NORDITA 2023-028, CTPU-PTC-23-22

aNordita, KTH Royal Institute of Technology and Stockholm University
Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
bParticle Theory and Cosmology Group, Center for Theoretical Physics of the
Universe,
Institute for Basic Science (IBS), Daejeon, 34126, Korea
cCosmology, Gravity and Astroparticle Physics Group,
Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS),
Daejeon, 34126, Korea
dDepartment of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

Under reasonable working assumptions including the polynomial boundedness, one
proves the well-known Cerulus-Martin lower bound on how fast an elastic
scattering amplitude can decrease in the hard-scattering regime. In this paper
we consider two non-trivial extensions of the previous bound. (i) We
generalize the assumption of polynomial boundedness by allowing amplitudes to
exponentially grow for some complex momenta and prove a more general lower
bound in the hard-scattering regime. (ii) We prove a new lower bound on
elastic scattering amplitudes in the Regge regime, in both cases of polynomial
and exponential boundedness. A bound on the Regge trajectory for negative
momentum transfer squared is also derived. We discuss the relevance of our
results for understanding gravitational scattering at the non-perturbative
level and for constraining ultraviolet completions. In particular, we use the
new bounds as probes of non-locality in black-hole formation, perturbative
string theory, classicalization, Galileons, and infinite-derivative field
theories, where both the polynomial boundedness and the Cerulus-Martin bound
are violated.

$\,{}^{\star}$<EMAIL_ADDRESS>
$\,{}^{\dagger}$<EMAIL_ADDRESS>
$\,{}^{\ddagger}$<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Locality, non-localizability, and S-matrix
    1. 2.1 Locality criterion by Jaffe
    2. 2.2 Locality criterion by S-matrix
      1. 2.2.1 Positivity bounds on the S-matrix
    3. 2.3 Candidates for non-localizable theories
      1. 2.3.1 Gravity and black-hole formation
      2. 2.3.2 Classicalization proposal
      3. 2.3.3 Galileon theories
      4. 2.3.4 Little String Theory
  3. 3 Basics & assumptions
    1. 3.1 Basics of kinematics
    2. 3.2 Analyticity domain in the $\cos\theta$-plane
    3. 3.3 Assumptions
  4. 4 Lower bound in the hard-scattering regime
    1. 4.1 Proof
    2. 4.2 Discussion
  5. 5 Lower bound in the Regge regime
    1. 5.1 Proof
    2. 5.2 Discussion
    3. 5.3 Bound on the Regge trajectory
      1. 5.3.1 Case of monotonically increasing Regge trajectory for $t<0$
  6. 6 Applications
    1. 6.1 Gravity and black-hole formation
    2. 6.2 Classicalization proposal
    3. 6.3 Galileon
    4. 6.4 Perturbative string theory
    5. 6.5 Infinite-derivative QFTs
  7. 7 Summary & conclusions
  8. A Analysis of the functions $C(x,a)$ and $C(x,\Delta)$

##  1 Introduction

The S-matrix operator is a central object in particle physics, through which
we can compute physical observables such as cross sections and decay rates.
From its mathematical properties we can typically derive interesting physical
implications. For example, the requirements of Lorentz invariance, polynomial
boundedness, unitarity and analyticity can be sufficient conditions to
guarantee the validity of several physical properties such as locality of
interactions, conservation of probabilities, micro- and macro-causality [1].

Given a Lagrangian in quantum field theory (QFT), we can compute the
corresponding S-matrix and check whether the aforementioned mathematical and
physical properties are verified. In the conventional framework of
perturbative QFT without gravity those properties are indeed satisfied.
However, some of the standard assumptions may be modified in the strong
gravity regime or due to some non-perturbative effects (not necessarily of
gravitational type). For instance, it is generally believed that the high-
energy scattering in gravity could be dominated by black-hole formation which
would cause an exponential suppression of the amplitudes and an exponential
increase of the spectral density, thus implying a violation of polynomial
boundedness [2, 3, 4, 5]. A similar violation of locality is expected to
appear in some ultraviolet (UV) completions such as string theory at the
perturbative level [6, 7, 8], and because of non-perturbative effects in the
classicalization proposal [9, 10] and in Galileon theories [10].

The assumption of specific fundamental principles for a given UV completion
results in general properties of scattering amplitudes and, thus, imposes
strong constraints on how the amplitudes behave in various kinematic regimes
including the physical regions. These constraints are usually expressed in the
form of upper or lower bounds [11]. If such bounds are not respected, it means
that at least one of the fundamental principles imposed at the level of UV
theory must be violated. Hence, these bounds can be very useful to gain
information on the fundamental principles of UV theory.

One of the most famous bounds in this context is the Froissart-Martin upper
bound on a $2\rightarrow 2$ elastic scattering amplitude in the forward limit
[12, 13], that was derived under the assumption of polynomial boundedness in
the unphysical domain in addition to unitarity and analyticity. Another
important example is a lower bound on elastic scattering amplitudes in the
high-energy limit and for fixed angle, that was derived in 1964 by Cerulus and
Martin [14]. Their result relies on the assumptions of unitarity, analyticity,
polynomial boundedness, and of a finite mass gap. In their formula, given an
amplitude $\mathcal{M}(s,\cos\theta),$ $s$ being the center-of-mass energy
squared and $\theta$ the scattering angle, the Cerulus-Martin bound reads [14]

$\max_{-a\leq\cos\theta\leq
a}\left|\mathcal{M}(s,\cos\theta)\right|\geq\mathcal{N}(s)\,e^{-f(a)\,\sqrt{s}\,\log(s/s_{0})}\,,$
(1)

where $\mathcal{N}(s)$ is a positive function of $s$ that is subdominant in
the $s\rightarrow\infty$ limit, $f(a)$ is a positive function of $a\in(0,1)$,
and $s_{0}$ is some energy-squared reference scale. Note that, the above
inequality is not a bound at some specific angle since any amplitude can in
principle vanish at given angle. Instead, it represents a lower bound on the
maximum of the modulus of the scattering amplitude for some finite interval of
the scattering angle. Because only the scatttering at small impact parameters
is relevant for the fixed-angle scattering, the Cerulus-Martin bound can be
used as a clean probe of UV physics.

In this work, we aim at investigating two non-trivial extensions of the
Cerulus-Martin bound. (i) Firstly, we replace the assumption of polynomial
boundedness with the more general condition of exponential boundedness that
would be generically respected even in theories with non-locality. Thus, we
derive a generalized lower bound on how fast an elastic scattering amplitude
can decrease in the high-energy limit and for fixed angles (i.e., in the so-
called hard-scattering limit). We propose to parameterize the exponential
boundedness and the degree of non-locality following Jaffe’s classification of
strictly localizable, quasi-local and non-localizable theories [15, 16]. In
this classification, the polynomial boundedness is included as a special case.
(ii) Secondly, we derive an analogue inequality in the high-energy limit and
for fixed momentum transfer (i.e., in the so-called Regge limit). This second
derivation provides new rigorous lower bounds in both cases of polynomial and
exponential boundedness. As a consequence, we also find a new bound on the
Regge trajectory for negative values of momentum transfer squared.

Despite being quite general and interesting in their own, our results can have
important implications for constraining the degree of (non-)locality of the
underlining theory in both gravitational and non-gravitational scenarios. We
discuss applications to gravitational scattering processes at high energy,
such as black-hole formation [2, 4, 17, 18], and in the context of various UV
completion scenarios. In particular, we use the new bounds as probes of non-
locality by analysing the behavior of scattering amplitudes in perturbative
string theory [6, 7, 19], classicalization proposal [9], Galileon theories
[10], and infinite-derivative QFTs [20, 21, 22, 23]. In all these
scenarios/theories the standard Cerulus-Martin bound (1) is violated. However,
this does not mean a pathology because polynomial boundedness is not satisfied
and, in fact, one has to verify whether the more general lower bound derived
under the assumption of exponential boundedness is respected.

The present paper is organized as follows.

sec. 2:

We introduce the idea of non-locality in field theory language and briefly
review Jaffe’s classification of strictly localizable, quasi-local and non-
localizable QFTs. We explain our parametrization of (non-)locality in terms of
the high-energy behavior of the scattering amplitude. In addition, we discuss
examples of possible candidates for non-localizable theories.

sec. 3:

We briefly review basics of kinematics and introduce the working setup. We
explain several mathematical details and introduce the main assumptions needed
for the derivations of the lower bounds.

sec. 4:

We extend the proof of Cerulus and Martin [14] by generalizing the assumption
of polynomial boundedness with exponential boundedness. We obtain a more
general lower bound on an elastic scattering amplitude in the hard-scattering
limit.

sec. 5:

We prove a general lower bound on an elastic scattering amplitude in the Regge
limit. As an implication we also obtain a bound on the Regge trajectory for
negative values of the momentum transfer squared.

sec. 6:

We use the newly derived bounds as probes of non-locality in gravity and in
some UV completion scenarios. In particular, we consider several
scenarios/theories in which the standard Cerulus-Martin bound can be violated,
such as gravitational scattering and black-hole formation, classicalization
proposal, Galileon theories, perturbative string theory, and infinite-
derivative QFTs.

sec. 7:

This section is devoted to summary of the main results, conclusions and
outlook.

sec. A:

We provide more mathematical details needed to understand some steps in the
derivations of the lower bounds.

Throughout the whole paper we use natural units $\hbar=1=c$ and adopt the
mostly positive convention for the metric signature $(\eta_{\mu\nu})={\rm
diag}(-1,+1,+1,+1).$

##  2 Locality, non-localizability, and S-matrix

In this work, we consider general S-matrices possessing certain features which
are expected to capture the (non-)local nature of the underlying theory. We
propose to quantify the degree of non-locality of a given theory by extending
the locality criterion introduced by Jaffe for the Wightman functions in [15,
16] to the behavior of the scattering amplitudes. We review Jaffe’s criterion
in sec. 2.1. Then, we briefly explain our parameterization of non-locality in
sec. 2.2 (in sec. 3.3 we give a more precise parameterization). We introduce
several concrete scenarios in which non-local features are expected to be
present in sec. 2.3. To introduce Jaffe’s locality criterion we are going to
work with the simplest case of a massive scalar field. However, the same
concept can also be generalized to massless fields and, indeed, in sec. 2.3 we
will consider examples of gapless theories.

###  2.1 Locality criterion by Jaffe

We now introduce the concepts of strictly localizable, quasi-local and non-
localizable fields through Jaffe’s classification [15, 16] according to which
the local or non-local nature of a quantum field is incorporated in the
properties of its $n$-point Wightman functions. We are going to present an
intuitive understanding of Jaffe’s idea of (non-)locality without providing a
rigorous treatment of (non-)localizable QFTs which, in fact, is not required
for the scope of this paper.

For illustration purposes, we focus on the two-point Wightman function of a
scalar field $\phi$ with a physical mass $m$. We may formally write its
expression in position space as

$\displaystyle W(x,y)=\bra{0}\phi(x)\phi(y)\ket{0}\,,$ (2)

where $\ket{0}$ denotes a Lorentz invariant vacuum, although strictly speaking
this object is generally understood as a distribution. Let us consider an
interacting QFT of $\phi$ which is subject to the Wightman axioms (such as
Lorentz invariance, the spectral condition, etc) and whose Wightman functions
are understood as tempered distributions [24]. Consequently, the two-point
Wightman function of $\phi$ in momentum space is positive, Lorentz invariant,
and supported in $V_{+}=\\{p:-p^{2}\geq m^{2},p^{0}>0\\}$, satisfying

$\displaystyle\int\mathrm{d}^{4}p\,\frac{\tilde{W}(p)}{(1+\|p\|^{2})^{n}}<\infty\,,$
(3)

for some finite $n$. Here, $\tilde{W}(p)$ is the Wigthman function in momentum
space and $\|p\|:=(\sum_{i=1}^{4}p_{i}^{2})^{1/2}$ denotes the Euclidean
length of a real four-vector $p$. It may be useful to write $\tilde{W}(p)$ in
terms of the Källén-Lehmann spectral density $\rho(-p^{2})$ as

$\displaystyle\tilde{W}(p)=2\pi\,\Theta(p^{0})\rho(-p^{2})\,,$ (4)

where $\Theta(p^{0})$ is the Heaviside step function and $\rho(-p^{2})$ is
required to be a non-negative distribution with support in $V_{+}$. In a free
theory, we have $\rho(-p^{2})=\delta(p^{2}+m^{2})$. The condition (3) requires
that $\rho(-p^{2})$ is polynomially bounded in the large momentum limit
$-p^{2}\to\infty$.

Jaffe pointed out that the restriction on the growth rate of the momentum-
space Wightman function (3) can be relaxed without spoiling the notion of
locality [15, 16].111Technically, this generalization is done by restricting
the space of test functions to the Gel’fand-Shilov space which is a subspace
of the Schwartz space; the latter space is used in the original Wightman’s
formulation. Consequently, Wightman functions in position space are not
necessarily tempered distributions in Jaffe’s formulation, and the bound (3)
on the momentum-space Wightman function can be relaxed. Let us define
$\mu\equiv-p^{2}$ and parameterize the growth rate of $\rho(\mu)$ as follows

$\displaystyle\rho(\mu)\leq A\,\mu^{N}e^{c\,\mu^{\alpha}}\,,$ (5)

where $N$ and $\alpha$ are real non-negative parameters, while $A$ and $c$
denotes some positive constants. According to the classification proposed by
Jaffe, a field $\phi$ is said to be a strictly localizable field or a non-
localizable field depending on the value of $\alpha$ as 222More precisely, the
condition of strict localizability is given by [15, 16]
$\displaystyle\int^{\infty}_{0}\mathrm{dt}\,\frac{\ln\rho(t^{2})}{1+t^{2}}<\infty\,.$
(6) If we parameterize the growth rate of $\rho$ as (5) with $\mu=t^{2}$, the
condition (6) gives $\alpha<1/2$.

$\begin{cases}0\leq\alpha<\frac{1}{2}&:\,\,\text{strictly
localizable}\,,\\\\[5.69054pt] \alpha\geq\frac{1}{2}&:\,\,\text{non-
localizable}\,.\end{cases}$ (7)

Note that the $\alpha=1/2$ case is special and is also called quasi-local; the
reason for this is explained below. The $\alpha=0$ case is the conventional
one in which the Wightman functions are understood as tempered distributions.
The generalization of this classification to $n$-point Wightman functions
$\tilde{W}(p_{1},\dots,p_{n})$ in momentum space is straightforward:

$\displaystyle|\tilde{W}(p_{1},\dots,p_{n})|\leq
A\left(\sum_{i=1}^{n}\|p_{i}\|^{2}\right)^{N}\exp\left[c_{n}\left(\sum_{i=1}^{n}\|p_{i}\|^{2}\right)^{\alpha}\right]\,,$
(8)

where $A$ and $c_{n}$ denote some other positive constants. The parameter
$\alpha$ is defined such that these conditions are satisfied for all $n$ (for
more details, see also [25, 26]). Then, the degree of non-locality of the
theory is defined according to (7).

To get some intuition of (7), following [10], let us define the Wightman
function $W(x,y)$ in position space by naively performing the Fourier
transform of $\tilde{W}(p)$ as

$\displaystyle
W(x,y)=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\tilde{W}(p)e^{ipz}=\int^{\infty}_{0}\mathrm{d}\mu\,\rho(\mu)W_{\text{free}}(z;\mu)\,,$
(9)

where $z:=x-y$ and $W_{\text{free}}(z;\mu)$ denotes a two-point correlation
function in position space for a free field $\phi$ with mass $m=\sqrt{\mu}$,
i.e.,

$\displaystyle
W_{\text{free}}(z;\mu):=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{3}}\Theta(p^{0})\delta(p^{2}+\mu)e^{ipz}\,.$
(10)

From the asymptotic behavior of the free Wightman function at large $|z^{2}|$,
i.e.,

$\displaystyle
W_{\text{free}}(z;\mu)\sim\begin{cases}\displaystyle\frac{(2\sqrt{\mu})^{1/2}}{\bigl{(}4\pi\sqrt{z^{2}}\bigr{)}^{3/2}}e^{-\sqrt{\mu
z^{2}}}&\text{for}\,\,\,\mu z^{2}\gg 1\,,\\\\[8.53581pt] \displaystyle-
ie^{-i\pi/4}\frac{(2\sqrt{\mu})^{1/2}}{\bigl{(}4\pi\sqrt{-z^{2}}\bigr{)}^{3/2}}e^{-i\sqrt{-\mu
z^{2}}}&\text{for}\,\,\,-\mu z^{2}\gg 1\,,\end{cases}$ (11)

we can notice that the integral in eq. (9) does not converge even for $x\neq
y$ when $\alpha\geq 1/2$ because the high-energy behavior of the spectral
density dominates. This intuitively explains why $\alpha<1/2$ gives a locality
criterion as in this case the Wightman function can be defined without the
need of smearing the fields. The $\alpha=1/2$ case is special in the sense
that (9) is convergent for sufficiently large but finite $|x-y|$. This is the
reason why $\alpha=1/2$ is called quasi-local; see also [25, 26].

