warn luatex=false

CERN-TH-2020-161
Saclay-t20/056

#  Positive Moments for Scattering Amplitudes

Brando Bellazzini  Institut de Physique Théorique, Université Paris Saclay,
CEA, CNRS, F-91191 Gif-sur-Yvette, France  CERN, Theoretical Physics
Department, Rte de Meyrin 385, CH-1211, Geneva, Switzerland  Joan Elias Miró
International Centre for Theoretical Physics (ICTP), Strada Costiera 11,
34135, Trieste, Italy  Riccardo Rattazzi  Theoretical Particle Physics
Laboratory (LPTP), Institute of Physics,EPFL, Lausanne, Switzerland  Marc
Riembau  Theoretical Particle Physics Laboratory (LPTP), Institute of
Physics,EPFL, Lausanne, Switzerland  Départment de Physique Théorique,
Université de Genève, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland
Francesco Riva  Départment de Physique Théorique, Université de Genève, 24
quai Ernest-Ansermet, 1211 Genève 4, Switzerland

###### Abstract

We find the complete set of conditions satisfied by the forward 2 $\to 2$
scattering amplitude in unitary and causal theories. These are based on an
infinite set of energy dependent quantities – the arcs – which are
dispersively expressed as moments of a positive measure defined at
(arbitrarily) higher energies. We identify optimal finite subsets of
constraints, suitable to bound Effective Field Theories (EFTs), at any finite
order in the energy expansion. At tree-level arcs are in one-to-one
correspondence with Wilson coefficients. We establish under which conditions
this approximation applies, identifying seemingly viable EFTs where it never
does. In all cases, we discuss the range of validity in both energy and
couplings, where the latter have to satisfy two-sided bounds. We also extend
our results to the case of small but finite $t$. A consequence of our study is
that EFTs in which the scattering amplitude in some regime grows in energy
faster than $E^{6}$ cannot be UV-completed.

Effective Field Theory (EFT) is the universal framework to describe particle
physics on the basis of the principles of quantum mechanics and relativity.
The EFT construction is nicely independent of the detailed features of the
microphysics lying above reachable energies. Yet unitarity, causality and
crossing place robust constraints on the structure of its couplings. In
particular, these constraints take the form of sharp positivity bounds from
dispersion relations for the scattering amplitude both forward Pham and Truong
(1985); Ananthanarayan  _et al._ (1995); Pennington and Portoles (1995); Adams
_et al._ (2006); Bellazzini (2017), and at finite angle Vecchi (2007); Nicolis
_et al._ (2010); de Rham  _et al._ (2017, 2018); Arkani-Hamed and Huang
(2019), with a multitude of interesting recent applications e.g. Komargodski
and Schwimmer (2011); Camanho  _et al._ (2016); Bellazzini  _et al._ (2016,
2014, 2018); Cheung and Remmen (2016); Luty  _et al._ (2013); Distler  _et
al._ (2007); Englert  _et al._ (2019); Bellazzini  _et al._ (2019a, b); Chen
_et al._ (2019); Remmen and Rodd (2019); Zhang and Zhou (2020), in addition to
the original studies in the context of the chiral Lagrangian Pham and Truong
(1985); Pennington and Portoles (1995); Ananthanarayan  _et al._ (1995).

In this article we extend the positivity conditions on the forward amplitude
to what appears to be a complete set. This is made possible by: _i)_ writing
the dispersion relations in terms of suitable energy dependent quantities in
the EFT, the _arcs_ , which are directly related to the Wilson coefficients at
tree level and capture features of the RG evolution at the quantum level;
_ii)_ noticing that by unitarity the arcs correspond to the sequence of
_moments_ of a positive measure over a compact interval. The resulting setup
precisely fulfils the hypotheses of Hausdorff’s moment problem, whose known
solution provides the set of necessary and sufficient positivity constraints
on the forward amplitude. We also extend some of our results beyond the
forward limit, using the classic result in Refs. Lehmann (1958); Martin (1965)
on the positivity of the $t$ derivatives of the imaginary part of the
amplitude.

Our results are related and partly overlap with those obtained by the
geometric approach put forward in Ref. Arkani-Hamed and Huang (2019). An
important difference is that our method enables to identify _optimal_
constraints that involve a finite number of arcs/Wilson coefficients only.
This is the situation closer to questions of phenomenological interest.

The constraints we present are most effective in derivatively coupled
theories, where the forward amplitude is finite in the massless limit. There,
the arcs are single scale quantities that purely depend on the running
couplings, and our constraints directly bound the RG flow (an aspect briefly
touched in Refs. Adams  _et al._ (2006); Distler  _et al._ (2007); Luty  _et
al._ (2013); Bellazzini  _et al._ (2019b, a)). One well formulated hypotheses
that can be tested in this context concerns the existence of symmetries –
exact, accidental or weakly broken – which may appear in the low-energy EFT.
In particular, the non-linear transformations

$\phi(x)\rightarrow\phi(x)+b+b_{\mu_{1}}x^{\mu_{1}}+\cdots+b_{\mu_{1}\cdots\mu_{N}}x^{\mu_{1}}\cdots
x^{\mu_{N}}\,,$ (1)

with $b^{\prime}s$ traceless, are symmetries in EFTs where interactions have
many derivatives and, therefore, deliver soft amplitudes, i.e. amplitudes with
a fast energy $E$ growth. The case $N=0$ is familiar: $U(1)$ Goldstone bosons
with $2\to 2$ amplitudes $\mathcal{M}\sim E^{4}$. The same behaviour arises
for the EFT of massless particles with spin, like for the Euler-Heisenberg
Lagrangian in QED, as well as the theories of Refs. Liu  _et al._ (2016);
Bellazzini  _et al._ (2017); Bellazzini and Riva (2018). The $N=1$ case
corresponds to Galileons Nicolis  _et al._ (2009), with $\mathcal{M}\sim
E^{6}$, and so on. We refer to theories with large exponent in
$\mathcal{M}\sim E^{2n}$, $n>2$, as super-soft Bellazzini (2017);
Hinterbichler and Joyce (2014). Similarly, longitudinal polarizations in
theories with massive spin-$J$ particles have supersoft amplitudes with
$2n\geq 3J$ Bellazzini  _et al._ (2019a). For example, approximate linear
diffeomorphisms suppress the self-interactions from the Einstein-Hilbert term,
relative to the super-soft linearised (Riemann)3 terms. Such a scenario for
gravity is incompatible with tree-level UV completions Camanho  _et al._
(2016). In this work we will show that supersoftness _in general_ cannot
emerge, neither by structure nor by accident, from any reasonable, weakly or
strongly-coupled UV completion.

This paper is organized as follows. In section I, focussing on the forward
limit, we construct the necessary general dispersion relations and derive
optimal bounds on the arc variables. In sections II.1 and II.2 we apply our
results, first assuming a weakly coupled UV completion and then lifting this
assumption while focussing on the more specific case of an abelian Goldstone
boson. In section III we consider a first foray of our methodology beyond the
forward limit. Finally in section IV we summarize our results and offer an
outlook.

##  I Arcs and Their Constraints

We study the $2\to 2$ scattering amplitude ${\cal M}$ of a single particle of
mass $m$ (including the limit $m^{2}\rightarrow 0$ if it exists). We discuss
first the forward limit $t\to 0$ in theories where it is finite, and define
${\cal M}(s)\equiv\lim_{t\rightarrow 0}{\cal M}(s,t)$; for concreteness we
focus on the spin-0 case, but our results carry over to arbitrary spin and
flavour structure Bellazzini (2017). Extensions to $t\neq 0$ are postponed to
section III. We will assume the following analytic properties of ${\cal M}$:

  1. $\circ$

Crossing symmetry in $s\leftrightarrow u$ with real analyticity, namely ${\cal
M}(s)={\cal M}^{*}(4m^{2}-s^{*})$.

  2. $\circ$

The singularities of ${\cal M}(s)$ in the complex $s$-plane consist solely of
a unitarity cut at physical energies $s>4m^{2}$, plus the crossing symmetric
one at $s<0$, see e.g. Ref. Martin (1969).111Resonances with masses
$M^{2}<4m^{2}$ below threshold, imply additional poles at $s=M^{2},3M^{2}$. We
ignore these for simplicity; their inclusion is straightforward.

  3. $\circ$

Unitarity, via the optical theorem, implies $\text{Im}{\cal M}(s)\\!>0$.

  4. $\circ$

The amplitude ${\cal M}(s)$ is polynomially bounded as $|s|\rightarrow\infty$.
In particular, we assume ${\cal M}(s)/s^{2}\rightarrow 0$ as
$s\rightarrow\infty$. In the gapped case, validity of this condition is
ensured by the Froissart bound Froissart (1961); Martin (1963).

Exploiting the analyticity of the amplitude in the upper half plane, we define
the arc variables (or simply “arcs”),

$a_{n}(\hat{s})\equiv\int_{\begin{minipage}[h]{26.01613pt} \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman[small] \vertex(x1) at
(0,.35); \vertex(x2) at (.1,.35); \vertex(i1) at (0,0); \vertex(i2) at
(.25,0); \vertex(i3) at (.25+.15,0.05){${}_{\hat{s}}$}; \diagram*{ (i1)
--[thick, half left, looseness = 2.7] (i2), (x1) --[white] (x2) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
\end{minipage}}\frac{d{\hat{s}^{\prime}}}{\pi i}\frac{{\hat{\cal
M}}({\hat{s}^{\prime}})}{\hat{s}^{\prime 2n+3}}\,,$ (2)

where $\hat{s}\equiv s-2m^{2}$ is the crossing-symmetric variable, ${\hat{\cal
M}}(\hat{s})\equiv{\cal M}(s)$, and ${}_{\hat{s}}$ represents a
counterclockwise semicircular path in the upper half plane of radius
$\hat{s}$, as shown in Fig. 1.

Figure 1: The semi-circle contour of Eq. (2) in the complex upper
${\hat{s}^{\prime}}$-plane. Wiggle lines denote the branch-cuts on the real
axis.

The Cauchy theorem implies that the integral over the closed contour
$C=\begin{minipage}[h]{26.01613pt} \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman[small] \vertex(x1) at
(0,.35); \vertex(x2) at (.1,.35); \vertex(i1) at (0,0); \vertex(i2) at
(.25,0); \vertex(i3) at (.25+.15,0.05){${}_{\hat{s}}$}; \diagram*{ (i1)
--[thick, half left, looseness = 2.7] (i2), (x1) --[white] (x2) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
\end{minipage}+\begin{minipage}[h]{28.18636pt} \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman[small] \vertex(x1) at
(0,.35); \vertex(x2) at (.1,.35); \vertex(i1) at (0,0); \vertex(i2) at
(.25,0); \vertex(i3) at (.25+.19,0.05){${}_{\infty}$}; \diagram*{ (i1)
--[thick, half left, looseness = 2.7] (i2), (x1) --[white] (x2) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
\end{minipage}+l_{1}+l_{2}$ vanishes. Thus, we deform the integral in Eq. (2)
along ${}_{\hat{s}}$ into an integral along $\begin{minipage}[h]{28.18636pt}
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman[small] \vertex(x1) at
(0,.35); \vertex(x2) at (.1,.35); \vertex(i1) at (0,0); \vertex(i2) at
(.25,0); \vertex(i3) at (.25+.19,0.05){${}_{\infty}$}; \diagram*{ (i1)
--[thick, half left, looseness = 2.7] (i2), (x1) --[white] (x2) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
\end{minipage}+l_{1}+l_{2}$. For integer $n$, we can use crossing symmetry and
real analyticity to relate the amplitude above the lefthand cut (path $l_{2}$)
to the one above the righthand cut (path $l_{1}$),

${\hat{\cal M}}(\hat{s}+i\epsilon)={\hat{\cal
M}}^{*}(\hat{s}-i\epsilon)={\hat{\cal M}}^{*}(-\hat{s}+i\epsilon)\,.$

Due to the Froissart bound, ${\hat{\cal M}}/\hat{s}^{2}\to 0$, the integral
along the semicircle of infinite radius vanishes for ${n\geq 0}$. Thus, we can
write Eq. (2) as

$a_{n}(\hat{s})=\frac{2}{\pi}\int_{\hat{s}}^{\infty}d{\hat{s}^{\prime}}\frac{\textrm{Im}{\hat{\cal
M}}({\hat{s}^{\prime}})}{\hat{s}^{\prime 2n+3}}\,,\quad n\geq 0\,.$ (3)

On the one hand, the arcs as defined via the IR representation Eq. (2) are IR
quantities that can be systematically computed as an expansion in powers of
$s$ in the domain of validity of the IR EFT $s\ll s_{max}$, with $s_{max}$ the
cutoff. In the simple case of a tree-level amplitude in the forward limit,
${{\hat{\cal M}}}(\hat{s})=\sum_{n=0}c_{2n}\hat{s}^{2n}$ and the arcs match
simply to Wilson coefficients, $a_{n}(\hat{s})=c_{2n+2}$. Moreover, the scale
dependence of arcs partly reflects the EFT RG flow, as we discuss in section
II.

