# Fate of the false vacuum in string-inspired nonlocal field theory

Anish Ghoshal, Florian Nortier

► To cite this version:

Anish Ghoshal, Florian Nortier. Fate of the false vacuum in string-inspired nonlocal field theory. Journal of Cosmology and Astroparticle Physics, 2022, 2022 (08), pp.047. 10.1088/1475-7516/2022/08/047. hal-04258170

HAL Id: hal-04258170

https://hal.science/hal-04258170v1

Submitted on 25 Oct 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Fate of the false vacuum
in string-inspired nonlocal field theory

Anish Ghoshal<sup>*a*</sup> and Florian Nortier<sup>*b*</sup>

E-mail: anish.ghoshal@fuw.edu.pl, florian.nortier@ipht.fr

**Abstract.** In this article, we study Coleman bounce in weakly nonlocal theories which are motivated from string field theory. The kinetic term is extended via an infinite series of high-order derivatives, which comes into play at an energy scale $M$, without introducing any new states or ghosts in the mass spectrum. We calculate the bubble nucleation in thin-wall approximation, treating the system in semi-classical manner. We find that the effect of nonlocal scale $M$ in the theory is to suppress the vacuum tunneling rate from false to true vacuum compared to the standard local bouncing scenario. Likewise, we show that as we move further away from the bubble wall, the effects of nonlocality gets reduced and this suppression is significant only around the wall of the nucleated bubble. From our investigations, we conclude that the main effect is due to the fact that the nonlocality smears the solution of the local bubble profile. However, the energy of the bubble wall remains unaffected by the microscopic nonlocal behavior of the theory in the thin-wall approximation. We also discuss the cases for Lee-Wick theories and applications of our result to cosmology.

**Keywords:** cosmological phase transitions; domain walls; particle physics - cosmology connection; string theory and cosmology

**ArXiv ePrint:** 2203.04438

<footnote><sup>*a*</sup>Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,
ul. Pasteura 5, 02-093 Warsaw, Poland</footnote>

<footnote><sup>*b*</sup>Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique,
91191, Gif-sur-Yvette, France</footnote>
# Contents

1 Introduction 1

2 Short review: String-inspired nonlocal scalar field theory 3

3 Model of the bounce 4

3.1 Weakly nonlocal scalar field theory 4

3.2 Thin-wall approximation 6

4 Solution of the soliton 8

4.1 Local solution 8

4.2 Nonlocal solution 9

5 Discussion and conclusion 15

## 1 Introduction

An inspiration from string theory often entails that strings are nonlocal objects [1–12]. However, the original motivation for weakly nonlocal field theory was to formulate a novel pathway for possible UV-regularized theories as inspired from string field theory [1, 13] in the context of weakly-coupled nonlocal Quantum Field Theory (QFT) as studied in Refs. [14–35]. This became an alternative way to address the divergence and the hierarchy problems<fnref n="1" /> in the Standard Model (SM) by generalizing the kinetic energy operators (which is second in order) to an infinite series of higher-order derivatives that are suppressed due the scale of nonlocality (M) [14, 37, 38]. Such modifications to the kinetic energy sector via introducing higher-derivatives are free of ghosts<fnref n="2" /> [33], and also cures the vacuum instability problem of the Standard Model (SM) Higgs [44] as have been studied by one of the authors. It was found out that the β-functions, reaches a conformal limit<fnref n="3" /> resolving the issue of Landau-poles [51]. Thus, via capturing the infinite derivatives by exponential of an entire function we get softened ultra-violate (UV) behavior in the most desirable manner, without introducing any new degrees of freedom that contribute to the particle mass spectrum, as they contain no new poles in the propagators (see Refs. [52, 53] for astrophysical implications, dimensional transmutation and dark matter phenomenology in these theories.). An estimated bound on the scale of nonlocality from such observations is $M \ge O(10)$ TeV (Refs. [38, 53]). The nonlocal theory further provided interesting implications in proton decay and Grand Unified Theories (GUT) recently in Ref. [54], as well as in braneworld models, [55]. On the other

<footnote n="1">¹In local QFT, renormalization leads to ultraviolet (UV) divergences, which in the Standard Model (SM) of particle physics, a similar issue infamously known as the hierarchy problem where the Electroweak (EW) scale or the Higgs mass and Planck scale ($M_{pl}$) are separated by 18 orders-of-magnitude apart leading to huge fine-tuning in the SM [36] which for example needs several new particles at the or near the EW scale, like in supersymmetry (SUSY) etc. but no new physics results from the Large Hadron Collider (LHC) has been obtained, so alternative solutions are actively being pursued.</footnote>

<footnote n="2">²The unitarity issues are well addressed and understood in Euclidean space (imaginary time/energy) and using Cutkosky rules, the results obtained there-in can be analytically continued to Minkowski spacetime (real time/energy) via the Pius-Sen prescription [39–43].</footnote>

<footnote n="3">³For other conformal theory applications to cosmology, see [45, 46] for dark matter and [47, 48] for inflation, [49, 50] for baryogenesis and detectable gravitational waves, just for some examples.</footnote>
hand, the nonperturbative strongly-coupled regimes and exact $\beta$-functions, and conditions
of confinement, in higher-derivative nonlocal theories are being actively investigated by one
of the authors [56–60]. The results obtained so far shows that the effect of the nonlocality
in the strong coupling limits is to dilute any mass gap in the UV that the system gener-
ates. In context to gravity theories, i.e. string-inspired nonlocal extensions of the Einstein
gravity, it has been shown that one can get rid of classical singularities, such as black hole
singularities [19, 24, 61–70], cosmological singularities [71–77], and topological defects [78–84].

Understanding tunneling and investigating the false vacuum of a scalar field theory
and the process of its decays via the nucleation of true vacuum bubbles have been an im-
portant direction in field theory with huge implications in the analysis of phase transitions
and Gravitational Waves (GW) signals in cosmology, however first studied in detail decades
ago [85, 86]. As we understand from the available studies, the bubble nucleation rate is expo-
nentially suppressed, with the exponent being twice the Brillouin-Kramers-Wentzel (WKB)
barrier penetration integral or, equivalently, one can say the Euclidean action of the bounce
solution to the Euclidean field equations [86]. Via the implementation of path integral meth-
ods to estimate the energy density of the false vacua, the pre-exponential factor of the decay
rate can be estimated [87], which is just the $[\det S''_E(\phi_{\text{bounce}})]^{-1/2}$, where $S''_E$ denotes the sec-
ond functional variation of the Euclidean action under consideration$^4$. However, an essential
ingredient of this calculation is the fact that $S''_E(\phi_{\text{bounce}})$ has a single negative eigenvalue,
which arises from a mode corresponding to expansion or contraction of the bounce. Because
of the square root, this negative mode leads to a factor of $i$ that makes the false vacuum
energy complex, with its imaginary part interpreted in terms of a rate of decay by bubble
nucleation; the bounce with the lowest action has one, and only one, negative mode as shown
in Ref. [90]. Recently false vacuum decay estimates have been investigated in the nonpertur-
bative regimes by one of the authors and others and has been a topic for great interest with
respect to GW aspects [88, 89, 91].

Coleman and De Luccia (CDL) [92] proposed that this formalism can be extended to
include gravitational effects by adding a Euclidean Einstein-Hilbert term to the action. Al-
though they did not address the issue of the pre-factor, it seems natural to assume that it
should be analogous to that for the non-gravitational case, and that the dominant bounce
should again have a single negative mode, however the analysis complicated as in gravity
one also needs to take into account the fluctuations about the bounce which enjoy a gauge
freedom, corresponding to the invariance under coordinate transformations of the underlying
theory, leading to some constraints that must be imposed on the possible fluctuations.

