Generalized Darmois-Israel junction conditions

Chong-Sun Chu1,2 and Hai Siong Tan1

1 Physics Division, National Center for Theoretical Sciences, National Tsing
Hua University
2 Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan

Abstract

We present a general method to derive the appropriate Darmois-Israel junction
conditions for gravitational theories with higher-order derivative terms by
integrating the bulk equations of motion across the singular hypersurface. In
higher derivative theories, the field equations can contain terms which are
more singular than the Dirac delta distribution. To handle them appropriately,
we formulate a regularization procedure based on representing the delta
function as the limit of a sequence of classical functions. This procedure
involves imposing suitable constraints on the extrinsic curvature such that
the field equations are compatible with the singular source being a delta
distribution. As explicit examples of our approach, we demonstrate in detail
how to obtain the generalized junction conditions for quadratic gravity,
$\mathcal{F}(R)$ theories, a 4D low-energy effective action in string theory
and action terms that are Euler densities. Our results are novel, and refine
the accuracy of previously claimed results in $\mathcal{F}(R)$ theories and
quadratic gravity. In particular, when the coupling constants of quadratic
gravity are those for the Gauss-Bonnet case, our junction conditions reduce to
the known ones for the latter obtained independently by boundary variation of
a surface term in the action. Finally, we briefly discuss a couple of
applications to thin-shell wormholes and stellar models.

###### Contents

  1. 1 Introduction
  2. 2 Generalized Junction Conditions - a general approach
    1. 2.1 Some Preliminaries
    2. 2.2 On the standard Darmois-Israel junction conditions in GR
    3. 2.3 Delta sequences and a double scaling limit
    4. 2.4 Generalized distributions and regularity constraints
      1. 2.4.1 Warm-up: more about $\delta(n),\delta^{\prime}(n)$
      2. 2.4.2 Regularity constraints for products of nascent delta functions & their derivatives
    5. 2.5 The relation to Hadamard Regularization
  3. 3 Generalized junction conditions for gravitational theories with quadratic terms
    1. 3.1 Junction terms from integrating $\tilde{G}_{ij}$
    2. 3.2 Junction terms from integrating $\tilde{G}_{in}$
    3. 3.3 Junction terms from integrating $\tilde{G}_{nn}$
    4. 3.4 Regularity constraints
    5. 3.5 A summary of results
  4. 4 Junction conditions for other examples of higher-derivative theories
    1. 4.1 $\mathcal{F}(R)$ theories
      1. 4.1.1 More about junction conditions of $R^{2}$ theory
      2. 4.1.2 Junction conditions for $R^{3}$ theory
      3. 4.1.3 Some general comments
    2. 4.2 Low-energy effective action from toroidal compactification of the Heterotic String
    3. 4.3 Higher-dimensional Euler densities
  5. 5 Applications
    1. 5.1 Thin-Shell wormholes in $R^{2}$ gravity
    2. 5.2 Implications for stellar models
  6. 6 Concluding Remarks
  7. A Some useful integral identities
  8. B From quadratic gravity to Gauss-Bonnet theory: some notes on the junction conditions

##  1 Introduction

In general relativity, the Darmois-Israel junction conditions prescribe the
appropriate boundary conditions across a singular hypersurface $(\Sigma)$
supported by a localized energy-momentum source that contains a Dirac delta
distribution. Apart from metric continuity at $\Sigma$, these ‘junction
conditions’ can be expressed simply as the following equation

$[K]h_{ij}-[K_{ij}]=8\pi S_{ij},\qquad K\equiv K^{m}_{m},$ (1.1)

where the square bracket indicates the jump discontinuity across $\Sigma$,
$\overbracket{h_{ij}}$ is the induced metric, $S_{ij}$ is the singular Dirac
delta source localized within $\Sigma$, and all indices pertain to coordinates
of $\Sigma$. The Darmois-Israel junction conditions in (1.1) were historically
obtained in [1, 2, 3] by integrating the field equations across the
infinitesimal width of $\Sigma$, and were later shown to equivalently follow
from the boundary variation of the Gibbons-Hawking-York term in the action [4,
5].

Since its discovery, (1.1) has found numerous applications in a plethora of
gravitational topics, being primarily used to govern the construction of
spacetime geometries obtained by a ‘cut-and-paste’ approach with $\Sigma$ as
the locus of identification. These geometries include thin-shell wormholes [6]
(and their other variants such as gravastars [7, 8]), static and dynamical
stellar models [9, 10] like the Oppenheimer-Snyder solution [11] describing
gravitational collapse, as well as domain wall configurations [12] separating
true and false vacua in cosmology.

In recent years, there have been interesting explorations of generalizing
(1.1) to the wider context of gravitational theories beyond Einstein’s theory
which we can consider to be a long-distance, effective description connected
to some presumably UV-complete framework of quantum gravity such as string
theory. When interpreted as such, from the effective field theory perspective,
it is natural to consider adding higher-derivative terms to the Einstein-
Hilbert action. Models that have attracted interest include supergravity,
$\mathcal{F}(R)$-theories, Gauss-Bonnet gravity, etc. Accordingly, there has
been a small number of different proposals for how (1.1) should be modified in
several contexts. For example, junction conditions were proposed in [13, 14,
15] for $\mathcal{F}(R)$ gravity, and in [16, 17] for quadratic gravity.

In this paper, we present a general method to derive the appropriate junction
conditions for gravitational theories with higher-order derivative terms by
integrating the bulk equations of motion across the infinitesimal width of the
singular surface $\Sigma$, in the same spirit as was done in the foundational
paper of Israel in [2]. Fundamentally, asserting an equivalence between both
sides of the field equations in the sense of distribution translates to
ensuring the convergence of the integral which is now much more complicated by
virtue of the higher-order differential operators. Completing the integral
then yields the generalized junction conditions compatible with a Dirac
$\delta$-distribution carried by the energy-momentum tensor.

We find that a combination of the following techniques furnishes an effective
toolbox for solving this problem. First, we choose to work in Gaussian normal
coordinates describing a local neighborhood of $\Sigma$ which is now defined
by taking $n=0$, where $n$ parametrizes the proper distance along a geodesic
orthogonal to $\Sigma$. In this chart, we invoke the Gauss-Codazzi relations
extensively to express various curvature terms in the equations of motion in
terms of the intrinsic geometric quantities of $\Sigma$, the extrinsic
curvature and their derivatives. The problem is now reduced to examining the
convergence of a one-dimensional integral. For this purpose, we find that
expressing the $\delta$-function as a limit of a sequence of classical
functions gives us a powerful language for organizing various integrand terms
according to their algebraic order of singularity.iiiWe note that in both the
math and physics literature, the representation of the $\delta$-distribution
by the limit of delta-convergent sequences has a long history (see e.g. [20]
for the context of the theory of distributions). A good, old example relevant
for both communities is the Kramers-Kronig or Sokhotski-Plemelj equation which
defines, in the distributional sense, $\delta(x)=\mp\frac{1}{\pi
i}\left(\lim_{\epsilon\rightarrow 0}\frac{1}{x\pm
i\epsilon}-\text{p.v.}\left(\frac{1}{x}\right)\right)$. Taking the sum of the
two choices of sign leads to the $\delta$-distribution being a limit of a
sequence of Cauchy distributions : $\delta(x)=\lim_{\epsilon\rightarrow
0}\frac{1}{\pi}\frac{\epsilon}{x^{2}+\epsilon^{2}}$, which is a concrete
example of a ‘nascent delta function’ that will feature frequently in our
narrative.  This allows us to identify what are the additional conditions on
the embedding geometry that naturally ensure the absence of singular terms
that are incompatible with a $\delta$-distribution source term. We call these
conditions ‘ _regularity constraints_ ’, and they are typically expressed as
the vanishing of some function of the extrinsic curvature components on each
side of $\Sigma$. As we shall elaborate later on, it turns out that they are
intimately related to the Hadamard regularization procedure [18] in the theory
of distributions (see e.g. [19]). In the absence of the regularity
constraints, the generalized junction conditions are precisely the Hadamard-
finite part of the integral. Our method can be readily adopted and applied to
a wide variety of gravitational theories, including matter and gauge
couplings, although in this paper, we focus almost exclusively on action terms
that are invariants constructed from the Riemann tensor as illustrative
examples.

Our results for $\mathcal{F}(R)$ theory and quadratic gravity refine the
accuracy of those reported earlier in literature. For example, in Section 4,
we point out a missing term in [14, 13, 16] for junctions conditions in the
theory with $R^{2}$ action term. In [16], the authors argued that the naive
appearance of products of delta function in the equations of motion leads to
inconsistency, and that there is no unambiguous manner to sum them up. On this
issue, we find that the use of delta-convergent sequences clearly quantifies
the growth property of each integrand term, and provides an organizational
principle for identifying regularity constraints and a well-defined procedure
to sum up the residual finite terms, eventually leading us to the junction
conditions. Another difference between our work and that of [16] is that in
[16], the authors posited that in quadratic gravity, the energy-momentum
tensor should contain the distributional derivative of the $\delta$-function
that they derived from the higher-order differential operators acting on a
discontinuous Riemann tensor.iiiiiiThey dubbed this component as the ‘double
layer’. Also, they postulated the regularity of the on-shell action as a
necessary condition, but here, we take the equations of motion as the physical
starting point of our analysis. For us, we preserve the meaning of
$T_{\mu\nu}$ as carrying only a $\delta$-source, and integrate the LHS of the
field equations to obtain the junction conditions directly, respecting the
equivalence between both sides of the field equations in the distributional
sense.

We should also mention that our method of derivation passes a stringent
consistency test that is noticeably absent in the past literature, namely that
when applied to quadratic gravity, i.e. taking the Lagrangian
$\mathcal{L}=\beta_{1}R^{2}+\beta_{2}R^{\alpha\beta}R_{\alpha\beta}+\beta_{3}R^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}$,
our junction conditions reduce to those of Gauss-Bonnet gravity
$(\beta_{2}=-4\beta_{3}=-4\beta_{1})$ that were derived some time ago in [21]
via a completely different approach — the boundary variation of a surface term
[23] that accompanies the gravitational action. As already argued in [23, 24]
and definitively shown in [25], the equations of motion of a generic higher-
derivative gravitational theory do not descend from a well-posed variational
principle with Dirichlet conditions, and hence the approach of obtaining their
appropriate junction conditions by boundary variation of surface terms is not
always applicable.iiiiiiiiiIn [17], an attempt was made to derive the junction
conditions for a generic quadratic theory using a variational principle, but
imposed a variation of the extrinsic curvature inconsistent with its
fundamental definition. See further comments in Appendix B.  An exception lies
in the family of theories defined by Euler density terms which are the linear
combinations of curvature invariants that generate only second-order field
equations, and thus, in principle, these exists appropriate surface terms for
them. The simplest example would be the Ricci scalar being the 2D Euler
density, with the Gibbons-Hawking-York term $S_{GHY}=\frac{1}{8\pi}\int
d^{d-1}x\,\,\sqrt{h}K$ being its surface term, and of which variation with
respect to the induced metric yields (1.1). The junction condition derived in
this manner nicely furnishes a consistency check for the other derivation
route obtained by integrating across $\Sigma$. In Appendix B, we will
demonstrate how the complicated junction conditions for a generic quadratic
gravity theory reduce to the known ones for Gauss-Bonnet gravity. The
regularity constraints need not be imposed precisely for
$\beta_{2}=-4\beta_{3}=-4\beta_{1}$, and this also serves as a consistency
check for the corresponding equations used in determining them. On this point,
we note that previously in [16] where the authors proposed junction conditions
for quadratic gravity, they imposed $[K_{ij}]=0$ even in the Gauss-Bonnet case
which contradicts the known ones derived by a boundary variation [21, 22].

Our exposition is structured as follows. In Section 2, we present the
background canvas of concepts underlying our general approach of deriving the
junction conditions via a bulk integration across $\Sigma$, including the
notion of delta-convergent sequences, regularity constraints and how our
method relates to Hadamard regularization. Section 3 contains a detailed
derivation of the junction conditions for quadratic gravity. This is
accompanied by Appendix A containing some integral identities that we
developed for our purpose, and by Appendix B containing details of how the
junction conditions in quadratic gravity reduce consistently to known ones for
Gauss-Bonnet gravity. Section 4 presents the junction conditions for three
types of theories: $\mathcal{F}(R)$ theories, a low-energy effective action
descending from string theory, and higher-dimensional Euler density terms that
appear in Lovelock gravity. In Section 5, we briefly discuss a couple of
applications by examining how thin-shell wormholes and stellar models are
governed by junction conditions in the context of $\mathcal{F}(R)=R+\beta
R^{2}$ theory. Finally, we end with some concluding remarks in Section 6. For
the reader who is uninterested in the illustrative examples of gravitational
theories we have chosen, Section 2 suffices as a stand-alone explanation of
the tapestry of techniques used in our method, but in our opinion, reviewing a
couple of concrete examples in Section 3 or 4 should still be useful towards
learning how to apply our method for other theories.

_Conventions_ : We work in natural units where $\mathcal{G}=c=1$. Greek
indices refer to all of spacetime, whereas Latin indices pertain to
coordinates of the singular hypersurface $\Sigma$, with ‘$n$’ reserved for the
direction normal to $\Sigma$. Our convention for the Riemann tensor is
${R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}+\ldots$.

##  2 Generalized Junction Conditions - a general approach

###  2.1 Some Preliminaries

For a generic gravitational theory with higher-order derivatives of the metric
in the action, let us denote the field equations as

$\tilde{G}_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\ldots=8\pi
T_{\mu\nu},$ (2.1)

where the ellipses represent the additional terms descending from the modified
action. In this work, we will mainly be studying gravitational theories where
these terms are products of various contractions of the Riemann tensor and
differential operators acting on them. On the RHS, we assume that the energy-
momentum tensor harbors a singular source localized on some codimension one
hypersurface $\Sigma$. If we integrate across the infinitesimal width of
$\Sigma$, only singular terms on both sides of (2.1) survive and the resulting
integral yields the appropriate _junction conditions_ for the gravitational
theory considered. The main purpose of this paper is to develop a conceptual
and computational basis for this integration procedure. Fundamentally, apart
from topological action terms which are not reflected in (2.1), the bulk
equations of motion suffice as the starting point for the derivation of the
junction conditions which can be regarded as consistency conditions for the
bulk dynamics induced on $\Sigma$.

In the full generality, the induced metric on $\Sigma$ $h_{ij}$ can be
timelike, spacelike or null. For the rest of the paper, we will mainly focus
on the timelike case which has garnered the most interest in the past
literature. When $\Sigma$ is endowed with a singular energy-momentum tensor
localized on it, it represents a singular thin shell of matter. (For a
spacelike $h_{ij}$, this would be a gravitational instanton whereas a null
$h_{ij}$ would represent a singular light cone.)

In the most generic setting, $\Sigma$ is a codimension-one hypersurface that
is embedded in a bulk manifold ($\mathcal{M}$) constructed from two distinct
ones ($\mathcal{M}_{1},\mathcal{M}_{2}$) that is joined by identifying

$\Sigma_{1}\sim\Sigma_{2}\sim\Sigma,$

where $\Sigma_{1,2}$ are hypersurfaces in $\mathcal{M}_{1,2}$ that can be
related via a homeomorphism. In this cut-and-paste procedure, the final
manifold $\mathcal{M}$ is the union of $\Sigma$ and the interiors of
$\mathcal{M}_{1,2}$ previously bounded by $\Sigma_{1,2}$. We can introduce a
set of coordinates $(\zeta^{i})$ intrinsic to $\Sigma$ which is defined
parametrically by

$f\left(x^{\mu}(\zeta^{i})\right)=0,$

with the induced metric

$h_{ij}=g_{\mu\nu}\frac{\partial x^{\mu}}{\partial\zeta^{i}}\frac{\partial
x^{\nu}}{\partial\zeta^{j}}.$ (2.2)

Denoting the sign of $n^{2}$ by $\xi$, the unit normal vectors to $\Sigma$
read

$n_{\mu}=\xi N\frac{\partial f}{\partial x^{\mu}},\qquad
N=\left|g^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f\right|^{-1/2},\,\,\,\,n^{2}=\xi,$

where $\xi=\\{+1,-1,0\\}$ for the cases of $\Sigma$ being timelike, spacelike
and null respectively. The second fundamental form or the extrinsic curvature
enacts an essential role in our discussion and can be defined as follows.

$K_{ab}=h^{m}_{a}h^{l}_{b}\nabla_{m}n_{l},\qquad n^{a}K_{ab}=0.$ (2.3)

For our purpose, we find it useful to work in the Gaussian normal coordinate
system in the local neighborhood of $\Sigma$ constructed from the family of
non-intersecting geodesics orthogonal to $\Sigma$.

$ds^{2}=\xi dn^{2}+h_{ij}(n,x^{k})dx^{i}dx^{j},$ (2.4)

with $\Sigma$ defined by taking $n=0$ and $x^{i}$ are the intrinsic
coordinates. Having briefly described the geometric characterization of
$\Sigma$, let us now return to the field equations in (2.1). On the RHS, we
can express the energy-momentum tensor as

$T_{\mu\nu}=S_{\mu\nu}\delta(n)+\ldots,$

where we have denoted the singular source by $S_{\mu\nu}$ and the ellipses
refer to other regular components of the energy-momentum tensor. Working in
the local Gaussian normal chart, we can integrate (2.1) across the
infinitesimal width of $\Sigma$ as follows.

$\lim_{\epsilon\rightarrow
0^{+}}\int^{\epsilon}_{-\epsilon}dn\,\,\tilde{G}_{\mu\nu}=8\pi S_{\mu\nu}.$
(2.5)

The infinitesimal integral domain picks up only the $\delta$-distribution part
of the energy-momentum tensor, since the regular components vanish in the
$\epsilon\rightarrow 0$ limit. Formally, we can see this by writing

$\lim_{\epsilon\rightarrow
0^{+}}\int^{\epsilon}_{-\epsilon}dn=\int^{\infty}_{-\infty}dn-\text{p.v.}\int^{\infty}_{-\infty}dn,$

where the second term on the RHS is the Cauchy principal value of the first
integral term. In local Gaussian normal coordinates, the Christoffel symbols
read

$\Gamma^{n}_{ij}=-\frac{1}{2}\partial_{n}h_{ij}=-K_{ij},\,\,\,\Gamma^{i}_{nj}=K^{i}_{j}=\frac{1}{2}h^{im}\partial_{n}h_{jm}.$
(2.6)

The discontinuity in $\partial_{n}h_{ij}$ or the extrinsic curvature induces
various singular terms in $\tilde{G}_{\mu\nu}$ which is then related to the
singular source $S_{\mu\nu}$ in (2.5).

