Two-pole structure of the $h_1(1415)$ axial-vector meson: resolving mass discrepancy
Samson Clymton Hyun-Chul Kim

INHA-NTG-03/2024
  


[E-mail: ]sclymton@inha.edu
Department of Physics, Inha University,
Incheon 22212, Republic of Korea 


[E-mail: ]hchkim@inha.ac.kr
Department of Physics, Inha University,
Incheon 22212, Republic of Korea 
School of Physics, Korea Institute for Advanced Study 
  (KIAS), Seoul 02455, Republic of Korea

§ ABSTRACT 
  We investigate isoscalar axial-vector mesons using a coupled-channel
  formalism. The kernel amplitudes are constructed from meson-exchange
  diagrams in the t- and u-channels, which are derived from
  effective Lagrangians based on hidden local symmetry. We incorporate
  six channels: πρ, ηω, KK̅^*, ηϕ,
  η'ω, and η'ϕ, and solve the off-shell coupled
  integral equations. We first discuss the dynamical generation of the
  h_1(1170). The pole diagram for h_1(1595) has a certain effect
  on the generation of h_1(1170). We observe two poles at
  (1387-i6) MeV and (1452-i51) MeV, which exhibit a two-pole
  structure of the h_1(1415) meson. This two-pole structure may
  resolve the discrepancy in the experimental data on the mass of
  h_1(1415). The results show that the lower pole couples strongly
  to the KK̅^* channel, while the higher pole couples
  predominantly to the ηϕ channel. This provides insights into
  the nature of h_1 mesons and explains possible discrepancies in
  the mass of h_1(1415). 

Two-pole structure of the h_1(1415) axial-vector meson:
  resolving mass discrepancy
    Hyun-Chul Kim
    September 9, 2024
====================================================================================






§ INTRODUCTION

Recently, the BESIII Collaboration reported new data on the excited
h_1(1415) meson with quantum numbers I^G(J^PC)=0^-(1^+-),
based on a partial-wave analysis of J/ψ→γη'η'. They found its mass to be m_h_1(1415)=(1384±
6_-0^+9) MeV and width Γ = (66± 10_-10^+12)
MeV <cit.>. These new results are consistent with the
earliest measurements of h_1(1415)<cit.>, which
reported m_h_1(1415)=(1380± 20) MeV and Γ=(80± 30) MeV 
from the K^-p→ K_S^0 K̅πΛ process. However, they
differ from several other experimental findings. The Crystal Barrel
experiment<cit.> measured respectively its mass
and width as (1440± 60) MeV and (170± 80) MeV from
pp̅→ K_LK_Sπ^0π^0. An earlier BESIII 
experiment <cit.> found m=(1412± 12) MeV and
Γ=(84±52) MeV from χ_1,2,J→ϕ KK̅π. In
2018, BESIII <cit.> reported m=(1423.2± 9.4) MeV
and Γ=(90.3±27.3) MeV from J/ψ→η'KK̅π, with 
interference effects yielding m=(1441.7±4.9) MeV and
Γ=(111.5±12.8) MeV.  
Notably, the KK̅^* threshold energy
(E_th^KK̅^*≈ 1390 MeV) lies between the 
h_1(1415) masses reported in Refs.<cit.> and those in Refs.<cit.>.
It is crucial to understand the origin of these discrepancies. While
h_1(1415) decays primarily into KK̅^̅*̅, other channels near
its mass likely contribute as well. The lowest h_1(1170) state
decays into πρ, suggesting this channel may also affect
h_1(1415) production. 

The Particle Data Group (PDG) classifies the h_1 mesons as
isoscalar qq̅ states, i.e. c_1(uu̅+dd̅) + c_2
ss̅ like η, η', ω, and ϕ <cit.>. This
indicates that the h_1 mesons are considered to be orbitally excited 
states as the isoscalar pseudoscalar or vector mesons with the same
quark content. Similarly, the a_1(1260) isovector axial-vector
meson is also regarded as the isovector qq̅ state. However, a
series of studies suggests that the a_1(1260) meson may be a
possible molecular state <cit.>.  Very recently, we
have investigated the b_1 isovector axial-vector mesons,
demonstrating that they can be dynamically generated by considering
the four different channels, i.e. πω, ηρ, πϕ,
and KK̅^* channels. Interestingly, it was shown that the
b_1(1235) arises from the b_1(1306) and
b_1(1356) <cit.>, indicating that the
b_1(1235) has a two pole structure. As will be shown in the current
work, the h_1(1415) meson originates from the two poles: 
h_1(1387) and h_1(1452). In fact, various hadrons exhibit the
two-pole structures. For example, the K_1(1270) may possibly be
regarded as the meson with the two-pole
structure <cit.>.  
Albaladejo et al. <cit.> showed that the
D^*(2400) arises as a two-pole structure, based on 
light pseudoscalar and D meson interactions in the coupled-channel
formalism (see a recent review <cit.> for detailed
discussion). The two-pole structure is also found in the baryonic
sector: Λ(1405) is now well established as the hyperon with
the two-pole structure <cit.>.  