On the other hand, according to the UV representation in Eq. (3), arcs receive
contributions from all microphysics scales up to the far UV. So, Eq. (2)
ideally represents something measurable in our low energy experiment, while
the representation in Eq. (3) requires knowledge of the theory at all scales.
Nevertheless, in this form, Eq. (3) has interesting properties that we now
discuss.

###  I.1 All Constraints

From Eq. (3) it follows that the arcs are positive,

$a_{n}(\hat{s})>0\,,$ (4)

since the imaginary part of the forward amplitude is positive. In fact,
convoluting in (3) a positive function $F$, we have

$\frac{2}{\pi}\int_{\hat{s}}^{\infty}d{\hat{s}^{\prime}}\frac{\textrm{Im}{\hat{\cal
M}}({\hat{s}^{\prime}})}{\hat{s}^{\prime
3}}F\left(\frac{\hat{s}}{{\hat{s}^{\prime}}}\right)>0\,.$ (5)

For instance,
$F(\hat{s}/{\hat{s}^{\prime}})=[\hat{s}/{\hat{s}^{\prime}}]^{2n}(1-[\hat{s}/{\hat{s}^{\prime}}]^{2})$,
with $\hat{s}<{\hat{s}^{\prime}}$, implies

$a_{n}-\hat{s}^{2}a_{n+1}>0\,.$ (6)

Clearly, to every function $F$, positive in
${(\hat{s}/{\hat{s}^{\prime}})^{2}\in[0,1]}$, correspond inequality
constraints relating arcs of different orders. Characterising the most general
such function will allow us to address,

Question 1: What is the complete set of constraints the arcs $a_{n}$ must
satisfy?

To answer this question we will relate our problem to the theory of moments.
The following change of variables

$x\equiv(\hat{s}/{\hat{s}^{\prime}})^{2}\quad
d\mu(x)=\frac{dx}{\pi}\,\textrm{Im}\,{\hat{\cal M}}(\hat{s}/\sqrt{x})\,$

simplifies the notation and defines a positive measure, so that we can write

$\hat{s}^{2n+2}a_{n}=\int_{0}^{1}x^{n}d\mu(x)\,\,.$ (7)

A sequence of dimensionless numbers, defined as in Eq. (7) with $d\mu$
positive, is called a _sequence of moments_. In fact, we have a one-parameter
family of sequences because each moment is a function of $\hat{s}$. We comment
on the $\hat{s}$ dependence below.

We introduce the discrete derivatives

$(\Delta a)_{n}=\hat{s}^{2}a_{n+1}-a_{n}\,,$ (8)

with higher order differences defined recursively,
$\Delta^{k}=\Delta(\Delta^{k-1})$ and $\Delta^{0}a_{n}=a_{n}$. For instance,
$(\Delta^{2}a)_{n}=a_{n}-2\hat{s}^{2}a_{n+1}+\hat{s}^{4}a_{2+n}$, etc. A
sequence of moments necessarily satisfies

$(-1)^{k}(\Delta^{k}a)_{n}=\frac{1}{\hat{s}^{2n+2}}\int_{0}^{1}x^{n}(1-x)^{k}d\mu(x)>0$

since the functions

$F(x)=x^{n}(1-x)^{k}$ (9)

are positive in the whole integration domain. The case $k=1$ corresponds to
Eq. (6).

The converse is also true, as implied by the Hausdorff moment theorem: if a
sequence satisfies

$(-1)^{k}(\Delta^{k}a)_{n}>0\quad\forall n,k\geq 0,$ (10)

then there exists a _unique_ measure $d\mu$ such that Eq. (7) is
satisfied.222Arcs probing the theory at finite $s$ are crucial for the mapping
to moments on a compact interval (Hausdorff’s problem). Instead, a sequence
made of amplitude’s residues at $s\to 0$ (equivalent to arcs with vanishing
radius) see e.g. Green and Wen (2019), maps to a non-compact domain (Stieltjes
half-moment problem), and the solution is not unique. This theorem, following
from the fact that the functions in Eq. (9) – called Bernstein polynomials –
are a basis of all positive functions in $[0,1]$, provides an _answer to our
Question 1_.

As mentioned above, both the arcs and their discrete derivatives depend
explicitly on the scale $\hat{s}$, as captured by,

$\\!\\!\\!\\!\\!\\!\frac{d}{d\hat{s}}[(-1)^{k}\Delta^{k}a_{n}]\\!\\!=\\!\\!\left\\{\begin{array}[]{ll}\\!\\!\\!-\frac{2}{\pi}\frac{\mathrm{I}m{\cal
M}(\hat{s}+2m^{2})}{\hat{s}^{3+2n}}&\\!\\!(k=0)\\\ &\\\
\\!\\!\\!2k\hat{s}[(-1)^{k}\Delta^{k-1}a_{n+1}]&\\!\\!(k\geq
1)\end{array}\right.$ (11)

which is negative for all $k$, because of Eq. (10). Therefore, as $\hat{s}$ is
increased, the arcs decrease – proportionally to Im${\cal M}$. This implies
that, given an EFT, the constraints Eq. (10) for $k\geq 1$ become more
stringent as $s$ increases. Conversely, if the conditions Eq. (10) are
satisfied at one scale $\hat{s}$ they are automatically satisfied at smaller
scales. This behaviour will play an important role later on, when we discuss
constraints on the arcs in specific EFTs.

###  I.2 Optimal Bounds for a Finite Set of Arcs

In practice we often focus on a finite number of arcs. For instance, in a
typical EFT only the first few powers of $s$ are phenomenologically
interesting – at tree-level this corresponds to the first few arcs. Thus it is
natural to ask,

Question 2: Considering only a _finite_ number $N$ of arcs, what are their
optimal constraints?

Here _optimal_ means the projection of all constraints on the finite set. In
the language of the previous section, we ask: assuming the components of
$\vec{a}=\\{\hat{s}^{2}a_{0},\dots,\hat{s}^{2+2N}a_{N}\\}$ are moments, what
is the subspace $A(N)\subset\mathbb{R}^{N}$ on which $\vec{a}$ takes values?

Of course Eq. (10) still holds and, for $k+n\leq N$, it involves only the
first $N$ arcs. The constraints with $k+n\geq N$ imply however additional
conditions on the subset $n\leq N$. In what follows we will show a simple
procedure to extract this information.

Similarly to Question 1 – that implied finding the most general positive
function in $[0,1]$ – Question 2 requires finding a parametrization for the
most general polynomial $p(x)=\sum_{i=1}^{N}\alpha_{i}x^{i}$ of finite degree
$\leq N$, positive in $x\in[0,1]$. Indeed, via Eq. (7), each such $p(x)$ leads
to a condition on arcs,

$\int_{0}^{1}p(x)d\mu(x)>0\,\quad\Rightarrow\quad\sum_{i=0}^{N}\alpha_{i}a_{i}>0\,.$
(12)

One can prove that any such polynomial can be written as g. see theorem 4.17.4
of (2015, Simon, B.),

$p=\sum_{J}\underbrace{q_{J,1}^{2}}_{\text{type-1}}+\underbrace{xq_{J,2}^{2}}_{\text{type-2}}+\underbrace{(1-x)q_{J,3}^{2}}_{\text{type-3}}+\underbrace{x(1-x)q_{J,4}^{2}}_{\text{type-4}}$
(13)

where $q_{J,k}(x)$’s are non-zero real polynomials – not necessarily positive
– such that $p(x)$ is degree $N$, i.e. $q_{J,k}$ has at most degree $d_{k}$,
with $d_{1}=\lfloor N/2\rfloor$, $d_{2}=d_{3}=\lfloor(N-1)/2\rfloor$ and
$d_{4}=\lfloor(N-2)/2\rfloor$, where $\lfloor k\rfloor$ is the integer part of
$k\geq 0$. Since Eq. (13) is a sum over positive terms, it is sufficient to
discuss them individually. Therefore we drop the index $J$ and consider one by
one generic polynomials $q_{k}$ of each type-$k$ in Eq. (13),

$q_{k}(x)=\sum_{j=0}^{d_{k}}\alpha_{kj}x^{j}$ (14)

with arbitrary real coefficients $\alpha_{kj}$.

We define the Hankel matrix $(H^{\ell}_{N})_{ij}=a_{i+j+\ell}$, for
$i,j=0,\dots,\lfloor(N-\ell)/2\rfloor$, so that $H^{\ell}_{N}$ involves arcs
up to $a_{N}$ for $N-\ell$ even and $a_{N-1}$ for $N-\ell$ odd. For instance,

$H^{0}_{4}=H^{0}_{5}\equiv\left(\begin{array}[]{ccc}a_{0}&a_{1}&a_{2}\\\
a_{1}&a_{2}&a_{3}\\\ a_{2}&a_{3}&a_{4}\\\ \end{array}\right)\,.$ (15)

Now, polynomials of type-1 imply,

$\int_{0}^{1}q_{1}(x)^{2}d\mu(x)=\hat{s}^{2}\sum_{i,j=0}^{\lfloor
N/2\rfloor}\hat{s}^{2(i+j)}\alpha_{1i}\,a_{i+j}\,\alpha_{1j}>0\,.$

Since the vector
$\\{\alpha_{10},\alpha_{11}\hat{s}^{2},\alpha_{12}\hat{s}^{4},\dots\alpha_{1,{N}/{2}}\hat{s}^{N}\\}$
is arbitrary, the following Hankel matrix must be positive definite,

$H^{0}_{N}\succ 0\,.$ (16)

Following similar steps one finds that the positiveness of
$\int_{0}^{1}xq^{2}_{2}(x)d\mu(x)>0$, $\int_{0}^{1}(1-x)q^{2}_{3}(x)d\mu(x)>0$
and $\int_{0}^{1}x(1-x)q^{2}_{4}(x)d\mu(x)>0$ imply,

$\displaystyle H^{1}_{N}$ $\displaystyle\succ$ $\displaystyle 0\,,$ (17)
$\displaystyle H^{0}_{N-1}-\hat{s}^{2}H^{1}_{N}$ $\displaystyle\succ$
$\displaystyle 0\,,$ (18) $\displaystyle H^{1}_{N-1}-\hat{s}^{2}H^{2}_{N}$
$\displaystyle\succ$ $\displaystyle 0\,,$ (19)

respectively. We refer to Eqs. (16-17) as _homogeneous_ and Eqs. (18,19) as
_inhomogeneous_ constraints.

Since (13) is an arbitrary positive polynomial for $x\in[0,1]$, Eqs. (16-19)
represent the optimal constraints, providing an _answer to Question 2_.

Figure 2: Allowed regions for the arcs $a_{0}$, $a_{1}$, $a_{2}$ (LEFT) and
including $a_{3}$ (RIGHT), according to Eqs. (20,28). LEFT: For fixed Wilson
coefficients, as energy is increased, the theory spans a trajectory in the
space of arcs: the blue trajectories (arrows in the direction of increasing
$s$) correspond to examples in the weak coupling limit Eq. (30), the red
trajectories are examples using Eq. (36) at strong coupling (with the explicit
values Eq. (35) and large $g_{2}$ from Eq. (38)). Values of $s^{4}a_{2}/a_{0}$
larger than the green solid/dashed/dotted lines are excluded by the conditions
Eq. (10) from Bernstein polynomials, up to $k+n=2,3,4$ respectively. RIGHT:
The projections into two dimensional planes correspond to optimal bounds when
only two coefficients are taken into account (the bottom projection
corresponds to the left panel). The volume of the allowed region is $1/180$
w.r.t. the volume of the unit cube.

