aainstitutetext: Department of Physics and IPAP, Yonsei University,
Seoul 03722, Republic of Koreabbinstitutetext: Theoretical Physics Department,
CERN,
1211 Geneva 23, Switzerlandccinstitutetext: Department of Physics and
Institute of Quantum Systems (IQS), Chungnam National University,
Daejeon 34134, Republic of Koreaddinstitutetext: Center for Theoretical
Physics of the Universe, Institute for Basic Science (IBS),
Daejeon, 34126, Koreaeeinstitutetext: Korea Institute for Advanced Study,
Seoul 02455, South Korea

# Effective Theory Approach for Axion Wormholes

Dhong Yeon Cheong a, e, 1 Seong Chan Park,111Corresponding author. c,d,e,1 and
Chang Sub Shin<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We employ the effective field theory approach to analyze the characteristics
of Euclidean wormholes within axion theories. Using this approach, we obtain
non-perturbative instantons in various complex scalar models with and without
a non-minimal coupling to gravity, as well as models featuring the $R^{2}$
term for a range of coupling values. This yields a series of analytical
expressions for the axion wormhole action, shedding light on the model
parameters and field dependencies of contributions in both the ultraviolet and
infrared domains. Consequently, model-dependent local operators that disrupt
axion shift symmetries are generated at lower energy levels. This, in turn,
provides crucial insights into the gravitational influences on the axion
quality problem.

††preprint: CERN-TH-2023-184

##  1 Introduction

The axion, a pseudo-Nambu-Goldstone Boson of a global $U(1)$ symmetry,
provides a novel solution to the strong CP problem Peccei:1977hh ;
Peccei:1977ur ; Wilczek:1977pj ; Weinberg:1977ma , and is associated with
various phenomenological puzzles, such as the dark matter problem
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah , dark energy problem or both
(see Kim:2008hd ; Marsh:2015xka ; DiLuzio:2020wdo ; Choi:2020rgn ;
Ferreira:2020fam  for recent reviews). However, gravity may jeopardize these
solutions, as no global symmetry is allowed in a gravitating system
Hawking:1987mz ; Giddings:1988cx ; Banks:2010zn ; Witten:2017hdv ;
Harlow:2018jwu .

The challenge of preserving the shift symmetry of the axion, the so-called
Peccei-Quinn (PQ) symmetry $U(1)_{\text{PQ}}$, against numerous potential
sources of global symmetry-breaking has been coined as the axion quality
problem Dine:1986bg ; Kamionkowski:1992mf ; Barr:1992qq ; Holman:1992us ;
Ghigna:1992iv . Many attempts have been made to resolve this issue
Witten:1984dg ; Randall:1992ut ; Cheng:2001ys ; Izawa:2002qk ; Choi:2003wr ;
Lillard:2018fdt ; Ardu:2020qmo ; Alvey:2020nyh ; Hamaguchi:2021mmt ;
Dvali:2022fdv ; Cheong:2022ikv ; Bonnefoy:2022vop ; Burgess:2023ifd ;
Choi:2023gin , but subtleties still exist as quantum gravitational
contributions are largely unknown and non-perturbative in their natures.

One particularly important direction to calculate a non-perturbative
contribution is to estimate the effect of the Euclidean axion wormhole, the
stationary point solution for the Euclidean action with gravity whose throat
is connected to an asymptotically flat region where the global PQ charge flows
Lee:1988ge ; Giddings:1989bq ; Abbott:1989jw ; Coleman:1989zu ; Kallosh:1995hi
; Hebecker:2016dsw ; Alonso:2017avz ; Hertog:2018kbz ; Hebecker:2018ofv ;
Loges:2022nuw ; Andriolo:2022rxc ; Loges:2023ypl ; Jonas:2023ipa . If the
wormhole action $S_{wh}$ is large enough, a semi-classical approximation can
be made without needing a complete UV theory of gravity. As the gravitational
impact on IR physics is suppressed by an exponential factor $e^{-S_{wh}}$, to
a good approximation, $S_{wh}\gtrsim 200$ guarantees the axion quality
Kallosh:1995hi . An early attempt by Giddings and Strominger Giddings:1989bq
suggested $S_{wh}\sim M_{P}/f_{a}$, thus, $f_{a}\lesssim M_{P}/200$ seemed to
define the preferred parameter region for the axion quality, however, it
turned out that the dynamics of the accompanying radial scalar partner of the
axion completely modifies the conclusion, as $S_{wh}\sim\log M_{P}/f_{a}$
Lee:1988ge ; Abbott:1989jw ; Coleman:1989zu ; Kallosh:1995hi .

Recently, it is noticed that a large non-minimal coupling of the complex
scalar to the Ricci scalar ($\xi\Phi^{*}\Phi R$) can make the wormhole action
large enough in both metric and Palatini formalisms of gravity
Hamaguchi:2021mmt ; Cheong:2022ikv . Here, the complex scalar
$\Phi=\frac{1}{\sqrt{2}}\phi e^{i\theta}$ has an axion as its complex phase
field and its radial partner. The non-minimal coupling term naturally arises
in an effective theory or is radiatively generated Ford:1981xj , and has been
actively investigated for inflationary cosmology in metric, Palatini or both
formulations Futamase:1987ua ; Bezrukov:2007ep ; Park:2008hz ; Bezrukov:2010jz
; Hamada:2014wna ; Hamada:2014iga ; Jinno:2019und .

On the other hand, both studies in Ref. Hamaguchi:2021mmt ; Cheong:2022ikv
heavily rely on numerical calculations and still need an analytic
understanding of the full parameter space. Therefore, in this paper, we
enlarge our scopes and attempt to study the analytical structure of these
Euclidean wormholes. In particular, we take an effective field theory approach
by imposing boundary conditions of wormholes with IR field degrees of freedom,
along with identifying the generation of associated PQ breaking operators at
low energies. This approach dramatically simplifies our analysis and allows us
to obtain an analytic expression of the wormhole action that is solely
determined by the IR fields. The solution provides a transparent understanding
of the parameter dependence of the action and perspectives on generalizations
with possible limitations.

This paper is organized as follows. We first present the generic Lagrangian
structure associated with axion theories, and take the effective theory
approach to obtain a generic formalism representing the Euclidean wormhole
solutions. We demonstrate that the wormhole action can be decomposed to its UV
and IR components, with the IR component containing the UV cutoff of the
theory. We then demonstrate this formalism in several different examples, each
representing a different class of models. We conclude with the implications of
this formalism.

##  2 Eulcidean Wormhole for General Axion Models

###  2.1 General Lagrangian

We introduce a set of axion fields
$\\{\theta\\}=\\{\theta^{I}(x)\\}_{I=1,2,\cdots,n_{a}}$, characterized as
‘angular fields’ due to their periodic and compact nature, adhering to the
discrete gauge symmetries $\theta^{I}(x)\equiv\theta^{I}(x)+2\pi$. Alongside
these symmetries, the axions exhibit continuous global symmetries, known as
the PQ symmetry, described by the transformation

$\displaystyle U(1)_{\rm
PQ}^{I}:\theta^{I}\rightarrow\theta^{I}+c^{I}\,,~{}~{}c^{I}\in\mathbb{R}.$ (1)

In conjunction with the axions, we introduce ‘radial fields’,
$\\{\phi\\}=\\{\phi^{A}(x)\\}_{A=1,2,\cdots,n_{s}}$, whose vacuum expectation
values (VEVs) determine the axion decay constants. These dynamic scalar fields
are commonly observed in various axion scenarios, including UV completions of
the PQ symmetry Abbott:1989jw ; Kallosh:1995hi ; Alonso:2017avz ;
Alvey:2020nyh . Notably, the number of radial scalars can differ from the
count of the axions, i.e., $n_{s}\neq n_{a}$. The general action for the
axions and their radial fields in the Einstein frame is given by