In the present work, we will show that the discrepancy in the
experimental data on the mass of h_1(1415) is rooted in its two-pole
structure. To this end, we formulate the off-shell
coupled-channel formalism <cit.>, introducing six different channels, i.e. πρ,
ηω, KK̅^*, ηϕ, η'ω, and η'ϕ
of which the threshold energies lie below 2 GeV. 
We first construct the kernel amplitude corresponding to each channel,
using the meson-exchange diagrams. We compute the coupled
Blankenbecler-Sugar (BbS) equations, which are obtained from the
three-dimensional (3D) reduction of the  Bethe-Salpeter
equations <cit.>. We have
already investigated how the a_1(1260) and 
b_1(1235) axial-vector mesons, D_s0^*(2317) and B_s0^*
mesons, and the hidden charm pentaquark states are generated
dynamically within the same framework <cit.>.
We will demonstrate that the h_1(1170) and h_1(1415) are
dynamically generated even without introducing the corresponding pole
diagrams. Remarkably, the two poles emerge in the second Riemann
sheet, which are related to the h_1(1415) meson, of which one lies
just below the KK̅^* threshold, and the other is found to have
a larger mass than its threshold energy. 

The outline of the current work is sketched as follows: In
Section <ref>, we first explain how the kernel amplitude for
each channel can be constructed by using the Feynman diagrams based on
the effective Lagrangian. Then, we perform the partial-wave expansion
for the coupled BbS integral equations, so that we examine the
relevant partial waves with proper quantum numbers corresponding to
the h_1 mesons.  In Section <ref>, we discuss the results for
the h_1 mesons. We examine the role of each channel in producing
them dynamically. In particular, we focus on the two-pole structure of
the h_1(1415) meson. The last section summarizes the present work. 




§ GENERAL FORMALISM


The scattering amplitude is defined as 

    𝒮_fi = δ_fi - i (2π)^4 δ(P_f - P_i)
      𝒯_fi,

where P_i and P_f stand for the total four momenta of the initial
and final states, respectively. The transition amplitude
𝒯_fi for a two-body reaction can be derived from the
Bethe-Salpeter integral equations  

    𝒯_fi (p',p;s) = 𝒱_fi(p',p;s) 
    + 1/(2π)^4∑_k ∫ d^4q 
    𝒱_fk(p',q;s)𝒢_k(q;s) 𝒯_ki(q,p;s),

where p and p' denote the relative four-momentum of the
initial and final states, respectively. q is the momentum transfer
for the intermediate states in the center of mass (CM) 
frame. s represents the square of the total energy, which is just
one of the Mandelstam variables, s=P_i^2=P_f^2. The coupled
integral equations given in Eq. (<ref>) can be illustrated as in
Fig. <ref>. The summation Σ represents the inclusion of
various coupled channels. 

To avoid the complexity due to the four-dimensional integral equations,
we make a 3D reduction. Among several methods for the 3D reduction, we
employ the BbS scheme <cit.>, which expresses the two-body propagator in the form of  

    𝒢_k(q) = δ(q_0-E_k1(q)-E_k2(q)/2)
    	π/E_k1(q)E_k2(q)E_k(q)/s-E_k^2(q).

Here, E_k represents the total on-mass-shell energy of the
intermediate state, E_k = E_k1+E_k2, and q
designates the three-momentum transfer of the intermediate
state. Utilizing Eq. (<ref>), we obtain the following coupled BbS
integral equations  

    𝒯_fi (p',p) = 𝒱_fi
    	(p',p)+1/(2π)^3∑_k∫d^3
    	q/2E_k1(q)E_k2(q)𝒱_fk
    	(p',q)E_k(q)/s-E_k^2(q)
    	+iε𝒯_ki(q,p),

where p and p' are the relative three-momenta of the 
initial and final states in the CM frame, respectively. In this
manner, the T matrix can be generated, the entire Hilbert space
being considered with the off-shell components. 

