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Nature of the vector resonance Yð2175Þ

S. S. Agaev,

K. Azizi ,

2,3

and H. Sundu

Institute for Physical Problems, Baku State University, Az-1148 Baku, Azerbaijan

Department of Physics, Univ ersity of Tehran, North Karegar Avenue, Tehran 14395-547, Iran

Department of Physics, Doˇguş University, Aci badem-Kadiköy, 34722 Istanbul, Turkey

Department of Physics, Kocaeli University, 41380 Izmit, Turkey

(Received 22 November 2019; accepted 30 March 2020; published 10 April 2020)

Spectroscopic parameters and decay channels of the vector resonance Yð2175Þ are studied by

considering it as a diquark-antidiquark state with the quark content su

u. The mass and coupling of

the tetraquark Yð2175Þ are calculated using the QCD two-point sum rules by taking into account various

quark, gluon and mixed condensates up to dimension 15. Partial widths of its strong decays to ϕf

ð980Þ,

ϕη, and ϕη

are computed as well. To this end, we explore the vertices Yϕf

ð980Þ, Yϕη, and Yϕη

, and

calculate the corresponding strong couplings by means of the QCD light-cone sum rule method. The

coupling G

Yϕf

of the vertex Yϕf

ð980Þ is found using the full version of this method, and by treating the

scalar meson f

ð980Þ as a diquark-antidiquark tetraquark state. The couplings g

Yϕη

and g

Yϕη

,however,are

calculated by applying the soft-meson approximation to the light-cone sum rule method. Prediction for the

mass of the resona nce m

¼ð2173  85Þ MeV is in excellent agreement with the data of the BABAR

Collaboration [Phys. Rev. D 74, 091103 (2006)], and within errors of calculations is compatible with the

result reported by BESIII [Phys. Rev. D 91, 052017 (2015)]. The full width Γ

full

¼ð91.1  20.5Þ MeV of

the Yð2175Þ saturated by its three strong decay channels is in a reasonable agreement with existing

experimental data.

DOI: 10.1103/PhysRevD.101.074012

I. INTRODUCTION

The resonances fYg with the quantum numbers J

−−

constitute two families of particles, interpretation of

which is one of interesting and yet unsettled problems of

the high energy physics. Members of the first family

populate the mass region m ¼ 4.2–4.7 GeV, and were

observed by different collaborations. These resonances

reside very close to each other, and are more numerous

than vector charmonia

cc from this mass range. Hence, at

least some of these resonances have different quark-gluon

structure, and are presumably states built of four valence

quarks. Besides a suggestion about the tetraquark nature of

heavy fYg states, there are various alternative models to

account for their parameters and decay channels.

Another family of the fYg resonances occupies the light

segment of meson spectroscopy and incorporates the

famous “old” state Yð2175Þ, and new ones Xð2239Þ and

Xð2100Þ seen recently. The structure Yð2175Þ was dis-

covered by the BABAR collaboration in the initial-state

radiation process e

−

→ γ

ISR

ϕf

ð980Þ as a resonance in

the ϕf

ð980Þ invariant mass spectrum [1]. The mass and

width of this resonance measured by BABAR amount to

m ¼ð2175  10  15Þ and Γ ¼ð58  16  20Þ MeV,

respectively. The same structure was seen also by the

BESIII collaboration in the exclusive decay J=ψ →

ηϕπ

−

[2]. The spectroscopic parameters of the Yð2175Þ

extracted in this experiment differ from original results and

are m ¼ð2200  6  5Þ and Γ ¼ð104  15  15Þ MeV.

Recently, anomalously high cross section at

ﬃﬃﬃ

2232 MeV was observed by the BESIII collaboration in

the channel e

−

→ ϕK

−

, which may be explained by

interference of different resonances [3]: more data are

necessary to decide whether Yð2175Þ contributes to

enhancement of this cross section or not. Because the

Yð2175Þ was seen by BABAR and confirmed by the BESII,

BESIII, and Belle collaborations [2,4,5], its existence is not

in doubt, but an uncertain situation with the mass and full

width of this resonance requires further experimental and

theoretical studies.

