# QCD axial anomaly enhances the $\eta\eta^{\prime}$ decay of the hybrid
candidate $\eta_{1}(1855)$

Hua-Xing Chen1<EMAIL_ADDRESS>Niu Su1<EMAIL_ADDRESS>Shi-Lin Zhu2
<EMAIL_ADDRESS>1School of Physics, Southeast University, Nanjing 210094,
China
2School of Physics and Center of High Energy Physics, Peking University,
Beijing 100871, China

###### Abstract

We study the hybrid mesons with the exotic quantum number
$I^{G}J^{PC}=0^{+}1^{-+}$ and investigate their decays into the
$\eta\eta^{\prime}$, $a_{1}(1260)\pi$, $f_{1}(1285)\eta$, $f_{1}(1420)\eta$,
$K^{*}(892)\bar{K}$, $K_{1}(1270)\bar{K}$, and $K_{1}(1400)\bar{K}$ channels.
We find that the QCD axial anomaly enhances the decay width of the
$\eta\eta^{\prime}$ channel although this mode is strongly suppressed by the
small $P$-wave phase space. Our results support the interpretation of the
$\eta_{1}(1855)$ recently observed by BESIII as the $\bar{s}sg$ hybrid meson
of $I^{G}J^{PC}=0^{+}1^{-+}$. The QCD axial anomaly ensures the
$\eta\eta^{\prime}$ decay mode to be a characteristic signal of the hybrid
nature of the $\eta_{1}(1855)$.

hybrid meson, QCD sum rules, QCD axial anomaly

##  I Introduction

Very recently, the BESIII collaboration performed a partial wave analysis of
the $J/\psi\to\gamma\eta\eta^{\prime}$ decay, and reported the first
observation of an abnormal state with the exotic quantum number
$I^{G}J^{PC}=0^{+}1^{-+}$ Ablikim:2022zze ; Ablikim:2022glj . It was observed
in the $\eta\eta^{\prime}$ invariant mass spectrum with a statistical
significance larger than $19\sigma$. Its mass and width were measured to be

$\displaystyle\eta_{1}(1855)$ $\displaystyle:$ $\displaystyle M=1855\pm
9^{+6}_{-1}{\rm~{}MeV}/c^{2}\,,$ $\displaystyle\Gamma=188\pm
18^{+3}_{-8}{\rm~{}MeV}\,.$

The $\eta_{1}(1855)$ is the isoscalar partner of the isovector state
$\pi_{1}(1600)$ E852:1998mbq  or $\pi_{1}(1400)$ IHEP-Brussels-LosAlamos-
AnnecyLAPP:1988iqi  with $I^{G}J^{PC}=1^{-}1^{-+}$. These states are of
particular interests, since their exotic quantum numbers can not be accessed
by conventional $\bar{q}q$ mesons and may arise from the gluon degree of
freedom pdg . One of their possible explanations is the hybrid meson composed
of one valence quark and one valence antiquark together with one valence
gluon, whose experimental confirmation is a direct test of QCD in the low
energy sector.

In the past half century there have been a lot of theoretical studies on
hybrid mesons, through the MIT bag model Barnes:1977hg ; Hasenfratz:1980jv ;
Chanowitz:1982qj , flux-tube model Isgur:1983wj ; Close:1994hc ; Page:1998gz ,
constituent gluon model Horn:1977rq ; Szczepaniak:2001rg ; Guo:2007sm ,
AdS/QCD model Andreev:2012hw ; Bellantuono:2014lra , lattice QCD
Michael:1985ne ; Juge:2002br ; Lacock:1996ny ; MILC:1997usn ; Dudek:2009qf ;
Dudek:2013yja , and QCD sum rules Balitsky:1982ps ; Govaerts:1983ka ;
Kisslinger:1995yw ; Jin:2002rw ; Narison:2009vj ; Li:2021fwk , etc. However,
their nature still remains elusive due to our poor understanding of the gluon
degree of freedom. It is not easy to experimentally identify hybrid mesons
unambiguously, and there is currently no definite experimental evidence on
their existence. It is also not easy to theoretically define the gluon degree
of freedom. A precise definition of the constituent gluon is still lacking,
although there have been some proposals to construct glueballs and hybrid
mesons using constituent gluons Horn:1977rq ; Coyne:1980zd ; Chanowitz:1980gu
; Barnes:1981ac ; Cornwall:1982zn ; Cho:2015rsa . We refer to the reviews pdg
; Klempt:2007cp ; Amsler:2004ps ; Bugg:2004xu ; Meyer:2010ku ; Meyer:2015eta ;
Chen:2016qju ; Briceno:2017max ; Ketzer:2019wmd ; Jin:2021vct ; Chen:2022asf
as well as the recent experimental analyses performed by the Compass and JPAC
collaborations COMPASS:2018uzl ; JPAC:2018zyd  for detailed discussions.

The $\eta_{1}(1855)$ was interpreted as a hybrid meson in a recent study
Qiu:2022ktc . There also exist some other possible explanations, e.g., the
$\eta_{1}(1855)$ may be explained as the
$K\bar{K}_{1}(1400)/K\bar{K}_{1}(1270)$ hadronic molecules in Refs.
Dong:2022cuw ; Zhang:2019ykd . In 2008 we systematically studied the
$qs\bar{q}\bar{s}$ ($q=u/d$) tetraquark state of $I^{G}J^{PC}=0^{+}1^{-+}$
using the QCD sum rule method, and we wrote in the abstract Chen:2008ne :“
$\cdots$ these currents lead to mass estimates around $1.8$-$2.1$ GeV, where
the uncertainty is due to the mixing of two single currents. Its possible
decay modes are $S$-wave $b_{1}(1235)\eta$ and $b_{1}(1235)\eta^{\prime}$, and
$P$-wave $KK$, $\eta\eta$, $\eta\eta^{\prime}$ and
$\eta^{\prime}\eta^{\prime}$, etc. The decay width is around 150 MeV through a
rough estimation.” Accordingly, the $\eta_{1}(1855)$ may also be interpreted
as a $qs\bar{q}\bar{s}$ tetraquark state. Note that we made two mistakes there
on its possible decay modes: a) the $b_{1}(1235)\eta$ and
$b_{1}(1235)\eta^{\prime}$ decay modes obtained by the Fierz rearrangement
should be replaced by the $a_{1}(1260)\pi$ and $f_{1}(1285)\eta$ decay modes;
b) the $\eta\eta$ and $\eta^{\prime}\eta^{\prime}$ decay modes are forbidden
due to the Bose-Einstein statistics.

(a) (a) (b) (b) (c) (c)

Figure 1: Possible decay mechanisms of the hybrid meson through (a) the normal
process with one quark-antiquark pair excited from the valence gluon as well
as the abnormal processes with the (b) $\eta^{\prime}$ and (c) $\eta$ mesons
produced by the QCD axial anomaly.

In 2010 we systematically studied decay properties of the $\bar{q}qg$
($q=u/d$) hybrid meson with $I^{G}J^{PC}=0^{+}1^{-+}$ using the methods of QCD
sum rules and light-cone sum rules, where we pointed out its
$\eta\eta^{\prime}$ decay mode Huang:2010dc ; Chen:2010ic . This is just the
discovery channel of the $\eta_{1}(1855)$ observed by BESIII Ablikim:2022zze ;
Ablikim:2022glj . However, the partial width of this decay mode was calculated
to be quite small in Ref. Chen:2010ic , where we only took into account the
normal decay process depicted in Fig. 1(a) with one quark-antiquark pair
excited from the valence gluon.

There exists a decades-long puzzle in the search of the hybrid mesons. Several
experimental groups reported the isovector states $\pi_{1}(1400)$ IHEP-
Brussels-LosAlamos-AnnecyLAPP:1988iqi  and $\pi_{1}(1600)$ E852:1998mbq  in
the $\eta\pi$ and $\eta^{\prime}\pi$ channels, respectively. However, a
“selection rule” was proposed in Refs. Page:1998gz ; Page:1996rj  through the
popular flux tube model that the $J^{PC}=1^{-+}$ hybrid meson does not decay
into two $S$-wave ground-state mesons. In other words, the $\eta\pi$ and
$\eta^{\prime}\pi$ modes are strictly forbidden in the flux tube model, which
is in strong contrast of the experimental fact that these two modes together
with the $\eta\eta^{\prime}$ are the discovery modes of $\pi_{1}(1400)$,
$\pi_{1}(1600)$, and $\eta_{1}(1885)$, respectively. The above selection rule
can be interpreted from the Feynman diagram depicted in Fig. 1(a) to some
extent, where the leading-order perturbative contribution vanishes in the
chiral limit. In our previous QCD sum rule calculation, the small $\eta\pi$
decay width arises from two chirality-violating sources: the $up/down$ current
quark mass and the non-perturbative QCD condensates.

In the present work, we shall further investigate the anomaly-assisted (or
“abnormal”) decay processes depicted in Fig. 1(b,c) with the $\eta^{(\prime)}$
mesons produced by the QCD axial anomaly. We shall find that these diagrams
enhance the $\eta\eta^{\prime}$ decay mode. We want to emphasize that this is
the characteristic decay mode of the hybrid mesons if we consider the QCD
axial anomaly 111A similar mechanism was investigated in [Phys. Lett. B 214,
463-466 (1988)] to study the decays of the $J^{PC}=1^{-+}$ hybrid mesons. We
thank Professor Jean-Marie Frere for informing us this work after our
manuscript was posted on the arXiv..

In this paper we shall also update our previous calculations of Ref.
Chen:2010ic  to investigate the $a_{1}(1260)\pi$, $f_{1}(1285)\eta$,
$f_{1}(1420)\eta$, $K^{*}(892)\bar{K}$, $K_{1}(1270)\bar{K}$, and
$K_{1}(1400)\bar{K}$ decay channels for both the $\bar{q}qg$ and $\bar{s}sg$
hybrid mesons of $I^{G}J^{PC}=0^{+}1^{-+}$. We shall find that the
$\eta_{1}(1855)$ recently observed by BESIII can be well explained as the
$\bar{s}sg$ hybrid meson. Moreover, the connection with the QCD axial anomaly
strongly suggests the hybrid nature of the $\eta_{1}(1855)$.

This paper is organized as follows. In Sec. II, we investigate the QCD axial
anomaly and study its contribution to the $\eta_{1}(1855)\to\eta\eta^{\prime}$
decay channel, based on the interpretation of the $\eta_{1}(1855)$ as the
$\bar{s}sg$ hybrid meson with $I^{G}J^{PC}=0^{+}1^{-+}$. In Sec. III we study
the $\eta_{1}(1855)$ decay into the $K^{*}(892)\bar{K}$ and
$K_{1}(1270)\bar{K}$ channels. The $a_{1}(1260)\pi$, $f_{1}(1285)\eta$,
$f_{1}(1420)\eta$, and $K_{1}(1400)\bar{K}$ decay channels are investigated in
Sec. IV for both the $\bar{s}sg$ and $\bar{q}qg$ hybrid mesons together with a
summary of the present study.

##  II The $\eta\eta^{\prime}$ decay channel

In this section we assume the $\eta_{1}(1855)$ to be the $\bar{s}sg$ hybrid
meson of $I^{G}J^{PC}=0^{+}1^{-+}$, and study its decay into the
$\eta\eta^{\prime}$ channel. We shall concentrate on the abnormal decay
process depicted in Fig. 1(b) with the $\eta^{\prime}$ meson produced by the
QCD axial anomaly. The other two decay processes depicted in Fig. 1(a,c) will
be shortly discussed at the end of this section.

