L_{R}\sim N/M_{p}$, and boosting yields $L_{N}^{-}\sim
L_{R}M_{p}Q/H\sim QN/H$. Retaining a common virtuality $\sim Q^{2}$, this is a
collinear-soft mode with

$\displaystyle p_{cs}\sim\frac{H}{Q}\left(Nx^{2},\frac{1}{N},x\right).$ (16)

We can consider two important regimes of the consequent EFT as we dial down
the value of $x$ with fixed $Q$, keeping $N$ to be large but fixed as shown in
Fig. 2. We see that at not so small $x\sim 1/N$, the soft and collinear-soft
modes are the same so that the two mode factorization formula discussed above,
Eq.(7), is accurate.

This means that while in a dense enough medium, $\lambda_{1}\gtrsim 1$ puts us
in the saturation regime, it does not necessarily lead to a non-linear
evolution. If the medium is large ($L^{-}_{N}\sim 1/Q$) we still have a linear
BFKL evolution.

Figure 2: Small $x$ regions for large $N$ nuclei (top panels) and small $N$
nuclei (bottom panel).

For smaller $x\ll 1/N\ll 1$, the collinear-soft and soft modes decouple, while
still being well separated from the collinear function. Even though we will
leave the explicit derivation of the factorization formula in this regime for
a future paper, we can still say a lot about the physics. The dipole needs to
interact at least twice with the nucleus to know its size, hence Eq.(7) is
modified for $m\geq$2\. In particular, the soft function $S_{m}^{\mu\nu}$ must
be further factorized into a $\bar{y}^{-}$(medium size) insensitive soft
function $\hat{S}_{m}^{\alpha\beta}$ (with $L_{N}^{-}\to 1/\nu$) and a
collinear-soft function sensitive to $L_{N}^{-}\nu$. The $m$ copies of the
function $\mathcal{B}_{\bar{n}}$ remains unchanged and still obey the BFKL
equation with the same natural rapidity scale $\nu\sim Q/x=H/Q$. The
refactorization will modify the rapidity RG equation for the soft function
away from BFKL with a natural rapidity scale $\sim Q$. However, by RG
consistency the combination of collinear-soft and soft function must still
obey the BFKL equation. The natural rapidity scale for the CS function is
$\nu\sim H/(QN)$, so evolution to the soft function sums logs of $Q^{2}N/H\sim
xN$. Thus $\ln x$ summation will now have two contributions: the linear BFKL
piece from $\mathcal{B}_{\bar{n}}$; and that from the CS function. At leading
order the latter modifies terms with at least two copies of
$\mathcal{B}_{\bar{n}}$, so we expect it to reproduce the non-linear BK/JIMWLK
equation. As a bonus, the CS function’s RG evolution also sums $\ln N$ terms.
Finally, as we approach $N=1$, the collinear soft mode merges with the
collinear mode. The $O(m)$ glauber exchange yields a two mode factorization
involving the same $\bar{y}^{-}$ insensitive Dipole function
$\hat{S}_{m}^{\alpha\beta}$ mentioned in the previous paragraph. However, in
this case the independent scattering approximation breaks down, so rather than
$m$ copies of ${\cal B}_{\bar{n}}$ two-point correlators, we now have $m$
point correlators ${\cal B}^{m}_{\bar{n}}$. This dependence on distinct
boundary conditions for each $m$ seems to complicate the resummation. To
determine from our EFT whether the $m$ dependence of these boundary conditions
is calculable at $Q$ with a small number of nonperturbative functions at
$\Lambda_{\rm QCD}$ requires a further scale separation $\Lambda_{\rm
QCD}^{2}\ll Q^{2}$ that we have not performed here. This assumption has been
implemented when the CGC formalism has been applied to a single hadron Iancu
_et al._ (2015); Beuf  _et al._ (2020).

We setup a top-down EFT for small $x$ DIS on a large inhomogeneous nucleus,
with specific power counting parameters that define its range of validity. We
gave a factorization formula which has an operator definition for the
universal physics of the medium, and separates out the probe and small-$x$
evolution. The structure at all orders in the number of independent probe-
medium interactions leads us to define the mean free path of the probe by an
operator matrix element in the nuclear state. This yields a power counting
parameter that controls the importance of multiple interactions with the
medium, and a means to define the onset of saturation. The case $x\ll 1/N\ll
1$ requires the introduction of a new soft-collinear radiation mode, and
causes a non-linear modification to the small-$x$ resummation.

###### Acknowledgements.

We thank Ira Rothstein and Raju Venugopalan for helpful discussions. This work
was supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, from DE-SC0011090 and in part by the Simons Foundation
through the Investigator grant 327942. V.V is supported by startup funds from
the University of South Dakota.

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