For instance, Eqs. (16-19) with $N=2$ define the $A(2)$ region:

$\left(\begin{array}[]{cc}a_{0}&a_{1}\\\ a_{1}&a_{2}\end{array}\right)\succ
0\,,\ \ a_{1}>0\,,\ \ a_{0}>\hat{s}^{2}a_{1}\,,\ \ a_{1}>\hat{s}^{2}a_{2}\,.$
(20)

This is illustrated in the left panel of Fig. 2. The first constraint in (20)
implies $a_{0}a_{2}>a_{1}^{2}$, and is saturated by the lowest parabola in
Fig. 2; the fourth constraint in (20) is saturated by the upper line, the
other constraints imply that the coordinates lie in the interval $[0,1]$. For
comparison, the green lines show the constraints obtained using Eq. (10) and
arcs up to $a_{2}$ (solid), $a_{3}$ (dashed), and $a_{4}$ (dotted), and then
projected onto the ($s^{2}a_{1}/a_{0}$,$s^{4}a_{2}/a_{0}$) plane: clearly
these converge to the optimal result A(2).

Similarly, evaluating Eqs. (16-19) for $N=3$, we find $A(3)$:

$\displaystyle\left(\begin{array}[]{cc}a_{0}&a_{1}\\\
a_{1}&a_{2}\end{array}\right)$ $\displaystyle\succ$ $\displaystyle 0\ ,\quad\
\left(\begin{array}[]{cc}a_{0}-a_{1}\hat{s}^{2}&a_{1}-a_{2}\hat{s}^{2}\\\
a_{1}-a_{2}\hat{s}^{2}&a_{2}-a_{3}\hat{s}^{2}\end{array}\right)\succ 0\,,$
(25) $\displaystyle\left(\begin{array}[]{cc}a_{1}&a_{2}\\\
a_{2}&a_{3}\end{array}\right)$ $\displaystyle\succ$ $\displaystyle 0\ ,\quad\
a_{1}>\hat{s}^{2}a_{2}\,,$ (28)

which we illustrate in the right panel of Fig. 2.

Eqs. (16-19) capture how the full constraint on the arc sequence is projected
on the first $N$ arcs. Of course we can consider the projection on any subset.
In particular for a sequence $a_{k},\cdots,a_{N}$ that starts at $k\neq 0$,
Eqs. (16-19) are generalised by the substitution $\ell\to\ell+k$, e.g. Eqs.
(16) becomes $H^{k}_{N}\succ 0$. Eq. (6) belongs in this class.

To conclude this section we compare our results to those of Ref. Arkani-Hamed
and Huang (2019), which – considering the forward amplitude – finds that
consistent EFTs must satisfy the set of homogeneous Hankel matrix positivity
constraints, Eqs. (16-17). Indeed, the whole set of homogeneous constraints
(i.e. for arbitrarily large $N$) implies the ensemble of constraints in Eq.
(10),333That is because, for $x\in[0,1]$, the Bernstein polynomials in Eq. (9)
are arbitrarily well approximated by a combination of polynomials of type-1
and type-2 in Eq. (13), with _arbitrarily large degree_ , which precisely
correspond to the complete set of homogeneous constraints in Eqs. (16-17). For
instance $F(x)=1-x$ is reproduced by
$q_{1}(x)=\sqrt{1-x}=1-x/2-x^{2}/8-x^{2}/16-\cdots$ and $q_{2}=0$, and
corresponds to an infinite Hankel matrix.  and thus by Hausdorff moment
theorem is a necessary and sufficient set. However, when only a finite number
$N$ of arcs/Wilson coefficients is considered, as it is often the case, our
equations Eqs. (16-19) represent the optimal constraints. For instance, if we
are interested in the allowed space for three arcs, as in Eq. (20), the first
two homogenous conditions correspond to simple Hankel determinants and can be
easily obtained with the methods of Arkani-Hamed and Huang (2019). The latter
two inhomogeneous conditions, instead, can only be obtained by considering
infinite many homogeneous Hankel matrices.

##  II Bounds on Wilson Coefficients

Constraints on arcs translate, in principle, into constraints on the
Lagrangian’s Wilson coefficients. In practice, this translation is complicated
by the fact that arcs might receive contributions from (infinitely) many
Wilson coefficients. In what follows we discuss under which circumstances this
translation is possible: section II.1 discusses the tree-level approximation
while section II.2 discusses sizeable quantum effects.

###  II.1 Tree Level

As a first application of our bounds, we focus on the forward limit in
situations where we can consider just the IR tree level amplitude – we will
discuss in the next section under which conditions this approximation holds.
At low energy this takes a polynomial form444In the single flavour case the
amplitude is a function of $s^{2}+t^{2}+u^{2}$ and $stu$, and odd powers of
$s$ vanish in the forward limit. See moreover footnote 1.

${\hat{\cal M}}(\hat{s})=\sum
c_{d}\hat{s}^{d}=c_{0}+c_{2}\hat{s}^{2}+c_{4}\hat{s}^{4}+\cdots\,.$ (29)

We compute the $a_{n}(\hat{s})$ using the definition Eq. (2) and Eq. (29) and
find,

$a_{n}=\frac{1}{(2n+2)!}\left.\frac{\partial^{2n+2}}{\partial\hat{s}^{2n+2}}\hat{\cal
M}(\hat{s})\right|_{\hat{s}=0}=c_{2n+2}\,,$ (30)

independently of $\hat{s}$. So the constraints of the previous sections can be
read directly in terms of the coefficients appearing in the amplitude. From a
practical point of view it is simpler to focus on a limited number of
coefficients (rather than the complete series), so that Eqs.(16-19) represent
the relevant constraints. Of these, the homogenous constraints of Eqs. (16-17)
do not depend on $\hat{s}$ and thus represents properties of the UV theory
that are intrinsic, i.e. independent of the overall scale of the dynamics. In
particular they include Eq. (4), which implies that all coefficients $c_{n}$
be strictly positive Adams  _et al._ (2006).

On the other hand, the energy scale $\hat{s}$ appears explicitly in Eqs.
(18-19) (this is translated in the normalisation of Figs. 2 being
$\hat{s}$-dependent). Given an EFT, in the form of a set of $c_{n}$ satisfying
Eqs. (16-17), we can think of Eqs. (18-19) as defining the highest possible
cutoff $\hat{s}_{max}$ where new dynamics must modify our EFT amplitude.

For the simple case of the first three coefficients, in addition to positivity
of each of them, Eq. (20) and Eq. (30) imply

$c_{2}-\hat{s}^{2}c_{4}>0\,,\quad c_{4}-\hat{s}^{2}c_{6}>0\,,\quad
c_{2}c_{6}>c_{4}^{2}\,.$ (31)

For fixed values of the Wilson coefficients, as $\hat{s}$ increases, the arcs
track trajectories in Fig. 2 (blue lines, for different values of $c_{n}$),
that start for $\hat{s}\to 0$ at the origin and evolve along parabolae. In
this case the cutoff must satisfy $\hat{s}^{2}_{max}<{c_{4}}/{c_{6}}$.

The simplest inhomogeneous constraints
${c_{2}\hat{s}^{2}-c_{n}\hat{s}^{n}>0}$, suffice to rule out any, even
approximate, supersoft behaviour, where the tree-level forward amplitude is
dominated by the $O(s^{n})$ growth, $n>2$. First of all, positivity already
implies that supersoft symmetries Eq. (1) are never exact: they are always
explicitly broken by a (possibly small) $c_{2}>0$. They could in principle
have been appreciable in the regime of energy
$\hat{s}\gtrsim\left({c_{2}}/{c_{n}}\right)^{\frac{1}{n-2}}$. However our
bounds forbid that: super-soft theories cannot consistently be UV-completed at
weak coupling, as the expansion in $s$ must be strictly decreasing Bellazzini
_et al._ (2019a); Englert  _et al._ (2019). We will see in the next section
that the same conclusion holds also beyond the weak coupling approximation.

As a concrete example, consider for instance interactions of the form
$(\partial\partial\phi)^{4}$, which is invariant under the Galilean symmetry
$\phi\to\phi+b+b_{\mu}x^{\mu}$, and giving $\mathcal{M}(s)\sim s^{4}$. Purely
on the basis of symmetries, these terms could have naturally dominated the
more relevant $(\partial\phi)^{4}$ interactions, which break the Galilean
symmetry. This is however inconsistent with the first inequality in (31) which
forces $(\partial\partial\phi)^{4}$ to be subdominant.

_Faster UV convergence._ So far the bounds in this section do not depend on
$c_{0}$ in (29). This changes if we assume that
$\lim_{s\rightarrow\infty}\hat{\cal M}(s)\equiv{\cal M}_{\infty}$ is finite.
This is the case, for instance, if the theory in the UV is described by a
finite number of resonances in the tree-level approximation.

For a finite ${\cal M}_{\infty}$ we can extend the definition of arcs in Eq.
(2) to $n\geq-1$. Then we can repeat an analysis similar to the one that we
did in the previous section for the subtracted amplitude $\hat{\cal M}-{\cal
M}_{\infty}$, finding that the arcs define a sequence of positive moments
$\\{a_{-1},a_{0},a_{1},\dots\\}$.

Note in particular that $a_{-1}=\hat{\cal M}(0)-{\cal M}_{\infty}$ can be
regarded as the difference of effective couplings defined by the value of the
forward amplitude in the IR and in the UV respectively, according to the weak
coupling intuition. Since the first of the homogeneous conditions on the arcs
is now $a_{-1}>0$, the IR forward amplitude is larger than the UV one.
Moreover, $a_{-1}a_{1}-a_{0}^{2}>0$ must be satisfied and can be regarded as
an upper bound on $c_{2}^{2}/c_{4}$ if the tree-level approximation of Eqs.
(29,30) holds.

An interesting case that we consider in detail in the next section is the
theory of a single Goldstone boson for which $c_{0}=0$. If the UV completion
is perturbative $-{\cal M}_{\infty}\ll 16\pi^{2}$, then we have

$0<\frac{c_{2}^{2}}{c_{4}}\leq-{\cal M}_{\infty}\ll 16\pi^{2}\,,$ (32)

on the Goldstone theory. 555As example, consider a potential
$V=\lambda/4(|\Phi|^{2}-v^{2})^{2}$ for a canonically normalized complex
scalar field $\Phi$. The tree-level forward scattering of the Goldstone bosons
gives ${\cal
M}(s)=\left[s^{2}/(s+m_{h}^{2})-s^{2}/(s-m_{h}^{2})\right]/(2v^{2})$ where
$m_{h}^{2}=\lambda v^{2}$. Therefore, ${\cal M}_{\infty}=-\lambda$ and
$c_{n}=\lambda/m_{h}^{2n}$, so that $c_{2}^{2}/c_{4}=\lambda$, which saturates
the bound (32).  The upper bound on $c_{2}^{2}/c_{4}$ is in agreement with the
expectation that RG running effects on $c_{4}$ – that we will see Eq. (35) –
are expected to be small in a weakly coupled theory.

_The boundary._ Which theories saturate Eqs. (16-19)? Hausdorff theorem
implies that the integration measure (the imaginary part of the UV forward
amplitude) is _uniquely_ determined if _all_ arcs are known. If the measure
has support on finitely many points, then a finite number of arcs suffices to
determine the measure uniquely.

Physically, a measure that consists of $P$ distinct delta functions,
$\textrm{Im}{\hat{\cal
M}}(\hat{s})=\pi/2\sum_{k=1}^{P}g_{k}^{2}M_{k}^{2}\delta(\hat{s}-M_{k}^{2})$,
is realised in the tree-level approximation when integrating out heavy
particles with $P$ distinct masses $M_{1}<M_{2}<\ldots<M_{P}$ and effective
squared couplings $g_{k}^{2}>0$. This situation corresponds to 666For
$P\rightarrow\infty$, finiteness of $a_{0}$ requires that
$g_{k}^{2}/M_{k}^{4n+4}$ decays sufficiently fast as $k\rightarrow\infty$.
Considering a large but finite number of particles is therefore a good
approximation.

$a_{n}=\sum_{k=1}^{P}\frac{g_{k}^{2}}{M_{k}^{4n+4}}\,.$ (33)

These arcs lie at the boundary of the $A(2P)$ region (defined by Eqs. (16-19)
with $N=2P$), as we now show.

