# An FPT Algorithm for the Exact Matching Problem and NP-hardness of Related
Problems

Hitoshi Murakami Independent Researcher (Graduated from Osaka University on
March 2024), Japan. Yutaro Yamaguchi Osaka University, Japan. Email:
<EMAIL_ADDRESS>
###### Abstract

The exact matching problem is a constrained variant of the maximum matching
problem: given a graph with each edge having a weight $0$ or $1$ and an
integer $k$, the goal is to find a perfect matching of weight exactly $k$.
Mulmuley, Vazirani, and Vazirani (1987) proposed a randomized polynomial-time
algorithm for this problem, and it is still open whether it can be
derandomized. Very recently, El Maalouly, Steiner, and Wulf (2023) showed that
for bipartite graphs there exists a deterministic FPT algorithm parameterized
by the (bipartite) independence number. In this paper, by extending a part of
their work, we propose a deterministic FPT algorithm in general parameterized
by the minimum size of an odd cycle transversal in addition to the (bipartite)
independence number. We also consider a relaxed problem called the correct
parity matching problem, and show that a slight generalization of an
equivalent problem is NP-hard.

#### Keywords

Exact matching problem, Fixed-parameter tractability, Alternating cycles,
Parity constraint.

##  1 Introduction

The _maximum matching problem_ is a fundamental problem in combinatorial
optimization: given a graph, the goal is to find a matching of maximum size.
This problem is well-known to be polynomial-time solvable by deterministic
algorithms, initiated by Edmonds [3].

The _exact matching problem (EM)_ is a constrained variant. We say that a
graph is _0/1-weighted_ if each edge has a weight $0$ or $1$, and define the
weight of an edge subset (a matching, a path, a cycle, etc.) as the sum of the
weights of edges in it. The problem is stated as follows.

######  Problem (Exact Matching (EM)).

Input:

A 0/1-weighted graph $G$ and an integer $k$.

Task:

Determine whether $G$ has a perfect matching of weight $k$.

This problem was introduced by Papadimitriou and Yannakakis [17], and is known
to be polynomial-time solvable by a _randomized_ algorithm proposed by
Mulmuley, Vazirani, and Vazirani [15]. It is, however, still widely open
whether there exists a _deterministic_ polynomial-time algorithm or not,
although several special cases have been solved, e.g., complete or comlete
bipartite graphs [11, 21, 10], $K_{3,3}$-minor-free graphs [22], and bounded-
genus graphs [9]. There are also several research directions such as
relaxation, approximation, and clarifying relations with other problems [2, 4,
6, 7, 22]; see [5] for more details on its history.

###  1.1 FPT Algorithms for Exact Matching Problem

For a parameter $k$ related to the input, a problem is said to be _fixed-
parameter tractable (FPT)_ if there exists and a computable function
$f\colon\mathbb{Z}\to\mathbb{Z}$ such that the problem can be solved in
$f(k)\cdot n^{O(1)}$ time, where $n$ is the input size. We also call an
algorithm _FPT (parameterized by $k$)_ if it admits such a computational time
bound.

Very recently, El Maalouly, Steiner, and Wulf [7] proposed a deterministic FPT
algorithm for EM in bipartite graphs parameterized by the bipartite
independence number. As the _independence number_ of a graph is defined as the
maximum size of an independent set, the _bipartite independence number_ of a
bipartite graph is defined as the maximum size of a _balanced_ independent set
divided by $2$, i.e., the maximum integer $\beta$ such that the bipartite
graph admits an independent set of size $2\beta$ taking exactly $\beta$
vertices from each color class. This result can be regarded as an extension of
the solution for complete bipartite graphs, which always have the bipartite
independence number $0$.

######  Theorem 1.1 (El Maalouly et al. [7]).

The exact matching problem in bipartite graphs can be solved by a
deterministic FPT algorithm parameterized by the bipartite independence
number.

Their result consists of two ingredients. One is an FPT reduction of EM to a
relaxed problem, the _bounded correct parity matching problem (BCPM)_ stated
as follows, parameterized by the independence number in general and by the
bipartite independence number for bipartite graphs.

######  Problem (Bounded Correct Parity Matching (BCPM)).

Input:

A 0/1-weighted graph $G$ and an integer $k$.

Task:

Determine whether $G$ has a perfect matching of weight $k^{\prime}$ such that
$k^{\prime}\leq k$ and $k^{\prime}\equiv k\pmod{2}$.

######  Theorem 1.2 (El Maalouly et al. [7]).

The exact matching problem can be reduced to the bounded correct parity
matching problem by a deterministic FPT algorithm parameterized by the
independence number in general and by the bipartite independence number for
bipartite graphs, where the input graph does not change.

The other is a deterministic polynomial-time algorithm for BCPM in bipartite
graphs.

######  Theorem 1.3 (El Maalouly et al. [7]).

There exists a deterministic polynomial-time algorithm for the bounded correct
parity matching problem in bipartite graphs.

Their algorithm is based on a standard dynamic programming approach for an
equivalent problem. It seems difficult to be generalized to general graphs
(due to existence of so-called _blossoms_), and it remains open whether BCPM
can be deterministically solved in polynomial time or not.

In this paper, we try to fill the gap between general graphs and bipartite
graphs by considering an _odd cycle transversal_ , which is a vertex subset
intersecting all the odd cycles in the graph, or equivalently, whose removal
makes the graph bipartite. Our results are stated as follows.

######  Theorem 1.4.

  * (1)

The bounded correct parity matching problem can be solved by a deterministic
FPT algorithm parameterized by the minimum size of an odd cycle transversal.

  * (2)

The exact matching problem can be solved by a deterministic FPT algorithm
parameterized by the independence number plus the minimum size of an odd cycle
transversal. The independence number can be sharpened as the bipartite
independence number after removing a minimum odd cycle transversal.

Since the empty set is an odd cycle transversal for any bipartite graph,
Theorem 1.4 extends Theorems 1.3 and 1.1.

###  1.2 On Correct Parity Matching Problem

We also consider a further relaxed problem, the _correct parity matching
problem (CPM)_ stated as follows.

######  Problem (Correct Parity Matching (CPM)).

Input:

A 0/1-weighted graph $G$ and an integer $k$.

Task:

Determine whether $G$ has a perfect matching of weight $k^{\prime}$ such that
$k^{\prime}\equiv k\pmod{2}$.

El Maalouly, Steiner, and Wulf [7] also proposed a deterministic polynomial-
time algorithm for this problem. Their algorithm utilizes a linear algebraic
trick with the aid of Lovász’ algorithm [14] for finding a basis of the linear
subspace spanned by perfect matchings. It is elegant but heavily depends on
the fact that we are only interested in the parity of the weight, and it seems
difficult to obtain from it a promising idea to tackle EM.

Let us consider a “purely graphic” approach to CPM, which may give some
hopeful idea for EM. We first find a perfect matching $M$. If the weight of
$M$ has the same parity as $k$, we are done. Otherwise, we solve the _odd
alternating cycle problem (OAC)_ stated as follows, where an _$M$ -alternating
cycle_ is a simple cycle that alternates between edges in $M$ and not in $M$.

######  Problem (Odd Alternating Cycle (OAC)).

Input:

A 0/1-weighted graph $G$ and a perfect matching $M$ in $G$.

Task:

Determine whether $G$ has an $M$-alternating cycle of odd weight.

It is not difficult to observe that the answers of the original CPM instance
and of the corresponding OAC instance are the same. If there exists an
$M$-alternating cycle $C$ of odd weight, then the symmetric difference
$M\triangle C=(M\setminus C)\cup(C\setminus M)$ is a desired perfect matching
(with its parity different from $M$). Conversely, if there exists a perfect
matching $M^{\prime}$ in $G$ with its parity different from $M$, then the
symmetric difference $M\triangle M^{\prime}$ forms disjoint $M$-alternating
cycles having an odd weight in total, which must contain at least one
$M$-alternating cycle of odd weight.

Thus, CPM and OAC are polynomial-time equivalent. One natural way to solve OAC
is testing for each edge $e\in M$ whether there exists such a cycle through
$e$. We call this subproblem the _odd alternating cycle through an edge
problem (OACe)_ , stated as follows.

######  Problem (Odd Alternating Cycle through an Edge (OACe)).

Input:

A 0/1-weighted graph $G$, a perfect matching $M$ in $G$, and a matching edge
$e\in M$.

Task:

Determine whether $G$ has an $M$-alternating cycle of odd weight and through
$e$.

Unfortunately, this problem turns out NP-hard. This result implies that, in
order to solve CPM, we should search an odd alternating cycle not locally but
globally.

######  Theorem 1.5.

The odd alternating cycle through an edge problem is NP-hard even if the input
graph is bipartite and contains exactly one (matching) edge of weight $1$.

The proof technique is based on a recent result of Schlotter and Sebő [19] on
the NP-hardness of finding a shortest odd path between two specified vertices
in an edge-weighted undirected graph with no negative cycle.

###  1.3 Organization

The rest of this paper is organized as follows. In Section 2, we describe
necessary definitions. In Section 3, we prove Theorem 1.4, and discuss
heuristic speeding-up of our FPT algorithm. In Section 4, we prove Theorem
1.5, and discuss further related problems.

##  2 Preliminaries

For basic concepts, terms, and notation on graphs and algorithms, see, e.g.,
[12, 20].

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$, which is not
necessarily simple. A _(simple) path_ $P$ in $G$ is a connected subgraph
$(V(P),E(P))$ of $G$, defined by an alternating sequence of vertices and edges
$(v_{0},e_{1},v_{1},e_{2},v_{2},\dots,v_{\ell-1},e_{\ell},v_{\ell})$ such that
all vertices $v_{0},v_{1},\dots,v_{\ell}$ are distinct and
$e_{i}=\\{v_{i-1},v_{i}\\}\in E$ for each $i=1,2,\dots,\ell$, where
$V(P)=\\{v_{0},v_{1},\dots,v_{\ell}\\}$ and
$E(P)=\\{e_{1},e_{2},\dots,e_{\ell}\\}$. A _(simple) cycle_ $C$ in $G$ is a
connected subgraph $(V(C),E(C))$ of $G$, defined by an alternating sequence of
vertices
$(v_{0},e_{1},v_{1},e_{2},v_{2},\dots,v_{\ell-1},e_{\ell},v_{\ell},e_{0},v_{0})$
such that
$P=(v_{0},e_{1},v_{1},e_{2},v_{2},\dots,v_{\ell-1},e_{\ell},v_{\ell})$ is a
path and $e_{0}=\\{v_{\ell},v_{0}\\}\in E\setminus E(P)$, where $V(C)=V(P)$
and $E(C)=E(P)\cup\\{e_{0}\\}$. We also deal with paths and cycles just as
edge subsets or as vertex sequences (if there is no confusion). For directed
graphs, we define them analogously by replacing undirected edges $\\{u,v\\}$
between $u$ and $v$ with directed edges $(u,v)$ from $u$ to $v$.

A _matching_ in $G$ is a subset of edges no two of which share a vertex. A
matching is said to be _perfect_ if it covers all vertices. A path or cycle is
said to be _$M$ -alternating_ if it alternates between edges in $M$ and not in
$M$.

An _independent set_ in $G$ is a subset of vertices no two of which are
adjacent in $G$. The maximum size of an independent set in a graph is called
the _independence number_. The complement of an independent set is called a
_vertex cover_ , i.e., for a vertex cover $X$ in $G$, any edge in $G$ is
incident to some vertex in $X$.

A graph is said to be _bipartite_ if there exists a (possibly trivial)
bipartition $(A,B)$ of its vertex set such that $A$ and $B$ are both
independent sets. When we fix such a bipartition, $A$ and $B$ are called the
_color classes_ of the bipartite graph. Then, a _balanced independent set_ in
the bipartite graph is an independent set $X$ such that $|X\cap A|=|X\cap B|$.
The maximum size of a balanced independent set in a bipartite graph divided by
$2$ (i.e., counting the vertices in either color class) is called the
_bipartite independence number_. An _odd cycle transversal_ in $G$ is a vertex
subset $X$ such that $G-X$ is bipartite, where $G-X$ denotes the subgraph of
$G$ obtained by removing the vertices in $X$ with its incident edges.

##  3 An FPT Algorithm for Exact Matching Problem

###  3.1 Proof of Theorem 1.4

In this section, we prove Theorem 1.4.

Let $G=(V,E)$ be a 0/1-weighted graph, $k$ be an integer, and $X\subseteq V$
be an odd cycle transversal of $G$. Let $(A,B)$ be a pair of the two color
classes of $G-X$ (if it is not unique, fix an arbitrary one). For each subset
$Y\subseteq X$, we define a bipartite graph $G_{Y}=(A\cup Y,B\cup(X\setminus
Y);E_{Y})$ as follows, which is a subgraph of $G$:

$E_{Y}\coloneq\\{\,\\{u,v\\}\in E\mid u\in A\cup Y,\ v\in B\cup(X\setminus
Y)\,\\}.$

We then have the following lemma.

######  Lemma 3.1.

$(G,k)$ is a yes-instance of BCPM (or EM) if and only if $(G_{Y},k)$ is a yes-
instance of BCPM (or EM, respectively) for some $Y\subseteq X$.

###### Proof.

Since each $G_{Y}$ is a subgraph of $G$, if $G_{Y}$ has a desired perfect
matching, then so does $G$.

Conversely, we show that, for any perfect matching $M$ in $G$, there exists a
subset $Y\subseteq X$ such that $G_{Y}$ has the same perfect matching $M$.
Since $A$ and $B$ are both independent sets in $G$ (as $G-X$ is bipartite),
each edge $e=\\{u,v\\}\in M$ satisfies $|\\{u,v\\}\cap A|\leq 1$ and
$|\\{u,v\\}\cap B|\leq 1$.

  * •

If $|\\{u,v\\}\cap A|=1$ and $|\\{u,v\\}\cap B|=1$, then $e$ always exists in
$G_{Y}$ by definition.

  * •

Suppose that $|\\{u,v\\}\cap A|=1$ and $|\\{u,v\\}\cap B|=0$. By symmetry, we
assume $u\in A$. In this case, $e\in E_{Y}$ if and only if $v\in X\setminus
Y$.

  * •

Suppose that $|\\{u,v\\}\cap A|=0$ and $|\\{u,v\\}\cap B|=1$. By symmetry, we
assume $u\in B$. In this case, $e\in E_{Y}$ if and only if $v\in Y$.

  * •

Suppose that $|\\{u,v\\}\cap A|=0$ and $|\\{u,v\\}\cap B|=0$. In this case,
$e\in E_{Y}$ if and only if $|\\{u,v\\}\cap Y|=1$.