Motivated by the nonlocal UV-completion of the QFT and the standard model, the goal
of this study is to compute the bubble nucleation and tunneling rate in a toy model with a
scalar field with 2 slightly non-degenerate vacua, such that one may perform the thin-wall
approximation. In order to get an approximate solution for the Coleman bounce with weak
nonlocality$^5$, we use the “diffusion” method of Refs. [78, 96–100], and compare with the local
case.

<footnote>$^4$Coleman’s method does not hold to be true in non-perturbative regimes which require a different strategy; such studies have been of recent interest, see Refs. [88, 89].</footnote>

<footnote>$^5$Other works treating the effects of time quantum mechanical aspects in nonlocal theories, cf. [93–95] for instance.</footnote>
The paper is organized as follows. In Section 2, we remind the main features of a string-inspired nonlocal theory for a scalar field. In Section 3, we present the toy model under study, and we express the vacuum decay rate and the energy of the bubble wall in the thin-wall approximation. In Section 4, we study the nonlocal soliton solution, which is used to compute the Coleman bounce in the thin-wall approximation. In Section 5, we summarize our results.

## 2 Short review: String-inspired nonlocal scalar field theory

Typically, the action for the string-inspired nonlocal theory given by [33, 97]:

$$S = \int d^4 x \left( \frac{1}{2} \phi e^{f(\Box)} (\Box - m^2) \phi - \frac{\lambda}{4!} \phi^4 \right) \quad (2.1)$$

where the normalization of $\phi$ is so chosen such that the residue at the $p^2 = -m^2$ pole is unity in this case, and $\Box = \eta_{\mu\nu}\partial^\mu\partial^\nu$ ($\mu, \nu = 0, 1, 2, 3$) with the convention of the metric signature $(-, +, +, +)$, $m$ is the mass of the scalar particle, and $M$ is the energy scale of the nonlocality which is usually taken to be below the Planck scale. The most important thing to note here is the fact that the kinetic energy terms are generalized with higher-derivatives suppressed by the nonlocal energy scale $M$, whilst the scalar self-interaction is the typical quartic potential one. This action is reduced into the standard local field theory taking the limit of $M \to \infty$.$^6$

In Euclidean space ($p^0 \to ip_E^0$), the propagator is given by,

$$\Pi(p^2) = -\frac{ie^{f(-p_E^2)}}{p_E^2 + m^2} \quad (2.2)$$

and the vertex factor is, as usual, given by $-i\lambda$. It can be clearly noted that the nonlocal extensions of the local theory leads to the exponential suppression of the propagators for $p_E^2 > M^2$ region. What this essentially means is that the quantum corrections in the ultra-violet (UV) will be frozen at energies higher than $M$ (basically all the $\beta$-functions vanish beyond the scale of nonlocality $M$ thereby making the theory reach an asymptotically conformal limit in the UV) [44, 51] which is a desirable feature in any QFT. In the other popular form factor considered in Refs. [28, 29], one can even weaken the interaction above $M$ and achieve asymptotic freedom (the coupling goes to zero in the UV).

An equivalent description of the action in Eq. (2.1) is the usual local Klein-Gordon kinetic operator via the following field re-definition [33]:

$$\begin{aligned} \tilde{\phi}(x) &= e^{-\frac{1}{2}f(\Box)}\phi(x) \\ &= \int d^4 y \mathcal{F}(x-y)\phi(y), \end{aligned} \quad (2.3)$$

where $\mathcal{F}(x-y) := e^{-\frac{1}{2}f(\Box)}\delta^{(4)}(x-y)$ is the kernel of the differential operator $e^{-\frac{1}{2}\Box}$. One gets the following action with an added source term $j(x)$:

$$S = \frac{1}{2} \int d^4 x \left( \tilde{\phi}(x)(\Box - m^2)\tilde{\phi}(x) - \int d^4 x \frac{\lambda}{4!} \left( e^{\frac{1}{2}f(\Box)}\tilde{\phi}(x) \right)^4 + j(x)e^{\frac{1}{2}f(\Box)}\tilde{\phi}(x) \right). \quad (2.4)$$

<footnote>$^6$Regarding the nonlocal form-factor $e^{f(\Box)}$, several possibilities have been popularly considered in the existing literature [101]. Here, however, for simplicity, we will use $e^{-\frac{1}{M^2}}$. Nonetheless, the essential criterion is that the nonlocal factor is an entire function without zeros, which allows us to avoid the appearance of possible ghosts in the physical spectrum of the theory.</footnote>
## 3 Model of the bounce

In this section we show the model of the bounce via the scalar field in local theory and then we show how the picture changes when nonlocality is introduced.

### 3.1 Weakly nonlocal scalar field theory

We consider a scalar field theory defined on a 4D Euclidean space $\mathbb{R}^4$:

$$ds^2 = \delta_{\mu\nu} dx^\mu dx^\nu, \quad \delta_{\mu\nu} = (+1, +1, +1, +1). \qquad (3.1)$$

Indeed, since the usual Wick rotation is not defined in string-inspired nonlocal theories, one must define the theory on an Euclidean space and then analytically continue to Minkowski spacetime only for the asymptotic states because of unitary issues [39–43]. In other words, the Lorentzian causal structure of spacetime emerges only at scales larger than the nonlocal length scale $\ell = 1/M$ [33, 102].

We introduce a real scalar field $\phi(x)$, whose potential is even under the $\mathbb{Z}_2$ transformation ($\phi \mapsto -\phi$):

$$V_0(\phi) = V_\mu(\phi) + V_I(\phi) + \Lambda_C, \qquad (3.2)$$

with the mass term ($\mu > 0$):

$$V_\mu(\phi) = -\frac{\mu^2}{2}\phi^2, \qquad (3.3)$$

the quartic (interaction) term ($\lambda > 0$):

$$V_I(\phi) = \frac{\lambda}{4!} \phi^4, \qquad (3.4)$$

and the "cosmological constant" term $\Lambda_C$. The 2 symmetric minima of $V_0(\phi)$ are

$$\phi_{\pm} = \pm v = \pm \sqrt{\frac{6}{\lambda}} \mu, \qquad (3.5)$$

and the "cosmological constant" $\Lambda_C$ is defined such that

$$V_0(\pm v) = 0 \quad \Rightarrow \quad \Lambda_C = \left(\frac{\mu v}{2}\right)^2 > 0. \qquad (3.6)$$

Let us focus on the Euclidean kinetic energy operator, which is now modified via infinite series of higher-order derivatives:

$$(\Box + \mu^2)e^{-\ell^2(\Box + \mu^2)}, \qquad (3.7)$$

with the Euclidean d'Alembertian

$$\Box = \delta^{\mu\nu} \partial_{\mu} \partial_{\nu}, \qquad (3.8)$$

such that the parameter $\ell$ interpolates between a local and a nonlocal field theory:

* in a local field theory: $\ell = 0$.

* in a string field theory-inspired nonlocal field theory: $\ell \neq 0$.
Let us define the parameter $\omega = (\mu\ell)^2$, such that $\sqrt{\omega}$ measures the separation between the energy scales $\mu$ and $M$.

