# Finite conformal quantum gravity and
spacetime singularities

Leonardo Modesto and Lesław Rachwał  Department of Physics, Southern
University of Science and Technology,
Shenzhen 518055, P. R. China lmodesto<EMAIL_ADDRESS>
###### Abstract

We show that a class of finite quantum non-local gravitational theories is
conformally invariant at classical as well as at quantum level. This is
actually a range of conformal anomaly-free theories in the spontaneously
broken phase of the Weyl symmetry. At classical level we show how the Weyl
conformal invariance is able to tame all the spacetime singularities that
plague not only Einstein gravity, but also local and weakly non-local higher
derivative theories. The latter statement is proved by a singularity theorem
that applies to a large class of weakly non-local theories. Therefore, we are
entitled to look for a solution of the spacetime singularity puzzle in a
missed symmetry of nature, namely the Weyl conformal symmetry. Following the
seminal paper by Narlikar and Kembhavi, we provide an explicit construction of
singularity-free black hole exact solutions in a class of conformally
invariant theories.

##  1 Introduction

There is one old symmetry that has always played a very crucial role in
fundamental physics. This is the conformal symmetry, which is enjoyed by
physical systems whose description does not depend on any characteristic
scale. Historically this symmetry dates back to Hermann Weyl who discovered
the concept of local gauge symmetries. At first, the gauge symmetries were
applied to the local change of scale of the length measuring rulers. Since the
measurements of space (and time) distances are the operational basis of
general relativity (GR), we can understand why the conformal symmetry as a
gauge symmetry first appeared in the gravitational setup. Only later the ideas
of gauge symmetries were used in the non-gravitational setting of matter field
theories, namely in the particle physics and condensed matter. Moreover, the
conformal symmetry was found back in gauge theories too but the necessary
condition here was to be in four spacetime dimensions. Indeed, the Maxwell’s
and Yang-Mills theories revealed to be exactly conformally invariant at the
classical level. In the classical notion of conformal invariance the spacetime
metric undergoes a rescaling by some always positive scalar function
eventually dependent on the spacetime points. This is also the meaning of
conformal symmetry used in classical GR, where it has been very useful to
study the black holes’ near-horizon geometry, the FRW cosmologies, and it was
also a crucial element for the construction of the Newman-Penrose formalism.

There is also a different meaning of the same symmetry having to do with some
properties of correlation functions on a given spacetime background. First of
all, the change of scale on flat spacetime can be very easily transmitted to a
change of coordinates and then the symmetry shows up in a somehow global
disguise. One can then study its extension to the full conformal group, which
on a flat 4-dimensional Minkowski spacetime acts as a realization of the
SO(2,3) group. This global symmetry initially finds its manifestation in some
conformally invariant classical field theory models, but its true power is in
constraining the correlation functions at the quantum level. Very soon, it was
proven that the requirement of scale invariance at the quantum level implies
that the beta functions of the couplings must exactly vanish. Moreover, the
scale-invariance must be promoted to conformal symmetry to have a fully
consistent field theory. The correlators of such very special theory show
amazing symmetries that are described by the conformal algebra and the
theories themselves are called conformal field theories (CFT’s). When the beta
functions are zero, then also the UV-divergences of the theory are absent, so
the theory in question is UV-finite. There are very special examples of such
theories and one of the most (super-) symmetric example is ${\cal N}=4$ Super-
Yang-Mills theory in four dimensions. This line of development found its
culmination in the AdS/CFT duality, where the ideas of CFT were also used to
study the gravitational physics on the anti-de Sitter background and vice
versa.

A natural question is whether these two ways of using conformal symmetry may
be joined together. Since the conformal transformations are of spacetime
character (not in the internal group of matter symmetries) then the merging
shoukd necessarily involve gravity. At the beginning, the crucial issue was
how to make finite quantum gravity. Unfortunately, the quantum gravitational
theories were from the beginning beset by UV-divergences and the first theory
of gravity invented by Einstein in 1915 revealed to be non-renormalizable at
the quantum level. There was completely no control over divergences, and so
the theory had to be modified. The addition of terms with higher derivatives
in the action helps with the renormalizability and even one-loop super-
renormalizability was achieved [1]-[7]. In this way a much bigger control over
divergences was gained. Additionally the inclusion of non-locality solved the
unitarity issue. The last piece in this quest for UV-completion required the
construction of a theory where all divergences were precisely cancelled. This
was achieved for the first time by the authors in [8, 9, 10]. This result
opened the possibility to have conformally invariant theories also at quantum
level because the condition for the cancellation of the conformal anomaly is
the vanishing of beta functions for all couplings. Now, we also see that the
implementation of conformal symmetry in a fully gravitational and fully
quantum framework has very profound and unexpected consequences. Indeed, not
only we have a theory free of pathologies in the UV, but also we have exact
spacetime solutions that are without singularities. This was another problem
that hampered Einstein’s GR, but now due to conformal quantum gravity we can
firmly predict and describe what is beyond the “hypothetical” singularity at
the black hole origin, Big Bang event, or in many others situations, which in
the past created problems with predictability and extendability of the
classical Cauchy evolution. The ideas of using conformal symmetry for solving
these problems originated from the seminal work by Narlikar and Kembhavi [11].

Below we describe in some details the explicit construction of the finite
quantum gravity with its natural conformal extension, and the mechanism for
the resolution of spacetime singularities.

##  2 UV-finite quantum gravity

As motivated in the introduction Einstein’s gravity must be modified to
achieve UV-finiteness. We here consider a general gravitational theory
described by the following Lagrangian density

$\mathcal{L}_{\rm
g}=-2\kappa_{4}^{-2}\,\sqrt{|g|}\left({R}+R\gamma_{0}(\Box)R+R_{\mu\nu}\gamma_{2}(\Box)R^{\mu\nu}+R_{\mu\nu\rho\sigma}\gamma_{4}(\Box)R^{\mu\nu\rho\sigma}+{\cal
V}({\cal R})\right),$ (1)

where the first two form-factors are defined in terms of entire functions
$H_{0}$, $H_{2}$ by

$\gamma_{0}(\Box)=-\frac{2\left(e^{H_{0}}-1\right)+4\left(e^{H_{2}}-1\right)}{12\Box}+\gamma_{4}(\Box)\quad{\rm
and}\quad\gamma_{2}(\Box)=\frac{e^{H_{2}}-1}{\Box}-4\gamma_{4}(\Box)\,.$ (2)

We remark that the third form-factor $\gamma_{4}(\Box)$ remains arbitrary and
is constrained only by renormalizability properties of the theory. The
functions $H_{0}$, $H_{2}$ are required to be analytic entire functions on the
whole complex plane without poles and with asymptotic polynomial behaviour.
The $\gamma_{4}$ is an analytic function without poles and with asymptotic
behaviour compatible with the renormalizability of the theory. In $\cal V(R)$
we collect terms, which are cubic or higher powers in generalized
gravitational curvatures $\cal R$. These terms are crucial in making the
theory finite as we will show below.

The theory described by the Lagrangian (1) has the same symmetry as Einstein’s
gravity (diffeomorphism invariance). However, it is non-local due to the non-
polynomial, but still analytic, structure in the form factors being functions
of the covariant box operator [13, 14]. The two entire functions $H_{0}$ and
$H_{2}$ fully specify the structure of the propagator for the gravitational
fluctuations. If they satisfied the conditions stipulated above there are only
two propagating degrees of freedom, namely the same as the two polarizations
of massless gravitons in quantum Einstein’s gravity. The theory is also
unitary at perturbative level in Minkowski space.

For questions about renormalizability and UV divergences we need to study the
behaviour of the theory for large values of momenta. We decided to concentrate
on the theory, which in the UV regime tends to a polynomial higher derivative
theory. This is achieved by requiring that for large values of the argument
around real axis the form-factors $\gamma_{i}(\Box)$ for $i=0,2$ or 4 have
monomial asymptotics. The first examples of such form-factors were given in
the seminal papers by Kuz’min and Tomboulis [2, 3, 12]. For simplicity we
assume that the exponent there is a natural number $\gamma$ the same for all
form-factors. It was found that for $\gamma\geqslant 3$ the theory is one-loop
superrenormalizable [5, 6, 7], which means that only one-loop divergences
survive and the theory from two loops on is completely convergent.

Having a super-renormalizable theory it is easy to look for an extension, in
which all one-loop divergences are absent. For this purpose we use the terms
in $\cal V(R)$. One can convince himself, using for example the covariant
Barvinsky-Vilkovisky (BV) formalism for the quantum effective action, that the
divergences can be cancelled by a precise choice of the front coefficients for
the following operators,

$s_{1}R^{2}\Box^{\gamma-2}R^{2},\quad
s_{2}R_{\mu\nu}^{2}\Box^{\gamma-2}R_{\rho\sigma}^{2},\quad
s_{3}R_{\mu\nu\rho\sigma}^{2}\Box^{\gamma-2}R_{\kappa\lambda\tau\omega}^{2}$
(3)

Indeed, in four dimensions (and for asymptotically monomial form-factors) we
have to make zero three beta functions for the dimensionless couplings in
front of the operators quadratic in the gravitational curvatures, namely

$\beta_{R^{2}},\quad\beta_{R_{\mu\nu}^{2}},\quad\beta_{R_{\mu\nu\rho\sigma}^{2}}$
(4)

Actually due to a topological relation, using Gauss-Bonnet density, these beta
functions are not independent (see also [15]). However, for full conformal
invariance we have to secure that there are no divergent terms also at the
boundary of our spacetime. It turns out that we have to cancel the fourth beta
function

$\beta_{\Box R}\quad\mbox{by using operators of the type}\quad s_{4}(\Box
R)\Box^{\gamma-2}R^{2}.$ (5)

By using dimensional arguments, counting powers of covariant curvatures, and
the power of BV formalism we see that the contribution of the listed above
terms (4), (5) in $\cal V(R)$ to the corresponding beta function is linear in
their coefficients $s_{i}$ ($i=1,2,3,4$). The matrix giving such linear
mapping can be easily shown to be non-degenerate hence the system of the
equations for cancellation of these beta functions has always real solutions
for the coefficients $s_{i}$. This construction bring us a UV-finite quantum
gravitational theory.

##  3 Finite conformal quantum gravity

When we have a theory that behaves well at quantum level (namely does not have
any divergences) we may compute its predictions to whatever accuracy with
special care of finite terms. But computing these observables we notice that
the theory possesses some hidden symmetry, which is revealed for example in
the form of scattering amplitudes for on-shell gravitons [16]. This hidden
symmetry is the Weyl’s conformal invariance, and the main effort now is only
to make such symmetry explicit in the action and on every step of the
computation. In other words, we want to rewrite our finite quantum gravity in
a conformally invariant way. We here use the compensating field method, which
can be employed in the Lagrangian density (1) making the replacement
$g^{\mu\nu}=\left(\phi^{2}\kappa_{4}^{2}\right)^{-1}\hat{g}^{\mu\nu}$, namely

$\displaystyle\mathcal{L}_{\rm
conf.}=-2\,\sqrt{|\hat{g}|}\left[\phi^{2}R(\hat{g})+6\hat{g}^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)\right]$
(6)
$\displaystyle-\frac{2}{\kappa_{4}^{2}}\sqrt{|g|}\left[{R}({g})\,\gamma_{0}({\Box}){R}(g)+R_{\mu\nu}(g)\,\gamma_{2}({\Box})R^{\mu\nu}(g)+R_{\mu\nu\rho\sigma}(g)\,\gamma_{4}({\Box})R^{\mu\nu\rho\sigma}(g)+{{\cal
V}({\cal R}(g))}\,\right]\Big{|}_{g=\phi\hat{g}}\,\,,$

where by $\big{|}_{\phi\hat{g}}$ we mean that the metric $g^{\mu\nu}$ must be
replaced with a product
$\left(\phi^{2}\kappa_{4}^{2}\right)^{-1}\hat{g}^{\mu\nu}$.