Other resonances that may be considered as candidates

to light exotic vector mesons were discovered by the

BESIII collaboration. The first of them, i.e., Xð2239Þ,

was fixed in the process e

−

→ K

−

as a resonant

structure in the cross section shape line [6]. The second

resonance Xð2100Þ was seen in the ϕη

mass spectrum in

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW D 101, 074012 (2020)

2470-0010=2020=101(7)=074012(12) 074012-1 Published by the American Physical Society

the process J=ψ → ϕηη

[7]. The quantum numbers of

Xð2239Þ were determined unambiguously, whereas a sit-

uation with Xð2100Þ remains unclear. Indeed, because of a

scarcity of experimental information the collaboration

could not clearly distinguish two 1

and 1

−

assumptions

for the spin-parity J

of the resonance Xð2100Þ. Hence,

BESIII extracted its mass and full width using both of these

options. Obtained results differ from each other and depend

on assumption about the parity of the state Xð2100Þ.

Theoretical interpretations of the light vector resonances

comprise all available models and approaches of the high

energy physics. Thus, the Yð2175Þ was considered as 2

excitation of the ordinary

ss meson [8,9]. It was explained

also as a dynamically generated state in the ϕK

K system

[10], or as a resonance appeared due to self-interaction

between ϕ and f

ð980Þ mesons [11]. A hybrid meson with

structure

ssg [12] and a baryon-antibaryon qqs

s state

which couples strongly to the Λ

Λ channel are among

alternative models of the Yð2175Þ resonance. There were

attempts to interpret Yð2175Þ as a vector tetraquark with

s or ss

s contents [13–15] (see Ref. [6] for other

models). The resonance Xð2100Þ was examined in the

framework of the QCD sum rule method in Refs. [16,17].

Recently, we explored the light resonances Xð2100Þ and

Xð2239Þ as the axial-vector and vector ss

s tetraquarks

[18], respectively. Besides spectroscopic parameters we

also inv estigated the strong decays Xð2100Þ → ϕη

and

Xð2100Þ → ϕη, and calculated their partial widths. Pre-

dictions obtained for the mass and width of the axial-vector

state allowed us to identify it with the resonance Xð2100Þ,

because our theoretical predictions are very close to its para-

meters measured by the BESIII collaboration. We classified

Xð2239Þ as the vector tetraquark ss

s and found a reason-

able agreement between theoretical and experimental results.

In the present work, we continue our investigations of the

light vector resonances and concentrate on features of the

state Yð2175Þ (hereafter, Y). Our treatment of this state dif-

fers from existing analyses. Thus, we consider it as a vector

tetraquark with content su

u rather than as a state ss

The traditional assumption about the quark content of the Y

is inspired by the fact that it was discovered in ϕf

ð980Þ

invariant mass distribution. Because in the standard model

of mesons one treats the ϕ and f

ð980Þ as vector and scalar

particles with the same

ss structure, then it is natural to

assume that Y is built of four valence s quarks.

But the conventional quark-antiquark model of mesons

in the case of light scalar nonets meets with evident

difficulties. In fact, the nonet of scalar mesons in the

model may be realized as 1

states. In accordance with

various computations, masses of the scalars 1

are higher

than 1 GeV. They were identified with the isoscalar mesons

ð1370Þ and f

ð1710Þ, the isovector a

ð1450Þ or iso-

spinor K



ð1430Þ states, i.e., with scalars from the second

light nonet. But masses of the mesons from the first nonet

are lower than 1 GeV, and they cannot be included into this

scheme. Therefore, to explain experimental information on

their masses, and an unusual mass hierarchy inside of the

nonet Jaffe made a suggestion on a four-quark nature of

these particles [19].