A main theme of this work is to seek the conditions characterizing the
embedding of $\Sigma$, under which the LHS of (2.5) is well-defined. This then
leads to an appropriate set of junction conditions on the LHS of (2.5) upon
performing the integral. _We take this to be the operational meaning of the
delta function source in the energy-momentum tensor, since the field equations
should be understood in the sense of distribution when the energy-momentum
tensor carries a delta-singular source._

Now in our derivation, the Gauss-Codazzi relations turn out to be crucial as
they express various curvature quantities in terms of the intrinsic Riemann
tensor and the extrinsic curvature tensor, elucidating the form of various
singular terms in $\tilde{G}_{\mu\nu}$ easily. Recall that the Gauss-Codazzi
equations read

$\displaystyle\hat{R}^{a}{}_{bcd}$ $\displaystyle=$
$\displaystyle\mathcal{P}\left({R^{a}}_{bcd}\right)+\xi\left(K^{a}_{c}K_{bd}-K^{a}_{d}K_{bc}\right),$
(2.7) $\displaystyle n^{a}R_{aqrs}h^{q}_{b}h^{r}_{c}h^{s}_{d}$
$\displaystyle=$ $\displaystyle\xi\left(D_{d}K_{bc}-D_{c}K_{bd}\right),$ (2.8)

where $\mathcal{P}$ refers to the indices being projected with the induced
metric $h_{ab}$, $D_{a}$ is the projected covariant derivative defined with
the induced metric and hatted variables are geometrical quantities intrinsic
to $\Sigma$. Using Gaussian normal coordinates and contracting some of the
indices, we further obtain the following useful set of equations.

$\displaystyle{R^{n}}_{snv}$ $\displaystyle=$
$\displaystyle\xi\left(-\partial_{n}K_{vs}+K_{av}K^{a}_{s}\right),\,\,\,R_{nijk}=D_{k}K_{ij}-D_{j}K_{ik},$
(2.9) $\displaystyle R_{ij}$ $\displaystyle=$
$\displaystyle\xi\left(-\partial_{n}K_{ij}+2K^{a}_{j}K_{ai}-KK_{ij}\right)+\hat{R}_{ij},$
(2.10) $\displaystyle R_{nn}$ $\displaystyle=$
$\displaystyle-\partial_{n}K-K^{ab}K_{ab}=-h^{ij}\partial_{n}K_{ij}+K^{ab}K_{ab},$
(2.11) $\displaystyle R_{in}$ $\displaystyle=$ $\displaystyle
D^{k}K_{ik}-D_{i}K,$ (2.12) $\displaystyle R$ $\displaystyle=$
$\displaystyle\hat{R}-\xi\left(2\partial_{n}K+K^{2}+K^{ab}K_{ab}\right).$
(2.13)

###  2.2 On the standard Darmois-Israel junction conditions in GR

It is instructive to first review how one can obtain the standard Darmois-
Israel junction conditions in ordinary GR before proceeding to more
complicated gravitational theories. In (2.5), we now simply take
$\tilde{G}_{\mu\nu}=G_{\mu\nu}$ and integrate

$\lim_{\epsilon\rightarrow
0^{+}}\int^{\epsilon}_{-\epsilon}dn\,\,G_{\mu\nu}=8\pi S_{\mu\nu}.$ (2.14)

On the LHS, only singular terms in the Einstein tensor survive in the limit
$\epsilon\rightarrow 0$. The singularity of these terms are of the Dirac delta
type and can be traced to a discontinuity in the derivative of the metric in
the direction normal to $\Sigma$ which, in the Gaussian normal chart, is
effectively the discontinuous extrinsic curvature. We keep track of these
terms appearing in the Einstein tensor which can be simplified to read

$\displaystyle\xi\left(\partial_{n}K^{i}_{j}-\delta^{i}_{j}\partial_{n}K\right)+\xi\left(-KK^{i}_{j}+\ldots\right)=8\pi
T^{i}_{j},$ (2.15)
$\displaystyle\xi\left(-\nabla_{i}K+\nabla_{j}K^{j}_{i}\right)=8\pi
T^{n}_{i},\,\,\,-\frac{1}{2}\hat{R}+\frac{1}{2}\xi\left(K^{2}-\text{Tr}(K^{2})\right)=8\pi
T^{n}_{n},$ (2.16)

where the ellipses are terms which do not involve $\partial_{n}$. Integrating
over $\Sigma$ as in (2.14) on both sides of the field equations then yields
the junction conditions

$\xi\left(-[K_{ij}]+[K]g_{ij}\right)=8\pi S_{ij},$ (2.17)

which is the standard Darmois-Israel junction conditions in GR, and we have
used the bracket $[...]$ to denote the jump or difference between the limiting
values of the bracketed quantity on each side of $\Sigma$, for example
$[K]\equiv K|_{n=0^{+}}-K|_{n=0^{-}}$. We also note that the junction
conditions in other directions are trivial, i.e. in (2.5),

$\lim_{\epsilon\rightarrow
0^{+}}\int^{\epsilon}_{-\epsilon}dn\,\,G_{in}=\lim_{\epsilon\rightarrow
0^{+}}\int^{\epsilon}_{-\epsilon}dn\,\,G_{nn}=0.$ (2.18)

Thus in GR, the localized singular source $S_{\mu\nu}$ cannot have components
non-parallel to $\Sigma$. As we shall see later, this is not generally true
for other gravitational theories. Another feature characterizing the junction
condition (2.17) that differentiates a generic higher-derivative theory from
GR is that the junction equation does not only involve bracketed terms but
also acquires terms that represent averaging across $\Sigma$.

A crucial ingredient in our formulation is a procedure that takes into account
singular terms more divergent than delta function terms in the LHS of the
field equations (2.1), that may appear in the equations of motion upon
assuming a discontinuous extrinsic curvature (note that this problem is absent
in (2.15)). We demonstrate how to impose suitable regularity constraints on
the extrinsic curvature such that the integral in (2.5) remains convergent,
leading to generalized junction conditions which are the appropriate boundary
conditions at $\Sigma$. We will find that an essential tool is to regard the
delta function as a limit of a sequence of absolutely convergent functions, a
delicate subject which we turn to next.

###  2.3 Delta sequences and a double scaling limit

Since the singularities in (2.1) essentially arise from derivatives of the
Heaviside step function parametrizing the discontinuity in the extrinsic
curvature, we first consider a suitable representation of it by defining it as
a limit $b\rightarrow 0$ of a sequence of classical functions $\Theta(n,b)$,
with

$\Theta(n)=\lim_{b\rightarrow
0}\Theta(n,b),\qquad\Theta(n,b)=\frac{1}{2}+\Theta_{X}\left(\frac{n}{b}\right),$
(2.19)

where we have chosen to parametrize it such that its derivative yields a
representation of the delta function as follows.

$\delta(n)=\lim_{b\rightarrow 0}\partial_{n}\Theta(n,b)=\lim_{b\rightarrow
0}\frac{1}{b}\Theta^{\prime}_{X}\left(X\right)\equiv\lim_{b\rightarrow
0}\frac{1}{b}F(X),\qquad X\equiv\frac{n}{b},$ (2.20)

The function $F(X)$ is sometimes known as a nascent delta function, with
$\Theta_{X}(X)$ being its antiderivative. As a generalized function or
distribution, we require that

$\lim_{b\rightarrow 0}\int^{\infty}_{-\infty}dn\frac{1}{b}F(X)f(n)=f(0),$
(2.21)

for all continuous $f(n)$ with compact support which implies that $F(X)$ is
normalized as

$\int^{\infty}_{-\infty}dX\,\,F(X)=1.$ (2.22)

Formally, each nascent delta function gives rise to a distribution of the form

$\tilde{f}_{1/b}=\partial_{n}\Theta(n,b)=\frac{1}{b}F(n/b),$ (2.23)

and the sequence $\tilde{f}_{1},\tilde{f}_{2},\ldots$ converges to the delta
function distribution $\delta(n)$ if the limit defined on the LHS on (2.21)
exists. Some popular choices of the nascent delta functionivivivSee for
example [26] for a deeper discussion. $F(X)$ appearing in related literature
are (i) $\frac{1}{\sqrt{\pi}}e^{-X^{2}}$ (Gaussian distribution), (ii)
$\frac{1}{\pi(1+X^{2})}$ (Cauchy distribution), (iii) $\frac{\sin(X)}{\pi X}$
(sinc function), etc. For the purpose of our work here, we adopt some choice
of $F(X)$ that is even in $n$: vvvWe note that the symmetry property (2.24)
and the ansatz for the step function in (2.19) imply that we are taking
$\Theta(0)=\frac{1}{2}$.

$F(X)=F(-X),\,\,\,\Theta_{X}\left(X\right)=-\Theta_{X}\left(-X\right),$ (2.24)

with $F(X)$ being infinitely differentiable, so that it is compatible with our
use of its antiderivative $\Theta_{X}(X)$ to describe functions of varying
degrees of smoothness.

Taking $F(X)$ to be an even function inherits the nature of $\delta(n)$ being
an even distribution, and retains its original physical attribute of being
symmetrical on either side of $\Sigma$. If we regard the nascent delta
function as a probability density function describing how energy-mass is
localized in $\Sigma$ (idealized by the $\delta$-distribution), then (2.24)
translates to $n=0$ being the mean of the energy-mass distribution, which is
expected on physical grounds.viviviSee e.g. [26, 27] for a more extensive
discussion of various applications of even nascent delta functions. As we
shall point out shortly in Section 2.4, the parity assignment (2.24) also
leads to a natural generalization of the derivatives of the $\delta$-function
when they are integrated against non-smooth functions. viiviiviiIn general,
any locally integrable function that can be normalized following (2.22) can
qualify as a nascent delta function. For the proof, we refer the reader to the
seminal texts of L.Schwartz [28], Jones [20], Gel’fand and Shilov [29], and a
more accessible version in [30]. Our assumption of an even nascent delta
function can in principle be relaxed, but does not alter the conceptual basis
of our approach in this paper.

Parametrizing the step function in this manner implies that for a function
$g(n)$ which may not be smooth at (and after) a certain order of its
derivative at $n=0$, we can consider it as the limit of a sequence of
functions as follows.

$g(n)=g_{1}(n)+\Theta(n)(g_{2}(n)-g_{1}(n))=\overline{g}(n)+\lim_{b\rightarrow
0}\Theta_{X}(X)[g(n)],$ (2.25)

where $g_{1}(n),g_{2}(n)$ are infinitely differentiable functions of $n$ each
analytically extending $g(n)$ beyond $n=0$ and we have defined

$\overline{g}(n)=\frac{1}{2}\left(g_{1}(n)+g_{2}(n)\right),\quad[g(n)]=g_{2}(n)-g_{1}(n),\quad
g_{2}(0)=g(0^{+}),\quad g_{1}(0)=g(0^{-}).$

For example, if $g(n)$ is of class $C^{1}$ with its second-order derivative
being discontinuous at $n=0$, then
$g_{1}(0)=g_{2}(0),g^{\prime}_{1}(0)=g^{\prime}_{2}(0),g^{\prime\prime}_{1}(0)\neq
g^{\prime\prime}_{2}(0)$, etc. One can perform ordinary derivatives on $g(n)$
before finally taking the $b\rightarrow 0$ limit.

In (2.5), the integrand is generally a rather complicated function of various
non-smooth curvature quantities, and it is crucial that we apply (2.25) for
all terms consistently before evaluating the integral. Formally, this
procedure yields a sequence of regular distributions (which is essentially a
complicated function of the nascent delta function and other functions of the
metric tensor analytic at $\Sigma$) that ensure the convergence of (2.5).

We will find that this manner of expressing the step function (and hence the
delta functions and their derivatives) lends us a powerful language for
simplifying and organizing various types of singular terms consistently in the
integrand of (2.5). It also necessitates a more precise prescription of the
infinitesimal integration procedure in (2.5) , one which takes into account
the relative scale separation between the sequence parameter $b$ and the
infinitesimal width $\epsilon$ of the surface $\Sigma$. We now address this
pivotal point that arises when we apply these considerations to our problem —
the relative scaling of the thin-width parameter $\epsilon$ and the nascent
delta function parameter $b$.

Recall that in deriving the junction equations, we perform an integral with
limits parametrizing the width of the infinitesimally thin $\Sigma$, and hence
we send the integral limits to zero after integrating. The integral preserves
only terms which accompany a delta function singularity. Consider again (2.21)
with an infinitely differentiable $f(n)$, but with the integral limits of
(2.5). We still expect to recover $f(0)$ on the RHS. Expanding $f(n)$ around
$n=0$ to obtain

$\lim_{\epsilon\rightarrow 0}\lim_{b\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,\delta_{b}(n)f(n)=\lim_{\epsilon\rightarrow
0}\lim_{b\rightarrow
0}\int^{\epsilon/b}_{-\epsilon/b}dy\,F(y)\left(f(0)+byf^{\prime}(0)+\frac{1}{2}b^{2}y^{2}f^{\prime\prime}(0)+\ldots\right),$
(2.26)

we find that we recover (2.21) upon taking the double scaling limit

$\epsilon\rightarrow 0,\qquad b/\epsilon\rightarrow 0.$ (2.27)

We note that on the contrary, if we specify $b$ to vanish such that
$b/\epsilon\rightarrow\mathcal{O}(1)$ or the converse $\epsilon/b\rightarrow
0$, then we do not recover $f(0)$. The physical reasoning behind (2.27) is
simple and intuitive: the thin-width limit must be kept away from/taken after
the limit of the sequence of nascent delta functions, or otherwise we are not
genuinely integrating a singular delta function across $\Sigma$.

In the field equations, we are seeking a distributional solution to

$\tilde{G}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\ldots=8\pi\left(T^{(reg)}_{\mu\nu}+S_{\mu\nu}\delta(n)\right),$
(2.28)

where for convenience of expression, we have taken $\Sigma$ to be at $n=0$ in
a local Gaussian normal chart, and we have grouped all regular components of
the energy-momentum tensor into $T^{(reg)}_{\mu\nu}$. By the distributional
definition of the delta function, any well-defined solution to (2.28) must
integrate to $S_{\mu\nu}$ in the following manner.

$\lim_{\epsilon\rightarrow 0}\lim_{b\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\left(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\ldots\right)=8\pi
S_{\mu\nu},$ (2.29)

where the limit in $b$ is taken for the sequence of distributions defined by
(2.23). We have to ensure that this sequence of integrals converges which
generically requires additional conditions to be imposed on the extrinsic
curvature for the integral in (2.29) to be well-defined.

This double scaling limit also preserves the fact that we can obtain a well-
defined operational meaning for the derivatives of the delta functions when
they are integrated against smooth functions, i.e. that we have, in a weak
distributional sense,

$\delta^{(k)}[g(n)]\sim(-1)^{k}g^{(k)}(0),$ (2.30)

for a $g(n)$ analytic at $n=0$. This formula needs to be refined generally for
a non-smooth $g(n)$ — an important yet tricky point that is relevant for us
since, as we shall see, the equations of motion of higher-derivative
gravitational theories typically contain terms which are products of non-
smooth functions and a number of delta functions each equipped with some order
of derivative.

###  2.4 Generalized distributions and regularity constraints

In this section, we present a well-defined procedure that extends the usual
notion of distribution for the delta function (and its derivatives) to one
that could be applied when they are integrated against non-smooth functions.
Essentially for our purpose here, this will ultimately turn out to furnish a
simple and clear method to identify and classify singular terms arising from
the junction equations, leading to regularity constraints which we can impose
on the extrinsic curvature to eliminate singular terms that render (2.5) to be
ill-defined.

We begin with the basic examples of the $\delta(n),\delta^{\prime}(n)$.

####  2.4.1 Warm-up: more about $\delta(n),\delta^{\prime}(n)$

Let us consider a discontinuous test function for the delta function which we
express as $f(n)=\overline{f}(n)+\Theta_{X}(X)[f(n)]$ where
$\overline{f}(n)=\frac{1}{2}\left(f_{1}(n)+f_{2}(n)\right),[f(n)]=f_{2}(n)-f_{1}(n)$,
with both $f_{1,2}(n)$ being smooth functions that extend $f(n)$ across
$\Sigma$. Within the infinitesimal domain width, we can expand $f(n)$ about
the origin. In the notations introduced in the previous section, we have

$\lim_{\epsilon\rightarrow 0}\lim_{b\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,f(n)\delta_{b}(n)=\lim_{\epsilon\rightarrow
0}\lim_{b\rightarrow
0}\sum_{l=0}\frac{b^{l}}{l!}\int^{\epsilon/b}_{-\epsilon/b}dX\,\,X^{l}F(X)\left(\overline{f}^{(l)}(0)+\Theta_{X}(X)[f^{(l)}(0)]\right)$
(2.31)

The only non-vanishing term is $l=0$, and since $F(X),\Theta_{X}(X)$ are even
and odd in $X$ respectively, we obtain

$\lim_{\epsilon\rightarrow 0}\lim_{b\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,f(n)\delta_{b}(n)=\overline{f}(0),$ (2.32)

a result which is naturally intuitive and reduces correctly to the expected
one in the continuous limit. Now we can extend this calculation to derivatives
of the delta function, for example, $\delta^{\prime}(n)$. Consider thus the
integral

$\lim_{\epsilon\rightarrow 0}\lim_{b\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,f(n)\delta^{\prime}_{b}(n)=\lim_{\epsilon\rightarrow
0}\lim_{b\rightarrow
0}\sum_{l=0}\frac{b^{l-1}}{l!}\int^{\epsilon/b}_{-\epsilon/b}dX\,\,X^{l}F^{\prime}(X)\left(\overline{f}^{(l)}(0)+\Theta_{X}(X)[f^{(l)}(0)]\right).$
(2.33)

For all $l>1$ this vanishes. For $l=1$, since $F^{\prime}(X),\Theta_{X}(X)$
are both odd in $X$, this term reads

$\int^{\infty}_{-\infty}dX\,XF^{\prime}(X)\overline{f}^{\prime}(0)=-\overline{f}^{\prime}(0),$
(2.34)

where we have integrated by parts. We are left with the $l=0$ term which is
singular since

$\lim_{b\rightarrow
0}\frac{1}{b}\int^{\infty}_{-\infty}dX\,\,F^{\prime}(X)\left(\overline{f}(0)+\Theta_{X}(X)[f(0)]\right)=\lim_{b\rightarrow
0}\frac{[f(0)]}{b}\int^{\infty}_{-\infty}dX\,\,F^{\prime}(X)\Theta_{X}(X).$
(2.35)

The integral is then well-defined only if we impose from the outset the
continuity of $f$ at $n=0$,viiiviiiviiiSee for example Section 1.3 of [31] for
some brief comments on smoothness conditions of test functions for
distributions.

$f_{2}(0)=f_{1}(0),$ (2.36)

leading to

$\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,f(n)\delta^{\prime}(n)=-\overline{f}^{\prime}(0).$
(2.37)

The continuity condition in (2.36) for $\delta^{\prime}(n)$ is the simplest
example of what we shall allude to as _regularity constraints_ that we would
need to solve for, and impose when we encounter some integrand in (2.5) that
is typically a product of some nascent delta functions each possibly equipped
with some order of derivative. Subject to (2.36), we can understand
$\delta^{\prime}(n)$ to yield (minus) the average of the first derivative of
$f(n)$ evaluated at $n=0$ should the function be non-differentiable at $n=0$.
In a similar vein, one can prove that more generally, we have

$\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,f(n)\delta^{(k)}(n)=(-1)^{k}\overline{f}^{(k)}(0),\,\,\,k\geq
1,$ (2.38)

which is an intuitive generalization of the case where the test functions
$f(n)$ are smooth. We note that the simple form of the RHS of (2.38) follows
by virtue of the symmetry property of the nascent delta function in (2.24)
which, if relaxed, leads to a more complicated relation. For example, if $F$
has no definite parity, then generally instead of (2.37) we have
$-\overline{f}^{\prime}(0)-[f^{\prime}(0)]\int^{\infty}_{-\infty}dXF^{2}(X)X$,
which depends on the precise form of $F$.

