NCQ scaling of $f_{0}(980)$ elliptic flow in 200 GeV Au+Au collisions by STAR and its constituent quark content

Searching for exotic state particles and studying their properties have furthered our understanding of quantum chromodynamics (QCD). The $f_{0}$(980) resonance is an exotic state with relatively high production rate in relativistic heavy-ion collisions, decaying primarily into $\pi\pi$. Currently, the structure and quark content of the $f_{0}$(980) are unknown with several predictions from theory being a $q\overline{q}$ state, a $qq\overline{q}\overline{q}$ state, a $K\overline{K}$ molecule state, or a gluonium state. We report the first $f_{0}$(980) elliptic flow ($v_{2}$) measurement from 200 GeV Au+Au collisions at STAR. The transverse momentum dependence of $v_{2}$ is examined and compared to those of other hadrons (baryons and mesons). The empirical number of constituent quark (NCQ) scaling is used to investigate the constituent quark content of $f_{0}$(980), which may potentially address an important question in QCD.


Introduction
Searching for exotic state particles and studying their properties have furthered our understanding of quantum chromodynamics (QCD). Currently the structure and quark content of f 0 (980) are unknown with several predictions being a qq state, a qqqq state, a KK molecule state, or a gluonium state [1][2][3][4][5][6]. In contrast to the vector and tensor mesons, the identification of the scalar mesons is a long-standing puzzle [7]. Previous preliminary experimental measurements [8] on the yield of f 0 (980) at RHIC and theoretical calculation [9] suggest that it could be a KK stat. In this analysis, the empirical number of constituent quark (NCQ) scaling [10][11][12] is used to investigate the constituent quark content of f 0 (980) [13].

Experiment setup and data analysis
The data reported here are from Au+Au collisions at a nucleon-nucleon center-of-mass energy of 200 GeV, collected by the STAR experiment [14] at Brookhaven National Laboratory in 2011, 2014 and 2016. A total of 2.4 billion minimum-bias (MB) events are selected for this analysis. The main subsystem used for the data analysis is the Time Projection Chamber (TPC) [15] with 2π azimuthal coverage at mid-rapidity. The TPC dE/dx is used to select π ± candidate with 0.2 < p T < 5.0 GeV/c.
The π + π − are used to reconstruct the f 0 (980). The combinatorial background subtraction is based on the mixed-event technique and the like-sign method [16]. The acceptance-corrected like-sign e-mail: zhao656@purdue.edu pairs [16,17] are used to subtract the combinatorial background after being normalized to unlikesign pairs in the invariant mass (m inv ) range beyond 1.5 GeV/c 2 . Figure 1 (left) shows the background subtracted π + π − invariant mass distribution. The resonance peaks are parametrized with the relativistic Breit-Wigner function [18,19]. The total fit function is given by: where is the phase space correction taking into account the ππ scattering during the hadronic phase [19][20][21][22], and bg(m inv ) is a third order polynomial function to describe the residual background. m X and Γ X are the mass and width of the corresponding resonances. Γ ρ 0 is set to 160 MeV, and m f 2 and Γ f 2 are set according to the PDG values [7]. T is the kinetic freeze-out temperature, set to 120 MeV [20].
The event-plane method [23] is used to study the elliptic flow (v 2 ) of f 0 (980). The event-plane is reconstructed by all charged particles in the TPC with pseudorapidity |η| < 1 and transverse momentum 0.2 < p T < 5.0 GeV/c. For each ππ pair, the two π candidates are removed from the event-plane reconstruction to avoid auto-correlation. The event-plane resolution is calculated by the correlation between two randomly divided sub-events from the full TPC [23]. Wide centrality bin effect is corrected by weighting the event-plane resolution with the f 0 (980) yield in each narrow centrality bin of 10% size [24]. Figure 1(right) shows the f 0 (980) yield as function of the azimuthal angle difference between the ππ pair (φ) and the event-plane direction (Ψ) in an example p T bin.

ρ 0 Mass and Yield Extraction
The ρ 0 (770), a resonance vector meson, has J = 1. The π + π − invariant mass distribution for ρ 0 is therefore fitted to a p-wave relativistic Breit-Weigner (BW) function [114] of the form: where A is a constant parameter proportional to the yield of the ρ 0 , Mππ is the π + π − invariant mass, M0 is the PDG ρ 0 mass, and Γ (Mππ) is the momentum dependent width [114,116].
In Equation 3.5, Γ0 is the ρ 0 full width at half maximum and mπ is the pion mass in PDG.
In heavy-ion collisions, besides the direct production of ρ 0 from partons [117, 118], a ρ 0 can also be produced through π + π − scattering in the hadronic medium via π + + π − → ρ 0 → π + π − [119, 120]. The π + π − invariant mass distribution for the ρ 0 generated in this way might be modified by the initial pions phase space distribution. Thus, the relativistic Breit-Wigner function should be multiplied by a Boltzmann factor [121-123] to account for Phase Space (PS). In case of p+p collisions, the hadronic medium is much smaller than that formed in heavy-ion collisions, but still the ρ 0 can be produced in the same process [110]. The functional form of the PS can be written as: where pT is the transverse momentum of the ρ 0 and T is the temperature at

