#  Pole position of the $a_{1}(1260)$ resonance in a three-body unitary
framework

Daniel Sadasivan<EMAIL_ADDRESS>Ave Maria University, Ave
Maria, FL 34142, USA  Andrei Alexandru<EMAIL_ADDRESS>The George Washington
University, Washington, DC 20052, USA  Department of Physics, University of
Maryland, College Park, MD 20742, USA  Hakan Akdag<EMAIL_ADDRESS>Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for
Theoretical Physics, Universität Bonn, 53115 Bonn, Germany  Felipe Amorim
<EMAIL_ADDRESS>Ave Maria University, Ave Maria, FL
34142, USA  Ruairí Brett<EMAIL_ADDRESS>The George Washington University,
Washington, DC 20052, USA  Chris Culver<EMAIL_ADDRESS>Department
of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United
Kingdom  Michael Döring<EMAIL_ADDRESS>The George Washington University,
Washington, DC 20052, USA  Thomas Jefferson National Accelerator Facility,
Newport News, VA 23606, USA  Frank X. Lee<EMAIL_ADDRESS>The George Washington
University, Washington, DC 20052, USA  Maxim Mai<EMAIL_ADDRESS>The George
Washington University, Washington, DC 20052, USA  Helmholtz-Institut für
Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics,
Universität Bonn, 53115 Bonn, Germany

###### Abstract

Masses, widths, and branching ratios of hadronic resonances are quantified by
their pole positions and residues with respect to transition amplitudes on the
Riemann sheets of the complex energy-plane. In this study we discuss the
analytic structure in the physical energy region of three-body scattering
amplitudes on such manifolds. As an application, we determine the pole
position of the $a_{1}(1260)$ meson from the ALEPH experiment by allowing for
$\pi\rho$ coupled channels in S- and D-wave. We find it to be
$\sqrt{s_{0}}=(1232^{+15+9}_{-0-11}-i266^{+0+15}_{-22-27})\leavevmode\nobreak\
\text{MeV}$.

††preprint: JLAB-THY-21-3533

##  I Introduction

Hadronic resonances often decay strongly into three particles. Especially in
the meson sector, three-body decays can be the dominant modes, e.g. for axial
mesons like the $a_{1}(1260)$ Zyla  _et al._ (2020). Excited mesons are
searched for in recent experimental efforts like GlueX Al Ghoul  _et al._
(2017), COMPASS Alekseev  _et al._ (2010), and at the BESIII accelerator Asner
_et al._ (2009) often in connection with exotic states that cannot consist of
two constituent quarks only. For example, an exotic $\pi_{1}(1600)$ was found
by COMPASS Alekseev  _et al._ (2010) in three-pion decays. These experiments
entail new partial-wave analysis (PWA) efforts, e.g. by COMPASS Adolph  _et
al._ (2017); Aghasyan  _et al._ (2018), BESIII Ablikim  _et al._ (2021, 2005),
CLEO Asner  _et al._ (2000), or in coupled channels using the PAWIAN framework
for $p\bar{p}$ induced meson production Albrecht  _et al._ (2020).

On the theory side, the final state interaction of three strongly interacting
particles has been studied with Khuri-Treiman equations and similar frameworks
by the Bonn group, JPAC, and others for light meson decays Pasquier and
Pasquier (1968); Aitchison (1977); Colangelo  _et al._ (2009); Kubis and
Schneider (2009); Schneider  _et al._ (2011); Kampf  _et al._ (2011); Niecknig
_et al._ (2012); Guo  _et al._ (2012); Danilkin  _et al._ (2015); Guo  _et
al._ (2015a, b); Daub  _et al._ (2016); Niecknig and Kubis (2015); Guo  _et
al._ (2017); Isken  _et al._ (2017); Albaladejo and Moussallam (2017);
Niecknig and Kubis (2018); Dax  _et al._ (2018); Jackura  _et al._ (2019);
Gasser and Rusetsky (2018); Albaladejo  _et al._ (2020); Mikhasenko  _et al._
(2020, 2019); Akdag  _et al._ (2021). Faddeev-type arrangements of chiral two-
body amplitudes were used to predict resonance states and study known ones
Martinez Torres  _et al._ (2008); Magalhaes  _et al._ (2011); Martinez Torres
_et al._ (2009, 2011); Aoude  _et al._ (2018). See also Ref. Aitchison (2015)
for a pedagogical introduction into dispersive methods and Ref. Aitchison and
Pasquier (1966) for connections between Khuri-Treiman equations and three-body
unitary methods.

One such method applies the principle of three-body unitarity to construct
three-to-three amplitudes Mai  _et al._ (2017), extending earlier work Aaron
_et al._ (1968); Aaron and Amado (1973) to the above-threshold regime. The
subthreshold behavior of this amplitude has been studied in Refs. Jackura  _et
al._ (2019); Dawid and Szczepaniak (2021) and new insights into covariant vs.
time-ordered formulations for the interaction kernel were obtained recently
Zhang  _et al._ (2022). The amplitude of Ref. Mai  _et al._ (2017) has been
extended to formulate three-body resonance decays including a fit to the
$a_{1}(1260)\to 3\pi$ lineshape and prediction of Dalitz plots Sadasivan  _et
al._ (2020). This study is the basis of the current work.

Experimentally, the $a_{1}(1260)$ resonance can be produced in $\tau$-decays
Asner  _et al._ (2000); Schael  _et al._ (2005) via
${\tau\rightarrow(3\pi)\nu_{\tau}}$. Therefore, its three-pion dynamics can be
separated off from the weak primordial interaction to be measured cleanly for
the $I^{G}(J^{PC})=1^{-}(1^{++})$ quantum numbers. This distinguishes this
semileptonic $\tau$ decay from some of the aforementioned experiments in which
multiple partial waves contribute to the final three-pion state. Of course,
the $a_{1}$ resonance still couples to various configurations of the 2+1
pions, dominated by $\rho\pi$ in S-wave and $\sigma\pi$ in P-wave ($\sigma$
standing for the $f_{0}(500)$ resonance), but also several subdominant waves,
see CLEO Asner  _et al._ (2000), COMPASS Adolph  _et al._ (2017), and BESIII
results Ablikim  _et al._ (2021). Recent calculations based on chiral unitary
methods predict that the $a_{1}(1260)\to\pi\sigma$ decay ratio is very small,
in the few percent range Molina  _et al._ (2021). This is in contrast to an
older phenomenological study Asner  _et al._ (2000) finding a more substantial
$\pi\sigma$ branching ratio. This shows that, despite the clean experimental
way to produce the $a_{1}(1260)$, its properties such as branching fractions
are under continued debate. The resonance is very wide (with very large
uncertainties quoted by the PDG Zyla  _et al._ (2020)), indicating strong and
non-trivial three-body effects which makes it a prime candidate to study few-
body dynamics. This is reflected in an increased interest in the properties
and structure of the $a_{1}(1260)$ Janssen  _et al._ (1993); Lutz and
Kolomeitsev (2004); Roca  _et al._ (2005); Geng  _et al._ (2007); Wagner and
Leupold (2008a, b); Lutz and Leupold (2008); Kamano  _et al._ (2011); Nagahiro
_et al._ (2011); Zhou  _et al._ (2014); Zhang and Xie (2018); Mikhasenko  _et
al._ (2018); Sadasivan  _et al._ (2020); Dai  _et al._ (2020); Dias  _et al._
(2021), as well as the related $\tau$-decay Bowler (1986); Kuhn and Mirkes
(1992); Isgur  _et al._ (1989); Dumm  _et al._ (2010); Nugent  _et al._
(2013); Dai  _et al._ (2019).

The study of the $a_{1}(1260)$ with the _ab-initio_ techniques of lattice QCD
has also made significant progress. For a pioneering calculation see Ref. Lang
_et al._ (2014) where the $\rho$-meson was treated as a stable particle,
motivated by the small box size. Recently, this approximation was lifted by
using up to three pion operators in combination with the finite-volume
unitarity (FVU) three-body quantization condition Mai and Döring (2017, 2019)
that allowed for the first pole extraction of a three-body resonance from
lattice QCD Mai  _et al._ (2021a). The infinite-volume version of that
formalism is very similar to the one of Ref. Sadasivan  _et al._ (2020)
featuring coupled channels and explicit sub-channel ($\rho$) dynamics. See
Refs. Mai  _et al._ (2021b); Hansen and Sharpe (2019); Rusetsky (2019) for
reviews on recent progress of three-body physics in lattice QCD.

In this work, we use the formalism of Ref. Mai  _et al._ (2021a) to determine
the $a_{1}(1260)$ pole position from experiment including statistical and some
systematic uncertainties. This work is related to older determinations of the
$a_{1}$ pole position Janssen  _et al._ (1993) but also to Ref. Mikhasenko
_et al._ (2018) (JPAC), in which the $S$-wave $\rho\pi$ channel was used to
fit the $a_{1}$ lineshape Schael  _et al._ (2005) with an approximately
unitary formalism. In contrast, our formalism is manifestly unitary, which
considerably complicates the analytic structure through the pertinent pion
exchange mechanism. This requires a thorough discussion in Sec. III based on
the formalism summarized in Sec. II. As such, it provides the only pole
determination in three-body unitary amplitudes except for Ref. Kamano  _et
al._ (2011) and Ref. Mai  _et al._ (2021a). However, in Ref. Kamano  _et al._
(2011) the PDG pole position of the $a_{1}$ was fitted, while in this study we
directly fit the lineshape from experiment. We therefore expect to extract the
most reliable pole position of the $a_{1}(1260)$ resonance to date, with our
results discussed in Sec. IV.

##  II Formalism

The $a_{1}(1260)$ couples to three-pion states in the
$I^{G}(J^{PC})=1^{-}(1^{++})$ channel that can be decomposed as $\pi\rho$ in
S/D-wave, $\pi\sigma$ and $\pi(\pi\pi)_{I=2}$ in P-waves and other channels.
Phenomenologically $(\pi\rho)_{\rm S}$ is dominant Kuhn  _et al._ (2004) with
the branching ratios into other channels quite uncertain Zyla  _et al._
(2020), see also Ref. Molina  _et al._ (2021). Therefore, we limit here the
channel space to $\pi\rho$ in S and D waves. Finally, we note that the isobar
formulation of the two-body sub-channel dynamics used in this study is not an
approximation but a re-parameterization of the full two-body amplitude Bedaque
and Grießhammer (2000); Hammer  _et al._ (2017).

Figure 1: Factorization of the (weak) production mechanism and (hadronic)
final state three-body interaction. Full-directed, dashed, and double-full
lines denote leptons, mesons and auxiliary $\rho$ fields, respectively. The
initial production mechanism is shown by a shaded circle and diamond
corresponding to Eq. (11). The three-body unitary dynamics of the final pion
states is depicted by the shaded rectangle, see Eq. (4).

Our formalism from Ref. Sadasivan  _et al._ (2020) is summarized in the
following. The $a_{1}$ lineshape with $\pi^{-}\pi^{-}\pi^{+}$ final states,

$\displaystyle\mathcal{L}(\sqrt{s})=N(m_{\tau}^{2}-s)^{2}\int\frac{\mathop{}\\!\mathrm{d}^{3}{\bm{q}}_{1}}{(2\pi)^{3}}\frac{\mathop{}\\!\mathrm{d}^{3}{\bm{q}}_{2}}{(2\pi)^{3}}\frac{\mathop{}\\!\mathrm{d}^{3}{\bm{q}}_{3}}{(2\pi)^{3}}\frac{(2\pi)^{4}}{8E_{q_{1}}E_{q_{2}}E_{q_{3}}}$
$\displaystyle\times\delta^{4}(P_{3}-q_{1}-q_{2}-q_{3})$
$\displaystyle\times\left(\big{|}\sum_{\lambda}\hat{\Gamma}_{-1\lambda}\big{|}^{2}+\frac{m_{\tau}^{2}}{s}\big{|}\sum_{\lambda}\hat{\Gamma}_{0\lambda}\big{|}^{2}+\big{|}\sum_{\lambda}\hat{\Gamma}_{+1\lambda}\big{|}^{2}\right)$
(1)

depends on the three-body energy $\sqrt{s}$ and scales with an irrelevant
normalization $N$. Here, $q_{1}$, and $q_{2}$ are outgoing $\pi^{-}$ momenta
that must be symmetrized later, $q_{3}$ is the outgoing $\pi^{+}$ momentum,
$m_{\tau}$ is the mass of the $\tau$, and
$E_{x}=\sqrt{\bm{x}^{2}+m_{\pi}^{2}}$ here and in the following. The term
$(m_{\tau}^{2}-s)^{2}$ accounts for the $\tau\rightarrow W^{-}\nu_{\tau}$
decay vertex and the two-body phase space of the $a_{1}$ and the $\nu_{\tau}$
of this process after integration over the neutrino angles Mikhasenko  _et
al._ (2018). See Fig. 1 for a graphical representation of the complete $\tau$
decay process. Furthermore, we chose the total four-momentum of the three-body
system $P_{3}=(\sqrt{s},{\bf 0})$.