Usually, when $\delta(n),\delta^{\prime}(n)$ are regarded as distributions on
some open subset of $\mathbb{R}$, the class of test functions are smooth,
compactly supported functions. As shown above, for test functions which are
non-smooth, $\delta(n),\delta^{\prime}(n)$ can still be regarded as
distributions with point support at $n=0$, but for $\delta^{\prime}(n)$ to be
a sensible distribution, we need to restrict the class of test functions to be
at least continuous at $n=0$.

####  2.4.2 Regularity constraints for products of nascent delta functions &
their derivatives

In the Gaussian normal chart, integrating the bulk equations of motion (of a
general higher-derivative gravitational theory) across $\Sigma$ is reduced to
evaluating a set of one-dimensional integrals, with the integrand being some
complicated product of various derivatives of the extrinsic curvature. Since
generally, we take the extrinsic curvature to be not necessarily continuous at
$\Sigma$ in response to a delta-singular energy source, this implies that in
the absence of some regularizing constraints, such an integral is generically
singular.

In the same vein by which we have studied the basic examples of
$\delta(n),\delta^{\prime}(n)$, we now explain how we can solve for regularity
constraints for products of non-smooth functions, delta functions and their
derivatives. As we have seen in the previous examples, it is useful to
describe a non-smooth function with the ansatz (2.25) which, in turn, contains
smooth functions that can be expanded around $n=0$ within the thin-width
domain of integral.

This implies that we can break down the integral in (2.5) into a linear
combination of integrals of the following form.

$I(l,\vec{k})=\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,n^{l}\partial^{k_{1}}_{n}\Theta(n)\partial^{k_{2}}_{n}\Theta(n)\ldots\partial^{k_{j}}_{n}\Theta(n),$
(2.39)

where $l$ and the indices $\vec{k}=\\{k_{1},k_{2},\ldots,k_{j}\\}$ are non-
negative integers, with each $k_{i}$ indicating the order of derivative.
Together with the scaling limit (2.27), we can express (2.39) as

$I(l,\vec{k})=\lim_{b\rightarrow
0}b^{l+1-\sum^{j}_{m=1}k_{m}}\int^{\infty}_{-\infty}dX\,\,X^{l}\Theta_{X}^{(k_{1})}(X)\ldots\Theta_{X}^{(k_{j})}(X).$
(2.40)

Such a term vanishes for $l\geq\sum_{m}k_{m}$, remains finite for
$l+1-\sum_{m}k_{m}=0$, and diverges for

$l+1-\sum^{j}_{m=1}k_{m}<0,\qquad
l=j-\sum^{j}_{m=1}k_{m}\,\,\,\,\,(\text{mod}\,\,2),$ (2.41)

with the second condition in (2.41) being due to the fact that the integral
$I(l,\vec{k})$ vanishes by virtue of $\Theta_{X}(X)=-\Theta_{X}(-X)$ for
$l=1+j-\sum_{m}k_{m}$ (mod 2).

After summing up the linear combination of $I(l,\vec{k})$ defining (2.5), for
various singular terms which diverge as $b^{s}$ for some negative index $s$,
we can now sum them up and set the overall coefficient to vanish. This
naturally leads to a regularity constraint that we have to impose separately
for _each order of singularity_ labelled by each distinct $s$. A caveat is
that the definite integral in (2.40) is generally representation-dependent.
Since we would like to eliminate all singular integrals for any choice of
nascent delta function, we should then impose a stronger condition: that for
each family of integrals labelled by the same $\vec{k}$, every singular $l$
satisfying (2.41) then leads to a regularity constraint. This procedure then
renders the integral in (2.5) to be well-defined for any choice of nascent
delta function. In the following, we elaborate on several illustrative
examples.

We begin with a basic example by taking $\vec{k}_{0}=\\{1,1,1\\}$, with

$I(l,\vec{k}_{0})=\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,n^{l}\,\left(\Theta^{\prime}(n)\right)^{3}=\lim_{b\rightarrow
0}b^{l-2}\int^{\infty}_{-\infty}dX\,\,X^{l}\,F^{3}(X),$ (2.42)

which diverges for $l=0$, vanishes for $l=1$ and $l\geq 3$, and for $l=2$, it
evaluates to $\int^{\infty}_{-\infty}dX\,\,X^{2}\,F^{3}(X)$. If we replace
$n^{l}$ in the integrand of (2.42) by a function $\phi(n)$ that is analytic at
$n=0$, then this implies that the integral diverges unless $\phi(0)=0$, in
which case we have

$I(\phi,\vec{k}_{0})\equiv\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,\phi(n)\,\left(\Theta^{\prime}(n)\right)^{3}=\frac{\phi^{\prime\prime}(0)}{2}\int^{\infty}_{-\infty}dX\,\,X^{2}\,F^{3}(X).$
(2.43)

In this case, $\phi(0)=0$ would be what we call as a _regularity constraint_
which stipulates how fast $\phi(n)$ should grow near the origin for the
integral to converge. The integral (2.43) also demonstrates that the naive
product of three $\delta$-functions can be understood as a proper
distribution, provided we restrict the space of test functions to be those
which grow at least as $\phi(n)\sim n+\mathcal{O}(n^{2})$ near the origin.
ixixixIn our context, the integral limits are different from the typical ones
($\mathbb{R}^{n}$) used, but if desired, they can be extended to $\mathbb{R}$
provided they decay sufficiently fast enough at infinity for the $F(X)$
chosen. Note that such a distribution has point-support, and is equivalent to
$\delta^{\prime\prime}(n)$ up to a normalization constant (which is
$\frac{1}{2}\int^{\infty}_{-\infty}dX\,\,X^{2}\,F^{3}(X)$), albeit with a
different space of test functions. This agrees with the well-known fact that
every distribution with point support is a finite linear combination of
$\delta$-function and its derivatives (see e.g. [31] for a semi-formal proof).
This serves as a simple example of how the use of nascent delta functions
resolves the ‘ambiguity’ that may arise in interpreting products of
$\delta$-functions as articulated in [16].

As an another example, consider the sum

$\sum_{l}C_{l}I(l,\vec{k}),$

with a fixed $\vec{k}$, the set of regularity constraints are simply

$C_{l}=0,\qquad\forall\,l\,\,\,\text{satisfying}\,\,\,\eqref{diverge}.$ (2.44)

In (2.5), as we shall see through explicit examples in later sections, it
involves a linear combination of integrals of the form $I(l,\vec{k})$ with a
finite set of $\vec{k}$. The entire set of regularity constraints is then the
union of all (2.44) associated with each $\vec{k}$.

It is also instructive to check how (2.41) applies to the previous basic
examples of $\delta(n),\delta^{\prime}(n)$ in (2.31) and (2.33). We note that
for the two terms on the RHS of (2.31), they correspond to $I(l,\\{1\\})$ and
$I(l,\\{0,1\\})$ in the notation of (2.39). With $\sum_{m}k_{m}=1$ for each
term, there is no solution to (2.41) and thus no regularity constraint is
needed. For (2.33), each of the two terms on the RHS are of the form
$I(l,\\{2\\})$ and $I(l,\\{0,2\\})$ respectively, with $\sum_{m}k_{m}=2$ for
each term. From (2.41), we thus see that there is a regularity constraint
needed for $I(l,\\{0,2\\})$ (where $j=2$) associated with $l=0$, and this is
simply (2.36).

After imposing the necessary regularity constraints in (2.5) following the
above approach, we are left with a linear combination of well-defined
integrals of the form

$I(l,\vec{k})=I\left(\sum^{j}_{m=1}k_{m}-1,\vec{k}\right)=\int^{\infty}_{-\infty}dX\,\,X^{(\sum^{j}_{m=1}k_{m}-1)}\Theta_{X}^{(k_{1})}(X)\ldots\Theta_{X}^{(k_{j})}(X).$
(2.45)

Since the integrand of (2.45) is odd for an even $j$, such a term only
contributes to the junction equations for odd values of $j$. For example, in
(2.31), the eventual expression corresponds to the first term within the RHS
bracket which is of the form (2.45) with $l=0,j=1,\sum_{m}k_{m}=1$, whereas
for (2.33), we note that (2.34) is of the form (2.45) with
$l=1,j=1,\sum_{m}k_{m}=2$. As another example, for the sum
$\sum_{l}C_{l}I(l,\vec{k})$, the only finite term (after imposing (2.44)) is

$C_{r}I\left(r,\vec{k}\right),\qquad r=\sum^{j}_{m=1}k_{m}-1,$ (2.46)

which is non-vanishing only if $j$ is odd. A caveat is that, like the definite
integral in (2.41), an expression like (2.45) is generally sensitive to the
choice of the nascent delta function, apart from terms like

$\int dX\,\,\Theta^{\prime}_{X},\qquad\int
dX\,\,X^{p}\Theta^{(p+1)}_{X},\qquad\text{or}\qquad\int
dX\,\,\left(\Theta_{X}\right)^{p}\Theta^{\prime}_{X},$ (2.47)

which are independent of the choice. For example, consider the integral

$I(2,\\{1,1,1\\})=\int^{\infty}_{-\infty}dX\,\,X^{2}\left(\Theta^{\prime}_{X}(X)\right)^{3}.$
(2.48)

If we pick the nascent delta function to be the Gaussian
$F(X)=\frac{1}{\sqrt{\pi}}e^{-X^{2}}$, then (2.48) evaluates to
$1/(6\pi\sqrt{3})$, whereas a choice of $F(X)=\frac{\sin(X)}{\pi X}$ yields
$1/(2\pi^{2})$ instead. It may seem like such junction terms are
‘regularization-dependent’. This feature is obviously absent in ordinary GR,
and here we see that the higher-order nature of the field equations may probe
the form of the nascent delta function. To write down consistent junction
conditions that are insensitive to the choice of nascent delta functions, we
could additionally set the coefficient $C_{r}$ in (2.46) to vanish each time
it appears in (2.5). We are then left with junction terms arising from
universal terms like those in (2.47).

In most of our working examples for the rest of the paper, these
representation-dependent terms turn out not to feature much. The only setting
where it arises non-trivially in this work is the case of $R^{3}$ theory for
which we find the terms

$\int dX\,\Theta^{\prime 2}_{X}\Theta_{X}X,\qquad\int
dX\,\Theta^{\prime\prime}_{X}\Theta^{\prime}_{X}\Theta_{X}X^{2},$

to be manifest in the junction equations. As we shall demonstrate later, these
terms would be absent if we further set $[K^{\prime}]=0$ as a regularity
constraint for the $R^{3}$ theory’s junction equations. In the general case,
our derivation procedure described above allows us to solve for the regularity
constraints that will yield the final junction conditions to be
representation-independent if desired. Nonetheless, it is noteworthy that the
convergence of the integral (2.5) is compatible with the presence of these
terms which, if allowed, implies that the specification of junction conditions
is only complete with a choice of nascent delta function.

To summarize, we can now state explicitly how to read off the regularity
constraints and junction conditions for (2.5). After parametrizing each
discontinuous geometric quantity and its derivatives by $\Theta_{X}$,
expanding the LHS of (2.5) would yield integrand terms typically of the form

$I_{\vec{k}}\equiv\phi(n)\Theta^{(k_{1})}_{X}\Theta^{(k_{2})}_{X}\ldots\Theta^{(k_{j})}_{X},$
(2.49)

for some vector index $\vec{k}$, with $\phi(n)$ analytic at $\Sigma$, then the
regularity constraints associated with $I_{\vec{k}}$ are

$\phi^{l}(0)=0,\,\,\,\forall\,\,l\leq\sum_{m=1}^{j}k_{m}-2,\,\,\,l=j-\sum_{m=1}^{j}k_{m}\,\,\,(\text{mod}\,\,2),$
(2.50)

with the junction term induced by integrating $I_{\vec{k}}$ across $\Sigma$
being

$J_{\vec{k}}=\frac{1}{((\sum_{m}k_{m})-1)!}\phi^{((\sum_{m}k_{m})-1)}(0)\left(\int^{\infty}_{-\infty}dX\,X^{(\sum_{m}k_{m})-1}\Theta^{(k_{1})}_{X}\ldots\Theta^{(k_{j})}_{X}\right).$
(2.51)

The final junction condition arising from (2.5) is then the sum of all the
junction terms, each of the form (2.51), subject to us imposing all the
regularity constraints, each of the form (2.50).

###  2.5 The relation to Hadamard Regularization

In this Section, we point out a relation between the regularity constraints
and Hadamard regularization [18] — a well-studied procedure to regularize
divergent integrals typically encountered in the theory of singular integral
operators, and also commonly invoked when one handles distributions defined by
divergent integrals (for an emphasis in the theory of distributions, see for
example [19]).

Let us first briefly review the notion of Hadamard regularization with a
simple example. Consider the (divergent) integral

$I_{H}=\int^{\infty}_{0}dx\,\frac{\phi(x)}{x^{3/2}}=\lim_{\epsilon\rightarrow
0}\int^{\infty}_{\epsilon}dx\,\frac{\phi(x)}{x^{3/2}},$

where $\phi(x)$ is regular and continuous at the origin. From the mean value
theorem, we can write
$\phi(x)=\phi(0)+x\phi^{\prime}(tx)\equiv\phi(0)+x\varphi(x),\,\,\,0<t<1$ and
thus we can express the above integral as

$I_{H}=\lim_{\epsilon\rightarrow
0}\frac{2\phi(0)}{\epsilon}+\lim_{\epsilon\rightarrow
0}\int^{\infty}_{\epsilon}dx\,\,\frac{\varphi(x)}{\sqrt{x}}.$ (2.52)

The second integral in (2.52) converges and we define it to be Hadamard finite
part of $I_{H}$, writing

$FP\int^{\infty}_{0}dx\,\,\frac{\phi(x)}{x^{3/2}}=\lim_{\epsilon\rightarrow
0}\int^{\infty}_{\epsilon}dx\frac{\phi(x)}{x^{3/2}}-\lim_{\epsilon\rightarrow
0}\frac{2\phi(0)}{\sqrt{\epsilon}}.$ (2.53)

We have thus extracted the singular piece in the integral, with the first term
of RHS of (2.53) being the residual finite integral. The above example can be
generalized to a similar regularization of the divergent integral

$I_{D}=\int^{\delta}_{-\delta}dx\,\,\,f(x)\phi(x),\qquad\delta>0,$

where we take $\phi(x)$ to be analytic at the origin, and $f(x)|x|^{m}$ has an
algebraic singularity of order $m$ at $x=0$, i.e. $m$ is some smallest
positive integer such that $f(x)|x|^{m}$ is locally integrable.

We can extend this notion to $f(x)$ being a distribution, the Hadamard
regularization of which is then defined as

$FP\langle
f,\phi\rangle=\int^{\delta}_{-\delta}dx\,\,f(x)\left(\phi(x)-\left(\phi(0)+\phi^{\prime}(0)x+\ldots+\frac{1}{m!}\phi^{(m-1)}(0)x^{m-1}\right)\right)\Theta\left(1-\frac{|x|}{\epsilon}\right).$
(2.54)

We now examine how (2.54) is relevant for our context. From (2.39) and (2.40),
we see that the sequence (in parameter $b$)

$\Delta_{\vec{k}}(n,b)\equiv\Theta^{(k_{1})}(n,b)\ldots\Theta^{(k_{j})}(n,b),$

converges to a distribution $\Delta_{\vec{k}}(n)$ that is of order of
singularity $\left(\sum_{m=1}^{j}k_{m}\right)-1$, and further, upon
integrating it against some function $\phi(n)$ that is analytic at $n=0$, the
Hadamard-regularized distribution reads

$\displaystyle FP\langle\Delta_{\vec{k}}(n),\phi(n)\rangle$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,\Delta_{\vec{k}}(n)$
$\displaystyle\times\Bigg{(}\phi(n)-\left(\phi(0)+\phi^{\prime}(0)n+\ldots+\frac{1}{(\sum_{m}k_{m}-2)!}\phi^{(\sum_{m}k_{m}-2)}(0)n^{\sum_{m}k_{m}-2}\right)\Bigg{)},$

where now $\delta=\epsilon=0^{+}$ for the relevant integral domain. Comparing
(2.5) against (2.50) and (2.51), and recalling that $I(l,\vec{k})$ vanishes
for $l=1+j-\sum_{m}k_{m}\,\,(\text{mod}\,\,2)$, it is then clear that:

  * •

_in the absence of the regularity constraints, the junction condition is
nothing but the Hadamard-finite part of the LHS of ( 2.5)_,

  * •

imposing the regularity constraints then ensures that the integral in (2.5)
converges, being trivially equivalent to its Hadamard regularization.

Although it is nice to have recognized that our derivation of the junction
conditions (and regularity constraints) admits a natural interpretation in
terms of Hadamard regularization, we stress that our method is independently
consistent, and can be understood and implemented without alluding to the
latter.

##  3 Generalized junction conditions for gravitational theories with
quadratic terms

In this Section, we derive the generalized junction conditions for the
following class of gravitational theories with action terms quadratic in
(various contractions of) the Riemann tensor.

$\mathcal{L}_{quad}=\frac{1}{16\pi}\left(R+\beta_{1}R^{2}+\beta_{2}R_{uv}R^{uv}+\beta_{3}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}\right).$
(3.1)

We integrate the equations of motion $\tilde{G}_{\mu\nu}=8\pi T_{\mu\nu}$
across the infinitesimally thin surface
$\int^{\epsilon}_{-\epsilon}dn\,\,\tilde{G}_{\mu\nu}$, identify the regularity
constraints and derive the final explicit covariant form of the generalized
junction condition. The equations of motion read

$\displaystyle
G_{\alpha\beta}+2\beta_{1}RR_{\alpha\beta}-4\beta_{3}R_{\alpha\mu}R^{\mu}_{\beta}+2\beta_{3}R_{\alpha\rho\mu\nu}{R_{\beta}}^{\rho\mu\nu}+(2\beta_{2}+4\beta_{3})R_{\alpha\mu\beta\nu}R^{\mu\nu}$
(3.2)
$\displaystyle-2(\beta_{1}+\frac{1}{2}\beta_{2}+\beta_{3})\nabla_{\alpha}\nabla_{\beta}R+(\beta_{2}+4\beta_{3})\Box
R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\bigg{(}-(4\beta_{1}+\beta_{2})\Box
R$ (3.3)
$\displaystyle+\beta_{1}R^{2}+\beta_{2}R_{\mu\nu}R^{\mu\nu}+\beta_{3}R_{\rho\sigma\mu\nu}R^{\rho\sigma\mu\nu}\bigg{)}=8\pi
T_{\alpha\beta}.$ (3.4)

In the following we consider $\tilde{G}_{ij},\tilde{G}_{in},\tilde{G}_{nn}$
separately. This Section is accompanied by the Appendix A which collects
several useful identities that we developed for evaluating the integrals
easily. We will find that in contrast to the case in GR, apart from bracketed
quantities, the junction terms also involve averaged quantities across
$\Sigma$. In particular, the expression
$\frac{1}{3}\left(\overline{fg}+2\overline{f}\overline{g}\right)[h]\equiv\overbracket{fg}[h]$
turns out to occur frequently (see (A.14) in Appendix A ).

