Bootstrapping the Chiral Anomaly at Large $N_{c}$

Teng Ma1,2, Alex Pomarol1,3 and Francesco Sciotti1

1IFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona
2 ICTP-AP, University of Chinese Academy of Sciences, 100190 Beijing, China
3Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona

##  1 Introduction

In the large-$N_{c}$ limit strongly-coupled gauge theories find a dual
description as weakly-coupled theories of mesons and glueballs [1, 2]. This
has allowed to obtain valuable information on QCD in the strongly-coupled
regime. In spite of this, the fact that the theory contains infinite number of
states with all spins has made difficult to get quantitative predictions on
physical quantities.

Recently [3, 4], it has been shown that a better quantitative understanding of
large-$N_{c}$ theories can be achieved by requiring consistency conditions,
such as unitarity and causality, to the scattering amplitudes [5, 6, 7, 8, 9,
10, 11, 12, 13]. Thanks to the simple analytical structure of the scattering
amplitudes at large-$N_{c}$, positivity constraints have provided precise and
strong bounds on low-energy quantities, such as Wilson coefficients of the
chiral Lagrangian as well as meson couplings. High-spin states contributions
to these low-energy physical quantities have shown to be small, giving an
understanding of the phenomenological successes of Vector Meson Dominance and
holographic QCD [4].

We extend these previous analysis to scattering amplitudes of Goldstones
($\pi$ and $\eta$) interacting with external gauge bosons ($W$). The main
purpose is to access information related to the chiral anomaly and other
physical quantities such as pion polarizabilities. In particular, we will
analyze the amplitudes $W\pi\to W\pi$ and $W\pi\to\eta\pi$, this later being
proportional, at low-energies, to the chiral anomaly. We will understand the
meson exchange selection rules that apply to these amplitudes and that will
help us to derive the sign of their on-shell residues. By demanding a good
high-energy behaviour of the scattering amplitudes, we will derive dispersion
relations that will allow us to obtain interesting bounds on low-energy
quantities.111Previous studies in the context of QCD can be found in [14, 15,
16, 17, 18, 19, 20, 21, 22].

Our main result will be the derivation of an upper bound on the coefficient of
the chiral anomaly $\kappa$:

$\frac{\kappa}{\sqrt{{\cal P}/F^{2}_{\pi}}}\leq\sqrt{\frac{1}{2}}\,,$ (1)

where $F_{\pi}$ is the pion decay constant, and ${\cal P}$, to be specified
later, is related to pion polarizabilities. This bound provides a non-trivial
connection between low-energy physical quantities and UV properties of the
theories.222Bounds on the $a$-anomaly from scattering amplitudes requirements
have been derived in [23, 24]. On the other hand, for phenomenological models
for QCD, such as a holographic or NJL models, the above upper bound can
provide a valuable consistency check, as well as for lattice simulations.

The analysis described here can also be useful to derive constraints on a
large number of low-energy physical constants, specially to those related to
the electromagnetic properties of pions and other mesons.

The paper is organized as follows. In Sec. 2 we discuss the properties of
amplitudes for pions and gauge bosons and derive the dispersion relations and
null constraints. In Sec. 3, we analytically derive an upper bound on the
chiral anomaly coefficient. We conclude in Sec. 4. The appendices contain
details about the classification of mesons and the signs of residues (A), pion
polarizabilities (B), numerical bounds on Wilson coefficients (C), and
$su$-models (D).

##  2 Amplitudes for pions and external gauge bosons in the large-$N_{c}$
limit

We are interested in studying $SU(N_{c})$ gauge theories in the large-$N_{c}$
limit. We will consider that the theory also contains $N_{f}$ massless quarks
($q_{L},q_{R}$) in the fundamental representation of $SU(N_{c})$, with the
following pattern of chiral symmetry breaking: $U(N_{f})_{L}\times
U(N_{f})_{R}\rightarrow SU(N_{f})\times U(1)$. The massless Goldstone bosons
associated with the breaking of this global symmetry are the pions ($\pi^{a}$
with $a=1,...,N_{f}^{2}-1$ in the adjoint representation of $SU(N_{f})$) and
the $\eta$ (singlet). The chiral axial symmetry $U(1)_{A}$ is anomalous but in
the large-$N_{c}$ limit the corresponding Goldstone, the $\eta$, remains
massless.

As usual, we will introduce the coupling to external gauge bosons by gauging
the global $SU(N_{f})\times U(1)$ symmetry. These gauge bosons will be
considered (non-propagating) external fields, sitting in the adjoint and
singlet representations.

For simplicity, we will concentrate in this paper in the $N_{f}=2$ case, but
the arguments we make can be extended straightforwardly to a general $N_{f}$.
For $N_{f}=2$ the pions and gauge bosons are isospin triplets, $\pi^{a}$ and
$W^{a}$, and isospin singlets, the $\eta$ and $B$.333We normalize the gauge
bosons as $W^{a}T^{a}+B\,\mathbb{I}/2$, where ${\rm Tr\hskip
2.0pt}[T^{a}T^{b}]=1/2$, and the gauge coupling $g=1$. We are interested in
the following amplitudes:

(2) $\displaystyle(a)\qquad\qquad\quad\ $ $\displaystyle(b)$
$\displaystyle\qquad\qquad\quad\ \ (c)$

The amplitude (a) is important as, at low-energies, becomes proportional to
the chiral anomaly coefficient that we want to bootstrap. For this purpose, as
we will see, we will also need the amplitudes (b) and (c). The amplitude (c)
was already discussed in detail in [3, 4]. In this section we will then only
discuss the amplitudes (b) and (a) in the large-$N_{c}$ limit for the
different polarizations of the gauge bosons $\lambda_{W}=\pm 1$. In the
following, we will derive the relevant dispersion relations necessary to
obtain sum rules for the chiral anomaly coefficient and other Wilson
coefficients, as well as null constraints necessary for obtaining bounds.

###  2.1 Exchanged meson states

In the large-$N_{c}$ limit, a $SU(N_{c})$ gauge theory reduces to a theory of
weakly-coupled mesons, whose trilinear couplings scale as $\sim
1/\sqrt{N_{c}}$ [1, 2]. Amplitudes are then mediated by tree-level meson
exchange. In the amplitudes (2), the exchanged mesons are colorless $q\bar{q}$
states that, as in the quark model, can be classified according to their
quantum numbers: Isospin ($I$), $G-$parity ($G$), parity ($P$) and spin ($J$).
The relation between these quantum numbers is discussed in Appendix
A.444Fixing $I$, $G$ and $P$, the spin $J$ is fixed to either be even or odd,
as explicitly shown in Table 2.1. We have six types of mesons as shown in
Table 2.1. It is important to say that in the large-$N_{c}$ limit, each $I=1$
state (left column) has an $I=0$ associated state (the one next in the right
column) of equal mass and related couplings due to the underlying $U(2)$
symmetry [1, 2].

[corners,hvlines]—c—c——c—c—c— yellow!201-2,1-3
blue!202-1,3-1,4-1,5-1,6-1,7-1,2-4,3-4,4-4,5-4,6-4,7-4 green!203-2,3-3,6-3,6-2
& Isospin $I=0$ Isospin $I=1$ $n$

3-1$G=+1$ $J_{\text{odd}}^{+}$ ($f_{J}$) $J_{\text{odd}}^{+}$ ($a_{J}$)
3-1$G=-1$ 1

$J_{\text{even}}^{+}$ ($f_{J}$) $J_{\text{even}}^{+}$ ($a_{J}$) 2

$J_{\text{even}}^{-}$ ($\eta_{J}$) $J_{\text{even}}^{-}$ ($\pi_{J}$) 3

3-1$G=-1$ $J_{\text{odd}}^{+}$ ($h_{J}$) $J_{\text{odd}}^{+}$ ($b_{J}$)
3-1$G=+1$ 4

$J_{\text{odd}}^{-}$ ($\omega_{J}$) $J_{\text{odd}}^{-}$ ($\rho_{J}$) 5

$J_{\text{even}}^{-}$ ($\omega_{J}$) $J_{\text{even}}^{-}$ ($\rho_{J}$) 6

Table 1: List of $q\bar{q}$ states classified in terms of their Isospin,
$G-$parity and $J^{P}$. We also provide the names as they are generally
referred to in QCD [25]. In green we highlight the states that enter in the
amplitude $W\pi\rightarrow\eta\pi$ which is related to the chiral anomaly.