###  II.1 QCD axial anomaly

The conversion of the gluons into the $\eta$ and $\eta^{\prime}$ mesons is
described by an amplitude relating to the $SU(3)$ symmetry and the QCD
$U(1)_{A}$ anomaly. Its matrix elements can be written as Voloshin:1980zf ;
Akhoury:1987ed ; Castoldi:1988dm ; Chao:1989yp ; Ball:1995zv ; Ali:1997ex :

$\displaystyle\langle 0|{\alpha_{s}\over
4\pi}G^{\alpha\beta}_{n}\tilde{G}_{n,\alpha\beta}|\eta\rangle$
$\displaystyle=$ $\displaystyle m^{2}_{\eta}f_{\eta}\,,$ (2)
$\displaystyle\langle 0|{\alpha_{s}\over
4\pi}G^{\alpha\beta}_{n}\tilde{G}_{n,\alpha\beta}|\eta^{\prime}\rangle$
$\displaystyle=$ $\displaystyle m^{2}_{\eta^{\prime}}f_{\eta^{\prime}}\,,$ (3)

where $G^{\alpha\beta}_{n}$ is the gluon field strength tensor and
$\tilde{G}_{n,\alpha\beta}=G_{n}^{\mu\nu}\times\epsilon_{\mu\nu\alpha\beta}/2$
is its dual field, with $n=1\cdots 8$ the color index and $\alpha,\beta$ the
Lorentz indices. The two decay constants $f_{\eta}$ and $f_{\eta^{\prime}}$
are defined as

$\displaystyle f_{\eta}$ $\displaystyle=$
$\displaystyle{f_{8}\over\sqrt{6}}\cos\theta_{8}-{f_{0}\over\sqrt{3}}\sin\theta_{0}\,,$
(4) $\displaystyle f_{\eta^{\prime}}$ $\displaystyle=$
$\displaystyle{f_{8}\over\sqrt{6}}\sin\theta_{8}+{f_{0}\over\sqrt{3}}\cos\theta_{0}\,.$

In the above expressions we have followed Refs. Leutwyler:1997yr ;
Kaiser:1998ds ; Escribano:2005qq ; Escribano:2015nra ; Escribano:2015yup ;
Schechter:1992iz ; Kiselev:1992ms ; Herrera-Siklody:1997pgy ; Bass:2018xmz ;
Bali:2021qem  and used the two-angle mixing formalism to describe the $\eta$
and $\eta^{\prime}$ mesons:

$\displaystyle|\eta\rangle$ $\displaystyle=$
$\displaystyle\cos\theta_{8}|\eta_{8}\rangle-\sin\theta_{0}|\eta_{0}\rangle+\cdots\,,$
(5) $\displaystyle|\eta^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\sin\theta_{8}|\eta_{8}\rangle+\cos\theta_{0}|\eta_{0}\rangle+\cdots\,,$

where

$\displaystyle|\eta_{8}\rangle$ $\displaystyle=$
$\displaystyle|u\bar{u}+d\bar{d}-2s\bar{s}\rangle/\sqrt{6}\,,$ (6)
$\displaystyle|\eta_{0}\rangle$ $\displaystyle=$
$\displaystyle|u\bar{u}+d\bar{d}+s\bar{s}\rangle/\sqrt{3}\,,$

and $\cdots$ are contributed by some other components such as the pseudoscalar
glueball and charmonium, etc.

The octet and singlet axial-vector currents mesons are defined as

$\displaystyle A_{\mu}^{8}$ $\displaystyle=$
$\displaystyle\left({\bar{u}\gamma_{\mu}\gamma_{5}u+\bar{d}\gamma_{\mu}\gamma_{5}d-2\bar{s}\gamma_{\mu}\gamma_{5}s}\right)/\sqrt{12}\,,$
(7) $\displaystyle A_{\mu}^{0}$ $\displaystyle=$
$\displaystyle\left({\bar{u}\gamma_{\mu}\gamma_{5}u+\bar{d}\gamma_{\mu}\gamma_{5}d+\bar{s}\gamma_{\mu}\gamma_{5}s}\right)/\sqrt{6}\,,$

which couple to the $\eta$ and $\eta^{\prime}$ mesons through

$\langle 0|A_{\mu}^{a}|P(k)\rangle=ik_{\mu}f_{P}^{a}\,,$ (8)

with the matrix for the decay constants $f_{P}^{a}$
($a=8,0;\,P=\eta,\eta^{\prime}$) defined as

$\left(\begin{array}[]{cc}f_{\eta}^{8}&f_{\eta}^{0}\\\
f_{\eta^{\prime}}^{8}&f_{\eta^{\prime}}^{0}\end{array}\right)=\left(\begin{array}[]{cc}f_{8}\cos\theta_{8}&-f_{0}\sin\theta_{0}\\\
f_{8}\sin\theta_{8}&f_{0}\cos\theta_{0}\end{array}\right)\,.$ (9)

To simply our calculations, we further construct the interpolating currents

$\displaystyle J_{\mu}^{\eta}=A_{\mu}^{8}+t_{\eta}A_{\mu}^{0}\,,$ (10)
$\displaystyle
J_{\mu}^{\eta\prime}=A_{\mu}^{8}+t_{\eta^{\prime}}A_{\mu}^{0}\,,$

which separately couple to the $\eta$ and $\eta^{\prime}$ mesons through

$\displaystyle\langle 0|J_{\mu}^{\eta}|\eta(k)\rangle$ $\displaystyle=$
$\displaystyle ik_{\mu}g_{\eta}\,,$ $\displaystyle\langle
0|J_{\mu}^{\eta\prime}|\eta^{\prime}(k)\rangle$ $\displaystyle=$
$\displaystyle ik_{\mu}g_{\eta^{\prime}}\,,$ (11) $\displaystyle\langle
0|J_{\mu}^{\eta}|\eta^{\prime}(k)\rangle$ $\displaystyle=$
$\displaystyle\langle 0|J_{\mu}^{\eta\prime}|\eta(k)\rangle=0\,,$

with the parameters

$\displaystyle g_{\eta}$ $\displaystyle=$ $\displaystyle
f_{\eta}^{8}-f_{\eta}^{0}f_{\eta^{\prime}}^{8}/f_{\eta^{\prime}}^{0}\,,$
$\displaystyle g_{\eta^{\prime}}$ $\displaystyle=$ $\displaystyle
f_{\eta^{\prime}}^{8}-f_{\eta^{\prime}}^{0}f_{\eta}^{8}/f_{\eta}^{0}\,,$ (12)
$\displaystyle t_{\eta}$ $\displaystyle=$ $\displaystyle-
f_{\eta^{\prime}}^{8}/f_{\eta^{\prime}}^{0}\,,$ $\displaystyle
t_{\eta^{\prime}}$ $\displaystyle=$ $\displaystyle-
f_{\eta}^{8}/f_{\eta}^{0}\,.$

The following values will be used for the parameters contained in the above
equations Ali:1997ex ; Feldmann:1997vc :

$\displaystyle\theta_{8}$ $\displaystyle=$ $\displaystyle-22.2^{\circ}\,,$
$\displaystyle\theta_{0}$ $\displaystyle=$ $\displaystyle-9.1^{\circ}\,,$
$\displaystyle f_{8}$ $\displaystyle=$ $\displaystyle 168{\rm~{}MeV}\,,$ (13)
$\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle 157{\rm~{}MeV}\,.$

At the same time we can reorganize the currents $A_{\mu}^{8}$, $A_{\mu}^{0}$,
$J_{\mu}^{\eta}$, and $J_{\mu}^{\eta\prime}$ to be

$\displaystyle J_{\mu}^{\bar{q}q}$ $\displaystyle=$
$\displaystyle\left({\bar{u}\gamma_{\mu}\gamma_{5}u+\bar{d}\gamma_{\mu}\gamma_{5}d}\right)/\sqrt{2}\,,$
(14) $\displaystyle J_{\mu}^{\bar{s}s}$ $\displaystyle=$
$\displaystyle\bar{s}\gamma_{\mu}\gamma_{5}s\,.$

They can also couple to the $f_{1}(1285)$ and $f_{1}(1420)$ through

$\displaystyle\langle 0|J_{\mu}^{\bar{q}q}|f_{1}(1285)\rangle$
$\displaystyle=$ $\displaystyle
f_{f_{1}(1285)}m_{f_{1}(1285)}\epsilon_{\mu}\,,$ (15) $\displaystyle\langle
0|J_{\mu}^{\bar{s}s}|f_{1}(1420)\rangle$ $\displaystyle=$ $\displaystyle
f_{f_{1}(1420)}m_{f_{1}(1420)}\epsilon_{\mu}\,,$

where the $f_{1}(1285)$ and $f_{1}(1420)$ mesons are assumed to be
predominated by their $\bar{q}q$ and $\bar{s}s$ components pdg ; Yang:2007zt .
The decay constants $f_{f_{1}(1285)}$ and $f_{f_{1}(1420)}$ were evaluated in
Ref. Yang:2007zt  to be:

$\displaystyle f_{f_{1}(1285)}$ $\displaystyle=$ $\displaystyle 173\pm
23{\rm~{}MeV}\,,$ (16) $\displaystyle f_{f_{1}(1420)}$ $\displaystyle=$
$\displaystyle 217\pm 27{\rm~{}MeV}\,.$

###  II.2 Three-point correlation function

The QCD sum rule method has proven to be a powerful and successful non-
perturbative method for the past decades Shifman:1978bx ; Reinders:1984sr ,
and we applied it to study three-gluon glueballs and double-gluon hybrid
mesons recently Chen:2021cjr ; Chen:2021bck ; Chen:2021smz . We consider the
following three-point correlation function to study the decay process depicted
in Fig. 1(b):

$T^{b}_{\mu\nu}(p,k,q)=\int d^{4}xe^{-ikx}\langle
0|{\mathbb{T}}J_{\mu}^{\eta_{1}}(0)J_{\nu}^{\eta\dagger}(x)|\eta^{\prime}(q)\rangle\,,$
(17)

where $p$, $k$, and $q$ are the momenta of the $\eta_{1}(1855)$, $\eta$, and
$\eta^{\prime}$, respectively. We also need Eq. (3) to relate $\eta^{\prime}$
to the QCD axial anomaly.

The $\bar{s}sg$ hybrid current $J_{\mu}^{\eta_{1}}$ of
$I^{G}J^{PC}=0^{+}1^{-+}$ is defined as:

$J^{\eta_{1}}_{\mu}=\bar{s}_{a}\gamma^{\nu}s_{b}{\lambda^{ab}_{n}\over
2}g_{s}G^{n}_{\mu\nu}\,,$ (18)

which couples to the $\eta_{1}(1855)$ through

$\langle
0|J^{\eta_{1}}_{\mu}|\eta_{1}(\epsilon,p)\rangle=\sqrt{2}f_{\eta_{1}}m_{\eta_{1}}^{3}\epsilon_{\mu}\,.$
(19)

We choose the value of $f_{\eta_{1}}$ to be the same as that of the
$\bar{q}qg$ hybrid meson Narison:1989aq ; Zhu:1999wg ; Zhu:1998bm :

$f_{\eta_{1}}\approx 0.026{\rm~{}GeV}\,,$ (20)

with the uncertainty $\pm 15\%$.

At the hadronic level we express Eq. (17) as

$\displaystyle T^{b}_{\mu\nu}(p,k,q)$ $\displaystyle=$ $\displaystyle
g^{b}_{\eta\eta^{\prime}}k_{\mu}k_{\nu}~{}{\sqrt{2}f_{\eta_{1}}m_{\eta_{1}}^{3}g_{\eta}\over(m_{\eta_{1}}^{2}-p^{2})(m_{\eta}^{2}-k^{2})}$
$\displaystyle+$ $\displaystyle
g_{f_{1}\eta}\theta_{s}\left(g_{\mu\nu}-{k_{\mu}k_{\nu}\over
m_{f_{1}}^{2}}\right){\sqrt{2}f_{\eta_{1}}m_{\eta_{1}}^{3}f_{f_{1}}m_{f_{1}}\over(m_{\eta_{1}}^{2}-p^{2})(m_{f_{1}}^{2}-k^{2})}+\cdots,$

where $\theta_{s}=-{1/\sqrt{3}}+{t_{\eta}/\sqrt{6}}$ describes the $s\bar{s}$
component contained in the current $J_{\nu}^{\eta}$. The coupling constants
$g^{b}_{\eta\eta^{\prime}}$ and $g_{f_{1}\eta}$ are defined through the
Lagrangians

$\displaystyle\mathcal{L}^{b}_{\eta\eta^{\prime}}$ $\displaystyle=$
$\displaystyle
ig^{b}_{\eta\eta^{\prime}}\eta_{1}^{\mu}(\partial_{\mu}\eta)\eta^{\prime}\,,$
(22) $\displaystyle\mathcal{L}_{f_{1}\eta}$ $\displaystyle=$ $\displaystyle
g_{f_{1}\eta}\eta_{1}^{\mu}f_{1\mu}\eta\,.$

At the quark-gluon level we calculate Eq. (17) using the method of operator
product expansion (OPE). Based on Eq. (3), we calculate the decay process
depicted in Fig. 1(b) to be

$\displaystyle T^{b}_{\mu\nu}(p,k,q)$ $\displaystyle=$
$\displaystyle\theta_{s}k_{\mu}k_{\nu}~{}\Big{(}-{2m^{2}_{\eta^{\prime}}f_{\eta^{\prime}}\over
3k^{2}}-{4\pi^{2}m^{2}_{\eta^{\prime}}f_{\eta^{\prime}}m_{s}\langle\bar{s}s\rangle\over
3k^{6}}\Big{)}$ $\displaystyle+$
$\displaystyle\theta_{s}\left(g_{\mu\nu}-{k_{\mu}k_{\nu}\over
k^{2}}\right){4\pi^{2}m^{2}_{\eta^{\prime}}f_{\eta^{\prime}}m_{s}\langle\bar{s}s\rangle\over
9k^{4}}+\cdots.$

Paying attention to the Lorentz structure $k_{\mu}k_{\nu}$, we perform the
Borel transformation to Eq. (II.2) and Eq. (II.2) at the limit of $p^{2}\sim
k^{2}\rightarrow\infty$, and obtain:

$\displaystyle
g^{b}_{\eta\eta^{\prime}}{\sqrt{2}m_{\eta_{1}}^{3}f_{\eta_{1}}g_{\eta}\over
m_{\eta_{1}}^{2}-m_{\eta}^{2}}\left(e^{-m_{\eta}^{2}/M_{B}^{2}}-e^{-m_{\eta_{1}}^{2}/M_{B}^{2}}\right)$
(24) $\displaystyle=$
$\displaystyle{2\theta_{s}m^{2}_{\eta^{\prime}}f_{\eta^{\prime}}\over
3}+{2\pi^{2}\theta_{s}m^{2}_{\eta^{\prime}}f_{\eta^{\prime}}m_{s}\langle\bar{s}s\rangle\over
3}{1\over M_{B}^{4}}\,.$

There are only two terms left, and the latter one proportional to $m_{s}$ is
negligible compared to the former one.

