aainstitutetext: Helmholtz-Institut für Strahlen- und Kernphysik and Bethe
Center for Theoretical Physics, Universität Bonn, D-53115 Bonn,
Germanybbinstitutetext: CAS Key Laboratory of Theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
Chinaccinstitutetext: School of Physical Sciences, University of Chinese
Academy of Sciences, Beijing 100049, Chinaddinstitutetext: Institute for
Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron
Physics, Forschungszentrum Jülich, D-52425 Jülich, Germanyeeinstitutetext:
Tbilisi State University, 0186 Tbilisi, Georgiaffinstitutetext: INPAC,
Shanghai Key Laboratory for Particle Physics and Cosmology, MOE Key Lab for
Particle Astrophysics and Cosmology, School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai, 200240, China

# Two-Meson Form Factors in Unitarized
Chiral Perturbation Theory

Yu-Ji Shi<EMAIL_ADDRESS>a Chien-Yeah Seng<EMAIL_ADDRESS>b,c
Feng-Kun Guo<EMAIL_ADDRESS>a Bastian Kubis<EMAIL_ADDRESS>a,d,e
Ulf-G. Meißner<EMAIL_ADDRESS>f
and Wei Wang<EMAIL_ADDRESS>
###### Abstract

We present a comprehensive analysis of form factors for two light pseudoscalar
mesons induced by scalar, vector, and tensor quark operators. The theoretical
framework is based on a combination of unitarized chiral perturbation theory
and dispersion relations. The low-energy constants in chiral perturbation
theory are fixed by a global fit to the available data of the two-meson
scattering phase shifts. Each form factor derived from unitarized chiral
perturbation theory is improved by iteratively applying a dispersion relation.
This study updates the existing results in the literature and explores those
that have not been systematically studied previously, in particular the two-
meson tensor form factors within unitarized chiral perturbation theory. We
also discuss the applications of these form factors as mandatory inputs for
low-energy phenomena, such as the semi-leptonic decays
$B_{s}\to\pi^{+}\pi^{-}\ell^{+}\ell^{-}$ and the $\tau$ lepton decay
$\tau\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$, in searches for physics beyond the
Standard Model.

##  1 Introduction

The study of multi-meson systems is an interesting problem as they are
universal in various physical processes. An example of this is the $B\to
K^{*}(\to K\pi)\mu^{+}\mu^{-}$ decay that is induced by the flavor-changing
neutral current. Such a process is highly suppressed in the Standard Model
(SM), and thus sensitive to physics beyond the Standard Model (BSM). As a
result it offers a large number of observables to test the SM ranging from
differential decay widths and polarizations to a full analysis of angular
distributions of the final-state particles. Recent experimental studies have
led to some hints for moderate deviations from the SM Aaij:2013qta ;
Aaij:2017vbb ; Abdesselam:2019wac . Note that this process is in fact a four-
body decay since the $K^{*}$ meson is reconstructed from the $K\pi$ final
state. Therefore, to handle such decay processes, the narrow-width
approximation is usually assumed in phenomenological studies. However, this
assumption may lead to sizable systematic uncertainties as it captures only
part of the $K\pi$ final-state interactions.

To solve this problem, one should use a complete factorization analysis that
can systematically separate the low-energy final-state interaction from the
short-ranged weak transition. In semi-leptonic processes like $B\to
M_{1}M_{2}\ell^{+}\ell^{-}$, the final two-meson state decouples from the
leptons to a good approximation. Thus, it is guaranteed by the Watson–Madigal
theorem Watson:1952ji ; Migdal:1956tc  that the phase of the hadronic
transition matrix element is equal to the phase of $M_{1}M_{2}$ elastic
scattering below the first inelastic threshold. More explicitly, as pointed
out, e.g., in Ref. Meissner:2013hya , the decay matrix element is proportional
to a two-meson form factor,

$\displaystyle\langle M_{1}M_{2}|\bar{q}(0)\Gamma q(0)|0\rangle~{},$ (1.1)

where the Dirac matrices $\Gamma=1,\gamma_{\mu},\sigma_{\mu\nu}$ correspond to
the scalar, vector, and antisymmetric tensor currents, respectively. The
choice depends on whether $M_{1}$ and $M_{2}$ are in relatively $S$-wave or
$P$-wave. The relation between vacuum-to-two-meson form factors as in Eq.
(1.1) and those appearing in heavy-meson decays can occasionally be sharpened
based on chiral-symmetry relations Albaladejo:2016mad .

One of the standard approaches to calculating these two-meson form factors is
using chiral perturbation theory (ChPT), which is a low-energy effective
theory of quantum chromodynamics (QCD) that describes the interaction among
light mesons and baryons. The next-to-leading-order (NLO) ChPT calculation for
the $\pi\pi$ scalar form factor was firstly given in Ref. Gasser:1984gg . Its
two-loop representation and some unitarization schemes were discussed in Ref.
Gasser:1990bv . After that, Refs. Meissner:2000bc ; Lahde:2006wr  performed
more complete studies of the scalar form factors in unitarized chiral
perturbation theory (uChPT), where the results of the NLO ChPT were extended
to a higher energy scale around $1\ \text{GeV}$, which was realized by
involving the channel coupling between the $\pi\pi$ and the $K\bar{K}$ systems
to impose unitarity constraints on the form factors. The reconstruction of the
scalar $\pi\pi$ and $K\bar{K}$ form factors based on a Muskhelishvili–Omnès
representation, relying on phenomenological phase shift input, has by now a
long history Donoghue:1990xh ; Moussallam:1999aq ; DescotesGenon:2000ct ;
Hoferichter:2012wf ; Daub:2015xja , which includes several dedicated
applications in the context of BSM physics searches Daub:2012mu ;
Celis:2013xja ; Monin:2018lee ; Winkler:2018qyg . Extensions beyond $1\
\text{GeV}$ with a new formalism including further inelastic channels were
discussed in Ref. Ropertz:2018stk . Studies of the $\pi\eta$ isovector scalar
form factor (also coupled to $K\bar{K}$) are much rarer Albaladejo:2015aca ;
Albaladejo:2016mad ; Lu:2020qeo , largely due to far less experimental
information on $\pi\eta$ scattering. The $K\pi$, $K\eta$ scalar form factors
up to $2\ \text{GeV}$ were given in Ref. Jamin:2001zq ; Jamin:2000wn ;
Jamin:2006tj ; Bernard:2009ds  with a coupled-channel dispersive analysis. The
two-meson vector form factors for $K\pi$ were first derived in ChPT
Gasser:1983yg , while it was mostly obtained by fitting to the data of semi-
leptonic $\tau$ decays in Refs. Boito:2011tp ; Boito:2010hq ; Boito:2010ak .
There is a number of works for the pion vector form factor based on the Omnès
dispersive representation Guerrero:1997ku ; DeTroconiz:2001rip ; Pich:2001pj ;
deTroconiz:2004yzs ; Guo:2008nc ; Hanhart:2012wi ; Dumm:2013zh ;
Colangelo:2018mtw ; Gonzalez-Solis:2019iod ; Colangelo:2020lcg , ChPT
calculations Gasser:1990bv ; Bijnens:1998fm ; Bijnens:2002hp ; Oller:2000ug ,
a model based on analyticity Bruch:2004py , and in the large-$N_{c}$ limit
Pich:2010sm . Throughout the present study, we will work in the isospin limit,
although also isospin-violating scalar and vector form factors have been
studied in uChPT or using dispersive methods, such as in the context of
$a_{0}$–$f_{0}$ mixing Hanhart:2007bd  or for studies of second-class currents
in $\tau$-decays Descotes-Genon:2014tla ; Escribano:2016ntp .

Unlike the scalar and vector cases, naturally the coupling with the
antisymmetric tensor current does not exist in the SM and conventional ChPT
(the energy-momentum tensor is symmetric, and has been built into ChPT up to
NLO Donoghue:1991qv ; Kubis:1999db ). However, in terms of the research for
BSM physics, for example in the Standard Model Effective Field Theory (SMEFT),
a number of high-dimensional operators including the tensor current are
necessary. Besides the conventional ChPT Lagrangian, additional terms with an
antisymmetric (antisymmetric is implicit in the following discussion of the
tensor part) tensor source was first given in Ref. Cata:2007ns , which is
crucial to calculating the tensor form factors. Recently, dispersive analyses
of tensor form factors in specific channels ($\pi\pi$ Miranda:2018cpf , $\pi
K$ Cirigliano:2017tqn , and for the nucleon Hoferichter:2018zwu ) have been
carried out.

In this work, we will perform a study of all three kinds of two-meson form
factors based on uChPT and dispersion relations. Section 2 gives a brief
introduction to ChPT and its unitarization, where we will discuss how
unitarized meson–meson scattering amplitudes can be obtained by the inverse
amplitude method (IAM). The coupled-channel IAM GomezNicola:2001as  is
modified by removing the imaginary parts of the $t$\- and $u$-channel loops in
order to restore unitarity in coupled-channel systems, which is otherwise
violated in particular around the $\rho$-meson region in the isospin-1 sector.
In Sect. 3, we will calculate the two-meson scalar form factors, which are
then unitarized by the IAM. There, unphysical sub-threshold singularities,
related to the so-called Adler zeros, will show up. To eliminate these
defects, an iteration procedure based on dispersion relations is performed for
each form factor, so that the improved form factors behave well in a wide
energy range $0\ldots 1.2\ \text{GeV}$. In Sects. 4 and 5, we will apply the
same procedure to the calculation of unitarized vector and tensor form
factors, respectively. Some of the form factors obtained in this work are
compared with those that have appeared in earlier works. For each kind of form
factor, we will also briefly introduce their applications in corresponding
phenomenological studies. This includes the application of the $\pi\pi$ form
factor for the $S$-wave-dominated decay $B_{s}\to
f_{0}(980)(\to\pi^{+}\pi^{-})\mu^{+}\mu^{-}$, the application of two-meson
vector form factors in the two-body hadronic decays of a charged lepton
$l\to\phi\phi^{\prime}\nu$, where $\phi$ and $\phi^{\prime}$ denote light
pseudoscalar mesons, and the application of two-meson tensor form factors for
the BSM effects in two-body $\tau$ decays
$\tau\to\phi\phi^{\prime}\nu_{\tau}$. Finally, Sect. 6 contains a summary.
Various technicalities are relegated to the appendices.

##  2 Framework

###  2.1 Chiral Perturbation Theory and Its Unitarization

ChPT Weinberg:1978kz ; Gasser:1983yg ; Gasser:1984gg  provides a systematic
framework to investigate the strong interaction at low energies. It is based
on the spontaneous breaking of the global
$G=\text{SU}(3)_{L}\times\text{SU}(3)_{R}$ symmetry of the QCD Lagrangian, in
the limit of vanishing $u$, $d$, and $s$ quark masses, down to the global
$H=\text{SU}(3)_{V}$ symmetry of the QCD vacuum. This spontaneous symmetry
breaking gives rise to eight pseudo-Nambu–Goldstone bosons (pNGBs), which are
the relevant degrees of freedom at low energies. In fact, it can be proven
that the eight pNGBs construct a space which is isomorphic to the quotient
space $G/H=\text{SU}(3)_{L}\times\text{SU}(3)_{R}/\text{SU}(3)_{V}$
Coleman:1969sm . This isomorphism enables one to parametrize any of these
quotient elements $U\in G/H$ by the eight pNGBs as

$\displaystyle U={\rm exp}\left[\frac{i\phi}{F_{0}}\right],$ (2.1)

where $F_{0}$ is the pion decay constant in the chiral limit, and

$\displaystyle\phi=\phi^{a}\lambda^{a}=\left(\begin{array}[]{ccc}\pi^{0}+\eta/\sqrt{3}&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\eta/\sqrt{3}&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-2\eta/\sqrt{3}\\\ \end{array}\right)$ (2.5)

contains the pNGB octet. Here, exact isospin symmetry is assumed, which turns
off the $\pi^{0}$-$\eta$ mixing for simplicity. We use the convention that
under $\text{SU}(3)_{L}\times\text{SU}(3)_{R}$ transformations $U$ behaves as
$U\to RUL^{\dagger}$, with $R\in\text{SU}(3)_{R}$ and $L\in\text{SU}(3)_{L}$.
With $U$ as the building block, the leading-order (LO) effective Lagrangian of
ChPT is constructed as

$\displaystyle\mathcal{L}^{(2)}=\frac{F_{0}^{2}}{4}\langle
D_{\mu}UD^{\mu}U^{\dagger}\rangle+\frac{F_{0}^{2}}{4}\langle\chi
U^{\dagger}+\chi^{\dagger}U\rangle~{},$ (2.6)

where $\langle\ldots\rangle$ denotes the trace in SU(3) flavor space,
$\chi=2B_{0}(M+s)$, with $M$ the quark mass matrix,
$D_{\mu}U\equiv\partial_{\mu}U-ir_{\mu}U+iUl_{\mu}$, and $s,l_{\mu},r_{\mu}$
are the scalar, the left-handed, and the right-handed external sources. The
parameter $B_{0}$ is proportional to the QCD quark condensate,
$3F_{0}^{2}B_{0}=-\langle{\bar{u}}u+\bar{d}d+\bar{s}s\rangle$.

Applying Eq. (2.6) at one-loop produces ultraviolet (UV) divergences that can
be regulated using dimensional regularization and then reabsorbed into the
low-energy constants (LECs) in the next-to-leading-order (NLO) Lagrangian
Gasser:1983yg ; Gasser:1984gg :

$\displaystyle{\cal L}^{(4)}$ $\displaystyle=$ $\displaystyle L_{1}\langle
D_{\mu}UD^{\mu}U^{\dagger}\rangle^{2}+L_{2}\langle
D_{\mu}U\left(D_{\nu}U\right)^{\dagger}\rangle\langle
D^{\mu}U(D^{\nu}U)^{\dagger}\rangle+L_{3}\langle
D_{\mu}U(D^{{\mu}}U)^{\dagger}D_{{\nu}}U(D^{\nu}U)^{\dagger}\rangle$ (2.7)
$\displaystyle+L_{4}\langle
D_{\mu}U(D^{{\mu}}U)^{\dagger}\rangle\langle\chi^{\dagger}U+U\chi^{\dagger}\rangle+L_{5}\langle
D_{\mu}UD^{{\mu}}U^{\dagger}(\chi^{\dagger}U+U\chi^{\dagger})\rangle$
$\displaystyle+L_{6}\langle\chi^{\dagger}U+U\chi^{\dagger}\rangle^{2}+L_{7}\langle\chi^{\dagger}U-U\chi^{\dagger}\rangle^{2}+L_{8}\langle\chi
U^{\dagger}\chi U^{\dagger}+U\chi^{\dagger}U\chi^{\dagger}\rangle$
$\displaystyle-iL_{9}\langle
R_{\mu\nu}D^{\mu}U(D^{\nu}U)^{\dagger}+L_{\mu\nu}(D^{\mu}U)^{\dagger}D^{\nu}U\rangle+L_{10}\langle
L_{\mu\nu}UR^{\mu\nu}U^{\dagger}\rangle~{},$

where $L_{\mu\nu}$ and $R_{\mu\nu}$ are field-strength tensors of the external
sources

$\displaystyle
L_{\mu\nu}=\partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}]~{},~{}~{}~{}~{}R_{\mu\nu}=\partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}]~{}.$
(2.8)

The UV-finite, scale-dependent renormalized LECs $\\{L_{i}^{r}\\}$ are defined
as $L_{i}^{r}=L_{i}-\Gamma_{i}\lambda$, with the UV-divergent parts
proportional to

$\displaystyle\lambda=\frac{1}{32\pi^{2}}\left[\frac{2}{d-4}-\ln(4\pi)+\gamma-1\right],$
(2.9)

and the nonzero values for their coefficients $\Gamma_{i}$ relevant to this
work are

$\displaystyle\Gamma_{1}=\frac{3}{32},\;\;\Gamma_{2}=\frac{3}{16},\;\;\Gamma_{4}=\frac{1}{8},\;\;\Gamma_{5}=\frac{3}{8},\;\;\Gamma_{6}=\frac{11}{144},\;\;\Gamma_{8}=\frac{5}{48},\;\;\Gamma_{9}=\frac{1}{4},\;\;\Gamma_{10}=-\frac{1}{4}.$
(2.10)

The scale dependence of these LECs is given by

$\displaystyle L_{i}^{r}(\mu_{2})$ $\displaystyle=$ $\displaystyle
L_{i}^{r}(\mu_{1})-\frac{\Gamma_{i}}{32\pi^{2}}\ln\left(\frac{\mu_{2}^{2}}{\mu_{1}^{2}}\right).$
(2.11)

Some details of the loop functions occurring in the one-loop ChPT calculations
can be found in Appendix A. Table 1 collects the numerical results for the
$L_{i}^{r}$ at the scale $\mu=M_{\rho}$ that were obtained previously. The
first column corresponds to the analysis up to ${\cal O}(p^{4})$ in ChPT
Gasser:1983yg ; Bijnens:1994ie , and the second refers to the fit with ${\cal
O}(p^{6})$ corrections Bijnens:2014lea . The third column corresponds to the
previous fit of meson–meson scattering phase shifts and inelasticities in the
coupled-channel IAM GomezNicola:2001as , which we will discuss further below.

Table 1: Low-energy constants $L_{i}^{r}$ (in units of $10^{-3}$) at $\mu=M_{\rho}$. For the description of the fits given in the last two columns, we refer to Sect. 2.3.  | $L_{i}^{r}$ |  ${\cal O}(p^{4})$ ChPT Gasser:1983yg ; Bijnens:1994ie  |  ${\cal O}(p^{6})$ ChPT Bijnens:2014lea  | uChPT GomezNicola:2001as  | uChPT Fit 1 | uChPT Fit 2
---|---|---|---|---|---|---
| $L_{1}^{r}$ | $0.4\pm 0.3$ | $1.11\pm 0.10$ | $0.56\pm 0.10$ | $0.52\pm 0.01$ | $0.55\pm 0.02$
| $L_{2}^{r}$ | $1.35\pm 0.3$ | $1.05\pm 0.17$ | $1.21\pm 0.10$ | $1.08\pm 0.01$ | $1.07\pm 0.03$
| $L_{3}$ | $-3.5\pm 1.1$ | $-3.82\pm 0.30$ | $-2.79\pm 0.24$ | $-2.72\pm 0.01$ | $-2.76\pm 0.03$
| $L_{4}^{r}$ | $-0.3\pm 0.5$ | $1.87\pm 0.53$ | $-0.36\pm 0.17$ | $-0.27\pm 0.01$ | $-0.20\pm 0.02$
| $L_{5}^{r}$ | $1.4\pm 0.5$ | $1.22\pm 0.06$ | $1.4\pm 0.5$ | $1.36\pm 0.19$ | $1.13\pm 0.61$
| $L_{6}^{r}$ | $-0.2\pm 0.3$ | $1.46\pm 0.46$ | $0.07\pm 0.08$ | 0.07 (fixed) | 0.07 (fixed)
| $L_{7}$ | $-0.4\pm 0.2$ | $-0.39\pm 0.08$ | $-0.44\pm 0.15$ | $-0.60\pm 0.04$ | $-0.52\pm 0.12$
| $L_{8}^{r}$ | $0.9\pm 0.3$ | $0.65\pm 0.07$ | $0.78\pm 0.18$ | $0.89\pm 0.09$ | $0.68\pm 0.29$
| $L_{9}^{r}$ | $6.9\pm 0.7$ | – | – | – | –
| $L_{10}^{r}$ | $-5.5\pm 0.7$ | – | – | – | –

The power counting of ChPT is organized according to the increasing power of
the ratio $p/\Lambda_{\chi}$ given in terms of a typical small pNGB momentum
$p$ of the order of the pNGB mass and the chiral symmetry breaking scale
$\Lambda_{\chi}\sim 4\pi F_{\pi}$ Manohar:1983md , where $F_{\pi}\approx 92.1\
\text{MeV}$ is the physical pion decay constant. Therefore, the perturbative
expansion in ChPT is expected to break down when $p/\Lambda_{\chi}\sim 1$.
Moreover, a perturbative expansion in powers of momenta to any finite order
cannot describe the physics of resonances, which are given by poles of the
$S$-matrix on unphysical Riemann sheets. Thus, the masses of the lowest
resonances in each meson–meson scattering channel limit the applicability
region of ChPT in the corresponding sector.

Unitarization (or resummation) is a systematic prescription intended to extend
the applicability of ChPT to higher energies, say $1\ \text{GeV}$, by
modifying the perturbative expression such that it satisfies the full instead
of only the perturbative unitarity requirement of quantum field theory. Since
unitarity is nonperturbative in its nature, in this way the lowest meson
resonances may also be described. Note, however, that unitarization comes with
a price: as there are various unitarization schemes, some scheme-dependence is
introduced. Also, crossing symmetry is often broken in such approaches.