Before we solve the coupled BbS integral equations, we need to
construct the kernel amplitudes 𝒱. We compute
𝒱_fi by using the effective Lagrangians for the
meson-meson interactions. Since the vector mesons are involved, we
consider hidden local symmetry, where the vector meson is considered
as a dynamic gauge boson <cit.>.  In
Ref. <cit.>, the effective interactions among vector,
vector, and pseudoscalar mesons were derived, based on the SU(3)
hidden local symmetry. The effective Lagrangians are then expressed as  

    ℒ_PPV   = -ig_PPV√(2) Tr([P,∂_μ P] V^μ),ℒ_VVV   = ig_VVV√(2) Tr((∂_μ V_ν
    - ∂_ν V_μ)V^μ V^ν),ℒ_PVV   =-g_PVV/m_V√(2) ε^
    μναβTr(∂_μ
    V_ν∂_α V_β P),

where P and V represent respectively the pseudoscalar and vector
matrices in flavor space  

    P    = [    1/√(2)π^0+1/√(6)η_8+1/√(2)η_1                              π^+                              K^+;                              π^- -1/√(2)π^0+1/√(6)η_8
    	+1/√(2)η_1                              K^0;                              K^-                             K̅^0             -2/√(6)η_8+1/√(2)η_1 ],
      V    = [  1/√(2)ρ^0_μ+1/√(2)ω_μ                  ρ_μ^+                 K_μ^*+;                  ρ_μ^- -1/√(2)ρ_μ^0+1/√(2)ω_μ                 K_μ^*0;                 K_μ^*-                K̅^*0_μ                    ϕ_μ ].

We assume the ideal mixing of the isoscalar vector meson singlet 
and octet. For η and η', we define them in terms of the 
pseudoscalar octet η_8 and singlet η_1: 

    η = η_8 cosθ_P - η_1sinθ_P,  η' = 
    	η_8 sinθ_P + η_1cosθ_P,

with mixing angle θ_P = -17^∘ taken from 
Ref. <cit.>. Note that the universal coupling constant is
given as g=g_PPV=g_VVV due to the hidden local symmetry.

As done in the previous works <cit.>, we consider the five different isoscalar axial-vector
channels coupled to the πρ channel, which are relevant to the
h_1 mesons, i.e., the ηω, KK̅^*, ηϕ,
η'ω, and η'ϕ channels. Thus, the kernel matrix is
now expressed as  

    𝒱   = [        𝒱_πρ→πρ        𝒱_ηω→πρ     𝒱_KK̅^*→πρ        𝒱_ηϕ→πρ       𝒱_η'ω→πρ       𝒱_η'ϕ→πρ;        𝒱_πρ→ηω        𝒱_ηω→ηω     𝒱_KK̅^*→ηω        𝒱_ηϕ→ηω       𝒱_η'ω→ηω       𝒱_η'ϕ→ηω;    𝒱_πρ→ KK̅^*    𝒱_ηω→ KK̅^* 𝒱_KK̅^*→ KK̅^*    𝒱_ηϕ→ KK̅^*   𝒱_η'ω→ KK̅^*   𝒱_η'ϕ→ KK̅^*;        𝒱_πρ→ηϕ        𝒱_ηω→ηϕ     𝒱_KK̅^*→ηϕ        𝒱_ηϕ→ηϕ       𝒱_η'ω→ηϕ       𝒱_η'ϕ→ηϕ;       𝒱_πρ→η'ω       𝒱_ηω→η'ω    𝒱_KK̅^*→η'ω       𝒱_ηϕ→η'ω      𝒱_η'ω→η'ω      𝒱_η'ϕ→η'ω;       𝒱_πρ→η'ϕ       𝒱_ηω→η'ϕ    𝒱_KK̅^*→η'ϕ       𝒱_ηϕ→η'ϕ      𝒱_η'ω→η'ϕ      𝒱_η'ϕ→η'ϕ;                ]


Each matrix element 𝒱_fi is obtained from the
corresponding tree-level Feynman diagram illustrated in
Fig. <ref>. Note that we do not have any pole diagrams in the
s-channel, which indicates that the h_1(1170) and h_1(1415)
mesons will be dynamically generated. However, the h_1(1595) meson
requires the s-channel pole digram, which we will discuss later its
physical implications. The Feynman amplitudes for the tree-level
diagrams are evaluated from the effective Lagrangians  
given in Eq.(<ref>). Imposing flavor SU(3) symmetry for the
coupling constants, we find that the following kernel 
amplitudes become null:  
𝒱_πρ→ηϕ, 𝒱_πρ→η'
ϕ, 𝒱_ηω→ηϕ, 𝒱_ηω→η'ϕ, 𝒱_ηϕ→η'ω and 
𝒱_η'ω→η'ϕ. 