###  II.3 Numerical analysis

In the present study we use the following values for various quark and gluon
parameters when performing numerical analyses pdg ; Ovchinnikov:1988gk ;
Yang:1993bp ; Ellis:1996xc ; Jamin:2002ev ; Ioffe:2002be ; Gimenez:2005nt ;
Narison:2011xe ; Narison:2018dcr :

$\displaystyle m_{s}(1\mbox{ GeV})$ $\displaystyle=$
$\displaystyle\left(93^{+11}_{-~{}5}\right)\times 1.35\mbox{ MeV}\,,$
$\displaystyle\langle\bar{q}q\rangle$ $\displaystyle=$ $\displaystyle-(0.24\pm
0.01)^{3}\mbox{ GeV}^{3}\,,$ $\displaystyle\langle\bar{s}s\rangle$
$\displaystyle=$ $\displaystyle(0.8\pm 0.1)\times\langle\bar{q}q\rangle\,,$
$\displaystyle\langle g_{s}\bar{q}\sigma Gq\rangle$ $\displaystyle=$
$\displaystyle-M_{0}^{2}\times\langle\bar{q}q\rangle\,,$ (25)
$\displaystyle\langle g_{s}\bar{s}\sigma Gs\rangle$ $\displaystyle=$
$\displaystyle-M_{0}^{2}\times\langle\bar{s}s\rangle\,,$ $\displaystyle
M_{0}^{2}$ $\displaystyle=$ $\displaystyle(0.8\pm 0.2)\mbox{ GeV}^{2}\,,$
$\displaystyle\langle\alpha_{s}GG\rangle$ $\displaystyle=$
$\displaystyle(6.35\pm 0.35)\times 10^{-2}\mbox{ GeV}^{4}\,,$
$\displaystyle\langle g_{s}^{3}G^{3}\rangle$ $\displaystyle=$
$\displaystyle\langle\alpha_{s}GG\rangle\times(8.2\pm 1.0)\mbox{ GeV}^{2}\,.$

Together with Eqs. (13) and (20), we calculate the coupling constant
$g^{b}_{\eta\eta^{\prime}}$ through Eq. (24). We show it in Fig. 2 as a
function of the Borel mass $M_{B}$. We obtain inside the Borel window
$M_{B}^{2}=1.5\pm 0.5$ GeV2 that

$|g^{b}_{\eta\eta^{\prime}}|=1.78\,.$ (26)

The above coupling constant is non-negligible. We further derive

$\Gamma(\eta_{1}(1855)\xrightarrow{b}\eta\eta^{\prime})=1.80^{+0.72}_{-0.44}{\rm~{}MeV}\,.$
(27)

Its uncertainty is due to the Borel mass and various parameters given in Eqs.
(13), (20), and (25). Because some of the parameters do not have
uncertainties, the uncertainties of our results should be eventually larger,
i.e., to be two times larger or smaller.

Figure 2: The coupling constant $g^{b}_{\eta\eta^{\prime}}$ as a function of
the Borel mass $M_{B}^{2}$, calculated using Eq. (24).

###  II.4 Other processes

Similarly, we investigate the abnormal decay process depicted in Fig. 1(c)
through the three-point correlation function

$T^{c}_{\mu\nu}(p,k,q)=\int d^{4}xe^{-ikx}\langle
0|{\mathbb{T}}J_{\mu}^{\eta_{1}}(0)J_{\nu}^{\eta^{\prime}\dagger}(x)|\eta(q)\rangle\,.$
(28)

In this case the $\eta$ meson is produced by the QCD axial anomaly through Eq.
(2), and the coupling constant $g^{c}_{\eta\eta^{\prime}}$ is defined through
the Lagrangian

$\mathcal{L}^{c}_{\eta\eta^{\prime}}=ig^{c}_{\eta\eta^{\prime}}\eta_{1}^{\mu}(\partial_{\mu}\eta^{\prime})\eta\,.$
(29)

We follow the same procedures to obtain

$|g^{c}_{\eta\eta^{\prime}}|=0.53\,,$ (30)

which is used to further derive

$\Gamma(\eta_{1}(1855)\xrightarrow{c}\eta\eta^{\prime})=0.74^{+0.42}_{-0.18}{\rm~{}MeV}\,.$
(31)

This value is at the same level as that given in Eq. (27). However, we are not
able to exactly derive the phase angle between their amplitudes, so we shall
simply sum over these partial decay widths in Sec. IV.

The normal decay process depicted in Fig. 1(a), with one quark-antiquark pair
excited from the valence gluon, has been studied in Ref. Chen:2010ic  for the
$\bar{q}qg$ hybrid meson. We replace the light $up/down$ quarks by the
$strange$ quarks, and evaluate its partial decay width to be

$\Gamma(\eta_{1}(1855)\xrightarrow{a}\eta\eta^{\prime})=0.68^{+2.26}_{-0.36}{\rm~{}MeV}\,.$
(32)

This value is smaller than that given in Eq. (27), so the abnormal decay
process depicted in Fig. 1(b) with the $\eta^{\prime}$ meson produced by the
QCD axial anomaly does enhance the $\eta\eta^{\prime}$ decay mode. Note that
this value 0.68 MeV is larger than that obtained in Ref. Chen:2010ic , because
the gluon condensates $\langle\alpha_{s}GG\rangle$ and $\langle
g_{s}^{3}G^{3}\rangle$ used here are larger than those used there.

Besides the Lorentz structure $k_{\mu}k_{\nu}$ investigated above, the three-
point correlation functions $T^{b}_{\mu\nu}(p,k,q)$ given in Eq. (II.2) and
Eq. (II.2) also contain the Lorentz structure
$\left(g_{\mu\nu}-{k_{\mu}k_{\nu}/k^{2}}\right)$. These components can be used
to study the contribution of the QCD axial anomaly to the $f_{1}\eta$ decay
channel. Assuming the mass of $\eta_{1}$ to be around 2100 MeV, its decay into
the $f_{1}(1420)\eta$ channel is kinematically allowed. However, the partial
decay width of this mode is evaluated to be quite small, so we shall not
discuss it any more.

##  III Other decay channels

###  III.1 The $K^{*}\bar{K}$ decay channel

In this subsection we assume the $\eta_{1}(1855)$ to be the $\bar{s}sg$ hybrid
meson of $I^{G}J^{PC}=0^{+}1^{-+}$, and study its decay into the
$K^{*}(892)\bar{K}$ channel through the normal decay process depicted in Fig.
1(a). We follow the same procedures as those used in Ref. Chen:2010ic  for the
$\pi_{1}\to\rho\pi$ decay, and study the three-point correlation function

$\displaystyle T^{K^{*}\bar{K}}_{\mu\nu}(p,k,q)$ $\displaystyle=$
$\displaystyle\int d^{4}xd^{4}ye^{ikx}e^{iqy}\langle
0|{\mathbb{T}}J^{K^{*}}_{\nu}(x)J^{\bar{K}}(y)J_{\mu}^{\eta_{1}\dagger}(0)|0\rangle\,.$

The currents $J^{K^{*}}_{\nu}$ and $J^{\bar{K}}$ are defined as

$\displaystyle J^{K^{*}}_{\nu}$ $\displaystyle=$
$\displaystyle\bar{q}\gamma_{\nu}s\,,$ (34) $\displaystyle J^{\bar{K}}$
$\displaystyle=$ $\displaystyle\bar{s}\gamma_{5}q\,,$

which couple to the $K^{*}(892)$ and $\bar{K}$ mesons through

$\displaystyle\langle 0|J^{K^{*}}_{\nu}|K^{*}(\epsilon,k)\rangle$
$\displaystyle=$ $\displaystyle f_{K^{*}}m_{K^{*}}\epsilon_{\nu}\,,$ (35)
$\displaystyle\langle 0|J^{\bar{K}}|\bar{K}(q)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{K}\,.$

We use the following values for the parameters in the above equations pdg ;
Becirevic:2003pn :

$\displaystyle f_{K^{*}}$ $\displaystyle=$ $\displaystyle 217{\rm~{}MeV}\,,$
$\displaystyle\lambda_{K}$ $\displaystyle=$
$\displaystyle{i(\langle\bar{q}q\rangle+\langle\bar{s}s\rangle)/f_{K}}\,,$
(36) $\displaystyle f_{K^{+}}$ $\displaystyle=$ $\displaystyle
155.7{\rm~{}MeV}\,,$

and obtain

$\Gamma(\eta_{1}(1855)\to
K^{*}(892)\bar{K}+c.c.)=98.1^{+82.9}_{-60.4}{\rm~{}MeV}\,.$ (37)

###  III.2 The $K_{1}(1270)\bar{K}$ decay channel

In this subsection we assume the $\eta_{1}(1855)$ to be the $\bar{s}sg$ hybrid
meson of $I^{G}J^{PC}=0^{+}1^{-+}$, and study its decay into the
$K_{1}(1270)\bar{K}$ channel through the normal decay process depicted in Fig.
1(a). We follow the same procedures as those used in Ref. Chen:2010ic  for the
$\pi_{1}\to f_{1}\pi$ decay, and study the three-point correlation function

$\displaystyle T^{K_{1}\bar{K}}_{\mu\nu}(p,k,q)$ $\displaystyle=$
$\displaystyle\int d^{4}xd^{4}ye^{ikx}e^{iqy}\langle
0|{\mathbb{T}}J^{K_{1}}_{\nu}(x)J^{\bar{K}}(y)J_{\mu}^{\eta_{1}\dagger}(0)|0\rangle\,.$

The current $J^{K_{1}}_{\nu}$ is defined as

$J^{K_{1}}_{\nu}=\bar{q}\gamma_{\nu}\gamma_{5}s\,,$ (39)

which couples to both the $K_{1}\equiv K_{1}(1270)$ and $K_{1}^{\prime}\equiv
K_{1}(1400)$ mesons through

$\displaystyle\langle 0|J^{K_{1}}_{\nu}|K_{1}(\epsilon,k)\rangle$
$\displaystyle=$ $\displaystyle f_{K_{1}}m_{K_{1}}\epsilon_{\nu}\,,$ (40)
$\displaystyle\langle 0|J^{K_{1}}_{\nu}|K_{1}^{\prime}(\epsilon,k)\rangle$
$\displaystyle=$ $\displaystyle
f_{K_{1}^{\prime}}m_{K_{1}^{\prime}}\epsilon_{\nu}\,.$

We use the following values for the parameters in the above equations pdg ;
Yang:2007zt :

$\displaystyle f_{K_{1}}$ $\displaystyle=$ $\displaystyle 197\pm
15{\rm~{}MeV}\,,$ (41) $\displaystyle f_{K_{1}^{\prime}}$ $\displaystyle=$
$\displaystyle 217\pm 27{\rm~{}MeV}\,,$

and obtain

$\Gamma(\eta_{1}(1855)\to
K_{1}(1270)\bar{K}+c.c.)=30.4^{+40.1}_{-19.3}{\rm~{}MeV}\,.$ (42)

##  IV Summary and Discussions

Table 1: Partial decay widths of the $\bar{q}qg$ ($q=u/d$) and $\bar{s}sg$ hybrid mesons with $I^{G}J^{PC}=0^{+}1^{-+}$. $\Gamma(\eta_{1}(1855)\xrightarrow{a,b,c}\eta\eta^{\prime})$ denote the partial decay widths of the processes depicted in Fig. 1(a,b,c), respectively. Masses and widths are in units of MeV. Channel | Scheme-I | Scheme-II
---|---|---
$\bar{q}qg$ with $M\approx 1600$ |  $\bar{s}sg$ with $M=1855$ |  $\bar{q}qg$ with $M=1855$ |  $\bar{s}sg$ with $M\approx 2100$
$\eta_{1}\xrightarrow{a}\eta\eta^{\prime}$ | $0.02^{+0.04}_{-0.02}$ | $0.7^{+2.3}_{-0.4}$ | $0.2^{+0.5}_{-0.1}$ | $2.5^{+1.8}_{-2.0}$
$\eta_{1}\xrightarrow{b}\eta\eta^{\prime}$ | $2.3^{+3.2}_{-2.4}$ | $1.8^{+0.7}_{-0.4}$ | $9.0^{+3.6}_{-2.2}$ | $2.5^{+1.0}_{-0.7}$
$\eta_{1}\xrightarrow{c}\eta\eta^{\prime}$ | $0.4^{+0.5}_{-0.4}$ | $0.7^{+0.4}_{-0.2}$ | $1.6^{+0.9}_{-0.4}$ | $1.1^{+0.7}_{-0.3}$
$\eta_{1}\rightarrow a_{1}(1260)\pi$ | $50.3^{+62.9}_{-40.6}$ | – | $145^{+182}_{-~{}88}$ | –
$\eta_{1}\rightarrow f_{1}(1285)\eta$ | – | – | $11.8^{+17.1}_{-~{}8.3}$ | –
$\eta_{1}\rightarrow f_{1}(1420)\eta$ | – | – | – | $24.4^{+55.2}_{-23.8}$
$\eta_{1}\rightarrow K^{*}(892)\bar{K}+c.c.$ | $25.6^{+25.3}_{-20.7}$ | $98.1^{+82.9}_{-60.4}$ | $63.0^{+51.7}_{-37.0}$ | $154^{+137.3}_{-~{}97.7}$
$\eta_{1}\rightarrow K_{1}(1270)\bar{K}+c.c.$ | – | $30.4^{+40.1}_{-19.3}$ | $20.2^{+26.0}_{-12.2}$ | $76.7^{+120.7}_{-~{}53.8}$
$\eta_{1}\rightarrow K_{1}(1400)\bar{K}+c.c.$ | – | – | – | $79.1^{+167.1}_{-~{}63.0}$
Sum | $79^{+92}_{-64}$ | $132^{+126}_{-~{}81}$ | $251^{+282}_{-148}$ | $340^{+484}_{-241}$