The amplitude
$\hat{\Gamma}_{\varLambda\lambda}\equiv\hat{\Gamma}_{\varLambda\lambda}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})$
describes the decay of the axial $a_{1}(1260)$ resonance at rest with helicity
$\varLambda$ measured along the $z$-axis into a $\pi^{-}$ and a
$\rho_{\lambda}^{0}\to\pi^{+}\pi^{-}$ with helicity $\lambda$,

$\displaystyle\hat{\Gamma}_{\varLambda\lambda}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})$
$\displaystyle=\frac{1}{\sqrt{2}}\big{[}\Gamma_{\varLambda\lambda}(\bm{q}_{1},\bm{q}_{2},\bm{q}_{3})-\left(\bm{q}_{1}\leftrightarrow\bm{q}_{2}\right)\big{]},$
(2)

where the minus sign in the exchange term comes from the overall odd intrinsic
parity of the process,

$\displaystyle\Gamma_{\varLambda\lambda}({\bm{q}}_{1},{\bm{q}}_{2},{\bm{q}}_{3})$
$\displaystyle=\sqrt{\frac{3}{4\pi}}\,{\mathfrak{D}}^{1*}_{\Lambda\lambda}(\phi_{1},\theta_{1},0)\times$
(3) $\displaystyle\hskip 60.00009ptv^{\pm}_{\lambda}(q_{2},q_{3})U_{\lambda
L}\breve{\Gamma}_{L}(q_{1})\,,$

and

$\displaystyle\breve{\Gamma}_{L}(q_{1})$
$\displaystyle=\tau(\sigma(q_{1}))\bigg{[}\,D_{L}(q_{1})+$ (4)
$\displaystyle\hskip
40.00006pt\int\limits_{0}^{\Lambda}\frac{\mathop{}\\!\mathrm{d}p\,p^{2}}{(2\pi)^{3}}\,\frac{1}{2E_{p}}\,T^{J}_{LL^{\prime}}(q_{1},p)\tau(\sigma(p))D_{L^{\prime}}(p)\bigg{]}\,.$

For readability (confusion with four-vector notation is excluded by context),
we have abbreviated $D(x):=D(|\bm{x}|)$,
$\breve{\Gamma}(x)=\breve{\Gamma}(|\bm{x}|)$,
$T(q_{1},p):=T(|\bm{q}_{1}|,|\bm{p}|)$, and $\sigma(x):=\sigma(|\bm{x}|)$
where the two-body invariant mass squared is denoted by

$\displaystyle\sigma(x)=s+m_{\pi}^{2}-2\sqrt{s}E_{x}\,.$ (5)

The angular structure of the final $\pi\rho$ state is conveyed by the usual
capital Wigner-D function,
$\mathfrak{D}^{J}_{\Lambda\lambda}(\phi_{1},\theta_{1},0)$ with angles
$\theta_{1}$ and $\phi_{1}$ giving the polar and azimuthal angles of
$\bm{q}_{1}$. Note that the third argument is set to zero in the current
convention, cf. Ref. Berman and Jacob (1965), which is consistent with the
$\rho$ polarization vectors of Appendix A obtained through a boost and two
rotations (no initial rotation about the $z$-axis).

Equation (3) contains the transformation from the $JLS$ basis to the helicity
basis, with $L$ denoting the orbital angular momentum between $\pi$ and $\rho$
and $J=S=1$ for total and $\rho$ spin, respectively. This transformation
involves the matrix

$\displaystyle U_{L\lambda}$
$\displaystyle=\sqrt{\frac{2L+1}{2J+1}}(L01\lambda|J\lambda)(1\lambda
00|1\lambda)\ ,$ $\displaystyle U$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\\
\frac{1}{\sqrt{6}}&-\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{6}}\\\
\end{array}\right)\,,$ (8)

expressed by Clebsch-Gordan coefficients Chung (1971), and ${U_{\lambda
L}=U_{L\lambda}}$, while we sum over identical indices $L$ and $L^{\prime}$ in
Eqs. (3) and (4), respectively.

The final decay vertex $v^{\pm}$ for $\rho^{0}\to\pi^{+}\pi^{-}$ in Eq. (3)
reads

$\displaystyle v^{\pm}_{\lambda}(q_{2},q_{3})$
$\displaystyle=I^{\prime}v_{\lambda}(q_{2},q_{3})\,,$ (9) $\displaystyle
v_{\lambda}(q_{2},q_{3})$
$\displaystyle=-ig_{1}\epsilon_{\lambda}^{\mu}(\bm{q}_{1})\leavevmode\nobreak\
(q_{2}-q_{3})_{\mu}\,,$ (10)

where $q_{2}$, $q_{3}$ denote four-momenta,
$\bm{q}_{1}=-\bm{q}_{2}-\bm{q}_{3}$, $g_{1}$ is the $\rho\to\pi\pi$ coupling,
$v_{\lambda}$ is the isospin-1 projected decay vertex, and $I^{\prime}$
describes the transition from isospin to particle basis as needed only in the
final $\rho$ decay. Note that the latter factor is irrelevant as long as there
is only one isobar ($\rho^{0}$). Then, this factor can be reabsorbed into the
overall normalization of the $a_{1}$ decay.

Continuing with the description of Eq. (4), the $\rho$ propagator $\tau$ is
discussed in more detail in Sec. II.1. Furthermore, the $a_{1}\to\rho\pi$
vertex, $D$, in Eq. (4) is directly parameterized in the $JLS$ basis as

$\displaystyle
D_{L^{\prime}}(p)=\left(D_{fL^{\prime}}+\frac{m^{2}_{\pi}\sqrt{c^{(-1)}_{L^{\prime}L^{\prime}}}D_{\tilde{f}}}{s-m^{2}_{a_{1}}}\right)\left(\frac{p}{m_{\pi}}\right)^{L^{\prime}}\,,$
(11)

where $D_{fL^{\prime}}$ for $L^{\prime}=0,2$ and $D_{\tilde{f}}$ are free
parameters that are fit to the lineshape accounting, as well, for its unknown
overall normalization. The quantity $m_{a_{1}}$ is the expansion point of the
three-body force and $c^{(-1)}_{L^{\prime}L^{\prime}}>0$ is an expansion
parameter (see below). Its square root may be understood as a bare
$a_{1}\pi\rho$ coupling.

The quantity $T^{J}_{LL^{\prime}}$ in Eq. (4) is the isobar-spectator
amplitude in the $JLS$ basis given by

$\displaystyle
T^{J}_{LL^{\prime}}(q_{1},p)=\left(B^{J}_{LL^{\prime}}(q_{1},p)+C_{LL^{\prime}}(q_{1},p)\right)+$
(12)
$\displaystyle\int\limits_{0}^{\Lambda}\frac{\mathop{}\\!\mathrm{d}l\,l^{2}}{(2\pi)^{3}2E_{l}}\left(B^{J}_{LL^{\prime\prime}}(q_{1},l)+C_{LL^{\prime\prime}}(q_{1},l)\right)\tau(\sigma(l))T^{J}_{L^{\prime\prime}L^{\prime}}(l,p)\,,$

where summation over $L^{\prime\prime}$ is implied. Note that the indices
correspond to matrix notation, i.e. the first indices $L$ and $q_{1}$ label
outgoing (angular) momentum while the second indices $p$ and $L^{\prime}$
label incoming (angular) momentum (similarly, in Eqs. (3) and (4)). The
integrations in Eq. (4) and (12) have been regularized by the same cutoff
$\Lambda$ in contrast to Ref. Sadasivan  _et al._ (2020) where covariant form
factors were used. We prefer here a hard cutoff because it simplifies the
analytic continuation as discussed in Sec. III which also contains the in-
depth description of the contours for the integrations in Eqs. (4) and (12).

In Eq. (12) the $\pi\rho$ interaction term $B$ is complex-valued as demanded
by three-body unitarity Mai  _et al._ (2017) and obtained from the plane-wave
expression in isospin $I=1$,

$\displaystyle
B_{\lambda\lambda^{\prime}}(\bm{p},\bm{p}^{\prime})=\frac{v^{*}_{\lambda}(P-p-p^{\prime},p)v_{\lambda^{\prime}}(P-p-p^{\prime},p^{\prime})}{2E_{p^{\prime}+p}(\sqrt{s}-E_{p}-E_{p^{\prime}}-E_{p^{\prime}+p}+i\epsilon)}\,,$
(13)

by projecting it to angular momenta ${L^{(\prime)}\in\\{S,D\\}}$ via

$\displaystyle
B^{J}_{\lambda\lambda^{\prime}}(q_{1},p)=2\pi\int\limits_{-1}^{+1}\mathop{}\\!\mathrm{d}x\,d^{J}_{\lambda\lambda^{\prime}}(x)B_{\lambda\lambda^{\prime}}($
$\displaystyle{\bm{q}}_{1},{\bm{p}})\,,$ (14)

where $d^{J}_{\lambda\lambda^{\prime}}(x=\cos{\theta})$ denotes the small
Wigner-d function and $\theta$ is the $\pi\rho$ scattering angle.
Subsequently, the $JLS$ expression is obtained by a linear transformation,

$\displaystyle
B^{J}_{LL^{\prime}}(q_{1},p)=U_{L\lambda}B^{J}_{\lambda\lambda^{\prime}}(q_{1},p)U_{\lambda^{\prime}L^{\prime}}\,,$
(15)

with $U_{L\lambda}$ from Eq. (8) and, as before,
$\lambda^{\prime},\,L^{\prime},\,p$ ($\lambda,\,L,\,q_{1}$) label the incoming
(outgoing) state.

Three-body unitarity allows for additional terms of the $\pi\rho$ interaction
that need to be real in the physical region Mai  _et al._ (2017). We refer to
such terms as contact terms or three-body forces that are generically
parameterized by a Laurent series in the $JLS$ basis
(${L^{(\prime)}\in\\{\text{S,D}\\}}$),

$\displaystyle C_{LL^{\prime}}(p,p^{\prime})$
$\displaystyle=\sum_{i=-1}^{\infty}c_{LL^{\prime}}^{(i)}\left(\frac{s-m_{a_{1}}^{2}}{m_{\pi}^{2}}\right)^{i}\frac{p^{L}\,p^{\prime
L^{\prime}}}{m_{\pi}^{L+L^{\prime}}}\,,$ (16)

including first-order poles to account for explicit resonances.

We fit the parameters $m_{a_{1}}$, $c_{00}^{(-1)}$, and $c_{00}^{(0)}$ with
all other parameters set to zero, meaning that the $C$-term couples directly
to the S$\,\to\,$S-wave transition but only indirectly to D-wave through the
$B$-term (15). The analysis shows that this restriction is sufficient to fit
the lineshape data as discussed in Sec. IV. Of course, in future fits to
Dalitz plots the data become more sensitive to the partial-wave content and we
expect that more fit parameters and channels are needed.

To fit the $a_{1}$ lineshape one needs to continue $\breve{\Gamma}$ to real
spectator momenta and perform the phase space integration over the final
three-pion state. This is described in detail in Ref. Sadasivan  _et al._
(2020) but is not repeated here.

Figure 2: Upper Plot: The phaseshift from Eq. (22) with free parameters
$a_{0}$ and $a_{1}$ fitted to data from Refs. Protopopescu  _et al._ (1973)
(black circles) and Estabrooks and Martin (1974) (gray circles) is indicated
in green (case (a)). The fit to only the data of Ref. Protopopescu  _et al._
(1973), case (b), is indicated in blue (to Estabrooks and Martin (1974) only,
case (c): in red). Lower Plot, case (d): The dispersive solution from Ref.
Pelaez  _et al._ (2019) (orange), and the three-parameter fit to it (purple).

###  II.1 Two-body input

The $\rho$ propagator $\tau$ of Eq. (4) in $n$-times subtratced form reads

$\displaystyle\tau^{-1}(\sigma)$
$\displaystyle=K_{n}^{-1}(\sigma)-\Sigma(\sigma)\,,$
$\displaystyle\Sigma(\sigma)$
$\displaystyle=\int\limits_{0}^{\infty}\frac{\mathop{}\\!\mathrm{d}k\,k^{2}}{(2\pi)^{3}}\,\frac{1}{2E_{k}}\,\left(\frac{\sigma}{\sigma^{\prime}}\right)^{n}\,\frac{\tilde{v}^{2}(k)}{\sigma-4E_{k}^{2}+i\epsilon}\,,$
$\displaystyle\sigma^{\prime}$
$\displaystyle=(2E_{k})^{2}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{v}(k)=\sqrt{\frac{16\pi}{3}}\,g_{1}k$ (17)

with a self-energy $\Sigma$, that converges for $n\geq 2$, and a regular
$K$-matrix like quantity. The former is expressed in terms of the vertex
$\tilde{v}$ projected to the ${I=1}$ P-wave (spin $S=1$) quantum numbers. This
vertex can be obtained by considering the first-order Born series for the
$\pi(k_{1})\pi(k_{2})\to\pi(k_{1}^{\prime})\pi(k_{2}^{\prime})$ scattering
amplitude in the two-body rest frame, i.e.
$k_{1}^{(\prime)}+k_{2}^{(\prime)}=(\sqrt{\sigma},\bm{0})$,
${k_{1}^{(\prime)\mu}-k_{2}^{(\prime)\mu}=(0,2\bm{k}^{(\prime)})}$. Using the
$\rho\pi\pi$ vertex from Eq. (10) this reads

$\displaystyle T_{\rho}(\sigma,z)$
$\displaystyle=I_{s}I_{\rho}\,\frac{\sum_{\lambda}v_{\lambda}(k_{1},k_{2})v^{*}_{\lambda}(k_{1}^{\prime},k_{2}^{\prime})}{\sigma-
m_{\rho}^{2}}=\frac{g_{1}^{2}}{\sigma-m_{\rho}^{2}}\,4kk^{\prime}z\,,$ (18)

where $z=\bm{k}\cdot\bm{k}^{\prime}/(kk^{\prime})$, $I_{\rho}=2$ is a factor
for isospin-1, and $I_{s}=\nicefrac{{1}}{{2}}$ is a symmetry factor. The
second equal sign in Eq. (18) is due to the general properties of the helicity
state vectors, cf. Eq. (38). Projecting this amplitude to the P-wave amounts
then to

$\displaystyle T_{\rho}^{1}(\sigma)$
$\displaystyle=2\pi\int\limits_{-1}^{1}\mathop{}\\!\mathrm{d}z\,P_{1}(z)\,T_{\rho}(\sigma,z)$
(19)
$\displaystyle=\left(\sqrt{\frac{16\pi}{3}}\,g_{1}k\right)\frac{1}{\sigma-
m_{\rho}^{2}}\left(\sqrt{\frac{16\pi}{3}}\,g_{1}k^{\prime}\right)$
$\displaystyle=:\frac{\tilde{v}(k)\tilde{v}(k^{\prime})}{\sigma-
m_{\rho}^{2}}\,,$

defining the projected vertex $\tilde{v}$ that automatically fulfills
$\tilde{v}=\tilde{v}^{*}$.