We wish to investigate the bubble nucleation when one of the two vacua is metastable through quantum effects, and decay to the stable one by barrier penetration. Therefore, one needs to introduce an additional term $V_\epsilon(\phi)$ to the potential $V_0(\phi)$, which is asymmetric with respect to the $\mathbb{Z}_2$ transformation, and thus leaves us with a certain degeneracy between the two vacua ($\phi_+ \neq \phi_-$). The magnitude of $V_\epsilon(\phi)$ is controlled by a small parameter

$$ \epsilon = V(\phi_+) - V(\phi_-) > 0, \qquad (3.9) $$

such that

$$ V_{\epsilon}(\phi) = V(\phi) - V_{0}(\phi) = \mathcal{O}(\epsilon), \qquad (3.10) $$

where $V(\phi)$ is the total scalar potential. Now, the Euclidean action of this scalar field theory is thus

$$ S = \int d^4 x \mathcal{L}. \qquad (3.11) $$

with the Euclidean Lagrangian

$$ \mathcal{L} = -\frac{1}{2} \phi (\Box + \mu^2) e^{-\ell^2 (\Box + \mu^2)} \phi + U(\phi), \qquad (3.12) $$

where the total potential is now

$$ U(\phi) = U_0(\phi) + V_{\epsilon}(\phi), \quad U_0(\phi) = V_0(\phi) - V_{\mu}(\phi). \qquad (3.13) $$

One can rescale the field $\phi \mapsto e^{\omega/2}\phi$, such that the quartic coupling is also rescaled: $\lambda \mapsto e^{-2\omega}\lambda$. It is also useful to define the rescaled VEV $v \mapsto e^{\omega}v$. By applying Hamilton's principle, one arrives at the Euler-Lagrange equation for $\phi$:

$$ (\Box + \mu^2) e^{-\ell^2 \Box} \phi = \partial_{\phi} U(\phi). \qquad (3.14) $$

First, let us assume that a bounce solution [86] to Eq. (3.14) exists also in the nonlocal case, at least for some region of the parameter space (we will come back to this assumption at the end of Section 4). At the semi-classical level, the probability of decay of the false vacuum per unit time per unit volume takes the form [86]

$$ \Gamma/V = Ae^{-B/\hbar} [1 + \mathcal{O}(\hbar)], \qquad (3.15) $$

where $\hbar$ is the reduced Planck constant$^7$. In the same spirit as in Ref. [86], one of the goals of this article is to compute the coefficient $B$ for $\epsilon \ll 1$. This coefficient $B$ is given by the on-shell Euclidean action

$$ B = S(\phi) - S(\phi_+), \qquad (3.16) $$

where $\phi(x)$ is a bounce solution of Eq. (3.14), i.e. satisfying the following properties:

* The bounce $\phi(x)$ is not constant;

* The bounce $\phi(x)$ approaches the false vacuum $\phi_+$ at Euclidean infinity;

<footnote>$^7$In the rest of the article, we work in the natural unit system, where $\hbar = 1$ and the speed of light in vacuum is $c = 1$.</footnote>
• The Euclidean action of the bounce $\phi(x)$ is less or equal to that of any other solution satisfying the two previous properties.

Under the assumption that the bounce $\phi(x)$ has a $O(4)$ symmetry, one introduces hyperspherical coordinates, such that $\phi(x) = \phi(\rho)$, where $\rho \ge 0$ is the radial direction coordinate:

$$ \rho = \sqrt{\delta^{\mu\nu} x_{\mu}x_{\nu}}. \qquad (3.17) $$

The Euclidean d'Alembertian applied on $\phi(\rho)$ gives us

$$ \Box\phi(\rho) = \left(\partial_{\rho}^{2} + \frac{3}{\rho}\partial_{\rho}\right)\phi(\rho). \qquad (3.18) $$

Therefore, the coefficient $B$ is the on-shell Euclidean action:

$$ \begin{aligned} B &= S = 2\pi^2 \int_{-\infty}^{+\infty} \rho^3 d\rho \mathcal{L}, \\ &= 2\pi^2 \int_{-\infty}^{+\infty} \rho^3 d\rho \left[ -\frac{1}{2}\phi \partial_{\phi}U(\phi) + U(\phi) \right]. \end{aligned} \qquad (3.19) $$

Thus, we arrive a generic expression for $B$, however in order to proceed we need to make some assumptions and compute the quantities in those limits.

## 3.2 Thin-wall approximation

As shown in Ref. [86] and what is famously known as the thin-wall approximation, which is usually valid when $\epsilon \ll 1$, one considers the bubble nucleates at a large time $\rho = R \gg 1/\mu$, $\ell$, $\rho$ traverses the real line, and the bounce $\phi(\rho)$ goes monotonically from $\phi_-$ to $\phi_+$. Here, the bounce looks like a large 4D bubble of radius $R$, with a thin wall separating the two vacua $\phi_\pm$. At the end of the computations, it is important to check if one gets $R \gg 1/\mu$, $\ell$ when $\epsilon \ll 1$. In the thin-wall approximation, $\phi(\rho)$ has the form:

$$ \phi(\rho) = \begin{cases} -\tilde{v} & \text{if } \rho \ll R, \\ \phi_w(\rho - R) & \text{if } \rho \sim R, \\ +\tilde{v} & \text{if } \rho \gg R. \end{cases} \qquad (3.20) $$

Indeed, far away from the bubble wall, the field theory is in the vacuum $\phi_\pm$, and the infinite series of derivatives play no role, since $\phi(\rho)$ varies only in the thin-shell of the wall, where $\rho \sim R$, where $\phi(\rho)$ can be approximated by a function $\phi_w(\rho - R)$. For $\rho \sim R$, one can solve the Euclidean Euler-Lagrange equation (3.14), with the approximate symmetric potential $U(\phi) \sim U_0(\phi)$, and the approximate kinetic operator

$$ (\Box + \mu^2)e^{-\ell^2\Box}\phi_w \sim (\partial_{\rho}^2 + \mu^2)e^{-\ell^2\partial_{\rho}^2}\phi_w, \qquad (3.21) $$

with the hypothesis that the viscous damping term in the d'Alembertian (3.18) can be neglected ($\Box\phi_w \sim \partial_{\rho}^2\phi_w$), which will be checked at the end of the computation. If one performs the change of variable $r = \mu(\rho - R)$, and one defines $\varphi(\omega, r) = \phi_w(r)/\tilde{v}$, the Euler-Lagrange equation for $\phi_w(r)$ is approximated by the equation of a soliton for $\varphi(\omega, r)$:

$$ (\partial_r^2 + 1)e^{-\omega\partial_r^2}\varphi(\omega, r) - \varphi^3(\omega, r) = 0, \qquad (3.22) $$
whose solution is odd with respect to $r = 0$, thus for $R \gg 1/\mu$, $\ell$, it is clear that the viscous damping term neglected in Eq. (3.21) will give a small perturbation to the approximate soliton profile. Eq. (3.22) can be obtained from Hamilton's principle applied to the Euclidean action

$$ \int_{-\mu R}^{+\infty} \frac{dr}{\mu} \mathcal{L}_w \underset{\mu R \gg 1}{\approx} \int_{-\infty}^{+\infty} dr \mathcal{L}_w, \qquad (3.23) $$

with

$$ \mathcal{L}_w = -\frac{1}{2} \varphi (\partial_r^2 + 1) e^{-\omega \partial_r^2} \varphi + \varphi^4 + \Lambda_C . \qquad (3.24) $$

The on-shell Euclidean action for the bounce near the bubble wall is then

$$ S_w(\omega) \approx \frac{\Lambda_C}{\mu} \int_{-\infty}^{+\infty} dr [1 - \varphi^4(\omega, r)]. \qquad (3.25) $$