Let us start with a simple example. In the case of $2\to 2$ multi-channel
scattering with the momenta of initial and final particles as $p_{1},p_{2}$
and $p_{3},p_{4}$, respectively, we can define the partial-wave amplitude
$T_{J}(s)$ with total angular momentum $J$ from the full scattering amplitude
$T(s,\cos\theta)$ by

$\displaystyle
T_{J}(s)=\frac{1}{2}\int_{-1}^{+1}P_{J}(\cos\theta)T(s,\cos\theta)d\cos\theta~{},$
(2.12)

where $s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2}$ is one of the usual Mandelstam
variables, $\theta$ is the angle between ${\vec{p}}_{3}$ and ${\vec{p}}_{1}$
in the center-of-mass (c.m.) frame, and $P_{J}(\cos\theta)$ is the Legendre
polynomial of order $J$. If we consider only two-particle intermediate states,
then the partial-wave amplitudes should satisfy the following unitarity
relation:

$\displaystyle\operatorname{Im}\left[T_{J}(s)\right]_{ij}=\sum_{kk^{\prime}}\left[T_{J}(s)\right]_{ik}\delta_{kk^{\prime}}\frac{\left|\vec{p}_{k}\right|}{8\pi\sqrt{s}}\Theta(s-s_{\mathrm{th}}^{i})\left[T_{J}(s)\right]_{k^{\prime}j}^{*},$
(2.13)

where time reversal invariance is assumed. The indices $i,j,k$, and
$k^{\prime}$ denote different scattering channels, $\left|\vec{p}_{k}\right|$
is the modulus of the c.m. 3-momentum in the $k$th channel, and
$s_{\mathrm{th}}^{i}=(M_{a_{i}}+M_{b_{i}})^{2}$ is the production threshold of
the $i$th channel particles. Equation (2.13) can be written in matrix form:

$\displaystyle\operatorname{Im}\left[T_{J}(s)\right]=T_{J}(s)\Sigma(s)T_{J}^{*}(s)~{},~{}~{}~{}\left[\Sigma(s)\right]_{ij}\equiv\delta_{ij}\frac{\left|\vec{p}_{i}\right|}{8\pi\sqrt{s}}\Theta(s-s_{\mathrm{th}}^{i})~{}.$
(2.14)

Multiplying $T_{J}^{-1}$ and $T_{J}^{*-1}$ on both sides leads to

$\displaystyle\frac{1}{2i}\left[T_{J}^{*-1}(s)-T_{J}^{-1}(s)\right]=-\operatorname{Im}T_{J}^{-1}(s)=\Sigma(s).$
(2.15)

In a fixed-order ChPT calculation, the relation (2.13) is only satisfied
perturbatively. For instance, substituting the ChPT expression of $T_{J}(s)$
at $\mathcal{O}(p^{4})$ to both sides will result in the breaking of the
equality at the $\mathcal{O}(p^{6})$ level. There are many ways to recover
exact unitarity, and in this work we mainly adopt the multi-channel IAM
outlined in Ref. GomezNicola:2001as . The procedure is as follows. First, the
partial-wave amplitude $T_{J}$ can be written as

$T_{J}=\left(\mathrm{Re}[T_{J}^{-1}]-i\Sigma\right)^{-1}.$ (2.16)

Since $\Sigma$ is a known matrix, we just need to calculate
$\mathrm{Re}[T_{J}^{-1}]$, i.e., the real part of the inverse matrix
$T_{J}^{-1}$ in order to obtain the full $T_{J}$. The IAM is a way to
calculate $\mathrm{Re}[T_{J}^{-1}]$ approximately Truong:1988zp ;
Dobado:1996ps ; GomezNicola:2001as . The idea is to start with the
perturbative expansion of $T_{J}$: $T_{J}=T_{J}^{(2)}+T_{J}^{(4)}+\ldots$,
where the superscripts $(2),(4)$ denote the order of the amplitude, which are
${\cal O}(p^{2})$ and ${\cal O}(p^{4})$, respectively. The corresponding
inverse matrix can be perturbatively expanded as

$\displaystyle
T_{J}^{-1}=T_{J}^{(2)-1}\left(1-T_{J}^{(4)}T_{J}^{(2)-1}+\ldots\right).$
(2.17)

Next, using the fact that $T_{J}^{(2)}$ is real (again assuming time reversal
invariance), we have

$\mathrm{Re}[T_{J}^{-1}]=T_{J}^{(2)-1}\left(1-\mathrm{Re}[T_{J}^{(4)}]T_{J}^{(2)-1}+\ldots\right).$
(2.18)

Since both $T_{J}^{(2)-1}$ and $\mathrm{Re}[T_{J}^{(4)}]$ are calculable in
ChPT, we have already achieved our goal. However, we can further simplify the
expression by plugging it into Eq. (2.16), which results in

$\displaystyle T_{J}$ $\displaystyle=$
$\displaystyle\left[T_{J}^{(2)-1}\left(1-\mathrm{Re}[T_{J}^{(4)}]T_{J}^{(2)-1}+\ldots\right)-i\Sigma\right]^{-1}$
(2.19) $\displaystyle=$ $\displaystyle
T_{J}^{(2)}\left(T_{J}^{(2)}-\mathrm{Re}[T_{J}^{(4)}]-iT_{J}^{(2)}\Sigma
T_{J}^{(2)}+\ldots\right)^{-1}T_{J}^{(2)}$ $\displaystyle=$ $\displaystyle
T_{J}^{(2)}\left(T_{J}^{(2)}-\mathrm{Re}[T_{J}^{(4)}]-i\mathrm{Im}[T_{J}^{(4)}]+\ldots\right)^{-1}T_{J}^{(2)}$
$\displaystyle\approx$ $\displaystyle
T_{J}^{(2)}\left(T_{J}^{(2)}-T_{J}^{(4)}\right)^{-1}T_{J}^{(2)}~{}.$

In the third line we have used the perturbative unitarity relation, which can
be obtained by power expansion of Eq. (2.14). The last line is the central
equation for the IAM at NLO. It effectively implies that we can use
perturbation theory to calculate $T_{J}^{(2)}$ and $T_{J}^{(4)}$, and resum
their effects to all orders. An advantage of this method is that the
resummation is a simple matrix inversion, and does not involve any extra
integral equation.

Figure 1: A Feynman diagram interpretation for the IAM method: summing up all
the $s$-channel ${\cal O}(p^{4})$ scattering amplitudes to all orders.

There is also a simple Feynman diagram interpretation of Eq. (2.19). As shown
in Fig. 1, we simply sum up all the $s$-channel ${\cal O}(p^{4})$ corrections
to all orders:

$\displaystyle T$ $\displaystyle=$ $\displaystyle
T^{(2)}+T^{(4)}+T^{(4)}(T^{(2)})^{-1}T^{(4)}+T^{(4)}(T^{(2)})^{-1}T^{(4)}(T^{(2)})^{-1}T^{(4)}+\ldots$
(2.20) $\displaystyle=$ $\displaystyle
T^{(2)}+\left[1+T^{(4)}(T^{(2)})^{-1}+T^{(4)}(T^{(2)})^{-1}T^{(4)}(T^{(2)})^{-1}+\ldots\right]T^{(4)}$
$\displaystyle=$ $\displaystyle
T^{(2)}+\left(1-T^{(4)}(T^{(2)})^{-1}\right)^{-1}T^{(4)}$ $\displaystyle=$
$\displaystyle\left(1-T^{(4)}(T^{(2)})^{-1}\right)^{-1}\left[\left(1-T^{(4)}(T^{(2)})^{-1}\right)T^{(2)}+T^{(4)}\right]$
$\displaystyle=$ $\displaystyle
T^{(2)}\left(T^{(2)}-T^{(4)}\right)^{-1}T^{(2)},$

which gives exactly the IAM result. The explicit expressions for the
scattering amplitudes classified by definite isospin states can be found in
Appendix B.

An obvious shortcoming of the IAM formula is that it leads to a peak when the
determinant $\mathrm{det}\left[T^{(2)}-T^{(4)}\right]$ approaches a minimum.
This peak may be unphysical and, in terms of dispersion relations, is due to
the failure to incorporate the pole contributions from the so-called Adler
zero of the partial wave in the sub-threshold region. This problem can be
satisfactorily resolved in the case of the single-channel IAM
GomezNicola:2007qj  but not for coupled channels, see Appendix C for a brief
explanation of the procedure. In Sect. 3.3, we will introduce an effective
solution based on dispersion relations for coupled-channel systems.

###  2.2 Unitarity

Figure 2: Check of the unitarity relations for several $T$-matrix elements in
coupled-channel uChPT, with the LECs taking the central values given in Ref.
GomezNicola:2001as . The indices of $T_{J}^{I}$ refer to the isospin $I$ and
angular momentum $J$ partial wave.

Although the uChPT was constructed to fulfill the unitarity relation
$\mathrm{Im}T=T\Sigma T^{\dagger}$, the one-step IAM solution of the partial
waves actually satisfies the exact relation only above the highest threshold.
In general, the unitarity relation below the highest threshold is broken due
to the mixing between the left-hand and right-hand cuts of different matrix
elements in $T$ during the process of matrix inversion. This phenomenon occurs
since all the particles in the initial and final states of the
$T^{(4)}$-matrix are treated as on shell Du:2018gyn . Such a problem is well-
known and also exists in other methods of unitarization Iagolnitzer:1973fq ;
Badalian:1981xj ; Xiao:2001pt ; Xiao:2001sw ; GomezNicola:2001as . However,
depending on the values of LECs, such unitarity violation is usually very
small and would not cause any real problem in practical applications of IAM
results. With the LEC values reported in Ref. GomezNicola:2001as , the only
exception is the $I=1,J=1$ channel, see Fig. 2. The imaginary part of the
partial-wave amplitude in this channel is peaked at $\sqrt{s}\approx 0.77\
\text{GeV}$ due to the existence of the $\rho$-resonance, and it turns out
that the IAM approach leads to a breaking of the unitarity relation by as much
as 20% around the $\rho$-peak.

As discussed in Ref. GomezNicola:2001as , this problem can in principle be
solved by adopting a multi-step unitarization approach, namely to take the
dimension of the $T$-matrix as a function of $s$, which changes by one unit
every time $s$ crosses a threshold. By doing so one explicitly avoids the
mixing of left-hand and right-hand cuts between different matrix elements
below the largest thresholds, and thus the unitarity relation is exactly
satisfied in all regions. One disadvantage of this approach is that one cannot
study the scattering amplitudes (and their associated form factors) below
their respective production thresholds because their corresponding matrix
elements simply do not exist in the scattering matrix. It is therefore highly
desirable to search for a prescription that allows for a simultaneous study of
form factors below and above thresholds, and at the same time avoids the
mixing of left-hand and right-hand cuts as much as possible. In the following,
we demonstrate a simple procedure that satisfies both requirements.

Figure 3: Unitarity relations are fulfilled exactly after removing the
imaginary parts of the troublesome $t$\- and $u$-channel loops. Here, the LECs
take the central values given in Ref. GomezNicola:2001as .

Let us focus on the $(I,J)=(1,1)$ $\pi\pi$ scattering, which couples to the
$K\bar{K}$ channel. At NLO ChPT, the $K\bar{K}\to K\bar{K}$ scattering
receives contributions from $t$\- and $u$-channel meson loops, where the
Mandelstam variables $t$ and $u$ are defined as $t=(p_{1}-p_{3})^{2}$ and
$u=(p_{1}-p_{4})^{2}$. If the initial and final kaons are on shell, the branch
point for the $K\bar{K}\to K\bar{K}$ partial wave due to the $\pi\pi$
$t$-channel loop is at $s=4(M_{K}^{2}-M_{\pi}^{2})$, which is below the
$K\bar{K}$ threshold, but above the $\pi\pi$ threshold. Similarly, the branch
point of the $t$-channel $\pi\eta$ loop occurs at
$s=4M_{K}^{2}-(M_{\pi}+M_{\eta})^{2}$, again above the $\pi\pi$ threshold. If
the kaons are off shell, such singularities would not be on the physical
Riemann sheet of $\pi\pi$ scattering Du:2018gyn , and would not cause any
problem. However, in the IAM treatment, the kaons are on shell, leading to an
overlap between such left-hand cuts with the $\pi\pi$ right-hand cut and thus
a violation of unitarity. We find that if we remove the imaginary part of the
troublesome $t$\- and $u$-channel loops, unitarity can be exactly maintained,
see Fig. 3, where the curves for ${\rm Im}T$ and $T\Sigma T^{\dagger}$
coincide. These loops include the $t$-channel $\pi\pi$ and $\pi\eta$ loops in
$K\bar{K}\to K\bar{K}$ (both $I=0$ and $I=1$), the $t$\- and $u$-channel
$\pi\pi$ loops in $\eta\eta\to\eta\eta$, the $t$\- and $u$-channel $\pi K$
loops in $K\bar{K}\to\eta\eta$, and the $t$-channel $\pi\pi$ and $u$-channel
$\pi K$ loops in $K\eta\to K\eta$.

###  2.3 Global Fit of LECs

Figure 4: After removing the unitarity-violating contributions, the LECs are
fitted to data. The narrow bands are the results from Fit 2 with errors of all
data pointed increased by adding 5% of the central values. The dotted lines
are from the best-fit values of Fit 1, i.e., fitting to data with the original
data errors. The results computed using the central values of LECs given in
Ref. GomezNicola:2001as  are shown as dashed lines. The $\pi\pi$ phase shift
data are from Ref. GarciaMartin:2011cn . For the $K\pi$ phase shifts, the data
are taken from Refs. Mercer:1971kn  (up triangles), Bingham:1972vy  (down
triangles), Linglin:1973ci  (circles), Baker:1974kr  (pentagons),
Estabrooks:1977xe  (rectangles), Aston:1987ir  (diamonds),
delAmoSanchez:2010fd  (left triangles), and Ablikim:2015mjo  (right
triangles). The $\pi\pi\to K\bar{K}$ data are from Refs. Martin:1979gm
(rectangles) and Cohen:1980cq  (circles). The $\eta_{0}^{0}$ data are from
Ref. Kaminski:1996da . The data for $d\sigma_{\pi\eta}/d\sqrt{s}$, as well as
the corresponding background, are taken from Ref. Armstrong:1991rg .

After the modification given above, the values of the LECs need to be refixed.
We fit to the following data sets using the MINUIT function minimization and
error analysis package James:1975dr ; iminuit1 ; iminuit2 : the $\pi\pi$
scattering phase shifts are taken from the dispersive analysis compiled in
Ref. GarciaMartin:2011cn  (which is perfectly compatible with the alternative
Roy equation analyses of Refs. Ananthanarayan:2000ht ; Colangelo:2001df ;
Caprini:2011ky  at the level of accuracy aimed for with the IAM);111The table
in Appendix D of Ref. GarciaMartin:2011cn  gives the $\pi\pi$ scattering phase
shifts up to $970\ \text{MeV}$. The $\delta_{0}^{0}$ data points above this
energy were read off from the band in Fig. 15 of this reference, and those of
$\delta_{1}^{1}$ were taken from Ref. Kaminski:2006qe . The latter reference
also provides an analysis of $\delta_{0}^{2}$, which, however, does not match
exactly to the data given in Ref. GarciaMartin:2011cn . Thus, for
$\delta_{0}^{2}$, we only use those of Ref. GarciaMartin:2011cn  up to $970\
\text{MeV}$. the data for the inelasiticity $\eta_{0}^{0}$ are taken from the
analysis of Ref. Kaminski:1996da ; the $\pi\pi\to K\bar{K}$ data are from
Refs. Martin:1979gm ; Cohen:1980cq  (cf. also Refs. Buettiker:2003pp ;
Pelaez:2018qny ); the $K\pi$ phase shifts are taken from Refs. Mercer:1971kn ;
Bingham:1972vy ; Linglin:1973ci ; Baker:1974kr ; Estabrooks:1977xe ;
Aston:1987ir ; delAmoSanchez:2010fd ; Ablikim:2015mjo  (cf. also the
corresponding dispersive analyses Buettiker:2003pp ; Pelaez:2016tgi ;
Pelaez:2020gnd ); the data for the $\pi\eta$ invariant mass distribution are
taken from Ref. Armstrong:1991rg , and the background is extracted from the
corresponding curve in that reference.

We notice that in the NLO ChPT amplitudes for $\pi\pi\to\pi\pi$, $K\pi\to
K\pi$, and $K\bar{K}\to K\bar{K}$, $L_{6}^{r}$ and $L_{8}^{r}$ always appear
as the same linear combination $2L_{6}^{r}+L_{8}^{r}$. Since most of the
available data are on these channels, it is difficult to fix $L_{6}^{r}$ and
$L_{8}^{r}$ independently. Thus, we fix $L_{6}^{r}$ to the central value given
in Ref. GomezNicola:2001as , and fit the other parameters to the above data.
The $\pi\eta$ invariant mass distribution is fitted with the following
expression Oller:1998hw ; GomezNicola:2001as :

$\frac{d\sigma_{\pi\eta}}{d\sqrt{s}}=c\,q_{\rm
cm}|(T_{0}^{1})_{\pi\eta\to\pi\eta}|^{2}+\text{background},$ (2.21)

where $c$ is a normalization constant to be fitted, $q_{\rm cm}$ is the
$\pi\eta$ c.m. momentum, and the background is extracted from the experimental
analysis Armstrong:1991rg . A direct fit to all these data sets leads to a
value of $\chi^{2}/\text{dof}=7.76$ with the LECs given in the column “Fit 1”
in Table 1. The large $\chi^{2}/\text{dof}$ value is due to the inconsistency
among the data sets. Following Refs. Oller:1998zr ; GomezNicola:2001as , we
increase the errors of the data points by hand, and find an additional error
of $5\%$ (of the central values) to all data points leads to
$\chi^{2}/\text{dof}=1.21$. The LECs from such a fit are listed in the last
column in Table 1, labelled as “Fit 2”. A comparison of these fits, as well as
the results using the central values of LECs in Ref. GomezNicola:2001as , to
the data is shown in Fig. 4. The errors propagated from the data in Fit 2 are
plotted as bands, which are rather narrow. One sees that the increase of
$\delta_{0}^{0}$ around the $K\bar{K}$ threshold is more abrupt in uChPT than
that from the dispersive analysis GarciaMartin:2011cn . Other than that, the
data are well described using these different sets of LEC values.

An additional remark is in order. We find that when the LECs take certain
values, $T^{(2)}-T^{(4)}$ in the $(I,J)=(0,0)$ channel can have zeros in the
physical region using this modified version of coupled-channel IAM. These are
not the Adler zeros in the single-channel scattering amplitudes, and can lead
to sharp kinks in phase shifts and other observables at the zeros. Such
unphysical singularities also exist in the original coupled-channel IAM.
However, in that case, due to the presence of nonvanishing imaginary parts
from the unphysical left-hand cuts, the singularities are in the complex
$s$-plane, and thus lead to smoother kinks.222See the kink around $800\
\text{MeV}$ in the solid line of the $\delta_{0}^{0}$ plot in Fig. 2 of Ref.
GomezNicola:2001as . Nevertheless, we checked that the best-fit LECs in both
Fit 1 and Fit 2, as well as those from Ref. GomezNicola:2001as , do not have
that problem. We will use the central values of these fits (the three last
columns in Table 1) in the study of form factors in the following to estimate
the uncertainty of this method.

In what follows, we shall apply the idea above to calculate the two-meson
scalar, vector, and tensor form factors. We shall also consider a dispersion-
theoretical improvement that will get rid of the unphysical sub-threshold
singularities due to Adler zeros.

##  3 Scalar form factors

Figure 5: The imaginary part of a form factor is caused by the on shell
configuration (cutting along the dashed line) of the intermediate states. In
the elastic region, the phase of a form factor is the same as that of the
scattering amplitude.

