Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/11/24
Proc. 14th Int. Conf. on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
JPS Conf. Proc. 13, 010010 (2017)https://doi.org/10.7566/JPSCP.13.010010

Formation of Possible η ′(958)-Nucleus Bound
States and η ′N Interaction
Hideko Nagahiro
Department of Physics, Nara Women’s University, Nara 630-8506, Japan
E-mail: nagahiro@cc.nara-wu.ac.jp(Received November 14, 2016)
The formation cross sections of η ′ (958)-nucleus systems by missing mass spectroscopies are
discussed for the study of in-medium modification of the mass of the η ′ (958) meson. Weconclude that the evidence of a possible mass reduction of η ′ can be observed as a peakstructure around the η ′ threshold. The spectroscopy of the (p, d) reaction has been performed
at GSI. The bound state formations by the (γ, p) reaction can be done at SPring-8.
KEYWORDS: η ′ (958) meson, mesic nuclei

1.

Introduction

The light pseudoscalar mesons (π, K, η) are considered as the Nambu-Goldstone bosons
associated with the dynamical chiral symmetry breaking. The ninth pseudoscalar mesonη ′ (958), however, has a heavier mass than the other pseudoscalar mesons, which is believed
to be explained by the breaking of the axial UA (1) symmetry at the quantum level referredas the UA (1) anomaly. There are many theoretical studies for the mass of the η ′ not only invacuum but also in finite temperature and/or finite density [1–4].
In Refs. [5–7], it has been pointed out theoretically that the anomaly eﬀect can contributethe mass of the η ′ meson only with the presence of the chiral symmetry breaking. In otherwords, if the chiral symmetry is partially restored in the nuclear medium, one can expect themass reduction of the η ′ meson in medium. For example, a model calculation [8] based on
the Nambu-Jona-Lasinio model shows a large mass reduction of the η ′ which is as large as
150 MeV at normal saturation density ρ0 . This mass reduction is coming from the decreasing⟨q̄q⟩ condensate in-medium, which makes UA (1) anomaly eﬀect smaller. If such a large mass
reduction takes place, we can observe the eﬀect as the η ′ -nucleus bound states.
The possible formation spectra of the η ′ -mesic nuclei was first investigated theoreticallyin Ref. [9] by the (γ, p) reaction. In Ref. [10] the formation spectra in the (p, d) reaction havebeen calculated theoretically, and the experiment has been performed at GSI [11], where no
distinct structures are observed near the η ′ threshold owing to the large mass reduction witha relatively small absorption width.
In this letter, we show the comprehensive calculation of the formation spectra of the η ′ mesic nuclei by the (γ, p) reaction with various set of the optical potentials and the emitted
proton angles. The corresponding experiment can be performed at SPring-8. We decompose the total spectra into two parts; the contribution from the η ′ -escape process and theconversion part of the possible absorption of η ′ . The decomposition helps to reduce possible
background contribution and also gives useful information for coincident measurements of thedecay particle from the η ′ . The detailed discussions are given in Ref. [10] and the references
therein.
1■■■010010-1©2017 The Author(s)This article is available under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work must maintain attribution to theauthor(s) and the title of the article, journal citation, and DOI.

Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/11/24

010010-2

JPS Conf. Proc. 13, 010010 (2017)2.

Formulation

In this article, we consider the (γ, p) reaction for the formation of the η ′ -mesic nuclei,
which can be performed at existing facilities like SPring-8. Missing mass spectroscopy is
considered here and it has been proved to be a powerful tool for the formation of the mesonbound states [12, 13]. In this spectroscopy, one observes only an emitted particle in a final
state, and obtains the double diﬀerential cross section d2 σ/dΩdE as a function of the energyof the η ′ -nucleus system which is uniquely determined by the emitted particle energy bymeans of the energy conservation law.
To evaluate the formation cross section, we use the Green’s function method [14]. In
this method, the reaction cross section is assumed to be separated into the nuclear response
function R(E) and the elementary cross section of the γp → pη ′ process with the impulse
approximation:( 2 )( )labd σdσ=× R(E),(1)dΩdE A(γ,p)(A−1)⊗η′dΩ p(γ,p)η′where the nuclear response function R(E) is given in terms of the in-medium Green’s function
G(E) as∑∫1
R(E) = − Im
drdr′ Tf† (r)G(E; r, r′ )Tf (r′ ) .
(2)πfHere, the summation is inclusively taken over all possible final states. The amplitude Tf
describes the transition of the incident photon to a proton hole and the outgoing proton:[]
Tf (r) = χ∗f (r) Yl∗η′ (r̂) ⊗ ψjp (r)χi (r)(3)
JM