In (6) we have an extra ghost-like degree of freedom. However, it can be
eliminated using the extra symmetry of the theory, namely conformal
invariance. Indeed, in this theory we have a dynamical metric tensor
$\hat{g}_{\mu\nu}$ and a scalar field $\phi$. When the unitary gauge
$\phi={\rm const.}$ is enplaned then we can come back to our finite quantum
theory. Moreover, the quantum properties of the theory can not change in
different gauges, and the theory must be UV-finite also when the scalar field
is not completely gauged away, but another gauge is implemented.

We notice that the metric
$g^{\mu\nu}=\left(\phi^{2}\kappa_{4}^{2}\right)^{-1}\hat{g}^{\mu\nu}$ (and
therefore the action (6)) is invariant under the conformal rescalings

$\hat{g}_{\mu\nu}\rightarrow\Omega^{2}(x)\,\hat{g}_{\mu\nu}\,,\quad\phi\rightarrow\Omega^{-1}(x)\,\phi,$
(7)

meaning that $\phi$ describes clocks and rulers (physical scales) while
$\hat{g}_{\mu\nu}$ contains exclusively information about the light-cone
causal structure of the spacetime. After gauge fixing, $\phi$ can be just a
constant and the distances are now measured by the metric $\hat{g}_{\mu\nu}$
or equivalently by the metric $g_{\mu\nu}$. The two metrics are now identified
and, therefore, they have the same number of degrees of freedom, namely $10$.
By choosing a conformal gauge we select a vacuum for the field $\phi$ hence
this process can be viewed as a spontaneous symmetry breaking. In other words
the “effective number of d.o.f.” is actually $10$ before and after breaking of
conformal invariance. The conformal “Higgs” mechanism here consists of moving
the d.o.f., measuring time and space, from $\phi$ to the metric
$\hat{g}_{\mu\nu}$. Such degree of freedom is the analog of the Higgs field
giving mass to particles in the standard model of particle physics (SM).

The theory defined by the Lagrangian (6) can be analyzed like any other QFT.
One can study the UV divergences of the theory when both the scalar and tensor
perturbations propagate in any conformal gauge. We consider the fluctuations
of the metric $\hat{g}_{\mu\nu}$ and of the scalar field $\phi$. Afterwords,
we expand the action around a given spacetime background and a given
background for the scalar field. The strict power counting and one-loop super-
renormalizability of this theory can be easily shown, for more details we
refer to [17]. Moreover, we have proved that the theory is unitary and all
divergences are cancelled due to terms in $\cal V(R)$ regardless of the chosen
conformal gauge in the quantization process. We also remind that the Fujikawa
integration measure in path integrals is naturally conformally invariant. In
the dimensional regularization scheme (DIMREG) some special care must be
exerted when the theory is defined away from $d=4$ spacetime dimensions. It is
a known fact that the only way how conformal anomaly can arise is due to non-
vanishing beta functions. If we do not have divergences, then we are also safe
against conformal anomaly. This claim is independent whether the vacuum is in
the broken or unbroken phase of the symmetry.

We showed that the theory is UV-finite (no divergences) and free of conformal
anomaly (all the beta functions vanish), therefore, conformal invariance is
preserved at the quantum level. The classical action of the theory (6) is also
manifestly invariant, hence we have found a class of conformally invariant
finite quantum gravity theories. These theories are in the spontaneously
broken phase of the quantum conformal Weyl symmetry.

##  4 Resolution of the black hole singularity in conformal gravity

We propose a solution of the black hole singularity problem based on conformal
symmetry. This idea was already explored in the 70’s by Narlikar and Kembhavi
[11], but we here reconsider and generalize their ideas for two reasons.
First, we can use conformal methods because we have conformal quantum gravity,
while they used it only at the classical level for a conformally invariant
theory. Secondly, we made this idea to work not only for Big Bang
singularities of FRW spacetimes (when the use of conformal methods seems
rather trivial, cf. [19]), but foremost for black holes. Moreover, we can also
get rid off of other pathologies like closed time-like curves, which by
conformal rescalings can be reduced to have zero time lapse. Therefore, we
conclude that we can make any singularity appearing in exact solutions of GR
innocuous by a proper conformal rescaling.

Spacetime singularities are properties of some exact solutions of Einstein’s
gravity, where the group of symmetry is the group of diffeomorphisms. We
remark that, contrary to some previous claims, theories with weak non-locality
are also not able to get rid out of the spacetime singularities [18, 20, 21].
Weak non-locality is not enough and we need to enlarge the underlying
diffeomorphism (Diff.) invariance of general relativity to include the Weyl
conformal invariance. Indeed, only an enhanced symmetry, here local conformal
symmetry is uniquely able to tame the singularities that plague not only
Einstein gravity, but also local and weakly non-local higher derivative
theories. We show this explicitly by analyzing the exact equations of motion
(EOM).

We start with a Riemannian spacetime manifold $\mathcal{M}$ with a metric
tensor $\hat{g}_{\mu\nu}$ and a scalar field $\phi$ as discussed in the
previous section. Then the EOM for the metric $\hat{g}_{\mu\nu}$ in the theory
(6) with the choice $\gamma_{4}=0$ read:

$\displaystyle\phi^{2}\hat{G}_{\mu\nu}=\nabla_{\nu}\partial_{\mu}\phi^{2}-\hat{g}_{\mu\nu}\hat{\Box}\phi^{2}-6\left((\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{1}{2}\hat{g}_{\mu\nu}g^{\alpha\beta}(\partial_{\alpha}\phi)(\partial_{\beta}\phi)\right)$
$\displaystyle\hskip
42.67912pt-\frac{\delta\left(\sqrt{|g|}\left(R\gamma_{0}(\Box)R+R_{\mu\nu}\gamma_{2}(\Box)R^{\mu\nu}+{\cal
V}\right)\right)\Big{|}_{\phi\hat{g}}}{\sqrt{|g_{\phi\hat{g}}|}\,\delta\hat{g}^{\mu\nu}},$
(8)

while the EOM for $\phi$ is:

$\hat{\Box}\phi=\frac{1}{6}\hat{R}\phi-\frac{1}{\sqrt{|g_{\phi\hat{g}}|}}\frac{\delta\left(\sqrt{|g|}\left(R\gamma_{0}(\Box)R+R_{\mu\nu}\gamma_{2}(\Box)R^{\mu\nu}+{\cal
V}\right)\right)\Big{|}_{\phi\hat{g}}}{\delta\phi}\,.$ (9)

We can choose the operators in $\cal V(R)$ to be function of Ricci tensors
only and not of the Riemann tensor. Since then all the operators resulting
from the variation are at least linear in the Ricci tensor $R_{\mu\nu}$ (note
that we here mean the Ricci tensor without a hat), then the Schwarzschild
metric is an exact solution of the conformally invariant theory (6). This can
be easily verified replacing the solution $\hat{g}_{\mu\nu}=g_{\mu\nu}^{{\rm
Sch}}$ and $\phi=\kappa_{4}^{-1}$ in the above EOM (8) and (9). We notice as
the result of the theorem that if the theory is general higher derivative or
non-local then the same applies and we find Schwarzschild (with its
singularity at the origin) as the exact solution of the theory [18].

However, the EOM are conformally invariant, hence if we consider another
manifold $\mathcal{M}^{*}$ obtained from $\mathcal{M}$ by a conformal
transformation (7) to ∗ quantities, then also $\hat{g}_{\mu\nu}^{*}$ and
$\phi^{*}$ satisfy the EOM (8) and (9). The transformation
$\phi\rightarrow\phi^{*}$ is mathematically valid provided $\Omega^{-1}$ does
not vanish (or become infinite). It is assumed that $\Omega=\Omega(x)$ is a
twice differentiable positive function (or a smooth function in a non-local
theory) of the spacetime coordinates $x$.

Starting from the exact Schwarzschild solution we can now construct an
infinite class of other exact solutions, all of them conformally equivalent to
the Schwarzschild metric, by doing rescalings (7) to both the scalar field
$\phi$ and the metric $\hat{g}_{\mu\nu}$ with some $\Omega$. For
$\phi=\kappa_{4}^{-1}$ and $\Omega=1$ we get the Schwarzschild spacetime
Moreover, $\hat{R}_{\mu\nu}(\hat{g}^{*}_{\mu\nu})\neq 0$, so the conformally
rescaled Schwarzschild metric is not any more Ricci-flat.

In this large class of solutions we can pick out a subclass of singularity-
free black hole spacetimes. One example of which is provided by the following
singularity-free exact black hole solution (in any conformally invariant
theory), obtained by rescaling the Schwarzschild metric by a suitable
$\Omega$:

$ds^{*2}=\left(1+\frac{L^{4}}{r^{4}}\right)\left[\left(1-\frac{2m}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2m}{r}}+r^{2}d\Omega^{(2)}\right],\quad\phi^{*}=\left(1+\frac{L^{4}}{r^{4}}\right)^{-1/2}\kappa_{4}^{-1}\,.$
(10)

If this metric is considered in GR, then one can easily show that the
spacetime is regular at $r=0$. This is supported by the regularity of all
curvature invariants (like Kretschmann scalar) and by the geodesic completion
for massive as well as massless particles moving in this geometry [22]. In the
solution (10) $L$ is an arbitrary length scale. This can be naturally chosen
either to be the Planck length or the mass of the black hole $m$. The warp
factor $\Omega(x)$ chosen in (10) is one of the infinitely many possible which
render the transformed spacetime completely regular and without any
singularities. An analogous rescalings can be found to make regular Kerr-
Newman rotating and charged black hole backgrounds. In a conformally invariant
theory this metric turns out to be physical and observable if the conformal
symmetry is spontaneously broken.

In conformal gravity the black hole thermodynamics and the Hawking evaporation
process had been presented in [23]. Similarly, the issue with the black hole
entanglement entropy in the quantum field theory framework had been solved in
[24]. Basically, the black hole entropy is finite because the theory (1) is
UV-finite at the quantum level.

## References

## References

  * [1] Asorey M, López J L , Shapiro I L, Int. J. Mod. Phys. A 12 5711 (1997) (Preprint hep-th/9610006)
  * [2] Tomboulis E T, (Preprint hep-th/9702146)
  * [3] Tomboulis E T, Mod. Phys. Lett. A 30, no. 03n04, 1540005 (2015)
  * [4] Khoury J, Phys. Rev. D 76, 123513 (2007) (Preprint hep-th/0612052)
  * [5] Modesto L, Phys. Rev. D 86, 044005 (2012) (Preprint hep-th/1107.2403)
  * [6] Modesto L, Astron. Rev. 8.2 (2013) 4-33 (Preprint hep-th/1202.3151)
  * [7] Modesto L, Preprint hep-th/1202.0008
  * [8] Modesto L and Rachwal L, Nucl. Phys. B 889, 228 (2014) (Preprint hep-th/1407.8036)
  * [9] Modesto L and Rachwal L, Nucl. Phys. B 900, 147 (2015) (Preprint hep-th/1503.00261)
  * [10] Modesto L, Piva M and Rachwal L, Phys. Rev. D 94, no. 2, 025021 (2016) (Preprint hep-th/1506.06227)
  * [11] Narlikar J V and Kembhavi A K, Lett. Nuovo Cim. 19, 517 (1977)
  * [12] Kuz’min Y V, Yad. Fiz. 50, 1630 (1989) [Sov. J. Nucl. Phys. 50, 1011 (1989)].
  * [13] Modesto L and Rachwal L, Int. J. Mod. Phys. D 26, 1730020 (2017)
  * [14] Calcagni G and Nardelli G, Phys. Rev. D 82, 123518 (2010) (Preprint hep-th/1004.5144)
  * [15] Modesto L, Rachwal L and Shapiro I L, Preprint hep-th/1704.03988
  * [16] Dona P, Giaccari S, Modesto L, Rachwal L and Zhu Y, J. High Energy Phys. JHEP08(2015)038
  * [17] Modesto L and Rachwal L, Preprint hep-th/1605.04173
  * [18] Modesto L and Rachwal L, J. High Energy Phys. JHEP1512(2015)173 (Preprint hep-th/1506.08619)
  * [19] Koshelev A S, Modesto L, Rachwal L and Starobinsky A A, J. High Energy Phys. JHEP1611(2016)067
  * [20] Calcagni G and Modesto L, Phys. Lett. B 773, 596 (2017) (Preprint gr-qc/1707.01119)
  * [21] Modesto L, Moffat J W and Nicolini P, Phys. Lett. B 695, 397 (2011) (Preprint gr-qc/1010.0680)
  * [22] Bambi C, Modesto L and Rachwal L, J. Cosmol. Astropart. Phys. JCAP 1705, no. 05, 003 (2017)
  * [23] Bambi C, Modesto L, Porey S and Rachwal L, J. Cosmol. Astropart. Phys. JCAP 1709, 033 (2017)
  * [24] Giaccari S, Modesto L, Rachwal L and Zhu Y, Preprint hep-th/1512.06206


