aainstitutetext: Princeton Gravity Initiative, Princeton University,
Princeton, NJ 08544, USAbbinstitutetext: Jefferson Physical Laboratory,
Harvard University, Cambridge, MA 02138, USA

# Density of States, Black Holes and the Emergent String Conjecture

Alek Bedroya b , Rashmish K. Mishra b , and Max Wiesner

###### Abstract

We study universal features of the density of one-particle states $\rho(E)$ in
weakly coupled theories of gravity at energies above the quantum gravity
cutoff $\Lambda$, defined as the scale suppressing higher-derivative
corrections to the Einstein–Hilbert action. Using thermodynamic properties of
black holes, we show that in asymptotically flat spacetimes, certain features
of $\rho(E)$ above the black hole threshold $M_{\rm min}$ are an indicator for
the existence of large extra dimensions, and cannot be reproduced by any
lower-dimensional field theory with finitely many fields satisfying the weak
energy condition. Based on the properties of gravitational scattering
amplitudes, we argue that there needs to exist a (possibly higher-dimensional)
effective description of gravity valid up to the cutoff $\Lambda$. Combining
this with thermodynamic arguments we demonstrate that $\rho(E)$ has to grow
exponentially for energies $\Lambda\ll E\ll M_{\rm min}$. Furthermore we show
that the tension of any weakly coupled $p$-brane with $p\geq 1$ is bounded
from below by $\Lambda^{p-1}$. We use this to argue that any tower of weakly
coupled states with mass below $\Lambda$ has to be a Kaluza–Klein (KK) tower.
Altogether these results indicate that in gravitational weak-coupling limits
the lightest tower of states is either a KK tower, or has an exponentially
growing degeneracy thereby resembling a string tower. This provides evidence
for the Emergent String Conjecture without explicitly relying on string theory
or supersymmetry.

##  1 Introduction and Summary

A characteristic feature of gravitational theories is the existence of massive
states beyond those describable by an effective field theory (EFT). Such
states could be strongly coupled (e.g. black hole microstates) or weakly
coupled (e.g. string excitations). Oftentimes, these states provide crucial
information about the theory of gravity and are believed to be required for
consistency of the theory at the quantum level. Whereas the exact spectrum of
the massive states depends on the details of the UV completion, one may expect
it to have some features that are universal for any theory of gravity.
Proposals for such universal features of the spectrum of the massive states in
quantum gravity have been formulated in the context of the Swampland program
Vafa:2005ui , e.g., through the Distance Conjecture Ooguri:2006in . These
proposals are based on evidence from string theory and have been tested to a
great extent in a plethora of explicit string compactifications (see
Palti:2019pca ; vanBeest:2021lhn ; Agmon:2022thq  for reviews). Whereas there
is strong evidence for such conjectures in the context of string theory, a
pressing question concerns their validity in general theories of quantum
gravity away from the string theory lamppost.

In this spirit, the goal of this work is to extract general properties of the
spectrum of massive states in quantum gravity regardless of the details of the
UV completion. The central object of our analysis is the density of one-
particle states, $\rho$, in quantum gravity. In particular we are interested
in extracting the universal behavior of $\rho$ as a function of energy
$E$.111For one-particle states we refer to the rest mass $m$ as “energy” in
the following. Concretely, we aim to study the properties of $\rho(E)$ at
energies above the quantum gravity cutoff. Combining results from
gravitational scattering amplitudes with thermodynamics for weakly coupled
theories of gravity, we relate the high-energy behavior of $\rho(E)$ to the
properties of towers of states below the black hole thresholds. This, in turn,
provides crucial insight into the general properties of weakly coupled
theories of gravity.

There are multiple notable energy scales that are relevant for the study of
high energy states in quantum gravity. The first energy scale is the Planck
scale $M_{\rm pl}$ which describes the strength of the gravitational coupling.
Another energy scale is the scale at which gravity becomes strongly coupled.
This scale is also known as the species scale Dvali:2007hz ; Dvali:2007wp ;
Dvali:2009ks ; Dvali:2010vm ; Dvali:2012uq  which, in theories with many
weakly coupled particles, can be parametrically below the Planck scale,
$\Lambda\ll M_{\rm pl}$. A precise proposal to define the species scale has
been given in vandeHeisteeg:2022btw ; vandeHeisteeg:2023ubh  that identifies
the species scale with the scale that suppresses gravitational higher-
derivative corrections to the Einstein–Hilbert action. Indeed, the higher-
derivative corrections have been confirmed to be a good measure for the
species scale in various explicit string theory setups Cribiori:2022cho ;
vandeHeisteeg:2023uxj ; Cribiori:2023sch ; vandeHeisteeg:2023dlw ;
Castellano:2023aum .222See Calderon-Infante:2023uhz  for a discussion of the
equivalence of the various definitions of the species scale in explicit string
theory setups.

It is often stated that the species scale, defined as the scale suppressing
the higher-derivative corrections, is also the curvature scale at the horizon
of the smallest black hole describable within effective field theory
Dvali:2007hz ; Dvali:2007wp ; vandeHeisteeg:2022btw . However, the inverse
radius of the smallest black hole describable in a given EFT can be
parametrically different from the species scale. The scale associated to the
inverse radius of this smallest black hole was recently conjectured in
Bedroya:2024uva  to be a universal scale in quantum gravity, called the black
hole scale $\Lambda_{\text{BH}}$. It was also suggested that $\Lambda_{\rm
BH}$—which is defined using black hole physics—is a well-defined substitute
for the mass scale of the lightest weakly coupled tower of particles at low-
energies. Let us briefly summarize this main observation of Bedroya:2024uva .
Consider the low-energy action of a theory given by the dimensional reduction
of a higher-dimensional theory on a compact manifold of large radius $R$. In
case the species scale of the higher-dimensional theory is the higher-
dimensional Planck scale, this scale will also be the species scale of the
lower-dimensional theory. If the radius of the horizon of the smallest black
hole describable in the lower-dimensional effective field theory was set by
the inverse species scale, one would conclude that any lower-dimensional black
hole solution with horizon size larger than the higher-dimensional Planck
scale is reliable. However, due to the Gregory–Laflamme instability
Gregory:1993vy , a black hole for which the length scale of the horizon is
much smaller than the radius, $R$, of the compact space is unstable and decays
into higher-dimensional black holes which are localized in the extra-
dimension. Therefore, the smallest black hole that can be described by the
lower-dimensional EFT has inverse-radius $\Lambda_{\rm BH}\sim 1/R$.

The above argument demonstrates two points: (1) There is often a rich
hierarchy of scales in quantum gravity ($\Lambda_{\rm BH}\ll\Lambda_{s}\ll
M_{\rm pl}$) which leaves an imprint both at low energies (existence of
particle towers) as well as high energies (Gregory–Laflamme transition). (2)
The connection between the higher-derivative corrections and the size of the
smallest black hole is more subtle than previously thought. In fact, the
hierarchy of scales could be extended to include more energy scales by having
multiple Gregory–Laflamme transitions in the presence of extra dimensions with
parametrically different length scales. In this work, we provide a unifying
framework to study the different energy scales and their connections to one
another by studying the behavior of $\rho(E)$ across a wide range of energies.
In particular, we shed light on the connection between the species scale and
the smallest black hole. The key point is to clarify what is meant by the
smallest black hole. For example if, in the presence of large extra
dimensions, “smallest black hole” refers to the smallest higher-dimensional
black hole, the radius of this black hole is indeed expected to be
$\mathcal{O}(\Lambda^{-1})$. In this paper we show that the smallest black
hole in this more general sense can be probed by gravitational amplitudes and
demonstrate how its size is related to the higher-curvature expansion of the
effective action. In that way we settle the problem of relating the size of
the smallest black holes to the species scale defined in terms of the higher-
curvature corrections. Our results can be summarized in the following way: the
inverse of the horizon size of the smallest black hole describable by any EFT
is given by the cutoff of this EFT. Importantly this EFT cutoff can differ
from the energy scale controlling the higher-curvature corrections. In
particular if the theory arises as compactification of a higher-dimensional
theory, the cutoff of the higher- and lower-dimensional EFTs are different.
The lower-dimensional theory has a tower of KK modes with mass $m\ll\Lambda$
that can never be included in an EFT in a parametrically controlled way.
Therefore, the cutoff of the lower-dimensional EFT is given by the mass scale
of the KK tower $\sim 1/R$. The lower-dimensional EFT describes black holes
with horizon radius $r_{H}\gg R$, while in the higher-dimensional EFT it is
possible to describe much smaller black holes with radius $R>r_{H}\gg
1/\Lambda$.

Since our arguments rely on the dependence of the density of one-particle
states $\rho$ on the energy, it is crucial to have a reliable way of
calculating $\rho$ at high energies above the EFT cutoff. By studying the
relation between scattering amplitudes and the density of one-particle states
for black holes and strings, we propose an approximate measure for $\rho(E)$
at energies $E\gg\Lambda$ in terms of gravitational scattering amplitudes.
Given this estimate of $\rho(E)$ we study its behavior for high energies which
we utilize to infer the structure of possible towers of states in quantum
gravity with mass below the black hole threshold. To that end, we work in
asymptotically flat spacetimes and focus on certain gravitational weak-
coupling limits. These are limits in the scalar field space of gravitational
theories where the gravitational backreaction vanishes for massive particles
with fixed mass (in units of $\Lambda$) below the black hole threshold.
Particles with vanishing gravitational backreaction are to be contrasted to
large black holes whose mass is entirely sourced by their gravitational self-
energy. For states with fixed mass (in units of $\Lambda$) to have vanishing
gravitational backreaction, we require $\Lambda\ll M_{\text{pl}}$. Therefore
for gravitational weak-coupling limits the quantum gravity cutoff, or species
scale, can be made parametrically smaller than the Planck scale, i.e.,
$\Lambda\ll M_{\rm pl}$. Since the species scale relates to the number of
light species in the theory, such limits feature towers of light states. Hence
gravitational weak-coupling limits are exactly those kinds of limits in moduli
space that are of interest in the context of the Distance Conjecture.

The nature of the possible towers of states in such infinite distance limits
is believed to be strongly constrained by the Emergent String Conjecture (ESC)
Lee:2019oct . This conjecture states that, at least in asymptotically flat
spacetime, any infinite distance limit of a theory of gravity is either a
limit in which a fundamental string becomes tensionless or a
decompactification limit to a higher-dimensional theory. According to the ESC
the lightest tower of states becoming asymptotically massless at an infinite
distance in the scalar field space either corresponds to a KK-tower or the
excitation tower of a light string. In particular, the ESC prohibits the
lightest tower to arise from membranes or other extended objects, see Alvarez-
Garcia:2021pxo  for a discussion on this in five-dimensional compactifications
of M-theory. The ESC has been tested and confirmed in different string theory
setups Lee:2018urn ; Lee:2018spm ; Lee:2019jan ; Lee:2019apr ; Baume:2019sry ;
Klaewer:2020lfg ; Lee:2021usk ; Wiesner:2022qys ; Bedroya:2023tch ;
Etheredge:2023odp ; Alvarez-Garcia:2023qqj  and has proven to be a powerful
ingredient for many recent bottom-up arguments (see e.g. Lanza:2021udy ;
Rudelius:2021oaz ; Etheredge:2022opl ; Montero:2022prj ; vandeHeisteeg:2022btw
; Rudelius:2022gbz ; vandeHeisteeg:2023ubh ; Bedroya:2023xue ; Bedroya:2023tch
). Still, a definite bottom-up proof for the stringent constraints on the
possible nature of the lightest tower of states predicted by the ESC is
lacking—even though Basile:2023blg  presented arguments in its favor based on
the concept of species scale thermodynamics Cribiori:2023ffn ; Basile:2024dqq
. In this work we take a different approach and use the properties of the
density of one-particle state and gravitational scattering amplitudes to argue
that the lightest tower of states in gravitational weak-coupling limits are
either KK-towers or a tower of states with degeneracy growing exponentially
fast in energy thereby resembling the excitation tower of a fundamental
string. Let us stress, however, that we do not identify the origin of this
tower of states with exponential degeneracy, and in particular cannot answer
whether or not they arise from the excitations of a fundamental string. Still,
we view our results as string theory independent evidence for the ESC.

This paper is structured as follows: the rest of this section provides an
overview of the main results of the paper. We review the definition of
vandeHeisteeg:2023dlw ; vandeHeisteeg:2023ubh  of the quantum gravity cutoff
in terms of higher-derivative corrections to the Einstein–Hilbert action in
section 2. In section 3 we introduce a way to estimate the density, $\rho$ of
one-particle states in terms of high-energy scattering amplitudes and discuss
some basic properties of the dependence of $\rho$ on $E$. In section 4 we use
gravitational scattering amplitudes to show the equivalence between the cutoff
of the gravitational EFT describing _all_ black holes in the theory and the
species scale, and present a bound on the tension of weakly coupled $p$-branes
in terms of the species scale. Section 5 studies the spectrum of weakly
coupled gravitational theories via the energy dependence of $\rho$ at energies
above the species scale which enables us to connect our results to the ESC. We
present our conclusions and future perspectives in section 6. The appendices
contain some background on effective strings, Gross–Mende saddles and
scattering amplitudes that are the basis of the arguments presented in the
main text.

###  1.1 Summary of Results

The central objects of our analysis are the species scale $\Lambda$ and the
density of one-particle states $\rho(E)$. Motivated by the behavior of
scattering amplitudes in the black hole and perturbative string regimes at
energies above the cutoff, $E>\Lambda$, we define an approximate notion for
$\rho$ in terms of $2\to 2$ scattering amplitudes at fixed impact parameter
$b\lesssim\Lambda^{-1}$. The precise identification is given in (10). The
energy dependence of $\rho$ can be parametrized as

$\rho(E)\sim\exp\left[\left(\frac{E}{\Lambda}\right)^{\alpha}\right]\,,$ (1)

for some $\alpha>0$ up to terms polynomial in $E$. The parameter $\alpha$ is
understood to be a piece-wise constant function of the energy $E$. As we show,
the value of $\alpha$ in a given energy window encodes crucial information
about the nature of the one-particle states with mass in that energy window.
For example, using basic properties of black hole thermodynamics we show that
in case $\alpha>1$ for some energy $E_{0}$, the corresponding one-particle
states with mass $m=E_{0}$ are necessarily black hole microstates, see
Proposition 1. In other words, the energy $M_{\rm min}$ corresponding to the
transition between particles and black holes can be detected in the density of
one-particle states as the energy at which $\alpha\leq 1$ changes to
$\alpha>1$. The energy scale $M_{\rm min}$ then corresponds to the minimal
mass of a black hole describable in the effective theory. Let us stress that
this black hole may be a higher-dimensional black hole, in case the effective
theory arises as a compactification of a higher-dimensional effective theory
of gravity.

The curvature at the horizon of the smallest black hole describable in the
(possibly higher-dimensional) EFT sets a scale $\Lambda_{\rm min}$ that serves
as cutoff of the EFT. Still, it is not obvious that this scale should be
identified with the scale $\Lambda$ appearing in the higher-derivative
corrections to the Einstein–Hilbert action. In this work we argue for the
identification of these two scales by using gravitational scattering
amplitudes. To that end, we focus on $2\to 2$ graviton scattering in the
unphysical regime in which the Mandelstam variable $t\gg 0$. To argue for
$\Lambda=\Lambda_{\rm min}$ we use that in the unphysical regime and for
center of mass energies above the black hole threshold the amplitude grows
exponentially in the radius of a black hole of that mass. We then show that
for energies $\Lambda_{\rm min}\ll\sqrt{s}\ll M_{\rm min}$ this radius has to
be replaced by $\Lambda_{\rm min}^{-1}$, i.e. the radius of the smallest black
hole describable in the theory. We use this amplitude to match the
coefficients of the higher-derivative expansion using the fact that below
$\Lambda_{\rm min}$ the full effective action including all higher-derivative
terms has to give the full amplitude. This leads to the identification
$\Lambda_{\rm min}=\Lambda$ as stated in Proposition 2 implying the existence
of a (possibly higher-dimensional) EFT description all the way up to
$\Lambda$. To the best of our knowledge, this provides the first general
gravitational argument as to why the scale suppressing the higher-derivative
corrections is the same as the scale of the curvature at the horizon of the
smallest black hole describable by some effective theory.