In general, a field $\phi$ is understood as an operator-valued distribution
which must be averaged with a smooth test function $f_{x_{0}}$ centered around
some reference point $x_{0},$ i.e.,

$\displaystyle\phi[f_{x_{0}}]:=\int\mathrm{d}^{4}x\,\phi(x)f_{x_{0}}(x)\,,$
(12)

whose Wightman function is everywhere finite including the coincident point
$z=0$. Here, $f_{x_{0}}$ is defined in terms of a smooth function
$\tilde{f}(p)$ as

$\displaystyle
f_{x_{0}}(x):=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\,\tilde{f}(p)e^{ip(x-x_{0})}\,.$
(13)

In what follows, we assume that $f(x)$ is real for any real four vector $x,$
thus we have $\tilde{f}(-p)=\tilde{f}^{*}(p)$. Now we define a two-point
Wightman function of an operator $\phi[f_{x}]$ as

$\displaystyle W[f_{x},f_{y}]:=\bra{0}\phi[f_{x}]\phi[f_{y}]\ket{0}$
$\displaystyle=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{3}}\,\Theta(p^{0})|\tilde{f}(p)|^{2}\rho(-p^{2})e^{ip(x-y)}\,.$
(14)

This expression is ensured to be finite when
$|\tilde{f}(p)|<e^{-c\|p\|^{2\alpha}/2}\times\text{(subdominant pieces)}$ in
the large momentum limit $\|p\|\to\infty$. Note that $-p^{2}\to\infty$ implies
$\|p\|\to\infty$.

A test function $f(x)$ has a compact support if and only if $\alpha<1/2$ [16].
To understand this, let us define $g(x)$ as the Fourier transform of a smooth
function $\tilde{g}(p)$,

$\displaystyle
g(x)=\int^{\infty}_{-\infty}\frac{\mathrm{d}p}{2\pi}\,\tilde{g}(p)e^{ipx}\,,$
(15)

and assume that $\tilde{g}(p)$ is bounded on the real axis as
$|\tilde{g}(p)|<b\,e^{-c|p|^{2\alpha}}$ for $p>L$ and $p<-L$ with a
sufficiently large but finite $L>0$. Here, $b$ and $c$ denote positive
constants. When $\alpha>1/2$, $g(x)$ is also analytic in the complex $x$-plane
because it can be shown to be finite for any $x\in\mathbb{C}$:

$\displaystyle|g(x)|<\left|\int^{L}_{-L}\frac{\mathrm{d}p}{2\pi}\,\tilde{g}(p)e^{ipx}\right|+b\int^{\infty}_{L}\frac{\mathrm{d}p}{\pi}\,e^{-c|p|^{2\alpha}}e^{-p\mathrm{Im}(x)}+b\int^{-L}_{-\infty}\frac{\mathrm{d}p}{\pi}\,e^{-c|p|^{2\alpha}}e^{-p\mathrm{Im}(x)}<\infty\,.$
(16)

Similarly, it is also possible to show that $g(x)$ is analytic in the strip
$\\{x:-\infty<\mathrm{Re}(x)<\infty,-c<\mathrm{Im}(x)<c\\}$ when $\alpha=1/2$.
In the $\alpha\geq 1/2$ case, the analyticity domain of $g(x)$ includes the
whole real axis. Hence, we conclude that any nontrivial test function $g(x)$
on the real axis cannot be compactly supported when $\alpha\geq 1/2$. A simple
example of a function with non-compact support is the Gaussian
$\tilde{g}(p)\sim e^{-c\,p^{2}}$ which corresponds to the $\alpha=1$ case. Its
Fourier transform is also a Gaussian $g(x)\sim e^{-x^{2}/4c}$ which is entire
and not compactly supported.

This explanation indicates that the Fourier transform of a function with
compact support on the real axis cannot decrease arbitrarily fast. In the
above discussion, it was shown that $|\tilde{g}(p)|$ cannot decrease faster
than $e^{-c|p|}$ along the real axis if its Fourier transform $g(x)$ is
compactly supported. Interestingly, this behavior for large $|p|$ is similar
to what happens with the Cerulus-Martin lower bound (1) on the $2\rightarrow
2$ scattering amplitude in the hard-scattering regime, i.e.,

$\displaystyle|\mathcal{M}(s,\cos\theta)|\gtrsim e^{-c\sqrt{s}}\,,$ (17)

which was derived for local ($\alpha=0$) QFTs [14].333However, according to
our proof in sec. 4, when $\alpha>0$ the lower bound gets smaller by an
additional factor $s^{\alpha}$ in the exponent, even in the case of strictly
localizable theories (see eq. (63)).

##### S-matrix.

It is worth mentioning that the formulation of quasi-local and non-localizable
theories has been developed. The Wightman formulation of these QFTs was given
in Refs. [25, 26]. Also, under some working assumptions, in Ref. [27] it was
proposed that one can construct a Lorentz invariant and unitary S-matrix which
exhibits standard properties such as cluster decomposition, LSZ formalism, and
CPT invariance.

##### Commutation rules.

Causality of local QFT with $\alpha=0$ is usually ensured by the local
commutation rule

$\displaystyle[\phi(x),\phi(y)]=0\quad\text{for}\quad(x-y)^{2}>0\,.$ (18)

This can also be expressed in terms of the Wightman function in position
space. For the two-point function, eq. (18) implies

$\displaystyle W(x-y)-W(y-x)=0\quad\text{for}\quad(x-y)^{2}>0\,.$ (19)

where $W(x-y)\equiv W(x,y)$. The generalization to $n$-point functions is
straightforward; see [24] for more details. For strictly localizable QFTs with
$0<\alpha<1/2$, it will be straightforward to impose the local commutativity
condition (18) because the test functions can have compact support [15, 16].

For non-localizable QFTs with $\alpha\geq 1/2$, instead, we expect that the
commutator $[\phi(x),\phi(y)]$ does not vanish even for space-like separated
spacetime points due to the non-local nature of the theory, i.e.,

$\left[\phi(x),\phi(y)\right]\neq 0\quad{\rm for}\,\,\,(x-y)^{2}>0\,.$ (20)

In such a case, it will be necessary to modify (19). Instead of (19), we need
to consider the following commutator smeared by some test function $f$ with
non-compact support:

$\displaystyle
W[f_{x},f_{y}]-W[f_{y},f_{x}]=\int\mathrm{d}^{4}\xi\,\left[W(\xi)-W(-\xi)\right]T^{f}_{x-y}(\xi)\,,$
(21)

where $T^{f}_{x-y}(\xi)$ denotes a test function defined in terms of an
original test function $f$ as

$\displaystyle
T^{f}_{x-y}(\xi):=\int\frac{\mathrm{d}^{4}p}{(2\pi)^{4}}\,\left|\tilde{f}(p)\right|^{2}e^{ip\left(\xi-(x-y)\right)}\,.$
(22)

Because $T^{f}_{x-y}(\xi)$ will be nonzero for some timelike $\xi$ even when
$(x-y)^{2}>0$ in general, we expect that the smeared commutator (21) does not
vanish even for spacelike separated points $(x,y)$. Nevertheless, one can
expect that (21) decays exponentially in the limit $(x-y)^{2}\to\infty$
because $|T^{f}_{x-y}(\xi)|$ decays exponentially for large $\|\xi-(x-y)\|$.
For instance, when $\alpha=1$ we have $|\tilde{f}(p)|^{2}\sim
e^{-\sigma\|p\|^{2}}$, thus $|T^{f}_{x-y}(\xi)|$ decays as fast as a Gaussian
$e^{-\sigma\|\xi-(x-y)\|^{2}}$.

Indeed, for the nonlocalizable case with $\alpha>1/2$, Ref. [28] proposed an
asymptotic commutativity condition instead of the local commutation rule (18)
or (19). Roughly speaking, the condition reads

$\displaystyle\left|\int\mathrm{d}^{4}\xi\,\left[W(\xi)-W(-\xi)\right]T^{f}_{x-y}(\xi)\right|\lesssim
A\,e^{-\sigma\|x-y\|^{2\alpha/(2\alpha-1)}}\quad\text{for}\quad(x-y)^{2}>0\,,$
(23)

where $A$ and $\sigma$ denote some positive constants whose precise values are
determined by the functional form of $T^{f}_{x-y}(\xi)$. The condition (23) is
imposed for all the allowed test functions $T^{f}_{x-y}(\xi)$ for a given non-
localizable theory. For a more precise definition of asymptotic commutativity
and its implications see [28, 29] and references therein.

In the case $\alpha=1/2$, the condition (23) can be improved thanks to the
quasi-local nature of the theory: $T^{f}_{x-y}(\xi)$ can vanish for some $\xi$
in this case. In particular, when $T^{f}_{x-y}(\xi)=0$ for any $\xi$
satisfying $\xi^{2}\leq\ell^{2}$, the quasi-locality (or “$\ell$-locality”)
condition holds [25],

$\displaystyle\int\mathrm{d}^{4}\xi\left[W(\xi)-W(-\xi)\right]T^{f}_{x-y}(\xi)=0\,.$
(24)

Here, $\ell$ is defined by the growth rate of spectral density as
$\ell/\sqrt{2}={\displaystyle\lim_{t\to\infty}}\ln\rho(t^{2})/t$. Since
$T^{f}_{x-y}(\xi)$ cannot vanish at $\xi=x-y$, the quasi-local commutativity
(24) can be satisfied only when $x$ and $y$ are sufficiently separated to
satisfy $(x-y)^{2}>\ell^{2}$. This result is natural from the perspective that
the non-locality appears only up to a certain finite distance $\ell$ for
quasi-local case as already mentioned below eq. (11). The generalization of
this condition to $n$-point functions is also given in [25].

##### Connection with low-energy EFTs.

Even though the smeared commutator (21) does not vanish even when $x$ and $y$
are space-like separated in non-localizable theories ($\alpha\geq 1/2$), it
becomes negligibly small for $(x-y)^{2}\gg\sigma^{(1/\alpha)-2}$. In this
sense, the parameter $\sigma$ may govern the scale of non-locality. This
suggests that, even when a UV completion is non-localizable, its effective
description at low energies below a certain non-locality scale would be given
in terms of a local QFT for which the usual local commutation rule (18) is
satisfied. This point may also be clarified by performing the low-energy
expansion of the exponentially-growing spectral density $\rho(\mu)$. For
instance, the function $\rho(\mu)=\mu^{N}e^{\mu/\mu_{0}}$ can be well
approximated by polynomials of some finite degree when focusing on the
sufficiently low-energy regime $\mu\ll\mu_{0}$. This means that, as far as
low-energy regimes are concerned, the theory can be well approximated by the
standard local QFT with $\alpha=0;$ see [10] for more details on the EFT
perspective of non-localizability.

##### Green’s functions.

For usual local QFTs with $\alpha=0$, the time-ordered two-point function
admits the Källén-Lehmann spectral representation which is analogous to (9)
for the Wightman function. Let us suppose that $\rho(\mu)$ decays faster than
$1/\log\mu$ for $\mu\to\infty$ as a toy example. In this case, the Wightman
function in position space is everywhere finite and (9) is correct. Similarly,
the spectral representation for the time-ordered two-point function
$\tilde{D}_{F}(p)$ in momentum space is given by [30]

$\displaystyle\tilde{D}_{F}(p)=\int^{\infty}_{0}\mathrm{d}\mu\,\rho(\mu)\frac{-i}{p^{2}+\mu-i\epsilon}\,.$
(25)

Substituting $\rho=\delta(\mu-m^{2})$ into this equation, we recover
$\tilde{D}_{F}(p)$ in the free theory. When $\rho$ grows polynomially as
$\rho(\mu)\sim\mu^{N}$, eq. (25) diverges. In such cases, the spectral
representation is derived as [30]

$\displaystyle\tilde{D}_{F}(p)=Q_{N+1}(-p^{2})\int^{\infty}_{0}\mathrm{d}\mu\,\frac{\rho(\mu)}{Q_{N+1}(\mu)}\frac{-i}{p^{2}+\mu-i\epsilon}+i\sum_{j=0}^{N}c_{j}(p^{2})^{j}\,.$
(26)

Here, a positive polynomial $Q_{N+1}(-p^{2})$ of $(N+1)$-th degree is
introduced such that the integral converges; $c_{j}$ are some constants. The
choice of $Q_{N+1}$ is not unique. This freedom leads to an ambiguity in the
definition of $\tilde{D}_{F}$ which can be renormalized by $c_{j}$. In
position space, these ambiguous terms correspond to the sum of derivatives of
the delta function. We can similarly define the spectral representation for
$\tilde{D}_{F}(p)$ even in the non-standard case $\alpha>0$ where
$\rho(\mu)\sim\mu^{N}e^{c\mu^{\alpha}}$ at large $\mu$. In such cases, we
should use a positive function $Q(-p^{2})$ which grows as fast as
$(-p^{2})^{N+1}e^{c(-p^{2})^{\alpha}}$ instead of a polynomial
$Q_{N+1}(-p^{2})$.

The spectral representation (26) suggests that when the Wightman functions
grow in the large momentum limit, the time-ordered Green’s function would also
grow accordingly. For instance, when the former grows polynomially as
$\rho(\mu)\sim\mu^{N}$ at large $\mu$, the growth rate of the latter may be
bounded in the complex $p^{2}$-plane as

$\displaystyle\lim_{|p|^{2}\to\infty}\frac{\tilde{D}_{F}(p)}{|p^{2}|^{N+1}}=0\,.$
(27)

A similar equation can also be written for $\alpha>0$ in which case the time-
ordered Green’s function may be exponentially bounded in the complex
$p^{2}$-plane.

The above discussion can also be extended to $n$-point functions with $n>2$;
see for example Ref. [31]. This indicates that the $n$-point scattering
amplitudes, which are related to the $n$-point time-ordered Green’s function
via the reduction formula, are also bounded in the large momentum limit
accordingly. Indeed, it is known that the four-point scattering amplitude is
polynomially bounded in the case $\alpha=0$, and this feature leads to the
dispersion relations with a finite number of subtractions. It is also found
that, even for the $\alpha>1/2$ case, the upper bound on the four-point
amplitude is given by the growth rate of the Wightman function as
$|\mathcal{M}(s,t)|<e^{s^{\alpha}}$ in the Regge limit ($s\to\infty$ with
fixed $t$) [32]. Here, $s$ and $t$ are usual Mandelstam variables
corresponding to the center-of-mass energy squared and the momentum transfer
squared, respectively.

###  2.2 Locality criterion by S-matrix

Based on the above observations, in this paper we propose to parameterize the
non-locality of given models in terms of the growth rate of the scattering
amplitude in the complex Mandelstam variable plane. We discuss the
implications of such a rapid growth in the next sections. In particular,
roughly speaking, our parameterization is the following:

$\displaystyle|\mathcal{M}(s,t(\cos\theta))|\leq
A\,s^{N}\,e^{c\,s^{\alpha}}\quad\text{for}\quad|s|\to\infty\,,$ (28)

where $\cos\theta$ takes values in a certain domain of the complex
$\cos\theta$-plane that we will define in the next section; e.g., see Fig. 1.
In sec. 3.3 we will provide more details on our assumptions that are needed to
prove the lower bounds in secs. 4 and 5.

Even though our parameterization of non-locality in eq. (28) is motivated by
the Jaffe’s criterion of (non-)localizability, we emphasize that our
formulation does not explicitly rely on Jaffe’s classification of QFTs as our
analysis is solely based on the properties of the S-matrix. This means that,
in principle, our study could be relevant even for theories which do not fall
into Jaffe’s QFTs but whose S-matrix has some peculiar features such as the
non-polynomial behavior in eq. (28).

##### Relation to causality.

The macro-causality condition (24) or (23) for non-localizable theories
implies that the retarded Green’s function $G_{\text{ret}}(x,y),$ which is
defined under the appropriate smearing by test functions, can be non-zero for
space-like separations but it must be sufficiently suppressed when $(x-y)^{2}$
is sufficiently larger than the square of the non-locality scale.
Interestingly, by considering the $(1+0)$-dimensional scattering model (the
so-called signal model discussed previously, e.g., in [33, 34]), Ref. [10]
claims that the analyticity of the S-matrix in the complex energy plane and
the exponential boundedness (28) can be sufficient to ensure the macro-
causality for non-localizable theories; whereas analyticity and the
boundedness (28) with $0\leq\alpha<1/2$ can imply causality for local and
strictly localizable theories, i.e., eq. (18). These observations motivate us
to assume suitable analytic properties and the condition of exponential
boundedness (28) on the behavior of the S-matrix.

