**Background**

Hadrons in the Standard Model of Particle Physics are first order composite particles bound by the strong force primarily governed by a part of the Standard Model known as QCD for quantum chromodynamics. Most hadrons, such as protons, neutrons, pions and kaons, are made up of quarks bound by the the strong force which is mediated by gluons.

But, in theory, hadrons that don't have quarks and are gluons bound to each other in first order composite structures are also possible and are called "glueballs". If they exist, glueballs are always bosons, because gluons are vector bosons (i.e. they have spin-1) which always have integer spins. The rules of QCD allow glueballs to exist, so they should be possible to create, although they might be ephemeral. The properties of glueballs in a pure state are among the easiest things to calculate analytically with QCD because the only physical constant involved in the calculation at the first order is the strong force coupling constant (currently known to about 1% accuracy), with all other properties determinable strictly from first principles.

But, no glueballs have ever been observed directly in experiments in the many billions of particle collider experiment collisions energetic enough to create them in theory that have been conducted and carefully measured with state of the art equipment since the 1970s when their existence was first predicted.

The non-observation of this hadron which is a basic prediction of QCD could be because some bosons, including hypothetical glueballs, are tricky to observe because bosons can (to oversimplify) blend into each other and may never appear in a free state (some pions and kaons are also "blended" boson states). In contrast, baryons (fermions with three valence quarks) and most mesons (usually with two valence quarks) that are either pseudo-scalar and vector in quantum numbers, tend to be fairly simple and don't mix with other kinds of bosons.

Scalar and axial vector mesons as well as a few light pseudo-scalar mesons, tend to mix with other mesons with the same quantum numbers and similar masses. There are observed scalar and axial vector hadrons, but they don't have structures that can be explained with simple valance quark structures like simpler quark-antiquark mesons, or three valence quark baryons.

This analysis is complicated because there are many different hadrons with similar properties in a given mass range and because in addition to the "ground" state of every single hadron there are, in principle, an infinite number of higher mass excited states of the same hadron with similar energies. The f0 meson family of scalar bosons are among them (they have the same quantum numbers, and the parenthetical number in their symbols is their approximate mass in MeV units) and might be explained with a glueball component. This is a possibility which the new paper below tries to evaluate.

**The Results Of The New Paper**

The study finds that the two lowest mass examples of this family of mesons, which are f0(500) and f0(980), have essentially no glueball contribution and are ground states with a quarkonium (a quark-antiquark pair or blend of pairs of the same flavor of quark in a bound structure that does not decay to pure energy) structure, the next two mesons in the family, which are f0(1370) and f0(1500), are excited states with only small glueball contributions, and that the f0 (1710) meson in this family is mostly a 0++ to be 1637 ∼ 1698 MeV with the balance of its mass due to mixing with quarkonium.

The data used to make these determinations comes from the decays of the J/ψ meson, which is a spin-1 vector meson with a charm quark and an anti-charm quark as valence quarks, with an experimentally measured mass of approximately, 3,098.92 MeV, which has a mean lifetime of about 7.09*10

The data used to make these determinations comes from the decays of the J/ψ meson, which is a spin-1 vector meson with a charm quark and an anti-charm quark as valence quarks, with an experimentally measured mass of approximately, 3,098.92 MeV, which has a mean lifetime of about 7.09*10

^{−21}seconds, into one or more photons and also hadrons, which are a subset of the "radiative decays" (i.e. decays with photons) of this meson.#
Revisiting the topic of determining fraction of glueball component in
f
0
mesons via radiative decays of J/ψ

f

#
(Submitted on 16 Mar 2020)
The QCD theory predicts existence of glueballs, but so far all experimental endeavor fails to identify any of such states. To remedy the obvious discrepancy between the QCD which is proved to be a successful theory for strong interaction and the failure of experimentally searching for glueballs, one is tempted to accept the most favorable interpretation that the glueballs mix with regular

q
q¯
states of the same quantum numbers. The lattice estimate on the masses of the pure 0++
glueballs ranges from 1 to 2 GeV which is the region of the
f
0
family. Thus many authors suggest that the
f
0
mesonic series is an ideal place to study possible mixtures of glueballs and
q
q¯
. In this paper following the strategy proposed by Close, Farrar and Li, we try to determine the fraction of glueball components in
f
0
mesons using the measured branching ratios of
J/ψ
radiative decays into
f
0
mesons. Since the pioneer papers by Close et al. more than 20 years elapsed and more accurate measurements have been done by several experimental collaborations, so that based on the new data, it is time to revisit this interesting topic. Our numerical results show that
f
0
(500)
,
f
0
(980)
, are almost pure quark anti-quark bound states, while for
f
0
(1370)
,
f
0
(1500)
and
f
0
(1710)
, to fit both the experimental data of
J/ψ
radiative decay and their mass spectra, glueball components are needed. Moreover, the mass of the pure 0++
glueball is phenomenologically determined.

(Submitted on 16 Mar 2020)

The QCD theory predicts existence of glueballs, but so far all experimental endeavor fails to identify any of such states. To remedy the obvious discrepancy between the QCD which is proved to be a successful theory for strong interaction and the failure of experimentally searching for glueballs, one is tempted to accept the most favorable interpretation that the glueballs mix with regular

f

f

q

q¯
f

J/ψ

f

f

(500)

f

(980)

f

(1370)

f

(1500)

f

(1710)

J/ψ

The results indicate that the experimentally measured masses of f0(500), f0(980) can correspond to the qq¯ states (ground states of uu¯+dd¯ √ 2 and ss¯), so can be considered as pure bound states of quark-antiquark.Whereas there are no values corresponding to the masses of f0(1370), f0(1500) and f0(1710). It signifies that they cannot be pure qq¯ states and extra components should be involved. To evaluate the fractions of glueballs in those states, diagonalizing the mass matrix whose eigen-values correspond to the masses of the physical states and the transformation unitary matrix determine the fractions of qq¯ and glueball in the mixtures. . . . After this introduction, in section II we calculate the mass spectra of qq¯ states of 0++ by solving the relativistic Schr¨odinger equations. The rest 0++ qq¯ states would have negligible probability to mix with glueballs because their masses are relatively apart from that of pure glueball. Now we propose that the physical states f0(1370), f0(1500) and f(1710) are mixtures of the second excited states of |Ni = | uu¯+dd¯ √ 2 i and |Si = |ss¯i with glueball state |Gi. . . .The solutions show that for f0(1370) and f0(1500), the main components are qq¯ bound states, whereas the glueball component in f0(1710) is overwhelmingly dominant. It also suggests the mass of a pure glueball of 0++ to be 1637 ∼ 1698 MeV.

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