###  2.2 The $\pi W\rightarrow\pi W$ Amplitudes

Let us start with the amplitude (b). For $N_{f}=2$, the most general structure
we can write for this amplitude takes the form

${\cal
M}(W^{a},\pi^{b},W^{c},\pi^{d})=A_{t}(s,u)\delta_{t}+\tilde{A}(s,u)\delta_{s}+\tilde{A}(u,s)\delta_{u}\,,$
(3)

where $s=(p_{a}+p_{b})^{2}$, $t=(p_{a}-p_{c})^{2}$, $u=(p_{a}-p_{d})^{2}$, and
$\delta_{s}=\delta_{ab}\delta_{cd}\ ,\ \delta_{t}=\delta_{ac}\delta_{bd}\ ,\
\delta_{u}=\delta_{ad}\delta_{cb}$. We have demanded the amplitude to be
invariant under the exchange of the two pions that also requires $A_{t}(s,u)$
to be $s\leftrightarrow u$ symmetric.

In the large-$N_{c}$ limit this amplitude is mediated by tree-level exchanges
of mesons, either in the $s-$channel, $t-$channel and/or $u-$channel. Since
$q\bar{q}$ states cannot have $I=2$, it will be important to focus in $I=2$
eigen-amplitudes. These are given by

$M^{I=2}_{s}=A_{t}(s,u)+\tilde{A}(u,s)\,,\qquad
M^{I=2}_{t}=\tilde{A}(s,u)+\tilde{A}(u,s)\,,\qquad
M^{I=2}_{u}=A_{t}(s,u)+\tilde{A}(s,u)\,,$ (4)

respectively for the $s$, $t$ and $u-$channel. These amplitudes cannot have
mesons exchanging in the $s$, $t$ and $u-$channel respectively.

In the amplitude Eq. (3) the $s-$channel and $u-$channel exchanged meson $i$
couples to $W\pi$, while the $t-$channel exchanged meson couples to $WW$ and
$W\pi$:

$\pi$$\pi$$W$$W$ $\propto g_{W\pi i}^{2}$ $\pi$$\pi$$W$$W$ $\propto g_{W\pi
i}^{2}$ $\pi$$\pi$$W$$W$ $\propto g_{WWi}g_{\pi\pi i}$ (5)

Since the couplings $g_{WWi}$ do not appear in the amplitude $W\pi\eta\pi$
(the one related to the chiral anomaly), we will restrict to situations in
which the $t-$channel meson exchange is not present such that we will not have
to deal with these couplings $g_{WWi}$ any longer. This can be achieved by
either taking $t-$fixed or working with $M^{I=2}_{t}$. This leaves only 3
possibilities to consider:

  1. 1.

$M^{I=2}_{t}$ at $u-$fixed ,

  2. 2.

$M^{I=2}_{t}$ at $t-$fixed ,

  3. 3.

$M^{I=2}_{u}$ at $t-$fixed .

Notice that we have not included $M^{I=2}_{s}$ since it contains the same
information as $M^{I=2}_{u}$ as far as our positivity arguments are concerned.

It is also of central importance to understand the sign of the on-shell
couplings of the meson being exchanged, or, equivalently, the sign of the
residues at the mass poles. Defining these as $R_{I=1,0}$ for the $I=1,0$
mesons respectively, we have

where the sum over $i$ runs over all the possible intermediate states. Knowing
that the amplitudes Eq. (4) cannot have poles at the $s$, $t$ and $u$–channel
respectively, we can derive the relative sign of the residues in the $s$ and
$u-$channel, which we show in Table 2.

Meson channel | $-A_{t}(s,u)$ | $-\tilde{A}(s,u)$ | $-\tilde{A}(u,s)$
---|---|---|---
$s-$channel | $R_{I=1}$ | $R_{I=0}$ | $-R_{I=1}$
$u-$channel | $R_{I=1}$ | $-R_{I=1}$ | $R_{I=0}$
Table 2: Residues in the $s$ and $u-$channel of the amplitudes $A_{t}$ and
$\tilde{A}$.

For $I=1\,(0)$ mesons, we have 3 types of states, $n=1,2,3\,(4,5,6)$, given in
Table 2.1. The signs of their residues depend on the helicity of the external
vector states. We have two independent possibilities, either an elastic
process ($W^{+}\pi\to W^{+}\pi$) which we will refer to as the $+-$ amplitude,
or inelastic ($W^{+}\pi\to W^{-}\pi$) which we will refer to as the $++$
amplitude. Following the discussion of Appendix A, one finds that the signs of
the residues are given by

$\displaystyle R_{I=0}^{+-}$ $\displaystyle=$
$\displaystyle+g_{4}^{2}+g_{5}^{2}+g_{6}^{2}\qquad\qquad
R_{I=1}^{+-}=+g_{1}^{2}+g_{2}^{2}+g_{3}^{2}\,,$ $\displaystyle R_{I=0}^{++}$
$\displaystyle=$ $\displaystyle-g_{4}^{2}+g_{5}^{2}-g_{6}^{2}\qquad\qquad
R_{I=1}^{++}=-g_{1}^{2}+g_{2}^{2}-g_{3}^{2}\,,$ (6)

where we have defined

$g_{n}^{2}\equiv g_{\pi W,n}^{2}>0\ ,\ \ \ n=1,...,6\,,$ (7)

and $n$ labels the different type of mesons as defined in Table 2.1.

The helicity configuration forces a particular kinematic pre-factor in the
amplitude which, in the spinor helicity formalism, can be written as

${\cal M}^{+-}\propto\langle 12\rangle^{2}[23]^{2}\propto su\quad,\quad{\cal
M}^{++}\propto[13]^{2}\propto s+u\,,$ (8)

where the numbers label the states as in Fig. 2 (b).

At energies below the mass $M$ of the lightest massive meson, we can expand
the amplitudes in powers of $s/M,u/M\to 0$. For the elastic scattering
amplitude we have

$\displaystyle M_{t}^{I=2}(s,u)|_{+-}$ $\displaystyle=$ $\displaystyle
2+su\bigg{(}\sum_{m=0}^{\infty}\sum_{l=0}^{[m/2]}\,g_{m,l}\,s^{\\{m-l}u^{l\\}}\bigg{)}$
$\displaystyle=$ $\displaystyle
2+su\big{(}g_{0,0}+g_{1,0}\,(s+u)+g_{2,0}\,(s^{2}+u^{2})+2g_{2,1}su+...)\ ,$
$\displaystyle M_{u}^{I=2}(s,u)|_{+-}$ $\displaystyle=$
$\displaystyle-1-su\bigg{(}\sum_{m=0}^{\infty}\sum_{l=0}^{[m/2]}\,h^{s}_{m,l}\,s^{\\{m-l}u^{l\\}}+\sum_{m=0}^{\infty}\sum_{l=0}^{[m/2]}\,h^{a}_{m,l}\,s^{[m-l}u^{l]}\bigg{)}$
$\displaystyle=$
$\displaystyle-1-su\big{(}h^{s}_{0,0}+h^{s}_{1,0}\,(s+u)+h^{a}_{1,0}\,(s-u)+h^{s}_{2,0}\,(s^{2}+u^{2})+...)\
,$

where Eq. (2.2) is fully $s\leftrightarrow u$ symmetric, and in Eq. (2.2) we
have separated the symmetric and the antisymmetric part. The first terms of
the expansion correspond to the pion exchange whose gauge coupling we
normalize to $1$. On the other hand, for the inelastic scattering amplitude we
have