Since $M$ is a matching, all the end vertices of the edges in $M$ are
distinct. Therefore, we can consider the above conditions independently for
each edge in $M$, and then some $Y\subseteq X$ satisfies the conditions of
$e\in E_{Y}$ for all the edges $e\in M$. Thus, we are done. ∎

By Lemma 3.1, we can solve BCPM by finding a minimum odd cycle transversal $X$
and by solving BCPM in the bipartite graphs $G_{Y}$ for all subsets
$Y\subseteq X$, i.e., $2^{|X|}$ times. Since a minimum odd cycle transversal
can be found by deterministic FPT algorithms parameterized by its size
(initiated by Reed, Smith, and Vetta [18]), combining with Theorem 1.3, we
obtain Theorem 1.4(1).

The first statement of Theorem 1.4(2) is an immediate consequence of Theorem
1.2 and Theorem 1.4(1) applied in this order. To see the second statement of
Theorem 1.4(2), let us swap the order, i.e., we first find a minimum odd cycle
transversal $X$ and apply Lemma 3.1, and then apply Theorem 1.2 to each
$G_{Y}$. We then obtain an algorithm for EM, which is FPT parameterized by the
maximum of the bipartite independence numbers $\beta_{Y}$ of $G_{Y}$ plus
$|X|$. As shown below, for every $Y\subseteq X$, the parameter $\beta_{Y}$ is
bounded by the bipartite independence number $\beta$ of $G-X$ plus $|X|$,
which concludes the second statement of Theorem 1.4(2).

######  Lemma 3.2.

$\beta_{Y}\leq\beta+|X|$ for any $Y\subseteq X$.

###### Proof.

Let $Z$ be a maximum balanced independent set in $G_{Y}$. Then, $Z\cap(A\cup
B)=(Z\cap A)\cup(Z\cap B)$ is an independent set in $G-X$, which includes a
balanced independent set of size $2\min\\{|Z\cap A|,|Z\cap B|\\}$ in $G-X$.
Also, as $Z$ is balanced, we have $|Z\cap A|+|Z\cap Y|=|Z\cap
B|+|Z\cap(X\setminus Y)|$. Thus,

$2\beta_{Y}=|Z|=2\bigl{(}\min\\{|Z\cap A|,|Z\cap B|\\}+\max\\{|Z\cap
Y|,|Z\cap(X\setminus Y)|\\}\bigr{)}\leq{}2(\beta+|X|).\qed$

######  Remark 3.3.

The bipartite independence number after removing a minimum odd cycle
transversal and the minimum size of an odd cycle transversal are somewhat
correlated, but can behave arbitrarily. For example, by starting with a
bipartite graph with small bipartite independence number, choosing a few
vertices, and add many edges incident to the chosen vertices, one can
construct a graph with both parameters small, for which our FPT algorithm
works.

###  3.2 Heuristic Speeding-Up

In this section, we consider heuristic speeding-up of the algorithm of Theorem
1.4.

Let $G=(V,E)$, $k$, $X$, and $(A,B)$ the same as the beginning of Proof of
Theorem 1.4, and let $n=|V|$. We assume that $G$ has a perfect matching, since
otherwise $(G,k)$ is clearly a no-instance. For $Y\subseteq X$, if
$|A|+|Y|\neq\frac{n}{2}$, then the bipartite graph $G_{Y}$ has no perfect
matching, and hence $(G_{Y},k)$ is a no-instance. Therefore, we can solve
BCPM/EM by solving BCPM/EM in the bipartite graphs $G_{Y}$ for all $Y\subseteq
X$ with $|Y|=\frac{n}{2}-|A|$, i.e., $\binom{|X|}{\frac{n}{2}-|A|}$ times
rather than $2^{|X|}$ times. To make $\binom{|X|}{\frac{n}{2}-|A|}$ small, we
consider the following problem, called the _unbalanced bipartization problem
(UB)_.

######  Problem (Unbalanced Bipartization (UB)).

Input:

A graph $G$ having a perfect matching, and two integers $k$ and $l$.

Task:

Determine whether $G$ has an odd cycle transversal $X\subseteq V$ satisfies
the following conditions:

  * •

$|X|\leq k$.

  * •

$G-X$ has two color classes $(A,B)$ such that $|A|\geq\frac{n}{2}-l$.

######  Lemma 3.4.

The unbalanced bipartization problem is NP-hard.

###### Proof.

We reduce to UB the _odd cycle transversal problem (OCT)_ stated as follows,
which is NP-hard and admits FPT algorithms parameterized by $k$ (initiated by
[18]).

######  Problem (Odd Cycle Transversal (OCT)).

Input:

A graph $G$ and an integer $k$.

Task:

Determine whether $G$ has an odd cycle transversal $X\subseteq V$ of size at
most $k$.

If $G$ has a perfect matching, then the OCT instance $(G,k)$ reduces to an UB
instance $(G,k,\frac{n}{2})$. Thus, it suffices to prove that OCT is NP-hard
even if the input graph has a perfect matching.

Suppose that we are given an OCT instance $(G=(V,E),k)$, where $G$ does not
necessarily have a perfect matching. We define a graph
$G^{\prime}=(V^{\prime},E^{\prime})$ as follows:

$\displaystyle V^{\prime}$ $\displaystyle\coloneq V\cup\\{\,v^{\prime}:v\in
V\,\\},$ $\displaystyle E^{\prime}$ $\displaystyle\coloneq
E\cup\\{\,\\{v,v^{\prime}\\}:v\in V\,\\}.$

Clearly, $G^{\prime}$ has a perfect matching consisting of the additional
edges. Also, since the degree of each additional vertex is $1$, no cycle
intersects them, and hence the two OCT instances $(G,k)$ and $(G^{\prime},k)$
are equivalent. Thus, we are done. ∎

If $l\geq\frac{k}{2}$, then the UB instance $(G,k,l)$ is equivalent to the OCT
instance $(G,k)$ ($|X|\leq k$ implies that $|A|\geq\frac{n-k}{2}$ or
$|B|\geq\frac{n-k}{2}$). Thus, we assume $l<\frac{k}{2}$. A naïve brute-force
algorithm for UB takes $n^{k+O(1)}$ time. We can design an XP algorithm with a
better computational time bound as follows.

######  Lemma 3.5.

The unbalanced bipartization problem can be solved in $2.3146^{k}n^{l+O(1)}$
time.

###### Proof.

This proof is based on an FPT algorithm for OCT parameterized by $k$ through a
reduction to the _unweighted vertex cover problem (UVC)_ (cf. [1]).

######  Problem (Unweighted Vertex Cover (UVC)).

Input:

A graph $G$ and an integer $k$.

Task:

Determine whether $G$ has a vertex cover of size at most $k$.

The _weighted vertex cover problem (WVC)_ is a weighted variant of UVC.

######  Problem (Weighted Vertex Cover (WVC)).

Input:

A graph $G$, a vertex-weight function $w\colon V\to\mathbb{Z}_{\geq 0}$, and
an integer $k$.

Task:

Determine whether $G$ has a vertex cover of weight at most $k$.

The proof is sketched as follows. We first reduce UB to WVC, and then reduce
WVC to UVC by the result of Niedermeier and Rossmonith [16] (Lemma 3.6).
Finally, we solve UVC by an FPT algorithm proposed by Lokshtanov,
Narayanaswamy, Raman, Ramanujan, and Saurabh [13] (Lemma 3.7).

######  Lemma 3.6 (Niedermeier and Rossmonith [16]).

Let $(G=(V,E),w,k)$ be a WVC instance. For each vertex $v\in V$, let $C(v)$ be
the set of $w(v)$ copies of $v$. We define a graph
$\tilde{G}=(\tilde{V},\tilde{E})$ as follows:

$\displaystyle\tilde{V}$ $\displaystyle\coloneq\bigcup_{v\in V}C(v),$
$\displaystyle\tilde{E}$
$\displaystyle\coloneq\\{\,\\{\tilde{u},\tilde{v}\\}:\\{u,v\\}\in E,\
\tilde{u}\in C(u),\ \tilde{v}\in C(v)\,\\}.$

Then, the WVC instance $(G,w,k)$ and the UVC instance $(\tilde{G},k)$ are
equivalent.

######  Lemma 3.7 (Lokshtanov et al. [13]).

There exists a deterministic algorithm for UVC running in
$2.3146^{k-\mu}n^{O(1)}$ time, where $\mu$ is the maximum size of a matching
in the graph.

Suppose that we are given a UB instance $G=(V,E)$, $k$, and $l$. Let $M$ be a
perfect matching in $G$. For each subset $F\subseteq M$, we construct a
corresponding WVC instance $(G^{F},w_{F},k_{F})$ as follows.

A graph $G^{F}=(V^{F},E^{F})$ is defined as follows, where $V(F)$ denotes the
set of end vertices of edges in $F$:

$\displaystyle V^{F}_{1}$ $\displaystyle\coloneq\\{\,v_{1}:v\in V\setminus
V(F)\,\\},$ $\displaystyle V^{F}_{2}$ $\displaystyle\coloneq\\{\,v_{2}:v\in
V\,\\},$ $\displaystyle V^{F}$ $\displaystyle\coloneq V^{F}_{1}\cup
V^{F}_{2},$ $\displaystyle E^{F}$
$\displaystyle\coloneq\\{\,\\{v_{1},v_{2}\\}:v\in V\setminus
V(F)\,\\}\cup\\{\,\\{u_{i},v_{i}\\}:\\{u,v\\}\in E,\ u_{i},v_{i}\in
V^{F}_{i},\ i\in\\{1,2\\}\,\\}.$

A vertex-weight function $w_{F}\colon V^{F}\to\mathbb{Z}_{\geq 0}$ is defined
as

$w_{F}(v)\coloneq\begin{cases}k+1&(v\in V^{F}_{1}),\\\ 1&(v\in V^{F}_{2}).\\\
\end{cases}$

Let us define

$k_{F}=\left(\frac{n}{2}-|F|\right)(k+2)+k.$

We then have the following lemma.

######  Lemma 3.8.

$(G,k,l)$ is a yes-instance of UB if and only if $(G^{F},w_{F},k_{F})$ is a
yes-instance of WVC for some $F\subseteq M$ with $|F|\leq l$.

###### Proof.

Suppose that $G$ has a desired odd cycle transversal $X\subseteq V$. Then,
$G-X$ has two color classes $(A,B)$ such that $|A|\geq\frac{n}{2}-l$. Let
$F=\\{\,e:e\in M,\ e\cap A=\emptyset\,\\}$, $A^{\prime}=\\{\,v_{1}:v\in
A\,\\}\subseteq V_{1}^{F}$, and $B^{\prime}=\\{\,v_{2}:v\in B\,\\}\subseteq
V_{2}^{F}$. As $A^{\prime}\cup B^{\prime}$ is an independent set in $G^{F}$,
its complement $V^{F}\setminus(A^{\prime}\cup B^{\prime})$ is a vertex cover
in $G^{F}$. Since $|A^{\prime}|=|A|=|M|-|F|=\frac{n}{2}-|F|$ and
$|B^{\prime}|=|B|=n-|X|-|A|\geq\frac{n}{2}-k+|F|$, its weight is

$\displaystyle w_{F}(V^{F})-w_{F}(A^{\prime}\cup B^{\prime})$
$\displaystyle=(|V_{1}^{F}|-|A^{\prime}|)(k+1)+|V_{2}^{F}|-|B^{\prime}|$
$\displaystyle\leq\left((n-2|F|)-\left(\frac{n}{2}-|F|\right)\right)(k+1)+n-\left(\frac{n}{2}-k+|F|\right)$
$\displaystyle=k_{F}.$

Thus, $V^{F}\setminus(A^{\prime}\cup B^{\prime})$ is a desired vertex cover.
As $|A|\geq\frac{n}{2}-l$, we have $|F|=|M|-|A|\leq l$.

Conversely, suppose that $G^{F}$ has a desired vertex cover $X$ for some
$F\subseteq M$ with $|F|\leq l$. We first show that

$\displaystyle|X\cap V_{1}^{F}|=\frac{n}{2}-|F|.$ (1)

Since $V_{1}^{F}$ has a perfect matching corresponding to $M\setminus F$, we
have $|X\cap V_{1}^{F}|\geq\frac{n}{2}-|F|$. To derive a contradiction,
suppose that $|X\cap V_{1}^{F}|>\frac{n}{2}-|F|$. Since $V_{2}^{F}$ also has a
perfect matching corresponding to $M$, we have $|X\cap
V_{2}^{F}|\geq\frac{n}{2}$. Thus, we obtain

$\displaystyle w_{F}(X)$ $\displaystyle=w_{F}(X\cap V_{1}^{F})+w_{F}(X\cap
V_{2}^{F})$ $\displaystyle\geq\left(\frac{n}{2}-|F|+1\right)(k+1)+\frac{n}{2}$
$\displaystyle=k_{F}+|F|+1,$

which contradicts $w_{F}(X)\leq k_{F}$. Thus, we have (1).

From (1) and $w_{F}(X)\leq k_{F}$, we obtain

$\displaystyle|X\cap V_{2}^{F}|\leq k_{F}-w_{F}(X\cap
V_{1}^{F})=\frac{n}{2}-|F|+k.$ (2)

Let $Z=V^{F}\setminus X$, $A^{\prime}=Z\cap V^{F}_{1}$, and $B^{\prime}=Z\cap
V^{F}_{2}$. Let $A$ and $B$ denote the vertex subsets in $G$ corresponding to
$A^{\prime}$ and $B^{\prime}$, respectively. We show that $V\setminus(A\cup
B)$ is a desired odd cycle transversal in the UB instance $(G,k,l)$.