In the variable $x\in[0,1]$ of section I.2, the measure $d\mu(x)$ consists of
strictly positive delta functions located at $x_{k}=\hat{s}^{2}/M_{k}^{4}$ for
$k=1,\dots,P$. In the EFT, i.e. $\hat{s}<M_{1}^{2}$, there are thus $P$ such
delta’s within $(0,1)$. Recalling Eq. (13), Hankel matrices of order
${\bar{P}}$ correspond to the quadratic forms generated by integrating
$x^{a}(1-x)^{b}q_{{\bar{P}}-1}(x)^{2}$ in $d\mu(x)$, with $q_{{\bar{P}}-1}(x)$
a generic polynomial of order ${\bar{P}}-1$. This integral gives a vanishing
result only if all the ${\bar{P}}-1$ zeroes of $q_{{\bar{P}}-1}(x)$ coincide
with the $P$ delta’s in the measure. This can only happen if ${\bar{P}}-1\geq
P$, in which case the corresponding Hankel matrix has order $>P$: Hankel
matrices of order $\leq P$ are strictly positive definite, while they are only
positive semi-definite if their order is $>P$. Considering then Eqs. (16-19),
we conclude that, for $\hat{s}$ within the EFT, the arcs are in the interior
of $A(N)$ for $N<2P$ and at the boundary of $A(2P)$.

This implies that – in weakly coupled theories – the measurements of the arcs
in the IR allows to indirectly count the number $P$ of resonances in the UV:
$P$ is just the smallest ${\bar{P}}$ for which $\det H^{0}_{2{\bar{P}}}=0$.

The same reasoning as above also implies that for $\hat{s}=M_{1}^{2}$, just at
the edge of the EFT, the arcs are at the boundary of the $A(2P-1)$ region.
Indeed, when $\hat{s}$ approaches $M_{1}^{2}$ from below,
$x_{1}=\hat{s}^{2}/M_{1}^{4}$ approaches the edge $x=1$ of the integration
region, and the contribution of the lightest resonance drops out when
integrated against the type-3 polynomial $(1-x)q_{{\bar{P}}-1}(x)^{2}$ of Eq.
(13). Then, for ${\bar{P}}=P$ the zeroes of $q_{{\bar{P}}-1}(x)$ can be chosen
to coincide with the location of the remaining $P-1$ heavier resonances, hence
$\det\left(H^{0}_{2P-2}-M_{1}^{4}H^{1}_{2P-1}\right)=0$. On the other hand,
for ${\bar{P}}<P$, $q_{{\bar{P}}-1}(x)$ has fewer zeroes than there are
resonances. This implies that for $N<2P-1$ Eq. (18) is still strictly
satisfied for $\hat{s}=M_{1}^{2}$ and that $\hat{s}>M_{1}^{2}$ is needed in
order to violate it: $N=2P-1$ arcs allows an optimal estimate of the EFT cut-
off $M_{1}$, while for $N<2P-1$ the estimate is always suboptimal.

In fact, the whole UV spectrum and couplings can be extracted in the IR by
determining the roots of the $P$-th order polynomial in $\hat{s}^{2}$
saturating Eq. (18),
$\det\left(H^{0}_{2P-2}-\hat{s}^{2}H^{1}_{2P-1}\right)=0$. 777According to eq.
(33), the $k$-th resonance contributes to the ${ij}$ entry of any Hankel
matrix a term proportional to $M_{k}^{-4(i+j)}$. The determinat associated to
Eq. (18) is thus given by sum of terms $\epsilon_{i_{1}\ldots
i_{P}}M_{k_{1}}^{-4i_{1}}M_{k_{P}}^{-4i_{P}}$ (hence fully antisymmetric in
the $k_{i}$) weighed by the product of
$g_{k_{i}}^{2}(1-\hat{s}^{2}/M_{k_{i}}^{4})$. The sum over $i_{k}$ therefore
vanishes for $s=M_{k^{2}}$ since only $P-1$ distinct $1/M_{k_{i}}^{4}$ terms
appear in the antisymmetric tensor. Then, the couplings $g_{k}^{2}$ can be
extracted by solving Eq. (33) in terms of the arcs.

As an example, consider two heavy particles $\varphi_{i=1,2}$ of masses
$M_{i}$ and trilinear vertex
$\frac{g_{i}}{\sqrt{2}}(\partial\pi)^{2}\varphi_{i}/M_{i}$, matching Eq. (33)
– here $\pi$ is the massless state associated with the $2\to 2$ amplitude. The
first 3 arcs $a_{0,1,2}$ populate the bulk of the allowed region $A(2)$ of Eq.
(20) and Fig. 2, even for $\hat{s}=M_{1}$. An estimate of the cutoff using
only these 3 arcs, $\hat{s}^{2}<{a_{1}}/{a_{2}}$, produces values that are
always above the true cutoff $\hat{s}=M_{1}^{2}$. By instead considering 4
arcs $a_{0,1,2,3}$, the estimated cutoff becomes exact: Eq. (18) is satisfied
for $\hat{s}=M_{1}^{2}$. Including 5 arcs, Eq. (16) is found to be marginally
satisfied, i.e. $\det H^{0}_{4}=0$, thus determining the number of states.

###  II.2 Beyond Tree Level

Under which circumstances is the tree-level formula, $a_{n}=c_{2n+2}$, a good
approximation, and when do the conclusions of the previous section hold?

To answer this, we study an EFT including RG effects, restricting for
simplicity to the massless case (where $\hat{s}=s$). Moreover, in order for
the forward limit to be well-defined, we focus on the case of a derivatively
coupled scalar, i.e. a Goldstone boson $\phi$ with symmetry $\phi\to\phi+b$.
Indeed, the explicit computation that we present below – up to two loops and
$O(s^{6})$ – does not exhibit any IR divergence. Moreover, for the Goldstone
theory, the tree-level amplitude is finite at $t=0$, and divergences could
originate only from collinear emissions (soft emission does not affect the
total cross section and hence neither the imaginary part of the amplitude).
Using collinear factorization (SCET, see e.g. Becher  _et al._ (2015)), it is
easy to see that these are finite, as the positive powers of collinear momenta
associated with the Goldstone derivative interactions always compensate
putative divergences in collinear propagators. We therefore assume that the
forward amplitude is well-defined.

In the upper half plane, up to $O(s^{6})$, the most general Goldstone boson
amplitude starts at order $O(s^{2})$ and reads,

$\displaystyle{\cal
M}(s)\\!=\\!c_{2}s^{2}+s^{4}\left[c_{4}\\!+\\!\beta_{4}\log(-is)\right]-i\pi
s^{5}\beta_{5}/2$ (34)
$\displaystyle+s^{6}\left[c_{6}+\beta_{6}\log(-is)+\beta_{6}^{\prime}\log^{2}(-is)\right]+O(s^{7})\,,$

where $\log$ is defined in the standard way, with the cut on the negative real
semi-axis. Then we have $2\log(-is)=\log(s)+\log(-s)$ for Im$s\geq 0$ and
$\log(s)-\log(-s)=i\pi$. For ease of notation we set the RG scale $\mu=1$, but
the generic choice is reinstated through $\log(-is)\to\log(-is/\mu^{2})$,
$c_{n}\to c_{n}(\mu)$. An explicit calculation in the Goldstone case gives
888By a slight abuse of notation, $c_{4}$ in Eq. (34) differs from the tree-
level amplitude in Eq. (29) by a finite one-loop piece:
$c_{4}^{(\ref{genF})}=c_{4}^{(\ref{ampforwardtree})}+449c_{2}^{2}/(300(16\pi^{2}))$.

$\displaystyle\beta_{4}$ $\displaystyle=$
$\displaystyle-\frac{7}{10}\frac{c_{2}^{2}}{16\pi^{2}}\
,\quad\beta_{5}=+\frac{4}{15}\frac{c_{2}c_{2,1}}{16\pi^{2}}\ ,$
$\displaystyle\beta_{6}$ $\displaystyle=$
$\displaystyle-\frac{83}{70}\frac{c_{4}c_{2}}{16\pi^{2}}-\frac{1}{30}\frac{c_{2,1}^{2}}{16\pi^{2}}-\frac{319}{175}\frac{c_{2}^{3}}{(16\pi^{2})^{2}}\
,\ \quad$ (35) $\displaystyle\beta_{6}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{83}{200}\frac{c_{2}^{3}}{(16\pi^{2})^{2}}\ ,$

where $c_{2,1}$ is the coefficient of $s^{2}t$ in the non-forward amplitude.
At $O(s^{6})$ in the energy-expansion, Eq. (34) does not receive any further
correction at any number of loops. From (34), the first three arcs read,

$\displaystyle a_{0}\\!$ $\displaystyle=\\!$ $\displaystyle
c_{2}+\\!\frac{s^{2}}{2}\beta_{4}\\!+\\!\frac{s^{3}}{3}\beta_{5}\\!+\\!\frac{s^{4}}{4}\Big{(}\beta_{6}\\!+\\!\frac{\beta_{6}^{\prime}}{2}(4\log
s\\!-\\!1)\Big{)}\\!+\\!\cdots\,,$ $\displaystyle a_{1}\\!$
$\displaystyle=\\!$ $\displaystyle
c_{4}(s)+s\beta_{5}+\frac{s^{2}}{2}\Big{(}\beta_{6}+\beta_{6}^{\prime}(2\log
s-1)\Big{)}+\cdots\,,$ $\displaystyle a_{2}\\!$ $\displaystyle=\\!$
$\displaystyle-\frac{\beta_{4}}{2s^{2}}-\frac{\beta_{5}}{s}+c_{6}(s)+\cdots\,,$
(36)

where

$\displaystyle c_{4}(s)$ $\displaystyle\equiv$ $\displaystyle
c_{4}+\beta_{4}\log s\,$ (37) $\displaystyle c_{6}(s)$ $\displaystyle\equiv$
$\displaystyle c_{6}+\beta_{6}\log
s+\beta_{6}^{\prime}(\log^{2}s-\frac{\pi^{2}}{12})\,.$

These functions are RG invariant by construction, and are the natural
extensions of $c_{4,6}$ in the interacting theory.

The dots in Eq. (36) denote higher powers of $s$, of which there are a priori
infinitely many. If all such contributions were important, the EFT would have
no predictive power. A meaningful EFT exists only under certain assumptions on
the convergence of the series. To this aim, a first unavoidable assumption, is
the perturbativity of the dimensionless couplings $g_{n}^{2}\equiv
c_{n}(s)s^{n}$ that control the IR loop expansion,

$g_{n}^{2}\equiv c_{n}(s)s^{n}\ll(4\pi)^{2}\,.$ (38)

This condition, which we assume throughout this work, implies that many of the
higher order terms that (could) enter Eq. (36) must be small. For instance it
implies that $\beta_{4}s^{2}$ and the contribution $\propto
s^{4}c_{4}c_{2}/16\pi^{2}$ in $\beta_{6}s^{4}$ – see (35) – are subleading in
$a_{0}$.

Nevertheless, Eq. (38) does not yet imply the validity of the tree-level
approximation for the arcs. Indeed, the corrections to the tree level result
$a_{n}=c_{2n+2}$, are controlled by a broader set of parameters given by the
ratios

$\frac{\beta_{n}s^{n}}{c_{m}(s)s^{m}}\,.$ (39)

We will refer to the situations where these parameters are small as _strong
perturbativity_. For $n<m$ these parameters grow in the IR and become smaller
in the UV, and viceversa for $n>m$: in the standard RG parlance they
respectively correspond to relevant and irrelevant deformations away from
tree-level. For $n=m$, they capture instead the logarithmic RG running of
Wilson coefficients (they measure the interaction strengths at the cutoff –
see (32) for the weakly coupled case). For instance, in the sigma-model
example of footnote 5, $c_{n}=\lambda/m_{h}^{2n}$ and
$\beta_{n}\sim(\lambda^{2}/16\pi^{2})/m_{h}^{2n}$ are characterised by one
scale and one coupling so that the ratios in Eq. (39) are of order
$(\lambda/16\pi^{2})(s/m_{h}^{2})^{n-m}$. Then, given $\lambda/16\pi^{2}\ll 1$
and $s/m_{h}^{2}<1$, for $n\geq m$ the parameters of Eq. (39) are always small
(strongly perturbative). However, for $n<m$, they can be larger than unity at
small enough energies,

$s\lesssim m_{h}^{2}\left(\frac{\lambda}{16\pi^{2}}\right)^{\frac{1}{m-n}}\,.$
(40)

In fact, these quantum effects always dominate the far IR $s\rightarrow 0$,
where the arcs asymptote to

$a_{0}\to c_{2}\,,\quad a_{1}\to\beta_{4}\log s\,,\quad a_{n\geq
2}\to-\frac{\beta_{4}}{(2n-2)s^{2n-2}}\,.$

For $c_{2}>0$, and given $\beta_{4}<0$, as implied by unitarity within the EFT

$0>-s^{-4}\frac{2}{\pi}\textrm{Im}{\cal M}(s)=\beta_{4}+O(s)\,,$ (41)

these arcs fulfil all the constraints. What happens is that for $s\to 0$, the
arcs are fully dominated by the IR tail of the spectral density, which is
positive and fully determined by the leading term $\propto c_{2}^{2}$ in the
$2\to 2$ cross-section. The Hausdorff condition is then trivially satisfied
and, as graphically represented in Fig. 2, all red trajectories flow to a
common attractor as $s\to 0$.