###  3.1 Junction terms from integrating $\tilde{G}_{ij}$

In the following, we present explicitly the result of integrating
$\tilde{G}_{ij}$ across $\Sigma$ for each term in the equation of motion.
Below, the hatted expressions refer to intrinsic quantities, whereas bracketed
ellipses refer to terms which do not contribute to (2.5). For definiteness, we
will focus on the case of $\Sigma$ being timelike unless explicitly stated
otherwise.

  * •

For $g_{ij}\Box
R=g_{ij}\left(\partial^{2}_{n}R+K\partial_{n}R+g^{al}\left(\partial_{l}\partial_{a}R-\Gamma^{k}_{la}\partial_{k}R\right)\right)$,
the Gauss-Codazzi equations enable us to express the first two terms as

$g_{ij}\left(\partial^{2}_{n}\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right)+K\partial_{n}\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right)\right),$

whereas the remaining terms are

$g_{ij}g^{al}\left(\partial_{l}\partial_{a}(-2\partial_{n}K)-\Gamma^{k}_{al}\partial_{k}(-2\partial_{n}K)\right)+\left(\ldots\right).$

The singular terms arise from $-2g_{ij}K\partial^{2}_{n}K$ and
$4K_{ij}\partial^{2}_{n}K$, the latter being derived after an integration by
parts. They sum up to be

$2\left(g_{ij}[K]^{2}-2[K_{ij}][K]\right)\int dn\,\,\,(\Theta^{\prime})^{2}.$

The remaining finite terms sum up to read

$\displaystyle-
g_{ij}\left[2\nabla^{2}_{\Sigma}K+2K^{\prime\prime}+4KK^{\prime}+K^{3}+KK^{ab}K_{ab}+(K^{ab}K_{ab})^{\prime}\right]$
(3.5)
$\displaystyle+8\overbracket{K_{ij}K}[K]+4\overbracket{K_{ij}K^{ab}}[K_{ab}]+g_{ij}[K]\left(\overbracket{K^{2}}+\overbracket{K^{ab}K_{ab}}\right)$
(3.6)
$\displaystyle+4\left(g_{ij}\overline{K^{\prime}}[K]+\overline{K_{ij}}[K^{\prime}]-\overline{K^{\prime}_{ij}}[K]\right)+8\left([K_{ij}][K]^{2}\right)\int^{\infty}_{0}dX\,XF^{2}(X),$
(3.7)

where $\nabla^{2}_{\Sigma}$ is the Laplacian defined on $\Sigma$, and we use
the superscript prime to denote $\partial_{n}=n^{\alpha}\nabla_{\alpha}$.

  * •

For $g_{ij}R^{2}$, the Gauss-Codazzi relations enable us to express it as

$g_{ij}\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right)^{2}.$

The singular term arises from $4g_{ij}\partial_{n}K\partial_{n}K$, and reads

$4g_{ij}[K]^{2}\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite terms sum up to read

$g_{ij}\left(-4[K](\hat{R}-\overbracket{K^{2}}-\overbracket{K^{ab}K_{ab}})+8\overline{K^{\prime}}[K]\right)+16[K_{ij}][K]^{2}\int^{\infty}_{0}dX\,XF^{2}(X).$
(3.8)

  * •

For
$g_{ij}R_{\mu\nu}R^{\mu\nu}=g_{ij}(R_{ab}R^{ab}+R_{nn}R^{nn})+\left(\ldots\right)$,
the Gauss-Codazzi relations enable us to express it as

$\displaystyle
g_{ij}\bigg{(}g^{mr}g^{ls}\left(-\partial_{n}K_{ml}+2K^{a}_{m}K_{la}-KK_{ml}+\hat{R}_{ml}\right)\left(-\partial_{n}K_{rs}+2K^{a}_{r}K_{sa}-KK_{rs}+\hat{R}_{rs}\right)$
(3.9)
$\displaystyle\left(\partial_{n}K+K^{ab}K_{ab}\right)\left(\partial_{n}K+K^{ab}K_{ab}\right)\bigg{)}.$
(3.10)

The singular term comes from the term

$g_{ij}g^{ml}g^{rs}\partial_{n}K_{ml}\partial_{n}K_{rs}=g_{ij}\partial_{n}K^{ab}\partial_{n}K_{ab}+4g_{ij}K^{ma}K_{m}^{b}\partial_{n}K_{ab},$
(3.11)

and also we have another singular term in

$g_{ij}\partial_{n}K\partial_{n}K,$

which yields the following overall singular term

$g_{ij}\left([K^{ab}][K_{ab}]+[K]^{2}\right)\int
dn\,\,\,(\Theta^{\prime})^{2}.$ (3.12)

The finite terms sum up to read

$\displaystyle
g_{ij}\left(2\overbracket{KK^{ab}}[K_{ab}]-2\hat{R}^{ab}[K_{ab}]+\overline{K^{ab^{\prime}}}[K_{ab}]+\overline{K_{ab}^{\prime}}[K^{ab}]+2\overline{K^{\prime}}[K]+2\overbracket{K^{ab}K_{ab}}[K]\right)$
(3.13)
$\displaystyle+4[K_{ij}]\left([K^{ab}][K_{ab}]+[K]^{2}\right)\int^{\infty}_{0}dX\,XF^{2}(X).$
(3.14)

  * •

For
$g_{ij}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}=4g_{ij}R_{ancn}g^{al}g^{cm}R_{lnmn}+\left(\ldots\right)$,
the Gauss-Codazzi relations enable us to express it as

$4g_{ij}g^{al}g^{mc}\left(-\partial_{n}K_{ac}+K_{ab}K^{b}_{c}\right)\left(-\partial_{n}K_{ml}+K_{ms}K^{s}_{l}\right).$
(3.15)

The singular term arises from
$4g_{ij}g^{al}g^{mc}\partial_{n}K_{ac}\partial_{n}K_{ml}$ which yields

$4g_{ij}[K^{lm}][K_{lm}]\int dn\,\,\,(\Theta^{\prime})^{2}.$ (3.16)

The finite terms sum up to read

$\displaystyle
g_{ij}\left(-8\overbracket{K_{l}^{m}K^{ln}}[K_{mn}]+4\overline{K^{mn^{\prime}}}[K_{mn}]+4\overline{K_{mn}^{\prime}}[K^{mn}]+16\overbracket{K^{mr}K_{r}^{l}}[K_{ml}]\right)$
(3.17)
$\displaystyle+16[K_{ij}][K^{lm}][K_{lm}]\int^{\infty}_{0}dX\,XF^{2}(X).$
(3.18)

  * •

For $RR_{ij}$, the Gauss-Codazzi relations enable us to express it as

$\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right)\left(-\partial_{n}K_{ij}+2K^{a}_{j}K_{ai}-KK_{ij}+\hat{R}_{ij}\right).$
(3.19)

The singular term arises from $2\partial_{n}K\partial_{n}K_{ij}$ and reads

$2[K][K_{ij}]\int dn\,\,\,(\Theta^{\prime})^{2},$ (3.20)

whereas the finite terms sum up to read

$2\overline{K^{\prime}}[K_{ij}]+2\overline{K_{ij}^{\prime}}[K]-2\left(2\overbracket{K_{i}^{m}K_{mj}}-\overbracket{KK_{ij}}+\hat{R}_{ij}\right)[K]-\left(\hat{R}-\overbracket{K^{2}}-\overbracket{K^{ab}K_{ab}}\right)[K_{ij}].$
(3.21)

  * •

For $R_{i\mu}R^{\mu}_{j}=R_{im}R^{m}_{j}+\left(\ldots\right)$, the Gauss-
Codazzi relations enable us to express it as

$g^{ml}\left(-\partial_{n}K_{im}+2K^{a}_{i}K_{am}-KK_{im}+\hat{R}_{im}\right)\left(-\partial_{n}K_{jl}+2K^{a}_{j}K_{al}-KK_{jl}+\hat{R}_{jl}\right).$
(3.22)

The singular term arises from $g^{ml}\partial_{n}K_{im}\partial_{n}K_{jl}$ and
reads

$[K^{a}_{i}][K_{ja}]\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite terms sum up to read

$\overbracket{K^{\prime}_{l(i}}[K_{j)}^{l}]-\left(2\overbracket{K_{m}^{l}K_{l(j}}-\overbracket{KK_{m(j}}+\hat{R}_{m(j}\right)[K_{i)}^{m}].$
(3.23)

  * •

For
$R_{i\mu\alpha\beta}{R_{j}}^{\mu\alpha\beta}=2R_{inrn}R_{jnsn}g^{rs}+\left(\ldots\right)$,
the Gauss-Codazzi relations enable us to express it as

$2g^{rs}\left(-\partial_{n}K_{ir}+K_{im}K^{m}_{r}\right)\left(-\partial_{n}K_{js}+K_{jm}K^{m}_{s}\right).$
(3.24)

The singular term arises from $2g^{rs}\partial_{n}K_{ir}\partial_{n}K_{js}$
and reads

$2[K_{i}^{s}][K_{js}]\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite terms read

$2\overline{K^{\prime}_{l(i}}[K_{j)}^{l}]-2\overbracket{K^{l}_{m}K_{l(i}}[K_{j)}^{m}].$
(3.25)

  * •

For $R_{i\mu
j\nu}R^{\mu\nu}=R_{iajb}R^{ab}+R_{injn}R^{nn}+\left(\ldots\right)$, the Gauss-
Codazzi relations enable us to express it as

$\displaystyle\left(\hat{R}_{iajb}-K_{ij}K_{ab}+K_{ib}K_{aj}\right)g^{al}g^{bm}\left(-\partial_{n}K_{lm}+2K^{r}_{l}K_{rm}-KK_{lm}+\hat{R}_{lm}\right)$
(3.26)
$\displaystyle+\left(-\partial_{n}K_{ij}+K_{ia}K^{a}_{j}\right)\left(-\partial_{n}K-K^{lm}K_{lm}\right).$
(3.27)

The singular term arises from $\partial_{n}K_{ij}\partial_{n}K$ and reads

$[K_{ij}][K]\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite terms sum up to read

$[K_{ab}]\left(-\overbracket{K_{i}^{b}K_{j}^{a}}-\hat{R}_{iajb}+\overbracket{K_{ij}K^{ab}}\right)+\overline{K^{\prime}_{ij}}[K]+\overline{K^{\prime}}[K_{ij}]-\overbracket{K_{im}K^{m}_{j}}[K]+\overbracket{K^{ab}K_{ab}}[K_{ij}].$
(3.28)

  * •

For $\nabla_{i}\nabla_{j}R$, the Gauss-Codazzi equation enables us to express
it as

$\partial_{i}\partial_{j}(-2\partial_{n}K)+K_{ij}\partial_{n}(-2\partial_{n}K-K^{2}-K^{ab}K_{ab})-\Gamma^{k}_{ij}\partial_{k}(-2\partial_{n}K)+\left(\ldots\right)$
(3.29)

The singular term comes from $-2K_{ij}\partial^{2}_{n}K$ and reads

$2[K_{ij}][K]\int dn\,\,(\Theta^{\prime})^{2}.$

The finite terms sum up to read

$-2\left[D_{i}D_{j}K\right]-\left[K_{ij}(K^{2}+K^{ab}K_{ab})\right]+[K_{ij}](\overbracket{K^{2}}+\overbracket{K^{ab}K_{ab}})-2\overline{K_{ij}}[K^{\prime}]+2\overline{K^{\prime}_{ij}}[K],$
(3.30)

where $D_{i}$ is the affine connection defined on $\Sigma$.

  * •

For $\Box R_{ij}$, it is useful to lay out explicitly various terms

$\displaystyle\Box R_{ij}$ $\displaystyle=$
$\displaystyle\partial_{n}(\partial_{n}R_{ij}-\Gamma^{l}_{n(i}R_{j)l})-K^{l}_{(i}\left(R^{\prime}_{j)l}-R_{j)m}\Gamma^{m}_{nl}-\Gamma^{m}_{j)n}R_{lm}\right)$
(3.32)
$\displaystyle+g^{rl}\left(\partial_{l}(\nabla_{r}R_{ij})-\Gamma^{k}_{lr}\nabla_{k}R_{ij}-\Gamma^{k}_{l(i}\nabla_{r}R_{j)k}-\Gamma^{n}_{l(i}\nabla_{r}R_{j)n}-\Gamma^{n}_{rl}\nabla_{n}R_{ij}\right).\;$

The singular terms arise from
$-K^{l}_{(i}R^{\prime}_{j)l}-\Gamma^{n}_{rl}\nabla_{n}R_{ij}$ and they sum up
to read

$\left(-2[K^{l}_{i}K_{lj}]+[K][K_{ij}]\right)\int
dn\,\,\,(\Theta^{\prime})^{2}.$

The finite terms sum up to read

$\displaystyle-[\nabla^{2}_{\Sigma}K_{ij}]+\left[2KK^{l}_{i}K_{lj}-K^{2}K_{ij}\right]+[K]\overbracket{KK_{ij}}+\overbracket{KK^{m}_{(i}}[K_{j)m}]-2\overbracket{K^{m}_{(i}K_{j)m}}[K]$
(3.33) $\displaystyle+$
$\displaystyle\overbracket{K^{m}_{l}K^{l}_{(j}}[K_{i)m}]+\overline{K^{l}_{(i}}[K_{j)l}^{\prime}]-\overline{K^{{}^{\prime}l}_{(i}}[K_{j)l}]+\overline{K^{\prime}}[K_{ij}]-\overline{K}[K^{\prime}_{ij}].$
(3.34)

###  3.2 Junction terms from integrating $\tilde{G}_{in}$

In the following, we present explicitly the result of integrating

$\displaystyle\tilde{G}_{in}$ $\displaystyle\equiv$ $\displaystyle
G_{in}+2\beta_{1}RR_{in}-4\beta_{3}R_{i\mu}R^{\mu}_{n}+2\beta_{3}R_{i\rho\mu\nu}{R_{n}}^{\rho\mu\nu}+(2\beta_{2}+4\beta_{3})R_{i\mu
n\nu}R^{\mu\nu}$ (3.36)
$\displaystyle\,\,\,-2(\beta_{1}+\frac{1}{2}\beta_{2}+\beta_{3})\nabla_{i}\nabla_{n}R+(\beta_{2}+4\beta_{3})\Box
R_{in},$

across $\Sigma$ for each term in the equation of motion. There are no singular
terms. In the following, we display the result of integrating each term across
$\Sigma$ after invoking Gauss-Codazzi relations.

  * •

From $RR_{in}$, we have

$-2\left(\overline{D^{a}K_{ia}}-\overline{D_{i}K}\right)[K].$ (3.37)

  * •

From $R_{i\mu}R^{\mu}_{n}$, we have

$-\left(\overline{D^{a}K_{ka}}-\overline{D_{k}K}\right)[K_{i}^{k}]-\left(\overline{D^{a}K_{ia}}-\overline{D_{i}K}\right)[K].$
(3.38)

  * •

From
$R_{i\rho\mu\nu}{R_{n}}^{\rho\mu\nu}=2R_{ikna}{R_{n}}^{kna}+\left(\ldots\right)$,
we have

$-2[K^{ak}]\left(\overline{D_{k}K_{ia}}-\overline{D_{i}K_{ak}}\right).$ (3.39)

  * •

From $R_{i\mu n\nu}R^{\mu\nu}=R_{ianb}R^{ab}+R_{innl}R^{nl}$, we have

$-[K^{ab}]\left(\overline{D_{a}K_{bi}}-\overline{D_{i}K_{ba}}\right)+[K_{il}]\left(\overline{D^{a}K^{l}_{a}}-\overline{D^{l}K}\right).$
(3.40)

  * •

For $\nabla_{i}\nabla_{n}R$, we first note that

$\nabla_{i}\nabla_{n}R=\partial_{i}\partial_{n}\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right)-K^{j}_{i}\partial_{j}\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right).$
(3.41)

After integrating across $\Sigma$, we have

$-D_{i}\left[2K^{\prime}+K^{2}+K^{ab}K_{ab}\right]+2\overline{K^{j}_{i}}[D_{j}K].$
(3.42)

  * •

For $\Box
R_{in}=\left(\nabla^{k}\nabla_{k}+\nabla^{n}\nabla_{n}\right)R_{in}$, we first
note that

$\nabla^{n}\nabla_{n}R_{in}=\partial_{n}(\nabla_{n}R_{in})-K^{k}_{i}(\partial_{n}R_{kn}-K^{l}_{k}R_{ln}),$
(3.43)

which, upon integrated across $\Sigma$, yields

$n^{\alpha}[\nabla_{\alpha}\left(D^{l}K_{il}-D_{i}K\right)]-\overline{K^{k}_{i}}[D^{l}K_{kl}-D_{k}K].$
(3.44)

For $\nabla^{k}\nabla_{k}R_{in}$, we note that

$\nabla^{k}\nabla_{k}R_{in}=g^{kl}\left(\partial_{l}(\nabla_{k}R_{in})-\Gamma^{j}_{lk}\nabla_{j}R_{in}-\Gamma^{n}_{lk}\nabla_{n}R_{in}-\Gamma^{j}_{li}\nabla_{k}R_{jn}-\Gamma^{n}_{li}\nabla_{k}R_{nn}-\Gamma^{j}_{ln}\nabla_{k}R_{ij}\right),$
(3.45)

and identify the terms which contribute to the integral to be

$\displaystyle\nabla_{k}R_{in}=-\Gamma^{n}_{ki}R_{nn}-\Gamma^{a}_{kn}R_{ia}+\left(\ldots\right),\quad\nabla_{j}R_{in}=-\Gamma^{n}_{ji}R_{nn}-\Gamma^{a}_{jn}R_{ia}+\left(\ldots\right),$
(3.46)
$\displaystyle\nabla_{n}R_{in}=\partial_{n}R_{in}+\left(\ldots\right),\quad\nabla_{k}R_{nn}=\partial_{k}R_{nn}+\left(\ldots\right),$
(3.47)
$\displaystyle\nabla_{k}R_{ij}=\partial_{k}R_{ij}-\Gamma^{l}_{k(i}R_{j)l}+\left(\ldots\right),\quad\nabla_{k}R_{jn}=-\Gamma^{n}_{kj}R_{nn}-\Gamma^{a}_{kn}R_{ja}+\left(\ldots\right).$

Some straightforward (but lengthy) algebra then gives the finite terms to be

$-\overline{D^{k}K_{ki}}[K]-2\overline{K^{l}_{i}}[D_{l}K]-\overline{K}[\nabla_{i}K]+\overline{D^{k}K^{a}_{k}}[K_{ia}]+2\overline{K^{ab}}[D_{a}K_{ib}]+\overline{K}[D^{a}K_{ia}].$
(3.48)

A useful consistency check lies in taking the Gauss-Bonnet limit
$\beta_{1}=\beta_{3}=-\frac{1}{4}\beta_{2}.$ where we find that the various
junction terms sum up to vanish, in accordance with the result reported in
earlier literature obtained by boundary variation of the surface term.