In the thin-wall approximation, from Eqs. (3.20) and (3.19), the coefficient $B$ is given by

$$ B = B_i + B_w + B_o, \qquad (3.26) $$

where the contribution inside the bubble is

$$ B_i = -\frac{\pi^2}{2} R^4 \epsilon, \qquad (3.27) $$

the contribution of the wall is

$$ B_w = 2\pi^2 R^3 S_w, \qquad (3.28) $$

the contribution outside the bubble is

$$ B_o = 0, \qquad (3.29) $$

and the value of $R$ is given by the one which minimizes the on-shell action $B$:

$$ \frac{dB}{dR} = 0 \quad \Rightarrow \quad R = \frac{3S_w}{\epsilon}. \qquad (3.30) $$

One can thus see that as long as $S_w(a) > 0$, we can choose $\epsilon$ sufficiently small in order to justify the thin-wall approximation, such that $R \gg 1/\mu$, $\ell$, which justified a posteriori all the previous approximations. In this case, one obtains

$$ B = \frac{27\pi^2 S_w^4}{2\epsilon^3}, \qquad (3.31) $$

which has the same qualitative form in both the local and nonlocal cases. In what follows, one thus needs to determine the quantity $S_w(\omega)$. In order to compare the on-shell action near the bubble wall $S_w$ for the nonlocal versus local field theory, it is useful to introduce a function

$$ Q(\omega) = \frac{S_w(\omega)}{S_w(0)}, \qquad (3.32) $$

such that with Eq. (3.31) becomes:

$$ B(\omega) = B(0) Q^4(\omega), \qquad (3.33) $$
and the thin-wall approximation means that we require $Q(\omega) > 0$. Note that we choose to normalize with a local model which have the same quartic coupling than the nonlocal one, since the quartic coupling has been rescaled ($\lambda \mapsto e^{-2\omega\lambda}$).

Before computing $Q(\omega)$ in the section 4, it is interesting to estimate the energy of the wall $E_w$. Outside the bubble wall, where nonlocality plays no role, one can perform an analytical continuation of the bounce in Minkowski space-time ($x^0 = it$):

$$ \phi(t, \vec{x}) = \phi\left(\rho = \sqrt{|\vec{x}|^2 - t^2}\right). \qquad (3.34) $$

In the limit where the bubble wall can be considered as a thin-shell, a section of this bubble wall at rest carries energy $S_w$, per unit area. Therefore, a section of the wall expands with a velocity

$$ v = \frac{d\vec{x}}{dt} = \frac{\sqrt{|\vec{x}|^2 - R^2}}{\vec{x}}, \qquad (3.35) $$

and carries an energy

$$ \frac{S_w}{\sqrt{1 - v^2}} \qquad (3.36) $$

per unit area. At a time when the radius of the bubble is $|\vec{x}|$, the energy of the wall is then

$$ \begin{aligned} E_w &= \frac{4\pi |\vec{x}|^2 S_w}{\sqrt{1 - v^2}} \\ &= \frac{4\pi |\vec{x}|^3 S_w}{R} \\ &= \frac{4\pi \epsilon |\vec{x}|^3}{3}, \end{aligned} \qquad (3.37) $$

where one used Eq. (3.35) to get the second and third lines, respectively. At the end, one sees that $E_w$ depends only on the parameter $\epsilon$, which controls the magnitude of the asymmetry of the scalar potential, and not on the microscopic scales $\mu$ and $M$, in the thin-wall approximation. It is not so surprising because the scales $\mu$ and $\Lambda$ mainly determines the width of the soliton-like profile of the bubble wall (cf. Eq. (3.22)), which we have considered as negligible in the computation.

# 4 Solution of the soliton

## 4.1 Local solution

In this subsection, we will determine $\varphi(0, r)$ which is the toy model of Ref. [86]. The exact solution of the local soliton equation is (cf. Fig. 1)

$$ \varphi(0, r) = \tanh\left(\frac{r}{\sqrt{2}}\right). \qquad (4.1) $$
Since the soliton equation gives the profile of the bubble wall in the approximation $|r| \gg 1$, one can only keep an asymptotic expansion (cf. Fig. 1):

$$ \varphi(0, r) \underset{|r| \ll 1}{\sim} \operatorname{sgn}(r) (1 - 2e^{-\sqrt{2}|r|}), \qquad (4.2) $$

with the sign function defined as

$$ \operatorname{sgn}(r) = \begin{cases} -1 & \text{if } r < 0, \\ 0 & \text{if } r = 0, \\ +1 & \text{if } r > 0. \end{cases} \qquad (4.3) $$

The on-shell action $S_w(0)$ for the local problem is

$$ S_w(0) = \frac{2\sqrt{2}}{3}\mu v^2, \qquad (4.4) $$

for both the exact and asymptotic profiles of the local soliton.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** In green: exact profile $\varphi(0, r)$ (Eq. (4.1)) of the local soliton. In blue: asymptotic profile $\varphi(0, r)$ (Eq. (4.2)) of the local soliton ($|r| \gg 1$).</figcaption></figure>

## 4.2 Nonlocal solution

In order to solve the nonlocal soliton equation, one can use the “diffusion” method of Refs. [78, 96–100] to get an approximate solution for $|r| \gg 1$. One considers that the field $\varphi(\omega, r)$, where $\omega$ is an auxiliary coordinate, satisfies the diffusion equation (with $\beta > 0$):

$$ \left( \partial_r^2 - \frac{\partial_\omega}{\beta} \right) \varphi(\omega, r) = 0 \quad \Rightarrow \quad \varphi(\omega, r) = e^{\beta\omega\partial_r^2} \varphi(0, r). \qquad (4.5) $$
By definition:

* $\varphi(0, r)$ is the profile of the soliton in the local case ($\omega = 0$);

* $\varphi(\omega, r)$ is the profile of the soliton in the nonlocal ($\omega \neq 0$).

One introduces the normalized Gaussian function

$$G(\sigma, r) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{r^2}{2\sigma^2}\right), \qquad (4.6)$$

such that it defines the heat kernel as

$$G(\sigma, r - r') = \exp\left(\frac{\sigma^2}{2} \partial_r^2\right) \delta(r - r'). \qquad (4.7)$$

One can then rewrite

$$\begin{aligned} \varphi(\omega, r) &= e^{\beta\omega\partial_r^2}\varphi(0, r), \\ &= \int dr' e^{\beta\omega\partial_r^2}\delta(r - r')\varphi(0, r'). \end{aligned} \qquad (4.8)$$

By performing the change of variable $\sigma^2 = 2\beta\omega$, one gets

$$\begin{aligned} \varphi(\sigma, r) &= \int dr' G(\sigma, r - r') \varphi(0, r'), \\ &= G(\sigma, r) * \varphi(0, r), \end{aligned} \qquad (4.9)$$

where * is the convolution product<fnref n="8" /> with respect to $r$. The new variable $\sigma \propto \sqrt{\omega}$ and is thus related to the separation of the scales $\mu$ and $M$. One can compute numerically the convolution of the exact soliton profile (4.1) with the Gaussian function (4.7), where one can see in Figs. 2 and 3 that nonlocality ($\sigma \neq 0$) smears the behavior of the local soliton near the kink when $\sigma$ increases: the soliton width is increased by the smearing effect of nonlocality. This effect is already known for zero-thickness topological defects [55, 78–83]. Far from the wall, the nonlocal soliton smoothly recovers the asymptotic behavior of the local one. Again, the effect of the infinite series of derivatives is important when spacetime variations of the field are important, such as near the bubble wall.