In this section, we will give a systematic calculation for the two-meson
scalar form factors in uChPT, where the IAM approach is applied. Note that the
unitarization of the two-loop scalar (and vector) pion form factor was already
discussed in Ref. Gasser:1990bv . The scalar form factor of a two-meson system
is defined by the matrix element

$B_{0}F_{S,i}^{\bar{q}^{\prime}q}(s)\equiv\left\langle
a_{i}(p_{a_{i}})b_{i}(p_{b_{i}})\right|\bar{q}^{\prime}q\left|0\right\rangle,$
(3.1)

where the subscript $i$ is again the channel index, $\\{a_{i},b_{i}\\}$ are
the two mesons in channel $i$, and $s=(p_{a_{i}}+p_{b_{i}})^{2}$. The
unitarity relation of the scalar form factor reads

$2\mathrm{Im}F_{S,i}^{\bar{q}^{\prime}q}(s)=\sum_{j}\frac{|\vec{p}_{j}|}{4\pi\sqrt{s}}\left[T_{0}^{*}(s)\right]_{ji}\Theta(s-s_{\mathrm{th}}^{j})F_{S,j}^{\bar{q}^{\prime}q}(s)~{},$
(3.2)

where the subscript “0” at the partial-wave amplitude denotes the $J=0$
component. A sketch is shown in Fig. 5, where the imaginary part of the form
factor is caused by the on-shell configuration of the intermediate states.
Furthermore, time reversal invariance leads to $(T_{0})_{ij}=(T_{0})_{ji}$.
Now, we can simplify the expression above by taking $F^{\bar{q}^{\prime}q}(s)$
as a column vector in the channel space and write Eq. (3.2) as a matrix
equation

$\mathrm{Im}F^{\bar{q}^{\prime}q}_{S}(s)=T_{0}^{*}(s)\Sigma(s)F^{\bar{q}^{\prime}q}_{S}(s)~{}.$
(3.3)

This is the exact unitarity relation of the multi-channel scalar form factor.
We can also derive the perturbative unitarity relation by expanding
$F^{\bar{q}^{\prime}q}(s)$ according to the chiral power counting

$F_{S}=F_{S}^{(0)}+F_{S}^{(2)}+\ldots~{},$ (3.4)

where the superscript $\bar{q}^{\prime}q$ and the argument $s$ are suppressed
for simplicity. Here we define the leading term of the expansion to be
$\mathcal{O}(p^{0})$ because it is not suppressed by the chiral expansion
parameter. Meanwhile, $T_{0}=T_{0}^{(2)}+T_{0}^{(4)}+\ldots$. Therefore, the
perturbative unitarity relation reads

$\mathrm{Im}F_{S}^{(0)}=0,\qquad\mathrm{Im}F_{S}^{(2)}=T_{0}^{(2)}\Sigma
F_{S}^{(0)},\qquad\ldots.$ (3.5)

###  3.1 ChPT Result

We have to first compute the ChPT results for the scalar form factor up to
${\cal O}(p^{2})$ as input to the IAM formula. For that purpose, we need to
express the scalar current in terms of ChPT fields. The scalar current in QCD
is defined as

$S_{ij}(x)\equiv\bar{q}_{i}(x)q_{j}(x)~{},$ (3.6)

where $q=(\begin{array}[]{ccc}u&d&s\end{array})^{T}$. To obtain the ChPT
version of this current, we start with the QCD Lagrangian and promote its
quark mass matrix $M$ to a general matrix $X_{q}$,

$\mathcal{L}_{QCD}=\bar{q}_{L}iD\\!\\!\\!\\!/\>q_{L}+\bar{q}_{R}iD\\!\\!\\!\\!/\>q_{R}-\bar{q}_{R}X_{q}q_{L}-\bar{q}_{L}X_{q}^{\dagger}q_{R}-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}.$
(3.7)

The scalar current defined in Eq. (3.6) is then obtained by taking the partial
derivative of the Lagrangian with respect to matrix elements of $X_{q}$, which
results in

$S_{ij}=-\left.\left(\frac{\partial\mathcal{L}_{QCD}}{\partial(X_{q})_{ij}}+\frac{\partial\mathcal{L}_{QCD}}{\partial(X_{q}^{\dagger})_{ij}}\right)\right|_{X_{q}=M}.$
(3.8)

We can now derive the scalar current in ChPT by applying the formula above to
the chiral Lagrangian. We can write the scalar current as
$S_{ij}=S_{ij}^{(2)}+S_{ij}^{(4)}+\ldots$, where $S_{ij}^{(n)}$ is defined as
the scalar current derived from the chiral Lagrangian $\mathcal{L}^{(n)}$. The
outcome for the scalar currents, up to $S^{(4)}$, is

$\displaystyle S_{ij}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{F_{0}^{2}B_{0}}{2}[U^{\dagger}+U]_{ji},$ $\displaystyle
S_{ij}^{(4)}$ $\displaystyle=$
$\displaystyle-2B_{0}L_{4}\langle\partial_{\mu}U\partial^{\mu}U^{\dagger}\rangle\big{[}U^{\dagger}+U\big{]}_{ji}-2B_{0}L_{5}\big{[}U^{\dagger}\partial_{\mu}U\partial^{\mu}U^{\dagger}+\partial_{\mu}U\partial^{\mu}U^{\dagger}U\big{]}_{ji}$
(3.9) $\displaystyle-8B_{0}^{2}L_{6}\langle
MU^{\dagger}+UM^{\dagger}\rangle\big{[}U^{\dagger}+U\big{]}_{ji}-8B_{0}^{2}L_{7}\langle
MU^{\dagger}-UM^{\dagger}\rangle\big{[}U^{\dagger}-U\big{]}_{ji}$
$\displaystyle-8B_{0}^{2}L_{8}\big{[}UM^{\dagger}U+U^{\dagger}MU^{\dagger}\big{]}_{ji}.$

In particular, the components of $S^{(2)}$ are

$\displaystyle\bar{u}u$ $\displaystyle=$ $\displaystyle
B_{0}\bigg{[}-F_{0}^{2}+K^{+}K^{-}+\frac{\pi^{0}\eta}{\sqrt{3}}+\frac{1}{6}\eta^{2}+\frac{1}{2}(\pi^{0})^{2}+\pi^{+}\pi^{-}\bigg{]}+\ldots,$
$\displaystyle\bar{d}d$ $\displaystyle=$ $\displaystyle
B_{0}\bigg{[}-F_{0}^{2}+K^{0}\bar{K}^{0}-\frac{\pi^{0}\eta}{\sqrt{3}}+\frac{1}{6}\eta^{2}+\frac{1}{2}(\pi^{0})^{2}+\pi^{+}\pi^{-}\bigg{]}+\ldots,$
$\displaystyle\bar{s}s$ $\displaystyle=$ $\displaystyle
B_{0}\bigg{[}-F_{0}^{2}+K^{0}\bar{K}^{0}+K^{+}K^{-}+\frac{2}{3}\eta^{2}\bigg{]}+\ldots,$
$\displaystyle\bar{u}d$ $\displaystyle=$ $\displaystyle
B_{0}\bigg{[}K^{0}K^{-}+\sqrt{\frac{2}{3}}\eta\pi^{-}\bigg{]}+\ldots,$
$\displaystyle\bar{u}s$ $\displaystyle=$ $\displaystyle
B_{0}\bigg{[}\bar{K}^{0}\pi^{-}-\frac{K^{-}\eta}{\sqrt{6}}+\frac{K^{-}\pi^{0}}{\sqrt{2}}\bigg{]}+\ldots,$
$\displaystyle\bar{d}s$ $\displaystyle=$ $\displaystyle
B_{0}\bigg{[}-\frac{\bar{K}^{0}\eta}{\sqrt{6}}-\frac{\bar{K}^{0}\pi^{0}}{\sqrt{2}}+K^{-}\pi^{+}\bigg{]}+\ldots.$
(3.10)

The ChPT prediction for scalar form factors is then obtained by calculating
the matrix elements for the currents in Eq. (3.10) with respect to two-meson
states up to one loop, as shown in Fig. 6. The full analytical results can be
found in Appendix D.

Figure 6: Feynman diagrams for a generic form factor at tree and one-loop
level. The wave function renormalization diagrams are not shown here.

###  3.2 Unitarization

If we restrict ourselves to one single channel, the IAM unitarization formula
for form factors (scalar, vector, and tensor) can be derived rigorously from a
dispersion relation in complete analogy to the derivation of the single-
channel IAM formula for partial waves, see Appendix C for more details (for an
early application of the single-channel IAM to scalar and vector form factors,
see Ref. Truong:1988zp ). For coupled-channel form factors, a dispersive
derivation of the IAM is not available, so here we offer a more empirical
derivation of the unitarization formula, which we expect at least to work well
above the highest production threshold of the coupled channels considered
here, which is sufficient for the applications to most of the interesting
processes we mentioned in the Introduction.

From Eq. (3.3), one notices that a possible solution to this unitarity
relation for the scalar form factor is $F_{S}=T_{0}A$, where $A$ is a real
vector. The proof is simple:

$T_{0}^{*}\Sigma F_{S}=T_{0}^{*}\Sigma
T_{0}A=(\mathrm{Im}T_{0})A=\mathrm{Im}F_{S}.$ (3.11)

In the second equality we have used the unitarity relation for $T_{0}$. So the
question is how to choose the form of the real vector $A$ such that its
expansion reproduces the ChPT result up to $\mathcal{O}(p^{4})$. The correct
choice turns out to be

$A=(T^{(2)}_{0})^{-1}\left[F^{(0)}_{S}+F^{(2)}_{S}-T_{0}^{(4)}(T_{0}^{(2)})^{-1}F^{(0)}_{S}\right].$
(3.12)

With this choice and a bit of algebra, we obtain the IAM formula for a
unitarized scalar form factor

$F_{S}=F_{S}^{(0)}+T_{0}^{(2)}\left(T_{0}^{(2)}-T_{0}^{(4)}\right)^{-1}F_{S}^{(2)}.$
(3.13)

Remember that as argued in Sect. 2.2, the LHCs of $T_{0}^{(4)}$ will be
transferred into $F_{S}$. Thus we also need to remove the imaginary part of
the troublesome $t$\- and $u$-channel loops of $T_{0}^{(4)}$ in this formula
manually.

###  3.3 Improvement by Dispersion Relation

It is well known that the IAM generates spurious structures such as peaks that
do not correspond to any physical resonance. This happens in particular in the
region below the lightest two-meson production threshold. In fact, since
$F_{S}=T_{0}A$ is only a possible solution for the unitarity equation (3.3)
above threshold (more rigorously, above the highest two-meson threshold since
we are using a one-step unitarization for a coupled-channel problem), it is
natural that the outcome can only be trusted above threshold. For example,
Fig. 7 shows the scalar nonstrange current form factor from the IAM
calculation, where the LECs are taken from Ref. GomezNicola:2001as  (gray
line), Fit 1 (red line), and Fit 2 (blue line), respectively. Obviously it
suffers from sub-threshold irregularities, such as unphysical peaks (the tiny
peaks near $0\ \text{GeV}$ in Fig. 7) and nonvanishing imaginary parts, in all
channels.

Figure 7: Real (solid lines) and imaginary (dashed lines) parts of
$F_{S,\pi\pi}^{n}$ from the original IAM calculation. The LECs are taken from
the original work GomezNicola:2001as  (gray line), Fit 1 (red line) and Fit 2
(blue line), respectively. In the sub-threshold region there exist unphysical
peaks due to Adler zeros in the scattering amplitude (pointed out by the black
arrows) and nonvanishing imaginary parts below the lowest threshold.

The unphysical sub-threshold singularities due to the IAM are studied
extensively in terms of dispersion relation for the case of single-channel
scattering amplitudes Hannah:1997sm ; GomezNicola:2007qj . There, the
existence of spurious poles in the scalar partial wave is identified as a
consequence of the failure to include the effect of the so-called Adler zero
in the dispersion integral of the inverse amplitude. This problem can be
solved by appropriately adding back such contributions in the IAM formula.
Unfortunately, a similar solution is not available in the coupled-channel case
because there is so far no dispersive derivation of the coupled-channel IAM
formula.

On the other hand, the dispersion relations of the form factors themselves are
much more straightforward. It simply takes the following form:

${\rm
Re}F(s)=\frac{1}{\pi}=\\!\\!\\!\\!\\!\\!\\!\int_{s_{\rm{th}}}^{\infty}dz\frac{{\rm
Im}F(z)}{z-s}~{},$ (3.14)

where $=\\!\\!\\!\\!\\!\\!\\!\int$ means the principal-value integration, and
$s_{\rm{th}}$ denotes the lowest threshold. Here it is sufficient to employ an
unsubtracted dispersion relation because $F(s)$ falls off as $1/s$ or faster
at large $s$ as suggested by perturbative QCD Brodsky:1973kr . This integral
equation suggests a better way to proceed: use the IAM-predicted imaginary
part of the form factor as the input to the dispersion integral, and obtain an
improved real part of the form factor. The obtained real part is then used to
modify the imaginary part, which will be the input of a subsequent dispersive
analysis. Such a procedure can be iterated until the curves of both the real
and imaginary parts of the form factor are stable. In the following we shall
depict the actual procedure and outline some of the details of such
iterations.

First, we use $F^{[n]}$ to denote the form factor after $n$ iterations (to
avoid confusion with the chiral order denoted by superscripts with
parentheses, here we use square brackets). Obviously, $F^{[0]}$ then
represents the original IAM result without undergoing any dispersive
correction. To start the iteration process, in the first step we set the
imaginary part of $F^{[1]}$ as

$\mathrm{Im}F^{[1]}\equiv T^{*}\Sigma F^{[0]},$ (3.15)

which will be used later as an input to the dispersion integral to obtain
$\mathrm{Re}F^{[1]}$. However, we have to apply one extra modification before
evaluating the dispersion integral: the IAM result $F^{[0]}$, which certainly
does not apply to arbitrarily large $s$ values, fails to reproduce the
asymptotic $1/s$-behavior. We therefore need to introduce a smooth transition
between the IAM-predicted $\mathrm{Im}F$ at small $s$ and the expected $1/s$
at large $s$ by hand. This can be achieved by defining a modified imaginary
part $\mathrm{Im}\tilde{F}^{[1]}$ as

$\mathrm{Im}\tilde{F}^{[1]}(s)\equiv\left[1-\sigma(s)\right]\mathrm{Im}F^{[1]}(s)+\sigma(s)\frac{\alpha^{[1]}}{s},$
(3.16)

where $\alpha^{[1]}$ is a constant to be determined later, and $\sigma(s)$ is
a monotonically increasing activation function that satisfies
$\sigma(-\infty)=0$ and $\sigma(+\infty)=1$. A simple choice of such a
function is

$\sigma(s)=\frac{1}{2}\left(\tanh\left\\{\frac{4(s-s_{0})}{\delta
s}\right\\}+1\right).$ (3.17)

This activation function is centered at $s_{0}$ and has a width of $\delta s$.
This means that $\mathrm{Im}\tilde{F}^{[1]}(s)$ can smoothly transform from
$\mathrm{Im}F^{[1]}(s)$ to $\alpha^{[1]}/s$ in the region $(s_{0}-\delta
s/2,s_{0}+\delta s/2)$.

Now, we should use the modified imaginary part $\mathrm{Im}\tilde{F}^{[1]}$,
instead of $\mathrm{Im}F^{[1]}$, in the dispersion integral to ensure the
convergence at infinity. The unknown constant $\alpha^{[1]}$ can be fixed by
requiring that $F(s)$ reproduces the NLO ChPT result at $s=0$, i.e.,
$F^{[1]}(0)=\mathrm{Re}F^{[1]}(0)=F_{\mathrm{ChPT}}(0)$. Once $\alpha^{[1]}$
is fixed, everything in the dispersion integral is known and we can use it to
numerically determine $\mathrm{Re}F^{[1]}(s)$. The dispersion relation
guarantees that the outcome makes sense both below and above threshold.

The whole procedure above can be iterated until a stable result is obtained.
At the second step, we define
$\mathrm{Im}F^{[2]}\equiv\mathrm{Re}\\{T^{*}\Sigma F^{[1]}\\}$ where
$F^{[1]}=\mathrm{Re}\tilde{F}^{[1]}+i\,\mathrm{Im}\tilde{F}^{[1]}$, and then
modify its UV-behavior by constructing $\mathrm{Im}\tilde{F}^{[2]}(s)$. Notice
that this construction will involve a new unknown constant $\alpha^{[2]}$
which is in general different from $\alpha^{[1]}$ so that it has to be re-
determined. After that, we can plug $\mathrm{Im}\tilde{F}^{[2]}$ into the
dispersion integral to obtain $\mathrm{Re}F^{[2]}$. This procedure will be
iterated for several times so that we can obtain a series of increasingly
refined form factors $F^{[3]},F^{[4]},\ldots$, which will eventually
stabilize. Finally, the unphysical peaks such as those in Fig. 7 are
completely wiped out after such a dispersive improvement.

It is worthwhile to stress that the Watson’s theorem is still fulfilled
perturbatively at the fixed point of the iteration procedure, and this
dispersive treatment is applicable to all scalar, vector, and tensor form
factors.

###  3.4 Numerical Results

Figure 8: Four iterations for the real and imaginary parts of
$F_{S,\pi\pi}^{n}$ with the LECs taking the central values of those given in
Ref. GomezNicola:2001as . Different colors distinguish various steps of the
iteration and the sub-figures zoom in to show the details.

In this section we show the numerical results of the unitarized scalar form
factors after the dispersive improvement. To show the convergence of the
iteration, as an example, Fig. 8 gives the first four iterations for the real
and imaginary parts of $F_{S,\pi\pi}^{n}$, with the LECs taken from Ref.
GomezNicola:2001as . It can be seen that the iteration successfully removes
the kink due to the Adler zeros of the scattering amplitudes, and the
imaginary part vanishes below the lowest threshold. The curves of all the
scalar form factors after the iteration are plotted in Figs. 9–12 within the
plot region $0\ \text{GeV}<\sqrt{s}<1.2\ \text{GeV}$. Here, we use three sets
of LECs: the original one from Ref. GomezNicola:2001as  (gray lines), the one
from Fit 1 (red lines), and the one from Fit 2 (blue lines) to plot the form
factors. The solid and dashed lines correspond to the real and imaginary
parts, respectively. The parameters for the activation function $\sigma(s)$
are taken to be $s_{0}=1.8\ \text{GeV}^{2}$ and $\delta s=0.6\
\text{GeV}^{2}$, which implies a smooth transition between the IAM result and
the $1/s$-behavior within the range $1.1\ \text{GeV}<\sqrt{s}<1.6\
\text{GeV}$.

Figure 9: Real (solid lines) and imaginary (dashed lines) parts of
$F_{S,\pi\pi}^{n}$, $F_{S,K\bar{K}}^{n}$, and $F_{S,\eta\eta}^{n}$,
respectively. Three sets of LECs are used to plot the form factors: the
original one from Ref. GomezNicola:2001as  (gray lines), one from Fit 1 (red
lines) and one from Fit 2 (blue lines).

Figure 10: Real and imaginary parts of $F_{S,\pi\pi}^{\bar{s}s}$,
$F_{S,K\bar{K}}^{\bar{s}s}$, and $F_{S,\eta\eta}^{\bar{s}s}$, respectively.
The meaning of different types of lines is the same as that in Fig. 9.

In the $n=(\bar{u}u+\bar{d}d)/\sqrt{2}$ channel and the $\bar{s}s$ channel,
the real parts of the form factors generate sharp peaks around $\sqrt{s}\sim
0.99\ \text{GeV}$ due to the simultaneous existence of the $f_{0}(980)$
resonance and the $K\bar{K}$-threshold within a narrow region. Our results for
$F_{S,\pi\pi}^{n}$, $F_{S,\pi\pi}^{\bar{s}s}$, and $F_{S,K\bar{K}}^{\bar{s}s}$
are consistent with those presented in Ref. Doring:2013wka  (barring
differences in overall normalization), the latter were obtained by a slightly
different version of algebraic unitarization formula, incorporating only the
$s$-channel cuts of the partial waves. However, the $f_{0}(980)$ region shown
in Fig. 9 has a narrower structure compared with that given in Refs.
Daub:2015xja ; Ropertz:2018stk . Other disagreement appears in the
$F_{S,K\bar{K}}^{n}$ form factor. In particular, the outcome of Ref.
Doring:2013wka  does not match the NLO ChPT value at $s=0$. Our result, on the
other hand, guarantees such a matching as it is implemented during the
determination of the coefficient $\alpha^{[i]}$. Also, we present the
$\eta\eta$ form factors that were not calculated in that paper.

In the $\bar{u}s$ channel, the strategy adopted in Ref. Doring:2013wka
therein is computationally involved as one has to first discretize
$s\rightarrow\\{s_{i}\\}$ and solve the dispersion relation by inverting a
huge rank matrix (in the $s$-space) to obtain the discretized form factor
$F(s_{i})$. Our approach is much simpler because we are simply taking the IAM
results above threshold as the input of the dispersion integral, and the
outcomes quickly stabilize after two or three iterations. In Fig. 13, our
results are compared with those from Refs. Jamin:2001zq ; Doring:2013wka . All
of them coincide with each other below $0.8\ \text{GeV}$. The mismatch at
larger $\sqrt{s}$ can be understood because the convergence of ChPT becomes
weaker at higher energies, which inevitably causes more uncertainties. One
notices that there is a cusp at the $K\pi$ threshold, signaling the opening of
the $K\pi$ channel in the $S$-wave. The second cusp appears at the $K\eta$
threshold, which enters through the coupled-channel treatment.333The strength
of the $K\eta$ threshold cusp in the $K\pi$ $S$-wave is likely overestimated
in uChPT, as the phenomenological inelasticity is very small below
$K\eta^{\prime}$ threshold Pelaez:2016tgi ; see also Refs. Noel:2020 ;
vonDetten:2020 .