NITEP 134

On the vacuum structure of gauge-Higgs unification models

Yuki Adachi, C.S. Lima and Nobuhito Marub,c

Department of Sciences, Matsue College of Technology, Matsue 690-8518, Japan.

a Tokyo Woman’s Christian University, Tokyo 167-8585, Japan

b Department of Mathematics and Physics, Osaka Metropolitan University, Osaka
558-8585, Japan

c Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),

Osaka Metropolitan University, Osaka 558-8585, Japan

Abstract

In this paper, we discuss the vacuum structure of the gauge-Higgs unification
theory, which is one of the attractive candidates of physics beyond the
standard model. This scenario has a remarkable feature, namely it has
infinitely degenerate vacua due to the characteristic periodic potential of
the Higgs filed, to be identified with the extra space component of the higher
dimensional gauge field. We address a question, whether to form the
superposition of such degenerate vacua, like the $\theta$-vacuum in QCD, is
necessary or not, in order to realize the true vacuum state of the theory. We
derive gauge field configuration which describes the transition between
neighboring vacua, like the instanton (or anti-instanton) solution in QCD, and
the corresponding Euclidean action in two models. In a simplified
2-dimensional U(1) model, the derived configuration to describe the transition
is shown to have finite Euclidean action, and accordingly the
“$\theta$-vacuum” and the resultant “$\theta$-term” are formulated. In a
realistic 5D U(1) model, however, the gauge field configuration to describe
the transition is shown to have infinite Euclidean action and therefore the
tunneling probability between the degenerate vacua vanishes. Thus the
superposition of the degenerate vacua is not necessary.

##  1 Introduction

In spite of its great success, the standard model of particle physics has some
theoretical problems, which led the efforts to seek the theories beyond the
standard model (BSM). One of such problems is well-known gauge hierarchy
problem, i.e. the problem that in order to keep the weak scale $M_{W}$ small
compared to the cutoff scale of the model under the quantum correction, some
unnatural tremendous fine tuning of the theory parameter is necessary. This
problem is recast into the one how to evade the quadratically divergent
quantum correction to the Higgs mass-squared.

In this paper we focus on the gauge-Higgs unification (GHU) scenario [1, 2],
where the Higgs field is identified with the extra-space component of some
higher dimensional gauge field. To be precise, for instance in the simplest
case of the gauge theory in the 5-dimensional (5D) space-time with gauge field
$A_{M}=(A_{\mu},A_{y})$, the Kaluza-Klein (KK) zero-mode of $A_{y}$,
$A^{(0)}_{y}$ behaves as the Higgs field. Since the mass term of gauge field
never suffers from the divergent quantum correction in general, the gauge
hierarchy problem is solved by virtue of the higher dimensional gauge symmetry
[3].

When the extra space is non-simply-connected space such as circle S1 in the
case of 5D gauge theory, novel features emerge. If the extra space is not
compactified, i.e. in the decompactification limit, the Higgs mass is never
generated, not only at the classical level, but also at the quantum level,
since the Higgs mass-squared term contradicts with the (higher dimensional)
local gauge invariance. Interestingly, however, when the extra space is
compactified on the S1, there appears a potential for $A^{(0)}_{y}$,
$V(A^{(0)}_{y})$, though the coefficient of each term is finite and the
solution to the hierarchy problem is still valid [3]. Furthermore, the
potential is known to be a periodic function of $A^{(0)}_{y}$ with a period
$1/gR$, as a specific feature of GHU. To be concrete, e.g. in the 5D QED
model, assuming the electron is massless for the sake of simplicity [3, 4, 5],
the potential is given by

$V(A^{(0)}_{y})=\frac{3}{32\pi^{7}}\frac{1}{R^{5}}\sum_{w=1}^{\infty}\
\frac{\cos(2\pi gA^{(0)}_{y}Rw)}{w^{5}},$ (1)

where the electric charge of the electron has been written as $g$ (instead of
$e$ in the ordinary 4D QED). We easily confirm that the potential vanishes at
the decompactification limit $R\to\infty$ as we expected.

At the first glance, these novel features seem to contradict with the local
gauge symmetry of the theory, since the potential, when regarded as a local
operator, is not invariant under the local gauge transformation. (This is why
the photon never gets its mass even under the quantum correction in the
ordinary 4D QED.) What’s really happening is that the potential
$V(A^{(0)}_{y})$ can be understood as a function of a non-local operator, i.e.
the Wilson-loop $W$ along the extra dimension S1:

$W=e^{i\int_{0}^{2\pi R}gA_{y}dy}=e^{i2\pi RgA^{(0)}_{y}},$ (2)

where $g$ is a generic gauge coupling constant. The second equality comes from
the fact that in the line integral of the KK mode expansion of $A_{y}$,
$A_{y}=\sum_{n}A^{(n)}_{y}(t)e^{i\frac{n}{R}y}$, only the KK zero mode
survives, since $\int_{0}^{2\pi R}e^{i\frac{n}{R}y}\ dy=0$ for $n\neq 0$. As
is well-known, the Wilson-loop $W$ is invariant under the local gauge
transformation, $gA_{y}\to gA^{\prime}_{y}=gA_{y}-\partial_{y}\lambda$ with
gauge transformation parameter $\lambda$, which is periodic in $y$:

$\lambda=\sum_{n}\lambda^{(n)}(t)e^{i\frac{n}{R}y}.$ (3)

This gauge transformation clearly leaves the KK zero mode $A^{(0)}_{y}$
intact. In this way the potential $V(A^{(0)}_{y})$ is induced, being
consistent with the gauge symmetry. Let us notice that, just as in the case of
Aharonov-Bohm effect, the Wilson-loop is physically meaningful since the
circle is non-simply-connected manifold and the line integral cannot be shrunk
to a point.

A remarkable thing in the GHU scenario is that the periodicity of the
potential with the period $\frac{1}{gR}$ implies the presence of infinitely
degenerate vacua, say $A^{(0)}_{y}=\nu\cdot\frac{1}{gR}\ (\nu:{\rm integer})$.
Such characteristic vacuum structure of GHU models reminds us the rich vacuum
structure of QCD and the resultant $\theta$-vacuum, which is obtained by the
superposition of the infinitely degenerate vacua with suitable weight factors.
The resultant “$\theta$-term” implies an interesting physical consequence
called strong CP problem. Thus, it will be natural to address a question
whether a similar superposition of the infinitely degenerate vacua is
necessary also in the GHU models of our interest, to realize the true vacuum.

The transitions between different vacua are described by instanton (or anti-
instanton) solutions in the case of QCD, whose Euclidean action determines the
tunneling probability between those different vacua. Therefore, the central
issues in this paper is what kind of field configuration describes the
transition between different vacua and whether the corresponding tunneling
probability is non-vanishing or not in the GHU models. If we get a non-
vanishing probability, we clearly have to perform the superposition of the
infinitely degenerate vacua as in the case of QCD to get a true vacuum of the
model. Otherwise, such procedure will not be necessary and it will suffice to
work in one of such infinitely degenerate vacua, the approach we usually take.

##  2 An analysis in a simplified 2D U(1) model

The novel structure of the vacuum state, attributed to the periodic potential
of the Higgs field in GHU scenario, already arises in a simplified Abelian
U(1) gauge model. Thus, in this section we investigate the vacuum structure in
the extremely simplified framework of U(1) GHU model in 2D space-time, with an
extra-space alone as the spatial dimension, which is a circle S1 of radius
$R$.

The 2D space-time is denoted by the coordinates $(x^{0},y)\ (x^{0}=t)$ with
$y$ being extra-space coordinate and the corresponding gauge fields are
$(A_{0},A_{y})$. The action $S$ is just the one for pure gauge theory
including the potential $V(A^{(0)}_{y})$:

$S=\int_{-\infty}^{\infty}dt\int_{0}^{2\pi R}dy\
\bigl{\\{}-\frac{1}{2}F_{0y}F^{0y}-V(A^{(0)}_{y})\bigr{\\}}.$ (4)

$V(A^{(0)}_{y})$ is expected to be induced radiatively, e.g., through 1-loop
quantum effect due to the charged fermion, as in the 5D QED [3]. Such matter
field, however, is not included in this action, since it has nothing to do
with the analysis of the vacuum structure of this theory given below.

The radiatively induced effective potential in the 2D GHU model is given by
[6]

$V(A^{(0)}_{y})=\frac{1}{2\pi^{3}R^{2}}\sum_{w=1}^{\infty}\ \frac{\cos(2\pi
gA^{(0)}_{y}Rw)}{w^{2}},$ (5)

where $1/R^{5}$ and $1/w^{5}$ in the potential for the case of 5D QED, Eq.(1),
are replaced by $1/R^{2}$ and $1/w^{2}$, respectively because of the
difference of the space-time dimension, together with a suitable change of the
coefficient factor.

For the convenience in the discussions below, we make a translation,
$A^{(0)}_{y}\to A^{(0)}_{y}+\frac{1}{2gR}$, so that the potential is now
written as

$V(A^{(0)}_{y})=\frac{1}{2\pi^{3}R^{2}}\sum_{w=1}^{\infty}\
(-1)^{w}\frac{\cos(2\pi gA^{(0)}_{y}Rw)}{w^{2}}.$ (6)

This potential is clearly periodic in $A^{(0)}_{y}$ with a period
$\frac{1}{gR}$ and we have infinitely degenerate vacua at
$A^{(0)}_{y}=\nu\cdot\frac{1}{gR}\ (\nu:{\rm integer})$. Thus, we address a
question whether the superposition of the infinitely degenerate vacua like the
$\theta$-vacuum in QCD is necessary in this GHU model.

In the case of QCD, the degenerate vacua are denoted by a winding number,
which specifies the homotopy class $\pi_{3}(SU(2))$ of the mapping ${\rm
S}^{3}\to$ SU(2). Similarly, in our model infinitely degenerate vacua are
denoted by a winding number $\nu$ given by a line integral, which is nothing
but the Wilson-line phase,

$\nu=\frac{1}{2\pi}\int_{0}^{2\pi R}gA_{y}\ dy.$ (7)

The integer $\nu$ specifies the homotopy class $\pi_{1}(U(1))$ of the mapping
${\rm S}^{1}\to$ U(1), where S1 is the extra space. In fact, the $\nu$-th
vacuum is given by a pure gauge configuration

$gA^{(0)}_{y}=U^{\dagger}i\partial_{y}U\ \ (U=e^{-i\frac{\nu}{R}y}),$ (8)

where $U=e^{-i\frac{\nu}{R}y}$ determines the mapping ${\rm S}^{1}\to$ U(1),
with $\nu$ standing for how many times the unit circle on the complex plane
representing U(1) is wrapped when $y$ goes from 0 to $2\pi R$.

