Cornering Large-$N_{c}$ QCD with Positivity Bounds

Clara Fernandez1, Alex Pomarol1,2,3, Francesco Riva4 and Francesco Sciotti1

1IFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona
2Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona
3CERN, Theory Division, Geneva, Switzerland
4Départment de Physique Théorique, Université de Genève, 24 quai Ernest-
Ansermet, 1211 Genève 4, Switzerland

##  1 Introduction

The understanding of theories at strong coupling is one of the most important
challenges of modern particle physics. Besides the pragmatic interest for QCD
hadronic physics, such understanding would broaden our perspective on theories
beyond the Standard Model and plausibly provide intuition on its shortcomings.

Dispersion relations distill the essential ingredients of quantum field
theory, unitarity and causality, into consistency conditions for scattering
amplitudes. They have been used as _positivity bounds_ to shape the parameter
space of effective field theories originating from healthy, albeit strongly
coupled, microscopic dynamics, see e.g. [1, 2, 3, 4, 5, 6, 7], and have had
important applications to QCD and the chiral Lagrangian [8, 9, 10, 11, 12, 13,
14, 15, 16].

At the same time, in the context of $SU(N_{c})$ gauge theories, the limit of
many colors $N_{c}\to\infty$ has provided one of the most insightful
approaches for understanding the strongly-coupled regime [17, 18], even though
real-world QCD has only $N_{c}=3$. The main consequence of this approximation
is that the theory has a dual description in terms of weakly-coupled mesons,
rather than quarks and gluons. In spite of this important step forward, the
predictions of large-$N_{c}$ QCD have been limited by the fact that the theory
contains an infinite number of mesons of any spin, whose Lagrangian is unknown
or contains an infinite number of terms.

Recently, Ref. [19] has combined these two approaches and derived important
constraints on the low energy $\pi\pi\to\pi\pi$ scattering amplitude. Despite
the higher-spin meson spectrum remains unknown, the simple analytic structure
of large-$N_{c}$ amplitudes – combined with certain assumptions on their high-
energy behavior – improves the predictiveness of dispersion relations.

In this article we push forward this approach, combining analytic and
numerical methods. One of the most important questions raised in Ref. [19]
concerns the understanding of which theories define the kinks (and the bulk)
of the allowed regions of Wilson coefficients. In this article we identify
these theories. We build UV-complete four-pions amplitudes, as variations of
the Lovelace-Shapiro amplitude [20, 21] and the Coon amplitude [22] in which
the spin-0 (and spin-1) poles have been removed, and show that they reside
close to the boundary. At the kinks, on the other hand, we find theories with
an infinitely degenerate higher-spin spectrum, or theories with a unique state
at finite mass, either of spin $J=0$, $J=1$ or $J=2$. While some of these
results are known [4, 6, 19], in this work we are able to exclude other
possibilities. In particular, while the numerical approach suggests the
existence of a new kink with a more exotic spectrum, we show that the kink
position can be reformulated as a 1D moment-problem whose solution converges
to the $J=1$ theory.

An important emerging property of low-energy QCD is Vector Meson Dominance
(VMD) [23], the hypothesis that the spin-1 $\rho$ meson gives the main
contribution to the chiral Lagrangian [24]. This property has an empirical
origin (it works!), but it lacks a theoretical explanation. We will show that
VMD finds its origin in the positivity bounds. Indeed, the chiral Lagrangian
coefficients can be written as a sum over positive quantities that depend on
the couplings and masses of the different mesons: when the theory contains a
$\rho$, this dominates over the other $J>1$ states. This is especially
prominent when the higher-spin states are heavier than the $\rho$, as it
occurs in real-world QCD. Our analysis also provides insight into why
holographic models, which contain only (charged) spin-0 and spin-1 states,
have been so successful in predicting low-energy properties of QCD, see e.g.
[25, 26]. Finally, we will show that this reasoning also extends to the heavy-
meson couplings to pions: the higher the spin, the smaller the coupling.

These arguments, as well as all those from Ref. [19], rely on assuming that
the high-energy amplitudes are particularly well behaved, ${\cal M}/s\to 0$ at
large $|s|$. In this article we will also discuss the implications of relaxing
this assumption, such that the $\pi\pi\to\pi\pi$ amplitude is limited only by
the Froissart-Martin bound ${\cal M}/s^{2}\to 0$ at large energies [27, 28].
In this case, the spin-1 and spin-0 contributions decouple from (and can
contribute more than) the $J\geq 2$ ones, that are now dominated by the spin-2
state instead, in a generalisation of VMD.

The paper is organised as follows. In Sec. 2 we review the analytical
structure of the $\pi\pi\rightarrow\pi\pi$ amplitude and the dispersion
relations that lead to positivity constraints. We also present possible UV
completions to the chiral Lagrangian, as they will play an important role to
understand the boundaries of the Wilson-coefficient allowed regions. In Sec. 3
we consider the case in which the four-pion amplitude at large $|s|$ satisfies
${\cal M}/s\to 0$. We determine the allowed regions of the leading Wilson
coefficients and show which theories reside at the kinks of these boundaries.
We also study the emergence of VMD and derive bounds on the couplings of meson
resonances to pions. In Sec. 4 we extend the analysis to the case in which the
four-pion amplitude only satisfies the Froissart-Martin bound at high
energies. We present several appendices with extended discussions on the
numerical bootstrap (Appendix A), on the analytical determination of the kinks
(Appendix B), the $su$-models (Appendix C), and the Lovelace-Shapiro and Coon
amplitudes (Appendix D and Appendix E respectively).

##  2 The $\pi\pi\rightarrow\pi\pi$ Amplitude in large-$N_{c}$ QCD

This section contains mostly a review of previous literature on the
$2\rightarrow 2$ pion amplitude, in particular the results of Ref. [19]. We
will work in the massless quark limit.

Pions are the massless Goldstone bosons associated to the spontaneous breaking
of the global $SU(N_{f})_{L}\times SU(N_{f})_{R}\to SU(N_{f})$, where $N_{f}$
is the number of quark flavors in QCD. They transform in the $Adj.$
representation of $SU(N_{f})$, which for the case $N_{f}=2$ corresponds to the
Isospin $I=1$, the triplet $\pi^{\pm}$ and $\pi^{0}$. This allows us to write
the $2\rightarrow 2$ pion amplitude as

${\cal
M}(\pi^{a}\pi^{b}\to\pi^{c}\pi^{d})=A(s|t,u)\left[\frac{2}{N_{f}}\delta_{s}+d_{s}\right]+A(t|u,s)\left[\frac{2}{N_{f}}\delta_{t}+d_{t}\right]+A(u|s,t)\left[\frac{2}{N_{f}}\delta_{u}+d_{u}\right]\,,$
(1)

where $s=(p_{a}+p_{b})^{2}$, $t=(p_{a}-p_{c})^{2}$, $u=(p_{a}-p_{d})^{2}$, and

$\displaystyle\delta_{s}=\delta_{ab}\delta_{cd}\ ,\
\delta_{t}=\delta_{s}(b\leftrightarrow c)\ ,\
\delta_{u}=\delta_{s}(b\leftrightarrow d)\,,$ (2) $\displaystyle
d_{s}=d_{abe}d_{cde}\ ,\ d_{t}=d_{s}(b\leftrightarrow c)\ ,\
d_{u}=d_{s}(b\leftrightarrow d)\,,$ (3)

correspond to the various ways of contracting $SU(N_{f})$ adjoint indices into
singlets. $A(s|t,u)$ is a function of $t,u$ symmetric under their interchange,
i.e. $A(s|t,u)=A(s|u,t)$.

In the large-$N_{c}$ limit, QCD reduces to a theory of weakly coupled mesons,
whose trilinear couplings scale as $\sim 1/\sqrt{N_{c}}$ [17, 18]. In this
limit, the $2\rightarrow 2$ pion amplitude is then dominated by a tree-level
meson exchange. Since these mesons are $q\bar{q}$ states with isospin $I=0,1$,
the isospin $I=2$ amplitude,

${\cal
M}^{I=2}_{s}(\pi^{a}\pi^{b}\to\pi^{c}\pi^{d})=A(t|u,s)+A(u|s,t)\equiv{\cal
M}(t,u)\,,$ (4)

(symmetric under $t\leftrightarrow u$) has no poles in the large-$N_{c}$
limit. All this leads to the following important implications for the
analytical structure of ${\cal M}$ in the large-$N_{c}$ limit:

  * •

The only singularities of ${\cal M}$ in the complex $s$-plane are simple poles
associated with the tree-level meson exchange (the branch cut along the
physical region is at least $O(1/N_{c}^{2}$)).

  * •

The absence of $I=2$ meson exchange in the $s$-channel implies that ${\cal
M}_{s}^{I=2}={\cal M}(t,u)$ has no poles for real $s>0$. Since $t=-s-u$, this
implies that for fixed $u<0$, there cannot be poles in ${\cal M}(t,u)$ on the
negative real $t$ axis. Now, by a simple exchange of arguments ($t\to s$), we
come to the conclusion that ${\cal M}(s,u)$, for fixed $u<0$, can only have
poles on the positive real $s$ axis.

  * •

For fixed $u<0$, ${\cal M}(s,t)$ can have poles either on the real positive or
negative $s$ axis.

This analytic structure of ${\cal M}$ is illustrated in Fig. 1.

$C_{\infty}$$C_{0}$$s$$\ M^{2}$ (a) ${\cal M}(s,u)$

$C_{\infty}$$C_{0}$$s$$\ \ \ M^{2}$$-M^{2}-u\ \ $ (b) ${\cal M}(s,t=-u-s)$

Figure 1: Analytic structure of ${\cal M}(s,u)$ and ${\cal M}(s,t=-s-u)$ for
fixed $u<0$. We denote by $C_{0}$, $C_{\infty}$ (to be taken at
$|s|\to\infty$) and the discontinuity along the real axis the relevant
contours of integration used for dispersion relations.

At energies below the mass $M$ of the lightest massive meson (corresponding to
the position of the first pole in Fig. 1), pions can be well described by an
effective theory, corresponding to an expansion in $s/M,u/M\to 0$,

$\displaystyle{\cal M}(s,u)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\sum_{l=0}^{[n/2]}\,g_{n,l}\,s^{\\{n-l}u^{l\\}}$
(5) $\displaystyle=$ $\displaystyle
g_{1,0}\,(s+u)+g_{2,0}\,(s^{2}+u^{2})+g_{2,1}su+g_{3,0}\,(s^{3}+u^{3})\,$
$\displaystyle+$ $\displaystyle
g_{3,1}\,(s^{2}u+su^{2})+g_{4,0}\,(s^{4}+u^{4})+g_{4,1}\,(s^{3}u+su^{3})+g_{4,2}\,s^{2}u^{2}+...\,,$

where for convenience we have defined the weighed symmetric tensor
$s^{\\{i}u^{j\\}}\equiv s^{i}u^{j}+(1-\delta_{ij})s^{j}u^{i}$ to avoid double
counting of terms with $n=2l$, such as $g_{2,1}$, $g_{4,2}$, etc. A constant
term is absent in Eq. (5) as the amplitude must go to zero for $s,u\to 0$ in
order to restore the Adler condition for pions. In the large-$N_{c}$ limit all
Wilson coefficients scale as $g_{n,l}\sim 1/N_{c}$. For the connection of Eq.
(5) with the QCD chiral Lagrangian, see Sec. 3.2.1.

###  2.1 Dispersion Relations and Sum Rules

Dispersion relations can be derived assuming that the amplitude ${\cal M}$
satisfies the following high-energy conditions, for fixed $u<0$ and for all
$k\geq{k_{\rm min}}$,

Different assumptions about the high-energy behavior of amplitudes are
associated with different values of ${k_{\rm min}}$. On general grounds, the
Froissart-Martin bound [27, 28, 29] ensures that in theories with a mass gap
Eqs. (6) are satisfied for $k_{\rm min}=2$, with similar results for the
massless case [7, 30, 31].

On the other hand, Ref. [19] advocated – invoking considerations on the
Pomeron Regge trajectory – that the four-pion amplitude might be bounded by
${\cal M}\lesssim s$ and therefore $k_{\rm min}=1$. In this article we will
study both, the case with ${k_{\rm min}}=1$ and ${k_{\rm min}}=2$.

####  2.1.1 IR-UV Relations

Taking $u<0$ fixed, we have that the integral of ${\cal M}(s,u)/s^{k+1}$ along
the contour $C_{\infty}$ of Fig. 1(a) vanishes for $k\geq{k_{\rm min}}$, due
to Eq. (6a). Because of amplitude’s analyticity, we can deform $C_{\infty}$
into the blue contour in Fig. 1(a),

$\oint_{\rm C_{0}}ds^{\prime}\ \frac{{\cal M}(s^{\prime},u)}{s^{\prime
k+1}}=2i\int_{M^{2}}^{\infty}ds^{\prime}\ \frac{{\rm Im}{\cal
M}(s^{\prime},u)}{s^{\prime k+1}}\,.$ (7)

The amplitude can be expanded in partial waves,

$\displaystyle{\rm Im}{\cal
M}(s,u)=\sum_{J}(2J+1)\rho_{J}(s)P_{J}\bigg{(}1+\frac{2u}{s}\bigg{)}\,,$ (8)

where $P_{J}$ are the Legendre polynomials and $\rho_{J}(s)$, the spectral
density, must be positive, $\rho_{J}(s)\geq 0$, due to unitarity of the
S-Matrix. Indeed, for large-$N_{c}$ theories, we have that the spectral
density is given by

$(2J+1)\rho_{J}(m^{2})=\pi\sum_{i}{g^{2}_{i\pi\pi}}m^{2}_{i}\,\delta(m^{2}-m^{2}_{i})\delta_{JJ_{i}}\,,$
(9)

where $i$ labels mesons of mass $m_{i}$, spin $J_{i}$ and coupling to pions
$g_{i\pi\pi}$.

Plugging Eq. (8) into Eq. (7), performing the contour integrals, and expanding
around small $u<0$ we find

$\displaystyle k=1:\qquad g_{1,0}+g_{2,1}u+g_{3,1}u^{2}+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{P_{J}(1)}{m^{2}}+2\frac{P^{\prime}_{J}(1)}{m^{4}}u+2\frac{P^{\prime\prime}_{J}(1)}{m^{6}}u^{2}+...\right\rangle\,,$
$\displaystyle k=2:\qquad g_{2,0}+g_{3,1}u+g_{4,2}u^{2}+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{P_{J}(1)}{m^{4}}+2\frac{P^{\prime}_{J}(1)}{m^{6}}u+2\frac{P^{\prime\prime}_{J}(1)}{m^{8}}u^{2}+...\right\rangle\,,$
$\displaystyle k=3:\qquad g_{3,0}+g_{4,1}u+g_{5,2}u^{2}+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{P_{J}(1)}{m^{6}}+2\frac{P^{\prime}_{J}(1)}{m^{8}}u+2\frac{P^{\prime\prime}_{J}(1)}{m^{10}}u^{2}+...\right\rangle\,,$
(10) $\displaystyle\vdots$

with the definition of the high-energy average [6],

$\langle(...)\rangle\equiv\frac{1}{\pi}\sum_{J}(2J+1)\int_{M^{2}}^{\infty}\frac{dm^{2}}{m^{2}}\rho_{J}(m^{2})(...)\,.$
(11)

Considering equations with $k\geq k_{\rm min}$ (that we will take later to be
$k_{\rm min}=1,2$), we can relate the IR Wilson coefficients with the UV-
averages of derivatives of $P_{J}$ in the following way:

$g_{n+l,l}=\frac{2^{l}}{l!}\left\langle\frac{P_{J}^{(l)}(1)}{m^{2(n+l)}}\right\rangle\
,\ \ \ n\geq k_{\rm min}\,\,\text{and}\ \
l=0,1,...,\left[\frac{n-1}{2}\right]\,.$ (12)

Since $P_{J}^{(l)}(1)\geq 0$, the contributions to Eq. (12) from the different
$J$-states are always additive, and therefore $g_{n+l,l}\geq 0$ – this is a
direct consequence of the lack of $s<0$ poles in ${\cal M}(s,u)$. Moreover,
$P_{J}^{(l)}(1)=0$ for $l>J$ implying that states with $J\leq l$ do not
contribute to $g_{n+l,l}$.