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left(-\frac{M_{P}^{2}}{2}R+\frac{1}{2}\sum_{A,B}^{n_{s}}G_{AB}(\phi)\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{B}+\frac{1}{2}\sum_{I,J}^{n_{a}}(f^{2}_{a}(\phi))_{IJ}\partial_{\mu}\theta^{I}\partial^{\mu}\theta^{J}\right),$
(2)

where $M_{P}=1/\sqrt{8\pi G}\approx 2.4\times 10^{18}~{}{\rm GeV}$ denotes the
reduced Planck mass, $G_{AB}(\phi)$ and $(f^{2}_{a}(\phi))_{IJ}$ are field
dependent metrics in the field spaces of the radial scalar fields and the
axion fields, respectively. Implicitly, we assumed that a potential for the
radial fields provides the vacuum expectation values of $\phi^{A}$ as
$\langle\phi^{A}\rangle$ so that
$(f_{a}^{2}(\langle\phi\rangle))_{IJ}=(f_{a}^{2})_{IJ}$; however, its explicit
form is not relevant for our discussions in the IR region Abbott:1989jw ;
Coleman:1989zu . The CP properties of the axions and the radial fields are
given as

$\displaystyle
CP:(\theta^{I},\phi^{A})\to(-\theta^{I},\phi^{A}),~{}~{}{}^{\forall}I,A,$

thus no kinetic mixing is allowed between any axion and raidal field.

The discrete gauge symmetry of the axions becomes more transparent by
dualizing the axion field to a three form field strength $H_{I\mu\nu\rho}$:

$\displaystyle\mathcal{S}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left(-\frac{M_{P}^{2}}{2}R+\frac{1}{2}G_{AB}(\phi)\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{B}\right.$
$\displaystyle\hskip
71.13188pt\left.+\,\frac{1}{12}(f^{-2}_{a}(\phi))^{IJ}H_{I\mu\nu\rho}H^{\mu\nu\rho}_{J}+\frac{1}{6}\theta^{I}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}H_{I\nu\rho\sigma}\right),$
(3)

where the covariant Levi-Civita tensor is given by
$\epsilon_{\mu\nu\rho\sigma}=(-1)^{\pi(\mu\nu\rho\sigma)}\sqrt{-g}=-(\epsilon^{\mu\nu\rho\sigma})^{-1}$
for the signature of the permutation $\pi(\mu\nu\rho\sigma)$. Using the
differential form notation, the equations of motion for $\theta^{I}$ give
$H_{I}=(1/6)H_{I\mu\nu\rho}dx^{\mu}\wedge dx^{\nu}\wedge dx^{\rho}=dB_{I}$,
where $B_{I}=(1/2)B_{I\mu\nu}dx^{\mu}\wedge dx^{\nu}$ is the 2-form gauge
field. The equations of motion for $H_{I}$ give
$H_{I}=(f^{2}_{a}(\phi))_{IJ}\star d\theta^{J}$. Inserting these solutions of
$H_{I}$ to the action, we recover the action for the axions in Eq. (2).

For the wormhole solution, we derive the Euclidean action after Wick rotation

$\displaystyle\mathcal{S}_{E}=$ $\displaystyle\int
d^{4}x\sqrt{g^{E}}\left(-\frac{M_{P}^{2}}{2}R_{E}+\frac{1}{2}G_{AB}(\phi)\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{B}+\frac{1}{12}(f^{-2}_{a}(\phi))^{IJ}H_{I\mu\nu\rho}H^{\mu\nu\rho}_{J}\right)$
$\displaystyle-i\int
d^{4}x\sqrt{g^{E}}\left(\frac{1}{6}\theta^{I}\epsilon_{E}^{\mu\nu\rho\sigma}\partial_{\mu}H_{I\nu\rho\sigma}\right),$
(4)

where the Euclidean metric is $g^{E}_{\mu\nu}=\delta_{\mu\nu}$ in the limit of
flat space, the Euclidean curvature scalar is $R_{E}$, and the Euclidean Levi-
Civita tensor is given by
$\epsilon^{E}_{\mu\nu\rho\sigma}=(-1)^{\pi(\mu\nu\rho\sigma)}\sqrt{g^{E}}=(\epsilon_{E}^{\mu\nu\rho\sigma})^{-1}$.

Figure 1: A schematic diagram of the Euclidean wormhole geometry

The geometry of the Euclidean wormhole is solved by varying $\delta S_{E}$:

$\displaystyle
ds^{2}_{wh}=g_{rr}^{E}dr^{2}+r^{2}d\Omega_{3}^{2}=\frac{dr^{2}}{\left(1-L_{0}^{4}/r^{4}\right)}+r^{2}d\Omega_{3}^{2},$
(5)

where $L_{0}$ denotes the size of the wormhole throat radius. The schematic
shape of the Euclidean wormhole is depicted in Fig. 1. It is intriguing to
notice that the Ricci scalar in the geometry Eq. (5) is solely determined by
$L_{0}$ and the distance from the center of the wormhole

$\displaystyle R_{E}=-\frac{6L_{0}^{4}}{r^{6}},~{}~{}~{}~{}~{}L_{0}<r<\infty.$
(6)

Effectively, the ‘interior’ of the wormhole is in $r<L_{0}$ and provides a
source of nonzero, integer Peccei-Quinn charge:

$\displaystyle\int_{\partial M}H_{I}=n_{I}\in\mathbb{Z},$ (7)

where $\partial M\sim S^{3}$ is the boundary of the wormhole.

Finally, the Euclidean action Eq. (2.1) with the wormhole geometry is given as

$\displaystyle S_{E}[\textrm{wormhole}]=S_{wh}[n,\phi]-in_{I}\theta^{I}$ (8)

for the background axion fields $\theta^{I}$. Here, the first term
$S_{wh}[n,\phi]$ is from the RHS in the first line of Eq. (2.1), and the
second term $-in_{I}\theta^{I}$ is from the RHS in the second
line.222Actually, $S_{wh}$ here is the half-wormhole action (or instanton
action) without the Gibbons-Hawking-York (GHY) boundary term York:1971hw ;
York:1972sj ; Gibbons:1976ue . The GHY surface term is less important in our
discussion, so we will not discuss its contribution explicitly. To make the
discussion more concise, we also use the term “wormhole action” to represent
$S_{wh}[n,\theta]$ in the whole discussion. The saddle point approximation in
the path integral yields PQ breaking local operators that are suppressed by
non-perturbative instanton factor

$\displaystyle e^{-S_{E}[\textrm{wormhole}]}=$ $\displaystyle
e^{-S_{wh}[n,\phi]+in_{I}\theta^{I}}.$ (9)

One can notice that the $2\pi$-periodicity of the axions provides the Dirac
quantization condition for the PQ charge.