An updated model of the light scalar nonets is based on

assumption about a diquark-antidiquark structure of these

particles, which appear as mixtures of spin-0 diquarks

from (

;

) representation with spin-1 diquarks from

;

Þ representation of the color-flavor group [20].In

Refs. [21,22] we investigated the scalar mesons f

ð500Þ

and f

ð980Þ as admixtures of the SU

ð3Þ basic light L ¼

½ud½

d and heavy H ¼ð½su½

uþ½ds½

dsÞ=

ﬃﬃﬃ

tetra-

quark states, and calculated their spectroscopic parameters

and full widths. Obtained predictions agree with existing

experimental data, therefore we consider the f

ð980Þ as the

exotic four-quark meson. Once we accept this model, a

treatment of the Y as a vector tetraquark Y ¼½su½

u

becomes quite reasonable.

We calculate the spectroscopic parameters of the vector

tetraquark Y ¼½su½

u and explore some of its decay

channels. The mass and coupling of the Y are evaluated

using the QCD two-point sum rule method [23,24].We

investigate the strong decays Y → ϕf

ð980Þ,Y → ϕη, and

Y → ϕη

, and find their partial widths. To this end, we use

the QCD light-cone sum rule (LCSR) method [25], and

calculate the couplings G

Yϕf

, g

Yϕη

, and g

Yϕη

corresponding

to the strong vertices Yϕf

ð980Þ, Yϕη, and Yϕη

, respec-

tively. The coupling G

Yϕf

is computed by employing the

full version of the LCSR method, whereas in the case of

Yϕη

, and g

Yϕη

this method is supplemented by a technique

of the soft-meson approximation [26–28]. Because the light

component of f

ð980Þis irrelevant for analysis of the decay

Y → ϕf

ð980Þ, we treat f

ð980Þ as a pure H state.

This article is organized as the following way: In Sec. II

we calculate the mass and coupling of the tetraquark Y. The

strong decays of this state are considered in Secs. III and IV.

In Sec. III we analyze the process Y → ϕf

ð980Þ using the

LCSR method and find the partial decay width of this

channel. The partial widths of the decay modes Y → ϕη,

and Y → ϕη

are calculated in Sec. IV. In Sec. V we analyze

the obtained results, and give our conclusions.

II. SPECTROSCOPIC PARAMETERS OF THE

TETRAQUARK Y: THE MASS m

AND

CURRENT COUPLING f

To evaluate the mass m

and coupling f

of the vector

tetraquark Y, we use the QCD two-point sum rule method

and start our calculations from analysis of the correlation

function,

μν

ðpÞ¼i

ipx

h0jT fJ

ðxÞJ

Y†

ð0Þgj0i; ð1Þ

where J

ðxÞ is the interpolating current for the Y state.

S. S. AGAEV, K. AZIZI, and H. SUNDU PHYS. REV. D 101, 074012 (2020)

074012-2

The current for a tetraquark with J

¼ 1

−

can be built of

a scalar diquark and vector antidiquark or/and a vector

diquark and scalar antidiquark. There are several options to

construct alternative currents with required spin-parities,

but because a scalar diquark (antidiquark) is a most stable

two-quark state [29], for J

we use the structure

Cγ

⊗ γ

C − Cγ

⊗ γ

C: ð2Þ

This current consists of two components, and each of

them describes a vector tetraquark. The whole structure

corresponds to a vector tetraquark with definite charge-

conjugation parity J

¼ 1

−−

. Indeed, the charge-conjuga-

tion transforms diquarks to antidiquarks and vice versa,

therefore the minus sign between two components in

Eq. (2) generates the current with C ¼ −1.

