c_{0}|\nabla_{i,j}\Phi_{i,j}|_{g_{i,j}}+C|A|_{g}|\Phi_{i,j}|_{g}\leq c_{1}.$

Similarly, for $\nabla^{2}{\Phi_{i,j}}$, we have

$\displaystyle\nabla^{2}\Phi_{i,j}=\nabla^{2}_{i,j}\Phi_{i,j}+(\nabla-\nabla_{i,j})\nabla_{i,j}\Phi_{i,j}+(\nabla(\nabla-\nabla_{i,j}))\Phi_{i,j},$

and so, since $A$, $\nabla A$ are uniformly bounded, there is a constant
$c_{2}$ such that

$\displaystyle|\nabla^{2}\Phi_{i,j}|_{g}\leq
C|\nabla^{2}_{i,j}\Phi_{i,j}|_{g_{i,j}}+C|A|_{g}|\nabla_{i,j}\Phi_{i,j}|_{g}+C|\nabla
A|_{g}|\Phi|_{g}\leq c_{2}.$

For $k\geq 2$, we have the estimate

$\displaystyle|\nabla^{k}\Phi_{i,j}|_{g}\leq
C\sum_{l=0}^{k}|A|^{l}_{g}|\nabla^{k-l}_{i,j}\Phi_{i,j}|_{g_{i,j}}+C\sum_{l=1}^{k-1}|\nabla^{l}A|_{g}|\nabla^{k-1-l}\Phi_{i,j}|_{g}.$

By the assumption (4.15), covariant derivatives of the torsion $T_{i}$ of all
orders are uniformly bounded, so using (3.29) and the estimate
$|\nabla^{k}A|_{g}\leq c_{k}$, by an induction argument, we can show the
existence of constants $c_{k}$ for $k\geq 0$ such that
$|\nabla^{k}\Phi_{i,j}|_{g}\leq c_{k}$ on $\Omega_{i}$ for all $j,k\geq 0$.

Since we have uniform bounds on covariant derivatives of all orders of
$\Phi_{i,j}$, the Arzelà-Ascoli Theorem (see eg. [AH11, Corollary 9.14])
implies that there exist a $4$-form $\Phi_{i,\infty}$ and a subsequence of
$\Phi_{i,j}$ in $j$, which we still denote by $\Phi_{i,j}$, that converges to
$\Phi_{i,\infty}$ smoothly on $\Omega_{i}$. In other words,

$\displaystyle|\nabla^{k}(\Phi_{i,j}-\Phi_{i,\infty})|_{g}\rightarrow
0,\quad\textup{as}\ j\rightarrow\infty,$ (4.17)

uniformly on $\Omega_{i}$, for all $k\geq 0$.

We prove that $\Phi_{i,\infty}$ is a $\mathrm{Spin}(7)$-structure on
$\Omega_{i}$. Since each $\Phi_{i,j}$ is a $\mathrm{Spin}(7)$-structure on
$\Omega_{i}$ with associated metric $g_{i,j}$, we have the following relation.
Let $u,v,w$ be any vector fields on $\Omega_{i}\subset M$, then

$\displaystyle(u\lrcorner v\lrcorner\Phi_{i,j})\wedge(u\lrcorner
w\lrcorner\Phi_{i,j})\wedge\Phi_{i,j}=6(g_{i,j}(u,u)g_{i,j}(v,w)-g_{i,j}(u,v)g_{i,j}(u,w))\operatorname{vol_{g}}_{i,j}.$

Letting $j\rightarrow\infty$ in the previous equation gives

$\displaystyle(u\lrcorner v\lrcorner\Phi_{i,\infty})\wedge(u\lrcorner
w\lrcorner\Phi_{i,\infty})\wedge\Phi_{i,\infty}=6(g_{i,\infty}(u,u)g_{i,\infty}(v,w)-g_{i,\infty}(u,v)g_{i,\infty}(u,w))\operatorname{vol_{g}}_{i,\infty}.$
(4.18)

Since, by the Cheeger-Gromov compactness Theorem, the limit $g_{i,\infty}$ is
a Riemannian metric on $\Omega_{i}$ and $\operatorname{vol_{g}}_{i,\infty}$ is
its volume form, (4.18) implies that $\Phi_{i,\infty}$ is an admissible or a
non-degenerate $4$-form and hence it is a $\mathrm{Spin}(7)$-structure on
$\Omega_{i}$ with the associated metric $g_{i,\infty}=g$.

We now go from $\Omega_{i}$ to $M$. Since $\\{\Omega_{i}\\}$ is nested, we
denote the inclusion map of $\Omega_{i}$ into $\Omega_{n}$ by

$\displaystyle I_{in}:\Omega_{i}\rightarrow\Omega_{n},\quad\text{for}\quad
n\geq i.$

As before, for each $\Omega_{n}$, we have $g_{n,j}$, $\Phi_{n,j}$ which, after
passing to a subsequence, converge to $g_{n,\infty}$, $\Phi_{n,\infty}$
respectively as $j\rightarrow\infty$. By definition,

$\displaystyle I^{*}_{in}g_{n,j}=g_{i,j}\quad\text{and}\quad
I^{*}_{in}\Phi_{n,j}=\Phi_{i,j}.$

Note that the pull-back $I^{*}_{in}$ is independent of $j$, hence by taking
$j\rightarrow\infty$ in the above expression, we get

$\displaystyle I^{*}_{in}g_{n,\infty}=g_{i,\infty}\quad\text{and}\quad
I^{*}_{in}\Phi_{n,\infty}=\Phi_{i,\infty}.$ (4.19)

Thus, from (4.19), there exists a $4$-form $\Phi$ on $M$, which is a
$\mathrm{Spin}(7)$-structure with associated metric $g$ such that

$\displaystyle I^{*}_{i}g=g_{i,\infty}\quad\text{and}\quad
I^{*}_{i}\Phi=\Phi_{i,\infty},$ (4.20)

where $I_{i}:\Omega_{i}\rightarrow M$ is the inclusion map.

Finally, we show that $(M_{i},\Phi_{i},p_{i})$ converges to $(M,\Phi,p)$. Let
$K$ be any compact subset of $M$. Since $\\{\Omega_{i}\\}$ exhausts $M$, there
exists $i_{0}$ such that $K$ is contained in $\Omega_{i}$ for all $i\geq
i_{0}$. Fix an $i$ such that $K\subset\Omega_{i}$. Then by (4.17), on $K$ we
have

$\displaystyle|\nabla^{k}(F^{*}_{l}\Phi_{l}-\Phi)|_{g}$
$\displaystyle=|\nabla^{k}(F^{*}_{i+j}\Phi_{i+j}-\Phi)|_{g},\ \ \ \
\textup{where}\ l=i+j,$
$\displaystyle=|\nabla^{k}(\Phi_{i,j}-\Phi_{i,\infty})|_{g}\rightarrow 0\ \ \
\ \textup{as}\ l\rightarrow\infty$

for all $k\geq 0$. This completes the proof. ∎

We note that, if all the metrics in the sequence $(M_{i},\Phi_{i},g_{i})$ are
the same, then the limiting $\mathrm{Spin}(7)$-structure $\Phi$ induces the
same metric.

####  4.4.2 Compactness for the harmonic $\mathrm{Spin}(7)$-flow

We now state and prove the compactness theorem for solutions of the harmonic
$\mathrm{Spin}(7)$-flow (HF).

######  Theorem 4.19.

Let $M_{i}$ be a sequence of compact $8$-manifolds and let $p_{i}\in M_{i}$
for each $i$. Let $\\{\Phi_{i}(t)\\}$ be a sequence of solutions to the
harmonic $\mathrm{Spin}(7)$-flow (HF) on $M_{i}$ for $t\in(a,b)$, with
$-\infty\leq a<0<b\leq\infty$. Suppose that

$\displaystyle\underset{i}{\textup{sup}}\ \underset{x\in
M_{i},t\in(a,b)}{\textup{sup}}|T_{i}(x,t)|_{g_{i}}<\infty,$ (4.21)

where $T_{i}$ denotes the torsion of $\Phi_{i}(t)$, and the injectivity radius
satisfies

$\displaystyle\underset{i}{\textup{inf}}\ inj(M_{i},g_{i}(0),p_{i})>0.$ (4.22)

Suppose further that there are uniform constants $C_{k}$, for all $k\geq 0$,
such that

$\displaystyle\underset{i}{\textup{sup}}\
|\nabla^{k}\mathrm{R}_{i}|_{g_{i}}\leq C_{k}.$ (4.23)

Then there exist an $8$-manifold $M$, a point $p\in M$ and a solution
$\Phi(t)$ of the flow (HF) on $M$ for $t\in(a,b)$ such that, after passing to
a subsequence,

$\displaystyle(M_{i},\Phi_{i}(t),p_{i})\longrightarrow(M,\Phi(t),p),\quad\textup{as
}i\rightarrow\infty.$

###### Proof.

The proof is similar in spirit to that of the compactness theorem for the
isometric flow of $\mathrm{G}_{2}$-structures in [DGK21, Theorem 3.13]. The
idea is to show that the bounds on the $\mathrm{Spin}(7)$-structure and on its
covariant derivatives and time derivatives at time $t=0$ extend to subsequent
times, in the presence of additional bounds on the torsion and its derivatives
for all time. We give the details below.

From the derivative estimates Corollary 4.10 and (4.21), we have

$|\nabla^{m}_{g_{i}(t)}T_{i}(x,t)|\leq C_{m}.$ (4.24)

Since each $M_{i}$ is compact, we know $|\mathrm{R}_{i}|_{g_{i}}$ is bounded,
and assumption (4.22) allows us to use Theorem 4.18 for $t=0$ to extract a
subsequence of $(M_{i},\Phi_{i}(0),p_{i})$ which converges to a complete limit
$(M,\Phi_{\infty}(0),p)$. That is, there exist compact subsets
$\Omega_{i}\subset M$ exhausting $M$ with $p\in\text{int}(\Omega_{i})$, for
each $i$, and diffeomorphisms $F_{i}:\Omega_{i}\rightarrow
F_{i}(\Omega_{i})\subset M_{i}$ with $F_{i}(p)=p_{i}$, such that
$F_{i}^{*}g_{i}(0)\rightarrow g_{\infty}(0)$ and
$F_{i}^{*}\Phi_{i}(0)\rightarrow\Phi_{\infty}(0)$ smoothly on any compact
subset $\Omega\subset M$ as $i\rightarrow\infty$.

Fix a compact subset $\Omega\times[c,d]\subset M\times(a,b)$ and let $i$ be
sufficiently large so that $\Omega\subset\Omega_{i}$. Let
$\bar{g}_{i}(t)=F_{i}^{*}g_{i}(t)$. Now since $\Phi_{i}(t)$ are all solutions
to the harmonic $\mathrm{Spin}(7)$-flow, we have $g_{i}(t)=g_{i}(0)$ for each
$i$. Thus we trivially have

$\sup_{\Omega\times[c,d]}|\nabla^{m}_{\bar{g}_{i}(0)}\bar{g}_{i}(t)|_{\bar{g}_{i}(0)}=0.$

Since the limit metric $g_{\infty}(0)$ is uniformly equivalent to $g_{i}(0)$,
we get

$\sup_{\Omega\times[c,d]}|\nabla^{m}_{\bar{g}_{i}(\infty)}\bar{g}_{i}(t)|_{\bar{g}_{i}(\infty)}\leq
C_{m}$

for some positive constants $C_{m}$ and similarly

$\sup_{\Omega\times[c,d]}\Big{|}\frac{\partial^{l}}{\partial
t^{l}}\nabla^{m}_{\bar{g}_{\infty}(0)}\bar{g}_{i}(t)\Big{|}_{\bar{g}_{\infty}(0)}\leq
C_{m,l}$

for some positive constants $C_{m,l}$.

Now let $\bar{\Phi}_{i}(t)=F_{i}^{*}\Phi_{i}(t)$. Then $\bar{\Phi}_{i}(t)$ is
a sequence of solutions of the harmonic $\mathrm{Spin}(7)$-flow on
$\Omega\subset M$ for $t\in[c,d]$. Using (4.24) and proceeding in a similar
way as in [DGK21, Claim 3.9] for $\mathrm{Spin}(7)$-structures, we deduce that
there exist constants $C_{m}$, independent of $i$, such that

$\sup_{\Omega\times[c,d]}|\nabla^{m}_{\bar{g}_{i}(0)}\bar{\Phi}_{i}(t)|_{\bar{g}_{i}(0)}\leq
C_{m}$ (4.25)

and since $\bar{g}_{i}(0)$ and $\bar{\Phi}(0)$ converge uniformly to
$g_{\infty}(0)$ and $\bar{\Phi}_{\infty}(0)$ with all their covariant
derivatives on $\Omega$, we have

$\sup_{\Omega\times[c,d]}|\nabla^{m}_{\bar{g}_{\infty}(0)}\bar{\Phi}_{i}(t)|_{\bar{g}_{\infty}(0)}\leq
C_{m}.$ (4.26)

Moreover, because the time derivatives can be written in terms of spatial
derivatives using the evolution equations of the harmonic
$\mathrm{Spin}(7)$-flow, we get, for some uniform constants $C_{m,l}$,

$\sup_{\Omega\times[c,d]}\Big{|}\frac{\partial^{l}}{\partial
t^{l}}\nabla^{m}_{\bar{g}_{\infty}(0)}\bar{\Phi}_{i}(t)\Big{|}_{\bar{g}_{\infty}(0)}\leq
C_{m,l}.$ (4.27)

It now follows from the Arzelà–Ascoli theorem that there exists a subsequence
of $\bar{\Phi}_{i}(t)$ that converges smoothly on $\Omega\times[c,d]$. A
diagonal subsequence argument then produces a subsequence that converges
smoothly on any compact subset of $M\times(a,b)$ to a solution
$\bar{\Phi}_{\infty}(t)$ of the isometric flow. ∎

## 5\. Monotonicity, entropy and $\varepsilon$-regularity of the
$\mathrm{Spin}(7)$-flow

In this section, just like in the $\mathrm{G}_{2}$ case, we first introduce a
quantity $\Theta$ that is _almost monotonic_ along (HF). We derive the
evolution of $\Theta$ in Lemma 5.1 below, and use it to prove the almost
monotonicity formula. We then introduce an _entropy functional_ , and use an
$\varepsilon$-regularity theorem to prove that small initial entropy implies
long-time existence and convergence to a critical point. The proofs of several
results in this section are similar to the corresponding theorems for
$\mathrm{G}_{2}$-structures in [DGK21], so in some cases we will only sketch
the proofs and refer the reader to that source for further details.