We further use gravitational scattering amplitudes to impose a bound on the
tension of weakly coupled strings in quantum gravity. Since these strings are
weakly coupled we argue that their worldsheet theory can be well-described in
terms of the Nambu–Goto action using the effective string theory of Polchinski
and Strominger Polchinski:1991ax  — even if the string is not a critical
string. The contribution of the Gross–Mende saddles Gross:1987ar  of these
weakly coupled strings to the scattering amplitude can then be compared to the
scattering amplitude expected from the full effective action including higher-
derivative terms. Using this comparison, we infer that the tension
$\mathcal{T}$ of any weakly coupled string has to be bounded as

$\mathcal{T}\gtrsim\Lambda^{2}\,.$ (2)

Since a weakly coupled $p$-brane with $p>1$ yields a weakly coupled effective
string upon dimensional reduction we can extend this bound to general
$p$-branes as claimed in Proposition 3.

Using this bound and the fact that there exists a (higher-dimensional) EFT
description for the theory valid up to the energy $\Lambda$ we then conclude
that in gravitational weak-coupling limits $\Lambda\ll M_{\rm pl,d}$ any tower
of states with mass below the species scale has to be a KK tower. This is
summarized in Proposition 4. For energies above the species scale but below
the black hole threshold, i.e. for $\Lambda\ll E\ll M_{\min}$, we employ basic
thermodynamic arguments to show that there exists a tower of one-particle
states with mass proportional to $\Lambda$ whose degeneracy grows
exponentially in the energy. In other words the Hagedorn behavior

$\rho(E)\propto\exp\left(\frac{E}{\Lambda}\right)\,,\quad\text{for}\quad\Lambda\ll
E\ll M_{\rm min}\,,$ (3)

is a universal property of the spectrum of quantum gravity as stated in
Proposition 5. Finally, we present an argument that indicates that above the
black hole threshold we can use the scaling of $\rho$ as a function of energy
to detect the existence of extra dimensions and their size by showing that
(higher-dimensional) Schwarzschild black hole are always the dominant
contribution to $\rho(E)$ and uniquely determine $\alpha$ in a given spacetime
dimension as stated in Proposition 6.

Using our results on the possible tower of states in gravitational weak-
coupling limits, we conclude in Corollary 1 that any tower in a weakly coupled
theory of gravity is either a KK tower or exhibits Hagedorn behavior which we
view as evidence for the key statement of the ESC that the lightest tower of
states is either a KK tower or the excitation tower of a weakly coupled
string.

##  2 Review: Species scale

Our analysis crucially relies on two quantities that we can define in theories
of quantum gravity: the quantum gravity cutoff (or species scale) $\Lambda$
and the density, $\rho$, of one-particle states. In this section we review the
definition of the species scale in terms of higher-derivative operators as
originally proposed in vandeHeisteeg:2022btw ; vandeHeisteeg:2023dlw ;
vandeHeisteeg:2023ubh . The relation of the species scale defined in this way
to the size of the smallest black hole will be made precise in section 4.1.

To define the species scale we start with a $d$-dimensional theory of gravity
that at low energies can be described by Einstein gravity with action

$S_{\rm EH}=\frac{M_{\rm pl,d}^{d-2}}{2}\int{\rm
d}^{d}x\sqrt{-g}\left(\mathcal{R}+\cdots\right)\,,$ (4)

where $M_{\rm pl,d}$ is the $d$-dimensional reduced Planck scale, $g$ is the
metric on the $d$-dimensional spacetime, $\mathcal{R}$ the Ricci scalar and
the $\cdots$ stand for terms involving additional fields coupled to gravity.
Radiative corrections to the effective action of the quantum theory are
expected to generate an infinite series of gravitational higher-derivative
operators involving the Riemann tensor and derivatives thereof. In other
words, the Einstein–Hilbert action above will be corrected by terms of the
form

$S_{\rm corr.}=\frac{M_{\rm pl,d}^{d-2}}{2}\int{\rm
d}^{d}x\sqrt{-g}\left(\sum_{n=1}^{\infty}a_{n}\frac{\mathcal{O}_{2n+2}(\mathcal{R})}{M_{\rm
pl,d}^{2n}}\right)\,.$ (5)

Here, $\mathcal{O}_{m}(\mathcal{R})$ are dimension-$m$ operators involving the
Riemann tensor $\mathcal{R}$ and its derivatives such as $\mathcal{R}^{2}$ and
$\mathcal{R}\,\Box\mathcal{R}$ which are examples for dimension-4 and -6
operators, respectively. The coefficients $a_{n}$ are the Wilson coefficients
of the respective operators that encode the imprints of quantum gravity on the
effective action. The coefficients $a_{n}$ in general depend on $n$ and, in
the presence of light scalar fields $\phi$, also on the vevs for these scalar
fields vandeHeisteeg:2022btw ; vandeHeisteeg:2023ubh . There typically exist
limits in the scalar field space of the theory of quantum gravity in which the
quantum gravity cutoff $\Lambda$ differs significantly from the Planck scale.
This manifests itself in the Wilson coefficient $a_{n}(\phi)$ being
parametrically large. Given a higher-derivative term one then expects the
relation

$\frac{\Lambda(\phi)}{M_{\rm pl,d}}\sim a_{n}(\phi)^{-1/2n}\,,$ (6)

In principle there can be some generic suppression of the Wilson coefficients
with $n$. For the string case, we discuss this $n$-dependence in more detail
around (31). Taking these $n$-dependent prefactors into account we conclude
that $\Lambda$ indeed replaces $M_{\rm pl,d}$ as cutoff scale in the effective
action (4). Notice that in practice this could give a slightly different
species scale depending on which term in the effective action is considered.
To avoid this ambiguity vandeHeisteeg:2023dlw  proposed to define the quantum
gravity cutoff as

$\Lambda(\phi)\sim\frac{M_{\rm pl,d}}{\sup_{n\geq
1}\left\\{a_{n}(\phi)^{\frac{1}{2n}}\right\\}}\,.$ (7)

For self-consistency the Wilson coefficients have to be evaluated at the scale
on LHS of the above equation.

Since the Wilson coefficient in the effective action (5) are energy-dependent,
we have to choose an appropriate energy scale $\mu$ to define $\Lambda$. If we
push $\mu$ to higher energies, we must integrate in all particles with mass
below $\mu$ into the EFT. However, this process cannot be continued
indefinitely due to the existence of black holes. In other words, there is an
energy threshold beyond which the particles do not admit a field theory
description. The energy scale $\mu$ at which we define $\Lambda$ is given by
the inverse radius of the smallest (potentially higher-dimensional) black hole
in the theory. This scale is a natural UV cutoff in quantum gravity, see
Brustein:2009ex  for a discussion on this. Equivalently we should consider the
EFT in which all black holes in the theory can be described. Only in this case
does the definition in (7) give the correct quantum gravity cutoff. To
illustrate this point let us, for the moment, consider a low-energy EFT in
which a light state with mass $m_{0}$ has been integrated out. From standard
EFT expectation, the Wilson coefficients of a general higher-derivative
operator of dimension $2n+2$ scale as

$a_{n}\sim\left(\frac{M_{\rm pl,d}}{m_{0}}\right)^{2n+2-d}\,.$ (8)

A low-energy observer unaware of the existence of the particle with mass
$m_{0}$ who measures the Wilson coefficient of the gravitational higher-
derivative operators for large enough $n$ and applies (7) would now
erroneously conclude that the species scale is set by $m_{0}$. However, if
$m_{0}\ll M_{\rm pl,d}$ the scaling of the higher-derivative Wilson
coefficients with $n$ is very different from what we expect for the actual
species scale. In the latter case, the value of $a_{n}^{-1/2n}$ for different
$n$ only differs by the aforementioned $n$-dependent constants that will be
discussed in section 4.1. In contrast, in case the $a_{n}$ are determined by
the light state with mass $m_{0}\ll M_{\rm pl,d}$, the coefficients
$a_{n}^{-1/2n}$ differ by powers of $M_{\rm pl,d}/m_{0}\gg 1$. The
$n$-dependence of $a_{n}^{-1/2n}$ is hence an indicator of whether the low-
energy effective action misses some light particles. The Wilson coefficients
$a_{n}$ only give the correct species scale for the EFT in which all light
particles have been integrated in.

As argued in Dvali:2007hz ; Dvali:2007wp ; Dvali:2010vm ; Dvali:2012uq , the
effective quantum gravity cutoff is expected to be significantly lowered
compared to the Planck scale in the presence of many species of light
particles. In this context, the quantum gravity cutoff is referred to as the
species scale as it effectively counts the number of light species in the
theory of gravity. According to the distance conjecture Ooguri:2006in  large
numbers of light species arise in asymptotic, infinite distance regimes in the
moduli space of gravitational theories. In these regimes the parametric
scaling of the species scale can then be extracted from the properties of the
tower of light states. As shown for concrete examples in vandeHeisteeg:2022btw
; vandeHeisteeg:2023dlw  the scaling of the species scale in these limits is
exactly reproduced by the coefficients of the higher-derivative corrections.
Therefore (7) indeed provides a definition of the species scale, $\Lambda$,
which furthermore is also valid in the interior of the moduli space. In our
work, we follow this proposal to define the species scale as it is more
suitable for our approach since it does not directly refer to the number of
light species.

As we stress in section 4.1, the fact that the species scale is identified
with the overall scale suppressing the higher-derivative expansion does,
however, not necessarily mean that the higher-derivative expansion diverges
for curvature scale of order $\Lambda^{2}$. Instead, the appearance of the
species scale in the full effective action

$S_{\rm eff.}=S_{\rm EH}+S_{\rm corr.}\,,$ (9)

is required so that it correctly computes certain high energy ($E\gg\Lambda$)
scattering amplitudes in the full theory of gravity.

##  3 Density of one-particle states

Defining a notion of the density of one-particle states in theories of quantum
gravity is a challenging task. Whereas for energies below the cutoff (i.e. the
species scale $\Lambda$), one can just count the number of states in the EFT,
it is not clear how to do this at energies above $\Lambda$. In section 3.1, we
discuss some challenges in defining the one-particle density of states and
provide a practical way to approximate the exact $\rho(E)$ at high energies
that addresses these issues. Basic properties of $\rho$ as a function of
energy are discussed in section 3.2 which are utilized in section 3.3 to infer
the energy of the particle-black hole transition from $\rho(E)$.

###  3.1 Defining $\rho(E)$: Challenges and Resolution

The analysis in this paper crucially relies on properties of the density of
one-particle states $\rho$ as a function of energy $E$. For a one-particle
state we consider its energy to be given by its rest mass $m$ and in this work
we use energy and mass interchangeably unless the distinction is important.
For energies below the cutoff, $E\ll\Lambda$, the density of states can be
determined simply by counting the states in the field theory. On the other
hand at energies $E\gg\Lambda$ computing the density of one-particle states is
more difficult due to some subtleties arising in defining an exact notion of
$\rho(E)$. Before we provide a practical way how to estimate $\rho(E)$ let us
briefly comment on these issues:

  1. 1.

Unstable particles: Unstable particles can contribute as intermediate states
in a scattering process, but do not show up as asymptotic states. Depending on
the decay rate $\Gamma$ of the unstable particles (in units of their mass
$M$), they may need to be included in the density of states to correctly
account for the thermodynamic properties. For example, consider systems with a
sufficiently long lifetime that can nevertheless be treated thermodynamically
in the adiabatic approximation (e.g. large black holes in flat space). For
such systems, unstable particles with sufficiently long lifetime are
thermodynamically relevant. However, this makes it challenging to define the
density of states in a precise way that includes such sufficiently long-lived
unstable particles.

  2. 2.

Bound states: A bound state is a state in the multi-particle Hilbert space of
stable particles which is captured in the density of states. We are interested
in using $\rho(E)$ to estimate the entropy, by mapping it to a system of free
particles. This raises the question whether including the bound states will
over-count the entropy. We note that the free particle approximation is
reasonable as long as the particles are sufficiently far from each other.
Consider a system with particles of type $p$, that can form a two-particles
bound state $b$, with size $r_{b}$. To calculate the entropy, we approximate a
state of the original system consisting of particles of type $p$ with a
collection of particles of type $p$ and $b$, that are separated by distances
larger than $r_{b}$. The inclusion of bound states is needed for a free-
particle counting to correctly capture the entropy of an interacting system.

  3. 3.

Mass averaging: Finally, even if we could define the mass of unstable
particles in a sharp way, the density of one-particle states would still be a
collection of $\delta$-functions. In this work we are, however, interested in
studying the behavior of some continuous function $\rho(E)$, i.e., an averaged
version of the one-particle density.

Given these challenges in defining $\rho(E)$ we now provide a way to estimate
the actual density of one-particle states. This estimate has to address the
points raised above and should correctly approximate the number of black hole
microstates at energies above the black hole threshold. As we argue below for
energies $E\gg\Lambda$ a good estimate of $\rho(E)$ is given in terms of the
$2\to 2$ scattering amplitude $\mathcal{A}_{2\to 2}(E)$ at fixed small impact
parameter $b\lesssim\Lambda^{-1}$ as

$-\log\left(\Lambda\rho(E)\right)\sim\log\left|\mathcal{A}_{2\to
2}(E)\right|^{2}+\mathcal{O}(\log(\Lambda/E))\,.$ (10)

Since $\rho$ is the density of states per unit energy, it has inverse mass
dimension. As indicated in the above expression, in this work we consider
$\rho$ in units of the inverse quantum gravity cutoff which in the following
we assume implicitly and drop the $\Lambda$ factor in front of $\rho(E)$.

Let us now explain why (10) is a reasonable estimate of the actual $\rho(E)$:
At high energies and low-impact parameters, the in-going particles have access
to $\sim\rho(E)$ many different intermediate states in the s-channel. Suppose
we estimate the intermediate state with a thermal system with entropy
$S\sim\log(\rho)$ then every particular outcome (such as a two particle out-
going state) has a probability of $\sim\exp(-\mathcal{O}\left(S)\right)$ which
leads to the estimate (10). This argument was first made for energies above
the black hole threshold in Banks:1999gd . The idea is that if $E$ is
sufficiently large and the impact parameter is sufficiently small, the in-
going particles in the $2\to 2$ scattering process are expected to form a
black hole with entropy $S_{\rm BH}(E)$. Then, the probability of the black
hole to decay into exactly two out-going particles is exponentially suppressed
$\left|\mathcal{A}_{2\to
2}(E)\right|^{2}\sim\exp\left(-\mathcal{O}\left(S_{\rm BH}(E)\right)\right)$.
This has also been confirmed using the Euclidean path integral Bah:2022uyz .

In addition to the motivation from black hole physics, the estimate (10) is
expected to hold even at energies much below the black hole formation. For
example, this identity was verified in string theory Bah:2022uyz  to be
compatible with the density of massive string states. We take these
observations as motivation to estimate $\rho(E)$ for energies above the cutoff
via (10) in general theories of gravity.