The amplitudes of the meson exchange diagrams shown in 
Fig. <ref> from left to right are written as

    𝒜_P^u(p',p) =    -IS g^2_PPV 
    	 F^2(p,p') (2p_2-p_3)·ϵ^*
    	𝒫(p_1-p_4) (2p_4-p_1)·ϵ,
    𝒜_V^u(p',p) =    -IS g^2_
    	PVV/m_V^2 F^2(p,p') ε_μναβ p_3^μϵ^
    	*ν(p_3-p_2)^α 𝒫^βδ(p_1-p_4) ε_γσηδp_1^γϵ^σ(p_1-p_4)^η,
    𝒜_V^t(p',p) =    -IS g^2_PPV 
    	F^2(p,p') (p_2+p_4)^μ𝒫_μν
            (p_1-p_3)   ×[(2p_1-p_3)·ϵ^*ϵ^ν+(2p_3-p_1)
    	·ϵϵ^*ν-ϵ·ϵ^*(p_1+p_3)^ν],

where the IS factor is related to the corresponding SU(3) Clebsch-Gordan
coefficient and isospin factor. In Table <ref>, we list the
values of the IS factors for all relevant processes. For the coupling
constants, we use the values for the coupling constants:
g_PPV^2/4π=0.72  and g_PVV^2/4π=1.88 from the previous
works <cit.>. The propagators for the
spin-0 and spin-1 mesons are expressed by  
  
    𝒫(p)    = 1/p^2-m^2,   𝒫_μν(p) = 1/p^2-m^2(-g_μν+p_μ p_ν/m^2),

where m denotes the mass corresponding to the exchange meson. As
done in the previous works <cit.>, we have turned off the energy-dependence in the
denominator of the propagator.


Since hadrons have finite sizes, we introduce a form factor
at each vertex. We employ the following
parametrization <cit.>: 

    F(p,p') =
      (nΛ^2-m^2/nΛ^2+p^2+p'^2)^n,

where n is the power of the form factor. The form given in
Eq. (<ref>) has a notable advantage: the value of Λ
remains constant regardless of change in n. As n approaches
infinity, Eq. (<ref>) converges to a Gaussian 
form. In most cases, we use n=1. However, we need to use n=2 for
vector-meson exchange, because they have stronger momentum
dependence. While the cut-off masses Λ in Eq. (<ref>)
are experimentally unknown for the current hadronic processes, we
minimize associated uncertainties using the following strategy as done
in the previous works <cit.>: we choose Λ by adding approximately
(500-700) MeV to the corresponding masses of the exchange
meson. Consequently, we define the cutoff mass as
Λ=Λ_0+m. We set Λ_0 to 
be 600 MeV for most cases, as listed in Table. <ref>. However,
to fit the total cross section for πρ scattering, we need to
use larger values of Λ_0 for some of the vector-meson exchange 
diagrams, as shown in Table <ref>.

To compute the coupled integral equations, we utilize a partial 
wave decomposition, transforming the equation into a one-dimensional 
integral equation as follows

    𝒯^J(fi)_λ'λ (p',p) = 
    	𝒱^J(fi)_λ'λ (p',p) + 1/(2π)^3∑_k,λ_k∫q^2dq/2E_k1E_k2𝒱^J(fk)_λ'λ_k(p',
    	q)E_k/
    	s-E_k^2𝒯^J(ki)_λ_kλ
    	(q,p).

Here, λ', λ and λ_k denote the relative 
helicity of the final, initial and intermediate two-body states, 
respectively. Their corresponding momenta are represented by 
p', p and q, respectively. The
partial-wave component is given by

    𝒱^J(fi)_λ'λ(p',p) = 
    	2π∫ d(
    	cosθ)  d^J_λ',λ(θ) 𝒱^fi_λ'λ(p',p,θ),

where θ stands for the scattering angle, and d^J_λ',
λ are the Wigner d-matrices.