###  2.2 Field dependent axion wormhole solutions

For the metric ansatz of the Euclidean wormhole Eq. (5), the field profiles
are also evaluated as the $O(4)$ symmetric solutions: $\phi^{A}(r)$ and
$H_{I\mu\nu\rho}(r)$. First of all, with the quantization condition Eq. (7),
the equation of motion for $H_{I\mu\nu\rho}$ gives the solution

$\displaystyle H_{I}=n_{I}\star\frac{dr}{2\pi^{2}}.$ (10)

Next, one can obtain the equations of motion for metric and scalar fields with
respect to the distance $r\in[L_{0},\infty)$ after inserting the solution of
$H_{I}$. Instead of using $r$, it is more convenient to take a new variable
$\tau$ defined as Arkani-Hamed:2007cpn

$\displaystyle\tau(r)=\frac{1}{4\pi^{2}L_{0}^{2}}\arctan\left(\sqrt{r^{4}/L_{0}^{4}-1}\right).$
(11)

The range of $\tau$ is from $\tau(L_{0})=0$ to $\tau(\infty)=1/8\pi
L_{0}^{2}=\tau_{\infty}$. The wormhole action $S_{wh}$ includes the Einstein-
Hilbert action, the kinetic term of the radial scalars, and the kinetic term
for the Euclidean 2-from gauge fields $B_{I\mu\nu}$ with the quantization
condition, Eq. (10). With the integral variable $\tau$, $S_{wh}$ can be
written as

$\displaystyle
S_{wh}=\int_{0}^{\tau_{\infty}}d\tau\left(12\pi^{4}M_{P}^{2}L_{0}^{4}+\frac{1}{2}G_{AB}(\phi)\frac{d\phi^{A}}{d\tau}\frac{d\phi^{B}}{d\tau}+\frac{1}{2}(f^{-2}_{a}(\phi))^{IJ}n_{I}n_{J}\right).$
(12)

The equations of motion for $\phi^{A}$ give

$\displaystyle
G_{AB}(\phi)\frac{d^{2}\phi^{B}}{d\tau^{2}}+\Gamma_{ABC}(\phi)\frac{d\phi^{B}}{d\tau}\frac{d\phi^{C}}{d\tau}=\frac{\partial}{\partial\phi^{A}}\left(\frac{1}{2}(f^{-2}_{a}(\phi))^{IJ}n_{I}n_{J}\right),$
(13)

where the ‘connection’ in the field space is

$\displaystyle\Gamma_{ABC}(\phi)=\frac{1}{2}\left(\frac{\partial
G_{AB}(\phi)}{\partial\phi^{C}}+\frac{\partial
G_{AC}(\phi)}{\partial\phi^{B}}-\frac{\partial
G_{BC}(\phi)}{\partial\phi^{A}}\right).$ (14)

The trace for the Einstein’s equation is calculated as

$\displaystyle
12\pi^{4}M_{P}^{2}L_{0}^{4}=\frac{1}{2}G_{AB}(\phi)\frac{d\phi^{A}}{d\tau}\frac{d\phi^{B}}{d\tau}-\frac{1}{2}(f^{-2}_{a}(\phi))^{IJ}n_{I}n_{J}.$
(15)

For given IR degrees of freedom $\phi^{A}$, the boundary conditions are given
by

$\displaystyle\left.\frac{d\phi^{A}(\tau)}{d\tau}\right|_{\tau=0}=0,\quad\phi^{A}(\tau_{\infty})=\phi^{A}.$
(16)

In general, the throat field values $\phi_{0}=\phi(\tau=0)$ are functions of
IR fields, with its sensitivity being quite model dependent. The throat radius
is directly given by the axion decay constant at the throat for given wormhole
charges $n_{I}$,

$\displaystyle
12\pi^{4}M_{P}^{2}L_{0}^{4}=\frac{1}{2}(f^{-2}_{a}(\phi_{0}))^{IJ}n_{I}n_{J}.$
(17)

One can easily notice that Eq. (15) with Eq. (17) resemble equations in
classical mechanics describing the motion of point particles. Using the
canonical momenta $p_{A}$ and the potential $V$ of $\phi$ given by

$\displaystyle p_{A}=G_{AB}(\phi)\frac{d\phi^{B}}{d\tau},\quad
V=\frac{1}{2}(f^{-2}_{a}(\phi_{0}))^{IJ}n_{I}n_{J}-\frac{1}{2}(f^{-2}_{a}(\phi))^{IJ}n_{I}n_{J},$
(18)

Eq. (13) can be written as

$\displaystyle\frac{dp_{A}}{d\tau}+\frac{1}{2}\left(\partial_{A}G^{BC}\right)p_{B}p_{C}+\partial_{A}V=0,\quad\frac{1}{2}G^{AB}p_{A}p_{B}+V=0,$
(19)

where $G^{AB}G_{BC}={\delta^{A}}_{C}$. The potential $V(\phi)$ and the metric
of the kinetic terms $G_{AB}(\phi)$ do not explicitely depend on $\tau$, so if
the solution exists, $p_{A}$s are given by the functions of $\phi^{A}$s, i.e.
$p_{A}(\phi)$. For given IR field values $\phi$, the throat values $\phi_{0}$
are determined by the equations,

$\displaystyle\sqrt{(f^{-2}_{a}(\phi_{0}))^{IJ}n_{I}n_{J}}\int_{\phi_{0}^{A}}^{\phi^{A}}\frac{d\varphi^{A}}{M_{P}}\frac{1}{G^{AB}(\varphi)p_{B}(\varphi)}=\frac{\pi\sqrt{6}}{4}\quad(\textrm{for
each $A$}),$ (20)

Utilizing the equations of motion, the wormhole action can be represented by
several ways.

$\displaystyle S_{wh}[n,\phi]$
$\displaystyle=\int_{0}^{\tau_{\infty}}d\tau(f^{-2}_{a}(\varphi))^{IJ}n_{I}n_{J}=3\pi^{3}M_{P}^{2}L_{0}^{2}+\int_{\phi_{0}}^{\phi}d\varphi^{A}\,p_{A}(\varphi)$
$\displaystyle=\frac{\pi\sqrt{6}}{4}M_{P}\sqrt{(f^{-2}_{a}(\phi_{0}))^{IJ}n_{I}n_{J}}+\int_{\phi_{0}}^{\phi}d\varphi^{A}\,p_{A}(\varphi).$
(21)

The last expression of Eq. (2.2) shows how $S_{wh}$ consists of the sum of the
UV (near the throat) and IR (the field dependent) contributions. The first
term of the RHS corresponds to the form of the Giddings-Strominger wormhole
Giddings:1989bq , and the second term represents the effect of the radial
field’s profile in the wormhole geometry. It contributes to the action
positively and determines how the wormhole action depends on the IR field
degrees of freedom.

##  3 Case Studies: Single Axion Models

###  3.1 Complex scalar model

The structure of the wormhole action becomes more transparent in single axion
models. We start with the minimal field content, a single axion plus a radial
mode associated with a single $U(1)_{\text{PQ}}$. In this case, we take
$\phi^{1}=\phi$, $\theta^{1}=\theta$, and $G_{11}=G(\phi)$,
$(f^{2}_{a}(\phi))_{11}=f^{2}_{a}(\phi)$. The action takes the form

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left(-\frac{M_{P}^{2}}{2}R+\frac{1}{2}G(\phi)\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}f^{2}_{a}(\phi)\partial_{\mu}\theta\partial^{\mu}\theta\right).$
(22)

Using the master formulas Eqs. (17) and (19), we obtain the throat radius
$L_{0}$ and the scalar field equation of motion as

$\displaystyle
L_{0}^{2}=\frac{1}{2\pi^{2}\sqrt{6}}\left(\frac{n}{M_{P}f_{a}(\phi_{0})}\right),\quad
p^{2}(\phi)=G(\phi)\left(\frac{n^{2}}{f^{2}_{a}(\phi)}-\frac{n^{2}}{f^{2}_{a}(\phi_{0})}\right).$
(23)

Using Eq. (20) we obtain the condition for the throat value of $\phi_{0}$ as
the function of IR field $\phi$.