Figure 11: Real and imaginary parts of $F_{S,K\pi}^{\bar{u}s}$ and
$F_{S,K\eta}^{\bar{u}s}$, respectively. For notations, see Fig. 9.

Figure 12: Real and imaginary parts of $F_{S,\pi\eta}^{\bar{u}d}$ and
$F_{S,K\bar{K}}^{\bar{u}d}$, respectively. For notations, see Fig. 9. Figure
13: Comparison of $F_{S,K\pi}^{\bar{u}s}$ in this work using the original LECs
in Ref. GomezNicola:2001as  (gray solid line), Fit 1 LECs (red dashed line)
and Fit 2 LECs (blue dot-dashed line), respectively, with that from a coupled-
channel dispersion relation analysis Jamin:2001zq  (magenta dot-dashed line)
as well as that from a Muskhelishvili–Omnès solution Doring:2013wka  (orange
dotted line).

###  3.5 Applications of the Scalar Form Factors

We end this section by discussing applications of the two-meson scalar form
factors, especially its $s$-dependence. Let us consider for example the decay
$B_{s}\to f_{0}(980)(\to\pi^{+}\pi^{-})\mu^{+}\mu^{-}$, which is a four-body
decay dominated by the $S$-wave contribution $f_{0}(980)\to\pi^{+}\pi^{-}$. To
study this decay process, the main task is to evaluate the
$B_{s}\to(\pi^{+}\pi^{-})_{S}$ transition matrix elements, which are
parameterized as

$\displaystyle\langle(\pi^{+}\pi^{-})_{S}|\bar{s}\gamma_{\mu}\gamma_{5}b|\bar{B}_{s}\rangle$
$\displaystyle=\frac{-i}{M_{B_{s}}}\Bigg{\\{}\bigg{[}P_{\mu}-\frac{M_{B_{s}}^{2}-M_{\pi\pi}^{2}}{q^{2}}q_{\mu}\bigg{]}{\cal
F}^{B_{s}\to\pi\pi}_{1}(M_{\pi\pi}^{2},q^{2})$
$\displaystyle\qquad\qquad+\frac{M_{B_{s}}^{2}-M_{\pi\pi}^{2}}{q^{2}}q_{\mu}{\cal
F}^{B_{s}\to\pi\pi}_{0}(M_{\pi\pi}^{2},q^{2})\Bigg{\\}},$
$\displaystyle\langle(\pi^{+}\pi^{-})_{S}|\bar{s}\sigma_{\mu\nu}q^{\nu}\gamma_{5}b|\bar{B}_{s}\rangle$
$\displaystyle=\frac{{\cal
F}^{B_{s}\to\pi\pi}_{T}(M_{\pi\pi}^{2},q^{2})}{M_{B_{s}}(M_{B_{s}}+M_{\pi\pi})}\bigg{[}({M_{B_{s}}^{2}-M_{\pi\pi}^{2}})q_{\mu}-q^{2}P_{\mu}\bigg{]},$
(3.18)

where $M_{\pi\pi}^{2}$ is the invariant mass square of the two-pion system.
These $B_{s}\to\pi^{+}\pi^{-}$ form factors can be calculated by light-cone
sum rules (LCSR) and expressed in terms of the $\pi^{+}\pi^{-}$ light-cone
distribution amplitudes (LCDAs) Mueller:1998fv ; Diehl:1998dk ;
Polyakov:1998ze ; Kivel:1999sd ; Diehl:2003ny ; Hagler:2002nh ; Pire:2008xe .

According to the Watson–Migdal theorem, since the $B_{s}\to\pi^{+}\pi^{-}$
transition totally decouples from the leptonic part at leading order, its
amplitude must share the same phase as that of the $\pi^{+}\pi^{-}$ scalar
form factor $F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})$ below the lowest
inelastic threshold (which is that of $K\bar{K}$ since the four-pion channel
is not considered, and the inelasiticity is known to be negligible below about
$1\ \text{GeV}$). Accordingly, in the framework of LCSR, the $S$-wave
$\pi^{+}\pi^{-}$ LCDAs are defined almost the same as those of a single scalar
meson $f_{0}$ but with the normalization factor taken as the scalar $\pi\pi$
form factor Diehl:1998dk ; Polyakov:1998ze ; Kivel:1999sd ; Diehl:2003ny ;
Meissner:2013hya . For example, the twist-2 LCDA is defined as

$\displaystyle\langle{(\pi^{+}\pi^{-})_{S}}|\bar{s}(x)\gamma_{\mu}s(0)|0\rangle$
$\displaystyle=$ $\displaystyle F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})B_{0}\
p_{{\pi\pi},\mu}\int_{0}^{1}du\,e^{iup_{\pi\pi}\cdot x}\phi_{\pi\pi}(u)~{},$
(3.19)

where $F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})B_{0}$ stands for the original
meson decay constant $f_{f_{0}}$ in the definition of the $f_{0}$ LCDAs. The
definition of all the twist-2 and 3 $S$-wave $\pi^{+}\pi^{-}$ LCDAs as well as
the explicit form of the resulting form factors Meissner:2013hya  can be found
in Appendix E. As a result, the form factors take the form

$\displaystyle{\cal F}_{i}(M_{\pi\pi}^{2},q^{2})$ $\displaystyle=$
$\displaystyle
B_{0}F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})\overline{F}_{i}(M_{\pi\pi}^{2},q^{2})~{}.$
(3.20)

Generally, the $\overline{F}_{i}$ depend on both $M_{\pi\pi}^{2}$ and $q^{2}$.
However, in the case of $B\to K^{*}(\to K\pi)$, as shown by Fig. 3 of Ref.
Meissner:2013hya , the $\overline{F}_{i}$’s have a much weaker dependence on
$M_{K\pi}^{2}$ than that on $q^{2}$. Such behavior is similar to the case of
$B_{s}\to\pi^{+}\pi^{-}$. This enables one to approximately suppress the
$M_{\pi\pi}^{2}$ dependence of $\overline{F}_{i}$ , which leads to a
factorized form

$\displaystyle{\cal F}_{i}(M_{\pi\pi}^{2},q^{2})$ $\displaystyle\approx$
$\displaystyle
B_{0}F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})\overline{F}_{i}(q^{2})~{},$
(3.21)

so that $\overline{F}_{i}$ can be described by a suitable parametrization
Cheng:2005nb ; Wang:2015paa ; Shi:2015kha ; Shi:2017pgh . Practically, one can
first fix $M_{\pi\pi}=M_{f_{0}(980)}$ to extract $\overline{F}_{i}(q^{2})$,
and then multiply it with $F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})$ again in
the form of Eq. (3.21) to recover the complete transition form factor ${\cal
F}_{i}(M_{\pi\pi}^{2},q^{2})$. The detailed calculation can be found in Ref.
Shi:2015kha . On the other hand, instead of the $S$-wave $\pi^{+}\pi^{-}$
LCDAs, one can firstly use the LCDAs of the $f_{0}$ to get the transition form
factors $\tilde{F}_{i}(q^{2})$ of $B_{s}\to f_{0}$, then due to the
approximation leading to Eq. (3.21), one can write the total form factor as

$\displaystyle{\cal F}_{i}(M_{\pi\pi}^{2},q^{2})$ $\displaystyle\approx$
$\displaystyle\frac{B_{0}}{f_{f_{0}}}F_{S,\pi\pi}^{\bar{s}s}(M_{\pi\pi}^{2})\tilde{F}_{i}(q^{2})~{}.$
(3.22)

Equations (3.21) and (3.22) explicitly reflect that the $\pi\pi$ distribution
of the $S$-wave-dominated decay $B_{s}\to
f_{0}(980)(\to\pi^{+}\pi^{-})\mu^{+}\mu^{-}$ is determined by the distribution
of the $\pi\pi$ scalar form factor.

All these scalar form factors are also required in the study of three-body $B$
decays, which have been studied using dispersively reconstructed form factors
or phenomenological parametrizations, see for instance Refs. Daub:2015xja ;
Ropertz:2018stk ; Chen:2002th ; Wang:2015uea ; Ma:2017idu ; Boito:2017jav ;
Liang:2018szw ; Xing:2019xti  and many references therein.

##  4 Vector form factors

Next we discuss the vector form factors. The matrix element of a vector
current with respect to a two-meson system can be parametrized as

$\left\langle
a_{i}(p_{a_{i}})b_{i}(p_{b_{i}})\right|\bar{q}^{\prime}\gamma^{\mu}q\left|0\right\rangle\equiv
F_{V+,i}^{\bar{q}^{\prime}q}(s)(p_{a_{i}}-p_{b_{i}})^{\mu}+F_{V-,i}^{\bar{q}^{\prime}q}(s)(p_{a_{i}}+p_{b_{i}})^{\mu}.$
(4.1)

Notice that we label the form factors $F_{V\pm}$ as above because they are
more commonly defined in the $t$-channel, where $p_{b_{i}}$ will switch sign.
From the equation of motion
$\partial_{\mu}(\bar{q}^{\prime}\gamma^{\mu}q)=i(m_{q^{\prime}}-m_{q})\bar{q}^{\prime}q$,
one sees that the form factor $F_{V-,i}$ is not independent since it can be
expressed in terms of $F_{V+,i}$ and the scalar form factor $F_{S,i}$
according to

$F_{V-,i}^{\bar{q}^{\prime}q}(s)=\frac{1}{s}\left[B_{0}(m_{q^{\prime}}-m_{q})F_{S,i}^{\bar{q}^{\prime}q}(s)-(m_{a_{i}}^{2}-m_{b_{i}}^{2})F_{V+,i}^{\bar{q}^{\prime}q}(s)\right].$
(4.2)

Therefore, it is sufficient to concentrate only on
$F_{V+,i}^{\bar{q}^{\prime}q}$.

The unitarity relation of $F_{V+,i}^{\bar{q}^{\prime}q}$ is most conveniently
expressed in terms of

$\displaystyle\tilde{F}_{V+}^{\bar{q}^{\prime}q}(s)$ $\displaystyle\equiv$
$\displaystyle\mathbb{P}F_{V+}^{\bar{q}^{\prime}q}(s)~{},$ (4.3)

where $\mathbb{P}_{ij}\equiv|\vec{p}_{i}|\delta_{ij}$ . One can then
straightforwardly express the unitarity relation as

$\displaystyle\mathrm{Im}\tilde{F}^{\bar{q}^{\prime}q}_{V+}$ $\displaystyle=$
$\displaystyle T_{1}^{*}\Sigma\tilde{F}^{\bar{q}^{\prime}q}_{V+}~{}.$ (4.4)

Notice that $\tilde{F}$ is associated with the $J=1$ partial-wave scattering
amplitude. However, the relation above is rigorously true only above the
highest threshold, where all elements of $\mathbb{P}$ are real. When $s$ is
between the lowest and highest thresholds, a more rigorous form of the
unitarity relation is

$\mathrm{Im}F^{\bar{q}^{\prime}q}_{V+}=(\mathbb{P}^{-1})^{*}T_{1}^{*}\Sigma\mathbb{P}F^{\bar{q}^{\prime}q}_{V+}.$
(4.5)

In particular, at the right-hand side of the equation above we have
$(\mathbb{P}^{-1})^{*}$ instead of $\mathbb{P}^{-1}$ so that the kinematical
imaginary part of $T_{1}^{*}$ (i.e., the imaginary part due to
$|\vec{p}_{i}|$) below the highest threshold can be canceled by that of
$(\mathbb{P}^{-1})^{*}$.

###  4.1 ChPT Result

There are two kinds of vector currents: the SU(3)-octet current $V_{\mu}^{a}$
($a=1,..,8$) and the singlet current $V_{\mu}^{0}$ due to the
$\text{SU}(3)_{V}$ and $\text{U}(1)_{B}$ ($B$ stands for baryon number)
symmetry, respectively. They can be defined as

$\displaystyle V_{\mu}^{0}=\bar{q}\gamma_{\mu}q,~{}~{}~{}V_{\mu}^{a}$
$\displaystyle=$ $\displaystyle\bar{q}T^{a}\gamma_{\mu}q,$ (4.6)

respectively. The easiest way to obtain such currents from the QCD Lagrangian
is to first promote $\text{SU}(3)_{V}$ and $\text{U}(1)_{B}$ to local
symmetries by introducing external fields $v_{\mu}=T^{a}v_{\mu}^{a}$ and
$v_{\mu}^{(s)}$:

${\cal L}={\cal L}^{0}_{\rm QCD}+{\cal L}_{\rm ext}={\cal L}^{0}_{\rm
QCD}+\bar{q}\gamma_{\mu}\bigg{(}v^{\mu}+\frac{1}{3}v^{\mu}_{(s)}+\gamma_{5}a^{\mu}\bigg{)}q-\bar{q}(s-i\gamma_{5}p)q.$
(4.7)

Taking the derivative of the Lagrangian with respect to the external fields
gives the currents

$\displaystyle V_{\mu}^{0}=\left.3\frac{\partial\mathcal{L}}{\partial
v_{(s)}^{\mu}}\right|_{v=v_{(s)}=0},\qquad
V_{\mu}^{a}=\left.\frac{\partial\mathcal{L}}{\partial
v^{\mu}_{a}}\right|_{v=v_{(s)}=0}.$ (4.8)

Again, we can apply the formulae above to obtain vector currents in ChPT. The
strict SU(3) symmetry of the ChPT Lagrangian up to ${\cal O}(p^{4})$ leads to
a vanishing $V_{\mu}^{0}$. It should be noted that at ${\cal O}(p^{6})$ a
certain SU(3) breaking term can be introduced so that $V_{\mu}^{0}$ no longer
vanishes. However, at ${\cal O}(p^{4})$ we will not consider this effect. For
the octet currents, we have

$\displaystyle V_{a\mu}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{iF_{0}^{2}}{4}\langle\lambda^{a}[U,\partial_{\mu}U^{\dagger}]\rangle~{},$
$\displaystyle V_{a\mu}^{(4)}$ $\displaystyle=$
$\displaystyle-2iL_{1}\langle\partial_{\nu}U\partial^{\nu}U^{\dagger}\rangle\langle\lambda^{a}[U,\partial_{\mu}U^{\dagger}]\rangle$
(4.9) $\displaystyle-
iL_{2}\big{\\{}\langle\partial_{\mu}U\partial_{\nu}U^{\dagger}\rangle\langle\lambda^{a}[U,\partial^{\nu}U^{\dagger}]\rangle+\langle\partial_{\nu}U\partial_{\mu}U^{\dagger}\rangle\langle\lambda^{a}[U^{\dagger},\partial^{\nu}U]\rangle\big{\\}}$
$\displaystyle-
iL_{3}\langle\big{(}[\lambda^{a},U]\partial_{\mu}U^{\dagger}+\partial_{\mu}U[\lambda_{a},U^{\dagger}]\big{)}\partial_{\nu}U\partial^{\nu}U^{\dagger}\rangle-2iB_{0}L_{4}\langle\lambda^{a}[U,\partial_{\mu}U^{\dagger}]\rangle\langle
MU^{\dagger}+UM^{\dagger}\rangle$ $\displaystyle-
iB_{0}L_{5}\langle([\lambda^{a},U]\partial_{\mu}U^{\dagger}+\partial_{\mu}U[\lambda_{a},U^{\dagger}])(MU^{\dagger}+UM^{\dagger})\rangle$
$\displaystyle+iL_{9}\langle\lambda^{a}\partial_{\nu}(\partial^{\nu}U\partial_{\mu}U^{\dagger}-\partial_{\mu}U\partial^{\nu}U^{\dagger})\rangle~{}.$

In particular, the components of $V^{(2)}_{a\mu}$ in ChPT are (making use of
the fact that
$V_{\mu}^{0}=\bar{u}\gamma_{\mu}u+\bar{d}\gamma_{\mu}d+\bar{s}\gamma_{\mu}s=0$
in the meson sector):

$\displaystyle\bar{u}\gamma^{\mu}d$ $\displaystyle=$ $\displaystyle
V_{1}^{\mu}+iV_{2}^{\mu}=-iK^{0}\overleftrightarrow{\partial^{\mu}}K^{-}+i\sqrt{2}\pi^{0}\overleftrightarrow{\partial^{\mu}}\pi^{-}+\ldots,$
$\displaystyle\bar{u}\gamma^{\mu}s$ $\displaystyle=$ $\displaystyle
V_{4}^{\mu}+iV_{5}^{\mu}=i\pi^{-}\overleftrightarrow{\partial^{\mu}}\bar{K}^{0}-i\sqrt{\frac{3}{2}}K^{-}\overleftrightarrow{\partial^{\mu}}\eta-\frac{i}{\sqrt{2}}K^{-}\overleftrightarrow{\partial^{\mu}}\pi^{0}+\ldots,$
$\displaystyle\bar{d}\gamma^{\mu}s$ $\displaystyle=$ $\displaystyle
V_{6}^{\mu}+iV_{7}^{\mu}=-iK^{-}\overleftrightarrow{\partial^{\mu}}\pi^{+}+i\sqrt{\frac{3}{2}}\eta\overleftrightarrow{\partial^{\mu}}\bar{K}^{0}-\frac{i}{\sqrt{2}}\pi^{0}\overleftrightarrow{\partial^{\mu}}\bar{K}^{0}+\ldots,$
$\displaystyle\bar{u}\gamma^{\mu}u$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}V_{8}^{\mu}+V_{3}^{\mu}=iK^{-}\overleftrightarrow{\partial^{\mu}}K^{+}+i\pi^{-}\overleftrightarrow{\partial^{\mu}}\pi^{+}+\ldots,$
$\displaystyle\bar{d}\gamma^{\mu}d$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}V_{8}^{\mu}-V_{3}^{\mu}=-iK^{0}\overleftrightarrow{\partial^{\mu}}\bar{K}^{0}-i\pi^{-}\overleftrightarrow{\partial^{\mu}}\pi^{+}+\ldots,$
$\displaystyle\bar{s}\gamma^{\mu}s$ $\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{3}}V_{8}^{\mu}=iK^{0}\overleftrightarrow{\partial^{\mu}}\bar{K}^{0}-iK^{-}\overleftrightarrow{\partial^{\mu}}K^{+}+\ldots.$
(4.10)

The one-loop ChPT results for the vector form factors are given in Appendix D.

###  4.2 Unitarization, Dispersive Improvement and Numerical Results

The IAM formula for the unitarized vector form factors can be obtained
directly from Eq. (3.13) by the replacements $T_{0}\rightarrow T_{1}$ and
$F_{S}\rightarrow\tilde{F}_{V+}$:

$\displaystyle\tilde{F}_{V+}$ $\displaystyle=$
$\displaystyle\tilde{F}_{V+}^{(0)}+T_{1}^{(2)}(T_{1}^{(2)}-T_{1}^{(4)})^{-1}\tilde{F}_{V+}^{(2)}$
$\displaystyle\Rightarrow F_{V+}$ $\displaystyle=$ $\displaystyle
F_{V+}^{(0)}+\mathbb{P}^{-1}T_{1}^{(2)}(T_{1}^{(2)}-T_{1}^{(4)})^{-1}\mathbb{P}F_{V+}^{(2)}~{}.$
(4.11)

This result is also required to be improved by a dispersion relation. The
whole procedure is identical to that of the scalar form factors, except that
now the imaginary parts of the vector form factors that enter the dispersion
integrals should be taken as

$\mathrm{Im}F^{[i]}_{V+}\equiv(\mathbb{P}^{-1})^{*}T_{1}^{*}\Sigma\mathbb{P}F^{[i-1]}_{V+}~{},$
(4.12)

where $i$ is the number of iteration following the unitarity relation of the
vector form factors.

Figure 14: Real and imaginary parts of $F_{V+,K\bar{K}}^{\bar{s}s}$,
respectively. The notations are the same as that of Fig. 9 but here the
imaginary parts are presented in the inset panel.

Figure 15: Real and imaginary parts of $F_{V+,K\pi}^{\bar{u}s}$ and
$F_{V+,K\eta}^{\bar{u}s}$, respectively. For notations, see Fig. 9. The insets
show that the Fit 1 and Fit 2 curves are very close to each other.