###  5.1 Almost monotonicity formula

Let $(M,g)$ be a complete Riemannian manifold. For $x_{0}\in M$, let $u$ be
the fundamental solution of the backward heat equation, starting with the
delta function at $x_{0}$ [Ham93]:

$\displaystyle\Big{(}\frac{\partial}{\partial
t}+\Delta\Big{)}u=0,\quad\lim_{t\to t_{0}}u=\delta_{x_{0}}$

and set $u=\frac{e^{-f}}{\big{(}4\pi(t_{0}-t)\big{)}^{4}}.$ For a solution
$\\{\Phi(t)\\}_{t\in[0,t_{0})}$ of the harmonic $\mathrm{Spin}(7)$-flow on
$(M,g)$, we define the function

$\displaystyle\Theta_{(x_{0},t_{0})}(\Phi(t))=(t_{0}-t)\int_{M}|T_{\Phi(t)}|^{2}u\,\operatorname{vol_{g}}.$
(5.1)

The following lemma is a direct adaptation of [DGK21, Lemma 5.2].

######  Lemma 5.1.

$\displaystyle\frac{d}{dt}\Theta_{(x_{0},t_{0})}(\Phi(t))=$
$\displaystyle-2(t_{0}-t)\int_{M}|\operatorname{div}T-\nabla f\lrcorner
T|^{2}u\operatorname{vol_{g}}$
$\displaystyle-2(t_{0}-t)\int_{M}\big{(}\nabla_{m}\nabla_{l}u-\frac{\nabla_{m}u\nabla_{l}u}{u}+\frac{ug_{ml}}{2(t_{0}-t)}\big{)}T_{m;is}T_{l;is}\operatorname{vol_{g}}$
$\displaystyle-(t_{0}-t)\int_{M}uR_{mlis}(2T_{l;ir}T_{m;rs}-2T_{m;ir}T_{l;rs}+\tfrac{1}{4}R_{mlis}-\tfrac{1}{8}R_{mlab}\Phi_{abis})$
$\displaystyle-2(t_{0}-t)\int_{M}T_{m;is}u\nabla_{l}R_{mlis}\operatorname{vol_{g}}.$
(5.2)

###### Proof.

By direct computation, and using (4.12), we have

$\displaystyle\frac{d}{dt}\Theta=$ $\displaystyle\
\int_{M}(t_{0}-t)u\frac{\partial}{\partial
t}|T|^{2}-|T|^{2}u+(t_{0}-t)|T|^{2}\frac{\partial}{\partial
t}u\,\operatorname{vol_{g}}$ $\displaystyle=$ $\displaystyle\
\int_{M}2(t_{0}-t)uT_{m;is}\Big{(}\nabla_{r}T_{r;ip}T_{m;ps}-\nabla_{r}T_{r;sp}T_{m;pi}+\pi_{7}(\nabla_{m}(\nabla_{r}T_{r;is})))\Big{)}$
$\displaystyle-|T|^{2}u-(t_{0}-t)|T|^{2}\Delta u\,\operatorname{vol_{g}},$

but $T_{m;is}\nabla_{r}T_{r;ip}T_{m;ps}=0$,
$T_{m;is}\nabla_{r}T_{r;sp}T_{m;pi}=0$ and $T_{m;is}\in\Omega^{2}_{7}$,
therefore

$\frac{d}{dt}\Theta=\int_{M}2(t_{0}-t)uT_{m;is}(\nabla_{m}(\nabla_{r}T_{r;is}))-|T|^{2}u-(t_{0}-t)|T|^{2}\Delta
u\,\operatorname{vol_{g}}.$

Integrating by parts and using the $\mathrm{Spin}(7)$-Bianchi identity (3.30),
we have

$\displaystyle\frac{d}{dt}\Theta=$ $\displaystyle\
\int_{M}-2(t_{0}-t)\Big{(}|\operatorname{div}T|^{2}u+\nabla_{m}uT_{m;is}\nabla_{r}T_{r;is}\Big{)}-|T|^{2}u+2(t_{0}-t)\Big{(}T_{m;is}\nabla_{l}u\nabla_{m}T_{l;is}$
$\displaystyle+2T_{m;is}\nabla_{l}uT_{l;ir}T_{m;rs}-2T_{m;is}\nabla_{l}uT_{m;ir}T_{l;rs}+\tfrac{1}{4}T_{m;is}\nabla_{l}uR_{mlis}-\tfrac{1}{8}T_{m;is}\nabla_{l}uR_{mlab}\Phi_{abis}\Big{)}\,\operatorname{vol_{g}},$

but, as before,

$T_{m;is}T_{l;ir}T_{m;rs}=0,\quad
T_{m;is}T_{m;ir}T_{l;rs}=0,\quad\text{and}\quad
T_{m;is}(\tfrac{1}{4}R_{mlis}-\tfrac{1}{8}R_{mlab}\Phi_{abis})=T_{m;is}R_{mlis},$

so

$\displaystyle\frac{d}{dt}\Theta=$ $\displaystyle\
\int_{M}-2(t_{0}-t)\Big{(}|\operatorname{div}T|^{2}u+\nabla_{m}uT_{m;is}\nabla_{r}T_{r;is}\Big{)}-|T|^{2}u$
$\displaystyle+2(t_{0}-t)\Big{(}T_{m;is}\nabla_{l}u\nabla_{m}T_{l;is}+T_{m;is}\nabla_{l}uR_{mlis}\Big{)}\,\operatorname{vol_{g}}$
$\displaystyle=$ $\displaystyle\
\int_{M}-2(t_{0}-t)\Big{(}|\operatorname{div}T|^{2}u-2\langle\operatorname{div}T,\nabla
f\lrcorner T\rangle u\Big{)}$
$\displaystyle-2(t_{0}-t)\int_{M}\big{(}\nabla_{m}\nabla_{l}u+\frac{ug_{ml}}{2(t_{0}-t)}\big{)}T_{m;is}T_{l;is}$
$\displaystyle-(t_{0}-t)\int_{M}uR_{mlis}(2T_{l;ir}T_{m;rs}-2T_{m;ir}T_{l;rs}+\tfrac{1}{4}R_{mlis}-\tfrac{1}{8}R_{mlab}\Phi_{abis})$
$\displaystyle-2(t_{0}-t)\int_{M}T_{m;is}u\nabla_{l}R_{mlis}\operatorname{vol_{g}}.\qed$

######  Theorem 5.2 (almost monotonicity formula).

Let $\\{\Phi(t)\\}$ be a solution of the harmonic $\mathrm{Spin}(7)$-flow (HF)
on $(M^{8},g)$.

  1. 1.

If $M$ is compact, then, for any $0<\tau_{1}<\tau_{2}<t_{0}$, there exist
$K_{1}$, $K_{2}>0$ depending only on the geometry of $(M,g)$ such that

$\Theta(\Phi(\tau_{2}))\leq
K_{1}\Theta(\Phi(\tau_{1}))+K_{2}(\tau_{1}-\tau_{2})(E(0)+1).$

  2. 2.

When $(M,g)=(\mathbb{R}^{8},g_{\mathrm{Eucl}})$, then, for any
$x_{0}\in\mathbb{R}^{8}$ and $0\leq\tau_{1}<\tau_{2}$ we have

$\Theta(\Phi(\tau_{2}))\leq\Theta(\Phi(\tau_{1})).$

###### Proof.

We sketch the proof following [DGK21, Theorem 5.3] and rely on Lemma 5.1.

  1. 1.

The third and fourth terms of (5.2) are bounded by

$C(1+\Theta(\Phi(t))),$

due to the bounded geometry of $(M,g)$, Young’s inequality and
$\int_{M}u\,\operatorname{vol_{g}}=1$.

For the second term of (5.2), use [Ham95] and the decreasing of $E(\Phi(t))$
along the harmonic $\mathrm{Spin}(7)$-flow to bound it by

$C\big{(}E(\Phi(0))+\log\frac{B}{(t_{0}-t)^{4}}\Theta(\Phi(t))\big{)},$

so that

$\displaystyle\frac{d}{dt}\Theta(\Phi(t))\leq$
$\displaystyle-2(t_{0}-t)\int_{M}|\operatorname{div}T-\nabla f\lrcorner
T|^{2}u$
$\displaystyle+C_{1}\Big{(}1+\log\big{(}\frac{B}{(t_{0}-t)^{4}}\big{)}\Big{)}\Theta(\Phi(t))+C_{2}(1+E(\Phi(0)))\operatorname{vol_{g}}.$

To control the logarithmic term, let $\xi(t)$ be any function satisfying

$\xi^{\prime}(t)=1+\log\frac{B}{(t_{0}-t)^{4}}.$

The claim is then obtained by integration over $[t_{0}-1,t_{0}[\,$ of

$\frac{d}{dt}\Big{[}e^{-C_{1}\xi(t)}\Theta(\Phi(t))\Big{]}\leq
K\big{(}E(\Phi(0))+1\big{)}.$

  2. 2.

On $(M^{8},g)=(\mathbb{R}^{8},g_{\mathrm{Eucl}})$, the backward heat kernel is

$u(x,t)=\frac{1}{(4\pi(t_{0}-t))^{4}}\exp\left\\{-\frac{|x-x_{0}|^{2}}{4(t_{0}-t)}\right\\}$

so indeed $\frac{d}{dt}\Theta(\Phi(t))\leq 0.$∎

One needs to introduce the quantity $\Theta$ because the energy functional $E$
is not scale-invariant, hence it is not sufficient to control the small-scale
behaviour of a $\mathrm{Spin}(7)$-structure. However, we would like to prove
long-time existence and convergence of the flow when the initial energy is
bounded (which is a reasonable assumption) and hence will need to bound
$\Theta(\Phi(t))$ in terms of the initial energy $E(\Phi(0))$ for $t<t_{0}$.
This fails for small times $t_{0}$, which can be observed from (5.1). Thus, we
require a quantity sturdier than $\Theta$. Motivated by analogous functionals
for the mean curvature flow [CM12], the high dimensional Yang–Mills flow
[BKS17], the harmonic map heat flow [KS16] and the isometric flow of
$\mathrm{G}_{2}$-structures [DGK21], we define the following _entropy_
functional.

######  Definition 5.3.

Let $(M^{8},\Phi,g)$ be a compact manifold with a
$\mathrm{Spin}(7)$-structure. Let $u_{(x,t)}(y,s)=u^{g}_{(x,t)}(y,s)$ be the
backwards heat kernel, starting from $\delta{(x,t)}$ as $s\rightarrow t$. For
$\sigma>0$ we define

$\displaystyle\lambda(\Phi,\sigma)=\underset{(x,t)\in
M\times(0,\sigma]}{\textup{max}}\left\\{t\int_{M}|T_{\Phi}|^{2}(y)u_{(x,t)}(y,0)\operatorname{vol_{g}}\right\\}.$
(5.3)

One should think of $\sigma$ as the “scale” at which we are analyzing the
flow. Since $M$ is compact, the maximum in (5.3) is achieved. We need the next
lemma for the proof of the $\varepsilon$-regularity theorem. We skip the
proof, as it is similar to the $\mathrm{G}_{2}$ case [DGK21, Lemma 5.6] and
uses the analysis on the backwards heat kernel from [Ham93].

######  Lemma 5.4.

Let $(M^{8},g)$ be compact and $E_{0}>0$. For every $\varepsilon>0$, there
exist $\delta=\delta(\varepsilon,g,E_{0})$ and
$\bar{r}=\bar{r}(\varepsilon,g,E_{0})>0$ such that the following holds:

Suppose $\\{\Phi(t)\\}_{t\in[0,t_{0})}$ is a solution of the harmonic
$\mathrm{Spin}(7)$-flow (HF), with induced metric $g$, satisfying
$E(\Phi(0))\leq E_{0}$. Whenever

$\Theta_{(x_{0},t_{0})}(\Phi(t_{1}))<\delta,\quad\text{for some
}(x_{0},t_{1})\in M\times[0,t_{0}),$

then, setting $r:=\textup{min}(\bar{r},\sqrt{t_{0}-t_{1}})$, we have

$\Theta_{(x,t)}(\Phi(t_{1}))<\varepsilon,\quad\forall\ (x,t)\in
B(x_{0},r)\times[t_{0}-r^{2},t_{0}].$

We can now state the $\varepsilon$-regularity theorem for the harmonic
$\mathrm{Spin}(7)$-flow.

######  Theorem 5.5 ($\varepsilon$-regularity).

Let $(M^{8},g)$ be compact and $E_{0}>0$. There exist $\varepsilon,\
\bar{\rho}>0$ such that, for every $\rho\in(0,\bar{\rho}]$, there exist
$r\in(0,\rho)$ and $C<\infty$ such that the following holds:

Suppose $\\{\Phi(t)\\}_{t\in[0,t_{0})}$ is a solution of the harmonic
$\mathrm{Spin}(7)$-flow (HF), with induced metric $g$, satisfying
$E(\Phi(0))\leq E_{0}$. Whenever

$\displaystyle\Theta_{(x_{0},t_{0})}(\Phi(t_{0}-\rho^{2}))<\varepsilon,\quad\text{for
some }x_{0}\in M,$

then, setting $\Lambda_{r}(x,t)=\textup{min}\
\left(1-r^{-1}d_{g}(x_{0},x),\sqrt{1-r^{-2}(t_{0}-t)}\right)$, we have

$\displaystyle\Lambda_{r}(x,t)|T_{\Phi}(x,t)|\leq\frac{C}{r},\quad\forall\
(x,t)\in B(x_{0},r)\times[t_{0}-r^{2},t_{0}].$

###### Proof.

Since the proof is similar to the $\mathrm{G}_{2}$ case in [DGK21, Theorem
5.7], we only sketch the main ideas. The proof is by contradiction. If the
claim were not true, then for any sequences $\varepsilon_{i}\rightarrow 0$ and
$\bar{\rho}_{i}\rightarrow 0$, there would exist
$\rho_{i}\in(0,\bar{\rho}_{i}]$ such that, for any $r_{i}\in(0,\rho_{i})$ and
$C_{i}\rightarrow\infty$, there exist counterexamples
$\\{(M,\Phi_{i}(t))\\}_{t\in[0,t_{i})}$ with $g_{\Phi_{i}(t)}=g$,
$E(\Phi(0))\leq E_{0}$, and $x_{i}\in M$, such that

$\displaystyle\Theta_{(x_{i},t_{i})}(\Phi_{i}(t_{i}-{\rho_{i}}^{2}))<\varepsilon_{i},$

while

$\displaystyle r_{i}\left(\underset{(x,t)\in
B(x_{i},r_{i})\times[t_{i}-{r_{i}}^{2},t_{i}]}{\textup{max}}\Lambda_{r_{i}}(x,t)|T_{\Phi_{i}}(x,t)|\right)>C_{i}.$
(5.4)

Let $(\bar{x},\bar{t})\in B(x_{i},r_{i})\times[t_{i}-{r_{i}}^{2},t_{i}]$ be
the point where the maximum is attained and set
$Q_{i}:=|T_{\Phi_{i}}(\bar{x_{i}},\bar{t_{i}})|$.