When higher energies are considered, predictivity is only retained if
contributions above a certain finite positive power of $s$ remain negligible,
in particular when considering the arcs (36). In view of that, we will now
discuss two scenarios for omitting higher order terms. We dub them _Simplest
EFT_ and _Next-to-Simplest EFT_. In the first case we assume that all
irrelevant contributions to the arcs are negligible, namely

$\beta_{n}s^{n}\ll c_{m}(s)s^{m}\quad\textrm{for}\,\,n>m\,.$ (42)

This is the standard situation in the context of EFTs. Instead, more complex
scenarios arise by allowing irrelevant parameters to sizeably contribute to
the arcs. In the _Next-to-Simplest EFTs_ , we will allow

$\beta_{n}s^{n}\sim c_{m}(s)s^{m}\quad\textrm{for some}\,\,n>m\,,$ (43)

while all the other $\beta$ coefficients still fulfil Eq. (42).

_Simplest EFT._ The IR relevant contribution to the arcs drastically modify
the allowed region for the running Wilson coefficients $c_{n}(s)$. This is
however not apparent in the first two arcs $a_{0}$ and $a_{1}$. Indeed, for
the first arc alone we have ${a_{0}\simeq c_{2}}$, since there are no possible
relevant nor marginal perturbations that enter. Therefore the positivity
constraint on $a_{0}$ gives $c_{2}>0$, equivalent to the tree-level case.

Consider now the first two arcs. The size of $\beta_{4}/c_{4}$ controls the
logarithmic running in $a_{1}\approx c_{4}(s)$. The bounds of section I.1, in
particular the optimal bounds in Eqs. (16-19), read

$\displaystyle c_{4}(s)>0$ (44) $\displaystyle{c_{2}}-c_{4}(s){s^{2}}>0\,,$
(45)

in addition to $c_{2}>0$. Written in term of the running coefficient
$c_{4}(s)$, Eqs. (44,45) have the same form as at tree-level constraints of
Eq. (31). Notice in particular that $c_{4}(s)$ (the lowest running Wilson
coefficient) cannot be negative. Moreover, the second inequality, together
with perturbativity defined in Eq. (38), justifies our neglect of the term
$\propto s^{6}c_{4}^{2}/16\pi^{2}c_{2}$ in $\beta_{8}s^{8}/c_{2}s^{2}$. In
other words, it implies that these terms automatically respect strong
perturbativity (42).

In a causal and unitary theory, violation of any of eqs. (44,45) must
correspond to the failure of the hypothesis, Eq. (42). This implies that
either the EFT is at its cutoff, or that it is non-standard, in the sense that
irrelevant higher derivative terms must be included, as we will discuss below
in an example.

Figure 3: _In colour the allowed area for (combinations of) Wilson
coefficients $c_{2}$, $c_{4}(s)$ and $c_{6}(s)$, evaluated at a scale $s$. In
all panels $\beta_{4}/c_{4}(s)=0.1$. Left and center panels: warmer colours
denote points where the distance to the cutoff (i.e. the energy $s_{max}$
where bounds are saturated) is larger; grey contour lines in the central plot
have $s_{max}/s=2,3,4,\ldots$ The black dashed curve denotes the tree-level
expectation Eq. (31). The region above the black lines have
$\beta_{4}/c_{6}(s_{max})s_{max}^{2}<0.1,0.2,0.4$ respectively. The inset in
the center panel shows a wider region of parameter space. Right panel: the
same as in center panel but with logarithmic scale. _

Starting with the third arc, the distinction between arcs and running Wilson
coefficients becomes apparent. Indeed, the third arc
$a_{2}\approx-{\beta_{4}}/{2s^{2}}+c_{6}(s)$,999For simplicity we neglect
$c_{2,1}$ – this is consistent because $c_{2,1}$ is not renormalised by the
other parameters. We include sizeable $c_{2,1}$ below, as a case study for the
Next to Simplest EFT. includes a relevant deformation from its tree-level
expectation $a_{2}=c_{6}$. The bounds read,

$\displaystyle{c_{4}(s)}-c_{6}(s)s^{2}>-\frac{\beta_{4}}{2}\,,$ (46)
$\displaystyle
c_{6}(s)-\frac{c_{4}(s)^{2}}{c_{2}}>\frac{\beta_{4}}{2s^{2}}\,.$ (47)

Given $c_{4}(s)$ and $c_{6}(s)$ at an energy $s$, the first expression is
stronger and the second is weaker than the tree-level conditions in (31). The
weaker condition implies that the determinant $c_{2}c_{6}(s)-c_{4}(s)^{2}$,
and indeed even $c_{6}(s)$, can be negative. In fact the RG effects in Eq.
(37) alone, already violate the naive tree-level bounds: in the far IR $s\to
0$, RG evolution leads to $c_{6}(s)\to\beta_{6}^{\prime}\log^{2}s>0$, but
makes the determinant negative,

$\det\left(\begin{array}[]{cc}c_{2}&c_{4}(s)\\\
c_{4}(s)&c_{6}(s)\end{array}\right)\to(\beta_{4}^{2}-c_{2}\beta_{6}^{\prime})\log^{2}s<0\,,$
(48)

where we have used the explicit values for $\beta_{4}$ and
$\beta_{6}^{\prime}$ from Eq. (35). Indeed it is now possible to have
consistent EFTs where the determinant is so negative that, as energy
increases, Eq. (47) is violated before Eq. (46): a radical difference w.r.t.
the tree-level approximation, where only the inhomogeneous conditions depend
on $s$. This is possible because quantum effects imply that arcs manifestly
depend on the energy scale: Eq. (11) implies that all constraints become more
stringent as $s$ increases, so that both homogeneous and inhomogeneous
conditions play now a role in defining the theory’s regime of validity.

This is illustrated in Fig. 3, where coloured regions correspond to Eqs.
(46,47) and black dashed curves report the naive tree-level expectation.

We can therefore identify two interesting classes of theories. First, theories
that possess a regime in which the parameters in Eq. (39) are small (in
particular $\beta_{4}/c_{4}\ll 1$ and $\beta_{4}/c_{6}s^{2}\ll 1$ in the case
of three arcs) and can be approximated by the tree-level expressions. Weakly
coupled theories where the EFT is obtained by integrating out massive
particles at tree-level (like discussed on page II.1), or at loop level (like
for the Euler-Heisenberg Lagrangian), belong in this class.101010The
discussion in section II.1 corresponds to the idealized limit where the
coupling goes to zero and the tree level regime for the arcs extends to
arbitrarily small energies.  In theories of this class, it is the
inhomogeneous bounds (Eqs. (45,46) for three arcs) that are violated first, as
$s$ increases. This state of things is illustrated in Fig. 3, where we have
chosen $\beta_{4}/c_{4}(s)=0.1$, and where the region above the solid black
lines correspond to varying sizes of $\beta_{4}/c_{6}(s_{max})s_{max}^{2}$, as
indicated in the figure.

On the other hand, in theories in which the relevant perturbation
$\beta_{4}/(c_{6}(s)s^{2})$ never becomes negligible, it is possible for
$c_{2}$, $c_{4}(s)$, $c_{6}(s)$ to lie at $O(1)$ outside of the tree-level
naive boundary Eq. (31). In the central and right plots of Fig. 3, these
theories feature at all energy scales a value of $c_{2}c_{6}(s)/c_{4}(s)^{2}$
significantly below its tree-level bound of 1, and include the option of a
negative value, implying $c_{6}(s)<0$. All in all, even in the _Simplest EFT_
scenario, quantum effects open a qualitatively new region in parameter space.
In these theories, as energy increases, it is the homogeneous conditions Eq.
(47) that are saturated first. This is illustrated by the red trajectories in
Fig. 2 exiting from the lower parabola.

_Next-to-Simplest EFT._ We now study the simplest case in which there exist a
sizeable effect associated with irrelevant parameters, as in Eq. (43)(i.e. a
sizeable $\beta_{n}s^{n}\sim c_{m}s^{m}$ for some $n>m$). The Galileon limit,
in which $c_{2,1}s\gg c_{2}$, is an example of this. Since $c_{2,1}$ doesn’t
enter in the forward amplitude at tree-level, it is not directly bounded by
our discussion in section II.1 (in section III we discuss the amplitude away
from the forward limit, but we anticipate that a positive $c_{2,1}$ is
unbounded by those arguments). In this limit, the part of $\beta_{6}$
involving $c_{2,1}$, which we denote
$\hat{\beta}_{6}=-c_{2,1}^{2}/(30(16\pi^{2}))$, can be potentially large and
depart from strong perturbativity at sufficiently high energy within the EFT –
though we still assume (strong) perturbativity for all higher coefficients.

Considering the first arc, keeping only the most important contributions in
the Galileon limit ($\beta_{5}$ in Eq. (II.2) is suppressed w.r.t.
$\hat{\beta}_{6}$ in this limit), positivity of $a_{0}\approx
c_{2}+{\hat{\beta}_{6}s^{4}}/{4}$ implies

$c_{2}\gtrsim s^{4}\frac{|\hat{\beta}_{6}|}{4}\,,$ (49)

thus $c_{2,1}$ can be at most a loop factor larger than $c_{2}$, i.e.
$c_{2,1}s^{2}\lesssim{8\pi\sqrt{30c_{2}}}$, as already discussed in Refs.
Bellazzini (2017); Bellazzini  _et al._ (2018); Nicolis  _et al._ (2010). Eq.
(49) dictates that strong perturbativity between $\beta_{6}$ and $c_{2}$ can
be violated only marginally.

Including also $a_{1}\approx c_{4}(s)+{\hat{\beta}_{6}s^{2}}/{2}$, we find the
further conditions

$-\frac{\hat{\beta}_{6}}{2}s^{4}\lesssim c_{4}(s)s^{2}\lesssim
c_{2}-s^{4}\frac{\hat{\beta}_{6}}{4}\,.$ (50)

The first inequality implies that $c_{4}(s)s^{2}$ must still be positive. It
also implies another bound of the form of Eq. (49), that can be written
explicitly as,

$c_{2,1}s\lesssim{8\pi\sqrt{15c_{4}(s)}}\,,$ (51)

and is stronger than the one implied by Eq. (49), for $c_{4}(s)s^{2}<2c_{2}$.

The second inequality in Eq. (50) shows that $c_{4}(s)s^{2}$ can now be larger
than $c_{2}$. However, compatibly with Eq. (49) it cannot exceed $2c_{2}$.
Therefore the violation of the bound $c_{2}>c_{4}(s)s^{2}$ is only marginal,
implying that supersoftness $c_{4}(s)s^{2}\gg c_{2}$ remains forbidden. These
results are summarised in Fig. 4.

If we further include information for the third arc, $a_{2}\approx c_{6}(s)$
(where $c_{6}(s)$ is dominated by $\hat{\beta}_{6}$ and we neglect the term
$\propto{\beta_{4}}/{s^{2}}\ll\hat{\beta}_{6}$), we have the inhomogeneous
bound,

$\frac{c_{6}(s)s^{4}}{c_{2}}\left(1+\frac{\hat{\beta}_{6}s^{4}}{2c_{2}}\right)\gtrsim\left(\frac{c_{4}(s)s^{2}}{c_{2}}+\frac{\hat{\beta}_{6}s^{4}}{4c_{2}}\right)^{2}$
(52)

So, a sizeable (negative) $\beta_{6}$ makes the lower bound on
${c_{6}(s)s^{4}}/{c_{2}}$ stronger, and shifts it towards larger values of
${c_{4}(s)s^{2}}/{c_{2}}$. These effects, however, appear in a relatively
uninteresting regime of the theory. Indeed, the quantum effects
$\propto\beta_{4}/s^{2}$ discussed above were sizeable in the IR and allowed
for theories that depart from the tree-level approximation, while still being
valid over a relatively large energy regime. Instead, the effects discussed
here, are controlled by the $s^{2}\hat{\beta}_{6}$ term, that is sizeable only
at high-energy, when the theory is already close to the cutoff.