Our task is to find out the field configuration in this model corresponding to
the instanton (or anti-instanton) solution in QCD, which is the classical
solution to the Euclidean equation of motion and provides the minimum
Euclidean action for a given homotopy class. To be more precise, we attempt to
find the classical solution which describes the transition from the vacuum
with the winding number $\nu=0$ at $t\to-\infty$ to another vacuum with, say
$\nu=1$ at $t\to\infty$. To make the analysis simple, relying on the gauge
invariance of the action, we first move to the gauge where $A_{y}=0$ except
for its KK zero mode $A^{(0)}_{y}$. This procedure is realized by a gauge
transformation, $gA_{y}\to gA^{\prime}_{y}=gA_{y}-\partial_{y}\lambda$ (see
Eq.(3)), by choosing

$\lambda^{(n)}=-i\frac{R}{n}gA^{(n)}_{y}\ \ (n\neq 0).$ (9)

Let us note that the KK zero mode $A^{(0)}_{y}$ cannot be “gauged away”.
Accordingly, $A_{0}$ also undergoes a gauge transformation $A_{0}\to
A^{\prime}_{0}$. Now the mixing between $A^{\prime}_{0}$ and $A^{\prime}_{y}$
through $F^{\prime}_{0y}$ disappears since the non-zero KK modes of $A_{y}$
have been eliminated and zero modes $A^{(0)}_{0}$ and $A^{(0)}_{y}$ do not mix
from the beginning as $\partial_{y}A^{(0)}_{0}=0$. Hence, rewriting
$A^{\prime}_{0},\ A^{\prime}_{y}$ simply as $A_{0}$ and
$A_{y}=A^{(0)}_{y}(t)$, the Euclidean action corresponding to the original
action Eq.(4) reduces to

$S_{E}=\int_{-\infty}^{\infty}dt\int_{0}^{2\pi R}dy\
\frac{1}{2}\left(\frac{\partial A_{0}}{\partial y}\right)^{2}+(2\pi
R)\int_{-\infty}^{\infty}\
\bigl{\\{}\frac{1}{2}\left(\frac{dA^{(0)}_{y}}{dt}\right)^{2}+V(A^{(0)}_{y})\bigr{\\}}\
dt.$ (10)

The (Euclidean) equation of motion for $A^{(0)}_{y}$ derived from this action
is equivalent to that of the 1-dimensional motion of a particle with an unit
mass under a potential

$-V(A^{(0)}_{y})\simeq\frac{1}{2\pi^{3}R^{2}}\ \left[\cos(2\pi
gA^{(0)}_{y}R)-1\right],$ (11)

where the potential is simplified by taking only the major contribution from
the sector of $w=1$ in Eq.(5) and some suitable constant has been added so
that $V(0)=0$, which does not affect the essence of our argument. (If
necessary, we can invoke numerical calculation by using the exact formula for
the potential.)

The transition from the vacuum $\nu=0$ at $t\to-\infty$ to the vacuum $\nu=1$
at $t\to\infty$ is described by a solution to the equation of motion with
initial conditions $A^{(0)}_{y}=0$ and $\frac{dA^{(0)}_{y}}{dt}=0$ at
$t=-\infty$. Therefore the conserved Euclidean energy (density) $\epsilon$
should vanish

$\epsilon=\frac{1}{2}\left(\frac{dA^{(0)}_{y}}{dt}\right)^{2}-V(A^{(0)}_{y})=0,$
(12)

since $-V(A^{(0)}_{y})=0$ for $A^{(0)}_{y}=0,\ \frac{1}{gR}$. Thus,
$\frac{dA^{(0)}_{y}}{dt}=\sqrt{2V(A^{(0)}_{y})}$, and the classical solution
to the equation of motion is obtained as follows:

$\int_{\frac{1}{2gR}}^{A^{(0)}_{y}}\
\frac{d\tilde{A}^{(0)}_{y}}{\sqrt{\frac{1}{\pi^{3}R^{2}}\ \left[1-\cos(2\pi
g\tilde{A}^{(0)}_{y}R)\right]}}\simeq t\ \to\ A^{(0)}_{y}\simeq\frac{2}{\pi
gR}\tan^{-1}e^{\sqrt{\frac{2}{\pi}}gt}.$ (13)

The solution Eq.(13) satisfies $A^{(0)}_{y}(-\infty)=0,\
A^{(0)}_{y}(\infty)=\frac{1}{gR}$ (by choosing the values of the multi-valued
function as $\tan^{-1}0=0,\ \tan^{-1}\infty=\frac{\pi}{2}$), and therefore is
confirmed to describe the transition between the neighboring vacua
$\nu=0\to\nu=1$. (By choosing other values for the multi-valued function, we
can get the solutions to describe the transitions between the vacua, whose
winding numbers differ not by 1 unit.)

The part of the action concerning $A^{(0)}_{y}$ given by this solution is
easily found by use of Eq.(12) (without using the explicit form of the
solution Eq.(13)) to be

$\displaystyle S_{E}$ $\displaystyle=$ $\displaystyle(2\pi
R)\int_{-\infty}^{\infty}\
\bigl{\\{}\frac{1}{2}\left(\frac{dA^{(0)}_{y}}{dt}\right)^{2}+V(A^{(0)}_{y})\bigr{\\}}\
dt$ (14) $\displaystyle=$ $\displaystyle(2\pi R)\int_{-\infty}^{\infty}\
\left(\frac{dA^{(0)}_{y}}{dt}\right)^{2}\ dt=(2\pi R)\int_{-\infty}^{\infty}\
\sqrt{2V(A^{(0)}_{y})}\frac{dA^{(0)}_{y}}{dt}\ dt$ $\displaystyle\simeq$
$\displaystyle(2\pi R)\int_{0}^{\frac{1}{gR}}\ \sqrt{\frac{1}{\pi^{3}R^{2}}\
\left[1-\cos(2\pi gA^{(0)}_{y}R)\right]}\ dA^{(0)}_{y}$ $\displaystyle=$
$\displaystyle\frac{4\sqrt{2}\pi^{-\frac{3}{2}}}{gR}.$

On the other hand, the solution to the equation of motion for $A_{0}$ is
clearly $A_{0}=0$, as its initial conditions are $A_{0}=\frac{\partial
A_{0}}{\partial t}=0$, and therefore the part of the action concerning $A_{0}$
also vanishes. Thus (14) is actually the Euclidean action for the whole
system, obtained using the solution to the equation of motion. As the
conclusion, the semi-classical (WKB) approximation of the tunneling amplitude
from the sector of $\nu=0$ to that of $\nu=1$ turns out to be non-vanishing:

$e^{-S_{E}}\simeq e^{-\frac{4\sqrt{2}\pi^{-\frac{3}{2}}}{gR}}.$ (15)

Since the Euclidean action is finite, we should form “$\theta$-vacuum” as the
real vacuum state mimicking the one in QCD:

$|\theta\rangle=\sum_{\nu}e^{-i\nu\theta}|\nu\rangle,$ (16)

where $|\nu\rangle$ stands for the vacuum state with winding number $\nu$.

As the solution Eq.(13) connects the vacua with $\nu$’s, different by one
unit, it itself has a winding number 1, which is denoted by the line integral
of the gauge field along the boundary of the 2D space-time (accompanied by a
factor $g/(2\pi)$), as is easily seen by noting
$A^{(0)}_{y}(t=\infty)-A^{(0)}_{y}(t=-\infty)=1/(gR)$ (see Eq.(13)) and
$A_{0}=0$. This argument can be generalized to the solution with arbitrary
winding number, and by use of the Stokes’ theorem, the winding number $\nu$ is
given by a gauge invariant 2D topological term (corresponding to the $\int
d^{4}x\ F_{\mu\nu}\tilde{F}^{\mu\nu}$ in 4D space-time):

$\nu=\frac{g}{2\pi}\int_{-\infty}^{\infty}dt\int_{0}^{2\pi R}dy\ F_{0y}.$ (17)

Hence, by use of Eq.(17), we easily see that the effect of adopting the theta
vacuum is equivalent to the introduction of the following “$\theta$-term” in
the lagrangian:

${\cal L}_{\theta}=\frac{g}{2\pi}\theta\int_{-\infty}^{\infty}dt\int_{0}^{2\pi
R}dy\ F_{0y}.$ (18)

Interestingly, if the space coordinate $y$ is replaced by $x_{1}=x$ and the
potential term is ignored, this system is equivalent to the well-known
Schwinger’s model [7], 2D QED, which has specific and remarkable features,
such as the “chiral” anomaly, given by exactly the same expression as the
r.h.s. of (17) and (18),
$\partial_{\mu}j_{5}^{\mu}=\frac{g}{2\pi}\epsilon^{\mu\nu}F_{\mu\nu}\
(\mu,\nu=0,1)$. Let us note that the potential for the gauge field is
meaningless for de-compactified space.

##  3 An analysis in a realistic 5D U(1) model

Now we move to the discussion on the vacuum structure of a realistic model, 5D
U(1) GHU model. In particular, we focus on the question whether the tunneling
probability between neighboring vacua with winding numbers different by 1 unit
stays finite as in the 2D GHU model or not.

The 5D space-time is denoted by the coordinates $(x^{\mu},y)\ (\mu=0,1,2,3)$,
with the extra-space being assumed to be a circle S1 of radius $R$. The
corresponding gauge fields are $(A_{\mu},A_{y})$. The 5D action for the gauge
fields is

$S=\int d^{4}x\int_{0}^{2\pi R}dy\
\bigl{\\{}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}F_{\mu y}F^{\mu
y}-V(A^{(0)}_{y})\bigr{\\}},$ (19)

where the radiatively induced potential $V(A^{(0)}_{y})$ due to the exchange
of U(1)-charged fermion (with vanishing bulk mass) is given by Eq.(1).

As in the previous section, we make a translation, $A^{(0)}_{y}\to
A^{(0)}_{y}+\frac{1}{2gR}$, so that the potential is written as

$V(A^{(0)}_{y})=\frac{3}{32\pi^{7}}\frac{1}{R^{5}}\sum_{w=1}^{\infty}\
(-1)^{w}\frac{\cos(2\pi gA^{(0)}_{y}Rw)}{w^{5}}.$ (20)

It is simplified by taking only the dominant contribution of the sector $w=1$
and adding a suitable constant so that $V(0)=0$:

$V(A^{(0)}_{y})\simeq\frac{3}{32\pi^{7}}\frac{1}{R^{5}}\ \left(1-\cos(2\pi
gA^{(0)}_{y}R)\right).$ (21)

Again we adopt a gauge where $A_{y}=0$ except for its KK zero-mode
$A^{(0)}_{y}$, which cannot be gauged away. Then, as in the case of the 2D
model discussed in the previous section, the mixing between $A_{\mu}$ and
$A_{y}$ disappears and the Euclidean action in this choice of gauge reduces to

$\displaystyle S_{E}$ $\displaystyle=$ $\displaystyle\int d^{4}x\int_{0}^{2\pi
R}dy\
\bigl{\\{}\frac{1}{2}(F_{0i})^{2}+\frac{1}{4}(F_{ij})^{2}+\frac{1}{2}\left(\frac{\partial
A_{\mu}}{\partial y}\right)^{2}\bigr{\\}}$ (22) $\displaystyle+$
$\displaystyle(2\pi R)\int d^{4}x\ \bigl{\\{}\frac{1}{2}\left(\frac{\partial
A^{(0)}_{y}}{\partial t}\right)^{2}+\frac{1}{2}\left(\frac{\partial
A^{(0)}_{y}}{\partial x^{i}}\right)^{2}+V(A^{(0)}_{y})\bigr{\\}},$

where $i,j=1,2,3$ are indices corresponding to the 3D space.

As in the case of 2D U(1) GHU model, this theory has infinitely degenerate
vacua denoted by $A^{(0)}_{y}=\nu\cdot\frac{1}{gR}\ (\nu:{\rm integer})$. Our
concern is the tunneling probability between these vacua. Especially, we focus
on the transition from the vacuum with the winding number $\nu=0$ at
$t\to-\infty$ to the neighboring vacuum $\nu=1$ at $t\to\infty$. Such
transition is described by a solution to the Euclidean equation of motion with
initial conditions $A^{(0)}_{y}=0$ and $\frac{\partial A^{(0)}_{y}}{\partial
t}=0$ at $t=-\infty$. Since the initial conditions do not depend of $x^{i}$,
the solution to the Euclidean equation of motion,

$\frac{\partial^{2}A^{(0)}_{y}}{\partial
t^{2}}+\frac{\partial^{2}A^{(0)}_{y}}{\partial
x^{i2}}-\frac{dV(A^{(0)}_{y})}{dA^{(0)}_{y}}=0,$ (23)

should depend only on $t$ and can be written in a form $A^{(0)}_{y}=f(t)$.
Plugging this into Eq.(23), $f(t)$ is known to satisfy an equation,

$\frac{d^{2}f}{dt^{2}}-\frac{dV}{df}=0,$ (24)

under the initial conditions $f=\frac{df}{dt}=0$ at $t=-\infty$. Just as in
the case of 2D model, this mechanical system is equivalent to the system of a
point particle with unit mass moving under the potential $-V$.