Since $X$ is a vertex cover in $G^{F}$, its complement $Z$ is an independent
set in $G^{F}$. Since $G^{F}$ has an edge $\\{v_{1},v_{2}\\}$ for every $v\in
V\setminus V(F)$, we have $A\cap B=\emptyset$. In addition, both $A$ and $B$
are independent sets in $G$, and hence $V\setminus(A\cup B)$ is an odd cycle
transversal in $G$. From (1) and $|F|\leq l$, we obtain

$\displaystyle|A|=|V^{F}_{1}|-|X\cap
V^{F}_{1}|=\frac{n}{2}-|F|\geq\frac{n}{2}-l.$

Combined with (2), we have

$\displaystyle|V\setminus(A\cup B)|$ $\displaystyle=|V|-|A|-|B|$
$\displaystyle=|V|-|A|-(|V^{F}_{2}|-|X\cap V^{F}_{2}|)$ $\displaystyle\leq
n-\left(\frac{n}{2}-|F|\right)-\left(n-\left(\frac{n}{2}-|F|+k\right)\right)$
$\displaystyle=k.$

Thus, $V\setminus(A\cup B)$ is a desired odd cycle transversal. ∎

By Lemma 3.8, we can solve a UB instance $(G,k,l)$ by solving the WVC
instances $(G_{F},w,k_{F})$ for all subsets $F\subseteq M$ with $|F|\leq l$,
i.e., $O(n^{l})$ times. By Lemma 3.6, we can define the corresponding UVC
instance $(\tilde{G}^{F}=(\tilde{V}^{F},\tilde{E}^{F}),k_{F})$ as follows:

$\displaystyle\tilde{V}^{F}$ $\displaystyle\coloneq\bigcup_{v\in V^{F}}C(v),$
$\displaystyle\tilde{E}^{F}$
$\displaystyle\coloneq\\{\,\\{\tilde{u},\tilde{v}\\}:\\{u,v\\}\in E^{F},\
\tilde{u}\in C(u),\ \tilde{v}\in C(v)\,\\}.$

As $G^{F}$ has a perfect matching $M^{F}=\\{\,\\{u_{1},v_{1}\\}:\\{u,v\\}\in
M\setminus F\,\\}\cup\\{\,\\{u_{2},v_{2}\\}:\\{u,v\\}\in M\,\\}$,
$\tilde{G}^{F}$ has a corresponding perfect matching $\tilde{M}^{F}$. By Lemma
3.7, we can solve each UVC instance $(G^{\prime}_{F},k_{F})$ in
$2.3146^{k_{F}-|\tilde{M}^{F}|}n^{O(1)}$ time, where

$\displaystyle k_{F}-|\tilde{M}^{F}|$
$\displaystyle=\left(\frac{n}{2}-|F|\right)(k+2)+k-\left(\left(\frac{n}{2}-|F|\right)(k+1)+\frac{n}{2}\right)$
$\displaystyle=k-|F|$ $\displaystyle\leq k.$

Thus, we are done. ∎

##  4 NP-hardness of Related Problems

###  4.1 Proof of Theorem 1.5

In this section, we prove Theorem 1.5. The proof is similar to that of [19,
Theorem 4.3]. We reduce to OACe the _back and forth paths problem (BFP)_
stated as follows, which is known to be NP-hard [8].

######  Problem (Back and Forth Paths (BFP)).

Input:

A directed graph $G$ and two vertices $s$ and $t$.

Task:

Determine whether $G$ has a simple cycle that contains both $s$ and $t$.

Suppose that we are given a BFP instance $G=(V,E)$ and $s,t\in V$. We
construct the corresponding OACe instance, i.e., a 0/1-weighted graph
$\hat{G}=(\hat{V},\hat{E})$, a perfect matching $\hat{M}$, and
$\hat{e}\in\hat{M}$ as follows.

We split each $v\in V$ into two vertices, _in-copy_ $v^{-}$ and _out-copy_
$v^{+}$. For each directed edge $(u,v)\in E$, we create an edge
$\\{u^{+},v^{-}\\}$ of weight $0$. For each vertex $v\in V$, we create an edge
$\\{v^{-},v^{+}\\}$, whose weight is $1$ if $v=t$ and $0$ otherwise. Define
$\hat{M}$ as those edges $\\{v^{-},v^{+}\\}\ (v\in V)$, and let $\hat{e}$ be
$\\{s^{-},s^{+}\\}$. To sum up, we define

$\displaystyle\hat{V}$ $\displaystyle\coloneq\\{\,v^{-}:v\in
V\,\\}\cup\\{\,v^{+}:v\in V\,\\},$ $\displaystyle\hat{E}$
$\displaystyle\coloneq\\{\,\\{u^{+},v^{-}\\}:(u,v)\in
E\,\\}\cup\\{\,\\{v^{-},v^{+}\\}:v\in V\,\\},$ $\displaystyle\hat{M}$
$\displaystyle\coloneq\\{\,\\{v^{-},v^{+}\\}:v\in V\,\\},$
$\displaystyle\hat{e}$ $\displaystyle\coloneq\\{s^{-},s^{+}\\},$

where only $\\{t^{-},t^{+}\\}$ has weight $1$ . Note that this construction
satisfies the additional conditions (bipartiteness and uniqueness of an edge
of weight 1) in Theorem 1.5.

We show that there exists a simple cycle containing $s$ and $t$ in $G$ if and
only if there exists an $\hat{M}$-alternating cycle of odd weight through
$\hat{e}=\\{s^{-},s^{+}\\}$ in $\hat{G}$.

Suppose that there exists a simple cycle
$C=(s,v_{1},v_{2},\dots,t,\dots,v_{\ell},s)$ containing $s$ and $t$ in $G$.
Then, there exists a corresponding simple cycle
$\hat{C}=(s^{-},s^{+},v_{1}^{-},v_{1}^{+},v_{2}^{-},v_{2}^{+},\dots,t^{-},t^{+},\dots,v_{\ell}^{-},v_{\ell}^{+},s^{-})$
in $\hat{G}$, which is clearly $\hat{M}$-alternating, of weight $1$ (due to
the edge $\\{t^{-},t^{+}\\}$), and through $\hat{e}=\\{s^{-},s^{+}\\}$.

Conversely, suppose that there exists an $\hat{M}$-alternating cycle $\hat{C}$
of odd weight through $\hat{e}=\\{s^{-},s^{+}\\}$ in $\hat{G}$. Since
$\\{t^{-},t^{+}\\}$ is the only edge having weight $1$, it must be traversed
by $\hat{C}$. In addition, since the graph $\hat{G}$ is bipartite and the two
color classes correspond to the signs, $\hat{C}$ alternately intersects
$u^{-}$ and $v^{+}$ for some vertices $u,v\in V$. That is, $\hat{C}$ is a form
of

$(s^{-},s^{+},v_{1}^{-},v_{1}^{+},v_{2}^{-},v_{2}^{+},\dots,t^{-},t^{+},\dots,v_{\ell}^{-},v_{\ell}^{+},s^{-}),$

and hence there exists a corresponding simple cycle
$C=(s,v_{1},v_{2},\dots,t,\dots,v_{\ell},s)$ in $G$. Thus, we are done.

###  4.2 Further Related Problems

In this section, we discuss several more problems related to OACe.

For a matching $M$, an _$M$ -augmenting path_ is a simple path between two
unmatched vertices that alternates between edges in $M$ and not in $M$. This
is an elementary but crucial structure in the maximum matching problem. In
particular, a matching $M$ is not of maximum size if and only if there exists
an $M$-augmenting path.

OACe reduces to the _odd augmenting path problem (OAP)_ stated as follows;
thus, OAP is also NP-hard.

######  Problem (Odd Augmenting Path (OAP)).

Input:

A 0/1-weighted graph $G$ and a matching $M$ in $G$.

Task:

Determine whether $G$ has an $M$-augmenting path of odd weight.

######  Lemma 4.1.

OACe reduces to OAP.

###### Proof.

Suppose that we are given an OACe instance, i.e., a 0/1-weighted graph
$G=(V,E)$, a perfect matching $M$, and a matching edge $e=\\{u,v\\}\in M$.
Consider a graph $G^{\prime}=(V,E\setminus\\{e\\})$ and a matching
$M^{\prime}=M\setminus\\{e\\}$ in $G^{\prime}$. In addition, if the weight of
$e$ is $1$, then we switch the weight (change it from $0$ to $1$ and from $1$
to $0$) of every edge incident to $u$ in $G^{\prime}$. Then, $G$ has an
$M$-alternating cycle of odd weight and through $e$ if and only if
$G^{\prime}$ has an $M^{\prime}$-augmenting path of odd weight. The former is
indeed transformed into the latter just by removing $e$ (note that both
traverse exactly one edge incident to $u$ in $G^{\prime}$, which preserves the
parity of their weights). Also, since $u$ and $v$ are the only unmatched
vertices with respect to $M^{\prime}$, the latter is transformed into the
former by adding $e$. Thus, we are done. ∎

The proof and Theorem 1.5 imply the following.

######  Corollary 4.2.

The odd augmenting path problem is NP-hard even if the input graph is
bipartite and contains exactly one (matching) edge of weight $1$ and the input
matching is of size $\frac{n}{2}-1$, where $n$ is the number of vertices.

OAP with this additional conditions reduces to the _disjoint augmenting path
problem (DAP)_ stated as follows; thus, DAP is also NP-hard.

######  Problem (Disjoint Augmenting Path (DAP)).

Input:

A graph $G$, a matching $M$ in $G$, and unmatched vertices
$s_{1},s_{2},t_{1},t_{2}$.

Task:

Determine whether $G$ has disjoint $M$-augmenting paths between
$\\{s_{1},s_{2}\\}$ and $\\{t_{1},t_{2}\\}$.

The reduction is easy. Let $s_{1},s_{2}$ be the original unmatched vertices,
let $t_{1},t_{2}$ be the end vertices of the unique matching edge $e$ of
weight $1$, and remove $e$.

######  Corollary 4.3.

The disjoint augmenting path problem is NP-hard even if the input graph is
bipartite.

In contrast, a relaxed problem, the _free disjoint augmenting path problem
(FDAP)_ stated as follows, is polynomial-time solvable.

######  Problem (Free Disjoint Augmenting Path (FDAP)).

Input:

A graph $G$, a matching $M$ in $G$, and unmatched vertices
$s_{1},s_{2},t_{1},t_{2}$.

Task:

Determine whether $G$ has disjoint $M$-augmenting paths with end vertices in
$\\{s_{1},s_{2},t_{1},t_{2}\\}$.

######  Lemma 4.4.

FDAP can be solved by a deterministic polynomial-time algorithm.

###### Proof.

Let $G^{\prime}$ be a subgraph of $G$ obtained by removing the unmatched
vertices other than $s_{1},s_{2},t_{1},t_{2}$. If $G^{\prime}$ has a perfect
matching $M^{\prime}$, then the symmetric difference $M\triangle M^{\prime}$
contains desired disjoint $M$-augmenting paths. Conversely, if $G$ contains
desired disjoint $M$-augmenting paths, so does $G^{\prime}$, which implies
that $G^{\prime}$ has a perfect matching. Thus, it suffices to solve the
maximum matching problem in $G^{\prime}$, for which a deterministic
polynomial-time algorithm exists. ∎

## Acknowledgments

The authors are grateful to anonymous reviewers for their helpful comments.
The second author was supported by JSPS KAKENHI Grant Numbers 20K19743 and
20H00605.

## References

  * [1] M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
  * [2] A. Dürr, N. El Maalouly, and L. Wulf. An approximation algorithm for the exact matching problem in bipartite graphs. In Proceedings of the 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2023), No. 18, 21pp., 2023.
  * [3] J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17 (1965), pp. 449–467.
  * [4] N. El Maalouly. Exact matching: algorithms and related problems. In Proceedings of the 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), No. 29, 17pp., 2023.
  * [5] N. El Maalouly, S. Haslebacher, and L. Wulf. On the exact matching problem in dense graphs. arXiv:2401.03924, 2024.
  * [6] N. El Maalouly and R. Steiner. Exact matching in graphs of bounded independence number. In Proceedings of the 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), No. 46, 14pp., 2022.
  * [7] N. El Maalouly, R. Steiner, and L. Wulf. Exact matching: correct parity and FPT parameterized by independence number. In Proceedings of the 34th International Symposium on Algorithms and Computation (ISAAC 2023), No. 28, 18pp., 2023.
  * [8] S. Fortune, J. Hopcroft, and J. Wyllie. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10 (1980), pp. 111–121.
  * [9] A. Galluccio and M. Loebl. On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electronic Journal of Combinatorics, 6 (1999), R6, 19pp.
  * [10] H.-F. Geerdes and J. Szabó. A unified proof for Karzanov’s exact matching theorem. EGRES Quick Proofs, No. 2011-02, 6pp., 2011.
  * [11] A. V. Karzanov. Maximum matching of given weight in complete and complete bipartite graphs. Cybernetics, 23 (1987), pp. 8–13.
  * [12] B. Korte and J. Vygen. Combinatorial Optimization: Theory and Algorithms. 6th Ed., Springer, 2018.
  * [13] D. Lokshtanov, N. S. Narayanaswamy, V. Raman, M. S. Ramanujan, and S. Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms, 11 (2014), No. 15, 31pp.
  * [14] L. Lovász. Matching structure and the matching lattice. Journal of Combinatorial Theory, Series B, 43 (1987), pp. 187–222.
  * [15] K. Mulmuley, U. V. Vazirani, and V. V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7 (1987), pp. 105–113.
  * [16] R. Niedermeier and P. Rossmonith. On efficient fixed-parameter algorithms for weighted vertex cover. Journal of Algorithms, 47 (2003), pp. 63–77.
  * [17] C. H. Papadimitriou and M. Yannakakis. The complexity of restricted spanning tree problem. Journal of the ACM, 29 (1982), pp. 285–309.
  * [18] B. Reed, K. Smith, and A. Vetta. Finding odd cycle transversals. Operations Research Letters, 32 (2004), pp. 299–301.
  * [19] I. Schlotter and A. Sebő. Odd paths, cycles and $T$-joins: connections and algorithms. arXiv:2211.12862, 2022.
  * [20] A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003.
  * [21] T. Yi, K. G. Murty, and C. Spera. Matchings in colored bipartite networks. Discrete Applied Mathematics, 121 (2002), pp. 261–277.
  * [22] R. Yuster. Almost exact matching. Algorithmica, 63 (2012), pp. 39–50.


aainstitutetext: Rudolf Peierls Centre for Theoretical Physics, Clarendon
Laboratory, Parks Road, University of Oxford, Oxford OX1 3PU, UK

# The Size of a Soliton

John F. Wheater a P. D. Xavier<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the recently proposed Bound State Conjecture for quantum field
theory in the context of solitons in two dimensional $\phi^{4}$ scalar field
theory. We calculate the quantum correction to the size of the soliton which,
taken together with the known mass correction, shows behaviour consistent with
regarding the soliton as a bound state, and with the conjecture.

###### Keywords:

soliton size, form-factor

##  1 Introduction

The requirement Arkani-Hamed:2006emk  of a consistent ultra-violet completion
including gravity is a powerful route to constraining field theories for low
energy phenomena and has led to many significant results – see Palti:2019pca
for a recent review and vanbeest:2021abc  for recent lecture notes on the
topic. In particular the Weak Gravity Conjecture leads to constraints on
scalar masses and coupling constants Palti:2017elp . These results motivated
the recent suggestion that the minimal size for a bound state system exists
_independently_ of an ultraviolet completion including gravity. The Bound
State Conjecture Hebecker2019  proposes that bound states in a renormalizable
quantum (effective) field theory (QFT) must have a minimum radius in the
following sense. Consider a bound state of radius $R$ in a theory where
$m\ll\Lambda$ is the mass of heaviest elementary particle and $\Lambda$ is the
cut-off scale; then the dimensionless quantity $mR=\Delta(\\{\lambda\\})$ is a
function of the parameters of the theory, $\\{\lambda\\}$, such as the
couplings and other particle masses. The proposal of Hebecker2019  is that
$\Delta(\\{\lambda\\})$ is bounded below by some positive constant, i.e.
$\Delta(\\{\lambda\\})\geq\delta>0$. The conjecture is non-trivial because, in
a theory with attractive interactions one can imagine turning-up the
appropriate coupling indefinitely and squeezing any bound state to an
arbitrarily small size. Clearly for systems that are only weakly bound, for
example the Hydrogen atom, the size will initially decrease as the coupling
strength is raised. However, as this happens, heuristically the constituents
become more energetic (momentum $p\sim R^{-1}$) and the process may become
unstable when there is enough kinetic energy for particle production (similar
to the mechanism involved in the Hagedorn limit Hagedorn ).