The idea now is to use the rescaled flow (see §4.3.1)

$\displaystyle\widetilde{\Phi_{i}}(t)$
$\displaystyle={Q_{i}}^{4}\Phi_{i}(\bar{t_{i}}+t{Q_{i}}^{-2}),\quad t\leq 0,$
$\displaystyle g_{i}$ $\displaystyle={Q_{i}}^{2}g,$

and the pointed sequence
$\\{(M,\widetilde{\Phi_{i}}(t),g_{i},\bar{x}_{i})\\}$. By the definition of
$Q_{i}$, we have

$|T_{\widetilde{\Phi_{i}}}(x,t)|\leq\frac{\Lambda_{i}(\bar{x_{i}},\bar{t_{i}})}{\Lambda_{i}(x,\bar{t_{i}}+t{Q_{i}}^{-2})}\leq
2\quad\text{and}\quad|T_{\widetilde{\Phi_{i}}}(\bar{x}_{i},0)|=1.$

We can then apply the derivative estimates and the compactness Theorem 4.19 to
deduce that the sequence of pointed solutions subsequentially converges to a
limit ancient solution of the harmonic $\mathrm{Spin}(7)$-flow
$({\mathbb{R}}^{8},\Phi_{\infty}(t),g_{\textup{Eucl}},0)_{t\in(-\infty,0]}$
with

$\displaystyle|T_{\Phi_{\infty}}(0,0)|=1.$ (5.5)

On the other hand, using the almost monotonicity formula Theorem 5.2, the
scale invariance of $\Theta$ combined with estimates on the backwards heat
kernel $u$ gives $|T_{\Phi_{\infty}}(0,0)|=0$, which is not possible due to
(5.5). ∎

An immediate corollary of the $\varepsilon$-regularity theorem is the
following result, which states that if the entropy of the initial
$\mathrm{Spin}(7)$-structure is small then the torsion is controlled at all
times. Again, the proof is similar to [DGK21, Cor. 5.8].

######  Corollary 5.6 (small initial entropy controls torsion).

Let $\\{\Phi(t)\\}$ be a solution of the harmonic $\mathrm{Spin}(7)$-flow (HF)
on compact $(M,g)$, starting at $\Phi_{0}$. For every $\sigma>0$, there exist
$\varepsilon,t_{0}>0$ and $C<\infty$ such that, if $\Phi_{0}$ induces $g$ and
its entropy (5.3) satisfies

$\displaystyle\lambda(\Phi_{0},\sigma)<\varepsilon,$

then

$\displaystyle\underset{M}{\textup{max}}\
|T_{\Phi(t)}|\leq\frac{C}{\sqrt{t}}.$

###  5.2 Long-time existence

In this section, we prove that the harmonic $\mathrm{Spin}(7)$-flow on a
compact manifold $M$, under smallness assumptions of the initial data, exists
for all time and converges to a harmonic $\mathrm{Spin}(7)$-structure. We
start with the following convexity result for the energy functional $E$ along
the flow.

######  Lemma 5.7.

Along a solution of the harmonic $\mathrm{Spin}(7)$-flow (HF) on compact
$(M,g)$,

$\displaystyle\frac{d^{2}}{dt^{2}}E(\Phi(t))$
$\displaystyle\geq\int_{M}(\Lambda-3|T|^{2})|\operatorname{div}T|^{2}\operatorname{vol_{g}}$
(5.6)

where $\Lambda$ is the first non-zero eigenvalue of the rough Laplacian of $g$
on two-forms.

###### Proof.

From (4.4),

$\displaystyle\frac{d^{2}}{dt^{2}}E(\Phi(t))$
$\displaystyle=-2\int_{M}\langle\frac{\partial}{\partial
t}\operatorname{div}T,\operatorname{div}T\rangle\operatorname{vol_{g}}$
$\displaystyle=-2\int_{M}\nabla_{m}(\nabla_{r}T_{r;ip}T_{m;ps}-\nabla_{r}T_{r;sp}T_{m;pi}+\pi_{7}(\nabla_{m}(\nabla_{r}T_{r;is})))\nabla_{a}T_{a;is}\operatorname{vol_{g}}$
$\displaystyle=2\int_{M}2\nabla_{r}T_{r;ip}T_{m;ps}\nabla_{m}(\nabla_{a}T_{a;is})+|\pi_{7}\nabla_{m}(\nabla_{r}T_{r;is})|^{2}\operatorname{vol_{g}}.$

Now $\operatorname{div}T\in\Omega^{2}_{7}$, so that
$(\operatorname{div}T)_{is}\Phi_{isab}=-6(\operatorname{div}T)_{ab}$, and
therefore

$\displaystyle-6\nabla_{m}(\operatorname{div}T)_{ab}=\nabla_{m}(\operatorname{div}T)_{is}\Phi_{isab}+2(\operatorname{div}T)_{is}T_{m;iq}\Phi_{qsab}-6(\operatorname{div}T)_{qb}T_{m;aq}-6(\operatorname{div}T)_{aq}T_{m;bq}.$

We use this fact to expand the second term in the integral:

$\displaystyle|\pi_{7}\nabla_{m}(\nabla_{r}T_{r;is})|^{2}=\langle\pi_{7}\nabla_{m}(\nabla_{r}T_{r;is}),\nabla_{m}(\nabla_{r}T_{r;is})\rangle$
$\displaystyle=\Big{(}\tfrac{1}{4}(\nabla_{m}\operatorname{div}T)_{ij}-\tfrac{1}{8}(\nabla_{m}\operatorname{div}T)_{ab}\Phi_{ijab}\Big{)}(\nabla_{m}\operatorname{div}T)_{ij}$
$\displaystyle=\tfrac{1}{4}|\nabla_{m}(\operatorname{div}T)|^{2}-\tfrac{1}{8}(\nabla_{m}\operatorname{div}T)_{ab}\Phi_{ijab}\nabla_{m}(\operatorname{div}T)_{ij}$
$\displaystyle=\tfrac{1}{4}|\nabla_{m}(\operatorname{div}T)|^{2}-\tfrac{1}{8}(\nabla_{m}\operatorname{div}T)_{ab}\Big{(}-2(\operatorname{div}T)_{is}T_{m;iq}\Phi_{qsab}+6(\operatorname{div}T)_{qb}T_{m;aq}+6(\operatorname{div}T)_{aq}T_{m;bq}$
$\displaystyle\qquad-6(\nabla_{m}\operatorname{div}T)_{ab}\Big{)}$
$\displaystyle=|\nabla_{m}(\operatorname{div}T)|^{2}+\tfrac{1}{4}(\nabla_{m}\operatorname{div}T)_{ab}(\operatorname{div}T)_{is}T_{m;iq}\Phi_{qsab}-\tfrac{3}{4}\Big{(}(\operatorname{div}T)_{qb}T_{m;aq}+(\operatorname{div}T)_{aq}T_{m;bq}\Big{)}(\nabla_{m}\operatorname{div}T)_{ab}$
$\displaystyle=|\nabla_{m}(\operatorname{div}T)|^{2}-\tfrac{3}{4}\Big{(}(\operatorname{div}T)_{qb}T_{m;aq}+(\operatorname{div}T)_{aq}T_{m;bq}\Big{)}(\nabla_{m}\operatorname{div}T)_{ab}$
$\displaystyle\qquad+\tfrac{1}{4}(\operatorname{div}T)_{is}T_{m;iq}\Big{(}-2(\operatorname{div}T)_{ir}T_{m;ip}\Phi_{prqs}+6(\operatorname{div}T)_{ps}T_{m;qp}+6(\operatorname{div}T)_{qp}T_{m;sp}-6(\nabla_{m}\operatorname{div}T)_{qs}\Big{)}$
$\displaystyle=|\nabla_{m}(\operatorname{div}T)|^{2}-\tfrac{3}{4}\Big{(}(\operatorname{div}T)_{qb}T_{m;aq}+(\operatorname{div}T)_{aq}T_{m;bq}\Big{)}(\nabla_{m}\operatorname{div}T)_{ab}$
$\displaystyle\qquad+\tfrac{3}{2}(\operatorname{div}T)_{is}T_{m;iq}(\operatorname{div}T)_{ps}T_{m;qp}+\tfrac{3}{2}(\operatorname{div}T)_{is}T_{m;iq}(\operatorname{div}T)_{qp}T_{m;sp}-\tfrac{3}{2}(\operatorname{div}T)_{is}T_{m;iq}(\nabla_{m}\operatorname{div}T)_{qs}$
$\displaystyle=|\nabla_{m}(\operatorname{div}T)|^{2}+\tfrac{3}{4}\Big{(}(\operatorname{div}T)_{qb}T_{m;aq}-(\operatorname{div}T)_{aq}T_{m;bq}\Big{)}(\nabla_{m}\operatorname{div}T)_{ab}$
$\displaystyle\qquad+\tfrac{3}{2}(\operatorname{div}T)_{is}T_{m;iq}(\operatorname{div}T)_{ps}T_{m;qp}+\tfrac{3}{2}(\operatorname{div}T)_{is}T_{m;iq}(\operatorname{div}T)_{qp}T_{m;sp}$
$\displaystyle=|\nabla_{m}(\operatorname{div}T)|^{2}+\tfrac{3}{2}(\operatorname{div}T)_{is}T_{m;iq}\Big{(}(\operatorname{div}T)_{ps}T_{m;qp}+(\operatorname{div}T)_{qp}T_{m;sp}\Big{)}$
$\displaystyle\geq|\nabla_{m}(\operatorname{div}T)|^{2}-3|\operatorname{div}T|^{2}|T|^{2}.$
$\displaystyle\therefore\qquad\frac{d^{2}}{dt^{2}}E(\Phi(t))$
$\displaystyle\geq\int_{M}|\nabla_{m}(\operatorname{div}T)|^{2}-3|\operatorname{div}T|^{2}|T|^{2}\operatorname{vol_{g}}$
$\displaystyle\geq\int_{M}(\Lambda-3|T|^{2})|\operatorname{div}T|^{2}\operatorname{vol_{g}},$

where $\Lambda$ is the first non-zero eigenvalue of the rough Laplacian on
$2$-forms. On a compact manifold, the kernel of that rough Laplacian consists
indeed of parallel $2$-forms, so it is orthogonal to $\operatorname{div}T$. ∎

The following general interpolation result can be proved in a similar way as
[DGK21, Lemma 5.12].

######  Lemma 5.8 (interpolation).

Let $(M,\Phi,g)$ be a compact $\mathrm{Spin}(7)$-structure manifold with
torsion $T=T_{\Phi}$. Suppose $|\nabla T|\leq C$ and non-collapsing at every
$x\in M$, i.e.,

$\displaystyle\operatorname{vol_{g}}(B(x,r))\geq
v_{0}r^{8},\quad\text{for}\quad 0<r\leq 1,$

for some constant $v_{0}(M,g)>0$. Then, for every $\varepsilon>0$, there
exists $\delta(\varepsilon,C,v_{0})\geq 0$ such that, if

$\displaystyle E(\Phi)<\delta,$ (5.7)

then $|T|<\varepsilon$.

We now prove Theorem C. First, we prove that if the torsion of the initial
$\mathrm{Spin}(7)$-structure is pointwise small, then (HF) exists for all time
and converges to a critical point of the energy functional. Then Corollary
5.6, the derivative estimates from Corollary 4.10, and the interpolation Lemma
5.8 imply that, if the initial entropy is small, then the torsion does
eventually become pointwise small.

######  Theorem 5.9 (small initial torsion gives long-time existence).

Let $(M,\Phi_{0},g)$ be a compact $\mathrm{Spin}(7)$-structure manifold. For
every $\delta>0$, there exists $\varepsilon(\delta,g)>0$ such that, if
$|T_{\Phi_{0}}|<\varepsilon$, then a harmonic $\mathrm{Spin}(7)$-flow (HF)
starting at $\Phi_{0}$ exists for all time and converges subsequentially
smoothly to a $\mathrm{Spin}(7)$-structure $\Phi_{\infty}$ such that

$\operatorname{div}T_{\Phi_{\infty}}=0,\quad|T_{\Phi_{\infty}}|<\delta.$

###### Proof.

We only sketch the proof, since it is very similar to that of the isometric
$\mathrm{G}_{2}$-flow [DGK21, Theorem 5.13]. By the doubling-time estimate
[Proposition 4.9], given $\delta>0$, there exists $\varepsilon_{0}>0$ such
that, whenever $|T_{\Phi_{0}}|<\varepsilon_{0}$,

$\displaystyle t_{*}:=\textup{max}\\{t\geq 0\ :\ |T_{\Phi(t)}|\leq
2\varepsilon_{0}\\}>\delta.$ (5.8)

We prove that $t_{*}=\infty$, hence that a flow with small initial torsion
exists for all time. By contradiction, suppose $t_{*}<\infty$. Then, by the
derivative estimates of Corollary 4.10 for the time interval
$[t_{*}-\delta,t_{*}]$, there is a constant $c_{0}=c_{0}(g,\varepsilon_{0})$
such that

$\displaystyle|\nabla T_{\Phi(t_{*})}|<c_{0}.$ (5.9)

Note that (HF) is the negative gradient flow of $E(\Phi(t))$ and hence
$E(\Phi(t_{*}))\leq E(\Phi(0))=E(\Phi_{0})$. Given $a>0$, the interpolation
Lemma 5.8 yields a constant $\gamma_{a}=\gamma_{a}(c_{0})>0$ such that, if
$E(\Phi_{0})<\gamma_{a}$, then $|T_{\Phi(t_{*})}|<a$. Now, if
$\varepsilon<\textup{min}\\{\varepsilon_{0},\gamma_{2\varepsilon_{0}}\\}$,
then $|T_{\Phi(t_{*})}|<2\varepsilon$, which contradicts the maximality of
$t_{*}$, as defined in (5.8). Thus $t_{*}=\infty$ and the flow exists for all
time.