$\displaystyle M_{t}^{I=2}(s,u)|_{++}$ $\displaystyle=$
$\displaystyle(s+u)\bigg{(}\sum_{m=0}^{\infty}\sum_{l=0}^{[m/2]}\,f_{m,l}\,s^{\\{m-l}u^{l\\}}\bigg{)}$
$\displaystyle=$
$\displaystyle(s+u)\big{(}f_{0,0}+f_{1,0}\,(s+u)+f_{2,0}\,(s^{2}+u^{2})+f_{2,1}su+...)\
\,.$

We do not need to give the expression for the low-energy expansion of
$M_{u}^{I=2}(s,u)|_{++}$ since, as we will see, this amplitude does not
provide any extra information for bounding the chiral anomaly. The low-energy
constants $g_{m,l}$, $h^{s}_{m,l}$, $h^{a}_{m,l}$ and $f_{m,l}$, referred here
as Wilson coefficients, are the low-energy parameters that we want to
constrain.

###  2.3 Dispersion Relations and Null Constraints

In order to derive sum rules and null constraints from our dispersion
relations, we will assume that for all $k\geq{k_{\rm min}}$, the amplitudes
behave as follows at high energies ($k_{\text{min}}$ will be indicated for the
various processes):

The amplitudes can be expanded in partial waves in the physical regions. For
the elastic process we have

$\displaystyle{\rm Im}M_{u,t}^{I=2}(s,u)\big{|}_{+-}$ $\displaystyle=$
$\displaystyle\sum_{i}(2J+1)\rho^{+-,\pm}_{J}(s)d_{1,1}^{J}\big{(}\cos\theta_{s}\big{)}\,,$
$\displaystyle(2J+1)\rho^{+-,\pm}_{J}(s)$ $\displaystyle=$
$\displaystyle\pi\sum_{i}\bigg{(}R_{I=0}^{+-}\pm
R_{I=1}^{+-}\bigg{)}_{i}m_{i}^{2}\delta(s-m_{i}^{2})\delta_{JJ_{i}}\,,$ (13)

where the $u,t$ subscript refers to the $\pm$ spectral density and
$\cos\theta_{s}=(t-u)/s$. For the inelastic amplitude similarly we have

$\displaystyle{\rm Im}M_{u,t}^{I=2}(s,u)\big{|}_{++}$ $\displaystyle=$
$\displaystyle\sum_{i}(2J+1)\rho^{++,\pm}_{J}(s)d_{1,-1}^{J}\big{(}\cos\theta_{s}\big{)}\,,$
$\displaystyle(2J+1)\rho^{++,\pm}_{J}(s)$ $\displaystyle=$
$\displaystyle\pi\sum_{i}\bigg{(}R_{I=0}^{++}\pm
R_{I=1}^{++}\bigg{)}_{i}m_{i}^{2}\delta(s-m_{i}^{2})\delta_{JJ_{i}}\,.$ (14)

$C_{\infty}$$C_{0}$$s$$\ M^{2}$ (a) $M_{t}^{I=2}(s,u)$

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

$C_{\infty}$$C_{0}$$s$$\ M^{2}$ (c) $M_{u}^{I=2}(s,-s-t)$

Figure 1: Analytic structure of $M_{t}^{I=2}$ at fixed $u<0$ (a), fixed $t<0$
(b) and of $M_{u}^{I=2}$ at fixed $t<0$ (c). 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 the dispersion
relations.

####  2.3.1 Elastic Process

For the elastic process the dispersion relations that we will consider involve
$M^{I=2}_{t}$ at fixed $u<0$ and fixed $t<0$, and $M^{I=2}_{u}$ at fixed $t<0$
(all in the $s-$plane). The analytic structure of these amplitudes and the
contours we choose for the dispersion relations can be found in Fig. 1. For
the elastic process we define the high-energy average along the lines of [11]
as

$\big{\langle}(...)\big{\rangle}_{+-}^{\pm}\equiv\frac{1}{\pi}\sum_{i}(2J+1)\int_{M^{2}}^{\infty}\frac{dm^{2}}{m^{2}}\rho^{+-,\pm}_{J}(m^{2})(...)\
.$ (15)

  * •

$M_{t}^{I=2}\ u-$fixed:

The integral of $M_{t}^{I=2}(s,u)/s^{k+1}$ along the contour $C_{\infty}$ of
Fig. 0(a) vanishes for $k\geq{k_{\rm min}}$, due to Eq. (12a). Because of the
amplitude’s analyticity, we can deform $C_{\infty}$ into the blue contour in
Fig. 0(a) giving

$\oint_{\rm C_{0}}ds^{\prime}\ \frac{M_{t}^{I=2}(s^{\prime},u)}{s^{\prime
k+1}}=2i\int_{M^{2}}^{\infty}ds^{\prime}\ \frac{{\rm
Im}M_{t}^{I=2}(s^{\prime},u)}{s^{\prime k+1}}\,.$ (16)

Assuming $k_{\rm min}=1$, we are able to derive, inserting Eq. (2.3) into Eq.
(16) and expanding for $u\rightarrow 0$, the following system of relations:

$\displaystyle(k=1)\quad g_{0,0}u+g_{1,0}u^{2}+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{(-1)^{J}{\cal
J}^{2}}{2m^{4}}u+\frac{(-1)^{J}{\cal
J}^{4}}{6m^{6}}u^{2}+...\right\rangle_{+-}^{-}\,,$ $\displaystyle(k=2)\quad
g_{1,0}u+g_{2,1}u^{2}+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{(-1)^{J}{\cal
J}^{2}}{2m^{6}}u+\frac{(-1)^{J}{\cal
J}^{4}}{6m^{8}}u^{2}+...\right\rangle_{+-}^{-}\ ,$ (17) $\displaystyle\vdots$

where ${\cal J}^{2}=J(J+1)$, ${\cal J}^{4}=(J-1)J(J+1)(J+2)$, etc… . The
system above allows us to relate the low-energy Wilson coefficients to the
high-energy UV-averages:

$g_{0,0}=\left\langle\frac{(-1)^{J}{\cal
J}^{2}}{2m^{4}}\right\rangle_{+-}^{-}\quad,\quad
g_{1,0}=\left\langle\frac{(-1)^{J}{\cal
J}^{2}}{2m^{6}}\right\rangle_{+-}^{-}\quad,\ ...\ .$ (18)

Furthermore one can notice that the system is over-constrained and therefore
it is possible to find a set of non trivial null constraints, the first of
which starts at $\mathcal{O}(1/m^{6})$:

$\left\langle\frac{(-1)^{J}{\cal
J}^{2}}{2m^{6}}\right\rangle_{+-}^{-}=\left\langle\frac{(-1)^{J}{\cal
J}^{4}}{6m^{6}}\right\rangle_{+-}^{-}.$ (19)

Nevertheless, as we will show later, the null constraints needed to bound the
chiral anomaly are only those at $\mathcal{O}(1/m^{4})$.

  * •

$M_{t}^{I=2}\ t-$fixed:

We can repeat the arguments explained above for $M^{I=2}_{t}(s,-s-t)$ at fixed
$t<0$. The integral identity one gets using Cauchy’s theorem, following Fig.
0(b), is

$\frac{1}{2i}\oint_{\rm C_{0}}ds^{\prime}\
\frac{M_{t}^{I=2}(s^{\prime},-t-s^{\prime})}{s^{\prime
k+1}}=\int_{M^{2}}^{\infty}ds^{\prime}\ \frac{{\rm
Im}M_{t}^{I=2}(s^{\prime},-t-s^{\prime})}{s^{\prime
k+1}}+(-1)^{k}\int_{M^{2}}^{\infty}ds^{\prime}\ \frac{{\rm
Im}M_{t}^{I=2}(s^{\prime},-t-s^{\prime})}{(s^{\prime}+t)^{k+1}}\,.$ (20)

This dispersion relation leads to a system of equations 555We can assume here
$k_{\rm min}=2$ as $k=1$ does not provide additional information.