The last question to be solved is a color structure of

constituent diquarks and antidiquarks. Thus, to get the

color-singlet current J

they should have the same color

structures and be either in color triplet ½

 ⊗ ½3

 or sextet

½6

 ⊗ ½

 configurations. The current of the type (2) and

built of color-sextet diquark-antidiquark has the following

form [30]:

1μ

¼ u

Cγ



− u

Cγ

: ð3Þ

The triplet current (2) is given by the expression

3μ

¼ u

Cγ

−



− u

Cγ

−

: ð4Þ

In Eqs. (3) and (4) a and b are color indices, and C is the

charge-conjugation matrix.

The J

1μ

and J

3μ

are color-singlet currents composed

of color-sextet and -triplet diquark-antidiquark pairs,

respectively. To see this, let us consider in a detailed form

1μ

. The color-sextet, i.e., color-symmetric a ↔ b nature

of the antidiquark fields in Eq. (3) is evident. The first

component of J

1μ

, for example, in the explicit color-singlet

form is

ðu

Cγ

þ u

Cγ

Þ½

; ð5Þ

where both the diquark and antidiquark are symmetric in

color indices. It is not difficult to see that diquarks u

Cγ

and u

Cγ

lead to identical results, hence it is enough in

1μ

to keep one of them. The similar analysis is valid for the

second component of J

1μ

as well. In the case of the current

3μ

, we see that the antidiquark fields in Eq. (4) are color-

triplet or color-antisymmetric constructions. The color-

triplet diquark field, for example, in the first component

of J

3μ

is ðu

Cγ

− u

Cγ

Þ, and both u

Cγ

and

−u

Cγ

give again the same results. Therefore, we use

one of them in the current J

3μ

and get (4).

An appropriate form of the current J

that ensures

stability and convergence of the sum rules, which are

actual in the case of light tetraquarks [31], is superposition

of J

1μ

and J

3μ

. In the present work we use J

ðJ

1μ

þ J

3μ

Þ=2, and get

ðxÞ¼½u

ðxÞCγ

ðxÞ½

ðxÞγ

ðxÞ

− ½u

ðxÞCγ

ðxÞ½

ðxÞγ

ðxÞ: ð6Þ

The J

ðxÞ is a sum of two colorless terms, but belongs

neither to sextet nor to triplet representations of the color

group being the admixture of such states J

1μ

and J

3μ

To obtain sum rules for the mass and coupling of Y,we

should express the correlation function in terms of these

spectral parameters, and also calculate Π

μν

ðpÞ using quark-

gluon degrees of freedom. The first expression forms the

physical side of the sum rules Π

Phys

μν

ðpÞ, whereas the second

one constitutes their QCD side Π

OPE

μν

ðpÞ. In terms of the

tetraquark’s parameters the correlation function has the

following form:

Phys

μν

ðpÞ¼

h0jJ

jYðpÞihYðpÞjJ

Y†

j0i

− p

þ: ð7Þ

Equation (7) is derived by saturating the correlation

function with a complete set of J

¼ 1

−−

states and

carrying out integration in Eq. (1) over x. As usual,

contributions arising from higher resonances and con-

tinuum states are denoted above by dots.

The correlator Π

Phys

μν

ðpÞ can be further simplified if one

introduces the matrix element,

h0jJ

jYðpÞi ¼ f

; ð8Þ

where ϵ

is the polarization vector of the Y state. Then the

correlation function Π

Phys

μν

ðpÞ takes the simple form

Phys

μν

ðpÞ¼

− p



−g

μν



þ; ð9Þ

and contains the Lorentz structure corresponding to the

vector state. Because a part of this structure proportional to

μν

receives contribution only from the vector states, we

work with this term and corresponding invariant ampli-

tude Π

Phys

ðp

Þ.

The QCD side of the sum rules is given by the same

correlation function Π

μν

ðpÞ but expressed in terms of the

quark propagators. Substituting the interpolating current

into Eq. (1), and contracting the quark fields, we get

NATURE OF THE VECTOR RESONANCE Yð2175Þ PHYS. REV. D 101, 074012 (2020)

074012-3

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