Denote by $\Lambda$ the first non-zero eigenvalue of the Laplacian acting on
$2$-forms. Note from (5.6) that, if $|T|^{2}\leq\frac{\Lambda}{6}$, then

$\displaystyle\frac{d}{dt}\int_{M}|\operatorname{div}T_{\Phi(t)}|^{2}\operatorname{vol_{g}}=-\frac{d^{2}}{dt^{2}}E(\Phi(t))\leq-\frac{\Lambda}{2}\int_{M}|\operatorname{div}T_{\Phi(t)}|^{2}\operatorname{vol_{g}}.$
(5.10)

Let $a_{0}=\sqrt{\frac{\Lambda}{6}}$. If we take
$\varepsilon<\textup{min}\\{\varepsilon,\gamma_{2\varepsilon_{0}},\gamma_{a_{0}}\\}$,
then from the previous paragraph we obtain existence of the flow for all time
and satisfying (5.10). Thus, we get the decay estimate

$\displaystyle\int_{M}|\operatorname{div}T_{\Phi(t)}|^{2}\operatorname{vol_{g}}\leq
e^{-\frac{\Lambda
t}{2}}\int_{M}|\operatorname{div}T_{\Phi(0)}|^{2}\operatorname{vol_{g}},\quad\forall\
t\geq 0.$ (5.11)

Now, for every $s_{1}<s_{2}$, we have

$\displaystyle\int_{M}|\Phi(s_{2})-\Phi(s_{1})|\operatorname{vol_{g}}$
$\displaystyle\leq\int_{M}\int_{s_{1}}^{s_{2}}\left|\partial_{t}\Phi(s)\right|ds\operatorname{vol_{g}}=\int_{s_{1}}^{s_{2}}\int_{M}|\operatorname{div}T_{\Phi(s)}|\operatorname{vol_{g}}ds$
(5.12) $\displaystyle\leq
c\int_{s_{1}}^{s_{2}}\left(\int_{M}|\operatorname{div}T_{\Phi(s)}|^{2}\operatorname{vol_{g}}\right)^{\frac{1}{2}}ds$
$\displaystyle\leq c\int_{s_{1}}^{s_{2}}e^{-\frac{\Lambda s}{4}}ds.$

Hence $\Phi(t)$ converges exponentially fast to a unique limit $\Phi_{\infty}$
in $L^{1}$ as $t\rightarrow+\infty$.

Moreover, the uniform torsion bound gives estimates on all derivatives of the
torsion. Given any sequence $t_{n}\rightarrow+\infty$, a subsequence of
$\\{\Phi(t_{n})\\}$ will converge smoothly to a limit, which must be
$\Phi_{\infty}$ by uniqueness. Therefore, the flow $\\{\Phi(t)\\}$ converges
smoothly to $\Phi_{\infty}$ as $t\rightarrow+\infty$. Inequality (5.11)
implies that $\operatorname{div}T_{\Phi_{\infty}}=0$, and choosing
$\varepsilon>0$ small enough, we can also achieve
$|T_{\Phi_{\infty}}|<\delta$, using the interpolation Lemma 5.8. ∎

We finally prove that small initial entropy implies long-time existence and
convergence.

######  Theorem 5.10 (small entropy).

On a compact $\mathrm{Spin}(7)$-structure manifold $(M,\Phi_{0},g)$, there
exist constants $C_{k}(M,g)<+\infty$, such that the following holds. For each
$\varepsilon>0$ and $\sigma>0$, there exists
$\lambda_{\varepsilon}(g,\sigma)>0$ such that, if the entropy (5.3) satisfies

$\displaystyle\lambda(\Phi_{0},\sigma)<\lambda_{\varepsilon},$ (5.13)

then the torsion becomes eventually pointwise small along the harmonic
$\mathrm{Spin}(7)$-flow (HF) starting at $\Phi_{0}$. Therefore the flow exists
for all time and subsequentially converges to a $\mathrm{Spin}(7)$-structure
$\Phi_{\infty}$ such that

$\operatorname{div}T_{\Phi_{\infty}}=0,\quad|T_{\Phi_{\infty}}|<\varepsilon\quad\text{and}\quad|\nabla^{k}T_{\Phi_{\infty}}|<C_{k},\,\forall\
k\geq 1.$

###### Proof.

By Corollary 5.6, if $\lambda_{\varepsilon}>0$ is small enough, then the
torsion along the flow satisfies $|T_{\Phi(t)}|\leq\frac{C}{\sqrt{t}}$, for
all $t\in(0,\tau]$. Thus, by the derivative estimates in Corollary 4.10,
$\Phi(\tau)$ satisfies $|\nabla T_{\Phi(\tau)}|\leq C^{\prime}$ for some
constant $C^{\prime}<\infty$. Hence by the interpolation Lemma 5.8, if
$\lambda_{\varepsilon}>0$ is even smaller then $|T_{\Phi(\tau)}|<\varepsilon$.
By Theorem 5.9, the flow then converges smoothly to a
$\mathrm{Spin}(7)$-structure $\Phi_{\infty}$ with divergence-free torsion and
derivative bounds. ∎

###  5.3 Singularity structure

We investigate the singularities of the harmonic $\mathrm{Spin}(7)$-flow. Let
$\\{\Phi(t)\\}_{t\in[0,\tau)}$ be a solution to (HF) on a compact manifold
$(M,g)$, with a finite time singularity at $\tau<+\infty$.

####  5.3.1 The singular set $S$

Let $\varepsilon$ and $\bar{\rho}$ be the quantities from the
$\varepsilon$-regularity Theorem 5.5. We define the _singular set_ of the flow
by

$\displaystyle S=\\{x\in M\ :\
\Theta_{(x,\tau)}(\Phi(\tau-\rho^{2}))\geq\varepsilon,\ \textup{for\ all}\
\rho\in(0,\bar{\rho}]\\}.$ (5.14)

The following lemma explains why $S$ is called the singular set of the flow.

######  Lemma 5.11.

The harmonic $\mathrm{Spin}(7)$-flow $\\{\Phi(t)\\}_{t\in[0,\tau)}$ restricted
to $M\setminus S$ converges as $t\rightarrow\tau$, smoothly and uniformly away
from $S$, to a smooth harmonic $\mathrm{Spin}(7)$-structure $\Phi(\tau)$ on
$M\setminus S$. Moreover, for every $x\in S$, there is a sequence
$(x_{i},t_{i})\to(x,\tau)$ such that

$\displaystyle\lim_{i}|T_{\Phi}(x_{i},t_{i})|=\infty.$

Thus, $S$ is indeed the singular set of the flow.

###### Proof.

By Theorem 5.5, for every $x\in M\setminus S$, there exist $r_{x}>0$ and
$C_{x}<\infty$ such that, setting

$\Lambda_{r_{x}}(y,t)=\textup{min}\
\left(1-\frac{d_{g}(x,y)}{r_{x}},\sqrt{1-\frac{\tau-t}{{r_{x}}^{2}}}\right),$

one has

$\Lambda_{r_{x}}(y,t)|T_{\Phi}(y,t)|\leq\frac{C_{x}}{r_{x}},\quad\forall\
(y,t)\in B(x,r_{x})\times[\tau-{r_{x}}^{2},\tau].$

So, for $\hat{r}_{x}=\frac{1}{2}r_{x}$, we have

$T_{\Phi}(y,t)\leq\widehat{C}_{x},\quad\forall\ (y,t)\in
B(x,\hat{r}_{x})\times[\tau-{\hat{r}_{x}}^{2},\tau].$

Hence, by the local derivative estimates of Corollary 4.16, there exist
constants $C_{x,j}$ such that

$|\nabla^{j}T_{\Phi}(y,t)|\leq C_{x,j},\quad\forall\ (y,t)\in
B(x,\frac{\hat{r}_{x}}{2})\times[\tau-\frac{{\hat{r}_{x}}^{2}}{4},\tau],\quad\forall\
j\geq 1.$

Thus as $t\rightarrow\tau$, the flow $\Phi(t)$ converges smoothly and
uniformly away from $S$ to a harmonic $\mathrm{Spin}(7)$-structure
$\Phi(\tau)$. ∎

####  5.3.2 Proof of Theorem D

Note from (5.14) that $S$ is a closed set. We therefore estimate
$\mathcal{H}^{6}(S)$, where $\mathcal{H}^{6}$ denotes the $6$-dimensional
Hausdorff measure on $(M,g)$.

Let $S^{\prime}$ be any subset of $S$. As in [GH96], there is
$S^{\prime\prime}\subset S^{\prime}$ such that

$\displaystyle\mathcal{H}^{6}(S^{\prime\prime})\geq\frac{1}{2}\mathcal{H}^{6}(S^{\prime})$
(5.15)

and

$\displaystyle
u_{S^{\prime\prime}}(y,s)=\int_{S^{\prime\prime}}u_{(x,\tau)}(y,s)\operatorname{vol_{g}}(x)$

satisfies

$\displaystyle u_{S^{\prime\prime}}(y,s)\leq\frac{C}{\tau-s},$ (5.16)

for some constant $C$ depending only on $g$. Thus, for every
$\rho\in(0,\bar{\rho})$, using (5.14) with the same $\varepsilon$ and
$S^{\prime\prime}\subset S$, we get

$\displaystyle\varepsilon\mathcal{H}^{6}(S^{\prime\prime})=\int_{S^{\prime\prime}}\varepsilon
d\mathcal{H}^{6}(x)\leq\int_{S^{\prime\prime}}\Theta_{(x,\tau)}(\Phi(\tau-\rho^{2}))d\mathcal{H}^{6}(x).$

Using the definition of $\Theta$, the estimate (5.16) and the hypothesis
(1.6), we estimate

$\displaystyle\varepsilon\mathcal{H}^{6}(S^{\prime\prime})$
$\displaystyle\leq\int_{S^{\prime\prime}}\int_{M}\rho^{2}|T(y,\tau-\rho^{2})|^{2}u_{(x,\tau)}(y,\tau-\rho^{2})\operatorname{vol_{g}}(y)d\mathcal{H}^{6}(x)$
$\displaystyle\leq\int_{M}\rho^{2}|T(y,\tau-\rho^{2})|^{2}u_{S^{\prime\prime}}(y,\tau-\rho^{2})\operatorname{vol_{g}}(y)$
$\displaystyle\leq C\int_{M}|T(y,\tau-\rho^{2})|^{2}\operatorname{vol_{g}}(y)$
$\displaystyle\leq CE_{0}.$

Since $S^{\prime}$ was arbitrary, the result follows.

######  Remark 5.12.

Theorem D says that the singular set is _at most_ $6$-dimensional. In order to
find a geometric interpretation of $S$ in terms of
$\mathrm{Spin}(7)$-geometry, it is likely that $S$ would be at most
$4$-dimensional, as there are no distinguished $6$-dimensional subspaces in
this context.

####  5.3.3 Soliton model at Type-i@ singularities

Let $\\{\Phi(t)\\}_{t\in[0,\tau]}$ be a solution to (HF) that exists for some
maximal time $\tau$. From the evolution of the norm of the torsion tensor $T$
along (HF) in Proposition 4.14, and the maximum principle, we see that the
rate of blow-up of the norm of the torsion tensor must be at least

$\displaystyle\underset{x\in
M}{\textup{sup}}|T(t)|\geq\frac{1}{\sqrt{C(\tau-t)}}.$

######  Definition 5.13.

The flow $\\{\Phi(t)\\}_{t\in[0,\tau]}$ is said to have a _Type-i@ singularity
at $\tau$_ if

$\displaystyle\underset{x\in
M}{\textup{sup}}|T(t)|\leq\frac{1}{\sqrt{C(\tau-t)}}.$

Using the almost monotonicity formula from Theorem 5.2 as in [GH96] and
[DGK21, Theorem 5.20], we see that a sequence of blow-ups at a Type-i@
singularity admits a subsequence that converges to a shrinking soliton (cf.
§4.2.1) of the flow:

######  Theorem 5.14.

Let $\\{\Phi(t)\\}_{t\in[0,\tau)}$ be the maximal smooth harmonic
$\mathrm{Spin}(7)$-flow on $(M^{8},\Phi_{0})$ starting at $\Phi_{0}$. Suppose
that the flow encounters a Type-i@ singularity at $\tau<\infty$. Let $x\in M$
and $\lambda_{i}\searrow 0$, and consider the rescaled sequence
$\Phi_{i}(t)={\lambda_{i}}^{-3}\Phi(\tau-{\lambda_{i}}^{2}t)$. Then, after
possibly passing to a subsequence, $(M,\Phi_{i}(t),x)$ converges smoothly to
an ancient harmonic $\mathrm{Spin}(7)$-flow $\\{\Phi_{\infty}(t)\\}_{t<0}$ on
$({\mathbb{R}}^{8},g_{\textup{Eucl}})$ induced by a shrinking soliton, i.e.

$\displaystyle\operatorname{div}T_{\Phi_{\infty}}(x,t)=-\frac{x}{2t}\lrcorner
T_{\Phi_{\infty}}.$

Moreover $x\in M\setminus S$ if, and only if, $\\{\Phi_{\infty}(t)\\}$ is the
stationary flow induced by a torsion-free $\mathrm{Spin}(7)$-structure on
$({\mathbb{R}}^{8},g_{\textup{Eucl}})$.

######  Remark 5.15.

It would be very interesting to get explicit examples of shrinking, steady or
expanding harmonic solitons, even for $\mathbb{R}^{8}$. On the contrary, if
one could prove that there do not exist any shrinking solitons on
$\mathbb{R}^{8}$, then the previous theorem tells us that the harmonic
$\mathrm{Spin}(7)$-flow has no Type-i@ singularities.

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(SD): Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower
Chaussee 25, 12489 Berlin.
<EMAIL_ADDRESS>
(EL): Univ. Brest, CNRS UMR 6205, LMBA, F-29238 Brest, France.
<EMAIL_ADDRESS>
(HSE): Institute of Mathematics, Statistics and Scientific Computing (IMECC),
University of Campinas (Unicamp), 13083-859 Campinas-SP, Brazil.
<EMAIL_ADDRESS>


# Instantons in $\maybebm{\phi^{4}}$ Theories: Transseries, Virial Theorems
and Numerical Aspects

Ludovico T. Giorgini  Nordita, Royal Institute of Technology and Stockholm
University, Stockholm 106 91, Sweden  Ulrich D. Jentschura  Department of
Physics, Missouri University of Science and Technology, Rolla, Missouri 65409,
USA  Enrico M. Malatesta  Department of Computing Sciences, Bocconi
University, via Sarfatti 25, 20136 Milan, Italy  Institute for Data Science
and Analytics, Bocconi University, 20136 Milano, Italy  Tommaso Rizzo
Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185
Rome, Italy  Institute of Complex Systems (ISC) - CNR, Rome unit, P.le A. Moro
5, 00185 Rome, Italy  Jean Zinn-Justin  IRFU/CEA, Paris-Saclay, 91191 Gif-sur-
Yvette Cedex, France

###### Abstract

We discuss numerical aspects of instantons in two- and three-dimensional
$\phi^{4}$ theories with an internal $O(N)$ symmetry group, the so-called
$N$-vector model. Combining asymptotic transseries expansions for large
argument with convergence acceleration techniques, we obtain high-precision
values for certain integrals of the instanton that naturally occur in loop
corrections around instanton configurations. Knowledge of these numerical
properties are necessary in order to evaluate corrections to the large-order
factorial growth of perturbation theory in $\phi^{4}$ theories. The results
contribute to the understanding of the mathematical structures underlying the
instanton configurations.