####  2.2.1 Positivity bounds on the S-matrix

The high-energy behavior of the scattering amplitudes in non-localizable
theories was first discussed in [32]. More recently, in Ref. [31] it was shown
that under some working assumptions the behavior of a scattering amplitude
$\mathcal{M}(s,t\leq 0)$ in the Regge limit is polynomially bounded as

$\displaystyle\left|\mathcal{M}(s,t\leq
0)\right|<s\left(A\,s^{2\alpha}+B\,s^{\alpha}\log(s/s_{0})+C\,\log^{2}(s/s_{0})\right)\,,$
(29)

where $A,$ $B,$ $C$ are some positive constants and $s_{0}$ some energy scale
squared; the Froissart bound [12] is consistently recovered in the local
$\alpha=0$ case. Combined with the Phragmén-Lindelöf theorem, the bound (29)
gives rise to the fixed-$t$ dispersion relations for the $\alpha<1$ case with
a finite number of subtractions, under the assumption of the sub-exponential
growth of $|\mathcal{M}(s,t)|$ in the limit $|s|\to\infty$ in the cut
$s$-plane. In particular, the number of subtractions can be two in the
strictly localizable case $\alpha<1/2$.444This does not mean that the twice-
subtracted dispersion relation has to be always violated in the $\alpha\geq
1/2$ case. The twice-subtracted dispersion relation leads to the so-called
positivity bound on the low-energy coefficient $c_{2,0}$ as [35, 36]

$\displaystyle
c_{2,0}=\frac{2}{\pi}\int^{\infty}_{4m^{2}}\mathrm{d}s\,\frac{\mathrm{Im}\,\mathcal{M}(s,0)}{s^{3}}>0\,,$
(30)

where $\\{c_{n,m}\\}$ are defined by

$\displaystyle\mathcal{M}(s,t)=\text{($s$,$t$,$u$)-poles}+\sum_{n,m=0}^{\infty}c_{n,m}s^{n}t^{m}\,;$
(31)

the condition $\mathrm{Im}\,\mathcal{M}(s,0)>0$ implied by unitarity is used
in (30). Hence, EFTs with negative $c_{2,0}$ cannot be embedded into strictly
localizable UV completions [31]. Recently, stronger bounds on low-energy
coefficients have been derived by making use of crossing symmetry [37, 38, 39,
40]. As suggestive observations, we will see in sec. 2.3 that the positivity
bound on $c_{2,0}$ derived from the twice-subtracted dispersion relation is
indeed violated in some candidates for non-localizable theories (with
$\alpha\geq 1/2$).

On the other hand, the bound (29) also shows that it is possible to derive
dispersion relations with a finite number of subtractions even for
$1/2\leq\alpha<1$. This means that it is still possible to derive positivity
bounds on higher-order coefficients such as $c_{n,0}$ with $n\geq 4$ for
theories with $1/2\leq\alpha<1$ [31]. It would be interesting to investigate
the consequences of these bounds on non-localizable theories. We leave this
task as a future work.

###  2.3 Candidates for non-localizable theories

Let us now discuss some examples of possible candidates for non-localizable
QFTs; see also [10] for a suggestive summary of these examples.

####  2.3.1 Gravity and black-hole formation

Black-hole (BH) physics provides a natural explanation to understand why
gravity is inherently non-local. Let us focus on the $2\rightarrow 2$
scattering amplitude $\mathcal{M}(s,t)$ of gravitons. In the usual
perturbative QFTs, it is possible to probe arbitrary short-distance scales
$L\sim E^{-1}$ by considering high-energy hard scattering processes with
$s,-t\sim E^{2}$. In Einstein’s General Relativity (GR), however, there exists
a lower limit on the distance scale $L$ that can be probed before BH formation
sets in, and this is given by $L\gtrsim r_{s}(E)=2E/M_{p}^{2}\,.$ Here,
$r_{s}(E)$ is the Schwarzschild radius of a BH of mass $E$. BH formation
prevents us from probing distances smaller than $r_{s}(E)$. The important
feature is that the length scale $r_{s}(E)$ grows with the energy, thus more
energetic probes will be affected by a larger uncertainty in resolving
distances.

In this case, the non-local nature of the theory may also be understood from
the exponential growth of the spectral density. In fact, if BH states dominate
at high energy, then the high-energy behavior of the spectral density will be
proportional to the number of BH states [2, 4], i.e.

$\rho(s)\sim e^{S_{\rm
BH}(\sqrt{s})}=e^{c\,(\sqrt{s}/M_{p})^{\frac{D-2}{D-3}}}\,,$ (32)

where $S_{\rm BH}\sim(\sqrt{s}/M_{p})^{\frac{D-2}{D-3}}$ is the Bekenstein-
Hawking entropy and $D$ is the spacetime dimension. Comparing this with
Jaffe’s criterion given by eqs. (5) and (7), we may identify the degree of
non-locality as

$\alpha=\frac{1}{2}\frac{D-2}{D-3}\,,$ (33)

which depends on the spacetime dimensions $D$. Interestingly, we have
$\alpha>1/2$ for $D>3$. This implies that the gravitational theory is non-
localizable. In particular, in $D=4$ we get $\alpha=1.$

The violation of locality can also be understood in the language of S-matrix
around flat spacetime. In fact, it is believed that a $2\rightarrow 2$
scattering amplitude at high energies will be dominated by BH production which
would make the scattering amplitude exponentially suppressed. For instance,
for fixed-angle (hard-scattering regime) one expects [17, 18]:555It is worth
mentioning that the behavior in eq. (34) may not only be typical of Einstein’s
GR, but it might also appear in some renormalizable higher-derivative theories
of gravity. The reason for this is that the tree-level amplitudes involving
massless gravitons as external legs are the same as in GR [41, 42, 43];
however, see [43, 44] for a different point of view. On the other hand, in
some other approaches to quantum gravity, graviton-graviton amplitudes could
still be polynomially boundeded and respect the Cerulus-Martin bound; e.g.,
this was claimed in the context of asymptotically safe gravity [45]. It would
be interesting to understand what happens to BH formation in this case; e.g.,
see [46, 47] for some discussions.

$E\gg M_{p}\quad\Rightarrow\quad\mathcal{M}(s,\cos\theta)\sim e^{-S_{\rm
BH}(\sqrt{s})}=e^{-c\,(\sqrt{s}/M_{p})^{\frac{D-2}{D-3}}}\,,$ (34)

where $\sqrt{s}$ is the center-of-mass energy of the incoming two-particle
state. In sec. 6.1 we will explicitly prove that the behavior (34) implies
that $\mathcal{M}(s,t)$ violates polynomial boundedness.

A few remarks are in order. First, the above estimates of the contributions
from BH formation to $\mathcal{M}(s,t)$ are suggestive, but to our knowledge
such non-perturbative contributions have not been rigorously evaluated yet.
There is a possibility that gravitational amplitudes $\mathcal{M}(s,t<0)$ with
fixed $t$ are actually polynomially bounded in any directions in the upper
half complex $s$-plane. Second, even though the above estimates indicate some
non-polynomial behavior of the amplitude, we can expect that
$\mathcal{M}(s,t<0)$ with fixed $t$ is polynomially bounded in any directions
in the upper half-plane in higher spacetime dimensions $D>4$. The reason for
this is the following. In $D>4$, partial-wave unitarity would imply the
polynomial boundedness of $\mathcal{M}(s,t<0)$ along the real-$s$ axis under
some physical assumptions [48], though we may have the exponential growth
$\mathcal{M}(s,t<0)\sim e^{|s|^{\alpha}}$ with eq. (33) in some directions in
the upper half-plane. Because we expect $\alpha<1$ when $D>4$, the Phragmén-
Lindelöf theorem implies that $\mathcal{M}(s,t<0)$ is polynomially bounded in
any directions in the upper half-plane. The discussion here is consistent with
Ref. [48] in which the high-energy behavior of gravitational amplitudes is
investigated.

Finally, in the S-matrix language the BH formation is described as a non-
perturbative scattering process and can be effective only in the super-
Planckian regime $s\gg M_{p}^{2}.$ If gravity is UV completed within its
weakly-coupled regime $s\lesssim M_{p}^{2}$, it is not necessary to concern
about BH formation and we may be interested in UV complete amplitudes in which
higher-order terms in $1/M_{p}^{2}$ are truncated. We can expect that such
amplitudes would satisfy the fixed-$t$ dispersion relation with finite number
of subtractions: e.g., tree-level string amplitudes satisfy them.

####  2.3.2 Classicalization proposal

In Ref. [49], the softening behavior of scattering amplitudes caused by BH
formation was proposed as a mechanism to achieve the UV completion of GR. The
same type of idea was generalized to other non-renormalizable theories in
which the high-energy behavior of scattering amplitudes gets weakened by the
formation of a macroscopic semi-classical object called classicalon (analog of
the BH). This type of UV completion was named classicalization [9].

An example of classicalizing theory was claimed to be the Goldstone model with
massless kinetic term and derivative potential [50]. Let us consider the
following scalar field theory

$\displaystyle\mathcal{L}=-\frac{1}{2}(\partial\phi)^{2}+\frac{\epsilon}{\Lambda^{4}}(\partial\phi)^{4}+\cdots\,,$
(35)

where $\epsilon=\pm 1$ and the ellipses stands for the higher-order terms.
This model violates perturbative unitarity at $s\sim\Lambda^{2}$, suggesting
the appearance of new degrees of freedom at around this scale. According to
the standard Wilsonian UV completion point of view, such degrees of freedom
are some new heavy particles. The model (35) is then regarded as an EFT. A
famous example of this type of scenario is the UV completion of electroweak
theory by the Higgs boson, where locality is mantained also in the UV. Hence,
the positivity bound (30) which requires $\epsilon=1$ for the model (35)
should be satisfied and, in known examples, this is indeed the case; see also
[36] for other examples.

However, in Ref. [9] another possibility for UV completion was proposed,
according to which the scalar field $\phi$ remains the only perturbative
degree of freedom at high energies, but new semi-classical states can be
formed non-perturbatively as a collective phenomena of $\phi$-particles. In
this non-Wilsonian scenario, the scattering amplitude is unitarized through
the formation of a classicalon which is a macroscopic bound state of
$\phi$-particles. This is analogous to the BH formation in GR and its radius
grows as the energy $\sqrt{s}$ increases with a certain positive power that is
model-dependent. As a result, distances shorter than the size of the
classicalon cannot be probed, thus classicalization is characterized by some
inherent non-locality. Then, we expect that the S-matrix of the classicalizing
models has some peculiar features reflecting non-locality.

In fact, because of classicalon formation, a $2\rightarrow 2$ scattering
amplitude in the model (35) is expected to be exponentially suppressed in the
hard scattering regime as $\mathcal{M}(s,\cos\theta)\sim e^{-c\,s^{2/3}}$. In
addition, the entropy of the classicalon was claimed to be given by $S\sim
c\,s^{2/3}$ [9]. As a consequence, the high-energy behavior of the spectral
density is expected to be dominated by the classicalon density of states,
i.e.,

$\rho(\mu)\sim e^{c\,\mu^{2/3}}\,.$ (36)

This indicates that the classicalizing Goldstone model in eq. (35) will be a
non-localizable QFT with $\alpha=2/3.$ Interestingly, it was found in Ref.
[50] that the classicalon solution exists only when $\epsilon=-1$, which
violates the positivity bound (30). This is consistent with the finding in
Ref. [31] according to which the violation of (30) excludes the strictly
localizable QFTs ($\alpha<1/2$) as candidates for UV completion.

More generally, the classicalon solution depends on the details of the higher-
derivative interaction terms. We refer to the $n$-point vertex with $2k$
derivatives as $\partial^{2k}\phi^{n}$; e.g., the quartic derivative term
$(\partial\phi)^{4}$ corresponds to $(k,n)=(2,4)$. Given an interaction term
$\partial^{2k}\phi^{n},$ it was argued that the density of states of the
classicalon should be proportional to $e^{c\,r_{*}\sqrt{s}},$ where
$r_{*}\sim(\sqrt{s})^{\frac{n-2}{n+4k-6}}$ is the size of the classicalon
(also called classicalization radius) in $D=4$ spacetime dimensions [9].
Notice that the largest radius corresponds to the lowest order interaction
term in a derivative expansion in the Lagrangian. Thus, in a classicalizing
model whose lowest order interaction term contains $2k$ derivatives and $n$
fields, we expect the spectral density to grow as

$\rho(\mu)\sim e^{c\,\mu^{\alpha}}\,,\quad\alpha=\frac{n+2k-4}{n+4k-6}\,.$
(37)

From the expression of $\alpha,$ it follows that $\alpha>1/2$ for $n\geq 3,$
which means that classicalizing theories are always non-localizable. Note
that, in the case $k=1$ we have $\alpha=1$ for any $n,$ in agreement with
four-dimensional Einstein’s gravity.

####  2.3.3 Galileon theories

Ref. [10] argued that the Galileon model belongs to the class of non-
localizable QFTs with $\alpha=3/5$ in $D=4$ spacetime dimensions. In
particular, the spectral density is found to grow exponentially as
$\rho(\mu)\sim e^{c\,\mu^{3/5}}$ in the Galileon theory which can be mapped to
the free theory via Galileon duality [51]. This behavior may be also
understood consistently from the viewpoint of the classicalization proposal as
suggested in [10]. The Lagrangian of Galileon theories is [52]

$\mathcal{L}=-\frac{1}{2}(\partial\phi)^{2}+\frac{g_{3}}{\Lambda^{3}}(\Box\phi)(\partial\phi)^{2}+\frac{g_{4}}{\Lambda^{4}}(\partial\phi)^{2}\left[(\Box\phi)^{2}-(\partial_{\mu}\partial_{\nu}\phi)^{2}\right]+\cdots\,,$
(38)

where $g_{3}$ and $g_{4}$ are dimensionless coupling constants, and the
higher-order terms are suppressed. The second term and the third term are the
lowest-order interaction terms called the cubic Galileon and the quartic
Galileon, respectively. These non-renormalizable interactions lead to the
violation of perturbative unitarity at energy scales above $\Lambda$. Since
the $(\partial\phi)^{4}$ term is absent in the Lagrangian, the positivity
bound (30) derived from the twice-subtracted dispersion relation is violated.
Even though the positivity bound is applicable only for gapped theories,
strictly speaking, this result suggests that the Galileon theories (38) may
not be embedded into standard local UV complete QFTs. Then, let us regard the
classicalization phenomenon as a mechanism for the UV completion of the
Galileon theories. The cubic and quartic Galileon terms are schematically
written as $\partial^{2k}\phi^{n}$ with $(k,n)=(2,3)$ and $(k,n)=(3,4)$,
respectively. Both of them predict $\alpha=3/5$ from eq. (37), thus implying
that the Galileon field is non-localizable with $\alpha=3/5$ in $D=4$
dimensions.

####  2.3.4 Little String Theory

Another possible candidate for non-localizable QFTs are Little String Theories
[53] which describe string theory or M-theory in a specific decoupling limit
in which gravitational and other bulk degrees of freedom are decoupled from
the fivebranes. The (non-gravitational) degrees of freedom living on the
fivebrane are interacting and, because of the involved decoupling limit, the
corresponding six-dimensional theory turns out to be different from the
conventional local QFTs. The decoupling limit is defined by taking
$g_{s}\rightarrow 0$ and $M_{p}\rightarrow\infty,$ while keeping the string
scale $M_{s}$ fixed [53]. This means that string effects are still present and
non-negligible although the string coupling $g_{s}$ and the gravitational
coupling $1/M_{p}$ vanish. Indeed, the name Little String Theory derives from
the fact that some of the properties of string theory that are not based on
standard local QFT are still mantained; e.g., $T$-duality is still respected.

It has been claimed that Little String Theories can be described as quasi-
local QFTs (i.e. with $\alpha=1/2$) because the Wightman function of the
corresponding fields can grow exponentially as $\rho(\mu)\sim e^{c\sqrt{\mu}}$
[54, 55]. In particular, this exponential growth was argued based on the
Hagedorn behavior of the density of energy states $\rho(E)\sim e^{c\,E},$
where $E=\sqrt{\mu}.$

Before concluding this section, it is worth mentioning that even in
perturbative string theory non-local features due to string effects naturally
emerge [56]. The non-local nature of perturbative string theory has been
studied by investigating the high-energy behavior of the scattering between
strings [6, 7, 19]; see sec. 6.4 for more discussions.

##  3 Basics & assumptions

In the next two sections we are going to prove two lower bounds on how fast an
elastic scattering amplitude can decrease in two different kinematic regimes:
in the hard-scattering limit (high center-of-mass energy and fixed angle) and
in the Regge limit (high center-of-mass energy and fixed momentum transfer).
Unlike the proof of the Cerulus-Martin bound [14], more generally we will
assume exponential boundedness instead of polynomial boundedness. To derive
such bounds several mathematical details need to be explained. In this section
we are going to introduce some basics of kinematics and discuss the
assumptions used for the proofs in the next sections.