Apart from reproducing the expected behavior of $\rho(E)$ in the black hole
regime and for perturbative string theory, our estimate in (10) also addresses
the earlier mentioned complications in defining the density of states.
Consider first the issue regarding unstable particles. To arrive at our
formula for $\rho(E)$, we required it to correctly estimate the density of
black hole microstates above the black hole threshold. The black hole
microstates, despite being long-lived, are unstable, and they need to be
included to give the correct answer. Given that the scattering amplitude
includes contributions from unstable particles as well, our estimate for
$\rho$ using the amplitude naturally captures the density of states of
sufficiently long-lived unstable particles. Note that in this work we are
interested in gravitational weak-coupling limits, $\Lambda\ll M_{\rm pl}$, in
which the gravitational backreaction of particles below the black hole
threshold (with fixed mass in units of $\Lambda$) becomes infinitely small. In
this limit, the gravitationally coupled particles become more and more stable
and increasingly relevant for any process. As for the issue of weakly coupled
bound states, our estimate for $\rho(E)$ captures the spectrum of such bound
states as well since they can also be produced as intermediate states in the
high-energy scattering process. Finally, regarding the issue of mass
averaging, by approximating $\rho(E)$ through scattering amplitudes our
estimate in (10) automatically provides a mass-averaged version of the density
of one-particle states.

###  3.2 Energy Dependence

Given our definition of the one-particle density of states we can now study
its dependence on the energy scale $E$. To that end it will prove useful to
consider the following ansatz

$\rho(E)\sim\exp\left[\left(\frac{E}{\Lambda}\right)^{\alpha}\right]\,.$ (11)

Here $\alpha$ can be viewed as an approximately piece-wise constant function
of energy. More precisely we are interested in the scaling of $\log\rho(E)$ up
to logarithmic corrections, i.e.

$\log\rho(E)=\widetilde{\mathcal{O}}\left[\left(\frac{E}{\Lambda}\right)^{\alpha}\right]\equiv\mathcal{O}\left[\left(\frac{E}{\Lambda}\right)^{\alpha}\cdot\log(E/\Lambda)^{k}\right]\,,$
(12)

where we have used that $\widetilde{\mathcal{O}}$ for a general function
$f(x)$ is given by

$\displaystyle\widetilde{\mathcal{O}}(f(x))=\mathcal{O}(f(x)\cdot\log(x)^{k})\,,$
(13)

for some positive constant $k$. Comparing with (10) we see that our
approximate definition of $\rho(E)$ in terms of scattering amplitudes is
indeed sufficient to capture this leading behavior. When considering particle
systems in the following sections it will be easier to first work with
$\Omega(E)$, the density of multiparticle states at energy $E$, instead of
$\rho$. Similar to $\rho$ we can make an ansatz

$\log\Omega(E)=\widetilde{\mathcal{O}}\left[\left(\frac{E}{\Lambda}\right)^{\alpha_{\Omega}}\right]\,.$
(14)

Again, $\alpha_{\Omega}$ is a piece-wise constant function of energy up to
logarithmic corrections. Note that, unlike for $\rho$, in order to work with
$\Omega$ we need to define an IR-cutoff to make the system finite. This IR-
cutoff regulates the infinity coming from the contribution of states obtained
by spatially translating a given single-particle state. In this work we
consider systems in a box and take this cutoff to be the size of the box.

Importantly, there is a useful connection between $\alpha_{\Omega}$ and
$\alpha$ which allows us to work with $\Omega$ instead of $\rho$ at many
points in this work: $\alpha_{\Omega}\geq 1$ implies $\alpha\geq 1$. In other
words, a sub-exponential growth of $\rho(E)$ cannot lead to an exponential
growth of $\Omega(E)$. Before giving a general proof for exponential growths
such as (12), it is instructive to study the case where $\rho$ grows
polynomially. This case is of particular interest given that KK states in the
presence of large extra dimensions have polynomial density. Suppose we have a
box of size $L$ with a gas of one-particle states with a density that behaves
as $\rho(E)\sim\left(E/E_{0}\right)^{p-1}$ (e.g. KK modes on a $p$-dimensional
compact manifold, with $E_{0}=m_{\rm KK}$). This system has a sub-exponential
growth of $\rho$ with energy. Note that the energy dependence of $\rho$ is the
same as that of the momentum states of a massless particle in a box with $p$
spatial dimensions of size $1/E_{0}$. We can use this equivalence to map our
computation of multi-particle states to a simpler higher-dimensional problem.
The number of multi-particle states in a $(d-1)$-dimensional box of size $L$
arising from a particle spectrum with $\rho(E)\sim\left(E/E_{0}\right)^{p-1}$
is equivalent to the number of microstates of radiation in a bigger,
$(p+d-1)$-dimensional, box consisting of the original $(d-1)$-dimensional box
of length $L$ and an additional $p$-dimensional box with sides of length
$1/E_{0}$. Note that in doing so we effectively traded a tower of states with
density $\rho(E)$ with $p$ extra compact dimensions that account for that
particle density in the $d$-dimensional theory. The entropy of radiation in
such a $(p+d-1)$-dimensional box is

$\displaystyle
S\sim\left(\frac{E}{E_{0}}\right)^{\frac{p+d-1}{p+d}}\left(E_{0}L\right)^{\frac{d-1}{p+d}}\,.$
(15)

Using $S\sim\log\Omega(E)$ we find

$\rho(E)\sim\left(\frac{E}{E_{0}}\right)^{p-1}\;\Rightarrow\quad\log\Omega(E)\sim\left(\frac{E}{E_{0}}\right)^{\frac{p+d-1}{p+d}}\,.$
(16)

Thus, a sub-exponential growth of $\rho(E)$ leads to
$\alpha_{\Omega}=(p+d-1)/(p+d)<1$.

The above argument does not yet exclude the possibility of an exponential
growth of $\Omega$ due to a one-particle density of states that is
superpolynomial but subexponential, i.e. $\rho\sim\exp(E^{\gamma})$ for
$0<\gamma<1$. We now give a more general argument that also addresses this
case.

Consider a particle system with total energy $E$ in a large volume $V$ such
that the system size is bigger than the size of any particle. Let $\bar{n}$ be
the average number of particles in the box. In the thermodynamic limit the
actual number of particles in the box for a generic state with energy $E$ will
deviate from $\bar{n}$ by

$\frac{\delta n}{\bar{n}}\sim\frac{1}{\sqrt{\bar{n}}}\,.$ (17)

We can now provide an upper bound for the entropy of the system at fixed
energy $E$. Since in the thermodynamic limit $\delta n\sim\sqrt{\bar{n}}$ the
majority of the entropy is accounted for by states with
$n\leq\bar{n}+\sqrt{\bar{n}}$ particles. For definiteness we therefore
consider states with $n\leq 2\bar{n}$ particles which will indeed account for
the majority of the entropy. On the other hand, the average energy of a
particle in the system is $\epsilon=\frac{E}{\bar{n}}$. Again, in the
thermodynamic limit the average energy of a particle in the box will deviate
from $\epsilon$ only by

$\delta\bar{\epsilon}\sim\frac{E}{\bar{n}^{2}}\,\delta{n}\sim\frac{\epsilon}{\sqrt{\bar{n}}}\,,$
(18)

where we used (17). An upper bound for the entropy is now given by considering
states with $n\leq 2\bar{n}$ particles such that the rest mass $m$ of each
particle satisfies $m\leq 2\epsilon$. In addition we allow the $n\leq
2\bar{n}$ particles to have any momentum inside the box. Since already the
rest mass can be of order $2\epsilon$ the total energy of most of these states
will exceed $E$. Therefore the number of the states we consider is much larger
than the microstates in the microcanonical ensemble at fixed $E$. Ignoring the
further overcounting from identical particles, we find the following upper
bound for the entropy

$\displaystyle
S<\log\left[\left(1+\int_{0}^{2\epsilon}dm\,\rho(m)\right)^{2\bar{n}}(V\Lambda^{d-1})^{2\bar{n}}\right]\lesssim
2\bar{n}\log\left[\left(1+\int_{0}^{2\epsilon}dm\,\rho(m)\right)\right]+\mathcal{O}(\bar{n})\,.$
(19)

Here $V\Lambda^{d-1}$ accounts for the number of choices for the momentum of a
single particle with a UV cutoff $\Lambda$ and IR-cutoff given by the size of
the box $V^{1/(d-1)}$. Moreover, $\int_{0}^{2\epsilon}dm\,\rho(m)$ represents
the number of particles with rest mass $m\leq 2\epsilon$ of which we choose
less than $2\bar{n}$.333This can also be viewed as counting the number of
lists with $2\bar{n}$ slots such that in each slot, there is either a particle
with mass less than $2\epsilon$, or no particle at all. The no-particle choice
for a given slot is accounted for by the additional $1$ in
$\left(1+\int_{0}^{2\epsilon}dm\,\rho(m)\right)$. If $\alpha_{\Omega}\geq 1$,
we have $\bar{n}\epsilon/\Lambda=E/\Lambda\lesssim S$ from which we infer

$\displaystyle\epsilon/\Lambda\lesssim\log\left[\left(1+\int_{0}^{2\epsilon}dm\,\rho(m)\right)\right]+\mathcal{O}(1)\,.$
(20)

Therefore, $\left(\int_{0}^{2\epsilon}dm\,\rho(m)\right)$ must have at least
an exponential growth in $\epsilon$ which implies that $\rho(m)$ also has at
least an exponential growth in its argument. We thus conclude

$\displaystyle\alpha_{\Omega}\geq 1\quad\Rightarrow\quad\alpha\geq 1\,.$ (21)

Note that since the number of multiparticle states $\Omega$ is greater than
the number of one-particle states $\rho$, we have $\alpha_{\Omega}\geq\alpha$.
Therefore, using (21) we find

$\displaystyle\alpha_{\Omega}=1\quad\Leftrightarrow\quad\alpha=1$ (22)

This result turns out to be crucial for our discussion in section 5.2. A well-
known example of a system that exhibits exponential growth for both, the
density of one-particle and the multi-particle states, with energy is given by
perturbative strings for which the exponential growth of the string
excitations translates into an exponential growth of $\Omega(E)$
Mitchell:1987hr ; Mitchell:1987th .

###  3.3 Connection to particle/black hole transition

Through their relation to the one-particle density of states in (10)
scattering amplitudes contain important information about the high-energy
regime of quantum gravitational theories. We now show that in fact they can be
used to extract information about the particle/black hole transition. More
precisely we show the following

###### Proposition 1

Consider a $d$-dimensional theory of gravity in asymptotically flat spacetime
with quantum gravity cutoff $\Lambda$. The states with mass $E$ are black hole
microstates if and only if at energy $E$ the density of one-particle states
scales as $\log\rho(E)\sim(E/\Lambda)^{\alpha}$ with $\alpha>1$.

In order to prove this Proposition we consider Schwarzschild black holes in
asymptotically flat $d$-dimensional spacetime. The entropy for these black
holes is given by

$S_{\rm BH,d}(E)=\log\rho(E)_{{\rm BH},d}\sim\left(\frac{E}{M_{\rm
pl,d}}\right)^{\frac{d-2}{d-3}}\>,$ (23)

where we have identified $E$ with the mass of the black hole. Comparing (23)
with (11), we get $\alpha=\frac{d-2}{d-3}>1$ for black holes. This
straightforwardly shows that for energies above the black hole threshold
$\alpha>1$. Note that in the argument above, we treated the black hole
microstates as one-particle states, which fits the definition of $\rho(E)$ in
section 3.1. This is because different microstates of a single black hole can
be viewed as different particles with a horizon (e.g. when their Schwarzschild
radius is greater than their Compton wavelength). This interpretation is
consistent with the computation of the entropy of supersymmetric black holes
via counting of BPS states Strominger:1996sh , which shows that the entropy of
black holes can microscopically be traced back to the large degeneracy of one-
particle states with fixed spin, charge, and mass.

To show the converse, i.e., $\alpha\leq 1$ implies that states have to be
particles, consider the free energy in the canonical ensemble at temperature
$T=1/\beta$ in a box of size $\Lambda_{\rm IR}^{-1}$, where $\Lambda_{\rm IR}$
is some IR-cutoff:

$\displaystyle e^{-\beta F}=\int_{\Lambda_{\rm IR}}{\rm
d}E\,\Omega(E)\exp(-\beta E)\sim\int_{\Lambda_{\rm IR}}{\rm
d}E\,\exp(\mathcal{O}(E^{\alpha_{\Omega}})-\beta E)\,.$ (24)

For any fixed total energy $E$ inside the box the gravitational backreaction
vanishes in the gravitational weak-coupling limit and the system is described
by particles. In other words, the particle description emerging in such limits
must make sense for arbitrarily high total energies which implies that the
unbounded partition function must converge. However, the convergence of the
integral in (24) requires $\alpha_{\Omega}\leq 1$. As we explained in section
3.2, $\alpha_{\Omega}\leq 1$ implies $\alpha\leq 1$ which concludes our proof
of Proposition 1.

Using this result the particle/black hole transition can be read off from the
scattering amplitudes in the following way. The scattering amplitudes are
related to the density of one-particle states via (10). Therefore if we track
the scattering amplitude as a function of energy we obtain an approximate form
of $\rho(E)$ as in (11). As a consequence of Proposition 1 the energy at which
the particle/black hole transition takes place corresponds to the energy at
which $\alpha$ in (11) switches from $\alpha\leq 1$ to $\alpha>1$. Therefore
knowledge of the scattering amplitudes as a function of $E$ allows us to read
off the energy scale $M_{\rm min}$ at which the particle/black hole transition
takes place. The energy $M_{\rm min}$ corresponds to the minimal mass of any
black hole in the theory. Importantly, as we discuss in more detail in section
5.3, in the presence of compact extra dimensions this minimal mass black hole
is in fact a higher-dimensional black hole.

##  4 Species Scale and Gravitational Amplitudes

In this section we discuss the species scale, i.e. the scale suppressing the
higher-derivative corrections to the gravitational effective action, from the
perspective of gravitational amplitudes. In section 4.1 we first argue that
the scale appearing in the higher-derivative corrections is indeed the cutoff
of the gravitational EFT defined as the inverse radius of the smallest black
hole in the theory. We do this by matching the $2\to 2$ scattering amplitude
of gravitons in a certain regime with the expansion of the effective action.
In section 4.2 we further use the contribution to gravitational amplitudes
from effective strings to constrain the tension of weakly coupled $p$-branes
with $p\geq 1$ in theories of quantum gravity in terms of the species scale.

Before we discuss the above points in detail, let us pause and clarify what we
mean by effective field theory in the following. To define an effective field
theory one typically fixes a spacetime dimension, the field content of the
theory and the symmetries of the theory. In addition one specifies a cutoff
$\Lambda_{\rm EFT}$ determining the regime of validity of the EFT description.
For example we can consider a $d$-dimensional theory of Einstein gravity
coupled to a set of light fields exhibiting certain symmetries. This theory
has a cutoff $\Lambda_{\rm EFT}$ beyond which the field theory description
breaks down. This breakdown can, for example, be due to the appearance of a
tower of states that cannot consistently be described in terms of an EFT. The
classical example for such a situation is a $d$-dimensional theory that arises
from a higher-dimensional theory. For energies above the KK scale, KK modes
need to be included in the EFT. This is however not possible in a
parametrically controlled way, since e.g. for scattering process with two
initial particles of mass $\Lambda_{\text{KK}}=1/R$, the amplitude becomes
non-unitary and one must include the second KK particle of mass $2/R$, even at
zero initial momentum.444Further, even if we were to include a finite number
of KK modes, they become strongly coupled at the scale $\Lambda_{\text{QG}}$,
so that the EFT cutoff cannot be made higher than $\Lambda_{\text{QG}}$.
However, this does not mean that there does not exist any EFT description at
energies above $\Lambda_{\rm KK}$ since we can describe the theory in terms of
the higher-dimensional EFT. Of course the field content and the symmetries of
this EFT are different from the original $d$-dimensional theory but this new
EFT allows us to describe the infinitely many KK-states in terms of finitely
many (higher-dimensional) fields. The cutoff of this new theory is expected to
be above $\Lambda_{\rm KK}$ but the quantum gravity cutoff $\Lambda$, defined
in terms of the higher-derivative corrections, does not change under
dimensional reduction. Hence even though the original $d$-dimensional
gravitational EFT is not valid above $\Lambda_{\rm KK}$ there may exist _some_
EFT description that is valid up to $\Lambda$ which – in the case of a KK-
reduction – is necessarily higher-dimensional. It is in this sense that in
this section we show that, indeed, there always exists some EFT description
with cutoff $\Lambda_{\rm EFT}=\Lambda_{\rm QG}$ even though this EFT may
significantly differ from the original $d$-dimensional EFT.