<footnote n="8">⁸See Refs. [103, 104] for weakly nonlocal theories in convolutional form.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** For $\sigma = 1/2$, in red: the gaussian function (4.7); in green: the exact convolution product (4.10) with the exact local soliton (4.1); in blue: the asymptotic nonlocal profile (4.13) of the soliton $\varphi(\sigma, r)$ with $|r| \gg 1$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** In (blue, red), exact convolution product (4.10) with the exact local soliton (4.1) for $\sigma = (0, 0.6)$.</figcaption></figure>
In this study, one needs only the asymptotic behavior for $|r| \gg 1$ to compute the coefficient $B$ in the thin-wall approximation. Therefore, from the asymptotic local solution (4.2) of $\varphi(0, r)$, one gets (cf. Fig. 4):

$$G(\sigma, r) * \varphi(0, r) = \operatorname{erf}\left(\frac{r}{\sqrt{2}\sigma}\right) - e^{\sigma^2 - \sqrt{2}r} \operatorname{erfc}\left(\frac{r}{\sqrt{2}\sigma} - \sigma\right) + e^{\sigma^2 + \sqrt{2}r} \operatorname{erfc}\left(\frac{r}{\sqrt{2}\sigma} + \sigma\right), \quad (4.10)$$

with the error function

$$\operatorname{erf}(r) = \frac{2}{\sqrt{\pi}} \int_{0}^{r} du \, e^{-u^2}, \quad (4.11)$$

and the complementary error function

$$\operatorname{erfc}(r) = 1 - \operatorname{erf}(r). \quad (4.12)$$

However, only the asymptotic behavior $|r| \gg 1$ of this result is meaningful (cf. Fig. 4):

$$\varphi(\sigma, r) \underset{|r| \ll 1}{\sim} \operatorname{sgn}(r) \left(1 - 2e^{\sigma^2 - \sqrt{2}|r|}\right). \quad (4.13)$$

We see on Fig. 5 that when $\sigma \neq 0$ increases, the departure of the nonlocal soliton profile from the local one is more important near the wall, since the nonlocal scale $M$ is closer to the mass scale $\mu$ of the scalar potential, and thus the nonlocal effects are more important.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** For $\sigma = 1/2$, in red: the Gaussian function (4.7); in green: the exact convolution product (4.10) with the asymptotic local soliton (4.2); in blue: the asymptotic nonlocal profile (4.13) of the soliton $\varphi(\sigma, r)$ with $|r| \gg 1$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 5.** In (blue, green, red), asymptotic nonlocal profile (4.13) of the soliton $\varphi(\sigma, r)$ with $|r| \gg 1$ for $\sigma = (0, 0.3, 0.6)$.</figcaption></figure>

After that, one has to determine for which value of $\beta$ the soliton profile (4.13) is the best approximation for a solution of the nonlocal soliton equation (3.22). To achieve this task, we introduce the function

$$ \forall r > 0, E(\beta, \omega, r) = \begin{cases} (\partial_r^2 + 1)\varphi(0, r) - e^{(1-\beta)\omega}\partial_r^2\varphi(0, r) \quad & \text{if } \beta < 1, \\ (\partial_r^2 + 1)e^{(\beta-1)\omega}\partial_r^2\varphi(0, r) - [e^{\beta\omega}\partial_r^2\varphi(0, r)]^3 \quad & \text{if } \beta \ge 1. \end{cases} \\ \underset{|r| \gg 1}{\sim} \begin{cases} 6(1 - e^{2(\beta-1)\omega})e^{-\sqrt{2}r-2(\beta-1)\omega} & \text{if } \beta < 1, \\ 6(e^{2\omega}-1)e^{-\sqrt{2}r+2(\beta-1)\omega} & \text{if } \beta \ge 1, \end{cases} \quad (4.14) $$

which evaluates the nonlocal soliton equation (3.22) for the soliton profile (4.13). It is straightforward to study the function $E(\beta, \omega, r) > 0$: there is only one minimum at $\beta = 1$ for $r \gg 1$, and one can check numerically that $E(1, \omega, r \gg 1) \ll 1$, such that $\varphi(1, r \gg 1)$ is a good approximation for the solution of Eq. (3.22). Since the soliton is an odd function with respect to $r = 0$, the same analysis holds for $r \ll -1$. In the following, one will thus take $\beta = 1$.

Now, one knows the bounce in the thin-wall approximation, which allows determining the function $Q(\sigma)$ from Eq. (3.32) (cf. Fig 6):

$$ Q(\sigma) = 3 - \frac{9e^{\sigma^2}}{2} + 4e^{2\sigma^2} - \frac{3e^{3\sigma^2}}{2}. \quad (4.15) $$

The thin-wall approximation is valid as long as $Q(\sigma) > 0$, i.e. for $\sigma \lesssim 0.67$. One can interpret this result by the fact that for larger values of $\sigma$, the bubble wall is so smeared by nonlocality that the thin-wall approximation is meaningless, such that one needs a finer
analysis to understand this region of the parameter space, which is beyond the scope of this first study. In the local case ($\sigma = 0$), one can always do a thin-wall approximation. When $M \gg \mu$,

$$Q(\sigma) \sim 1 - \sigma^2, \qquad (4.16)$$

and we see in Fig 6 that the behavior is closed to the local case, as expected. From Eq. (4.15) and Fig 6, we see that the effect of weak nonlocality is to exponentially suppress the vacuum decay rate with respect to the local theory. As discussed previously, weak nonlocality increases the bubble width, such as it is more difficult for the system to cross the potential barrier by quantum tunneling, in analogy with the well-known results in 1D quantum mechanics where the transmission amplitude of the incoming particle is exponentially suppressed when one increases the width of the potential barrier. Note that the height of the potential barrier is kept fixed, since we choose to compare the nonlocal model with a local theory which has the same effective quartic coupling below the nonlocal scale.

<figure><img src="image_7.png" /><figcaption>Figure 6. In green: $Q(\sigma)$ in the thin-wall approximation; in blue: $Q(\sigma)$ for $\sigma \ll 1$.</figcaption></figure>

Finally, let us come back to our original assumption that a bounce solution to Eq. (3.14) exists. In Ref. [86], Coleman shows that a bounce solution always exists in his toy scalar model. Note that if $M \gg \mu$ in our weakly nonlocal case, it is clear from the decoupling of scales that such deep-UV nonlocal scale will not spoil the existence of a bounce solution for the scalar field with a deep-IR mass scale in the potential. Now, if the nonlocal scale is closer to the mass scale of the potential with $M \ge \mu$, and that the thin-wall approximation is reliable, we showed that Eq. (3.14) reduces to the soliton equation (3.22), for which we were able to give an approximate solution by the “diffusion” method developed in Refs. [78, 96–100]. Therefore, we showed that a bounce solution exists for $M \ge \mu$ in the region of parameter space where the thin-wall approximation is valid. The analysis of this paper does not treat
the general case (for which the existence of a bounce solution may not be guaranteed), that needs to be investigated beyond the thin-wall approximation.

# 5 Discussion and conclusion

In a simple quartic scalar field theory generalized to infinite sets of higher-order derivatives, as motivated from string field theory approaches, we investigated the false vacuum decay rate and bubble nucleation. We found that the effect of nonlocality completely changes the rate of bubble nucleation. We summarize the main findings of our paper as follows:

* We investigated the Coleman bounce in string-inspired nonlocal scalar field theory in the thin-wall approximation. We used the method where nonlocality is treated as a diffusion process in an auxiliary extra dimension (cf. Eq. (4.5)) in order to get an approximate solution for the bounce.

* In the limit $M \to \infty$, the false vacuum decay rate becomes just like in the local theory (cf. Eq. (4.16)), which is consistent in any nonlocal theory in weak coupling limits.

* We observed that the nonlocal effects, implemented via a string-inspired nonlocal kinetic term for the scalar field smear the width of the bubble wall, and have no effect far away from it (cf. Figs. 3 and 5).