###  3.3 Junction terms from integrating $\tilde{G}_{nn}$

In the following, we present explicitly the result of integrating
$\tilde{G}_{nn}$ across $\Sigma$ for each term in the equation of motion.
There are no singular terms. In the following, we display the result of
integrating each term across $\Sigma$ after invoking Gauss-Codazzi relations.

  * •

For $RR_{nn}$, the Gauss-Codazzi relations imply that we can write it as

$\left(\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}\right)\left(-\partial_{n}K-K^{ab}K_{ab}\right),$
(3.49)

from which we can read off the singular term to be

$2[K]^{2}\int dn\,\,\,(\Theta^{\prime})^{2},$

and the finite terms to be

$3\overbracket{K^{ab}K_{ab}}[K]-[K]\left(\hat{R}-\overbracket{K^{2}}\right)+4\overline{K^{\prime}}[K].$
(3.50)

  * •

For $R_{n\mu}R^{\mu}_{n}$, the Gauss-Codazzi relations imply that we can write
it as

$(\partial_{n}K+K^{ab}K_{ab})(\partial_{n}K+K^{ab}K_{ab}),$ (3.51)

from which we can read off the singular term to be

$[K]^{2}\int dn\,\,\,(\Theta^{\prime})^{2},$

and the finite terms to be

$2[K]\left(\overline{K^{\prime}}+\overbracket{K^{ab}K_{ab}}\right).$ (3.52)

  * •

For $R_{n\rho\mu\nu}{R_{n}}^{\rho\mu\nu}$, the Gauss-Codazzi relations imply
that we can write it as

$2(-\partial_{n}K_{kl}+K_{ka}K^{a}_{l})g^{kr}g^{ls}(-\partial_{n}K_{rs}+K_{rb}K^{b}_{s}),$
(3.53)

from which we can read off the singular term to be

$2[K^{ab}][K_{ab}]\int dn\,\,\,(\Theta^{\prime})^{2},$

and the finite terms to be

$-4[K_{kl}]\overbracket{K^{k}_{b}K^{bl}}+4\overline{K^{\prime}_{kl}}[K^{kl}].$
(3.54)

  * •

For $R_{n\mu n\nu}R^{\mu\nu}$, the Gauss-Codazzi relations imply that we can
write it as

$\left(-\partial_{n}K_{ij}+K_{ia}K^{a}_{j}\right)g^{ir}g^{js}\left(-\partial_{n}K_{rs}+2K^{b}_{r}K_{sb}-KK_{rs}+\hat{R}_{rs}\right),$
(3.55)

from which we can read off the singular term to be

$[K_{ab}][K^{ab}]\int dn\,\,(\Theta^{\prime})^{2},$ (3.56)

and the finite terms to be

$-[K_{ij}]\left(2\overbracket{K^{ib}K^{j}_{b}}-\overbracket{KK^{ij}}+\hat{R}^{ij}\right)-[K_{rs}]\overbracket{K^{r}_{a}K^{as}}+2\overline{K^{\prime}_{ij}}[K^{ij}].$
(3.57)

  * •

For $\nabla_{n}\nabla_{n}R$, after invoking the Gauss-Codazzi relations, we
have the finite terms

$[-2\partial^{2}_{n}K-\partial_{n}(K^{2})-\partial_{n}(K^{ab}K_{ab})].$

  * •

For $\Box
R_{nn}=-[\partial_{n}(\partial_{n}K+K^{ab}K_{ab})]+g^{ab}\nabla_{a}\nabla_{b}R_{nn}$,
we first expand the second term to read

$g^{kl}\left(\partial_{l}\partial_{k}(-\partial_{n}K+\ldots)-\Gamma^{i}_{lk}\partial_{i}(-\partial_{n}K+\ldots)\right)-K\partial_{n}(\partial_{n}K+K^{ab}K_{ab})+2K^{ik}(K_{ki}\partial_{n}K-K^{j}_{k}\partial_{n}K_{ij}+\ldots)$

after which it is easier to read off the singular term to be

$[K]^{2}\int dn\,\,\,(\Theta^{\prime})^{2},$

and the finite terms to be

$-[KK^{\prime}]-2\overbracket{KK_{ab}}[K^{ab}]+2\overbracket{K^{ab}K_{ab}}[K]-2\overbracket{K^{ik}K^{j}_{k}}[K_{ij}]-[\nabla^{2}_{\Sigma}K]-[\partial_{n}(K^{\prime}+K^{ab}K_{ab})].$
(3.58)

  * •

From $g_{nn}\Box R$, the singular term reads

$2[K]^{2}\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite term reads

$-\left[2\nabla^{2}_{\Sigma}K+2K^{\prime\prime}+4KK^{\prime}+K^{3}+KK^{ab}K_{ab}+(K^{ab}K_{ab})^{\prime}\right]+[K](\overbracket{K^{2}}+\overbracket{K^{ab}K_{ab}})+4\overline{K}^{\prime}[K].$
(3.59)

  * •

From $g_{nn}R^{2}$, the singular term reads

$4[K]^{2}\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite term reads

$-4[K]\left(\hat{R}-\overbracket{K^{2}}-\overbracket{K^{ab}K_{ab}}\right)+8\overline{K^{\prime}}[K].$
(3.60)

  * •

From $g_{nn}R_{\mu\nu}R^{\mu\nu}$, the singular term reads

$\left([K^{ab}][K_{ab}]+[K]^{2}\right)\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite term reads

$2\overbracket{KK^{ab}}[K_{ab}]-2\hat{R}^{ab}[K_{ab}]+\overline{K^{{}^{\prime}ab}}[K_{ab}]+\overline{K^{\prime}_{ab}}[K^{ab}]+2\overline{K^{\prime}}[K]+2\overbracket{K^{ab}K_{ab}}[K].$
(3.61)

  * •

From $g_{nn}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$, we have the singular
term

$4[K_{ab}][K^{ab}]\int dn\,\,\,(\Theta^{\prime})^{2},$

whereas the finite term reads

$8\overbracket{K^{\mu}_{\alpha}K^{\alpha\nu}}[K_{\mu\nu}]+4\overline{K^{{}^{\prime}\mu\nu}}[K_{\mu\nu}]+4\overline{K^{\prime}_{\mu\nu}}[K^{\mu\nu}].$
(3.62)

In the Gauss-Bonnet limit, we find that all terms from integrating
$\tilde{G}_{nn}$ across $\Sigma$, both singular and finite, vanish
identically.

###  3.4 Regularity constraints

We now examine the conditions under which the integration is well-defined by
summing up all singular terms arising from the integration in all the field
equations. As shown above, in this family of theories, every such term is
associated with the divergent term

$\lim_{b\rightarrow 0}\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}dn\,\,(\partial_{n}\Theta(n,b))^{2}=\lim_{b\rightarrow
0}\frac{1}{b}\int^{\infty}_{-\infty}dXF^{2}(X).$ (3.63)

In the following, we gather the coefficients from each set of components of
the field equations which are various functions of jumps of the extrinsic
curvature.

From $\tilde{G}_{ij}$, the various singular terms sum up to read

$\displaystyle I^{(sing)}_{ij}$ $\displaystyle=$
$\displaystyle\left(4\beta_{3}-\beta_{2}-8\beta_{1}\right)[K][K_{ij}]-2\left(\beta_{2}+4\beta_{3}\right)[K^{l}_{i}][K_{lj}]$
(3.65)
$\displaystyle+g_{ij}\left((2\beta_{1}+\frac{1}{2}\beta_{2})[K]^{2}-(\frac{1}{2}\beta_{2}+2\beta_{3})[K^{ab}][K_{ab}]\right),$

whereas from $\tilde{G}_{nn}$, the various singular terms sum up to be

$I^{(sing)}_{nn}=\frac{3}{2}\left([K]^{2}(4\beta_{1}+\beta_{2})+[K^{ab}][K_{ab}](\beta_{2}+4\beta_{3})\right).$
(3.66)

Since there are no other singular terms from $\tilde{G}_{in}=0$, we proceed to
set both (3.65) and (3.66) to vanish. From (3.65), we can take its trace to
obtain

$\left(4\beta_{3}+(2d-8)\beta_{1}+(\frac{d}{2}-1)\beta_{2}\right)[K]^{2}-(\beta_{2}+4\beta_{3})(2+\frac{d}{2})[K^{ab}][K_{ab}]=0.$
(3.67)

We can classify solutions to $I^{(sing)}_{ij}=I^{(sing)}_{nn}=0$ as follows:

  1. (I)

We first search for points in the moduli space such that there is no
additional constraints to be imposed on the extrinsic curvature. From (3.65),
we can read off

$\beta_{2}+4\beta_{3}=0,\qquad 4\beta_{3}-\beta_{2}-8\beta_{1}=0,$ (3.68)

which also solves (3.66) and leads to

$\beta_{2}=-4\beta_{3}=-4\beta_{1}.$ (3.69)

This is precisly the combination for the Gauss-Bonnet theory! Recall that the
Einstein-Gauss-Bonnet theory is defined with the following addition

$\alpha\left(R^{2}-4R_{\alpha\beta}R^{\alpha\beta}+R^{\alpha\beta\mu\nu}R_{\alpha\beta\mu\nu}\right),$
(3.70)

to the Einstein-Hilbert action, with $\alpha$ being some constant parameter
(that could be $\sim l^{2}_{s}$ where $l_{s}$ is the string length if there is
a string-theoretic origin [23]). In [16], the authors imposed the smoothness
condition that terms naively containing product of delta-functions should be
forbidden in the action from the outset. This implies that the extrinsic
curvature has to be continuous and it was argued that the junction conditions
are identical to GR for the theory with an added Gauss-Bonnet term in the
action. Further, since $[K_{ab}]=0$, no thin-shell singular sources should be
permitted. However, we find that in the related past literaturexixixiFor
example, in [32], a similar result was obtained through a similar derivation
formulated in terms of differential forms and in the Gaussian chart. Using a
specific example (4D cosmological brane in a 5D spacetime with negative
cosmological constant) where a $\mathbb{Z}_{2}$ symmetry was further imposed,
they showed how this junction condition can be equivalently derived by
integrating over $\Sigma$ starting from the bulk equations and assuming a
delta-singular source (eqn. 28 of [32]). xxxAs mentioned earlier, a common way
of denoting the sum in (B.16) is to simply replace it with taking the
difference of each quantity across $\Sigma$ once we take into account the
switch of sign in the normal vector across $\Sigma_{\pm}$. In [33], a similar
derivation was made, and the authors showed how in the Gaussian chart, we can
simply use the Gauss-Codazzi relations to read off the extra junction
condition term in (B.16) (term in $\alpha$) starting from the bulk field
equations. In equation B8, one can find a formula for this extra terms in
Gaussian coordinates and we have checked that it is equivalent to its
expression in differential forms as defined in eqn. 12 of [32]. , a consistent
junction condition for Einstein-Gauss-Bonnet theory has been presented in a
few papers (that is different from the above conclusion).In the next Section,
we will show that our results recover those of [23, 21, 22, 33] in the
topological limit. This serves as a stringent consistency check for many of
our equations, and demonstrates definitively that the Gauss-Bonnet theory does
have non-trivial junction conditions.

  2. (II)

Since imposing $[K_{ij}]=0$ naturally removes all singular terms, we look for
less stringent conditions on the extrinsic curvature. From the form of (3.65)
and (3.66), we find the following class of solutions

$[K]=0,\qquad\beta_{2}+4\beta_{3}=0,$ (3.71)

corresponding to a family of theories for which we can set just the trace of
the extrinsic curvature instead of all its components to vanish. In the space
of the couplings $(\beta_{1},\beta_{2},\beta_{3})$, this is a plane containing
the Gauss-Bonnet ‘line’ ((3.69)).

  3. (III)

Finally, we have the trivial solution

$[K_{ij}]=0,$ (3.72)

which is accompanied by no other constraints on the coupling parameters. There
are still non-trivial junction equations to write down even in this case,
since as we have seen earlier, in general, these equations sometimes involve
normal derivatives of the extrinsic curvature, i.e. while the first (normal)
derivative of the metric has to be continuous, the higher-order ones need not
be.

We have focussed on the case where $\Sigma$ is timelike for definiteness.
Nonetheless, this derivation can be repeated in an identical fashion for a
spacelike $\Sigma$ since the difference lies in a few signs to be switched on
in the Gauss-Codazzi relations. In particular we find that the regularity
constraints for the spacelike case remain the same, and the classification of
junction conditions presented above is also preserved. Explicitly, restoring
$\xi$, the sign of $n^{2}$, we find that (3.65) and (3.66) generalize to

$\displaystyle I^{(sing)}_{ij}$ $\displaystyle=$
$\displaystyle\left(4\beta_{3}-\beta_{2}-8\beta_{1}\right)[K][K_{ij}]-2\left(\beta_{2}+4\beta_{3}\right)[K^{l}_{i}][K_{lj}]$
(3.75)
$\displaystyle+g_{ij}\left((2\beta_{1}+\frac{1}{2}\beta_{2})[K]^{2}-\left(\frac{1}{2}\beta_{2}+2\beta_{3}\right)[K^{ab}][K_{ab}]\right)$
$\displaystyle+(\xi-1)(4\beta_{1}+\beta_{2})[K]\left([K]g_{ij}-[K_{ij}]\right).$

There are no singular terms identically from $\tilde{G}_{in}$, whereas for
$\tilde{G}_{nn}$, the various singular terms sum up to be

$\displaystyle I^{(sing)}_{nn}$ $\displaystyle=$
$\displaystyle\frac{3\xi}{2}\left([K]^{2}(4\beta_{1}+\beta_{2})+[K^{ab}][K_{ab}](\beta_{2}+4\beta_{3})\right)$
(3.77)
$\displaystyle+(1-\xi)[K]^{2}\left(2\beta_{1}+\frac{3}{2}\beta_{2}+4\beta_{3}\right).$

It is then straightforward to show that setting $I^{(sing)}_{ij}$ and
$I^{(sing)}_{nn}$ to vanish gives the same classification of junction
conditions as described earlier.

In the following section, we simplify and present the final form of the
junction conditions for Class II and III theories with their respective
regularity constraints implied. It is worthwhile to note that the finite terms
multiplied to the representation-dependent factor
$\int^{\infty}_{-\infty}dX\,\Theta XF^{2}(X)$ sum up to zero in the junction
equations since from integrating $\tilde{G}_{ij}$ across $\Sigma$, we find
that these terms assemble to read

$2\left((4\beta_{1}+\beta_{2})[K]^{2}-(\beta_{2}+4\beta_{3})[K^{ab}][K_{ab}]\right)[K_{ij}]\times\int^{\infty}_{0}dX\,XF^{2}(X),$
(3.78)

and one can check that for each of the above classes of theories, upon
imposing their respective smoothness conditions, (3.78) vanishes exactly. The
final junction equations are thus insensitive to the choice of the nascent
delta function for all three families of theories.

###  3.5 A summary of results

Having solved for the appropriate regularity constraints, we can now impose
them on the set of general finite junction terms derived earlier, and obtain
the generalized Darmois-Israel junction conditions. For a generic choice of
coupling parameters the resulting junction equations can be rather elaborate.
In the following, we present the explicit junction equations for each of the
three classes of theories defined in the previous section, including the
ordinary Darmois-Israel junction terms. On the RHS of the each equation, the
boundary energy-momentum tensor is defined as the singular part of the bulk
energy-momentum tensor localized within $\Sigma$, denoted by
$S_{\alpha\beta}=\lim_{\epsilon\rightarrow
0}\int^{\epsilon}_{-\epsilon}T_{\alpha\beta}$.

For Case (I), we find that our results reduce nicely to those for Gauss-Bonnet
obtained independently by the method of boundary variation. This topological
theory is defined by the line of couplings
$\beta_{1}=\beta_{3}=-\frac{1}{4}\beta_{2}$ and we find that the junction
conditions simplify to read

$-2\beta_{1}\left[3J_{ij}-Jh_{ij}\right]-4\beta_{1}[K^{cd}]P_{icdj}+[K]h_{ij}-[K_{ij}]=8\pi
S_{ij},$ (3.79)

where

$\displaystyle J_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{3}\left(2KK_{ic}K^{c}_{j}+K_{cd}K^{cd}K_{ij}-2K_{ic}K^{cd}K_{dj}-K^{2}K_{ij}\right),\quad
J\equiv J^{k}_{k}$ (3.80) $\displaystyle P_{icdj}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\hat{R}g_{i[d}g_{j]c}+\hat{R}_{icdj}+\hat{R}_{c[d}g_{j]i}-\hat{R}_{i[d}g_{j]c}.$
(3.81)

We note that the bracketed rank-4 curvature tensor in the second line turns
out to be the divergence-free component of the intrinsic Riemann tensor. There
are no junction terms arising from integrating
$\tilde{G}_{in},\tilde{G}_{nn}$. We also note that (3.79) is equivalent to the
junction condition derived in earlier literature [23, 21, 22, 33] by taking
the boundary variation of the Gauss-Bonnet surface term. Thus, this gives a
strong consistency check of our general derivation which is noticeably absent
in other previous proposals for junction conditions in quadratic gravity [16,
17]. Since this is an important point, in Appendix B, we present a detailed
proof of how our equations reduce to (3.79).

We should mention that a previous work in [16], which proposes junction
conditions for quadratic gravity, fails this consistency test. The authors
argued one has to take the extrinsic curvature to be continuous at $\Sigma$
even for such the Gauss-Bonnet case, and its junction conditions should read
simply as $[K_{ij}]=0$. This contradicts the result obtained by either the
boundary variation of the Gauss-Bonnet surface term, or integrating the
equations of motion across $\Sigma$. In contrast, our derivation elucidates
how various junction terms involving both averaged and jump quantities
assemble nicely in the Gauss-Bonnet limit to yield (3.79), consistent with
what we expect from the action principle.