As we did in the previous section, we impose a condition that the conserved
Euclidean energy (density) $\epsilon$ vanishes:

$\epsilon=\frac{1}{2}\left(\frac{df}{dt}\right)^{2}-V(f)=0,$ (25)

which leads to $\frac{df}{dt}=\sqrt{2V(f)}$, and the solution to the equation
of motion, satisfying $f=\frac{df}{dt}=0$ at $t=-\infty$, is easily obtained
as follows:

$\int_{\frac{1}{2gR}}^{f}\
\frac{d\tilde{f}}{\sqrt{\frac{3}{16\pi^{7}}\frac{1}{R^{5}}\ \left[1-\cos(2\pi
g\tilde{f}R)\right]}}\simeq t\ \to\ A^{(0)}_{y}=f(t)\simeq\frac{2}{\pi
gR}\tan^{-1}e^{\sqrt{\frac{3}{8\pi^{5}R^{3}}}gt}.$ (26)

Since $A^{(0)}_{y}(-\infty)=0$ and $A^{(0)}_{y}(\infty)=\frac{1}{gR}$, this
solution describes the transition $\nu=0\to\nu=1$.

The part of the action concerning $A^{(0)}_{y}$ given by this solution is
easily found by use of Eq.(25) to be

$\displaystyle S_{E}$ $\displaystyle=$ $\displaystyle(2\pi
R)V_{3D}\int_{-\infty}^{\infty}\
\bigl{\\{}\frac{1}{2}\left(\frac{df}{dt}\right)^{2}+V(f)\bigr{\\}}\ dt$ (27)
$\displaystyle=$ $\displaystyle(2\pi R)V_{3D}\int_{-\infty}^{\infty}\
\left(\frac{df}{dt}\right)^{2}\ dt$ $\displaystyle\simeq$ $\displaystyle(2\pi
R)V_{3D}\int_{0}^{\frac{1}{gR}}\ \sqrt{\frac{3}{16\pi^{7}}\frac{1}{R^{5}}\
\left[1-\cos(2\pi gfR)\right]}\ df$ $\displaystyle=$
$\displaystyle\sqrt{\frac{6}{\pi^{7}R^{5}}}\frac{1}{g}V_{3D}=\frac{\sqrt{3}}{\pi^{4}}\frac{1}{g_{4}}\frac{V_{3D}}{R^{3}},$

where $V_{3D}$ denotes the volume of the 3D space and
$g_{4}=\frac{g}{\sqrt{2\pi R}}$ is the 4D (dimension-less) gauge coupling
constant.

On the other hand, the solution to the equation of motion for $A_{\mu}$ is
clearly $A_{\mu}=0$, since its initial conditions are $A_{\mu}=\frac{\partial
A_{\mu}}{\partial t}=0$, and the part of the action concerning $A_{\mu}$ also
vanishes. Thus Eq.(27) is exactly the Euclidean action for the whole system
due to the solution to the equation of motion. Thus, in the limit
$V_{3D}\to\infty$, $S_{E}$ also gets infinity, and therefore the tunneling
amplitude between the neighboring vacua vanishes:

$\lim_{V_{3D}\to\infty}e^{-S_{E}}=0.$ (28)

Thus, the procedure we usually take, i.e. to assume that the vacuum state is
one of the degenerate vacua with, say $\nu=0$, is justified.

##  4 Summary and discussion

In this paper, we discussed the novel feature of the GHU scenario, one of the
attractive candidates of physics BSM, namely the infinitely degenerate vacua
due to the characteristic periodic potential of the Higgs filed, to be
identified with the extra space component of the higher dimensional gauge
field. We discussed that the periodicity is the inevitable consequence of the
fact that the Higgs field is interpreted to be a Wilson-loop phase along the
extra dimension, which has a physical meaning when the extra space is non-
simply-connected manifold like a circle S1.

We addressed a question, whether to form the superposition of such degenerate
vacua, like the “$\theta$-vacuum” in QCD, is necessary or not, in order to
realize the true vacuum state of the theory. Especially, we derived gauge
field configuration which describes the transition between neighboring vacua,
like instanton (or anti-instanton) solution in QCD, and the corresponding
Euclidean action, which determines the tunneling probability between the
neighboring vacua.

First, as the extremely simplified model, we discussed 2D U(1) GHU model. Each
of the degenerate vacua is denoted by a topological winding number,
corresponding to the homotopy class $\pi_{1}(U(1))$. We derived a
configuration to describe the transition between neighboring vacua, and have
shown that it has finite Euclidean action and therefore non-vanishing
tunneling probability. Accordingly, the $\theta$-vacuum was formulated and the
resultant “$\theta$-term” was presented.

Next, we discussed a realistic 5D U(1) GHU model. In contrast to the case of
the 2D U(1) GHU model, though we could find the gauge field configuration to
describe the transition between neighboring vacua, the corresponding Euclidean
action turns out to be infinite. This means the superposition of the
degenerate vacua is not necessary, and we can take one of the degenerate vacua
as our vacuum state, the attitude we usually take.

Though we have concentrated on the U(1) GHU models, the periodic Higgs
potential is the common feature of GHU, shared also by non-Abelian GHU models,
as long as the extra space is non-simply-connected manifold, such as S1. The
winding number $\nu$ in order to characterize the degenerate vacua is still
given by the Wilson-loop phase. For instance, in the case of the minimal SU(3)
electro-weak unified GHU model [5], $\nu$ is written in terms of the line-
integral of $A^{6}_{y}$ associated with the Gell-Mann matrix $\lambda_{6}$,
which develops the VEV. Note that in this case the homotopy class
$\pi_{1}$(U(1)) is for the mapping from S1 to the U(1) generated by
$\lambda_{6}$, the subgroup of SU(3) (If necessary, $\lambda_{6}$ can be
diagonalized by a suitable gauge transformation).

### Acknowledgments

This work was supported in part by Japan Society for the Promotion of Science,
Grants-in-Aid for Scientific Research, No. 15K05062.

## References

  * [1] N. S. Manton, Nucl. Phys. B158 (1979) 141.
  * [2] Y. Hosotani, Phys. Lett. B126 (1983) 309; Y. Hosotani, Phys. Lett. B129 (1983) 193; Y. Hosotani, Annals Phys. 190 (1989) 233.
  * [3] H. Hatanaka, T. Inami and C. S. Lim, Mod. Phys. Lett. A13 (1998) 2601\.
  * [4] I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3 (2001), 20.
  * [5] M. Kubo, C. S. Lim, and H. Yamashita, Mod. Phys. Lett. A17 (2002) 2249; C. A. Scrucca, M. Serone, and L. Silvestrini, Nucl. Phys. B669 (2003) 128.
  * [6] Y. Hosotani, N. Maru, K. Takenaga and T. Yamashita, Prog. Theor. Phys. 118, 1053-1068 (2007).
  * [7] J. Schwinger, Phys. Rev. 128, 2425 (1962).


# Supplemental Material for the Paper
Verification of Wave Turbulence in the Kinetic Limit

Alexander Hrabski  Yulin Pan  Department of Naval Architecture and Marine
Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

##  I Derivation of the Kolmogorov-Zakharov (KZ) Spectrum

The derivation of the KZ spectrum begins with the Wave Kinetic Equation (WKE),
provided again for convenience:

$\begin{split}\frac{\partial n_{k}}{\partial
t}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}4\pi
n_{1}n_{2}n_{3}n_{k}\left(\frac{1}{n_{k}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right)\\\
\times\delta(k_{1}+k_{2}-k_{3}-k)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{k})dk_{1}dk_{2}dk_{3}.\end{split}$
(1)

While it isn’t in general necessary [1], we adopt the popular assumption of an
isotropic spectrum. In one-dimensional systems, this takes the form of
$n_{k}=n_{-k}$. Each of $k_{1}$, $k_{2}$, and $k_{3}$ can take a positive or
negative sign in the delta function of (1), producing $8$ possibilities.
However, not every combination of these signs produces a non-trivial
resonance. A trivial resonance is one that satisfies the resonance conditions
by having $k_{1}=k_{3}$ and $k_{2}=k$, or $k_{2}=k_{3}$ and $k_{1}=k$. For
these cases, the integrand of (1) takes a zero value, so they may be ignored.
After these trivial cases are removed, we are left with

$\begin{split}\frac{\partial n_{k}}{\partial
t}=4\pi\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}n_{1}n_{2}n_{3}n_{k}\left(\frac{1}{n_{k}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{k})\\\
\times\left(\delta(k_{1}+k_{2}+k_{3}-k)+\delta(k_{1}-k_{2}-k_{3}-k)+\delta(-k_{1}+k_{2}-k_{3}-k)\right.\\\
+\left.\delta(-k_{1}-k_{2}+k_{3}-k)\right)dk_{1}dk_{2}dk_{3}.\end{split}$ (2)

It will later become important to integrate over the resonant manifold
(described by the $\delta$-functions), which is much easier to interpret as
quadratic functions in $\omega=k^{1/2}$. Therefore, we next rewrite the above
equation as an integral over $\omega$, and replace (on the LHS) $n_{k}$, the
spectral density in $k$, with $\mathcal{N}_{\omega}$, the spectral density in
$\omega$. This leads to

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial
t}=128\pi\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}(\omega_{1}\omega_{2}\omega_{3}\omega)n_{1}n_{2}n_{3}n_{\omega}\left(\frac{1}{n_{\omega}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\\\
\times\left(\delta(\omega^{2}_{1}+\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})+\delta(\omega^{2}_{1}-\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})+\delta(-\omega^{2}_{1}+\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})\right.\\\
+\left.\delta(-\omega^{2}_{1}-\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})\right)d\omega_{1}d\omega_{2}d\omega_{3},\end{split}$
(3)

where $n_{\omega}=n(k(\omega))$, and we have used the facts that $dk=2\omega
d\omega$ and $\mathcal{N}(\omega)d\omega=n(k)dk+n(-k)dk=2n(k)dk$. We note that
the factor of $2$ on the spectral element relation is necessary for a
consistent and correct flux definition.