In this paper we apply the method of QCD sum rules to study the
$\eta_{1}(1855)$ recently observed by BESIII as the $\bar{s}sg$ hybrid meson
of $I^{G}J^{PC}=0^{+}1^{-+}$, and calculate its partial decay widths into the
$\eta\eta^{\prime}$, $K^{*}(892)\bar{K}$, and $K_{1}(1270)\bar{K}$ channels.
We have considered the normal decay process depicted in Fig. 1(a) with one
quark-antiquark pair excited from the valence gluon. We have also considered
the abnormal decay processes depicted in Fig. 1(b,c) with the
$\eta^{(\prime)}$ mesons produced by the QCD axial anomaly. The results are

$\displaystyle\Gamma(\eta_{1}(1855)\xrightarrow{a}\eta\eta^{\prime})$
$\displaystyle=$ $\displaystyle 0.7^{+2.3}_{-0.4}{\rm~{}MeV}\,,$
$\displaystyle\Gamma(\eta_{1}(1855)\xrightarrow{b}\eta\eta^{\prime})$
$\displaystyle=$ $\displaystyle 1.8^{+0.7}_{-0.4}{\rm~{}MeV}\,,$
$\displaystyle\Gamma(\eta_{1}(1855)\xrightarrow{c}\eta\eta^{\prime})$
$\displaystyle=$ $\displaystyle 0.7^{+0.4}_{-0.2}{\rm~{}MeV}\,,$
$\displaystyle\Gamma(\eta_{1}(1855)\to K^{*}(892)\bar{K}+c.c.)$
$\displaystyle=$ $\displaystyle 98.1^{+82.9}_{-60.4}{\rm~{}MeV}\,,$
$\displaystyle\Gamma(\eta_{1}(1855)\to K_{1}(1270)\bar{K}+c.c.)$
$\displaystyle=$ $\displaystyle 30.4^{+40.1}_{-19.3}{\rm~{}MeV}\,,$

where $\Gamma(\eta_{1}(1855)\xrightarrow{a,b,c}\eta\eta^{\prime})$ denote the
partial decay widths of the processes depicted in Fig. 1(a,b,c), respectively.
We simply sum over them to obtain $\Gamma_{\rm sum}\approx
132^{+126}_{-~{}81}$ MeV. Considering that there still exist some other
possible decay channels, our results support the interpretation of the
$\eta_{1}(1855)$ as the $\bar{s}sg$ hybrid meson of $I^{G}J^{PC}=0^{+}1^{-+}$.
Moreover, the decay processes depicted in Fig. 1(b,c) with the
$\eta^{(\prime)}$ mesons produced by the QCD axial anomaly enhance the
$\eta\eta^{\prime}$ decay mode, and this possible relationship to the QCD
axial anomaly implies the hybrid nature of the $\eta_{1}(1855)$.

It is interesting to compare our results to those of Ref. Dong:2022cuw . In
this paper we explain the $\eta_{1}(1855)$ as the $\bar{s}sg$ hybrid meson of
$I^{G}J^{PC}=0^{+}1^{-+}$, and its partial decay width into the
$K^{*}(892)\bar{K}$ channel was estimated to be around 98.1 MeV. In Ref.
Dong:2022cuw  the authors explained the $\eta_{1}(1855)$ as the
$K\bar{K}_{1}(1400)$ hadronic molecule, and this partial decay width was
estimated to be only around 0.9 MeV. This significant difference may be useful
when examining the nature of the $\eta_{1}(1855)$.

Based on the interpretation of the $\eta_{1}(1855)$ as the $\bar{s}sg$ hybrid
meson of $I^{G}J^{PC}=0^{+}1^{-+}$, we have also investigated its partner
state, that is the $\bar{q}qg$ ($q=u/d$) hybrid meson of
$I^{G}J^{PC}=0^{+}1^{-+}$. Assuming its mass to be around $1600\pm 100$ MeV,
we calculate its partial decay widths into the $\eta\eta^{\prime}$ and
$a_{1}(1260)\pi$ channels. The obtained results are summarized in Table 1, and
this is our first scheme, labelled “Scheme-I”.

Besides, we have considered another scheme, labelled “Scheme-II”, where the
$\eta_{1}(1855)$ is interpreted as the $\bar{q}qg$ hybrid meson of
$I^{G}J^{PC}=0^{+}1^{-+}$. In this case its partner state is the $\bar{s}sg$
hybrid meson of $I^{G}J^{PC}=0^{+}1^{-+}$. Assuming their masses to be around
$1855\pm 9^{+6}_{-1}$ MeV and $2100\pm 100$ MeV, we calculate their partial
decay widths into the $\eta\eta^{\prime}$, $a_{1}(1260)\pi$,
$f_{1}(1285)\eta$, $f_{1}(1420)\eta$, $K^{*}(892)\bar{K}$,
$K_{1}(1270)\bar{K}$, and $K_{1}(1400)\bar{K}$ channels. During the
calculations we have used $f_{a_{1}(1260)}=238\pm 10$ MeV Yang:2007zt . The
obtained results are summarized in Table 1. It seems also possible to
interpret the $\eta_{1}(1855)$ as the $\bar{q}qg$ ($q=u/d$) hybrid meson of
$I^{G}J^{PC}=0^{+}1^{-+}$, but this explanation is less favored considering:
a) its total width is already larger than 250 MeV and there still exist some
other possible decay channels not investigated, and b) the $\pi_{1}(1600)$ is
a good isovector hybrid candidate whose isoscalar partner may have a similar
mass.

To differentiate the above two schemes, and more important, to verify whether
the $\eta_{1}(1855)$ is a hybrid meson or not, we propose to further search
the other decay modes and its partner states in future BESIII, Belle-II,
GlueX, LHC, and PANDA experiments.

## Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grants No. 12075019, No. 11975033, and No. 12070131001, the Jiangsu
Provincial Double-Innovation Program under Grant No. JSSCRC2021488, and the
Fundamental Research Funds for the Central Universities.