For Case (II) where we take $[K]=0$, parametrizing this class of theories by
$\\{\beta_{1},\beta_{3}\\}$, we have the junction equations

$\displaystyle
2(\beta_{1}-\beta_{3})\Bigg{(}-g_{ij}\left[2K^{\prime\prime}+4KK^{\prime}+KK^{ab}K_{ab}+n^{\alpha}\nabla_{\alpha}(\text{Tr}(K^{2}))\right]+4\overbracket{K_{ij}K^{ab}}[K_{ab}]$
(3.82)
$\displaystyle-[K_{ij}]\overline{R}+\overline{K_{ij}}[K^{ab}K_{ab}+6K^{\prime}]\Bigg{)}-2\beta_{3}\Bigg{[}J_{ij}-\frac{1}{3}\left(\left(3KK_{cd}K^{cd}-2K^{cd}K_{ac}K_{d}^{a}\right)h_{ij}\right)$
(3.83)
$\displaystyle+2K^{cd}\left(\frac{1}{2}\hat{R}g_{i[d}g_{j]c}+\hat{R}_{icdj}+\hat{R}_{c[d}g_{j]i}-\hat{R}_{i[d}g_{j]c}\right)-K_{ij}\Bigg{]}=8\pi
S_{ij},$ (3.84)
$2(\beta_{1}-\beta_{3})h^{\alpha}_{i}\nabla_{\alpha}[2K^{\prime}+K^{ab}K_{ab}]=8\pi
S_{in},$ (3.85)
$-2(\beta_{1}-\beta_{3})\left[2KK^{\prime}+KK^{ab}K_{ab}\right]=8\pi S_{nn},$
(3.86)

where
$K^{\prime}=n^{\alpha}\nabla_{\alpha}K,K^{\prime\prime}=n^{\alpha}n^{\beta}\nabla_{\alpha}\nabla_{\beta}K$,
restoring covariance in notation. In the $\beta_{1}\rightarrow\beta_{3}$
limit, we recover the Gauss-Bonnet junction equations with the additional
constraint $[K]=0$. In the $\beta_{3}=0$ limit, we recover the simplest
example of $\mathcal{F}(R)$-type gravitational theories of which Lagrangian is
an analytic function of the Ricci scalar $R$.

For Case (III) where $[K_{ij}]=0$, we have the set of equations

$\displaystyle-
h_{ij}(4\beta_{1}+\beta_{2})\left([K^{\prime\prime}]+2K[K^{\prime}]+\frac{1}{2}[(\text{Tr}(K^{2}))^{\prime}]\right)$
$\displaystyle+$ $\displaystyle
4(3\beta_{1}+\beta_{2}+\beta_{3})K_{ij}[K^{\prime}]$ (3.87)
$\displaystyle+(\beta_{2}+4\beta_{3})\left(K^{l}_{(i}[K^{\prime}_{j)l}]-K[K^{\prime}_{ij}]\right)$
$\displaystyle=$ $\displaystyle 8\pi S_{ij},$ (3.88) $\displaystyle
4(\beta_{1}+\frac{1}{2}\beta_{2}+\beta_{3})h^{\alpha}_{i}\nabla_{\alpha}[K^{\prime}]+(\beta_{2}+4\beta_{3})n^{\alpha}[\nabla_{\alpha}R_{in}]$
$\displaystyle=$ $\displaystyle 8\pi S_{in},$ (3.89)
$\displaystyle-(4\beta_{1}+\beta_{2})[KK^{\prime}]-\frac{1}{2}(\beta_{2}+4\beta_{3})[n^{\alpha}\nabla_{\alpha}(\text{Tr}(K^{2}))]$
$\displaystyle=$ $\displaystyle 8\pi S_{nn}.$ (3.90)

As we see here, away from the Gauss-Bonnet limit, it could have non-vanishing
components in orthogonal directions. Note that we expect all terms to vanish
in the Gauss-Bonnet limit in which setting $[K_{ij}]$ to vanish implies the
absence of junction conditions.

##  4 Junction conditions for other examples of higher-derivative theories

###  4.1 $\mathcal{F}(R)$ theories

A class of gravitational theories considered in literature on modified gravity
are $\mathcal{F}(R)$ theories which refer to a Lagrangian that is an analytic
function of the Ricci scalar. The equations of motion read

$\partial_{R}\mathcal{F}(R)R_{\mu\nu}-\frac{1}{2}\mathcal{F}(R)g_{\mu\nu}+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)\partial_{R}\mathcal{F}(R)=8\pi
T_{\mu\nu}.$ (4.1)

####  4.1.1 More about junction conditions of $R^{2}$ theory

In the previous section, we covered a simple example of such theories: the
$\mathcal{F}(R)=R+\beta_{1}R^{2}$ Lagrangian which belongs to the class of
theories with quadratic curvature invariants that admit the regularity
condition $[K]=0$. Setting $\beta_{3}=0$ in equations (3.82)—(3.86), we obtain
the junction conditions to be

$\displaystyle-[K_{ij}]+2\beta_{1}\bigg{(}-g_{ij}\left[2K^{\prime\prime}+4KK^{\prime}+KK^{ab}K_{ab}+(\text{Tr}(K^{2}))^{\prime}\right]$
(4.2)
$\displaystyle\qquad+4\overbracket{K_{ij}K^{ab}}[K_{ab}]-[K_{ij}]\overline{R}+\overline{K_{ij}}[K^{ab}K_{ab}+6K^{\prime}]\bigg{)}=8\pi
S_{ij},$ (4.3) $\displaystyle
2\beta_{1}\nabla_{i}[2K^{\prime}+K^{ab}K_{ab}]=8\pi S_{in},$ (4.4)
$\displaystyle-2\beta_{1}\left[2KK^{\prime}+KK^{ab}K_{ab}\right]=8\pi S_{nn}.$
(4.5)

Before we briefly comment on some typical features of the junction conditions
for a general $\mathcal{F}(R)$ theory, let us review in detail some aspects of
the junction equations in the $R^{2}$ theory which would serve to highlight
certain useful points. The equations of motion for the $R^{2}$ theory reads

$G_{\mu\nu}+\beta_{1}\left(2RR_{\mu\nu}-\frac{1}{2}R^{2}g_{\mu\nu}+2(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})R\right)=8\pi
T_{\mu\nu}.$ (4.6)

Recall that $R=\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}$. Setting $[K]=0$
thus ensures that $R$ is non-singular. From inspection, there is no singular
term arising from integrating (4.6) across $\Sigma$. The $R^{2}g_{\mu\nu}$
term contains no delta-singularity and hence it integrates to zero. Consider
the term $2g_{\mu\nu}\Box R$, and take the indices to be parallel to $\Sigma$.
The terms which could be non-vanishing after integrating are
$g_{ij}\partial^{2}_{n}R+g_{ij}g^{kl}\Gamma^{n}_{kl}\partial_{n}R=g_{ij}\partial^{2}_{n}R+g_{ij}K\partial_{n}R=\partial_{n}(g_{ij}\partial_{n}R)-\partial_{n}g_{ij}\partial_{n}R+g_{ij}K\partial_{n}R$.
Integrating $\partial_{n}(g_{ij}\partial_{n}R)$ across $\Sigma$ yields
$[g_{ij}\partial_{n}R]$. The other two terms require some work. As we
discussed in Section 3, we can proceed by expanding all terms using the Gauss-
Codazzi relations and then integrating the terms ensuring that the various
discontinuities present in each term and its derivatives are handled
appropriately (e.g. using the integral identities in Appendix A). We find

$\displaystyle-\int dn\,\,\partial_{n}g_{ij}\partial_{n}R$ $\displaystyle=$
$\displaystyle
2\overline{K_{ij}}\left(2[K^{\prime}]+[K^{ab}K_{ab}]\right)+\frac{1}{3}[K_{ij}][K^{ab}][K_{ab}]=-2\overline{K_{ij}}[R]+\frac{1}{3}[K_{ij}][K^{ab}][K_{ab}],$
(4.7) $\displaystyle\int dn\,\,g_{ij}K\partial_{n}R$ $\displaystyle=$
$\displaystyle-g_{ij}K(2[K^{\prime}]+[K^{ab}K_{ab}])=g_{ij}K[R].$ (4.8)

The remaining terms in (4.6) integrate to yield

$\displaystyle\int dn\,\,RR_{ij}$ $\displaystyle=$ $\displaystyle
2\overline{K^{\prime}}[K_{ij}]-\left(\hat{R}-K^{2}-\overbracket{K_{ab}K^{ab}}\right)[K_{ij}]=-\overline{R}[K_{ij}]-\frac{1}{6}[K^{ab}][K_{ab}][K_{ij}],$
(4.9) $\displaystyle\int dn\,\,\nabla_{i}\nabla_{j}R$ $\displaystyle=$
$\displaystyle\overline{K_{ij}}[R]-\frac{1}{6}[K^{ab}][K_{ab}][K_{ij}].$
(4.10)

Summing up all terms, we find the junction condition

$-[K_{ij}]+2\beta_{1}\left(g_{ij}[n^{\alpha}\partial_{\alpha}R]-\overline{R}[K_{ij}]-3\overline{K_{ij}}[R]+[R]Kg_{ij}+\frac{1}{3}[K_{ij}][K^{ab}][K_{ab}]\right)=8\pi
S_{ij},$ (4.11)

which is equivalent to (4.2) but with some expressions succintly expressed in
terms of the Ricci scalar. We can also express (4.4) and (4.5) as

$2\beta_{1}\nabla_{i}[R]=-8\pi S_{in},\qquad 2\beta_{1}K[R]=8\pi S_{nn}.$
(4.12)

Incidentally, in [13] and [14], the authors attempted to study the junction
conditions for $\mathcal{F}(R)$ gravity, and derived them under the assumption
of a continuous Ricci scalar. For comparison sake, consider the case where we
impose $R$ to be continuous at $\Sigma$. This renders the junction conditions
in directions not parallel to $\Sigma$ trivial, whereas (4.2) or (4.11) reduce
to

$2\beta_{1}\left(g_{ij}[n^{\alpha}\partial_{\alpha}R]-R[K_{ij}]+\frac{1}{3}[K_{ij}][K^{ab}][K_{ab}]\right)=8\pi
S_{ij}.$ (4.13)

The first two terms on the LHS of (4.13) are naively expected from integrating
$2g_{ij}\partial^{2}_{n}R$ and $2RR_{ij}$ respectively after using Gauss-
Codazzi relations, and were also obtained in [13] (see Appendix, eqn. (A17))
and [14] (see eqn. (3.11)). But the third term requires a more careful
evaluation of the integrals as we have pinpointed in each of the integrals in
(4.7) and (4.9). We note that the appearance of the ‘anomalous’ term
$[K_{ij}][K^{ab}][K_{ab}]$ in (4.7) and (4.9) has been missed in [13, 14]. In
[14], the authors also employed the technique of integrating the equations of
motion across $\Sigma$ but they missed this term due to presumably an
incorrect evaluation of the integral. In [13], a different method for deriving
the junction conditions was proposed which, as mentioned earlier, appears to
be afflicted with not having a consistent Gauss-Bonnet limit when it is
applied to quadratic gravity. Again, we note that fundamentally, our use of
nascent delta functions enables us to compute the integral across $\Sigma$
accurately, leading to the appropriate junction conditions.

####  4.1.2 Junction conditions for $R^{3}$ theory

In the following, we derive the junction conditions explicitly for the
gravitational theory with additional $\beta R^{3}$ term in the action with the
equation of motion

$G_{\mu\nu}+\beta\left(3R^{2}R_{\mu\nu}-\frac{1}{2}R^{3}g_{\mu\nu}+3(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})R^{2}\right)=8\pi
T_{\mu\nu},$ (4.14)

where
$g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}=g_{\mu\nu}\left(\partial^{2}_{n}+K\partial_{n}+D^{2}\right)-\left(K_{\mu\nu}\partial_{n}+D_{\mu}D_{\nu}\right)$,
and $D_{\alpha}$ is the covariant derivative associated with $\Sigma$. We
impose the constraint that $R$ be non-singular, thus taking $[K]=0$ at $n=0$
as our regularity condition. From inspection, since the highest order
$n$-derivative is a $\partial^{2}_{n}$ acting on $R$ which would yield at most
a delta-like singularity, this suffices to be our only regularity condition.

This time, since we only have the Ricci scalar and no other curvature
invariants, we express the Ricci scalar and its derivative explicitly in terms
of the nascent delta functions, their derivatives and functions of the
extrinsic curvature right from the outset. We first note that we can express
the Ricci scalar and its derivative as

$\displaystyle R$ $\displaystyle=$
$\displaystyle\left(\hat{R}-2\overline{K^{\prime}}-\overline{K}^{2}-\overline{K^{ab}}\overline{K_{ab}}\right)+\Theta_{X}(-2[K^{\prime}]-2[K]\overline{K}-2[K^{ab}]\overline{K_{ab}})$
(4.16)
$\displaystyle\;\;+\Theta^{2}_{X}(-[K]^{2}-[K^{ab}][K_{ab}])-2\Theta^{\prime}[K]\equiv
R_{avg}+\Theta_{X}[R]+\Theta^{2}_{X}R_{(2)}-2\Theta^{\prime}[K],\;\;$
$\displaystyle\partial_{n}R$ $\displaystyle=$
$\displaystyle\Theta^{\prime}[2K^{\prime}-R]-2\Theta^{\prime\prime}[K]+2\Theta_{X}\Theta^{\prime}R_{(2)}+\left(\ldots\right),$
(4.17)

where the ellipses represent finite quantities which vanish in the double
scaling limit when we integrate (4.14) across $\Sigma$. We note that in
(4.14), the terms with the covariant derivative intrinsic to $\Sigma$ do not
generate non-vanishing terms in the junction equation, and we can evaluate the
normal derivatives by the usual chain rule
$\partial_{n}f=R^{\prime}\partial_{R}f$, etc. Another pertinent point is that
while we impose $[K]=0$ at $n=0$ as our regularity condition, we should keep
factors of $[K]$ in all our expressions first before evaluating the integral
since as a function of $n$, $[K]\neq 0$ and various terms in (4.14) are
generalized distributions which may probe the non-vanishing derivatives of
$[K]$ at $n=0$ in the integral.

We begin by considering the indices to be those of $\Sigma$, with
$\\{\mu,\nu\\}=\\{i,j\\}$. Using
$R_{ij}=-\partial_{n}K_{ij}+2K^{a}_{j}K_{ai}-KK_{ij}+\hat{R}_{ij}$, and
substituting (4.16) and (4.17) into (4.14), we organize all terms according to
the order of singularity defined by various powers and derivatives of
$\Theta^{\prime}$. In (4.14) only $R^{2}R_{\mu\nu}$ and
$\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)R^{2}$ terms give
junction terms, which we collate below. In the following, all curvature
quantities are understood to be evaluated at $n=0$.

  1. (i)

From the $3R^{2}R_{ij}$ term:

     * •

$\Theta^{\prime}$:

$-3\int
dn\,\,\Theta^{\prime}[K_{ij}]\left(R^{2}_{avg}+\Theta^{2}_{X}([R]^{2}+2R_{avg}R_{(2)})+\Theta^{4}_{X}R^{2}_{(2)}\right),$
(4.18)

     * •

$(\Theta^{\prime})^{2}$:

$\int dn\,\,n\Theta^{\prime 2}\Theta_{X}(12[K^{\prime}][R][K_{ij}]),$ (4.19)

     * •

$(\Theta^{\prime})^{3}$:

$\int dn\,\,n^{2}\Theta^{\prime 3}(-12[K_{ij}][K^{\prime}]^{2}).$ (4.20)

  2. (ii)

From the $3\left(g_{ij}\Box-\nabla_{i}\nabla_{j}\right)R^{2}$ term:

We first note that after integration by parts,

$\int
dn\,\,\,3\left(g_{ij}\Box-\nabla_{i}\nabla_{j}\right)R^{2}=6[g_{ij}R\partial_{n}R]+6\left(\int
dn\,\,\left(g_{ij}K-3K_{ij}\right)R\partial_{n}R\right).$ (4.21)

For a more compact notation at this point, we introduce
$c_{1}=g_{ij}K-3\overline{K_{ij}}$. Organizing all terms according to their
order of singularity, we have

     * •

$\Theta^{\prime}$:

$\displaystyle 6\int
dn\,\,\,\Theta^{\prime}\Bigg{(}([R]-2[K^{\prime}])c_{1}R_{avg}$ (4.22)
$\displaystyle+\Theta^{2}_{X}(([R]-2[K^{\prime}])(c_{1}R_{(2)}+c_{2}[R])+2R_{(2)}(c_{1}[R]+c_{2}R_{avg}))+2\Theta^{4}c_{2}R^{2}_{(2)}\Bigg{)},$

     * •

$(\Theta^{\prime})^{2}$:

$6\int
dn\,\,\,n(\Theta^{\prime})^{2}\Theta_{X}\left((-2[K^{\prime}])(-3[K_{ij}]([R]-2[K^{\prime}])+2c_{1}R_{(2)})\right),$
(4.23)

     * •

$\Theta^{\prime\prime}$:

$6\int
dn\,\,\Theta^{\prime\prime}n(-2[K^{\prime}])\left(c_{1}R_{avg}+\Theta^{2}_{X}(c_{1}R_{(2)}-3[K_{ij}][R])\right),$
(4.24)

     * •

$\Theta^{\prime}\Theta^{\prime\prime}$:

$6\int
dn\,\,n^{2}\Theta_{X}\Theta^{\prime}\Theta^{\prime\prime}(-12)[K_{ij}][K^{\prime}]^{2}.$
(4.25)

The junction equation is then the sum of all above terms. We note that there
are representation-dependent integrals which are not all independent as they
satisfy the following two constraints.

$\displaystyle\int
dn\,\,\left((\Theta^{\prime})^{3}n^{2}+2\Theta^{\prime\prime}\Theta^{\prime}\Theta_{X}n^{2}+2\Theta^{\prime
2}\Theta_{X}n\right)=0,$ (4.26) $\displaystyle\int
dn\,\,\left(\Theta^{\prime\prime}\Theta^{2}_{X}n+2\Theta^{\prime
2}\Theta_{X}n\right)=-\frac{1}{12},$

which straightforwardly follows by integrating $\partial_{n}(\Theta^{\prime
2}\Theta_{X}n^{2}),\partial_{n}(\Theta^{2}_{X}\Theta^{\prime}n)$. After some
simplification, we find that the representation-dependent terms remain and
they sum up to be

$\mathcal{T}_{rep}\equiv 24[K^{\prime}][K_{ij}]\left([K_{ab}K^{ab}](\int
dn\,\,\Theta^{\prime 2}\Theta_{X}n)-2[K^{\prime}](\int
dn\,\,\Theta^{\prime\prime}\Theta^{\prime}\Theta_{X}n^{2})\right).$ (4.27)

Gathering all terms, we find the junction equation in the $ij$-directions to
be

$\displaystyle 8\pi S_{ij}$ $\displaystyle=$
$\displaystyle\beta\Bigg{(}6[g_{ij}RR^{\prime}]-3R^{2}_{avg}[K_{ij}]-\frac{7}{4}[K_{ij}]\left(2R_{avg}R_{(2)}+[R]^{2}\right)-\frac{39}{80}[K_{ij}]R^{2}_{(2)}$
(4.28) $\displaystyle+$ $\displaystyle
6c_{1}[R](R_{avg}+\frac{1}{4}R_{(2)})+\mathcal{T}_{rep}\Bigg{)}-[K_{ij}].$
(4.29)

We can also express (4.28) in terms of the mean
$\overline{R}=\hat{R}-2\overline{K^{\prime}}-K^{2}-\overline{K^{ab}K_{ab}}$,
and difference of the Ricci scalar $[R]$ across $\Sigma$, using the identity

$\overline{K^{ab}K_{ab}}-\overline{K^{ab}}\overline{K_{ab}}=\frac{1}{4}[K^{ab}][K_{ab}]\equiv-\frac{1}{4}R_{(2)},\,\,\,\overline{R}=R_{avg}+\frac{1}{4}R_{(2)}.$

Then (4.28) can also be written as

$\displaystyle 8\pi S_{ij}$ $\displaystyle=$
$\displaystyle\beta\left(6[g_{ij}RR^{\prime}]-3\overline{R}^{2}[K_{ij}]+[K_{ij}]R_{(2)}(-2\overline{R}+\frac{1}{5}R_{(2)})+6c_{1}[R]\overline{R}+\mathcal{T}_{rep}\right)-[K_{ij}],$
(4.30) $\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
g_{ij}K-3\overline{K}_{ij},\qquad R_{(2)}=-[K]^{2}-[K^{ab}][K_{ab}].$ (4.31)

The first two terms on the RHS would be what we naively expect from
integrating $3g_{ij}\partial^{2}_{n}R^{2}+3R^{2}R_{ij}$, whereas the remaining
terms arise from a more careful evaluation.