$\displaystyle\int_{\phi}^{\phi_{0}}\frac{d\varphi}{M_{P}}\,\frac{\sqrt{G(\varphi)}}{\sqrt{f^{2}_{a}(\phi_{0})/f^{2}_{a}(\varphi)-1}}=\frac{\pi\sqrt{6}}{4}.$
(24)

Note that the solution $\phi_{0}$ is independent of $n$, while the radial
profile of $\phi(\tau)$ depends on $n$ as Eq. (23). The corresponding wormhole
action Eq. (2.2) becomes

$\displaystyle S_{wh}[n,\phi]$
$\displaystyle=\int_{0}^{\tau_{\infty}}d\tau\frac{n^{2}}{f^{2}_{a}(\varphi)}=\int_{\phi_{0}}^{\phi}\frac{d\varphi}{p(\varphi)}\frac{G(\varphi)n^{2}}{f^{2}_{a}(\varphi)}$
$\displaystyle=n\int_{\phi}^{\phi_{0}}\frac{d\varphi}{f_{a}(\varphi)}\frac{\sqrt{G(\varphi)}}{\sqrt{1-f^{2}_{a}(\varphi)/f^{2}_{a}(\phi_{0})}}.$
(25)

In the second line of the equation, we assume $\phi_{0}>\phi$. Because
$\phi_{0}$ is independent of $n$, the wormhole action is linearly dependent on
the wormhole charge $n$.

In the following sections, we derive several analytic results for the wormhole
action in various models of single axion cases.

####  3.1.1 Model with a non-minimal coupling to gravity

We consider an example of a complex scalar field. Here we allow a non-minimal
coupling to gravity Coule:1989xu ; Coule:1992pz . The action is

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left[-\left(\frac{M_{P}^{2}}{2}+\xi|\Phi|^{2}\right)R+\partial_{\mu}\Phi\partial^{\mu}\Phi^{*}\right].$
(26)

Decomposing the complex scalar into the radial mode $\phi$, and the angular
mode (axion) $\theta$,

$\displaystyle\Phi(x)=\frac{1}{\sqrt{2}}\phi(x)e^{i\theta(x)}$ (27)

the action in the Einstein frame becomes

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left(-\frac{M_{P}^{2}}{2}R+\frac{1}{2}G(\phi)\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}f^{2}_{a}(\phi)\partial_{\mu}\theta\partial^{\mu}\theta\right),$
(28)

where

$\displaystyle
G(\phi)=\frac{M_{P}^{2}(M_{P}^{2}+\xi\phi^{2}(1+\alpha\xi))}{(M_{P}^{2}+\xi\phi^{2})^{2}},\quad
f_{a}(\phi)=\frac{M_{P}\phi}{\sqrt{M_{P}^{2}+\xi\phi^{2}}}.$ (29)

We introduce a constant $\alpha$ that represent different gravity formalisms,

$\displaystyle\alpha$ $\displaystyle=6\quad\textrm{(Metric formalism)}$
$\displaystyle=0\quad\textrm{(Palatini formalism)}$ (30)

Following the equations of motion for Euclidean action, we get two master
formulae for the boundary conditions and the wormhole action. For boundary
conditions, Eq. (24) gives

$\displaystyle\sqrt{\frac{(1+\alpha\xi)(1+\xi\phi_{0}^{2}/M_{P}^{2})}{\xi}}\arcsin\sqrt{\frac{1+\xi\phi^{2}(1+\alpha\xi)/M_{P}^{2}}{1+\xi\phi_{0}^{2}(1+\alpha\xi)/M_{P}^{2}}}$
$\displaystyle-\sqrt{\alpha}\arctan\sqrt{\frac{(1+\xi\phi^{2}(1+\alpha\xi)/M_{P}^{2})(1+\xi\phi_{0}^{2}/M_{P}^{2})}{(\xi\phi_{0}^{2}-\xi\phi^{2})\alpha\xi/M_{P}^{2}}}=\frac{\pi\sqrt{6}}{4}.$
(31)

We expect that $\phi_{0}$ is proportional to the Planck mass $M_{P}$.
Therefore, one can safely take an approximation of $\phi\ll\phi_{0}$, where
$\phi$ is the IR field whose actual value is hierarchically smaller than the
cut-off scale of the model. Ignoring the ${\cal O}(\phi/\phi_{0})$
contributions, $\phi_{0}$ is provided by

$\displaystyle\frac{\phi_{0}}{M_{P}}$
$\displaystyle=\frac{\pi\sqrt{6}}{4}\left(1+{\cal O}(\xi)\right)\hskip
56.9055pt(\xi\ll 1)$ $\displaystyle=\sqrt{\frac{5}{4\hskip
1.13791pt\xi}}\left(1+{\cal O}(1/\xi)\right)\hskip 45.52458pt(\xi\gg 1,\
\alpha=6)$ $\displaystyle=\sqrt{\frac{3}{2}}\left(1+{\cal
O}(1/\sqrt{\xi})\right)\hskip 35.56593pt(\xi\gg 1,\ \alpha=0).$ (32)

We note that the throat field value $\phi_{0}$ in the Palatini formalism is
different from that in the metric formalism. However, the wormhole throat size
is insensitive to $\phi_{0}$ because the axion decay constant $f_{a}(\phi)$
becomes independent of $\phi$ as $\xi\phi^{2}\gtrsim M_{P}^{2}$, and converges
to the form $M_{P}/\sqrt{\xi}$. The throat radius is given by

$\displaystyle
L_{0}^{2}=\frac{1}{2\pi^{2}\sqrt{6}}\left(\frac{n}{M_{P}f_{a}(\phi_{0})}\right)$
$\displaystyle=\frac{n}{3\pi^{3}}\left(\frac{1}{M_{P}^{2}}\right)\hskip
48.36958pt(\xi\ll 1)$
$\displaystyle=\frac{n\sqrt{3}}{2\pi^{2}\sqrt{10}}\left(\frac{\sqrt{\xi}}{M_{P}^{2}}\right)\hskip
28.45274pt(\xi\gg 1,\ \alpha=6)$
$\displaystyle=\frac{n}{2\pi^{2}\sqrt{6}}\left(\frac{\sqrt{\xi}}{M_{P}^{2}}\right)\hskip
34.14322pt(\xi\gg 1,\ \alpha=0).$ (33)

Finally, the corresponding wormhole action becomes

$\displaystyle S_{wh}[n,\phi]$
$\displaystyle=n\left[\sqrt{\Delta_{\xi,\alpha}}\arccos\Big{(}\frac{1}{\sqrt{1+\Delta_{\xi,\alpha}}}\Big{)}+\ln\frac{2\phi_{0}}{\phi\sqrt{1+\Delta_{\xi,\alpha}}}\right],$
(34)

where $\Delta_{\xi,\alpha}=\xi\phi_{0}^{2}(1+\alpha\xi)/M_{P}^{2}$.