Figure 16: Real and imaginary parts of $F_{V+,\pi\pi}^{\bar{u}d}$ and
$F_{V+,K\bar{K}}^{\bar{u}d}$, respectively. For notations, see Fig. 9.

Our final results for the vector form factors are summarized in Figs. 14–16.
In the $\bar{s}s$ channel, there is a pole on the real axis below the
$K\bar{K}$ threshold, which corresponds to the $\omega_{8}$ resonance as
pointed out in Ref. GomezNicola:2001as . Its mass is between the masses of
$\omega(782)$ and $\phi(1020)$, which is reasonable since the $\omega_{8}$ is
a mixture of $\omega$ and $\phi$. Note that for this channel we have removed
the unphysical left-hand cuts of the $\pi\pi$ and $\pi\eta$ loops in
$T_{I=0,J=1}^{(4)}$ as we did for the $I=1$, $J=1$ channel in Sect. 2.2. Since
we do not consider $3\pi$ intermediate states here, this sub-threshold
resonance must be a bound state with zero width. Keeping the left-hand cuts of
the scattering amplitude results in an unphysical imaginary part of the form
factor below the $K\bar{K}$ threshold, which is due to the on-shell
approximation of the IAM, see Sect. 2.2. On the other hand, due to the absence
of Adler zeros and with the unphysical imaginary parts cut off, further
improvement by dispersive iteration is unnecessary for this channel.

In the $\bar{u}s$ channel, it turns out that the IAM result is almost
invariant under the dispersive treatment; the imaginary parts of the $K\pi$
and $K\eta$ form factors peak around $\sqrt{s}\approx 0.89\ \text{GeV}$,
reflecting the existence of the $K^{*}(892)$ resonance.

As explained in Sect. 2.2, there is a breaking of the unitarity relation below
the highest threshold due to the mixing between the left-hand and right-hand
cuts using the coupled-channel IAM as in Ref. GomezNicola:2001as . Such a
violation is especially serious for the $I=1,J=1$ channel, see Fig. 2, and
thus must happen as well when calculating the $u\bar{d}$ vector form factors.
It is found that only after the modification proposed in Sect. 2.2, with the
imaginary part of the troublesome $t$\- and $u$-channel loops removed, the
iteration procedure for the $u\bar{d}$ vector form factors converges. The
result is shown in Fig. 16, and agrees well with the existing literature (see
the discussion below).

The $\pi\pi$ vector form factor is calculated in Ref. Guo:2008nc  through the
Omnès representation

$\displaystyle
F(s)=\exp\left(P_{1}s+\frac{s^{2}}{\pi}\int_{4M_{\pi}^{2}}^{\infty}ds^{\prime}\frac{\delta_{1}^{1}(s^{\prime})}{(s^{\prime})^{2}(s^{\prime}-s-i\epsilon)}\right),$
(4.13)

where $P_{1}=\left\langle r^{2}\right\rangle/6$ with $\left\langle
r^{2}\right\rangle$ the pion radius squared (see Refs. Hanhart:2016pcd ;
Ananthanarayan:2017efc ; Colangelo:2018mtw  for recent determinations of the
pion charge radius from data), and a twice-subtracted dispersion relation is
applied. The Omnès representation relates the form factor with the
corresponding phase shift $\delta_{1}^{1}(s)$, which can be extracted in a
rough approximation from the mass and decay width of the $\rho$ meson using a
Breit–Wigner parametrization. The scattering amplitude dominated by the
$s$-channel $\rho$ resonance reads

$\displaystyle
a_{1}^{1}(s)=\frac{c}{s-M_{\rho}^{2}-iM_{\rho}\Gamma_{\mathrm{tot}}(s)}~{},\ \
\ \Gamma_{\mathrm{tot}}=\frac{g_{\rho\pi\pi}^{2}p^{3}}{6\pi
M_{\rho}^{2}}=\frac{g_{\rho\pi\pi}^{2}\left(\frac{s}{4}-M_{\pi}^{2}\right)^{3/2}}{6\pi
M_{\rho}^{2}}~{},$ (4.14)

where $c$ is an irrelevant constant. The phase shift is derived as

$\displaystyle\delta_{1}^{1}(s)=\arctan\frac{\operatorname{Im}a_{1}^{1}(s)}{\operatorname{Re}a_{1}^{1}(s)}=\arctan\frac{M_{\rho}\Gamma_{\operatorname{tot}}(s)}{M_{\rho}^{2}-s}~{}.$
(4.15)

The left panel in Fig. 17

Figure 17: Comparison between the $\pi\pi$ vector form factor derived in this
work and that from the Omnès representation Guo:2008nc  (left), comparison
between the $K^{+}K^{-}$ EM form factor derived in this work and that from an
earlier uChPT work Oller:2000ug  (right).

shows a comparison between the $\pi\pi$ vector form factor derived in this
work and that from the Omnès representation, and we observe good agreement. In
the right panel, instead of $F_{V,K^{+}K^{-}}^{\bar{u}d}$ we compare our
result for the kaon electromagnetic (EM) form factor with that from the
earlier literature also using uChPT Oller:2000ug , since the explicit result
for $F_{V,K^{+}K^{-}}^{\bar{u}d}$ is not available from that source. The EM
form factor is defined as

$\left\langle
K^{+}(p)K^{-}(p^{\prime})\right|\sum_{i}e_{i}\bar{q}_{i}^{\prime}\gamma^{\mu}A_{\mu}q_{i}\left|\gamma\right\rangle\equiv
F_{V,\ K^{+}K^{-}}^{EM}(s)\epsilon\cdot(p-p^{\prime}),$ (4.16)

where $i=u,d,s$ and $e_{i}=2/3,-1/3,-1/3$. Generally, due to isospin symmetry,
our vector form factors derived above are related to this EM form factor
according to

$\displaystyle
F_{V,K^{+}K^{-}}^{EM}(s)=\frac{1}{2}F_{V+,K\bar{K}}^{\bar{u}d}(s)+\frac{1}{3\sqrt{2}}F_{V+,K\bar{K}}^{\bar{s}s}(s)-\frac{1}{6\sqrt{2}}F_{V+,K\bar{K}}^{\bar{u}u+\bar{d}d}(s).$
(4.17)

Note that due to charge conservation we must have
$F_{V,K^{+}K^{-}}^{EM}(0)=1$, which can be checked by our results (using the
U(1) symmetry constraint:
$\bar{u}\gamma_{\mu}u+\bar{d}\gamma_{\mu}d=-\bar{s}\gamma_{\mu}s$). The EM
form factor from Eq. (4.17) is shown by the blue solid line in the right panel
of Fig. 17, where the divergent bound state peak around 935 ${\rm MeV}$
corresponds to the $\omega_{8}$. However, in Ref. Oller:2000ug , instead of
the $\omega_{8}$ two physical resonances $\omega$ and $\phi$ were introduced
manually, which are mixtures of $\omega_{1}$ and $\omega_{8}$. Thus, to
reproduce these two distinct resonances for better comparison, we replace the
last two terms in Eq. (4.17) with the standard Breit–Wigner distributions of
the $\omega$ and $\phi$ :

$\frac{1}{\sqrt{2}}\frac{g}{g_{\phi}}\frac{M_{\phi}^{2}}{s-M_{\phi}^{2}+i\
M_{\phi}\Gamma_{\phi}}-\frac{1}{2}\frac{g}{g_{\omega}}\frac{M_{\omega}^{2}}{s-M_{\omega}^{2}+i\
M_{\omega}\Gamma_{\omega}},$ (4.18)

where $g$ is the SU(3)-symmetric vector-to-two-pseudoscalars coupling
constant, and $g_{\phi}$, $g_{\omega}$ refer to the coupling constants of
$\phi$ and $\omega$ to the electromagnetic current. Their explicit definitions
can be found in Refs. Meissner:1987ge ; Klingl:1996by , where
$g_{\phi}=-12.89,\ g_{\omega}=17.05$ and $g=6.05$. The decay widths are taken
as $\Gamma_{\phi}=4$ MeV and $\Gamma_{\omega}=8.5$ MeV. The corresponding
curve is the red dashed line shown in the right plot of Fig. 17. Finally, the
magenta dotted line in the right panel is the EM form factor extracted from
Fig. 4 of Ref. Oller:2000ug . From this comparison, we find that regardless
the region where the un-mixed resonant $\omega_{8}$ or the physical mixtures
$\omega$ and $\phi$ emerge, the agreement is good at both low (below 0.7 GeV)
and high (above 1.1 GeV) energies.

###  4.3 Applications of the Vector Form Factors

We end this section by discussing an application of the two-meson vector form
factors in the two-body hadronic decays of a charged lepton
$l\to\phi\phi^{\prime}\nu$. The leading contribution to
$l\to\phi\phi^{\prime}\nu$ is due to a single exchange of a $W$-boson, which,
at low energies, can be approximated by the Fermi interaction. The
corresponding amplitude is

$iM=-\frac{iG_{F}}{\sqrt{2}}V_{qq^{\prime}}^{*}\bar{u}_{\nu}\gamma^{\mu}(1-\gamma_{5})u_{l}\left\langle\phi\phi^{\prime}|\bar{q}^{\prime}\gamma_{\mu}q|0\right\rangle,$
(4.19)

where $G_{F}$ is Fermi’s constant and $V_{qq^{\prime}}$ is the
Cabibbo–Kobayashi–Maskawa (CKM) matrix element. Since the axial vector
component in the hadronic matrix element vanishes due to parity, the matrix
element can be expressed in terms of the vector form factors defined in Eq.
(4.1). Considering only the spin-averaged decay, the differential decay width
is a function of two kinematic variables. They can be chosen as
$s=(p_{\phi}+p_{\phi^{\prime}})^{2}$ and $\theta$, which is the angle between
$\vec{p}_{\phi}$ and $\vec{p}_{l}$ in the c.m. frame of $\phi\phi^{\prime}$.
After carrying out the phase space integration, one obtains

$\frac{d\Gamma}{ds\>d\cos\theta}=G_{F}^{2}|V_{qq^{\prime}}|^{2}\frac{(m_{l}^{2}-s)|\vec{p}_{\phi}|}{(8\pi)^{3}m_{l}^{3}\sqrt{s}}\left[A(s)+\frac{|\vec{p}_{\phi}|}{\sqrt{s}}B(s)\cos\theta+\frac{|\vec{p}_{\phi}|^{2}}{s}C(s)\cos^{2}\theta\right],$
(4.20)

where the $\phi\phi^{\prime}$ subscript and $\bar{q}^{\prime}q$ superscript in
the form factors have been suppressed for notational simplicity, and

$\displaystyle A(s)$ $\displaystyle\equiv$
$\displaystyle\frac{2(m_{l}^{2}-s)}{s^{2}}\Big{[}m_{l}^{2}s\big{(}s|F_{V-}|^{2}+2(M_{\phi}^{2}-M_{\phi^{\prime}}^{2})\mathrm{Re}\\{F_{V+}^{*}F_{V-}\\}\big{)}$
$\displaystyle+|F_{V+}|^{2}\left(M_{\phi}^{4}(m_{l}^{2}+s)-2m_{\phi}^{2}(M_{\phi^{\prime}}^{2}m_{l}^{2}+M_{\phi^{\prime}}^{2}s+s^{2})+M_{\phi^{\prime}}^{4}(m_{l}^{2}+s)-2M_{\phi^{\prime}}^{2}s^{2}+s^{3}\right)\Big{]},$
$\displaystyle B(s)$ $\displaystyle=$
$\displaystyle-\frac{8m_{l}^{2}(m_{l}^{2}-s)}{s}\left(|F_{V+}|^{2}(M_{\phi}^{2}-M_{\phi^{\prime}}^{2})+s\,\mathrm{Re}\\{F_{V+}^{*}F_{V-}\\}\right),$
$\displaystyle C(s)$ $\displaystyle=$ $\displaystyle
8|F_{V+}|^{2}(m_{l}^{2}-s)^{2}.$ (4.21)

Two-body hadronic decay of charged leptons is one of the commonly studied
processes in the extraction of the CKM matrix elements, for example $\tau\to
K\pi\nu_{\tau}$ for $V_{us}$ Amhis:2019ckw . Therefore, an improved
understanding of the $s$-dependence in the vector form factors may improve on
the $V_{us}$ precision Bernard:2011ae  and lead to a better reconciliation of
the same quantity measured in other processes such as the kaon leptonic/semi-
leptonic decays.

##  5 Tensor form factors

Next, we study the tensor form factors of a two-meson system, which were so
far only investigated in limited channels. We define the tensor form factors
through the following matrix elements:

$\left\langle
a_{i}(p_{a_{i}})b_{i}(p_{b_{i}})\right|\bar{q}^{\prime}\sigma^{\mu\nu}q\left|0\right\rangle\equiv
i\frac{\Lambda_{2}}{F_{\pi}^{2}}(p_{a_{i}}^{\mu}p_{b_{i}}^{\nu}-p_{a_{i}}^{\nu}p_{b_{i}}^{\mu})F_{T,i}^{\bar{q}^{\prime}q}(s)~{},$
(5.1)

where $\Lambda_{2}$ is an LEC that appears when introducing external tensor
sources to the chiral Lagrangian, which we shall discuss later. The unitarity
relation obeyed by the tensor form factor is identical with that of the vector
form factor $F_{V+}$ Hoferichter:2018zwu :

$\mathrm{Im}F^{\bar{q}^{\prime}q}_{T}=(\mathbb{P}^{-1})^{*}T_{1}^{*}\Sigma\mathbb{P}F^{\bar{q}^{\prime}q}_{T}~{}.$
(5.2)

Again, it is associated with the $J=1$ partial-wave amplitude.

###  5.1 ChPT Result

The derivation of tensor currents in ChPT requires the introduction of an
antisymmetric Hermitian tensor source $\bar{t}_{\mu\nu}$ into the QCD
Lagrangian:

$\mathcal{L}=\bar{q}\sigma^{\mu\nu}\bar{t}_{\mu\nu}q~{}.$ (5.3)

The corresponding effective field theory was first investigated in Ref.
Cata:2007ns . The LO chiral Lagrangian coupled to the tensor source scales as
${\cal O}(p^{4})$ and is given by

$\displaystyle{\cal L}_{T}^{(4)}=-i\Lambda_{2}\langle
t_{+}^{\mu\nu}u_{\mu}u_{\nu}\rangle~{},$ (5.4)

where

$\displaystyle
u_{\mu}=i\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{\dagger}\right]~{},~{}~{}~{}t_{\pm}^{\mu\nu}=u^{\dagger}t^{\mu\nu}u^{\dagger}\pm
u\ t^{\mu\nu\dagger}u~{}.$ (5.5)

$t^{\mu\nu}$ and $t^{\mu\nu\dagger}$ are given as

$\displaystyle
t^{\mu\nu}=\frac{1}{4}\big{(}g^{\mu\lambda}g^{\nu\rho}-g^{\nu\lambda}g^{\mu\rho}-i\epsilon^{\mu\nu\lambda\rho}\big{)}\bar{t}_{\lambda\rho}~{},$
$\displaystyle
t^{\mu\nu\dagger}=\frac{1}{4}\big{(}g^{\mu\lambda}g^{\nu\rho}-g^{\nu\lambda}g^{\mu\rho}+i\epsilon^{\mu\nu\lambda\rho}\big{)}\bar{t}_{\lambda\rho}~{},$
(5.6)

and the convention $\epsilon^{0123}=1$ has been used for the Levi-Civita
tensor.

At NLO, there exist quite a number of corresponding operators, among which the
ones contributing to tensor form factors are given as

$\displaystyle{\cal L}_{T}^{(6)}$ $\displaystyle=$ $\displaystyle
iC_{34}\langle
t_{+}^{\mu\nu}\\{\chi_{+},u_{\mu}u_{\nu}\\}\rangle+iC_{35}\langle
t_{+}^{\mu\nu}u_{\mu}\chi_{+}u_{\nu}\rangle+iC_{36}\langle\chi_{+}\rangle\langle
t_{+}^{\mu\nu}u_{\mu}u_{\nu}\rangle$ (5.7) $\displaystyle+iC_{37}\langle
t_{+}^{\mu\nu}\rangle\langle\chi_{+}u_{\mu}u_{\nu}\rangle+iC_{88}\langle\nabla^{\rho}t_{+}^{\mu\nu}[h_{\mu\rho},u_{\nu}]\rangle+iC_{89}\langle\nabla_{\mu}t_{+}^{\mu\nu}[h_{\nu\rho},u^{\rho}]\rangle$
$\displaystyle+C_{106}\langle
t_{+}^{\mu\nu}[\chi_{-\mu},u_{\nu}]\rangle+iC_{107}\langle
t_{+_{\mu\nu}}h^{\mu\alpha}h^{\nu}_{\>\>\alpha}\rangle,$

with the covariant derivative defined as

$\nabla_{\mu}X=\partial_{\mu}X+\left[\Gamma_{\mu},X\right],~{}~{}~{}~{}\Gamma_{\mu}=\frac{1}{2}\left(u^{\dagger}\partial_{\mu}u+u\partial_{\mu}u^{\dagger}\right).$
(5.8)

They are used to cancel the divergence that occurs in the one-loop corrections
to the tensor form factors. The renormalized LECs $C_{i}^{r}$ and divergence
coefficients $\gamma_{i}^{T}$ are defined as:

$C_{i}^{r}=C_{i}-\frac{\Lambda_{2}}{F_{0}^{2}}\lambda\gamma_{i}^{T}.$ (5.9)

In fact, it turns out that the requirement to cancel all divergences in tensor
form factors does not fix all the $\\{\gamma_{i}^{T}\\}$ independently, but
only a subset of them, which are determined to be the following constraints:

$\displaystyle\gamma_{34}^{T}+\frac{1}{2}\left(\gamma_{89}^{T}+\gamma_{106}^{T}+\gamma_{107}^{T}\right)$
$\displaystyle=-\frac{1}{24},$
$\displaystyle\gamma_{35}^{T}+\gamma_{89}^{T}+\gamma_{106}^{T}+\gamma_{107}^{T}$
$\displaystyle=-\frac{3}{4},$
$\displaystyle\gamma_{36}^{T}=-\frac{11}{18},\qquad\qquad\gamma_{37}^{T}$
$\displaystyle=-\frac{1}{2},$
$\displaystyle\gamma_{88}^{T}-\gamma_{89}^{T}-\gamma_{107}^{T}$
$\displaystyle=-\frac{1}{4}.$ (5.10)

The numerical values of $\Lambda_{2}$ and $C_{i}^{r}$ at a given
renormalization scale $\mu$ are obviously required to make definite
predictions on tensor form factors. Unfortunately, as far as we know, no
lattice data are available for these LECs. We therefore make use of the
results in Ref. Jiang:2012ir  that attempted to evaluate the effective action
from first principles (under certain uncontrollable approximations). However,
a critical issue in that approach is that one cannot study the scale
dependence of the renormalized LECs, and therefore the issue of how one should
match their results to the renormalized LECs with the standard
Gasser–Leutwyler subtraction scheme at a given scale, say $\mu=M_{\rho}$,
remains ambiguous. To account for this issue, we assume that the LECs in Ref.
Jiang:2012ir  are given at some unknown scale $\tilde{\mu}$, and could be run
to $\mu=M_{\rho}$ by a renormalization group (RG) running

$C_{i}^{r}(M_{\rho})=C_{i}^{r}(\tilde{\mu})-\frac{\gamma_{i}^{T}}{32\pi^{2}}\frac{\Lambda_{2}}{F_{0}^{2}}\ln\frac{M_{\rho}^{2}}{\tilde{\mu}^{2}}~{}.$
(5.11)

which still involves an unknown coefficient
$\ln(M_{\rho}^{2}/\tilde{\mu}^{2})$. To fix this coefficient, we refer to Ref.
Jiang:2015dba , in which the $\mathcal{O}(p^{4})$ LECs $L_{i}$ are calculated
within the same formalism. We perform a RG running of those LECs and compare
them to $\\{L_{i}^{r}\\}$ at $\mu=M_{\rho}$ that are fitted to experimental
data Bijnens:2014lea . That allows us to get a best-fit value of
$\ln(M_{\rho}^{2}/\tilde{\mu}^{2})$ by minimizing the $\chi^{2}$. We find
that, with this best-fit value, the changes in the numerical values of
$\\{L_{i}\\}$ are relatively small. Therefore, for the case of tensor LECs, we
shall simply cite the results in Ref. Jiang:2012ir , assuming that systematic
errors due to the ambiguity in the renormalization scale are much smaller than
the other theoretical errors quoted in the paper. Within such a framework, the
coefficients $C_{36}^{r}$ and $C_{37}^{r}$ vanish in the large-$N_{c}$ limit,
where $N_{c}$ refers to the number of colors in the QCD Lagrangian, while
other nonzero coefficients are collected in Table 2 (readers should be alerted
to certain differences in the definition of operators between Refs.
Cata:2007ns ; Jiang:2012ir  that lead to changes in numerical values of LECs
and have been properly taken into account in Table 2).