* We notice that when the scale of nonlocality is close to the mass scale of the potential $\mu$, the false vacuum decay rate is exponentially suppressed (cf. Eq. (4.15) and Fig. 6).

* For any given nonlocal scale, we showed that the thin-wall approximation breaks down when $M \sim \mu$, because of the smearing effect of nonlocality on the bubble wall (cf. Fig. 6).

* We computed the bubble wall energy density and found that it is independent of the microscopic scales $\mu$ and $M$, as long as the thin-wall approximation is valid (cf. Eq. (3.37)).

* Our bounce solution should be also valid for higher-derivative scalar field theories (Lee-Wick theories [105, 106]), with derivatives of finite largest order $N \gg 1$. Indeed, it was shown in Ref. [107–109] that such theory is asymptotically nonlocal when $N \to \infty$, such that it is equivalent to our nonlocal model. For $N \sim 1$, a specific analysis is needed that is beyond the scope of the present draft.

* Recently, a new recipe was proposed to build (from a “mother” local QFT) a UV-complete weakly nonlocal QFT that one can apply to gravity coupled to gauge and matter fields [34]. Such construction is ghost-free in the nonlocal regime even in presence of a shift of vacuum, like in the Higgs mechanism [35], without relying on UV-completion in string theory. Other particularities are: (i) the tree-level scattering amplitudes are identical to the ones of the mother local QFT [110], with the same causal properties [111]; (ii) any solution of the classical field equations of the mother local theory is also solution of the nonlocal one. Such properties do not hold in the class of string-inspired nonlocal scalar theory [33] studied in our article. An important consequence is that, for these new theories, our semi-classical analysis will give the same results as the Coleman analysis in the local theory [86], such that one needs to compute the quantum corrections to the bounce [87] in order to quantify the effect of weak nonlocality on the vacuum decay rate.
Therefore, from our comparison between approximate analytical solutions and numerical solutions with the boundary conditions can help to understand the diligence of the diffusion method that have been employed. Particularly, we have found that the kink-type profiles used to describe non-perturbative transitions between non-degenerate vacua of the theory. Now since string-inspired nonlocal theories provide simplified playgrounds for the study of nonlocality in inflationary models, the search for background solutions could be of practical importance for alternative cosmological scenarios and also extend to strongly-coupled regimes. We envisage that our results will pave the way to understand the finite temperature effects on nonlocal solitons and understanding of the critical temperatures of such systems, with further applications in phase transitions and early universe cosmology including predictions of GW (from inflation, strong first-order phase transitions, etc.) in current and future GW detectors and recently searched by LIGO [112–114]. However, such study is beyond the scope of the present manuscript and will be taken up in future publication.

## Acknowledgments

Authors acknowledge Jens Boos, Gianluca Calcagni, Anupam Mazumdar, Leonardo Modesto, Marco Frasca and Masahide Yamaguchi for useful comments.

## References

[1] E. Witten, *Noncommutative Geometry and String Field Theory*, Nucl. Phys. B 268 (1986) 253.

[2] P.G.O. Freund and M. Olson, *Non-archimedean strings*, Phys. Lett. B 199 (1987) 186.

[3] P.G.O. Freund and E. Witten, *Adelic string amplitudes*, Phys. Lett. B 199 (1987) 191.

[4] L. Brekke, P.G.O. Freund, M. Olson and E. Witten, *Nonarchimedean String Dynamics*, Nucl. Phys. B 302 (1988) 365.

[5] P.H. Frampton and Y. Okada, *Effective scalar field theory of p-adic string*, Phys. Rev. D 37 (1988) 3077.

[6] V.A. Kostelecky and S. Samuel, *The Static Tachyon Potential in the Open Bosonic String Theory*, Phys. Lett. B 207 (1988) 169.

[7] V.A. Kostelecky and S. Samuel, *On a Nonperturbative Vacuum for the Open Bosonic String*, Nucl. Phys. B 336 (1990) 263.

[8] A.A. Tseytlin, *On singularities of spherically symmetric backgrounds in string theory*, Phys. Lett. B 363 (1995) 223 [hep-th/9509050].

[9] W. Siegel, *Stringy gravity at short distances*, hep-th/0309093.

[10] T. Biswas, M. Grisaru and W. Siegel, *Linear Regge trajectories from worldsheet lattice parton field theory*, Nucl. Phys. B 708 (2005) 317 [hep-th/0409089].

[11] G. Calcagni and L. Modesto, *Nonlocality in string theory*, J. Phys. A 47 (2014) 355402 [1310.4957].

[12] G. Calcagni and L. Modesto, *Nonlocal quantum gravity and M-theory*, Phys. Rev. D 91 (2015) 124059 [1404.2137].

[13] W. Taylor and B. Zwiebach, *D-Branes, Tachyons, and String Field Theory*, in *Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2001): Strings, Branes and Extra Dimensions*, 2003, DOI [hep-th/0311017].
[14] J.W. Moffat, Finite nonlocal gauge field theory, Phys. Rev. D 41 (1990) 1177.

[15] D. Evens, J.W. Moffat, G. Kleppe and R.P. Woodard, Nonlocal regularizations of gauge theories, Phys. Rev. D 43 (1991) 499.

[16] G. Kleppe and R.P. Woodard, Nonlocal Yang-Mills, Nucl. Phys. B 388 (1992) 81 [hep-th/9203016].

[17] E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146.

[18] M.A. Clayton, Nonlocal regularization of Abelian models with spontaneous symmetry breaking, Mod. Phys. Lett. A 16 (2001) 1117 [hep-th/0106025].

[19] T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards Singularity- and Ghost-Free Theories of Gravity, Phys. Rev. Lett. 108 (2012) 031101 [1110.5249].

[20] L. Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86 (2012) 044005 [1107.2403].

[21] L. Modesto, Super-renormalizable Higher-Derivative Quantum Gravity, 1202.0008.

[22] L. Modesto, Super-renormalizable Multidimensional Quantum Gravity, Astron. Rev. 8 (2013) 4 [1202.3151].

[23] L. Modesto, Finite Quantum Gravity, 1305.6741.

[24] T. Biswas, A. Conroy, A.S. Koshelev and A. Mazumdar, Generalized ghost-free quadratic curvature gravity, Class. Quant. Grav. 31 (2014) 015022 [1308.2319].

[25] L. Modesto, Multidimensional finite quantum gravity, 1402.6795.

[26] L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [1407.8036].

[27] S. Talaganis, T. Biswas and A. Mazumdar, Towards understanding the ultraviolet behavior of quantum loops in infinite-derivative theories of gravity, Class. Quant. Grav. 32 (2015) 215017 [1412.3467].

[28] L. Modesto and L. Rachwał, Universally finite gravitational and gauge theories, Nucl. Phys. B 900 (2015) 147 [1503.00261].

[29] L. Modesto, M. Piva and L. Rachwal, Finite quantum gauge theories, Phys. Rev. D 94 (2016) 025021 [1506.06227].

[30] E.T. Tomboulis, Nonlocal and quasilocal field theories, Phys. Rev. D 92 (2015) 125037 [1507.00981].

[31] L. Modesto, L. Rachwał and I.L. Shapiro, Renormalization group in super-renormalizable quantum gravity, Eur. Phys. J. C 78 (2018) 555 [1704.03988].

[32] M.N. Hashi, H. Isono, T. Noumi, G. Shiu and P. Soler, Higgs mechanism in nonlocal field theories, JHEP 08 (2018) 064 [1805.02676].