To obtain the transition amplitudes numerically, it is crucial to 
deal with the singularities in the BbS propagator. The
one-dimensional integral, free from energy singularities, takes the
form  

    𝒯^fi_λ'λ (p',p) = 
    	𝒱^fi_λ'λ (p',p) + 1/(2π)^3∑_k,λ_k[∫_0^∞dqqE_k/E_k1E_k2ℱ(q)
    	-ℱ(q̃_k)/s-E_k^2+ 1/2√(s)(ln|√(s)-E_k^thr/√(s)
    	+E_k^thr|-iπ)ℱ
    	(q̃_k)],

where q̃_k represents the momentum when 
E_k1+E_k2=√(s) and ℱ(q) is defined as 

    ℱ(q)=1/2q 𝒱^fk_λ'λ_k(p',
    	q)𝒯^ki_λ_kλ(q,
    	p).
 

Regularization is applied only when the total energy √(s) 
surpasses the threshold energy of the k-th channel, denoted as 
E_k^thr. Notably, the transition amplitudes derived from
these equations can be analytically continued to the complex
energy plane directly, as the energy singularities have been
eliminated. Moreover, Eq. <ref> enables the use  
of the matrix inversion method to compute the transition
amplitudes. The partial-wave transition amplitudes are then obtained
by transforming the helicity basis into the LSJ basis. For the
specific case of pseudoscalar and vector meson scattering, this
transformation is represented as 

    𝒯^JS_L'L = √((2L+1)(2L'+1))/2J+1∑_λ'λ(L'0Sλ'|Jλ')
    	(1λ' 00|1λ')(L0Sλ|Jλ)
    	(1λ 00|Sλ)𝒯^J_λ',
    	λ.

Here, (j_1m_1j_2m_2|JM) denotes the Clebsch-Gordan coefficient.


So far, we have focused on using only the t and u channel  
exchange diagrams. However, it is necessary to include the pole
diagram for the h_1(1595) meson in the s channel. It is   
essential to improve the results in comparison with the experimental
data on πρ scattering. A similar approach was demonstrated in
the case of ππ scattering <cit.>, where the
explicit inclusion of the f_0(1400) pole was necessary to reproduce
the experimental phase shift of the scalar-isoscalar channel in  
the energy region below 2 GeV. The presence of f_0(1400) influenced
both the high-energy region and the structure below 1 GeV. Similarly, we
incorporate the s-channel pole diagram for h_1(1595) in πρ
scattering, as illustrated in Fig. <ref>. The interaction
vertices are determined by the following effective Lagrangian  
  
    ℒ_h_1πρ = g_h_1πρ/m_h_1^0(∂_μρ_ν-∂_νρ_μ)
    	π∂^μ h_1^ν.

The bare mass and coupling of h_1(1595) are set to be  
m_h_1(1595)^(0)=1320 MeV and
g_h_1(1595)πρ^2/4π=0.54. Additionally,  
we employ the form factor in the s channel 

    F_s(p) = (Λ^2+(m_h_1^0)^2/Λ^2
    	+p^2)^4,

to ensure that the contribution of the pole diagram in the high 
momentum region is negligible. Consequently, the pole diagram has a 
small impact on the dynamical generation of the h_1 meson. We set 
the cutoff mass, denoted as Λ, to be 1920 MeV, following the 
previously mentioned rule. As a result, the dressed mass and width of 
h_1(1595) evaluated in the complex energy plane match the PDG 
average value <cit.>. We found the u-channel  
contribution to the transition amplitude to be negligible and omitted it  
for simplicity. The t-channel diagram is not allowed due to G
parity and isospin symmetry. The necessity of including the pole
diagram suggests that the h_1(1595) may be predominantly a 
genuine qq̅ state.  




§ RESULTS AND DISCUSSIONS 




 §.§ h_1 resonances

The ground state of the h_1 axial-vector meson was first observed in 
the 3π mass spectra of the charge exchange reaction π^- p →π^+π^-π^0 n <cit.>. Its existence was later 
confirmed by only one other experiment using the same
reaction <cit.>. According to the PDG, there
are two excited states of h_1 meson: h_1(1415) and
h_1(1595) <cit.>. The latter was observed in the ηω
mass distribution <cit.>, while the former  
exhibits an intriguing structure similar to the renowned Λ
(1405). Initially, referred to as h_1(1380), 
subsequent experimental studies determined its mass to be
approximately 1.42-1.44 GeV, leading to its renaming as
h_1(1415). However, a very recent experiment again suggests that the
mass of h_1(1415) is roughly 1.38 GeV. Currently, there is no
explanation for the conflicting mass measurements of h_1(1415).
Interestingly, h_1(1380) is located just below the KK̅^*
threshold, whereas the renamed h_1(1415) is found to be above
 KK̅^* threshold. This implies that h_1(1380) (h_1(1415))
 may have a two-pole structure.  In this work, we will examine each
 h_1 meson within the coupled-channel framework. Specifically, we
 will demonstrate that the conflicting mass measurements of
 h_1(1415) can be explained by a two-pole structure. 