###### Contents

  1. I Orientation
  2. II Fundamentals of Instantons in $\maybebm{\phi^{4}}$ Theories
    1. II.1 Instanton Equations
    2. II.2 Instantons and Large–Order Behavior
  3. III Three–Dimensional Instanton
    1. III.1 Large Argument
    2. III.2 Small Argument
  4. IV Two–Dimensional Instanton
    1. IV.1 Large Argument
    2. IV.2 Small Argument
  5. V Instantons and Virial Theorems
    1. V.1 Derivation of the Virial Theorems
    2. V.2 Summary of the Virial Theorems
    3. V.3 Asymptotic Behavior
  6. VI Instanton Integrals
  7. VII Conclusions
  8. A Four–Dimensional Massless Quartic Theory
  9. B Six–Dimensional Massless Cubic Theory

##  I Orientation

The $N$-vector model (the self-interacting $\phi^{4}$ field theory in $D=2$
and $D=3$ dimensions) gives rise to instanton configurations, whose structure
is more complicated than the corresponding configurations in quantum mechanics
(in one space dimension), which is equivalent to a $D=1$ dimensional field
theory (see Figs. 2 and 3 of Ref. JeSuZJ2009sigma ). The instantons provide a
nontrivial saddle point of the Euclidean action, about which we expand
partition functions, and generating functions BrPa1978 ; MaPaRi2017 ;
GiEtAl2020 ; GiEtAl2022 ; GiEtAl2024ii . Instantons also constitute
fundamental objects in statistical and optimization problems possessing hard
phases (see Refs. Ma2023 ; MaKi2023 ; Ri2021 . Here, we derive a semi-analytic
representation which can be used to describe the instanton uniformly over the
radial variable, to a relative accuracy of $10^{-22}$ or better.

In one dimension (1D), one canonically identifies the argument of the
instanton as the Euclidean “time” $t$, with the notion that $-\infty<t<\infty$
(see Ref. JeSuZJ2009sigma ). In 2D and 3D, this is not so easy, because the
angular symmetry dictates that one should choose a radial variable. The radial
variable $r$, in turn, can only take values in the range $0<r<\infty$. The
connection to the 1D case GiEtAl2020  is found if we consider that in 1D, we
can interpret the “radial” variable with the $\mathbbm{Z}_{2}$ symmetry
(positive and negative real numbers). The surface area of the zero-dimensional
unit sphere embedded in one-dimensional space is
$2\pi^{(D=1)/2}/\Gamma((D=1)/2)=2$; the result confirms the $\mathbbm{Z}_{2}$
symmetry of the (analytically known) instantons in one-dimensional theories
JeSuZJ2009sigma ; GiEtAl2020 .

In two-dimensional and three-dimensional $\phi^{4}$ theories, the instanton is
not known analytically. Here, we aim to demonstrate that the analytic
structure of the instanton is shown to be linked to the concept of transseries
and resurgent expansions (see Refs. Ph1989 ; CaNoPh1993 ; Bo1994 ; Ed2009 ;
AnBaSc2019 ; Do2019 ; vSVo2022 ; DuHa2021 ; BoBr2022 ). Specifically, we
derive an asymptotic representation of the instanton, for large argument, in
the form of a transseries (resurgent expansion) in the variables $\chi=1/r$
and $\exp(-1/\chi)=\exp(-r)$, where $r$ is the distance from the origin. The
transseries representation for large $r$ is complemented by a power-series
representation for small $r$, which is augmented by Padé approximants and
nonlinear sequence transformations to enhance its applicability for
intermediate values of the radial variable. The goal is to match the large-$r$
and small-$r$ representations at a suitable intermediate transition value of
the radial variable, to obtain a uniform, high-precision representation of the
instanton in 2D and 3D.

We organized the paper as follows. Fundamentals of instantons in $\phi^{4}$
theories are discussed in Sec. II. The three-dimensional instanton in a three-
dimensional $\phi^{4}$ theory is analyzed in Sec. III. Our analysis of the
instanton configuration in a two-dimensional field theory follows in Sec. IV.
Virial theorems and the asymptotic behavior of the instanton are discussed in
Sec. V. The high-precision evaluation of instanton integrals and of instanton
actions is discussed in Sec. VI. Conclusions are drawn in Sec. VII.

##  II Fundamentals of Instantons in $\maybebm{\phi^{4}}$ Theories

###  II.1 Instanton Equations

For the consideration of the instanton configuration, it is sufficient to
consider the $D$-dimensional scalar theory, with the action

$S[\phi]=\int{\mathrm{d}}^{D}x\,\left[\frac{1}{2}\,\left(\vec{\nabla}\phi(\vec{x})\right)^{2}+\frac{1}{2}\,\phi(\vec{x})^{2}+\frac{g}{4}\,\phi(\vec{x})^{4}\right]\,,$
(1)

where $\vec{x}$ is a $D$-dimensional vector. Consideration of the variation
$\delta S[\phi]$ leads to the defining equation of the instanton,

$-\vec{\nabla}^{2}\phi_{\mathrm{cl}}(\vec{x})+\phi_{\mathrm{cl}}(\vec{x})\,+g\,\phi_{\mathrm{cl}}(\vec{x})^{3}=0\,.$
(2)

Differentiation with respect to a coordinate leads to the equation of the zero
mode $\partial_{\mu}\phi_{\mathrm{cl}}(\vec{x})$,

$\left(-\vec{\nabla}^{2}+1+3\,g\,\phi_{\mathrm{cl}}(\vec{x})^{2}\right)\partial_{\mu}\phi_{\mathrm{cl}}(\vec{x})=0\,.$
(3)

where $\mu=1,\dots,D$. In a quartic theory, the instanton solution exists only
for negative $g$, because the tunneling can proceed only through a barrier.
Therefore, with the scaling

$\phi_{\mathrm{cl}}(\vec{x})=\sqrt{-\frac{1}{g}}\,\xi_{\mathrm{cl}}(\vec{x})\,,$
(4)

the equations for the instanton and the zero mode are, respectively,

$\displaystyle\left(-\vec{\nabla}^{2}+1-\xi_{\mathrm{cl}}(\vec{x})^{2}\right)\xi_{\mathrm{cl}}(\vec{x})=$
$\displaystyle\;0\,,$ (5)
$\displaystyle\left(-\vec{\nabla}^{2}+1-3\,\xi_{\mathrm{cl}}(\vec{x})^{2}\right)\partial_{\mu}\xi_{\mathrm{cl}}(\vec{x})=$
$\displaystyle\;0\,.$ (6)

In a theory with an internal $O(N)$ symmetry group, one has the following
instanton,

$\underline{\phi}_{\mathrm{cl}}(\vec{x})=\phi_{\mathrm{cl}}(\vec{x})\,\underline{u}=\sqrt{-\frac{1}{g}}\,\xi_{\mathrm{cl}}(\vec{x})\,\underline{u}\,,$
(7)

where vectors in the internal space are designated by underlining, and we can
choose

$\underline{u}=\\{1,0,\cdots,0\\}^{\rm T}\,.$ (8)

As is evident from Figs. 2 and 3 of Ref. JeSuZJ2009sigma , there is a sign
ambiguity in the choice of the instanton, and the degeneracy under the
operation $\xi_{\mathrm{cl}}\to-\xi_{\mathrm{cl}}$ needs to be taken into
account when using dispersion relations. Indeed, via dispersion relations, one
can establish that the instanton action $A$, defined via

$S[\phi_{\mathrm{cl}}]=-A/g\,,$ (9)

governs the large-order behavior of the perturbative coefficients ${\sf
G}_{K}$ in the $K$th order of the expansion in $g$ of the $n$-point
correlation functions in a $D$-dimensional $O(N)$ theory GiEtAl2020 . In the
notation adopted in Eq. (1.9) of Ref. GiEtAl2020 , we have

$\displaystyle{\sf G}_{K}=$
$\displaystyle\;\frac{c(N,D)}{\pi}\,\left(\frac{1}{A}\right)^{(n+N+D-1)/2}\,\left(-\frac{1}{A}\right)^{K}\,$
(10)
$\displaystyle\;\times\Gamma\left(K+\frac{n+N+D-1}{2}\right)\,[1+{\mathcal{O}}(1/K)]\,.$
(11)

Here, $c(N,D)$ is a constant coefficient to be determined separately for each
$N$ and $D$.

Figure 1:  Tree diagram illustrating the emergence of the instanton.

###  II.2 Instantons and Large–Order Behavior

The connection between instantons and large-order behavior is usually obtained
by saddle-point evaluations of contour integrals ZJ2002 . In the following, we
will mention a less known derivation Pa1977  that has the advantage of being
simpler and more intuitive. The basic idea is that Feynman diagrams of the
$\phi^{4}$ theory at large orders $K\gg 1$ are essentially random regular
graphs with connectivity 4 and size $K$. For a large number of vertices, it is
known that random regular graphs have a locally tree-like structure (with the
size of the loops growing as $\log K$). This allows us to write an iterative
equation that turns out to be equivalent to the instanton equation (2). The
Feynman rules imply that there is a factor $1/(\vec{k}_{i}^{2}+m^{2})$ for
each line $i$ in the graph and a Dirac-$\delta$ function on each vertex
ensuring the conservation of momentum. Invoking the tree-like structure, one
can then write the following equation (see also Fig. 1)

$g(\vec{k})=\frac{1}{\vec{k}^{2}+m^{2}}\int\frac{{\mathrm{d}}^{D}k_{1}}{(2\pi)^{D}}\int\frac{{\mathrm{d}}^{D}k_{2}}{(2\pi)^{D}}\int\frac{{\mathrm{d}}^{D}k_{3}}{(2\pi)^{D}}\\\
g(\vec{k}_{1})g(\vec{k}_{2})g(\vec{k}_{3})\delta^{(D)}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3}-\vec{k})\,.$
(12)

So, multiplying everything by $\vec{k}^{2}+m^{2}$, the previous equation in
real space is equivalent to

$(-\vec{\nabla}^{2}+m^{2})\,g(\vec{x})=g^{3}(\vec{x})\,,$ (13)

which is the instanton equation for the $\phi^{4}$ theory for
$g(\vec{x})=\xi_{\mathrm{cl}}(\vec{x})$.

Using a standard procedure (similar in spirit to the cavity method from spin-
glass theory MePaVi1986 ), one can identify the action in Eq. (1) as the Bethe
free energy of the problem. This allows us to derive in a simple way the
instanton equation and the action; the $1/K$ correction about the instanton
GiEtAl2020 ; GiEtAl2022  corresponds to the $1/K$ finite size correction to
the Bethe lattice random graphs.

##  III Three–Dimensional Instanton

###  III.1 Large Argument

We use the fact that
$\xi_{\mathrm{cl}}(\vec{x})=\xi_{\mathrm{cl}}(|\vec{x}|)=\xi_{\mathrm{cl}}(r)$
is radially symmetric and this constitutes am “$S$ state” in the formalism
adopted in atomic physics ItZu1980 ; JeAd2022book . The equation fulfilled by
the instanton $\xi_{\mathrm{cl}}(r)=\xi^{(3)}_{\mathrm{cl}}(r)$ (including the
dimension $D$ in the superscript) in three dimensions is [see Eq. (5)],

$-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(3)}_{\mathrm{cl}}(r)-\frac{2}{r}\frac{\partial}{\partial
r}\xi^{(3)}_{\mathrm{cl}}(r)+\xi^{(3)}_{\mathrm{cl}}(r)-[\xi^{(3)}_{\mathrm{cl}}(r)]^{3}=0\,.$
(14)

We are attempting to find a systematic expansion of the solution of Eq. (14),
and do so for large argument $r\to\infty$ in the current section. A remark
might be in order. Namely, linear second-order differential equations
typically have solutions regular and irregular at infinite argument. The
equations defining the instanton, by contrast, are highly nonlinear, and hence
this consideration does not apply. In fact, the asymptotics for large argument
uniquely determine the instanton solution. In this context, it is instructive
to recall St2015  that the uniqueness of the solution for a nonlinear
differential equation, determined by a given initial condition or asymptotic
behavior, constitutes a pivotal factor in the emergence of a range of complex
phenomena, including chaos. This uniqueness leads to behaviors which are
sensitive to the small variations in the initial conditions Lo1963Nonperiodic
; Cl2021Nonlinear ; Gl1994Stability . Furthermore, this intrinsic uniqueness
in nonlinear differential equations is analogous to the sensitive dependency
on initial conditions observed in fluid dynamics, particularly in the
transition from laminar to turbulent flow, where even minor perturbations can
drastically alter the flow patterns, echoing the underlying chaotic dynamics
described in fluid mechanics research RuTa1971Turbulence .