with the proton hole wave function ψjp , the distorted waves of initial and final particles χiand χf , and the η ′ angular wave function Ylη′ (r̂). For the proton hole, we use the harmonic
oscillator wave function for simplicity. The Green’s function G(E) contains the η ′ -nucleus
optical potential in the Hamiltonian as
G(E; r, r′ ) = ⟨p−1 |ϕη′ (r)1ϕ†η′ (r′ )|p−1 ⟩′E − Hη + iϵ(4)where ϕ†η′ is the η ′ creation operator and |p−1 ⟩ is the proton hole state.
The elementary cross section for in center-of-mass frame is roughly estimated to be 50[nb/sr] for Eγ = 2.5 GeV [15]. The elementary cross section in the laboratory frame in Eq. (1)are then evaluated to be 519 [nb/sr], 504 [nb/sr] and 472 [nb/sr] for the proton angle 1, 6
and 11 degrees.
The Green’s function G(E, r, r′ ) is calculated by solving the Klein-Gordon equation withthe appropriate boundary condition. Thus, the Green’s function represents both the η ′ meson
scattering states and bound states. If there are a quasi-bound state of the η ′ meson, thespectral function has a peak structure at the corresponding energy. This can be seen in theformation spectra as a signal of the bound state.
The optical potential Uη′ (r) can be written as
U (r) = V (r) + iW (r)(5)

where V and W denote the real and imaginary parts of the optical potential. We employ theWoods-Saxon type potential as
V (r) = V0

ρ(r)
,ρ0

ρ(r) =2■■■

ρN()
1 + exp r−Ra(6)Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/11/24010010-3

JPS Conf. Proc. 13, 010010 (2017)

where ρ0 is the normal saturation density, and R = 1.18A1/3 − 0.48 fm, a ∫= 0.5 fm with
nuclear mass number A. A normalization factor ρN is determined such that d3 rρ(r) = A.
In this letter, we show the (γ, p) spectra with potential depth V0 = −20, −50 and −100
MeV to discuss the observation possibility. The imaginary part W0 is fixed to be 12 MeV
corresponding to the observed in-medium width of η ′ Γ ∼ 25 MeV [16, 17].

3.

Numerical results

In Fig. 1, we show the calculated spectra of the η ′ -nucleus system for the 12 C target
with the photon energy Eγ = 2.5 GeV. The emitted proton angles are 1, 6 and 11 degrees
from top to bottom as indicated in the figure. The potential strength (V0 , W0 ) = −(20, 12)
MeV, −(50, 12) MeV and −(100, 12) MeV from left to right. The horizontal axis indicatesthe excitation energy Eex − E0 defined asEex − E0 = −Bη′ + [Sp (jp ) − Sp (ground)],(7)