[33] L. Buoninfante, G. Lambiase and A. Mazumdar, Ghost-free infinite derivative quantum field theory, Nucl. Phys. B 944 (2019) 114646 [1805.03559].

[34] L. Modesto, Nonlocal Spacetime-Matter, 2103.04936.

[35] L. Modesto, The Higgs mechanism in nonlocal field theory, JHEP 06 (2021) 049 [2103.05536].

[36] H.E. Haber and G.L. Kane, The Search for Supersymmetry: Probing Physics Beyond the Standard Model, Phys. Rept. 117 (1985) 75.

[37] N.V. Krasnikov, Nonlocal gauge theories, Theor. Math. Phys. 73 (1987) 1184.

[38] T. Biswas and N. Okada, Towards LHC physics with nonlocal Standard Model, Nucl. Phys. B 898 (2015) 113 [1407.3331].
[39] R. Pius and A. Sen, *Cutkosky rules for superstring field theory*, JHEP 10 (2016) 024
[1604.01783].

[40] F. Briscese and L. Modesto, *Cutkosky rules and perturbative unitarity in Euclidean nonlocal
quantum field theories*, Phys. Rev. D 99 (2019) 104043 [1803.08827].

[41] P. Chin and E.T. Tomboulis, *Nonlocal vertices and analyticity: Landau equations and general
Cutkosky rule*, JHEP 06 (2018) 014 [1803.08899].

[42] F. Briscese and L. Modesto, *Non-unitarity of Minkowskian non-local quantum field theories*,
Eur. Phys. J. C 81 (2021) 730 [2103.00353].

[43] A.S. Koshelev and A. Tokareva, *Unitarity of Minkowski nonlocal theories made explicit*, Phys.
Rev. D 104 (2021) 025016 [2103.01945].

[44] A. Ghoshal, A. Mazumdar, N. Okada and D. Villalba, *Stability of infinite derivative Abelian
Higgs models*, Phys. Rev. D 97 (2018) 076011 [1709.09222].

[45] B. Barman and A. Ghoshal, *Scale invariant FIMP miracle*, JCAP 03 (2022) 003
[2109.03259].

[46] B. Barman and A. Ghoshal, *Probing pre-BBN era with Scale Invariant FIMP*, 2203.13269.

[47] A. Ghoshal, N. Okada and A. Paul, *Radiative Plateau Inflation with Conformal Invariance:
Dynamical Generation of Electroweak and Seesaw Scales*, 2203.00677.

[48] A. Ghoshal, D. Mukherjee and M. Rinaldi, *Inflation and primordial gravitational waves in
scale-invariant quadratic gravity*, 2205.06475.

[49] A. Dasgupta, P.S.B. Dev, A. Ghoshal and A. Mazumdar, *Gravitational Wave Pathway to
Testable Leptogenesis*, 2206.07032.

[50] A. Ghoshal and A. Salvio, *Gravitational waves from fundamental axion dynamics*, JHEP 12
(2020) 049 [2007.00005].

[51] A. Ghoshal, A. Mazumdar, N. Okada and D. Villalba, *Nonlocal non-Abelian gauge theory:
Conformal invariance and $\beta$-function*, Phys. Rev. D 104 (2021) 015003 [2010.15919].

[52] L. Buoninfante, A. Ghoshal, G. Lambiase and A. Mazumdar, *Transmutation of nonlocal scale
in infinite derivative field theories*, Phys. Rev. D 99 (2019) 044032 [1812.01441].

[53] A. Ghoshal, *Scalar dark matter probes the scale of nonlocality*, Int. J. Mod. Phys. A 34 (2019)
1950130 [1812.02314].

[54] N.V. Krasnikov, *Nonlocal SU(5) GUT, Mod. Phys. Lett. A 36* (2021) 2150104 [2012.10161].

[55] F. Nortier, *Shadow Extra Dimensions & Fuzzy Branes with an Exorcized Spectrum*,
2112.15592.

[56] M. Frasca and A. Ghoshal, *Strongly Coupled String-inspired Infinite Derivative Non-local
Yang-Mills: Diluted Mass Gap*, JHEP 21 (2020) 226 [2102.10665].

[57] M. Frasca and A. Ghoshal, *Mass gap in strongly coupled infinite derivative non-local Higgs:
Dyson-Schwinger approach*, Class. Quant. Grav. 38 (2021) 175013 [2011.10586].

[58] M. Frasca, A. Ghoshal and N. Okada, *Confinement and renormalization group equations in
string-inspired nonlocal gauge theories*, Phys. Rev. D 104 (2021) 096010 [2106.07629].

[59] M. Frasca, A. Ghoshal and A.S. Koshelev, *Non-perturbative Lee-Wick gauge theory: BRST,
confinement & RGE with strong couplings*, 2202.09578.

[60] M. Frasca, A. Ghoshal and S. Groote, *Confinement in QCD and generic Yang-Mills theories
with matter representations*, 2202.14023.

[61] V.P. Frolov, A. Zelnikov and T. de Paula Netto, *Spherical collapse of small masses in the
ghost-free gravity*, JHEP 06 (2015) 107 [1504.00412].
[62] V.P. Frolov and A. Zelnikov, *Head-on collision of ultrarelativistic particles in ghost-free theories of gravity*, Phys. Rev. D 93 (2016) 064048 [1509.03336].

[63] A.S. Koshelev, J.a. Marto and A. Mazumdar, *Schwarzschild 1/r-singularity is not permissible in ghost free quadratic curvature infinite derivative gravity*, Phys. Rev. D 98 (2018) 064023 [1803.00309].

[64] A.S. Koshelev and A. Mazumdar, *Do massive compact objects without event horizon exist in infinite derivative gravity?*, Phys. Rev. D 96 (2017) 084069 [1707.00273].

[65] L. Buoninfante, A.S. Koshelev, G. Lambiase and A. Mazumdar, *Classical properties of non-local, ghost- and singularity-free gravity*, JCAP 09 (2018) 034 [1802.00399].

[66] A.S. Cornell, G. Harmsen, G. Lambiase and A. Mazumdar, *Rotating metric in nonsingular infinite derivative theories of gravity*, Phys. Rev. D 97 (2018) 104006 [1710.02162].

[67] L. Buoninfante, A.S. Koshelev, G. Lambiase, J.a. Marto and A. Mazumdar, *Conformally-flat, non-singular static metric in infinite derivative gravity*, JCAP 06 (2018) 014 [1804.08195].

[68] L. Buoninfante, G. Harmsen, S. Maheshwari and A. Mazumdar, *Nonsingular metric for an electrically charged point-source in ghost-free infinite derivative gravity*, Phys. Rev. D 98 (2018) 084009 [1804.09624].

[69] S. Abel, L. Buoninfante and A. Mazumdar, *Nonlocal gravity with worldline inversion symmetry*, JHEP 01 (2020) 003 [1911.06697].

[70] L. Buoninfante, G. Lambiase, Y. Miyashita, W. Takebe and M. Yamaguchi, *Generalized ghost-free propagators in nonlocal field theories*, Phys. Rev. D 101 (2020) 084019 [2001.07830].

[71] T. Biswas, A. Mazumdar and W. Siegel, *Bouncing universes in string-inspired gravity*, JCAP 03 (2006) 009 [hep-th/0508194].

[72] T. Biswas, R. Brandenberger, A. Mazumdar and W. Siegel, *Non-perturbative Gravity, Hagedorn Bounce* (& CMB, JCAP 12 (2007) 011 [hep-th/0610274]).

[73] T. Biswas, T. Koivisto and A. Mazumdar, *Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity*, JCAP 11 (2010) 008 [1005.0590].