The instanton goes to zero as $r\to\infty$, and so one can neglect the term
$[\xi^{(3)}_{\mathrm{cl}}(r)]^{3}\ll\xi^{(3)}_{\mathrm{cl}}(r)$ in a first
approximation. Then, one obtains the relation

$-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(3)}_{\mathrm{cl}}(r)-\frac{2}{r}\frac{\partial}{\partial
r}\xi^{(3)}_{\mathrm{cl}}(r)+\xi^{(3)}_{\mathrm{cl}}(r)\approx 0\,.$ (15)

Our ansatz

$\xi^{(3)}_{\mathrm{cl}}(r)=\frac{\exp(-r)}{r}\sum_{n=0}^{\infty}\frac{a_{n}}{r^{n}}\,,$
(16)

is a nonanalytic expansion in the variable $1/r$, for large $r$. In fact, in
The variable $\chi=1/r$, the expansion (16) constitutes a nonanalytic
(resurgent, transseries) expansion in the variables $\chi$ and $\exp(-1/\chi)$
(see Refs. Ph1989 ; CaNoPh1993 ; Bo1994 ), and $\chi=0$ becomes a singular
point of the differential equation. The importance of nonanalytic exponentials
(resurgent expansions) in the solution of differential equations with singular
points has been stressed in Ref. Bo1994 . The substitution
$\xi^{(3)}_{\mathrm{cl}}(r)=g(r)/r$ takes Eq. (15) into the form

$-\frac{\partial^{2}}{\partial r^{2}}g(r)+g(r)=0\,,\qquad
g(r)={\mathcal{C}}\,\exp(-r)\,,$ (17)

for which the solution regular at infinity is just $\exp(-r)$. Hence, the
ansatz (16) collapses to a single term, with ${\mathcal{C}}$ being an overall
constant, and reads

$\xi^{(3)}_{\mathrm{cl}}(r)={\mathcal{C}}\,\frac{\exp(-r)}{r}\,,$ (18)

for which the approximate equality in Eq. (15) becomes an exact equality.
Here, ${\mathcal{C}}$ is a coefficient which can be determined numerically. A
60-figure result for ${\mathcal{C}}$ is

$\displaystyle{\mathcal{C}}=$
$\displaystyle\;2.712\,808\,360\,940\,844\,770\,465\,994\,573\,657$
$\displaystyle\;\phantom{2.}808\,840\,265\,350\,950\,750\,281\,746\,458\,229(1)\,.$
(19)

One can now approximate, in Eq. (14),

$-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(3)}_{\mathrm{cl}}(r)-\frac{2}{r}\frac{\partial}{\partial
r}\xi^{(3)}_{\mathrm{cl}}(r)+\xi^{(3)}_{\mathrm{cl}}(r)-\xi^{(3)}_{\mathrm{cl}}(r)^{3}\\\
\approx-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(3)}_{\mathrm{cl}}(r)-\frac{2}{r}\frac{\partial}{\partial
r}\xi^{(3)}_{\mathrm{cl}}(r)+\xi^{(3)}_{\mathrm{cl}}(r)-\frac{\exp(-3r)}{r^{3}}=0\,.$
(20)

The structure of this equation justifies the ansatz

$\xi^{(3)}_{\mathrm{cl}}(r)={\mathcal{C}}\,\frac{\exp(-r)}{r}+\frac{\exp(-3r)}{r}\,\sum_{n=0}^{\infty}\frac{b_{n}}{r^{n}}+\dots\,.$
(21)

Matching of the $b_{n}$ coefficients leads to the result,

$\xi^{(3)}_{\mathrm{cl}}(r)={\mathcal{C}}\,\frac{\exp(-r)}{r}-{\mathcal{C}}^{3}\,\frac{\exp(-3r)}{8r^{3}}\,\left(1-\frac{3}{2r}+\frac{21}{8\,r^{2}}-\frac{45}{8r^{3}}+\frac{465}{32\,r^{4}}-\frac{2835}{64\,r^{5}}+\frac{40005}{256\,r^{6}}+{\mathcal{O}}(r^{-7})\right)+\dots\,.$
(22)

This expression, cubed, generates terms proportional to
$\left[\frac{\exp(-r)}{r}+\frac{\exp(-3r)}{r^{3}}\right]^{3}\to\left(\frac{\exp(-r)}{r}\right)^{2}\times\frac{\exp(-3r)}{r^{3}}=\frac{\exp(-5r)}{r^{5}}$.
Now, we enter with the ansatz

$\displaystyle\xi^{(3)}_{\mathrm{cl}}(r)=$
$\displaystyle\;{\mathcal{C}}\,\frac{\exp(-r)}{r}-{\mathcal{C}}^{3}\,\frac{\exp(-3r)}{8r^{3}}\,\left(1-\frac{3}{2r}+\frac{21}{8\,r^{2}}-\frac{45}{8r^{3}}+\frac{465}{32\,r^{4}}-\frac{2835}{64\,r^{5}}+\frac{40005}{256\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;+\frac{\exp(-5r)}{r}\,\sum_{n=0}^{\infty}\frac{c_{n}}{r^{n}}+\dots\,,$
(23)

again into Eq. (14), match the coefficients $c_{n}$, and find

$\displaystyle\xi^{(3)}_{\mathrm{cl}}(r)=$
$\displaystyle\;{\mathcal{C}}\,\frac{\exp(-r)}{r}-{\mathcal{C}}^{3}\,\frac{\exp(-3r)}{8r^{3}}\,\left(1-\frac{3}{2r}+\frac{21}{8\,r^{2}}-\frac{45}{8r^{3}}+\frac{465}{32\,r^{4}}-\frac{2835}{64\,r^{5}}+\frac{40005}{256\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;+{\mathcal{C}}^{5}\,\frac{\exp(-5r)}{64r^{5}}\,\left(1-\frac{19}{6r}+\frac{151}{18\,r^{2}}-\frac{815}{36r^{3}}+\frac{56921}{864\,r^{4}}-\frac{1094215}{5184\,r^{5}}+\frac{2592553}{3456\,r^{6}}+{\mathcal{O}}(r^{-7})\right)\,.$
(24)

Finally, with the contribution of order $\exp(-7r)$ included, we have

$\displaystyle\xi^{(3)}_{\mathrm{cl}}(r)=$
$\displaystyle\;{\mathcal{C}}\,\frac{\exp(-r)}{r}-{\mathcal{C}}^{3}\,\frac{\exp(-3r)}{8r^{3}}\,\left(1-\frac{3}{2r}+\frac{21}{8\,r^{2}}-\frac{45}{8r^{3}}+\frac{465}{32\,r^{4}}-\frac{2835}{64\,r^{5}}+\frac{40005}{256\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;+{\mathcal{C}}^{5}\,\frac{\exp(-5r)}{64r^{5}}\,\left(1-\frac{19}{6r}+\frac{151}{18\,r^{2}}-\frac{815}{36r^{3}}+\frac{56921}{864\,r^{4}}-\frac{1094215}{5184\,r^{5}}+\frac{2592553}{3456\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;-{\mathcal{C}}^{7}\,\frac{\exp(-7r)}{512r^{7}}\,\left(1-\frac{29}{6r}+\frac{271}{16\,r^{2}}-\frac{3943}{72r^{3}}+\frac{614143}{3456\,r^{4}}-\frac{8322275}{13824\,r^{5}}+\frac{80215771}{36864\,r^{6}}+{\mathcal{O}}(r^{-7})\right)\,.$
(25)

For the term proportional to $\exp(-3r)$, we find the compact formula,

$-{\mathcal{C}}^{3}\,\frac{\exp(-3r)}{8r^{3}}\,\left(1-\frac{3}{2r}+\frac{21}{8\,r^{2}}-\frac{45}{8r^{3}}+\frac{465}{32\,r^{4}}+{\mathcal{O}}(r^{-5})\right)=-{\mathcal{C}}^{3}\,\frac{\exp(-r)}{r}\,\left(\textrm{Ei}(-2r)-2\exp(2r)\textrm{Ei}(-4r)\right)\,\,,$
(26)

where $\textrm{Ei}(r)$ the exponential integral function, but we were unable
to find general expressions for the terms in the series multiplying the
exponential factors $\exp(-5r)$ and $\exp(-7r)$.

Transseries in the variable $\chi=1/r$ have been encountered in the study of
anharmonic oscillators ZJJe2004i ; ZJJe2004ii ; JeSuZJ2010 ; JeZJ2011 . They
have also been investigated mathematically Ph1989 ; CaNoPh1993 ; Bo1994 . We
see that only “odd-transseries” orders of the form $\exp[-(2n+1)/\chi]$
contribute. The “one-transseries” contribution to the instanton wave function
is found to read as ${\mathcal{C}}\,\chi\,\exp(-1/\chi)$, without correction
terms. The expansion (III.1) shows that the large-argument expansion of the
instanton wave function $\xi^{(3)}_{\mathrm{cl}}(r)$ is determined by a single
constant ${\mathcal{C}}$, whose numerical value is given in Eq. (III.1).

An inspection shows that the perturbative coefficients in the variable
$\chi=1/r$ grow factorially. In order to match the resurgent expansion for
large argument with the Taylor expansion for small argument $r$, we have
calculated terms up to the 13-instanton contribution and summed the series,
starting from large values of $r$, down to $r=3$,

$\xi^{(3)}_{\mathrm{cl}}(r=3.0)=0.045\,013\,219\,071\dots\,,$ (27)

where a more precise result for $\xi^{(3)}_{\mathrm{cl}}$ at the matching
point is given in Eq. (63). In the summation process, we have used
$[40/40]$–Padé approximations CaEtAl2007  in order to sum the divergent
perturbative series decorating the instanton contributions of order
$\exp(-nr)$, where $n$ is an odd integer, and $77$-order Weniger–Levin
transformations We1989 , in order to verify the accuracy of the result (27) in
the intermediate region near $r\approx 3.0$.

###  III.2 Small Argument

We recall the equation fulfilled by the instanton [see Eq. (14)]

$-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(3)}_{\mathrm{cl}}(r)-\frac{2}{r}\frac{\partial}{\partial
r}\xi^{(3)}_{\mathrm{cl}}(r)+\xi^{(3)}_{\mathrm{cl}}(r)-\xi^{(3)}_{\mathrm{cl}}(r)^{3}=0\,.$
(28)

Plugging in a polynomial ansatz into Eq. (28), with

$\xi^{(3)}_{\mathrm{cl}}(0)={\mathcal{F}}=4.337\,387\,679\,976\dots$ (29)

[see also Eq. (62)], one finds

$\displaystyle\xi^{(3)}_{\mathrm{cl}}(r)=$
$\displaystyle\;{\mathcal{F}}+\frac{1}{6}\,({\mathcal{F}}-{\mathcal{F}}^{3})\,r^{2}+\frac{1}{120}({\mathcal{F}}-4\,{\mathcal{F}}^{3}+3\,{\mathcal{F}}^{5})\,r^{4}$
$\displaystyle\;+\frac{{\mathcal{F}}-17\,{\mathcal{F}}^{3}+35\,{\mathcal{F}}^{5}-19\,{\mathcal{F}}^{7}}{5040}\,r^{4}+{\mathcal{O}}(r^{6})\,.$
(30)

Only even powers of $r$ contribute. Using computer algebra Wo1999 , one can
easily determine all coefficients up to order $r^{80}$, and write

$\xi^{(3)}_{\mathrm{cl}}(r)=\sum_{n=0}^{\infty}a_{2n}\,r^{2n}\,.$ (31)

A closer inspection reveals that the series of the $a_{2n}$ is factorially
divergent and alternating. Still, one can use summation techniques to confirm
the result (27) at the matching point $\xi^{(3)}_{\mathrm{cl}}(r=3.0)$ [see
also Eq. (63)]. In the summation process, we have used $[62/62]$–Padé
approximations CaEtAl2007  in order to sum the divergent perturbative series
at $r=3$, or alternatively $117$-order Weniger–Levin transformations We1989 .
This leads to the desired accuracy in the intermediate region. Improvements of
the numerical accuracy are possible when one expands the instanton about
additional reference points (e.g., where $r$ assumes the value of a small
integer) and concatenates the expansions in regions of overlap.

##  IV Two–Dimensional Instanton

###  IV.1 Large Argument

In two dimensions, the instanton is equally radially symmetric (see Fig. 2),
and we can write
$\xi^{(2)}_{\mathrm{cl}}(\vec{x})=\xi^{(2)}_{\mathrm{cl}}(r)$. The equation
fulfilled by the instanton is

$-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(2)}_{\mathrm{cl}}(r)-\frac{1}{r}\frac{\partial}{\partial
r}\xi^{(2)}_{\mathrm{cl}}(r)+\xi^{(2)}_{\mathrm{cl}}(r)-\xi^{(2)}_{\mathrm{cl}}(r)^{3}=0\,.$
(32)

Just like in the three-dimensional case (see Sec. III.1), the instanton goes
exponentially to zero as $r\to\infty$, and so one can neglect the term
$[\xi^{(2)}_{\mathrm{cl}}(r)]^{3}$ in a first approximation. Then, one obtains
the relation [see also Eq. (15)]

$-\frac{\partial^{2}}{\partial
r^{2}}\xi^{(2)}_{\mathrm{cl}}(r)-\frac{1}{r}\frac{\partial}{\partial
r}\xi^{(2)}_{\mathrm{cl}}(r)+\xi^{(2)}_{\mathrm{cl}}(r)\approx 0\,.$ (33)

By a similar analysis as described for the three-dimensional case, one obtains

$\displaystyle\xi^{(2)}_{\mathrm{cl}}(r)=$
$\displaystyle\;{\mathcal{D}}\,\frac{\exp(-r)}{r^{1/2}}\,\left(1-\frac{1}{8r}+\frac{9}{128\,r^{2}}-\frac{75}{1024\,r^{3}}+\frac{3675}{32768\,r^{4}}-\frac{59535}{262144\,r^{5}}+\frac{2401245}{4194304\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;-{\mathcal{D}}^{3}\,\frac{\exp(-3r)}{8r^{3/2}}\,\left(1-\frac{9}{8r}+\frac{213}{128\,r^{2}}-\frac{3215}{1024r^{3}}+\frac{238563}{32768\,r^{4}}-\frac{5283711}{262144\,r^{5}}+\frac{273186513}{4194304\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;+{\mathcal{D}}^{5}\,\frac{\exp(-5r)}{64r^{5/2}}\,\left(1-\frac{53}{24r}+\frac{589}{128\,r^{2}}-\frac{96193}{9216r^{3}}+\frac{23551553}{884736\,r^{4}}-\frac{544320827}{7077888\,r^{5}}+\frac{85429251785}{339738624\,r^{6}}+{\mathcal{O}}(r^{-7})\right)$
$\displaystyle\;-{\mathcal{D}}^{7}\,\frac{\exp(-7r)}{512r^{7/2}}\,\left(1-\frac{79}{24r}+\frac{10037}{1152\,r^{2}}-\frac{629833}{27648r^{3}}+\frac{55541473}{884736\,r^{4}}-\frac{1326870785}{7077888\,r^{5}}+\frac{23212812833}{37748736\,r^{6}}+{\mathcal{O}}(r^{-7})\right)\,.$
(34)

Just as in the three-dimensional case, we can find a compact expression for
the leading term, which for $D=2$ is of order $\exp(-r)/\sqrt{r}$,

${\mathcal{D}}\,\frac{\exp(-r)}{r^{1/2}}\,\left(1-\frac{1}{8r}+\frac{9}{128\,r^{2}}-\frac{75}{1024\,r^{3}}+\frac{3675}{32768\,r^{4}}+{\mathcal{O}}(r^{-5})\right)={\mathrm{i}}\,{\mathcal{D}}\,H^{(1)}_{0}({\mathrm{i}}r)\,.$
(35)

Here, $H^{(1)}_{\alpha}(r)$ is the Hankel function AbSt1972  of the first kind
of order $\alpha$. However, we were unable to find general expressions for the
terms in the series multiplying the exponential factors $\exp(-3r)$,
$\exp(-5r)$ and $\exp(-7r)$.