The two-body dynamics is encoded in $\tilde{v}$ but also in $K$ of Eq. (17)
that is very similar to a $K$-matrix, up to the selfenergy $\Sigma$ that
contains also a real part. The subtraction polynomial $K$ reads

$\displaystyle
K_{n}^{-1}(\sigma)=m_{\pi}^{2}\sum_{i=0}^{n-1}a_{i}\left(\frac{\sigma}{m_{\pi}^{2}}\right)^{i}\,.$
(20)

The parameters $a_{i}$ are fitted to the $\pi\pi$ phaseshift data by
introducing the two-to-two on-shell $T$-matrix for $I=S=1$,

$\displaystyle
T_{22}(\sigma)=\tilde{v}(k_{\text{cm}})\tau(\sigma)\tilde{v}(k_{\text{cm}})\
,$ (21)

where $k_{\text{cm}}=\sqrt{\sigma/4-m_{\pi}^{2}}$. The connection to the
vector, isovector phaseshift $\delta^{11}$ is given by

$\displaystyle\delta^{11}(\sigma)=\tan^{-1}\left(\frac{\text{Im}\,T_{22}(\sigma)}{\text{Re}\,T_{22}(\sigma)}\right)\,,$
(22)

which depends on $g_{1}$ from Eq. (10) and the $a_{i}$ from Eq. (20). However,
$g_{1}$ is fully correlated with the $a_{i}$ so we fix it at $g_{1}=1$.

To assess systematic effects from the two-body input we perform different
fits: (a) In a twice-subtracted fit ($n=2$), data from both experimental
phaseshifts from Refs. Protopopescu  _et al._ (1973); Estabrooks and Martin
(1974) are fitted as shown in Fig. 2 with the green line. As can be seen in
the figure, the two sets of data are not in perfect agreement. In order to
account for this source of systematic uncertainty we perform two additional
fits: (b) only to data from Ref. Protopopescu  _et al._ (1973), and (c) only
to the data of Ref. Estabrooks and Martin (1974). (d) Additionally, it has
been shown in Refs. Ananthanarayan  _et al._ (2001); Colangelo  _et al._
(2001); Garcia-Martin  _et al._ (2011); Colangelo  _et al._ (2019); Pelaez
_et al._ (2019) that these data have certain inconsistencies and that by
imposing S-matrix principles such as crossing symmetry the $\rho$ phase shift
can be improved. Therefore, we perform an additional fit to the recent phase-
shift parameterization from Ref. Pelaez  _et al._ (2019) using three
subtractions $(n=3)$.

The parameters and the corresponding $\rho$ pole positions for cases (a) to
(d) are given in Table 1. The pole position for case (d) is close to the one
of Ref. Pelaez  _et al._ (2019) itself, of about $\sqrt{\sigma_{\rho}}=(763\pm
1.5-i(73\pm 1.5))$ MeV. As the table shows the $\rho$ pole positions for cases
(a) to (d) are further apart than the statistical uncertainties. The latter
was calculated for case (a) from resampling the phase shift data as indicated
in the table. We will propagate these statistical errors and also the
systematic differences between cases (a) to (d) to the three-body sector as
described in Sec. IV.2.

Input: | (a)Protopopescu  _et al._ (1973); Estabrooks and Martin (1974) | (b)Protopopescu  _et al._ (1973) | (c)Estabrooks and Martin (1974) | (d)Pelaez  _et al._ (2019)
---|---|---|---|---
$a_{0}$ | $-0.460(2)$ | $-0.471$ | $-0.464$ |  $-0.327$
$a_{1}\cdot 10$ | $+0.156(1)$ | $+0.157$ | $+0.157$ |  $0.062$
$a_{2}\cdot 10^{3}$ | - | - | - | 0.259
Re $\sqrt{\sigma_{\rho}}$ [MeV] | $754(<1)$ | $758$ | $753$ | 766
Im $\sqrt{\sigma_{\rho}}$ [MeV] | $-72(<1)$ | $-71$ | $-71$ | -74
Table 1: Fitted parameters of the two-body subsystem according to Eq. (20) and
$\rho(770)$ meson pole positions $\sqrt{\sigma_{\rho}}$. Statistical
uncertainties are only quoted for the combined fit to both data sets and show
that systematic effects are larger (i.e., fits to the individual data sets).
Figure 3: Contours in two-body scattering. Upper row: Momentum integration
(left) and physical amplitude (right) at $\sqrt{\sigma}+i\epsilon$ on the
first (“I”) Riemann sheet. Lower row: Momentum integration and amplitude on
unphysical sheet II with Im $\sqrt{\sigma}<0$. See text for additional
explanations.

##  III Analytic continuation

The key ingredient of the discussed $a_{1}$-production mechanism is the
integral equation (12) solved for the $\pi\rho$ scattering amplitude. This
equation is solved by replacing the integrations over the real-valued
magnitude of the meson momenta with complex values along certain contours
described in the following. Additionally, and in view of the final goal of
this study – determination of the $a_{1}(1260)$ resonance pole – one needs to
analytically continue the scattering amplitude (12) in the three-body energy
$\sqrt{s}$ to complex values.

Specifically, two types of integrations occur: (1) in $l:=|\bm{l}|$ within the
integral equation (12); and (2) in $k:=|\bm{k}|$ within the selfenergy term of
the two-body subsystem (17). The corresponding complex contours can be chosen
individually and are referred to in the following as ‘spectator momentum
contour’ (SMC) and ‘selfenergy contour’ (SEC), respectively. Both contours
start at the respective origins, $l=k=0$, and end at $l=\Lambda$ and
$k=\infty$, respectively. In between these limits, different choices for the
contours define different Riemann sheets in $\sqrt{s}$ as discussed in the
following.

A similar discussion of the analytic structure in the context of dynamical
coupled-channel approaches can be found in Ref. Döring  _et al._ (2009) for
the Jülich/Bonn/Washington approach Rönchen  _et al._ (2013, 2014); Mai  _et
al._ (2021c) and in Ref. Suzuki  _et al._ (2010) for the EBAC/ANL-Osaka
approach Julia-Diaz  _et al._ (2009); Kamano  _et al._ (2013). There is also a
discussion in Ref. Mikhasenko  _et al._ (2018) on analytic continuation, but
the structure of the scattering equation is substantially different because it
does not contain three-body cuts from pion exchange as demanded by three-body
unitarity Mai  _et al._ (2017). In Ref. Döring  _et al._ (2009), a
continuation obtained by certain approximations for three-body cuts was
discussed, but the method proposed here is rigorous. Regularization is often
achieved with form factors, and the SMC is given by a straight line from $l=0$
into the lower complex-momentum half-plane; see, e.g. Refs. Rönchen  _et al._
(2013); Huang  _et al._ (2012); Rönchen  _et al._ (2018); Sadasivan  _et al._
(2020). We refrain from the use of form factors because they make the analytic
structure of the amplitude unnecessarily complicated and use a cutoff
$\Lambda$ instead.

Figure 4: Example for SMC and SEC contours in the complex plane of the two-
body energy-squared, $\sigma$. Typical Gauss node distributions on the
contours are indicated with turquoise and black dots, respectively. The
shading and coloring indicate magnitude and phase $\phi$ of
$\tau=|\tau|e^{i\phi}$, respectively. The $\rho$ pole at
$\sigma=\sigma_{\rho}$ is highlighted with the white star. See text for
further explanations.

###  III.1 Two-body scattering

To discuss the analytic structure, we recall that the placement of cuts is a
choice, and that only the branch point marking the energy at which a cut
begins is fixed. Cuts are the curves along which different Riemann sheets are
analytically “glued” together. For example, a common choice in two-body
scattering is to run the physical, right-hand cut along the real axis from
threshold to $\infty$ in the two-body energy-squared $\sigma$. This simplifies
the formulation of the dispersion relations and provides a convenient
definition of the first and second Riemann sheet. With that definition,
resonance poles can only be found on the second, “unphysical” Riemann sheets
as demanded by causality (see, e.g. Ref. Gribov (2008) for a proof).

In contrast to cuts, the position of a branch point is fixed in $s$ and
$\sigma$ for three- and two-particle scattering, respectively. Branch points
define thresholds and arise whenever the pertinent momentum integrations begin
in singularities or branch points themselves Döring  _et al._ (2009). This is
illustrated in Fig. 3 for the two-body case above threshold. The placement of
the SEC producing the physical amplitude is constrained by the $+i\epsilon$
term in Eq. (17). In the figure, two possible integration contours are
depicted, passing the singularity at $k=k_{\text{cm}}+i\epsilon$ either on the
right ($\Gamma_{R}$) or left ($\Gamma_{L}$). The former does not change the
sign of (${\rm Im}\,\Sigma$) and, thus, yields the physical amplitude (21)
describing experimental measurements at real energies. In contrast, choosing
$\Gamma_{L}$ leads to a sign change in $({\rm Im}\,\Sigma)$ and an unphysical
$T_{22}$ on sheet II (still, at the same
${\sigma=\sigma_{\text{phys}}+i\epsilon}$).

The physical and unphysical scattering amplitudes $T_{22}$ are connected to
each other, smoothly building the $2^{N}$ Riemann sheets ($N$ being the number
of two-body thresholds). By convention, the physical amplitude on sheet I in
the upper half-plane of $\sigma\in\mathds{C}$ is connected along the real
axis, $\sigma\in[4m_{\pi}^{2},\infty)$, to the unphysical sheet II in the
lower half-plane. For energies with Im $\sigma<0$ (see Fig. 3, lower right),
the amplitude on sheet II can be obtained by deforming the SEC as shown to the
lower left. In particular, the two-body singularity at
$k_{\text{cm}}=\sqrt{\sigma/4-m_{\pi}^{2}}$ also acquires a negative imaginary
part, but a smooth deformation of $\Gamma_{R}$ ensures that the SEC still
passes the two-body singularity to the right. This guarantees that the
amplitude has been analytically continued from physical scattering energies
$\sigma$ to the second sheet in the lower half-plane, where resonance poles
can be found.

In summary, passing the two-body singularity to the left or to the right
($\Gamma_{L}$ vs. $\Gamma_{R}$) defines the Riemann sheet, except for one
point at $\sigma=4m_{\pi}^{2}$. There, the two-body singularity coincides with
the lower limit of the integration, $k=k_{\text{cm}}=0$. Consequently, at this
point there is no distinction between sheets, i.e. one is at the branch point
that defines the two-pion threshold. We stress the (otherwise trivial) fact
that a singularity in an integration limit induces a branch point, because it
helps identifying branch points for the more complicated three-body case
discussed in Sec. III.3; see also Ref. Döring  _et al._ (2009).

In regards to the present application to the $a_{1}(1260)$ channel, we chose
the SEC for the $\pi\pi$ subsystem in the $\rho$ channel as

$\displaystyle\Gamma_{\text{SEC}}=\left\\{k|k=t+\frac{ic_{1}}{2}\,\arctan(c_{2}t),t\in[0,\infty)\right\\}\,,$
(23)

with shape parameters $c_{1}$ and $c_{2}$ chosen such that this contour lies
in the lower right quadrant of the $k$ plane and always avoids the two-body
singularity except at threshold,

$\displaystyle\Gamma_{\text{SEC}}\cap\\{k_{\text{cm}}\\}\backslash\\{0\\}=\O\,.$
(24)

To display the SEC and the $\rho(770)$ resonance pole in the same plot,
$\Gamma_{\text{SEC}}$ is mapped to the $\sigma$ plane according to
$\sigma=4(m_{\pi}^{2}+k^{2})$. The result is labeled “SEC” with the black
circles in Fig. 4 indicating the Gauss nodes chosen for numerical integration.
As the figure shows, the chosen $\Gamma^{\prime}_{R}$ is sufficiently deformed
to not only allow for the calculation of the physical amplitude but also for
the calculation of $\tau$ in a large portion of unphysical sheet II, bound by
the mapped SEC. This portion includes the $\rho$ pole (white star). In other
words, the so-defined two-body amplitude has its actual cut along the mapped
SEC. Furthermore, instead of a discontinuity in $\sigma$ along that cut, the
amplitude exhibits a series of poles which is a consequence of the
discretization in a finite number of Gauss nodes. This is made visible in the
figure through the shading (repeated transitions from transparent to dark gray
indicating increasing values of $|\tau|$), in addition to the color coding
that indicates the phase $\phi$ of $\tau=|\tau|e^{i\phi}$. Notably, the $\rho$
pole exhibits one full cycle $-\pi\to\phi\to+\pi$ (blue $\to$ red $\to$
green), indicating that the $\rho$ pole is indeed a first-order singularity as
required for a resonance.

The idea of suitably constructing contours to access the Riemann sheet(s) of
interest, where resonance poles are situated, can be generalized to three-body
scattering as discussed in the following. In particular, contour deformation
replaces other methods of analytic continuation in which explicit
discontinuities have to be added to the amplitude.