In particular,

$\displaystyle g_{n,0}$ $\displaystyle=$
$\displaystyle\left\langle\frac{1}{m^{2n}}\right\rangle=\sum_{i}\frac{g^{2}_{i\pi\pi}}{m^{2n}_{i}}\
,\ \ \ \ $ $\displaystyle g_{n+1,1}$ $\displaystyle=$
$\displaystyle\left\langle\frac{{\cal
J}^{2}}{m^{2(n+1)}}\right\rangle=\sum_{i}\frac{g^{2}_{i\pi\pi}\,J_{i}(J_{i}+1)}{m^{2(n+1)}_{i}}\
,\ \ \ \ $ $\displaystyle g_{n+2,2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\langle\frac{{\cal J}^{4}-2{\cal
J}^{2}}{m^{2(n+1)}}\right\rangle=\sum_{i}\frac{g^{2}_{i\pi\pi}\,J_{i}(J_{i}-1)(J_{i}+1)(J_{i}+2)}{4m^{2(n+1)}_{i}}\,,$
(13)

where $\mathcal{J}^{2}\equiv J(J+1)$. Notice that only for $k_{\rm min}=1$ all
Wilson coefficients have a dispersive representation in terms of Eq. (12). For
$k_{\rm min}=2$, the couplings $g_{1,0}$ and $g_{2,1}$ are not captured by
these dispersion relations.

In a similar way, we can obtain dispersion relations for the ${\cal M}(s,t)$
amplitude, whose analytic structure is given in Fig. 1(b),

$\frac{1}{2i}\oint_{\rm C_{0}}ds^{\prime}\ \frac{{\cal
M}(s^{\prime},-u-s^{\prime})}{s^{\prime
k+1}}=\int_{M^{2}}^{\infty}ds^{\prime}\ \frac{{\rm Im}{\cal
M}(s^{\prime},-u-s^{\prime})}{s^{\prime
k+1}}+(-1)^{k}\int_{M^{2}}^{\infty}ds^{\prime}\ \frac{{\rm Im}{\cal
M}(s^{\prime},-u-s^{\prime})}{(s^{\prime}+u)^{k+1}}.$ (14)

These can be expanded as in the previous section, and provide yet more
relations [19]. In the case $k_{\rm min}>1$ these new relations are crucial,
as they give access to Wilson coefficients that do not have a dispersive
representation in terms of ${\cal M}(s,u)$. In particular, for $k_{\rm
min}=2$, the coupling $g_{2,1}$ is not determined by Eq. (12) but appears in
Eq. (14),

$g_{2,1}=2g_{2,0}-2\left\langle\frac{(-1)^{J}}{m^{4}}\right\rangle\,,$ (15)

while for $k_{\rm min}=3$, $g_{3,1}$ can only be determined by

$g_{3,1}=3g_{3,0}+\left\langle\frac{(-1)^{J}(2{\cal
J}^{2}-3)}{m^{6}}\right\rangle\,.$ (16)

###  2.2 Null Constraints

The dispersion relations in Eq. (10), and the small-$u$ expansion of Eq. (14),
over-determine the Wilson coefficients. This leads to a set of _null
constraints_ ,

$\big{\langle}{\cal X}_{n,k}(J,m^{2})\big{\rangle}=0\,,\quad\big{\langle}{\cal
Y}_{n,k}(J,m^{2})\big{\rangle}=0\,,$ (17)

on the high-energy spectral density, with $m^{2n}{\cal X}_{n,k}$ and
$m^{2n}{\cal Y}_{n,k}$ functions of $J$ only. Their compact expression at all
orders is provided in Ref. [19]. For the analytic arguments in this article we
are only interested in the most relevant null constraints (those involving
less powers of $1/m$) and in those with the leading asymptotic $J\to\infty$
behavior at a fixed order $n$ in $1/m^{2n}$.

For $k_{\min}=1$, there is one (and only one) null constraint $\sim
O(J^{2(n-1)}/m^{2n})$ at each order $n$,111We use a slightly different
normalization w.r.t. Ref. [19], which has no impact on Eq. (17).

$\displaystyle n=2:\qquad m^{4}\,\mathcal{Y}_{2,1}$ $\displaystyle=$
$\displaystyle-2(1-(-1)^{J})+\mathcal{J}^{2}\,,$ $\displaystyle n=3:\qquad
m^{6}\,\mathcal{X}_{3,1}$ $\displaystyle=$
$\displaystyle-6\mathcal{J}^{2}+\mathcal{J}^{4}\,,$ $\displaystyle n=4:\qquad
m^{8}\,\mathcal{X}_{4,1}$ $\displaystyle=$
$\displaystyle-24\mathcal{J}^{2}-8\mathcal{J}^{4}+\mathcal{J}^{6}\,,$
$\displaystyle\vdots$ $\displaystyle\qquad(n-1)!^{2}\
m^{2n}\,\mathcal{X}_{n,1}$ $\displaystyle=$
$\displaystyle\frac{2^{n-1}}{(n-1)!}{P_{J}^{(n-1)}(1)}-{\cal J}^{2}\,.$ (18)

The other null constraints have only subleading terms in powers of $J$ w.r.t.
these.

When we study larger $k_{\rm min}$, the constraints in Eq. (18) disappear, and
subleading null constraints now dominate. For $k_{\rm min}=2$ this involves
null constraints that grow as $O(J^{2(n-2)}/m^{2n})$. There are two of them at
each order $n$, and can be separated into those where the sign of the term
$O(J^{2(n-2)}/m^{2n})$ is fixed, and those where this sign oscillates between
$J$-odd and $J$-even. In the first class we have,

$\displaystyle n=4:\qquad
m^{8}\big{(}\mathcal{Y}_{4,2}-\mathcal{Y}_{4,1}\big{)}$ $\displaystyle=$
$\displaystyle 8(1-(-1)^{J})-10\mathcal{J}^{2}+\mathcal{J}^{4}\,,$
$\displaystyle n=5:\qquad\qquad\ \ \ \ m^{10}\,\mathcal{X}_{5,2}$
$\displaystyle=$ $\displaystyle
30\mathcal{J}^{2}-17\mathcal{J}^{4}+\mathcal{J}^{6}\,,$ $\displaystyle
n=6:\qquad\qquad\ \ \ \ m^{12}\,\mathcal{X}_{6,2}$ $\displaystyle=$
$\displaystyle
144\mathcal{J}^{2}-46\mathcal{J}^{4}-20\mathcal{J}^{6}+\mathcal{J}^{8}\,,$
$\displaystyle\vdots$ $\displaystyle\qquad(n-2)!^{2}\
m^{2n}\,\mathcal{X}_{n,2}$ $\displaystyle=$
$\displaystyle\frac{2^{n-2}}{(n-2)!}{P_{J}^{(n-2)}(1)}-2{P_{J}^{(2)}(1)}\,.$
(19)

In both cases, Eq. (18) and Eq. (19), the $\mathcal{Y}_{n,k}$ null constraints
(originating from ${\cal M}(s,t)$ dispersion relations) appear only at the
lowest order in $1/m^{2}$, and at higher order the dominant $J$ behavior is
controlled by the $\mathcal{X}_{n,k}$ null constraints (originating from
${\cal M}(s,u)$ dispersion relations). On the other hand, the most relevant
oscillating null constraint is,

$n=3:\qquad m^{6}\
\mathcal{Y}_{3,1}\,=\,-6(1-(-1)^{J})+2(1-2(-1)^{J})\mathcal{J}^{2}\,,$ (20)

where the sign of the $\mathcal{J}^{2}$ term oscillates with $J$.

Figure 2: Null constraints from Eq. (18), as a function of $J$, for fixed $m$.

Notice that for $J=0$ the arguments of all null constraints vanish, as can be
easily seen in Eq. (18) and Eq. (19) (and more generally by the expressions in
Ref. [19]). This implies that the spin-0 component of the UV spectrum
decouples and is not restricted by null constraints. This is related to the
fact that models with only $J=0$ states can provide a consistent UV completion
of the pion amplitude, satisfying Eq. (6a) and Eq. (6b) for $k_{\rm min}=1$,
as we will discuss in section 2.3. For $k_{\rm min}=2$, also $J=1$ states give
zero contributions and decouple from the null constraints. This again can be
understood from the fact that models with only $J=1$ states provide a pion
amplitude that consistently satisfies the Froissart-Martin bound. This pattern
persists: for $k_{\rm min}=3$, one finds that the $J=0,1,2$ states decouple
from the null constraints, and so on.

It is also instructive to understand how the null constraints can be
satisfied, as this tells us information about the mass spectrum of the
theories. In Fig. 2 we show the first few null constraints in Eq. (18) as a
function of $J$. We see that for the first two null constraints the
contribution from the $J=1$ is the only opposite in sign to the other ones. As
these expressions must average to zero, Eq. (11), this implies that the theory
must contain $J=1$ states. For ${\cal X}_{4,1}$, also the $J=2$ contribution
is negative, implying that also $J=2$ states are needed. As the order $n$ of
the null constraint increases, one finds that the number of states with
negative coefficients increases: eventually all $J$ are needed to satisfy the
null constraints. So, for $k_{\rm min}=1$, theories in the large-$N_{c}$ limit
either have no $J>0$ state, or have states with all values of $J$ from $1$ to
$\infty$. Similarly, for $k_{\rm min}=2$, states with spin $J\geq 2$ are
either absent, or are all present.

###  2.3 UV completions of the pion amplitude

Before proceeding to examine the implications from positivity, we would like
to discuss simple UV completions for a theory of pions. By this we mean
theories that generate consistent crossing-symmetric amplitudes ${\cal
M}(s,u)$, with simple poles at real $s>0$, positive spectral density,
satisfying the high-energy behavior of Eqs. (6) for some $k_{\rm min}$.
Interestingly, these amplitudes will turn out to reside at the kinks of the
allowed parameter space, as we will discuss in the next section.

Simple amplitudes describing exchange of a single spin-$J$ particle with mass
$m_{J}$ are characterised by a pole-structure with residue on the associated
partial wave,

${\cal
M}^{J}(s,u)=\frac{m_{J}^{2}P_{J}(1+2u/m_{J}^{2})}{m_{J}^{2}-s}+F(s,u)+(s\leftrightarrow
u)\,,$ (21)

where $F$ is an analytic function that defines the theory, but does not
contribute to the residue. We will fix $F$ by imposing the Adler’s zero ${\cal
M}^{J}(0,0)=0$ and by requiring that the amplitude satisfies the large-$s$
behavior of Eqs. (6) for the lowest possible value of $k_{\rm min}$.

####  2.3.1 Theory of scalars

Scalars can provide a consistent UV completion to a theory of pions, via the
Higgs mechanism in the linear sigma-model. The $2\rightarrow 2$ pion amplitude
mediated by a spin-0 state with $I=0$ is given by,

${\cal M}^{(s)}(s,u)=\frac{g^{2}_{s\pi\pi}s}{m^{2}_{s}-s}+(s\leftrightarrow
u)\,.$ (22)

Expanding Eq. (22) at low energies, $s,u\ll m^{2}_{s}$, we obtain,

$g_{n,0}=\frac{g^{2}_{s\pi\pi}}{m^{2n}_{s}}\ \ \ (n\geq 1)\ ,\quad\quad
g_{n,l}=0\ \ \ (l\neq 0)\,.$ (23)

An important property of Eq. (22) is that it satisfies the high-energy
behavior in Eqs. (6a,6b) with $k_{\rm min}=1$. Therefore the Wilson
coefficients Eq. (23) obey the sum rules Eq. (13), as can be easily checked.
The fact that a model of only scalars does not need higher-spin states to
satisfy Eqs. (6a,6b) with $k_{\rm min}=1$, explains why the $J=0$ states
decouple from the null constraints, as explained above.

####  2.3.2 Theory of vectors

Let us now consider a (weakly coupled) spin-1 resonance with isospin $I=1$
(or, in general, in the $Adj.$ representation of $SU(N_{f})$), which we will
refer to as $\rho$, in analogy with QCD. From Eq. (21) we have,

${\cal
M}^{(\rho)}(s,u)=\frac{g^{2}_{\rho\pi\pi}m^{2}_{\rho}}{m^{2}_{\rho}-s}P_{1}\left(1+\frac{2u}{m^{2}_{\rho}}\right)+(s\leftrightarrow
u)\,,$ (24)

corresponding to the contribution from the transverse components of a massive
vector coupling to pions via
$g_{\rho\pi\pi}f_{abc}\rho^{a}_{\mu}\pi^{b}\partial^{\mu}\pi^{c}$ (minimal
coupling) where $f_{abc}$ are the $SU(N_{f})$ structure constants.222A more
general function $F$ in Eq. (21) would be associated with the exchange of
longitudinal modes (contact terms to the Lagrangian) which would yield in the
amplitude multiplicative factors $(s/m_{\rho}^{2})^{n}$ that worsen the high-
energy behavior.  This amplitude can arise in models in which the $\rho$ gets
its mass from the Higgs mechanism, or in holographic models where the $\rho$
arises as a Kaluza-Klein state.

Eq. (24) satisfies the Froissart-Martin bound, Eqs. (6a,6b) but only for
$k_{\rm min}=2$. Nevertheless, the high-energy behaviour of Eq. (24) can be
improved by the following deformation [19]:

$\widehat{\cal
M}^{(\rho)}(s,u)=\frac{g^{2}_{\rho\pi\pi}m^{2}_{\rho}}{m^{2}_{\rho}-s}P_{1}\left(1+\frac{2u}{m^{2}_{\rho}}\right)\frac{m^{2}_{\infty}}{m^{2}_{\infty}-u}+(s\leftrightarrow
u)\,,$ (25)

that at high energy $s\gg m^{2}_{\infty}$ satisfies Eqs. (6a,6b) with $k_{\rm
min}=1$. By studying the pole structure of Eq. (25), one can see that the
amplitude is mediated by states of any $J$ with masses $m_{\infty}$, but also
at $s=m_{\rho}$ we have now states with $J\not=1$. Nevertheless, by taking the
limit $m_{\infty}/m_{\rho}\to\infty$ in Eq. (25), we recover Eq. (24) as well
as its low-energy predictions. So, as long as we are only interested in the
Wilson coefficients, we can safely use Eq. (24).

At low energies, Eq. (24) leads to

$g_{1,0}=3\,\frac{g^{2}_{\rho\pi\pi}}{m^{2}_{\rho}}\ ,\ \
g_{2,1}=\frac{4}{3}\,\frac{g_{1,0}}{m^{2}_{\rho}}\ ,\ \
g_{n,0}=\frac{1}{3}\,\frac{g_{1,0}}{m^{2n}_{\rho}}\ ,\ \
g_{n+1,1}=\frac{2}{3}\,\frac{g_{1,0}}{m^{2(n+1)}_{\rho}}\ \ (n\geq 2)\,,$ (26)

while $g_{n,l}=0$ for $l\geq 2$. Alternatively, we could compute the Wilson
coefficients Eq. (26) from dispersion relations, by using the explicit form of
the UV spectral density Eq. (9), with support on $J=1$ and $m_{i}=m_{\rho}$
only. This can help us to appreciate the difference between ${\cal
M}^{(\rho)}$ and its improved version $\widehat{\cal M}^{(\rho)}$. Indeed,
since the vector amplitude Eq. (24) fulfils Eq. (6a) only for $k_{\rm min}\geq
2$, we can not use the sum rules Eq. (12) for $k_{\rm min}=1$. In particular,
the expressions for $g_{1,0}$ and $g_{2,1}$ from Eq. (13) do not hold – indeed
they differ from those obtained directly from the amplitude Eq. (26). On the
other hand, Eq. (24) fulfills Eq. (6b) for $k_{\rm min}=1$ and one can use the
prediction for $g_{2,1}$ from Eq. (15), that agrees with Eq. (26).

If instead we use $\widehat{\cal M}^{(\rho)}$ from Eq. (25) – which has
$k_{\rm min}=1$ high-energy behavior – the extra states beyond the $\rho$ give
a nonzero contribution to $g_{1,0}$ and $g_{2,1}$ that makes it to coincide
with Eq. (26). These contributions tend to zero in Eq. (15).

####  2.3.3 Theory of spin-2 states

The pole structure of four-pion amplitude mediated by a spin-2 state only (in
analogy with QCD we refer to it as $f_{2}$), can be written as,

${\cal
M}^{(f_{2})}=\frac{g^{2}_{f_{2}\pi\pi}m^{2}_{f_{2}}}{m^{2}_{f_{2}}-s}P_{2}\left(1+\frac{2u}{m^{2}_{f_{2}}}\right)+(s\leftrightarrow
u)\,.$ (27)

Since $P_{2}(x)=(3x^{2}-1)/2$, this amplitude grows like $\sim s^{2}/u$ for
large $s$, violating even the Froissart-Martin $k_{\rm min}=2$ bound.
Nevertheless, Eq. (27) can be deformed as in Eq. (25) to improve its high-
energy behavior:

$\widehat{\cal
M}^{(f_{2})}=\frac{g^{2}_{f_{2}\pi\pi}m^{2}_{f_{2}}}{m^{2}_{f_{2}}-s}P_{2}\left(1+\frac{2u}{m^{2}_{f_{2}}}\right)\frac{m^{2}_{\infty}}{m^{2}_{\infty}-u}+(s\leftrightarrow
u)\,,$ (28)

that indeed satisfies the Froissart-Martin bound for $s\gg m_{\infty}^{2}$,
and leads to Eq. (27) in the limit $m_{\infty}/m_{f_{2}}\to\infty$. It is not
possible to improve this further, as the resulting amplitude would have
negative spectral density, and violates unitarity. For this reason amplitudes
with light $J\geq 3$ (which grow as $s^{J}$ from $P_{J}(1+2s/m_{J}^{2})$ can
not be completed into amplitudes that satisfy the Froissart-Martin bound, see
also [32].