The tensor currents are defined as

$T_{ij}^{\mu\nu}\equiv\bar{q}_{i}\sigma^{\mu\nu}q_{j}.$ (5.12)

Using the $\mathcal{O}(p^{4})$ and $\mathcal{O}(p^{6})$ chiral Lagrangian with
tensor sources, we are able to derive the tensor currents $T^{(4)}_{\mu\nu}$
and $T^{(6)}_{\mu\nu}$ in ChPT, respectively, which are

$\displaystyle T_{ij}^{(4)\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{i\Lambda_{2}}{4}\left[u^{\dagger}[u^{\mu},u^{\nu}]u^{\dagger}-i\varepsilon^{\mu\nu\lambda\rho}u^{\dagger}u_{\lambda}u_{\rho}u^{\dagger}+u[u^{\mu},u^{\nu}]u+i\varepsilon^{\mu\nu\lambda\rho}uu_{\lambda}u_{\rho}u\right]_{ji}~{},$
$\displaystyle T_{ij}^{(6)\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{F_{0}^{2}}\left(\partial^{[\mu}\phi_{a}\right)\left(\partial^{\nu]}\phi_{b}\right)\biggl{[}C_{34}\\{\chi,\lambda^{a}\lambda^{b}\\}+C_{35}\lambda^{a}\chi\lambda^{b}+C_{36}\langle\chi\rangle\lambda^{a}\lambda^{b}$
$\displaystyle\left.+C_{37}\langle\chi\lambda^{a}\lambda^{b}\rangle+\frac{1}{2}C_{106}[\\{\chi,\lambda^{a}\\},\lambda^{b}]\right]_{ji}+\frac{2i}{F_{0}^{2}}C_{107}\left(\partial^{\alpha}\partial^{[\mu}\phi_{a}\right)\left(\partial^{\nu]}\partial_{\alpha}\phi_{b}\right)\left(\lambda^{a}\lambda^{b}\right)_{ji}$
$\displaystyle-\frac{i}{F_{0}^{2}}[\lambda^{a},\lambda^{b}]_{ji}\left[C_{88}\partial^{\rho}\left((\partial^{[\mu}\partial_{\rho}\phi_{a})\partial^{\nu]}\phi_{b}\right)+C_{89}\partial^{[\mu}\left((\partial^{\nu]}\partial_{\rho}\phi_{a})\partial^{\rho}\phi_{b}\right)\right]+O(\phi^{3})~{},$

where $A^{[\mu}B^{\nu]}\equiv A^{\mu}B^{\nu}-A^{\nu}B^{\mu}$. In particular,
the components of $T^{(4)\mu\nu}$ read

$\displaystyle\bar{u}\sigma_{\mu\nu}u$ $\displaystyle=$
$\displaystyle\frac{i\Lambda_{2}}{F_{0}^{2}}(-\partial_{\mu}K^{+}\partial_{\nu}K^{-}+\partial_{\mu}K^{-}\partial_{\nu}K^{+}-\partial_{\mu}\pi^{+}\partial_{\nu}\pi^{-}+\partial_{\mu}\pi^{-}\partial_{\nu}\pi^{+})+\ldots,$
$\displaystyle\bar{d}\sigma_{\mu\nu}d$ $\displaystyle=$
$\displaystyle\frac{i\Lambda_{2}}{F_{0}^{2}}(\partial_{\mu}\bar{K}^{0}\partial_{\nu}K^{0}-\partial_{\mu}K^{0}\partial_{\nu}\bar{K}^{0}+\partial_{\mu}\pi^{+}\partial_{\nu}\pi^{-}-\partial_{\mu}\pi^{-}\partial_{\nu}\pi^{+})+\ldots,$
$\displaystyle\bar{s}\sigma_{\mu\nu}s$ $\displaystyle=$
$\displaystyle\frac{i\Lambda_{2}}{F_{0}^{2}}(-\partial_{\mu}\bar{K}^{0}\partial_{\nu}K^{0}+\partial_{\mu}K^{0}\partial_{\nu}\bar{K}^{0}+\partial_{\mu}K^{+}\partial_{\nu}K^{-}-\partial_{\mu}K^{-}\partial_{\nu}K^{+})+\ldots,$
$\displaystyle\bar{u}\sigma_{\mu\nu}d$ $\displaystyle=$
$\displaystyle\frac{i\Lambda_{2}}{F_{0}^{2}}(\partial_{\mu}K^{-}\partial_{\nu}K^{0}-\partial_{\mu}K^{0}\partial_{\nu}K^{-}+\sqrt{2}\partial_{\mu}\pi^{0}\partial_{\nu}\pi^{-}-\sqrt{2}\partial_{\mu}\pi^{-}\partial_{\nu}\pi^{0})+\ldots,$
$\displaystyle\bar{u}\sigma_{\mu\nu}s$ $\displaystyle=$
$\displaystyle\frac{i\Lambda_{2}}{2F_{0}^{2}}(-2\partial_{\mu}\bar{K}^{0}\partial_{\nu}\pi^{-}+2\partial_{\mu}\pi^{-}\partial_{\nu}\bar{K}^{0}+\sqrt{6}\partial_{\mu}\eta\partial_{\nu}K^{-}+\sqrt{2}\partial_{\mu}\pi^{0}\partial_{\nu}K^{-}$
$\displaystyle-\sqrt{6}\partial_{\mu}K^{-}\partial_{\nu}\eta-\sqrt{2}\partial_{\mu}K^{-}\partial_{\nu}\pi^{0})+\ldots,$
$\displaystyle\bar{d}\sigma_{\mu\nu}s$ $\displaystyle=$
$\displaystyle\frac{i\Lambda_{2}}{2F_{0}^{2}}(\sqrt{6}\partial_{\mu}\eta\partial_{\nu}\bar{K}^{0}-\sqrt{2}\partial_{\mu}\pi^{0}\partial_{\nu}\bar{K}^{0}-\sqrt{6}\partial_{\mu}\bar{K}^{0}\partial_{\nu}\eta+\sqrt{2}\partial_{\mu}\bar{K}^{0}\partial_{\nu}\pi^{0}$
(5.14)
$\displaystyle+2\partial_{\mu}\pi^{+}\partial_{\nu}K^{-}-2\partial_{\mu}K^{-}\partial_{\nu}\pi^{+})+\ldots.$

The one-loop ChPT results for the tensor form factors are given similarly in
Appendix D.

Table 2: Low-energy constants for the chiral Lagrangian with tensor sources derived from Ref. Jiang:2012ir . $\Lambda_{2}$ is given in units of $10^{-3}b_{0}$ while the remainders are given in units of $10^{-3}\text{GeV}^{-2}b_{0}$, where $b_{0}=1.32\ \text{GeV}$. The renormalization scale is assumed to be $\mu=M_{\rho}$ (see the discussion in the text). $\Lambda_{2}$ | $C_{34}^{r}$ | $C_{35}^{r}$ | $C_{88}^{r}$ | $C_{89}^{r}$ | $C_{106}^{r}$ | $C_{107}^{r}$
---|---|---|---|---|---|---
$13.79$ | $0.01$ | $-4.14$ | $-1.44$ | $10.26$ | $-9.04$ | $-0.09$

###  5.2 Unitarization, Dispersive Improvement and Numerical Results

The unitarity relation for tensor form factors $F_{T}$ is identical to that of
vector form factors, so their IAM formulae should also take the same form

$\displaystyle F_{T}$ $\displaystyle=$ $\displaystyle
F_{T}^{(0)}+\mathbb{P}^{-1}T_{1}^{(2)}(T_{1}^{(2)}-T_{1}^{(4)})^{-1}\mathbb{P}F_{T}^{(2)}.$
(5.15)

The results are given in Figs. 18–20, where due to the same reason as that for
$F_{V}^{\bar{s}s}$, $F_{T}^{n}$ and $F_{T}^{\bar{s}s}$ are also calculated by
cutting off the left-hand cuts and are not improved by dispersive iteration.
The iteration procedures for $F_{T}^{\bar{u}d}$ and $F_{T}^{\bar{u}s}$ are
exactly the same as the one for the corresponding vector form factors, so we
shall not repeat all the details here.

Figure 18: Real and imaginary parts of $F_{T-,K\bar{K}}^{n},$ respectively
(left), and real and imaginary parts of $F_{T-,K\bar{K}}^{\bar{s}s}$,
respectively (right). For notations, see Fig. 14.

Figure 19: Real and imaginary parts of $F_{T-,KK}^{\bar{u}d}$ and
$F_{T-,\pi\pi}^{\bar{u}d}$, respectively. For notations, see Fig. 9.

Figure 20: Real and imaginary parts of $F_{T-,K\pi}^{\bar{u}s}$ and
$F_{T-,K\eta}^{\bar{u}s}$, respectively. For notations, see Fig. 9.

For the form factor $F_{T,\pi\pi}^{ud}$, we compare it with that derived in
Ref. Miranda:2018cpf , where $F_{T,\pi\pi}^{ud}$ was obtained using the Omnès
representation. Since according to the Watson–Migdal theorem, in the elastic
region, the phases of the tensor form factors equal those of the vector form
factors, $\delta_{T}(s)=\delta_{+}(s)$, one can use the dispersion relation

$\displaystyle\frac{F_{T}(s)}{F_{T}(0)}=\exp\left\\{\frac{s}{\pi}\int_{4M_{\pi}^{2}}^{\infty}ds^{\prime}\frac{\delta_{T}\left(s^{\prime}\right)}{s^{\prime}\left(s^{\prime}-s-i\epsilon\right)}\right\\}$
(5.16)

to obtain the normalized tensor form factor. The comparison is shown in Fig.
21, and we observe a significant difference. For instance, the sizes of the
peak at $s=M_{\rho}^{2}$ are quite different in the two calculations, and
there exists a zero point in our curve above $1\ \text{GeV}$ that does not
occur in the phase dispersive representation. The differences are exclusively
due to the SU(3)-breaking LECs in the tensor form factors at NLO, and
therefore probably a rather large uncertainty should be associated with them.
An independent cross-check is therefore highly desirable, and in Appendix F we
argue that this is in principle doable through a comparison with future
lattice QCD calculations of the tensor charge of the $\rho$-meson.

Figure 21: Comparison of $F_{T,\pi\pi}^{\bar{u}d}$ derived in this work (gray
solid, red dashed and blue dot-dashed lines corresponding to the three sets of
LECs) and that from Ref. Miranda:2018cpf  by phase dispersive representation
(magenta dotted line).

###  5.3 Applications of the Tensor Form Factors

In the previous section, we have discussed the application of the two-meson
vector form factors that characterize the SM contribution to the hadronic
charged-lepton decay. At the same time, these processes also provide a
suitable platform for searching for the BSM physics due to their large phase
space. In the literature, the BSM physics effects are included by introducing
a general set of higher-dimensional operators beyond the SM. For instance, the
dimension-6 effective operators (with only left-handed neutrinos) responsible
for $\tau\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$ are given by Miranda:2018cpf

$\displaystyle\mathcal{L}_{CC}=$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}\left[\left(1+\left[v_{L}\right]_{\ell\ell}\right)\bar{\ell}_{L}\gamma_{\mu}\nu_{\ell
L}\bar{u}_{L}\gamma^{\mu}d_{L}+\left[v_{R}\right]_{\ell\ell}\bar{\ell}_{L}\gamma_{\mu}\nu_{\ell
L}\bar{u}_{R}\gamma^{\mu}d_{R}\right.$
$\displaystyle+\left[s_{L}\right]_{\ell\ell}\bar{\ell}_{R}\nu_{\ell
L}\bar{u}_{R}d_{L}+\left[s_{R}\right]_{\ell\ell}\bar{\ell}_{R}\nu_{\ell
L}\bar{u}_{L}d_{R}\left.+\left[t_{L}\right]_{\ell\ell}\bar{\ell}_{R}\sigma_{\mu\nu}\nu_{\ell
L}\bar{u}_{R}\sigma^{\mu\nu}d_{L}\right]+\text{ h.c.},$ (5.17)

where the SM Lagrangian is recovered by setting
$v_{L}=v_{R}=s_{L}=s_{R}=t_{L}=0$. In particular, the $t_{L}$ term contains
the tensor interaction. The decay amplitude for
$\tau^{-}(P)\rightarrow\pi^{-}\left(P_{\pi^{-}}\right)\pi^{0}\left(P_{\pi^{0}}\right)\nu_{\tau}\left(P^{\prime}\right)$
reads

$\displaystyle\mathcal{M}$
$\displaystyle=\mathcal{M}_{V}+\mathcal{M}_{S}+\mathcal{M}_{T}$
$\displaystyle=\frac{G_{F}V_{ud}\sqrt{S_{EW}}}{\sqrt{2}}\left(1+\epsilon_{L}+\epsilon_{R}\right)\left[L_{\mu}H^{\mu}+\hat{\epsilon}_{S}LH+2\hat{\epsilon}_{T}L_{\mu\nu}H^{\mu\nu}\right],$
(5.18)

with the leptonic and hadronic sectors, respectively, given by

$\displaystyle L_{\mu}$
$\displaystyle=\bar{u}_{\nu_{\tau}}\left(P^{\prime}\right)\gamma^{\mu}\left(1-\gamma^{5}\right)u_{\tau}(P)~{},$
$\displaystyle L$
$\displaystyle=\bar{u}_{\nu_{\tau}}\left(P^{\prime}\right)\left(1+\gamma^{5}\right)u_{\tau}(P)~{},$
$\displaystyle L_{\mu\nu}$
$\displaystyle=\bar{u}_{\nu_{\tau}}\left(P^{\prime}\right)\sigma_{\mu\nu}\left(1+\gamma^{5}\right)u_{\tau}(P)~{},$
(5.19)

and

$\displaystyle H$
$\displaystyle=\left\langle\pi^{0}\pi^{-}|\bar{d}u|0\right\rangle~{},$
$\displaystyle H^{\mu}$
$\displaystyle=\left\langle\pi^{0}\pi^{-}\left|\bar{d}\gamma^{\mu}u\right|0\right\rangle~{},$
$\displaystyle H^{\mu\nu}$
$\displaystyle=\left\langle\pi^{0}\pi^{-}\left|\bar{d}\sigma^{\mu\nu}u\right|0\right\rangle~{}.$
(5.20)

Therefore an improved understanding of the tensor form factors (as well as the
scalar and vector form factors) will better constrain the strength of the BSM
physics interactions. The same applies to other decay processes such as
$\tau^{-}\to(K\pi)^{-}\nu_{\tau}$ Cirigliano:2017tqn ; Rendon:2019awg ;
Chen:2019vbr  and $\tau^{-}\to(\pi\eta)^{-}\nu_{\tau}$ Garces:2017jpz ;
Castro:2018cot .

##  6 Conclusions

In this work, we have performed a complete study of the scalar, vector, and
tensor two-meson form factors based on uChPT improved by dispersion relations
that remove the unphysical sub-threshold singularities. The slight unitarity
violation in the coupled-channel IAM as in Ref. GomezNicola:2001as  is thus
removed. The low-energy constants used for the uChPT calculation are fixed by
a global fit to the available data of two-meson scattering observables. The
resulting form factors are expected to be applicable in a wide energy region
$0\ldots 1.2\ \text{GeV}$. For the scalar and vector form factors, our results
may be regarded as an update to those given in the existing literature, but
with a much simpler realization of the dispersion relation improvement. On the
other hand, this work provides the first-ever systematic analysis of the full
two-meson tensor form factors. These form factors play important roles in
precision tests of SM and searches for the BSM physics.

## Acknowledgements

The authors are very grateful to José R. Peláez and Michael Döring for
providing data and for useful discussions. We also thank the anonymous referee
for pointing out a defect in the first version of this manuscript. BK thanks
Gilberto Colangelo for helpful e-mail communication. This work is supported in
part by Natural Science Foundation of China under Grant Nos. 11735010,
11835015, 12047503, 11911530088, 11961141012 and U2032102, by Natural Science
Foundation of Shanghai under Grant No. 15DZ2272100, by the DFG and the NSFC
through funds provided to the Sino-German Collaborative Research Center TRR110
“Symmetries and the Emergence of Structure in QCD” (NSFC Grant No.
12070131001, DFG Project-ID 196253076), by the Chinese Academy of Sciences
(CAS) under Grant Nos. XDB34030000 and QYZDB-SSW-SYS013, and by the CAS Center
for Excellence in Particle Physics (CCEPP). The work of UGM was also supported
by the CAS President’s International Fellowship Initiative (PIFI) (Grant No.
2018DM0034), by the VolkswagenStiftung (Grant No. 93562) and by the EU
(Strong2020). CYS is also supported by the Alexander von Humboldt Foundation
through a Humboldt Research Fellowship.

##  Appendix A Loop Functions

In this appendix we summarize the relevant loop functions that appear in the
one-loop calculations within ChPT. First, from the tadpole diagrams we
encounter the following integral:

$\displaystyle\int\frac{d^{d}q}{(2\pi)^{d}}\frac{i}{q^{2}-M_{i}^{2}}=2M_{i}^{2}\lambda+\frac{M_{i}^{2}}{16\pi^{2}}\ln\frac{M_{i}^{2}}{\mu^{2}}\equiv
2M_{i}^{2}\lambda+2F_{0}^{2}\mu_{i}~{},$ (A.1)

where $\mu$ is the renormalization scale and
$\mu_{i}=(M_{i}^{2}/32\pi^{2}F_{0}^{2})\ln(M_{i}^{2}/\mu^{2})$, with
$i=\pi,K,\eta$, and $F_{0}$ is the pion decay constant in the three-flavor
chiral limit. Next, in a loop integral with two propagators we encounter the
following two-point function:

$\displaystyle-i\int\frac{d^{d}q}{(2\pi)^{d}}\frac{1}{[q^{2}-M_{P}^{2}][(q-p)^{2}-M_{Q}^{2}]}=-2\lambda+J_{PQ}^{r}(s)=J_{PQ}(0)+{\bar{J}}_{PQ}(s)~{},$
(A.2)

where $P,Q=\pi$, $K$, and $\eta$, $s=p^{2}$, and

$\displaystyle J_{PQ}(0)$ $\displaystyle=$
$\displaystyle-2\lambda-\frac{2F_{0}^{2}}{\Delta}(\mu_{P}-\mu_{Q})~{},$
$\displaystyle\bar{J}_{PQ}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}\left[2+\left(\frac{\Delta}{s}-\frac{\Sigma}{\Delta}\right)\ln\frac{M_{Q}^{2}}{M_{P}^{2}}+\frac{\nu}{s}\left\\{\ln\left(\frac{(s-\Sigma-\nu-i\varepsilon)^{2}}{\Delta^{2}-\Sigma^{2}}\right)+i\pi\right\\}\right]~{},$
(A.3)

with

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
M_{P}^{2}-M_{Q}^{2}~{},\;\;\Sigma=M_{P}^{2}+M_{Q}^{2}~{},$
$\displaystyle\nu(s)$ $\displaystyle=$
$\displaystyle\sqrt{\left(s-\left(M_{P}+M_{Q}\right)^{2}\right)\left(s-\left(M_{P}-M_{Q}\right)^{2}\right)}~{}.$
(A.4)

For the case of a single mass $M_{P}=M_{Q}$, the $\bar{J}$-function reads

$\displaystyle J_{PP}(0)$ $\displaystyle=$
$\displaystyle-2\lambda-\frac{1}{16\pi^{2}}-\frac{2F_{0}^{2}}{M_{P}^{2}}\mu_{P}~{},$
$\displaystyle{\bar{J}}_{PP}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}\left[4+\frac{\sqrt{s(s-4M_{P}^{2})}}{s}\left\\{\ln\left(-\frac{(s-2M_{P}^{2}-\sqrt{s(s-4M_{P}^{2})}-i\varepsilon)^{2}}{4M_{P}^{2}}\right)+i\pi\right\\}\right]~{}.$

Note that the above integrals have the correct unitarity structure along the
right-hand cut, which extends on the real axis from $s=(M_{P}+M_{Q})^{2}$ to
infinity.

##  Appendix B Isospin Decomposition of the Scattering Amplitudes and Form
Factors

In this appendix we state our conventions in defining one- and two-particle
isospin eigenstates, and show how to construct scattering amplitudes of
definite isospin.