The couplings $c_{2,1}$, $c_{4}$ and $c_{6}$ are all compatible with Galilean
symmetry; $c_{4}$ and $c_{6}$ are exactly invariant while $c_{2,1}$ is a sort
of Wess-Zumino-Witten (WZW) term Nicolis  _et al._ (2009); Goon  _et al._
(2012). Instead, $c_{2}$ violates the symmetry. Our analysis shows that
$c_{2,1}$ can be at most a loop factor larger than $c_{4}$, while the exactly
invariant couplings $c_{4}$ and $c_{6}$ are directly limited by $c_{2}$. In
some sense, our constraints privilege the WZW couplings over the exactly
symmetric ones.

Figure 4: _The Galileon case $c_{2,1}\neq 0$, for fixed
${-\beta_{4}/c_{4}(s)=0.1}$. Orange Region: allowed from bounds on RGE of the
forward $(t=0)$ amplitude (for comparison the upper dashed curve shows the
upper bound from Ref. Bellazzini  _et al._ (2018)). Blue region: allowed by
the tree-level $t\neq 0$ bounds of section III (for comparison the lower
dashed curve reports the results of Ref. de Rham  _et al._ (2017)). The solid
black curve shows the intersection. Notice that upper and lower parts of the
plot have different scales._

##  III Beyond Forward

In this section we extend our methodology to the class of amplitudes that are
also analytic in $t$ in a finite region around $t=0$. This trivially includes
the case of tree level amplitudes generated by the exchange of massive states,
but it also includes the general case of finite mass. Indeed, for fixed
physical $s\geq 4m^{2}$ the amplitude ${\cal M}(s,t)$ is analytic for complex
$\cos\theta\equiv 1+\frac{2t}{s-4m^{2}}$ inside the so called Lehmann ellipse,
namely an ellipse with foci at $\cos\theta=\pm 1$ Lehmann (1958), see also
e.g. Itzykson and Zuber (1980).

For $t$ finite and real, $s\leftrightarrow u$ crossing symmetry and real
analyticity dictate ${\cal M}(s,t)={\cal M}^{*}(4m^{2}-s^{*}-t,t)$. Further
constraints on the $t$ and $s$ dependence are given by the partial wave
expansion, which diagonalizes the unitarity condition on the S-matrix. In the
physical $s$-channel region the expansion reads

${\cal M}(s,t)=\sum_{l=0}^{\infty}P_{\ell}(\cos\theta)f_{\ell}(s)$ (53)

while a similar relation also holds in the $u$-channel. Then, unitarity of the
partial waves

$\textrm{Im}(f_{\ell}(s))>0\quad\forall{\ell}\qquad\mbox{for }s\geq 4m^{2}\,,$
(54)

together with positivity of the Legendre polynomials and their derivatives at
$t=0$, Eq. (54) imply Martin (1965)

$\left.\partial^{k}_{t}\textrm{Im}{\cal
M}(s,t)\right|_{t=0}=\sum_{\ell=0}^{\infty}\partial^{k}_{t}P_{\ell}(\cos\theta)|_{t=0}\textrm{Im}f_{\ell}(s)>0\,,$
(55)

for all $k$ and for $s$ along the $s$-channel cut. The above additional
positivity conditions can be exploited as we now discuss.

Considering the above properties we extend the definition of the arcs (2) to

$a_{n}(\hat{s},t)\equiv\int_{\begin{minipage}[h]{26.01613pt} \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman[small] \vertex(x1) at
(0,.35); \vertex(x2) at (.1,.35); \vertex(i1) at (0,0); \vertex(i2) at
(.25,0); \vertex(i3) at (.25+.15,0.05){${}_{\hat{s}}$}; \diagram*{ (i1)
--[thick, half left, looseness = 2.7] (i2), (x1) --[white] (x2) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
\end{minipage}}\frac{d{\hat{s}^{\prime}}}{\pi i}\frac{{\hat{\cal
M}}({\hat{s}^{\prime}},t)}{(\hat{s}^{\prime}+\frac{t}{2})^{2n+3}}$ (56)

where we recall $\hat{s}\equiv s-2m^{2}$, ${\hat{\cal M}}(\hat{s},t)={\cal
M}(s,t)$ and ${}_{\hat{s}}$ is now a contour with radius $\hat{s}+t/2$ centred
at $-t/2$. The condition ${\cal M}(s,t)={\cal M}^{*}(4m^{2}-s^{*}-t,t)$
ensures the reality of the arcs. Furthermore using that ${\hat{\cal
M}}(\hat{s},t)/s^{2}\to 0$ for $s\to\infty$ (as dictated by the analog of the
Froissart bound at finite $t$ Jin and Martin (1964)), the arcs can be
expressed by a dispersive integral like in Eq. (3), 111111Since the amplitude
is analytic also for $0\leq t\leq 4m^{2}$ Martin (1965),
$\textrm{Im}{\hat{\cal M}}(\hat{s},t)=\sum_{n}\partial^{n}_{t}\textrm{Im}{\cal
M}(s,t)|_{t=0}t^{n}/n!>0$ is positive there Vecchi (2007); Nicolis  _et al._
(2010), so that the constraints of section I apply up to replacing
$a_{n}(\hat{s})\rightarrow a_{n}(\hat{s},t)$ for $0\leq t\leq 4m^{2}$. In what
follows we provide stronger constraints than these, by using that each
derivative in Eq. (55) is separately positive. Notice also that it is in
principle possible to build dispersion relations for other combinations that
respect real analyticity, such as $\partial_{t}{\hat{\cal
M}}-\partial_{\hat{s}}{\hat{\cal M}}/2$. These also lead to Eq. (58).

$a_{n}(\hat{s},t)=\frac{2}{\pi}\int_{\hat{s}}^{\infty}\\!d{\hat{s}^{\prime}}\frac{\textrm{Im}{\hat{\cal
M}}({\hat{s}^{\prime}},t)}{(\hat{s}^{\prime}+\frac{t}{2})^{2n+3}}\quad n\geq
0\,.$ (57)

At this point we can take $t$-derivatives at $t=0$ and use Eq. (55) to obtain
positivity conditions

$\displaystyle\partial_{t}a_{n}(\hat{s},t)|_{t=0}$ $\displaystyle=$
$\displaystyle a^{(1)}_{n}(\hat{s})-\frac{2n+3}{2}a_{n+1/2}(\hat{s})\,,$ (58)
$\displaystyle\partial_{t}^{2}a_{n}(\hat{s},t)|_{t=0}$ $\displaystyle=$
$\displaystyle a^{(2)}_{n}(\hat{s})-(2n+3)a^{(1)}_{n+1/2}(\hat{s})$
$\displaystyle+\frac{(2n+3)(2n+4)}{4}a_{n+1}(\hat{s})\,,$

and so on; where we have defined

$a^{(k)}_{n}(\hat{s})=\frac{2}{\pi}\int_{\hat{s}}^{\infty}d{\hat{s}^{\prime}}\frac{\partial^{k}_{t}\mathrm{Im}{\hat{\cal
M}}(\hat{s},t)|_{t=0}}{\hat{s}^{\prime 2n+3}}\,,$ (59)

and $a^{(0)}_{n}\equiv a_{n}$. Notice that while the arcs $a_{n}(\hat{s},t)$
are defined for integer $n$ and can be written purely in terms of IR data
according to Eq. (56), the $a^{(k)}_{n}(\hat{s})$ of (58), are defined for
half-integer $n\geq 0$ and only through the UV representation in (59).

For every $k$, $\\{\hat{s}^{2n+2}a_{n}^{(k)}\\}$, with half-integer $n\geq 0$,
is a series of moments because the measure in (59) is positive according to
(55). Therefore, they fullfil versions of the positivity constraints,
analogous to Eqs. (16-19), but including half-integer arcs. More precisely,
half-integer arcs fulfil Eqs. (16-19), recast for new Hankel matrices
$\left(h^{\ell}_{N}\right)_{2i2j}=a_{{i+j}+\ell}$ where
$2i,2j=0,1,\ldots,\lfloor N-\ell\rfloor$ (here both $N$ and $\ell$ can be
half-integers). The matrices $h^{\ell}_{N}$ contain both integer and half
integer arcs, e.g.

$h^{0}_{1}=\left(\begin{array}[]{cc}a_{0}&a_{1/2}\\\
a_{1/2}&a_{1}\end{array}\right)\,.$ (60)

Half-integer arcs $a_{n+1/2}$ ($n$ integer) and all $a^{(k)}_{m}$ ($m$ integer
or half-integer for $k>0$) are not calculable in the EFT because they are
defined in terms of UV integrals (59), and have no IR counterpart like (56).
On the contrary, $a_{n}$ and $\partial_{t}a_{n}$, with integer $n$, can be
computed directly within the IR. Our goal is therefore to understand the
constraints on $a_{n}$ and $\partial_{t}a_{n}$ for arbitrary values of
$a^{(k)}_{m}$ and $a_{n+1/2}$, compatible with their being moments. In
appendix A, we outline an analytic procedure to do so, from which the bounds
below are derived, but that applies in principle to all $N$. For more
$t$-derivatives this procedure becomes cumbersome: in Ref. Bellazzini  _et
al._ (to appear) we propose a numerical technique, based on semi-definite
programming, to extract the bounds efficiently.

For instance, for $N=0$ we find the constraint on arcs $t$-derivatives to be
$\partial_{t}a_{0}>-\frac{3}{2}\frac{a_{0}}{\hat{s}}$, a condition that
appears in different form already in Ref. de Rham  _et al._ (2017).

For $N=1$ our conditions constrain the space
$\\{\partial_{t}a_{0},\partial_{t}a_{1}\\}$ as,

$\displaystyle\partial_{t}a_{0}>-\frac{3}{2}\sqrt{a_{0}a_{1}}\,,\qquad\partial_{t}a_{1}>-\frac{5}{2}\,\frac{a_{1}}{\hat{s}}\,,$
$\displaystyle\partial_{t}a_{0}-s^{2}\partial_{t}a_{1}>\frac{5}{2}\sqrt{\frac{a_{1}}{a_{0}}}(\hat{s}^{2}a_{1}-\frac{3}{5}a_{0})\,,$
(61)

which we illustrate in Fig. 5 for negative ${\partial_{t}a_{0}}$ (for
${\partial_{t}a_{0}>0}$, the allowed region is unbounded: the distance between
the upper and lower boundaries of the projection on the
$\\{\hat{s}^{2}a_{1}/a_{0},\hat{s}^{3}\partial_{t}a_{1}/a_{0}\\}$ plane
increases with $\hat{s}\partial_{t}a_{0}$). These conditions are more
stringent than just $\partial_{t}a_{0}>-\frac{3}{2}\frac{a_{0}}{\hat{s}}$
(which corresponds to the left boundary of the box in Fig. 5), as shown also
in Fig. 4.

Figure 5: _Allowed region in the space of arcs and their first $t$-derivative
for ${\partial_{t}a_{0}<0}$, according to Eq. (61); 2D projections in grey. At
tree level we have $a_{0}=c_{2}$, $a_{1}=c_{4}$, $\partial_{t}a_{0}=c_{2,1}$
and $\partial_{t}a_{1}=c_{4,1}$. _

We illustrate an application of these bounds to the Wilson coefficients at
tree-level121212At one loop, for a $U(1)$ Goldstone boson (i.e. with
$c_{0}=0$),  $\left.\partial_{t}{\cal M}\right|_{\begin{subarray}{c}t\to 0\\\
m\to 0\end{subarray}}\\!\\!=c_{2,1}s^{2}-\frac{i41\pi
c_{2}^{2}}{60(16\pi^{2})}s^{3}+\left(\\!\\!c_{4,1}\\!-\\!\frac{11c_{2}c_{2,1}}{15(16\pi^{2})}\log(-is)\\!\right)\\!s^{4}$
with $t\rightarrow 0$ before taking $m\rightarrow 0$, is finite. Therefore,
for statements up to $O(s^{4}t)$, the use of the tree-level expressions is
justified. We postpone a more refined discussion of these loop effects to Ref.
Bellazzini  _et al._ (to appear).