Contact terms (the function $F$ in Eq. (21)) would modify Eq. (27).
Nevertheless, demanding that they do not grow faster than Eq. (27), leaves
only terms up to $O(s)$. These terms can only affect $g_{1,0}$, telling us
that this Wilson coefficient cannot be predicted in a theory of spin-2 states.
The rest of the Wilson coefficients, however, can be unambiguously derived
from Eq. (27) in the low-energy limit:

$\displaystyle g_{2,0}=7\,\frac{g^{2}_{f_{2}\pi\pi}}{m_{f2}^{4}}\ ,\ \ \
g_{2,1}=\frac{12}{7}g_{2,0}=g_{3,1}m^{2}_{f_{2}}=g_{4,2}m^{4}_{f_{2}}\ ,\ \ \
g_{n,0}=\frac{1}{7}\,\frac{g_{2,0}}{m^{2n-4}_{f_{2}}}\ \ \ (n\geq 3)\,,$
$\displaystyle g_{n,1}=\frac{6}{7}\,\frac{g_{2,0}}{m^{2n-4}_{f_{2}}}\ \ (n\geq
4)\ ,\ \ \ g_{n,2}=\frac{6}{7}\,\frac{g_{2,0}}{m^{2n-4}_{f_{2}}}\ \ (n\geq
5)\,,$ (29)

with $g_{n,l}=0$ for $l>2$.

Similarly to the spin-1 case, we could calculate some of the Wilson
coefficients from dispersion relations, with spectral density for $J=2$ only.
As Eq. (27) satisfies the high-energy limit Eqs. (6) for $k\geq k_{\rm
min}=3$, we can use the sum rules Eq. (12) with $k_{\rm min}=3$ to obtain all
the Wilson coefficients except $g_{1,0},\ g_{2,0},\ g_{2,1},\ g_{3,1},\
g_{4,2}$ and $g_{5,3}$ – indeed their value via Eq. (12) disagrees with Eq.
(29). This can also be understood by realizing that the states of mass
$m_{\infty}$ present in the consistent amplitude Eq. (28) give finite
contributions to $g_{1,0},\ g_{2,0},\ g_{2,1},\ g_{3,1},\ g_{4,2}$ and
$g_{5,3}$ when Eq. (12) is used, even in the limit $m_{\infty}\to\infty$. For
the coefficient $g_{3,1}$, however, we can alternatively determine it by using
Eq. (16) that indeed agrees with Eq. (29).

####  2.3.4 The $su$-models

The $su$-models [6, 7] give the simplest four-pion amplitude mediated entirely
by higher-spin states. The particularity of these models is that their
spectrum is fully degenerate, ${\cal M}(s,u)\propto 1/(s-m^{2})(u-m^{2})$, a
condition that naturally places these models at the boundary of the allowed
parameter space, as we shall see. These $su$-amplitudes and the associated
Wilson coefficients are discussed in detail in Appendix C.

####  2.3.5 The Lovelace-Shapiro and Coon amplitude

There are other four-pion amplitudes mediated by higher-spin states: the
Lovelace-Shapiro [20, 21] amplitude and its generalization, the Coon amplitude
[22]. These amplitudes originate in the context of string theory, and can
provide therefore fully consistent UV completions to a theory of pions. We
will see that the Wilson coefficients predicted by these amplitudes lie at the
closest point to one of the boundaries of the allowed regions. Therefore they
provide information about the mass spectrum of the theories residing on these
boundaries. The details of these amplitudes are given in Appendices D and E.

##  3 Implications of ${\cal M}(s,u)/s\to 0$ at large $s$

In this section, we assume that the pion amplitude ${\cal M}(s,u)$ satisfies
the conditions Eqs. (6a,6b) for $k_{\rm min}=1$, as argued in Ref. [19]. Since
all Wilson coefficients scale like $1/N_{c}$, and because positivity bounds
are inherently projective (i.e. only ratios of Wilson coefficients are
constrained), we will work with

$\tilde{g}_{n,l}\equiv\frac{g_{n,l}}{g_{1,0}}M^{2(n-1)}\,,$ (30)

where $g_{1,0}$ is the leading Wilson coefficient and $M$ the EFT cutoff
defined in Fig 1. Unless stated, we will take $M$ as the lowest resonance
mass. The $\tilde{g}_{n,l}$ are independent of $N_{c}$ in the large-$N_{c}$
limit.

We can use the sum rules Eq. (12) to determine the Wilson coefficients as a
function of the mass spectrum, which is itself constrained by the null
constraints. Our goal will be to shape the boundaries of the EFT parameter
space, and identify the UV theories that generate it. When possible we shall
use analytic arguments, complemented when necessary by numerics.

###  3.1 Bounds on the leading Wilson coefficients

Figure 3: Allowed regions in the $\tilde{g}_{2,1}$–$\,\tilde{g}_{2,0}$ plane,
divided into regions with $J=0$ states only (red line) and $J>0$ states (blue)
– using null constraints with $n_{\rm max}=3$ (dashed light-blue line) and
$n_{\rm max}=11$ (dashed blue line). The dark blue area spans Eq. (47) for
$0\leq m_{\rho}\leq M$ and $m_{\rho}\leq m\leq\infty$. Moreover, $J=1$ models
lie on the blue line, $J>0$ $su$-models on the black line (Eq. (92) with Eq.
(94) and $0\leq m\leq M$), and the magenta line corresponds to Eq. (48). The
brown and yellow dot correspond to the Lovelace-Shapiro amplitude with and
without scalars (Eq. (113))) respectively. The dashed brown and yellow line
are the Coon amplitude (Eq. (118) with $C=1$) with and without $J=0$ states
respectively.

Let us start by studying the implications of positivity bounds for $\pi\pi$
scattering at order $O(s^{2})$, i.e. for the coefficients $g_{2,0}$ and
$g_{2,1}$. As explained in Ref. [19], and recalled in Appendix A, the allowed
regions can be obtained by numerical methods, see Fig. 3. These methods work
extremely well as explorative tools and give conclusive answers when they
rapidly approach known theories, e.g. [33]. Nevertheless, they are limited by
computer power, and leave the question open of whether the extrapolation from
finite resolution truly reveals a physically meaningful result. Here we will
show that for some questions the numerical convergence is too slow, and we
proceed by using analytic methods to map as much of the parameter space as
possible.

##### Scalar theories and the $\tilde{g}_{2,1}>0$ boundary.

The smallest value of $\tilde{g}_{2,1}=0$ is saturated by an amplitude
mediated by $J=0$ states, as discussed in section 2.3.1 – see also Ref. [19].
In particular, from Eq. (23), identifying $M=m_{s}$, we have
$(\tilde{g}_{2,1},\tilde{g}_{2,0})=(0,1)$, depicted as a red dot in Fig. 3.
This point must clearly be a corner (kink) of the full allowed region, since
from Eq. (13) one can see that $\tilde{g}_{2,1}\geq 0$ and
$\tilde{g}_{2,0}\leq 1$. When more (non-degenerate) scalars are present, the
value of $g_{2,0}/g_{1,0}$ always decreases, since $g_{2,0}$ scales as $\sim
1/m_{s}^{4}$ and therefore extra heavy scalars contribute more to $g_{1,0}$
than $g_{2,0}$ (the same is true for $g_{n,0}/g_{n-1,0}$). This is shown in
Fig. 3 by a red line.

As discussed in section 2.1, states with $J<k_{\rm min}$ decouple from the
null constraints. For $k_{\rm min}=1$, only the $J=0$ states decouple, meaning
that it is the only simple theory that can provide a standalone UV completion
of the chiral Lagrangian. All other UV completions must involve infinitely
many states with all spins. It is therefore interesting to study the boundary
of the region with $J\geq 1$ independently, as we discuss in what follows.

##### Spin-1 theories, the ${\tilde{g}_{2,1}}/{\tilde{g}_{2,0}}<4$ boundary
and its kink.

The largest value of $\tilde{g}_{2,1}/\tilde{g}_{2,0}$ can be determined from
Eq. (13) and Eq. (15), which gives

$\frac{\tilde{g}_{2,1}}{\tilde{g}_{2,0}}=\frac{4\left\langle\frac{1}{m^{4}}\right\rangle_{J-\rm
odd}}{\left\langle\frac{1}{m^{4}}\right\rangle_{J-\rm
odd}+\left\langle\frac{1}{m^{4}}\right\rangle_{J-\rm even}}\,,$ (31)

where the subscript denotes average over even or odd $J$ only. This is
saturated by ${\tilde{g}_{2,1}}/{\tilde{g}_{2,0}}=4$ and corresponds to
theories with only $J$-odd states. The simplest theory of this type is that
described in Sec. 2.3.2 consisting of a light spin-1 state only, whose
amplitude with the improved high-energy behavior is given in Eq. (25) with
$m_{\infty}/m_{\rho}\to\infty$. From Eq. (26) we have

$(\tilde{g}_{2,1},\tilde{g}_{2,0})_{\rm vector}=(4/3,1/3)\,,$ (32)

shown in Fig. 3 as a blue dot. Adding extra spin-1 states allows us to move
down from Eq. (32) to the origin, along the blue line of Fig. 3. This line
must be part of the boundary of the allowed region for
$({\tilde{g}_{2,1}},{\tilde{g}_{2,0}})$, since the ratio
${\tilde{g}_{2,1}}/{\tilde{g}_{2,0}}=4$ takes the largest possible value.

The important question to address now is whether the allowed region can extend
along the blue line beyond Eq. (32) or not. The numerical analysis of [19] was
able to show that as one increases the number of null constraints, the kink
moves towards Eq. (32), but appeared to tend asymptotically to a larger value.
Here we will show that the kink resides at Eq. (32), i.e. the extremal theory
along the boundary contains only a spin-1 light state.

To show this, we will first argue that at the boundary,

$\tilde{g}_{2,0}=M^{2}\frac{\left\langle\frac{1}{m^{4}}\right\rangle}{\left\langle\frac{1}{m^{2}}\right\rangle}\quad\to\quad
M^{2}\frac{\left\langle\frac{1}{m^{4}}\right\rangle_{J=1}}{\left\langle\frac{1}{m^{2}}\right\rangle_{J=1}}\,\frac{1}{1+\frac{\left\langle\frac{1}{m^{2}}\right\rangle_{J>1}}{\left\langle\frac{1}{m^{2}}\right\rangle_{J=1}}}\,.$
(33)

This can be proven as follows. Eq. (31) implies that at the boundary even-$J$
states must decouple from the average in $g_{2,0}$,

$\big{\langle}\frac{1}{m^{4}}\big{\rangle}_{J-\rm even}\to 0\,.$ (34)

Using the Cauchy-Schwarz inequality (where $\big{\langle}{\cal
J}^{4}/m^{8}\big{\rangle}$ must be finite because it enters in the null
constraint in the first expression of Eq. (19)), we have

$\sqrt{\left\langle\frac{{\cal J}^{4}}{m^{8}}\right\rangle_{J-\rm
even}\left\langle\frac{1}{m^{4}}\right\rangle_{J-\rm
even}}\geq\left\langle\frac{{\cal J}^{2}}{m^{6}}\right\rangle_{J-\rm even}\to
0\,.$ (35)

This, together with the null constraint $\big{\langle}{\cal
Y}_{3,1}\big{\rangle}=0$ Eq. (20), implies,

$\left\langle\frac{{\cal J}^{2}}{m^{6}}\right\rangle_{J-\rm
even}=3\left\langle\frac{{\cal J}^{2}-2}{m^{6}}\right\rangle_{J-\rm odd}\geq
12\left\langle\frac{1}{m^{6}}\right\rangle_{J>1-\rm odd}\to 0\,,$ (36)

because ${\cal J}^{2}-2=0$ on $J=1$ and ${\cal J}^{2}-2\geq 4$ for $J\geq 2$.
Using Cauchy-Schwarz again (with measure on odd $J>1$ only), and the fact that
$\big{\langle}\frac{1}{m^{2}}\big{\rangle}_{J>1-\rm odd}$ is finite, we find

$\sqrt{\left\langle\frac{1}{m^{6}}\right\rangle_{J>1-\rm
odd}\left\langle\frac{1}{m^{2}}\right\rangle_{J>1-\rm
odd}}\geq\left\langle\frac{1}{m^{4}}\right\rangle_{J>1-\rm odd}\to 0\,,$ (37)

thus proving Eq. (33). The theory that lives on the boundary must therefore
consist of spin-1 states at finite mass accompanied by higher-spin states at
infinite mass.

Unfortunately, this is not yet enough to claim that it is _exactly_ the theory
in Eq. (25), since there might be different ways in which the $J\to\infty$
states enter in the spectral density.

To prove that it is, we reformulate the question of finding the maximum value
of the kink, as a 1D moment problem. Indeed, the kink is positioned at the
maximum value of $\tilde{g}_{2,0}$ along the boundary given by Eq. (33). The
first factor
${\big{\langle}\frac{1}{m^{4}}\rangle_{J=1}}/{\langle\frac{1}{m^{2}}\big{\rangle}_{J=1}}\leq
1$ is maximally saturated when the spectrum contains only one spin-1 particle
at the mass $M$,
$M^{2}\langle\frac{1}{m^{4}}\rangle_{J=1}=\langle\frac{1}{m^{2}}\rangle_{J=1}\equiv
g_{\rho\pi\pi}^{2}/M^{2}$. Then, from the second factor in Eq. (33) we read
that the kink is located at the _minimum_ of,

$\frac{M^{2}}{g_{\rho\pi\pi}^{2}}\left\langle\frac{1}{m^{2}}\right\rangle_{J>1}\,.$
(38)

This minimum cannot be zero, because the null constraints relate the high-
energy averages to $g_{\rho\pi\pi}^{2}$, in such a way that the ratio is
finite. Indeed, along the boundary, the dominant null constraints Eq. (18) can
be written as,

$2(n-1)!^{2}=\frac{M^{2n}}{g_{\rho\pi\pi}^{2}}\left\langle\frac{J^{2(n-1)}}{m^{2n}}\right\rangle_{J>1}\
,\ \ n=\underline{2},3,4,...\,,\\\ $ (39)

i.e. all subleading terms $J^{k}$ with $k<2(n-1)$ can be neglected. This can
be understood from the comment below Eq. (37) – for states with infinite mass,
if in the average $\lim_{m\to\infty}J^{2(n-1)}/m^{2n}$ is finite, then all
subleading powers of $J$ must vanish.333It can also be seen by using the
Hölder inequality and $\big{\langle}{1}/{m^{2k}}\big{\rangle}_{J>1}\to 0$ (for
$k\geq 1$) from Eq. (37),
$\left\langle\frac{1}{m^{2k}}\right\rangle_{J>1}^{\frac{1}{k}}\left\langle\frac{J^{2(n-k-1)}}{m^{2(n-k)}}\right\rangle_{J>1}^{1-\frac{1}{k}}\geq\left\langle\frac{J^{2(n-k-1)}}{m^{2n}}\right\rangle_{J>1}\to
0\,.$ (40) With a change of variables to (the square of) impact parameter
$x\equiv J^{2}M^{2}/m^{2}$, and redefining $n$, we can write Eq. (39) as

$2n!^{2}=\int^{\infty}_{0}d\mu(x)x^{n}dx\equiv\mu_{n}\ \ ,\ \ n=1,2,3,...\,,$
(41)

with $d\mu(x)$ a positive distribution. In this language Eq. (38) is
$\mu_{0}$, and the problem of finding the kink position translates into a
1-dimensional moment problem:

finding the minimum of $\mu_{0}$ such that
$\\{\mu_{0},\mu_{1},\mu_{2},\cdots\\}$ is a moment series from a positive
distribution, with $\mu_{n}=2n!^{2}$ for $n\geq 1$. (42)

A sufficient condition for this, is that the Hankel matrix $H^{0}_{N}$, with
$(H^{k}_{N})_{ij}=\mu_{i+j+k}$, for $i,j=0,\dots,\lfloor N/2\rfloor$, be
asymptotically positive definite [34, 4],

$\lim_{N\to\infty}H^{0}_{N}=2\lim_{N\to\infty}\left(\begin{matrix}\mu_{0}/2&1!^{2}&2!^{2}&\cdots&n!^{2}\\\
1!^{2}&2!^{2}&3!^{2}&\cdots&(n+1)!^{2}\\\
2!^{2}&3!^{2}&4!^{2}&\cdots&(n+2)!^{2}\\\
\cdots&\cdots&\cdots&\ddots&\vdots\\\
n!^{2}&(n+1)!^{2}&(n+2)!^{2}&\cdots&(2n)!^{2}\end{matrix}\right)\succ 0\,.$
(43)

Equivalently (using Silvester’s criterion) this can be rewritten as,

$1-\mu_{0}/2\leq\lim_{N\to\infty}\frac{\det\left.H^{0}_{N}\right|_{\mu_{0}=1}}{\det
H^{2}_{N}}\,.$ (44)

Explicit evaluation of Eq. (44) for fixed $N$ enables us to reach smaller and
smaller values, going from $\mu_{0}\approx 0.95$ for $N=10$ (equivalent to
$\tilde{g}_{2,0}\approx 0.51$) to $\mu_{0}\approx 1.54$ for $N=700$
(equivalent to $\tilde{g}_{2,0}\approx 0.39$) – to be compared with
$\tilde{g}_{2,0}\approx 0.42$ of Ref. [19]. Computing the asymptotic behavior
of determinants of this type is an interesting open problem in mathematics,
see e.g. [35], motivated by their appearance in random matrix theory
(interestingly, also in relation with QCD and chiral perturbation theory [36,
37]). Leaving this for future work, in Appendix B we take a shortcut and,
rather that computing the individual determinants, we focus on the most
efficient way of computing the ratio Eq. (44), and show that as $n\to\infty$,

$\mu_{0}\rightarrow
2\quad\quad\textrm{and}\quad\quad(\tilde{g}_{2,1},\tilde{g}_{2,0})\to(4/3,1/3)\,.$
(45)

At the kink resides the theory of a single spin-1 state, with the improved
high-energy behavior amplitude Eq. (25) with $m_{\infty}\gg m_{\rho}$.