We now assume that $n_{\omega}=A\omega^{\gamma}$, where $\gamma$ refers to an
arbitrary exponent that will later be used to determine the KZ exponents.
Substituting this into (3), we are left with

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}=128\pi
A^{3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\omega^{\gamma+1}_{1}\omega^{\gamma+1}_{2}\omega^{\gamma+1}_{3}\omega^{\gamma+1}\left(\omega^{-\gamma}+\omega^{-\gamma}_{3}-\omega^{-\gamma}_{1}-\omega^{-\gamma}_{2}\right)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\\\
\times\left(\delta(\omega^{2}_{1}+\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})+\delta(\omega^{2}_{1}-\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})+\delta(-\omega^{2}_{1}+\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})\right.\\\
+\left.\delta(-\omega^{2}_{1}-\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})\right)d\omega_{1}d\omega_{2}d\omega_{3},\end{split}$
(4)

Next we employ the Zakharov transformations [2, 1], which are a set of
conformal transformations one applies to the integrand that result in the
reduction of the sum of delta functions to a single delta function. This new
structure of the integrand will allow us to (a) explicitly see the zeros of
the equation and (b) explicitly compute the Kolmogorov constant $C$. See [3]
for an intuitive, geometric description of how these transformations achieve
these effects assuming only a self-similar spectrum. We distribute the sum of
delta functions to expand the integrand into 4 terms, and we handle each
separately. The first term,

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}^{(1)}=128\pi
A^{3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\omega^{\gamma+1}_{1}\omega^{\gamma+1}_{2}\omega^{\gamma+1}_{3}\omega^{\gamma+1}\left(\omega^{-\gamma}+\omega^{-\gamma}_{3}-\omega^{-\gamma}_{1}-\omega^{-\gamma}_{2}\right)\\\
\times\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\delta(\omega^{2}_{1}+\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})d\omega_{1}d\omega_{2}d\omega_{3},\end{split}$
(5)

is the form onto which we will map the other terms. We have used a superscript
$(1)$ to denote we are referring to the first term. Now, as an example, we
manipulate the second term in full,

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}^{(2)}=128\pi
A^{3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\omega^{\gamma+1}_{1}\omega^{\gamma+1}_{2}\omega^{\gamma+1}_{3}\omega^{\gamma+1}\left(\omega^{-\gamma}+\omega^{-\gamma}_{3}-\omega^{-\gamma}_{1}-\omega^{-\gamma}_{2}\right)\\\
\times\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\delta(\omega^{2}_{1}-\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})d\omega_{1}d\omega_{2}d\omega_{3},\end{split}$
(6)

to which we apply the following transformations:
$\omega_{1}=\omega^{2}/\omega_{1}^{\prime}$,
$\omega_{2}=\omega\omega_{2}^{\prime}/\omega_{1}^{\prime}$, and
$\omega_{3}=\omega\omega_{2}^{\prime}/\omega_{1}^{\prime}$. Under these
transformations,
$d\omega_{1}d\omega_{2}d\omega_{3}=\left(\frac{\omega}{\omega_{1}^{\prime}}\right)^{4}d\omega_{1}^{\prime}d\omega_{2}^{\prime}d\omega_{3}^{\prime}$,
and (6) becomes, after some reduction,

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}^{(2)}=128\pi
A^{3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\left(\frac{\omega}{\omega_{1}^{\prime}}\right)^{3\gamma+5}\omega^{\prime\gamma+1}_{1}\omega^{\prime\gamma+1}_{2}\omega^{\prime\gamma+1}_{3}\omega^{\prime\gamma+1}\left(\omega^{\prime-\gamma}_{1}+\omega^{\prime-\gamma}_{3}-\omega^{\prime-\gamma}-\omega^{\prime-\gamma}_{2}\right)\\\
\times\delta(\omega^{\prime}+\omega_{2}^{\prime}-\omega_{3}^{\prime}-\omega_{1}^{\prime})\delta(\omega^{\prime
2}_{1}+\omega^{\prime 2}_{2}+\omega^{\prime 2}_{3}-\omega^{\prime
2})d\omega_{1}^{\prime}d\omega_{2}^{\prime}d\omega_{3}^{\prime}.\end{split}$
(7)

We note that the identity $\int\delta(ax)dx=\int\delta(x)/|a|dx$ is used to
simply the above expression. The integrand of (7) _almost_ reflects (5) with
an additional factor of $(\omega/\omega_{1}^{\prime})^{3\gamma+5}$. If one
carefully looks at the signed terms in the equation, however, it becomes
apparent that certain indices have become switched as a result of our
transformation. The choice of indices is arbitrary, so we renumber them
according to $123\rightarrow 132$. This leaves us with

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}^{(2)}=-128\pi
A^{3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\left(\frac{\omega}{\omega_{1}^{\prime}}\right)^{3\gamma+5}\omega^{\prime\gamma+1}_{1}\omega^{\prime\gamma+1}_{2}\omega^{\prime\gamma+1}_{3}\omega^{\prime\gamma+1}\left(\omega^{\prime-\gamma}+\omega^{\prime-\gamma}_{3}-\omega^{\prime-\gamma}_{1}-\omega^{\prime-\gamma}_{2}\right)\\\
\times\delta(\omega^{\prime}_{1}+\omega_{2}^{\prime}-\omega_{3}^{\prime}-\omega)\delta(\omega^{\prime
2}_{1}+\omega^{\prime 2}_{2}+\omega^{\prime 2}_{3}-\omega^{\prime
2})d\omega_{1}^{\prime}d\omega_{2}^{\prime}d\omega_{3}^{\prime}.\end{split}$
(8)

In this form, the symmetry with (5) is obvious. We perform the remaining
Zakharov transformations (see [2] for the forms of the other transformations),
sum the four terms, and drop the primes from our notation. This results in

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}=128\pi
A^{3}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\omega^{\gamma+1}_{1}\omega^{\gamma+1}_{2}\omega^{\gamma+1}_{3}\omega^{\gamma+1}\left(\omega^{-\gamma}+\omega^{-\gamma}_{3}-\omega^{-\gamma}_{1}-\omega^{-\gamma}_{2}\right)\\\
\times\left(1+\left(\frac{\omega_{3}}{\omega}\right)^{y}-\left(\frac{\omega_{1}}{\omega}\right)^{y}-\left(\frac{\omega_{2}}{\omega}\right)^{y}\right)\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega)\delta(\omega^{2}_{1}+\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})d\omega_{1}d\omega_{2}d\omega_{3},\end{split}$
(9)

with $y\equiv-3\gamma-5$. Careful consideration of the resonance conditions
reveals that (9) is an integral over the intersection of the plane
$\omega_{1}+\omega_{2}-\omega_{3}-\omega=0$ and the sphere
$\omega^{2}_{1}+\omega^{2}_{2}+\omega^{2}_{3}=\omega^{2}$ for any given
$\omega$. Thus, we do not need to consider integrating over any
$\omega_{i}>\omega$. This enables a reparameterization in terms of some
$\xi_{i}=\omega_{i}/\omega\in[0,1]$, so that (9) becomes

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial t}=128\pi
A^{3}\omega^{-y-1}I(y)=128\pi
A^{3}\omega^{-y-1}\int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}\left(\xi_{1}\xi_{2}\xi_{3}\right)^{\gamma+1}\left(1+\xi^{-\gamma}_{3}-\xi^{-\gamma}_{1}-\xi^{-\gamma}_{2}\right)\\\
\times(1+\xi_{3}^{y}-\xi_{1}^{y}-\xi_{2}^{y})\delta(\xi_{1}+\xi_{2}-\xi_{3}-1)\delta(\xi^{2}_{1}+\xi^{2}_{2}+\xi^{2}_{3}-1)d\xi_{1}d\xi_{2}d\xi_{3}.\end{split}$
(10)

This is the form of the WKE that allows for the solution of $\gamma$
corresponding to stationary solutions.

One can see by the second product in the integrand that if $\gamma=0$, the RHS
of (10) is identically $0$. This corresponds to equipartition of wave action,
i.e., $n_{\omega}$ is constant. Also, if $\gamma=-1$, then the second product
is identically $0$ whenever the resonance condition is satisfied, also leading
to a $0$ of the collision integral. This solution $n_{\omega}=A/\omega$
corresponds to the Rayleigh-Jeans spectrum. Neither of these solutions
correspond to wave turbulence, but rather are equilibrium solutions. The KZ
solutions are given by $y=0$ and $y=1$, which produce $0$’s of the collision
integral by the same arguments as the equilibrium solutions. Setting $y=0$,
one obtains $n_{\omega}=A\omega^{-5/3}$, and for $y=1$, one obtains
$n_{\omega}=A\omega^{-2}$. To determine which of these out-of-equilibrium
spectra correspond to the forward cascade of energy and which corresponds to
the inverse cascade of wave action, one may use the ordering of the exponents
$\gamma$ relative to the equilibrium spectra [1], or Zakharov’s method via
evaluating (10) [4], while in both cases being careful to ensure the flux
directions are not non-physical via comparison to the equilibrium spectra [2,
1, 4]. In [4], it is demonstrated that the forward and inverse cascade for our
system have physical cascade directions and that $y=1$ (with $\gamma=-2$)
corresponds to the forward cascade.

Next, we derive the Kolmogorov constant $C$. The energy flux through frequency
$\omega$ is defined by a control volume argument in spectral space as

$P(\omega)\equiv-\int\limits_{0}^{\omega}\omega^{\prime}\frac{\partial\mathcal{N}}{\partial
t}(\omega^{\prime})d\omega^{\prime}=-128\pi
A^{3}\frac{\omega^{1-y}}{1-y}I(y),$ (11)

where an negative sign is introduced to ensure that a positive flux
corresponds to a cascade of energy from large to small scales (long to short
time scales via the dispersion relation). For the forward cascade with $y=1$,
the computation of $P(\omega)$ involves the limit of an indeterminate
quantity, which can be obtained via L’Hospital’s rule to be

$P(\omega)=128\pi A^{3}\lim_{y\rightarrow 1}\frac{dI(y)}{dy}.$ (12)

The limit of the desired derivative, $S$, is given by

$\begin{split}S=\lim_{y\rightarrow
1}\frac{dI(y)}{dy}=\int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}\left(\xi_{1}\xi_{2}\xi_{3}\right)^{-1}\left(1+\xi^{2}_{3}-\xi^{2}_{1}-\xi^{2}_{2}\right)\\\
\times(\xi_{3}\ln{\xi_{3}}-\xi_{1}\ln{\xi_{1}}-\xi_{2}\ln{\xi_{2}})\delta(\xi_{1}+\xi_{2}-\xi_{3}-1)\delta(\xi^{2}_{1}+\xi^{2}_{2}+\xi^{2}_{3}-1)d\xi_{1}d\xi_{2}d\xi_{3},\end{split}$
(13)

where we have used the fact that $\lim_{y\rightarrow
1}\frac{dx^{y}}{dy}=x\ln{x}$. In the next section, we develop a precise method
to numerically evaluate (13) to find $S=0.09353$. We now compute the
relationship between $A$ and $P(\omega)$, revealing both the Kolmogorov
constant $C$ via

$A=CP^{1/3}=\left(128\pi S\right)^{-1/3}P^{1/3},$ (14)

resulting in $C=0.2984$. Given that $C$ is positive, we can now be sure that
the cascade direction is correct. Thus, the KZ spectrum associated with the
forward cascade process in our MMT equation is given by
$n_{\omega}=0.2984P^{1/3}\omega^{-2}$, or, via the linear dispersion relation,
$n_{k}=0.2984P^{1/3}k^{-1}$. We note that this value of $C$ is different from
the result of Zakharov et al. [4], due to their missing factor of $2$ in the
spectral element relation $\mathcal{N}(\omega)d\omega=2n(k)dk$.