The authors of Hebecker2019  considered a number of examples, mostly in scalar
field theories in four dimensions, where they were able to show mechanisms
that lead to the conjecture being satisfied. Field theories containing non-
elementary states where the full dynamics can be closely controlled are not so
common, but theories containing topological solitons are one example. Such
solitons are known to possess a particle-like interpretation in QFT
GervaisExtended1975 ; on the other hand they are also extended objects for
which stability against finite energy perturbations is guaranteed so $R$
should remain a well-defined quantity independent of coupling. Solitons can
also be viewed as a bound state of the fundamental point-like particles; for
example baryons in $N_{f}>1$ QCD are described by the Skyrme model
Zahed:1986qz , ma:2016abc , and in $N_{f}=1$ QCD by a quantum hall droplet
Komargodski:2018abc . In this paper we investigate the behaviour of the radius
of kink solution of two-dimensional $\phi^{4}$ field theory (hereafter the
$\phi_{2}^{4}$ model) by computing its first quantum correction as a first
step toward understanding whether the conjecture applies for this model.

Let us briefly summarise the arguments and contents of this paper. Solitons
are finite energy, non-dissipative solutions to the classical field equations
– non-dissipative means that
$\lim_{t\to\infty}\max_{\bm{x}}\mathcal{H}(t,\bm{x})\neq 0$ Coleman1985  (Sec.
6.1), where $\mathcal{H}$ is the energy density. Intuitively, this means that
the field stays lumped-up for all time. Classically, the radius of a soliton,
$R_{cl}$, is easy to define and compute: we adopt the definition that it is
the width of the energy density, normalized by the total energy Lee1987 ,

$R^{2}_{cl}=\int d^{d}x\,\,|\bm{x}|^{2}\mathcal{H}(\bm{x})/\int
d^{d}x\,\mathcal{H}(\bm{x}).$ (1)

If the soliton solution is known then (1) can be evaluated straightforwardly.
To derive the quantum correction requires a quantum theory of solitons and a
method to compute matrix elements involving a single soliton in the initial
and final states. The collective coordinate quantization developed in the
1970’s GervaisPoint1976 ; Raj  enables this.

The configuration space of QFT in $\mathbb{R}^{d+1}$ is the set of field
configurations on $\mathbb{R}^{d}$. (Let $\bm{x}$ denote positions in
$\mathbb{R}^{d}$.) An elementary particle state of the QFT is a plane-wave
impulse of the field. This can be made precise: take the initial data
$\phi_{cl}(\bm{x})=\varepsilon\,e^{i\bm{p}.\bm{x}}$,
$\pi_{cl}(\bm{x})=\varepsilon\,ip^{0}e^{i\bm{p}.\bm{x}}$ and define an
operator $O=\exp i\int d^{d}\bm{x}\left(\pi\phi_{cl}-\phi\pi_{cl}\right)$.
This operator has the property that $O^{-1}\phi O=\phi+\phi_{cl}$ and
$O^{-1}\pi O=\pi+\pi_{cl}$. Therefore, it causes a shift by the classical
configurations $\phi_{cl}$, $\pi_{cl}$. In the limit that $\varepsilon$
becomes infinitesimal,
$\partial_{\varepsilon}O|_{\varepsilon=0}=a^{\dagger}_{\bm{p}}$. Therefore,
$a^{\dagger}_{\bm{p}}$ creates a plane wave impulse. In just the same way, if
the QFT contains a soliton, described by some initial data $\phi_{cl}$ and
$\pi_{cl}$, then we can construct a soliton creation operator (essentially a
coherent state operator) – as was first done in Cahill1974 .

To obtain a state that _approximately_ describes an elementary particle, one
usually applies $a^{\dagger}$ to the vacuum state $|0\rangle$. Similarly, to
obtain a state approximately describing a soliton, one should act with $O$ on
the vacuum. In an interacting theory these approximate states (hereafter
called ‘proxy states’) are not eigenstates of the Hamiltonian (which we will
call ‘exact states’). However, the proxy states are designed to have non-zero
overlap with the corresponding exact state with the same quantum numbers.
Assuming that we are dealing with the lowest energy state in the channel, the
exact state can be extracted by a limiting procedure:
$\ket{\text{exact}}=\lim_{T\to\infty}e^{ET}e^{-HT}\ket{\text{proxy}}/\braket{\text{exact}}{\text{proxy}}$
where $E$ is the energy of the state. With this procedure in hand, one can
study matrix elements between exact states. For example, given an exact
elementary particle state $\ket{p^{\mu}}$ consider
$\bra{p^{\prime}}\mathcal{H}\ket{p}$. Using the limiting procedure, this can
be written as a matrix element between proxy states:
$\sim\bra{0}a_{\bm{p}^{\prime}}e^{-HT}\,\mathcal{H}\,e^{-HT}a^{\dagger}_{\bm{p}}\ket{0}$.
In the case of solitons, $a$ and $a^{\dagger}$ will be replaced by
$O^{\dagger}$ and $O$ respectively, but everything else is the same.

Since the classical radius is defined as the second moment of the energy
density, (1), to define a quantum analogue, we need to consider the form
factor $\bra{p^{\prime}}\mathcal{H}(0)\ket{p}$. Taking derivatives of this
w.r.t. to the momentum transfer $p^{\prime}-p$ gives moments of the energy
density; in particular, taking two derivatives gives the second moment, so we
define the quantum radius by

$\displaystyle
R^{2}=M^{-2}\partial_{\ell^{2}}\bra{k}\mathcal{H}(0)\ket{k+\ell}\big{|}_{\ell=0}$
(2)

which we will show gives
$R^{2}=\frac{1}{2M^{4}}\bra{k}\mathcal{H}(0)K^{2}\ket{k}$ in two dimensions,
where $K$ is the boost generator and $k^{\mu}$ is the rest momentum $(M,0)$.
This is a matrix element between an initial and final soliton at rest.
Therefore it can be directly evaluated by the methods of collective coordinate
quantization, and the result can be organised in powers of the coupling
constant. The leading term recovers the classical definition (1), and we will
then go on to calculate the first quantum correction.

The layout of the paper is as follows. In §2 we discuss soliton operators and
soliton states. In §3 we introduce the relevant aspects of the classical
$\phi_{2}^{4}$ model and its kink. In §4 we define the radius of the soliton,
quantum mechanically. In §5 we review the method of collective coordinate
quantization and use it to compute the first quantum correction to the kink
radius.

A note on notation: in general spacetime dimensions, we will write spatial
positions as $\bm{x}$ and spacetime positions as $x$ or $x^{\mu}$; in two
dimensions, however, we write spatial positions as $x$ and spacetime positions
as $x^{\mu}$.

##  2 Soliton States and Operators

Consider a QFT of a real scalar, $\phi(\bm{x})$, in $d+1$ dimensional
Minkowksi space. The orthonormal basis states of the Hilbert space are
$\ket{f}_{\phi}$ (where $f\equiv f(\bm{x})$), which are eigenstates of
$\phi(\bm{x})$: $\phi(\bm{x})\ket{f}_{\phi}=f(\bm{x})\ket{f}_{\phi}$.
Similarly, there is a dual orthonormal basis $\ket{g}_{\pi}$ of eigenstates of
$\pi(\bm{x})$, the conjugate momentum operator:
$\pi(\bm{x})\ket{g}_{\pi}=g(\bm{x})\ket{g}_{\pi}$.

Now let the classical equations of motion possess a soliton solution. This
means that in the rest frame of the soliton there exist initial data
$\phi_{cl}(\bm{x})$ and $\pi_{cl}(\bm{x})$ such that the energy density
doesn’t dissipate under classical time evolution. The soliton creation
operator (first written down by Cahill Cahill1974 ) is

$\mathcal{O}(\bm{x})=\exp\left(-i\int\pi(\bm{x}+\bm{a})\phi_{cl}(\bm{a})d^{d}\bm{a}+i\int\phi(\bm{x}+\bm{a})\pi_{cl}(\bm{a})d^{d}\bm{a}\right),$
(3)

and the annihilation (equivalently, anti-soliton) operator is
$\mathcal{O}(\bm{x})^{\dagger}$.

As a consequence of the canonical commutation relations, the soliton operators
cause a shift in the field value by the classical configurations,

$\displaystyle\mathcal{O}(\bm{x})^{-1}\phi(\bm{y})\mathcal{O}(\bm{x})$
$\displaystyle=\phi(\bm{y})+\phi_{cl}(\bm{y}-\bm{x}),$ (4)
$\displaystyle\mathcal{O}(\bm{x})^{-1}\pi(\bm{y})\mathcal{O}(\bm{x})$
$\displaystyle=\pi(\bm{y})+\pi_{cl}(\bm{y}-\bm{x}).$ (5)

From this we deduce that
$\mathcal{O}(\bm{x})\ket{f(\bm{y})}_{\phi}=\text{const.}\ket{f(\bm{y})+\phi_{cl}(\bm{y}-\bm{x})}_{\phi}$
i.e. a soliton centred on $\bm{x}$ has been added to the field
configuration;111Alternatively, this follows from the fact that
$\exp\left(-\int
d^{d}\bm{x}\,g(\bm{x})\pi(\bm{x})\right)\ket{f}_{\phi}=\ket{f+g}_{\phi}$. and
a similar result applies in the dual basis.

$\mathcal{O}(\bm{x})$ transforms as expected under a rotation $\bm{R}$. Assume
that $\phi_{cl}(\bm{x})$ and $\pi_{cl}(\bm{x})$ are centered on $\bm{x}=0$ and
are spherically symmetric. Then, denoting by $U$ the unitary operator acting
on states,

$U(\bm{R})\mathcal{O}(\bm{x})U^{-1}(\bm{R})=\mathcal{O}(\bm{R}\bm{x}),$ (6)

and similarly for a translation $\bm{a}$,

$U(\bm{a})\mathcal{O}(\bm{x})U^{-1}(\bm{a})=\mathcal{O}(\bm{x}+\bm{a}).$ (7)

As mentioned in the introduction, an eigenstate of the Hamiltonian can be
constructed by preparing a state at $t=0$ and time translating it to past
Euclidean infinity. The state at $t=0$ can be thought of as a proxy to the
exact state. Because time translation commutes with $P_{i}$ and $J_{ij}$ (the
spatial-momentum and angular-momentum generators), the proxy state must
already have good spatial-momentum and angular-momentum quantum numbers.
Further, the exact state (if it contains several particles) inherits the
exchange statistics of the proxy state. In conventional field theory (where
one is interested in elementary particles), proxy states are Fock states.
Given a suitable proxy, $\ket{n,\sim}$, the exact state $\ket{n}$ is,222This
follows from inserting a complete set of energy eigenstates and picking out
the state with lowest energy and non-vanishing overlap with $\ket{n,\sim}$.

$\ket{n}=\lim_{T\to\infty}e^{ET}e^{-HT}\ket{n,\sim}/\braket{n}{n,\sim},$ (8)

where $E$ is the energy $H\ket{n}=E\ket{n}$. (8) projects the proxy state onto
the lowest-energy exact-state (with non-vanishing overlap with the proxy-
state) with the spatial-momentum and angular-momentum quantum numbers of the
proxy-state. The factor $\braket{n}{n,\sim}^{-1}$ ensures $\ket{n}$ is
normalized to unity. We should note that (8) avoids Haag’s theorem
Streater1989  because $e^{ET}e^{-HT}$ is not unitary. The factor $e^{ET}$
amputates external legs.333For a simple demonstration of this, imagine
$\ket{n,\sim}$ was a Fock state. Then $e^{ET}$ is equivalent to $e^{H_{0}T}$
since $H_{0}$ is assumed to have the same spectrum as $H$ (Weinberg2005  Sec
3.1). Then the difference between $H$ and $H_{0}$ will contain the difference
between the bare mass (contained in $H$) and the exact mass (contained in
$H_{0}$). These interaction vertices will then completely remove radiative
corrections.

Although (8) looks like it is in Euclidean signature, it is actually physical.
As an example, consider $\phi^{4}$ theory in 3+1 dimensions. Let
$\braket{m}{n}$ be a scattering amplitude, with
$\ket{n,\sim}=\prod_{i=1}^{n}a^{\dagger}_{\bm{k}_{i}}\ket{0}$ and
$\ket{m,\sim}=\prod_{j=1}^{m}a^{\dagger}_{\bm{p}_{j}}\ket{0}$. Using (8), we
get

$\braket{m}{n}=C\int[d\phi]e^{-S[\phi]}\int\prod_{i=1}^{n}d^{4}x_{i}e^{i\bm{k}_{i}\bm{x}_{i}-E_{\bm{k}_{i}}x_{i}^{0}}\Box_{x_{i}}\phi(x_{i})\int\prod_{j=1}^{m}d^{4}y_{j}e^{-i\bm{p}_{j}\bm{y}_{j}+E_{\bm{p}_{j}}y_{j}^{0}}\Box_{y_{j}}\phi(y_{j}),$
(9)

where $C=\braket{n}{n,\sim}^{-1}\braket{m,\sim}{m}^{-1}$,
$\Box_{x}\equiv(\partial_{x^{0}}^{2}+\bm{\nabla}_{\bm{x}}^{2}-m^{2})$,
$E_{\bm{k}}=\sqrt{\bm{k}^{2}+m^{2}}$ and $S=\int
d^{4}x(\dot{\phi}^{2}+(\bm{\nabla}\phi)^{2}+m^{2}\phi^{2})/2+\lambda\phi^{4}$.
Up to the factor $C$, this is the amputated, Euclidean Greens function
$G_{E}(k_{1},...,k_{n};-p_{1},...,-p_{m})$ with
$k_{i}=(iE_{\bm{k}_{i}},\bm{k}_{i})$ and $p_{j}=(iE_{\bm{p}_{j}},\bm{p}_{j})$.
The net result is that the external momenta are in Lorentz signature and the
loop-momenta (i.e. unfixed momenta) are in Euclidean signature, which is
exactly the prescription used to calculate physical amplitudes. Further, the
missing factors of $i$ in the Feynman rules only affect the amplitude by an
overall factor of $i$ (Ramond  Sec. 4.8).