##### The su-model and the boundary for $J\geq 1$ with minimal
$\tilde{g}_{2,1}/\tilde{g}_{2,0}$.

At the largest value of $\tilde{g}_{2,0}=1$ must lie theories with a
degenerate spectrum, see Eq. (13). Apart from a theory of a scalar (discussed
before), the only amplitude with this property is the $su$-model discussed in
Appendix C, with amplitude Eq. (92). This amplitude can also be obtained
analytically by solving the null constraints. Indeed, for a degenerate
spectrum, the null constraints reduce to a system of equations for the
couplings $g_{i\pi\pi}^{2}$. The dominant null constraints Eq. (18), for
instance, are linearly independent, and can be solved explicitly for a fixed
number of couplings $g_{i\pi\pi}^{2}$ with $i=1,\cdots,n$. The solution is a
function that can be resummed and converges into the $su$-model prediction.

This $su$-model contains a fraction of scalar residues, controlled by the
value of $\lambda$ in Eq. (93); for the value in Eq. (94) the theory has no
scalars. Its amplitude lies at,

$(\tilde{g}_{2,1},\tilde{g}_{2,0})_{J>0\ su\rm-model}=(\approx 3.26,1)\,,$
(46)

shown by the black dot in Fig. 3. The uniqueness of this amplitude naturally
puts it at kink of the $J\geq 1$ region (and its linear combination with the
scalar amplitude at the boundary of the $J\geq 0$ region).

As the spectrum becomes heavier, $M/m\to 0$, the $su$-model morphs into the
free theory (at the origin of the plot in Fig. 3). Interestingly, this line
defines the boundary of the allowed region for theories characterised by
resonances of spin $J\geq 1$. In Appendix C we show that this line is indeed a
boundary by considering the most generic deformation of the $su$-model and
showing that in order to not spoil positivity these deformations must push you
in the bulk of the exclusion region (excluding the contribution from scalars).

##### Boundary between kinks.

Therefore we are left with the boundary line joining the two kinks, Eq. (32)
and Eq. (46). We have not found an analytical expression for this curve.
Nevertheless, as proposed in Ref. [19], we can have a reasonable analytical
formula by considering a model that interpolates between a spin-1 model and
the $J>0$ $su$-model. The interpolating amplitude is given by444There are
other possible interpolating amplitudes, but Eq. (47) is the one with the
smallest number of states that we have found. Adding more states will give
predictions for $(\tilde{g}_{2,1},\tilde{g}_{2,0})$ that will lie on the left
of the magenta line of Fig. 3.

${\cal M}={\cal M}^{(su)}_{1}-\frac{3(\ln
8-2)}{g^{2}_{\rho\pi\pi}}\left(\widehat{\cal M}^{(\rho)}(m_{\rho}\to
m)-\widehat{\cal M}^{(\rho)}\right)\,,$ (47)

which corresponds to a $J>0$ $su$-model in which the spin-1 state of mass $m$
has been subtracted and replaced by a spin-1 state of mass $m_{\rho}$. We have
used the corrected amplitude $\widehat{\cal M}^{(\rho)}$ from Eq. (25), to
assure that Eq. (47) satisfies the high-energy conditions with $k_{\rm min}=1$
for all values of its parameters; nevertheless we will be taking the limit
$m_{\infty}/m_{\rho}\to\infty$ that corresponds to $\widehat{\cal
M}^{(\rho)}\to{\cal M}^{(\rho)}$. From Eq. (47) it is clear that by varying
the mass $m$ from $m_{\rho}$ to infinity, we are effectively pushing up the
masses of the $J>1$ states in the $su$-model, leaving only a $J=1$ state at
low energy. The corresponding Wilson coefficients of Eq. (47) can be easily
calculated and one obtains, for $M=m_{\rho}$,

$\tilde{g}_{2,0}=\frac{a-(3-10a)(r^{2}-1)}{a-(9-28a)(r-1)}\ ,\ \ \ \
\tilde{g}_{2,1}=\frac{1+(36a-11)(r^{2}-1)}{a-(9-28a)(r-1)}\,,$ (48)

where $r={m^{2}_{\rho}}/{m^{2}}$ and $a=1-\ln 2$. The magenta line in Fig. 3
is obtained by varying $r\in[0,1]$.

We have determined this boundary numerically (following Ref. [19], see our
Appendix A for details), including null constraints with $n_{\rm max}=3$ and
$n_{\rm max}=11$, shown by dashed lines in Fig. 3. Due to the lack of
computational power, however, we have not been able to understand how much the
true boundary approaches the analytic boundary Eq. (48) as $n\to\infty$.
Nevertheless, we can claim that Eq. (48) cannot coincide with the true bound,
as we know of another consistent amplitude that lies on the RHS of this line.
This is the Lovelace-Shapiro amplitude in which the scalar contribution has
been removed (see Appendix D for details). The prediction for this amplitude
(see Eq. (115)) is shown by a yellow dot in Fig. 3. In spite of this, the
prediction of this amplitude is impressively close to Eq. (47) as can be
appreciated by the zoom area in Fig. 3. We also show a version of the scalar-
subtracted Coon amplitude, Eq. (118) in Appendix E, with $C=1$ and $q$ varied
from $0$ to $1$. This line starts at the $J>0$ $su$-model and goes down to the
Lovelace-Shapiro model. It slightly improves Eq. (48) but only close to the
Lovelace-Shapiro values ($q\simeq 1$). It could be interesting to know if
there is another deformation of the Lovelace-Shapiro amplitude that saturates
the true boundary. We also show in Fig. 3 the prediction from the full
Lovelace-Shapiro and Coon amplitudes (with $C=1$) without subtracting the
scalar contributions.

For models with both $J=0$ and $J>0$ states, the allowed region corresponds to
the convex hull spanned by the individual boundaries, shown in light red (plus
blue) in Fig. 3.

###  3.2 Emergence of Vector Meson Dominance

We have seen that a theory of a spin-1 state does not satisfy ${\cal
M}(s,u)/s\to 0$ at large $s$, fixed $u<0$, and requires higher-spin states to
soften the high-energy behavior. The converse is also true: any model of
higher-spin states must contain spin-1 mesons. This can be made explicit by
looking at null constraints. For example, the first two null constraints,
$\left\langle\mathcal{Y}_{2,1}\right\rangle=0$ and
$\left\langle\mathcal{X}_{3,1}\right\rangle=0$ of Eq. (18), lead to

$\displaystyle\left\langle\frac{1}{m^{4}}\right\rangle\bigg{|}_{J=1}$
$\displaystyle=$ $\displaystyle
3\left\langle\frac{1}{m^{4}}\right\rangle\bigg{|}_{J=2}+4\left\langle\frac{1}{m^{4}}\right\rangle\bigg{|}_{J=3}+\cdots\,,$
$\displaystyle\left\langle\frac{1}{m^{6}}\right\rangle\bigg{|}_{J=1}$
$\displaystyle=$ $\displaystyle
9\left\langle\frac{1}{m^{6}}\right\rangle\bigg{|}_{J=3}+35\left\langle\frac{1}{m^{6}}\right\rangle\bigg{|}_{J=4}+\cdots\,.$
(49)

Since the RHS is always positive, this identity can only be fulfilled if there
are spin-1 states in the theory. These equations also tell us that, at any
order in $1/m^{n}$ in the average, the contributions from any individual $J>1$
state must always be smaller than the $\rho$ contribution, since the
coefficients appearing on the RHS are always bigger than one. Moreover, these
coefficients scale with large ${J}$ as $\sim{J}^{2n-2}/m^{2n}$, which is
faster than how they appear in the low-energy couplings such as
$g_{n,0}\sim\langle 1/m^{2n}\rangle$ or $g_{n,1}\sim\langle
J^{2}/m^{2n}\rangle$ (but $g_{2,1}\sim\langle 1/m^{4}\rangle$) from Eq.
(12).555For couplings $g_{n,l}$ with larger and larger $l$ this argument is
more involved, since the $J=1$ contribution cancels from Eq. (12) and is
restored only via null constraints.

The property that the $\rho$ meson dominates the low-energy amplitude of pions
(or at least that amplitudes with $\rho$ mesons populate the space of
consistent pion amplitudes) is referred to as Vector Meson Dominance (VMD)
[23, 24]. Despite its poor theoretical motivation, VMD is known to lead to
good agreement with QCD experimental data. Here we see that VMD emerges from
unitarity and crossing symmetry. For the most relevant couplings, it is
illustrated by the alignment between the $J>0$ allowed region and the spin-1
line observed in Fig. 4. The QCD experimental value, as determined in Ref.
[38], is denoted by the red cross in the figure.

As the masses of the $J>1$ states increase, VMD becomes more manifest. Indeed,
from Eqs. (13,15), the contributions from the high-spin states at large mass
$m_{J>1}\to\infty$ go as,

$\frac{g_{2,0}}{g_{1,0}}\sim\frac{g_{2,1}}{g_{1,0}}\sim\frac{1}{m^{2}_{J>1}}\to
0\,,$

while the $J=1$ contribution remains finite. This is also true for the rest of
the Wilson coefficients, since for large scale separations between the $\rho$
and higher spins $m_{\rho}/m_{J>1}\to 0$, The null constraints Eq. (18)
require a finite contribution from the $J\to\infty$ states, in such a way that
$J^{2}/m^{2}_{J>1}$ remains fixed in units of $g_{1,0}\sim 1/m^{2}_{J>1}$.
Using Eq. (13), this implies that the $J>1$ contribution to the Wilson
coefficients scales as,

$\Delta
g_{n+l,l}\sim\frac{J^{2l}}{m^{2(n+l)}_{J>1}}\sim\frac{1}{m^{2n}_{J>1}}\,,$
(50)

and therefore tends to zero for large $m_{J>1}$.

At finite masses, we can study this effect numerically, requiring a finite
mass gap between the $\rho$ meson and other resonances $M^{\prime}>m_{\rho}$,
i.e. we work with the spectral density [19],

$(2J+1)\rho_{J}(s)\to\frac{\pi}{2}g^{2}_{\rho\pi\pi}\delta_{J,1}\delta(s-m^{2}_{\rho})+(2J+1)\rho^{\prime}_{J}(s)\,,$
(51)

where the last term corresponds to extra states with $s\in[M^{\prime},\infty)$
(see Appendix A). In QCD, the lightest higher-spin resonce is a spin-2 meson
$f_{2}$ with a mass $m_{f_{2}}\approx 1.3$ GeV, which implies
$M^{\prime}/m_{\rho}\sim 1.65$. With this mass gap, the allowed region reduces
to the small green strip in Fig. 4, which is more strongly aligned with the
the spin-1 prediction, evidencing VMD.

Figure 4: Allowed regions from positivity with $J\geq 1$ (as in Fig. 3). The
green regions have heavier resonances of all spin $J\geq 1$ with masses
$M^{\prime}\geq 1.65\,m_{\rho}$: the dashed green line uses null constraints
up to $n_{\rm max}=11$. In orange the allowed region for a fixed
$\tilde{g}_{\rho}=1/2$ (left) and $\tilde{g}_{\rho}=1/3$ (right), with still
$M^{\prime}/m_{\rho}=1.65$. In purple, the allowed region from Lattice
simulations [39, 40, 41]. In red the QCD experimental value from Ref. [38].

It is instructive to further divide the allowed region of parameters in terms
of the contribution of the $\rho$ to the leading Wilson coefficient $g_{1,0}$,
i.e.

$\tilde{g}^{2}_{\rho}\equiv\frac{g^{2}_{\rho\pi\pi}}{g_{1,0}m_{\rho}^{2}}\,,$
(52)

which can be thought to quantify VMD. In Fig. 4 we show in orange the allowed
regions in which $\tilde{g}^{2}_{\rho}$ matches the experimental QCD value and
the value taken in the vector model Eq. (24),

$(a)\ \tilde{g}_{\rho}^{2}\simeq\frac{1}{2}\ \ \ (\text{QCD})\ ,\ \ \ \ (b)\
\tilde{g}_{\rho}^{2}=\frac{1}{3}\ \ \ (\text{spin-1 model})\,.$ (53)

As we will discuss in Sec. 3.4, $\tilde{g}_{\rho}^{2}={1}/{2}$ is close to its
maximal value $\tilde{g}^{2}_{\rho}\sim 0.78$, which is also where the
associated spectral density has the $J>1$ contributions maximised (saturated
by the $su$-model). On the other hand, even for $\tilde{g}_{\rho}^{2}={1}/{3}$
(for which $2/3$ of the leading effects are taken care by higher spin-mesons),
the allowed region for $\tilde{g}_{2,0}$ and $\tilde{g}_{2,1}$ still sits
close to the spin-1 contribution (blue dot), showing a small effect from the
$J>1$ states.

Our discussion of VMD so far focused on quantifying the contributions from
$J>1$ mesons: these are the difficult ones to model, and for which our
arguments are particularly important. On the other hand, since $J=0$ states
decouple from the null constraint, they could indeed dominate
$\tilde{g}_{2,0}$ and $\tilde{g}_{2,1}$, as it happens in the Higgs model.
Nevertheless, scalars can be easily accommodated in any phenomenological
analyses as they have simple UV completions. It is worth noticing, however,
that when a spin-1 $\rho$ is assumed to be the lightest meson in the spectrum,
as in QCD, the scalar contribution becomes smaller. This property is tied to
the fact that contributions to the Wilson coefficients are always positive.
For example, taking scalars with masses $\gtrsim 1.65\ m_{\rho}$, while still
fixing $\tilde{g}_{\rho}^{2}$ to the values considered above, we find that the
resulting allowed regions depicted in Fig. 4 increase in size by only
$10-25\%$ along the $\tilde{g}_{2,0}$ direction.