where Bη′ is the η ′ binding energy and Sp (jp ) the proton separation energy from the protonsingle-particle level jp . Sp (ground) indicates the separation energy from the proton level
corresponding to the ground state of the daughter nucleus. E0 is the η ′ production thresholdenergy.
As shown in Fig. 1, the case of (V0 , W0 ) = −(100, 12) MeV with the forward proton angle(θp = 1 deg.), we can see the clear peak structure corresponding to the η ′ -nucleus boundstate. We can see also the significant enhancement at the η ′ threshold. This peak structure
at the threshold is not a bound state but is the so-called threshold enhancement due to theattractive potential. In the shallow potential case of (V0 , W0 ) = −(20, 12) MeV, althoughsome shallow bound states exist, one cannot see isolated peak structure in the total spectra.
This is because of finite momentum transfer in this reaction [9] which enhances higher angularmomentum components of η ′ giving a large contribution in quasi-free region.
In the middle and lower panels in Fig. 1, we show the formation spectra with the finite
proton angles θp = 6 degrees and 11 degrees to see the angular dependence of the spectra.
We can see some peak structures around the threshold in some cases of the finite angles, but
it is clear that the observation possibility of the peak structure becomes smaller for larger
proton angle. We can see also that the cross sections for larger proton angle becomes smaller
due to the larger momentum transfer for the larger proton angle.
In the Green’s function method [14], one can separately evaluate each contribution to thespectrum coming from the diﬀerent η ′ process. In Fig. 2, we show the decomposed spectrawith (V0 , W0 ) = −(0, 12) MeV (no attraction case) and −(100, 12) MeV (large attractioncase) with θp = 1 degree. We decompose the total spectra into two parts; the contribution
from the η ′ -escape process and the conversion part of the possible absorption of η ′ . In thepresent formalism, we cannot specify the source of the imaginary part.
From this figure, we can imagine a spectral shape if one measures coincidently two photon
in the final state coming from η ′ → γγ decay which is possible by using LEPS2/BGOegg [18,19]. Although our formulation does not include the natural decay process of η ′ , but the coincident measurement of η ′ → γγ together with the forward proton might give the contributionas (η ′ -escape part)× BR(η ′ → γγ). The branching ratio BR(η ′ → γγ) is 2%, and then thecross section can be roughly 0.2 [nb/sr]. In fact, even below the η ′ threshold, the virtual η ′ candecay into γγ. This contribution can be estimated as (η ′ -conversion part)× BR∗ (η ′ → γγ).
Here, BR∗ is the branching ratio in medium and is very small as 5 keV/24 MeV∼ 10−4 . Therefore, by the coincident measurement of η ′ → γγ, we expect that one can obtain information
mainly on the quasi-free η ′ contribution above the threshold.
3■■■

Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/11/24010010-4

JPS Conf. Proc. 13, 010010 (2017)

161412total

108642098Fig. 1. Calculated spectra of the 12 C(γ, p)11 B ⊗ η ′ reaction for the formation of the η ′ -nucleus
systems with the photon energy Eγ = 2.5 GeV and the proton angles θp = 1, 6 and 11 degrees asfunctions of the excited energy Eex . E0 is the η ′ production threshold. The η ′ -optical potential andthe proton angles are indicated in the figure. The thick black lines show the total spectra and thecolored solid and dashed lines indicate subcomponents of (nℓp )−1
p ⊗ ℓη ′ as indicated in the figure,′where (nℓp )−1is
theproton-holestateandℓistheangularmomentumof η ′ .
ηpAs we can see from the figure, the shape of the η ′ -escape part is distorted due to theexistence of the attractive potential. The diﬀerence between the spectra with and without theattractive potential might become clearer if one plots them as a function of η ′ energy insteadof the excitation energy Eex − E0 , the latter includes the proton separation energy [Sp (jp ) −Sp (ground)] [20]. To discuss the observation possibility with the coincident measurement
more in detail, one needs further studies in future.

4.

Discussion

In the preset letter, we fix the strength of the imaginary part of the optical potential
W0 = 12 MeV to reproduce the observed in-medium η ′ width Γ ∼ 25 MeV [16,17]. In Ref. [21],η ′ -nucleus optical potential based on the chiral unitary approach [22] has been evaluated up
to the second order terms accounting for η ′ absorption by two nucleons. There the possible
contribution from the UA (1) axial anomaly eﬀect [23, 24] to the η ′ -nucleus optical potential
has been considered using the eﬀective hadronic model. Recently, η ′ N interaction has been
discussed in an eﬀective theory based on the on the linear realization of the SU (3) chiral4■■■Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/11/24

010010-5

JPS Conf. Proc. 13, 010010 (2017)

Fig. 2. Calculated spectra of the 12 C(γ, p)11 B ⊗ η ′ reaction for the formation of the η ′ -nucleus
systems with the photon energy Eγ = 2.5 GeV and the proton angle θp = 1 degree as functions ofthe excited energy Eex . E0 is the η ′ production threshold. The η ′ -optical potential is (a) (V0 , W0 ) =−(100, 12) MeV and (b) (V0 , W0 ) = −(0, 12) MeV. The thick black lines show the total spectra, thered dashed lines denote the η ′ escape part and the blue dot-dashed lines the conversion part.

symmetry [25]. Furthermore, some eﬀective models predict the possible η ′ N bound state [25].
If such a two-body bound state exists, it must aﬀect the η ′ -nucleus many body system. Suchapproaches to η ′ N interaction described in hadron eﬀective theories are important when
we discuss the possibilities of the coincident measurement of the decay particle from theη ′ -nucleus bound states.