In Ref. <cit.>, two axial-vector resonances, a_1 
and h_1, were identified in the charge exchange
reaction. Previously, we analyzed experimental data on the former one,
and in this current study, we extend our analysis to the  
latter one, keeping values of the parameters the same as the earlier 
investigation <cit.>. The assumption of mixing 
between η and η' alters the isospin value of η exchange 
in the KK̅^* elastic channel. However, the mixing has 
a negligible effect on the previous results for the a_1 meson, given
the fact that η exchange barely contributes to KK̅^* elastic 
scattering.


To describe the experimental data in the isoscalar channel, we relate
the total cross section to the transition amplitude 𝒯 using a 
constant factor C, expressed as

    σ≡ -C Im[𝒯_πρ(M_πρ)
    ],

where C has a different value from that in the isovector channel
due to the absorption of the isospin factor into C. The pole diagram
for h_1(1595) comes into essential play in enhancing the width of
h_1(1170), resulting in good agreement with the experimental
data. The total cross section for πρ scattering in the
isoscalar channel clearly reveals the resonance of the h_1(1170)
meson, of which the mass and width are determined to be m_h_1(1170)
= (1.19±0.06) GeV and Γ = (0.32± 0.05) GeV,
respectively <cit.>. 
The present results describe the experimental data well. We also
compare them with those in Ref. <cit.> that utilized a
single πρ channel including explicitly the h_1(1170)
pole diagram. As shown in Fig. <ref>, the single channel with
the explicit pole of the h_1(1170) is insufficient to describe the 
experimental data, specifically the h_1(1170) peak structure. In
contrast, we find that the h_1(1595) pole diagram has an important
contribution to explain the h_1(1170) resonance. Moreover, 
various coupled channels play essential roles in dynamically
generating the h_1 (1170) resonance. The dynamical generation of
h_1 (1170) has also been observed in Ref. <cit.>.  
So, we conclude that the h_1(1170) does not solely originate from the
qq̅ state but contains a substantial component of the molecular
state. Furthermore, we predict a small peak structure near the
KK̅^* threshold, which is barely seen in the experimental
data.


To identify dynamically generated resonances in the current approach,
we examine the T amplitude in the second Riemann sheet.   
Since the T amplitude generated by Eq. (<ref>) is
a meromorphic function in the complex energy plane, we can directly
identify the h_1 resonances in the second Riemann sheet. In
Fig. <ref>, the contour plot of the modulus of T clearly
exhibits the existence of four poles. The first pole below the
ηω threshold is positioned at (1080-i118) MeV, so its mass
and width are respectively given as 1080 MeV and 236 MeV.
The value of the width given in the PDG is (375± 35)
MeV <cit.>, which was taken from the fit by using the Bowler
model. Compared with it, the current result is smaller than the
empirical data. In Ref. <cit.>, the corresponding pole for
h_1(1170) was found at √(s)=(919-i17) MeV, which deviates
significantly from the experimental data.
The second and third poles are located at (1387-i6) MeV and
(1452-i51) MeV, respectively. These two poles are related to 
h_1(1415), once called as h_1(1380). Since the second pole lies
just below the KK̅^* threshold, it may be considered as the
KK̅^* molecular state in the isoscalar channel. Interestingly,
its width is very small. The third pole is located at about 60 MeV above 
the KK̅^* threshold, and its width is 102 MeV. While the
average value from the PDG is given as (78± 11) MeV, the
experimental data on its width ranges from (66± 10_-10^+12)
MeV <cit.>  to (170± 80)
MeV <cit.>. Thus, the width of h_1(1415)
should be measured more precisely. If h_1(1415) has a two-pole
structure, its width may be determined more precisely. 