Figure 5: Domain of real solutions $p_{\pm}^{\prime}(p,x)$ according to Eq.
(25) for $\sqrt{s}=7.6\,m_{\pi}$. Gray and red dots represent individual
solutions for some $x\in[-1,+1]$ which are enclosed by the
$p_{\pm}^{\prime}(p,x=\pm 1)$ boundary.

###  III.2 Three-body cuts

Before turning to the construction of a suitable spectator momentum contour
(SMC) to access the $a_{1}(1260)$ pole, one needs to discuss three-body cuts
because they must be avoided by the SMC. This has been known for a long time
and is discussed in the context of the $a_{1}$ resonance in Ref. Janssen  _et
al._ (1994) (we adapt and extend the discussion here). These cuts arise from
the pion-exchange term of Eq. (13) that is a direct consequence of three-body
unitarity Mai  _et al._ (2017). We note that this term corresponds to the
forward-going part of pion exchange only. If one adds the backward-going part,
one recovers the covariant denominator $u-m_{\pi}^{2}+i\epsilon$ Mai  _et al._
(2017) but we refrain from using this term as it can induce unphysical
unitarity-violating imaginary parts above threshold if the regularization is
not chosen correctly. Note also the related but different discussions on sub-
threshold behavior of this denominator vs. triangle graph in Ref. Jackura  _et
al._ (2019), and the comparison of the Feynman denominator and time-ordered
perturbation theory in Ref. Zhang  _et al._ (2022) where it was shown that the
breaking of covariance in the latter is rather small.

It should also be noted that there will be a much more complicated analytic
structure in unphysical regions of the amplitude which $s$-channel unitarity
alone cannot fix (analogous to two-body amplitudes). However, these structures
are far away from the region in which we search for the $a_{1}$-pole and one
can safely neglect them, i.e. the expansion of the $C$-term in the Laurent
series of Eq. (16) contains the dominant contributions.

The denominator of Eq. (13) vanishes for any $x=\cos\theta\in[-1,1]$ according
to the partial-wave decomposition of Eq. (14). For a fixed three-body energy
$\sqrt{s}$ and incoming spectator momentum $p$ the singularities are given by

$\displaystyle p_{\pm}^{\prime}=$
$\displaystyle\frac{px(p^{2}-\alpha^{2})\pm\alpha\sqrt{\left(\beta+p^{2}\left(x^{2}-1\right)\right)^{2}-4m_{\pi}^{2}\beta}}{2\beta}\,,$
$\displaystyle\alpha(p)=\sqrt{s}-E_{p}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \beta(p,x)=\alpha^{2}(p)-p^{2}x^{2}\,.$ (25)

The domain of real solutions is indicated in Fig. 5 and bound by
$p_{\pm}^{\prime}(p,x=\pm 1)$.

Figure 6: Position of three-body singularities in $p^{\prime}(p,x)$ for fixed
$p$, $\sqrt{s}=7.6\,m_{\pi}$, and $x\in[-1,1]$ for solutions
$p_{\pm}^{\prime}$ according to Eq. (25).

Not all solutions $p^{\prime}$ are real, as Fig. 6 shows. Notably, there are
kinematic regions (e.g. $p=3.4m_{\pi}$) in which the singularities fully
enclose the origin which renders a naive integration from $0$ to $\infty$, or
to any physically required cutoff $\Lambda$, impossible. For the pions in the
$\rho$ selfenergy in Eq. (17) to be on-shell, the smallest physically required
cutoff $p_{\text{min}}$ is given by the condition $\sigma(p)>4m_{\pi}^{2}$,
this leads to

$\displaystyle
p^{2}_{\text{min}}=\frac{9m_{\pi}^{4}-10m_{\pi}^{2}s+s^{2}}{4s}\,.$ (26)

Simultaneously, $p_{\text{min}}$ must be large enough to cover all physically
allowed momenta in the pion-exchange, given by the extension of the domain
shown in Fig. 5. This can be determined through the vanishing argument of the
square root of Eq. (25) at $x=1$,

$\displaystyle\beta^{2}(p_{\text{min}},1)-4m_{\pi}^{2}\beta(p_{\text{min}},1)=0\,.$
(27)

The solution of this equation is also given by Eq. (26) as expected.

The crucial point is that the positions of the three-body singularities in
$p^{\prime}$ depend on the value of $p$ itself. For a suitably chosen contour
SMC with $p\in\Gamma_{\rm SMC}$ and $p^{\prime}\in\Gamma_{\rm SMC}$, the
three-body cuts “open up” and allow for the integration of the scattering
equation (12) which has been known for a long time Haftel and Tabakin (1970).
See also Ref. Döring  _et al._ (2013) for a similar numerical scheme in the
context of Muskhelishvili-Omnés equations.

Figure 7: Three-body singularities at fixed, complex 3-body energy $\sqrt{s}$.
The color-coded dots show both solutions $p^{\prime}$ of Eq. (25) for
different $p\in\Gamma_{1}$ (blue), $p\in\Gamma_{2}$ (green), $p\in\Gamma_{3}$
(red), and $x\in[-1,1]$. Solutions $p^{\prime}(p,x)$ for $E_{p+p^{\prime}}=0$
from Eq. (13), $p\in\Gamma_{\text{SMC}}$, and $x\in[-1,1]$ are indicated in
light brown.

The precise form of the SMC is not fixed. We choose the smooth contour
depicted in Fig. 7 that is split into different color-coded segments,
$\Gamma_{\text{SMC}}=\bigcup_{i=1}^{3}\Gamma_{i}$, to show which parts of the
SMC correspond to which positions of three-body singularities, indicated with
the dot clouds for both $p_{+}$ and $p_{-}$. Additionally, the singularities
$p^{\prime}(p,x)$ of the $1/E_{p^{\prime}+p}$ term in Eq. (13) are indicated
by brown dots irrespective of the value of $p\in\Gamma_{\text{SMC}}$. The SMC
is parameterized as

$\displaystyle\Gamma_{\text{SMC}}=\\{p|p=$ $\displaystyle
t+iV_{0}(1-e^{-t/w})$
$\displaystyle\times(1-e^{(t-\Lambda)/w}),\,t\in[0,\Lambda]\\}\,.$ (28)

This expression contains a parameter for the initial and final slope, $w$, and
another one for the extension of the SMC into the lower half-plane, $V_{0}$.
In general, a larger $V_{0}$ allows one to go further into the complex
$\sqrt{s}$ plane to look for poles; piecewise-straight contours are also
possible, in general, but require more integration nodes than smooth paths for
a given precision. An example of integration nodes is shown in Fig. 7 with the
gray circles on top of $\Gamma_{\text{SMC}}$.

To avoid singularities of the $B$-term one simply ensures that
$\Gamma_{\text{SMC}}$ never overlaps with the solutions of Eq. (25),

$\displaystyle\Gamma_{\text{SMC}}\cap\\{p^{\prime}(p\in\Gamma_{\text{SMC}},x\in[-1,1])\\}=\O\,;$
(29)

similarly, for the $E_{p^{\prime}+p}$ term,

$\displaystyle\Gamma_{\text{SMC}}\cap\\{p^{\prime}|E_{p^{\prime}+p}=0,p\in\Gamma_{\text{SMC}},x\in[-1,1]\\}=\O\,.$
(30)

There is a region of $\sqrt{s}$ in the lower complex half-plane for which this
is the case, and the extent of that region depends on $\Gamma_{\text{SMC}}$.
We have made sure that with the SMC of Eq. (28) the corresponding
$\sqrt{s}$-region covers the pole region of the $a_{1}(1260)$.

###  III.3 Real and complex threshold openings

In Sec. III.1 we have already discussed the analytic structure of the two-body
amplitude, its (threshold) branch point at $\sigma=4m_{\pi}^{2}$, where the
two Riemann sheets coincide, and the $\rho$ pole on the second Riemann sheet.
We now discuss this amplitude in the presence of the SMC, i.e. the two-body
system being a subsystem of the three-body amplitude with the spectator
momentum being on the SMC.

Figure 4 shows the SEC mapped to the $\sigma$ plane by $\sigma=4E_{k}^{2}$. It
also shows the SMC mapped to this plane via Eq. (5). This representation has
the advantage that a crossing of SEC and SMC in the figure directly indicates
a zero of the selfenergy denominator ($\sigma-4E_{k}^{2}$) of Eq. (17) that
has to be avoided,

$\displaystyle\sigma(p^{2})\neq 4E_{k}^{2}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ p\in\Gamma_{\text{SMC}}\wedge
k\in\Gamma_{\text{SEC}}\,.$ (31)

The last condition is that neither contour can cross the $\rho$ pole at
$\sigma_{\rho}$, except for $p=0$,

$\displaystyle\sigma(p^{2})\neq\sigma_{\rho}\leavevmode\nobreak\ \forall
p\in\Gamma_{\text{SMC}}\backslash\\{0\\}\leavevmode\nobreak\
\wedge\leavevmode\nobreak\ 4E_{k}^{2}\neq\sigma_{\rho}\leavevmode\nobreak\
\forall k\in\Gamma_{\text{SEC}}\,.$ (32)

The conditions (24, 29-32) constitute the complete set of rules to access all
Riemann sheets in the problem.

Figure 8: Analytic structure of the 3-body amplitude in the $\sqrt{s}$ plane.
The real and complex branch points (thick black dots) are shown together with
their respective cuts (red dashed lines). The $\pi\rho$ branch points lie on
sheet II and induce additional sheets III and IV (not shown). The insets show
the SMC mapped to the $\sigma$ plane (blue lines), in a qualitative way. Its
position changes according to the approximate value of $\sqrt{s}$ where the
insets are placed. In the $\sigma$ plane, the SEC (black lines) starting at
the $\pi\pi$ threshold (small black dots) does not change if $\sqrt{s}$
changes, but the SMC does.

The exclusion of $p=0$ in Eq. (32) can be understood in the context of branch
points. According to Eq. (5), the condition $p=0$ and $\sigma=\sigma_{\rho}$
corresponds to $\sqrt{s}=\pm\sqrt{\sigma_{\rho}}+m_{\pi}$. In other words, at
these complex three-body energies the spectator momentum integration starts at
the $\rho$ pole. According to Sec. III.1, if an integration limit coincides
with a singularity, a branch point is generated. Therefore, taking only the
square root of interest (positive $\text{Re}\,\sqrt{\sigma_{\rho}}$) and
invoking the Schwarz reflection principle, we conclude that the three-body
amplitude has branch points at
$\sqrt{s}=\sqrt{s_{\rho}}:=\sqrt{\sigma_{\rho}}+m_{\pi}$ and
$\sqrt{s}=\sqrt{s_{\rho}}^{\,*}=\sqrt{\sigma_{\rho}}^{\,*}+m_{\pi}$. We refer
to them as $\pi\rho$ branch points in the following.

There is a third branch point: the two-body threshold induces the real-valued
3-body threshold at $s=(3m_{\pi})^{2}$ because at that energy and $p=0$, we
have $\sigma=4m_{\pi}^{2}$ according to Eq. (5). The spectator momentum
integration starts at the two-body branch point, which induces another branch
point in the three-body amplitude. In Ref. Ceci  _et al._ (2011) additional
properties of these branch points were discussed.

The overall analytic structure of the three-body amplitude of Eq. (12) is
visualized in Fig. 8. It shows the real branch point at $\sqrt{s}=3m_{\pi}$
with its associated cut chosen along the real $\sqrt{s}$ axis defining sheets
I and II. Also, the figure shows one of the complex branch points at
$\sqrt{s}=\sqrt{s_{\rho}}$ which is situated on sheet II. The cut associated
with the complex branch point is conveniently run into the negative imaginary
$\sqrt{s}$-direction so that the shown Riemann sheet is the region closest to
the physical axis. If regions behind that cut ought to be explored (defined as
sheet III), more complicated contours must be chosen Döring  _et al._ (2009).

In addition, the insets in Fig. 8 show the $\sigma$ plane with the SMC and SEC
similar as in Fig. 4. The position of the insets in the $\sqrt{s}$ plane
qualitatively corresponds to the $\sqrt{s}$ used to map the SMC to the
$\sigma$ plane, according to Eq. (5). Note how the position of the SMC changes
relative to the $\rho$ pole at $\sigma_{\rho}$. For example, for $\sqrt{s}$ to
the left (right) of the $\pi\rho$ branch cut, the SMC passes the $\rho$ pole
to the left (right).

Figure 9: Typical amplitude $|T_{00}|$ of Eq. (12), color coded from small
values (dark blue) to large values (red; maxed-out values at white). The
$a_{1}(1260)$ pole, $\pi\rho$ branch point at $\sqrt{s}=\sqrt{s_{\rho}}$ and
its associated cut are also indicated. See text for further explanations.

In Fig. 9 we show a typical picture of $|T_{00}|$ of Eq. (12) with the
integration contours defined in Eqs. (23) and (28). The shape parameters that
allow access to a sufficiently large region in the broad vicinity of the
$a_{1}(1260)$ pole, which make the cut of the $\pi\rho$ branch point run
approximately in the negative-imaginary $\sqrt{s}$ direction, are given in
Table 2.

$c_{1}\,[m_{\pi}]$ | $c_{2}\,[m_{\pi}^{-1}]$ | $w\,[m_{\pi}]$ | $V_{0}\,[m_{\pi}]$
---|---|---|---
-7.16 | 0.418 | 1.433 | -3.58
Table 2: Shape parameters for the SEC in Eq. (23) and SMC in Eq. (28).