ρ 0 Mass and Yield Extraction
The ρ 0 (770), a resonance vector meson, has J = 1. The π + π − invariant mass distribution for ρ 0 is therefore fitted to a p-wave relativistic Breit-Weigner (BW) function [114] of the form: where A is a constant parameter proportional to the yield of the ρ 0 , M ππ is the π + π − invariant mass, M 0 is the PDG ρ 0 mass, and Γ (M ππ ) is the momentum dependent width [114,116].
In Equation 3.5, Γ 0 is the ρ 0 full width at half maximum and m π is the pion mass in PDG.
In heavy-ion collisions, besides the direct production of ρ 0 from partons [117, 118], a ρ 0 can also be produced through π + π − scattering in the hadronic medium via π + + π − → ρ 0 → π + π − [119, 120]. The π + π − invariant mass distribution for the ρ 0 generated in this way might be modified by the initial pions phase space distribution. Thus, the relativistic Breit-Wigner function should be multiplied by a Boltzmann factor [121-123] to account for Phase Space (PS). In case of p+p collisions, the hadronic medium is much smaller than that formed in heavy-ion collisions, but still the ρ 0 can be produced in the same process [110]. The functional form of the PS can be written as: where p T is the transverse momentum of the ρ 0 and T is the temperature at

ρ 0 Mass and Yield Extraction
The ρ 0 (770), a resonance vector meson, has J = 1. The π + π − invariant mass distribution for ρ 0 is therefore fitted to a p-wave relativistic Breit-Weigner (BW) function [114] of the form: where A is a constant parameter proportional to the yield of the ρ 0 , M ππ is the π + π − invariant mass, M 0 is the PDG ρ 0 mass, and Γ (M ππ ) is the momentum dependent width [114,116].
In Equation 3.5, Γ 0 is the ρ 0 full width at half maximum and m π is the pion mass in PDG.
In heavy-ion collisions, besides the direct production of ρ 0 from partons [117, 118], a ρ 0 can also be produced through π + π − scattering in the hadronic medium via π + + π − → ρ 0 → π + π − [119, 120]. The π + π − invariant mass distribution for the ρ 0 generated in this way might be modified by the initial pions phase space distribution. Thus, the relativistic Breit-Wigner function should be multiplied by a Boltzmann factor [121-123] to account for Phase Space (PS). In case of p+p collisions, the hadronic medium is much smaller than that formed in heavy-ion collisions, but still the ρ 0 can be produced in the same process [110]. The functional form of the PS can be written as: where p T is the transverse momentum of the ρ 0 and T is the temperature at  Results are compared with other identified particles: π, K, p, K 0 s , Λ, Ξ, Ω, φ [24]. In the low p T region, the f 0 (980) v 2 seems to follow the mass ordering. In the higher p T region, the f 0 (980) v 2 seems closer to the baryon band. Results of other particles are taken from Ref. [24]. Black line in the right panel represents a fit to results of other particles using a NCQ scaling inspired function (Eq. 2).  compared to the fit of other particles [24] using a NCQ scaling inspired function [25]: The 2-quarks (4-quarks) scaled f 0 (980) v 2 seems to deviate from the fit, above (below) the fit by ∼ 1σ for the last one or two points at high (m T − m 0 )/n q . Figure 4 shows f 0 (980) v 2 as a function of m T − m 0 with a fit according to the function shown in Eq. 2. In the fit, only the n q of f 0 (980) is treated as a free parameter and all other parameters are fixed according to the fit in the right panel of the Fig. 3. This NCQ scaling fit of the f 0 (980) v 2 yields n q = 3.0 ± 0.7 (stat) ± 0.5 (syst).
With the current uncertainty, our result is not able to determine whether f 0 (980) is a qq, qqqq, KK molecule, gluonium state, or produced through ππ coalescence. It could also be given by some combined states as well. Future measurements, e.g. the f 0 (980) yields, could also provide different aspect to understand it.

Summary
Preliminary results on the f 0 (980) v 2 in 30-80% centrality Au+Au collisions at √ s NN = 200 GeV are presented. In the low p T region (p T <2 GeV/c), the f 0 (980) v 2 seems to follow the mass ordering.
In the higher p T region (p T >2 GeV/c), the f 0 (980) v 2 seems closer to the baryon band. A NCQ scaling inspired function was used to fit the f 0 (980) v 2 . The extracted quark content of f 0 (980) is n q = 3.0 ± 0.7 (stat) ± 0.5 (syst). More data are needed to understand whether f 0 (980) is a qq, qqqq, KK molecule, gluonium state, or produced through ππ coalescence. Our study indicates that heavyion collisions can be a useful place to examine the quark content of scalar mesons. The isobar data taken in 2018 at RHIC and the 8-fold increase in Au+Au data expected in 2023-2025 would provide more insights.