5.

Summary

We have calculated the formation spectra of the η ′ (958)-nucleus systems for the missing
mass spectroscopies to investigate the expected mass reduction of the η ′ meson in nuclear
medium. In this letter, we show the spectra with (γ, p) reaction with various optical potential
and the proton angles to see the observation possibility of the possible peak structure. Sofar, a relatively small scattering length of the η ′ -nucleon has been reported [26] and no clear
peak structure of the η ′ -nucleus bound state has been observed at GSI [11]. To understandthem, we have to develop an advanced understanding of η ′ -nucleon interaction from themicroscopic and fundamental point of view. We believe that the present theoretical results
and/or advanced developments are very important to obtain deeper insight into the propertiesof the η ′ (958) meson and symmetries of QCD.
References[1] R. D. Pisarski and F. Wilczek: Phys. Rev. D 29 (1984) 338.
[2] T. Kunihiro and T. Hatsuda: Phys. Lett. B 206 (1988) 385 Erratum: [Phys. Lett. B 210 (1988)
278].
[3] S. D. Bass and A. W. Thomas: Phys. Lett. B 634 (2006) 368.
[4] K. Tsushima: Nucl. Phys. A 670 (2000) 198.
[5] S. H. Lee and T. Hatsuda: Phys. Rev. D 54 (1996) R1871.
[6] T. D. Cohen: Phys. Rev. D 54 (1996) R1867.
[7] D. Jido, H. Nagahiro and S. Hirenzaki: Phys. Rev. C 85 (2012) 032201.
[8] H. Nagahiro, M. Takizawa and S. Hirenzaki: Phys. Rev. C 74 (2006) 045203.
[9] H. Nagahiro and S. Hirenzaki: Phys. Rev. Lett. 94 (2005) 232503.
[10] H. Nagahiro, D. Jido, H. Fujioka, K. Itahashi and S. Hirenzaki: Phys. Rev. C 87 (2013) 045201.
[11] Y. K. Tanaka et al. (η-PRiME/Super-FRS Collaboration): Phys. Rev. Lett. 117 (2016) 202501.
5■■■

Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/11/24010010-6

JPS Conf. Proc. 13, 010010 (2017)[12] H. Toki, S. Hirenzaki and T. Yamazaki: Nucl. Phys. A 530 (1991) 679.
[13] S. Hirenzaki, H. Toki and T. Yamazaki: Phys. Rev. C 44 (1991) 2472.
[14] O. Morimatsu and K. Yazaki: Nucl. Phys. A 435 (1985) 727.
[15] V. Crede et al. [CBELSA/TAPS Collaboration]: Phys. Rev. C 80 (2009) 055202.
[16] M. Nanova et al. [CBELSA/TAPS Collaboration]: Phys. Lett. B 710 (2012) 600.
[17] M. Nanova [CBELSA/TAPS Collaboration]: Prog. Part. Nucl. Phys. 67 (2012) 424.
[18] N. Muramatsu et al. [LEPS Collaboration]: Nucl. Instrum. Meth. A 737 (2014) 184[19] H. Shimizu et al., Letter of Intent for the BGOegg Experiment, (2012)
http://www.lns.tohoku.ac.jp/~hadron/img/BGOegg_proposal-E_121125.pdf[20] S. Friedrich et al. [CBELSA/TAPS Collaboration]: Phys. Lett. B 736 (2014) 26.
[21] H. Nagahiro, S. Hirenzaki, E. Oset and A. Ramos: Phys. Lett. B 709 (2012) 87.
[22] E. Oset and A. Ramos: Phys. Lett. B 704 (2011) 334.
[23] K. Kawarabayashi and N. Ohta: Prog. Theor. Phys. 66 (1981) 1789.
[24] B. Borasoy: Phys. Rev. D 61 (2000) 014011.
[25] S. Sakai and D. Jido: Phys. Rev. C 88 (2013) no.6, 064906.
[26] E. Czerwinski et al.: Phys. Rev. Lett. 113 (2014) 062004.

6■■■