The $a_{1}$ pole is always to the lower right of the $\pi\rho$ branch point in
the $\sqrt{s}$ plane as Fig. 9 shows. Therefore, the qualitative positions of
SEC and SMC in the $\sigma$ plane, corresponding to $\sqrt{s}$ taking the
value of the $a_{1}$ pole, are given by the lower right inset of Fig. 8 which
is also the situation shown in Fig. 4. Similar to Fig. 4, the cut induced by
$\sqrt{s_{\rho}}$ is approximated by a series of poles due to the numerical
discretizations, as Fig. 9 shows. While in the former case this was due to the
self-energy integration, in the latter case it is due to the integration over
the spectator momentum.

While Fig. 9 and all results in this paper have been obtained using the shape
parameters of Table 2, the figure also shows that this choice is not
universally valid for all three-body energies. In the lower right-hand corner
(highest energies, farthest into the complex plane), we observe numerical
fluctuations. These are poles induced by three-body singularities coinciding
with the SMC as illustrated in Fig. 7; they correspond to violations of Eqs.
(29) or (30). If the analytic continuation in such regions of $\sqrt{s}$ is
desired, one needs to choose a different SMC.

$\Lambda$ [GeV] | $+0.73$ | $+0.90$ | $+1.05$ | $+1.2$ |  $+0.73$ (no $B$) | +0.73 (Protopopescu  _et al._ (1973) data) | +0.73 (Estabrooks and Martin (1974) data) | +0.73 (Pelaez  _et al._ (2019) input)
---|---|---|---|---|---|---|---|---
Re $\sqrt{s_{0}}$ [MeV] | $+1232^{+15}_{-<1}$ | $+1223$ | $+1231$ | $+1240$ | $+1174$ | $+1233$ | $+1230$ |  $+1226$
Im $\sqrt{s_{0}}$ [MeV] | $-266^{+<1}_{-22}$ | $-269$ | $-244$ | $-251$ | $-252$ | $-278$ | $-261$ |  $-253$
$\chi^{2}/(65-6)$ | $0.99$ | $1.32$ | $1.60$ | $1.90$ | $2.56$ | $0.99$ | $0.98$ | 1.09
$c_{00}^{-1}$ | $+16.48^{+0.005}_{-0.007}$ | $+14.59$ | $+12.67$ | $+11.53$ | $+20.16$ | $+16.74$ | $+16.49$ | +15.83
$c_{00}^{0}$ | $+1.729^{+0.008}_{-0.005}$ | $+1.750$ | $+1.843$ | $+2.073$ | $+0.019$ | $+1.712$ | $+1.720$ | +2.077
$m_{a_{1}}$ [GeV] | $+1.293^{+0.001}_{-0.000}$ | $+1.287$ | $+1.281$ | $+1.278$ | $+1.391$ | $+1.296$ | $+1.294$ | +1.324
$D_{f0}\times 10^{7}$ [a.u.] | $-1.841^{+0.049}_{-0.027}$ | $-2.371$ | $-2.126$ | $-2.250$ | $-0.925$ | $-1.887$ | $-1.829$ |  $-2.002$
$D_{f2}\times 10^{8}$ [a.u.] | $+6.462^{+0.451}_{-0.149}$ | $+3.094$ | $+1.567$ | $+0.837$ | $-6.824$ | $+6.718$ | $+6.512$ |  $+2.073$
$D_{\tilde{f}}\times 10^{6}$ [a.u.] | $-1.319^{+0.002}_{-0.000}$ | $-1.358$ | $-1.338$ | $+1.372$ | $-1.235$ | $-1.329$ | $-1.318$ | $-1.366$
Table 3: Pole positions $\sqrt{s_{0}}$ of the $a_{1}(1260)$, $\chi^{2}$, and
fit parameters. Statistical uncertainties are only quoted for the first
column. The column labeled “no $B$” shows a fit for which we set the pion
exchange term $B=0$. The last three columns show variation from different two-
body input as referenced in the labels. The $c$-terms are unitless while the
abbreviation “a.u.” for the $D$-terms stands for “arbitrary units” because
they contain the factor that connects to the un-normalized line shape data.
Figure 10:  Fit to the line shape data from the ALEPH experiment Davier  _et
al._ (2014) for different cutoff values (top) and the normalized residuals for
the $\Lambda=0.73$ GeV case (bottom), where $D-\mathcal{L}$ is a given
residual and $\Delta$ the pertinent data uncertainty. In the upper figure, the
blue band shows the statistical uncertainties of the $\Lambda=0.73$ GeV fit,
multiplied by ten for visibility. The red dashed line shows the $\Lambda=0.73$
GeV fit with all D-wave terms set to 0.

##  IV Results

###  IV.1 Fit

The free parameters of the model are fixed by a fit to the lineshape for the
decay $\tau^{-}\rightarrow\pi^{-}\pi^{-}\pi^{+}\nu_{\tau}$. Data for this
process measured in the ALEPH experiment were originally published in Ref.
Schael  _et al._ (2005). In Ref. Davier  _et al._ (2014) the unfolding method
was improved and an error was fixed (see Ref. web  for numerical values) . The
data include correlations that correspond to both systematic and statistical
uncertainty. However, the systematic uncertainties are small relative to the
statistical uncertainties, thus we neglect them. The $\chi^{2}$ can then be
calculated with the formula

$\displaystyle\chi^{2}=(\vec{\mathcal{L}}-\vec{D})^{T}\Sigma^{-1}(\vec{\mathcal{L}}-\vec{D})\
,$ (33)

where $\Sigma$ is the data covariance matrix, $\vec{D}$ is a vector containing
the central values of the ALEPH line shape data and $\vec{\mathcal{L}}$ is a
vector containing our prediction of the lineshape calculated with Eq. (II) as
a function of $\sqrt{s}$ at each of the central values of energy for the ALEPH
data. We fit the 65 data points in the range $0.55$ GeV $<\sqrt{s}<1.50$ GeV
but do not include all correlations in $\Sigma$; instead we set all
correlations to 0 except those between nearest neighbors. This choice is
justified in Appendix B.

We fit the parameters $c_{00}^{-1}$ and $c_{00}^{0}$ from the expansion of the
three-body term in Eq. (16). We do not include $c_{00}^{1}$ or any higher
$c_{00}$ terms and we do include any $c_{10}$, $c_{01}$, or $c_{11}$ terms
because including these terms in the fit when $\Lambda=0.73$ GeV increases the
$\chi^{2}_{\text{dof}}$. Fits for other values of $\Lambda$ just serve to
assess systematic effects and we do not try to change their parameterization.
Thus, our fit of the lineshape has a total of 6 free parameters:
$c_{00}^{-1}$, $c_{00}^{0}$, and $m_{a_{1}}$ from Eq. (16), and $D_{f0}$,
$D_{f2}$, and $D_{\tilde{f}}$ from Eq. (11).

The line shape depends on $\Lambda$ in Eqs. (4) and (12). A cutoff of
$\Lambda=0.73$ GeV is the lowest possible value allowed by Eq. (26) if an
upper limit of $\sqrt{s}=1.5$ GeV is chosen for the fit. We consider the case
$\Lambda=0.73$ GeV to be our primary fit because it leads to the best
$\chi^{2}$ as shown in Table 3. However, to study systematic effects, we also
vary $\Lambda$, leaving the two-body input encoded in the parameters $a_{0}$
and $a_{1}$ unchanged, and perform several fits. We list the pole position and
free parameters of these fits in Tab. 3. As we increase $\Lambda$, $m_{a_{1}}$
and $c_{00}^{-1}$ decrease, whereas $c_{00}^{0}$ $m_{a_{1}}$ increases. The
pole position remains relatively unchanged, indicating the physical pole
position does not depend on the cutoff.

Figure 11:  Left: Compilation of pole positions determined in this work
including statistical and systematic uncertainties. For convenience, the PDG
Zyla  _et al._ (2020) average and a result by JPAC Mikhasenko  _et al._ (2018)
are quoted as well. See text for further explanations. Right: Pole couplings
according to Eq. (34) as a function of a real spectator momentum $p$. These
quantities play the role of spectator-momentum dependent branching ratios.
Results for S-wave are shown at the top and for D-wave at the bottom, for
different cutoffs $\Lambda$ as indicated. The solid (dashed) lines show the
real (imaginary) parts.

###  IV.2 Discussion

The $a_{1}(1260)$ lineshape data and best fits for different cutoffs are shown
in Fig. 10 with the solid lines. Fit uncertainties are indicated with the blue
band for our main result ($\Lambda=0.73$ GeV). For better visibility, the band
width is multiplied by a factor of 10. The plot of reduced residuals for the
$\Lambda=0.73$ GeV case (bottom of Fig. 10) shows that there are no obvious
systematic deviations of the fit from data, except maybe for a structure at
$\sqrt{s}\approx 0.8$ GeV. Also, around that energy the fits for different
$\Lambda$ differ from each other more than in other energy regions, and a
substantial fraction of the $\chi^{2}$ for larger cutoffs $\Lambda$ (see Table
3) arises in this energy region. In general, the pertinent fits are all very
close together which is reflected in the very small variation in pole
positions indicated in Table 3.

We show with a red dashed line in Fig. 10 the line shape with $\Lambda=0.73$
GeV calculated with the D-wave contributions set to 0. The difference in this
case from the blue line shows the contributions of the D-wave term. This
difference is small, which is qualitatively in line with the PDG Zyla  _et
al._ (2020). Note that this is a prediction at this point; the D-wave can only
be determined more quantitatively once Dalitz plots are analyzed, as measured,
e.g. by CLEO Asner  _et al._ (2000).

We test the impact of the rescattering term by performing a fit in which we
exclude $B^{J}_{LL^{\prime}}(q_{1},p)$ from Eq. (12). This pion-exchange term
is required by three-body unitarity Mai  _et al._ (2017) and omitted in Ref.
Mikhasenko  _et al._ (2018). As Table 3 (last row) and Fig. 11 (open red
circle) show, the $a_{1}(1260)$ pole position is significantly shifted in the
refit without the B-term. Still, Fig. 11 also shows that, even without the
$B$-term, our model is not identical to the one of JPAC Mikhasenko  _et al._
(2018) which might be due to a slightly different treatment of the two-body
input, different cutoffs, or the fact that in Ref. Mikhasenko  _et al._ (2018)
the lineshape data of Ref. Schael  _et al._ (2005) is fitted, while we fit the
data of Ref. Davier  _et al._ (2014).

We show our pole predictions in Fig. 11. Statistical uncertainties are
calculated through a re-sampling procedure (blue dots) in a two-step process.
Firstly, the two-body data from Refs. Protopopescu  _et al._ (1973);
Estabrooks and Martin (1974) are resampled 20 times with a normal distribution
using the given data uncertainties. Parameters $a_{0}$ and $a_{1}$ (given in
Eq. (20)) are fit to each resampled set. Secondly, 20 sets of resampled ALEPH
data Davier  _et al._ (2014); web  are generated. A fit is performed for each
resampled set using the different values of $a_{0}$ and $a_{1}$ calculated in
the first step. From each of these fits, a pole position is calculated, shown
with blue dots in Fig. 11. We also show the pertinent error ellipse keeping in
mind that this is a non-linear fit problem.

The green rectangle in Fig. 11 shows the region of pole positions from
different cutoffs $\Lambda$ according to Table 3. To that we add a second
source of systematic uncertainties from the different two-body input according
to the last two columns of Table 3. Such variations in $\Lambda$ help assess
the influence of inherent model uncertainties so we add then as systematic
error to our result for the $a_{1}(1260)$ pole position quoted in the next
section. Our entire confidence region, including both statistical and
systematic uncertainties, lies entirely within the PDG estimate of the
$a_{1}(1260)$ denoted with the gray rectangle, and it is not in strong tension
with the JPAC result Mikhasenko  _et al._ (2018) (orange rectangle).

The purple dot in Fig. 11 shows the predicted pole position using the improved
two-body input from Ref. Pelaez  _et al._ (2019), see Sec. II.1. The value of
$\sqrt{s_{0}}=(1226-253i)$ MeV lies within our systematic uncertainty which
demonstrates that there is not a very strong dependence on the two-body input.

We extract the residues of the pole position using the coupled-channel
partial-wave amplitude of Eq. (12). It can be expanded in $\sqrt{s}$ around
the pole position $\sqrt{s_{0}}$ of the $a_{1}(1260)$,

$\displaystyle
T^{J}_{LL^{\prime}}(p,p^{\prime})=\frac{\tilde{g}_{L}(p)\,\tilde{g}_{L^{\prime}}(p^{\prime})}{\sqrt{s}-\sqrt{s_{0}}}+{\cal
O}(1)\,,$ (34)

with $\tilde{g}_{L}$ playing the role of (Breit-Wigner) branching ratios, but
defined at the pole Zyla  _et al._ (2020). In addition, for the current case
of $\pi\rho$ scattering in S and D-waves, the $\tilde{g}$ are necessarily
functions of spectator momentum, $\tilde{g}_{L}\equiv\tilde{g}_{L}(p)$.
Analogously one might think of resonance transition form factors that are
closely related to pole residues depending on photon virtuality Kamano (2018);
Mai  _et al._ (2021c). For a numerically stable method to calculate residues
see Appendix C of Ref. Döring  _et al._ (2011). We show the $\tilde{g}_{L}(p)$
for real spectator momenta $p$ in Fig. 11, which requires another analytic
extrapolation from the complex $p$ on the spectator momentum contour (SMC) at
which the solution is calculated.

As Fig. 11 shows, the $a_{1}$ resonance does couple to the $\pi\rho$ D-wave
channel even if the corresponding coupling term appearing in Eq. (11) is not
fitted, $c_{22}^{-1}=0$, and, similarly, $c_{20}^{-1}=c_{02}^{-1}=0$ in Eq.
(16). This is due to the $B$-term which always allows for non-diagonal
transitions between S and D-wave channels. The D-wave decay is clearly smaller
than the S-wave decay, and the contribution to the lineshape from D-wave (at
real energies) is very small, see Fig. 10. While our complex pole couplings
are a prediction at this point, in future work they can be tested and even
extracted from data by analyzing Dalitz plots of the $a_{1}$ decay such as
measured at CLEO Asner  _et al._ (2000).