${\hat{\cal M}}(\hat{s},t)=\sum_{n+m>0}c_{n,m}\hat{s}^{n}t^{m}\,.$ (62)

Eq. (61) reads at $t=0$,

$\displaystyle
c_{2,1}>-\frac{3}{2}\sqrt{c_{4}c_{2}}\,\quad\quad\hat{s}\,c_{4,1}>-\frac{5}{2}c_{4}$
$\displaystyle
c_{2,1}-\hat{s}^{2}c_{4,1}>\frac{5}{2}\sqrt{\frac{c_{4}}{c_{2}}}(\hat{s}^{2}c_{4}-\frac{3}{5}c_{2})$
(63)

where we have identified $c_{n}\equiv c_{n,0}$ to match the notation of the
previous sections.

Eq. (III) implies that $c_{2,1}$ can be negative but not arbitrarily so, as it
is limited in magnitude by $\sqrt{c_{4}c_{2}}$.

To further illustrate the constraining power of Eq. (63), we can for instance
use it to test the consistency of a theory where the amplitude is dominated by
$\propto E^{10}$ terms for sufficiently large $E$ within the EFT domain of
validity. By Lorentz invariance the only option is ${\cal M}\simeq
stu(s^{2}+t^{2}+u^{2})$, corresponding to a $c_{4,1}s^{4}t$ term dominating
$c_{2}s^{2}$, $c_{2,1}s^{2}t$ and $c_{4}s^{4}$. This hierarchy appears
natural, since it is protected by one of the approximate symmetries in Eq.
(1), see Hinterbichler and Joyce (2014). Such example is not constrained by
our tree-level forward bounds in section II.1, simply because the largest
contribution to the amplitude vanishes at $t=0$. However, Eq. (III) provides
upper and lower bounds on $c_{4,1}\hat{s}^{5}$, controlled by more relevant
Wilson coefficients, see Fig. 5 and its caption. In the physical region
$|t|<s$, these bounds imply that $c_{4,1}s^{4}t$ can never dominate the terms
with lower powers of $E$. In other words, supersoft amplitudes that vanish in
the forward limit and have more powers of energy than $c_{2,1}s^{2}t$, are
excluded by our bounds.

The same arguments lead to constraints on higher $t$-derivatives of arcs. The
structure is always the same: $\partial_{t}^{k}a_{0}$ are bound from below,
but can be arbitrarily large when positive. Instead $\partial_{t}^{k}a_{n}$,
$n>0$ are bound from below and from above.

At tree level, this means that $c_{n,k}$, with $n>2$ for any $k$, fulfil two-
sided bounds. Moreover, in the single flavour case considered here, crossing
symmetry implies that $c_{2,k}$ with $k\geq 2$ are related to $c_{2+k,0}$ (for
$k$ even) or $c_{1+k,1}$ (for $k$ odd), which are already bounded from above
from our constraints (grey area in table 1). For instance, the amplitude
${\cal M}(s,t)\propto(s^{2}+t^{2}+u^{2})^{2}$ implies $c_{2,2}=3c_{4}$, and
$c_{4}$ is bounded from above. In other words, only $c_{2}$ and $c_{2,1}$ are
unbounded from above by tree-level arguments; all other coefficients are
instead bounded from below and above. We illustrate this in table 1.

Table 1: _Schematic summary of tree-level results. The grey area encompasses
coefficients which are not independent because of crossing symmetry in the
single-flavour case. $c_{0}$ is unconstrained, $c_{2}$ is constrained to be
positive, $c_{2,1}$ is constrained to be larger than a combination of the
other coefficients. All other coefficients are bounded below and above._

As pointed out in Ref. Arkani-Hamed and Huang (2019), Eq. (55) contains more
information than just positivity, which is what we have exploited so far.
Indeed, the $\partial^{k}_{t}\textrm{Im}{\cal M}(s,t)|_{t=0}$ (and their
integrals $a^{(k)}_{n}(\hat{s})$) are not merely positive, but are the sum of
positive parameters with known coefficients:
$(s-4m^{2})^{k}\partial^{k}_{t}P_{\ell}(\cos\theta)|_{t=0}={(\ell+k)!}/{(\ell-k)!k!}$.
This implies more bounds involving at least three $a^{(k)}_{n}(\hat{s})$, with
$k+n=constant$ Arkani-Hamed and Huang (2019). Because they involve at least
three different $t$-derivatives, and because they are saturated by the
crossing symmetry condition in the simplest cases, these bounds didn’t play a
role in our discussion of supersoftness; for a relevant application see Huang
_et al._ (2020). In Ref. Bellazzini  _et al._ (to appear) we will show how
these bounds emerge in the language of moments and discuss quantitatively the
impact of IR divergences.

##  IV Summary and Outlook

In this paper we introduced a set of energy dependent quantities, the arcs
$a_{n}(s)$ of Eq. (2), that conveniently encode the constraints of causality,
unitarity and crossing on the forward $2\to 2$ scattering amplitude. A
dispersion relation, Eq. (3), allows to express the arcs as the moments of a
positive measure in the range $[s,\infty)$. Hausdorff’s moment theorem then
establishes the set of necessary and sufficient conditions the arcs must
satisfy, given the positivity of the measure. Both the arcs and the
constraints are infinite sets. However we derive the projection of the full
set of constraints on the subsets of the lowest arcs, ${a_{0},\dots,a_{N}}$
for any $N$. These are expressed by Eqs. (16-19) and fall into two classes,
in-homogeneous and homogeneous, according to their explicit, or only implicit,
dependence on $s$. These projections are interesting within EFTs, because the
lowest $a_{n}$’s encode the effects of the correspondingly lowest Wilson
coefficients.

Our result is particularly relevant to determine the acceptable range of
validity of EFTs in both couplings and energy. Concerning the latter and as
implied by Eq. (11), the energy dependence of the arcs leads to stronger
bounds on the EFT parameters as the energy is increased. Satisfaction of the
positivity constraints at a certain $s$, automatically implies satisfaction at
lower $s$ but not at higher $s$. When the parameters of a given EFT violate
the constraints above a certain scale, the only option to respect positivity
is the breakdown of the EFT description at or below that scale.

In the idealized limit where the tree level approximation holds exactly, the
arc series is in one to one correspondence with the series of Wilson
coefficients in the expansion of the forward amplitude. The energy dependence
of the arcs arises at the quantum level from two sources: the RG evolution of
the Wilson coefficients and collinear radiation from the initial state. The
latter induce effects that typically go like powers of $\ln s/m^{2}$, and thus
diverge in the massless limit, corresponding to the IR divergence of the total
cross section. There are however situations, in particular when the
interactions are purely derivative, where these IR divergences are absent and
the arcs energy dependence is purely controlled by RG evolution. It is this
simpler situation that we have mostly considered for illustrative purposes,
focussing on the theory of one abelian Goldstone boson. We could have
similarly considered the case of massless vectors or fermions, where gauge
invariance or supersymmetry mandate derivative interactions.

The constraints are conveniently described by grouping EFTs into two broad
classes. The first are EFTs that emerge from a weakly coupled UV completion,
either at tree level or from loops. Here, for $s$ not too much below the
physical EFT cut-off, the arcs are reliably approximated by the Wilson
coefficients at tree level. Eqs. (16-19) then translate directly into sharp
constraints on the Wilson coefficients. In particular, the energy dependent
inhomogeneous constraints dictate the strict convergence of the $s$ expansion
of the forward amplitude ${\cal M}(s)$ and rule out the possibility for
supersoft EFTs, where ${\cal M}(s)$ grows faster than $s^{2}$. Indeed the
energy dependent constraints, through their violation, also allow to infer the
maximal cut-off of the EFT. As we discussed, even in these weakly coupled EFTs
the tree level approximation for the arcs breaks down at sufficiently low $s$
because of mixing at the quantum level with more relevant Wilson coefficients.
Consequently in the far IR the allowed region for the running Wilson
coefficients differs significantly with respect to the near cut-off region.
This fact is also directly related to the existence of the second class of
EFTs, for which the tree level approximation for the arcs is never realized in
their domain of validity. Positivity still implies strict constraints in
couplings space and in particular rules out supersoft EFTs, at least under the
simplest assumptions we could check. In this second class of theories the
homogeneous constraints, now energy dependent because of quantum effects, can
control the maximal allowed UV cut-off.

Our study mostly concerned the forward amplitude, but in section III we extend
it to $t\not=0$. We studied the arc first $t$-derivative; in derivatively
coupled theories, this too is free of IR-divergences. The bounds we obtain
complete our analysis into the realm of theories with suppressed forward
amplitude. In particular they show that ${\cal M}(s)$ can be dominated over a
limited range of energies by $\propto s^{2}t$ behaviour, but that anything
softer is forbidden. More bounds than those presented here can be derived
using the explicit form of Legendre polynomials (in addition to positivity of
their derivatives) Arkani-Hamed and Huang (2019), but always involve at least
two $t$-derivatives. These involve quantities that are IR divergent in the
$m\to 0$ limit, and goes beyond the scope of the present study (see comments
further down).

Our investigation could be furthered in a number of ways. One could be to try
and connect to the S-matrix bootstrap Paulos  _et al._ (2017); Guerrieri  _et
al._ (2019); Correia  _et al._ (2020). The latter approach exploits the full
$2\rightarrow 2$ unitarity equation, schematically $2\text{Im}T>|T|^{2}$,
while our analytical bounds purely exploit positivity $\text{Im}T>0$. Besides
trying to implement full unitarity one could perhaps use an ansatz for the
$2\rightarrow 2$ amplitude similar to the S-matrix bootstrap approach of
Paulos  _et al._ (2017).

An obvious way to extend our work would be to more systematically study other
instances of derivatively coupled theories. In particular one could consider
cases involving states of different helicity and with a flavor structure. Here
it would be interesting to consider the forward amplitude for superpositions
of helicity and flavor. A less obvious one would be to consider non
derivatively coupled EFTs where the cross section is affected by collinear
divergences. In this case the bounds on the Wilson coefficients at some scale
$s$ will seemingly have a dependence, to be determined, on the mass $m$
providing the IR regulation. Alternatively one could treat the mass $m$ itself
as the RG scale and work with $s\sim m^{2}$.

### Acknowledgements

We thank Nima Arkani-Hamed, Thomas Becher, Gabriel Cuomo, Brian Henning, Yu-
Tin Huang, Paolo Nason and Andrea Wulzer for useful discussions, and Alberto
Nicolis, Joao Penedones, Francesco Sgarlata for comments on the manuscript.
The work of MR, FR and RR is supported by the Swiss National Science
Foundation under grants no. PP00P2-170578, no. 200021-178999, no.
200020-169696 and through the National Center of Competence in Research
SwissMAP. BB and FR were also supported by the Munich Institute for Astro- and
Particle Physics (MIAPP) which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany
Excellence Strategy D EXC-2094 D 390783311. JEM, RR and FR thank the Simons
Collaboration on the Non-perturbative Bootstrap for hospitality in ICTP-SAIFR.

##  Appendix A Analytic Results at Finite-$t$

It is convenient to rearrange the Hankel matrices $\left(h^{\ell}_{N}\right)$
defined above Eq. (60), in terms of blocks of either half-integer or integer
arcs, by extending the definition of section I to
$\left(H^{\ell}_{N}\right)_{ij}=a_{i+j+\ell}$ where now $\ell$ and $N$ can be
half-integer or integer, and $i,j=0,1,\ldots,\lfloor(N-\ell)/2\rfloor$, e.g.