Finally, the fourth pole is observed at (1603-i158) MeV that
corresponds to the h_1(1595) meson, placed above the ηϕ
threshold. Note that we have included the pole diagram for h_1(1595)
in the s channel with the bare mass m_h_1(1595)^(0)=1320
MeV. It is dressed to be 1603 MeV,   
as the pole diagram for h_1(1595) is “renormalized” by
the coupled integral equations. When considering only the
πρ elastic channel, the dressed mass is smaller than the bare
one. However, the pole diagram gains additional mass after being
coupled to all channels. A similar effect is observed in ππ
elastic scattering, where the scalar-isoscalar meson f_0(1400) has a
bare mass of 1300 MeV with the scalar coupling, and after coupling to
other channels, the physical mass becomes 1400
MeV <cit.>. While a recent study shows that the
h_1(1595) is a ground state of a light tetraquark
state <cit.>, the current result suggests that the
structure of the h_1(1595) has a large component of the qq̅,
being dressed by various channels. 

The coupling strengths of the h_1 resonances can be derived from the
residues of the 𝒯 matrix, defined as 

    ℛ_a,b   := lim_s→ s_R(s-s_R)
    	𝒯_a,b/4π=g_ag_b.

It is impossible to determine the signs of the coupling
strengths, so we choose the coupling strengths to the πρ  
channel to be positive. Then, we can determine the relative signs for
other coupling strengths. Table <ref> lists the results for the
coupling strengths to all possible channels.

The first resonance, h_1(1170), couples most strongly to the
πρ channel, which is in line with the experimental 
observations of the h_1(1170)→πρ
decay <cit.>.  
The next strongest coupling comes from the ηϕ
channel. Considering that the ηϕ threshold  energy is
around 1570 MeV, it is remarkable that the ηϕ channel is the
second most dominant one for h_1(1170). This indicates that the
h_1(1170) contains a significant ss̅ component, resembling
the case of dynamically generated a_1(1260) and
b_1(1235) <cit.>. The ηϕ
channel couples most strongly to the second and third
resonances, located at (1387-i6) MeV and (1452-i51) MeV, 
which are considered to be h_1(1415). As depicted in
Fig. <ref>, one can clearly observe the strongest peak structure 
below the KK̅^* threshold in the squared modulus of the
ηϕ→ηϕ transition amplitude multiplied by
q_πρ, q_πρ|T|^2, where q_πρ is the
magnitude of the momentum of the πρ system. 
Interestingly, we also have nonvanishing πρ→ηϕ
transition amplitudes, as shown in Fig. <ref>. 
Although we impose the flavor SU(3) symmetry, which results in the
absence of the πρ→ηϕ kernel amplitude, the
πρ→ηϕ transition amplitude is generated through the
KK̅^* intermediate states, revealing a resonance
structure below the KK̅^* threshold.  
As expected, the second resonance has a strong coupling strength to
the KK̅^* channel (see also the dash-dotted curve in
Fig. <ref>). On the other hand, the coupling strengths to 
the S-wave ηω, KK̅^*, η'ω, and
η'ϕ channels are of the same order. It is worth noting that
the D-wave πρ channel has a sizable coupling strength. 

As for the h_1(1595), many channels are strongly coupled to it. As
explained eralier, the mass of h_1(1595) would have been less than
the bare mass had we considered the πρ channel only. As shown
in Table <ref>, the πρ, ηω, KK̅^*, and
η'ω contribute to the generation of the h_1(1595) meson.  



 §.§ Two-pole structure of the h_1(1415)


Table <ref> presents the mass of the h_1(1415) as measured by
various experiments. The LASS Collaboration initially detected a
signal of h_1(1415) in the KK̅π system using the K^-p→
K_S^0K̅πΛ reaction <cit.>. The data
exhibited an enhancement near 1.4 GeV, diminishing rapidly as the
energy increased. Using the Breit-Wigner parameterization, the mass and
width were determined to be M=1380± 20 MeV and Γ= 80± 30
MeV, respectively. 
Subsequent experiments, however, reported different results. The
Crystal Barrel Collaboration <cit.> performed
proton annihilation to produce the KK̅π system and discovered
the h_1 meson above the KK̅^* threshold, with a mass of
M=(1440± 60) MeV and a width of Γ=(170± 80) MeV. Two
experimental results from the BESIII Collaboration, obtained from
different charmonium state decays, confirmed these findings with a
slightly lower mass. It is noteworthy that interference influences the
determination of the mass and width of the
h_1(1415) <cit.>, and these results align 
with those of the Crystal Barrel experiment. 
In a recent experiment conducted by the
BESIII <cit.> Collaboration, the mass and width of the
h_1(1415), measured in the γη' invariant mass, were found
to be consistent with those obtained by the LASS Collaboration but
contradictory to the two previous BESIII results. This discrepancy
strongly suggests that the h_1(1415) does not originate from a
single pole, indicating a more complex structure. Evidently, this
structure cannot be explained by the quark model alone. Consequently,
based on the results of this study, we propose a two-pole structure to
account for the discrepancy in the measurements of the h_1(1415)
meson mass. 