Figure 2: (Left) The full wormhole action $S_{wh}$ for a complex scalar with a
non-minimal coupling for both formalisms. The solid lines correspond to the
full analytic computations with $\phi=10^{15}\,{\rm GeV}$, and the numerically
obtained values correspond to the markers with $f_{a}=10^{15}\,{\rm GeV}$ as
in Ref. Cheong:2022ikv . It clearly shows the behavior of the action
$S_{wh}\sim\sqrt{\xi}$ for $\xi\gg 1$. (Right) The UV field value $\phi_{0}$
for both metric and Palatini scenarios. The values correspond to the minimal
coupled case in the $\xi\ll 1$ regime, whereas in the $\xi\gg 1$ regime the
behaviors deviate, with the Palatini case saturating and the metric case
decreasing as $1/\sqrt{\xi}$

Fig. 2 compares the actions calculated numerically with potential terms that
give the vacuum value of the axion decay constant $f_{a}=10^{15}\,{\rm GeV}$
and the analytic results from our effective theory approach with
$\phi=10^{15}\,{\rm GeV}$. It clearly shows the validity of our approximation
in which the parameter dependence becomes transparent. Depending on the values
of $\xi$ and $\alpha$, the form of the wormhole action can be decomposed into
the UV part mostly given by the throat regime, and the IR part logarithmically
dependent on the radial field $\phi$:

$\displaystyle S_{wh}[n,\phi]$ $\displaystyle\simeq n\left(S_{wh}^{\rm
UV}[\xi]+\ln\frac{\Lambda_{\rm eff}[\xi]}{\phi}\right).$ (35)

The UV part has the form $S_{wh}^{\rm UV}[\xi]=c_{0}+c_{1}\sqrt{\xi}$:

$\displaystyle S_{wh}^{\rm UV}[\xi]$
$\displaystyle=\ln\frac{\pi\sqrt{6}}{2}\hskip 85.35826pt{\rm for}\ \xi\ll 1$
$\displaystyle=\ln\frac{\sqrt{6}}{3}+\frac{\pi\sqrt{30}}{4}\sqrt{\xi}\hskip
32.72049pt{\rm for}\ \xi\gg 1,\alpha=6$ $\displaystyle=\ln
2+\frac{\pi\sqrt{6}}{4}\sqrt{\xi}\hskip 51.21504pt{\rm for}\ \xi\gg
1,\alpha=0.$ (36)

The logarithmic contribution of $\xi$ can be naturally parametrized by the
cut-off $\Lambda_{\rm eff}[\xi]$, whose dependence on the value of $\xi$ is
given as

$\displaystyle\Lambda_{\rm eff}[\xi]$ $\displaystyle=M_{P}\hskip
35.56593pt{\rm for}\ \xi\ll 1$ $\displaystyle=\frac{M_{P}}{\xi}\hskip
32.72049pt{\rm for}\ \xi\gg 1,\alpha=6$
$\displaystyle=\frac{M_{P}}{\sqrt{\xi}}\hskip 32.72049pt{\rm for}\ \xi\gg
1,\alpha=0.$ (37)

It behaves as $\Lambda_{\rm eff}[\xi]\sim M_{P}/\xi$ in the metric formalism,
while $\sim M_{P}/\sqrt{\xi}$ in the Palatini formalism. They coincide with
the perturbative cut-off scales of the model around the vacuum in each
formalism Barbon:2009ya ; Hertzberg:2010dc ; Bezrukov:2010jz ; Bauer:2010jg ;
Antoniadis:2021axu .

Figure 3: Comparison between the field dependent cutoff $\Lambda(\phi)$ in
Eqs. (3.1.1), (3.1.1) and the effective energy scale of the wormhole
$\sqrt{|R|}$, where $R$ is the scalar curvature Eq. (6), at a given position
$r\in[L_{0},\infty)$ for both metric (left) and Palatini (right) formalisms.
The non-minimal coupling is taken as $\xi=10^{4}$.

We note that, for a large non-minimal coupling, the perturbative cut-off scale
is sensitive to the background field value of $\phi$, especially when the
axions originate from the underlying $U(1)$ symmetries (not a non-abelian
symmetry) Bezrukov:2010jz ; Antoniadis:2021axu . We have

$\displaystyle\Lambda(\phi)$ $\displaystyle=\frac{M_{P}}{\xi}\quad{\rm for}\
\phi\ll\frac{M_{P}}{\xi},$ $\displaystyle=\frac{\xi\phi^{2}}{M_{P}}\quad{\rm
for}\ \frac{M_{P}}{\xi}\ll\phi\ll\frac{M_{P}}{\sqrt{\xi}},$
$\displaystyle=M_{P}\quad\hskip 2.84544pt{\rm for}\
\frac{M_{P}}{\sqrt{\xi}}\ll\phi$ (38)

for the metric formalism ($\alpha=6$), while

$\displaystyle\Lambda(\phi)$ $\displaystyle=\frac{M_{P}}{\sqrt{\xi}}\quad{\rm
for}\ \phi\ll\frac{M_{P}}{\sqrt{\xi}},$ $\displaystyle=\phi\quad\hskip
12.80365pt{\rm for}\ \frac{M_{P}}{\sqrt{\xi}}\ll\phi\ll M_{P},$
$\displaystyle=M_{P}\quad\hskip 1.9919pt{\rm for}\ M_{P}\ll\phi$ (39)

for the Palatini formalism ($\alpha=0$) in the Einstein frame. Therefore one
needs to estimate the validity of the solution by comparing the field
dependent cut-off scale $\Lambda(\phi(r))$ and a typical energy scale
represented by the scalar curvature $\sqrt{|R(r)|}$ in the range of
$r=[L_{0},\infty)$. In Fig. 3, we compare the two scales in the large $\xi$
limit, along with profile of the field $\phi(r)$. We identify that the energy
scale is always smaller than the local cut-off scale, so we confirm that our
results are safe from various quantum corrections.

The combination of $\ln\Lambda_{\rm eff}[\xi]/\phi$ in the wormhole action Eq.
(35) can provide a more closed expression of the $U(1)_{\rm PQ}$ breaking non-
perturbative factor as

$\displaystyle e^{-S_{wh}+in\theta}$
$\displaystyle=\exp\left[-n\left(S_{wh}^{\rm UV}[\xi]+\ln\frac{\Lambda_{\rm
eff}[\xi]}{\phi}-i\theta\right)\right]=\left(\sqrt{2}\,e^{-S_{wh}^{\rm
UV}[\xi]}\frac{\Phi}{\Lambda_{\rm eff}[\xi]}\right)^{n}.$ (40)

This combination will enter the effective Lagrangian of $\Phi$. At this
moment, the overall scale of the prefactor $M_{*}^{4}$ that will be multiplied
by the local operator Eq. (40) is not trivial. Conservatively we can think of
it as just ${\cal O}(M_{P}^{4})$, but it can change greatly depending on the
cut-off scales in the vacuum like ${\cal O}(L_{0}^{-4},\Lambda^{4}_{\rm
eff}[\xi])$. We leave the detailed estimation to future work.

The form of Eq. (40) clearly shows that (i) the contribution of a wormhole
with a PQ charge of $n$ is equivalent to $n$ times the contribution of a unit-
charged wormhole, (ii) when combined with an axion-dependent term, the primary
contribution is the holomorphic function of $\Phi$. Each properties are not
independent of each other. In our example, the UV contribution is nearly
independent of the IR field value, so the IR contribution is quite similar to
the case of an extremal instanton with $L_{0}=0$. Since $f^{2}_{a}(\phi)\ll
f^{2}_{a}(\phi_{0})$ with $\phi\ll\phi_{0}$, we have
$p(\phi)\simeq-n/f_{a}(\phi)$ and

$\displaystyle S^{\rm
IR}_{wh}[n,\phi]\simeq\int^{\phi}_{\phi_{0}}d\varphi\,p(\varphi)\simeq
n\int_{\phi}^{\phi_{0}}\frac{d\varphi}{f_{a}(\varphi)}.$ (41)

The holomorphicity of the local operator is quite general even for multiple
axion cases if the UV contribution of the wormhole (contribution near the
throat) is nearly independent of the IR field values for the radial scalar
partners.