$H^{1/2}_{5/2}=\left(\begin{array}[]{cc}a_{1/2}&a_{3/2}\\\
a_{3/2}&a_{5/2}\end{array}\right)\,.$ (64)

In this way, for $N$ integer, the constraints Eqs. (16-19) on $h^{\ell}_{N}$
(involving both integer and half-integer arcs) can be written in terms of
constraints on $H^{\ell^{\prime}}_{N}$ (integer arcs only for $\ell^{\prime}$
integer) and $H^{\ell^{\prime}+1/2}_{N}$ (half-integer arcs only):

$\displaystyle H^{1/2}\succ 0\,,\qquad H^{1}\succ
H^{1/2}(H^{0})^{-1}H^{1/2}\,,$ $\displaystyle H^{3/2}\succ
H^{1}(H^{1/2})^{-1}H^{1}$ $\displaystyle\Delta H^{0}\succ 0\,,\qquad\Delta
H^{1/2}\succ 0\,,$ $\displaystyle\Delta H^{1}\succ\Delta H^{1/2}(\Delta
H^{0})^{-1}\Delta H^{1/2}\,,$ $\displaystyle\Delta H^{3/2}\succ\Delta
H^{1}(\Delta H^{1/2})^{-1}\Delta H^{1}\,,$ (65)

where we defined $\Delta H^{\ell}_{N}\equiv
H^{\ell}_{N-1/2}-\hat{s}H^{\ell+1/2}_{N}$ and used Schur’s complement. The
lower label of Hankel matrices is implicit and corresponds to $N$.

All half-integer arcs in these conditions are replaced using Eq. (58),

$a_{n+1/2}=\frac{2}{2n+3}\left(a^{(1)}_{n}-\partial_{t}a_{n}\right)\,.$ (66)

Together with the positivity bounds for $a^{(1)}_{n}$, involving $a^{(1)}_{n}$
up to $\lfloor N-1/2\rfloor$, they define the space of allowed arc
derivatives, once we eliminate all the $a^{(1)}_{n}$.

## References

  * Pham and Truong (1985) T. N. Pham and Tran N. Truong, “Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian from Forward Dispersion Relation,” Phys. Rev. D31, 3027 (1985).
  * Ananthanarayan  _et al._ (1995) B. Ananthanarayan, D. Toublan, and G. Wanders, “Consistency of the Chiral Pion Pion Scattering Amplitudes with Axiomatic Constraints,” Phys. Rev. D51, 1093–1100 (1995), arXiv:hep-ph/9410302 [hep-ph] .
  * Pennington and Portoles (1995) M. R. Pennington and J. Portoles, “The Chiral Lagrangian Parameters, L1, L2, are Determined by the Rho Resonance,” Phys. Lett. B344, 399–406 (1995), arXiv:hep-ph/9409426 [hep-ph] .
  * Adams  _et al._ (2006) Allan Adams, Nima Arkani-Hamed, Sergei Dubovsky, Alberto Nicolis, and Riccardo Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” JHEP 10, 014 (2006), arXiv:hep-th/0602178 [hep-th] .
  * Bellazzini (2017) Brando Bellazzini, “Softness and amplitudes? positivity for spinning particles,” JHEP 02, 034 (2017), arXiv:1605.06111 [hep-th] .
  * Vecchi (2007) Luca Vecchi, “Causal versus analytic constraints on anomalous quartic gauge couplings,” JHEP 11, 054 (2007), arXiv:0704.1900 [hep-ph] .
  * Nicolis  _et al._ (2010) Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini, “Energy’s and amplitudes’ positivity,” JHEP 05, 095 (2010), [Erratum: JHEP11,128(2011)], arXiv:0912.4258 [hep-th] .
  * de Rham  _et al._ (2017) Claudia de Rham, Scott Melville, Andrew J. Tolley, and Shuang-Yong Zhou, “Positivity Bounds for Scalar Field Theories,” Phys. Rev. D96, 081702 (2017), arXiv:1702.06134 [hep-th] .
  * de Rham  _et al._ (2018) Claudia de Rham, Scott Melville, Andrew J. Tolley, and Shuang-Yong Zhou, “UV Complete Me: Positivity Bounds for Particles with Spin,” JHEP 03, 011 (2018), arXiv:1706.02712 [hep-th] .
  * Arkani-Hamed and Huang (2019) Nima Arkani-Hamed and Yu-tin Huang, “,” respectively, Lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2019; Talk at the 24th Rencontres Itzykson of the IPhT of CEA-Saclay (2019).
  * Komargodski and Schwimmer (2011) Zohar Komargodski and Adam Schwimmer, “On Renormalization Group Flows in Four Dimensions,” JHEP 12, 099 (2011), arXiv:1107.3987 [hep-th] .
  * Camanho  _et al._ (2016) Xian O. Camanho, Jose D. Edelstein, Juan Maldacena, and Alexander Zhiboedov, “Causality Constraints on Corrections to the Graviton Three-Point Coupling,” JHEP 02, 020 (2016), arXiv:1407.5597 [hep-th] .
  * Bellazzini  _et al._ (2016) Brando Bellazzini, Clifford Cheung, and Grant N. Remmen, “Quantum Gravity Constraints from Unitarity and Analyticity,” Phys. Rev. D93, 064076 (2016), arXiv:1509.00851 [hep-th] .
  * Bellazzini  _et al._ (2014) Brando Bellazzini, Luca Martucci, and Riccardo Torre, “Symmetries, Sum Rules and Constraints on Effective Field Theories,” JHEP 09, 100 (2014), arXiv:1405.2960 [hep-th] .
  * Bellazzini  _et al._ (2018) Brando Bellazzini, Francesco Riva, Javi Serra, and Francesco Sgarlata, “Beyond Positivity Bounds and the Fate of Massive Gravity,” Phys. Rev. Lett. 120, 161101 (2018), arXiv:1710.02539 [hep-th] .
  * Cheung and Remmen (2016) Clifford Cheung and Grant N. Remmen, “Positive Signs in Massive Gravity,” JHEP 04, 002 (2016), arXiv:1601.04068 [hep-th] .
  * Luty  _et al._ (2013) Markus A. Luty, Joseph Polchinski, and Riccardo Rattazzi, “The $a$-theorem and the Asymptotics of 4D Quantum Field Theory,” JHEP 01, 152 (2013), arXiv:1204.5221 [hep-th] .
  * Distler  _et al._ (2007) Jacques Distler, Benjamin Grinstein, Rafael A. Porto, and Ira Z. Rothstein, “Falsifying Models of New Physics via Ww Scattering,” Phys. Rev. Lett. 98, 041601 (2007), arXiv:hep-ph/0604255 [hep-ph] .
  * Englert  _et al._ (2019) Christoph Englert, Gian F. Giudice, Admir Greljo, and Matthew Mccullough, “The $\hat{H}$-Parameter: An Oblique Higgs View,” JHEP 09, 041 (2019), arXiv:1903.07725 [hep-ph] .
  * Bellazzini  _et al._ (2019a) Brando Bellazzini, Francesco Riva, Javi Serra, and Francesco Sgarlata, “Massive Higher Spins: Effective Theory and Consistency,” JHEP 10, 189 (2019a), arXiv:1903.08664 [hep-th] .
  * Bellazzini  _et al._ (2019b) Brando Bellazzini, Matthew Lewandowski, and Javi Serra, “Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity,” Phys. Rev. Lett. 123, 251103 (2019b), arXiv:1902.03250 [hep-th] .
  * Chen  _et al._ (2019) Wei-Ming Chen, Yu-Tin Huang, Toshifumi Noumi, and Congkao Wen, “Unitarity bounds on charged/neutral state mass ratios,” Phys. Rev. D100, 025016 (2019), arXiv:1901.11480 [hep-th] .
  * Remmen and Rodd (2019) Grant N. Remmen and Nicholas L. Rodd, “Consistency of the Standard Model Effective Field Theory,” JHEP 12, 032 (2019), arXiv:1908.09845 [hep-ph] .
  * Zhang and Zhou (2020) Cen Zhang and Shuang-Yong Zhou, “Convex Geometry Perspective to the (Standard Model) Effective Field Theory Space,” Phys. Rev. Lett. 125, 201601 (2020), arXiv:2005.03047 [hep-ph] .
  * Lehmann (1958) H. Lehmann, “Analytic properties of scattering amplitudes as functions of momentum transfer,” Nuovo Cim. 10, 579–589 (1958).
  * Martin (1965) Andre Martin, “Extension of the Axiomatic Analyticity Domain of Scattering Amplitudes by Unitarity. 1.” Nuovo Cim. A42, 930–953 (1965).
  * Liu  _et al._ (2016) Da Liu, Alex Pomarol, Riccardo Rattazzi, and Francesco Riva, “Patterns of Strong Coupling for Lhc Searches,” JHEP 11, 141 (2016), arXiv:1603.03064 [hep-ph] .
  * Bellazzini  _et al._ (2017) Brando Bellazzini, Francesco Riva, Javi Serra, and Francesco Sgarlata, “The other effective fermion compositeness,” JHEP 11, 020 (2017), arXiv:1706.03070 [hep-ph] .
  * Bellazzini and Riva (2018) Brando Bellazzini and Francesco Riva, “New phenomenological and theoretical perspective on anomalous ZZ and Z? processes,” Phys. Rev. D98, 095021 (2018), arXiv:1806.09640 [hep-ph] .
  * Nicolis  _et al._ (2009) Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini, “The Galileon as a local modification of gravity,” Phys. Rev. D79, 064036 (2009), arXiv:0811.2197 [hep-th] .
  * Hinterbichler and Joyce (2014) Kurt Hinterbichler and Austin Joyce, “Goldstones with Extended Shift Symmetries,” Int. J. Mod. Phys. D23, 1443001 (2014), arXiv:1404.4047 [hep-th] .
  * Martin (1969) André Martin,  _Scattering Theory: Unitarity, Analyticity and Crossing_, Vol. 3 (1969).
  * Froissart (1961) Marcel Froissart, “Asymptotic Behavior and Subtractions in the Mandelstam Representation,” Phys. Rev. 123, 1053–1057 (1961).
  * Martin (1963) A. Martin, “Unitarity and High-Energy Behavior of Scattering Amplitudes,” Phys. Rev. 129, 1432–1436 (1963).
  * Green and Wen (2019) Michael B. Green and Congkao Wen, “Superstring amplitudes, unitarily, and Hankel determinants of multiple zeta values,” JHEP 11, 079 (2019), arXiv:1908.08426 [hep-th] .
  * g. see theorem 4.17.4 of (2015, Simon, B.) E g. see theorem 4.17.4 of,  _Real Analysis. A Comprehensive Course in Analysis, Part 1_ (American Mathematical Society, 2015, Simon, B.).
  * Becher  _et al._ (2015) Thomas Becher, Alessandro Broggio, and Andrea Ferroglia,  _Introduction to Soft-Collinear Effective Theory_, Vol. 896 (Springer, 2015) arXiv:1410.1892 [hep-ph] .
  * Goon  _et al._ (2012) Garrett Goon, Kurt Hinterbichler, Austin Joyce, and Mark Trodden, “Galileons as Wess-Zumino Terms,” JHEP 06, 004 (2012), arXiv:1203.3191 [hep-th] .
  * Itzykson and Zuber (1980) C. Itzykson and J.B. Zuber,  _Quantum Field Theory_ , International Series In Pure and Applied Physics (McGraw-Hill, New York, 1980).
  * Jin and Martin (1964) Y. S. Jin and A. Martin, “Number of Subtractions in Fixed-Transfer Dispersion Relations,” Phys. Rev. 135, B1375–B1377 (1964).
  * Bellazzini  _et al._ (to appear) Brando Bellazzini, Joan Elias Miro, Riccardo Rattazzi, Marc Riembau, and Francesco Riva, “Bootstrapping EFTs,” (to appear).
  * Huang  _et al._ (2020) Yu-tin Huang, Jin-Yu Liu, Laurentiu Rodina, and Yihong Wang, “Carving Out the Space of Open-String S-Matrix,” (2020), arXiv:2008.02293 [hep-th] .
  * Paulos  _et al._ (2017) Miguel F. Paulos, Joao Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira, “The S-matrix Bootstrap III: Higher Dimensional Amplitudes,” (2017), arXiv:1708.06765 [hep-th] .
  * Guerrieri  _et al._ (2019) Andrea L. Guerrieri, Joao Penedones, and Pedro Vieira, “Bootstrapping QCD Using Pion Scattering Amplitudes,” Phys. Rev. Lett. 122, 241604 (2019), arXiv:1810.12849 [hep-th] .
  * Correia  _et al._ (2020) Miguel Correia, Amit Sever, and Alexander Zhiboedov, “An Analytical Toolkit for the S-Matrix Bootstrap,” (2020), arXiv:2006.08221 [hep-th] .