##  II Numerical Integration over the Resonant Manifold

###  II.1 Evaluation of $S$

We are interested in integrating (13). For simplicity, we will refer the non-
delta part of the integrand by $f(\xi_{1},\xi_{2},\xi_{3}$) so that

$f(\xi_{1},\xi_{2},\xi_{3})\equiv\left(\xi_{1}\xi_{2}\xi_{3}\right)^{-1}\left(1+\xi^{2}_{3}-\xi^{2}_{1}-\xi^{2}_{2}\right)(\xi_{3}\ln{\xi_{3}}-\xi_{1}\ln{\xi_{1}}-\xi_{2}\ln{\xi_{2}}).$
(15)

This leaves

$S=\int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}f(\xi_{1},\xi_{2},\xi_{3})\delta(\xi_{1}+\xi_{2}-\xi_{3}-1)\delta(\xi^{2}_{1}+\xi^{2}_{2}+\xi^{2}_{3}-1)d\xi_{1}d\xi_{2}d\xi_{3}.$
(16)

The first of these delta functions is linear in $\xi_{1}$. Making use of the
property

$\int\limits_{0}^{1}g(x)\delta(x-a)dx=g(a)\ \ \text{for}\ \ 0\leq a\leq 1,$
(17)

we can integrate over $\xi_{3}$ to obtain

$S=\iint\limits_{\Delta(\xi_{1},\xi_{2})}f(\xi_{1},\xi_{2},\xi_{1}+\xi_{2}-1)\delta\left(\xi_{1}^{2}+\xi^{2}_{2}+(\xi_{1}+\xi_{2}-1)^{2}-1\right)d\xi_{1}d\xi_{2}.$
(18)

Integrating over the region
$\Delta(\xi_{1},\xi_{2})\equiv\\{0<\xi_{1}<1,0<\xi_{2}<1,1<\xi_{1}+\xi_{2}<2\\}$
simply ensures $0<\xi_{3}<1$. We would like to now apply (17) again, however
we require a transformation so that the argument
$\xi^{2}_{1}+\xi^{2}_{2}+\xi^{2}_{3}-1$ is of the required form. To do this,
we transform the inner integral to one with respect to $du$, where
$u=\xi_{1}^{2}+\xi_{2}^{2}+(\xi_{1}+\xi_{2}-1)^{2}-1$ and
$du=2(2\xi_{1}+\xi_{2}-1)d\xi_{1}$. This leaves

$S=\iint\limits_{\Delta(u,\xi_{2})}\frac{f(\xi_{1}(u),\xi_{2},\xi_{1}(u)+\xi_{2}-1)}{2(2\xi_{1}(u)+\xi_{2}-1)}\delta\left(u\right)dud\xi_{2}.$
(19)

Now, we may apply (17), being careful to include only the part of
$u(\xi_{1},\xi_{2})=0$ that lies in $\Delta(\xi_{1},\xi_{2})$. After some
manipulation of our definition of $u$ we find that, of the two branches for
which $u=0$, the one with

$\xi_{1}=\frac{1-\xi_{2}+\sqrt{(1-\xi_{2})(3\xi_{2}+1)}}{2},\ \ 0<\xi_{2}<1$
(20)

is in the region $\Delta(\xi_{1},\xi_{2})$. After applying (17),

$S=\int\limits_{0}^{1}\frac{f(\xi_{1}(\xi_{2}),\xi_{2},\xi_{1}(\xi_{2})+\xi_{2}-1)}{2\sqrt{(1-\xi_{2})(3\xi_{2}+1)}}d\xi_{2}.$
(21)

This form of $S$ is suitable for numerical integration, where the integrand as
$\xi_{2}\rightarrow 1$ (from below) can be shown to approach zero via
L’Hospital’s rule.

###  II.2 Evaluation of the WKE Collision Integral

Just as in the evaluation of $\frac{dI(y)}{dy}$, this explicit algebraic
approach works to evaluate (3). To start, we rewrite (3) in terms of the
function
$f(\omega_{1},\omega_{2},\omega_{3},\omega)=\omega_{1}\omega_{2}\omega_{3}\omega
n_{1}n_{2}n_{3}n_{\omega}\left(\frac{1}{n_{\omega}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right)$:

$\begin{split}\frac{\partial\mathcal{N}_{\omega}}{\partial
t}=128\pi\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}f(\omega_{1},\omega_{2},\omega_{3},\omega)\\\
\times\left(\delta(\omega^{2}_{1}+\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})+\delta(\omega^{2}_{1}-\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})+\delta(-\omega^{2}_{1}+\omega^{2}_{2}-\omega^{2}_{3}-\omega^{2})\right.\\\
+\left.\delta(-\omega^{2}_{1}-\omega^{2}_{2}+\omega^{2}_{3}-\omega^{2})\right)d\omega_{1}d\omega_{2}d\omega_{3},\end{split}$
(22)

For each $\delta$-function with quadratic arguments in $\omega$, we perform
the procedure outlined in the previous subsection. After a good deal of
simplification, this results in the following expression for
$\frac{\partial\mathcal{N}_{\omega}}{\partial t}$:

$\displaystyle\frac{\partial\mathcal{N}_{\omega}}{\partial t}=$
$\displaystyle\frac{\partial\mathcal{N}_{\omega}^{(1a)}}{\partial
t}+\frac{\partial\mathcal{N}_{\omega}^{(1b)}}{\partial
t}+\frac{\partial\mathcal{N}_{\omega}^{(2)}}{\partial
t}+\frac{\partial\mathcal{N}_{\omega}^{(3a)}}{\partial
t}+\frac{\partial\mathcal{N}_{\omega}^{(3b)}}{\partial t},\text{ \ where}$
(23) $\displaystyle\frac{\partial\mathcal{N}_{\omega}^{(1a)}}{\partial t}=$
$\displaystyle\int\limits_{0}^{\sqrt{4/3}\omega}\frac{f(\omega_{1}^{(1a)}(u),\omega_{2}^{(1a)}(u),\omega_{3}^{(1a)}(u),\omega)}{2\sqrt{4\omega^{2}-3u^{2}}}du$
(24) $\displaystyle\frac{\partial\mathcal{N}_{\omega}^{(1b)}}{\partial t}=$
$\displaystyle\int\limits_{\omega}^{\sqrt{4/3}\omega}\frac{f(\omega_{1}^{(1b)}(u),\omega_{2}^{(1b)}(u),\omega_{3}^{(1b)}(u),\omega)}{2\sqrt{4\omega^{2}-3u^{2}}}du$
(25) $\displaystyle\frac{\partial\mathcal{N}_{\omega}^{(2)}}{\partial t}=$
$\displaystyle
2\int\limits_{0}^{\infty}\frac{f(\omega_{1}^{(2)}(u),\omega_{2}^{(2)}(u),\omega_{3}^{(2)}(u),\omega)}{2\sqrt{u^{2}+4\omega^{2}}}du$
(26) $\displaystyle\frac{\partial\mathcal{N}_{\omega}^{(3a)}}{\partial t}=$
$\displaystyle\int\limits_{0}^{\infty}\frac{f(\omega_{1}^{(3a)}(u),\omega_{2}^{(3a)}(u),\omega_{3}^{(3a)}(u),\omega)}{2\sqrt{u^{2}+4\omega^{2}}}du$
(27) $\displaystyle\frac{\partial\mathcal{N}_{\omega}^{(3b)}}{\partial t}=$
$\displaystyle\int\limits_{0}^{\infty}\frac{f(\omega_{1}^{(3b)}(u),\omega_{2}^{(3b)}(u),\omega_{3}^{(3b)}(u),\omega)}{2\sqrt{u^{2}+4\omega^{2}}}du.$
(28)

The first delta function on the RHS of (22) corresponds to
$\partial\mathcal{N}_{\omega}^{(1)}/\partial t$, the next two delta functions
(due to symmetry) produce identical contributions and are combine in
$\partial\mathcal{N}_{\omega}^{(2)}/\partial t$, and the last delta function
corresponds to $\partial\mathcal{N}_{\omega}^{(3)}/\partial t$. The separate
$a$ and $b$ contributions result from multiple branches of the resonant
manifold parameterized by $u$. The corresponding functions of
$\omega_{i}^{j}(u)$ are given by:

$\displaystyle(1a)$
$\displaystyle\left\\{\begin{array}[]{l}\omega_{2}(u)=\frac{1}{3}\left(\omega+\sqrt{4\omega^{2}-3u^{2}}\right)\\\
\omega_{1}(u)=\frac{1}{2}\left(\omega-\omega_{2}(u)+\sqrt{(\omega-\omega_{2}(u))(3\omega_{2}(u)+\omega)}\right)\\\
\omega_{3}(u)=\omega_{1}(u)+\omega_{2}(u)-\omega\end{array}\right.$ (32)
$\displaystyle(1b)$
$\displaystyle\left\\{\begin{array}[]{l}\omega_{2}(u)=\frac{1}{3}\left(\omega-\sqrt{4\omega^{2}-3u^{2}}\right)\\\
\omega_{1}(u)=\frac{1}{2}\left(\omega-\omega_{2}(u)+\sqrt{(\omega-\omega_{2}(u))(3\omega_{2}(u)+\omega)}\right)\\\
\omega_{3}(u)=\omega_{1}(u)+\omega_{2}(u)-\omega\end{array}\right.$ (36)
$\displaystyle(2)$
$\displaystyle\left\\{\begin{array}[]{l}\omega_{2}(u)=-\omega+\sqrt{u^{2}+4\omega^{2}}\\\
\omega_{1}(u)=\frac{1}{2}\left(\omega-\omega_{2}+\sqrt{(3\omega+\omega_{2}(u))(\omega_{2}(u)-\omega)}\right)\\\
\omega_{3}(u)=\omega_{1}(u)+\omega_{2}(u)-\omega\end{array}\right.$ (40)
$\displaystyle(3a)$
$\displaystyle\left\\{\begin{array}[]{l}\omega_{3}(u)=\omega+\sqrt{u^{2}+4\omega^{2}}\\\
\omega_{2}(u)=\frac{1}{2}\left(\omega+\omega_{3}(u)+\sqrt{(\omega_{3}(u)+\omega)(\omega_{3}(u)-3\omega)}\right)\\\
\omega_{1}(u)=\omega_{3}(u)+\omega-\omega_{2}(u)\end{array}\right.$ (44)
$\displaystyle(3b)$
$\displaystyle\left\\{\begin{array}[]{l}\omega_{3}(u)=\omega+\sqrt{u^{2}+4\omega^{2}}\\\
\omega_{2}(u)=\frac{1}{2}\left(\omega+\omega_{3}(u)-\sqrt{(\omega_{3}(u)+\omega)(\omega_{3}(u)-3\omega)}\right)\\\
\omega_{1}(u)=\omega_{3}(u)+\omega-\omega_{2}(u)\end{array}\right.$ (48)

This is a form of the collision integral that is suitable for numerical
integration. When computing $P_{WKE}$ directly from the collision integral in
the main text, $n_{k}\equiv n(k(\omega))$ is evaluated explicitly from the
idealized, truncated KZ spectrum given by

$n_{k}=\left\\{\begin{array}[]{l}k^{-1},\ k_{a}\leq k\leq k_{c}\\\ 0,\
\text{otherwise}\end{array}\right.\\\ $ (50)

When the simulated spectrum is instead used in this evaluation (for the two
points in figure 3b of the main text), we use piecewise-linear interpolation
of the simulated spectrum to evaluate $n_{k}$ for $k\notin\Lambda_{L}$. In
both of these cases, $n_{k}=0$ after some large value of $k$ with a
corresponding $u$, meaning that the upper integral bounds of
$\partial\mathcal{N}_{\omega}^{(3a)}/\partial t$ and
$\partial\mathcal{N}_{\omega}^{(3b)}/\partial t$ in our numerical evaluations
can be taken to be finite. All integrals are evaluated via adaptive
quadrature.

##  III Selection of Numerical Parameters

For a fair comparison between domains of different $L$, we would like to keep
key quantities like the length-averaged Hamiltonian density $H$,
$\varepsilon$, and $P$ approximately constant as $L$ increases, while allowing
other quantities to vary. Choosing the parameters of each numerical simulation
to achieve this goal turns out to be a non-trivial task. Given the definition
of our Fourier series

$\hat{\psi}_{k}=\frac{1}{L}\int\limits_{0}^{L}\psi_{\boldsymbol{x}}e^{-ik\cdot\boldsymbol{x}}d\boldsymbol{x},$
(51)

the spectral amplitude $\hat{\psi}_{k}$ is normalized such that the length-
averaged total action does not depend on $L$, as can be seen via Parseval’s
theorem [1],

$\frac{1}{L}\int\limits_{0}^{L}|\psi_{\boldsymbol{x}}|^{2}d\boldsymbol{x}=\sum_{k\in\Lambda_{L}^{2}}|\hat{\psi}_{k}|^{2}.$
(52)

The key parameters that control the energy content of our simulations are
forcing and dissipation. Thus, we attempt to keep the action injection rate,
controlled by the standard deviation of the Gaussian forcing $\sigma_{F}$, and
the dissipation rate, controlled by $\nu_{k}$, the same across domains of
different $L$ at each $\varepsilon$ of interest. After a careful scaling
analysis, we determine that this is achieved by $\nu_{k}$ remaining constant
in $L$, while $\sigma_{F}\sim 1/\sqrt{L}$. Under this scheme, we find $H$,
$\varepsilon$, $P$, and other related quantities remain close to constant as
$L$ increases.

##  IV On the Interscale-flux and Dissipation Rate Distributions

Figure 1: The distributions of $P(k_{b},t)$ (a) and $P_{d}(t)$ (b) for the
highest $\varepsilon$ data, for $L=2\pi$ (blue), $L=2\pi$ (red), $L=2\pi$
(magenta), and $L=2\pi$ (green). The distributions given by (——) represent
Gaussian (a) and Log-normal (b) distributions of equal mean and variance to
the data they outline.