$\displaystyle(k=2)\quad-g_{0,0}+g_{1,0}t+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{2}{m^{4}}\right\rangle_{+-}^{-}+\left\langle\frac{2{\cal
J}^{2}-5}{m^{6}}\right\rangle_{+-}^{-}t+...\,,$ $\displaystyle\vdots$

that is not over-constrained. Therefore, by themselves, these equations do not
lead to any null constraint. Neverthless, we can combine it with the $u-$fixed
system in Eq. (17) to get a new set of null constraints. We will be interested
in the one at $\mathcal{O}(1/m^{4})$ which we will need to constrain the
anomaly. This is given by

$\left\langle\frac{(-1)^{J}{\cal J}^{2}+4}{2m^{4}}\right\rangle_{+-}^{-}=0\,.$
(22)

Notice that the averages are taken with $R_{I=0}^{+-}-R_{I=1}^{+-}$, and
therefore positivity is not guaranteed.

  * •

$M_{u}^{I=2}\ t-$fixed:

The amplitude $M_{u}^{I=2}=A_{t}(s,u)+\tilde{A}(s,u)$ at $t-$fixed is the only
one whose residues are all positive, as can be seen from Table 2. Its has
poles only in the $s-$channel, as can be seen in Fig. 0(c). According to Eq.
(2.3) the high-energy average will now have a positive spectral density
$\rho_{J}^{+-,+}(s)>0$. The dispersion relations look identical to Eq. (16),
but with the low-energy amplitude given by Eq. (2.2). Following the steps
explained in the previous sections, we can find sum rules for the Wilson
coefficients of Eq. (2.2), as for example (for $k=2$),

$h_{0,0}^{s}=\left\langle\frac{1}{m^{4}}\right\rangle_{+-}^{+}\,.$ (23)

The conditions one gets from these dispersion relations (analogous to Eq.
(17)) is however not over-constrained, and therefore no additional null
constraints are obtained.

####  2.3.2 Inelastic Process

Let us now focus on the inelastic process. In particular, out of the three
possible dispersion relations listed in Fig. 1, we will only consider
$M_{t}^{I=2}$ at fixed $u<0$ and fixed $t<0$ in the $s-$plane.666 We will not
consider $M_{u}^{I=2}$ in the inelastic case. The corresponding dispersion
relations were only useful for the elastic case to relate Wilson coefficients,
such as $h^{s}_{0,0}$, to sums over positive residues. We can proceed
similarly as in the previous sections, but with the low-energy amplitude given
in Eq. (2.2), the partial-wave decomposition given in Eq. (2.3), and the high-
energy average defined by

$\big{\langle}(...)\big{\rangle}_{++}^{\pm}\equiv\frac{1}{\pi}\sum_{i}(2J+1)\int_{M^{2}}^{\infty}\frac{dm^{2}}{m^{2}}\rho^{++,\pm}_{J}(m^{2})(...)\
.$ (24)

  * •

$M_{t}^{I=2}\ u-$fixed:

Once again the integral of $M_{t}^{I=2}(s,u)/s^{k+1}$ along the contour
$C_{\infty}$ of Fig. 0(a) vanishes for $k\geq{k_{\rm min}}$, due to Eq. (12a).
We have again Eq. (16) that expanded for small $u$ gives

$\displaystyle(k=1)\quad\quad\quad\ \ \ f_{0,0}+2f_{1,0}u+...$
$\displaystyle=$
$\displaystyle\left\langle\frac{(-1)^{J+1}}{m^{2}}-\frac{(-1)^{J}(J^{2}+J-1)}{m^{4}}u+...\right\rangle_{++}^{-}\,,$
$\displaystyle(k=2)\quad f_{1,0}+(f_{2,1}+f_{2,0})u+...$ $\displaystyle=$
$\displaystyle\left\langle\frac{(-1)^{J+1}}{m^{4}}-\frac{(-1)^{J}(J^{2}+J-1)}{m^{6}}u+...\right\rangle_{++}^{-}\
,$ (25) $\displaystyle\vdots$

leading to a new set of null constraints. At $\mathcal{O}(1/m^{4})$, we have

$\left\langle\frac{(-1)^{J}({\cal
J}^{2}-3)}{2m^{4}}\right\rangle_{++}^{-}=0\,.$ (26)

  * •

$M_{t}^{I=2}\ t-$fixed:

For fixed $t<0$, we can obtain new dispersion relations that combined with the
system Eq. (25) leads to a second set of null constraints. By doing this, one
can notice that the first new null constraint enters at order
$\mathcal{O}(1/m^{6})$.

####  2.3.3 Bounds on Wilson coefficients

From the above dispersion relations it is already possible to obtain
interesting bounds on the low-energy Wilson coefficients of the $W\pi W\pi$
amplitude, that are related to physical quantities such as the dipole and
quadrupole polarizabilities of the pions (see Appendix B). A more detailed
discussion is given in Appendix C. For example, at ${\cal O}(s^{2})$, we find
the bounds

$-2\leq\frac{g_{0,0}}{h^{s}_{0,0}}\leq 2\,,\ \ \ \
-1\leq\frac{f_{1,0}}{h^{s}_{0,0}}\leq 1\,.$ (27)

For higher-oder Wilson coefficients a numerical analysis is sometimes needed
in order to find the allowed values. In Figure 2 we provide an example (see
Appendix C for details).

Figure 2: Exclusion plot for the Wilsons $g_{2}$ and $g_{2}^{\prime}$ given in
Eq. (62).

###  2.4 The $W\pi\to\eta\pi$ Amplitude and the Chiral Anomaly

Let us now consider the $W\pi\to\eta\pi$ amplitude. For $\lambda_{W}=+1$
helicity, we have

$\mathcal{M}(W^{a+},\pi^{b},\eta,\pi^{c})\propto[12]\langle
24\rangle[41]\propto\sqrt{stu}\,.$ (28)

At low-energy, this amplitude can be written as

$|\mathcal{M}(W^{a+},\pi^{b},\eta,\pi^{c})|=|f_{abc}|\mathcal{M}_{W\pi\eta\pi}=\frac{\kappa}{2\sqrt{2}}|f_{abc}|\sqrt{stu}+...\,.$
(29)

The Wilson coefficient $\kappa$ is related to the chiral anomaly. Indeed, from
the WZW term [26, 27] we have

$-\frac{N_{c}}{48\pi^{2}}\varepsilon^{\mu\nu\alpha\beta}{\rm Tr\hskip
2.0pt}[A_{\mu L}U_{\nu L}U_{\alpha L}U_{\beta
L}+\text{L}\to\text{R}]\subset\mathcal{L}_{WZW}\,,$ (30)

where $U_{\nu L}=(\partial_{\nu}U)U^{\dagger}$, $U_{\nu
R}=U^{\dagger}(\partial_{\nu}U)$, and
$U=\text{Exp}(i\eta/F_{\pi})\text{Exp}(2i\pi^{a}\tau^{a}/F_{\pi})$ with
$\tau^{a}=\sigma^{a}/2$ and $A^{a}_{L}=A^{a}_{R}=W^{a}$. By matching Eq. (29)
with Eq. (30), we obtain

$\kappa=\frac{N_{c}}{12\pi^{2}F_{\pi}^{3}}\,.$ (31)

The $W\pi\eta\pi$ amplitude is mediated by the following meson states:

$s/u-\text{channel:}$ $G=-1$$I=1$$\pi$$\pi$$W$$\eta$ $t-\text{channel:}$
$G=1$$I=1$$\pi$$\pi$$W$$\eta$ (32)

For the $s/u-$channel the exchanged mesons must have $G=-1$, $I=1$,
$J=\text{even}$ and parity $P=(-1)^{J}=+1$. These states are classified as
$n=2$ in Table. 2.1 (highlighted in green). On the other hand, in the
$t-$channel the states must have $G=+1$, $I=1$, $J=\text{odd}$ and
$P=(-1)^{J}=-1$. These states are classified as $n=5$ in Table. 2.1 (also
highlighted in green). We notice therefore that in this amplitude, only 2 out
of the 6 possible types of mesons contribute.