###  4.1 Existence of a (higher-dimensional) EFT with cutoff $\Lambda$

Consider a theory of gravity at some energy $E>M_{\rm min}$ where $M_{\rm
min}$ is the energy scale at which the scaling of the density of one-particle
states in (11) changes from $\alpha\leq 1$ to $\alpha>1$. From Proposition 1
we recall that for energies $E>M_{\rm min}$ the one-particle states are
gravitationally strongly coupled states whose mass is dominated by their
gravitational self-energy. In other words these states correspond to black
holes. In Lorentzian signature such states are hence expected to be
describable as solutions of a Lorentzian gravitational field theory with a
horizon. In order for the gravitational field theory description to be
applicable, the curvature of the horizon furthermore needs to be smaller than
the cutoff of the field theory. Consequentially, there needs to exist a field
theory with cutoff $\Lambda_{\rm min}$ such that all states with mass
$M>M_{\rm min}$ are describable as field theory solutions for which the
curvature at the horizon does not exceed $\Lambda_{\rm min}^{2}$. This implies
that the horizon radius of the smallest black hole in the theory should not be
smaller than $\Lambda_{\rm min}^{-1}$. As mentioned at the beginning of this
section let us stress again that if we start with a $d$-dimensional EFT, the
actual EFT with cutoff $\Lambda_{\min}$ can be higher-dimensional. In this
case the black hole with minimal mass $M_{\rm min}$ is a black hole of this
higher-dimensional EFT and is localized in the extra dimension.

On the other hand, as reviewed in section 2, the quantum gravity cutoff, or
the species scale, $\Lambda$, is defined as the scale that suppresses the
higher-derivative terms in the effective gravitational action (5). Based on
EFT arguments, one expects the higher derivative corrections at cut-off scale
$\Lambda_{\min}$ to be controlled by the same energy scale
($\Lambda\sim\Lambda_{\min}$). In the following, we give an independent
argument based on scattering amplitudes and the properties of black hole
physics that makes this connection more clear.

###### Proposition 2

Consider a theory of quantum gravity in which the smallest black hole (which
could be higher-dimensional) has radius $\Lambda_{\rm min}^{-1}$ and mass
$M_{\rm min}$. Then the higher-derivative corrections to the Einstein–Hilbert
action are of the form

$\mathcal{L}_{\rm corr}\supset\frac{M_{\rm pl,d}^{d-2}}{\Lambda_{\rm
min}^{2n}}\mathcal{O}_{2n+2}(\mathcal{R})\,,$ (25)

where $\mathcal{O}_{m}(\mathcal{R})$ is a dimension-$m$ operator involving
$\mathcal{R}$ and its derivatives.

Let us make an important clarifying remark. As pointed out in Bedroya:2024uva
, the species scale $\Lambda$, and the inverse radius of the smallest black
hole described by a given EFT can be vastly different. However, here we allow
for black hole solutions to any effective action that is valid at low
energies. In particular this includes higher-dimensional black holes that are
localized in possible extra dimensions.

Given the definition of the quantum gravity cutoff $\Lambda$ in terms of the
higher-derivative corrections in section 2, the main takeaway of Proposition 2
is that there indeed exists some effective field theory description up to
energies of order of the quantum gravity cutoff $\Lambda$. Let us stress that
the statement of Proposition 2 has previously appeared in the literature.
However, as we will discuss in the following, there are some subtleties in the
justification of the identification between $\Lambda$ and $\Lambda_{\rm min}$.
Typically one argues in the following way vandeHeisteeg:2023dlw ;
vandeHeisteeg:2023ubh : suppose there is a perturbative series of higher-
derivative corrections, each proportional to $M_{\rm
pl,d}^{d-2}\Lambda^{-2n}\mathcal{R}\Box^{n-1}\mathcal{R}$ or $M_{\rm
pl,d}^{d-2}\Lambda^{-2n}\mathcal{R}^{n+1}$, then the effective action
containing this series diverges for $\mathcal{R}\gg\Lambda^{2}$. Since
$\Lambda_{\rm min}^{2}$ is the curvature scale at the horizon of the smallest
black hole that can be described in effective field theory, one expects the
breakdown of the higher-derivative expansion to occur for curvatures of the
order of $\Lambda_{\rm min}^{2}$ which would imply $\Lambda=\Lambda_{\rm
min}$. This reasoning is, however, flawed for two reasons:

  1. 1.

The same argument could be applied to non-gravitational field theories with a
series of irrelevant operators to argue for an ultimate cutoff for the EFT
description. However, that conclusion would be incorrect. For example, the
effective action of quantum electrodynamics at energies below the electron
mass $m_{e}$ has irrelevant operators $\propto(m_{e})^{-4n+d}(F^{2})^{n}$.
Nevertheless in this case the breakdown of the EFT at energies $\sim m_{e}$ is
attributed to the fact that additional massive modes which were previously
ignored (in this case the electron) need to be integrated in. In this way, in
field theory the cutoff can be raised by integrating in additional states. The
same is, however, not possible in theories of gravity where, e.g., black hole
microstates are expected to be strongly coupled and cannot be simply
integrated into the EFT. This implies that the quantum gravity cutoff is
qualitatively different from some field theory cutoffs.

  2. 2.

The effective action does not necessarily diverge at curvature scales
$\mathcal{R}\gg\Lambda^{2}$ as one might have expected. For example, consider
the higher-derivative term

$\mathcal{L}_{\rm corr}\supset\hat{a}_{n}\frac{M_{\rm
pl,d}^{d-2}}{\Lambda^{2n}}\mathcal{R}\Box^{n-1}\mathcal{R}\,,$ (26)

where we have included a constant $\hat{a}_{n}$ here that accounts for a
possible $n$-dependence of the Wilson coefficient of the above term and does
not include any dependence on the scalar fields in the theory. Near the
horizon of a black hole with radius $r_{H}$, we have
$\mathcal{R}\Box^{n-1}\mathcal{R}\propto r_{H}^{-2n-2}$. One may expect that
the sum of such terms diverges for $r_{H}\ll\Lambda^{-1}$. However, as we will
shortly see, the coefficients $\hat{a}_{n}$ are exponentially small in $n$
which can make the series convergent. This is due to the fact that
gravitational $2\rightarrow 2$ scattering amplitudes are always exponentially
suppressed at high energies and the higher-derivative terms are just a Taylor
expansion of such amplitudes in energy, see Appendix C. The higher-derivative
expansion of the gravitational effective action hence does not necessarily
diverge at curvature scales $\mathcal{O}(\Lambda^{2})$.

Having explained the shortcomings of the usual argument we need to work a bit
harder in order to establish $\Lambda=\Lambda_{\rm min}$ and argue for
Proposition 2 correctly. To that end we make use of certain properties of
gravitational amplitudes which we summarize in Appendix C. In short, we show
that the imprint of the smallest black hole on high-energy $2\rightarrow 2$
gravitational amplitude can be reproduced only by higher-derivative
corrections of the form

$\displaystyle\mathcal{L}_{\rm corr}\supset\hat{a}_{n}\frac{M_{\rm
pl,d}^{d-2}}{\Lambda_{\min}^{2n}}\mathcal{R}\Box^{n-1}\mathcal{R}\,,$ (27)

which, by comparison to (26) implies $\Lambda_{\rm min}=\Lambda$.

To argue for (27) we consider $2\to 2$ scattering amplitudes for gravitons. At
center of mass energies $E\gg M_{\text{min}}$ and impact parameter $b\ll
r_{\text{min}}=1/\Lambda_{\rm min}$ the scattering process involves a black
hole formation/evaporation process. The exclusive $2\rightarrow 2$ amplitude
corresponds to the case where the Hawking radiation consists of only two
particles. Therefore, the amplitude is inversely proportional to the number of
black hole microstates and is exponentially suppressed in the black hole
entropy. Instead of the impact parameter $b$ we are interested in the
dependence of the amplitude in terms of the Mandelstam variables $s,t$ defined
in (119). To that end we need to perform a Fourier transform on the impact
parameter $b$. At fixed angle and large energies, the contribution of black
holes to the scattering amplitudes is exponentially suppressed and the
dominant contribution comes from particles whose separation is much larger
than the Schwarzschild radius, $r_{s}(E)$, of a black hole with mass $E$.
Nonetheless, as explained in appendix C.3, the exponential suppression at low-
impact parameters still leaves a phase factor in the amplitude which becomes
exponentially large upon analytic continuation to the unphysical regime $t\gg
0$ (cf. the discussion around (132))

$\mathcal{A}\sim\exp\left(2r_{s}(E)\sqrt{t}\right)\,,\qquad\text{for}\quad
t\gg 0\,.$ (28)

As we detail in appendix C.3, we expect the above behavior to persist at
energies $E\ll M_{\min}$. However, $r_{s}$ will be replaced by
$b_{c}(E)=\Lambda_{\min}^{-1}\mathcal{O}(E/\Lambda_{\min})$.

$\displaystyle\mathcal{A}\sim\exp\left(\Lambda_{\rm
min}^{-1}\sqrt{t}\times\mathcal{O}(\log E/\Lambda_{\rm min})\right)\,,$ (29)

To reach this conclusion we use that Regge behavior dictates that the energy
dependence of $b_{c}(E)$ at energies $E\ll M_{\rm min}$ can be at most
logarithmic and that the amplitude needs to correctly match the black hole
result (28) at the black hole threshold $E\sim M_{\min}$.

Since black holes are saddles of the Euclidean gravitational action they can
be thought of as non-perturbative contributions to gravitational amplitudes.
Nevertheless at energies below $\Lambda_{\rm min}$ the amplitude has to be
computable perturbatively from the full effective action including the higher-
derivative corrections. The amplitude computed from the full effective action
has to match the high-energy amplitude at
$\sqrt{s}\sim\mathcal{O}(\Lambda_{\rm min})$ even in the unphysical regime for
$t$. To reproduce the exponential amplitudes of the form (29) we need an
infinite series of higher-derivative terms. For the particular case of the
$2\to 2$ scattering amplitude, we hence require terms in the effective action
of the form

$\mathcal{L}_{\rm eff}\supset M_{\rm pl}^{d-2}\frac{\hat{a}_{n}}{\Lambda_{\rm
min}^{2n}}\mathcal{R}\Box^{n-1}\mathcal{R}\,,$ (30)

in order to reproduce the amplitude in (29). As before, the $\hat{a}_{n}$
encode the $n$-dependence of the coefficients of the higher-derivative terms.
In order to make sure that the high-derivative terms correctly reproduce the
Taylor expansion of (29) we need to have

$\hat{a}_{n}\sim\frac{1}{(2n)!(n-1)!}\,.$ (31)

The $2n!$ comes from the Taylor expansion of the exponential in (29) and the
$(n-1)!$ is to counter the repetition resulted from having $n-1$ identical
d’Alembertians in (30). From (30) we conclude that the higher-derivative terms
are controlled by $\Lambda_{\rm min}$. However, as reviewed in section 2, the
energy scale controlling the higher-derivative expansions was defined to be
the species scale or quantum gravity cutoff $\Lambda$ vandeHeisteeg:2022btw .
Therefore, using amplitudes we showed that $\Lambda=\Lambda_{\rm min}$ and
hence Proposition 2. Note that compared to the definition (7), here we fixed
the overall $n$-dependence of the Wilson coefficients $a_{n}(\phi)$ such that
the scale appearing in the individual higher-derivative terms in (30) is
indeed always the species scale.

Let us further note that, as advertised above, contrary to the field theory
expectation the sum over the terms in (30) does not diverge at curvature
scales $\mathcal{R}\sim\Lambda^{2}$ since the $n$-dependence of the
coefficients $\hat{a}_{n}$ ensures that the radius of convergence of the
higher-derivative expansion in (30) is larger than
$\mathcal{R}\sim\Lambda^{2}$. In the context of string theory the suppression
in $n$ for instance ensures that string amplitudes are analytic at high
energies.

###  4.2 Bound on the tension of weakly coupled $p$-branes
$\mathcal{T}\gtrsim\Lambda^{p+1}$

In section 2 we reviewed the definition of the species scale $\Lambda$ in
terms of the Wilson coefficients of the higher-derivative corrections to the
Einstein–Hilbert action. In this section, we use this definition for $\Lambda$
to bound the tension of all weakly coupled $p$-branes with $p\geq 1$ as

###### Proposition 3

In an effective theory of gravity, the tension $\mathcal{T}$ of any weakly
coupled $p$-brane with $p\geq 1$ is bounded by the quantum gravity cutoff
$\Lambda$ as

$\mathcal{T}\gtrsim\Lambda^{p+1}\,.$ (32)

In the following we always implicitly assume $p\geq 1$. In this work, we adapt
the following definition for a weakly coupled brane: A weakly coupled brane
has a self-energy that is negligible compared to its tension. For example, for
a $p$-brane coupled to a $(p+1)$-form gauge potential, this condition implies
that the gauge coupling in the units of the tension of the brane is
negligible. To be more precise, let us consider such a $p$-brane coupled to a
$(p+1)$-form gauge potential $\mathcal{A}_{p+1}$ for which the kinetic term in
the $d$-dimensional bulk action reads

$S_{p}=-\frac{1}{2g_{p+1}^{2}}\int\mathcal{F}_{p+2}\wedge*\mathcal{F}_{p+2}\,,$
(33)

where $\mathcal{F}_{p+2}$ is the field strength of $\mathcal{A}_{p+1}$. The
gauge coupling $g_{p+1}$ has mass dimension $p+2-d/2$. A typical configuration
of the gauge potential on the $p$-brane is of the form

$\mathcal{A}_{p+1}\sim\frac{g_{p+1}^{2}}{L^{d-p-3}}\,,$ (34)

where $L$ is the typical length scale of the $p$-brane. The self-energy
associated to the coupling of the $p$-brane to $\mathcal{A}_{p+1}$ can now be
estimated as

$\delta E\propto\mathcal{A}_{p+1}L^{p}\sim\frac{g_{p+1}^{2}}{L^{d-2p-3}}\,.$
(35)

The requirement that the self-energy induced by the coupling of the $p$-brane
to $\mathcal{A}_{p+1}$ is negligible compared to the tension then amounts to

$\delta
E<\mathcal{T}L^{p}\qquad\Rightarrow\qquad\frac{g_{p+1}^{2}}{L^{d-p-3}}<\mathcal{T}\,.$
(36)

Given our definition of weakly coupled $p$-branes we start by considering the
case of weakly coupled strings with tension $\mathcal{T}$. It is instructive
to review why the inequality $\mathcal{T}\gtrsim\Lambda^{2}$ holds in the
string landscape. The effective action of any perturbative string theory, at
tree-level, contains a series of higher-derivative corrections

$\displaystyle S_{\rm tree}=\frac{M_{\rm pl,d}^{d-2}}{2}\int{\rm
d}^{d}x\sqrt{-g}\left(\mathcal{R}+\sum_{n=4}^{\infty}\hat{a}_{n}\frac{\mathcal{O}_{n}(\mathcal{R})}{\mathcal{T}^{\frac{n-2}{2}}}\right)+\ldots\,,$
(37)

where $\hat{a}_{n}$ are dimensionless constants that do not depend on any
scalar fields.555Any dependence of the Wilson coefficients of the higher-
derivative terms on the scalar fields in the theory will be encoded in
$\mathcal{T}$ directly. This expansion is known as the
$\alpha^{\prime}$-expansion in string theory. By matching the above action
with the expression (26), we can identify the species scale as
$\Lambda\sim\sqrt{\mathcal{T}}$ in any perturbative string theory. Note that
the effective action in string theory also contains a sub-leading series of
loop corrections which are suppressed by positive powers of the string
coupling $g_{s}$. Therefore, in the weak-coupling limit, $g_{s}\ll 1$, the
effect of the loop expansion on $\Lambda$ is negligible.