##  V Conclusions

In this work we have detailed how the pole position of the $a_{1}(1260)$ meson
can be determined using a manifestly unitary three-body formalism. The three-
body dynamics of the decay are fully taken into account, including the line
shape corrections due to pion exchange (sometimes referred to as
“rearrangement“ graph). This process is a direct consequence of unitarity. It
ensures that, apart from the usual isobar-spectator propagation in the
$s$-channel, this is the only possible on-shell arrangement of three pions.
Also, the amplitude necessarily exhibits two independent integrations that
cannot be simply recast and factorized into the phase space calculation.

Three-body cuts and the two integrations imply problems for the analytic
continuation of the amplitude to the complex pole position of the
$a_{1}(1260)$. We explain in detail how the continuation is achieved by
contour deformation and how different Riemann sheets are induced by an
appropriate choice of integration contours.

Upon implementation, we find that the pion exchange term does have significant
influence on the pole position of the $a_{1}(1260)$; taking into account
nearest-neighbor correlations in the data from ALEPH Davier  _et al._ (2014),
the pole position is determined to be

$\displaystyle\sqrt{s_{0}}=(1232^{+15+9}_{-0-11}-i266^{+0+15}_{-22-27})\leavevmode\nobreak\
\text{MeV}\,,$ (35)

where the first errors are statistical (including nearest-neighbor
correlations) and the second are systematic. The systematic uncertainties stem
from the cut-off dependence and two-body input as shown in Table 3.

The current calculation is restricted to $\rho\pi$ channels in S and D-wave.
Future upgrades to include more coupled channels, like sub-dominant
$\sigma\pi$, will enable accurate simultaneous fits to line shape and Dalitz
plot data to exploit unitarity which relates them.

Acknowledgments — This material is based upon work supported by the National
Science Foundation under Grant No. PHY-2012289 and the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics under Award Number DE-
SC0016582, DE-AC05-06OR23177, and DE-FG02-95ER40907. Work of MM is supported
in part by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) and the NSFC through the funds provided to the Sino-German
Collaborative Research Center TRR110 “Symmetries and the Emergence of
Structure in QCD” (DFG Project ID 196253076 - TRR 110, NSFC Grant No.
12070131001).CC is supported by UK Research and Innovation grant MR/S015418/1.
HA thanks the Avicenna-Studienwerk e.V. for financial support with funds from
the BMBF.