###  B.1 One-Particle Isospin Eigenstates

The one-particle isospin eigenstate is generically denoted as
$\left|\phi,I,I_{3}\right\rangle$. The phases of such states are chosen such
that they satisfy standard results when acted on by isospin-raising and
-lowering operators:

$\displaystyle\hat{J}_{+}\left|I,I_{3}\right\rangle$ $\displaystyle=$
$\displaystyle\sqrt{(I-I_{3})(I+I_{3}+1)}\left|I,I_{3}+1\right\rangle,$
$\displaystyle\hat{J}_{-}\left|I,I_{3}\right\rangle$ $\displaystyle=$
$\displaystyle\sqrt{(I+I_{3})(I-I_{3}+1)}\left|I,I_{3}-1\right\rangle.$ (B.1)

For the pion triplet, we choose:

$\displaystyle\left|\pi,1,+1\right\rangle=-\left|\pi^{+}\right\rangle,~{}~{}~{}~{}\left|\pi,1,0\right\rangle=\left|\pi^{0}\right\rangle,~{}~{}~{}~{}\left|\pi,1,-1\right\rangle=\left|\pi^{-}\right\rangle.$
(B.2)

For the kaon doublet $K^{+},K^{0}$, we choose:

$\displaystyle\left|K,\frac{1}{2},+\frac{1}{2}\right\rangle=\left|K^{+}\right\rangle,~{}~{}~{}~{}\left|K,\frac{1}{2},-\frac{1}{2}\right\rangle=\left|K^{0}\right\rangle.$
(B.3)

For the anti-kaon doublet $\bar{K}^{0}$, $K^{-}$, we choose:

$\displaystyle\left|\bar{K},\frac{1}{2},+\frac{1}{2}\right\rangle=\left|\bar{K}^{0}\right\rangle,~{}~{}~{}~{}\left|\bar{K},\frac{1}{2},-\frac{1}{2}\right\rangle=-\left|K^{-}\right\rangle.$
(B.4)

Finally, the $\eta$-particle is simply an isospin singlet:
$\left|\eta,0,0\right\rangle=\left|\eta\right\rangle$.

###  B.2 Two-Particle Isospin Eigenstates

Two-particle isospin eigenstates, denoted generically as
$\left|\phi\phi^{\prime},I,I_{3}\right\rangle$, are simply obtained by
combining one-particle isospin eigenstates with appropriate Clebsch–Gordan
coefficients. There is one complication, namely isospin eigenstates that are
constructed by two particles in the same isospin multiplet should be
multiplied by a factor $1/\sqrt{2}$ so that the completeness relation they
satisfied is properly normalized as
$\sum_{I,I_{3}}\left|I,I_{3}\right\rangle\left\langle
I,I_{3}\right|=1$.444Such a factor is sometimes defined into the partial-wave
expansion, see, e.g., Ref. GomezNicola:2001as . In the following we present
the two-particle isospin eigenstates that are relevant to this work:

$\displaystyle\left|\pi\pi,0,0\right\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}\left[\left|\pi^{+}\pi^{-}\right\rangle+\left|\pi^{0}\pi^{0}\right\rangle+\left|\pi^{-}\pi^{+}\right\rangle\right]~{},$
$\displaystyle\left|\pi\pi,1,0\right\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\left|\pi^{+}\pi^{-}\right\rangle-\left|\pi^{-}\pi^{+}\right\rangle\right]~{},$
$\displaystyle\left|\pi\pi,2,2\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|\pi^{+}\pi^{+}\right\rangle~{},$
$\displaystyle\left|K\pi,\frac{3}{2},\frac{3}{2}\right\rangle$
$\displaystyle=$ $\displaystyle-\left|K^{+}\pi^{+}\right\rangle~{},$
$\displaystyle\left|K\pi,\frac{1}{2},-\frac{1}{2}\right\rangle$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\left|K^{+}\pi^{-}\right\rangle-\sqrt{\frac{1}{3}}\left|K^{0}\pi^{0}\right\rangle~{},$
$\displaystyle\left|K\bar{K},1,0\right\rangle$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}\left|K^{+}K^{-}\right\rangle+\sqrt{\frac{1}{2}}\left|K^{0}\bar{K}^{0}\right\rangle~{},$
$\displaystyle\left|K\bar{K},0,0\right\rangle$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{1}{2}}\left|K^{+}K^{-}\right\rangle-\sqrt{\frac{1}{2}}\left|K^{0}\bar{K}^{0}\right\rangle~{},$
$\displaystyle\left|K\eta,\frac{1}{2},-\frac{1}{2}\right\rangle$
$\displaystyle=$ $\displaystyle\left|K^{0}\eta\right\rangle~{},$
$\displaystyle\left|\pi\eta,1,0\right\rangle$ $\displaystyle=$
$\displaystyle\left|\pi^{0}\eta\right\rangle~{},$
$\displaystyle\left|\eta\eta,0,0\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|\eta\eta\right\rangle~{}.$ (B.5)

###  B.3 Scattering Amplitudes of Definite Isospin

Following Ref. GomezNicola:2001as , one may choose the eight independent
$\phi\phi$ scattering amplitudes as follows:

$\displaystyle T_{1}(s,t,u)$ $\displaystyle\equiv$ $\displaystyle
T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(s,t,u)~{},$ $\displaystyle
T_{2}(s,t,u)$ $\displaystyle\equiv$ $\displaystyle T_{K^{+}\pi^{+}\rightarrow
K^{+}\pi^{+}}(s,t,u)~{},$ $\displaystyle T_{3}(s,t,u)$ $\displaystyle\equiv$
$\displaystyle T_{K^{+}K^{-}\rightarrow K^{+}K^{-}}(s,t,u)~{},$ $\displaystyle
T_{4}(s,t,u)$ $\displaystyle\equiv$ $\displaystyle
T_{K^{0}\bar{K}^{0}\rightarrow K^{+}K^{-}}(s,t,u)~{},$ $\displaystyle
T_{5}(s,t,u)$ $\displaystyle\equiv$ $\displaystyle T_{K^{0}\eta\rightarrow
K^{0}\eta}(s,t,u)~{},$ $\displaystyle T_{6}(s,t,u)$ $\displaystyle\equiv$
$\displaystyle T_{K^{0}\pi^{0}\rightarrow K^{0}\eta}(s,t,u)~{},$
$\displaystyle T_{7}(s,t,u)$ $\displaystyle\equiv$ $\displaystyle
T_{\pi^{0}\eta\rightarrow\pi^{0}\eta}(s,t,u)~{},$ $\displaystyle T_{8}(s,t,u)$
$\displaystyle\equiv$ $\displaystyle
T_{\eta\eta\rightarrow\eta\eta}(s,t,u)~{}.$ (B.6)

All scattering amplitudes of definite isospin can be expressed in terms of
these eight amplitudes and their crossings. The results are as follows. For
$I=0$, we have

$\displaystyle T^{I=0}_{\pi\pi\rightarrow\pi\pi}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[3T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(s,t,u)+T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(t,s,u)+T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(u,t,s)\right],$
$\displaystyle T^{I=0}_{\pi\pi\rightarrow K\bar{K}}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\left[T_{K^{+}\pi^{+}\rightarrow
K^{+}\pi^{+}}(u,s,t)+T_{K^{+}\pi^{+}\rightarrow
K^{+}\pi^{+}}(t,s,u)\right]~{},$ $\displaystyle
T^{I=0}_{\pi\pi\rightarrow\eta\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{2}T_{\pi^{0}\eta\rightarrow\pi^{0}\eta}(t,s,u)~{},$
$\displaystyle T^{I=0}_{K\bar{K}\rightarrow K\bar{K}}(s,t,u)$ $\displaystyle=$
$\displaystyle T_{K^{+}K^{-}\rightarrow
K^{+}K^{-}}(s,t,u)+T_{K^{0}\bar{K}^{0}\rightarrow K^{+}K^{-}}(s,t,u)~{},$
$\displaystyle T^{I=0}_{K\bar{K}\rightarrow\eta\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle-T_{K^{0}\eta\rightarrow K^{0}\eta}(t,s,u)~{},$ $\displaystyle
T^{I=0}_{\eta\eta\rightarrow\eta\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}T_{\eta\eta\rightarrow\eta\eta}(s,t,u)~{}.$ (B.7)

For $I=1/2$, we have

$\displaystyle T^{I=1/2}_{K\pi\rightarrow K\pi}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{3}{2}T_{K^{+}\pi^{+}\rightarrow
K^{+}\pi^{+}}(u,t,s)-\frac{1}{2}T_{K^{+}\pi^{+}\rightarrow
K^{+}\pi^{+}}(s,t,u)~{},$ $\displaystyle T^{I=1/2}_{K\pi\rightarrow
K\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle-\sqrt{3}T_{K^{0}\pi^{0}\rightarrow K^{0}\eta}(s,t,u)~{},$
$\displaystyle T^{I=1/2}_{K\eta\rightarrow K\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle T_{K^{0}\eta\rightarrow K^{0}\eta}(s,t,u)~{}.$ (B.8)

For $I=1$, we have

$\displaystyle T^{I=1}_{\pi\pi\rightarrow\pi\pi}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(t,s,u)-T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(u,t,s))~{},$
$\displaystyle T^{I=1}_{\pi\pi\rightarrow K\bar{K}}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(T_{K^{+}\pi^{+}\rightarrow
K^{+}\pi^{+}}(u,s,t)-T_{K^{+}\pi^{+}\rightarrow K^{+}\pi^{+}}(t,s,u)~{},$
$\displaystyle T^{I=1}_{\pi\eta\rightarrow\pi\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle T_{\pi^{0}\eta\rightarrow\pi^{0}\eta}(s,t,u)~{},$
$\displaystyle T^{I=1}_{K\bar{K}\rightarrow K\bar{K}}(s,t,u)$ $\displaystyle=$
$\displaystyle T_{K^{+}K^{-}\rightarrow
K^{+}K^{-}}(s,t,u)-T_{K^{0}\bar{K}^{0}\rightarrow K^{+}K^{-}}(s,t,u)~{},$
$\displaystyle T^{I=1}_{K\bar{K}\rightarrow\pi\eta}(s,t,u)$ $\displaystyle=$
$\displaystyle\sqrt{2}T_{K^{0}\pi^{0}\rightarrow K^{0}\eta}(t,s,u)~{}.$ (B.9)

For $I=3/2$, we have

$\displaystyle T^{I=3/2}_{K\pi\rightarrow K\pi}(s,t,u)$ $\displaystyle=$
$\displaystyle T_{K^{+}\pi^{+}\rightarrow K^{+}\pi^{+}}(s,t,u)~{}.$ (B.10)

Finally, for $I=2$, we have

$\displaystyle T^{I=2}_{\pi\pi\rightarrow\pi\pi}(s,t,u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(t,s,u)-T_{\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}}(u,t,s))~{}.$
(B.11)

###  B.4 Form Factors

Finally, we discuss how the two-meson form factors are classified according to
the isospin eigenstates. The form factors of interest have the following
general form

$\left\langle\phi\phi^{\prime}\right|\bar{q}\Gamma
q^{\prime}\left|0\right\rangle,$ (B.12)

where $\Gamma$ is any matrix in the non-flavor space. It is obvious that with
different choices of $\bar{q}q^{\prime}$ and $\phi\phi^{\prime}$ there will be
different form factors. However, not all of them are independent because some
of them are related via charge conjugation and isospin symmetry. In this
appendix, we will extract all the independent form factors in order to
minimize the calculation.

We shall start from the classification of $\phi\phi^{\prime}$ states according
to isospin. There are 10 groups of them: $\pi\pi$, $\pi K$, $\pi\bar{K}$,
$\pi\eta$, $KK$, $K\bar{K}$, $K\eta$, $\bar{K}\bar{K}$, $\bar{K}\eta$, and
$\eta\eta$. However, $KK$ and $\bar{K}\bar{K}$ states have no nonvanishing
form factors because they have strangeness $\pm 2$, which cannot be obtained
from a quark bilinear. Next, $\pi K$ and $\pi\bar{K}$ are related by charge
conjugation (the same holds for $K\eta$ and $\bar{K}\eta$), so we just need to
choose one of them. Therefore, there are only 6 groups of independent
$\phi\phi^{\prime}$ states that will give nonvanishing form factors. They can
be chosen as

$\pi\pi,\quad K\pi,\quad\pi\eta,\quad K\bar{K},\quad
K\eta,\quad\text{and}\quad\eta\eta.$ (B.13)

Next we study the quark bilinear operators (we shall ignore the matrix
$\Gamma$ for notational simplicity). At the first sight, there are 9 possible
combinations: $\bar{u}u$, $\bar{d}d$, $\bar{s}s$, $\bar{u}d$, $\bar{u}s$,
$\bar{d}u$, $\bar{d}s$, $\bar{s}u$, and $\bar{s}d$. However, some of them are
related by isospin and charge conjugation (C.C). For example,

$\displaystyle\bar{u}d~{}\stackrel{{\scriptstyle\mathrm{isospin}}}{{\longleftrightarrow}}~{}\bar{u}u-\bar{d}d~{}\stackrel{{\scriptstyle\mathrm{isospin}}}{{\longleftrightarrow}}~{}\bar{d}u~{},$
$\displaystyle\bar{u}s~{}\stackrel{{\scriptstyle\mathrm{isospin}}}{{\longleftrightarrow}}~{}\bar{d}s~{}\stackrel{{\scriptstyle\mathrm{C.C}}}{{\longleftrightarrow}}~{}\bar{s}d~{}\stackrel{{\scriptstyle\mathrm{isospin}}}{{\longleftrightarrow}}~{}\bar{s}u~{}.$
(B.14)

For two bilinears related by isospin, their matrix elements are related to
each other by the Wigner–Eckart theorem. For bilinears related by charge
conjugation, their matrix elements are related by charge-conjugating the
outcoming mesons. Therefore, there are only four independent quark bilinears,
which can be chosen as

$\displaystyle I=0:~{}n\equiv\frac{\bar{u}u+\bar{d}d}{\sqrt{2}}~{},\quad
I=0:~{}\bar{s}s~{},\quad I=1:~{}\bar{u}d~{},\quad
I=\frac{1}{2}:~{}\bar{u}s~{}.$ (B.15)

Now,

Table 3: Quark bilinears and meson fields that form independent form factors. Quark bilinear | Two-particle isospin eigenstates
---|---
$n=({\bar{u}u+\bar{d}d})/{\sqrt{2}}$ |  $\pi\pi$, $K\bar{K}$, $\eta\eta$
$\bar{s}s$ |  $\pi\pi$, $K\bar{K}$, $\eta\eta$
$\bar{u}d$ |  $\pi\pi$, $\pi\eta$, $K\bar{K}$
$\bar{u}s$ |  $K\pi$, $K\eta$

with each independent quark bilinear, we just need to compute one matrix
element for each independent $\phi\phi^{\prime}$ group; the others are related
by Wigner–Eckart theorem. Therefore, the independent form factors can be
chosen as the matrix elements of the quark bilinears between the vacuum and
two-particle isospin eigenstates given in Table 3.

##  Appendix C Subtraction of Sub-Threshold Poles

It is well known that the naïve application of the IAM will lead to spurious
poles in the sub-threshold region for quantities in the $J=0$ channel, which
is related to the existence of the Adler zero in the $S$-wave Hannah:1997sm ;
GomezNicola:2007qj . In this section, we briefly outline the method one can
adopt to subtract such spurious pole in both unitarized partial waves and form
factors. One may refer to Ref. GomezNicola:2007qj  for detailed discussions of
the topic.

Let us restrict ourselves to a single-channel unitarization of partial waves
and form factors. Also, we shall simply use $T_{2}$ and $T_{4}$ to denote the
$\mathcal{O}(p^{2})$ and $\mathcal{O}(p^{4})$ $J=0$ partial wave for
notational simplicity. First, let us recall the naïve IAM formula for partial
waves,

$T=\frac{T_{2}^{2}}{T_{2}-T_{4}}~{}.$ (C.1)

The $\mathcal{O}(p^{2})$ amplitude $T_{2}$ has a zero at $s=s_{2}$ below the
production threshold: $T_{2}(s_{2})=0$. This is nothing but the Adler zero at
the $\mathcal{O}(p^{2})$ level for the $S$-wave. Meanwhile, the combination
$T_{2}-T_{4}$ has a different zero at $s=s^{\prime}$. Since in general
$s^{\prime}\neq s_{2}$, the naïve IAM formula (C.1) has a spurious pole at
$s=s^{\prime}$. Similarly, the naïve IAM formula for the scalar form factor

$F_{S}=F_{S}^{(0)}+\frac{T_{2}}{T_{2}-T_{4}}F_{S}^{(2)}$ (C.2)

suffers from a pole at $s=s^{\prime}$.

From the dispersive point of view, the existence of such spurious poles is due
to the negligence of the pole contribution of various inverse amplitudes in
the derivation of the IAM formula through a dispersion relation. Therefore,
the problem can be resolved by appropriately adding back these contributions.
Unfortunately, a dispersive derivation of the multi-channel IAM is still
missing, so we can only stick to the single-channel case in this discussion.

To derive the single-channel IAM formula for partial waves, we shall consider
the dispersion relation of $1/T$, $1/T_{2}$ and $T_{4}/T_{2}^{2}$
respectively. Also, here we shall simplify our discussion by considering an
unsubtracted dispersion relation of each quantity, the outcome turns out to be
equivalent to choosing the subtraction point at the Adler zero of the full
amplitude GomezNicola:2007qj . The dispersion relations we obtain are

$\displaystyle\frac{1}{T(s)}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{s_{\rm
th}}dz\frac{\mathrm{Im}T^{-1}(z)}{z-s}+LC(1/T)+PC(1/T)~{},$
$\displaystyle\frac{1}{T_{2}(s)}$ $\displaystyle=$ $\displaystyle
PC(1/T_{2})~{},$ $\displaystyle\frac{T_{4}(s)}{T_{2}^{2}(s)}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{s_{\rm
th}}dz\frac{\mathrm{Im}\\{T_{4}(z)/T_{2}^{2}(z)\\}}{z-s}+LC(T_{4}/T_{2}^{2})+PC(T_{4}/T_{2}^{2})~{}.$
(C.3)

Here $LC$ and $PC$ denote the left-hand cut and the pole contributions to the
dispersion integral, respectively. Since the full and perturbative amplitude
satisfies the following unitarity relation on the right-hand cut:

$-\mathrm{Im}T^{-1}(s)=\mathrm{Im}\\{T_{4}(s)/T_{2}^{2}(s)\\}=\frac{|\vec{p}_{\text{cm}}|}{8\pi\sqrt{s}}~{},$
(C.4)

the right-hand cut contributions to $1/T$ and $T_{4}/T_{2}^{2}$ simply differ
by a sign. Furthermore, one approximates $LC(1/T)\approx LC(T_{4}/T_{2}^{2})$
by arguing that the left-hand cut contribution is weighted at low energies
where the usual ChPT expansion is appropriate. With these, we can write

$\frac{1}{T(s)}\approx\frac{1}{T_{2}(s)}-\frac{T_{4}(s)}{T_{2}^{2}(s)}-PC(1/T_{2})+PC(T_{4}/T_{2}^{2})+PC(1/T)~{},$
(C.5)

where the explicit expressions for the pole contributions are given by

$\displaystyle PC(1/T)$ $\displaystyle=$
$\displaystyle\frac{1}{(s-s_{A})T^{\prime}(s_{A})}~{},$ $\displaystyle
PC(1/T_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{(s-s_{2})T^{\prime}_{2}(s_{2})}~{},$ $\displaystyle
PC(T_{4}/T_{2}^{2})$ $\displaystyle=$
$\displaystyle\frac{T^{\prime}_{4}(s_{2})}{(s-s_{2})(T^{\prime}_{2}(s_{2}))^{2}}+\frac{T_{4}(s_{2})}{(s-s_{2})^{2}(T^{\prime}_{2}(s_{2}))^{2}}-\frac{T_{4}(s_{2})T^{\prime\prime}_{2}(s_{2})}{(s-s_{2})(T^{\prime}_{2}(s_{2}))^{3}}~{}.$
(C.6)

Here $s_{A}$ is the Adler zero of the full partial-wave $T$-matrix. Of course
its exact value is unknown, but we can approximate it by the Adler zero of
$T_{2}+T_{4}$, which is $s_{A}\approx s_{2}+s_{4}$ where $s_{4}\approx-
T_{4}(s_{2})/T_{2}^{\prime}(s_{2})$. If one neglects the pole contributions
then the naïve IAM formula (C.1) is recovered, the inclusion of the pole
contributions will eliminate the spurious pole in the sub-threshold region.