Let us revisit how the tree-level string amplitude gives rise to the
perturbative series in the effective action (37). We first identify the
imprint of higher-derivative corrections on high-energy $2\rightarrow 2$
amplitudes from which we read off the corresponding Wilson coefficients.
Consider for example a term of the form $\frac{M_{\rm
pl,d}^{d-2}}{M^{2k}}\mathcal{R}\Box^{k-1}\mathcal{R}$ in the effective action
where $M$ is some mass scale. We expand the spacetime metric $g_{\mu\nu}$
around Minkowski background $\eta_{\mu\nu}$ as

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+\frac{h_{\mu\nu}}{M_{\rm
pl,d}^{\frac{d-2}{2}}},$ (38)

where the normalization of $h$ ensures that it has a canonical kinetic term.
The higher-derivative term gives rise to a four-point interactions of the form

$\mathcal{R}\Box^{k-1}\mathcal{R}\;\longrightarrow\;M_{\rm
pl,d}^{4-2d}\partial_{\gamma}h_{\alpha\beta}\partial^{\beta}h^{\alpha\gamma}\Box^{k-1}\partial_{\gamma^{\prime}}h_{\alpha^{\prime}\beta^{\prime}}\partial^{\beta^{\prime}}h^{\alpha^{\prime}\gamma^{\prime}}$
(39)

which has $2k+2$ derivatives and for which the coefficient in the effective
action is proportional to $M_{\rm pl,d}^{d-2}/M^{2k}$. Therefore, such a term
contributes to the fixed-angle $2\rightarrow 2$ graviton amplitude in the
following way

$\displaystyle\frac{M_{\rm
pl,d}^{d-2}}{M^{2k}}\mathcal{R}\Box^{k-1}\mathcal{R}\in\mathcal{L}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{E^{2k+2}}{M^{2k}M_{\rm
pl,d}^{d-2}}f(\theta)\in\mathcal{A}(E,\theta),$ (40)

where $E$ is the center of mass energy and $\theta$ is the scattering angle.
We have neglected the polarization-dependence for the sake of simplicity.
Knowing how each term contributes to the gravitational amplitudes, allows us
to read off the Wilson coefficient $M_{\rm pl,d}^{d-2}/M^{2k}$ from the
amplitude in string theory. In string theory, the $2\rightarrow 2$ amplitude
at tree-level is given by the high-energy limit of the Virasoro-Shapiro
amplitude Polchinski:1998rq

$\mathcal{A}(E,\theta)\sim\frac{\mathcal{T}}{M_{\rm
pl,d}^{d-2}}\,\exp\left[-\frac{1}{16\pi}\left(\frac{s/\mathcal{T}}{\log(s/\mathcal{T})}+\frac{t/\mathcal{T}}{\log(t/\mathcal{T})}+\frac{u/\mathcal{T}}{\log(u/\mathcal{T})}\right)\right].$
(41)

where $(s,t,u)$ are the Mandelstam variables and $\mathcal{T}$ is the string
tension. Note that, if we go to energies much higher than $\sqrt{\mathcal{T}}$
the effect of higher-genus contributions will become dominant and change the
exponential behavior of the amplitude from $\exp(cE^{2}/\log(E))$ to
$\exp(-\mathcal{O}(E))$ (see appendix B). However, our conclusion about the
perturbative expansion of the amplitude in terms of $E/\sqrt{\mathcal{T}}$
will not be affected by the higher-genus corrections. The contribution in (41)
comes from the sphere amplitude which can also be understood as a saddle of
the Polyakov action ($\mathcal{A}\propto\exp(S_{\text{saddle}})$) Gross:1987ar
. If one expands the above amplitude in terms of the contribution in (40) of
local terms to the full amplitude one finds

$\mathcal{A}(E,\theta)=\sum_{k\geq
1}f_{k}(\theta)\frac{E^{2k+2}}{\mathcal{T}^{k}M_{\rm pl,d}^{d-2}}\,,$ (42)

where we absorbed the $k$-dependend expansion coefficients into
$f_{k}(\theta)$. Therefore, the Wilson coefficient of
$\mathcal{R}\Box^{k-1}\mathcal{R}$ in any perturbative string theory is
proportional to $(M_{\rm pl,d}^{d-2}/\mathcal{T}^{k})$. The above argument
shows how the sphere amplitude ensures the presence of higher-order
corrections of the type $\frac{M_{\rm
pl,d}^{d-2}}{\mathcal{T}^{k}}\mathcal{R}\Box^{k-1}\mathcal{R}$ consistent with
(37). Therefore in critical string theory the species scale is indeed set by
$\Lambda\sim\sqrt{\mathcal{T}}$.

Having reviewed how the tension of a critical string is related to the species
scale, let us revisit the above argument and understand to what extent the
argument depends on the criticality of the string. There are three assumptions
underlying the argument which are met by the critical string but not
necessarily true for a generic string:

  1. 1.

The graviton is a string state and therefore the gravitational amplitude is
given by a string amplitude.

  2. 2.

The action of the string is known exactly and does not receive corrections
based on the embedding of the string worldsheet in the spacetime.

  3. 3.

The tree-level amplitude (i.e. sphere diagram) is the dominant contribution.

In the following, we argue that one can reproduce the same argument for any
_weakly coupled_ string. More precisely, we show that any weakly coupled
string with tension $\mathcal{T}$ will generate higher-derivative corrections

$S_{\rm corr}\propto\int d^{d}x\sqrt{g}\frac{M_{\rm
pl,d}^{d-2}}{\mathcal{T}^{k}}\mathcal{R}\Box^{k-1}\mathcal{R}\,.$ (43)

As reviewed in section 2, the overall magnitude of the Wilson coefficients of
the higher-derivative terms sets the species scale. Therefore the term in (43)
has to be at least suppressed by the species scale leading to the bound

$\displaystyle\frac{1}{\Lambda^{2k}}\gtrsim\frac{1}{\mathcal{T}^{k}}\;\Rightarrow\;\mathcal{T}\gtrsim\Lambda^{2}\,,$
(44)

which is (32) for the case of strings, i.e. $p=1$. Let us stress that this
bound is consistent with the analysis in Cota:2022yjw  where it was found that
in F-theory compactification on elliptically fibered Calabi–Yau fourfolds the
tension of any weakly coupled string, dubbed EFT strings in Lanza:2021udy , is
bounded from below by the species scale in asymptotic limits; see also
Martucci:2024trp  for a discussion of the relation between species scale and
EFT string tensions in such setups.

Let us start with the definition of a weakly coupled string (we refer the
reader to appendix A for some details on general effective strings). A weakly
coupled string is a string that has a Nambu–Goto action with corrections that
are suppressed as long as the extrinsic curvatures $L^{-2}$ of the worldsheet
are smaller than $\mathcal{T}$. In other words the coupling of a weakly
coupled string to itself via any gauge field (e.g. graviton or a two-form) can
be ignored as long as the string is in a sufficiently flat, or long wave-
length, configuration.666For example for the QCD string, this assumption is
violated close to the confinement scale where the string self-intersects and
becomes highly curved. Moreover, the effective action of the QCD string is
expected to depend strongly on its extrinsic curvature Gross:1992tu . In such
a limit, the physics of the string is described by the IR limit of the
worldsheet theory. Therefore a weakly coupled string is a string that, in
configurations with curvature smaller than its tension, is described in the IR
by a two-dimensional CFT. Since we expect the string action to contain a
Nambu–Goto term, it is crucial to understand the two-dimensional CFT
corresponding to the Nambu–Goto action for non-critical strings.

For the pure Nambu–Goto action this CFT was worked out in Polchinski:1991ax ,
by adding a term to the Polyakov action that cancels the conformal anomaly. In
the light-cone gauge the action is given by

$\displaystyle S=\int{\rm d}\sigma^{+}{\rm
d}\sigma^{-}\left[\mathcal{T}\partial_{+}X^{\mu}\partial_{-}X_{\mu}+\frac{d-26}{12}\frac{(\partial_{+}^{2}X^{\mu}\partial_{-}X_{\mu})(\partial_{-}^{2}X^{\nu}\partial_{+}X_{\nu})}{(\partial_{+}X^{\rho}\partial_{-}X_{\rho})^{2}}+\mathcal{O}(L^{2}/\mathcal{T})\right]\,.$

For a mildly curved string ($L\sim\partial_{\pm}X^{\mu}\ll\mathcal{T}^{-1/2}$)
the second term is also negligible. We are interested in the contribution of
the weakly coupled effective strings to the gravitational scattering
amplitudes. For the fundamental strings in string theory, the high energy
behavior can be deduced from the worldsheet saddle with the dominant
contribution.

In the following, we apply the same analysis to the effective string in the
regime of its validity.777Our considerations do not apply to the weakly
coupled string description of large N QCD, since the strings are not
fundamental and are resolved into the constituent field theory states above
the confinement scale. For the effective description to be valid, we focus on
worldsheet saddles with weak extrinsic curvatures
$L\sim\partial_{\pm}X^{\mu}\ll\mathcal{T}^{-1/2}$ for which the Polyakov
action is a good classical approximation. Some saddles of the Polyakov action,
which are $N$-covering of a sphere with punctures, were studied by Gross and
Mende Gross:1987ar , see appendix B for a review. The tree-level saddle is
given by the following embedding

$X^{\mu}=\sum_{j=1}^{4}\frac{ip_{j}^{\mu}}{4\pi
N\mathcal{T}}\log|z-z_{j}|+\mathcal{O}\left(\frac{\sqrt{\mathcal{T}}}{s}\right)\,,$
(45)

where $p_{j}$ are the momenta of ingoing/outgoing particles and $z$ is a
complex coordinate parametrizing a complex sphere with a point removed and for
which the worldsheet metric has been Weyl rescaled to be flat. Moreover, the
locations $z_{j}$ of the external legs on the sphere satisfy the following
relation in the saddle configuration

$\displaystyle\frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{1}-z_{2})(z_{3}-z_{4})}\simeq\frac{-t}{s}\,.$
(46)

As explained in appendix B, there is a range of $N$ for which we have

$L\gg\frac{1}{\sqrt{\mathcal{T}}}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ S_{\rm
Saddle}\sim\frac{E}{\sqrt{\mathcal{T}}}\widetilde{\mathcal{O}}\left(L\sqrt{\mathcal{T}}\right)\,,$
(47)

where $S_{\rm Saddle}$ is the action of the saddle. For these saddles, the
external legs of the worldsheet are string states (see figure -479). However,
since we do not assume that the graviton is a string state, we need to cut the
saddle and add a graviton vertex according to its interaction through the
Nambu–Goto term

$\int{\rm d}\sigma^{+}{\rm d}\sigma^{-}\frac{\mathcal{T}}{M_{\rm
pl,d}^{\frac{d-2}{2}}}h_{\mu\nu}\partial_{+}X^{\mu}\partial_{-}X^{\nu}\,,$
(48)

where we used the normalization in (38). The above term gives an interaction
vertex

$V_{\rm graviton-worldsheet}\sim\frac{\mathcal{T}L^{2}}{M_{\rm
pl,d}^{\frac{d-2}{2}}}\,.$ (49)

Since we need to insert four such vertices, we find that the contribution of
the worldsheet saddle to the gravitational amplitude is

$\displaystyle\mathcal{A}_{\rm string}\sim V_{\rm graviton-
worldsheet}^{4}\mathcal{T}^{\frac{d}{2}}e^{-S_{\rm saddle}}\,,$ (50)

where the factor $\mathcal{T}^{\frac{d}{2}}$ comes from the integration
measure of the worldsheet action.888For example, if we were to calculate the
contribution of the worldsheet saddle to the cosmological constant, it would
go like $\mathcal{T}^{\frac{d}{2}}e^{-S_{\rm saddle}}$. Plugging the saddle
action (47) and the vertex amplitude (49) into the above expression for
$\mathcal{A}_{\rm string}$ leads to

$\displaystyle\mathcal{A}_{\rm
string}\sim\frac{\mathcal{T}^{\frac{d+8}{2}}L^{8}}{M_{\rm
pl,d}^{2(d-2)}}\exp\left[-\frac{E}{\sqrt{\mathcal{T}}}\widetilde{\mathcal{O}}\left(L\sqrt{\mathcal{T}}\right)\right]\,.$
(51)

$L\sim E/\mathcal{T}$$L\sim E/\mathcal{T}$ Figure -479: Left: The Gross-Mende
saddle with external legs that extend to infinity. Right: The external legs
are cut-off and a graviton vertex is added to regulate the extrinsic curvature
of the saddle and study the gravitational amplitude.

Notice that the above calculation can only be trusted if the saddle is mildly
curved ($L\sqrt{\mathcal{T}}\gg 1$). Therefore, by decreasing $L$ to $\sim
1/\sqrt{\mathcal{T}}$ the calculation starts breaking down and we can estimate
the largest trustable contribution to the high-energy gravitational amplitude
to satisfy999We write an inequality because there could be contributions from
other channels to the gravitational amplitude.

$\displaystyle\log\left(|\mathcal{A}|\right)\gtrsim-\widetilde{\mathcal{O}}\left(\frac{E}{\sqrt{\mathcal{T}}}\right)\,.$
(52)

A Taylor expansion of the exponential amplitude now proceeds in analogy to the
discussion below (41) such that we can conclude that any weakly coupled string
with tension $\mathcal{T}$is expected to give rise to a higher-derivative
correction as in (43). Comparing to the prediction of the species scale we
obtain the bound (32) for $p=1$.

Let us now turn to higher-dimensional weakly coupled branes. Similar to weakly
coupled strings we consider a brane to be weakly coupled if the extrinsic
curvature of the brane configuration is small compared to the tension. In this
case the higher-derivative corrections to the worldvolume theory on the brane
are suppressed and we can trust the worldvolume theory. Consider such a weakly
coupled $p$-brane with co-dimension $d-p-1\geq 1$ and tension $\mathcal{T}$.
To show $\mathcal{T}\gtrsim\Lambda^{p+1}$ we proceed via contradiction.
Therefore suppose there exists $\epsilon\ll 1$ such that
$\mathcal{T}=\epsilon^{p+1}\Lambda^{p+1}\ll\Lambda^{p+1}$. Then, one can
compactify the theory on a $(p-1)$-dimensional torus of radius $R$ such that

$R^{p-1}=\frac{1}{\epsilon^{p}\Lambda^{p-1}}\,.$ (53)

Since $\epsilon\ll 1$ the radius of the compactification $R\gg\Lambda^{-1}$
such that the compactification is within the regime of validity of the target
space effective field theory which has a cutoff $\Lambda$. We can now wrap the
brane on this $(p-1)$-torus to obtain an effective string in the extended
directions. Notice that the extrinsic curvature of this brane configuration is
set by the inverse radius of the torus. On the other hand we have

$\mathcal{T}^{\frac{1}{p+1}}=\epsilon\Lambda\gg\epsilon^{\frac{p}{p-1}}\Lambda=R^{-1}\,,\quad\text{for}\quad
p<\infty$ (54)

such that the extrinsic curvature of this brane configuration is small
compared to the brane tension. Hence we have a weakly coupled brane that
yields a weakly coupled string upon dimensional reduction on the
$(p-1)$-torus. The tension $\widetilde{\mathcal{T}}$ of this string is given
by

$\widetilde{\mathcal{T}}=\mathcal{T}R^{p-1}=\epsilon^{p+1}\epsilon^{-p}\Lambda^{2}=\epsilon\Lambda^{2}\,.$
(55)

Since $\epsilon\ll 1$ we hence get a weakly coupled string with tension
$\widetilde{\mathcal{T}}\ll\Lambda^{2}$ in contradiction to our previous
result. Therefore, our assumption $\mathcal{T}\ll\Lambda^{p+1}$ was incorrect
and we find that any weakly coupled $p$-brane satisfies (32) which proves
Proposition 3.