For the other directions not completely parallel to $\Sigma$, we need to
integrate

$\displaystyle\partial_{R}\mathcal{F}R_{nn}-\frac{1}{2}\mathcal{F}+(\Box-\nabla_{n}\nabla_{n})\partial_{R}\mathcal{F},$
(4.32)
$\displaystyle\partial_{R}\mathcal{F}R_{in}-\nabla_{i}\nabla_{n}\partial_{R}\mathcal{F},$
(4.33)

across $\Sigma$. For (4.33), we can integrate straightforwardly and obtain an
explicit expression for a general $\mathcal{F}(R)$. Since
$R_{in}=D^{k}K_{ik}-D_{i}K$ is non-singular and so is
$\partial_{R}\mathcal{F}$, the only term that could contain a delta-
singularity is

$-\nabla_{i}\nabla_{n}(\partial_{R}\mathcal{F})=\partial_{n}\left(-\partial_{i}\partial_{R}\mathcal{F})+K^{j}_{i}\partial_{j}\partial_{R}\mathcal{F}\right).$
(4.34)

This integrates to $-[\partial_{i}R\partial^{2}_{R}\mathcal{F}]$, and so the
junction condition reads simply as

$8\pi S_{in}=-\beta[\partial_{i}R\partial^{2}_{R}\mathcal{F}].$ (4.35)

For (4.32), since $\partial_{R}\mathcal{F}R_{nn}-\frac{1}{2}\mathcal{F}(R)$ is
non-singular, it vanishes upon integration and we are left with

$g^{kl}\nabla_{k}\nabla_{l}\partial_{R}\mathcal{F}=-g^{kl}\Gamma^{n}_{kl}R^{\prime}\partial^{2}_{R}\mathcal{F}+\left(\ldots\right)=K\partial^{2}_{R}\mathcal{F}\partial_{n}R+\left(\ldots\right)$
(4.36)

to consider, with the ellipses representating other terms that would not
survive the integral. If $R$ is continuous, then $\partial_{n}R$ is no longer
singular, so clearly this junction condition probes the discontinuity of $R$.
For the $R^{3}$ theory, we need to integrate

$\displaystyle\int
dn\,\,6(\overline{K}+\Theta_{X}[K])\left(R_{avg}+\Theta_{X}[R]+\Theta^{2}_{X}R_{(2)}-2\Theta^{\prime}[K]\right)$
(4.37)
$\displaystyle\qquad\times\left((R^{\prime}_{avg}+\Theta_{X}[R^{\prime}]+\Theta^{2}_{X}R^{\prime}_{(2)})+\Theta^{\prime}([R]-2[K^{\prime}]+2\Theta_{X}R_{(2)})-2\Theta^{\prime\prime}[K]\right).$
(4.38)

Only terms containing derivatives of $\Theta^{\prime}$ survive the
integration. In the following, we organize various contributions as we have
done earlier for the junction equation in the $ij$-directions.

  * •

$\Theta^{\prime}$:

$([R]-2[K^{\prime}])KR_{avg}+\frac{1}{12}\left(2R_{(2)}\overline{K}[R]+([R]-2[K^{\prime}])KR_{(2)}\right),$

  * •

$(\Theta^{\prime})^{2}$:

$-4K[K^{\prime}]R_{(2)}\int dn\,\,(\Theta^{\prime})^{2}\Theta_{X}\,n,$

  * •

$\Theta^{\prime\prime}$:

$-2[K^{\prime}]KR_{avg}\int
dn\,\Theta^{\prime\prime}n-2K[K^{\prime}]R_{(2)}\int
dn\,\,\Theta^{\prime\prime}\Theta^{2}_{X}n.$

Note that the term containing $\Theta^{\prime}\Theta^{\prime\prime}$
integrates to zero since $[K]=0$. After some algebra, and invoking the
relation $\int
dn\,\,\left(\Theta^{\prime\prime}\Theta^{2}_{X}n+2\Theta^{\prime
2}\Theta_{X}n\right)=-\frac{1}{12}$, we obtain the junction equation

$8\pi S_{nn}=\beta[R]K\left(R_{avg}+\frac{1}{4}R_{(2)}\right).$ (4.39)

####  4.1.3 Some general comments

Having obtained the explicit junction equations for the $R^{2}$ and $R^{3}$
theory via a rather intricate integration procedure, let us briefly comments
on some features that we can deduce for the general $\mathcal{F}(R)$ case
without alluding to some specific form of $\mathcal{F}$.

  * •

$[K]=0$ as the regularity condition

We have seen that $[K]=0$ is an appropriate regularity condition for the
$R^{2}$ and $R^{3}$ theories, and it is easy to see that it is valid for a
general analytic $\mathcal{F}$. If $[K]=0$, then $R$ is non-singular, so
$\partial_{R}\mathcal{F}R_{\mu\nu}$ cannot generate divergent terms upon
integration, since there is at most a delta-singularity carried by
$R_{\mu\nu}$. The term $\mathcal{F}g_{\mu\nu}$ vanishes after integration
leaving us with only
$(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})\partial_{R}\mathcal{F}$. Consider
the normal derivatives — after an integration by parts, we are left with only
$\partial_{n}$ acting on $\partial_{R}\mathcal{F}$ which generate at most a
delta-singularity that yields finite quantities after integration. This
applies to the junction equations in all directions. Hence, $[K]=0$ is a valid
regularity condition generally.

When we discuss the case of quadratic gravity earlier, we obtained the general
equations of which solutions give all possible regularity constraints. For the
$R^{2}$ theory, we found no regularity condition apart from $[K]=0$ and
similarly, one can show that this is the case for the $R^{3}$ theory as well.

  * •

On taking $R$ to be continuous:

Another constraint that we can impose on top of $[K]=0$ is the continuity of
$R$ at $n=0$. This implies that we take

$2[K^{\prime}]+[K^{ab}K_{ab}]=0,\qquad[K]=0.$ (4.40)

As we observed earlier, the junction equations in the orthogonal directions
would be trivial in this limit leaving only those parallel to $\Sigma$. In the
specific cases of $R^{2}$ and $R^{3}$ theories, this is evident in equations
(4.12), (4.39) and (4.35). Thus, if we further take $[R]=0$, the discontinuity
in the extrinsic curvature can be physically supported purely by a singular
source that only has non-vanishing components parallel to $\Sigma$. For the
$R^{2}$ theory, the junction equation in the case of a continuous $R$ can be
found in (4.13) whereas for the $R^{3}$ theory, we have explicitly

$\displaystyle 8\pi S_{ij}$ $\displaystyle=$
$\displaystyle\beta\Bigg{(}6g_{ij}R[R^{\prime}]-3R^{2}[K_{ij}]+[K_{ij}]R_{(2)}(-2R+\frac{1}{5}R_{(2)})$
(4.42) $\displaystyle\,\,\,+12[K^{\prime}][K_{ij}]R_{(2)}\int
dn\,\,(\Theta^{\prime})^{3}n^{2}\Bigg{)}-[K_{ij}].$

In both (4.13) and (4.42), we see that although the first two terms on the RHS
may be naively expected from integrating
$g_{ij}\partial^{2}_{n}\partial_{R}\mathcal{F}$ and
$(\partial_{R}\mathcal{F})R_{ij}$ respectively, there are non-trivial terms
which arise from the intricacies of the integral.

  * •

On representation-dependent terms and $[K^{\prime}]$

In (4.30), we note the appearance of $\mathcal{T}_{rep}$ which is sensitive to
the choice of the nascent delta function. We note that such terms always come
with at least a factor of $[K^{\prime}]$. To see this, from (4.16) and (4.17),
we see that the $\Theta^{\prime},\Theta^{\prime\prime}$ terms in
$R,\partial_{n}R$ are each multiplied to a factor of $[K]$. Now the
representation-dependent terms can always be traced to a product of them. Let
$m$ be the total order of derivatives defined as the sum of the order of
derivative on each $\Theta_{X}$ in some product. Any such term is generically
representation-dependent, with the junction term arising from integrating them
against the $(m-1)^{\text{th}}$ derivative of the coefficient. These
derivatives must act only on factors of $[K]$ as otherwise, the term will
vanish since we have imposed $[K]=0$ at $n=0$. Thus, we always have
$[K^{\prime}]$ as part of the overall coefficient of any representation-
dependent term. Although the actual form of $\mathcal{T}_{rep}$ depends on
what $\mathcal{F}(R)$ is as a function of $R$, this implies that universally
across $\mathcal{F}(R)$-theories, setting

$[K^{\prime}]=0,$ (4.43)

implies the absence of these representation-dependent terms. We stress however
that such a constraint is not necessary for the regularity of the junction
equations.

  * •

A restrictive set of regularity constraints

We consider a set of regularity constraints at $n=0$ which allows us to
explicitly derive the appropriate junction equations for a generic analytic
$\mathcal{F}(R)$. This amounts to simply taking _all_ components of $R$ to be
continuous at $\Sigma$. Since $R=\hat{R}-2\partial_{n}K-K^{2}-K^{ab}K_{ab}$,
this implies that we impose

$[K_{ij}]=0,\qquad[K^{\prime}]=0.$ (4.44)

From (4.40) and (4.43), this set of regularity constraints can also be
understood as the one that leads to an absence of representation-dependent
terms and a singular energy source that doesn’t have components orthogonal to
$\Sigma$.

Keeping only terms involving normal derivatives, the equations of motion
simplify to read

$\displaystyle-(\partial_{R}\mathcal{F})\partial_{n}K_{ij}+g_{ij}\partial_{n}\left(\partial_{n}R(\partial^{2}_{R}\mathcal{F})\right)+g_{ij}R^{\prime}K(\partial^{2}_{R}\mathcal{F})+K_{ij}R^{\prime}(\partial^{2}_{R}\mathcal{F})+\left(\ldots\right)=8\pi
T_{ij},$ (4.45) (4.46)
$\displaystyle-\partial^{2}_{R}\mathcal{F}\partial_{n}\partial_{i}R-\partial_{n}R\,\partial_{i}R\,\partial^{3}_{R}\mathcal{F}-2(\partial^{2}_{R}\mathcal{F})K^{m}_{i}\partial_{m}\partial_{n}K+\left(\ldots\right)=8\pi
T_{in},$ (4.47) (4.48)
$\displaystyle-(\partial_{R}\mathcal{F})\partial_{n}K+K\partial_{n}R\,(\partial^{2}_{R}\mathcal{F})+\left(\ldots\right)=8\pi
T_{nn},\qquad$ (4.49)

where we use the Gaussian normal chart in which
$\Gamma^{n}_{ij}=-K_{ij},\Gamma^{i}_{nj}=K^{i}_{j}$, the Gauss-Codazzi
relations and note that the components $\tilde{G}_{ni}=0$ identically.
Integrating across $\Sigma$ in the double scaling limit then yields the
junction conditions to be

$g_{ij}(\partial^{2}_{R}\mathcal{F})\left(2[K^{\prime\prime}]+(K^{ab}K_{ab})^{\prime}\right)=-8\pi
S_{ij},$ (4.50)

with no other junction conditions arising from other components
$\tilde{G}_{\alpha\beta}$. Restoring covariance, we note that
$[K^{\prime\prime}]=n^{\alpha}n^{\beta}[\nabla_{\alpha}\nabla_{\beta}K]$ and
similarly $(K^{ab}K_{ab})^{\prime}=n^{\alpha}\nabla_{\alpha}(K^{ab}K_{ab})$.

Again, we wish to emphasize that the smoothness conditions
$[K_{ij}]=0,[\partial_{n}K]=0$ that arise from a continuous Ricci scalar are
not the least restrictive ones. As we have seen in the previous examples of
$R^{2},R^{3}$ theories, one could just impose $[K]=0$, leading to a more
complicated set of junction conditions. In particular, there are non-trivial
ones which generically require the singular source at $\Sigma$ to have non-
vanishing orthogonal components.

###  4.2 Low-energy effective action from toroidal compactification of the
Heterotic String

As an illustration of how our method could be applied straightforwardly to the
presence of matter couplings in higher-derivative gravitational theories, we
consider a simple example motivated by string phenomenology - the low energy
effective action arising from a particular compactification of a ten-
dimensional string theory. This effective action involves two scalar fields
coupled to the Riemann tensor in a certain manner.

To first-order in $\alpha^{\prime}$ expansion, suppressing all gauge fields
for simplicity, the 10D effective action of the Heterotic Superstring reads
(see e.g. Chapter 12 of [34])

$S=\frac{g^{2}_{s}}{\kappa^{(10)}}\int
d^{10}x\sqrt{|g|}e^{-2\phi}\left(R+4\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{12}H^{2}+\frac{\alpha^{\prime}}{8}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right),$
(4.51)

where $\kappa^{(10)}=16\pi G^{(10)}$, $g_{s}$ is the string coupling and the
Riemann-squared term is required for supersymmetry to first order in
$\alpha^{\prime}$ (essentially this follows from the Chern-Simons terms in the
3-form field strength). It was shown in [35] that upon compactification on a
$T^{6}$, up to leading order in the string coupling, the effective $4D$ action
can be simplified to read

$S_{eff}=\frac{1}{\kappa}\int
d^{4}x\sqrt{|g|}\left(R-\frac{1}{2}\left(\partial^{\mu}\phi\partial_{\mu}\phi+\partial^{\mu}\varphi\partial_{\mu}\varphi\right)-\frac{\alpha^{\prime}}{8}\phi\left(R^{2}-4R^{\mu\nu}R_{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\right)+\frac{\alpha^{\prime}}{8}\varphi
R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}\right),$ (4.52)

with $\kappa\sim\kappa^{(10)}/\text{Vol}(T^{6})$,
$\tilde{R}_{\mu\nu\alpha\beta}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}R^{\rho\sigma}_{\alpha\beta}R^{\mu\nu\alpha\beta}$,
and $\phi,\varphi$ can be interpreted as the dilaton and axion fields
respectively. We see that the effective action is a sum of the Gauss-Bonnet
and Chern-Simons terms, each coupled to a scalar field. Since our method works
for any coefficients of the interaction terms, in the following, we derive the
junction conditions for the above theory with the factors of
$\pm\frac{\alpha^{\prime}}{8}$ being replaced by $\beta_{2,1}$ respectively
which are arbitrary constants in units of $\alpha^{\prime}$.

The equations of motion read

$\displaystyle
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-\beta_{1}g_{\nu\lambda}\delta^{\lambda\sigma\alpha\beta}_{\mu\rho\gamma\delta}{R^{\gamma\delta}}_{\alpha\beta}\nabla^{\rho}\nabla_{\sigma}\phi+2\beta_{2}\nabla^{\rho}\nabla^{\sigma}\left(\tilde{R}_{\rho(\mu\nu)\sigma}\varphi\right)+\beta_{1}\phi
H_{\mu\nu}$ (4.53)
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\left(\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}(\partial\phi)^{2}\right)+\frac{1}{2}\left(\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}(\partial\varphi)^{2}\right)=8\pi
T_{\mu\nu},$ (4.54)
$\displaystyle\nabla^{2}\phi=-\beta_{1}\left(R^{2}-4R^{\mu\nu}R_{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\right),$
(4.55)
$\displaystyle\nabla^{2}\varphi=-\beta_{2}R_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma},$
(4.56)

where $T_{\mu\nu}$ represents the external energy-momentum tensor field, and

$H_{\mu\nu}=2\left(RR_{\mu\nu}-2R_{\alpha\mu}R^{\alpha}_{\nu}+R_{\mu\alpha\beta\gamma}{R_{\nu}}^{\alpha\beta\gamma}-2R_{\mu\alpha\nu\beta}R^{\alpha\beta}\right)-\frac{1}{2}g_{\mu\nu}\left(R^{2}-4R_{\alpha\beta}R^{\alpha\beta}+R^{\rho\sigma\alpha\beta}R_{\rho\sigma\alpha\beta}\right)$

is the second Lovelock tensor. If all quantities are differentiable, then it
vanishes in 4D by Bach-Lanczos identity, reflecting the topological nature of
the Gauss-Bonnet term. But its presence here simply indicates that in the
limit where we freeze the dynamics of the scalar fields to be constants, the
junction conditions reduce to those of Einstein-Gauss-Bonnet theory. For the
term in $\beta_{2}$, it is useful to invoke Bianchi identities to rewrite it
as

$\beta_{2}\nabla^{\rho}\nabla^{\sigma}\left(\tilde{R}_{\rho(nn)\sigma}\varphi\right)=\beta_{2}\nabla_{k}\left(\partial^{m}\varphi\,\epsilon_{mefn}{R_{n}}^{kef}\right).$
(4.57)

before proceeding to derive the junction equations below.

We first examine the singular terms in (4.53) — (4.56). Let us begin with the
matter fields’ equations of motion. Since there is no term more singular than
the delta function in the Gauss-Bonnet nor the Chern-Simons term, this implies
that integrating (4.55), (4.56) across $\Sigma$ will yield a finite quantity.
In the Gaussian normal chart,
$\nabla^{2}=\partial^{2}_{n}+\nabla^{2}_{\Sigma}$. Hence, we have the
continuity constraints for the scalar fields:

$[\phi]=0,\qquad[\varphi]=0,$ (4.58)

since otherwise, the action of $\partial^{2}_{n}$ on the fields would lead to
a singular integral across $\Sigma$. To preserve as much generality as
possible, we do not assume however that their normal derivatives are
continuous at $\Sigma$. We can integrate (4.55) and (4.56) across $\Sigma$
obtain

$[\phi^{\prime}]=-4\hat{R}[K]+8\hat{R}^{ab}[K_{ab}]-4[KK^{ab}K_{ab}]+\frac{4}{3}[K^{3}]+\frac{8}{3}[K^{m}_{l}K^{ln}K_{mn}],$
(4.59)
$[\varphi^{\prime}]=8\beta_{2}\epsilon_{ijk}D^{j}\overline{K^{kb}}K^{i}_{b}.$
(4.60)

Let us now study if (4.53) yields any non-trivial regularity constraints.
Consider first $\tilde{G}_{ij}$, suppressing the Gauss-Bonnet and scalar field
terms,

$\tilde{G}_{ij}=-\beta_{1}g_{j\lambda}\delta^{\lambda\sigma\alpha\beta}_{i\rho\gamma\delta}R^{\gamma\delta}_{\alpha\beta}\nabla^{\rho}\nabla_{\sigma}\phi+\beta_{2}\nabla_{k}(\partial^{m}\varphi\epsilon_{mef(j}{R_{i)}}^{kef})+\ldots,$
(4.61)

where the ellipses refer to finite terms in (2.5). For a $1/b$-type
singularity, we need terms of the form $R^{na}_{nb}$ and
$\partial^{2}_{n}\phi$ to be present simultaneously or a term that goes as
$\nabla_{n}R^{na}_{nb}$. By observation, this cannot arise from the dilaton
term where for the axion interaction term, consider the term where we take the
indices $k=f=n$ in (4.61). After some algebra, we can simplify this term to
read
$-2\beta_{2}\partial^{m}\varphi\epsilon_{me(j}\partial^{2}_{n}K^{e}_{i)}$. But
integrating this term across $\Sigma$ yields

$-{\epsilon^{me}}_{j}[\partial_{m}\varphi][K_{ei}]\lim_{b\rightarrow
0}\frac{1}{b}\int dXF^{2}(X)+\ldots$

where we suppress the manifestly finite quantities. Since $\varphi$ and thus
$\partial_{m}\varphi$ is continuous across $\Sigma$, we find no singularity
here. Finally, we note that as shown in the previous section, there is no
singular term descending from $H_{\mu\nu}$ (no additional regularity
constraints required for pure Einstein-Gauss-Bonnet theory). Thus, from
inspection, we can see that there is no singular term from integrating
$\tilde{G}_{ij}$ across $\Sigma$.