The current work reveals two poles around 1.4 GeV, one at (1387
-i6) MeV and the other at (1452-i51) MeV, as shown in
Table <ref>. Figure <ref> demonstrates how these two poles
appear in the complex energy plane. The lower pole is located close to
the KK̅^* threshold, while the higher pole is situated 80 MeV
above it. Due to its narrowness, detecting the lower pole
experimentally may pose a challenge. 
Examining the coupling strengths in Table <ref>, we observe that
both poles couple strongly to open strange and hidden strange
channels. However, the higher pole couples to the ηϕ channel
far more strongly than to other channels. This suggests that the
higher pole might be an ηϕ molecular state, while the lower
pole, being located very close to the KK̅^* threshold, could be
a KK̅^* molecular state. 
Notably, the g_KK̅^* for the higher pole has a larger
imaginary part than its real part, causing destructive
interference. This results in the disappearance of the higher pole in
KK̅^* elastic scattering, a feature also observed in the case of
the higher pole of the b_1 meson in a previous
study <cit.>. This characteristic is the essential
clue to explaining the absence of the higher pole in 
the LASS Collaboration experiment. 

Based on this analysis, we propose an explanation for the conflicting 
mass measurements of h_1(1415). As shown in Fig. <ref>, the
crucial aspect is that the higher pole vanishes in KK̅^* elastic
scattering, leaving only the KK̅^* threshold enhancement. The
LASS collaboration investigated the h_1(1415) in the K^-p→
K_S^0K̅πΛ process, which can effectively be represented
as KK̅^* elastic scattering. They observed a threshold
enhancement near the KK̅^* threshold, aligning with the current
results. In contrast, the Crystal Barrel <cit.>
and BESIII experiments <cit.> measured
the KK̅π state as a final state, which can be considered as
the KK̅^* state. Consequently, both collaborations measured the 
h_1(1415) above the KK̅^* threshold. The most recent
measurement from the BESIII Collaboration comes from the J/ψ→γη'η' decay. 
In conclusion, we propose that the h_1(1415) can be constructed from
two different resonances: the h_1(1380) and h_1(1440) mesons, with
the higher pole vanishing in KK̅^* elastic scattering. This
two-pole structure accounts for the seemingly conflicting experimental
results. 



§ SUMMARY AND CONCLUSIONS

In this work, we aimed at investigating the isoscalar axial-vector
h_1 mesons using a coupled-channel formalism. We first constructed
kernel amplitudes from meson-exchange diagrams in the t- and
u-channels, derived from effective Lagrangians based on hidden local
symmetry. In addition, we introduced the pole diagram for
h_1(1595) to generate the h_1(1595) resonance. We found that 
the h_1(1595) pole diagram also has a significant effect on the
generation of h_1(1170). Six channels were incorporated: πρ,
ηω, KK̅^*, ηϕ, η'ω, and η'ϕ
channels. We solved the off-shell coupled integral equations and
discussed the dynamical generation of the h_1(1170).  
The present analysis revealed two poles at (1387-i6)
MeV and (1452-i51) MeV, exhibiting a two-pole structure of the
h_1(1415) meson. This two-pole structure may resolve the discrepancy
in the experimental data on the mass of h_1(1415). The results
showed that the lower pole couples strongly to the KK̅^*
channel, while the higher pole couples predominantly to the ηϕ
channel. This provides an essential clue to understand the nature of
h_1 mesons and explains possible discrepancies in the mass of
h_1(1415). The two-pole structure of h_1(1415) can account for the
seemingly conflicting experimental results, with the lower pole
corresponding to h_1(1380) and the higher pole to
h_1(1440). Notably, the higher pole vanishes in KK̅^* elastic
scattering, which explains why some experiments observe only the lower
pole.  
 

  The authors are grateful to Terry Mart for valuable discussions. Part of
  the work was done at Department of Physics, University of Indonesia.
  HCK wants to express his gratitude to Tetsuo Hyodo, Makoto Oka, and
  Qian Wang for valuable discussions. The present work is supported by
  Inha University Research Grant in 2024 (No.73014-1).   



apsrev4-2