####  3.2.1 Comparison with Lattice QCD

The Wilson coefficients $L_{1,2,3}$, traditionally defined in the $SU(3)$
chiral Lagrangian [42],666Following the chiral Lagrangian definition in Ref.
[42],  ${\cal
L}=\frac{F_{\pi}^{2}}{4}\text{Tr}\left(\partial_{\mu}U^{\dagger}\partial^{\mu}U\right)+L_{1}\text{Tr}^{2}\left(\partial_{\mu}U^{\dagger}\partial^{\mu}U\right)+L_{2}\text{Tr}\left(\partial_{\mu}U^{\dagger}\partial_{\nu}U\right)\text{Tr}\left(\partial^{\mu}U^{\dagger}\partial^{\nu}U\right)+L_{3}\text{Tr}\left(\partial_{\mu}U^{\dagger}\partial^{\mu}U\partial_{\nu}U^{\dagger}\partial^{\nu}U\right)\,.$
are related to ours by

$\tilde{g}_{2,0}=4(2L_{1}+3L_{2}+L_{3})\frac{M^{2}}{F_{\pi}^{2}}\ ,\
\quad\tilde{g}_{2,1}=16L_{2}\frac{M^{2}}{F_{\pi}^{2}}\,.$ (54)

In the large-$N_{c}$ limit, $2L_{1}=L_{2}$ [42], which leads to

$\frac{\tilde{g}_{2,0}}{\tilde{g}_{2,1}}=\frac{1}{4}\left(1+\frac{\Delta_{L}}{L_{2}}\right),$
(55)

where $\Delta_{L}=3L_{2}+L_{3}$. This quantity vanishes for theories with only
spin-1 resonances, so VMD predicts $\Delta_{L}\sim 0$. Moreover, the
positivity of the Wilson coefficients implies that $L_{2},\Delta_{L}\geq 0$.

A combination of recent lattice simulation results for large-$N_{c}$ QCD [39,
40, 41], gives $\Delta_{L}=(-0.12\pm 0.22)\cdot 10^{-3}$. Unfortunately, we
have not found any lattice determination of $L_{2}$. Nevertheless, for any
given value of $\tilde{g}_{\rho}$, we have a minimal value for $L_{2}\propto
g_{2,1}$, which plugged into Eq. (55) with the lattice value of $\Delta_{L}$,
can provide a bound on ${\tilde{g}_{2,0}}/{\tilde{g}_{2,1}}$. For the two
values of Eq. (53), we find

$(a)\ 0.15\lesssim\frac{\tilde{g}_{2,0}}{\tilde{g}_{2,1}}\lesssim 0.28\ ,\ \ \
(b)\ 0.10\lesssim\frac{\tilde{g}_{2,0}}{\tilde{g}_{2,1}}\lesssim 0.30\,,$ (56)

that correspond to the purple areas in Fig. 4. We notice that in both cases
positivity bounds are complementary to bounds from lattice, ruling out
different regions. In the particular case $(a)$, the upper bound coming from
lattice seems to be more restrictive than the one from positivity, but this is
not the case as we decrease $\tilde{g}_{\rho}^{2}$. In the future a
combination of both approaches can lead to a better determination of the
geometry of the allowed regions in parameter space.

####  3.2.2 Holography

The results in Figs. 4 provide also an explanation for the success of
holography for predicting QCD properties [25, 26]. Holographic models consist
in weakly-coupled 5D constructions describing 4D strongly-coupled dynamics in
both the $N_{c}\to\infty$ limit, and the limit of a large mass gap between
spin-0,1 states and other higher-spin states.777The model also has 5D
gravitons, but these correspond to glueballs of spin $J\leq 2$, which decouple
from the pion amplitudes.

In real QCD, however, the mass ratio between the $q\bar{q}$ mesons of spin-2
and spin-1 is not large, $m_{f_{2}}/m_{\rho}\sim 1.65$. Therefore, one would
expect holographic models not to provide a good description of low-energy QCD,
contrary to what is observed [25, 26]. Nevertheless, the above analysis shows
that unitarity, causality and crossing-symmetry suppress the effects of
higher-spin states in the QCD Wilson coefficients. Therefore, even if the mass
gap $m_{f_{2}}/m_{\rho}$ is not large, the low-energy QCD quantities are
mostly affected by only spin-0 and spin-1 states, which are the ones captured
by holographic models. For this reason they can provide a good fit to real-
world QCD.

In particular, in the holographic model of Ref. [43], one can show that the
predictions for $\tilde{g}_{2,0}$ and $\tilde{g}_{2,1}$ are very close to
those of the vector model $(\tilde{g}_{2,1},\tilde{g}_{2,0})=(1.32,0.33)$.

###  3.3 Higher order Wilson coefficients

The features that sculpt the allowed region of $\tilde{g}_{2,0}$ and
$\tilde{g}_{2,1}$ play a dominant role also in understanding higher-order
Wilson coefficients, $\tilde{g}_{n,0}$ versus $\,\tilde{g}_{n,1}$. From Eq.
(13) we have,

$\frac{\tilde{g}_{n,1}}{\tilde{g}_{n,0}}=\frac{\big{\langle}\frac{{\cal
J}^{2}}{m^{2n}}\big{\rangle}}{\big{\langle}\frac{1}{m^{2n}}\big{\rangle}}\,,$
(57)

whose minimal value (zero) corresponds to a model with $J=0$. Focusing instead
on $J>0$ theories, the minimal value arises for models of spin-1 that give
${\tilde{g}_{n,1}}/{\tilde{g}_{n,0}}=2$.888Notice that we could not use this
argument for the case $n=2$, since we cannot use the sum rules in Eq. (57)
with $n=2$ for a theory of $J=1$ states only, as explained at the end of Sec.
2.3.2. In other words, the infinitely heavy $J>1$ states give zero
contribution to Eq. (57) only when $n>2$.

Figure 5: Allowed region in the $\tilde{g}_{3,1}$–$\,\tilde{g}_{3,0}$ plane
from positivity. Same labelling as in Fig. 3, with $n_{\rm max}=11$ null
constraints.

We illustrate this in Fig. 5, where a blue dot corresponds to a model with a
single $J=1$ state:

$(\tilde{g}_{3,1},\tilde{g}_{3,0})_{\rm vector}=(2/3,1/3)\,,$ (58)

while theories with many spin-1 states populate the blue line.

As for $\tilde{g}_{2,0},\tilde{g}_{2,1}$, we can show that Eq. (58)
corresponds to a kink of the boundary, see Appendix B. The other kink
corresponds again to the $J>0$ $su$-model (the only one with a degenerate
spectrum) that gives

$(\tilde{g}_{3,1},\tilde{g}_{3,0})_{su\rm-model}=(\approx 3.26,1)\,.$ (59)

We have not been able to find an analytic formula for the boundary connecting
the two kinks, Eq. (58) and Eq. (59); we illustrate the numerical analysis in
Fig. 5. We believe that by adding more null constraints the boundary must
approach, but not reach, Eq. (47), consisting of a theory connecting the two
kinks (the magenta line in Fig. 5). Nevertheless, as in Sec. 3.1, this line
cannot be the true boundary since the Lovelace-Sphapiro model with $J>0$
states lies at the left of this line, and so does part of the Coon amplitude,
Eq. (118) (with $C=1$ and $q\in[0,1]$, after subtracting all scalars).

###  3.4 Bounding the couplings of mesons to pions

Figure 6: Upper bound on $\tilde{g}_{\rho}^{2}$ (dashed blue line),
$\tilde{g}^{2}_{f_{2}}$ (dashed green line) and $\tilde{g}^{2}_{\rho_{3}}$
(dashed magenta line) as a function of $M^{\prime}/m_{\rho}$ using null
constraints with $n_{\rm max}=7$. The solid lines correspond to the prediction
from the interpolating model Eq. (47). The dots correspond to the values of
the Lovelace-Shapiro amplitude without scalars, and the diamonds to the QCD
experimental values.

So far, we have phrased dispersion relations as UV$\to$ IR vehicles to
reformulate microscopic unitarity, causality and crossing-symmetry as
predictions for low-energy coefficients. Null constraints, however, provide
genuine UV-UV relations, inspired by the same principles. As such, they
contain information on the UV meson spectrum and couplings to pions. We define
the latter, normalized as,

$\tilde{g}^{2}_{i}=\frac{g^{2}_{i\pi\pi}}{g_{1,0}\,m_{i}^{2}}\,,$ (60)

where $i=s,\rho,f_{2},\rho_{3},...$ labels $J=0,1,2,3,...$ mesons, following
the QCD notation of Ref. [44].

Since spin-0 mesons decouple from the null constraints, it is easy to
understand that $\tilde{g}^{2}_{s}$ is maximised by the smallest possible
value of $g_{1,0}$ that, due to its additive property, occurs when the
spectrum contains one scalar only:

$\tilde{g}^{2}_{s}\leq 1\,.$ (61)

On the other hand, bounds on the couplings of $J\geq 1$ mesons, involve all
null constraints, which we explore numerically, as explained in Appendix A.
The results are illustrated in Fig. 6. For instance, the bound on the $\rho$
coupling $\tilde{g}_{\rho}^{2}$ (dashed blue line), is obtained as a function
of $M^{\prime}\geq m_{\rho}$ by singling out this state from the spectral
density as in Eq. (51). The bound goes from a maximal value
$\tilde{g}^{2}_{\rho}\simeq 0.78$, corresponding to the $J>0$ $su$-model where
$M^{\prime}=m_{\rho}$, to the minimal value corresponding to the vector model,
$\tilde{g}^{2}_{\rho}=1/3$ where $M^{\prime}\to\infty$. This can be compared
with the interpolating model Eq. (47) shown by the solid blue line and with
the Lovelace-Shapiro model without scalars Eq. (113) (shown by dots) that lies
between the two lines. Interestingly, while all these models give similar
predictions in terms of the Wilson coefficients – see Fig. 3 – they differ
substantially at the quantitative level in Fig. 6. This provides an
interesting experimental handle to differentiate these theories by testing the
couplings of pions to the accessible resonances (amplitude’s residues).

Similarly for the spin-2 meson $f_{2}$, we rewrite the spectral density as,

$(2J+1)\rho_{J}(s)\to\frac{\pi}{2}g^{2}_{\rho\pi\pi}\delta_{J,1}\delta(s-m^{2}_{\rho})+\frac{\pi}{2}g^{2}_{f_{2}\pi\pi}\delta_{J,2}\delta(s-M^{\prime
2})+(2J+1)\rho^{\prime}_{J}(s)\,,$ (62)

and look for the upper bound on $\tilde{g}_{f_{2}}^{2}$ as a function of
$M^{\prime}/m_{\rho}$ for any value of $g^{2}_{\rho\pi\pi}>0$. This is shown
by the green dashed line in Fig. 6. The resulting bound is much stricter than
that for the $\rho$ – another manifestation of VMD.

As we study mesons of higher and higher spin $J$, the bound on
$\tilde{g}^{2}_{i}$ becomes stronger and stronger – see the magenta dashed
line for $\tilde{g}_{\rho_{3}}^{2}$. Interestingly, this pattern is also
tracked by the QCD experimental values, $\tilde{g}_{\rho}^{2}=0.51\pm 0.01$
and $\tilde{g}_{f_{2}}^{2}=0.18\pm 0.01$, extracted from the widths
$\rho,f_{2}\to\pi\pi$ [44], as shown by the diamonds in Fig. 6.

##  4 Implications of ${\cal M}(s,u)/s^{2}\to 0$ at large $s$

A more conservative assumption is that the four-pion amplitude satisfies the
Froissart-Martin bound, Eqs. (6a,6b) for $k_{\rm min}=2$, rather than $k_{\rm
min}=1$ as in the previous section. Unfortunately in this case we lose the sum
rule for the Wilson coefficient $g_{1,0}$ (and $g_{2,1}$ is no longer
expressible by Eq. (12) – which comes from $k_{\rm min}=1$ – but requires Eq.
(15)). So, we will need to choose another Wilson coefficient to normalize the
rest and make the predictions $N_{c}$-independent. We will use $g_{2,0}$ and
define,

$\bar{g}_{n,l}\equiv\frac{g_{n,l}}{g_{2,0}}M^{2(n-2)}\,.$ (63)

There is another important difference w.r.t. Sec. 3. The amplitude mediated by
a spin-1 state, Eq. (24), fulfils ${\cal M}^{(\rho)}/s^{2}\to 0$ for large
$|s|$, and therefore now provides a good UV description of the four-pion
amplitude. As a consequence, models of spin-1 states do not require anymore
higher-spin states. From the perspective of null constraints this is realised
by the decoupling of $J=1$, when setting $k_{\rm min}=2$, as we explained in
Sec. 2.1

It is the spin-2 state that now plays an analog role to that of the
$\rho$-meson in Sec. 3. Indeed, the
$\langle\mathcal{Y}_{4,2}-\mathcal{Y}_{4,1}\rangle=0$ null constraint reads,

$6\left\langle\frac{1}{m^{8}}\right\rangle\bigg{|}_{J=2}=10\left\langle\frac{1}{m^{8}}\right\rangle\bigg{|}_{J=3}+50\left\langle\frac{1}{m^{8}}\right\rangle\bigg{|}_{J=4}+\cdots,$
(64)

which tells us that spin-2 states need $J>2$ states and viceversa (for
instance, the amplitude Eq. (27) requires $J\geq 3$ states as in Eq. (28), to
comply with the Froissart-Martin bound).

###  4.1 Bounds on Wilson coefficients

Figure 7: Allowed region in the $\bar{g}_{3,1}$–$\,\bar{g}_{3,0}$ plane (left)
and $\bar{g}_{4,1}$–$\,\bar{g}_{4,0}$ plane (right) from positivity. The black
lines correspond to the $J>1$ $su$-model (Eq. (97) with Eq. (98) and $0\leq
m\leq M$), the green lines to $J=2$ models, and the magenta lines to Eq. (69)
and Eq. (70) for the left and right plot respectively. The green areas limited
by the dashed line corresponds to the allowed region with $n_{\rm max}=7$ null
constraints.

At the leading order $O(s^{2})$, we only have $\bar{g}_{2,1}$. As discussed
before, we have $\bar{g}_{2,1}=0$ ($\bar{g}_{2,1}=4$) for $J=0$ ($J=1$)
models, that are decoupled from higher-spin states. Focusing instead on
theories with $J\geq 2$ states, the largest value of $\bar{g}_{2,1}$ comes
from the $su$-model, once we have subtracted not only the scalar but also the
$J=1$ state, whose amplitude is given by Eq. (97) in appendix C, using Eq.
(98). Then, from Eq. (100) we find,

$\bar{g}_{2,1}\leq\frac{18\ln 2-13}{10\ln 2-7}\simeq 7.6\,.$ (65)

At order $O(s^{3})$ and $O(s^{4})$ we can also consider
$(\bar{g}_{3,1},\bar{g}_{3,0})$ and $(\bar{g}_{4,1},\bar{g}_{4,0})$. The
contributions from models of scalars and vectors are given by the lines going
respectively from the points $(0,1)$ and $(2,1)$ to the origin, illustrated in
Fig. 7 in red and blue.

The allowed regions for $J\geq 2$ are less trivial, and correspond to the
green areas in Fig. 7. Again the upper kink is associated with the $su$-model,
with its degenerate spectrum that makes $\bar{g}_{n,0}$ maximal. The
coefficients are

$(\bar{g}_{n,1},\bar{g}_{n,0})_{J>1\ su\rm-model}=(\approx 7.6,1)\,,$ (66)

as illustrated by the black dots in Fig. 7. For $m\in[0,M]$, the $su$-model
spans the minimum and maximum of the $J\geq 2$ region of
$(\bar{g}_{3,1},\bar{g}_{3,0})$ and $(\bar{g}_{4,1},\bar{g}_{4,0})$
respectively (black lines in the figures).

From Eq. (57) the minimal values for $\bar{g}_{n,1}/\bar{g}_{n,0}$ correspond
to models with the lowest possible spin. This would mean a theory with $J=2$
mesons only, but we have seen that this is incompatible with the Froissart-
Martin bound: $J\geq 2$ states are also needed to satisfy the null
constraints. Although these states can be made infinitely heavy (see section
2.3.3), they only decouple in $\bar{g}_{n,1}$ for $n\geq 4$. Therefore the
$J=2$ theory would give the minimal value for
$\bar{g}_{4,1}/\bar{g}_{4,0}=J(J+1)|_{J=2}=6$, but not for
$\bar{g}_{3,1}/\bar{g}_{3,0}$. The amplitude of a single spin-2 meson gives
from Eq. (27),

$(\bar{g}_{4,1},\bar{g}_{4,0})_{\rm spin-2}=(6/7,1/7)\,.$ (67)

Following a similar analysis as in Appendix B, we expect this to be also a
kink in the limit where all null constraints are taken into account. The
boundary between this kink Eq. (67) and the $J>1$ $su$-model has be obtained
numerically and is illustrated by the green dashed line in Fig. 7.

For $(\bar{g}_{3,1},\bar{g}_{3,0})$, we cannot assure the existence of a kink
at the $J=2$ value $(12/7,1/7)$, but the numerical bound (green dashed line)
seems to approach this point.