## References

  * (1) M. Ablikim et al. [BESIII Collaboration], Observation of an isoscalar resonance with exotic $J^{PC}=1^{-+}$ quantum numbers in $J/\psi\rightarrow\gamma\eta\eta^{\prime}$, [arXiv:2202.00621 [hep-ex]].
  * (2) M. Ablikim et al. [BESIII Collaboration], Partial wave analysis of $J/\psi\rightarrow\gamma\eta\eta^{\prime}$, [arXiv:2202.00623 [hep-ex]].
  * (3) G. S. Adams et al. [E852 Collaboration], Observation of a New $J^{PC}=1^{-+}$ Exotic State in the Reaction $\pi^{-}p\to\pi^{+}\pi^{-}\pi^{-}p$ at 18 GeV$/c$, Phys. Rev. Lett. 81, 5760-5763 (1998).
  * (4) D. Alde et al. [IHEP-Brussels-Los Alamos-Annecy(LAPP) Collaboration], Evidence for a $1^{-+}$ exotic meson, Phys. Lett. B 205, 397 (1988).
  * (5) P. A. Zyla et al. [Particle Data Group], Review of Particle Physics, PTEP 2020, no.8, 083C01 (2020).
  * (6) T. Barnes, Coloured quark and gluon constituents in the MIT bag model: A model of mesons, Nucl. Phys. B 158, 171-188 (1979).
  * (7) P. Hasenfratz, R. R. Horgan, J. Kuti and J. M. Richard, The effects of coloured glue in the QCD motivated bag of heavy quark-antiquark systems, Phys. Lett. B 95, 299-305 (1980).
  * (8) M. S. Chanowitz and S. R. Sharpe, Hybrids: Mixed states of quarks and gluons, Nucl. Phys. B 222, 211-244 (1983), [erratum: Nucl. Phys. B 228, 588-588 (1983)].
  * (9) N. Isgur and J. E. Paton, A flux tube model for hadrons, Phys. Lett. B 124, 247-251 (1983).
  * (10) F. E. Close and P. R. Page, The production and decay of hybrid mesons by flux-tube breaking, Nucl. Phys. B 443, 233-254 (1995).
  * (11) P. R. Page, E. S. Swanson and A. P. Szczepaniak, Hybrid meson decay phenomenology, Phys. Rev. D 59, 034016 (1999).
  * (12) D. Horn and J. Mandula, Model of mesons with constituent gluons, Phys. Rev. D 17, 898 (1978).
  * (13) A. P. Szczepaniak and E. S. Swanson, Coulomb gauge QCD, confinement, and the constituent representation, Phys. Rev. D 65, 025012 (2001).
  * (14) P. Guo, A. P. Szczepaniak, G. Galata, A. Vassallo and E. Santopinto, Gluelump spectrum from Coulomb gauge QCD, Phys. Rev. D 77, 056005 (2008).
  * (15) O. Andreev, Exotic hybrid pseudopotentials and gauge/string duality, Phys. Rev. D 87, no.6, 065006 (2013).
  * (16) L. Bellantuono, P. Colangelo and F. Giannuzzi, Exotic $J^{PC}=1^{-+}$ mesons in a holographic model of QCD, Eur. Phys. J. C 74, no.4, 2830 (2014).
  * (17) C. Michael, Adjoint sources in lattice gauge theory, Nucl. Phys. B 259, 58-76 (1985).
  * (18) P. Lacock et al. [UKQCD Collaboration], Hybrid mesons from quenched QCD, Phys. Lett. B 401, 308-312 (1997).
  * (19) C. W. Bernard et al. [MILC Collaboration], Exotic mesons in quenched lattice QCD, Phys. Rev. D 56, 7039-7051 (1997).
  * (20) K. J. Juge, J. Kuti and C. Morningstar, Fine Structure of the QCD String Spectrum, Phys. Rev. Lett. 90, 161601 (2003).
  * (21) J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards and C. E. Thomas, Highly Excited and Exotic Meson Spectrum from Dynamical Lattice QCD, Phys. Rev. Lett. 103, 262001 (2009).
  * (22) J. J. Dudek et al. [Hadron Spectrum Collaboration], Toward the excited isoscalar meson spectrum from lattice QCD, Phys. Rev. D 88, no.9, 094505 (2013).
  * (23) I. I. Balitsky, D. Diakonov and A. V. Yung, Exotic mesons with $J^{PC}=1^{-+}$ from QCD sum rules, Phys. Lett. B 112, 71-75 (1982).
  * (24) J. Govaerts, F. de Viron, D. Gusbin and J. Weyers, Exotic mesons from QCD sum rules, Phys. Lett. B 128, 262 (1983), [erratum: Phys. Lett. B 136, 445-445 (1984)].
  * (25) L. S. Kisslinger and Z. P. Li, Hybrid baryons via QCD sum rules, Phys. Rev. D 51, R5986-R5989 (1995).
  * (26) H. Y. Jin, J. G. Korner and T. G. Steele, Improved determination of the mass of the $1^{-+}$ light hybrid meson from QCD sum rules, Phys. Rev. D 67, 014025 (2003).
  * (27) S. Narison, $1^{-+}$ light exotic mesons in QCD, Phys. Lett. B 675, 319-325 (2009).
  * (28) S. H. Li, Z. S. Chen, H. Y. Jin and W. Chen, Mass of $1^{-+}$ fourqurk-hyrbird mixed states, [arXiv:2111.13897 [hep-ph]].
  * (29) J. J. Coyne, P. M. Fishbane and S. Meshkov, Glueballs: Their spectra, production and decay, Phys. Lett. B 91, 259 (1980).
  * (30) M. S. Chanowitz, Have We Seen Our First Glueball? Phys. Rev. Lett. 46, 981 (1981).
  * (31) T. Barnes, A transverse gluonium potential model with Breit-fermi hyperfine effects, Z. Phys. C 10, 275 (1981).
  * (32) J. M. Cornwall and A. Soni, Glueballs as bound states of massive gluons, Phys. Lett. B 120, 431 (1983).
  * (33) Y. M. Cho, X. Y. Pham, P. Zhang, J. J. Xie and L. P. Zou, Glueball physics in QCD, Phys. Rev. D 91, no.11, 114020 (2015).
  * (34) E. Klempt and A. Zaitsev, Glueballs, hybrids, multiquarks: Experimental facts versus QCD inspired concepts, Phys. Rept. 454, 1-202 (2007).
  * (35) C. Amsler and N. A. Tornqvist, Mesons beyond the naive quark model, Phys. Rept. 389, 61-117 (2004).
  * (36) D. V. Bugg, Four sorts of meson, Phys. Rept. 397, 257-358 (2004).
  * (37) C. A. Meyer and Y. Van Haarlem, Status of exotic-quantum-number mesons, Phys. Rev. C 82, 025208 (2010).
  * (38) C. A. Meyer and E. S. Swanson, Hybrid mesons, Prog. Part. Nucl. Phys. 82, 21-58 (2015).
  * (39) H. X. Chen, W. Chen, X. Liu and S. L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639, 1-121 (2016).
  * (40) R. A. Briceno, J. J. Dudek and R. D. Young, Scattering processes and resonances from lattice QCD, Rev. Mod. Phys. 90, no.2, 025001 (2018).
  * (41) B. Ketzer, B. Grube and D. Ryabchikov, Light-meson spectroscopy with COMPASS, Prog. Part. Nucl. Phys. 113, 103755 (2020).
  * (42) S. Jin and X. Shen, Highlights of light meson spectroscopy at the BESIII experiment, Natl. Sci. Rev. 8, no.11, nwab198 (2021).
  * (43) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, An updated review of the new hadron states, [arXiv:2204.02649 [hep-ph]].
  * (44) M. Aghasyan et al. [COMPASS Collaboration], Light isovector resonances in $\pi^{-}p\to\pi^{-}\pi^{-}\pi^{+}p$ at 190 GeV/${\it c}$, Phys. Rev. D 98, no.9, 092003 (2018).
  * (45) A. Rodas et al. [JPAC Collaboration], Determination of the Pole Position of the Lightest Hybrid Meson Candidate, Phys. Rev. Lett. 122, no.4, 042002 (2019).
  * (46) L. Qiu and Q. Zhao, Towards the establishment of the light $J^{P(C)}=1^{-(+)}$ hybrid nonet, [arXiv:2202.00904 [hep-ph]].
  * (47) X. K. Dong, Y. H. Lin and B. S. Zou, Interpretation of the $\eta_{1}(1855)$ as a $K\bar{K}_{1}(1400)+c.c.$ molecule, [arXiv:2202.00863 [hep-ph]].
  * (48) X. Zhang and J. J. Xie, Prediction of possible exotic states in the $\eta\bar{K}K^{*}$ system, Chin. Phys. C 44, no.5, 054104 (2020).
  * (49) H. X. Chen, A. Hosaka and S. L. Zhu, $I^{G}J^{PC}=0^{+}1^{-+}$ tetraquark states, Phys. Rev. D 78, 117502 (2008).
  * (50) P. Z. Huang, H. X. Chen and S. L. Zhu, Strong decay patterns of the $1^{-+}$ exotic hybrid mesons, Phys. Rev. D 83, 014021 (2011).
  * (51) H. X. Chen, Z. X. Cai, P. Z. Huang and S. L. Zhu, Decay properties of the $1^{-+}$ hybrid state, Phys. Rev. D 83, 014006 (2011).
  * (52) P. R. Page, Why hybrid meson coupling to two $S$-wave mesons is suppressed? Phys. Lett. B 402, 183-188 (1997).
  * (53) M. B. Voloshin and V. I. Zakharov, Measuring Quantum-Chromodynamic Anomalies in Hadronic Transitions between Quarkonium States, Phys. Rev. Lett. 45 (1980), 688.
  * (54) R. Akhoury and J. M. Frere, $\eta$, $\eta^{\prime}$ mixing and anomalies, Phys. Lett. B 220, 258-264 (1989).
  * (55) P. Castoldi and J. M. Frere, $\eta^{\prime}\to\pi^{+}\pi^{-}\pi^{0}$: a key to understanding the $\eta$-$\eta^{\prime}$ system, Z. Phys. C 40, 283 (1988).
  * (56) K. T. Chao, QCD axial anomaly and mixing of pseudoscalars, Nucl. Phys. B 317, 597-616 (1989).
  * (57) P. Ball, J. M. Frere and M. Tytgat, Phenomenological evidence for the gluon content of $\eta$ and $\eta^{\prime}$, Phys. Lett. B 365 (1996), 367-376.
  * (58) A. Ali, J. Chay, C. Greub and P. Ko, Contribution of $b\to sgg$ through the QCD anomaly in exclusive decays $B^{\pm}\to(\eta^{\prime},\eta)(K^{\pm},K^{*\pm})$ and $B^{0}\to(\eta^{\prime},\eta)(K^{0},K^{*0})$, Phys. Lett. B 424 (1998), 161-174.
  * (59) H. Leutwyler, On the $1/N$-expansion in chiral perturbation theory, Nucl. Phys. B Proc. Suppl. 64 (1998), 223-231.
  * (60) R. Kaiser and H. Leutwyler, Pseudoscalar decay constants at large $N_{c}$, [arXiv:hep-ph/9806336 [hep-ph]].
  * (61) R. Escribano and J. M. Frere, Study of the $\eta$-$\eta^{\prime}$ system in the two mixing angle scheme, JHEP 06, 029 (2005).
  * (62) R. Escribano, P. Masjuan and P. Sanchez-Puertas, The $\eta$ transition form factor from space- and time-like experimental data, Eur. Phys. J. C 75, no.9, 414 (2015).
  * (63) R. Escribano, S. Gonzàlez-Solís, P. Masjuan and P. Sanchez-Puertas, $\eta^{\prime}$ transition form factor from space- and timelike experimental data, Phys. Rev. D 94, no.5, 054033 (2016).
  * (64) J. Schechter, A. Subbaraman and H. Weigel, Effective hadron dynamics: From meson masses to the proton spin puzzle, Phys. Rev. D 48, 339-355 (1993).
  * (65) A. V. Kiselev and V. A. Petrov, Two schemes of $\eta$-$\eta^{\prime}$-mixing, Z. Phys. C 58, 595-600 (1993).
  * (66) P. Herrera-Siklody, J. I. Latorre, P. Pascual and J. Taron, $\eta$-$\eta^{\prime}$ mixing from $U(3)_{L}\otimes U(3)_{R}$ chiral perturbation theory, Phys. Lett. B 419 (1998), 326-332.
  * (67) S. D. Bass and P. Moskal, $\eta^{\prime}$ and $\eta$ mesons with connection to anomalous glue, Rev. Mod. Phys. 91, no.1, 015003 (2019).
  * (68) G. S. Bali et al. [RQCD Collaboration], Masses and decay constants of the $\eta$ and $\eta^{\prime}$ mesons from lattice QCD, JHEP 08, 137 (2021).
  * (69) T. Feldmann and P. Kroll, Flavor symmetry breaking and mixing effects in the $\eta\gamma$ and $\eta^{\prime}\gamma$ transition form factors, Eur. Phys. J. C 5 (1998), 327-335.
  * (70) K. C. Yang, Light-cone distribution amplitudes of axial-vector mesons, Nucl. Phys. B 776, 187-257 (2007).
  * (71) M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCD and resonance physics. theoretical foundations, Nucl. Phys. B 147, 385-447 (1979).
  * (72) L. J. Reinders, H. Rubinstein and S. Yazaki, Hadron properties from QCD sum rules, Phys. Rept. 127, 1 (1985).
  * (73) H. X. Chen, W. Chen and S. L. Zhu, Toward the existence of the odderon as a three-gluon bound state, Phys. Rev. D 103, no.9, L091503 (2021).
  * (74) H. X. Chen, W. Chen and S. L. Zhu, Two- and three-gluon glueballs of $C=+$, Phys. Rev. D 104, no.9, 094050 (2021).
  * (75) H. X. Chen, W. Chen and S. L. Zhu, New hadron configuration: The double-gluon hybrid state, [arXiv:2111.04514 [hep-ph]].
  * (76) S. Narison, QCD spectral sum rules, World Sci. Lect. Notes Phys. 26 (1989), 1-527.
  * (77) S. L. Zhu, Some decay modes of the $1^{-+}$ hybrid meson in QCD sum rules reexamined, Phys. Rev. D 60 (1999), 097502.
  * (78) S. L. Zhu, W. Y. P. Hwang and Z. S. Yang, Electromagnetic decay of vector mesons as derived from QCD sum rules, Phys. Lett. B 420 (1998), 8-12.
  * (79) A. A. Ovchinnikov and A. A. Pivovarov, QCD sum rule calculation of the quark gluon condensate, Sov. J. Nucl. Phys. 48, 721-723 (1988).
  * (80) K. C. Yang, W. Y. P. Hwang, E. M. Henley and L. S. Kisslinger, QCD sum rules and neutron-proton mass difference, Phys. Rev. D 47, 3001-3012 (1993).
  * (81) J. R. Ellis, E. Gardi, M. Karliner and M. A. Samuel, Renormalization-scheme dependence of Padé summation in QCD, Phys. Rev. D 54, 6986-6996 (1996).
  * (82) M. Jamin, Flavour-symmetry breaking of the quark condensate and chiral corrections to the Gell-Mann-Oakes-Renner relation, Phys. Lett. B 538, 71-76 (2002).
  * (83) B. L. Ioffe and K. N. Zyablyuk, Gluon condensate in charmonium sum rules with three-loop corrections, Eur. Phys. J. C 27, 229-241 (2003).
  * (84) V. Gimenez, V. Lubicz, F. Mescia, V. Porretti and J. Reyes, Operator product expansion and quark condensate from lattice QCD in coordinate space, Eur. Phys. J. C 41, 535-544 (2005).
  * (85) S. Narison, Gluon condensates and precise $\overline{m}_{c,b}$ from QCD-moments and their ratios to order $\alpha_{s}^{3}$ and $\langle G^{4}\rangle$, Phys. Lett. B 706, 412-422 (2012).
  * (86) S. Narison, QCD parameter correlations from heavy quarkonia, Int. J. Mod. Phys. A 33, no.10, 1850045 (2018).
  * (87) D. Becirevic, V. Lubicz, F. Mescia and C. Tarantino, Coupling of the light vector meson to the vector and to the tensor current, JHEP 05 (2003), 007.


can display kinks or discontinuities (Fig. 7) that produce characteristic
jumps in its signature in the visibility domain (Fig. 6). Understanding these
patterns is crucial for devising a test of the Kerr hypothesis based on the
photon ring shape that remains valid at high inclinations.

Reversing the logic, one can assume the Kerr nature of the source and try to
infer its parameters from the photon ring shape. We hope to further explore
the question of spin inference by solving the inverse problem described in
Sec. 6.3. This approach could most likely also be used to infer the BH mass
with much greater precision than the EHT has so far achieved without being
able to probe higher spatial frequencies.

As discussed in Sec. 7.1, it is important to determine whether our conclusions
generalize to models with geometrically thick emission, with noise created by
inevitable source fluctuations, with different astrophysical conditions (e.g.,
a radially infalling flow rather than a circularized one), and also with
tilted disks that are misaligned with the BH spin. For instance, our
observation that the measured diameters $(d_{+},d_{-})$ of the $n=2$ ring lie
on a straight line connecting the inner and outer edges of the $n=2$ lensing
band could well depend sensitively on any one of these assumptions, with
direct consequences for estimates of BH spin.

Finally, the most pressing task is to revisit all these questions in the
context of the first ($n=1$) subring, which is likely to be observed first.
What is the predicted shape of the $n=1$ ring? How well can the BH parameters
be recovered from its shape?

From a theoretical standpoint, these questions are harder to tackle because
the $n=1$ ring is likely too thick to display a universal regime (20); that
is, its width-to-diameter ratio $w/d$ is likely not small enough for the
separation of scales (20) to arise. If so, then there is no baseline regime in
which its interferometric signature follows the universal form (22), and it
may well be necessary to compute the subleading correction in $w/d$ to this
visibility amplitude. We take a first stab at these issues in App. C.

##  8 Summary

Images of a BH are typically dominated by a bright, narrow ring encircling a
central brightness depression. This “photon ring” decomposes into a sequence
of self-similar subrings indexed by half-orbit number $n$, with each subring
consisting of a full, lensed image of the main emission surrounding the BH
(Sec. 2). Here, we analyzed a method proposed by Gralla et al. (2020) to test
the Kerr hypothesis via an interferometric measurement of the $n=2$ ring
shape. To check the viability of this shape test, we applied it to a wide
range of simulated images of a Kerr BH surrounded by a thin, equatorial disk
(Sec. 3), varying both the emission profile (Sec. 5) and the BH spin and
inclination (Sec. 6). These two parameter surveys confirm the robustness of
the test at low-to-moderate inclinations $i\lesssim 45^{\circ}$, indicating
that M87* (which likely lies at an inclination $i\approx 17^{\circ}$ from
Earth) is a promising target for such a test via space-VLBI observation.