Here, we need to construct a 1-particle soliton state. It is enough to
construct the rest-state, from which the moving states can be obtained by a
boost. Let $k^{\mu}=(M,0,...,0)$ be the rest momentum and let $\ket{k,\sim}$
be the proxy to the rest-state. It is Cahill1974

$\ket{k,\sim}=\int d^{d}\bm{x}\,\,\mathcal{O}(\bm{x})\ket{\Omega,\sim},$ (10)

with

$\ket{\Omega,\sim}=\ket{\phi(\bm{x})=\phi_{0}}_{\phi},$ (11)

where $\phi_{0}$ is the minimum of the classical potential.
$\ket{\Omega,\sim}$ is a proxy state to the true vacuum state, $\ket{\Omega}$,
since it has zero spatial-momentum and angular-momentum quantum numbers.

From (6), (7), it follows that $\ket{k,\sim}$ has zero spatial and angular
momentum. The exact state $\ket{k}$ is then

$\ket{k}=\text{const.}\frac{\lim_{T\to\infty}e^{MT}e^{-HT}\ket{k,\sim}}{\braket{k}{k,\sim}},$
(12)

where $M$ is the pole-mass of the particle (i.e. renormalized and quantum
corrected) and ‘$\text{const}.$’ allows for the possibility of adopting some
different convention for the normalization of the particle state. The moving
state $\ket{p}$ is

$\ket{p}=U(\Lambda)\ket{k},$ (13)

where $\Lambda:k\to p$. Note that this definition is self-consistent: if we
choose a different boost $\Lambda^{\prime}:k\to p$, it will be related to
$\Lambda$ by a rotation and rotations leave $\ket{k}$ unchanged.

It is instructive to understand why the soliton operators take the form (3)
from a different point of view. Consider the matrix element
$\bra{\psi_{out}}e^{-2HT}\mathcal{O}(\bm{x}_{0})\ket{\psi_{in}}$. Transforming
to the path integral, this is equal to

$\int[d\phi][d\pi]\psi_{out}^{*}[\phi]\,\psi_{in}[\phi]\,e^{S^{\prime}[\phi,\pi]},$
(14)

where

$\displaystyle
S^{\prime}=S-i\int\pi(-T,\bm{x}_{0}+\bm{a})\phi_{cl}(\bm{a})d^{d}\bm{a}+i\int\phi(-T,\bm{x}_{0}+\bm{a})\pi_{cl}(\bm{a})d^{d}\bm{a}$
(15)

and $S=\int_{-T}^{T}dt\int
d^{d}\bm{x}\left(i\pi\dot{\phi}-\mathcal{H}(\pi,\phi)\right)$.

If we evaluate (14) perturbatively, we obtain an infinite sum of tree diagrams
because the ‘free’ part of $S^{\prime}$ contains terms linear in the fields.
These tree diagrams actually sum up to the classical background Cheung2020 ;
Riccardo2019 ; Riccardo2020  which is determined by the equations of motion of
$S^{\prime}$ ($\delta S^{\prime}/\delta\phi=\delta S^{\prime}/\delta\pi=0$):

$\displaystyle i\,\dot{\pi}(x)$
$\displaystyle=i\,\delta(t+T)\pi_{cl}(\bm{x}-\bm{x}_{0})-\frac{\partial\mathcal{H}}{\partial\phi(x)},$
(16) $\displaystyle i\,\dot{\phi}(x)$
$\displaystyle=i\,\delta(t+T)\phi_{cl}(\bm{x}-\bm{x}_{0})+\frac{\partial\mathcal{H}}{\partial\pi(x)}.$
(17)

Note that the usual background $\phi=\pi=0$ is not a solution. To build a
solution, consider initial data $\phi(\bm{x})=\pi(\bm{x})=0$ at a very early
time slice. Then at $t=-T$ there is an abrupt jump to values
$\phi_{cl}(\bm{x}-\bm{x}_{0})$ and $\pi_{cl}(\bm{x}-\bm{x}_{0})$ respectively
and then there is standard evolution (i.e. governed by $S$). In other words,
let $\phi_{cl}(t,\bm{x})$ and $\pi_{cl}(t,\bm{x})$ be the solutions to the
_unsourced_ EOM (i.e. the EOM derived from $S$) such that
$\phi_{cl}(-T,\bm{x})=\phi_{cl}(\bm{x})$ and
$\pi_{cl}(-T,\bm{x})=\pi_{cl}(\bm{x})$ is an initial condition. Then the
solution to (16), (17) is

$\displaystyle\phi(x)$
$\displaystyle=\theta(t+T)\phi_{cl}(t,\bm{x}-\bm{x}_{0}),$ (18)
$\displaystyle\pi(x)$
$\displaystyle=\theta(t+T)\pi_{cl}(t,\bm{x}-\bm{x}_{0}).$ (19)

The step function indicates the creation of a soliton at $t=-T$. The operator
(3) is the $1+1$ dimensional topological soliton analogue of t’Hooft’s
topology changing operators in $2+1$ dimensional gauge theory Hooft1978 .

##  3 The classical $\phi_{2}^{4}$ model and the kink

We will deal in this section with the purely classical aspects of the
$\phi_{2}^{4}$ model, a theory of one real scalar field $\phi$ in 1+1
dimensions. In the conventions of DHN1974  the Lagrangian is

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{4\lambda^{2}}\phi^{4}+\frac{1}{2}m^{2}\phi^{2}-\frac{m^{4}\lambda^{2}}{4}.$
(20)

With $m^{2}>0$, the potential is a Mexican hat with two absolute minima of
zero at $\phi=\pm\lambda m$. We will follow GervaisPert1975  to set $m=1$ and
work instead with the Lagrangian

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{4\lambda^{2}}\phi^{4}+\frac{1}{2}\phi^{2}-\frac{\lambda^{2}}{4}.$
(21)

It is easy to restore factors of $m$ at the end using the fact that the mass
dimensions of $\phi$ and $\lambda$ are $0$ and $-1$ respectively. The equation
of motion (EOM) which follows is

$\Box\phi+\lambda^{-2}\phi^{3}-\phi=0.$ (22)

This admits a static soliton solution:

$\phi_{cl}(x)=\lambda\tanh(x/\sqrt{2}).$ (23)

The solution interpolates from one vacuum value $-\lambda$ at $x=-\infty$ to
the other $\lambda$ at $x=\infty$. The energy of a static solution is $\int
dx~{}(\phi^{\prime 2}/2+V(\phi))$ (where
$V=\phi^{4}/(4\lambda^{2})-\phi^{2}/2+\lambda^{4}/4$). However, a virial
relation Goldstone1975  ensures that $\int dx~{}V=\int dx~{}\phi^{\prime
2}/2$. Therefore the energy is $\int dx~{}\phi^{\prime 2}$. This gives us the
classical mass of the soliton:

$M_{0}=2\sqrt{2}\lambda^{2}/3.$ (24)

We also have

$\int_{\infty}^{\infty}dx\,\phi_{cl}^{\prime}(x)\phi_{cl}(x)=\phi_{cl}^{2}\big{|}_{-\infty}^{\infty}=0.$
(25)

To establish the spectrum of perturbations, we expand around $\phi_{cl}$:
$\phi=\phi_{cl}+\Psi$. Using this in the EOM (22), the perturbation satisfies

$\Box\Psi+\left(3\tanh^{2}\left(x/\sqrt{2}\right)-1\right)\Psi=0,$ (26)

to linear order. Letting $\Psi(t,x)=\exp(i\omega t)\psi(x)$, (26) becomes the
Schrödinger equation

$\psi^{\prime\prime}+\left(\omega^{2}+1-3\tanh^{2}\left(x/\sqrt{2}\right)\right)\psi=0.$
(27)

The spectrum $\mathcal{S}$ consists of two discrete eigenvalues with
$\omega=0$, which is the translational zero mode, and $\omega=\sqrt{3/2}$,
which is localised, plus a continuum labelled by $k\in\mathbb{R}$ with
$\omega_{k}=\sqrt{k^{2}+2}$. The eigenfunctions are

$\displaystyle\psi_{b}$
$\displaystyle=\frac{\phi_{cl}^{\prime}}{\sqrt{M_{0}}}\hskip
28.45274pt\omega_{b}=0,$ (28) $\displaystyle\psi_{a}$
$\displaystyle=\sqrt{\frac{3}{2\sqrt{2}}}\,\frac{\sinh(x/\sqrt{2})}{\cosh^{2}(x/\sqrt{2})}\hskip
28.45274pt\omega_{a}=\sqrt{3/2},$ (29) $\displaystyle\psi_{k}$
$\displaystyle=N_{k}^{-1}\exp(ikx)\left(3\tanh^{2}(x/\sqrt{2})-3\sqrt{2}ik\tanh(x/\sqrt{2})-1-2k^{2}\right),$
(30) $\displaystyle\omega_{k}$ $\displaystyle=\sqrt{k^{2}+2},\quad
N_{k}=\sqrt{2(k^{2}+2)(2k^{2}+1)}.$ (31)

Note that $\psi_{k}^{*}=\psi_{-k}$. These functions are orthonormal with the
measure $dx$ and satisfy the completeness relation

$\displaystyle\delta(x-x^{\prime})=\psi_{b}(x)\psi_{b}(x^{\prime})+\psi_{a}(x)\psi_{a}(x^{\prime})+\int\frac{dk}{2\pi}\psi^{*}_{k}(x^{\prime})\psi_{k}(x).$
(32)

We record for later the useful relationship

$\displaystyle\psi_{k}(x)\psi_{k}^{*}(x^{\prime})$
$\displaystyle=e^{ik(x-x^{\prime})}\left(1+N_{k}^{-2}\sum_{l=0}^{3}k^{l}\,{A}_{l}(x,x^{\prime})\right),$
(33)

where

$\displaystyle{A}_{3}(x,x^{\prime})$
$\displaystyle=6i\sqrt{2}(\tau-\tau^{\prime}),$ (34)
$\displaystyle{A}_{2}(x,x^{\prime})$ $\displaystyle=-6\tau^{2}-6\tau^{\prime
2}+18\tau\tau^{\prime}-6,$ (35) $\displaystyle{A}_{1}(x,x^{\prime})$
$\displaystyle=3i\sqrt{2}(\tau-\tau^{\prime})(3\tau\tau^{\prime}+1),$ (36)
$\displaystyle A_{0}(x,x^{\prime})$ $\displaystyle=(3\tau^{2}-1)(3\tau^{\prime
2}-1)-4,$ (37)

with $\tau\equiv\tanh(x/\sqrt{2})$ and
$\tau^{\prime}\equiv\tanh(x^{\prime}/\sqrt{2})$.

Of course (22) also has the trivial static solution

$\phi(x)=\lambda.$ (38)

The spectrum of fluctuations about this solution is labelled by
$k\in\mathbb{R}$, with orthonormal functions

$\psi_{k}^{0}(x)=e^{ikx},$ (39)

and completeness relation

$\displaystyle\delta(x-x^{\prime})=\int\frac{dk}{2\pi}\psi^{0*}_{k}(x^{\prime})\psi^{0}_{k}(x).$
(40)

##  4 Form Factors and the Radius

In this section we relate the radius of the soliton of physical mass $M$ to a
form factor, in two dimensions.

For a particle state $\ket{p^{\mu}}$, a form factor is
$\bra{p^{\prime}}\mathcal{O}\ket{p}$, for some operator $\mathcal{O}$. A form
factor, when expanded in powers of the momentum transfer, $p^{\prime}-p$,
reveals information about the internal structure of the particle in question,
and with different operators, $\mathcal{O}$, one can probe different aspects
of the internal structure. As discussed in the introduction, our goal will be
to calculate the radius of the soliton by computing the ‘width’ of the energy
density, i.e. by computing $\bra{p^{\prime}}\mathcal{H}\ket{p}$, where
$\ket{p}$ is the soliton state and $\mathcal{H}$ is the energy density.

We use the single-particle normalization conditions

$\displaystyle\braket{k}{k^{\prime}}$ $\displaystyle=4\pi
k^{0}\,\delta(k-k^{\prime}),$ (41) $\displaystyle U(\Lambda)\ket{k}$
$\displaystyle=\ket{\Lambda k},$ (42)

so that the mass dimension of $\ket{k}$ is $0$. Consider

$\Gamma^{\mu\nu}(k,k^{\prime}):=\bra{k}T^{\mu\nu}(0,0)\ket{k^{\prime}},$ (43)

where $T^{\mu\nu}(t,x)$ is the energy-momentum tensor. The mass dimension of
$\Gamma$ is $+2$. A standard set of manipulations Weinberg2005  (Sec. 10.6)
shows that

$\displaystyle\Gamma^{\mu\nu}(k,k^{\prime})$
$\displaystyle=\frac{1}{2}\left(F_{1}(l^{2})(k+k^{\prime})^{\mu}(k+k^{\prime})^{\nu}+F_{5}(l^{2})\left(\eta^{\mu\nu}-l^{-2}l^{\mu}l^{\nu}\right)\right),$
(44)

where $l^{\mu}\equiv(k-k^{\prime})^{\mu}$. The form factors $F_{1}$ and
$F_{5}$, which encode everything non-trivial about $\Gamma^{\mu\nu}$, are
undetermined, real functions of $l^{2}$ and $M$, and satisfy $F_{1}(0)=1$ and
$F_{5}(0)=0$.

The radius of the particle is related to the first derivatives of the form
factors, $F_{1}^{\prime}(0)$ and $F_{5}^{\prime}(0)$. The form factors depend
only on the momentum difference, so we can conveniently choose
$k^{\mu}=(M,0)$, $k^{\prime\mu}=(\sqrt{M^{2}+q^{2}},q)$ and expand in powers
of $q$. The momentum transfer satisfies $l^{2}=-q^{2}+{O}(q^{4})$. We define
the radius, $R$, by

$R^{2}=\frac{1}{M^{2}}\partial_{-q^{2}}\Gamma^{00}(k,k^{\prime})\big{|}_{q=0}.$
(45)

As we will show later, (82), to leading order in $\hbar$ (i.e. classically),
this definition gives $R^{2}=\int dx\,x^{2}\mathcal{H}_{cl}(x)/\int
dx\,\mathcal{H}_{cl}(x)$ which coincides with the discussion in §1. Now, let
$\Lambda$ be the Lorentz transformation that takes $k\to k^{\prime}$; the
corresponding operator $U(\Lambda)$ expanded in powers of $q$ is,444This
follows from $U=\exp(i\omega_{10}K)$ where
$\omega_{\mu\nu}=\eta_{\mu\rho}\omega^{\rho}_{\nu}$ and $\omega^{\mu}_{\nu}$
is determined from
$\Lambda^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\omega^{\mu}_{\nu}+O(\omega^{2})$ –
which gives $\omega_{10}=-q/M$.