###  3.1 Basics of kinematics

Let us decompose the S-matrix $\bm{S}$ into an identity matrix $\bm{1}$ and
the transfer matrix $\bm{T}$ as $\bm{S}=\bm{1}+i\bm{T}$. We define the
scattering amplitude $\mathcal{M}$ from the initial state $\ket{i}$ to the
final state $\ket{f}$ as

$\displaystyle\bra{f}\bm{T}\ket{i}=(2\pi)^{4}\delta^{(4)}(p_{i}-p_{f})\mathcal{M}(i\to
f)\,,$ (39)

where $p_{i}$ and $p_{f}$ are momentum eigenvalues of $\ket{i}$ and $\ket{f}$,
respectively. We now consider a $2\rightarrow 2$ elastic scattering with
incoming momenta $p_{1},p_{2}$ and outgoing momenta $p_{3},p_{4},$ such that
$p_{1}+p_{2}=p_{3}+p_{4}$. The corresponding scattering amplitude
$\mathcal{M}$ is defined by

$\displaystyle\langle
p_{3},p_{4}|\bm{T}|p_{1},p_{2}\rangle=(2\pi)^{4}\delta(p_{1}+p_{2}-p_{3}-p_{4})\mathcal{M}(s,t,u)\,,$
(40)

where $s$, $t$, and $u$ are the Mandelstam variables,

$\displaystyle s=-(p_{1}+p_{2})^{2}\,,\quad t=-(p_{1}-p_{3})^{2}\,,\quad
u=-(p_{1}-p_{4})^{2}\,.$ (41)

For simplicity we assume that all masses are equal, $p_{i}^{2}=-m^{2},$ thus
the relation $s+t+u=4m^{2}$ holds. By working in the center-of-mass frame,
i.e., $\vec{p}_{1}+\vec{p}_{2}=\vec{0}=\vec{p}_{3}+\vec{p}_{4},$ and defining
the scattering angle $\theta$ through the relation
$\vec{p}_{1}\cdot\vec{p}_{3}=|\vec{p}_{1}||\vec{p}_{3}|\cos\theta,$ the
variable $s$ coincides with the center-of-mass energy squared,
$s=(p^{0}_{1}+p_{2}^{0})^{2}=(p^{0}_{3}+p_{4}^{0})^{2},$ whereas the other two
Mandelstam variables can be written as

$t=-2p^{2}_{s}(1-\cos\theta)\,,\quad u=-2p_{s}^{2}(1+\cos\theta)\,,$ (42)

where $p_{s}$ is the modulus of the spatial momentum in the center-of-mass
frame $p_{s}=|\vec{p}_{1}|=|\vec{p}_{3}|=\frac{1}{2}\sqrt{s-4m^{2}}.$ The
variables $t$ and $u$ have the physical meaning of momentum transfer squared.

Because of the condition $s+t+u=4m^{2}$, we can regard $\mathcal{M}(s,t,u)$ as
a function of two kinematic invariants. In this paper we will consider two
different choices: we will work either with the variables $s$ and
$z=\cos\theta=1+t/2p^{2}_{s},$ or with $s$ and $t.$ In the former case we
write $\mathcal{M}$ as $\mathcal{M}(s,z);$ whereas by using an abuse of
notation we write $\mathcal{M}(s,t)$ in the latter case. These two functions
are related as $\mathcal{M}(s,z)=\mathcal{M}(s,t(z))$ with
$t(z)=-2p_{s}^{2}(1-z)$. We use $\mathcal{M}(s,z)$ when we analyze the
amplitude in the high-energy limit $s\to\infty$ with fixed $z$. This includes
the hard-scattering regime defined as

$\text{hard-scattering regime:}\qquad
s\rightarrow\infty\,,\quad-1<\cos\theta<1={\rm fixed}\,.$ (43)

Whereas, we use $\mathcal{M}(s,t)$ when we consider the Regge regime:

$\text{Regge regime:}\qquad s\rightarrow\infty\,,\quad t={\rm fixed}\,.$ (44)

This limit includes the high-energy scattering processes at small angles
$|\theta|\ll 1$. For large but finite $s$, we define the Regge limit as $s\gg
m^{2},|t|$, whereas the hard-scattering limit is $s\sim-t\gg m^{2}$.

We assume that the amplitude $\mathcal{M}(s,z)$ is analytic in the complex
$z$-plane for $s\geq 4m^{2}$, except for the presence of usual branch cuts on
the real axis for $s\geq 4m^{2},$ $z\geq 1+2m^{2}/p_{s}^{2}$ and
$z\leq-1-2m^{2}/p_{s}^{2}.$ These details will be important for the next
subsection.

###  3.2 Analyticity domain in the $\cos\theta$-plane

Let us now introduce some mathematical details that are needed for both
derivations in secs. 4 and 5. We define a subdomain $\mathcal{D}_{R}$ in a
finite region of the $z$-cut plane. Though the precise shape of
$\mathcal{D}_{R}$ is given in the paper by Cerulus and Martin [14], here we
will proceed in several steps and provide more details as its definition is
not straightforward. To define it we first make two variable transformations
$z\rightarrow w\rightarrow\tau,$ and by imposing some conditions in the
complex $\tau$-plane we fix the shape of $\mathcal{D}_{R}.$ Let us show how
this can be done.

As we mentioned in sec. 3.1, $\mathcal{M}(s,z)$ has two branch cuts in the
$z$-plane which run from $-\infty$ to $-\rho$ and from $\rho$ to $+\infty,$
respectively; we defined $\rho:=1+2m^{2}/p_{s}^{2}$. Let us also introduce two
points $z=-a$ and $z=a$ on the real axis such that $0<a<1.$ Then, we have
$\rho>1>a.$ Now we make the first transformation:

$w(z)=\frac{\rho}{z}\left(\rho-\sqrt{\rho^{2}-z^{2}}\right)\,,\quad
w\in\mathbb{C}\,,$ (45)

and define

$\displaystyle w(z=\pm
1)=\pm\rho\left(\rho-\sqrt{\rho^{2}-1}\right)\equiv\pm\varepsilon\,,$ (46)
$\displaystyle w(z=\pm
a)=\pm\frac{\rho}{a}\left(\rho-\sqrt{\rho^{2}-a^{2}}\right)\equiv\pm
a^{\prime}\,.$ (47)

From these definitions, it is easy to verify the inequalities
$\rho>1>\varepsilon>a^{\prime}>0$.

We now make a second transformation from $w$ to a new variable $\tau$ that is
defined such that the segment $[-a^{\prime},a^{\prime}]$ in the $w$-plane is
mapped into a unit circle in the $\tau$-plane. This map is given by

$\displaystyle\tau(w)=\frac{1}{a^{\prime}}\left(w+\sqrt{w^{2}-a^{\prime
2}}\right)\,,\quad\tau\in\mathbb{C}\,.$ (48)

Let us check that the segment $w\in[-a^{\prime},a^{\prime}]$, i.e., the domain
$\mathcal{W}_{\text{sg}}=\mathcal{W}_{\text{sg}}^{+}\cup\mathcal{W}_{\text{sg}}^{-}$
where
$\mathcal{W}_{\text{sg}}^{\pm}\equiv\\{w\in\mathbb{C}:\,-a^{\prime}<\mathrm{Re}\,w<a^{\prime}\,,\,\mathrm{Im}\,w=\pm
0^{+}\\},$ is mapped into the unit circle in the $\tau$-plane. For
$w\in\mathcal{W}_{\text{sg}}^{\pm}$, we have

$\displaystyle\sqrt{w^{2}-a^{\prime 2}}=\sqrt{a^{\prime 2}-w^{2}}e^{\pm
i\pi/2}=\pm i\sqrt{a^{\prime 2}-w^{2}}\,,$ (49)

and by writing $\mathrm{Re}\,w=a^{\prime}\cos\vartheta$ we obtain a simple
expression for $\tau(w):$

$\displaystyle\tau(w)=\cos\vartheta\pm i\sin\vartheta=e^{\pm
i\vartheta}\qquad(w\in\mathcal{W}_{\text{sg}})\,.$ (50)

This concludes that the domain $\mathcal{W}_{\text{sg}}$ in the $w$-plane is
mapped to the unit circle in the complex $\tau$-plane.

We now define

$\displaystyle\tau(w=\rho)=\frac{1}{a^{\prime}}\left(\rho+\sqrt{\rho^{2}-a^{\prime
2}}\right)\equiv R\,,$ (51)
$\displaystyle\tau(w=\varepsilon)=\frac{1}{a^{\prime}}\left(\varepsilon+\sqrt{\varepsilon^{2}-a^{\prime
2}}\right)\equiv E\,.$ (52)

Since $\rho>1>\varepsilon>a^{\prime},$ it follows that $R>E>1.$

We choose the subdomain $\mathcal{D}_{R}$ in the $z$-plane by requiring that
its boundary corresponds to the circle of radius $R$ in the $\tau$-plane,
i.e.,

$\displaystyle\partial\mathcal{D}_{R}=\left\\{z\in\mathbb{C}:\,|\tau(w(z))|^{2}=R^{2}\right\\}\,.$
(53)

The shape of $\partial\mathcal{D}_{R}$ is the dumb-bell shown in Fig. 1, and
we will explicitly determine it below.

Figure 1: This figure shows the boundary domain $\partial\mathcal{D}_{R}$
(blue line) in the $z$-plane defined in eq. (53). Below we show that
$\partial\mathcal{D}_{R}$ is mapped to the ellipse $\mathcal{W}_{\beta_{R}}$
in the $w$-plane and to the circle of radius $R$ in the $\tau$-plane.
Moreover, the red line corresponds to the segment $[-a,a]$, while the orange
line to the boundary domain $\partial\mathcal{D}_{E}$ that is mapped to the
ellipse $\mathcal{W}_{\beta_{E}}$ in the $w$-plane and to the circle of radius
$E$ in the $\tau$-plane as we will show below. To draw this figure we set
$\rho=1.1$ and $a=0.8,$ which also give $a^{\prime}\simeq 0.474$.

##### From $w$-plane to $\tau$-plane.

We now clarify how the domains in the $w$-plane look like in the $\tau$-plane.
For this purpose, we introduce two real variables $\beta$ and $\varphi$ which
are related to $w$ as $w=a^{\prime}\cosh(\beta+i\varphi)$. With this
parametrization, the domain $\mathcal{W}_{\beta}$ defined as

$\mathcal{W}_{\beta}\equiv\\{w\in\mathbb{C}:w=a^{\prime}\cosh(\beta+i\varphi),\,-\pi<\varphi\leq\pi\\}$
(54)

is an ellipse666This can be easily checked by writing
$w=a^{\prime}\cosh(\beta+i\varphi)=x+iy$, with
$x=a^{\prime}\cosh\beta\cos\varphi$ and $y=a^{\prime}\sinh\beta\sin\varphi,$
from which one can derive the ellipse equation
$x^{2}/(a^{\prime}\cosh\beta)^{2}+y^{2}/(a^{\prime}\sinh\beta)^{2}=1$. with
foci $w=\pm a^{\prime},$ semi-major axis $a^{\prime}\cosh\beta$ and semi-minor
axis $a^{\prime}\sinh\beta$. In the limit $\beta\to 0^{+}$, the ellipse
$\mathcal{W}_{\beta}$ shrinks and coincides with the domain
$\mathcal{W}_{\text{sg}}$ because we have $\mathrm{Im}\,w\to 0^{+}\sin\varphi$
and $\mathrm{Re}\,w\to a^{\prime}\cos\varphi$. In terms of the variables
$\beta$ and $\varphi$, we have a simple expression for $\tau(w)$ of the form

$\displaystyle\tau=e^{\beta+i\varphi}\,.$ (55)

This relation makes it manifest that the domain $\mathcal{W}_{\beta}$ in the
$w$-plane is mapped to the circle with radius $e^{\beta}$ in the $\tau$-plane.
In the limit $\beta\to 0^{+}$, this result correctly reproduces the statement
that the domain $\mathcal{W}_{\text{sg}}$ in the $w$-plane is mapped to the
unit circle in the $\tau$-plane. In particular, we define $\beta_{E}$ and
$\beta_{R}$ by

$\displaystyle\cosh\beta_{E}=\frac{\varepsilon}{a^{\prime}}\,,\qquad\cosh\beta_{R}=\frac{\rho}{a^{\prime}}\,.$
(56)

Note that we have $\beta_{R}>\beta_{E}>0$ as a consequence of the inequality
$\rho>\varepsilon>a^{\prime}$. Then, the ellipses $\mathcal{W}_{\beta_{R}}$
and $\mathcal{W}_{\beta_{E}}$ in the $w$-plane are mapped to the circles with
radius $R$ and $E$ in the $\tau$-plane, respectively. The ellipse
$\mathcal{W}_{\beta_{R}}$ is tangent to and lies within the circumference of
radius $\rho$ in the $w$-plane. Because $|\tau|=e^{\beta}$ is a monotonically
increasing function of $\beta$, we conclude that the domain inside the circle
with radius $R$ is the image of the domain inside the ellipse
$\mathcal{W}_{\beta_{R}}$ in the $w$-plane. In Fig. 2a we showed the shapes of
the domains $\mathcal{W}_{\text{sg}}$, $\mathcal{W}_{\beta_{E}}$ and
$\mathcal{W}_{\beta_{R}}$ in the $w$-plane, and in Fig. 2b the three
corresponding circles of radius $|\tau|=1,E,R$ in the $\tau$-plane.

##### From $w$-plane to $z$-plane.

Next, we investigate how the ellipses $\mathcal{W}_{\beta}$ look like in
$z$-plane, i.e., we derive the shape of $\partial\mathcal{D}_{R}$. To do so,
it is convenient to solve (45) in terms of $z,$ thus we obtain

$\displaystyle z(w)=\frac{2\rho^{2}w}{w^{2}+\rho^{2}}\,.$ (57)

which is analytic except for the poles at $w=\pm i\rho$.

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

Figure 2: (a) Ellipses $\mathcal{W}_{\beta_{R}}$ (bue line),
$\mathcal{W}_{\beta_{E}}$ (orange line) and segment $\mathcal{W}_{sg}$ (red
line) in the $w$-plane. The ellipse $\mathcal{W}_{\beta_{R}}$ is tangent to
the circumference of radius $\rho$ (dashed black line). (b) The respective
corresponding three circles of radius $|\tau|=R$ (blue line), $|\tau|=E$
(orange line) and $|\tau|=1$ (red line) in the $\tau$-plane. To draw these
figures we set $\rho=1.1$ and $a=0.8,$ which also give $a^{\prime}\simeq
0.474$ and $\varepsilon\simeq 0.7$.

The expression for $z(w)$ allows us to investigate how the ellipses
$\mathcal{W}_{\beta}$ look like in $z$-plane by simply substituting
$w=a^{\prime}\cosh(\beta+i\varphi)$ into (57). By writing
$w=a^{\prime}\cosh(\beta+i\varphi)=x+iy$, with
$x=a^{\prime}\cosh\beta\cos\varphi$ and $y=a^{\prime}\sinh\beta\sin\varphi,$
we obtain the following parametric equations:

$\displaystyle\mathrm{Re}\,z(w)=\frac{2\rho^{2}x(x^{2}+y^{2}+\rho^{2})}{(x^{2}-y^{2}+\rho^{2})^{2}+4x^{2}y^{2}}\,,\qquad\mathrm{Im}\,z(w)=-\frac{2\rho^{2}y(y^{2}+x^{2}-\rho^{2})}{(x^{2}-y^{2}+\rho^{2})^{2}+4x^{2}y^{2}}\,.$
(58)

In particular, we find that the ellipse $\mathcal{W}_{\beta_{R}}$ in the
$w$-plane is the image of the dumb-bell shaped curve $\partial\mathcal{D}_{R}$
defined in eq. (53) and shown in Fig. 1, which is parameterized by

$\displaystyle\mathrm{Re}\,z|_{\beta_{R}}=\frac{2\rho^{3}\cos\varphi\left(2\rho^{2}-a^{\prime
2}\sin^{2}\varphi\right)}{a^{\prime
4}\sin^{4}\varphi+4\rho^{4}\cos^{2}\varphi}\,,\qquad\mathrm{Im}\,z|_{\beta_{R}}=\frac{2\rho^{2}a^{\prime
2}\sqrt{\rho^{2}-a^{\prime 2}}\sin^{3}\varphi}{a^{\prime
4}\sin^{4}\varphi+4\rho^{4}\cos^{2}\varphi}\,,$ (59)

with $-\pi<\varphi\leq\pi$. The parametric equations in eq. (59) reproduce the
curve in Fig. 1 for $\rho=1.1$ and $a=0.8.$ The intersection points of
$\partial\mathcal{D}_{R}$ with the imaginary $z$-axis can be obtained from the
condition $\mathrm{Re}\,z|_{\beta_{R}}=0,$ i.e., by substituting
$\varphi=\pm\pi/2$ into (59). Writing these points as $z=\pm iz_{0}$, we have

$\displaystyle z_{0}=\frac{2\rho^{2}\sqrt{\rho^{2}-a^{\prime 2}}}{a^{\prime
2}}\,.$ (60)

As a consistencty check, we can also verify that for $\varphi=0$ and
$\varphi=\pi$ we have $\mathrm{Re}\,z|_{\beta_{R}}=\rho$ and
$\mathrm{Re}\,z|_{\beta_{R}}=-\rho,$ respectively.

Similarly, by evaluating $x$ and $y$ at $\beta=\beta_{E}$ in eq. (58) we find
that the ellipse $\mathcal{W}_{\beta_{E}}$ in the $w$-plane is the image of
the dumb-bell shaped curve $\partial\mathcal{D}_{E}$ shown in Fig. 1, where
$\mathcal{D}_{E}$ is the domain bounded by $\partial\mathcal{D}_{E}.$ In
addition, one can easily see that the segment $\mathcal{W}_{\rm sg}$ in the
$w$-plane corresponds to the segment $[-a,a]$ in the $z$-plane with
$a=2\rho^{2}a^{\prime}/(a^{\prime^{2}}+\rho^{2}).$

###  3.3 Assumptions

We now list the set of hypotheses that we assume in order to prove the lower
bounds on the maximum of the modulus of the elastic scattering amplitude in
the hard-scattering and Regge limits:

  1. 1.