At higher inclinations $i\gtrsim 45^{\circ}$, our study uncovers the emergence
of qualitatively new phenomena that complicate ring shape measurements (Sec.
4). These effects arise because the ring width develops significant angular
variation, which impacts the range of baselines dominated by the ring’s
interferometric signature. In particular, we find that observations made at
fixed baseline length in the visibility domain cannot measure the shape of a
single ring at every angle: such observations are sensitive to a fixed ring
width and must therefore either jump between the $n=1$ and $n=2$ rings (at
moderately high inclinations; Fig. 6), or else pick up the signature of a
“hybrid $n=1\&2$ ring” of roughly uniform width (at very high inclinations;
Fig. 7). A more refined, sophisticated shape test has yet to be devised for
these situations.

Our investigation also reveals that for a given choice of BH spin and
inclination, the observable photon ring shape depends noticeably on the
astrophysical properties of the emitting source. In particular, the most
directly accessible ($n=1$ and $n=2$) rings can differ significantly from the
theoretical critical curve, and in many models they are in fact closer to the
critical curves of neighboring BH spins and inclinations (Figs. 10 & 11). As
such (and as discussed throughout Secs. 5 and 6), the lensing bands introduced
in Sec. 2.5 are more relevant than the critical curve to the problem of BH
spin inference from the photon ring shape. We have outlined a promising
approach for tackling this problem in Sec. 6.3, with some encouraging
preliminary results (Fig. 12).

## Acknowledgements

All of the images in this paper were ray traced on the tycho computer cluster
of the Paris Observatory. We thank Samuel Gralla, Alejandro Càrdenas-Avendaño,
Andrei Lobanov, and Hengrui Zhu for helpful comments and discussions. A.L.
also gratefully acknowledges support from Will and Kacie Snellings, and M.W.
thanks Alexandra Elbakyan for her contributions to the open science
initiative.

## References

  * Bardeen (1973) Bardeen, J. M. 1973, in Black Holes (Les Astres Occlus), ed. C. Dewitt & B. S. Dewitt (Gordon and Breach Science Publishers), 215–239
  * Bauer et al. (2022) Bauer, A. M., Cárdenas-Avendaño, A., Gammie, C. F., & Yunes, N. 2022, ApJ, 925, 119
  * Beckwith & Done (2005) Beckwith, K. & Done, C. 2005, MNRAS, 359, 1217
  * Broderick et al. (2022) Broderick, A. E., Tiede, P., Pesce, D. W., & Gold, R. 2022, ApJ, 927, 6
  * Cárdenas-Avendaño et al. (2022) Cárdenas-Avendaño, A., Lupsasca, A., & Zhu, H. 2022, arXiv e-prints, arXiv:2211.07469
  * Chael et al. (2021) Chael, A., Johnson, M. D., & Lupsasca, A. 2021, ApJ, 918, 6
  * Cunha & Herdeiro (2018) Cunha, P. V. P. & Herdeiro, C. A. R. 2018, General Relativity and Gravitation, 50, 42
  * Cunningham (1975) Cunningham, C. T. 1975, ApJ, 202, 788
  * Doeleman et al. (2019) Doeleman, S., Blackburn, L., Dexter, J., et al. 2019, in Bulletin of the American Astronomical Society, Vol. 51, 256
  * Event Horizon Telescope Collaboration (2019a) Event Horizon Telescope Collaboration. 2019a, ApJ, 875, L1
  * Event Horizon Telescope Collaboration (2019b) Event Horizon Telescope Collaboration. 2019b, ApJ, 875, L2
  * Event Horizon Telescope Collaboration (2019c) Event Horizon Telescope Collaboration. 2019c, ApJ, 875, L3
  * Event Horizon Telescope Collaboration (2019d) Event Horizon Telescope Collaboration. 2019d, ApJ, 875, L4
  * Event Horizon Telescope Collaboration (2019e) Event Horizon Telescope Collaboration. 2019e, ApJ, 875, L5
  * Event Horizon Telescope Collaboration (2019f) Event Horizon Telescope Collaboration. 2019f, ApJ, 875, L6
  * Event Horizon Telescope Collaboration (2021a) Event Horizon Telescope Collaboration. 2021a, ApJ, 910, L12
  * Event Horizon Telescope Collaboration (2021b) Event Horizon Telescope Collaboration. 2021b, ApJ, 910, L13
  * Falcke et al. (2000) Falcke, H., Melia, F., & Agol, E. 2000, ApJ, 528, L13
  * Farah et al. (2020) Farah, J. R., Pesce, D. W., Johnson, M. D., & Blackburn, L. 2020, ApJ, 900, 77
  * Gates et al. (2021) Gates, D. E. A., Hadar, S., & Lupsasca, A. 2021, Phys. Rev. D, 103, 044050
  * Gebhardt et al. (2011) Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119
  * Gelles et al. (2021) Gelles, Z., Prather, B. S., Palumbo, D. C. M., et al. 2021, ApJ, 912, 39
  * Gold et al. (2020) Gold, R., Broderick, A. E., Younsi, Z., et al. 2020, ApJ, 897, 148
  * Gralla (2020) Gralla, S. E. 2020, Phys. Rev. D, 102, 044017
  * Gralla et al. (2019) Gralla, S. E., Holz, D. E., & Wald, R. M. 2019, Phys. Rev. D, 100, 024018
  * Gralla & Lupsasca (2020a) Gralla, S. E. & Lupsasca, A. 2020a, Phys. Rev. D, 101, 044031
  * Gralla & Lupsasca (2020b) Gralla, S. E. & Lupsasca, A. 2020b, Phys. Rev. D, 101, 044032
  * Gralla & Lupsasca (2020c) Gralla, S. E. & Lupsasca, A. 2020c, Phys. Rev. D, 102, 124003
  * Gralla et al. (2020) Gralla, S. E., Lupsasca, A., & Marrone, D. P. 2020, Phys. Rev. D, 102, 124004
  * Gravity Collaboration (2018) Gravity Collaboration. 2018, A&A, 615, L15
  * Gurvits et al. (2022) Gurvits, L. I., Paragi, Z., Amils, R. I., et al. 2022, Acta Astronautica, 196, 314
  * Hadar et al. (2021) Hadar, S., Johnson, M. D., Lupsasca, A., & Wong, G. N. 2021, Phys. Rev. D, 103, 104038
  * Haworth et al. (2019) Haworth, K., Johnson, M., Pesce, D. W., et al. 2019, in Bulletin of the American Astronomical Society, Vol. 51, 235
  * Herdeiro et al. (2021) Herdeiro, C. A. R., Pombo, A. M., Radu, E., Cunha, P. V. P., & Sanchis-Gual, N. 2021, J. Cosmology Astropart. Phys., 2021, 051
  * Himwich et al. (2020) Himwich, E., Johnson, M. D., Lupsasca, A., & Strominger, A. 2020, Phys. Rev. D, 101, 084020
  * Igumenshchev et al. (2003) Igumenshchev, I. V., Narayan, R., & Abramowicz, M. A. 2003, ApJ, 592, 1042
  * James et al. (2015) James, O., Tunzelmann, E. v., Franklin, P., & Thorne, K. S. 2015, Classical and Quantum Gravity, 32, 065001
  * Janssen et al. (2021) Janssen, M., Falcke, H., Kadler, M., et al. 2021, Nature Astronomy, 5, 1017
  * Johannsen (2013) Johannsen, T. 2013, ApJ, 777, 170
  * Johannsen & Psaltis (2010) Johannsen, T. & Psaltis, D. 2010, ApJ, 718, 446
  * Johnson et al. (2020) Johnson, M. D., Lupsasca, A., Strominger, A., et al. 2020, Science Advances, 6, eaaz1310
  * Lamy et al. (2018) Lamy, F., Gourgoulhon, E., Paumard, T., & Vincent, F. H. 2018, Classical and Quantum Gravity, 35, 115009
  * LIGO Scientific Collaboration & Virgo Collaboration (2016) LIGO Scientific Collaboration & Virgo Collaboration. 2016, Phys. Rev. Lett., 116, 061102
  * Luminet (1979) Luminet, J. P. 1979, A&A, 75, 228
  * Medeiros et al. (2022) Medeiros, L., Chan, C.-K., Narayan, R., Özel, F., & Psaltis, D. 2022, ApJ, 924, 46
  * Medeiros et al. (2020) Medeiros, L., Psaltis, D., & Özel, F. 2020, ApJ, 896, 7
  * Narayan et al. (2003) Narayan, R., Igumenshchev, I. V., & Abramowicz, M. A. 2003, PASJ, 55, L69
  * Narayan et al. (2019) Narayan, R., Johnson, M. D., & Gammie, C. F. 2019, ApJ, 885, L33
  * Okounkova et al. (2019) Okounkova, M., Scheel, M. A., & Teukolsky, S. A. 2019, Classical and Quantum Gravity, 36, 054001
  * Pesce et al. (2019) Pesce, D., Haworth, K., Melnick, G. J., et al. 2019, in Bulletin of the American Astronomical Society, Vol. 51, 176
  * Psaltis (2019) Psaltis, D. 2019, General Relativity and Gravitation, 51, 137
  * Teo (2003) Teo, E. 2003, General Relativity and Gravitation, 35, 1909
  * Vincent et al. (2022) Vincent, F. H., Gralla, S. E., Lupsasca, A., & Wielgus, M. 2022, A&A, 667, A170
  * Vincent et al. (2016) Vincent, F. H., Meliani, Z., Grandclément, P., Gourgoulhon, E., & Straub, O. 2016, Classical and Quantum Gravity, 33, 105015
  * Vincent et al. (2011) Vincent, F. H., Paumard, T., Gourgoulhon, E., & Perrin, G. 2011, Classical and Quantum Gravity, 28, 225011
  * Vincent et al. (2021) Vincent, F. H., Wielgus, M., Abramowicz, M. A., et al. 2021, A&A, 646, A37
  * Walker et al. (2018) Walker, R. C., Hardee, P. E., Davies, F. B., Ly, C., & Junor, W. 2018, ApJ, 855, 128
  * Wielgus (2021) Wielgus, M. 2021, Phys. Rev. D, 104, 124058
  * Wielgus et al. (2020) Wielgus, M., Horák, J., Vincent, F., & Abramowicz, M. 2020, Phys. Rev. D, 102, 084044
  * Wong (2021) Wong, G. N. 2021, ApJ, 909, 217

##  Appendix A Computing Kerr lensing bands

We consider a null geodesic connecting a source at
$(t_{s},r_{s},\theta_{s},\varphi_{s})$ to an observer at
$(t_{o},r_{o},\theta_{o},\varphi_{o})$. Its behavior can be characterized by
its conserved quantities $(\lambda,\eta)$ [Eq. (2) above] via the radial and
angular potentials (e.g., Gralla & Lupsasca 2020b)

$\displaystyle\mathcal{R}(r)$
$\displaystyle=\left(r^{2}+a_{*}^{2}-a_{*}\lambda\right)^{2}-\Delta(r)\left[\eta+\left(\lambda-
a_{*}\right)^{2}\right],$ (57a) $\displaystyle\Theta(\theta)$
$\displaystyle=\eta+a_{*}^{2}\cos^{2}{\theta}-\lambda^{2}\cot^{2}{\theta},$
(57b)

whose zeros define the radial and angular turning points of the ray.
Integrating the geodesic equation along the trajectory gives

$\displaystyle
I_{r}\equiv\fint_{r_{s}}^{r_{o}}\frac{\mathop{}\\!\mathrm{d}r}{\pm_{r}\sqrt{\mathcal{R}(r)}}=\fint_{\theta_{s}}^{\theta_{o}}\frac{\mathop{}\\!\mathrm{d}\theta}{\pm_{\theta}\sqrt{\Theta(\theta)}}\equiv
G_{\theta},$ (58)

where the signs $\pm_{r}$ and $\pm_{\theta}$ switch at radial and angular
turning points, respectively. These expressions keep track of the total Mino
time $\tau=I_{r}=G_{\theta}$ elapsed along the trajectory.