[74] T. Biswas, A.S. Koshelev, A. Mazumdar and S.Y. Vernov, *Stable bounce and inflation in non-local higher derivative cosmology*, JCAP 08 (2012) 024 [1206.6374].

[75] A.S. Koshelev and S.Y. Vernov, *On bouncing solutions in non-local gravity*, Phys. Part. Nucl. 43 (2012) 666 [1202.1289].

[76] A.S. Koshelev, J.a. Marto and A. Mazumdar, *Towards resolution of anisotropic cosmological singularity in infinite derivative gravity*, JCAP 02 (2019) 020 [1803.07072].

[77] I. Kolář, F.J.M. Torralba and A. Mazumdar, *New nonsingular cosmological solution of nonlocal gravity*, Phys. Rev. D 105 (2022) 044045 [2109.02143].

[78] G. Calcagni and G. Nardelli, *Nonlocal instantons and solitons in string models*, Phys. Lett. B 669 (2008) 102 [0802.4395].

[79] J. Boos, V.P. Frolov and A. Zelnikov, *Gravitational field of static p-branes in linearized ghost-free gravity*, Phys. Rev. D 97 (2018) 084021 [1802.09573].

[80] J. Boos, *Gravitational Friedel oscillations in higher-derivative and infinite-derivative gravity?*, Int. J. Mod. Phys. D 27 (2018) 1847022 [1804.00225].

[81] I. Kolář and A. Mazumdar, *NUT charge in linearized infinite derivative gravity*, Phys. Rev. D 101 (2020) 124005 [2004.07613].

[82] J. Boos, *Angle deficit and nonlocal gravitoelectromagnetism around a slowly spinning cosmic string*, Int. J. Mod. Phys. D 29 (2020) 2043027 [2003.13847].
[83] I. Kolar, F.J. Maldonado Torralba and A. Mazumdar, *Junction conditions in infinite derivative gravity*, Phys. Rev. D **103** (2021) 064078 [2012.10349].

[84] I. Kolář, T. Málek, S. Dengiz and E. Kilicarslan, *Exact gyratons in higher and infinite derivative gravity*, Phys. Rev. D **105** (2022) 044018 [2107.11884].

[85] I.Y. Kobzarev, L.B. Okun and M.B. Voloshin, *Bubbles in Metastable Vacuum*, Yad. Fiz. **20** (1974) 1229.

[86] S.R. Coleman, *The Fate of the False Vacuum. 1. Semiclassical Theory*, Phys. Rev. D **15** (1977) 2929.

[87] C.G. Callan, Jr. and S.R. Coleman, *The Fate of the False Vacuum. 2. First Quantum Corrections*, Phys. Rev. D **16** (1977) 1762.

[88] D. Croon, E. Hall and H. Murayama, *Non-perturbative methods for false vacuum decay*, 2104.10687.

[89] M. Frasca, A. Ghoshal and N. Okada, *Fate of False Vacuum in Non-perturbative Regimes*, 2201.12267.

[90] S.R. Coleman, *Quantum Tunneling and Negative Eigenvalues*, Nucl. Phys. B **298** (1988) 178.

[91] G. Calcagni, M. Frasca and A. Ghoshal, *Fate of false vacuum in non-perturbative regimes: Gravity effects*, 2206.09965.

[92] S.R. Coleman and F. De Luccia, *Gravitational Effects on and of Vacuum Decay*, Phys. Rev. D **21** (1980) 3305.

[93] L. Buoninfante, G. Lambiase and A. Mazumdar, *Quantum solitonic wave-packet of a meso-scopic system in singularity free gravity*, Nucl. Phys. B **931** (2018) 250 [1708.06731].

[94] L. Buoninfante, G. Lambiase and A. Mazumdar, *Quantum spreading of a self-gravitating wave-packet in singularity free gravity*, Eur. Phys. J. C **78** (2018) 73 [1709.09263].

[95] L. Buoninfante, A. Mazumdar and J. Peng, *Nonlocality amplifies echoes*, Phys. Rev. D **100** (2019) 104059 [1906.03624].

[96] G. Calcagni, M. Montobbio and G. Nardelli, *Localization of nonlocal theories*, Phys. Lett. B **662** (2008) 285 [0712.2237].

[97] G. Calcagni and G. Nardelli, *String theory as a diffusing system*, JHEP **02** (2010) 093 [0910.2160].

[98] J.A.R. Cembranos, S.E.R. Ciarreta and L.J. Garay, *Scale Holography*, Eur. Phys. J. C **78** (2018) 732 [1611.00853].

[99] G. Calcagni, L. Modesto and G. Nardelli, *Initial conditions and degrees of freedom of non-local gravity*, JHEP **05** (2018) 087 [1803.00561].

[100] G. Calcagni, *Taming the Beast: Diffusion Method in Nonlocal Gravity*, Universe **4** (2018) 95 [1809.04292].

[101] J. Edholm, A.S. Koshelev and A. Mazumdar, *Behavior of the Newtonian potential for ghost-free gravity and singularity-free gravity*, Phys. Rev. D **94** (2016) 104033 [1604.01989].

[102] C.D. Carone, *Unitarity and microscopic acausality in a nonlocal theory*, Phys. Rev. D **95** (2017) 045009 [1605.02030].

[103] C. Heredia, I. Kolář, J. Llosa, F.J.M. Torralba and A. Mazumdar, *Infinite-derivative linearized gravity in convolutional form*, Class. Quant. Grav. **39** (2022) 085001 [2112.05397].

[104] I. Kolář, *Non-local scalar fields in static spacetimes via heat kernels*, 2201.09908.

[105] B. Grinstein, D. O'Connell and M.B. Wise, *The Lee-Wick standard model*, Phys. Rev. D **77** (2008) 025012 [0704.1845].
[106] C.D. Carone and R.F. Lebed, *A Higher-Derivative Lee-Wick Standard Model*, JHEP 01 (2009) 043 [0811.4150].

[107] J. Boos and C.D. Carone, *Asymptotic nonlocality*, Phys. Rev. D 104 (2021) 015028 [2104.11195].

[108] J. Boos and C.D. Carone, *Asymptotic nonlocality in gauge theories*, Phys. Rev. D 104 (2021) 095020 [2109.06261].

[109] J. Boos and C.D. Carone, *Asymptotic nonlocality in non-Abelian gauge theories*, Phys. Rev. D 105 (2022) 035034 [2112.05270].

[110] L. Modesto and G. Calcagni, *Tree-level scattering amplitudes in nonlocal field theories*, JHEP 10 (2021) 169 [2107.04558].

[111] S. Giaccari and L. Modesto, *Causality in Nonlocal Gravity*, in 10th MATHEMATICAL PHYSICS MEETING: School and Conference on Modern Mathematical Physics, pp. 121–136, 2020 [1803.08748].

[112] LIGO SCIENTIFIC, VIRGO collaboration, *Search for the isotropic stochastic background using data from Advanced LIGO's second observing run*, Phys. Rev. D 100 (2019) 061101 [1903.02886].

[113] KAGRA, VIRGO, LIGO SCIENTIFIC collaboration, *Search for anisotropic gravitational-wave backgrounds using data from Advanced LIGO and Advanced Virgo's first three observing runs*, Phys. Rev. D 104 (2021) 022005 [2103.08520].

[114] A. Romero, K. Martinovic, T.A. Callister, H.-K. Guo, M. Martínez, M. Sakellariadou et al., *Implications for First-Order Cosmological Phase Transitions from the Third LIGO-Virgo Observing Run*, Phys. Rev. Lett. 126 (2021) 151301 [2102.01714].