Note that our argument for the Proposition 3 is an example of UV/IR connection
in quantum gravity. The branes with a mass scale much smaller than the Planck
mass can form well-controlled saddles in processes that involve energies much
higher than their mass scale and therefore contribute to the high energy
gravitational amplitudes. Moreover, as explained below the equation (114), a
saddle of a given topology becomes less curved and more trustable at higher
energies. In this way, these saddles are similar to black holes which are
under better perturbative control in the semiclassical gravity when they are
heavier. This is due to the fact that the heavier black holes have smaller
horizon curvatures.

Let us close by considering the case of D$p$-branes in critical string theory.
The tension $\mathcal{T}_{p}$ of these branes and the coupling $g_{p+1}$ to
the RR-form $C_{p+1}$ are given in terms of the string scale $M_{s}$ and the
string coupling $g_{s}$ as

$\mathcal{T}_{p}\sim\frac{M_{s}^{p+1}}{g_{s}}\,,\qquad g_{p+1}\sim
g_{s}M_{s}^{p+2-d/2}\,.$ (56)

Using these expression we find that for a D$p$-brane with
$L^{p+1}\sim\mathcal{T}_{p}$ the constraint (36) implies

$g_{s}M_{s}^{p+4-d/2}<\left(\frac{M_{s}^{p+1}}{g_{s}}\right)^{1+\frac{2-d}{2(p+1)}}\qquad\Rightarrow\qquad
g_{s}^{2+\frac{d-2}{2(p+1)}}<1\,.$ (57)

Hence in the weak-coupling limit for the critical string, $g_{s}\ll 1$, the
D$p$-branes are indeed weakly coupled under the RR-forms. On the other hand,
for small string coupling the quantum gravity cutoff is set by the string
scale, i.e. $\Lambda\sim M_{s}$ such that for $g_{s}\ll 1$ the tension of the
weakly coupled D$p$-branes satisfies $\mathcal{T}\gg\Lambda^{p+1}$ such that
our Proposition 3 is indeed satisfied.

##  5 Spectrum of weakly coupled gravitational theories

In the previous section we obtained two central results of this work. First we
showed that there exists an effective field theory with cutoff given by the
species scale $\Lambda$. Second, we argued that the tension of any weakly
coupled $p$-brane is constrained to be bigger than $\Lambda^{p+1}$. In this
section we utilize these crucial results to constrain the spectrum of states
in a weakly coupled theory of gravity. In section 5.1 we first constrain the
kind of tower of weakly coupled states with mass below $\Lambda$. We then show
in section 5.2 that for energies $\Lambda\ll E\ll M_{\rm min}$ the density of
one-particle state needs to grow exponentially in energy reproducing a
Hagedorn-like behavior. In section 5.3 we present an argument that the
existence of extra dimensions and their size is directly related to the growth
of $\rho(E)$ above $M_{\rm min}$. Finally, in section 5.4 we show how these
results, that were extracted from the properties of gravitational amplitudes,
provide bottom-up evidence for a central statement of the Emergent String
Conjecture Lee:2019oct .

###  5.1 Constraining towers lighter than $\Lambda$

In the previous section we argued that there needs to exist an EFT that is
valid up to energies $\Lambda$. Importantly, this is the EFT that calculates
the entropy of all black holes in the theory. On the other hand, we know that
any field couples universally to gravity and therefore corrects the
gravitational action. For the sake of generality, we also allow the effective
theory to include weakly coupled defects whose worldvolume theory is described
by a weakly coupled field theory.101010Defects, for which the worldvolume
theory is strongly coupled, and that do not admit a Lagrangian description at
low energies are not relevant for our discussion since they cannot provide a
tower of weakly coupled excitations. Closed configurations of these defects
can lead to additional particle states in the theory. In the following, we
collectively refer to the weakly coupled particles and the weakly coupled
defect states of the EFT that is valid up to energy $\Lambda$ as weakly
coupled states. Note that these are the only states that can perturbatively
couple to gravity. On the other hand, strongly coupled states, even if stable
and light, might not be described by the EFT. As an example of such states,
consider BPS states in Calabi–Yau threefold compactifications of M-theory. For
these theories there exist finite distance limits along which an effective
divisor shrinks to zero size leading to a theory containing a tensionless
string. In these limits there exist an infinite tower of BPS states with
vanishing central charge. However, since these states are strongly-coupled and
associated to a strongly coupled, tensionless strings they do not allow for a
description in terms of a weakly coupled EFT.

In this section we wish to constrain the tower of such weakly coupled states
with mass $m$ in the regime of validity of the EFT with cutoff $\Lambda$, i.e.
states with $m<\Lambda$. More precisely we show the following

###### Proposition 4

Consider a $d$-dimensional theory of gravity with quantum gravity cutoff
$\Lambda$. Then in the weak-coupling limit $\Lambda\ll M_{\rm pl,d}$ the only
tower of weakly coupled states with mass below $\Lambda$ can be a KK-tower.

Since we are interested in states with mass parametrically below $\Lambda$,
such states allow for a description in terms of an EFT, which may be higher-
dimensional. This means that there exists a description of the theory
involving only finitely many fields and defects that collectively account for
all the states in the theory with energies below $\Lambda$. These states can
arise as one-particle states in the field theory, as bound states of the field
excitations, or from the weakly coupled defects which includes states obtained
by wrapping this defect along the compact dimensions. Note that Proposition 4
is also satisfied by confining large-$N$ gauge theories which have a tower of
confined states. In the large $N$-limit, the energy scale $\Lambda$ at which
the higher-derivative corrections become important matches the mass scale of
the ’t Hooft string which is also the mass scale of the tower of confined
states.

Let us assume we identified the effective theory that describes the theory of
gravity up to $\Lambda$ as a $D$-dimensional EFT. As we showed in section 4.2
any weakly coupled $p$-brane has tension at or above the species scale
$\mathcal{T}\gtrsim\Lambda^{p+1}$. As a consequence these states cannot give
rise a tower of states with mass below $\Lambda$. Still, upon compactification
on a $p$-dimensional manifold of radius $R$ such $p$-branes may give rise to a
tower of particle states in the $(D-p)$-dimensional theory with mass
$m\sim\mathcal{T}R^{p}$. Since the cutoff of the EFT sets the minimal length
scale $R_{\rm min}\sim\Lambda^{-1}$ for which the effective theory can be
described in terms of the $D$-dimensional theory, the mass of this tower of
states is bounded by $m\gtrsim\Lambda$. Therefore also after dimensional
reduction a defect theory cannot give rise to a tower of states describable
within a gravitational EFT with cutoff $\Lambda$. In addition we now argue
that there can be only a finite number of weakly coupled particles (including
bound states) in the D-dimensional EFT at weak coupling. This EFT is defined
in the following way: for every finite value of $\Lambda/M_{\rm pl,d}$ we
consider the special EFT that describes all black holes in the theory. As we
vary $\Lambda/M_{\rm pl,d}$ this defines a family of EFTs. The EFT at weak
coupling is defined as the limit of this family as $\Lambda/M_{\rm
pl,d}\rightarrow 0$.111111The existence of a well-defined theory as we take
$\Lambda/M_{\rm pl,d}\rightarrow 0$ is an assumption about how this limit is
taken. For example, in string theory, this achieved by moving on an infinite
geodesic in the moduli space. However, one can easily consider a non-geodesic
path towards the asymptotic boundary of the moduli space (corresponding to
$\Lambda/M_{\rm pl,d}\rightarrow 0$) that oscillates between different weak
coupling descriptions. We would not consider such a trajectory in the space of
theories a weak-coupling limit. Since by definition this EFT is valid up to
energies $\Lambda$, its partition function has to be finite when the theory is
put on a thermal circle with circumference $\beta>\Lambda^{-1}$ which we keep
fixed in units of $\Lambda$ as we take the weak-coupling limit. Finiteness of
the partition function implies that the number of weakly coupled states with
masses below $\Lambda$, which we denote by $N$, is also finite since

$\displaystyle\mathcal{Z}(\beta)\geq\int_{0}^{\Lambda}dm\leavevmode\nobreak\
e^{-\beta m}\rho(m)\geq
e^{-\beta\Lambda}\int_{0}^{\Lambda}dm\leavevmode\nobreak\
\rho(m)=e^{-\beta\Lambda}N\,.$ (58)

In the first inequality we bound the full partition function from below by
only considering the contribution of the single-particle states with mass
below $\Lambda$. Therefore, assuming that $\mathcal{Z}$ is finite, we conclude
that the number of particles is bounded from above as

$\displaystyle N\leq\mathcal{Z}(\beta)e^{\beta\Lambda}\,,$ (59)

for any $\beta>\Lambda^{-1}$ and is hence finite.

This leaves us with a finite number of particles in the $D$-dimensional EFT.
Still these states can acquire different momenta in the $D-d$ compact
dimensions (KK modes). As is well known, the compactification of the
$D$-dimensional theory on a $(D-d)$-dimensional manifold of size $R$ leads to
a KK spectrum with mass $m_{\rm KK}\sim 1/R$ which in the limit
$R\rightarrow\infty$ (corresponding to the weak-coupling limit
$\Lambda/M_{\text{pl,d}}\to 0$) indeed leads to an infinite tower of states
with mass below $\Lambda$. Therefore the only possible tower of weakly coupled
states with mass below $\Lambda$ is a KK-tower as claimed in Proposition 4.

###  5.2 Universality of Hagedorn behavior

Our analysis so far has revealed that for a theory of quantum gravity there
are at least two important energy scales, the quantum gravity cutoff $\Lambda$
and the mass of the minimal black hole $M_{\rm min}$. In general these two
energy scales do not coincide. In this section we want to argue that the
behavior of the density of one-particle states for energies between $\Lambda$
and $M_{\rm min}$ has universal properties in any theory of gravity with weak-
coupling limit. Our main claim is summarized in

###### Proposition 5

Consider a $d$-dimensional theory of gravity in asymptotically flat spacetime
in its weak-coupling limit $\Lambda/M_{\rm pl,d}\rightarrow 0$ with $\Lambda$
the quantum gravity cutoff as defined in section 2 and let $M_{\rm min}$ be
the energy scale of the particle/black hole transition. Then for energies
$\Lambda\ll E\ll M_{\rm min}$ the density of one-particle states scales as
$\log\rho(E)\sim\widetilde{\mathcal{O}}\left(E/\Lambda\right)$.

To prove this Proposition we first recall from section 4.1 that in the
effective theory of gravity the shortest meaningful length scale is set by
$\Lambda^{-1}$. On the other hand we can consider the partition function for a
given system in our theory at temperature $T$ which admits a Euclidean path
integral definition with a thermal circle of size $\beta=1/T$. Since the
shortest length scale is set by $\Lambda^{-1}$ the Euclidean effective field
theory is only valid for $\beta\gtrsim\Lambda^{-1}$. For effective theories of
quantum gravity we therefore expect the temperature to be bounded from above
by $T\lesssim\Lambda$. For black holes, which are described by this effective
theory, this relation is indeed satisfied. For example for Schwarzschild black
holes the temperature is inversely proportional to the horizon
radius—independent of the dimension of the spacetime. For the smallest black
hole with mass $M_{\rm min}$ we therefore have $T(M_{\rm min})\sim 1/r_{\rm
min}=\Lambda$.

Let us now turn our attention to particle systems. In section 4.1 we argued
that there exists an EFT description that is valid up to energies $\Lambda$
and showed in section 5.1 that all towers of states below $\Lambda$ can be
understood as KK-towers. Let us assume that the EFT, which describes the
theory up to $\Lambda$, lives in $D$ dimensions. Consider a box of size
$L\gg\Lambda^{-1}$ and energy $E$ in $D$ dimensions. We are interested in the
weak gravitational limits where $L\Lambda$ and $E/\Lambda$ are kept fixed
while $\Lambda/M_{\rm pl,D}\rightarrow 0$. Since the $D$-dimensional theory
obtained in the gravitational weak-coupling limit has no tower of particles
with mass below $\Lambda$, the mass $m_{\rm max}$ of the heaviest state in
this EFT is much smaller than $\Lambda$, $m_{\rm max}\ll\Lambda$. By inserting
energy into the box we increase the temperature of the system in the box above
$m_{\rm max}$ such that the light particles in the EFT can be treated as
effectively massless. Therefore, for temperatures $m_{\rm max}\ll T<\Lambda$,
we have

$E\sim T^{D}L^{D-1},$ (60)

which is the energy of radiation at temperature $T$ inside a
$(D-1)$-dimensional spatial box. At energy $E_{c}=\Lambda^{D}L^{D-1}$ the
temperature will be proportional to $\Lambda$. Importantly, in the
gravitational weak-coupling limit we keep $L\Lambda$ fixed such that
$E_{c}\propto\Lambda$. Therefore, this energy will be much smaller than the
mass of a black hole of the same size as the box (i.e. radius $L$) which is
given by $E_{\max}=L^{D-3}M_{\rm pl,D}^{D-2}$. Since $E_{c}\ll E_{\max}$ the
system has not collapsed into black hole and hence is indeed gravitationally
weakly coupled.

At $E_{c}$ the temperature is already of the order of the maximal temperature
$T_{\max}\propto\Lambda$ such that increasing the energy beyond $E_{c}$ cannot
lead to a further parametric increase of the temperature. And indeed, from
Proposition 1 we know that below $M_{\rm min}$, $\Omega(E)$ cannot grow super-
exponentially, i.e. $\Omega(E)$ is given by (14) with $\alpha_{\Omega}\leq 1$.
For energies $E_{c}<E\ll E_{\max}$ the temperature in the microcanonical
ensemble thus remains essentially constant:

$\displaystyle T(E)^{-1}=\frac{{\rm d}\log\Omega}{{\rm
d}E}\sim\widetilde{\mathcal{O}}\left(\Lambda^{-1}\right)\,,\quad\text{for}\quad
E_{c}<E\ll E_{\max}\,.$ (61)

Integrating the above equation leads to

$\displaystyle\log\Omega(E)\propto\widetilde{\mathcal{O}}\left(\frac{E}{\Lambda}\right)\,,\quad\text{for}\quad
E_{c}<E\ll E_{\max}\,.$ (62)

Comparing with (14) this implies $\alpha_{\Omega}=1$ in the energy window
above. Suppose the average energy of _single_ particles near the upper end of
the energy interval is given by $\epsilon_{\max}\gg\Lambda$. On the other
hand, since at the lower end of the energy interval we are dealing with
radiation in a box of temperature $\mathcal{O}(\Lambda)$, the average energy
of a single particle at the lower end the interval $\epsilon_{\rm min}$ is
proportional to $\Lambda$, $\epsilon_{\min}\propto\Lambda$. In the weak-
coupling limit $\epsilon_{\max}\rightarrow\infty$ though its exact value in
terms of $\Lambda$ and $M_{\rm pl,D}$ in this limit depends on the profile of
$\rho(E)$. From (22) we recall that an exponential growth of $\Omega$ implies
an exponential growth for $\rho(E)$. This, in turn, implies

$\displaystyle\log\rho(E)\sim\widetilde{\mathcal{O}}\left(\frac{E}{\Lambda}\right)\,,\quad\text{for}\quad\Lambda\ll
E\ll\epsilon_{\max}\,.$ (63)

We thus conclude that the density of states $\rho(E)$ at energy $E=x\Lambda$
converges to an exponential function in $E/\Lambda$ in the weak-coupling limit
for any $x\gg 1$. The very fact that the density of particles converges means
that in the gravitational weak-coupling limit there is a tower of states with
exponentially growing density for which the masses are proportional to
$\Lambda$.121212In string theory, these states are the string excitations and
$\Lambda$ is the string scale.