We consider the case of a distant observer ($r_{o}=D\gg M$) at an inclination
$\theta_{o}=i$ from the BH spin axis who shoots photons backwards into the
geometry from a polar position $(\rho,\varphi_{\rho})$ in the sky. Such a
photon has conserved quantities

$\displaystyle\lambda=-D\rho\cos{\varphi_{\rho}}\sin{i},\qquad\eta=(D\rho)^{2}-\lambda^{2}-a_{*}^{2}\cos^{2}{i},$
(59)

in terms of which the four roots $r_{j}$ of the quartic radial potential
$\mathcal{R}(r)$ are given by [Eqs. (79)–(95) of Gralla & Lupsasca (2020b)]:

$\displaystyle
r_{j}=-z-\epsilon_{j}\sqrt{-\frac{\mathcal{A}}{2}-z^{2}+\tilde{\epsilon}_{j}\frac{\mathcal{B}}{4z}},$
$\displaystyle\epsilon_{j}=\left\\{\begin{array}[]{ll}1&\mbox{if
}j\in\left\\{1,3\right\\},\\\ -1&\mbox{if
}j\in\left\\{2,4\right\\},\end{array}\right.\qquad\tilde{\epsilon}_{j}=\left\\{\begin{array}[]{ll}1&\mbox{if
}j\in\left\\{1,2\right\\},\\\ -1&\mbox{if
}j\in\left\\{3,4\right\\},\end{array}\right.$
$\displaystyle\mathcal{A}=a_{*}^{2}-\eta-\lambda^{2},\quad\mathcal{B}=2M\left[\eta+\left(\lambda-
a_{*}\right)^{2}\right],\quad\mathcal{C}=-a_{*}^{2}\eta,$
$\displaystyle\mathcal{P}=-\frac{\mathcal{A}^{2}}{12}-\mathcal{C},\quad\mathcal{Q}=-\frac{\mathcal{A}}{3}\left[\left(\frac{\mathcal{A}}{6}\right)^{2}-\mathcal{C}\right]-\frac{\mathcal{B}^{2}}{8},$
$\displaystyle\omega_{\pm}=\left(-\frac{\mathcal{Q}}{2}\pm\sqrt{\frac{\mathcal{P}^{3}}{27}+\frac{\mathcal{Q}^{2}}{4}}\right)^{1/3},\quad
z=\sqrt{\frac{\omega_{+}+\omega_{-}-\mathcal{A}/3}{2}}.$

Letting $r_{ij}=r_{i}-r_{j}$, we define $A=\sqrt{r_{32}r_{42}}$,
$B=\sqrt{r_{31}r_{41}}$, and introduce the antiderivatives of $I_{r}$ (Gralla
& Lupsasca 2020a, b):

$\displaystyle\mathcal{I}_{r}^{+}(r)$
$\displaystyle=\frac{2}{\sqrt{r_{31}r_{42}}}F\left(\left.\arcsin{\sqrt{\frac{r-r_{4}}{r-r_{3}}\frac{r_{31}}{r_{41}}}}\right|\frac{r_{32}r_{41}}{r_{31}r_{42}}\right),$
(60) $\displaystyle\mathcal{I}_{r}^{-}(r)$
$\displaystyle=\frac{1}{\sqrt{AB}}F\left(\left.\arccos\left(\frac{1-\frac{r-r_{2}}{r-r_{1}}\frac{B}{A}}{1+\frac{r-r_{2}}{r-r_{1}}\frac{B}{A}}\right)\right|\frac{(A+B)^{2}-r_{21}^{2}}{4AB}\right).$
(61)

  * •

If $\rho>\tilde{\rho}(\varphi_{\rho})$, so that the photon is shot from a
position outside the critical curve, then all its roots are real and ordered
as $r_{4}>r_{3}>r_{+}>r_{-}>r_{2}>r_{1}$. As such, it must encounter a radial
turning point at $r=r_{4}$ and eventually be deflected back to infinity. The
Mino time $\tau(r)$ elapsed along its trajectory is

$\displaystyle
I_{r}^{\mathcal{C}^{+}}(r)=\mathcal{I}_{r}^{+}(r_{o})\mp\mathcal{I}_{r}^{+}(r),$
(62)

with upper/lower sign before/after reaching the turning point. The total Mino
time elapsed along the full light ray is

$\displaystyle I_{r}^{\rm
total}=\frac{4}{\sqrt{r_{31}r_{42}}}F\left(\left.\arcsin{\sqrt{\frac{r_{31}}{r_{41}}}}\right|\frac{r_{32}r_{41}}{r_{31}r_{42}}\right).$
(63)

  * •

If $\rho<\tilde{\rho}(\varphi_{\rho})$, so that the photon is shot from a
position inside the critical curve, then it cannot encounter a radial turning
point and must therefore asymptotically reach the horizon at $r=r_{+}$. The
Mino time $\tau(r)$ elapsed along its trajectory is

$\displaystyle
I_{r}^{\mathcal{C}^{-}}(r)=\mathcal{I}_{r}^{\pm}(r_{o})-\mathcal{I}_{r}^{\pm}(r),$
(64)

with the upper sign when all roots are real (in which case
$r_{+}>r_{-}>r_{4}>r_{3}>r_{2}>r_{1}$) and the lower sign when
$r_{3}=\bar{r}_{4}$ are complex conjugate roots with
$r_{+}>r_{-}>r_{2}>r_{1}$.131313There also exist vortical ($\eta<0$) rays with
both $r_{1}=\bar{r}_{2}$ and $r_{3}=\bar{r}_{4}$, but we may ignore them here
as they cannot reach the equatorial plane.  The total Mino time elapsed along
the full light ray is

$\displaystyle I_{r}^{\rm total}=I_{r}^{\mathcal{C}^{-}}(r_{+}).$ (65)

Regardless of where the photon is shot back from, it crosses the equatorial
plane a maximum number $N+1$ of times given by

$\displaystyle N=\left\lfloor\frac{I_{r}^{\rm
total}\sqrt{-u_{-}a_{*}^{2}}+\operatorname{sign}(\sin{\varphi_{\rho}})F_{o}}{2K}\right\rfloor-H(\sin{\varphi_{\rho}}),$
(66) $\displaystyle
u_{\pm}=\Delta_{\theta}\pm\sqrt{\Delta_{\theta}^{2}+\frac{\eta}{a_{*}^{2}}},\qquad\Delta_{\theta}=\frac{1}{2}\left(1-\frac{\eta+\lambda^{2}}{a_{*}^{2}}\right),$
$\displaystyle K=K\left(\frac{u_{+}}{u_{-}}\right),\qquad
F_{o}=F\left(\left.\arcsin\left(\frac{\cos(i)}{\sqrt{u_{+}}}\right)\right|\frac{u_{+}}{u_{-}}\right),$

where $H$, $F$, and $K$ respectively denote the Heaviside function, the
incomplete elliptic integral of the first kind $F(u|k)$, and its completion
$K(k)\equiv F(\pi/2|k)$. $u_{\pm}$ are zeros $u=\cos^{2}{\theta}$ of the
angular potential $\Theta(\theta)$; when evaluated on critical geodesics with
$(\lambda,\eta)=(\tilde{\lambda},\tilde{\eta})$ [Eq. (6)], they reproduce
$\tilde{u}_{\pm}$ in Eq. (2.2).

The photon intersects the equatorial plane for the $(n+1)^{\text{th}}$ time
(for $n\leq N$) when the Mino time $\tau(\theta)$ elapsed along its trajectory
is [Eq. (81) of Gralla & Lupsasca (2020a)]

$\displaystyle
G_{\theta}(n)=\frac{2mK-\operatorname{sign}(\sin{\varphi_{\rho}})F_{o}}{\sqrt{-u_{-}a_{*}^{2}}},$
(67)

where $m=n+H(\sin{\varphi_{\rho}})$ is the number of angular turning points
encountered along the trajectory. We note that $N$ is by definition the non-
negative integer such that $G_{\theta}(N)\leq I_{r}^{\rm total}\leq
G_{\theta}(N+1)$.

The $n^{\text{th}}$ image of an equatorial source ring of constant radius
$r_{s}$ is the set of points $(\rho,\varphi_{\rho})$ in the observer sky that
satisfy

$\displaystyle\boxed{I_{r}^{\mathcal{C}^{\pm}}(r_{s})=G_{\theta}(n)}$ (68)

with $G_{\theta}(n)$ given in Eq. (67) and $I_{r}^{\mathcal{C}^{\pm}}(r_{s})$
given in Eqs. (62) and (64) for points outside and inside the critical curve.
These points always trace a closed, continuous curve $\mathcal{C}_{n}(r_{s})$.

The contours $\mathcal{C}_{0}(r_{s})$ and $\mathcal{C}_{1}(r_{s})$
corresponding to a selection of radii $r_{s}$ are shown for various BH spins
and inclinations in Fig. 6 of Gralla & Lupsasca (2020a). While all these
curves are continuous, some of them may cross the critical curve, in which
case they are stitched together from segments solving Eq. (68) with opposite
signs. Aside from this minor inconvenience, it is straightforward to
numerically solve Eq. (68) for any $\mathcal{C}_{n}(r_{s})$.

In particular, as explained in Sec. 2.5, the $n^{\text{th}}$ lensing band is
the annular region in the sky bounded within the contours
$\mathcal{C}_{n}(r_{+})$ and $\mathcal{C}_{n}(\infty)$, which respectively
trace its inner and outer edges. These contours are relatively simple to
compute because they can never cross the critical curve: trajectories that are
deflected back to infinity must obey Eq. (62), so the outer edge
$\mathcal{C}_{n}(\infty)$ is obtained by solving Eq. (68) with a $+$ sign
only; likewise, trajectories that reach the BH must obey Eq. (64), so the
inner edge $\mathcal{C}_{n}(r_{+})$ is obtained by solving Eq. (68) with a $-$
sign only.

We note that when crossing the inner and outer edges $\mathcal{C}_{n}(r_{+})$
and $\mathcal{C}_{n}(\infty)$ of the $n^{\text{th}}$ lensing band, the maximal
number $N+1$ of equatorial crossings (66) must by definition change by 1. This
provides a means of checking that $\mathcal{C}_{n}(r_{+})$ and
$\mathcal{C}_{n}(\infty)$ have been correctly computed (or an alternative way
to compute them).

Finally, we note that Eq. (58) may also be directly solved for $r_{s}(I_{r})$
using a single “unified inversion formula” given in Gralla & Lupsasca (2020b).
Then $r_{\rm eq}^{(n)}(\rho,\varphi_{\rho})\equiv r_{s}(G_{\theta}(n))$ are
transfer functions that give the radii at which light rays shot back from
position $(\rho,\varphi_{\rho})$ cross the equatorial plane for the
$(n+1)^{\text{th}}$ time. Contours of these functions are precisely the
$\mathcal{C}_{n}(r_{s})$ defined above. Hence, lensing bands may also be
computed using these transfer functions, as described in App. A of Chael et
al. (2021).

##  Appendix B Computing the range of allowed phovals within a lensing band

Here, we provide a detailed solution to the following problem. Given a lensing
band $\mathcal{B}\subsetneq\mathbb{R}^{2}$, how do we compute the subregion
$\mathcal{A}$ of the $(d_{+},d_{-})$ plane spanned by the phovals that fit
within $\mathcal{B}$?

We glossed over this question after introducing it in Sec. 5.4, but its answer
is crucial for obtaining the blue regions in Figs. 10, 11, and 12. At first
glance, this problem seems straightforward: we could scan the 6D space of
phovals (55) and ask, for each
$\@vec{R}=\left\\{R_{0},R_{1},R_{2},X,\chi,\varphi_{0}\right\\}$ in this
parameter space $\mathcal{P}_{6}$, whether the associated phoval fits inside
the lensing band $\mathcal{B}$. The phovals that fit inside the lensing band
would form a 6D region $\mathcal{A}_{6}\subsetneq\mathcal{P}_{6}$ whose 2D
projection $\Pi:\mathcal{P}_{6}\to\mathbb{R}^{2}$ via the map (54),

$\displaystyle\Pi:\@vec{R}\mapsto(d_{+},d_{-})=2(R_{0}+R_{1},R_{0}+R_{2}),$
(69)

would give precisely the sought-after region in the $(d_{+},d_{-})$ plane:

$\displaystyle\mathcal{A}=\Pi(\mathcal{A}_{6}).$ (70)

In practice, however, the computational cost of this “brute force” approach is
too high to achieve sufficient precision in reasonable time: the high-
dimensionality of the parameter space $\mathcal{P}_{6}$ makes any detailed
scan too slow. Hence, another approach is needed.

The tractable method we came up with is quite general and consists of two
parts. First, we recast the question of whether a given point $(d_{+},d_{-})$
belongs in $\mathcal{A}$ into a minimization problem, which is easier to
solve. Second, we devised an efficient way to scan across regions of the
$(d_{+},d_{-})$ plane while solving this minimization problem. We now describe
each part separately.

### Part 1: Minimization method

Let $\mathcal{B}\subsetneq\mathbb{R}^{2}$ denote an annular subregion of the
plane with inner edge $\mathcal{I}$ and outer edge $\mathcal{O}$. We can
define a separation function characterizing the distance from a closed curve
$\mathcal{C}$ to $\mathcal{B}$ as

$\displaystyle\delta_{\mathcal{B}}(\mathcal{C})=\max_{z\in\mathcal{C}}d(z,\mathcal{B}),$
(71)

where $d$ is the usual distance between a point and a set, namely:

$\displaystyle
d(z,\mathcal{B})=\begin{cases}\min_{a\in\mathcal{I}}|z-a|&\mbox{if $z$ lies
inside of }\mathcal{I},\\\ \min_{a\in\mathcal{O}}|z-a|&\mbox{if $z$ lies
outside of }\mathcal{O},\\\ 0&\mbox{if }z\in\mathcal{B}.\end{cases}$ (72)

For us, $\mathcal{C}$ will be a phoval shape [Eq. (55)] and $\mathcal{B}$ a
Kerr lensing band with inner edge $\mathcal{I}=\mathcal{C}_{n}(r_{+})$ and
outer edge $\mathcal{O}=\mathcal{C}_{n}(\infty)$, as in App. A.