## References

  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * Al Ghoul  _et al._ (2017) H. Al Ghoul _et al._ (GlueX), “Measurement of the beam asymmetry $\Sigma$ for $\pi^{0}$ and $\eta$ photoproduction on the proton at $E_{\gamma}=9$ GeV,” Phys. Rev. C95, 042201 (2017), arXiv:1701.08123 [nucl-ex] .
  * Alekseev  _et al._ (2010) M. Alekseev _et al._ (COMPASS), “Observation of a $J^{PC}=1^{-+}$ exotic resonance in diffractive dissociation of 190-GeV/c $\pi^{-}$ into $\pi^{-}\pi^{-}\pi^{+}$,” Phys. Rev. Lett. 104, 241803 (2010), arXiv:0910.5842 [hep-ex] .
  * Asner  _et al._ (2009) D. M. Asner _et al._ , “Physics at BES-III,” Int. J. Mod. Phys. A24, S1–794 (2009), arXiv:0809.1869 [hep-ex] .
  * Adolph  _et al._ (2017) C Adolph _et al._ (COMPASS), “Resonance Production and $\pi\pi$ S-wave in $\pi^{-}+p\rightarrow\pi^{-}\pi^{-}\pi^{+}+p_{recoil}$ at 190 GeV$/c$,” Phys. Rev. D 95, 032004 (2017), arXiv:1509.00992 [hep-ex] .
  * Aghasyan  _et al._ (2018) M. Aghasyan _et al._ (COMPASS), “Light isovector resonances in $\pi^{-}p\to\pi^{-}\pi^{-}\pi^{+}p$ at 190 GeV/${\it c}$,” Phys. Rev. D 98, 092003 (2018), arXiv:1802.05913 [hep-ex] .
  * Ablikim  _et al._ (2021) M. Ablikim _et al._ (BESIII), “Study of the Decay $D^{+}_{s}\rightarrow\pi^{+}\pi^{+}\pi^{-}\eta$ and Observation of the W-annihilation Decay $D^{+}_{s}\rightarrow a_{0}(980)^{+}\rho^{0}$,” Phys. Rev. D 104, 071101 (2021), arXiv:2106.13536 [hep-ex] .
  * Ablikim  _et al._ (2005) M. Ablikim _et al._ (BES), “Resonances in $J/\psi\to\phi\pi^{+}\pi^{-}$ and $\phi K^{+}K^{-}$,” Phys. Lett. B 607, 243–253 (2005), arXiv:hep-ex/0411001 .
  * Asner  _et al._ (2000) D. M. Asner _et al._ (CLEO), “Hadronic structure in the decay $\tau^{-}\to\nu_{\tau}\pi^{-}\pi^{0}\pi^{0}$ and the sign of the tau-neutrino helicity,” Phys. Rev. D 61, 012002 (2000), arXiv:hep-ex/9902022 .
  * Albrecht  _et al._ (2020) M. Albrecht _et al._ (Crystal Barrel), “Coupled channel analysis of ${\bar{p}p}\,\rightarrow\,\pi^{0}\pi^{0}\eta$, ${\pi^{0}\eta\eta}$ and ${K^{+}K^{-}\pi^{0}}$ at 900 MeV/c and of ${\pi\pi}$-scattering data,” Eur. Phys. J. C 80, 453 (2020), arXiv:1909.07091 [hep-ex] .
  * Pasquier and Pasquier (1968) R. Pasquier and J. Y. Pasquier, “Khuri-Treiman-Type Equations for Three-Body Decay and Production Processes,” Phys. Rev. 170, 1294–1309 (1968).
  * Aitchison (1977) I. J. R. Aitchison, “Relativistic Three Pion Dynamics Generated by Two-Body Unitarity and Analyticity,” J. Phys. G 3, 121 (1977).
  * Colangelo  _et al._ (2009) Gilberto Colangelo, Stefan Lanz, and Emilie Passemar, “A New Dispersive Analysis of $\eta\to 3\pi$,” PoS CD09, 047 (2009), arXiv:0910.0765 [hep-ph] .
  * Kubis and Schneider (2009) Bastian Kubis and Sebastian P. Schneider, “The Cusp effect in $\eta^{\prime}\to\eta\pi\pi$ decays,” Eur. Phys. J. C 62, 511–523 (2009), arXiv:0904.1320 [hep-ph] .
  * Schneider  _et al._ (2011) Sebastian P. Schneider, Bastian Kubis, and Christoph Ditsche, “Rescattering effects in $\eta\to 3\pi$ decays,” JHEP 02, 028 (2011), arXiv:1010.3946 [hep-ph] .
  * Kampf  _et al._ (2011) Karol Kampf, Marc Knecht, Jiri Novotny, and Martin Zdrahal, “Analytical dispersive construction of $\eta\to 3\pi$ amplitude: first order in isospin breaking,” Phys. Rev. D 84, 114015 (2011), arXiv:1103.0982 [hep-ph] .
  * Niecknig  _et al._ (2012) Franz Niecknig, Bastian Kubis, and Sebastian P. Schneider, “Dispersive analysis of $\omega\to 3\pi$ and $\phi\to 3\pi$ decays,” Eur. Phys. J. C 72, 2014 (2012), arXiv:1203.2501 [hep-ph] .
  * Guo  _et al._ (2012) Peng Guo, Ryan Mitchell, Matthew Shepherd, and Adam P. Szczepaniak, “Amplitudes for the analysis of the decay $J/\psi\to K^{+}K^{-}\pi^{0}$,” Phys. Rev. D 85, 056003 (2012), arXiv:1112.3284 [hep-ph] .
  * Danilkin  _et al._ (2015) I. V. Danilkin, C. Fernández-Ramírez, P. Guo, V. Mathieu, D. Schott, M. Shi, and A. P. Szczepaniak, “Dispersive analysis of $\omega$/$\phi$→3$\pi$,$\pi$$\gamma$*,” Phys. Rev. D 91, 094029 (2015), arXiv:1409.7708 [hep-ph] .
  * Guo  _et al._ (2015a) Peng Guo, I. V. Danilkin, and Adam P. Szczepaniak, “Dispersive approaches for three-particle final state interaction,” Eur. Phys. J. A 51, 135 (2015a), arXiv:1409.8652 [hep-ph] .
  * Guo  _et al._ (2015b) Peng Guo, Igor V. Danilkin, Diane Schott, C. Fernández-Ramírez, V. Mathieu, and Adam P. Szczepaniak, “Three-body final state interaction in $\eta\to 3\pi$,” Phys. Rev. D 92, 054016 (2015b), arXiv:1505.01715 [hep-ph] .
  * Daub  _et al._ (2016) J. T. Daub, C. Hanhart, and B. Kubis, “A model-independent analysis of final-state interactions in ${\overline{B}}_{d/s}^{0}\to J/\psi\pi\pi$,” JHEP 02, 009 (2016), arXiv:1508.06841 [hep-ph] .
  * Niecknig and Kubis (2015) Franz Niecknig and Bastian Kubis, “Dispersion-theoretical analysis of the D+ → K- $\pi$+ $\pi$+ Dalitz plot,” JHEP 10, 142 (2015), arXiv:1509.03188 [hep-ph] .
  * Guo  _et al._ (2017) P. Guo, I. V. Danilkin, C. Fernández-Ramírez, V. Mathieu, and A. P. Szczepaniak, “Three-body final state interaction in $\eta$ → 3 $\pi$ updated,” Phys. Lett. B 771, 497–502 (2017), arXiv:1608.01447 [hep-ph] .
  * Isken  _et al._ (2017) Tobias Isken, Bastian Kubis, Sebastian P. Schneider, and Peter Stoffer, “Dispersion relations for $\eta^{\prime}\rightarrow\eta\pi\pi$,” Eur. Phys. J. C 77, 489 (2017), arXiv:1705.04339 [hep-ph] .
  * Albaladejo and Moussallam (2017) M. Albaladejo and B. Moussallam, “Extended chiral Khuri-Treiman formalism for $\eta\to 3\pi$ and the role of the $a_{0}(980)$, $f_{0}(980)$ resonances,” Eur. Phys. J. C 77, 508 (2017), arXiv:1702.04931 [hep-ph] .
  * Niecknig and Kubis (2018) Franz Niecknig and Bastian Kubis, “Consistent Dalitz plot analysis of Cabibbo-favored $D^{+}\to\bar{K}\pi\pi^{+}$ decays,” Phys. Lett. B 780, 471–478 (2018), arXiv:1708.00446 [hep-ph] .
  * Dax  _et al._ (2018) Maximilian Dax, Tobias Isken, and Bastian Kubis, “Quark-mass dependence in $\omega\rightarrow 3\pi$ decays,” Eur. Phys. J. C 78, 859 (2018), arXiv:1808.08957 [hep-ph] .
  * Jackura  _et al._ (2019) A. Jackura, C. Fernández-Ramírez, V. Mathieu, M. Mikhasenko, J. Nys, A. Pilloni, K. Saldaña, N. Sherrill, and A.P. Szczepaniak (JPAC), “Phenomenology of Relativistic $3\to 3$ Reaction Amplitudes within the Isobar Approximation,” Eur. Phys. J. C 79, 56 (2019), arXiv:1809.10523 [hep-ph] .
  * Gasser and Rusetsky (2018) Juerg Gasser and Akaki Rusetsky, “Solving integral equations in $\eta\to 3\pi$,” Eur. Phys. J. C 78, 906 (2018), arXiv:1809.06399 [hep-ph] .
  * Albaladejo  _et al._ (2020) M. Albaladejo, D. Winney, I. V. Danilkin, C. Fernández-Ramírez, V. Mathieu, M. Mikhasenko, A. Pilloni, J. A. Silva-Castro, and A. P. Szczepaniak (JPAC), “Khuri-Treiman equations for $3\pi$ decays of particles with spin,” Phys. Rev. D 101, 054018 (2020), arXiv:1910.03107 [hep-ph] .
  * Mikhasenko  _et al._ (2020) M. Mikhasenko _et al._ (JPAC), “Dalitz-plot decomposition for three-body decays,” Phys. Rev. D 101, 034033 (2020), arXiv:1910.04566 [hep-ph] .
  * Mikhasenko  _et al._ (2019) M. Mikhasenko, Y. Wunderlich, A. Jackura, V. Mathieu, A. Pilloni, B. Ketzer, and A.P. Szczepaniak, “Three-body scattering: Ladders and Resonances,” JHEP 08, 080 (2019), arXiv:1904.11894 [hep-ph] .
  * Akdag  _et al._ (2021) Hakan Akdag, Tobias Isken, and Bastian Kubis, “Patterns of $C$\- and $CP$-violation in hadronic $\eta$ and $\eta^{\prime}$ three-body decays,” (2021), arXiv:2111.02417 [hep-ph] .
  * Martinez Torres  _et al._ (2008) A. Martinez Torres, K. P. Khemchandani, and E. Oset, “Three body resonances in two meson-one baryon systems,” Phys. Rev. C 77, 042203 (2008), arXiv:0706.2330 [nucl-th] .
  * Magalhaes  _et al._ (2011) P. C. Magalhaes, M. R. Robilotta, K. S. F. F. Guimaraes, T. Frederico, W. de Paula, I. Bediaga, A. C. dos Reis, C. M. Maekawa, and G. R. S. Zarnauskas, “Towards three-body unitarity in $D^{+}\to K^{-}\pi^{+}\pi^{+}$,” Phys. Rev. D 84, 094001 (2011), arXiv:1105.5120 [hep-ph] .
  * Martinez Torres  _et al._ (2009) A. Martinez Torres, K. P. Khemchandani, and E. Oset, “Solution to Faddeev equations with two-body experimental amplitudes as input and application to $J^{P}=1/2^{+}$, $S=0$ baryon resonances,” Phys. Rev. C 79, 065207 (2009), arXiv:0812.2235 [nucl-th] .
  * Martinez Torres  _et al._ (2011) A. Martinez Torres, K. P. Khemchandani, D. Jido, and A. Hosaka, “Theoretical support for the $\pi(1300)$ and the recently claimed $f_{0}(1790)$ as molecular resonances,” Phys. Rev. D 84, 074027 (2011), arXiv:1106.6101 [nucl-th] .
  * Aoude  _et al._ (2018) R. T. Aoude, P. C. Magalhães, A. C. Dos Reis, and M. R. Robilotta, “Multimeson model for the $D^{+}\to K^{+}K^{-}K^{+}$ decay amplitude,” Phys. Rev. D 98, 056021 (2018), arXiv:1805.11764 [hep-ph] .
  * Aitchison (2015) Ian J. R. Aitchison, “Unitarity, Analyticity and Crossing Symmetry in Two- and Three-hadron Final State Interactions,” (2015), arXiv:1507.02697 [hep-ph] .
  * Aitchison and Pasquier (1966) I. J. R. Aitchison and R. Pasquier, “Three-Body Unitarity and Khuri-Treiman Amplitudes,” Phys. Rev. 152, 1274 (1966).
  * Mai  _et al._ (2017) M. Mai, B. Hu, M. Döring, A. Pilloni, and A. Szczepaniak, “Three-body Unitarity with Isobars Revisited,” Eur. Phys. J. A 53, 177 (2017), arXiv:1706.06118 [nucl-th] .
  * Aaron  _et al._ (1968) R. Aaron, R. D. Amado, and J. E. Young, “Relativistic three-body theory with applications to $\pi-N$ scattering,” Phys. Rev. 174, 2022–2032 (1968).
  * Aaron and Amado (1973) R. Aaron and R. D. Amado, “Analysis of three-hadron final states,” Phys. Rev. Lett. 31, 1157–1159 (1973).
  * Dawid and Szczepaniak (2021) Sebastian M. Dawid and Adam P. Szczepaniak, “Bound states in the B-matrix formalism for the three-body scattering,” Phys. Rev. D 103, 014009 (2021), arXiv:2010.08084 [nucl-th] .
  * Zhang  _et al._ (2022) Xu Zhang, Christoph Hanhart, Ulf-G. Meißner, and Ju-Jun Xie, “Remarks on non-perturbative three–body dynamics and its application to the $KK{\bar{K}}$ system,” Eur. Phys. J. A 58, 20 (2022), arXiv:2107.03168 [hep-ph] .
  * Sadasivan  _et al._ (2020) Daniel Sadasivan, Maxim Mai, Hakan Akdag, and Michael Döring, “Dalitz plots and lineshape of $a_{1}(1260)$ from a relativistic three-body unitary approach,” Phys. Rev. D 101, 094018 (2020), arXiv:2002.12431 [nucl-th] .
  * Schael  _et al._ (2005) S. Schael _et al._ (ALEPH), “Branching ratios and spectral functions of tau decays: Final ALEPH measurements and physics implications,” Phys. Rept. 421, 191–284 (2005), arXiv:hep-ex/0506072 .
  * Molina  _et al._ (2021) R. Molina, M. Doering, W. H. Liang, and E. Oset, “The $\pi f_{0}(500)$ decay of the $a_{1}(1260)$,” Eur. Phys. J. C 81, 782 (2021), arXiv:2107.07439 [hep-ph] .
  * Janssen  _et al._ (1993) G. Janssen, J. W. Durso, K. Holinde, B. C. Pearce, and J. Speth, “$\pi\rho$ scattering and the $\pi NN$ form-factor,” Phys. Rev. Lett. 71, 1978–1981 (1993).
  * Lutz and Kolomeitsev (2004) M. F. M. Lutz and E. E. Kolomeitsev, “On meson resonances and chiral symmetry,” Nucl. Phys. A 730, 392–416 (2004), arXiv:nucl-th/0307039 .
  * Roca  _et al._ (2005) L. Roca, E. Oset, and J. Singh, “Low lying axial-vector mesons as dynamically generated resonances,” Phys. Rev. D 72, 014002 (2005), arXiv:hep-ph/0503273 .
  * Geng  _et al._ (2007) L. S. Geng, E. Oset, L. Roca, and J. A. Oller, “Clues for the existence of two $K_{1}(1270)$ resonances,” Phys. Rev. D 75, 014017 (2007), arXiv:hep-ph/0610217 .
  * Wagner and Leupold (2008a) Markus Wagner and Stefan Leupold, “Tau decay and the structure of the $a_{1}$,” Phys. Lett. B 670, 22–26 (2008a), arXiv:0708.2223 [hep-ph] .
  * Wagner and Leupold (2008b) Markus Wagner and Stefan Leupold, “Information on the structure of the $a_{1}$ from tau decay,” Phys. Rev. D 78, 053001 (2008b), arXiv:0801.0814 [hep-ph] .
  * Lutz and Leupold (2008) M. F. M. Lutz and S. Leupold, “On the radiative decays of light vector and axial-vector mesons,” Nucl. Phys. A 813, 96–170 (2008), arXiv:0801.3821 [nucl-th] .
  * Kamano  _et al._ (2011) H. Kamano, S. X. Nakamura, T. S. H. Lee, and T. Sato, “Unitary coupled-channels model for three-mesons decays of heavy mesons,” Phys. Rev. D 84, 114019 (2011), arXiv:1106.4523 [hep-ph] .
  * Nagahiro  _et al._ (2011) H. Nagahiro, K. Nawa, S. Ozaki, D. Jido, and A. Hosaka, “Composite and elementary natures of $a_{1}(1260)$ meson,” Phys. Rev. D 83, 111504 (2011), arXiv:1101.3623 [hep-ph] .
  * Zhou  _et al._ (2014) Yu Zhou, Xiu-Lei Ren, Hua-Xing Chen, and Li-Sheng Geng, “Pseudoscalar meson and vector meson interactions and dynamically generated axial-vector mesons,” Phys. Rev. D 90, 014020 (2014), arXiv:1404.6847 [nucl-th] .
  * Zhang and Xie (2018) Xu Zhang and Ju-Jun Xie, “The three-pion decays of the $a_{1}(1260)$,” Commun. Theor. Phys. 70, 060 (2018), arXiv:1712.05572 [nucl-th] .
  * Mikhasenko  _et al._ (2018) M. Mikhasenko, A. Pilloni, M. Albaladejo, C. Fernández-Ramírez, A. Jackura, V. Mathieu, J. Nys, A. Rodas, B. Ketzer, and A. P. Szczepaniak (JPAC), “Pole position of the $a_{1}(1260)$ from $\tau$-decay,” Phys. Rev. D 98, 096021 (2018), arXiv:1810.00016 [hep-ph] .
  * Dai  _et al._ (2020) L. R. Dai, L. Roca, and E. Oset, “Tau decay into $\nu_{\tau}$ and $a_{1}(1260)$, $b_{1}(1235)$, and two $K_{1}(1270)$,” Eur. Phys. J. C 80, 673 (2020), arXiv:2005.02653 [hep-ph] .
  * Dias  _et al._ (2021) J. M. Dias, G. Toledo, L. Roca, and E. Oset, “Unveiling the K1(1270) double-pole structure in the B¯→J/$\psi$$\rho$K¯ and B¯→J/$\psi$K¯*$\pi$ decays,” Phys. Rev. D 103, 116019 (2021), arXiv:2102.08402 [hep-ph] .
  * Bowler (1986) M. G. Bowler, “The A1 Revisited,” Phys. Lett. B 182, 400–404 (1986).
  * Kuhn and Mirkes (1992) Johann H. Kuhn and E. Mirkes, “Structure functions in tau decays,” Z. Phys. C 56, 661–672 (1992), [Erratum: Z.Phys.C 67, 364 (1995)].
  * Isgur  _et al._ (1989) Nathan Isgur, Colin Morningstar, and Cathy Reader, “The a1 in tau Decay,” Phys. Rev. D 39, 1357 (1989).
  * Dumm  _et al._ (2010) D. Gomez Dumm, P. Roig, A. Pich, and J. Portoles, “$\tau\to\pi\pi\pi\nu_{\tau}$ decays and the $a_{1}(1260)$ off-shell width revisited,” Phys. Lett. B 685, 158–164 (2010), arXiv:0911.4436 [hep-ph] .
  * Nugent  _et al._ (2013) I. M. Nugent, T. Przedzinski, P. Roig, O. Shekhovtsova, and Z. Was, “Resonance chiral Lagrangian currents and experimental data for $\tau^{-}\to\pi^{-}\pi^{-}\pi^{+}\nu_{\tau}$,” Phys. Rev. D 88, 093012 (2013), arXiv:1310.1053 [hep-ph] .
  * Dai  _et al._ (2019) L. R. Dai, L. Roca, and E. Oset, “$\tau$ decay into a pseudoscalar and an axial-vector meson,” Phys. Rev. D 99, 096003 (2019), arXiv:1811.06875 [hep-ph] .
  * Lang  _et al._ (2014) C. B. Lang, Luka Leskovec, Daniel Mohler, and Sasa Prelovsek, “Axial resonances a1(1260), b1(1235) and their decays from the lattice,” JHEP 04, 162 (2014), arXiv:1401.2088 [hep-lat] .
  * Mai and Döring (2017) M. Mai and M. Döring, “Three-body Unitarity in the Finite Volume,” Eur. Phys. J. A53, 240 (2017), arXiv:1709.08222 [hep-lat] .
  * Mai and Döring (2019) Maxim Mai and Michael Döring, “Finite-Volume Spectrum of $\pi^{+}\pi^{+}$ and $\pi^{+}\pi^{+}\pi^{+}$ Systems,” Phys. Rev. Lett. 122, 062503 (2019), arXiv:1807.04746 [hep-lat] .
  * Mai  _et al._ (2021a) Maxim Mai, Andrei Alexandru, Ruairí Brett, Chris Culver, Michael Döring, Frank X. Lee, and Daniel Sadasivan (GWQCD), “Three-Body Dynamics of the a1(1260) Resonance from Lattice QCD,” Phys. Rev. Lett. 127, 222001 (2021a), arXiv:2107.03973 [hep-lat] .
  * Mai  _et al._ (2021b) Maxim Mai, Michael Döring, and Akaki Rusetsky, “Multi-particle systems on the lattice and chiral extrapolations: a brief review,” Eur. Phys. J. Spec. Top. (2021b), 10.1140/epjs/s11734-021-00146-5, arXiv:2103.00577 [hep-lat] .
  * Hansen and Sharpe (2019) Maxwell T. Hansen and Stephen R. Sharpe, “Lattice QCD and Three-particle Decays of Resonances,” Ann. Rev. Nucl. Part. Sci. 69, 65–107 (2019), arXiv:1901.00483 [hep-lat] .
  * Rusetsky (2019) Akaki Rusetsky, “Three particles on the lattice,” PoS LATTICE2019, 281 (2019), arXiv:1911.01253 [hep-lat] .
  * Kuhn  _et al._ (2004) Joachim Kuhn _et al._ (E852), “Exotic meson production in the $f_{1}(1285)\pi^{-}$ system observed in the reaction $\pi^{-}p\to\eta\pi^{+}\pi^{-}\pi^{-}p$ at 18 GeV/c,” Phys. Lett. B 595, 109–117 (2004), arXiv:hep-ex/0401004 .
  * Bedaque and Grießhammer (2000) Paulo F. Bedaque and Harald W. Grießhammer, “Quartet S wave neutron deuteron scattering in effective field theory,” Nucl. Phys. A671, 357–379 (2000), arXiv:nucl-th/9907077 [nucl-th] .
  * Hammer  _et al._ (2017) H. W. Hammer, J. Y. Pang, and A. Rusetsky, “Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data,” JHEP 10, 115 (2017), arXiv:1707.02176 [hep-lat] .
  * Berman and Jacob (1965) S. M. Berman and Maurice Jacob, “Systematics of angular Polarization Distributions in Three-Body Decays,” Phys. Rev. 139, B1023–B1038 (1965).
  * Chung (1971) Suh Urk Chung, “Spin formalism,”  (1971), 10.5170/CERN-1971-008.
  * Protopopescu  _et al._ (1973) S. D. Protopopescu, M. Alston-Garnjost, A. Barbaro-Galtieri, Stanley M. Flatte, J. H. Friedman, T. A. Lasinski, G. R. Lynch, M. S. Rabin, and F. T. Solmitz, “$\pi\pi$ Partial Wave Analysis from Reactions $\pi^{+}p\to\pi^{+}\pi^{-}\Delta^{++}$ and $\pi^{+}p\to K^{+}K^{-}\Delta^{++}$ at 7.1-GeV/c,” Phys. Rev. D7, 1279 (1973).
  * Estabrooks and Martin (1974) P. Estabrooks and Alan D. Martin, “$\pi\pi$ Phase Shift Analysis Below the$K\bar{K}$ Threshold,” Nucl. Phys. B79, 301–316 (1974).
  * Pelaez  _et al._ (2019) J. R. Pelaez, A. Rodas, and J. Ruiz De Elvira, “Global parameterization of $\pi\pi$ scattering up to 2 ${\mathrm{\,GeV}}$,” Eur. Phys. J. C 79, 1008 (2019), arXiv:1907.13162 [hep-ph] .
  * Ananthanarayan  _et al._ (2001) B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler, “Roy equation analysis of pi pi scattering,” Phys. Rept. 353, 207–279 (2001), arXiv:hep-ph/0005297 [hep-ph] .
  * Colangelo  _et al._ (2001) G. Colangelo, J. Gasser, and H. Leutwyler, “$\pi\pi$ scattering,” Nucl. Phys. B603, 125–179 (2001), arXiv:hep-ph/0103088 [hep-ph] .
  * Garcia-Martin  _et al._ (2011) R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira, and F. J. Yndurain, “The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV,” Phys. Rev. D 83, 074004 (2011), arXiv:1102.2183 [hep-ph] .
  * Colangelo  _et al._ (2019) Gilberto Colangelo, Martin Hoferichter, and Peter Stoffer, “Two-pion contribution to hadronic vacuum polarization,” JHEP 02, 006 (2019), arXiv:1810.00007 [hep-ph] .
  * Döring  _et al._ (2009) M. Döring, C. Hanhart, F. Huang, S. Krewald, and U.-G. Meißner, “Analytic properties of the scattering amplitude and resonances parameters in a meson exchange model,” Nucl. Phys. A 829, 170–209 (2009), arXiv:0903.4337 [nucl-th] .
  * Rönchen  _et al._ (2013) D. Rönchen, M. Döring, F. Huang, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald, U.-G. Meißner, and K. Nakayama, “Coupled-channel dynamics in the reactions $\pi N\to\pi N,\,\eta N,\,K\Lambda,\,K\Sigma$,” Eur. Phys. J. A 49, 44 (2013), arXiv:1211.6998 [nucl-th] .
  * Rönchen  _et al._ (2014) D. Rönchen, M. Döring, F. Huang, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald, U. G. Meißner, and K. Nakayama, “Photocouplings at the Pole from Pion Photoproduction,” Eur. Phys. J. A 50, 101 (2014), [Erratum: Eur.Phys.J.A 51, 63 (2015)], arXiv:1401.0634 [nucl-th] .
  * Mai  _et al._ (2021c) Maxim Mai, Michael Döring, Carlos Granados, Helmut Haberzettl, Ulf-G. Meißner, Deborah Rönchen, Igor Strakovsky, and Ron Workman (Jülich-Bonn-Washington), “Jülich-Bonn-Washington model for pion electroproduction multipoles,” Phys. Rev. C 103, 065204 (2021c), arXiv:2104.07312 [nucl-th] .
  * Suzuki  _et al._ (2010) N. Suzuki, T. Sato, and T. S. H. Lee, “Extraction of Electromagnetic Transition Form Factors for Nucleon Resonances within a Dynamical Coupled-Channels Model,” Phys. Rev. C 82, 045206 (2010), arXiv:1006.2196 [nucl-th] .
  * Julia-Diaz  _et al._ (2009) B. Julia-Diaz, H. Kamano, T. S. H. Lee, A. Matsuyama, T. Sato, and N. Suzuki, “Dynamical coupled-channels analysis of p(e,e-prime pi)N reactions,” Phys. Rev. C 80, 025207 (2009), arXiv:0904.1918 [nucl-th] .
  * Kamano  _et al._ (2013) H. Kamano, S. X. Nakamura, T. S. H. Lee, and T. Sato, “Nucleon resonances within a dynamical coupled-channels model of $\pi N$ and $\gamma N$ reactions,” Phys. Rev. C 88, 035209 (2013), arXiv:1305.4351 [nucl-th] .
  * Huang  _et al._ (2012) F. Huang, M. Döring, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald, Ulf-G. Meißner, and K. Nakayama, “Pion photoproduction in a dynamical coupled-channels model,” Phys. Rev. C 85, 054003 (2012), arXiv:1110.3833 [nucl-th] .
  * Rönchen  _et al._ (2018) D. Rönchen, M. Döring, and Ulf-G. Meißner, “The impact of $K^{+}\Lambda$ photoproduction on the resonance spectrum,” Eur. Phys. J. A 54, 110 (2018), arXiv:1801.10458 [nucl-th] .
  * Gribov (2008) Vladimir Gribov,  _Strong Interactions of Hadrons at High Energies : Gribov Lectures on Theoretical Physics_ (Cambridge University Press, Leiden, 2008).
  * Janssen  _et al._ (1994) G. Janssen, K. Holinde, and J. Speth, “A Meson exchange model for pi rho scattering,” Phys. Rev. C 49, 2763–2776 (1994).
  * Haftel and Tabakin (1970) Michael I. Haftel and Frank Tabakin, “Nuclear Saturation and smoothness of nucleon-nucleon potentials,” Nucl. Phys. A 158, 1–42 (1970).
  * Döring  _et al._ (2013) Michael Döring, Ulf-G. Meißner, and Wei Wang, “Chiral Dynamics and S-wave Contributions in Semileptonic B decays,” JHEP 10, 011 (2013), arXiv:1307.0947 [hep-ph] .
  * Ceci  _et al._ (2011) S. Ceci, M. Döring, C. Hanhart, S. Krewald, U.-G. Meißner, and A. Švarc, “Relevance of complex branch points for partial wave analysis,” Phys. Rev. C 84, 015205 (2011), arXiv:1104.3490 [nucl-th] .
  * Davier  _et al._ (2014) Michel Davier, Andreas Höcker, Bogdan Malaescu, Chang-Zheng Yuan, and Zhiqing Zhang, “Update of the ALEPH non-strange spectral functions from hadronic $\tau$ decays,” Eur. Phys. J. C 74, 2803 (2014), arXiv:1312.1501 [hep-ex] .
  * (104) “Invariant mass squared distributions from ALEPH,” http://aleph.web.lal.in2p3.fr/tau/specfun13.html.
  * Kamano (2018) Hiroyuki Kamano, “Electromagnetic $N^{*}$ Transition Form Factors in the ANL-Osaka Dynamical Coupled-Channels Approach,” Few Body Syst. 59, 24 (2018).
  * Döring  _et al._ (2011) M. Döring, C. Hanhart, F. Huang, S. Krewald, U.-G. Meißner, and D. Rönchen, “The reaction $\pi^{+}p\to K^{+}\Sigma^{+}$ in a unitary coupled-channels model,” Nucl. Phys. A 851, 58–98 (2011), arXiv:1009.3781 [nucl-th] .
  * Bruns  _et al._ (2013) Peter C. Bruns, Ludwig Greil, and Andreas Schäfer, “Chiral behavior of vector meson self energies,” Phys. Rev. D 88, 114503 (2013), arXiv:1309.3976 [hep-ph] .