$U(\Lambda)=1-\frac{iq}{M}K-\frac{q^{2}}{2M^{2}}K^{2}+{O}(q^{3}),$ (46)

where $K$ is the boost generator,

$K=\int dx~{}x\,T^{00}(0,x).$ (47)

Using (42), (46) in (45) shows that

$R^{2}=\frac{1}{2M^{4}}\bra{k}\mathcal{H}(0,0)K^{2}\ket{k},$ (48)

where the energy density $\mathcal{H}(t,x)=T^{00}(t,x)$, and then using (47)
gives

$R^{2}=\bra{k}O\ket{k},$ (49)

where

$O\equiv\frac{1}{2M^{4}}\int
x\,x^{\prime}\,\mathcal{H}(0,0)\mathcal{H}(0,x)\mathcal{H}(0,x^{\prime})dx\,dx^{\prime}.$
(50)

Taking this definition of the radius and applying it to the soliton case using
(10), (12) we then have,555The denominator of (51) follows from applying (10),
(12) to $\braket{k}{k}$ and eliminating the common factor of
$|\braket{k}{k,\sim}|^{2}$ between this and $R^{2}$.

$R^{2}=\frac{\lim_{T\to\infty}e^{2MT}\int\bra{\Omega,\sim}\mathcal{O}^{\dagger}(y)e^{-HT}Oe^{-HT}\mathcal{O}(z)\ket{\Omega,\sim}dz\,dy}{\braket{k}{k}^{-1}\lim_{T\to\infty}e^{2MT}\int\bra{\Omega,\sim}\mathcal{O}^{\dagger}(y)e^{-2HT}\mathcal{O}(z)\ket{\Omega,\sim}dz\,dy}.$
(51)

For the moment, we will suppose that the system is in a spatial box of size
$L$. The factor of $\braket{k}{k}$ in (51) is fixed since (41) reads
$\braket{k}{k}=2ML$ in a box of size $L$. Now, since $\pi_{cl}=0$, the soliton
operator is $\mathcal{O}(x)=\exp(-i\int da~{}\phi_{cl}(a)\pi(x+a))$. Using
$\ket{\Omega,\sim}=\ket{0}_{\phi}$ we then have
$\mathcal{O}(x)\ket{\Omega,\sim}=\ket{\phi_{cl}(\cdot-x)}_{\phi}$ (this is the
configuration of a soliton centered on $x$). For convenience we will denote it
as $\ket{x}\equiv\ket{\phi_{cl}(\cdot-x)}_{\phi}$. Therefore

$R^{2}=\frac{\lim_{T\to\infty}e^{2MT}\int\bra{y}e^{-HT}Oe^{-HT}\ket{z}dz\,dy}{\frac{1}{2ML}\lim_{T\to\infty}e^{2MT}\int\bra{y}e^{-2HT}\ket{z}dz\,dy}.$
(52)

##  5 Evaluation of the radius for the $\phi_{2}^{4}$ kink

The Hamiltonian of the $1+1$ dimensional $\phi^{4}$ model in the conventions
of (21) is

$\displaystyle H(\pi,\phi)$ $\displaystyle=\int\mathcal{H}(x)\,dx$
$\displaystyle=\int\frac{1}{2}\left(\pi^{2}+\phi^{\prime
2}-(1+\delta_{m^{2}})\phi^{2}+\frac{\lambda^{2}}{2}(1+\delta_{m^{2}})^{2}+\frac{\phi^{4}}{2\lambda^{2}}+2C(x)\right)dx,$
(53)

where $\delta_{m^{2}}$ is the mass counterterm, and $C(x)$ is the vacuum
energy density counterterm chosen so that $H\ket{\Omega}=0$ and formally given
by

$C(x)=-\bra{\Omega}\frac{1}{2}\left(\pi^{2}+\phi^{\prime
2}-(1+\delta_{m^{2}})\phi^{2}+\frac{\lambda^{2}}{2}(1+\delta_{m^{2}})^{2}+\frac{\phi^{4}}{2\lambda^{2}}\right)\ket{\Omega}.$
(54)

Note that the renormalized mass parameter is $m=1$ (as observed in §3 factors
of $m$ can be restored by dimensional analysis). In this convention, the
Feynman rules are given in GervaisPert1975 .

To evaluate $R^{2}$ from (52) requires several steps: the first is to quantize
using the collective coordinate formalism; then we expand to obtain an
expression equivalent to (52) to first sub-leading order in $\lambda^{-1}$;
and we complete the calculation using point-splitting regularization.

###  5.1 Collective Coordinate Quantization

$R^{2}$, given in (52), involves matrix elements between initial and final
states in which there is a single soliton at rest. The method of computing
such matrix elements is collective-coordinate-quantization (CCQ)
GervaisPoint1976 .

CCQ consists of a change of variables from the usual field variables, to a
soliton position variable and a perturbation field around the soliton:
$\phi(x)\to(X,\chi(x))$. The domains of the new variables are
$\chi(x)\in\mathbb{R}$ and $X\in\mathbb{R}$. $X(\phi)$ is the soliton position
and it is defined by

$\int_{-\infty}^{\infty}\phi_{cl}^{\prime}(x)\phi(x+X(\phi))\,dx=0.$ (55)

For example, if $\phi(x)=\phi_{cl}(x-x_{0})$, then $X(\phi)=x_{0}$ due to
(25). There is no proof that (55) has only one solution for each $\phi$, but
this is a global issue which will not be important when we work in
perturbation theory Callan1975  (p. 36). $\chi(x)$ is defined as

$\chi(x)\equiv\phi(x+X(\phi))-\phi_{cl}(x).$ (56)

The inverse of this relation is

$\phi(x)=\phi_{cl}(x-X)+\chi(x-X).$ (57)

As a consequence of (25), (55), $\chi$ satisfies the constraint

$Q[\chi]=0,$ (58)

where

$Q[\phi]\equiv\int_{-\infty}^{\infty}M_{0}^{-1/2}\phi_{cl}^{\prime}(x)\phi(x)\,dx.$
(59)

It can be checked that this change of variables is a bijection. It can be
shown that the Jacobian of the transformation is

$J=\frac{\sqrt{M_{0}}}{\int
dx(\phi_{cl}^{\prime}+\chi^{\prime})\phi_{cl}^{\prime}}.$ (60)

We define $P$, dual to $X$, such that $[P,X]=-i$; and $\bar{\pi}(x)$ dual to
$\chi(x)$ satisfying Tomboulis1975

$Q[\bar{\pi}]=0,$ (61)

and

$i[\bar{\pi}(x),\chi(y)]=\delta(x-y)-\frac{\phi_{cl}^{\prime}(x)\phi_{cl}^{\prime}(y)}{M_{0}}.$
(62)

There is the following relationship between the old and new variables
Tomboulis1975 :

$\displaystyle\pi(x)=\bar{\pi}(x-X)-\frac{1}{2}\left\\{\frac{\phi_{cl}^{\prime}(x-X)}{\int\phi_{cl}^{\prime}(\chi^{\prime}+\phi_{cl}^{\prime})},P+\int\bar{\pi}\chi^{\prime}\right\\},$
(63)

where $\\{,\\}$ is the anti-commutator.

By substituting $\pi(x)$ and $\phi(x)$, from (63) and (57) respectively, into
the Hamiltonian $H(\phi,\pi)$, we obtain the dual Hamiltonian,
$H(X,\chi,P,\bar{\pi})$. (Abusing notation we have denoted the dual
Hamiltonian with the same symbol.)

We can also construct path integrals in the new variables. Consider a matrix
element, $\bra{\psi_{out}}e^{-HT}{O}e^{-HT}\ket{\psi_{in}}$, where $O$ is an
operator at time zero that is function of $X,\chi(x),P,\bar{\pi}(x)$. It can
be written as a path integral:

$\displaystyle\bra{\psi_{out}}e^{-HT}{O}e^{-HT}\ket{\psi_{in}}$
$\displaystyle=\int[dX][d\chi][dP][d\bar{\pi}]\,\psi_{in}\left(X,\chi\right)\psi^{*}_{out}\left(X,\chi\right)\delta\left(Q[\chi]\right)\,\delta\left(Q[\bar{\pi}]\right)$
$\displaystyle\times O(P,\bar{\pi},X,\chi)$
$\displaystyle\exp\int_{-T}^{T}dt\left(iP\dot{X}+i\int
dx(\bar{\pi}\dot{\chi})-H(X,\chi,P,\bar{\pi})\right),$ (64)

where the integral is over configurations

$\displaystyle
X(t),\chi(t,x),P(t),\bar{\pi}(t,x):t\in[-T,T],x\in[-\infty,\infty].$ (65)

The delta-functions arise due to (58), (61), and enforce
$Q[\chi]=Q[\bar{\pi}]=0$, $\forall\,t$. The time argument of the fields in
(64) have been suppressed: in $O$, all fields have time argument $t=0$ and in
$\psi_{in/out}$ they are $t=\mp\infty$ respectively. Also in (64), we have
defined the wavefunctions
$\psi_{in/out}(X,\chi)=\braket{X,\chi}{\psi_{in/out}}$, where $\ket{X,\chi}$
are the simultaneous eigenstates of the operators $X$ and $\chi(x)$,
normalized such that $1=\int dX[d\chi]\,\ket{X,\chi}\bra{X,\chi}$ (where the
integrals are over configurations $X,\chi(x)$ at fixed time). For example, if
$\ket{\psi}=\ket{\phi_{cl}(\cdot-x_{0})}_{\phi}$, then
$\psi=\sqrt{J}\,\delta(X-x_{0})\delta(\chi)=M_{0}^{-1/2}\delta(X-x_{0})\delta(\chi)$
using (60).

###  5.2 Radius to ${\mathcal{O}}(\lambda^{-2})$

In this section we evaluate $R^{2}$ in powers of $\lambda^{-1}$, determining
the leading and subleading terms. (We should note that the expansion in powers
of $\lambda^{-1}$ is equivalent to the expansion in powers of $\hbar$ (korepin
Sec. 1.1), so what we are working out is the first quantum correction to $R$.)

We begin by expanding (63):666The order of $\lambda$ can be counted by
recalling $\phi_{cl}\propto\lambda$, $M_{0}\propto\lambda^{2}$, and
$\delta_{m^{2}}\propto\lambda^{-2}$.

$\pi(x)=\bar{\pi}(x-X)-\frac{1}{2M_{0}}\left\\{\phi_{cl}^{\prime}(x-X),P+\int\chi^{\prime}\bar{\pi}\right\\}+O(\lambda^{-2}).$
(66)

The energy density, from (53), is

$\mathcal{H}(x)=\frac{1}{2}\left(\pi(x)^{2}+\phi^{\prime}(x)^{2}-(1+\delta_{m^{2}})\phi(x)^{2}+\frac{\phi^{4}(x)}{2\lambda^{2}}+\frac{\lambda^{2}}{2}(1+\delta_{m^{2}})^{2}+2C(x)\right).$
(67)

Substituting (66), (57) into this, we get
$\mathcal{H}(x)=\tilde{\mathcal{H}}(x-X)$, where

$\displaystyle\tilde{\mathcal{H}}(x)$
$\displaystyle=a(x)+b(x)+c(x)+d(x)+{O}(\lambda^{-2}),$ (68)

where $a,b,c,d$ are ${\mathcal{O}}(\lambda^{2},\lambda,1,\lambda^{-1})$
respectively and are given by

$\displaystyle a(x)$ $\displaystyle=\frac{1}{2}\left(\phi_{cl}^{\prime
2}-\phi_{cl}^{2}+\frac{\phi_{cl}^{4}}{2\lambda^{2}}+\frac{\lambda^{2}}{2}\right),$
$\displaystyle b(x)$ $\displaystyle=(\phi_{cl}^{\prime}\chi)^{\prime},$
$\displaystyle c(x)$
$\displaystyle=\frac{1}{2}\left(\bar{\pi}^{2}+\chi^{\prime
2}-\chi^{2}+\frac{3}{\lambda^{2}}\phi_{cl}^{2}\chi^{2}-\delta_{m^{2}}(\phi_{cl}^{2}-\lambda^{2})\right)+C_{0}(x),$
$\displaystyle d(x)$
$\displaystyle=-\frac{1}{4M_{0}}\left\\{\bar{\pi},\left\\{\phi_{cl}^{\prime},P+\int\chi^{\prime}(y)\bar{\pi}(y)\,dy\right\\}\right\\}-\delta_{m^{2}}\phi_{cl}\chi+\frac{\chi^{3}\phi_{cl}}{\lambda^{2}}+C_{-1}(x).$
(69)

Here $C_{0}(x)$ and $C_{-1}(x)$ are the ${O}(\lambda^{0})$ and
${O}(\lambda^{-1})$ components respectively of $C(x)$. Integrating
$\tilde{\mathcal{H}}(x-X)$ w.r.t. $x$, gives

$\displaystyle H=M_{0}+H_{free}+H_{int}+O(\lambda^{-2})$ (70)

where GervaisPert1975

$\displaystyle H_{free}$ $\displaystyle=\int
dx~{}\frac{1}{2}\left(\bar{\pi}^{2}+\chi^{\prime
2}-\chi^{2}+\frac{3}{\lambda^{2}}\phi_{cl}^{2}\chi^{2}-\delta_{m^{2}}(\phi_{cl}^{2}-\lambda^{2})\right)+\int
dx\,C_{0}(x),$ (71) $\displaystyle H_{int}$ $\displaystyle=\int
dx\left(-\delta_{m^{2}}\phi_{cl}\chi+\frac{\chi^{3}\phi_{cl}}{\lambda^{2}}\right)+\int
dx\,C_{-1}(x),$ (72)

where we have used the constraint (61) and the fact that integration variables
can be shifted by $X$ provided that every factor under the integral commutes
with $X$.

The numerator and denominator of $R^{2}$ (52) is of the form of the matrix
element, $\bra{\psi_{out}}e^{-HT}{O}e^{-HT}\ket{\psi_{in}}$, of the previous
section, with $\ket{\psi_{in}}=\ket{z}$, $\ket{\psi_{out}}=\ket{y}$. The
wavefunctions of the ‘in’ and ‘out’ states are then (as remarked in the
previous section) $M_{0}^{-1/2}\delta(X-z)\delta(\chi)$ and
$M_{0}^{-1/2}\delta(X-y)\delta(\chi)$ respectively. Applying (64) we thus
have,777To obtain this formula we have made use of the fact that the
$\delta(\chi(t=\pm\infty))$ factors affect the amplitude by only an overall
constant factor.

$\displaystyle\int\bra{y}e^{-HT}\mathcal{H}(x_{1})\mathcal{H}(x_{2})\mathcal{H}(x_{3})e^{-HT}\ket{x}dx\,dy$
$\displaystyle=\xi\int[dX][d\chi][dP][d\bar{\pi}]$
$\displaystyle\quad\quad\times\tilde{\mathcal{H}}(x_{1}-X)\tilde{\mathcal{H}}(x_{2}-X)\tilde{\mathcal{H}}(x_{3}-X)\exp\int_{-T}^{T}dt\left(iP\dot{X}+i\int
dx(\bar{\pi}\dot{\chi})-H\right),$ (73)

where $\xi$ is an overall constant.