A consistent amplitude that interpolates between the $J>1$ $su$-model and the
spin-2 model is given by

$\widehat{\cal M}={\cal M}^{(su)}_{2}-\frac{5(13\ln
2-9)}{g^{2}_{f_{2}\pi\pi}}\left(\widehat{\cal M}^{(f_{2})}(m_{f_{2}}\to
m)-\widehat{\cal M}^{(f_{2})}\right)\,,$ (68)

and corresponds to the $J>1$ $su$-model with the spin-2 state of mass $m$
removed and added back at mass $m_{f_{2}}$. In the limit
$m_{\infty}\to\infty$, Eq. (68) leads to

$\displaystyle\bar{g}_{3,0}$ $\displaystyle=$
$\displaystyle\frac{20-23r^{3}+a(-65+75r^{3})}{140-143r^{2}+5a(-91+93r^{2})}\
,\ \ \ \
\bar{g}_{3,1}=\frac{240-245r^{3}+a(-780+798r^{3})}{140-143r^{2}+5a(-91+93r^{2})}\,,$
(69) $\displaystyle\bar{g}_{4,0}$ $\displaystyle=$
$\displaystyle\frac{20-23r^{4}+a(-65+75r^{4})}{140-143r^{2}+5a(-91+93r^{2})}\
,\ \ \ \
\bar{g}_{4,1}=\frac{120-125r^{4}+6a(-65+68r^{4})}{140-143r^{2}+5a(-91+93r^{2})}\,.$
(70)

where $r={m^{2}_{f_{2}}}/{m^{2}}$ and $a=1-\ln 2$, shown in Fig. 7 as magenta
lines, and compared to the predictions from the Lovelace-Shapiro amplitude
(yellow dot), that lies – again – outside of the area limited by the magenta
lines.

###  4.2 VMD and Spin-2 dominance

As we have seen, for amplitudes satisfying ${\cal M}/s^{2}\to 0$ at large
energies, spin-1 states decouple and our previous arguments for VMD no longer
hold. In this case, spin-1, spin-0 and higher spin contributions are
independent phenomenological quantities and VMD would be a mere accident of
Nature. In spite of this, we still expect the $f_{2}$ coupling to pions to be
larger than that for $J\geq 3$ states, in analogy to VMD. To quantify this
statement, we define

$\bar{g}^{2}_{i}=\frac{g^{2}_{i\pi\pi}}{g_{2,0}\,m_{i}^{4}}\,,$ (71)

and look for their largest allowed values compatible with positivity. For
scalars and vectors, that decouple from the null constraints, we have

$\bar{g}^{2}_{s},\bar{g}^{2}_{\rho}\leq 1\,.$ (72)

For the spin-2 state, instead, we obtain

$\bar{g}_{f_{2}}^{2}\lesssim 0.80\,.$ (73)

Similarly to the spin-1 meson in the case ${\cal M}/s^{2}\rightarrow 0$, this
latter bound is saturated by the $J>1$ $su$-model. The QCD experimental value
(extracted from data in [44]) is $\bar{g}_{f_{2}}^{2}=0.41\pm 0.25$ and is
much smaller than Eq. (73). This is due to the presence of the $\rho$-meson
that also contributes to the normalization factor $g_{2,0}$ and therefore
makes $\bar{g}_{f_{2}}^{2}$ smaller. For higher-spin states, we find, taking
$m_{i}/m_{f_{2}}\sim 1.3$,

$\bar{g}_{\rho_{3}}^{2}\lesssim 0.14\ ,\quad\bar{g}_{f_{4}}^{2}\lesssim
0.04\,.$ (74)

Interestingly, if we assume $\bar{g}_{\rho}\neq 0$ (which is not implied by
the null constraints), we can still find a bound on the maximal contribution
to the Wilson coefficients arising from $J>1$ states:

$\frac{\bar{g}_{2,1}}{\bar{g}_{2,1}|_{\rho}}\lesssim
1+0.17\,\frac{1-\bar{g}^{2}_{\rho}}{\bar{g}^{2}_{\rho}}\,.$ (75)

Here we have used that theories containing $J\geq 2$ mesons have $\langle
1/m^{4}\rangle_{J-\rm odd}/\langle 1/m^{4}\rangle\lesssim 0.17$, with the
bound saturated by the $J>1$ $su$-model. Taking for instance
$\bar{g}^{2}_{\rho}\sim 0.5$, we obtain from Eq. (75) that the $\rho$
contributes $\gtrsim 80\%$: it still dominates, in line with VMD.

##  5 Conclusions

We have studied pion scattering amplitudes in the large-$N_{c}$ limit, using
dispersion relations based on crossing-symmetry, unitarity and causality of
the QCD dynamics. Under different assumptions about the high-energy behavior
of amplitudes, but agnostic of the specific meson spectral density, we have
identified the allowed regions of parameter space for the low-energy Wilson
coefficients.

For amplitudes with ${\cal M}(s)\lesssim s$ at high-energy, building upon Ref.
[19], we have made progress in several directions. By separating theories with
$J=0$ mesons only, from theories with $J\geq 1$, we have revealed a pattern of
kinks that characterizes both the leading and the more irrelevant Wilson
coefficients. Some kinks are populated by known theories, such as the linear
sigma model (involving only a spin-0 meson in the UV completion) and the
(scalarless) $su$-model, that involves a degenerate spectrum of infinitely
many higher-spin resonances. The other kinks appear to be associated with a
more complex high-energy spectrum, but a numerical exploration of the
constraints converges slowly. In these points, we have been able to solve the
null constraints analytically, by translating them into a 1D moment problem,
and shown that at the kink sits a theory with a single light spin-1 resonance,
the $\rho$ meson.

There is one part of the boundary of the allowed region of Wilson coefficients
that still eludes analytic methods: here we have found that the scalar-
subtracted Lovelace-Shapiro (and part of the Coon) amplitude, provides the
best analytical approximation of the constraints – within the numerical bounds
but outside the region spanned by simpler theories. This suggests that
possibly another deformation of the Lovelace-Shapiro amplitude exists which
better approximates the entire boundary.

We have also shown how the soft high-energy behavior of amplitudes implies,
via positivity bounds, Vector Meson Dominance and explains the success of
holographic QCD. The reason is that although these models do not incorporate
higher-spin states, positivity constraints tell us that their contributions in
low-energy quantities have to be small.

For amplitudes that are not as soft at high-energy, but that still respect the
Froissart-Martin bound ${\cal M}(s)\lesssim s\log^{2}s$, we have found that
analogous results hold. Now, theories with states with $J=0$ or $J=1$ decouple
from the rest and can be studied in isolation. The remaining theories with
$J\geq 2$ also have kinks, populated by the (scalarless and vectorless)
$su$-model or by a theory with a single light spin-2 meson. In this context,
we have highlighted a version of “spin-2 meson dominance”.

The comparison between our results and large-$N_{c}$ lattice simulations (see
Fig. 4) emphasizes the complementarity between the two methods and lies down
an exciting symbiotic program that combines analytic and lattice methods to
corner QCD.

The physics captured by the chiral Lagrangian is rich and describes many
processes beyond $\pi\pi$ scattering: the program of cornering large-$N_{c}$
QCD with positivity bounds can move forward in many directions. An important
direction for further study would be processes involving pions interacting
with external sources, scalars or photons. Here we expect that including full
unitarity of the QCD amplitude, as done by Ref. [45] in the context of the
$a$-anomaly, would reveal an interesting interplay between the chiral anomaly,
the QED minimal coupling, and the parameters studied in this article. This
direction of research is interesting also in the framework of physics beyond
the Standard Model, in particular for the question of whether the Higgs is
composite or not. There, Ref. [46] proposed a set of power-counting rules (on
of which was dubbed _minimal coupling_) which are realized at weak coupling
and in string theory – it would be interesting to prove this in the wider
context of dispersion relations.

On a different front, it would be interesting to incorporate more
systematically real-world QCD data to shape the meson spectrum, or vice-versa,
see e.g. [47]. To this goal, it is essential to first understand finite
$N_{c}$ effects, as discussed in the single-flavor case in Refs. [4, 48] – a
task that we will leave for future work [49, 50].

From a more technical point of view, it is important to bring the analytic
methods we have used in this work into a more systematic tool to identify all
theories at the boundary, e.g. via a clever use of Lagrange multipliers and
resummations at all orders. This would allow us to understand if the Lovelace-
Shapiro amplitude (with subtracted scalars, or scalars and vectors) is indeed
extremal or not. Here it would be useful to further understand which
deformations of known amplitudes are consistent with unitarity, along the
lines of [51, 52, 53].

## Acknowledgments

F.R. thanks Jan Albert, Adam Falkowski, Felipe Figueroa, Davide Lombardo,
Marcos Marino, Leonardo Rastelli, Antonio Rodriguez, Jullian Sonner, Matteo
Tacchi and Piotr Tourkine for important discussions. F.S. thanks Filippo Nardi
for useful conversations. F.S. and F.R. acknowledge the support of the
European Consortium for Astroparticle Theory in the form of an Exchange Travel
Grant. We are grateful to the Mainz Institute for Theoretical Physics (MITP)
of the Cluster of Excellence PRISMA+ (Project ID 39083149), for its
hospitality and support during the workshop _Amplitudes meet BSM_. C.F. is
supported by the fellowship FPU18/04733 from the Spanish Ministry of Science,
Innovation and Universities. A.P. has been supported by the Catalan ICREA
Academia Program, and the grants 2014-SGR-1450,
PID2020-115845GB-I00/AEI/10.13039/501100011033. The work of F.R. is supported
by the Swiss National Science Foundation under grants no. 200021-205016 and
PP00P2-206149.

##  Appendix A Numerical Bootstrap

In this appendix, we briefly explain the numerical optimization procedure used
in this article, following [6, 7, 19]. From Eq. (17) we know that,

$\big{\langle}\mathcal{X}_{n,l}(m^{2},J)\big{\rangle}=0\
,\quad\big{\langle}\mathcal{Y}_{n,l}(m^{2},J)\big{\rangle}=0\ ,$ (76)

while Eq. (12) implies,

$g_{n+l,l}=\big{\langle}g_{n+l,l}(m^{2},J)\big{\rangle}\
,\qquad\text{where}\qquad
g_{n+l,l}(m^{2},J)=\frac{2^{l}}{l!}\frac{P_{J}^{(l)}(1)}{m^{2(n+l)}}\ .$ (77)

If we define the vectors

$\vec{v}_{1}=\begin{pmatrix}\ 1\ \\\ 0\\\ \vdots\\\ 0\end{pmatrix}\
,\quad\vec{v}_{n}=\begin{pmatrix}\ 0\ \\\ 1\\\ \vdots\\\ 0\end{pmatrix}\
,\quad\vec{v}_{\text{HE}}(m^{2},J)=\begin{pmatrix}\ -g_{1,0}(m^{2},J)M^{2}\
\\\ -g_{n,l}(m^{2},J)M^{2n}\\\ \mathcal{Y}_{2,1}(m^{2},J)\\\
\vdots\end{pmatrix},$ (78)

then the following equation holds

$g_{1,0}M^{2}\vec{v}_{1}+g_{n,l}M^{2n}\vec{v}_{n}+\big{\langle}\vec{v}_{\text{HE}}(m^{2},J)\big{\rangle}=0\
.$ (79)

This equation can be adjusted to include more Wilson coefficients.Now,
multiplying everything by a vector $\vec{\alpha}$ we can solve the following
optimization problem:

maximize $\displaystyle\qquad\qquad\vec{\alpha}\cdot\vec{v}_{1}$ such that
$\displaystyle\qquad\qquad\vec{\alpha}\cdot\vec{v}_{\text{HE}}(m^{2},J)\geq 0$
$\displaystyle\qquad\qquad\vec{\alpha}\cdot\vec{v}_{n}=\pm 1\ ,$

where positivity of the high-energy average and the normalization
$\vec{\alpha}\cdot\vec{v}_{n}=+1$ will yield the upper bound
$\tilde{g}_{n,l}\leq-\vec{\alpha}_{(+)}\cdot\vec{v}_{1}$, while
$\vec{\alpha}\cdot\vec{v}_{n}=-1$ the lower bound $\tilde{g}_{n,l}\geq\
\vec{\alpha}_{(-)}\cdot\vec{v}_{1}$.

A problem of this kind is solved using a semidefinite problem solver such as
SDPB [54]. Notice that in this procedure, we divided everything by the
positive term $g_{1,0}M^{2}$ and for the last term in Eq. (79) we reabsorbed
it in the definition of the high-energy average. Furthermore SDPB can only
impose the positivity condition in Eq. (A) at the level of polynomials,
therefore we must perform the change of variables $m^{2}\rightarrow
M^{2}(1+x)$ with $x\geq 0$ and absorb the common denominator once again in the
definition of the high-energy average. We can also implement a mass gap simply
by modifying the above change of variables to the one $m^{2}\rightarrow
M^{\prime 2}(1+x)$, where $M^{\prime}/M$ is the mass gap. Numerically, Eq.
(79) is evaluated on a grid in $x$ and $J$. The choice of the maximum value
$J_{\text{max}}$ depends on the number of null constraints we want to
consider; as more null constraints are included, a bigger $J_{\text{max}}$
must be chosen in order to ensure the convergence of the bounds.

In the specific case of constraining $\tilde{g}_{2,0},\tilde{g}_{2,1}$, we
input fixed values of $\tilde{g}_{2,0}$, and use

$\vec{v}_{1}+\tilde{g}_{2,0}\vec{v}_{2}+\big{\langle}\vec{v}_{\text{HE}}(m^{2},J)\big{\rangle}=0\
.$ (81)

Then we probe the allowed values of $\tilde{g}_{2,0}$ to find bounds on
$\tilde{g}_{2,1}$,

$(1+\tilde{g}_{2,0})\vec{v}_{1}+\tilde{g}_{2,1}\vec{v}_{2}+\big{\langle}\vec{v}_{\text{HE}}(m^{2},J)\big{\rangle}=0\
.$ (82)

To extract a $\rho$ state from the high-energy average and impose a mass gap
between the $\rho$ and the spectrum in the UV (as explained in the paragraph
above), we define $\vec{v}_{\rho}=\vec{v}_{\text{HE}}(m_{\rho}^{2},1)$. Then
we start by bounding $\tilde{g}_{2,0}$ via,

$\vec{v}_{1}+\tilde{g}_{\rho}^{2}\vec{v}_{\rho}+\tilde{g}_{2,0}\vec{v}_{2}+\big{\langle}\vec{v}_{\text{HE}}(m^{2},J)^{\prime}\big{\rangle}=0\
.$ (83)

We can both reabsorb the term $\tilde{g}_{\rho}^{2}\vec{v}_{\rho}$ in
$\vec{v}_{1}$ if we want to assume a value for the coupling, otherwise we can
add it to the high-energy average in the positivity condition Eq. (A). Then we
use the allowed values of $\tilde{g}_{2,0}$ to bound $\tilde{g}_{2,1}$ with
the equation

$(1+\tilde{g}_{2,0})\vec{v}_{1}+\tilde{g}_{\rho}^{2}\vec{v}_{\rho}+\tilde{g}_{2,1}\vec{v}_{2}+\big{\langle}\vec{v}_{\text{HE}}(m^{2},J)^{\prime}\big{\rangle}=0\
.$ (84)

The procedure described above gives the plots shown in Figs. 3,4. As we vary
$n_{\text{max}}$ (the number of null constraints included), features of the
exclusion plots vary.

##  Appendix B Minimum of $\mu_{0}$

_A_ measure that reproduces Eq. (41) is $d\tilde{\mu}(x)\equiv
4K_{0}(2\sqrt{x})dx$, where $K_{0}$ is the Bessel function of the second kind.
The (Stieltjes) moment problem associated with Eq. (42) is indeterminate, in
the sense that there exist multiple measures $d\mu(x)$ that give rise to this
sequence of moments. For instance, asymptotically
$K_{0}(\sqrt{x})\to~{}e^{-\sqrt{x}}$ has the same moments as
$e^{-\sqrt{x}}(1+w\cos\sqrt{x})$ for any $w$. For this reason, as far as
$\mu_{0}$ is concerned, we will treat the measure as unknown, and employ
$d\tilde{\mu}(x)$ only for $\mu_{n}$, $n\geq 1$.

We separate the integration domain $I\equiv[0,\infty[$ into the origin $0$ and
$I_{L}\equiv]0,L]$, where eventually we will take $L\to\infty$. Physically
this corresponds to separate out the contribution of states with infinite mass
but fixed spin $J\geq 1$. As noted already on page 3.1, heavy states
contribute more to lower, rather than higher, moments; in what follows we will
see that in this case infinitely heavy states ($x=0$) contribute only to the
first moment. Indeed, moments $\mu_{n}$ with $n\geq 1$ have no support on 0
and therefore $\mu_{n}=\mu_{n}^{I_{L}}$, where
$\mu_{n}^{I_{L}}=\int_{I_{L}}x^{n}d\mu(x)>0$ are also moments. Instead
$\mu_{0}$ has support in 0 (we call it $\mu_{0}^{0}\geq 0$), and we can write
$\mu_{0}=\mu_{0}^{0}+\mu_{0}^{I_{L}}\geq\mu_{0}^{I_{L}}$.