$\mathcal{M}(s,z)$ is analytic in the subdomain $\mathcal{D}_{R}$ of the cut
$z$-plane.

  2. 2.

$\mathcal{M}(s,z)$ is exponentially bounded on the boundary
$\partial\mathcal{D}_{R}$ as

$|\mathcal{M}(s,z)|\leq
A\left(\frac{s}{s_{0}}\right)^{N}\,e^{\sigma(s/s_{0})^{\alpha}}\,,\qquad\frac{s}{s_{0}}\gg
1\,,$ (61)

where $N$ and $\alpha$ are positive constants independent of $s$ and $z$,
$\sigma$ and $A$ are positive parameters independent of $s$ but can in general
depend on $z$, while $s_{0}$ denotes some energy-squared reference scale.

  3. 3.

The modulus of the forward scattering amplitude ($z=\cos\theta=1$) is bounded
from below as

$\left|\mathcal{M}(s,1)\right|\geq\frac{B}{(s/s_{0})^{\beta}}\,,\qquad\frac{s}{s_{0}}\gg
1\,,$ (62)

where $\beta$ and $B$ are positive constants independent of $s$. The condition
$\beta+N>0$ must be satisfied when $\alpha=0$.

##### Remark.

Assumption 1 was also assumed to prove the standard Cerulus-Martin bound in
the hard-scattering regime [14]. Assumption 61 is more general than the one
used in Ref. [14] where polynomial boundedness was assumed (i.e., $\alpha=0$).
Assumption 62 was assumed in the initial work [14] by Cerulus and Martin, but
then in Ref. [57] it was proved that unitarity, dispersion relations and
polynomial boundedness together imply the lower bound on the forward
scattering amplitude: if we parameterize the forward scattering amplitude as
$|\mathcal{M}(s,1)|\sim s^{-\beta}$, then one can show eq. (62) with
$\beta\leq 2$ [57, 58]. Hence, the Cerulus-Martin bound can be essentially
derived by replacing assumption 62 with unitarity. It is however still unclear
whether eq. (62) can be derived from unitarity for general amplitudes
satisfying exponential boundedness. Therefore, in what follows we take the
inequality (62) as a starting assumption to prove the lower bounds, without
using unitarity, as done in the initial proof by Cerulus and Martin [14].

##  4 Lower bound in the hard-scattering regime

###  4.1 Proof

We now prove the following statement:777In our derivation we assume the
presence of a finite mass gap as originally done by Cerulus and Martin in
their proof [14]. However, it is worth to mention that the case of vanishing
mass gap might be taken into account by suitably changing the definitions of
the domains in the complex $z$-plane and slightly changing assumption 62 away
from the forward limit. For instance, this has been done in Ref. [59] at least
for the Cerulus-Martin bound, i.e., for gapless theories with $\alpha=0$. See
the end of sec. 7 for more discussions on this point.

If the assumptions 1, 61, and 62 on the elastic scattering amplitude
$\mathcal{M}(s,\cos\theta)$ are satisfied, then in the presence of a finite
mass gap it follows that in the high-energy limit and for fixed scattering
angle (hard-scattering regime) the maximum of the modulus
$\left|\mathcal{M}(s,\cos\theta)\right|$ is bounded as

$\boxed{\max_{-a\leq\cos\theta\leq
a}\left|\mathcal{M}(s,\cos\theta)\right|\geq\mathcal{N}(s)\,e^{-f(a)\,\sqrt{s}\,\log(s/s_{0})}e^{-g(a)\,s^{\alpha+\frac{1}{2}}}\,,}$
(63)

where $\mathcal{N}(s)$ is a positive function of $s$ that is subdominant in
the $s\rightarrow\infty$ limit, and $f(a)$ and $g(a)$ are some positive
functions of $a\in(0,1).$

###### Proof.

Let us consider the three circles of radii $1,$ $E$ and $R$ in the
$\tau$-plane (see Fig. 2b). From the remarks made in the previous section we
know that $1<E<R$, therefore the circle $|\tau|^{2}=R^{2}$ is the outer one,
$|\tau|^{2}=E^{2}$ the intermediate, and $|\tau|^{2}=1$ the inner. Let us call
the maxima of $\left|\mathcal{M}(s,z)\right|$ on the three circles as

$\displaystyle
M_{R}\equiv\max\limits_{z\in\partial\mathcal{D}_{R}}|\mathcal{M}(s,z)|\,,\quad
M_{E}\equiv\max\limits_{z\in\partial\mathcal{D}_{E}}|\mathcal{M}(s,z)|\,,\quad
M_{1}\equiv\max\limits_{-a\leq z\leq a}|\mathcal{M}(s,z)|\,,$ (64)

respectively.

Since $\mathcal{M}(s,z)$ is analytic in $\mathcal{D}_{R}$ (assumption 1), it
will also be analytic in the annulus $1\leq|\tau|\leq R.$ Thus, we can apply
the Hadamard-three-circle theorem [60] on the three circles in the
$\tau$-plane, and obtain the following inequality:

$\displaystyle\log R\log M_{E}\leq\log\frac{R}{E}\log M_{1}+\log E\log M_{R}$
$\displaystyle\Leftrightarrow$ $\displaystyle\log M_{E}\leq\left(1-\frac{\log
E}{\log R}\right)\log M_{1}+\frac{\log E}{\log R}\log M_{R}\,.$ (65)

We are interested in the high-energy behavior, i.e., in the large $s$ limit
for which $\rho\rightarrow 1,$ $R\to E$ and $\log E/\log R\to 1,$ thus it is
convenient to introduce a positive function $C(x,a)$ defined by the following
equation:

$\displaystyle\frac{\log E}{\log R}=1-\frac{2m}{p_{s}}C(x,a)\,.$ (66)

The explicit expression of $C$ is

$\displaystyle C(x,a)$
$\displaystyle=\frac{1}{\sqrt{x}}\frac{\log\left(\frac{\sqrt{2(x+2)\left(\sqrt{(x+2)^{2}-4a^{2}}-x-2\right)+8a^{2}}+2a}{\sqrt{2}\sqrt{(x+2)\left(a^{2}\left(x-\sqrt{x(x+4)}+2\right)+\sqrt{(x+2)^{2}-4a^{2}}-x-2\right)}+a\left(x-\sqrt{x(x+4)}+2\right)}\right)}{\log\left(\frac{\sqrt{2(x+2)\left(\sqrt{(x+2)^{2}-4a^{2}}-x-2\right)+8a^{2}}+2a}{x+2-\sqrt{(x+2)^{2}-4a^{2}}}\right)}\,,$
(67)

where $x\equiv(2m/p_{s})^{2},$ and its positivity follows from the inequality
$R>E$.

In terms of $C$, eq. (65) becomes

$\displaystyle\log M_{E}\leq\frac{2mC}{p_{s}}\log
M_{1}+\left(1-\frac{2mC}{p_{s}}\right)\log M_{R}\,,$ (68)

and exponentiating we get

$\displaystyle M_{E}\leq M_{1}^{2mC/p_{s}}\,M_{R}^{1-2mC/p_{s}}\,,$ (69)

which can be written as

$\displaystyle M_{1}^{2mC/p_{s}}\geq M_{R}^{-1+2mC/p_{s}}M_{E}\,.$ (70)

Because the exponent of $M_{R}$, i.e., the term $-1+2mC/p_{s}$, is always
negative, the assumption 61 gives a lower bound on the left-hand-side of (70).
We then get

$\displaystyle
M_{1}\geq\exp\left\\{\left(-\frac{p_{s}}{2mC}+1\right)\log\left[A(s/s_{0})^{N}e^{\sigma(s/s_{0})^{\alpha}}\right]\right\\}M_{E}^{p_{s}/2mC}\,.$
(71)

Since $M_{E}\geq|\mathcal{M}(s,z=1)|$, we notice that (71) provides a lower
bound on the high-energy behavior of the absolute value of the modulus of the
amplitude ($M_{1}=\max_{-a\leq z\leq a}|\mathcal{M}(s,z)|$) in terms of the
forward-limit scattering amplitude. Then, by using the assumption 62, we
obtain

$\displaystyle\max_{-a\leq\cos\theta\leq
a}|\mathcal{M}(s,\cos\theta)|\geq\exp\left[-F(s,a)\right]\,,$ (72)

where $F(s,a)$ is given by

$\displaystyle F(s,a)$
$\displaystyle\equiv\left(\frac{p_{s}}{2mC}-1\right)\log\left[A(s/s_{0})^{N}e^{\sigma(s/s_{0})^{\alpha}}\right]-\frac{p_{s}}{2mC}\log\left[B(s/s_{0})^{-\beta}\right]$
$\displaystyle=\frac{p_{s}}{2mC}\log\left[\frac{A}{B}\left(s/s_{0}\right)^{N+\beta}e^{\sigma(s/s_{0})^{\alpha}}\right]-\log\left[A\left(s/s_{0}\right)^{N}e^{\sigma(s/s_{0})^{\alpha}}\right]\,.$
(73)

The inequality in eq. (72) holds for large but finite values of $s$ for which
the assumptions we imposed in sec. 3.3 are satisfied. To understand the bound
(72) more clearly, we need the behavior of the function $C(x,a)$ in the
$x\rightarrow 0$ limit. Its expansion around $x=0$ strongly depends on the
type of kinematic regime under investigation. In this section, we are
interested in the hard-scattering limit ($x\to 0$, $a=$ fixed), while the
Regge limit will be discussed in the next section. Hence, we consider the
expansion of $C(x,a)$ around $x=0$ by keeping $0<a<1$ fixed and obtain
$C(x,a)=C_{0}(a)+\mathcal{O}(\sqrt{x}),$ where888We can numerically confirm
that this expansion works well for sufficiently small $x\lesssim 0.01$ unless
$a$ is very close to unity to satisfy $1-x\lesssim a\leq 1$.

$\displaystyle C_{0}(a)$
$\displaystyle=\frac{a}{\sqrt{2(\sqrt{1-a^{2}}+a^{2}-1)}\left\\{\log\left[\frac{1}{a}(\sqrt{1-a^{2}}+1)\left(1+\frac{\sqrt{2}}{a}\sqrt{\sqrt{1-a^{2}}+a^{2}-1}\right)\right]\right\\}}\,.$
(74)

We can numerically verify the inequality $C(x,a)\geq C_{0}(a)$ for
sufficiently small $x$ satisfying the condition $x\leq x_{*}(a)$ for given
$a\in(0,1)$. Here, $x_{*}(a)\neq 0$ is defined by the equation
$C(x_{*}(a),a)=C_{0}(a)$; see App. A for more details. Since $x_{*}(a)$ is
approximated as $x_{*}(a)\simeq 8(1-a)/9$ for $a\simeq 1$ (see eq. (133) in
App. A), the condition $x\leq x_{*}(a)$ is equivalent to $a\leq
1-\frac{9}{8}x$, approximately.

Therefore, for $x\leq x_{*}(a)$ we have $C(x,a)\geq C_{0}(a)$ which implies

$F(s,a)\leq F(s,a)|_{C\rightarrow C_{0}(a)}\,,$ (75)

where we have used

$\log\left[\frac{A}{B}\left(s/s_{0}\right)^{N+\beta}e^{\sigma(s/s_{0})^{\alpha}}\right]>0\,,$
(76)

which is always valid at sufficiently high energies. Thus, from eq. (72) we
obtain

$\displaystyle\max_{-a\leq\cos\theta\leq
a}|\mathcal{M}(s,\cos\theta)|\geq\exp\left[-F(s,a)|_{C\rightarrow
C_{0}(a)}\right]\,,$ (77)

or, equivalently,

$\displaystyle\max_{-a\leq\cos\theta\leq
a}|\mathcal{M}(s,\cos\theta)|\geq\mathcal{N}(s)\,\exp\left\\{-\frac{p_{s}}{2mC_{0}(a)}\left[(N+\beta)\log\left(\frac{s}{s_{0}}\right)+\sigma\left(\frac{s}{s_{0}}\right)^{\alpha}\right]\right\\}\,,$
(78)

where

$\displaystyle\mathcal{N}(s)$ $\displaystyle\equiv
A\,\left(\frac{B}{A}\right)^{\frac{p_{s}}{2mC_{0}(a)}}\left(\frac{s}{s_{0}}\right)^{N}e^{\sigma(s/s_{0})^{\alpha}}$
(79)

is a positive function of $s$ that is subdominant in the large $s$ limit.

Finally, by using999Note that, for $m\neq 0$ and for finite values of $s$ we
have the strict inequality $p_{s}<\sqrt{s}/2.$ However, this inequality can be
saturated in the strict infinite energy limit $s\rightarrow\infty.$ This is
the reason why we wrote $p_{s}\leq\sqrt{s}/2.$ $p_{s}\leq\sqrt{s}/2,$ from the
inequality (78) we obtain the lower bound

$\displaystyle\max_{-a\leq\cos\theta\leq
a}|\mathcal{M}(s,\cos\theta)|\geq\mathcal{N}(s)\,e^{-f(a)\,\sqrt{s}\,\log(s/s_{0})-g(a)\,s^{\alpha+\frac{1}{2}}}\,,$
(80)

which coincides with eq. (63); we have defined
$f(a)\equiv\frac{N+\beta}{4mC_{0}(a)}>0$ and
$g(a)\equiv\frac{\sigma}{4mC_{0}(a)s_{0}^{\alpha}}>0.$ Thus, we have showed
the existence of a lower bound on the maximum of the modulus of an elastic
scattering amplitude in the hard-scattering regime.

∎

##### Remarks.

We now summarize several remarks on our results.

  * •

In the case $\alpha=0$ the scattering amplitude is polynomially bounded (see
assumption 61 above) and we consistently recover the Cerulus-Martin bound
[14], i.e., eq. (63) reduces to the inequality (1). The lower bound in eq.
(77), (78), or (80) is valid for large but finite values of $s$ for which the
condition $x\leq x_{*}(a)$ is satisfied thanks to the inequality $C(x,a)\geq
C_{0}(a)$, rather than an approximation $C(x,a)\simeq C_{0}(a)$ which becomes
exact only in the limit $s\to\infty$. In particular, our result in eq. (78) is
more precise and even for $\alpha=0$ can be seen as an update of the original
derivation of the Cerulus-Martin bound in [14].

  * •

The lower bound becomes trivial in the limit $a\to 0$ because we have
$C(x,0)=C_{0}(0)=0$, which is also mentioned in [61]. This can be understood
intuitively as follows: the left-hand-side of (72) becomes
$|\mathcal{M}(s,\cos\theta=0)|$ in this limit, and there is a possibility that
$\mathcal{M}(s,\cos\theta=0)$ exactly vanishes at least for some discrete set
of points on the positive real $s$-axis.

###  4.2 Discussion

The important new result is that we have obtained a new lower bound in eq.
(77), (78), or (80) with $\alpha>0$ in the hard-scattering regime that applies
to amplitudes for which polynomial boundedness is violated. The
$\alpha$-dependent term is dominant at sufficiently high energies and the
lower bound effectively reads

$\max_{-a\leq\cos\theta\leq
a}\left|\mathcal{M}(s,\cos\theta)\right|\geq\tilde{\mathcal{N}}(s)e^{-g(a)\,s^{\alpha+\frac{1}{2}}}\qquad(\alpha>0)\,,$
(81)

where $\tilde{\mathcal{N}}(s)$ is a subdominant function of $s$ in the
$s\rightarrow\infty$ limit. This shows that a violation of the Cerulus-Martin
bound does not always signal a pathology because in the absence of polynomial
boundedness a more general lower bound holds.

Our result admits a violation of the original Cerulus-Martin bound even when
the theory is strictly localizable $0<\alpha<1/2$. This is interesting since
the Cerulus-Martin bound has been used as a test of locality in the past.
However, it may also be that there is a stronger bound than ours. This is
logically possible since our bound provides only a necessary condition. In
fact, it would be interesting to understand whether one can find another way
to prove the lower bound of the Cerulus-Martin type (1) even for
$0<\alpha<1/2$ by changing some of the assumptions; see also sec. 6 for more
discussions on this point. We leave this aspect for future work.

##  5 Lower bound in the Regge regime

We now focus on the Regge limit ($s\rightarrow\infty,$
$t=-2p_{s}^{2}(1-\cos\theta)={\rm fixed}$) and show the existence of a lower
bound in this regime for both cases of polynomial ($\alpha=0$) and exponential
($\alpha>0$) boundedness. To our knowledge, our lower bound on the amplitude
in the Regge limit is new in the literature, even in the $\alpha=0$ case.