For $D=2$, one finds for the ${\mathcal{D}}$ coefficient the following
60-figure result [cf. Eq. (III.1)]

$\displaystyle{\mathcal{D}}=$
$\displaystyle\;3.518\,062\,198\,025\,031\,180\,209\,129\,887\,741$
$\displaystyle\;\phantom{3.}356\,933\,215\,813\,390\,992\,384\,663\,366\,560(1)\,.$
(36)

Furthermore, it is clear that the perturbative coefficients in the variable
$\chi=1/r$ grow very fast, and in fact, they grow factorially. We have
calculated terms up to the contribution of order $\exp(-13\,r)$ and summed the
series, starting from large values of $r$, down to $r=3$, with the result

$\xi^{(2)}_{\mathrm{cl}}(r=3.0)=0.097\,418\,218\,653\dots\,,$ (37)

where the most precise result for the value of
$\xi^{(2)}_{\mathrm{cl}}(r=3.0)$ at the matching point is given in Eq. (63).
Just as in the three-dimensional case, in the summation process, we have used
$[40/40]$–Padé approximations in order to sum the divergent perturbative
series decorating the instanton contributions of order $\exp(-nr)$, where
$n\leq 13$ is an odd integer, or $77$-order Weniger–Levin transformations
We1989 , in order to achieve the accuracy in the intermediate region.

Figure 2:  The two-dimensional instanton
$\xi_{\mathrm{cl}}(r)=\xi^{(2)}_{\mathrm{cl}}(r)$ is radially symmetric Its
value at the origin is
$\xi^{(2)}_{\mathrm{cl}}(0)={\mathcal{G}}=2.206\,200\,864\,650\ldots$,
according to Eqs. (39) and (62).

###  IV.2 Small Argument

We now need to repeat the analysis from Sec. III.2, for the two-dimensional
case. Plugging in a polynomial ansatz into Eq. (32),

$\left(-\frac{\partial^{2}}{\partial
r^{2}}-\frac{1}{r}\frac{\partial}{\partial
r}+1-\xi^{(2)}_{\mathrm{cl}}(r)^{2}\right)\,\xi^{(2)}_{\mathrm{cl}}(r)=0$ (38)

with

$\xi^{(2)}_{\mathrm{cl}}(0)={\mathcal{G}}=2.206\,200\,864\,650\ldots\,,$ (39)

[see also Eq. (62)], one finds

$\displaystyle\xi^{(2)}_{\mathrm{cl}}(r)=$
$\displaystyle\;{\mathcal{G}}+\frac{1}{4}\,({\mathcal{G}}-{\mathcal{G}}^{3})\,r^{2}+\frac{1}{64}({\mathcal{G}}-4\,{\mathcal{G}}^{3}+3\,{\mathcal{G}}^{5})\,r^{4}$
$\displaystyle\;+\frac{{\mathcal{G}}-19\,{\mathcal{G}}^{3}+39\,{\mathcal{G}}^{5}-21\,{\mathcal{G}}^{7}}{2304}\,r^{4}+{\mathcal{O}}(r^{6})\,.$
(40)

Using computer algebra Wo1999 , one can easily determine all coefficients up
to order $r^{80}$, say, and write the divergent, asymptotic expansion

$\xi^{(2)}_{\mathrm{cl}}(r)=\sum_{n=0}^{\infty}a_{2n}\,r^{2n}\,.$ (41)

Here, too, a closer inspection reveals that the series is divergent, because
of factorial divergence of the magnitude of the (alternating-in-sign) power
series about $r=0$. One confirms the result given in Eq. (37). In the
summation process, we have used $[62/62]$–Padé approximations in order to sum
the divergent perturbative series at $r=3$, or alternatively $117$-order
Weniger–Levin transformations We1989 . This yields the desired accuracy in the
intermediate region.

##  V Instantons and Virial Theorems

###  V.1 Derivation of the Virial Theorems

For the following investigations, it is instrumental to derive several
relations fulfilled by the instanton; these constitute virial theorems. We
consider the general action

${\mathcal{S}}(\phi)=\int{\mathrm{d}}^{D}x\left[\frac{1}{2}\sum_{\mu}\left(\vec{\nabla}\phi(\vec{x})\right)^{2}+{\mathcal{V}}\bigl{(}\phi(\vec{x})\bigr{)}\right]\,,$
(42)

where in the case of the action (1), one has

${\mathcal{V}}\bigl{(}\phi(\vec{x})\bigr{)}=\frac{1}{2}\phi(\vec{x})+\frac{1}{4}g\,\phi^{4}(\vec{x})\,.$
(43)

We assume that the field equation, obtained by variational calculus, has a
finite action solution $\phi_{{\mathrm{cl}}}(\vec{x})$. If the action
${\mathcal{S}}(\phi_{{\mathrm{cl}}})$ is finite so is the action
${\mathcal{S}}(\phi_{{\mathrm{cl}}},\lambda)$, obtained from
${\mathcal{S}}(\phi_{{\mathrm{cl}}})={\mathcal{S}}(\phi_{{\mathrm{cl}}},\lambda=1)$
by the replacement $\phi(\vec{x})\to\phi_{{\mathrm{cl}}}(\lambda\,\vec{x})$.
If we change variables in the action, setting
$\lambda\vec{x}=\vec{x}^{\prime}$, we find

${\mathcal{S}}[\phi_{{\mathrm{cl}}},\lambda]=\lambda^{2-D}\int{\mathrm{d}}^{D}x\,\left[\frac{1}{2}\,\sum_{\mu}\left(\vec{\nabla}\phi_{{\mathrm{cl}}}(\vec{x})\right)^{2}\right]\\\
+\lambda^{-D}\,\int{\mathrm{d}}^{D}x\,{\mathcal{V}}\bigl{(}\phi_{{\mathrm{cl}}}(\vec{x})\bigr{)}\,.$
(44)

Because $\phi_{{\mathrm{cl}}}(x)$ satisfies the field equation, the variation
of the action vanishes for $\lambda=1$, i.e., we have the equation
$\left.\frac{{\mathrm{d}}}{{\mathrm{d}}\lambda}{\mathcal{S}}[\phi_{{\mathrm{cl}}},\lambda]\right|_{\lambda=1}=0$.

$\int{\mathrm{d}}^{D}x^{\prime}\,\left[\frac{D-2}{2}\left(\vec{\nabla}\phi_{{\mathrm{cl}}}(\vec{x}^{\prime})\right)^{2}+D\,{\mathcal{V}}\bigl{(}\phi_{{\mathrm{cl}}}(\vec{x}^{\prime})\bigr{)}\right]=0\,.$
(45)

This relation allows us to express the kinetic term [integral of
$(\vec{\nabla}\phi_{{\mathrm{cl}}})^{2}$] in terms of the potential term
(integral of ${\mathcal{V}}(\phi_{{\mathrm{cl}}})$ and vice versa. The
classical action ${\mathcal{S}}[\phi_{{\mathrm{cl}}}]$ can thus be expressed
in terms of the kinetic term only:

${\mathcal{S}}[\phi_{{\mathrm{cl}}}]=\frac{1}{D}\;\int{\mathrm{d}}^{D}x\,\left(\vec{\nabla}\phi_{{\mathrm{cl}}}(\vec{x})\right)^{2}\,,$
(46)

a form that shows that ${\mathcal{S}}(\phi_{{\mathrm{cl}}})$ is always
positive. The second derivative of
${\mathcal{S}}(\phi_{{\mathrm{cl}}},\lambda)$ reads as

$\left.\frac{{\mathrm{d}}^{2}}{\left({\mathrm{d}}\lambda\right)^{2}}{\mathcal{S}}[\phi_{c},\lambda]\right|_{\lambda=1}=\left(2-D\right)\,\int\left[\partial_{\mu}\phi_{c}(x)\right]^{2}{\mathrm{d}}^{D}x\,.$
(47)

For $D\geq 2$, this result shows that the solution is not a local minimum of
the action and, thus, the so-called longitudinal fluctuation operator ${{\bf
M}}_{\mathrm{L}}$, defined as

${{\bf
M}}_{\mathrm{L}}(\vec{x},\vec{x}^{\prime})=\left.\frac{\delta^{2}{\mathcal{S}}}{\delta\phi(\vec{x})\,\delta\phi(\vec{x}^{\prime})}\right|_{\phi=\phi_{\mathrm{cl}}}\,,$
(48)

has at least one negative eigenvalue JeZJ2011 ; GiEtAl2020 .

In the example of potentials of special form

${\mathcal{V}}(\phi)=\frac{1}{2}\,\phi^{2}+\frac{1}{4}\,g\,\phi^{M}\,,$ (49)

one can derive an additional relation. If the action
${\mathcal{S}}[\phi_{{\mathrm{cl}}}]$ is finite, so is the following action
obtained by the replacement
$\phi_{\mathrm{cl}}\to\lambda\,\phi_{\mathrm{cl}}$,

${\mathcal{S}}[\lambda\phi_{{\mathrm{cl}}}]=\lambda^{2}\,\int{\mathrm{d}}^{D}x\,\frac{1}{2}\left[\left(\vec{\nabla}\phi_{{\mathrm{cl}}}(\vec{x})\right)^{2}+\phi_{{\mathrm{cl}}}^{2}(\vec{x})\right]\\\
+\frac{1}{4}\,\lambda^{M}\,g\int{\mathrm{d}}^{D}x\,\phi_{{\mathrm{cl}}}^{M}(\vec{x})\,.$
(50)

Again, if $\phi_{{\mathrm{cl}}}$ is the instanton solution, then the
derivative with respect to $\lambda$ must vanish for $\lambda=1$. One obtains
further relations, in addition to (46),

${\mathcal{S}}(\phi_{{\mathrm{cl}}})=-\frac{g}{8}(M-2)\,\int{\mathrm{d}}^{D}x\,\phi^{M}_{{\mathrm{cl}}}(\vec{x})\\\
\mathop{=}^{M=4}-\frac{g}{4}\int{\mathrm{d}}^{D}x\,\phi^{4}_{{\mathrm{cl}}}(\vec{x})\qquad(M=4)\,.$
(51)

This relation is consistent with the fact that the instanton exists only for
negative $g$. Thus, one can express the instanton action as follows,

${\mathcal{S}}(\phi_{{\mathrm{cl}}})=\frac{M-2}{2D-M(D-2)}\,\int{\mathrm{d}}^{D}x\,\phi_{{\mathrm{cl}}}^{2}(x)\\\
\mathop{=}^{M=4}\frac{1}{4-D}\,\int{\mathrm{d}}^{D}x\,\phi_{{\mathrm{cl}}}^{2}(x)\,.$
(52)

We have used Eqs. (46) and (51). In particular, Eqs. (51) and (52) are
consistent only if the denominator in the expression on the right-hand side of
Eq. (52) is positive, which implies

$M\leq\frac{2D}{D-2}\,,$ (53)

and, therefore, the field theory must be super-renormalizable or at least
renormalizable. At the special dimension $D=2M/(M-2)$, where the theory is
renormalizable, one finds the paradoxical result
$\int{\mathrm{d}}^{D}x\,\phi_{{\mathrm{cl}}}^{2}(x)=0$. This implies that only
the massless equation (where the coefficient of $\phi^{2}$ in the original
action vanishes), has instanton solutions (see also Appendix A). Finally, one
verifies that the second derivative at $\lambda=1$ is negative, confirming the
existence of a negative eigenvalue of ${\bf M}_{\mathrm{L}}$ for all
dimensions.

###  V.2 Summary of the Virial Theorems

We summarize. From Eqs. (46), (51) and (52), we have for the $\phi^{4}$ theory
with the action (1),

${\mathcal{S}}(\phi_{{\mathrm{cl}}})=\frac{1}{D}\;\int{\mathrm{d}}^{D}x\,\left(\vec{\nabla}\phi_{{\mathrm{cl}}}(\vec{x})\right)^{2}=-\frac{g}{4}\int{\mathrm{d}}^{D}x\,\phi^{4}_{{\mathrm{cl}}}(\vec{x})\\\
=\frac{1}{4-D}\,\int{\mathrm{d}}^{D}x\,\phi_{{\mathrm{cl}}}^{2}(\vec{x})\,.$
(54)

With the scaling given by Eqs. (4), the action of the instanton becomes
($g<0$),

$\displaystyle\phi_{{\mathrm{cl}}}(\vec{x})=$
$\displaystyle\;\sqrt{-\frac{1}{g}}\,\xi_{{\mathrm{cl}}}(r)\,,\quad
r=|\vec{x}|\,,\quad{\mathcal{S}}(\phi_{{\mathrm{cl}}})=-\frac{A}{g}>0\,.$
(55a) We have three equivalent representations of the action $A$,
$A=\frac{1}{D}\,\int{\mathrm{d}}^{D}x\left(\vec{\nabla}\xi_{{\mathrm{cl}}}(r)\right)^{2}=\frac{1}{4}\,\int{\mathrm{d}}^{D}x\,\xi^{4}_{{\mathrm{cl}}}(r)\\\
=\frac{1}{4-D}\,\int{\mathrm{d}}^{D}x\,\xi^{2}_{{\mathrm{cl}}}(r)\,.$ (55b)

Using the radial symmetry of the solution, we can establish that

$A=\frac{\Omega_{D}}{D}\;\int_{0}^{\infty}{\mathrm{d}}r\,r^{D-1}\,[\xi^{\prime}_{{\mathrm{cl}}}(r)]^{2}\,,$
(56)

where $\Omega_{D}=2\pi^{D/2}/\Gamma({\textstyle{\frac{1}{2}}}D)$ is the
generalized surface of the $(D-1)$-dimensional unit sphere embedded in
$D$-dimensional space.

###  V.3 Asymptotic Behavior

The asymptotic behavior of the radial instanton equation

$\left[-\left(\frac{{\mathrm{d}}}{{\mathrm{d}}r}\right)^{2}-\frac{D-1}{r}\frac{{\mathrm{d}}}{{\mathrm{d}}r}+1\right]\xi_{{\mathrm{cl}}}(r)-\xi_{{\mathrm{cl}}}^{3}(r)=0$
(57)

is of interest for large $r$. Asymptotically, one can show that, for
$r\to\infty$,

$\xi_{\mathrm{cl}}(r)={C\sqrt{\frac{2}{\pi}}}\;r^{1-D/2}K_{D/2-1}(r)+{\mathcal{O}}({\mathrm{e}}^{-3r})$
(58)

where $K_{\nu}$ is a modified Bessel function of the second kind normalized
such that $K_{\nu}(r)\sim\sqrt{\pi/(2r)}\;{\mathrm{e}}^{-r}$ for large $r$.
This would mean that

$\xi^{(D)}_{\mathrm{cl}}(r)\propto{\sqrt{\frac{2}{\pi}}}\;r^{1-D/2}\sqrt{\frac{\pi}{2r}}\;{\mathrm{e}}^{-r}=\frac{{\mathrm{e}}^{-r}}{r^{D/2-1/2}}\,.$
(59)

Our formulas (III.1) and (IV.1) confirm this asymptotic behavior.