###  2.5 Dispersion Relations

Let us consider the $W\pi\eta\pi$ amplitude without the non-analytical pre-
factor coming from the polarization structure,
$\mathcal{M}_{W\pi\eta\pi}/\sqrt{stu}$, and assume at fixed $u<0$

$\lim_{|s|\to\infty}\frac{\mathcal{M}_{W\pi\eta\pi}}{\sqrt{stu}}=0\,,$ (33)

and similarly at fixed $t<0$. In the $s$-plane with $u$ fixed, we get the
following dispersion relation:

$\oint\frac{ds}{2\pi
i}\frac{\mathcal{M}_{W\pi\eta\pi}(s,u)}{\sqrt{-s(s+u)u}}=\sum_{i}^{J^{+}_{\text{even}}}\frac{g_{W\pi
i}g_{\pi\eta
i}\,d^{J}_{1,0}(-1-\frac{2u}{m_{i}^{2}})}{\sqrt{-m_{i}^{2}(m_{i}^{2}+u)u}}+\sum_{i}^{J^{-}_{\text{odd}}}\frac{g_{W\eta
i}g_{\pi\pi
i}\,d^{J}_{1,0}(-1-\frac{2u}{m_{i}^{2}})}{\sqrt{-m_{i}^{2}(m_{i}^{2}+u)u}}\,,$
(34)

where we specified that only $J^{P}$ mesons with $J$ odd or even are summed
over. Expanding the above dispersion relation for $u\to 0$, we obtain

$\displaystyle\frac{\kappa}{2\sqrt{2}}=\sum_{i}^{J^{-}_{\text{odd}}}\frac{g_{W\eta
i}g_{\pi\pi i}\,{\cal
J}}{m_{i}^{3}}-\sum_{i}^{J^{+}_{\text{even}}}\frac{g_{W\pi i}g_{\pi\eta
i}\,{\cal J}}{m_{i}^{3}}\,,$ (35)

where ${\cal J}=\sqrt{{\cal J}^{2}}$.

Similarly, we can also get another dispersion relation by fixing $t<0$ in the
$s$-plane. We obtain in this case

$\frac{\kappa}{2\sqrt{2}}=2\sum_{i}^{J^{+}_{\text{even}}}\frac{g_{W\pi
i}g_{\pi\eta i}\,{\cal J}}{m_{i}^{3}}\,.$ (36)

From Eq. (35) and Eq. (36) we get the null constraint

$\sum_{i}^{J^{-}_{\text{odd}}}\frac{g_{W\eta i}g_{\pi\pi i}\,{\cal
J}}{m_{i}^{3}}=3\sum_{i}^{J^{+}_{\text{even}}}\frac{g_{W\pi i}g_{\pi\eta
i}\,{\cal J}}{m_{i}^{3}}\,.$ (37)

At large $N_{c}$, the coupings of singlet $\eta$ can be related with the
$\pi^{a}$ couplings:

$\displaystyle g_{W\eta i}=(g_{5})_{i}\ \ \ \ (J^{-}_{\text{odd}})\quad,\quad
g_{\pi\eta i}=g_{\pi\pi i}\ \ \ \ (J^{+}_{\text{even}})\,.$ (38)

Using these relations, we can express $\kappa$ and the null constraint Eq.
(37) as

$\displaystyle\frac{\kappa}{4\sqrt{2}}$ $\displaystyle=$
$\displaystyle\sum_{i}^{J^{+}_{\text{even}}}\frac{(g_{2}g_{\pi\pi})_{i}\mathcal{J}}{m_{i}^{3}}$
(39) $\displaystyle=$
$\displaystyle\sum_{i}^{J^{-}_{\text{odd}}}\frac{(g_{5}g_{\pi\pi})_{i}\mathcal{J}}{3m_{i}^{3}}\
.$ (40)

##  3 Bounding the Chiral Anomaly

In this section we analytically derive an upper bound on the anomaly
coefficient $\kappa$ using the $W\pi\to W\pi$ null constraints Eq. (22) and
Eq. (26). These two null constraints are enough to get a bound on $\kappa$. We
will however improve the bound by incorporating the null constraint Eq. (37)
from $W\pi\to\eta\pi$.

Using the identity (for $a_{i},b_{i}\in\mathbb{R}$)

$\sum_{i}a_{i}b_{i}\leq\sqrt{\sum_{i}a_{i}^{2}b_{i}^{2}}\leq\sqrt{\big{(}\sum_{i}a_{i}^{2}\big{)}\big{(}\sum_{i}b_{i}^{2}\big{)}}\
,$ (41)

we can obtain a bound on $\kappa$ using Eq. (39) and Eq. (40) respectively

$\displaystyle\frac{\kappa}{4\sqrt{2}}$ $\displaystyle\leq$
$\displaystyle\sqrt{\bigg{(}\sum^{J^{+}_{\text{even}}}_{i}\frac{g_{\pi\pi
i}^{2}}{m_{i}^{2}}\bigg{)}\bigg{\langle}\frac{\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{2}}\equiv\kappa_{2}^{\text{UB}}\,,$
$\displaystyle\frac{\kappa}{4\sqrt{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{3}\sqrt{\bigg{(}\sum^{J^{-}_{\text{odd}}}_{i}\frac{g_{\pi\pi
i}^{2}}{m_{i}^{2}}\bigg{)}\bigg{\langle}\frac{\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{5}}\equiv\kappa_{5}^{\text{UB}}\,.$
(42)

where we have defined

$\langle(...)\rangle_{n}\equiv\frac{1}{\pi}\sum_{i}(2J+1)\int_{M^{2}}^{\infty}\frac{dm^{2}}{m^{2}}\rho^{n}_{J}(m^{2})(...)\,,$
(43)

with

$(2J+1)\rho^{n}_{J}(m^{2})=\pi\sum_{i}(g_{n}^{2})_{i}m_{i}^{2}\delta(m^{2}-m_{i}^{2})\delta_{JJ_{i}}\
,$ (44)

and $g_{n}^{2}\geq 0$ defined in Eq. (7). The bounds Eq. (42) depend on ${\cal
J}^{2}/m^{4}$ high-energy averages which can be written as a function of
$1/m^{4}$ averages by using the null constraints Eq. (22) and Eq. (26),
rewritten as

$\displaystyle+\bigg{\langle}\frac{\mathcal{J}^{2}-4}{m^{4}}\bigg{\rangle}_{1}-\bigg{\langle}\frac{\mathcal{J}^{2}+4}{m^{4}}\bigg{\rangle}_{2}-\bigg{\langle}\frac{\mathcal{J}^{2}+4}{m^{4}}\bigg{\rangle}_{3}-\bigg{\langle}\frac{\mathcal{J}^{2}-4}{m^{4}}\bigg{\rangle}_{4}-\bigg{\langle}\frac{\mathcal{J}^{2}-4}{m^{4}}\bigg{\rangle}_{5}+\bigg{\langle}\frac{\mathcal{J}^{2}+4}{m^{4}}\bigg{\rangle}_{6}=0\,,\
$
$\displaystyle-\bigg{\langle}\frac{\mathcal{J}^{2}-3}{m^{4}}\bigg{\rangle}_{1}-\bigg{\langle}\frac{\mathcal{J}^{2}-3}{m^{4}}\bigg{\rangle}_{2}+\bigg{\langle}\frac{\mathcal{J}^{2}-3}{m^{4}}\bigg{\rangle}_{3}+\bigg{\langle}\frac{\mathcal{J}^{2}-3}{m^{4}}\bigg{\rangle}_{4}-\bigg{\langle}\frac{\mathcal{J}^{2}-3}{m^{4}}\bigg{\rangle}_{5}-\bigg{\langle}\frac{\mathcal{J}^{2}-3}{m^{4}}\bigg{\rangle}_{6}=0\,.\
$ (45)