Before closing this section, let us briefly discuss different types of axion
models. When an axion comes from string theory Svrcek:2006yi , the radial
scalar partner usually has a geometrical meaning. In that case, for
$G(\phi)=1$ in Eq. (22), the axion decay constant has the form
$f_{a}(\phi)=f_{a}e^{\beta\phi/M_{P}}$ with $\beta={\cal O}(1)$, a dilaton-
like coupling. It is well known from many studies Giddings:1987cg ;
Kallosh:1995hi ; Heidenreich:2015nta ; Hebecker:2016dsw ; Brax:2023now  that
the corresponding axion wormhole action depends less on the dynamics of the
scalar partner than in the case we discussed. For the change of the radial
field value $\Delta\phi\sim M_{P}$ in the wormhole background, the axion decay
constant does not change significantly, i.e. $\Delta
f_{a}(\phi)/f_{a}(\phi)={\cal O}(1)$. Therefore, the parameter dependence of
the action is more similar to the Giddings-Strominger wormhole action,
$S_{wh}\sim M_{P}/f_{a}$. Since the throat size is sensitive to the IR field
value, the holomorphicity of the wormhole induced operator does not hold in
general. We can first define a complex scalar field

$\displaystyle T(x)=S(x)-i\theta(x)$ (42)

to represent the kinetic term of the axion and its radial partner as

$\displaystyle\frac{1}{2}(\partial_{\mu}\phi)^{2}+\frac{1}{2}f_{a}^{2}(\phi)(\partial_{\mu}\theta)^{2}=\frac{2M_{P}^{2}}{\beta^{2}(T+T^{*})^{2}}(\partial_{\mu}T)(\partial^{\mu}T^{*}).$
(43)

It can be easily proved that the wormhole solution exists only for
$|\beta|<\sqrt{2/3}$. From Eqs. (24), (3.1) we have

$\displaystyle S_{wh}=\frac{nM_{P}\sin(\frac{\pi\sqrt{6}}{4}\beta)}{\beta
f_{a}(\phi)}=n\sin\left(\frac{\pi\sqrt{6}}{4}\beta\right)S,$ (44)

where $S={\rm Re}(T)$. This gives

$\displaystyle
e^{-S_{wh}+in\theta}=e^{-nT}\times\exp\left[\frac{n}{2}\Big{(}1-\sin\Big{(}\frac{\pi\sqrt{6}}{4}\beta\Big{)}\Big{)}(T+T^{*})\right].$
(45)

The non-holomorphic piece that exists in addition to the holomorphic part
$e^{-nT}$ disappears when $\beta=\sqrt{2/3}$, i.e. $L_{0}=0$. In this sense,
the fact that many examples for axions in string theory give
$\beta\geq\sqrt{2/3}$ could be related to the condition of preserving
supersymmetry.

###  3.2 Complex scalar model with an additional scalar field

Apart from a complex scalar $\Phi$, other real scalar fields $\chi$ that
couple to $\Phi$ can exist. One of the well-motivated forms of the coupling is
that the additional scalar couples to the kinetic term of the complex scalar
as a product form $Z(\chi)\times K_{\rm kin}(\Phi,\Phi^{*})$.

In this section, we discuss how such an additional scalar field changes the
wormhole action. Our starting action is given by

$\displaystyle\mathcal{S}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left(-\frac{M_{P}^{2}}{2}R+\frac{1}{2}Z(\chi)K_{\rm
kin}(\Phi,\Phi^{*})+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\right)$
$\displaystyle=\int
d^{4}x\sqrt{-g}\left[-\frac{M_{P}^{2}}{2}R+\frac{1}{2}Z(\chi)\Big{(}\partial_{\mu}\phi\partial^{\mu}\phi+f^{2}(\phi)\partial_{\mu}\theta\partial^{\mu}\theta\Big{)}+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\right].$
(46)

For a given complex scalar, $\phi$ is the radial partner of the axion
$\theta$. $\phi$ and $\chi$ are properly normalized without introducing a
kinetic mixing between them. Considering Eq. (2.1), we identify fields as
$\phi^{1}=\phi$, $\phi^{2}=\chi$, $\theta^{1}=\theta$, and the metric of the
kinetic term as $G_{11}=Z(\chi)$, $G_{12}=G_{21}=0$, $G_{22}=1$,
$(f^{2}_{a})_{11}=Z(\chi)f^{2}(\phi)$.

From the equations of motion for the Euclidean action, we have the relation
between the throat radius and the throat field values as

$\displaystyle
L_{0}^{2}=\frac{1}{2\pi^{2}\sqrt{6}}\left(\frac{n}{M_{P}\sqrt{Z(\chi_{0})}f(\phi_{0})}\right).$
(47)

From Eqs. (13), (15), the equations of motion for $\phi$ and $\chi$ are

$\displaystyle\left(\frac{d\chi}{d\tau}\right)^{2}+Z(\chi)\left(\frac{d\phi}{d\tau}\right)^{2}=\frac{n^{2}}{Z(\chi)f^{2}(\phi)}-\frac{n^{2}}{Z(\chi_{0})f^{2}(\phi_{0})},$
$\displaystyle\hskip
5.69046pt\frac{d^{2}\chi}{d\tau^{2}}-\frac{1}{2}\frac{dZ}{d\chi}\left(\frac{d\phi}{d\tau}\right)^{2}+\frac{1}{2}\frac{dZ}{d\chi}\left(\frac{n^{2}}{Z^{2}(\chi)f^{2}(\phi)}\right)=0.$
(48)

With the boundary conditions $\phi^{\prime}(0)=\chi^{\prime}(0)=0$, these
equations are simplified as

$\displaystyle\left(\frac{d\phi}{d\tau}\right)^{2}$
$\displaystyle=\frac{n^{2}}{Z^{2}(\chi)}\left(\frac{1}{f^{2}(\phi)}-\frac{1}{f^{2}(\phi_{0})}\right),$
$\displaystyle\left(\frac{d\chi}{d\tau}\right)^{2}$
$\displaystyle=\frac{n^{2}}{f^{2}(\phi_{0})}\left(\frac{1}{Z(\chi)}-\frac{1}{Z(\chi_{0})}\right).$
(49)

For given IR fields $\chi$ and $\phi$, the throat field values $\phi_{0}$ and
$\chi_{0}$ are determined by

$\displaystyle\int_{\phi}^{\phi_{0}}d\phi\frac{1}{\sqrt{f^{2}(\phi_{0})/f^{2}(\varphi)-1}}$
$\displaystyle=\int_{\chi}^{\chi_{0}}\frac{dx}{\sqrt{Z(x)}}\frac{1}{\sqrt{1-Z(x)/Z(\chi_{0})}},$
$\displaystyle\int_{\chi}^{\chi_{0}}\frac{dx}{M_{P}}\frac{1}{\sqrt{Z(\chi_{0})/Z(x)-1}}$
$\displaystyle=\frac{\pi\sqrt{6}}{4},$ (50)

assuming $\phi_{0}>\phi$ and $\chi_{0}>\chi$. With the solutions, the wormhole
action becomes

$\displaystyle
S_{wh}[n,\phi,\chi]=\int_{0}^{\tau_{\infty}}d\tau\frac{n^{2}}{Z(x)f^{2}(\varphi)}=n\int_{\phi}^{\phi_{0}}\frac{d\varphi}{f(\varphi)}\frac{1}{\sqrt{1-f^{2}(\varphi)/f^{2}(\phi_{0})}}.$
(51)

Notice that the general expression for the wormhole action does not explicitly
depend on $Z(\chi)$. However, the existence of $\chi$ affects the value of
$\phi_{0}$, so the form of $S_{wh}$ can have a nontrivial dependence on
$\chi$. Being equipped with the general formulation for this case, we now
present some specific examples and examine their implications.