To argue for a tower of states with masses proportional to $\Lambda$ and
exponential density in the energy window $\Lambda\ll E\ll M_{\rm min}$,
consider a state for which the mass converges to $x\Lambda$ in the weak-
coupling limit for some $x\gg 1$. If we move away from the weak-coupling
limit, the gravitational coupling for this state increases until the
gravitational self-energy becomes dominant and it collapses into a black hole.
However, as long as we are at a point along the weak-coupling limit where the
mass of the state is much smaller than the mass, $M_{\rm min}$, of the
smallest black hole, the gravitational corrections to the mass of the state
are small. Therefore for energies $\Lambda\ll E\ll M_{\rm min}$ there indeed
exist states with exponential density and mass proportional to $\Lambda$. Note
that the tower we considered here might not include all particles. For
example, there could be a tower of particles for which the mass in the weak-
coupling limit is not bounded in units of $\Lambda$.131313An example of such
tower is $D0$ branes in type IIA string theory in the limit where the string
coupling goes to zero. Still, we showed that there exists a subset of
particles with mass scale $\Lambda$ with exponentially growing density
$\log\rho(E)\propto E/\Lambda$ up to logarithmic corrections in $E/\Lambda$.
In fact the contribution to $\rho$ from other states has to be of the same
order or subleading. If this was not the case, the number of one-particle
states at energy $E=M_{\rm min}$ would be parametrically larger than
$\exp(M_{\rm min}/\Lambda)$ which is the density of states of the smallest
black hole with entropy $M_{\rm min}/\Lambda$. Altogether we can hence
conclude that the tower of states with masses between $\Lambda$ and $M_{\rm
min}$ leads to $\log\rho(E)\propto E/\Lambda$ up to logarithmic corrections as
claimed in Proposition 5.

Note that our results are consistent with the existence of the
Horowitz–Polchinski solution Horowitz:1997jc  as an additional phase before
reaching $M_{\rm min}$. These solutions describe selfgravitating string
backgrounds and their entropy to leading order is linear in energy
Chen:2021dsw . Therefore, they satisfy Proposition 5.

###  5.3 High-energy signature of extra dimensions

In section 3.3 we argued that the states at energy $E$ are black hole
microstates if and only if $\log\rho(E)\sim(E/\Lambda)^{\alpha}$ for
$\alpha>1$ (see Proposition 1). In particular we saw that in a $d$-dimensional
theory of gravity Schwarzschild black holes lead to
$\alpha=\frac{d-2}{d-3}>1$. In this section, we show that the weak energy
condition at low energies implies that the values for $\alpha$ in the range
$1<\alpha<\frac{d-2}{d-3}$ can only be attained in the presence of extra
dimensions.

###### Proposition 6

Consider a $d$-dimensional theory of gravity in asymptotically flat spacetime
with quantum gravity cutoff $\Lambda$. In theories that the weak energy
condition is satisfied by the low energy field theory, the density, $\rho$, of
one-particle states as a function of energy, $E$, can grow like
$\log\rho(E)\propto(E/\Lambda)^{\alpha}$ with $1<\alpha<\frac{d-2}{d-3}$ only
in the presence of extra dimensions. The one-particle states leading to such a
growth of $\rho$ are the microstates of black holes localized in the compact
dimensions with horizon size smaller than the size of the extra dimensions.

Notice that for large energies $\alpha=\frac{d-2}{d-3}$ since for any compact
space there exists a mass threshold beyond which the horizon of any black hole
is larger than the size of this compact space. Therefore for large energies
all black holes will be $d$-dimensional. Still at lower energies the effective
scaling of $\rho(E)$ parametrized by $\alpha$ can change. In order to show
that a change from $\alpha=\frac{d-2}{d-3}$ to $1<\alpha<\frac{d-2}{d-3}$
implies the presence of extra dimensions, we first need to argue such a change
cannot be caused by adding some arbitrary but finite number of states to the
theory.

To that end we use the weak energy condition (which is expected to hold in
asymptotically flat spacetimes) to show that for a fixed energy one cannot
construct a state with entropy higher than the Schwarzschild black hole by
adding a finite number of arbitrary fields and interactions to the theory. To
see that consider a $d$-dimensional spherically symmetric black hole with
fixed mass $M$ described by a corrected version of the Schwarzschild metric
for which the corrections arise due to the various gravitational couplings in
the action. Using the weak energy condition, we show that the corrections
decrease the entropy of the black hole. Suppose the metric is given by

$\displaystyle
ds^{2}=-e^{2\nu(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}d\Omega_{d-2}^{2}\,,$
(64)

where $\nu(r)$ and $\lambda(r)$ are functions encoding the corrections to the
Schwarzschild metric. The uncorrected Schwarzschild solution corresponds to

$e^{-2\lambda_{S}(r)}=e^{2\nu_{S}(r)}=1-\frac{\kappa M}{4\pi r^{d-3}}\,,$ (65)

where $\kappa=M_{\rm pl,d}^{2-d}$. The entropy of neutral large black holes at
leading order is given by $S=\frac{A}{4G}+\mathcal{O}(\log A)$ Sen:2012dw .
Therefore, to show that the corrections to the Schwarzschild metric in (64)
decrease the entropy, we only need to show that the radius of the black hole
described by (64) is smaller than the radius of a Schwarzschild black hole
with the same mass.

We can define coordinates such that $\lim_{r\rightarrow\infty}\nu(r)=0$ which,
via the Einstein equations, also implies
$\lim_{r\rightarrow\infty}\lambda(r)=0$. Suppose the horizon radius is given
by $r_{H}$ and the mass of the ADM black hole is given by $M$. Then we have

$\displaystyle\lim_{r\rightarrow r_{H}}e^{-\lambda(r)}=\lim_{r\rightarrow
r_{H}}e^{\nu(r)}=0\,,$ (66)

and

$\displaystyle\lim_{r\rightarrow\infty}(1-e^{2\nu(r)})r^{d-3}=\lim_{r\rightarrow\infty}(1-e^{-2\lambda(r)})r^{d-3}=\frac{\kappa
M}{4\pi}\,.$ (67)

Note that the relationship between $M$ and $r_{H}$ is no longer
$r_{H}^{d-3}=\kappa M/(4\pi)$ due to the corrections. The Einstein equations
give

$\kappa
T^{0}_{0}=-(d-2)\frac{e^{-2\lambda}}{r^{2}}\left[\frac{(d-3)}{2}(-1+e^{2\lambda})+\,r\lambda^{\prime}\right]\,.$
(68)

where ′ denotes derivatives with respect to $r$. On the other hand, the weak
energy condition implies

$T^{0}_{0}\leq 0\,.$ (69)

Hence, we find

$\displaystyle\frac{(d-3)}{2}\left(-1+e^{2\lambda}\right)+\,r\lambda^{\prime}$
$\displaystyle\geq 0\>.$ (70)

The above inequality is saturated for the Schwarzschild solution with
$\lambda=\lambda_{S}$ defined in (65). We therefore find

$\displaystyle\lambda^{\prime}-\lambda^{\prime}_{S}\geq\frac{(d-3)}{2}\frac{e^{2\lambda_{S}}-e^{2\lambda}}{r}\,.$
(71)

We now show that for a black hole with ADM mass given in (67) the above
inequality implies $e^{2\lambda}\leq e^{2\lambda_{S}}$ where $\lambda_{S}$
corresponds to the metric of a Schwarzschild metric of the _same_ mass. Since
we fix the ADM mass $M$, for large $r$ we have

$\displaystyle e^{2\lambda}$ $\displaystyle=1+\frac{\kappa M}{4\pi
r^{d-3}}+\mathcal{O}(r^{6-2d})\>,\qquad e^{2\lambda_{S}}=1+\frac{\kappa
M}{4\pi r^{d-3}}+\mathcal{O}(r^{6-2d})\>,$ (72)

such that the difference between $e^{2\lambda}$ and $e^{2\lambda_{S}}$ is
$\mathcal{O}(r^{6-2d})$. In terms of

$\delta\lambda\equiv\lambda-\lambda_{S}\,,$ (73)

we can rewrite the inequality (71) in the $r\to\infty$ limit as

$(\delta\lambda)^{\prime}\geq\frac{(d-3)}{2r}\left(1+\frac{\kappa M}{4\pi
r^{d-3}}+\mathcal{O}(r^{6-2d})\right)\left(-2\delta\lambda+\mathcal{O}(\delta\lambda)^{2}\right)\,.$
(74)

Keeping only the leading orders in $1/r$ and $\delta\lambda$ we find

$(\delta\lambda)^{\prime}\geq-\frac{(d-3)}{r}\delta\lambda\,.$ (75)

This form of the inequality suggests the ansatz

$\delta\lambda=\frac{f}{r^{d-3}}\,,$ (76)

for some function $f=f(r)$. Via (72) the requirement that the ADM mass of the
corrected black hole is equal to the mass of a Schwarzschild black hole
translates into

$\lim_{r\to\infty}f=0\>.$ (77)

From (75) we further find that at large radius

$f^{\prime}\geq 0\>.$ (78)

These conditions on $f$ imply $\delta$ approaches zero from below as
$r\to\infty$, which means $e^{2\lambda}\leq e^{2\lambda_{S}}$ for sufficiently
large $r$. The condition (71) then translates to
$0>\lambda^{\prime}\geq\lambda^{\prime}_{S}$ and $e^{2\lambda}\leq
e^{2\lambda_{S}}$ everywhere. Since $e^{2\lambda}\leq e^{2\lambda_{S}}$,
$\lambda$ is finite whenever $\lambda_{S}$ is finite. Therefore, $\lambda(r)$
is finite for all radii $r\geq r_{S}(M)=(\kappa M/(4\pi))^{1/(d-3)}$. Hence,
we find that the horizon radius $r_{H}$ at which $\lambda$ diverges satisfies
$r_{H}\leq r_{S}$, which is the desired result.

In order to obtain a black object whose entropy exceeds that of a
Schwarzschild black hole of the same mass, we need to add an _infinite_ number
of states to the theory. Such states can be provided by an infinite tower of
states. However, as we argued in section 4.1, black holes with mass $E>M_{\rm
min}$ have to be describable in terms of some EFT. From Proposition 4 we
recall that the only infinite tower of weakly coupled states with mass below
$\Lambda$ can be a KK-tower arising from dimensional reduction of a higher-
dimensional theory. The objects with entropy higher than a $d$-dimensional
Schwarzschild black hole (at fixed energy) are therefore black holes in higher
dimensions.

Concretely, consider a $D$-dimensional theory compactified on a
$(D-d)$-dimensional manifold. For simplicity, let us consider a homogeneous
internal geometry characterized by a length scale $R$ and let $M_{\rm pl,D}$
and $M_{\rm pl,d}$ be, respectively, the $D$\- and $d$-dimensional Planck
scales which are related via

$\displaystyle M_{{\rm pl,D}}^{D-2}=M_{{\rm pl,d}}^{d-2}R^{d-D}.$ (79)

In this construction, $D$-dimensional black holes correspond to localized
objects inside the compact geometry which, however, only exist for $r_{H}<R$.
If we increase the energy and therefore the horizon radius of this black hole
to $r_{H}>R$, the black hole eventually reaches the size of the compact
geometry and the backreaction causes the horizon to form a black brane
wrapping the internal geometry. In the lower dimensional theory, this black
brane corresponds to a black hole. The condition $r_{H}\geq R$ translates to

$\displaystyle E\geq m_{c}\equiv R^{d-3}M_{{\rm pl,d}}^{d-2}\,.$ (80)

Note that for $E\leq m_{c}$, the black hole and the black brane solutions are
both sensible classical solutions, but the black brane solution has higher
entropy and is more stable thermodynamically. Consider for example the entropy
of the localized $D$-dimensional and the $d$-dimensional Schwarzschild black
holes in the microcanonical ensemble

$\displaystyle S_{\rm BH,d}(E)\sim\left(\frac{E}{M_{{\rm
pl,d}}}\right)^{\frac{d-2}{d-3}}$
$\displaystyle=\left(\frac{E}{m_{c}}\right)^{\frac{d-2}{d-3}}(RM_{{\rm
pl,d}})^{d-2}\>,$ (81) $\displaystyle S_{\rm
BH,D}(E)\sim\left(\frac{E}{M_{{\rm pl,D}}}\right)^{\frac{D-2}{D-3}}$
$\displaystyle=\left(\frac{E}{m_{c}}\right)^{\frac{D-2}{D-3}}(RM_{{\rm
pl,d}})^{d-2}\>,$

where in the last equalities we used (79) and (80). At $E\gg m_{c}$ the most
entropic states are the $d$-dimensional Schwarzschild black holes. These,
however, become unstable at energies $E\sim m_{c}$ corresponding to the
Gregory–Laflamme transition Gregory:1993vy . At energies $E\ll m_{c}$ the
entropically favored black hole is the $D$-dimensional one, as is clear from
(81) since $\frac{D-2}{D-3}<\frac{d-2}{d-3}$. As a consequence, the smallest-
sized black hole trustable in _any_ effective theory is $D$-dimensional black
hole, with $D$ the highest possible dimension. We see that, indeed,
$1<\alpha<\frac{d-2}{d-3}$ can be achieved in the presence of extra dimensions
at energies $E$ corresponding to black holes with a horizon radius smaller
than the extra dimensions. According to our previous discussion this is also
the only way to obtain such a value for $\alpha$ which is the statement of
Proposition 6.

Let us close with some comments about the above result:

  1. 1.

If both the lower- and the higher-dimensional black holes contributed to the
amplitude as $\exp(-\mathcal{O}(S))$, where $S$ is their respective entropy,
the contribution from the black hole with smaller entropy would dominate. This
would undermine the claim that the gravitational amplitude captures the more
stable saddle. However, this does not happen. Suppose we are in an energy
window in which the higher- and the lower-dimensional black hole solutions are
both well-defined, but the lower-dimensional black hole is unstable due to the
Gregory–Laflamme instability Gregory:1993vy . The inverse of the growth rate
of the tachyonic mode which sets the time scale for this transition is $\sim
r_{H}$. This time scale is much smaller than the lifetime $\sim
r_{H}^{d-1}M_{\rm pl,d}^{d-2}$ set by the Hawking radiation. Therefore, the
black hole transforms into the more stable higher-dimensional configuration
before Hawking radiating into two particles. This was expected since the
Gregory–Laflamme instability would only be meaningful if the decay time
between the two black holes is short enough to be seen before the black hole
evaporates.

  2. 2.

The scale $m_{c}$ that occurs in the context of extra compact dimensions is an
example of a more general scale, the black hole scale $\Lambda_{\rm BH}$, that
was recently conjectured to be universally present in theories of quantum
gravity where the black hole undergoes a phase transition to a entropically
favored solution Bedroya:2024uva . As argued in there, in the case of a string
tower, the scale $\Lambda_{\rm BH}$ is identified with the scale of the
Horowitz-Polchinski transition.