By summing these two relations we get

$\bigg{\langle}\frac{2\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{2}+\bigg{\langle}\frac{2\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{5}=-\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{1}-\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{2}-\bigg{\langle}\frac{7}{m^{4}}\bigg{\rangle}_{3}+\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{4}+\bigg{\langle}\frac{7}{m^{4}}\bigg{\rangle}_{5}+\bigg{\langle}\frac{7}{m^{4}}\bigg{\rangle}_{6}\
.$ (46)

Eq. (46) gives an interesting relation between the two $\mathcal{J}^{2}/m^{4}$
averages involved in Eq. (42) that allows to find the following relation

$\bigg{\langle}\frac{\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{2}+\bigg{\langle}\frac{\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{5}=3f_{1,0}-2g_{0,0}\equiv{\cal
P}\geq 0\,,$ (47)

where we have used

$\displaystyle g_{0,0}$ $\displaystyle=$
$\displaystyle\bigg{\langle}\frac{2}{m^{4}}\bigg{\rangle}_{1}+\bigg{\langle}\frac{2}{m^{4}}\bigg{\rangle}_{2}+\bigg{\langle}\frac{2}{m^{4}}\bigg{\rangle}_{3}-\bigg{\langle}\frac{2}{m^{4}}\bigg{\rangle}_{4}-\bigg{\langle}\frac{2}{m^{4}}\bigg{\rangle}_{5}-\bigg{\langle}\frac{2}{m^{4}}\bigg{\rangle}_{6}\,,$
(48) $\displaystyle f_{1,0}$ $\displaystyle=$
$\displaystyle\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{1}+\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{2}-\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{3}-\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{4}+\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{5}+\bigg{\langle}\frac{1}{m^{4}}\bigg{\rangle}_{6}\,,$
(49)

obtained from the sum rules Eq. (2.3.1) and Eq. (25) respectively.

We can also obtain a relation between the sum over $g^{2}_{\pi\pi
i}/m_{i}^{2}$ in Eq. (42) by using sum rules derived from dispersion relations
for the $\pi\pi\rightarrow\pi\pi$ process in [3, 4]. In particular, we need

$F_{\pi}^{-2}=2\bigg{(}\sum^{J^{+}_{\text{even}}}_{i}\frac{g_{\pi\pi
i}^{2}}{m_{i}^{2}}+\sum^{J^{-}_{\text{odd}}}_{i}\frac{g_{\pi\pi
i}^{2}}{m_{i}^{2}}\bigg{)}\,.$ (50)

Armed with Eq. (47) and Eq. (50), we can now obtain a bound on $\kappa$ as a
function of $F^{2}_{\pi}$ and $\mathcal{P}$. Defining

$X\equiv F_{\pi}^{2}\sum^{J^{-}_{\text{odd}}}_{i}\frac{g_{\pi\pi
i}^{2}}{m_{i}^{2}}\quad\text{and}\quad
Y\equiv\frac{1}{\mathcal{P}}\bigg{\langle}\frac{\mathcal{J}^{2}}{m^{4}}\bigg{\rangle}_{5}\
,$ (51)

we can rewrite $\kappa_{5}^{\text{UB}}$ and $\kappa_{2}^{\text{UB}}$ as

$\frac{\kappa_{5}^{\text{UB}}}{\sqrt{\mathcal{P}/F_{\pi}^{2}}}=\frac{1}{3}\sqrt{XY}\
,\ \ \ \
\frac{\kappa_{2}^{\text{UB}}}{\sqrt{\mathcal{P}/F_{\pi}^{2}}}\leq\sqrt{\bigg{(}\frac{1}{2}-X\bigg{)}\bigg{(}\frac{1}{2}-Y\bigg{)}}\,.$
(52)

Written in this way we notice that as we decrease $X$ and $Y$ to make
$\kappa_{2}^{\text{UB}}$ larger, $\kappa_{5}^{\text{UB}}$ becomes smaller, and
viceversa. Since $\kappa$ must be smaller than both bounds, the largest value
of $\kappa$ is reached when $\kappa_{2}^{\text{UB}}=\kappa_{5}^{\text{UB}}$.
This is achieved when

$\frac{1}{9}XY=\left(\frac{1}{2}-X\right)\left(\frac{1}{2}-Y\right)\quad\rightarrow\quad
Y=\frac{X-1/2}{16X/9-1}\,,$ (53)

that inserted in Eq. (52) gives us $\kappa_{5}^{\text{UB}}$ as a function of
only $X$ that is maximized for $X=3/8$. This corresponds to $Y=3/8$, and gives
$\kappa_{5}^{\text{UB}}|_{\rm max}=1/8$. Taking this value in Eq. (42), we
finally get the following upper bound on the chiral anomaly:

$\frac{\kappa}{\sqrt{\mathcal{P}/F_{\pi}^{2}}}\leq\frac{1}{\sqrt{2}}\,.$ (54)

This is the main result of the paper. This tells us that the anomaly
coefficient is bounded by $\mathcal{P}$, a quantity related with the
polarizabilities of the pions (see Appendix B). Using the constraints Eq.
(27), we can write Eq. (54) as a function of other Wilson coefficients. For
example, we get that ${\cal P}\leq 7h^{s}_{0,0}$ that leads to

$\frac{\kappa}{\sqrt{h_{0,0}^{s}/F_{\pi}^{2}}}\leq\sqrt{\frac{7}{2}}\,.$ (55)

In the case of $SU(N_{c})$ gauge theories with $N_{f}$ quarks, the anomaly
coefficient $\kappa$ is known, and Eq. (54) provides a bound on the Wilson
coefficients. For example, using Eq. (31) and
$F_{\pi}\simeq\sqrt{N_{c}/3}\,m_{\rho}/7$ [28], we obtain from Eq. (55)

$h^{s}_{0,0}\gtrsim\frac{0.44}{m_{\rho}^{4}}\,.$ (56)

As in [3, 4], it is also interesting to obtain the predictions of different
models to the ratio Eq. (54), in order to understand how close these are to
the upper bound. For example, in models of only scalars, we have that
$h^{s}_{0,0}$ is zero and then $\kappa=0$. For $su$-models (introduced in [11]
for the first time, and studied in [12, 3, 4] for four-pion amplitudes), we
show in Appendix D that

$\frac{\kappa}{\sqrt{{\cal
P}/F_{\pi}^{2}}}\Bigg{|}_{su-\text{model}}\lesssim\frac{0.97}{\sqrt{2}}\,,$
(57)

which falls within the bound established in Eq. (54), only $\sim 3\%$ away
from saturation.

Eq. (54) gives a non-trivial constraint on phenomenological models for QCD,
such as NJL or holographic models. For example, in holographic models the
chiral anomaly arises from a Chern-Simons (CS) term. When the corresponding
dual 4D theory is not known, the CS coefficient cannot be matched to the UV
theory and therefore cannot be determined. Eq. (54) provides in this case a
bound on the the size of the CS term. This could also be useful for models of
axions.