The pole subtraction for the scalar form factor follows a similar logic. Let
us start by considering the dispersion relations of $(F_{S}-F^{(0)}_{S})/T$
and $F^{(2)}_{S}/T_{2}$, respectively,

$\displaystyle\frac{F_{S}(s)-F_{S}^{(0)}(s)}{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{s_{\rm
th}}dz\frac{\mathrm{Im}\\{(F_{S}(z)-F_{S}^{(0)}(z))/T(z)\\}}{z-s}+LC((F_{S}-F_{S}^{(0)})/T)$
$\displaystyle+PC((F_{S}-F_{S}^{(0)})/T),$
$\displaystyle\frac{F_{S}^{(2)}(s)}{T_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\infty}_{s_{\rm
th}}dz\frac{\mathrm{Im}\\{F_{S}^{(2)}(z)/T_{2}(z)\\}}{z-s}+LC(F_{S}^{(2)}/T_{2})+PC(F_{S}^{(2)}/T_{2})~{}.$
(C.7)

Along the right-hand cut, we have

$\mathrm{Im}\left\\{\frac{F_{S}(s)-F^{(0)}_{S}(s)}{T(s)}\right\\}=\mathrm{Im}\left\\{\frac{F_{S}^{(2)}(s)}{T_{2}(s)}\right\\}=F_{S}^{(0)}(s)\frac{|\vec{p}_{\rm
cm}|}{8\pi\sqrt{s}}~{}.$ (C.8)

Therefore, the right-hand-cut contributions for $(F_{S}-F_{S}^{(0)})/T$ and
$F_{S}^{(2)}/T_{2}$ are the same. Furthermore, we assume that the two left-
hand-cut contributions are also approximately the same, following an argument
similar to the case of the partial-wave scattering amplitude. With this we
obtain

$F_{S}(s)\approx
F_{S}^{(0)}(s)+\frac{T(s)}{T_{2}(s)}F_{S}^{(2)}(s)+T(s)\left\\{PC((F_{S}-F_{S}^{(0)})/T)-PC(F_{S}^{(2)}/T_{2})\right\\}~{},$
(C.9)

where the explicit expressions for the pole contributions are given by

$\displaystyle PC((F_{S}-F_{S}^{(0)})/T)$ $\displaystyle=$
$\displaystyle\frac{F_{S}(s_{A})-F_{S}^{(0)}(s_{A})}{(s-s_{A})T^{\prime}(s_{A})}~{},$
$\displaystyle PC(F_{S}^{(2)}/T)$ $\displaystyle=$
$\displaystyle\frac{F^{(2)}_{S}(s_{2})}{(s-s_{2})T_{2}^{\prime}(s_{2})}~{}.$
(C.10)

Again, we may approximate $F_{S}$ and $T$ in the formulae above by their
respective ChPT expressions up to NLO.

Notice that if we neglect the pole contributions in Eq. (C.9) and substitute
$T(s)$ by the naïve IAM formula for the partial-wave $T$-matrix, then we re-
obtain the naïve IAM unitarized scalar form factor. This expression works fine
above threshold but suffers from a spurious pole below threshold. On the other
hand, if we use Eq. (C.9) with the expression of $T(s)$ given in Eq. (C.5),
then the spurious pole will be smoothly eliminated.

##  Appendix D Form Factors in ChPT to One Loop

In this section, we list the NLO ChPT results for scalar, vector, and tensor
form factors. Notice that we express our result in terms of the physical pion
decay constant $F_{\pi}$, which is related to $F_{0}$ by

$F_{\pi}=F_{0}\left\\{1-2\mu_{\pi}-\mu_{K}+\frac{4}{F_{0}^{2}}\left[2L_{4}^{r}M_{K}^{2}+(L_{4}^{r}+L_{5}^{r})M_{\pi}^{2}\right]\right\\}.$
(D.1)

###  D.1 Scalar Form Factors for the $I=0$ System

The two-meson states are chosen to be exact isospin eigenstates. At NLO,
calculating the Feynman diagrams shown in Fig. 6 gives

$\displaystyle-\frac{F^{n}_{S,\pi\pi}(s)}{\sqrt{3}}$ $\displaystyle=$
$\displaystyle
1+\mu_{\pi}-\frac{1}{3}\mu_{\eta}-\frac{8(2M_{K}^{2}+3M_{\pi}^{2}-s)}{F_{\pi}^{2}}L_{4}^{r}+\frac{4(s-4M_{\pi}^{2})}{F_{\pi}^{2}}L_{5}^{r}+\frac{16(2M_{K}^{2}+3M_{\pi}^{2})}{F_{\pi}^{2}}L_{6}^{r}$
$\displaystyle+\frac{32M_{\pi}^{2}}{F_{\pi}^{2}}L_{8}^{r}+\frac{2s-M_{\pi}^{2}}{2F_{\pi}^{2}}J^{r}_{\pi\pi}(s)+\frac{s}{4F_{\pi}^{2}}J^{r}_{KK}(s)+\frac{M_{\pi}^{2}}{18F_{\pi}^{2}}J^{r}_{\eta\eta}(s)~{},$
$\displaystyle-F^{n}_{S,K\bar{K}}(s)$ $\displaystyle=$ $\displaystyle
1+\frac{2}{3}\mu_{\eta}-\frac{8(6M_{K}^{2}+M_{\pi}^{2}-2s)}{F_{\pi}^{2}}L_{4}^{r}+\frac{4(s-4M_{K}^{2})}{F_{\pi}^{2}}L_{5}^{r}+\frac{16(6M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}L_{6}^{r}$
$\displaystyle+\frac{32M_{K}^{2}}{F_{\pi}^{2}}L_{8}^{r}+\frac{3s}{4F_{\pi}^{2}}J_{\pi\pi}^{r}(s)+\frac{3s}{4F_{\pi}^{2}}J^{r}_{KK}(s)+\frac{9s-2M_{\pi}^{2}-6M_{\eta}^{2}}{36F_{\pi}^{2}}J^{r}_{\eta\eta}(s)~{},$
$\displaystyle 3F^{n}_{S,\eta\eta}(s)$ $\displaystyle=$ $\displaystyle
1-3\mu_{\pi}+4\mu_{K}-\frac{1}{3}\mu_{\eta}+\frac{8(-10M_{K}^{2}+M_{\pi}^{2}+3s)}{F_{\pi}^{2}}L_{4}^{r}-\frac{4(16M_{K}^{2}-4M_{\pi}^{2}-3s)}{3F_{\pi}^{2}}L_{5}^{r}$
(D.2)
$\displaystyle+\frac{16(10M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{6}^{r}-\frac{128(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{7}^{r}+\frac{32M_{\pi}^{2}}{F_{\pi}^{2}}L_{8}^{r}+\frac{3M_{\pi}^{2}}{2F_{\pi}^{2}}J^{r}_{\pi\pi}(s)$
$\displaystyle+\frac{9s-2M_{\pi}^{2}-6M_{\eta}^{2}}{4F_{\pi}^{2}}J^{r}_{KK}(s)+\frac{16M_{K}^{2}-7M_{\pi}^{2}}{18F_{\pi}^{2}}J^{r}_{\eta\eta}(s)$

for the nonstrange form factors and

$\displaystyle-\sqrt{\frac{2}{3}}F^{\bar{s}s}_{S,\pi\pi}(s)$ $\displaystyle=$
$\displaystyle\frac{8(s-2M_{\pi}^{2})}{F_{\pi}^{2}}L_{4}^{r}+\frac{32M_{\pi}^{2}}{F_{\pi}^{2}}L_{6}^{r}+\frac{s}{2F_{\pi}^{2}}J^{r}_{KK}(s)+\frac{2M_{\pi}^{2}}{9F_{\pi}^{2}}J^{r}_{\eta\eta}(s)~{},$
$\displaystyle-\frac{F^{\bar{s}s}_{S,K\bar{K}}(s)}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle
1+\frac{2}{3}\mu_{\eta}-\frac{8(4M_{K}^{2}+M_{\pi}^{2}-s)}{F_{\pi}^{2}}L_{4}^{r}+\frac{4(s-4M_{K}^{2})}{F_{\pi}^{2}}L_{5}^{r}+\frac{16(4M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}L_{6}^{r}$
$\displaystyle+\frac{32M_{K}^{2}}{F_{\pi}^{2}}L_{8}^{r}+\frac{3s}{4F_{\pi}^{2}}J_{KK}^{r}(s)+\frac{9s-2M_{\pi}^{2}-6M_{\eta}^{2}}{18F_{\pi}^{2}}J^{r}_{\eta\eta}(s)~{},$
$\displaystyle\frac{3\sqrt{2}F^{\bar{s}s}_{S,\eta\eta}(s)}{4}$
$\displaystyle=$ $\displaystyle
1+2\mu_{K}-\frac{4}{3}\mu_{\eta}+\frac{2(-16M_{K}^{2}-2M_{\pi}^{2}+3s)}{F_{\pi}^{2}}L_{4}^{r}-\frac{4(16M_{K}^{2}-4M_{\pi}^{2}-3s)}{3F_{\pi}^{2}}L_{5}^{r}$
(D.3)
$\displaystyle+\frac{8(8M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}L_{6}^{r}+\frac{64(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{7}^{r}+\frac{32(2M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{8}^{r}$
$\displaystyle+\frac{9s-2M_{\pi}^{2}-6M_{\eta}^{2}}{8F_{\pi}^{2}}J^{r}_{KK}(s)+\frac{16M_{K}^{2}-7M_{\pi}^{2}}{18F_{\pi}^{2}}J^{r}_{\eta\eta}(s)$

for the hidden-strangeness form factors.

###  D.2 Scalar Form Factors for the $I=\frac{1}{2}$ System

The scalar form factors for the $I=1/2$ meson–meson systems up to NLO in ChPT
are given by

$\displaystyle\sqrt{\frac{2}{3}}F^{\bar{u}s}_{S,K\pi}(s)$ $\displaystyle=$
$\displaystyle
1+\frac{5s-4M_{\pi}^{2}}{4(M_{K}^{2}-M_{\pi}^{2})}\mu_{\pi}-\frac{s}{2(M_{K}^{2}-M_{\pi}^{2})}\mu_{K}+\frac{8M_{K}^{2}+4M_{\pi}^{2}-9s}{12(M_{K}^{2}-M_{\pi}^{2})}\mu_{\eta}$
$\displaystyle-\frac{8(2M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}\left(L_{4}^{r}-2L_{6}^{r}\right)+\frac{4(-2M_{K}^{2}-2M_{\pi}^{2}+s)}{F_{\pi}^{2}}L_{5}^{r}+\frac{16(M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}L_{8}^{r}$
$\displaystyle-\frac{3M_{\eta}^{2}(3M_{K}^{2}-3M_{\pi}^{2}+s)-9M_{K}^{4}+M_{K}^{2}(9M_{\pi}^{2}+2s)+s(7M_{\pi}^{2}-9s)}{72F_{\pi}^{2}s}\bar{J}_{K\eta}(s)$
$\displaystyle-\frac{3M_{K}^{4}+M_{K}^{2}(2s-6M_{\pi}^{2})+3M_{\pi}^{4}+2M_{\pi}^{2}s-5s^{2}}{8F_{\pi}^{2}s}\bar{J}_{\pi
K}(s)~{},$ $\displaystyle-\sqrt{6}F^{\bar{u}s}_{S,K\eta}(s)$ $\displaystyle=$
$\displaystyle
1-\frac{3(4M_{\pi}^{2}-3s)}{4(M_{K}^{2}-M_{\pi}^{2})}\mu_{\pi}+\frac{16M_{K}^{2}-9s}{2(M_{K}^{2}-M_{\pi}^{2})}\mu_{K}+\frac{-88M_{K}^{2}+28M_{\pi}^{2}+27s}{12(M_{K}^{2}-M_{\pi}^{2})}\mu_{\eta}$
(D.4)
$\displaystyle-\frac{8(2M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}\left(L_{4}^{r}-2L_{6}^{r}\right)+\frac{4(-14M_{K}^{2}+2M_{\pi}^{2}+3s)}{3F_{\pi}^{2}}L_{5}^{r}+\frac{128(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{7}^{r}$
$\displaystyle-\frac{3M_{\eta}^{2}(3M_{K}^{2}-3M_{\pi}^{2}+s)-9M_{K}^{4}+M_{K}^{2}(9M_{\pi}^{2}+2s)+s(7M_{\pi}^{2}-9s)}{8F_{\pi}^{2}s}\bar{J}_{\pi
K}(s)$
$\displaystyle-\frac{9M_{\eta}^{4}-6M_{\eta}^{2}(3M_{K}^{2}+s)+9M_{K}^{4}-18M_{K}^{2}s+s(4M_{\pi}^{2}+9s)}{24F_{\pi}^{2}s}\bar{J}_{K\eta}(s)$
$\displaystyle+\frac{16(5M_{K}^{2}-3M_{\pi}^{2})}{F_{\pi}^{2}}L_{8}^{r}~{}.$

###  D.3 Scalar Form Factors for the $I=1$ System

The scalar form factors for the $I=1$ meson–meson systems up to NLO in ChPT
are given by

$\displaystyle F^{\bar{u}d}_{S,K\bar{K}}(s)$ $\displaystyle=$ $\displaystyle
1+\frac{2}{3}\mu_{\eta}-\frac{8(2M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}\left(L_{4}^{r}-2L_{6}^{r}\right)+\frac{4(s-4M_{K}^{2})}{F_{\pi}^{2}}L_{5}^{r}+\frac{32M_{K}^{2}}{F_{\pi}^{2}}L_{8}^{r}$
$\displaystyle+\frac{s}{4F_{\pi}^{2}}J^{r}_{KK}(s)-\frac{8M_{K}^{2}+M_{\pi}^{2}+3M_{\eta}^{2}-9s}{18F_{\pi}^{2}}J^{r}_{\pi\eta}(s)~{},$
$\displaystyle-\sqrt{\frac{3}{2}}F^{\bar{u}d}_{S,\pi\eta}(s)$ $\displaystyle=$
$\displaystyle
1-\mu_{\pi}+2\mu_{K}-\frac{1}{3}\mu_{\eta}-\frac{8(2M_{K}^{2}+M_{\pi}^{2})}{F_{\pi}^{2}}\left(L_{4}^{r}-2L_{6}^{r}\right)-\frac{4(8M_{K}^{2}+4M_{\pi}^{2}-3s)}{3F_{\pi}^{2}}L_{5}^{r}$
$\displaystyle-\frac{64(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{7}^{r}+\frac{32M_{\pi}^{2}}{F_{\pi}^{2}}L_{8}^{r}+\frac{9s-8M_{K}^{2}-M_{\pi}^{2}-3M_{\eta}^{2}}{12F_{\pi}^{2}}J^{r}_{KK}(s)$
$\displaystyle+\frac{M_{\pi}^{2}}{3F_{\pi}^{2}}J^{r}_{\pi\eta}(s)~{},$
$\displaystyle F^{\bar{u}d}_{S,\pi\pi}(s)$ $\displaystyle=$ $\displaystyle
0\quad(\text{by isospin and Bose symmetry}).$ (D.5)

###  D.4 Vector Form Factors for the $I=0$ System

As mentioned in Sect. 4.1, the SU(3) singlet vector form factor of
$\bar{u}\gamma^{\mu}u+\bar{d}\gamma^{\mu}d+\bar{s}\gamma^{\mu}s$ is zero up to
$\mathcal{O}(p^{4})$. Therefore, it is sufficient to present the form factors
of $\bar{s}\gamma^{\mu}s$ only:

$\displaystyle\frac{F^{\bar{s}s}_{V+,K\bar{K}}(s)}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle
1-2\mu_{K}+\frac{s-6M_{K}^{2}}{96\pi^{2}F_{\pi}^{2}}+\frac{2s}{F_{\pi}^{2}}L_{9}^{r}+\frac{s-4M_{K}^{2}}{4F_{\pi}^{2}}J^{r}_{KK}(s)~{},$
$\displaystyle F^{\bar{s}s}_{V+,\pi\pi}(s)$ $\displaystyle=$ $\displaystyle
0\quad(\text{by isospin and Bose symmetry}),$ $\displaystyle
F^{\bar{s}s}_{V+,\eta\eta}(s)$ $\displaystyle=$ $\displaystyle 0\quad(\text{by
Bose symmetry}).$ (D.6)

###  D.5 Vector Form Factors for the $I=\frac{1}{2}$ System

The vector form factors for the $I=1/2$ meson–meson systems up to NLO in ChPT
are given by

$\displaystyle-\sqrt{\frac{2}{3}}F^{\bar{u}s}_{V+,K\pi}(s)$ $\displaystyle=$
$\displaystyle
1+\frac{s-4M_{K}^{2}}{4(M_{K}^{2}-M_{\pi}^{2})}\mu_{\pi}+\frac{-8M_{K}^{2}+4M_{\pi}^{2}+s}{2(M_{K}^{2}-M_{\pi}^{2})}\mu_{K}+\frac{3(4M_{K}^{2}-s)}{4(M_{K}^{2}-M_{\pi}^{2})}\mu_{\eta}$
(D.7)
$\displaystyle-\frac{5M_{K}^{2}+M_{\pi}^{2}-s}{96\pi^{2}F_{\pi}^{2}}+\frac{2s}{F_{\pi}^{2}}L_{9}^{r}+\frac{M_{\eta}^{4}-2M_{\eta}^{2}(M_{K}^{2}+s)+(M_{K}^{2}-s)^{2}}{8F_{\pi}^{2}s}\bar{J}_{K\eta}(s)$
$\displaystyle+\frac{M_{K}^{4}-2M_{K}^{2}(M_{\pi}^{2}+s)+(M_{\pi}^{2}-s)^{2}}{8F_{\pi}^{2}s}\bar{J}_{\pi
K}(s)~{}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2}{3}}F^{\bar{u}s}_{V+,K\eta}(s)~{}.$

###  D.6 Vector Form Factors for the $I=1$ System

The vector form factors for the $I=1$ meson–meson systems up to NLO in ChPT
are given by

$\displaystyle\frac{F^{\bar{u}d}_{V+,\pi\pi}(s)}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle
1-\frac{4}{3}\mu_{\pi}-\frac{2}{3}\mu_{K}+\frac{s-2M_{K}^{2}-4M_{\pi}^{2}}{96\pi^{2}F_{\pi}^{2}}+\frac{2s}{F_{\pi}^{2}}L_{9}^{r}+\frac{s-4M_{\pi}^{2}}{6F_{\pi}^{2}}J^{r}_{\pi\pi}(s)+\frac{s-4M_{K}^{2}}{12F_{\pi}^{2}}J^{r}_{KK}(s)$
$\displaystyle=$ $\displaystyle F^{\bar{u}d}_{V+,K\bar{K}}(s)~{},$
$\displaystyle F^{\bar{u}d}_{V+,\pi\eta}(s)$ $\displaystyle=$ $\displaystyle
0\quad(\text{by C parity}).$ (D.8)

###  D.7 Tensor Form Factors for the $I=0$ System

For simplicity of notation, we will define $\tilde{C}_{i}^{r}\equiv
C_{i}^{r}/\Lambda_{2}$. The tensor form factors for the isoscalar systems up
to NLO in ChPT are given by

$\displaystyle F^{n}_{T,K\bar{K}}(s)$ $\displaystyle=$ $\displaystyle
1-7\mu_{\pi}-\frac{1}{3}\mu_{\eta}+\frac{s-6M_{K}^{2}}{96\pi^{2}F_{\pi}^{2}}-\frac{8(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L^{r}_{5}-4M_{\pi}^{2}\tilde{C}^{r}_{34}+2(M_{\pi}^{2}-2M_{K}^{2})\tilde{C}^{r}_{35}$
$\displaystyle-2(2M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{36}+8(M_{K}^{2}-M_{\pi}^{2})\tilde{C}^{r}_{37}-2s\tilde{C}^{r}_{88}+2(s-2M_{K}^{2})\tilde{C}^{r}_{89}-4M_{K}^{2}\tilde{C}^{r}_{106}$
$\displaystyle+2(s-2M_{K}^{2})\tilde{C}^{r}_{107}+\frac{s-4M_{K}^{2}}{4F_{\pi}^{2}}J^{r}_{KK}(s)~{},$
$\displaystyle F^{n}_{T,\pi\pi}(s)$ $\displaystyle=$ $\displaystyle
0\quad(\text{by isospin and Bose symmetry}),$ $\displaystyle
F^{n}_{T,\eta\eta}(s)$ $\displaystyle=$ $\displaystyle 0\quad(\text{by isospin
and Bose symmetry}),$ (D.9)
$\displaystyle-\frac{F^{\bar{s}s}_{T,K\bar{K}}(s)}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle
1-4\mu_{\pi}-2\mu_{K}-\frac{4}{3}\mu_{\eta}+\frac{s-6M_{K}^{2}}{96\pi^{2}F_{\pi}^{2}}-\frac{8(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L^{r}_{5}+4(M_{\pi}^{2}-2M_{K}^{2})\tilde{C}^{r}_{34}$