To prove that $\mu_{0}^{I_{L}}$ can be as small as $1$, we define a sequence
of functions built using positive powers of $x$,
$f_{n}(x)=\sum_{k=1}^{n}a_{k}x^{k}$, so that

$0\leq 1-f_{n}(x)\leq 1\quad x\in I_{L}\,,$ (85)

and such that $f_{n}$ converges pointwise to unity
$\lim_{n\to\infty}f_{n}(x)\to 1$ on $I_{L}$ (for instance, a candidate for
$f_{n}(x)$ is $1/x$ times the Taylor expansion of $1/x$ in $L$). Then,
integrating over the measure,

$0\leq\int_{I_{L}}(1-f_{n}(x))d\mu(x)=\mu_{0}^{I_{L}}-\int_{I_{L}}f_{n}(x)d\tilde{\mu}(x)\,,$
(86)

where the last integral can be written in terms of higher moments
$\sum_{k=1}^{n}a_{k}\mu_{k}^{I_{L}}$ and provides an expression for the
minimum of $\mu_{0}^{I_{L}}$ in terms of higher-moments only. Now we can use
the dominated convergence theorem to write

$\mu_{0}\geq\mu_{0}^{L}\geq\lim_{n\to\infty}\int_{I_{L}}f_{n}(x)d\tilde{\mu}(x)=\int_{I_{L}}d\tilde{\mu}(x)\,.$
(87)

Since $x$ corresponds to (the square of) impact parameter, in theories with a
mass gap, the measure $\mu(x)$ falls off exponentially at large $x$.

Since the measure decreases exponentially at large $x$, the limit $L\to\infty$
is regular, and the arguments still hold. For $L\to\infty$, the integral in
Eq. (87) is known, and we find, $\mu_{0}\geq\int d\tilde{\mu}(x)=2$ which is
equivalent to

$\tilde{g}_{2,0}\leq\frac{1}{3}\,.$ (88)

This implies that the kink must lie at
$(\tilde{g}_{2,1},\tilde{g}_{2,0})=({4}/{3},{1}/{3})$, corresponding to a
theory of a spin-1 particle at mass $M$, and higher-spin states at $\infty$,
as in Eq. (25).

Higher kinks. These arguments can be exploited also to obtain the kink
positions in the planes of other Wilson coefficients. In particular,
_considering only spins $J\geq 1$_ (i.e. singling out the spin-0
contribution), one finds that the allowed regions of
$(\tilde{g}_{n,1},\tilde{g}_{n,0})$ have also kinks, see Fig. 5. Again,
numerically the position of such kinks converges very slowly, but analytically
we can prove that their position is at,

$(\tilde{g}_{n,1},\tilde{g}_{n,0})=\left(2/3,1/3\right)\,,\qquad\text{for
}n>2\ .$ (89)

Indeed, from Eq. (57), we have for $J\geq 1$ that $\frac{g_{n,1}}{g_{n,0}}\geq
2$. Moving along the extremal line ${g_{n,1}}=2{g_{n,0}}$, corresponding to a
spin-1 state, the kink resides at the largest value of $g_{n,0}/g_{1,0}$. To
find this value one has to ask whether it is possible add to the spin-1 state
a spectrum of states that enhance $g_{n,0}/g_{1,0}$ without affecting the
ratio ${g_{n,1}}/{g_{n,0}}$ ($n=3,4,...$). This latter condition implies, from
Eq. (57), $\big{\langle}\frac{{\cal J}^{2}}{m^{2n}}\big{\rangle}\to 0$ and
therefore also $\big{\langle}{1}/{m^{2n}}\big{\rangle}\to 0$. This leads to

$\tilde{g}_{n,0}=\frac{1}{1+\frac{M^{2}}{g^{2}_{\rho\pi\pi}}\big{\langle}\frac{1}{m^{2}}\rangle}\,.$
(90)

Following the same reasoning as before, one can obtain that
$\frac{M^{2}}{g^{2}_{\rho\pi\pi}}\big{\langle}\frac{1}{\hat{m}^{2}}\rangle\geq
2$, and therefore $\tilde{g}_{n,0}\leq 1/3$, leading to Eq. (89). Also this
kink lies at the extremum of the spin-1 line.

##  Appendix C The $su$-models

Let us consider the most general theory of a degenerate spectrum that
contributes to the four-pion amplitude ${\cal M}(s,u)$ [6, 7]. This means that
all states have equal mass $m$, and therefore the denominator of this
amplitude is fixed to be ${\cal M}(s,u)\propto 1/((s-m^{2})(u-m^{2}))$. If we
further demand that Eq. (6a) and Eq. (6b) are satisfied for $k_{\rm min}=1$,
we are led to

${\cal
M}(s,u)=\frac{a_{1}m^{4}+a_{2}m^{2}(s+u)+a_{3}su}{(s-m^{2})(u-m^{2})}\,,$ (91)

where $a_{i}$ are constants. The Adler’s zero condition fixes $a_{1}=0$. Then,
aside from a global multiplicative factor, the amplitude has only one free
parameter. We can write it as

${\cal M}^{(su)}_{1}(s,u)=\frac{m^{2}(s+u)+\lambda su}{(s-m^{2})(u-m^{2})}\,,$
(92)

where the possible values of $\lambda$ are determined by unitarity. Indeed,
imposing the positivity of the residues of Eq. (92), we obtain

$-2\leq\lambda\leq\frac{2\ln 2-1}{1-\ln 2}\,.$ (93)

In the limiting case $\lambda=-2$, the residues of all $J>0$ states are zero,
and we are left with the scalar amplitude Eq. (22). In the other limit,

$\lambda=\frac{2\ln 2-1}{1-\ln 2}\simeq 1.26\,,$ (94)

the residue of the spin-0 state is zero, leading to an amplitude mediated by
an infinite tower of states of spin $J>0$ and mass $m$. We will refer to this
latter case as the $J>0$ $su$-model.

Expanding Eq. (92) for $s,u\ll m^{2}$, we can obtain the Wilson coefficients:

$g_{n,0}=\frac{1}{m^{2n}}\ ,\qquad g_{n,l}=\frac{2+\lambda}{m^{2n}}\ \ \
(n,l>0)\,.$ (95)

For Eq. (94), the Wilson coefficients, normalized as in Eq. (30) for $M=m$,
are given by

$\tilde{g}_{n,0}=1\ ,\qquad\tilde{g}_{n,l}=\frac{1}{1-\ln 2}\simeq 3.26\ \ \
(n>1,l>0)\,.$ (96)

These are shown in Figs. 3–5 as a black dot.

We can proceed in a similar way to construct the most general amplitude of
degenerate high-spin states, but now satisfying Eq. (6a) and Eq. (6b) for
$k_{\rm min}=2$. Imposing the Adler’s zero condition, such amplitude is given
by (up to a multiplicative constant factor)

${\cal
M}^{(su)}_{2}(s,u)=\frac{m^{2}(s+u)+\lambda_{1}su+\lambda_{2}(s^{2}+u^{2})}{(s-m^{2})(u-m^{2})}\,.$
(97)

The values of the constants $\lambda_{1}$ and $\lambda_{2}$ determine
different theories according to:

  * •

For $2+\lambda_{1}+2\lambda_{2}=0$, the residues for $J>1$ vanish and we have
a theory with only a spin-0 and spin-1 state.

  * •

For $\lambda_{2}=(2+\lambda_{1})\frac{6\ln 8-12}{25-36\ln 2}$, the residue for
the spin-1 state is zero and we have a theory with $J=0$ and $J>1$.

  * •

For $\lambda_{2}=-1+\frac{1-\lambda_{1}(\ln 4-2)}{\ln 16-1}$, the residue for
the spin-0 vanishes and we have a theory with spins $J>0$.

Furthermore, for $\lambda_{1}=-2$, $\lambda_{2}=0$, we recover the scalar
amplitude Eq. (22), while for $\lambda_{1}=\lambda_{2}=-{2}/{3}$ we obtain the
spin-1 amplitude Eq. (24). We are interested in the case where the amplitude
Eq. (97) contains only $J>1$ states which corresponds to taking

$\lambda_{1}=\frac{20\ln 2-13}{19-28\ln 2}\ ,\ \ \ \lambda_{2}=\frac{6\ln
8-12}{19-28\ln 2}\,.$ (98)

We will refer to this case as the $J>1$ $su$-model.

The Wilson coefficients of Eq. (97) are given by

$\begin{split}g_{1,0}=\frac{1}{m^{2}}\ ,\qquad
g_{2,1}=\frac{2+\lambda_{1}}{m^{4}}\ ,\qquad
g_{n,0}=\frac{1+\lambda_{2}}{m^{2n}}\ \ \ (n>1)\,,\\\
g_{n,1}=\frac{2+\lambda_{1}+\lambda_{2}}{m^{2n}}\ ,\qquad
g_{n,l}=\frac{2+\lambda_{1}+2\lambda_{2}}{m^{2n}}\ \ \ (n,l>1)\,.\end{split}$
(99)

For the case Eq. (98), normalizing the coefficients as given in Eq. (63), we
have

$\bar{g}_{n,0}=1\ ,\ \ \ \bar{g}_{n,1}=\frac{18\ln 2-13}{10\ln 2-7}\simeq
7.64\ ,\ \ \ \bar{g}_{n,l}=\frac{1}{7-10\ln 2}\simeq 14.59\ \ \ (n>2,l>1)\,.$
(100)

These are shown in Fig. 7 as a black dot.

###  C.1 Two-mass $su$-model

The $su$-models define part of the boundaries of the allowed regions of the
Wilson coefficients. To see this, we can deform the above $su$-models by
introducing an additional pole in the amplitude, i.e., ${\cal M}(s,u)\propto
1/((s-m^{2})(u-m^{2})(s-M^{2})(u-M^{2}))$. In this case the most general
amplitude can be written as

${\cal M}(s,u)={\cal M}^{(su)}_{1}(m)+\alpha{\cal
M}^{(su)}_{1}(M)+\frac{\beta\,s^{2}u^{2}}{(s-m^{2})(u-m^{2})(s-M^{2})(u-M^{2})}\
,$ (101)

that corresponds to two $su$-models (Eq. (92)) with mass $m$ and $M$
respectively, and an extra term. Apart from the masses, the amplitude has 4
parameters: the two $\lambda$ of the $su$-models, $\alpha$ and $\beta$. We are
interested in this model without the scalars. Removing the scalars in the two
$su$-models fixes the $\lambda$’s to the value Eq. (94). Removing the scalar
from the last term of Eq. (101) corresponds to adding to the amplitude the
term

$\beta\left[f(m,M)\left(\frac{1}{s-m^{2}}+\frac{1}{u-m^{2}}\right)+(M\leftrightarrow
m)\right]\,,$ (102)

where

$f(m,M)=\frac{m^{4}M^{2}+m^{6}\left(\ln
2-1\right)+m^{2}M^{4}\ln\frac{M^{2}}{m^{2}+M^{2}}}{\left(m^{2}-M^{2}\right)^{2}}\,.$
(103)

Requiring the positivity of the spectral function for the $J>0$ states in Eq.
(101) leads to $\beta\geq 0$.

Eq. (101) with Eq. (102) leads to

$\frac{\tilde{g}_{2,1}}{\tilde{g}_{2,0}}=\frac{3.26\left(\frac{1}{m^{4}}+\frac{a}{M^{4}}\right)}{\left(\frac{1}{m^{4}}+\frac{a}{M^{4}}\right)-\beta(\frac{f(m,M)}{m^{4}}+\frac{f(M,m)}{M^{4}})}\,.$
(104)

Since $\beta(\frac{f(m,M)}{m^{4}}+\frac{f(M,m)}{M^{4}})$ is a positive-
definite function, we see that the ratio ${\tilde{g}_{2,1}}/{\tilde{g}_{2,0}}$
is bounded from below by the $su$-model.

##  Appendix D The Lovelace-Shapiro amplitude

The Lovelace-Shapiro (LS) amplitude for the scattering of four pions is
defined as [20, 21]

${\cal
M}^{(\text{LS})}(s,u)=\frac{\Gamma(1-\alpha(s))\Gamma(1-\alpha(u))}{\Gamma(1-\alpha(s)-\alpha(u))}\
,$ (105)

where $\alpha(s)=\alpha_{0}+\alpha^{\prime}s$ is referred as the Regge
trajectory. We will fix the values of $\alpha_{0}$ and $\alpha^{\prime}$ by
requiring that Eq. (106) satisfies the Adler zero condition, ${\cal
M}^{(\text{LS})}(s,u)\to 0$ for $s,u\to 0$, and that the first pole of Eq.
(106) occurs for $s=m_{\rho}^{2}$. These two conditions lead to
$\alpha_{0}=1/2$ and $\alpha^{\prime}=1/(2m_{\rho}^{2})$ [55] and then we can
write

${\cal
M}^{(\text{LS})}(s,u)=\frac{\Gamma\left(\frac{1}{2}-\frac{s}{2m^{2}_{\rho}}\right)\Gamma\left(\frac{1}{2}-\frac{u}{2m^{2}_{\rho}}\right)}{\Gamma\left(\frac{t}{2m^{2}_{\rho}}\right)}\
.$ (106)

By looking at the poles of Eq. (106), one can see that the LS amplitude
corresponds to a theory of higher-spin states with masses

$m_{n}^{2}=m_{\rho}^{2}(2n+1)\,,\ \ n=0,1,2,...\,.$ (107)

For a given $n$, there are at most $n+1$ states with spin $J=0,1,...,n+1$.
Furthermore, Eq. (106) satisfies the condition Eq. (6a) and Eq. (6b) with
$k_{\rm min}=1$.

The first Wilson coefficients arising from Eq. (106) in a low-energy expansion
are given by

$\displaystyle g_{1,0}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2m_{\rho}^{2}}\ ,\qquad
g_{2,0}=\frac{1}{2}g_{2,1}=\frac{\pi\ln 2}{2m_{\rho}^{4}}\ ,\qquad
g_{3,0}=\frac{\pi^{3}+12\pi\ln^{2}2}{48m_{\rho}^{6}}\ ,$ (108) $\displaystyle
g_{3,1}$ $\displaystyle=$ $\displaystyle\frac{3\pi\ln^{2}2}{4m_{\rho}^{6}}\
,\qquad g_{4,0}=\frac{\pi\left(\pi^{2}\ln
2+4\ln^{3}2+6\zeta(3)\right)}{48m_{\rho}^{8}}\ ,$ (109) $\displaystyle
g_{4,1}$ $\displaystyle=$ $\displaystyle\frac{\pi\left(\pi^{2}\ln
2+16\ln^{3}2+3\zeta(3)\right)}{48m_{\rho}^{8}}\ ,\qquad
g_{4,2}=\frac{\pi\left(4\ln^{3}2-\zeta(3)\right)}{8m_{\rho}^{8}}\,,$ (110)

with $\zeta$ the Riemann zeta function. For the normalized Wilson coefficients
defined in Eq. (30) we have, taking $M=m_{\rho}$,

$\tilde{g}_{2,0}\simeq 0.69\,,\ \tilde{g}_{2,1}\simeq 1.39\,,\
\tilde{g}_{3,0}\simeq 0.65\,,\ \tilde{g}_{3,1}\simeq 0.72\,,\
\tilde{g}_{4,0}\simeq 0.64\,,\ \tilde{g}_{4,1}\simeq 0.66\,,\
\tilde{g}_{4,2}\simeq 0.03\,,$ (111)

while for the normalized coefficients in Eq. (63), we have

$\bar{g}_{2,1}\simeq 2\,,\ \bar{g}_{3,0}\simeq 0.94\,,\ \bar{g}_{3,1}\simeq
1.04\,,\ \bar{g}_{4,0}\simeq 0.92\,,\ \bar{g}_{4,1}\simeq 0.95\,,\
\bar{g}_{4,2}\simeq 0.05\,.$ (112)

Since a theory of scalars provides a consistent UV completion of the pion
amplitude ${\cal M}$, satisfying Eq. (6a) and Eq. (6b) with $k_{\rm min}=1$,
we can find a new consistent amplitude by subtracting the scalars from Eq.
(106). This leads to