##  4 Conclusions

The analyticity, unitarity and the good high-energy behaviour of scattering
amplitudes provide severe constraints on the low-energy physical quantities of
gauge theories like QCD. In this work we have studied scattering amplitudes
involving external gauge fields and goldstones ($\pi$ and $\eta$) in the
large-$N_{c}$ limit to obtain several new constraints.

Our main result has been to provide an analytical bound on the chiral anomaly,
Eq. (54). To derive this bound we have first analyzed the $W\pi\rightarrow
W\pi$ amplitude, providing the selection rules for the mesons exchanges, as
well as the sign of their on-shell residues. We have considered the elastic
($W^{+}\pi\rightarrow W^{+}\pi$) and inelastic ($W^{+}\pi\rightarrow
W^{-}\pi$) processes, as both are needed to derive $\mathcal{O}(1/m^{4})$ null
constraints on the meson couplings.777We have found that adding more null
constraints does not improve the bound. We have also derived sum rules for the
Wilson coefficients of these amplitudes. Of especial use has been the sum rule
for $h^{s}_{0,0}$ that has shown to have all its terms positive.

We have later considered the $W\pi\to\eta\pi$ amplitude, which at low energy
yields the chiral anomaly coefficient $\kappa$. We have shown, once again via
dispersion relations, that there are two different ways to determine $\kappa$,
implying a non-trivial constraint among the corresponding meson couplings.

Putting all this together, we have been able to derive analytically Eq. (54).
This bound has interesting implications for UV completions of these
amplitudes, as we have shown for the case of large-$N_{c}$ QCD. We have also
considered $su$-models (see Appendix D) that seem to almost saturate the
bound. We hope that in the future this bound could be tested in lattice
simulations. Finally, we have also briefly study the bounds on other Wilson
coefficients, showing the EFT-hedron geometry [6] of their allowed parameter
space -see for example Fig. 2.

In the future, it could also be interesting to study in more detail which
theories saturate these bounds. Similarly, our analysis can be extended to
understand implications on composite Higgs models and bounds on dimension six
operators such as $H^{\dagger}HF_{\mu\nu}F^{\mu\nu}$.

Note added: While preparing this article, it was submitted to the archives
Ref. [29] that also uses dispersion relations to numerically obtain bounds on
the chiral anomaly coefficient.

## Acknowledgments

We are very grateful to Francesco Riva and Pyungwon Ko for their valuable
insights. This work has been partly supported by the research grants
2021-SGR-00649 and PID2020-115845GB-I00/AEI/10.13039/501100011033.

##  Appendix A $q\bar{q}$ mesons and the sign of on-shell residues

The $q\bar{q}$ mesons can be classified according to their quantum numbers:
mass $m$, spin $J$, parity $P$, Isospin $I$, and $G$-parity. Defining by
$\ell$ the orbital angular momentum of the $q\bar{q}$ system, $s=0,1$ the
total spin, and $I=0,1$ the isospin, we have

$\displaystyle J$ $\displaystyle=$ $\displaystyle\ell+s,...,\ell-s\,,$
$\displaystyle P$ $\displaystyle=$ $\displaystyle(-1)^{\ell+1}\,,$
$\displaystyle G$ $\displaystyle=$ $\displaystyle(-1)^{I+\ell+s}\,.$ (58)

By taking $\ell=0,1,2,...$, we can build the meson list of Table 2.1. Notice
that we have only six types of mesons that we label with $n=1,...,6$.

To understand the sign of the residues in Eq. (6) arising from meson exchanges
in the process $W\pi\to W\pi$, we proceed in the following way (recalling that
the couplings $g_{W\pi i}$ are real):

  * •

For the elastic case $W^{+}\pi\to W^{+}\pi$ (amplitude ${\cal M}^{+-}$), we
obviously have that these are proportional to $g_{W\pi i}^{2}>0$ so the sign
is always positive.

  * •

For $W^{+}\pi\to W^{-}\pi$ (amplitude ${\cal M}^{++}$) we can consider the
process $W^{+}\pi\to R_{i}\to W^{-}\pi$ in the forward limit $t\to 0$. By
performing a $P$ transformation and a spacial rotation of 1800 degrees, the
coupling in $W^{+}\pi\to R_{i}$ can be related to that of $R_{i}\to W^{-}\pi$.
Using that $d_{1,0}^{J}(-\cos\theta)=(-1)^{J}d_{1,0}^{J}(\cos\theta)$, we have
then that $W^{+}\pi\to R_{i}\to W^{-}\pi$ is proportional to $g_{W\pi
i}^{2}\times(-1)^{J_{i}}\times P_{i}$.

This leads to the signs in Eq. (6) for the different type of mesons.

##  Appendix B Polarizabilities of the pions

Given the $\gamma\gamma\to\pi\pi$ scattering amplitude with the pion exchange
subtracted, ${\cal M}(\gamma^{+},\gamma^{\pm},\pi,\pi)$, the pion
polarizabilities are defined as the coefficients coming from an expansion in
$s$ at fixed $t=m^{2}_{\pi}$ [30, 18, 31]:

$\frac{\alpha}{m_{\pi}}{\cal
M}(\gamma^{+},\gamma^{\pm},\pi,\pi)(s,t=m^{2}_{\pi})=(\alpha_{1}\mp\beta_{1})_{\pi}+\frac{s}{12}(\alpha_{2}\mp\beta_{2})_{\pi}+...\,.$
(59)

In particular, $(\alpha_{1}\mp\beta_{1})_{\pi}$ and
$(\alpha_{2}\mp\beta_{2})_{\pi}$ are respectively the dipole and quadrupole
polarizabilities of the pions $\pi=\pi^{\pm},\pi^{0}$. These quantities have
been measured experimentally [30, 18, 31] and also analyzed in lattice
simulations [32]. In our convention the photon corresponds to the gauging of
$Q=T_{3}+B/2$, a subgroup of the global $U(2)$, and we must change
$s\leftrightarrow t$ in Eq. (59). The dipole polarizabilities receive then
contributions from our $g_{0,0}$ and $h^{s}_{0,0}$ for the $+-$ helicities of
the photon (although they are suppressed by $m^{2}_{\pi}$), and $f_{0,0}$ for
the $++$. On the other hand, $f_{1,0}$ contributes to the quadrupole
polarizability.

##  Appendix C Bounds on the Elastic $W\pi W\pi$ Wilson coefficients

Let us show here how to bound the parameter space of the Wilson coefficients
appearing in Eq. (2.2) and Eq. (2.2). We are considering the elastic process
since better bounds can be obtained in this case due to positivity.
Nevertheless, the same approach can also be used for the inelastic case.

Let us begin with the simplest analytical bound arising from the sum rules of
$g_{0,0}$ and $h^{s}_{0,0}$, Eq. (2.3.1) and Eq. (23) respectively, that gives

$\frac{g_{0,0}}{h^{s}_{0,0}}=-\frac{\left\langle\frac{2}{m^{4}}\right\rangle_{+-}^{-}}{\left\langle\frac{1}{m^{4}}\right\rangle_{+-}^{+}}\,.$
(60)

The absolute value of this ratio must always be smaller or equal to 2, since
the states that enter in $g_{0,0}$ and in $h^{s}_{0,0}$ are the same, the only
difference being that in $h^{s}_{0,0}$ everything enters additively, while
some of these states enter with a minus sign in $g_{0,0}$. The bound will be
therefore maximized when the only states exchanged are those who contribute
negatively to $g_{0,0}$ (namely states with $n=1,2,3$), while it is minimized
when the states that contribute positively are the only ones being exchanged
($n=4,5,6$). This leads to the bound Eq. (27), and similarly for $f_{1,0}$
from its sum rule in Eq. (25).