To $O(\lambda^{-2})$ the highest order surviving term in
$\tilde{\mathcal{H}}(x)$ is $c(x)$, and in $H$ is $H_{int}$. Then, to the
order of our calculation, $H$ or $\tilde{\mathcal{H}}$ doesn’t depend on $P$;
so the integral over $P$ gives $\delta(\dot{X})$. The integral over $X$ is
then restricted to constant paths $X(t)=x_{0}$, giving

$\displaystyle\xi\int[d\chi][d\bar{\pi}]\int
dx_{0}~{}\tilde{\mathcal{H}}(x_{1}-x_{0})\tilde{\mathcal{H}}(x_{2}-x_{0})\tilde{\mathcal{H}}(x_{3}-x_{0})~{}\exp\int_{-T}^{T}dt\left(i\int
dx(\bar{\pi}\dot{\chi})-H\right).$ (74)

Similarly we have

$\displaystyle\int\bra{y}e^{-2HT}\ket{x}dx\,dy=\xi\,L\int[d\chi][d\bar{\pi}]~{}\exp\int_{-T}^{T}dt\left(i\int
dx(\bar{\pi}\dot{\chi})-H\right),$ (75)

where the factor of $L$ arises from the $x_{0}$ integral. Using these results
in (52) gives

$\displaystyle R^{2}$
$\displaystyle=\frac{\int[d\chi][d\bar{\pi}]e^{S[\bar{\pi},\chi]}\int
dx\,dx^{\prime}\,dx^{\prime\prime}\left(x-x^{\prime\prime}\right)\left(x^{\prime}-x^{\prime\prime}\right)\,\tilde{\mathcal{H}}(x^{\prime\prime})\,\tilde{\mathcal{H}}(x)\,\tilde{\mathcal{H}}(x^{\prime})}{M^{3}\int[d\chi][d\bar{\pi}]e^{S[\bar{\pi},\chi]}},$
(76)

where $S=S_{free}+S_{int}$ and

$\displaystyle S_{free}=\int
d^{2}x\left(i\bar{\pi}\dot{\chi}-\frac{1}{2}\bar{\pi}^{2}-\frac{1}{2}\chi^{\prime
2}+\frac{1}{2}\chi^{2}-\frac{3}{2\lambda^{2}}\phi_{cl}^{2}\chi^{2}\right),$
(77) $\displaystyle S_{int}=\int
d^{2}x\left(-\frac{\phi_{cl}\chi^{3}}{\lambda^{2}}+\delta_{m^{2}}\phi_{cl}\chi\right).$
(78)

Dropping everything beyond the first sub-leading order, we obtain

$\displaystyle R^{2}$ $\displaystyle=M^{-3}(\eta+\gamma+{O}(\lambda^{3}))$
(79)

where $\eta,\gamma$ are ${O}(\lambda^{6})$ and ${O}(\lambda^{4})$ respectively
and are given by

$\displaystyle\eta$ $\displaystyle=\int a_{1}a_{2}a_{3}\,d\mu,$ (80)
$\displaystyle\gamma$ $\displaystyle=\int d\mu\big{(}a_{1}a_{2}\langle
S_{int}\,b_{3}\rangle+a_{1}\langle b_{2}b_{3}\rangle+\langle c_{1}\rangle
a_{2}a_{3}+\text{cyclic permutations of {1,2,3}}\big{)}.~{}~{}~{}~{}$ (81)

Here $a_{1,2,3}$ stand for $a(x^{\prime\prime}),a(x),a(x^{\prime})$
respectively, and similarly for $b_{1,2,3}$ and $c_{1,2,3}$. The integration
measure is $d\mu\equiv
dx\,dx^{\prime}\,dx^{\prime\prime}(x-x^{\prime\prime})(x^{\prime}-x^{\prime\prime})$,
and $\langle\cdot\rangle$ denotes expectation with respect to $S_{free}$.

In fact $a(x)$ is the classical energy density, so $\int
a(x)dx=M_{0}=2\sqrt{2}\lambda^{2}/3$, $\int x\,a(x)dx=0$ and the leading term
is given by

$\eta=M_{0}^{2}a_{(2)},$ (82)

where

$a_{(2)}\equiv\int
x^{2}a(x)=\frac{\sqrt{2}m\lambda^{2}}{9}\left(\pi^{2}-6\right).$ (83)

This leading contribution to $R^{2}$ (79) is simply the classical radius
squared $R_{cl}^{2}$ (1).

The quantum correction simplifies to

$\gamma=c_{(2)}M_{0}^{2}+2c_{(0)}M_{0}\,a_{(2)}-2M_{0}^{2}\,\int\langle
S_{int}\,\chi(x)\rangle\,\phi_{cl}^{\prime}(x)\,x\,dx$ (84)

where $c_{(n)}\equiv\int x^{n}\langle c(x)\rangle dx$ and we have used

  1. 1.

$\int b=\int(\phi_{cl}^{\prime}\chi)^{\prime}=0$ because $\phi_{cl}^{\prime}$
is exponentially damped at large $|x|$,

  2. 2.

$\int x\,b=\int
x(\phi_{cl}^{\prime}\chi)^{\prime}=[x\phi_{cl}^{\prime}\chi]^{\infty}_{-\infty}+\int\phi_{cl}^{\prime}\chi=0$
because of the constraint on $\chi$ (see (58)).

A similar, though less involved, calculation gives the quantum corrected mass
of the kink. This follows from $M=\braket{k}{k}^{-1}\bra{k}H\ket{k}$ and using
all the steps that led from (48) to (79)-(81), which gives

$\displaystyle M$ $\displaystyle=\int_{-\infty}^{\infty}dx(a(x)+\langle
c(x)\rangle)+{O}(\lambda^{-1})=M_{0}+c_{(0)}+{O}(\lambda^{-1}).$ (85)

###  5.3 Free field expectation values

As usual, expectation values of polynomials of $\bar{\pi}$, $\chi$ w.r.t.
$S_{free}$ can be constructed from the two-point functions, which are
GervaisPert1975 :

$\displaystyle\langle\chi(t,x)\chi(t^{\prime},x^{\prime})\rangle$
$\displaystyle=G(t-t^{\prime};x,x^{\prime}),$ (86)
$\displaystyle\langle\chi(t,x)\bar{\pi}(t^{\prime},x^{\prime})\rangle$
$\displaystyle=i\partial_{t^{\prime}}G(t-t^{\prime};x,x^{\prime}),$ (87)
$\displaystyle\langle\bar{\pi}(t,x)\bar{\pi}(t^{\prime},x^{\prime})\rangle$
$\displaystyle=-\partial_{t}\partial_{t^{\prime}}G(t-t^{\prime};x,x^{\prime})+\Delta(t-t^{\prime};x,x^{\prime}),$
(88)

with

$\displaystyle G(t-t^{\prime};x,x^{\prime})$
$\displaystyle=\int\frac{d\omega}{2\pi}e^{i\omega(t-t^{\prime})}\left(\frac{\psi_{a}(x)\psi_{a}^{*}(x^{\prime})}{\omega^{2}+\omega_{a}^{2}}+\int\frac{dk}{2\pi}\frac{\psi_{k}(x)\psi_{k}^{*}(x^{\prime})}{\omega^{2}+k^{2}+2}\right),$
(89) $\displaystyle\Delta(t-t^{\prime};x,x^{\prime})$
$\displaystyle=\delta(t-t^{\prime})\left(\psi_{a}(x)\psi_{a}^{*}(x^{\prime})+\int\frac{dk}{2\pi}\psi_{k}(x)\psi_{k}^{*}(x^{\prime})\right).$
(90)

To obtain $C_{0}(x)$ ($C_{-1}(x)$ makes no contribution to the calculation of
$R^{2}$ at one loop order), we use
$\ket{\Omega}=\braket{\Omega}{\lambda}_{\phi}^{-1}\lim_{T\to\infty}e^{-HT}\ket{\lambda}_{\phi}$
in (54), and expand about the vacuum state by setting
$\phi(t,x)=\lambda+\varphi(t,x)$ in the path integral. This gives

$\displaystyle C_{0}(x)$
$\displaystyle=-\frac{1}{2}\frac{\int[d\varphi][d\pi]\left(\pi(x)\pi(x^{\prime})+\left(\partial_{x}\partial_{x^{\prime}}+2\right)\varphi(x)\varphi(x^{\prime})\right)e^{S^{\prime}_{free}}}{\int[d\varphi][d\pi]e^{S^{\prime}_{free}}},$
(91)

with $S^{\prime}_{free}=\int
dxdt\left(i\pi\dot{\varphi}-\pi^{2}/2-\varphi^{\prime
2}/2-\varphi^{2}\right)$. This can be evaluated using

$\displaystyle\langle\varphi(t,x)\varphi(t^{\prime},x^{\prime})\rangle^{\prime}$
$\displaystyle=G^{0}(t-t^{\prime};x,x^{\prime}),$ (92)
$\displaystyle\langle\varphi(t,x)\pi(t^{\prime},x^{\prime})\rangle^{\prime}$
$\displaystyle=i\partial_{t^{\prime}}G^{0}(t-t^{\prime};x,x^{\prime}),$ (93)
$\displaystyle\langle\pi(t,x)\pi(t^{\prime},x^{\prime})\rangle^{\prime}$
$\displaystyle=-\partial_{t}\partial_{t^{\prime}}G^{0}(t-t^{\prime};x,x^{\prime})+\Delta^{0}(t-t^{\prime};x,x^{\prime}),$
(94)

where $\langle\cdot\rangle^{\prime}$ is expectation w.r.t.
$S_{free}^{\prime}$, and

$\displaystyle G^{0}(t-t^{\prime};x,x^{\prime})$
$\displaystyle=\int\frac{d\omega\,dk}{4\pi^{2}}\frac{e^{i\omega(t-t^{\prime})}e^{ik(x-x^{\prime})}}{\omega^{2}+k^{2}+2},$
(95) $\displaystyle\Delta^{0}(t-t^{\prime};x,x^{\prime})$
$\displaystyle=\delta(t-t^{\prime})\int\frac{dk}{2\pi}e^{ik(x-x^{\prime})}.$
(96)

Finally it is useful to define

$\displaystyle G^{s}(t-t^{\prime};x,x^{\prime})$
$\displaystyle=G(t-t^{\prime};x,x^{\prime})-G^{0}(t-t^{\prime};x,x^{\prime})$
$\displaystyle=\int\frac{d\omega}{2\pi}e^{i\omega(t-t^{\prime})}\frac{\psi_{a}(x)\psi_{a}^{*}(x^{\prime})}{\omega^{2}+\frac{3}{2}}$
$\displaystyle\quad+\int\frac{d\omega\,dk}{4\pi^{2}}\frac{e^{i\omega(t-t^{\prime})}e^{ik(x-x^{\prime})}}{\omega^{2}+k^{2}+2}\frac{\sum_{l=0}^{3}k^{l}\,{A}_{l}(x,x^{\prime})}{2(k^{2}+2)(2k^{2}+1)},$
(97) $\displaystyle\Delta^{s}(t-t^{\prime};x,x^{\prime})$
$\displaystyle=\Delta(t-t^{\prime};x,x^{\prime})-\Delta^{0}(t-t^{\prime};x,x^{\prime})$
$\displaystyle=-\delta(t-t^{\prime})\,\psi_{b}(x)\psi_{b}(x^{\prime}).$ (98)

###  5.4 Regularization

To complete the evaluation of $R^{2}$ to one loop order requires the
regularization of divergences both for the mass counterterm and for the vacuum
energy subtraction. In previous calculations of the mass shift DHN1974 , this
was done by working at finite volume with a momentum cut-off and taking the
volume and cut-off to infinity at the end of the calculation. However here we
need the density distributions, not just their spatial integrals, and it is
more convenient to use point-splitting regularisation. Divergences associated
with products of two fields at the same space-time points are regularised by
point-splitting field products $\Phi(0,x)^{2}$ to
$\Phi(0,x)\,\Phi(t,x^{\prime})$ and taking the coincidence limit, $t\to
0,\,x^{\prime}\to x$, at the end of the calculation once all the divergences
have cancelled. In this scheme the mass counter-term, which is computed in the
trivial vacuum, is given by

$\displaystyle\delta_{m^{2}}=\frac{3}{\lambda^{2}}G^{0}(0,x,x^{\prime}),$ (99)

where $G^{0}$ is the propagator for $\phi$ computed in the vacuum and is given
in (92).