The main difference with the hard-scattering regime is that, in the Regge
limit, we should not expand the function $C(x,a)$ around $x=0$ for fixed
values of $a,$ because the variable that now has to be kept fixed is the
momentum transfer squared $t.$ To correctly take into account the Regge limit,
we introduce the quantity

$\Delta\equiv\frac{1-a}{x}=\frac{p_{s}^{2}}{4m^{2}}(1-a)\,,$ (82)

such that to derive a lower bound we now need to expand the function
$C(x,\Delta)\equiv C(x,a)|_{a\rightarrow 1-x\Delta}$ around $x=0$ and for
fixed values of $\Delta.$ In terms of $\Delta,$ the momentum transfer squared
can be expressed as $t=-8m^{2}\Delta|_{a=\cos\theta}.$ Moreover, in terms of
$t$ and $\Delta$ the inequalities $-a\leq\cos\theta\leq a$ become
$8m^{2}\Delta-4p_{s}^{2}\leq t\leq-8m^{2}\Delta.$

###  5.1 Proof

We now prove the following statement:

If the assumptions 1, 61, and 62 on the elastic scattering amplitude
$\mathcal{M}(s,t(\cos\theta))$ are satisfied, then in the presence of a finite
mass gap it follows that in the high-energy limit and for fixed negative
momentum transfer squared (Regge regime) the maximum of the modulus
$\left|\mathcal{M}(s,t)\right|$ is bounded as

$\boxed{\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left|\mathcal{M}(s,t)\right|\geq
h(\Delta)\,e^{-\tilde{f}(\Delta)\log(s/s_{0})-\tilde{g}(\Delta)s^{\alpha}}\,,}$
(83)

where $h(\Delta),$ $\tilde{f}(\Delta)$ and $\tilde{g}(\Delta)$ are positive
functions of $0<\Delta\leq p_{s}^{2}/4m^{2}$ defined in (90).

###### Proof.

The starting point is eq. (72) rewritten for fixed momentum transfer $t<0,$
i.e.,

$\displaystyle\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}|\mathcal{M}(s,t)|\geq\exp\left[-F(s,\Delta)\right]\,,$
(84)

where $F(s,\Delta)\equiv F(s,a)|_{a\rightarrow 1-x\Delta},$ and $F(s,a)$ was
defined in eq. (73). Because we are now interested in the Regge limit, we
expand the function $C(x,\Delta)\equiv C(x,a)|_{a\to 1-x\Delta}$ around $x=0$
by keeping $\Delta$ fixed as101010We note that some details on the behavior of
the function $C(x,a)$ in the Regge limit (i.e., of $C(x,\Delta)$ in the limit
$x\rightarrow 0$ and for fixed $\Delta$) are also given in the footnote 11 of
[62].

$\displaystyle
C(x,\Delta)=\left[1-6C_{0}(\Delta)\right]\frac{1}{\sqrt{x}}+C_{0}(\Delta)+\mathcal{O}\left(\sqrt{x}\right)\,,$
(85)

where

$C_{0}(\Delta)=\frac{\sqrt{\sqrt{2\Delta+1}-1}}{6({2\Delta+1})^{1/4}}\,,$ (86)

and we have $1-6C_{0}(\Delta)>0$ for $\Delta>0.$ We can notice that the small
$x$ behavior of the function $C(x,\Delta)$ in the Regge regime (i.e., small
angles) is qualitatively different from the one in the hard-scattering regime
(i.e., large angles) derived in the previous section.

We can numerically verify the inequality
$C(x,\Delta)\geq[1-6C_{0}(\Delta)]x^{-1/2}$ for sufficiently small $x$
satisfying the condition $x\leq x_{*}(\Delta)$. Here, $x_{*}(\Delta)\neq 0$ is
defined by the equation
$C(x_{*}(\Delta),\Delta)=[1-6C_{0}(\Delta)]x_{*}^{-1/2}$; see App. A and Fig.
4a for more details. Therefore, for $x\leq x_{*}(\Delta)$ we can write

$F(s,\Delta)\leq F(s,\Delta)|_{C\rightarrow[1-6C_{0}(\Delta)]x^{-1/2}}\,.$
(87)

Thus, from eq. (84) we obtain

$\displaystyle\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}|\mathcal{M}(s,t)|\geq\exp\left[-F(s,\Delta)|_{C\rightarrow[1-6C_{0}(\Delta)]x^{-1/2}}\right]\,,$
(88)

or, equivalently,

$\displaystyle\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left|\mathcal{M}(s,t)\right|\geq
h(\Delta)\,e^{-\tilde{f}(\Delta)\log(s/s_{0})-\tilde{g}(\Delta)s^{\alpha}}\,,$
(89)

where we have defined

$h(\Delta)\equiv\left(\frac{B}{A^{6C_{0}(\Delta)}}\right)^{\frac{1}{1-6C_{0}(\Delta)}}\,,\quad\tilde{f}(\Delta)\equiv\frac{6C_{0}(\Delta)N+\beta}{1-6C_{0}(\Delta)}\,,\quad\tilde{g}(\Delta)\equiv\frac{6C_{0}(\Delta)}{1-6C_{0}(\Delta)}\frac{\sigma}{s_{0}^{\alpha}}\,,$
(90)

which are positive functions of $\Delta$.

∎

##### Remarks.

We now summarize several remarks on our results.

  * •

In the forward limit $t=0$ we have $\Delta=0$ (i.e., $a=1$),
$C_{0}(\Delta=0)=0$ and $C(x,\Delta=0)=1/\sqrt{x}.$ As a consequence, we
obtain $h(0)=B,$ $\tilde{f}(0)=\beta,$ $\tilde{g}(0)=0,$ and

$|\mathcal{M}(s,t=0)|=|\mathcal{M}(s,\cos\theta=1)|\geq\frac{B}{(s/s_{0})^{\beta}}\,,$
(91)

which, as a consistency check, coincides with the assumption 62. The
importance of the newly derived inequality (83) is that it provides a lower
bound for fixed momentum transfer and beyond the forward limit (i.e., for
$t<0$ and $\Delta>0$).

  * •

The $s$-dependence in the lower bound for fixed momentum transfer differs from
the one for fixed scattering angle by a factor of $\sqrt{s}$ in the exponent.
This means that amplitudes in the hard-scattering regime (large angles) can be
more suppressed as compared to the ones in the Regge regime (small angles).
This could be understood physically by observing that the probability of two
particles scattering should be greater for smaller scattering angles.

###  5.2 Discussion

We have derived a new bound lower bound (83) on elastic scattering amplitudes
in the Regge limit. To our knowledge, this result is new in the literature
even in the case of polynomial boundedness $\alpha=0$; the only work that is
close to our analysis in this section is Ref. [62] where some details on the
Regge limit of the function $C(x,\Delta)$ were discussed but no bound was
derived.

In the case $\alpha=0$, the lower bound (83) reduces to

$\displaystyle\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left|\mathcal{M}(s,t)\right|\geq\tilde{h}(\Delta)\,\left(\frac{s_{0}}{s}\right)^{\tilde{f}(\Delta)}\qquad(\alpha=0)\,,$
(92)

where $\tilde{h}(\Delta)\equiv h(\Delta)e^{-\tilde{g}(\Delta)}.$ Thus, under
the assumption of locality ($\alpha=0$), in the Regge limit the elastic
scattering amplitude is bounded from below by inverse powers of $s$.

In the case $\alpha>0,$ the inequality (83) reduces to

$\displaystyle\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left|\mathcal{M}(s,t)\right|\geq
H(s,\Delta)\,e^{-\tilde{g}(\Delta)s^{\alpha}}\qquad(\alpha>0)\,,$ (93)

where $H(s,\Delta)\equiv h(\Delta)e^{-\tilde{f}(\Delta)\log(s/s_{0})}$ is a
positive function that is subdominant in the $s\rightarrow\infty$ limit.
Therefore, unlike the case $\alpha=0,$ for $\alpha>0$ the elastic scattering
amplitude in the Regge limit turns out to be exponentially bounded from below.

We can also ask whether there is a regime where the bound in the Regge limit
overlaps with the one in the the hard-scattering limit. This actually happens
when $\Delta\gg 1$ which allows to expand
$\tilde{f}(\Delta)=2(N+\beta)\sqrt{2\Delta}-(3N+\beta)/2+\mathcal{O}(\Delta^{-1/2})$
and
$\tilde{g}(\Delta)=\sigma[2\sqrt{2\Delta}-3/2+\mathcal{O}(\Delta^{-1/2})]/s_{0}^{\alpha}.$
Thus, substituting these expansions in eq. (83) and using
$\sqrt{\Delta}\sim\sqrt{s}$ we get the same $s$-dependence in the exponent as
in eq. (63).

###  5.3 Bound on the Regge trajectory

As an application of the newly derived bound (83), it is interesting to
constrain the Regge trajectory $j(t)$ in the regime $-s\ll t\leq 0$.

Suppose that the amplitude in the Regge regime is dominated by the
contribution from the leading Regge pole. In such a case we can parametrize
the behavior of the amplitude in the regime $-s/t\gg 1$ as [63, 64]

$\displaystyle\mathcal{M}(s,t)=B(s,t)\left(\frac{s}{s_{0}}\right)^{j(t)}\,,$
(94)

where $B(s,t)$ is a subdominant multiplicative factor, and $j(t)$ is the so-
called Regge trajectory which, in general, is real for $t\in\mathbb{R}$ but
can take complex values for $t\in\mathbb{C}$. Suppose that the amplitude (94)
satisfies the requirement of local QFT and is polynomially bounded on the
domain $\partial\mathcal{D}_{R}$ in the complex $z$-plane. Then it has to
satisfy the lower bound with $\alpha=0$ in eq. (92):

$\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left[B(s,t)\left(\frac{s}{s_{0}}\right)^{j(t)}\right]\geq\tilde{h}(\Delta)\left(\frac{s_{0}}{s}\right)^{\tilde{f}(\Delta)}\,,$
(95)

where $j(t)$ takes only real values as the maximum is taken for negative real
values of $t.$ If $j(t)>0,$ the lower bound is always satisfied because
$\tilde{f}(\Delta)>0.$ The interesting case is when the Regge trajectory is
negative. Indeed, for $j(t)<0$ the inequality (95) can be recast as a non-
trivial bound on the Regge trajectory:111111Note that, the inequality (96) can
be saturated only in the strict $s\rightarrow\infty$ limit, in which
subdominant $(s,t)$-dependent terms, such as $\log B(s,t)/\log(s/s_{0})$ and
$\log\tilde{h}(\Delta)/\log(s/s_{0}),$ become completely negligible on both
sides of the inequality.

$\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}j(t)\geq-\tilde{f}(\Delta)=-\frac{6C_{0}(\Delta)N+\beta}{1-6C_{0}(\Delta)}\,.$
(96)

Since the amplitude (94) is a polynomial in $s$ we can easily link $\beta>0$
and $N>0$ to $j(t)$ evaluated at some specific values of $t.$ From assumption
62 we know that $\beta$ is related to the lower bound in the forward limit
($t=0$) that can be taken into account in eq. (96) by including $a=1$ which
corresponds to $\Delta=0$ and $C_{0}(\Delta=0)=0.$ Therefore, we have
$\beta=-j(0).$ Moreover, the amplitude (94) is a polynomially growing function
of $s$ for $j(t)>0.$ Thus, from assumption 61 we have
$N=\max\limits_{z\in\partial\mathcal{D}_{R}}[{\rm Re}\,j(t(z))]$. In what
follows, to keep the notation simple we just write $N$ instead of its explicit
expression.

We can recast the inequality (96) in a more useful form by noticing that for
sufficiently large values of $\Delta$ the function $\tilde{f}(\Delta)$ is
smaller than its leading order in the $\Delta\gg 1$ expansion, i.e., we have

$\tilde{f}(\Delta)\leq 2[N-j(0)]\sqrt{2\Delta}\,,$ (97)

where $N-j(0)=N+\beta>0$ by construction. More precisely, eq. (97) holds true
as long as $\Delta\geq\Delta_{*},$ where $\Delta_{*}>0$ is defined as the
solution of the equation $\tilde{f}(\Delta_{*})=2[N-j(0)]\sqrt{2\Delta_{*}}.$

Therefore, by using (97) we obtain the following bound on the Regge
trajectory:

$\boxed{\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}j(t)\geq-2[N-j(0)]\sqrt{2\Delta}}\quad\text{for}\quad\Delta\geq\Delta_{*}\,,$
(98)

which is non-trivial only for scenarios in which $j(t)<0$ for
$8m^{2}\Delta-4p_{s}^{2}\leq t\leq-8m^{2}\Delta.$

##### Remarks.

Let us now make the following observations.

  * •

An approximate analytic expression for $\Delta_{*}$ can be found in the limit
$\Delta\gg 1$ which also implies $t\ll-m^{2}.$ In such a regime we can write

$\tilde{f}(\Delta)=2[N-j(0)]-\frac{1}{2}[3N-j(0)]+\frac{7}{8}[N-j(0)]\frac{1}{\sqrt{2\Delta}}+\mathcal{O}(1/\Delta)\,,$
(99)

and solve the equation $\tilde{f}(\Delta_{*})=2[N-j(0)]\sqrt{2\Delta_{*}}$ up
to this order in the expansion. By doing so, we obtain

$\Delta_{*}\simeq\frac{49}{32}\left[\frac{N-j(0)}{3N-j(0)}\right]^{2}\,.$
(100)

  * •

In the regime $s\gg-t\geq 8m^{2}\Delta\geq 8m^{2}\Delta_{*}$ the bound (98) is
consistent with the original Cerulus-Martin bound (1). Indeed, since
$s>8m^{2}\Delta$ we have $\sqrt{s}>\sqrt{8m^{2}\Delta}$ and

$\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}j(t)>-[N-j(0)]\sqrt{s}\,,$ (101)

which implies

$\displaystyle\max\limits_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left(\frac{s}{s_{0}}\right)^{j(t)}>\left(\frac{s}{s_{0}}\right)^{-2[N-j(0)]\sqrt{2\Delta}}>e^{-[N-j(0)]\sqrt{s/m^{2}}\ln(s/s_{0})}\,.$
(102)

####  5.3.1 Case of monotonically increasing Regge trajectory for $t<0$

We now apply the above result to the case of Regge trajectories that are
monotonically increasing for negative $t,$ i.e., we assume121212Strictly
speaking, we only need (104) to obtain (105). Monotonicity (103) is a
sufficient condition for (104).

$\displaystyle\frac{{\rm d}j(t)}{{\rm d}t}\geq 0\quad\text{for}\quad t<0\,.$
(103)

If eq. (103) is valid, we have

$\max\limits_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}j(t)=j(-8m^{2}\Delta),$ (104)

and thus we can recast the bound (98) in the following more suggestive form:

$\boxed{j(t)\geq-[N-j(0)]\sqrt{-t/m^{2}}\,}\quad\text{for}\quad
t\leq-8m^{2}\Delta_{*}\,,$ (105)

where we recall that $N-j(0)>0$ by construction and $\Delta_{*}>0$ was defined
by the equation $\tilde{f}(\Delta_{*})=2[N-j(0)]\sqrt{2\Delta_{*}}.$

The inequality (105) implies that in the case of local theories with
$\alpha=0$ a negative Regge trajectory $j(t)$ cannot decrease faster than
$\sqrt{-t}$ for momentum transfer squared in the range
$-\infty<t\leq-8m^{2}\Delta_{*}.$ For example, linear Regge trajectories for
negative $t$ are not allowed. On the other hand, the lower bound (105) does
not constrain the Regge trajectory in the regime $t>-8m^{2}\Delta_{*};$ in
particular, it does not say anything about the Regge trajectory for positive
$t.$

Note that, the Veneziano amplitude [65] in perturbative string theory violates
both the bound (105) and the Cerulus-Martin bound. Indeed, the Regge
trajectory of the Veneziano amplitude is monotonically increasing and linear
in the regime $-s\ll t\leq 0,$ thus it violates the lower bound (105).
Moreover, in the hard-scattering regime the Veneziano amplitude behaves as
$\sim e^{-\xi(\cos\theta)\,s},$ thus signaling a violation of (1). See sec.
6.4 for more discussions on this point and for the explicit expression of
$\xi(\cos\theta)$.

##  6 Applications

We are now ready to consider applications of our results in various contexts.
Since the newly derived bounds on elastic scattering amplitudes concern the
high-energy regime, they can be used as powerful probes of non-locality and as
tools to constrain UV physics.

Given a scenario in which the high-energy behavior of an elastic scattering
amplitude is known, then we can use the newly derived bounds to check whether
the starting assumptions are satisfied. In particular, the degree of
(non-)localizability of the underlining UV theory can be constrained as
follows. Assume that in a certain framework/scenario the high-energy behavior
of an elastic scattering amplitude in the hard-scattering regime goes as

$|\mathcal{M}(s,z)|\sim e^{-c\,s^{\gamma}}\,,$ (106)

where $c>0$ and $\gamma>0$ are independent of $s.$ Then, to be consistent with
the lower bound in eq. (63) we must impose the condition

$\alpha\geq\gamma-\frac{1}{2}\quad(\text{hard-scattering regime})\,,$ (107)

which represents a non-trivial constraint on the degree of non-locality of the
UV theory that predicts the high-energy behavior in (106). Indeed, if
$\gamma\leq 1/2$ the Cerulus-Martin bound and the property of polynomial
boundedness can still be respected. On the other hand, if $\gamma>1/2$ we must
have $\alpha>0$ which implies that the underlining theory is either strictly
localizable, quasi-local or non-localizable (see eq. (7)).

A similar discussion can be made in the Regge regime. Assume that the high-
energy behavior of an elastic scattering amplitude in the Regge limit behaves
as (106), then to be consistent with the lower bound in eq. (83) we must
impose the condition

$\alpha\geq\gamma\quad(\text{Regge regime})\,.$ (108)

It follows that the lower bound in the Regge limit (83) can be more
restrictive than the one in the hard-scattering regime (63), i.e., eq. (108)
is stronger than eq. (107). Of course, the constraint (108) can only be useful
if the elastic scattering amplitude is exponentially suppressed in the Regge
regime, which however is usually not the case.