With this definition, $\delta_{\mathcal{B}}(\mathcal{C})=0$ if and only if
$\mathcal{C}$ lies within the lensing band $\mathcal{B}$. Thus, to check if a
point $(d_{+},d_{-})$ corresponds to a phoval $\mathcal{C}$ in the lensing
band $\mathcal{B}$, we can minimize the function

$\displaystyle\Delta:(d_{+},d_{-},\@vec{r})\mapsto\delta_{\mathcal{B}}\left(\mathcal{C}\left(R_{0},\frac{d_{+}}{2}-R_{0},\frac{d_{-}}{2}-R_{0},X,\chi,\varphi_{0}\right)\right),$

where $\mathcal{C}(\@vec{R})$ is the phoval with projected position (55), over
the 4D space $\mathcal{P}_{4}$ of parameters
$\@vec{r}=(R_{0},X,\chi,\varphi_{0})$. In other words, we compute
$m:\mathbb{R}^{2}\to\mathbb{R}$ defined as

$\displaystyle
m:(d_{+},d_{-})\mapsto\min_{\@vec{r}\in\mathcal{P}_{4}}\Delta(d_{+},d_{-},\@vec{r}).$
(73)

If $m(d_{+},d_{-})$ vanishes (or stays below a tolerance threshold that
accounts for numerical artifacts), then there exists a phoval with diameters
$(d_{+},d_{-})$ that lies within the lensing band, and thus,

$\displaystyle\mathcal{A}=\left\\{(d_{+},d_{-})\in\mathbb{R}^{2}:m(d_{+},d_{-})=0\right\\}.$
(74)

We have thus recast the problem of determining whether a given $(d_{+},d_{-})$
belongs to $\mathcal{A}$ to a problem of minimization over $\mathcal{P}_{4}$,
which is only a 4D space. This completes part 1.

### Part 2: Random walk

To compute $\mathcal{A}$, it now suffices to define a grid of values
$(d_{+},d_{-})$ (typically, between the values found for $\mathcal{I}$ and
$\mathcal{O}$) and compute $m(d_{+},d_{-})$ for each point on the grid. Since
the computational cost of the minimization method in part 1 is still quite
high, we must still devise an efficient method to scan across a large grid,
ideally without having to evaluate $m$ for each point. This can be done using
a random walk with a reflecting barrier:

  1. 1.

We start from the parameters
$\@vec{R}\in\mathcal{A}_{6}\subset\mathcal{P}_{6}$ of some phoval that we know
to lie in the lensing band: for instance, some intermediate phoval between
those approximating $\mathcal{I}$ and $\mathcal{O}$.

  2. 2.

We randomly pick one of the radii $\left\\{R_{0},R_{1},R_{2}\right\\}$ and add
to this parameter a predefined increment $\pm b$, with the sign picked at
random (we recall our convention that $R_{1}\geq R_{2}$; if this changes, then
we must interchange $R_{1}\leftrightarrow R_{2}$ to maintain it).

  3. 3.

After the increment, we test if the resulting phoval (with the same values of
$X$, $\chi$, and $\varphi_{0}$) still lies within the lensing band. If that is
the case, then we store $(d_{+},d_{-})=2(R_{0}+R_{1},R_{0}+R_{2})$ as an
accepted value in the region $\mathcal{A}$ and then proceed with the random
walk by repeating step 2. If not, then we compute $m(d_{+},d_{-})$, with two
possible outcomes:

     * •

$m(d_{+},d_{-})=0$ (or the minimum is under threshold), in which case we store
$(d_{+},d_{-})=2(R_{0}+R_{1},R_{0}+R_{2})$ as an accepted value in the region
$\mathcal{A}$ and return to step 2 to continue the random walk; otherwise,

     * •

$m(d_{+},d_{-})>0$ (or the minimum is above threshold), in which case we store
$(d_{+},d_{-})=2(R_{0}+R_{1},R_{0}+R_{2})$ as a rejected value, then restore
the values of $\left\\{R_{0},R_{1},R_{2}\right\\}$ to their previous state and
return to step 2 to take another step of the random walk in a new direction.

Steps 2 and 3 can be repeated any number of times, resulting in a set of
rejected values $\left\\{(d_{+},d_{-})\right\\}_{\rm rejected}$ and a set of
accepted values

$\displaystyle\mathcal{A}=\left\\{(d_{+},d_{-})\right\\}_{\rm accepted}.$ (75)

One advantage is that the minimization routine $m$ is only evaluated in the
rejected region of the $(d_{+},d_{-})$ plane, or inside the allowed region but
very near its boundary. This boundary acts as a reflecting barrier: each time
it is crossed, the random walk returns to a previous position in
$\mathcal{A}$. Thus, it cannot escape far in the rejected region, where the
computational cost is higher.

The boundaries of the region $\mathcal{A}$ (which corresponds to the blue
boxes in Figs. 10, 11, and 12) can be computed to arbitrary precision by
tuning the value of the increment $b$ that controls the step size in the
random walk, and also by running several instances of the random walk starting
from different positions.

##  Appendix C Shape of the $n=1$ photon ring

Figure 13:  Kerr hypothesis test using the GLM method applied to the $n=1$
subring. An image was simulated for a BH of spin $a=0.94$ and inclination
$i=17^{\circ}$, with equatorial emission profile parameters $\mu=r_{-}$,
$\gamma=-\frac{3}{2}$, $\sigma=0.5$. (Upper left) Amplitude $|V(u,\varphi)|$
of the visibility at $\varphi=135^{\circ}$, with baselines colored according
to which subring dominates the signal (Sec. 2.7). (Upper Right) Visibility
amplitude fit (with envelope) in the $n=1$-dominated baseline window
$u_{w}\in[30,70]\,$G$\lambda$ (Sec. 3). (Lower left) Goodness of fit (38) as a
function of fitting diameter $d_{\varphi}^{(1)}$. The dot corresponds to the
diameter with maximal goodness of fit $g_{\rm
max}=e^{-\operatorname{RMSD}_{u,\rm min}}$. The error bar includes nearby
diameters such that $\operatorname{RMSD}_{u}(d_{\varphi})\leq
2\operatorname{RMSD}_{u,\rm min}$, or equivalently $g(d_{\varphi})\geq g_{\rm
max}^{2}$. (Lower Right) Fitting $d_{\varphi}^{(1)}$ to a circlipse using the
multi-fit method (Sec. 3.2.4), with the same prescription for the error bars.
Angles $\varphi=125^{\circ}$ and $130^{\circ}$ were removed because the visamp
fit in the chosen baseline window was too poor.

The $n=1$ ring of M87* will almost certainly be the first photon subring to be
measured (since its signal dominates the visibility on the shortest baselines
to space), and it is therefore interesting to predict its geometric shape and
interferometric signature. Does the $n=1$ ring also follow the circlipse shape
(41)? If so, to what precision? Can its projected diameter $d_{\varphi}^{(1)}$
be recovered from the periodicity of the visibility amplitude
$|V(u,\varphi)|$?

Answering these questions in detail would be the subject of a whole other
paper. Here, we merely record some observations that arose from our study and
should inform future analyses:

  1. 1.

Only narrow rings with a very small width-to-diameter ratio $w/d\ll 1$ admit a
universal regime (20). For thicker rings, there is not a large enough
separation between the scales $1/d$ and $1/w$ for universality to set in. The
$n=1$ ring tends to be rather thick in most models (though not all, since
there can be substantial variation in its width with the choice of
astrophysical profile, as discussed below Eq. (48) and in Sec. 5), with a
typical width of $w_{1}\approx 1M$ and diameter $d\approx 10M$. Hence
$w_{1}/d\approx 10\%$, implying that subleading corrections to the universal
visamp profile (22) may be substantial. Indeed, in most of our $n=1$ visamps,
we observed a much faster decay than the predicted $u^{-1/2}$ universal power-
law fall-off.

  2. 2.

Geometrically, this also means that the periodicity $d_{\varphi}^{(1)}$ of the
$n=1$ visamp no longer admits a simple interpretation as the projected
diameter of the $n=1$ ring in the image, as its diameter is not sharply
defined to better than $w_{1}/d\approx 10\%$.

  3. 3.

Even in the absence of a universal regime $1/d\ll u\ll 1/w$ with simple
interferometric signature (22), the $n=1$ ring can still dominate the visamp
on some baselines $b_{1}\ll u\ll b_{2}$ where we may hope to measure the
periodicity of $|V(u,\varphi)|$, and thus $d_{\varphi}^{(1)}$, but only so
long as the visamp displays enough periods within this baseline domain of
width $\Delta u=b_{2}-b_{1}$.

While these issues may complicate a measurement of the $n=1$ ring and its
interpretation, none of them seems insurmountable (see Fig. 13), and we
briefly address them in reverse order.

Regarding point 3: a baseline region of width $\Delta u$ contains a number of
visamp hops $N=\Delta u/T$, where $T$ is the period of a hop, which for a ring
is $T=1/d_{\varphi}$. Hence, the number of visamp hops in the $n=1$-dominated
regime $b_{1}\ll u\ll b_{2}$ is

$\displaystyle
N_{1}(\varphi)\approx\left[b_{2}(\varphi)-b_{1}(\varphi)\right]d_{\varphi}.$
(76)

Likewise, for higher-order rings $n\gtrsim 2$, using Eq. (47), we have

$\displaystyle N_{n}(\varphi)$
$\displaystyle\approx\left[b_{n+1}(\varphi)-b_{n}(\varphi)\right]d_{\varphi}\approx
e^{\gamma(\varphi)}N_{n-1}(\varphi),$ (77)

so that the number of visamp hops within the regime dominated by the
$n^{\text{th}}$ ring grows exponentially with $n$ at a rate controlled by the
image demagnification factor $e^{\gamma}$. Thus, the numbers of hops
$N_{n}(\varphi)$ for higher-order rings are predicted by Kerr lensing once the
number of hops $N_{1}(\varphi)$ for the first ring is known, but this latter
number depends on astrophysical details. For instance, for the model in Fig.
13, we have $\Delta u=b_{2}-b_{1}\approx 80\,$G$\lambda$ at
$\varphi=135^{\circ}$. Since the photon ring of M87* has an expected diameter
of

$\displaystyle d_{\varphi}\approx
40\,\mu\rm{as}\approx\frac{1}{5\,\rm{G}\lambda},$ (78)

where we manipulated units as in Eq. (39), it follows that

$\displaystyle N_{1}(135^{\circ})\approx 16,$ (79)

and indeed, we count 16 visamp nulls within the gray region in Fig. 13. This
is in principle a sufficient number of hops to obtain a good estimate of the
periodicity, and hence of $d_{\varphi}^{(1)}$, which in the example of Fig. 13
follows a circlipse shape, albeit with larger error bars than for the $n=2$
ring.

Regarding point 2: in the limit $w/d\to 0$, a ring’s projected diameter
$d_{\varphi}$ becomes sharply defined in its image. Otherwise, it is only
defined up to ambiguities of order $w/d$. If the $n=1$ ring displays a visamp
periodicity $1/d_{\varphi}^{(1)}$, which feature of the ring image does that
correspond to? A natural guess would be the distance $d_{\varphi}^{\rm peak}$
between peaks of the ring’s intensity profile, but in simple models (such as
the Gaussian ring below) one can prove that
$d_{\varphi}^{(1)}<d_{\varphi}^{\rm peak}$, with the difference vanishing as
$w/d\to 0$. This effect can be attributed to the everywhere-positive curvature
of a ring, which implies that its Radon transform has its center of mass
shifted inward relative to its peaks, with the corresponding 1D Fourier
transform $|V(u,\varphi)|$ displaying a slightly smaller $d_{\varphi}^{(1)}$.

Regarding point 1: it is helpful to consider an axisymmetric ring with purely
radial intensity profile $I(\rho,\varphi)=I(\rho)$, such as

$\displaystyle I(\rho)=\frac{1}{2\pi
w^{2}}e^{-\frac{d^{2}}{8w^{2}}}I_{0}\left(\frac{d\rho}{2w^{2}}\right)e^{-\frac{\rho^{2}}{2w^{2}}},$
(80)

with $I_{0}$ denoting the $0^{\text{th}}$ modified Bessel function of the
first kind. This is known as the Gaussian ring because
$I_{0}(x)\stackrel{{\scriptstyle x\to\infty}}{{\approx}}(2\pi
x)^{-\frac{1}{2}}e^{x}$, so that as $d\to\infty$ the radial profile approaches
a Gaussian of width $w$ centered as $\rho=d/2$, with normalization chosen such
that the total flux is unity, $V(0)=1$. The visibility is exactly

$\displaystyle V(u,\varphi)=V(u)=J_{0}(\pi du)e^{-2\pi^{2}w^{2}u^{2}}.$ (81)

The key point is the following. If the ring is very thin, then there is a wide
range of baselines $u\ll 1/w$ in which the exponential is negligible and the
visibility is simply $V(u)\approx J_{0}(\pi du)$, which matches the universal
prediction (22) as soon as $du\gtrsim 1\gg wu$. If the ring is thick, then we
may never ignore that exponential term, yet we still expect to see the
universal visamp (22), only multiplied by that exponential; in general, we
expect $|V(u,\varphi)|$ to take the form (22) times $e^{-c(\varphi)u^{p}}$
(with $p=2$ for a Gaussian ring).