${\cal M}^{(\text{LS})}_{J>0}(s,u)={\cal
M}^{(\text{LS})}(s,u)-\sum_{n=0}^{\infty}\left[\frac{m_{n}^{2}}{s-m_{n}^{2}}\kappa_{s,0}^{\text{LS}}+(s\leftrightarrow
u)\right]\,,$ (113)

with

$\kappa_{s,J}^{\text{LS}}=\frac{2J+1}{2}\int_{-1}^{1}dx\
P_{J}(x)\underset{s=m_{n}^{2}}{\text{Res}}\left[{\cal
M}^{(\text{LS})}(s,u(x))\right]\,,$ (114)

where $u(x)=-s(1-x)/2$. From Eq. (113), we obtain by expanding at small $s,u$:

$\tilde{g}_{2,0}\simeq 0.55\,,\ \tilde{g}_{2,1}\simeq 2.05\,,\
\tilde{g}_{3,0}\simeq 0.49\,,\ \tilde{g}_{3,1}\simeq 1.07\,,\
\tilde{g}_{4,0}\simeq 0.48\,,\ \tilde{g}_{4,1}\simeq 0.97\,,\
\tilde{g}_{4,2}\simeq 0.05\,.$ (115)

In a similar way, we can also remove the (infinite) spin-1 states of Eq. (113)
to obtain an amplitude that still satisfies the Froissart-Martin condition,
Eq. (6a) and Eq. (6b) with $k_{\rm min}=2$:

${\cal M}^{(\text{LS})}_{J>1}(s,u)={\cal
M}^{(\text{LS})}_{J>0}(s,u)-\sum_{n=0}^{\infty}\left[\frac{m_{n}^{2}+2u}{s-m_{n}^{2}}\kappa_{s,1}^{\text{LS}}+(s\leftrightarrow
u)\right]\ .$ (116)

From Eq. (116) we obtain

$\bar{g}_{2,1}\simeq 0.97\,,\ \bar{g}_{3,0}\simeq 0.71\,,\ \bar{g}_{3,1}\simeq
5.99\ ,\ \bar{g}_{4,0}\simeq 0.60\,,\ \bar{g}_{4,1}\simeq 4.18\,,\
\bar{g}_{4,2}\simeq 8.15\,.$ (117)

##  Appendix E The Coon amplitude

The Lovelace-Shapiro amplitude presented in Appendix D can be generalized to a
larger class of amplitudes depending on an additional parameter $q$. This is
the so-called Coon amplitude, which was first proposed in [22]999The idea of
the Coon amplitude goes back to an earlier work by Coon [56], where he defined
a generalization of the Veneziano amplitude which was slightly different from
Eq. (118). Shortly after that he proposed the Coon version of Lovelace-Shapiro
amplitude together with Sukhatme and Tran Thanh Van in [22].:

${\cal
M}_{q}(s,u)=C(\sigma,\tau,q)\prod_{n=0}^{\infty}\frac{\left(1-q^{n+1}\right)\left(\sigma\tau-q^{n+1}\right)}{\left(\sigma-q^{n+1}\right)\left(\tau-q^{n+1}\right)}\
,$ (118)

where $\sigma=1+(q-1)(\alpha_{0}+\alpha^{\prime}s)$ and
$\tau=1+(q-1)(\alpha_{0}+\alpha^{\prime}u)$. As explained in Appendix D, we
take $\alpha_{0}=1/2$ and $\alpha^{\prime}=1/(2m_{\rho}^{2})$. The parameter
$q$ takes values between 0 and 1, and in the limit $q\rightarrow 1$ we recover
the LS amplitude Eq. (106). There is some freedom in the choice of the
prefactor $C$, as long as it satisfies $\lim_{q\rightarrow
1}C(\sigma,\tau,q)=1$.

The Coon amplitude has an infinite number of simple poles at

$s_{n}=m_{\rho}^{2}\frac{1+q-2q^{n+1}}{1-q}\ ,\qquad n=0,1,2,...\ .$ (119)

The corresponding residues are

$\underset{s=s_{n}}{\text{Res}}\ {\cal M}_{q}(s,u)=C(\sigma_{n},\tau,q)\
\frac{2q^{n+1}}{1-q}\ \frac{\tau-1}{\tau^{n+1}}\
m_{\rho}^{2}\prod_{l=0}^{n-1}\frac{(\tau-q^{l-n})}{(1-q^{l-n})}\ ,$ (120)

where $\sigma_{n}=\sigma(s=s_{n})$. It is important to remark that the
spectrum has an accumulation point at
$s_{*}=\lim_{n\rightarrow\infty}s_{n}=m_{\rho}^{2}\frac{1+q}{1-q}$. In the
limit $q\rightarrow 1$, the accumulation point is located at infinity and we
recover the evenly-spaced spectrum of the LS amplitude.

It is customary to fix the prefactor $C(\sigma,\tau,q)$ with the further
assumption that the residues of the Coon amplitude are polynomials in $u$,
since it is believed that non-polynomial residues lead to problems with the
locality of the theory. The prefactor is in this case set to

$C(\sigma,\tau,q)=q^{\frac{\ln\sigma\ln\tau}{\ln q\ln q}}\ ,$ (121)

which reduces to $C(\sigma_{n},\tau,q)=\tau^{n+1}$ at the $s_{n}$ pole. This
term cancels the factor $\tau^{n+1}$ in the denominator of Eq. (120) and
ensures that the residues are polynomials. In this case, we have that for any
$n$, there are $n+1$ states with spin $J=0,1,...,n+1$, as in the LS amplitude.

Using the prefactor Eq. (121) makes however the Coon amplitude Eq. (118) non-
meromorphic. In addition to the simple poles, there is a branch cut starting
at the accumulation point $s_{*}$. Although the physical meaning of this kind
of singularities is unclear, amplitudes with branch cuts can still obey the
requirements of unitarity, crossing symmetry and Regge boundedness, so it is
interesting to include them in our study.

Regarding the high-energy behavior, the amplitude with prefactor Eq. (121)
grows at fixed $u$ like ${\cal M}_{q}(s,u)\sim f(u)\ s^{\ln\tau/\ln q}$. For
negative $u$, $\ln\tau/\ln q<0.5$, so the amplitude obeys Eq. (6a) for $k_{\rm
min}=1$. At fixed $t$, the amplitude grows like ${\cal M}_{q}(s,u)\sim
s^{\ln((1-q)\alpha^{\prime}s)/\ln q}$. Since $\ln((1-q)\alpha^{\prime}s)/\ln
q<0.5$, it also obeys Eq. (6b) for $k_{\rm min}=1$.

The last point to address is the unitarity of the Coon amplitude. We have
found that the $J=0$ states have negative residues for all $n>0$, making the
amplitude non-unitary. This result is in agreement with the early study
[57]101010Some recent works [58, 59, 60] on the Veneziano version of the Coon
amplitude have shown that it is unitary in $D=4$ dimensions. Our result agrees
with the results of Coon and Yu [57], who considered the Lovelace-Shapiro
version of the Coon amplitude., which pointed out the presence of ghosts. On
the other hand, we have found that the rest of the $J$ states have positive
residues inside the scope of our numerical searches. As for the branch cut
discontinuity, it can also be expanded in partial waves according to Eq. (8).
We have obtained that the spectral density $\rho_{J}(s)$ is positive for all
$J$ except from $J=0$. Due to this problems, we will not discuss this case
further.

On the other hand, relaxing the assumption of polynomial residues, we can
simply take $C(\sigma,\tau,q)=1$, and the amplitude becomes meromorphic (there
is no branch cut). We now have an infinite number of spins present for each
$n$. At large $s$, taking both fixed $u$ and $t$, we have ${\cal
M}_{q}(s,u)\sim{\text{c}onstant}$. Therefore it satisfies both Eq. (6a) and
Eq. (6b) for $k_{\rm min}=1$. For $C(\sigma,\tau,q)=1$ we have also found that
the only negative residue corresponds to the $J=0$, $n=1$ state only in the
small region $0.94\lesssim q<1$.

The contribution of the Coon amplitude with $C=1$ is shown in Fig. 3 for
$q\in[0,1]$. We also show the case in which all scalars are removed from the
spectrum. In the limit $q\rightarrow 1$ the Coon amplitude approaches the
$J>0$ Lovelace-Shapiro model, while for $q\rightarrow 1$, the Coon amplitude
is dominated by the $n=0$ level (the $n>0$ levels decouple), giving the $J>0$
$su$-model.

## References

  * Adams et al. [2006] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, JHEP 10, 014 (2006), arXiv: hep-th/0602178.
  * Arkani-Hamed et al. [2021] N. Arkani-Hamed, T.-C. Huang, and Y.-T. Huang, JHEP 05, 259 (2021), arXiv: 2012.15849.
  * de Rham et al. [2017] C. de Rham, S. Melville, A. J. Tolley, and S.-Y. Zhou, Phys. Rev. D96, 081702 (2017), arXiv: 1702.06134.
  * Bellazzini et al. [2021a] B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau, and F. Riva, Phys. Rev. D 104, 036006 (2021a), arXiv: 2011.00037.
  * Tolley et al. [2020] A. J. Tolley, Z.-Y. Wang, and S.-Y. Zhou (2020), arXiv: 2011.02400.
  * Caron-Huot and Van Duong [2021] S. Caron-Huot and V. Van Duong, JHEP 05, 280 (2021), arXiv: 2011.02957.
  * Caron-Huot et al. [2021] S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin (2021), arXiv: 2102.08951.
  * Pham and Truong [1985] T. N. Pham and T. N. Truong, Phys. Rev. D31, 3027 (1985).
  * Ananthanarayan et al. [1995] B. Ananthanarayan, D. Toublan, and G. Wanders, Phys. Rev. D 51, 1093 (1995), arXiv: hep-ph/9410302.
  * Pennington and Portoles [1995] M. R. Pennington and J. Portoles, Phys. Lett. B 344, 399 (1995), arXiv: hep-ph/9409426.
  * Bijnens et al. [1997] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, and M. E. Sainio, Nucl. Phys. B 508, 263 (1997), [Erratum: Nucl.Phys.B 517, 639–639 (1998)], arXiv: hep-ph/9707291.
  * Ananthanarayan et al. [2001] B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler, Phys. Rept. 353, 207 (2001), arXiv: hep-ph/0005297.
  * Colangelo et al. [2001] G. Colangelo, J. Gasser, and H. Leutwyler, Nucl. Phys. B 603, 125 (2001), arXiv: hep-ph/0103088.
  * Manohar and Mateu [2008] A. V. Manohar and V. Mateu, Phys. Rev. D 77, 094019 (2008), arXiv: 0801.3222.
  * Guerrieri et al. [2019] A. L. Guerrieri, J. Penedones, and P. Vieira, Phys. Rev. Lett. 122, 241604 (2019), arXiv: 1810.12849.
  * Guerrieri et al. [2020] A. Guerrieri, J. Penedones, and P. Vieira (2020), arXiv: 2011.02802.
  * ’t Hooft [1974] G. ’t Hooft, Nucl. Phys. B 72, 461 (1974).
  * Witten [1979] E. Witten, Nucl. Phys. B 160, 57 (1979).
  * Albert and Rastelli [2022] J. Albert and L. Rastelli, JHEP 08, 151 (2022), arXiv: 2203.11950.
  * Lovelace [1968] C. Lovelace, Phys. Lett. B 28, 264 (1968).
  * Shapiro [1969] J. A. Shapiro, Phys. Rev. 179, 1345 (1969).
  * Coon et al. [1973] D. D. Coon, U. P. Sukhatme, and J. Tran Thanh Van, Phys. Lett. B 45, 287 (1973).
  * Sakurai [1969] J. J. Sakurai, _Currents and mesons_ (Chicago U. P., 1969).
  * Ecker et al. [1989] G. Ecker, J. Gasser, A. Pich, and E. de Rafael, Nucl. Phys. B 321, 311 (1989).
  * Erlich et al. [2005] J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett. 95, 261602 (2005), arXiv: hep-ph/0501128.
  * Da Rold and Pomarol [2005] L. Da Rold and A. Pomarol, Nucl. Phys. B 721, 79 (2005), arXiv: hep-ph/0501218.
  * Froissart [1961] M. Froissart, Phys. Rev. 123, 1053 (1961).
  * Martin [1963] A. Martin, Phys. Rev. 129, 1432 (1963).
  * Jin and Martin [1964] Y. S. Jin and A. Martin, Phys. Rev. 135, B1375 (1964).
  * Häring and Zhiboedov [2022] K. Häring and A. Zhiboedov (2022), arXiv: 2202.08280.
  * Herrero-Valea et al. [2021] M. Herrero-Valea, R. Santos-Garcia, and A. Tokareva, Phys. Rev. D 104, 085022 (2021), arXiv: 2011.11652.
  * Bellazzini et al. [2019] B. Bellazzini, F. Riva, J. Serra, and F. Sgarlata, JHEP 10, 189 (2019), arXiv: 1903.08664.
  * El-Showk et al. [2012] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Phys. Rev. D 86, 025022 (2012), arXiv: 1203.6064.
  * Lasserre [2009] J. Lasserre, Imperial College Press (2009).
  * Basor et al. [2001] E. L. Basor, Y. Chen, and H. Widom, Journal of Functional Analysis 179, 214 (2001), ISSN 0022-1236, URL https://www.sciencedirect.com/science/article/pii/S0022123600936723.
  * Toublan and Verbaarschot [2001] D. Toublan and J. J. M. Verbaarschot, Nucl. Phys. B 603, 343 (2001), arXiv: hep-th/0012144.
  * Akemann [2016] G. Akemann (2016), arXiv: 1603.06011.
  * Bijnens et al. [1994] J. Bijnens, G. Colangelo, and J. Gasser, Nucl. Phys. B 427, 427 (1994), arXiv: hep-ph/9403390.
  * Bali et al. [2013] G. S. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio, B. Lucini, and M. Panero, JHEP 06, 071 (2013), arXiv: 1304.4437.
  * Hernández et al. [2019] P. Hernández, C. Pena, and F. Romero-López, Eur. Phys. J. C 79, 865 (2019), arXiv: 1907.11511.
  * Baeza-Ballesteros et al. [2022] J. Baeza-Ballesteros, P. Hernández, and F. Romero-López, JHEP 06, 049 (2022), arXiv: 2202.02291.
  * Gasser and Leutwyler [1985] J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 (1985).
  * Panico and Wulzer [2007] G. Panico and A. Wulzer, JHEP 05, 060 (2007), arXiv: hep-th/0703287.
  * Workman et al. [2022] R. L. Workman et al. (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Karateev et al. [2022] D. Karateev, J. Marucha, J. a. Penedones, and B. Sahoo (2022), arXiv: 2204.01786.
  * Giudice et al. [2007] G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, JHEP 06, 045 (2007), arXiv: hep-ph/0703164.
  * Caprini et al. [2012] I. Caprini, G. Colangelo, and H. Leutwyler, Eur. Phys. J. C72, 1860 (2012), arXiv: 1111.7160.
  * Bellazzini et al. [2021b] B. Bellazzini, M. Riembau, and F. Riva (2021b), arXiv: 2112.12561.
  * Lombardo [2022] D. M. Lombardo, Ph.D. thesis, Geneva U., Dept. Theor. Phys. (2022).
  * Falkowski et al. [In Preparation] A. Falkowski, D. Lombardo, F. Riva, and A. Rodriguez (In Preparation).
  * Cheung and Remmen [2022] C. Cheung and G. N. Remmen (2022), arXiv: 2210.12163.
  * Geiser and Lindwasser [2022a] N. Geiser and L. W. Lindwasser (2022a), arXiv: 2210.14920.
  * Huang and Remmen [2022] Y.-t. Huang and G. N. Remmen, Phys. Rev. D 106, L021902 (2022), arXiv: 2203.00696.
  * Simmons-Duffin [2015] D. Simmons-Duffin, JHEP 06, 174 (2015), arXiv: 1502.02033.
  * Bianchi et al. [2021] M. Bianchi, D. Consoli, and P. Di Vecchia, JHEP 03, 119 (2021), arXiv: 2002.05419.
  * Coon [1969] D. D. Coon, Phys. Lett. B 29, 669 (1969).
  * Coon and Yu [1974] D. D. Coon and S. Yu, Phys. Rev. D 10, 3780 (1974).
  * Figueroa and Tourkine [2022] F. Figueroa and P. Tourkine, Phys. Rev. Lett. 129, 121602 (2022), arXiv: 2201.12331.
  * Geiser and Lindwasser [2022b] N. Geiser and L. W. Lindwasser (2022b), arXiv: 2207.08855.
  * Chakravarty et al. [2022] J. Chakravarty, P. Maity, and A. Mishra, JHEP 10, 043 (2022), arXiv: 2208.02735.