####  3.2.1 Model with a light dilaton

Let us consider a dilatonic coupling for the complex scalar field
$\Phi=\frac{1}{\sqrt{2}}\phi e^{i\theta}$ Alvey:2020nyh , so

$\displaystyle Z(\chi)=e^{2\beta\chi/M_{P}},\quad f(\phi)=\phi.$ (52)

Here we take a convention $\beta\geq 0$. Using Eq. (3.2), we get

$\displaystyle\exp\left(-\frac{\beta(\chi_{0}-\chi)}{M_{P}}\right)=\cos\left(\frac{\pi\sqrt{6}}{4}\beta\right).$
(53)

The wormhole solution is only available for $\beta<\sqrt{2/3}$. We also obtain

$\displaystyle\frac{\phi_{0}}{M_{P}}\sqrt{1-\frac{\phi^{2}}{\phi_{0}^{2}}}=\frac{e^{-\beta\chi}}{\beta}\sin\left(\frac{\pi\sqrt{6}}{4}\beta\right)=\frac{e^{-\beta\chi_{0}}}{\beta}\tan\left(\frac{\pi\sqrt{6}}{4}\beta\right).$
(54)

The wormhole throat radius becomes

$\displaystyle L_{0}^{2}$
$\displaystyle=\frac{1}{2\pi^{2}\sqrt{6}}\left(\frac{n}{M_{P}e^{\beta\chi_{0}}\phi_{0}}\right)=\frac{1}{2\pi^{2}\sqrt{6}}\left(\frac{n}{M_{P}^{2}\tan(\frac{\pi\sqrt{6}}{4}\beta)/\beta}\right).$
(55)

It is confirmed that the wormhole throat size is small $L_{0}\lesssim
1/M_{P}$, so the UV contribution is just $O(1)$. For $\phi\ll\phi_{0}$, the
corresponding wormhole action can be represented by the function of $\phi$ and
$\chi$ as

$\displaystyle S_{wh}[n,\phi,\chi]$
$\displaystyle=n\int_{\phi}^{\phi_{0}}\frac{d\phi}{\phi}\frac{1}{\sqrt{1-\phi^{2}/\phi_{0}^{2}}}\simeq
n\ln\left(\frac{2M_{P}\sin(\frac{\pi\sqrt{6}}{4}\beta)/\beta}{e^{\beta\chi}\phi}\right).$
(56)

The IR dominated wormhole action has the form of $S_{wh}\simeq n\ln
M_{P}/f_{a}(\chi,\phi)$, where the effective axion decay constant is provided
by $f_{a}(\phi,\chi)=e^{\beta\chi}\phi$. Therefore, the overall value is log-
enhanced, almost always leading to an at most $\mathcal{O}(10)$ value. This
particular case can also be interpreted in the sense that the dilaton field
$\chi$ alters the axion decay constant to an effective value $f_{a}$. The PQ
breaking local operator is proportional to

$\displaystyle e^{-S_{wh}+in\theta}$
$\displaystyle=\left(\frac{e^{\beta\chi}\Phi}{\sqrt{2}M_{P}\sin(\frac{\pi\sqrt{6}}{4}\beta)/\beta}\right)^{n}.$
(57)

####  3.2.2 Model with a $R^{2}$-term

The previous type of action can naturally arise from a theory consisting of a
complex scalar non-minimally coupled to gravity, along with a
$R^{2}$-term333This class of theories are also considered as a UV extension to
theories incorporating non-minimal couplings to gravity Ema:2017rqn ;
Gorbunov:2018llf ; He:2018gyf ; He:2018mgb . :

$\displaystyle\hskip-2.84544pt\mathcal{S}=\int
d^{4}x\sqrt{-g}\left[-\left(\frac{M_{P}^{2}}{2}+\xi_{\phi}|\Phi|^{2}\right)R+\frac{\xi_{s}}{4}R^{2}+\partial_{\mu}\Phi\partial^{\mu}\Phi^{*}-V(|\Phi|)\right].$
(58)

In metric formalism, introducing the $R^{2}$ term leads to a new degree of
freedom in the Einstein frame. The equivalent form of the action becomes

$\displaystyle\hskip-4.26773pt{S}=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[-\frac{M_{P}^{2}}{2}R+\frac{1}{2}e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}}\left(\partial_{\mu}\phi\partial^{\mu}\phi+\phi^{2}\partial_{\mu}\theta\partial^{\mu}\theta\right)\right.$
$\displaystyle\hskip
59.75095pt\left.+\,\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi-U(\chi,\phi)\right]$
(59)

with the potential term containing the $\chi,\phi$ dependence as the form

$\displaystyle
U(\chi,\phi)=\frac{M_{P}^{2}}{4\xi_{s}}\left[1-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}}\left(1+\frac{\xi_{\phi}\phi^{2}}{M_{P}^{2}}\right)\right]^{2}+e^{2\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}}V(\phi).$
(60)

For the following discussion, we neglect the effect of $V(\phi)$ as we did in
the previous sections. However, the first term of $U(\chi,\phi)$ cannot be
simply ignored. Around the minimum, the mass of $\chi$ is $m_{s}={\cal
O}(M_{P}/\sqrt{\xi_{s}})$. If the mass of $\chi$ is smaller than
$M_{P}/\xi_{\phi}$, $\chi$ plays a role of unitarizing the action up to the
Planck scale at the vacuum.

Let us consider cases of large $\xi_{\phi}$. When $\xi_{s}\to 0$, $\chi$ is
supermassive, therefore it is just frozen with the condition
$\partial_{\chi}U(\chi,\phi)=0$. In this limit, we recover the single complex
scalar model, Eq. (26). Thus, $S_{wh}\sim\sqrt{\xi_{\phi}}+\ln
M_{P}/(\xi_{\phi}\phi)$. On the other hand, in the opposite limit
$\xi_{s}\to\infty$, $\chi$ is effectively massless. Hence, the action becomes
the form of Eq. (3.2). In this case, even for a large value of $\xi_{\phi}$,
$S_{wh}\sim\ln M_{P}/(e^{\beta\chi/M_{P}}\phi)$ with $\beta=1/\sqrt{6}$. This
implies that there is a screening effect of $\xi_{\phi}$ as $\xi_{s}$
increases. More specifically, we notice that in the regime where
$\xi_{\phi}\phi^{2}\gtrsim M_{P}^{2}$, $U(\chi,\phi)$ resembles a quartic
potential

$\displaystyle
U(\chi,\phi)\simeq\frac{1}{4}e^{2\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}}\left(\xi_{\phi}^{2}/\xi_{s}\right)\phi^{4}.$
(61)

From the IR value of $\chi$, the displacement of $\chi$ in the wormhole
background is at most of ${\cal O}(M_{P})$, so that
$e^{2\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}}={\cal O}(1)$. When the quartic
coupling $\lambda=\xi_{\phi}^{2}/\xi_{s}\ll 1$, this potential term is
irrelevant and the system reduces to the previous example in Sec. 3.2.1. It
results in a small wormhole action as Eq. (56). As $\lambda$ gradually
increases ($\xi_{s}$ decreases), the potential is approximated by a quartic
term of $\phi$ with a very large self coupling. In this case, the throat size
explicitly depends on the values of $\xi_{s}$ and $\xi_{\phi}$ Abbott:1989jw ,
giving the form

$\displaystyle
L_{0}^{2}\simeq\frac{n}{8\pi^{2}M_{P}^{2}}\left(\frac{n\,\xi_{\phi}^{2}}{2\pi^{2}\xi_{s}}\right)^{1/3}\quad{\rm
for}\ \sqrt{\xi_{\phi}}\ll\xi_{s}\ll\xi_{\phi}^{2}.$ (62)

In turn, this leads to the throat part of the wormhole action

$\displaystyle
S_{wh}^{(1)}=3\pi^{3}M_{P}^{2}L_{0}^{2}\simeq\frac{3n}{8}\left(\frac{n\pi\xi_{\phi}^{2}}{2\xi_{s}}\right)^{1/3}\quad{\rm
for}\ \sqrt{\xi_{\phi}}\ll\xi_{s}\ll\xi_{\phi}^{2}.$ (63)

This $S_{wh}^{(1)}$ is depicted in Fig. 4. For a large value of $\xi_{\phi}$,
the analytic form of $S_{wh}^{(1)}$ are different in three limiting cases, (I)
$\xi_{s}\ll\sqrt{\xi_{\phi}}$ (Green), (II)
$\sqrt{\xi_{\phi}}\ll\xi_{s}\ll\xi_{\phi}^{2}$ (Orange), (III)
$\xi_{\phi}^{2}\ll\xi_{s}$ (Blue).