The same result can be found numerically using SDPB [33]. To find this bound
we must initially redefine the Wilson coefficients and null constraints (of
both the elastic and inelastic processes) in terms of the high-energy average
defined Eq. (43), similarly to what was done in Eq. (45). We can then
construct the bootstrap equation

$h_{0,0}\vec{v}_{1}+g_{0,0}\vec{v}_{2}+\sum_{n=1..6}\langle\vec{v}_{n}(m^{2},J)\rangle_{n}=0\,,$
(61)

which holds true if $\vec{v}_{1}=(1,0,0,..),\vec{v}_{2}=(0,1,0,...)$ and for
an appropriate choice of the vectors $\vec{v}_{n}(m^{2},J)$ (which contain the
null constraints). Following canonical the optimization procedure explained in
detail in [11, 12, 3], one can bound numerically the Wilson coefficients. In
particular, this method can be replicated for higher-order Wilson
coefficients. For example, defining

$g_{2}=\frac{g_{2,0}}{h_{2,0}^{a}+2h_{2,0}^{s}+h_{2,1}^{s}/2}\qquad\text{and}\qquad
g_{2}^{\prime}=\frac{g_{2,1}}{h_{2,0}^{a}+2h_{2,0}^{s}+h_{2,1}^{s}/2}\,,$ (62)

where the normalization has been chosen opportunely for positivity arguments
as

$h_{2,0}^{a}+2h_{2,0}^{s}+h_{2,1}^{s}/2=\bigg{\langle}\frac{(\mathcal{J}^{2}-8)\mathcal{J}^{2}+14}{4m^{8}}\bigg{\rangle}_{+-}^{+}\,,$
(63)

we can repeat the numerical procedure and find the exclusion plot shown in
Figure 2.

##  Appendix D The $su$-model

The $su$-models are defined as those leading to healthy amplitudes with a
single mass scale $m$. They often predict Wilson coefficients at the
boundaries of the allowed regions [11, 12, 3, 4]. For the $\pi\pi\to\pi\pi$
process the $su$-amplitude was already presented in [4] where it was shown to
take the general form

$\mathcal{M}(s,u)\bigg{|}_{4\pi}=\frac{m^{2}(s+u)+\lambda
su}{(s-m^{2})(u-m^{2})}\,,$ (64)

with -$2\leq\lambda\leq\frac{2\ln 2-1}{1-\ln 2}$. In the limiting case
$\lambda=\frac{2\ln 2-1}{1-\ln 2}$, the amplitude Eq. (64) contains no poles
associated to $J=0$ states [4]. In this case, the residues in the $s$-channel
of Eq. (64) are given by

$g_{\pi\pi i}^{2}\big{|}_{4\pi}=\\{0.78,0.18,0.04,...\\}\,,$ (65)

and correspond to states of $J^{P}=1^{-},2^{+},...$ respectively ($n=5$ and
$n=2$ of states $G=1$ alternating). The amplitude Eq. (64) predicts the Wilson
coefficient

$F_{\pi}^{-2}=\frac{2}{m^{2}}\,.$ (66)

Notice that we have absorbed an overall factor in Eq. (64) into
$F_{\pi}^{-2}$.

An $su$-amplitude for $M^{I=2}_{t}(s,u)\big{|}_{+\pm}$ can also be constructed
following the conditions:

  * •

It must have a single mass scale $m$.

  * •

No $t-$channel poles.

  * •

It must be proportional to $t$ and $su$ for $++$ and $+-$ amplitudes
respectively.

  * •

It must drop as $M^{I=2}_{t}(s,u)/s\rightarrow 0$ for $|s|\rightarrow\infty$
at $t-$fixed, and similarly for $u-$fixed.

The most general amplitude following these criteria takes the form

$M^{I=2}_{t}(s,u)\big{|}_{++}=-\frac{m^{2}\,t}{(s-m^{2})(u-m^{2})}\,,\ \ \ \
M^{I=2}_{t}(s,u)\big{|}_{+-}=-\alpha\frac{su}{(s-m^{2})(u-m^{2})}\,,$ (67)

where $\alpha$ is a constant. The residues of Eq. (67) in the $s$-channel are

$R^{++}_{I=0}-R^{++}_{I=1}=\\{0.82,-0.15,0.03,...\\}\,,\ \ \
R^{+-}_{I=0}-R^{+-}_{I=1}=\alpha\,\\{0.58,-0.09,0.02,...\\}\,.$ (68)

corresponding to $J=1,2,3,...$ states, where for each $J_{\rm odd}$ ($J_{\rm
even}$) they can be of type $n=1,4,5$ ($n=2,3,6$) of Table 2.1.

From Eq. (67) we obtain

$f_{1,0}=\frac{1}{m^{4}}\,,\ \ \ g_{1,0}=-\frac{\alpha}{m^{4}}\ \ \ \to\ \
{\cal P}=\frac{3+2\alpha}{m^{4}}\,.$ (69)

Notice that we have absorbed an overall factor in the $++$ amplitude in Eq.
(67) into $f_{1,0}$.

According to Eq. (6), we can obtain the couplings $g_{2}^{2}$ and $g_{5}^{2}$
(which are the ones that enter in the anomaly), by adding the two sets of
residues of Eq. (68). We obtain

$g_{W\pi
i}^{2}\big{|}_{W^{2}\pi^{2}}=\\{0.41+0.29\alpha,0.07+0.05\alpha,...\\}\,,$
(70)

corresponding to $n=5$ and $n=2$ states alternating. To have positive
$g^{2}_{5,2}$, we must demand $-1.41\leq\alpha\leq 1.41$. The value
$\alpha=1.41\ (-1.41)$ maximizes (minimizes) $g^{2}_{5,2}$.

Finally, the $W\pi\eta\pi$ amplitude can be constructed with the following
conditions:

  * •

Single mass scale $m$.

  * •

$s\leftrightarrow u$ crossing symmetric.

  * •

Proportional to $\sqrt{stu}$.

  * •

$s/u-$channel poles associated only to even-spin states (see Eq. (32)).

  * •

It must drop as ${\cal M}_{W\pi\eta\pi}/\sqrt{stu}\rightarrow 0$ for
$|s|\rightarrow\infty$ at $t-$fixed, and similarly for $u-$fixed.

The most general amplitude one can construct following these conditions is

${\cal M}_{W\pi\eta\pi}=-\beta
m\sqrt{stu}\,\frac{m^{2}/2+t}{(s-m^{2})(u-m^{2})(t-m^{2})}\ ,$ (71)

where $\beta>0$ is a constant. We obtain the following residues in the
$s$-channel:

$|g_{W\pi i}g_{\pi\pi i}|=\beta\,\\{0.48,0.10,0.02,...\\}\,,$ (72)

corresponding to states $n=5$ and $n=2$ of Table 2.1 alternating.

From Eq. (29) and Eq. (71) we get the anomaly coefficient

$\kappa=\frac{\sqrt{2}\beta}{m^{3}}\,.$ (73)

To maximize this value, we must take the largest possible value of $\beta$.
Nevertheless, this is constrained by the fact that the couplings in Eq. (72)
cannot overcome

$\sqrt{g_{W\pi i}^{2}\big{|}_{W^{2}\pi^{2}}\,g_{\pi\pi
i}^{2}\big{|}_{4\pi}}=\\{\sqrt{0.32+0.23\alpha},\sqrt{0.01+0.01\alpha},...\\}\,,$
(74)

coming from Eq. (65) and Eq. (70), as there can always be more states $n=5,2$
(for a given $J$) in Eq. (64) and Eq. (67) than in Eq. (71) (as this latter
requires that the interchanged mesons must have both $g_{W\pi i}$ and
$g_{\pi\pi i}$ nonzero). This gives

$\beta_{\rm max}=\sqrt{1.37+0.97\alpha}\,.$ (75)

With Eq. (66), Eq. (69), Eq. (73) and Eq. (75), one finds that
$\kappa/\sqrt{{\cal P}/F^{2}_{\pi}}$ is maximized for $\alpha=1.41$, leading
to Eq. (57).

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