##  VI Instanton Integrals

We give a collection of numerical results for integrals of the instanton in
quartic theories, with enhanced accuracy. Our aim is to give, for 2D and 3D,
results approaching the realm of applicability of the PSLQ algorithm FeBa1992
; BaPl1997 ; FeBaAr1999 ; BaBr2001  which is designed to search for analytic
expressions of integrals in terms of known constants. First, for completeness,
in one dimension, we recall that the instanton solution is JeSuZJ2009sigma

$\xi_{\mathrm{cl}}(r)=\frac{2}{\sqrt{\cosh(r)+1}}\,,\qquad\xi_{\mathrm{cl}}^{(D=1)}(r=0)=\sqrt{2}\,,$
(60)

where the one-dimensional action is

$S[\phi]=2\int_{0}^{\infty}{\mathrm{d}}r\,\left[\frac{1}{2}\,\left(\partial_{r}\phi(r)\right)^{2}+\frac{1}{2}\,\phi(r)^{2}+\frac{g}{4}\,\phi(r)^{4}\right]\,.$
(61)

The prefactor $2$ reflects on the angular factor
$\Omega_{D}=2\pi^{D/2}/\Gamma({\textstyle{\frac{1}{2}}}D)$ which evaluates to
$2$ for $D=1$. The prefactor matters because the instanton action is
normalized to $S[\phi_{\mathrm{cl}}(r)]=-A/g$, according to Eq. (9).
Analytically known instantons in a four-dimensional $\phi^{4}$ theory and in a
six-dimensional $\phi^{3}$ theory are given in Appendixes A and B,
respectively.

We have noticed that instanton solutions are determined by the value at the
origin, given in Eqs. (29) and (39). Numerically more accurate results can be
obtained via convergence acceleration algorithms, starting from a linear
lattice of radial coordinates JeGi2024 . Explicit results, to 78 decimal
figures, are given as follows,

$\displaystyle\xi_{\mathrm{cl}}^{(D=1)}(r=0)=$ $\displaystyle\;\sqrt{2}\,,$
(62a) $\displaystyle\xi^{(D=2)}_{\mathrm{cl}}(r=0)={\mathcal{G}}=$
$\displaystyle\;2.206\,200\,864\,650\,746\,074\,783\,634\,064\,578\,940\,196\,610$
$\displaystyle\;\phantom{2.}274\,520\,602\,192\,125\,757\,262\,456\,450\,184\,032\,518\,642(1)\,,$
(62b) $\displaystyle\xi_{\mathrm{cl}}^{(D=3)}(r=0)={\mathcal{F}}=$
$\displaystyle\;4.337\,387\,679\,976\,994\,356\,522\,109\,173\,841\,761\,465\,745$
$\displaystyle\;\phantom{4.}284\,082\,970\,785\,762\,761\,882\,558\,415\,947\,364\,399\,341(1)\,.$
(62c)

The values at the matching point $r=3$ are interesting for $D=2$ and $D=3$,

$\displaystyle\xi^{(D=2)}_{\mathrm{cl}}(r=3.0)=$
$\displaystyle\;0.097\,418\,218\,653\,642\,217\,741\,513\,024\,960\,584\,546\,095$
$\displaystyle\;\phantom{0.}157\,618\,276\,680\,772\,556\,932\,915\,093\,354\,850\,219\,044(1)\,.$
(63a) $\displaystyle\xi_{\mathrm{cl}}^{(D=3)}(r=3.0)=$
$\displaystyle\;0.045\,013\,219\,071\,010\,523\,997\,989\,047\,723\,112\,322\,109$
$\displaystyle\;\phantom{0.}014\,075\,244\,317\,789\,103\,014\,970\,206\,885\,072\,459\,490(1)\,.$
(63b)

Numerical results for the instanton action $A$ are

$\displaystyle A(D=1)=$ $\displaystyle\;4/3\,,$ (64a) $\displaystyle A(D=2)=$
$\displaystyle\;\phantom{1}5.850\,448\,262\,279\,826\,939\,326\,986\,338\,934\,453\,868\,499$
$\displaystyle\;\phantom{18.}064\,115\,959\,470\,267\,545\,644\,043\,014\,800\,957\,116\,007(1)\,.$
(64b) $\displaystyle A(D=3)=$
$\displaystyle\;18.897\,251\,302\,546\,190\,505\,297\,247\,993\,763\,227\,763\,807$
$\displaystyle\;\phantom{18.}178\,891\,316\,289\,857\,028\,151\,589\,245\,449\,182\,127\,167(1)\,.$
(64c)

These results are essential for large-order perturbation theory [see Eq.
(10)]. For what follows, it is convenient to introduce the notation

$I_{n}=\int{\mathrm{d}}^{D}x\,[\xi_{\mathrm{cl}}(r)]^{n}\,,\qquad
I_{2}=\frac{4-D}{4}\,I_{4}\,,\qquad
I_{4}=\frac{4}{D}\,\int{\mathrm{d}}^{D}x\,[\vec{\nabla}\xi_{\mathrm{cl}}(r)]^{2}\,,\qquad
A=\frac{1}{4}I_{4}\,,$ (65)

where we recall that the (generalized) surface area of the $(D-1)$-dimensional
unit sphere, embedded in $D$ dimensions, is
$\Omega_{D}=2\pi^{D/2}/\Gamma(D/2)$. Results for $I_{2}$ and $I_{4}$ follow
from the above results for the instanton action $A$. Results for $I_{3}$ and
$I_{6}$ are given as follows,

$\displaystyle I_{3}(D=1)=$ $\displaystyle\;\sqrt{2}\,\pi\,,\qquad
I_{6}(D=1)=128/15\,,$ (66a) $\displaystyle I_{3}(D=2)=$
$\displaystyle\;\phantom{6}15.109\,669\,726\,889\,195\,199\,613\,754\,001\,702\,125\,888\,865$
$\displaystyle\;\phantom{615.}874\,563\,104\,430\,202\,476\,703\,241\,753\,965\,063\,516\,331(1)\,.$
(66b) $\displaystyle I_{3}(D=3)=$
$\displaystyle\;\phantom{6}31.691\,521\,838\,323\,486\,451\,591\,907\,257\,120\,270\,170\,457$
$\displaystyle\;\phantom{631.}351\,985\,790\,745\,758\,122\,769\,708\,466\,866\,938\,412\,287(1)\,.$
(66c) $\displaystyle I_{6}(D=2)=$
$\displaystyle\;\phantom{6}71.080\,171\,542\,041\,917\,440\,792\,898\,285\,323\,353\,751\,992$
$\displaystyle\;\phantom{615.}546\,589\,483\,197\,579\,092\,397\,461\,668\,330\,495\,246\,091(1)\,.$
(66d) $\displaystyle I_{6}(D=3)=$
$\displaystyle\;659.868\,351\,544\,567\,238\,188\,639\,540\,582\,544\,719\,267\,515$
$\displaystyle\;\phantom{615.}748\,898\,263\,457\,303\,680\,826\,218\,470\,215\,371\,739\,493(1)\,.$
(66e)

##  VII Conclusions

We have analyzed the properties of instanton solutions in $O(N)$-symmetric
quartic field theories in $D=2$ and $D=3$ dimensions. The basic formulation
for the quartic instanton has been given in in Sec. II. We concentrate on the
three-dimensional instanton ($D=3$) in Sec. III, which is phenomenologically
the most interesting case. We derive asymptotic expansions for large argument
in the form of a transseries [Eq. (III.1)] and in the form of an asymptotic
power series [Eq. (III.2)] for small argument. The quartic instanton in $D=2$
is discussed in Sec. IV. Virial theorems are derived in Sec. V. Instanton
integrals are given in Sec. VI, with a precision approaching the realm of
applicability of the PSLQ algorithm FeBa1992 ; BaPl1997 ; FeBaAr1999 ;
BaBr2001  which is designed to search for analytic expressions of integrals in
terms of known constants such as the Euler constant $\gamma_{E}=0.57721\dots$,
various Riemann zeta functions, powers of $\pi$, and multiplicative
combinations of these constants. We can report that we have carried out a
limited set of searches with the same constants that were used in Eq. (A11) of
Ref. GiEtAl2020  without success. A more detailed search might constituent a
possible direction for the future.

In a quartic theory, the instanton solution exists only for negative $g$,
because the tunneling can proceed only through a barrier, and the latter
exists only negative coupling $g<0$. The imaginary part of the partition
function, and of correlation functions, obtained by expanding about the
instanton solution, is proportional (see Ref. GiEtAl2020 ) to
$\exp(-(-A/g))=\exp(A/g)$, where ${\mathcal{S}}[\phi_{\mathrm{cl}}]=-A/g$ and
$A$ is given for $D=2$ in Eq. (64) for $D=3$ in Eq. (64). The instanton action
$A$ universally enters large-order formulas for the perturbative coefficients
of Green functions [see Eq. (10)].

Our calculations suggest that instanton solutions in quartic theories cannot
be expressed in closed analytic form, except for the case $D=1$. We also note
that the general properties of the instanton in massive theories are valid
only for dimensions $D<4$. The dimension 4 is singular, as is evident, e.g.,
from Eq. (54). The cases of a massless quartic theory in four dimensions, and
of a cubic theory in six dimensions, are treated in Appendixes A and B.

Our investigations indicate that, with the exception of known singular cases
(see Appendixes A and B), instanton configurations cannot be calculated
analytically for general field theories, notably, for the two- and three-
dimensional $\phi^{4}$ theories. Nevertheless, in view of the nonlinear nature
of the defining differential equations, they admit transseries solutions and
asymptotic expansions which can be used for accurate numerical calculations.
These results are useful in expansions of partition and correlation functions
about instanton configurations GiEtAl2024ii .

Let us conclude by mentioning open problems, which could inspire future
research. The first of these concerns the possibility of analytic expressions
for the 78-figure results reported here for particular function values and
integrals of the two- and three-dimensional instantons, notably, those
communicated in Eqs. (62)—(66). As already mentioned, we have performed a
limited search based on the PSLQ algorithm FeBa1992 ; BaPl1997 ; FeBaAr1999 ;
BaBr2001  using various Riemann zeta functions, logarithms, and
polylogarithms, without finding suitable analytic formulas. Our inability to
find fully analytic representations is mirrored in recent, somewhat related
investigations BoBr2022 . The second open problem concerns the search for
closed-form representations of the higher-order terms in the transseries
solution (III.1) and (IV.1), generalizing the results given in Eqs. (26) and
(35) to higher orders of the exponential factor $\exp(-r)$.

## Acknowledgements

The authors acknowledge insightful conversations with Professor Giorgio
Parisi. This work has been supported by the National Science Foundation (Grant
PHY–2110294) and by the Swedish Research Council (Grant No. 638-2013-9243).
Support from the Simons Foundation (Grant 454949) also is gratefully
acknowledged. E.M.M. acknowledges the MUR-Prin 2022 funding Prot. 20229T9EAT,
financed by the European Union (Next Generation EU).

##  Appendix A Four–Dimensional Massless Quartic Theory

The existence of instantons in the renormalizable (but not super-
renormalizable) quartic theory in four dimensions has been anticipated in Sec.
V.1. We consider the action

${\mathcal{S}}[\phi]=\int{\mathrm{d}}^{4}x\left[\frac{1}{2}\left(\vec{\nabla}\phi\right)^{2}+\frac{1}{4}\,g\,\phi^{4}\right]\,.$
(67)

The corresponding field equation is
$-\vec{\nabla}^{2}\phi_{\mathrm{cl}}(\vec{x})+g\,\phi_{\mathrm{cl}}^{3}(\vec{x})=0$.
We know that the solution of minimal action is spherically symmetric, thus we
set

$\phi_{\mathrm{cl}}(x)=\sqrt{-\frac{1}{g}}\,\xi_{\mathrm{cl}}(r)\,,$ (68)

where $r=|\vec{x}|$. We then obtain a differential equation
$\left[\left(\frac{{\mathrm{d}}}{{\mathrm{d}}r}\right)^{2}+\frac{3}{r}\frac{{\mathrm{d}}}{{\mathrm{d}}r}+\xi_{\mathrm{cl}}^{2}(r)\right]\xi_{\mathrm{cl}}(r)=0$.
The solution is

$\xi_{\mathrm{cl}}(r)=\frac{2\sqrt{2}}{1+r^{2}}$ (69)

The instanton action is

$S[\phi_{\mathrm{cl}}]=-\frac{A}{g}\,,\qquad A=\frac{8\pi^{2}}{3}\,.$ (70)

The instanton integrals,
$I_{n}=\int{\mathrm{d}}^{4}x\,\xi_{\mathrm{cl}}(r)^{n}$, for $n=3,4,6$, are
$I_{3}=8\sqrt{2}\,\pi^{2}$, $I_{4}=\frac{32\,\pi^{2}}{3}$,
$I_{6}=\frac{128\,\pi^{2}}{5}$.

##  Appendix B Six–Dimensional Massless Cubic Theory

Another example of the existence of analytically calculable instantons is the
six-dimensional massless cubic theory BoDuMe2021 . We consider the action

${\mathcal{S}}[\phi]=\int{\mathrm{d}}^{6}x\left[\frac{1}{2}\left(\vec{\nabla}\phi\right)^{2}+\frac{1}{3}\,g\,\phi^{3}\right]\,.$
(71)

The corresponding field equation for the instanton is
$-\vec{\nabla}^{2}\phi_{\mathrm{cl}}(\vec{x})+g\,\phi_{\mathrm{cl}}^{2}(\vec{x})=0$.
We know that the solution of minimal action is spherically symmetric, thus we
set

$\phi_{\mathrm{cl}}(x)=\frac{1}{g}\,\xi_{\mathrm{cl}}(r)\,,$ (72)

where $r=|\vec{x}|$ and we observe that the instanton exists for positive $g$.
We then obtain a differential equation
$\left[\left(\frac{{\mathrm{d}}}{{\mathrm{d}}r}\right)^{2}+\frac{5}{r}\frac{{\mathrm{d}}}{{\mathrm{d}}r}+\xi_{\mathrm{cl}}(r)\right]\xi_{\mathrm{cl}}(r)=0$.
The solution is

$\xi_{\mathrm{cl}}(r)=-\frac{24}{(1+r^{2})^{2}}$ (73)

The instanton action is

$S[\phi_{\mathrm{cl}}]=\frac{A}{g^{2}}\,,\qquad A=\frac{192\pi^{3}}{5}\,.$
(74)

The instanton integrals,
$I_{n}=\int{\mathrm{d}}^{4}x\,\xi_{\mathrm{cl}}(r)^{n}$, for $n=3,4,6$, are
$I_{3}=-\frac{1152\,\pi^{3}}{5}$, $I_{4}=\frac{55\,296\,\pi^{3}}{35}$,
$I_{6}=\frac{10\,616\,832\,\pi^{3}}{55}$.

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