Figure 4: $S_{wh}^{(1)}$ for $\xi_{\phi}=10^{4}$. The black dots represent the
numerical computation, the orange line corresponds to Eq. (63), the green line
corresponds to the value when $\xi_{s}\ll 1$, and the blue line corresponds to
the value obtained with Eq. (47). We note a transition around
$\xi_{s}\simeq\xi_{\phi}^{2}$, and another transition expected around
$\xi_{s}\lesssim\sqrt{\xi_{\phi}}$, both depicted in dashed - gray lines.

Therefore, the value of $\xi_{s}$ is quite important to determine the size of
axion wormhole action when $\xi_{\phi}\gg 1$. Actually, in the metric
formalism, the natural value of $\xi_{s}$ is proportional to $\xi_{\phi}^{2}$
because of the quantum corrections from the following renormalization group
equation at one-loop level Elizalde:1993ee ; Elizalde:1993ew ; Salvio:2014soa
; Salvio:2015kka ; Codello:2015mba ; Markkanen:2018bfx ; Ema:2019fdd ;
Ema:2020evi .

$\displaystyle\mu\frac{d\xi_{s}}{d\mu}=-\frac{1}{4\pi^{2}}\left(\xi_{\phi}+\frac{1}{6}\right)^{2}.$
(64)

This implies that even if the tree-level value of $\xi_{s}$ is vanishing, we
expect $\xi_{s}$ is naturally generated as $\xi_{s}\sim
0.02\,\xi_{\phi}^{2}\ln\Lambda/\mu\sim 0.02\xi_{\phi}^{2}$. Inserting this to
the wormhole action, Eq. (63), we have $S_{wh}^{(1)}\simeq 1.5$ ($n=1$). This
shows the difficulty of solving the axion quality problem with the help of the
non-minimal coupling to gravity for the complex scalar field that includes the
axion as $\Phi=\frac{1}{\sqrt{2}}\phi e^{i\theta}$.

Before closing this section, we briefly comment on the case of Palatini
formalism. In the Palatini formalism, there is no new scalar degree of freedom
even if we introduce $R^{2}$-term (see Sotiriou:2008rp  for a review). In the
Einstein frame, Eq. (58) becomes

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left(-\frac{M_{P}^{2}}{2}R+\frac{|\partial_{\mu}\Phi|^{2}}{(1+\xi_{\phi}\phi^{2}/M_{P}^{2})}+\frac{\xi_{s}|\partial_{\mu}\Phi|^{4}}{2(1+\xi_{\phi}\phi^{2}/M_{P}^{2})^{2}M_{P}^{4}}\right).$
(65)

Because of nontrivial higher derivative terms, it is more difficult to figure
out the effect of large $\xi_{s}$. From a naive dimensional analysis,
$\partial\phi\sim\phi/r$, and for $\xi_{\phi}\phi^{2}\gtrsim M_{P}^{2}$, the
higher derivative term can be important when the distance becomes scale as
$r\lesssim\sqrt{\xi_{s}/\xi_{\phi}}M_{P}^{-1}$. By comparing it with the
throat radius in the small $\xi_{s}$ limit
$L_{0}\sim\xi_{\phi}^{1/4}M_{P}^{-1}$, we see that the higher derivative terms
are not effective in the regime $L_{0}\leq r<\infty$ for
$\xi_{s}^{2}\ll\xi_{\phi}^{3}$, while it can provide a nontrivial effect in
the opposite limit. We leave the study for the Palatini formalism to a future
work.

##  4 Discussion

In this study, we employ the effective theory approach with a complex scalar
field, which incorporates the axion as its complex phase degree, to
investigate the Euclidean wormhole action, where the stability of the wormhole
throat relies on the presence of a non-zero Peccei-Quinn charge associated
with axions. Our analysis yielded a clear understanding of both the parameter
dependence of the UV contribution and the IR behavior of the wormhole action.
We achieve this understanding by explicitly calculating the dependency of the
infrared (IR) degrees of freedom in axion models featuring radial scalar
partners. In particular, when a significant non-minimal coupling of the scalar
field with gravity is present, our analysis reveals that the UV contribution
of the wormhole action becomes substantial, reaching the order of $S_{wh}^{\rm
UV}=\mathcal{O}(\sqrt{\xi})$. This substantial wormhole action, surpassing a
value of approximately $\mathcal{O}(200)$ with $\xi\gtrsim 10^{4}$, offers a
straightforward solution to the axion quality problem.

The reason behind the substantial magnitude of the action in the case of a
large non-minimal coupling lies in the fact that, unlike the minimally coupled
model with $\xi=0$, the dynamics of the radial field becomes independent of
that of the axion as the radial field surpasses a critical threshold.
Consequently, as we approach the wormhole throat from the asymptotic region,
the axion’s decay constant simply converges to $M_{P}/\sqrt{\xi}$. This
decoupling phenomenon between the radial scalar field and the axion field
appears to be rooted in symmetry considerations and may also be associated
with the conditions of the scalar potential required for inflation. However,
we demonstrated that the effects of a large non-minimal coupling can be
mitigated by introducing the $R^{2}$-term with a specific range of coupling
values. The true impact of a large $\xi$ warrants a more detailed
investigation in both the metric and Palatini formalisms.

If additional gauge symmetries are present, the QCD axion must necessarily be
a gauge singlet, formed as a combination of pseudo-Nambu-Goldstone bosons,
denoted as $\theta(x)=\sum_{I}q_{I}\theta^{I}(x)$. Other combinations are
absorbed by the gauge bosons. For complex scalar fields
$\Phi_{I}=\frac{1}{\sqrt{2}}\phi_{I}e^{i\theta^{I}}$, which carry charges
under these gauge symmetries, this results in local operators proportional to
$e^{-S_{wh}+iq_{I}\theta^{I}}\simeq e^{-S_{wh}^{\rm
UV}}\prod_{I}(\Phi_{I}/\Lambda_{\rm eff})^{q_{I}}$. On one hand, if the axion
is a composite particle, evaluating the wormhole solution becomes highly
intricate. This complexity arises because we must account for strongly
interacting fermions in the equations of motion. Subsequent research will
provide a deeper connection between more fundamental theory and axion
phenomenology.

###### Acknowledgements.

We thank Miguel Escudero, Koichi Hamaguchi, Hyun Min Lee, Natsumi Nagata, and
Hyeonseok Seong for valuable discussions. SCP is supported by National
Research Foundation grants funded by the Korean government
(NRF-2021R1A4A2001897) and (NRF-2019R1A2C1089334). CSS acknowledges support
from (NRF-2022R1C1C1011840, NRF-2022R1A4A5030362) and IBS under the project
code, IBS-R018-D1. DYC also acknowledges support from the CERN-Korea
Theoretical Physics Collaboration and Developing Young High-Energy Theorists
fellowship program (NRF-2012K1A3A2A0105178151).

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