In this section, we present our measurements of inter-scale energy flux,
defined explicitly as $P(k_{b}=300,t)$

$P(k_{b},t)=\sum_{|k|<k_{b}}\omega_{k}\frac{d|\hat{\psi}_{k}|^{2}}{dt},$ (53)

where the derivative on the RHS may be computed from the MMT equation given
some statistically stationary field $\hat{\psi}_{k}(t)$. We begin with the
flux distribution $P(k_{b}=300,t)$, depicted in figure 1a of this supplemental
material. Results only for the highest nonlinearity case are shown, however
they are representative of the other cases. $P(t)$ takes a Gaussian
distribution whose standard deviation is much larger than its mean. This is a
similar result to our previous work in a 2D MMT model with a Nonlinear
Schrödinger Equation-like dispersion relation [5], suggesting a Gaussian
energy cascade is a general feature of wave turbulence. In fact, a simple
argument supports a Gaussian energy cascade: first, consider the form of (53),
which expresses the instantaneous inter-scale flux as a sum over a large
number of modes. If the the instantaneous time rate of change of energy at
each of these modes is assumed to be independent, then the Central Limit
Theorem (CLT) can be used to show $P(k_{b},t)$ approaches a Gaussian
distribution as the number of modes in the domain becomes large. Similarly,
the inter-scale flux can be though of as a sum over a huge number of
interactions, where again the CLT may be applied.

As $L$ is increased, the standard deviation of $P$ is decreased. Measurements
of the standard deviation reveal that $\sigma(P(k_{b},t))\sim 1/\sqrt{L}$.
This too can be explained in the context of the CLT, as the variance of
$P(k_{b},t)$ is given by a large sum over
$\sigma^{2}(\frac{d|\hat{\psi}_{k}|^{2}}{dt})$.
$\sigma^{2}(\frac{d|\hat{\psi}_{k}|^{2}}{dt})$ scales linearly with the number
of modes with $k<k_{b}$, which scales linearly with $L$. On the other hand,
one can use the MMT equation to show that
$\sigma^{2}(\frac{d|\hat{\psi}_{k}|^{2}}{dt})\sim 1/L^{2}$, given that
$|\hat{\psi}_{k}|\sim 1/\sqrt{L}$ by Parseval’s theorem (52). The end result
is that $\sigma^{2}(P(k_{b},t))\sim 1/L$, leading to the desired
$\sigma(P(k_{b},t))\sim 1/\sqrt{L}$.

In figure 1b, we provide the steady time distribution of high wave number
dissipation rate, derived directly from the dissipation term to be

$P_{d}(t)\equiv-\sum_{|k|>900}2\omega_{k}\nu_{k}|\hat{\psi}_{k}|^{2}.$ (54)

The distribution is log-normal, and the standard deviation exhibits the same
$1/\sqrt{L}$ scaling due to similar arguments. Log-normal distributions of
dissipation rate have been described in flow turbulence [6, 7], however we are
not aware of similar results in WT.

##  V A Fast Direct Method for Computing Non-local Interactions

To compute the non-local contribution to the small-scale energy evolution rate
for a stationary spectrum, we begin by using a control volume argument to
express the average dissipation rate directly in terms of the nonlinearity of
the MMT model. This is achieved by

$P_{d}=J=-\sum_{|k|\geq 900}2\omega_{k}\nu_{k}\frac{dn_{k}}{dt}=-\sum_{|k|\geq
900}2\omega_{k}\nu_{k}\sum_{123}2\text{Im}\langle\hat{\psi}_{1}\hat{\psi}_{2}\hat{\psi}_{3}^{*}\hat{\psi}_{k}^{*}\rangle\delta_{K}(k_{1}+k_{2}-k_{3}-k).$
(55)

We note that for this section (in contrast to the previous section) $P_{d}$
indicates a time-averaged quantity. To proceed, we would like to compute the
spectral evolution at some wavenumber $k\in\Lambda_{L}$ to interactions with
the set of wave modes that reside in the forcing range. Let us call this set
of forced modes $A\subseteq\Lambda_{L}$. We will consider an interaction to
involve $A$ if _one or more_ waves in the interaction are part of $A$. To
express this clearly, it is useful to define the following set of 3-wave
triples $(k_{1},k_{2},k_{3})$:

$S^{(i)}_{A}\equiv\left\\{(k_{1},k_{2},k_{3})\ |\ (k_{i}\in
A)\cup\left(\bigcup_{j\neq i}(k_{j}\in\Lambda_{L})\right)\right\\}.$ (56)

$S_{A}^{(i)}$ denotes the set of all 3-wave triples where the $i$-th component
is part of $A$ and the other two components are explicitly arbitrary. We are
now ready to formally define $\left.\frac{dn(k)}{dt}\right|_{A}$, the spectral
evolution involving interactions with $A$:

$\left.\frac{dn_{k}}{dt}\right|_{A}\equiv\sum_{(k_{1},k_{2},k_{3})\in
S_{A}^{(1)}\cup S_{A}^{(2)}\cup
S_{A}^{(3)}}2\text{Im}\langle\hat{\psi}_{1}\hat{\psi}_{2}\hat{\psi}^{*}_{3}\hat{\psi}^{*}_{k}\rangle\delta_{K}(k_{1}+k_{2}-k_{3}-k).$
(57)

When substituted into (55), we will obtain the average dissipation rate due to
interactions with the forcing range. While the definition of
$\left.\frac{dn_{k}}{dt}\right|_{A}$ is clear in this form, it is expensive to
compute. However, it is possible to rewrite the expression for the spectral
evolution due interactions with $A$ as a carefully constructed series of sums
which can be computed much more quickly.

We start by reformulating (57) in terms of the evolution of $\psi(k)$,

$\left.\frac{\partial n_{k}}{\partial
t}\right|_{A}=\left\langle\psi_{k}\left.\frac{\partial\hat{\psi}^{*}_{k}}{\partial
t}\right|_{A}+\hat{\psi}^{*}_{k}\left.\frac{\partial\hat{\psi}_{k}}{\partial
t}\right|_{A}\right\rangle,$ (58)

with

$\left.i\frac{\partial\hat{\psi}_{k}}{\partial
t}\right|_{A}\equiv\omega_{k}\psi_{k}+\sum_{(k_{1},k_{2},k_{3})\in
S_{A}^{(1)}\cup S_{A}^{(2)}\cup
S_{A}^{(3)}}\psi_{1}\psi_{2}\hat{\psi}^{*}_{3}\delta^{12}_{3k}.$ (59)

It follows from the inclusion-exclusion principle that the sum in the
nonlinear term of (59) can be rewritten as

$\displaystyle\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(1)}\cup S_{A}^{(2)}\cup
S_{A}^{(3)}}=$ $\displaystyle\sum_{(k_{1},k_{2},k_{3})\in
S_{A}^{(1)}}+\sum_{(k_{1},k_{2},k_{3})\in
S_{A}^{(2)}}+\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(3)}}$ (60) $\displaystyle-$
$\displaystyle\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(1)}\cap
S_{A}^{(2)}}-\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(2)}\cap
S_{A}^{(3)}}-\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(1)}\cap S_{A}^{(3)}}$
$\displaystyle+$ $\displaystyle\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(1)}\cap
S_{A}^{(2)}\cap S_{A}^{(3)}}.$

The advantage of this form is that sums over $S_{A}^{(i)}$ and intersections
of $S_{A}^{(i)}$ can be computed quickly via Fourier transforms. Consider the
term

$\hat{NL}(k,S_{A}^{(1)})=\sum_{(k_{1},k_{2},k_{3})\in
S_{A}^{(1)}}\hat{\psi}_{1}\hat{\psi}_{2}\hat{\psi}^{*}_{3}\delta^{12}_{3k}.$
(61)

We already know that $\hat{NL}(k,\Lambda_{L}^{3})$ (where the sum is taken
over $(k_{1},k_{2},k_{3})\in\Lambda_{L}^{3}$) is the Fourier domain
representation of a quantity that is readily computed in physical domain, as
this is exactly the basis of the pseudospectral method used to solve the MMT
equation. Let’s call the inverse Fourier transform of (61)
$NL(x,S_{A}^{(1)})$. This can be written as

$NL(x,S_{A}^{(1)})=\psi_{x}\psi_{x}^{*}\sum_{k_{1}\in
A}\psi(k_{1})e^{ik_{1}x}.$ (62)

However, the last term in the product (62) is simply the expression of an
ideal band-pass filter applied to $\psi_{x}$ that admits only those modes in
set $A$, and sets to $0$ any mode with $k\notin A$. Let $B_{A}$ be such a
filter, so that

$NL(x,S_{A}^{(1)})=\psi_{x}\psi_{x}^{*}B_{A}\left(\psi_{x}\right).$ (63)

By exactly the same argument, the sums over intersections may be computed. For
example:

$NL(x,S_{A}^{(1)}\cap
S_{A}^{(3)})=\psi_{x}B_{A}\left(\psi_{x}^{*}\right)B_{A}\left(\psi_{x}\right).$
(64)

The inverse Fourier transform can then be used to obtain the desired term in
the sum (60). Every term in the sum (60) may be computed in physical domain
according to this method, meaning (58) may also be computed in this way. For
$N$ the number of modes, this method has $O(N\log N)$ complexity, as opposed
to the $O(N^{2})$ complexity of computing (57) directly. The desired quantity
$J_{nl}\equiv J_{A}$ for $A=[20,30]$ can then be computed, with

$J_{A}=-\sum_{|k|\geq
900}2\omega_{k}\nu_{k}\left.\frac{dn_{k}}{dt}\right|_{A}$ (65)

Finally, we note that if one considers the set of interactions in
$S_{A^{C}}^{(1)}\cap S_{A^{C}}^{(2)}\cap S_{A^{C}}^{(3)}$, where $A^{C}$ is
the compliment to $A$, one can directly obtain

$\sum_{(k_{1},k_{2},k_{3})\in S_{A}^{(1)}\cup S_{A}^{(2)}\cup
S_{A}^{(3)}}=\sum_{(k_{1},k_{2},k_{3})\in\Lambda_{L}^{3}}-\sum_{(k_{1},k_{2},k_{3})\in
S_{A^{C}}^{(1)}\cap S_{A^{C}}^{(2)}\cap S_{A^{C}}^{(3)}},$ (66)

where the RHS can also be evaluated in a pseudospectral way according to the
method we have described.

## References

  * Nazarenko [2011] S. Nazarenko,  _Wave Turbulence_, Lecture Notes in Physics, Vol. 825 (Springer, Berlin, Heidelberg, 2011).
  * Zakharov  _et al._ [1992] V. E. Zakharov, V. S. L’vov, and G. Falkovich,  _Kolmogorov Spectra of Turbulence I_, edited by F. Calogero, B. Fuchssteiner, G. Rowlands, H. Segur, M. Wadati, and V. E. Zakharov, Springer Series in Nonlinear Dynamics (Springer, Berlin, Heidelberg, 1992).
  * Majda  _et al._ [1997] A. J. Majda, D. W. McLaughlin, and E. G. Tabak, A one-dimensional model for dispersive wave turbulence, Journal of Nonlinear Science 7, 9 (1997), number: 1.
  * Zakharov  _et al._ [2004] V. Zakharov, F. Dias, and A. Pushkarev, One-dimensional wave turbulence, Physics Reports 398, 1 (2004).
  * Hrabski and Pan [2022] A. Hrabski and Y. Pan, On the properties of energy flux in wave turbulence, Journal of Fluid Mechanics 936, 10.1017/jfm.2022.106 (2022), publisher: Cambridge University Press.
  * Mandelbrot [1974] B. B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, Journal of Fluid Mechanics 62, 331 (1974), publisher: Cambridge University Press.
  * Pearson and Fox-Kemper [2018] B. Pearson and B. Fox-Kemper, Log-Normal Turbulence Dissipation in Global Ocean Models, Physical Review Letters 120, 094501 (2018), publisher: American Physical Society.