To compute $\langle c(x)\rangle$ (69), point-splitting the products and
performing the functional integral gives

$\displaystyle
c_{ps}(x,x^{\prime})=\left(c^{1}_{ps}(t,x,x^{\prime})-c^{0}_{ps}(t,x,x^{\prime})\right)|_{t=0}\,,$
(100)

where, from (69) and (91),

$\displaystyle c^{1}_{ps}(t,x,x^{\prime})$
$\displaystyle=\frac{1}{2}(\partial_{t}^{2}+\partial_{x}\partial_{x^{\prime}}+\frac{3}{\lambda^{2}}\phi_{cl}^{2}{(x)}-1)\,G(t,x,x^{\prime})+\frac{1}{2}\Delta(t,x,x^{\prime})+\frac{1}{2}(\lambda^{2}-\phi_{cl}^{2}{(x)})\delta_{m^{2}},$
$\displaystyle c^{0}_{ps}(t,x,x^{\prime})$
$\displaystyle=\frac{1}{2}(\partial_{t}^{2}+\partial_{x}\partial_{x^{\prime}}+2)\,G^{0}(t,x,x^{\prime})+\frac{1}{2}\Delta^{0}(t,x,x^{\prime}).$
(102)

Note that we can set $t=0$ in (100) because all the $\delta(t)$ components
have cancelled once the subtraction on the r.h.s. has been computed. Using
(100), (LABEL:eqn:cps1) and (102) gives

$\displaystyle c_{ps}(x,x^{\prime})$
$\displaystyle=\left.\frac{1}{2}(\partial_{t}^{2}+\partial_{x}\partial_{x^{\prime}}+\frac{3}{\lambda^{2}}\phi_{cl}^{2}{(x)}-1)G^{s}(t,x,x^{\prime})+\frac{1}{2}\Delta^{s}(t,x,x^{\prime})\right|_{t=0}$
$\displaystyle\quad+\frac{3}{2}\left(\frac{\phi_{cl}^{2}{(x)}}{\lambda^{2}}-1\right)\left(G^{0}(0,x,x^{\prime})-\frac{\lambda^{2}}{3}\delta_{m^{2}}\right).$
(103)

The last term in (103) vanishes by the definition of the mass counter-term
(99). The point spectrum contribution to $c_{ps}$ is

$\displaystyle c_{ps}^{P}(x,x^{\prime})=$
$\displaystyle\left.\frac{1}{2}\int\frac{d\omega}{2\pi}e^{i\omega(t-t^{\prime})}\,\left(\frac{-\omega^{2}+\partial_{x}\partial_{x^{\prime}}+3\lambda^{-2}\phi_{cl}^{2}{(x)}-1}{\omega^{2}+\frac{3}{2}}+1\right)\psi_{a}(x)\psi_{a}(x^{\prime})\right|_{t=0}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sqrt{\frac{2}{3}}\left(\partial_{x}\partial_{x^{\prime}}+3\lambda^{-2}\phi_{cl}^{2}{(x)}+\frac{1}{2}\right)\psi_{a}(x)\psi_{a}(x^{\prime})$
(104)

which is finite when $x\to x^{\prime}$. The continuous spectrum contribution
is

$\displaystyle c_{ps}^{C}(x,x^{\prime})=$
$\displaystyle\frac{1}{2}\int\frac{d\omega
dk}{(2\pi)^{2}}e^{i\omega(t-t^{\prime})}$
$\displaystyle\qquad\left.\left(\frac{-\omega^{2}+\partial_{x}\partial_{x^{\prime}}+3\lambda^{-2}\phi_{cl}^{2}{(x)}-1}{\omega^{2}+k^{2}+2}+1\right)\left(\psi_{k}(x)\psi^{*}_{k}(x^{\prime})-e^{ik(x-x^{\prime})}\right)\right|_{t=0}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int\frac{dk}{2\pi\sqrt{k^{2}+2}}\left(\partial_{x}\partial_{x^{\prime}}+k^{2}+3\lambda^{-2}\phi_{cl}^{2}{(x)}+1\right)\left(\psi_{k}(x)\psi^{*}_{k}(x^{\prime})-e^{ik(x-x^{\prime})}\right).$
(105)

Using the explicit form (30) for $\psi_{k}(x)$ gives

$c_{ps}^{C}(x,x^{\prime})=\sum_{l=0}^{5}B_{l}(x,x^{\prime})\mathcal{K}_{l}(x,x^{\prime}),$
(106)

where

$\mathcal{K}_{l}(x,x^{\prime})=\int\frac{k^{l}\,dk}{16\pi}\frac{e^{ik(x-x^{\prime})}}{(k^{2}+2)^{3/2}(2k^{2}+1)},$
(107)

and the functions $B_{\ell}(x,x^{\prime})$ are regular at $x=x^{\prime}$. As
the ${\mathcal{K}}_{\ell}(x,x^{\prime})$ are finite at $x=x^{\prime}$ for
$\ell<4$, it is sufficient to evaluate them there,

$\displaystyle\mathcal{K}_{0}(x,x)$
$\displaystyle=\frac{4\sqrt{3}\pi-9}{432\pi},$
$\displaystyle\mathcal{K}_{1}(x,x)$ $\displaystyle=\mathcal{K}_{3}(x,x)=0,$
$\displaystyle\mathcal{K}_{2}(x,x)$
$\displaystyle=\frac{9-\sqrt{3}\pi}{216\pi}.$ (108)

For $l=4,5$, we have,888To show this, we expand $k^{4}(2k^{2}+1)^{-1}$, in
(107), in powers of $(k^{2}+2)^{-1}$ which gives
$\mathcal{K}_{4}(x,x^{\prime})=\frac{1}{32\pi}\int
dk\,e^{ik(x-x^{\prime})}\left\\{\frac{1}{(k^{2}+2)^{3/2}}-{O}\left(\frac{1}{(k^{2}+2)^{5/2}}\right)\right\\}.$
(109) The sub-leading terms give a finite contribution when $x^{\prime}\to x$,
so we will not be interested in these; the leading term, on the other hand, is
divergent when $x^{\prime}\to x$ and is given by
$\frac{1}{16\pi}K_{0}((x-x^{\prime})/\sqrt{2})$ where $K_{0}$ is a modified
Bessel function of the second kind. ${\mathcal{K}}_{5}$ is then obtained by
acting with $-i\partial_{x}$ on ${\mathcal{K}}_{4}$.

$\displaystyle\mathcal{K}_{4}(x,x^{\prime})$
$\displaystyle=-\frac{\log|x-x^{\prime}|}{16\pi}\left(1+{O}(x-x^{\prime})\right),$
(110) $\displaystyle\mathcal{K}_{5}(x,x^{\prime})$
$\displaystyle=\frac{i}{16\pi(x-x^{\prime})}+{O}(1).$ (111)

Although ${\mathcal{K}}_{4}$ is logarithmically divergent,
$B_{4}(x,x^{\prime})={O}(x-x^{\prime})$, so there is no $\ell=4$ contribution
to $\langle{c(x)}\rangle$. However there is a contribution for $\ell=5$
because

$\displaystyle B_{5}(x,x^{\prime})$
$\displaystyle=12i\,\text{sech}^{2}\left(\frac{x}{\sqrt{2}}\right)(x-x^{\prime})+{O}\left((x-x^{\prime})^{2}\right),$
(112)

and the linear divergence in ${\mathcal{K}}_{5}$ exposes the leading term.
Finally, using

$\displaystyle B_{0}(x,x^{\prime})$
$\displaystyle=-3\text{sech}^{2}\left(\frac{x}{\sqrt{2}}\right)\left(1+15\tanh^{4}\left(\frac{x}{\sqrt{2}}\right)\right)+{O}(x-x^{\prime}),$
$\displaystyle B_{2}(x,x^{\prime})$
$\displaystyle=3\text{sech}^{2}\left(\frac{x}{\sqrt{2}}\right)\left(1-9\tanh^{2}\left(\frac{x}{\sqrt{2}}\right)\right)+{O}(x-x^{\prime}),$
(113)

we find

$\displaystyle\langle c(x)\rangle$ $\displaystyle=\lim_{x^{\prime}\to
x}\left(c_{ps}^{P}(x,x^{\prime})+c_{ps}^{C}(x,x^{\prime})\right),$
$\displaystyle=c_{ps}^{P}(x,x)+\sum_{l=0,2}B_{l}(x,x)\mathcal{K}_{l}(x,x)+\lim_{x^{\prime}\to
x}B_{5}(x,x^{\prime})\mathcal{K}_{5}(x,x^{\prime})$
$\displaystyle=\frac{\text{sech}^{6}(\frac{x}{\sqrt{2}})}{96\pi}\left((-72-9\sqrt{3}\pi)\cosh(\sqrt{2}x)+(2\sqrt{3}\pi-9)\cosh(2\sqrt{2}x)+27+9\sqrt{3}\pi\right).$
(114)

The moments are

$c_{(0)}=\frac{1}{2\sqrt{6}}-\frac{3}{\pi\sqrt{2}},\quad\quad\quad
c_{(2)}=\frac{\pi\left(14\sqrt{3}+\pi\left(\sqrt{3}\pi-18\right)\right)-18}{36\sqrt{2}\pi}.$
(115)

After mass renormalization, the final contribution to (84) contains only
convergent $k$ integrals so point splitting is not necessary. We find

$\displaystyle-2M_{0}^{2}\,\int\langle
S_{int}\,\chi(x)\rangle\,\phi_{cl}^{\prime}(x)\,x\,dx$
$\displaystyle=6M_{0}^{2}\lambda^{-2}\int
x\,dt\,dy\,dx\,\phi_{cl}^{\prime}(x)\phi_{cl}(y)G^{s}(0;y,y)G(t;y,x)$
$\displaystyle=\frac{M_{0}^{2}(-9-8\sqrt{3}\pi+\sqrt{3}\pi^{3})}{18\sqrt{2}\pi}.$
(116)

##  6 Discussion

Combining the results obtained above, restoring factors of $m$, and
introducing the dimensionless quantity $\bar{\lambda}\equiv(m\lambda)^{-1}$,
the kink mass (85) is given by

$\displaystyle
Mm^{-1}=\frac{2\sqrt{2}}{3}\bar{\lambda}^{-2}\left(1+\bar{\lambda}^{2}\underbrace{\left(\frac{\sqrt{3}}{8}-\frac{9}{4\pi}\right)}_{\approx-0.50}+{O}(\bar{\lambda}^{4})\right),$
(117)

in agreement with the standard result DHN1974 ; Wiedig ; Raj . The radius $R$
(79) is given by

$mR=\sqrt{\frac{1}{6}\left(\pi^{2}-6\right)}\left(1+\bar{\lambda}^{2}\underbrace{\frac{\sqrt{3}\pi\left(2+\pi^{2}\right)-72}{8\pi\left(\pi^{2}-6\right)}}_{\approx-0.076}+{O}(\bar{\lambda}^{4})\right),$
(118)

or by

$MR=\frac{2\bar{\lambda}^{-2}}{3}\sqrt{\frac{1}{3}\left(\pi^{2}-6\right)}+\underbrace{\frac{\left(\pi^{2}-2\right)(\pi-3\sqrt{3})}{6\pi\sqrt{\left(\pi^{2}-6\right)}}}_{\approx-0.44}+{O}(\bar{\lambda}^{2}).$
(119)

In the perturbative, $0<\bar{\lambda}\ll 1$, regime, $R$ is very much greater
than the Compton wavelength of the soliton so it makes sense to regard the
soliton as an extended object with the characteristics of a bound state. In
this regime the topologically trivial excitations of the scalar field are
scalar elementary (ie point-like) particles with a mass $m\ll M$; despite its
excess mass the soliton in its ground state is stable against particle
emission because it is topologically protected. As $\bar{\lambda}$ increases,
keeping $m$ fixed the scalar potential becomes more attractive so the binding
energy increases which we expect to see manifested in reduced $M$, and size,
$R$, of the soliton, and this is what the perturbation theory result shows.

According to the Bound State Conjecture Hebecker2019 , if we can regard the
soliton as a bound state, we expect $mR\geq\delta>0$ for all couplings. At
least at first order in the perturbation expansion the relative size of the
quantum correction for $M$ (117) is seven times larger than that for $R$
(118). Taking this literally, the solitons will become very light while $mR$
is still $O(1)$. At this point we might expect the creation of kink / anti-
kink pairs to become important and the state cease to be a single identifiable
kink; as these pairs can separate without energy cost, the decrease in $R$
will be turned off in a manner that is consistent with the conjecture.
Clearly, we would not be able to make this argument if the relative size of
the quantum correction for $R$ were to be larger than for $M$.

The expansion in $\bar{\lambda}$ gives us a clue about what might happen as
the coupling strength increases. However to show that the mechanism described
above actually prevents the soliton from violating the conjecture of
Hebecker2019  we need to go beyond perturbation theory to include a full
quantum description of kinks and anti-kinks; in which multi-soliton matrix
elements and pair creation of solitons are calculationally accessible. The
question is probably best pursued in an integrable theory such as the sine-
Gordon model.

###### Acknowledgements.

PDX acknowledges the support of an Oxford Berman Scholarship. This research
was funded in whole, or in part, by Research England. For the purpose of Open
Access, the authors have applied a CC BY public copyright licence to any
Author Accepted Manuscript version arising from this submission.

## References

  * (1) N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060, [hep-th/0601001].
  * (2) E. Palti, The Swampland: Introduction and Review, Fortsch. Phys. 67 (2019), no. 6 1900037, [arXiv:1903.06239].
  * (3) M. van Beest, J. Calderón-Infante, D. Mirfendereski, and I. Valenzuela, Lectures on the swampland program in string compactifications, 2021.
  * (4) E. Palti, The Weak Gravity Conjecture and Scalar Fields, JHEP 08 (2017) 034, [arXiv:1705.04328].
  * (5) B. Freivogel, T. Gasenzer, A. Hebecker, and S. Leonhardt, A conjecture on the minimal size of bound states, SciPost Phys. 8, 058 (2020) [arXiv:1912.09485].
  * (6) R. Hagedorn, Statistical thermodynamics of strong interactions at high energies, Nuovo Cimento, Suppl. 3 (1965), no. CERN-TH-520 147–186.
  * (7) J. L. Gervais and B. Sakita, Extended particles in quantum field theories, Phys. Rev. D 11 (1975) 2943.
  * (8) I. Zahed and G. E. Brown, The Skyrme Model, Phys. Rept. 142 (1986) 1–102.
  * (9) Y.-L. Ma and M. Harada, Lecture notes on the skyrme model, 2016.
  * (10) Z. Komargodski, Baryons as quantum hall droplets, 2018.
  * (11) S. R. Coleman, Aspects of Symmetry: Selected Erice Lectures.  Cambridge University Press, 1985.
  * (12) R. Friedberg, T. D. Lee, and Y. Pang, Mini-soliton stars, Phys. Rev. D 35 (1987) 3640.
  * (13) J. L. Gervais and A. Jevicki, Point canonical transformations in path integral, Nucl. Phys. B 110 (1976) 93–112.
  * (14) R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory.  North-Holland Publishing Company, 1982.
  * (15) K. E. Cahill, Extended particles and solitons, Phys. Lett. B 53 (1974) 174–176.
  * (16) R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that.  Princeton University Press, 2016.
  * (17) S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations.  Cambridge University Press, 2005.
  * (18) P. Ramond, Field theory: a modern primer.  Addison-Wesley Publishing, 1990.
  * (19) C. Cheung and J. Mangan, Scattering amplitudes and the navier-stokes equation, arXiv preprint arXiv:2010.15970 (2020).
  * (20) G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, The epsilon expansion meets semiclassics, Journal of High Energy Physics 110 (2019).
  * (21) G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, Feynman diagrams and the large charge expansion in 3-$\varepsilon$ dimensions, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics 802 (2020).
  * (22) G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978) 1–25.
  * (23) R. F. Dashen, B. Hasslacher, and A. Neveu, Nonperturbative methods and extended hadron models in field theory. ii. two-dimensional models and extended hadrons, Phys. Rev. D 10 (1974) 4130–4138.
  * (24) J. L. Gervais, A. Jevicki, and B. Sakita, Perturbation expansion around extended particle states in quantum field theory. 1., Phys. Rev. D 12 (1975) 1038.
  * (25) J. Goldstone and R. Jackiw, Quantization of nonlinear waves, Phys. Rev. D 11 (1975).
  * (26) C. G. Callan Jr. and D. J. Gross, Quantum perturbation theory of solitons, Nucl. Phys. B 93 (1975) 29–55.
  * (27) E. Tomboulis, Canonical quantization of nonlinear waves, Phys. Rev. D 12 (1975) 1678–1683.
  * (28) L. D. Faddeev and V. E. Korepin, Quantum theory of solitons, Physics Reports 42 (1978).
  * (29) T. Weidig, Quantum mass correction of solitons in (1+ 1) d via numerical methods, arXiv preprint hep-th/9912005 (1999).