Hence, the newly derived bounds turn out to be very powerful tools to
constrain approaches to UV completion such as string theory [6, 7, 19] and
classicalization [9, 10]. In addition, our results can be important to deepen
our understanding of gravitational processes at high energies [4, 5], e.g.,
black-hole formation. In what follows we are going to discuss applications of
our results to specific theories/scenarios. While in sec. 2.3 we have shown
examples of non-localizable theories by mainly looking at the spectral
densities, in the next subsection we will analyse the behavior of the
scattering amplitudes and get information about the degree of non-
localizability (i.e., the value of $\alpha$) by using our lower bounds as
probes.

###  6.1 Gravity and black-hole formation

As discussed in sec. 2.3.1, in Einstein’s gravity one expects that the high-
energy behavior of an elastic scattering amplitude is characterized by an
exponential suppression caused by BH production. In the hard-scattering regime
(i.e., fixed angle) and in $D$ dimensions we have [17, 18]

$\sqrt{s}\gg M_{p}\quad\Rightarrow\quad|\mathcal{M}(s,z)|\sim
e^{-c\,(s/M_{p}^{2})^{\frac{1}{2}\frac{D-2}{D-3}}}\,,$ (109)

where $c$ is some positive parameter independent of $s.$ By comparing with
(1), we can immediately notice that the Cerulus-Martin bound is always
violated for any $D>3.$ Thus, it is very important to understand which of the
assumptions in the derivation by Cerulus and Martin is violated.

We expect analyticity (assumption 1) and the lower bound on the forward limit
in eq. (62) (assumption 62) to be satisfied [4]. Whereas, we already know from
sec. 2.3.1 that polynomial boundedness (assumption 61 with $\alpha=0$) should
be violated in gravitational theories. In addition, because of the massless
nature of the graviton the assumption of a finite mass gap is also not
satisfied. However, the high-energy behavior in eq. (109) should be
insensitive to the presence of massless poles. In fact, as shown in [59] at
least for $\alpha=0$ the derivation of (63) can be generalized to the case of
gapless theories by suitably changing the definitions of domains in the
complex $z$-plane (see also footnote 7). Therefore, it is reasonable to
conclude that the violation of the Cerulus-Martin bound by (109) is due to the
lack of polynomial boundedness or, in other words, to a violation of locality
in gravity; see also [3] for a similar reasoning. This means that the
consistency of eq. (109) must be checked by making a comparison with the more
general lower bound (63) derived under the assumption of exponential
boundedness (assumption 61).

In the hard-scattering regime, we have to impose the consistency condition
(107) with $\gamma=\frac{1}{2}\frac{D-2}{D-3},$ thus we obtain the following
constraint on the degree of non-locality in gravity with $D>3$:

$\alpha\geq\frac{1}{2(D-3)}\,.$ (110)

In $D=4,$ we have $\alpha\geq 1/2,$ which means that Einstein’s theory is
either quasi-local or non-localizable. However, in higher dimensions $D>4$,
the bound (110) also admits the value $\alpha<1/2$ and the strictly
localizable case is not excluded. By contrast, the discussion in sec. 2.3.1
based on the BH entropy argument suggest that Einstein’s GR in asymptotically
flat spacetime is always non-localizable with $\alpha>1/2$ for any dimensions
$D>3$. This discrepancy between the conclusion based on our bound (107) and
the BH entropy argument is due to the $1/2$ term on the RHS of (107). It would
be then interesting to study if we can actually refine our current lower bound
on the fixed angle scattering to replace (107) by the stronger constraint
$\alpha\geq\gamma$. We leave this task for future work.

###  6.2 Classicalization proposal

In the more general case of the classicalization proposal, we can analyse the
high-energy behavior of the S-matrix for a generic interaction potential of
the form $\partial^{2k}\phi^{n}.$ As mentioned in sec. 2.3.2, in this case it
was argued that the high-energy behavior of the amplitude in the hard-
scattering regime and in $D=4$ is suppressed as [9]

$\sqrt{s}\gg\Lambda\quad\Rightarrow\quad|\mathcal{M}(s,z)|\sim
e^{-c\,(s/\Lambda^{2})^{\frac{n+2k-4}{n+4k-6}}}\,,$ (111)

where $\Lambda$ is a cutoff energy scale at which the perturbative EFT
description breaks down. In this case the Cerulus-Martin bound is always
violated for any $n\geq 3,$ and it is still reasonable to assume that the
reason lies in the lack of polynomial boundedness.

Therefore, we expect eq. (111) to be consistent with the more general bound
(63). By imposing the consistency condition (107), we obtain the following
non-locality constraint:

$\alpha\geq\frac{1}{2}\frac{n-2}{n+4k-6}\,.$ (112)

Note that for $k=1$ we recover the gravitational case $\alpha\geq 1/2$ in eq.
(110) with $D=4$ dimensions. For other values of $k$ the constraint (112) is
not very strong because it would also allow for values $\alpha<1/2.$ However,
from the discussion in sec. 2.3.2 we expect that $\alpha>1/2$ for $n\geq 3.$
As we discussed in sec. 6.1, this discrepancy is due to the $1/2$ term on the
RHS of (107).

###  6.3 Galileon

In Ref. [10] it was argued that at the non-perturbative level the Galileon
violates the Cerulus-Martin bound because the fixed-angle elastic amplitude
should behave as

$\sqrt{s}\gg\Lambda\quad\Rightarrow\quad|\mathcal{M}(s,z)|\sim
e^{-c\,(s/\Lambda^{2})^{\frac{3}{5}}}\,.$ (113)

This conjectured behavior was hinted by looking at the behavior of the non-
perturbative spectral density which was shown to be exponentially growing with
$\alpha=3/5$ [10], as mentioned in sec. 2.3.3 and in agreement with a UV
completion by classicalization.

Hence, the consistency constraint on the exponential behavior of the Galileon
model is a special case of eq. (112) obtained for the classicalization
proposal. We have

$\alpha\geq\frac{1}{10}\,,$ (114)

which turns out to be a weak constraint on the degree of non-locality because
it would also allow for strict localizability. This is again due to the $1/2$
term on the RHS of (107). In fact, as discussed in sec. 2.3.3 we expect the
Galileon field to belong to the class of non-localizable QFT with
$\alpha=3/5.$

###  6.4 Perturbative string theory

Let us now analyse scattering amplitudes in perturbative string theory and ask
whether the Cerulus-Martin bound and the polynomial boundedness condition are
violated. Let us consider the tree-level elastic scattering amplitude between
tachyons in open string theory of the form [66]

$\mathcal{M}(s,t)=\mathcal{A}(s,t)+\mathcal{A}(s,u)+\mathcal{A}(t,u)\,,$ (115)

where

$\mathcal{A}(s,t)=\frac{\Gamma(-j(s))\Gamma(-j(t))}{\Gamma(-j(s)-j(t))}\,,$
(116)

is the Veneziano amplitude [65]; similar formulas can be written for
$\mathcal{A}(s,u)$ and $\mathcal{A}(t,u).$ $\Gamma(x)$ is the Euler gamma
function, and the argument $x=-j(s)$ is linear in $s$,

$j(s)=j_{0}+\alpha^{\prime}s\,,$ (117)

where $j_{0}\equiv j(0)$ and $\alpha^{\prime}$ are positive constants.
Remember that $u=-s-t+4m^{2}$ where, in this case, $m^{2}=-1/\alpha^{\prime}$
is the tachyon mass. The Veneziano amplitude has an infinite number of poles
located on the real axis, e.g., the first term (116) gives $s,t$-poles at

$s,t=\frac{n-j_{0}}{\alpha^{\prime}}\,,\quad n=0,1,2,\dots\,.$ (118)

One usually has $j_{0}=1,$ thus both massless (for $n=1$) and massive
particles are included in the physical spectrum.

Now we study the high-energy behavior of the amplitude (115) in both hard-
scattering (fixed $z=\cos\theta$) and Regge (fixed $t$) regimes. It is
sufficient to focus on $\mathcal{A}(s,t)$ because similar results can be found
for the other two terms. For fixed angle we have:

$\text{hard-scattering regime:}\quad\mathcal{A}(s,t(z))\sim
g(s,z)\,e^{-\xi(z)\,j(s)}\sim e^{-\xi(z)\,\alpha^{\prime}\,s}\,,$ (119)

where $g(s,z)$ is subdominant in the large $s$ limit and
$\xi(z)=\frac{1-z}{2}\log(\frac{2}{1-z})+\frac{1+z}{2}\log(\frac{2}{1+z})$ is
positive for physical scattering angles. For fixed momentum transfer, instead,
we obtain the following polynomial behavior:

$\text{Regge regime:}\quad\mathcal{A}(s,t)\sim B(s,t)\,s^{j(t)}\,,$ (120)

where $B(s,t)$ is subdominant in the large $s$ limit, and
$j(t)=j_{0}+\alpha^{\prime}t$ is the Regge trajectory.

As it is well known, eq. (119) violates the Cerulus-Martin bound (1). In
addition, the linear Regge trajectory in eq. (120) violates the bound in (105)
that we derived for negative $t$ in the case $\alpha=0$. As it is pointed out
in [6], the reason behind the violation of the Cerulus-Martin bound would lie
in the lack of polynomial boundedness. This is because, even if we tune the
parameters $j_{0}$ and $\alpha^{\prime}$ so that no massless pole appears
(i.e., $j_{0}-n\neq 0$) we will still have the same high-energy behaviors in
the two regimes. Moreover, both assumptions 1 and 62 can be satisfied by the
amplitude (116). Indeed, we can verify that there exists some value $z_{*}$
for which the amplitude $\mathcal{M}(s,t(z))$ grows exponentially as $e^{+{\rm
Re}[\xi(z_{*})]\,s}$ in the limit $s\rightarrow\infty,$ where ${\rm
Re}[\xi(z_{*})]>0.$ This means that (perturbative) string theory is
characterized by features that cannot be described in the context of local QFT
and falls into the class $\alpha=1$ according to our parameterization.

A similar analysis can be performed in the case of closed string theory for
the tree-level closed string exchange, e.g., for the Virasoro-Shapiro
amplitude [67, 66]. Also in this case the high-energy behavior of the
scattering amplitude is not consistent with polynomial boundedness. In fact,
the hard-scattering and Regge behaviors are the same as in the case of
Veneziano amplitude.

It is worth mentioning that the violation of the Cerulus-Martin bound in
perturbative string theory was initially discussed in Ref. [6, 7]. Here, by
using the more general lower bound (63) we can understand the reason behind
such a violation more concretely and that it does not imply a pathology
because the high-energy behavior of the Veneziano amplitude must be checked to
be consistent with a different lower bound, i.e., the one in eq. (63) with
$\alpha=1,$ which is indeed satisfied by (119).

Despite the non-local nature of scattering amplitudes evaluated
perturbatively, Ref. [19] claims that the Cerulus-Martin bound is restored in
the non-perturbative amplitude obtained after the resummation of divergent
perturbative expansion. This result is interpreted as an improvement of the
locality of strings [19]. This implies that string amplitudes may be
consistent with the $\alpha=0$ case non-perturbatively. If so, the linearity
of the Regge trajectory $j(t)$ will be modified at least for complex $t$ with
large modulus such that $\max\limits_{z\in\partial\mathcal{D}_{R}}[{\rm
Re}\,j(t(z))]$ is finite.

###  6.5 Infinite-derivative QFTs

To conclude the list of applications, we now consider a non-local QFT in which
the Lagrangian contains infinite-order differential operators. In particular,
we focus on the following type of Lagrangian [68, 23]

$\mathcal{L}=\frac{1}{2}\phi\,e^{(-\Box/M^{2})^{n}}(-\Box+m^{2})\,\phi-\frac{\lambda}{3!}\phi^{3}\,,$
(121)

where $n$ is a positive integer. For these models the corresponding S-matrix
respects unitarity (i.e., the optical theorem) at any loop-order in
perturbation [69, 70, 71, 72, 73]. It is also worth to mention that
differential operators such as $e^{c\,\Box}$ are typical of string field
theory [74, 56, 69] and p-adic string [75, 76].

We are interested in the behavior of the $2\rightarrow 2$ elastic scattering
amplitude. At tree level we have

$\mathcal{M}^{(n)}(s,t)=\lambda^{2}\left(\frac{e^{-(-s)^{n}/M^{2n}}}{s-m^{2}}+\frac{e^{-(-t)^{n}/M^{2n}}}{t-m^{2}}+\frac{e^{-(-u)^{n}/M^{2n}}}{u-m^{2}}\right)\,.$
(122)

Let us note that infinite-derivative QFTs with odd powers $n$ can be
pathological because the scattering amplitude grows exponentially in $s$ in
the Regge limit ($s\to\infty$ with fixed $t$) even when $t<0,s\in\mathbb{R}.$
In the case of string field theory similar exponential operators with $n=1$
are present but the form of the interaction vertices is sufficiently
complicated such that no pathological behavior arises [69].

As an example, we explicitly analyse the case $n=2$ in which the perturbation
theory is well-defined at high energy for physical values of the momenta. In
this case, the tree-level amplitude (122) reduces to

$\mathcal{M}^{(2)}(s,t)=\lambda^{2}\left(\frac{e^{-s^{2}/M^{4}}}{s-m^{2}}+\frac{e^{-t^{2}/M^{4}}}{t-m^{2}}+\frac{e^{-u^{2}/M^{4}}}{u-m^{2}}\right)\,.$
(123)

Obviously, the high-energy behavior of this amplitude for fixed angle is in
contradiction with the Cerulus-Martin bound (1). Indeed, in the hard-
scattering regime we need to take the limit $s\rightarrow\infty$ with
$z=\cos\theta=\text{fixed},$ thus we obtain

$\text{hard-scattering
regime:}\quad\mathcal{M}^{(2)}(s,t(z))\sim\lambda^{2}\left(\frac{e^{-s^{2}/M^{4}}}{s}-2\frac{e^{-s^{2}(1-z)^{2}/4M^{4}}}{s(1-z)}-2\frac{e^{-s^{2}(1+z)^{2}/4M^{4}}}{s(1+z)}\right)\,,$
(124)

which decays faster than $e^{-c\,\sqrt{s}\log s}.$

We now want to check whether all the assumptions 1, 61 and 62 are satisfied
and, in particular, that polynomial boundedness is violated. First of all, the
analytic structure of the amplitude is the same as the one in the standard
local case ($M\rightarrow\infty$) besides the presence of essential
singularities at infinity. In the forward limit ($t=0$) and at high energy the
amplitude tends to a constant, thus respecting eq. (62). Moreover, we can
check that the amplitude is exponentially bounded for $z\in\mathbb{C}$ and
large $s\in\mathbb{R}.$ Since there exists some value $z_{*}$ such that ${\rm
Re}[(1+z_{*})^{2}]<0,$ the amplitude (123) can diverge in the complex
$z$-plane for large $s$ as

$\mathcal{M}^{(2)}(s,t(z_{*}))\sim\lambda^{2}\frac{e^{+c\,s^{2}|{\rm
Re}[(1+z_{*})^{2}]|/M^{4}}}{s}\,,$ (125)

where $c$ is some positive constant. This behavior suggests that the infinite-
derivative theory (121) with $n=2$ is non-localizable with degree of non-
locality given by $\alpha=n=2.$

Hence, we can conclude that the violation of the Cerulus-Martin bound is not a
pathology of the model (121) because polynomial boundedness is not satisfied.
In fact, the amplitude (123) must satisfy a more general lower bound derived
under the assumption of exponential boundedness and, indeed, we can easily
check that the high-energy behavior for fixed angle is consistent with the
lower bound (63) with $\alpha=2$ derived in the hard-scattering regime.

##  7 Summary & conclusions

In this work we derived new rigorous lower bounds on elastic scattering
amplitudes in two different kinematic regimes: hard-scattering and Regge
limits. The working assumptions are (see also sec. 3.3): (1) analyticity in a
certain subdomain $\mathcal{D}_{R}$ of the complex $z$-plane (where
$z=\cos\theta$), (61) exponential boundedness of the amplitute in
$\mathcal{D}_{R}$, (62) forward amplitude bounded from below by inverse powers
of the center-of-mass energy, and (4) a finite mass gap. The assumption 61 on
the exponential growth of the amplitude is physically related to the non-local
nature of the corresponding underlining theory. In particular, in our study we
proposed to define and parametrize non-locality in terms of the S-matrix, and
our parameterization in eq. (28) or (61) is motivated by Jaffe’s criterion (7)
for the Wightman functions.