##  Appendix A Technical details on spin-1 systems

For each helicity state $\lambda\in\\{-1,0,+1\\}$ of the spin-1 field, the
four-vector $\varepsilon$ depends on the direction of the propagation Chung
(1971) as

$\displaystyle\varepsilon_{0}(\bm{p})$
$\displaystyle=\frac{1}{m_{\rho}}\begin{pmatrix}p\\\
E^{\rho}_{p}\cos\phi_{\bm{p}}\sin\theta_{\bm{p}}\\\
E^{\rho}_{p}\sin\phi_{\bm{p}}\sin\theta_{\bm{p}}\\\
E^{\rho}_{p}\cos\theta_{\bm{p}}\end{pmatrix}\,,$ (36)
$\displaystyle\varepsilon_{\pm 1}(\bm{p})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
\mp\cos{\theta_{{\bm{p}}}}\cos{\phi_{{\bm{p}}}}+i\sin{\phi_{{\bm{p}}}}\\\
\mp\cos{\theta_{{\bm{p}}}}\sin{\phi_{{\bm{p}}}}-i\cos{\phi_{{\bm{p}}}}\\\
\pm\sin{\theta_{{\bm{p}}}}\end{pmatrix}\,,$ (37)

where $E^{\rho}_{p}:=\sqrt{m_{\rho}^{2}+\bm{p}^{2}}$ and we chose
$m_{\rho}=6.44\,m_{\pi}$. On-shell, this fulfills required properties, such as
the transversality, i.e., $p_{\mu}\varepsilon^{\mu}=0$ exactly, see Ref. Chung
(1971). Away from the on-shell point one can generalize the above definitions
using $m_{\rho}\to\sqrt{E_{p}^{2}-p^{2}}$. However, as the difference between
both versions does not lead to new singularities of the spin-1 propagator,
perturbation theory is viable, allowing one to reabsorb it into the local
terms Bruns  _et al._ (2013).

Equation (18) requires the calculation of the helicity sum for the $s$-channel
$\rho$ propagation,

$\displaystyle\sum_{\lambda}\epsilon_{\lambda,\mu}(\bm{p})\epsilon^{*}_{\lambda,\nu}(\bm{p})=-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{m_{\rho}^{2}}\,.$
(38)

Figure 12: The fits (upper row) and normalized residuals (lower row) using a Legendre Polynomial Expansion to the ALEPH data ($n=10$ in Eq. (39)). The color coding indicates the cases where the covariance matrix includes all correlations (blue), no correlations (orange), and nearest-neighbor correlations (green).  |  |  |  |
---|---|---|---|---
$\chi^{2}$ | 64 | 62 | 127 | 16
No. data  | 63 | 63 | 63 | 75
Table 4: Total $\chi^{2}$ for fits to the ALEPH data while various
correlations were included. The first three entries refer to the 2013 data of
Ref. Davier  _et al._ (2014) while the last column refers to the 2005 data of
Ref. Schael  _et al._ (2005). Figure 13: Older ALEPH data Schael  _et al._
(2005), pertinent fit, and normalized fit residuals. These data are not used
in the analysis. See Fig. 10 for notation.

##  Appendix B Data consistency tests

Rather than considering all data correlations from Ref. Davier  _et al._
(2014) in the covariance matrix for the calculation of $\chi^{2}$ using Eq.
(33), we include only nearest-neighbor correlations. All other correlations
are neglected because we find that no reasonably smooth curve can describe the
data when they are included as shown in the following.

We fit the ALEPH data with a Legendre Polynomial expansion,
$f(\sqrt{s},x_{i})$ given by

$\displaystyle
f(\sqrt{s},x_{i})=\sum_{i=0}^{n}x_{i}P_{n}\left(\frac{\sqrt{s}-\sqrt{s_{0}}}{\sqrt{s_{0}}}\right),$
(39)

where $\sqrt{s_{0}}=1$ GeV is chosen such that the argument is always in the
interval $[-1,1]$ and the $x_{i}$ are fitted by minimizing the $\chi^{2}$,

$\displaystyle\chi^{2}=(\vec{f}(x_{i})-\vec{\mathcal{L}})^{T}\Sigma^{-1}(\vec{f}(x_{i})-\vec{\mathcal{L}})\
,$ (40)

where $\vec{f}$ is constructed from the data at $\sqrt{s_{i}}$,
$(\vec{f})_{j}=f(\sqrt{s_{j}},x_{i})$. Similarly, the central values of the
data are collected in the vector $\vec{\mathcal{L}}$. When we fit in the range
$0.6\leavevmode\nobreak\ \text{GeV}<\sqrt{s}<1.5$ GeV, with $n=10$ including
all correlations we find that the minimum $\chi^{2}$ for the 63 data points is
127 ($\chi^{2}_{\text{dof}}=2.43$) as Table 4 shows. Increasing the number of
polynomials in this expansion will decrease the total $\chi^{2}$ but does not
decrease the $\chi^{2}_{\text{dof}}$. For example, when $n=15$, $\chi^{2}=122$
and, therefore, $\chi^{2}_{\text{dof}}=2.58$. We show the $n=10$ fit and the
residuals in Fig. 12. These results indicate that it is very difficult for any
smooth curve, regardless of whether or not it is theoretically justifed, to
describe the data.

In contrast, when we use the same function to perform a fit where we include
only the uncorrelated uncertainties and set all correlations to 0, we find the
total $\chi^{2}$ for the 63 data points to be 62 indicating that the
uncorrelated data can be easily fitted with a smooth function. The orange line
and data of Fig. 12 show the fit and residuals when none of the correlations
are included in the data. This fit is quite similar to the one that includes
all correlations even though the $\chi^{2}$ is very different.

We note that while this case demonstrates that the uncorrelated data can be
reasonably described by a smooth function, the residuals still display some
noticeable correlation. In order to account for these correlations we
introduce another case, shown in blue in Fig. 12. Here, we include only the
nearest-neighbor correlations. When we fit the data with $n=10$ to this case,
we obtain $\chi^{2}=64$. As this case includes the maximum of correlations
that can be reconciled with a statistically sound description of the data, we
regard this case as the data set for the analysis described in the main text.

We also apply this phenomenological test to the data as they were originally
published in Ref. Schael  _et al._ (2005). These data, shown in Fig. 13, have
since been updated in Ref. Davier  _et al._ (2014). We find that the older
data are considerably over-fit for $n=10$ with a total $\chi^{2}$ of 16 as
shown in Table 4. We therefore discard these data.

In summary, all three fits shown in Fig. 12 are quite similar, even for the
residuals. This implies that the best-fit parameters for each case will be
quite similar regardless of which correlations are included. Thus, our choice
of data (nearest-neighbor correlations only) affects the value of our
$\chi^{2}$, but it does not have much effect on the best values for pole
position or residues.