  3. 3.

Finally, we note that an important consequence of our results is that the
higher-dimensional black hole cannot be dimensionally reduced. In other words,
no EFT with finitely many KK particles can approximate the higher-dimensional
black hole and its entropy. Instead, the higher-dimensional black hole must be
understood as a condensate of infinitely many KK states. Our argument provides
a sharp explanation for the top–down observation made in Bedroya:2024uva  that
a thermodynamic transition of the black hole to a more stable phase is
correlated with the existence of an infinite tower of states. The known
examples include the Gregory–Laflamme transition which is accompanied by the
KK tower and the the Horowitz–Polchinski solution Horowitz:1997jc  in $d<7$
which is accompanied by a tower of string excitations.

$E$$\Lambda$$M_{\rm
min}$$m_{c}$$\log\rho(E)\sim\widetilde{\mathcal{O}}\left(\left(E/\Lambda\right)^{\alpha}\right)$$\alpha<1$$\alpha=1$$\alpha=\frac{D-2}{D-3}$$\alpha=\frac{d-2}{d-3}$

Figure -478: The density of one-particle states $\rho$ as a function of energy
$\log\rho(E)\sim\widetilde{\mathcal{O}}(E^{\alpha})$ in a $D$-dimensional
theory with $(D-d)$ compact dimensions of radius $R$ in the weak-coupling
limit $\Lambda\ll M_{\rm pl,d}$. According to the Propositions 1, 5 and 6 the
parametric scaling of $\log\rho(E)$ is universal in the energy windows set by
the quantum gravity cutoff, $\Lambda$, the mass of the minimal black hole
$M_{\rm min}$ and the mass scale $m_{c}$ in (80).

###  5.4 Relation to Emergent String Conjecture

The results of the previous sections provide strong constraints on the
spectrum of massive one-particle states in theories of quantum gravity. In
particular the Propositions 1, 5, and 6 constrain the scaling of the density
of one-particle states $\rho$ as a function of energy in the gravitational
weak-coupling limits $\Lambda\ll M_{\rm pl}$. A sketch of this behavior is
provided in Figure -478.

Together with Proposition 4 these results nicely relate to the Emergent String
Conjecture (ESC) Lee:2019oct  which, as reviewed in the introduction, states
that the lightest tower of states in any asymptotic limit of quantum gravity
is either a KK-tower or consists of the excitations of a fundamental string.
In light of the ESC, it is important to notice that our results as summarized
in Propositions 1-6 have a built-in UV-IR connection that allows us to extract
the properties of towers of light states from the behavior of $\rho$ at much
higher energies. It follows from Propositions 1 and 5 that we can use
$\rho(E)$ to read off $\Lambda$ and $M_{\rm min}$. Working from now on in
$d$-dimensional Planck units, we can differentiate the cases141414By
Proposition 5 the third possibility $M_{\rm min}\gg\Lambda^{3-d}$ (in Planck
units) can never be realized since the exponential degeneracy in the interval
$[\Lambda,M_{\rm min}]$ causes states to form black holes at energies above
$\Lambda^{3-d}$.

$a):\;M_{\rm min}\ll\Lambda^{3-d}\,,\quad b):\;M_{\rm
min}\sim\Lambda^{3-d}\>.$ (82)

For a Schwarzschild black hole in $d$ dimensions its mass and temperature are
related via

$M_{\rm BH,d}\sim T^{3-d}\,.$ (83)

Recall that the minimal mass black hole with mass $M_{\rm min}$ has
temperature $T=\Lambda$. In case $a)$ this minimal mass black hole therefore
cannot be $d$-dimensional, and needs to be a different state which is
entropically favored over the $d$-dimensional Schwarzschild black hole. By
Proposition 6 this has to be due to the existence of $(D-d)\geq 1$ extra
dimensions with radius $R$ such that the smallest mass black hole is a
$D$-dimensional black hole with mass (in $d$-dimensional Planck units)

$M_{\rm min}=\frac{R^{d-3}}{(\Lambda R)^{D-3}}\,.$ (84)

Since this black hole only exists for energies $E<m_{c}$, with $m_{c}$ defined
in (80), we need to have $M_{\rm min}<m_{c}$ leading to the constraint

$R^{-1}<\Lambda\,,\qquad\Leftrightarrow\qquad m_{\rm KK}<\Lambda\,,$ (85)

implying that there is a KK-tower with mass below $\Lambda$. By Proposition 4
this KK-tower has to be the lightest tower of states since there cannot exist
any other tower with mass below $\Lambda$. Case $a)$ is the situation depicted
in Figure -478. On the other hand in case $b)$ the minimal mass black hole is
a $d$-dimensional black hole. Even in the presence of extra dimensions,
$m_{c}\leq M_{\rm min}$ and higher-dimensional black holes do not exist.
Accordingly there is no KK tower with mass below $\Lambda$.

Due to the presence of a tower of light states, infinite distance limits in
field space precisely correspond to weak-coupling limits for which $\Lambda\ll
M_{\rm pl}$. Hence our results concerning the properties of the density of
one-particle states apply in particular to these limits. The statements of
Proposition 4 and 5 can then be reinterpreted as bottom-up evidence for the
ESC since they can be reformulated into the following

###### Corollary 1

In any weak-coupling limit ($\Lambda\ll M_{\rm pl}$) of an Einstein theory of
gravity in asymptotically flat space, the lightest tower of weakly coupled
states is either $a)$ a KK-tower, or $b)$ a tower of states with exponential
density of states $\rho(E)\sim e^{E/\Lambda}$.

Notice that fundamental strings predicted by the ESC precisely have
degeneracies that grow exponentially in the energy level $\log\rho(E)\sim
E/M_{s}$, and for fundamental strings the species scale is given by the string
scale $\Lambda=M_{s}$. The statement in Corollary 1 is agnostic about the
existence of a duality frame in which this tower arises from a perturbative
fundamental string. Still our result implies that (in asymptotically flat
space) there are essentially just two options for the structure of the light
states in quantum gravity. Even though in case the tower is not a KK-tower we
cannot show that the states arise from fundamental strings, our results imply
that in this situation the Hagedorn-growth for these states is universal.

Let us close by noticing that by (16) an exponentially growing $\rho(E)$ can
be viewed as the limit $p\rightarrow\infty$ for the density of KK-states on a
$p$-dimensional compact manifold. Corollary 1 therefore implies that the
lightest tower of states either behaves like a KK-tower for finite $p$ or like
a KK-tower in the limit $p\rightarrow\infty$. This is interesting since the
$\mathcal{O}(1)$ coefficients appearing in the Distance Conjecture are bounded
by the values obtained for emergent string limits Etheredge:2022opl ;
vandeHeisteeg:2023ubh ; vandeHeisteeg:2023dlw  which always correspond to the
$p\rightarrow\infty$ limit for the values of decompactification limits. As our
results imply that the most extreme density of a light tower of states can be
viewed as the $p\rightarrow\infty$ limit for a KK-tower this provides further
evidence for the bounds on $\mathcal{O}(1)$ parameters derived from the
Emergent String Conjecture to hold in general.

##  6 Discussion

In this work we investigated universal features of the density, $\rho$, of
one-particle states in theories of quantum gravity. More precisely we studied
the universal behavior of $\rho$ as a function of energy in weak-coupling
limits in which the quantum gravity cutoff can be made parametrically smaller
than the Planck scale, $\Lambda\ll M_{\rm pl}$. Central to our analysis was
our estimate of the density of one-particle states $\rho$ discussed in section
3. The behavior of scattering amplitudes in perturbative string theory and the
black hole regime served as the motivation to estimate $\rho$ via scattering
amplitudes for energies above the cutoff $\Lambda$. Let us stress that we did
not provide a first-principle derivation of the identification of these two
quantities. It would be interesting to further scrutinize this identification
in future work.

Scattering amplitudes played a crucial role in clarifying the appearance of
the EFT cutoff in the higher-derivative terms correcting the Einstein–Hilbert
action. Even though it has been argued previously vandeHeisteeg:2023dlw ;
vandeHeisteeg:2023ubh  that the species scale $\Lambda$ should be the scale
suppressing these higher-derivative terms in the effective action, the
equivalence of this scale to the EFT cutoff set by the radius of the smallest
black holes describable in the EFT had not been demonstrated explicitly. In
this work, we provided an argument for the identification between this
equivalence based on the properties of gravitational scattering amplitudes. In
particular our argument crucially made use of the features of scattering
amplitudes in the black hole regime thereby highlighting the gravitational
nature of the effective theory. We believe that our approach of exploiting
gravitational scattering amplitudes to investigate important quantities for
recent developments within the Swampland program (in this case the species
scale) is a promising avenue to further pursue.

Gravitational scattering amplitudes in combination with effective string
theory Polchinski:1991ax  further led to the bound on the tension of weakly
coupled $p$-branes in terms of the species scale. Whereas such a bound has
previously been observed in explicit string theory compactifications (see,
e.g., Cota:2022yjw ) our result gives the first bottom-up argument for such a
bound. In the context of our analysis this result proved useful in arguing
that the only tower of weakly coupled states below $\Lambda$ can be a KK-tower
indicating strong constraints on such possible towers in an EFT. On the other
hand, at energies $\Lambda\ll E\ll M_{\rm min}$ we showed that the density of
one-particle states scales exponentially in the energy leading to an
approximately constant temperature in that energy window, up to logarithmic
corrections. From the perspective of the Swampland program, our result
provides evidence for the Emergent String Conjecture Lee:2019oct  by showing
that in asymptotically flat spacetime the lightest tower of states has to
either be a KK tower or a tower with Hagedorn growth for the one-particle
states, a behavior familiar from fundamental strings. Let us stress that with
our argument we are not able to infer the exact origin of the tower of states
with exponential degeneracy, i.e., whether the states are always excitations
of fundamental strings as predicted by the ESC. It would be interesting to see
whether one can argue for this feature of the tower utilizing only bottom-up
arguments. Even in explicit string theory constructions the identification of
the fundamental string requires detailed knowledge of the intricate dualities
of string theory. Hence, addressing this aspect of the ESC requires a bottom-
up derivation of string dualities, see e.g. Bedroya:2023xue  for a recent
discussion of deriving string dualities using Swampland principles.

Still, since the ESC is at the bedrock of many other arguments for non-
trivial, universal features of quantum gravity, we believe that our result
provides an important step towards consolidating the validity of the ESC. At
its heart the ESC quantifies the observation in string theory, that the
leading tower of light states in quantum gravity always collectively describes
a richer semi-classical theory. The fact that in this work we are able to
provide bottom-up evidence for one of the prediction of the ESC can be
understood as a consequence of the emergence of such a semi-classical
description. Another semi-classical signature of quantum gravity is the
validity of the Euclidean Path Integral (EPI) in uncovering the UV properties
of gravity. In fact we used the validity of the EPI description in our
analysis in section 5, when identifying the maximal temperature describable
within the EFT with the quantum gravity cutoff, and when discussing the effect
of additional fields on corrections to the entropy of Schwarzschild black
holes. It would be interesting to explore the relation between the semi-
classical description of gravity predicted by the ESC and the EPI in more
detail. Another important aspect of the ESC that we have not addressed in this
work is the prediction that KK towers or towers of excitations of fundamental
string necessarily have to arise at infinite distance in the scalar field
space. In other words, we have not addressed the part of the ESC corresponding
to the Distance Conjecture. It would be very interesting to see whether
methods akin to those exploited in this work can also provide bottom-up
evidence for the Distance Conjecture itself, see Calderon-Infante:2023ler  for
a related discussion.

In any event, we believe that our results demonstrate the significance of
combining different perspectives on quantum gravity. For example our work
suggests a natural connection to the amplitude bootstrap program, given our
non-trivial results for the density of massive states in quantum gravity. Even
though our considerations here have been for black holes in asymptotically
flat spacetimes, we can envision extending some of the arguments to small
black holes in asymptotically Anti-de Sitter space, where such black holes are
not eternal due to Hawking radiation, and are qualitatively similar to the
flat case. In this way it might also be possible to provide bottom-up evidence
for Swampland constraints without assuming asymptotically flat spacetimes.

#### Acknowledgments

We have greatly benefited from discussions with Nima Arkani-Hamed, Alberto
Castellano, Eduardo Garcia-Valdecasas, Damian van de Heisteeg, Dieter Lüst,
John Stout, and Cumrun Vafa. We are especially grateful to Matt Reece for
comments on the draft. The work of AB is supported in part by the Simons
Foundation grant number 654561 and by the Princeton Gravity Initiative at
Princeton University. The work of MW is supported by a grant from the Simons
Foundation (602883,CV), the DellaPietra Foundation, and by the NSF grant
PHY-2013858. The work of RKM is supported by NSF grants PHY-1620806,
PHY-1748958 and PHY-1915071, the Chau Foundation HS Chau postdoc award, the
Kavli Foundation grant “Kavli Dream Team”, and the Moore Foundation Award
8342.

##  Appendix A Effective strings

In this section we review the theory of effective strings which is the
framework that studies non-fundamental strings Polchinski:1991ax . Let us
start by explaining precisely what is different between a generic effective
string, and the fundamental string that is quantized in string theory. In the
following we focus for simplicity on bosonic strings.

The common meaning of a string is a $(1+1)$-dimensional defect that couples to
a bulk effective field theory. In other words, in the presence of the string,
the total action can be written as

$\displaystyle S_{\text{total}}=S_{\text{Bulk}}+S_{\text{String}},$ (86)

where $S_{\text{String}}$ is a 2d action defined on the locus of the string
and which can depend on the values of the spacetime fields in the vicinity of
the string. Even the critical string, that is quantized in string theory, fits
this description since the critical string has a two-dimensional bosonic
action of the form

$\displaystyle
S_{\text{Critical}}=-\mathcal{T}\int\sqrt{-g_{\text{ind}}}-\frac{1}{4\pi}\int\sqrt{-g_{\text{ind}}}\mathcal{R}_{\text{ind}}\phi+i\int
B$ (87)

where $\sqrt{g_{\text{ind}}}$ and $\mathcal{R}_{\text{ind}}$ are,
respectively, the induced metric on the string worldsheet and its
corresponding scalar curvature in the string frame, and $B_{\mu\nu}$ and
$\phi$ are spacetime gauge potentials and the dilaton. The difference between
the critical string and an effective string is the regime of validity of this
action. Suppose the initial state for the string is a closed configuration.
Due to its non-zero tension, the string wants to shrink to zero size such that
this configuration is unstable. As the string shrinks, the extrinsic curvature
of the string, which is inversely related to the size of the string,
increases. Since the 2d action of the string depends on the embedding of the
string in the spacetime, it can contain a series of higher-order terms in the
extrinsic curvature. In fact, as we will discuss below, this is typically
required and the series is controlled by the string tension. With that in
mind, the higher-order corrections of the string action become important once
the size of the string reaches the inverse square root of the tension, i.e.
the string length. However, if we were to quantize the string, the average
size of the string would be exactly of this order. This is the reason why
quantizing a generic string is difficult and why the critical string in string
theory is special. In string theory, we assume that we know the full action of
the string at weak coupling which allows us to quantize the string in small,
closed configurations.

Nonetheless, if one wants to study the string in configurations with small
extrinsic curvature, the effective action is sufficient and very useful. Let
us mention two examples of such configurations.

  * •

Long string: Consider a string in a planar configuration in spacetime. The
extrinsic curvature on the worldsheet is zero. Therefore, the use of a finite-
term effective action is justified.

  * •

Wrapped string: In the presence of a large extra dimension, one can consider a
string that wraps the extra dimension. If the string coupling is small (such
that the string does not break up into smaller pieces), the topology of the
circle makes this string configuration stable and does not allow it to shrink
to zero size. Moreover, if the size of the circle is large, the extrinsic