To clearly summarize our results we now write down again the newly derived
lower bounds. In the hard-scattering limit we obtained (in eq. (63))

$\max_{-a\leq\cos\theta\leq
a}\left|\mathcal{M}(s,\cos\theta)\right|\geq\mathcal{N}(s)\,e^{-f(a)\,\sqrt{s}\,\log(s/s_{0})}e^{-g(a)\,s^{\alpha+\frac{1}{2}}}\,,$
(126)

where $\mathcal{N}(s)$ is a positive function of $s$ that is subdominant in
the $s\rightarrow\infty$ limit, and $f(a)$ and $g(a)$ are some positive
functions of $a\in(0,1)$. In the case of polynomial boundedness ($\alpha=0$)
we consistently recovered the Cerulus-Martin lower bound [14], i.e., eq. (1).
Whereas, in the case of exponential boundedness ($\alpha>0$) the scattering
amplitude is more suppressed in the high-energy limit and thus the violation
of the Cerulus-Martin bound is allowed.

In the Regge limit we obtained (in eq. (83))

$\max_{8m^{2}\Delta-4p_{s}^{2}\leq
t\leq-8m^{2}\Delta}\left|\mathcal{M}(s,t)\right|\geq
h(\Delta)\,e^{-\tilde{f}(\Delta)\log(s/s_{0})-\tilde{g}(\Delta)s^{\alpha}}\,,$
(127)

where $h(\Delta),$ $\tilde{f}(\Delta)$ and $\tilde{g}(\Delta)$ are positive
functions of $0<\Delta\leq p_{s}^{2}/4m^{2},$ see eq. (90). This lower bound
turns out to be a new result in both cases of polynomial and exponential
boundedness. In particular, in the former case ($\alpha=0$) the lower bound is
given by inverse powers of $s,$ whereas in the latter case ($\alpha>0$) an
exponential suppression of the amplitude is allowed also in the Regge limit.
Note that, the bound in the Regge limit is stronger than the one in the hard-
scattering regime due to a missing factor $\sqrt{s}$ in the exponent. This
means that, in principle, scattering amplitudes at large angles are allowed to
be more suppressed as compared to the ones at small angles.

It should be emphasized that the above two inequalities are not bounds at some
specific $\theta$ or $t$ because any amplitude can in principle vanish at
given $\theta$ or $t$. Indeed, they are lower bounds on the maximum of the
modulus of the amplitude for some finite intervals of $\theta$ or $t$.

Furthermore, as a consequence of the bound in the Regge regime, we studied the
implications for the Regge trajectory. Given a Regge behavior of the form
$s^{j(t)},$ where $j(t)$ is the Regge trajectory, in eq. (98) we obtained a
new bound on $j(t)$ for negative $t.$ In particular, for a monotonically
increasing and negative Regge trajectory, in eq. (105) we obtained the
following suggestive bound:

$j(t)\geq-[N-j(0)]\sqrt{-t/m^{2}}\,,$ (128)

for $j(t)<0$ and $t\leq-8m^{2}\Delta_{*},$ where $N-j(0)>0$ by construction
and $\Delta_{*}>0$ was defined by the equation
$\tilde{f}(\Delta_{*})=2[N-j(0)]\sqrt{2\Delta_{*}};$ see sec. 5.3.1 for more
details. Such a bound on the Regge trajectory is derived under the assumption
of polynomial boundedness ($\alpha=0$) and is very important because it
forbids linear behaviors for negative $t.$ This also provides a clear
explanation why the linear behavior of the Regge trajectory for the Veneziano
amplitude in perturbative string theory causes a violation of the Cerulus-
Martin bound, as we discussed in detail in sec. 6.4.

Our results provide rigorous and quite general lower bounds on elastic
scattering amplitudes. Moreover, they can be useful tools to constrain and/or
gain information on the underlining UV theories. In sec. 6 we investigated
several applications of the newly derived lower bounds. By analysing the high-
energy behavior of scattering amplitudes and BH formation in gravity we used
the lower bound in the hard-scattering regime to constrain the degree of non-
locality of gravitational theories. In particular, we found a new non-locality
bound given by $\alpha\geq 1/2$ in $D=4$ spacetime dimensions, which implies
that gravitational theories (such as Einstein’s general relativity) are non-
localizable (or quasi-local) according to Jaffe’s classification. In addition,
we studied the high-energy behavior of scattering amplitudes in the
classicalization proposal, the Galileon model, perturbative string theory, and
infinite-derivative QFTs. By means of our lower bounds, we constrained their
degree of non-locality.

The non-locality bound turns out to be quite weak for the Galileon model, but
it can be stronger in the more general case of classicalization for certain
values of the number of fields and derivatives in the interaction potential.
In perturbative string theory we obtained $\alpha\geq 1/2,$ implying that
tree-level string amplitudes cannot be captured by a local QFT description.

Hence, the newly derived bounds can act as probes of non-locality in various
contexts. This also means that they can be used as consistency checks: they
must be respected in any future approach to UV completion which violates the
polynomial boundedness condition. In this paper we mainly used the bound in
the hard-scattering regime to constrain the degree of non-locality. As a
future work, it would be interesting to consider scenarios in which the
amplitude is exponentially suppressed even in the Regge limit. In such a case,
we would be able to obtain a stronger constraint on the degree of non-locality
of the underlining UV theory, as explained at the beginning of sec. 6.

Finally, let us emphasize that the derivations of the lower bounds elaborated
here are only valid in the presence of a finite mass gap and that all the
scenarios considered as applications do not respect such an assumption.
However, as explained in sec. 6, it is reasonable to expect that the real
reason why the Cerulus-Martin bound is violated is the lack of polynomial
boundedness. In fact, as already mentioned in several places in the paper,
e.g., in the footnote 7, one could try to generalize our derivations to the
case of vanishing mass gap by suitably changing the definitions of the dumb-
bell domains in the complex $z$-plane. This has been done for the Cerulus-
Martin bound (i.e., $\alpha=0$ and hard-scattering regime) in Ref. [59]. It
would be interesting to explicitly generalize our derivations under the
assumption of zero mass gap for any $\alpha\geq 0$ and in both cases of hard-
scattering and Regge limits. This task will be part of a future work.

### Acknowledgements

L. B. is grateful to Anna Tokareva for her hospitality at Imperial College
London during the final stages of this work and would also like to thank The
Royal Society for financial support. Nordita is supported in part by
NordForsk. J.T. is supported by IBS under the project code, IBS-R018-D1. M.Y.
is supported by IBS under the project code, IBS-R018-D3, and by JSPS Grant-in-
Aid for Scientific Research Number JP21H01080.

##  Appendix A Analysis of the functions $C(x,a)$ and $C(x,\Delta)$

In this appendix we wish to show more details on the functions131313With an
abuse of notation we are calling the functions $C(x,a)$ and $C(x,\Delta)$ with
the same letter $C.$ However, it should be understood that the functional
dependence on $(x,a)$ is different from the one on $(x,\Delta)$. Indeed, as
already explained in the main text, the latter is defined in terms of the
former as $C(x,\Delta)\equiv C(x,a)|_{a=1-x\Delta}$. Below we will make a
similar abuse of notation when we will define $x_{*}(a)$ and $x_{*}(\Delta)$.
$C(x,a)$ and $C(x,\Delta)$ whose behaviors were crucial for the derivations of
the lower bounds in the hard-scattering (sec. 4) and in the Regge (sec. 5)
limits; recall that $x=(2m/p_{s})^{2}.$ Let us consider the two regimes
separately.

##### Hard-scattering regime.

In the limit of high energy and fixed-angle, we expanded the function (67)
around $x=0$ (i.e., $s\rightarrow\infty$) and for fixed values of $a\in(0,1),$
and as a result we obtained $C(x,a)=C_{0}(a)+\mathcal{O}(\sqrt{x}),$ where
$C_{0}(a)$ is given in eq. (74). A crucial step to derive the lower bound (63)
is the use of the inequality

$C(x,a)\geq C_{0}(a)\,.$ (129)

We can numerically show that (129) holds for all $x\leq x_{*}(a),$ where
$x_{*}(a)\neq 0$ is defined as the solution of the equation
$C(x_{*}(a),a)=C_{0}(a).$ In Fig. 3a we have shown the behavior of the
function $C(x,a)$ as compared to $C_{0}(a)$ for $a=0.6.$ The figure suggests
that the derivative $\partial_{x}C(x,a)$ is positive at $x=0$ while it becomes
negative at some point in the interval $0<x<x_{*}(a)$.

This behavior of $C(x,a)$ can be understood analytically by working with an
approximate analytic expression for $x_{*}(a)$. Let us expand $C(x,a)$ and its
derivative as

$\displaystyle C(x,a)$ $\displaystyle=$ $\displaystyle
C_{0}(a)+C_{1/2}(a)\sqrt{x}+C_{1}(a)x+\mathcal{O}(x^{3/2})\,,$ (130)
$\displaystyle\frac{\partial}{\partial x}C(x,a)$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{1/2}(a)\frac{1}{\sqrt{x}}+C_{1}(a)+\mathcal{O}(\sqrt{x})\,,$
(131)

where the expressions of the coefficients $C_{1/2}(a)$ and $C_{1}(a)$ are
lengthy and we do not write them explicitly. Here, it is only important that
$C_{1/2}(a)$ and $C_{1}(a)$ are always positive and negative for $0<a<1$,
respectively. Now we solve the equation $C(x_{*}(a),a)=C_{0}(a)$ by using
(130) and obtain

$x_{*}(a)\simeq\left(-\frac{C_{1/2}(a)}{C_{1}(a)}\right)^{2}\,.$ (132)

The analytic estimate (132) is plotted in Fig. (3b). We can notice that for
$a=0.6$ we get $x_{*}(0.6)\simeq 0.019,$ which gives a good quantitative
estimation of the intersection point between $C(x,a)$ and $C_{0}(a).$ The
function $x_{*}(a)$ vanishes at $a=0$ and $a=1,$ i.e.,
$x_{*}(a=0)=0=x_{*}(a=1).$ Around these points, $x_{*}(a)$ is approximated as

$\displaystyle x_{*}(a)|_{a\sim
0}\simeq\frac{9a^{4}\log^{2}\left(\frac{4}{a}\right)}{\left(\log\left(\frac{4}{a}\right)+12\right)^{2}}\,,\qquad
x_{*}(a)|_{a\sim 1}\simeq\frac{8(1-a)}{9}\,.$ (133)

Moreover, from eq. (131) we can understand that $\partial_{x}C(x,a)$ is
positive for $0\leq x\lesssim\frac{x_{*}(a)}{4}$, while it becomes negative
for $\frac{x_{*}(a)}{4}\lesssim x\leq x_{*}(a)$. This explains the behavior of
$C(x,a)$ for $0\leq x\leq x_{*}(a)$ shown in Fig. 3a.

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

Figure 3: (a) Behavior of $C(x,a)$ in eq. (67) as a function of
$x=(2m/p_{s})^{2}$ (blue line). We can notice that for sufficiently small
values of $x$ (sufficiently high energies) the inequality $C(x,a)\geq
C_{0}(a)$ holds true, where $C_{0}(a)$ is given in eq. (74). We have set
$a=0.6,$ which implies $C_{0}(a)\simeq 0.601$ (red dashed line) and
$x_{*}(a)\simeq 0.019$. (b) Plot of the analytic estimate (132) for
$x_{*}(a)$.

##### Regge regime.

In the limit of high energy and fixed momentum transfer in sec. 5 we
considered an expansion of the function $C(x,\Delta)$ around $x=0$ and for
fixed values of $\Delta.$ Also in this case, the behavior of the function at
high energies was crucial for the derivation of the bound (83).

First of all, by inspecting the structure of $C(x,\Delta)$ we can notice that
it is real valued for $x\leq 1/\Delta,$ while it becomes complex valued for
$x>1/\Delta.$ This happens because for $x>1/\Delta$ we get a logarithm with a
negative argument, as it can be explicitly checked by taking $C(x,a)$ in eq.
(67) and replacing $a\rightarrow 1-x\Delta.$ For completeness, here we show
its full expression:

$C(x,\Delta)=\frac{1}{\sqrt{x}}\frac{\log\left(\frac{2\sqrt{(x+2)^{2}\left(1-\frac{\left(-\sqrt{x\left(8\Delta-4\Delta^{2}x+x+4\right)}+x+2\right)^{2}}{4(\Delta
x-1)^{2}}\right)}+2x+4}{\sqrt{(x+2)^{2}\left(\left(x-\sqrt{x(x+4)}+2\right)^{2}-\frac{\left(-\sqrt{x\left(8\Delta-4\Delta^{2}x+x+4\right)}+x+2\right)^{2}}{(\Delta
x-1)^{2}}\right)}+(x+2)\left(x-\sqrt{x(x+4)}+2\right)}\right)}{\log\left(\frac{(1-x\Delta)\left(2\sqrt{(x+2)^{2}\left(1-\frac{\left(-\sqrt{x\left(8\Delta-4\Delta^{2}x+x+4\right)}+x+2\right)^{2}}{4(\Delta
x-1)^{2}}\right)}+2x+4\right)}{(x+2)\left(-\sqrt{x\left(8\Delta-4\Delta^{2}x+x+4\right)}+x+2\right)}\right)}\,.$
(134)

This does not cause any problem in our analysis because we are interested in
the parameter region $0\leq a\leq 1$ that is equivalent to $x\leq 1/\Delta$.
Thus, in what follows we assume the condition $x\leq 1/\Delta$.

To prove (83) it was crucial the use of the inequality

$C(x,\Delta)\geq[1-6C_{0}(\Delta)]\frac{1}{\sqrt{x}}\,.$ (135)

We can numerically show that (135) holds for all $x\leq x_{*}(\Delta),$ where
$x_{*}(\Delta)\neq 0$ is defined as the solution of the equation
$C(x_{*}(\Delta))=[1-6C_{0}(\Delta)](x_{*}(\Delta))^{-1/2}.$ In Fig. 4a we
have shown the behavior of the function $C(x,\Delta)$ as compared to
$[1-6C_{0}(\Delta)]x^{-1/2}$ for $\Delta=1.$ We can explicitly see that the
sub-leading term $C_{\text{sub}}(x,\Delta)\equiv
C(x,\Delta)-[1-6C_{0}(\Delta)]x^{-1/2}$ is non-negative and monotonically
decreases in $x$ for $x\leq x_{*}(\Delta)$.

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

Figure 4: (a) Behavior of $C(x,\Delta)$ in eq. (134) as a function of
$x=(2m/p_{s})^{2}$ (blue line) as compared to $[1-6C_{0}(\Delta)]x^{-1/2}$
(red dashed line), where $C_{0}(\Delta)$ is given in eq. (86). We can notice
that for sufficiently small values of $x$ (i.e., sufficiently high energies)
the inequality $C(x,\Delta)\geq[1-6C_{0}(\Delta)]x^{-1/2}$ holds true. We have
set $\Delta=1,$ which implies $C_{0}(\Delta)\simeq 0.1083$ and
$x_{*}(\Delta)\simeq 0.2966$. Note that the curves stop at $x=1$ because the
function $C(x,\Delta)$ is real only for $x\leq 1/\Delta;$ it becomes complex
valued for $x>1/\Delta.$ (b) Plot of the approximate solution for
$x_{*}(\Delta)$ in eq. (138).

This behavior can be understood analytically by working with an approximate
expression for $x_{*}(\Delta)$. Let us expand $C(x,\Delta)$ around $x=0$ as

$\displaystyle
C(x,\Delta)=C_{-1/2}(\Delta)\frac{1}{\sqrt{x}}+C_{0}(\Delta)+C_{1/2}(\Delta)\sqrt{x}+\mathcal{O}(x)\,,$
(136)

where $C_{0}(\Delta)$ was defined in eq. (86), whereas

$C_{-1/2}(\Delta)\equiv 1-6C_{0}(\Delta)\,,\,\quad
C_{1/2}(\Delta)\equiv\frac{\Delta\left(85-83\sqrt{2\Delta+1}\right)-2\sqrt{2\Delta+1}+2}{360(2\Delta+1)^{1/4}\left(\sqrt{2\Delta+1}-1\right)^{3/2}}\,.$
(137)

From eq. (136) also follows that $C_{\text{sub}}$ can be approximated as
$C_{\text{sub}}\simeq C_{0}(\Delta)+C_{1/2}(\Delta)\sqrt{x}$. This implies
that $C_{\text{sub}}$ is a monotonically decreasing function of $x$ because
$C_{1/2}(\Delta)$ is negative for $\Delta>0$.

The equation $C(x_{*}(\Delta),\Delta)=C_{-1/2}(\Delta)x_{*}^{-1/2}$ can be
solved in the approximation (136), and we obtain

$x_{*}(\Delta)\simeq\left(-\frac{C_{0}(\Delta)}{C_{1/2}(\Delta)}\right)^{2}=\frac{7200}{6889\Delta+6723\sqrt{2\Delta+1}+6725}\,.$
(138)

We have shown the behavior of the analytic estimate (138) for $x_{*}(\Delta)$
in Fig. 4b. $x_{*}(\Delta)$ is a monotonically decreasing function, it becomes
maximum at $\Delta=0$ and vanishes in the limit $\Delta\rightarrow\infty.$
These qualitative features can be verified by looking at the intersection
point of the full (non-expanded) $C(x,\Delta)$ and $C_{-1/2}(\Delta)x^{-1/2}$
by varying the values of $\Delta.$ In addition, we can also check that eq.
(138) provides a good quantitative estimation: for $\Delta=1$ we have
$x_{*}(\Delta=1)\simeq(-C_{0}(1)/C_{1/2}(1))^{2}\simeq 0.28$ (see Fig. 4a).

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