Similar arguments apply for the other components of the field equations,
explicitly,

$\displaystyle\tilde{G}_{nn}$ $\displaystyle=$
$\displaystyle-\beta_{1}g_{nn}\delta^{n\sigma\alpha\beta}_{n\rho\gamma\delta}R^{\gamma\delta}_{\alpha\beta}\nabla^{\rho}\nabla_{\sigma}\phi+2\beta_{2}\nabla_{k}(\partial^{m}\varphi\epsilon_{mef(n}{R_{n)}}^{kef})+\ldots$
(4.62) $\displaystyle\tilde{G}_{in}$ $\displaystyle=$
$\displaystyle-\beta_{1}g_{ik}\delta^{k\sigma\alpha\beta}_{n\rho\gamma\delta}R^{\gamma\delta}_{\alpha\beta}\nabla^{\rho}\nabla_{\sigma}\phi+\beta_{2}\nabla_{k}(\partial^{m}\varphi\epsilon_{mef(i}{R_{n)}}^{kef})+\ldots$
(4.64)

By inspection, one can again draw the conclusion that no singular terms remain
after integrating (4.62) and (4.64) across $\Sigma$ by checking the absence of
terms of the form $\nabla_{n}R^{na}_{nb}$ or products of $R^{na}_{nb}$ and
$\partial^{2}_{n}\phi$. Altogether, the above considerations reveal that just
like the pure Gauss-Bonnet theory, there is no additional regularity
constraints that we need to impose here. This fits intuitively well with the
fact that the interaction terms mixing the otherwise free scalar fields and
the graviton happen to be linear in the Gauss-Bonnet and Chern-Simons terms.

We now proceed to derive the junction equations. From the form of (4.62), we
find that there are no terms which carry delta-like singularities so
integrating (4.62) across $\Sigma$ cannot give any junction condition. For
(4.64), the dilaton interaction term could not give rise to any junction term
since the generalized Kronecker delta symbol already contains an $n$-index,
and introducing some pair of $n$-indices (for either $R^{na}_{nb}$ or
$\partial^{2}_{n}\phi$ ) would annihilate the term by symmetry of the symbol.

For the axion interaction term in (4.64) which reads

$-\beta_{2}\left(\nabla_{a}\left(\partial^{\beta}\varphi\epsilon_{\beta\mu\nu
i}{R_{n}}^{a\mu\nu}\right)+\nabla_{\alpha}\left(\partial^{b}\varphi\epsilon_{buvn}{R_{i}}^{\alpha
uv}\right)\right).$ (4.65)

For the second term in (4.65), the only non-vanishing term arises from taking
the dummy index $\alpha$ to be $n$, generating the following junction term
after integration:

$\beta_{2}\epsilon_{buvn}\partial^{b}\varphi\left[{R_{i}}^{nuv}\right].$

For the first term in (4.65), it is helpful to first expand all terms that
come with the covariant derivative.

$\displaystyle\partial^{\beta}\varphi\epsilon_{\beta\mu\nu
i}\left(\partial_{a}{R_{n}}^{a\mu\nu}+\Gamma^{a}_{al}{R_{n}}^{l\mu\nu}+\Gamma^{\mu}_{ak}{R_{n}}^{ak\nu}+\Gamma^{\nu}_{ak}{R_{n}}^{a\mu
k}+\Gamma^{\mu}_{an}{R_{n}}^{an\nu}+\Gamma^{\nu}_{an}{R_{n}}^{a\mu n}\right)$
$\displaystyle=$ $\displaystyle 2\partial^{\beta}\varphi\epsilon_{\beta
nvi}\left(\partial_{a}{R_{n}}^{anv}+\Gamma^{a}_{al}{R_{n}}^{lnv}\right)+2\epsilon_{nuvi}\varphi^{\prime}K^{u}_{a}{R_{n}}^{anv}+2\epsilon_{buni}\partial^{b}\varphi\Gamma^{u}_{ak}{R_{n}}^{akn}+\left(\ldots\right),$

where we have only displayed terms that will survive the integral. Invoking
the Gauss-Codazzi relation
${R^{n}}_{snv}=-\partial_{n}K_{sv}+K_{bv}K^{b}_{s}$, and assembling all terms,
we integrate to obtain the junction condition for $\tilde{G}_{in}$ to be

$8\pi
S_{in}=\beta_{2}\left(\epsilon_{buvn}\partial^{b}\varphi\left[{R_{i}}^{nuv}\right]+2\epsilon_{nmri}\left(\partial^{m}\varphi[D_{a}K^{ar}]-\overbracket{\varphi^{\prime}K^{m}_{a}}[K^{ar}]\right)\right).$
(4.68)

We now consider the junction condition for $\tilde{G}_{ij}$. Apart from the
Gauss-Bonnet term (multiplied to $\phi$) which integrates to yield $\phi
J^{(GB)}_{ij}$ where $J^{(GB)}_{ij}$ is the junction equation of pure Gauss-
Bonnet theory, the dilaton and axion interaction terms contribute to the
junction condition as well.

Let us begin with the dilaton interaction term which could contribute to the
junction equation with the following two terms from (4.61):

$\tilde{G}_{ij}=-\beta_{1}g_{jl}\delta^{lnab}_{ingd}R^{gd}_{ab}\nabla^{n}\nabla_{n}\phi-\beta_{1}g_{jl}\delta^{lsnb}_{irnd}R^{nd}_{nb}\nabla^{r}\nabla_{s}\phi+\left(\ldots\right)$
(4.69)

For the first term, expanding the generalized Kronecker symbol, we obtain

$\beta_{1}\left(g_{ji}R^{gd}_{gd}-g_{jg}R^{gd}_{id}+g_{jd}R^{gd}_{ig}-g_{jd}R^{gd}_{gi}+g_{jg}R^{gd}_{di}-g_{ij}R^{gd}_{dg}\right)\nabla^{n}\nabla_{n}\phi=-4\beta_{1}G_{ij}\nabla^{n}\nabla_{n}\phi$
(4.70)

Integrating across $\Sigma$ then gives

$-4\beta_{1}\overbracket{G_{ij}}[\partial_{n}\phi]\equiv-4\beta_{1}\left(-\overline{K^{\prime}_{ij}}+2\overbracket{K^{a}_{j}K_{ai}}-\overbracket{KK_{ij}}+\hat{R}_{ij}-\frac{1}{2}g_{ij}\left(\hat{R}-2\overline{K^{\prime}}-\overbracket{K^{2}}-\overbracket{K^{ab}K_{ab}}\right)\right)[\partial_{n}\phi]$
(4.71)

For the second term in (4.69), we first note that

$g_{jl}\delta^{lab}_{igd}\nabla^{g}\nabla_{a}\phi\,K^{d}_{b}=g_{ji}\nabla^{2}_{\Sigma}\phi
K-\nabla_{j}\nabla_{i}\phi K+\nabla^{m}\nabla_{i}\phi
K_{jm}-\nabla^{2}_{\Sigma}\phi K_{ji}+\nabla_{j}\nabla_{d}\phi
K^{d}_{i}-g_{ji}\nabla^{g}\nabla_{d}\phi K^{d}_{g}$ (4.72)

after expanding the generalized Kronecker symbol. Integrating the second term
of (4.69) across $\Sigma$ then gives, after some algebra,

$\beta_{1}\left(\nabla^{2}_{\Sigma}\phi(g_{ij}[K]-[K_{ij}])+(D_{m}D_{a}\phi+\overbracket{K_{ma}\phi^{\prime}})\mathcal{C}^{mars}_{ij}[K_{rs}]\right),$
(4.73)

where we have defined

$\mathcal{C}^{mars}_{ij}\equiv\delta^{a}_{i}\delta^{r}_{j}g^{ms}+\delta^{m}_{j}\delta^{s}_{i}g^{ar}-\delta^{m}_{j}\delta^{a}_{i}g^{rs}-g^{ar}g^{ms}g_{ij}.$

For the axion interaction term, it is helpful to first write out explicitly
three types of terms which contribute to the junction equation. From the
second term of (4.61), we have

$\beta_{2}\left(2\nabla_{n}(\partial^{m}\varphi\epsilon_{men(j}{R_{i)}}^{nen})+\nabla_{n}(\partial^{n}\varphi\epsilon_{nef(j}{R_{i)}}^{nef})+2\nabla_{k}(\partial^{m}\varphi\epsilon_{mnf(j}{R_{i)}}^{knf})\right)+\left(\ldots\right)$
(4.74)

Expanding the first term of (4.74) and using Gauss-Codazzi relation to express
the Christoffel symbols in terms of the extrinsic curvature gives

$\nabla_{n}(\partial^{m}\varphi\epsilon_{men(j}{R_{i)}}^{nen})=\partial_{n}(\partial^{m}\varphi\epsilon_{men(j}{R_{i)}}^{nen})+K^{m}_{l}\partial^{l}\varphi\epsilon_{men(j}{R_{i)}}^{nen}+\partial^{m}\varphi\epsilon_{men(j}\left({R_{i)}}^{nln}K^{e}_{l}-K^{a}_{i)}{R_{a}}^{nen}\right)$
(4.75)

After integration this gives the junction terms

$[\partial^{m}\varphi\epsilon_{men(j}{R_{i)}}^{nen}]+\overline{K^{m}_{l}}\partial^{l}\varphi\epsilon_{men(j}[K^{e}_{i)}]+\overline{K^{e}_{l}}\partial^{m}\varphi\epsilon_{men(j}[K^{l}_{i)}]-\partial^{m}\varphi\epsilon_{men(j}\overline{K^{a}_{i)}}[K^{e}_{a}]$
(4.76)

The second term of (4.74) simply gives, after integration,

$\left[\partial^{n}\varphi\epsilon_{nef(j}{R_{i)}}^{nef}\right],$

whereas the third term of (4.74) yields

$\partial^{m}\varphi\epsilon_{mnf(j}\left(-\overline{K}_{i)k}[K^{kf}]+\overline{K}[K^{f}_{i)}]\right),$

after we note that
$\nabla_{k}\left(\partial^{m}\varphi\epsilon_{mnf(j}{R_{i)}}^{knf}\right)=\partial^{m}\varphi\epsilon_{mnf(j}\left(K_{i)k}{R_{n}}^{knf}-K{R_{i)}}^{nnf}+\ldots\right)$.
Assembling all terms together, we find the junction condition associated with
$\tilde{G}_{ij}$ to be

$\displaystyle 8\pi S_{ij}$ $\displaystyle=$
$\displaystyle\beta_{2}\left(\left[\partial^{n}\varphi\epsilon_{nef(j}{R_{i)}}^{nef}\right]+2\partial^{m}\varphi\epsilon_{mnf(j}\big{(}-\overline{K}_{ki)}[K^{kf}]+\overline{K}[K^{f}_{i)}]\big{)}\right)$
$\displaystyle+2\beta_{2}\left([\partial^{m}\varphi\epsilon_{men(j}{R_{i)}}^{nen}]+\partial^{l}\varphi\epsilon_{men(j}\overline{K^{m}_{l}}[K^{e}_{i)}]+\partial^{m}\varphi\epsilon_{men(j}\overline{K^{e}_{l}}[K^{l}_{i)}]\right)$
$\displaystyle+$
$\displaystyle\beta_{1}\left(\nabla^{2}_{\Sigma}\phi(g_{ij}[K]-[K_{ij}])+(D_{m}D_{a}\phi+\overbracket{K_{ma}\phi^{\prime}})\mathcal{C}^{mars}_{ij}[K_{rs}]-4\overbracket{G_{ij}}[\phi^{\prime}]+\phi[J^{(GB)}_{ij}]\right)$
$\displaystyle+$ $\displaystyle[K]h_{ij}-[K_{ij}].$ (4.79)

Together with (4.68), (4.2) specify the appropriate junction condition for the
low-energy effective theory defined by the action (4.52). As a simple
consistency check, we note that taking the scalar fields to be constant give
the junction conditions for the pure Gauss-Bonnet theory (for a continuous
axion, the pure Chern-Simons gravity theory has a trivial junction condition).
If we assume a stronger constraint $[K_{ij}]=0$, then (4.2) simplifies to read

$8\pi
S_{ij}=-2\beta_{2}n_{\alpha}\epsilon_{mln(j}\left[\nabla^{\alpha}K_{i)}^{l}\right]\partial^{m}\varphi,$
(4.80)

where only the axion-coupling remains in the junction condition.

###  4.3 Higher-dimensional Euler densities

The Gauss-Bonnet term is topological in four dimensions, yet in higher
dimensions where it bears a non-topological nature, the junction conditions
are still valid. More broadly speaking, it is an example of the Euler
characteristic that can be defined in higher dimensions and which is the most
general extension of the Einstein-Hilbert action that yield at most second-
order field equations. The appropriate surface terms were derived some time
ago and their variation with respect to the induced metric of $\Sigma$ gives
the junction conditions - as was shown explicitly in the Gauss-Bonnet case in
[23, 21, 22].

The topological Euler density term for a $2m$-dimensional manifold is defined
as

$\mathcal{L}_{m}=\Omega^{a_{1}b_{1}}\wedge\ldots\wedge\Omega^{a_{m}b_{m}}\wedge\epsilon_{a_{1}b_{1}\ldots
a_{m}b_{m}}=\frac{1}{2^{m}}\delta^{[c_{1}d_{1}\ldots
c_{m}d_{m}]}_{a_{1}b_{1}\ldots
a_{m}b_{m}}{R^{a_{1}b_{1}}}_{c_{1}d_{1}}\ldots{R^{a_{m}b_{m}}}_{c_{m}d_{m}}$
(4.81)

where the Kronecker $\delta$-function above is totally antisymmetric in both
sets of indices, $\Omega$ is the curvature two-form, and we have normalized it
such that the anti-symmetrization symbol in (4.81) has no other normalization
factor. Note that $\mathcal{L}_{1},\mathcal{L}_{2}$ are the Ricci scalar and
Gauss-Bonnet term respectively. One can consider extending (4.81) to other
dimensions apart from $2m$. For dimensions less than $2m$, it simply vanishes
whereas for dimensions higher, it will be non-topological.

Now the Euler-Lagrange equations of motion of (4.81) read

$\tilde{G}_{ij}=-\frac{1}{2^{j+1}}g_{i\mu}\delta^{[\mu a_{1}\ldots
a_{2m}]}_{jb_{1}\ldots
b_{2m}}{R_{a_{1}a_{2}}}^{b_{1}b_{2}}\ldots{R_{a_{2m-1}a_{2m}}}^{b_{2m-1}b_{2m}}.$
(4.82)

This follows from a well-posed variational principle if appropriate surface
terms can be added at $\Sigma$ such that setting $\delta g_{ij}=0$ at $\Sigma$
is sufficient for the vanishing of the action variation and no terms of the
form $\nabla_{k}\delta g_{ij}$ survive. In the following, we briefly review
how such a surface term was derived in [23] in the language of differential
forms. One begins by defining a Chern-Simons form $Q_{m}$ such that

$\displaystyle\mathcal{L}_{m}(\omega)-\mathcal{L}_{m}(\omega_{0})$
$\displaystyle=$ $\displaystyle dQ_{m}(\omega,\omega_{0}),$ (4.83)
$\displaystyle Q_{m}$ $\displaystyle=$ $\displaystyle
m\int^{1}_{0}ds\,\,\theta^{a_{1}b_{1}}\wedge\Omega^{a_{2}b_{2}}_{s}\wedge\ldots\wedge\Omega^{a_{m}b_{m}}_{s}\wedge\epsilon_{a_{1}b_{1}\ldots
a_{m}b_{m}},$ (4.84)

where $\omega$ is the connection one-form, $\omega_{0}$ is the conection one-
form defined on $\Sigma$, $\theta=\omega-\omega_{0}$ is the extrinsic
curvature/second fundamental form, and finally
$\Omega_{s}=d\omega_{s}+\omega_{s}\wedge\omega_{s}$ is the curvature two-form
defined with $\omega_{s}=\omega-s\theta$. In [23], it was shown that taking
the variation of (4.83) implies
$\delta_{\omega}\int_{M}\mathcal{L}_{m}=\int_{M}d(\delta_{\omega}Q_{m}),$
which leads naturally to the surface term in the action

$I_{m}=\int_{M}\mathcal{L}_{m}-\int_{\partial M}Q_{m}.$ (4.85)

In coordinate form, the Chern-Simons form that enacts the surface term can be
written as [23]

$Q_{m}=\int^{1}_{0}ds\,\,\delta^{\mu_{1}\ldots\mu_{2m-1}}_{\nu_{1}\ldots\nu_{2m-1}}K^{\nu_{1}}_{\mu_{1}}\times\left(\frac{1}{2}{R^{\nu_{2}\nu_{3}}}_{\mu_{2}\mu_{3}}-s^{2}K^{\nu_{2}}_{\mu_{2}}K^{\nu_{3}}_{\mu_{3}}\right)\ldots\left(\frac{1}{2}{R^{\nu_{2m-2}\nu_{2m-1}}}_{\mu_{2m-2}\mu_{2m-1}}-s^{2}K^{\nu_{2m-2}}_{\mu_{2m-2}}K^{\nu_{2m-1}}_{\mu_{2m-1}}\right)$
(4.86)

In principle, one could take the variation of $Q_{m}$ with respect to the
induced metric to obtain the junction condtions. Yet even for the Gauss-Bonnet
case, this can be a rather elaborate calculation as shown in [21, 22].

In the following, we will integrate the equations of motion $\tilde{G}_{ij}$
across $\Sigma$ and obtain the junction conditions for the Euler density term
valid in dimension $\geq 2m$. We first check that there are no singular terms
and hence no regularity constraints to impose. This is manifest in the form of
(4.82). Consider again the Gauss-Bonnet term as an example. The singular terms
can only arise from a product of two Riemann tensors each of which carries two