Tests of constituent-quark generation methods which maintain both the nucleon center of mass and the desired radial distribution in Monte Carlo Glauber models

Several methods of generating three constituent quarks in a nucleon are evaluated which explicitly maintain the nucleon's center of mass and desired radial distribution and can be used within Monte Carlo Glauber frameworks. The geometric models provided by each method are used to generate distributions over the number of constituent quark participants $\left(N_{\mathrm{qp}}\right)$ in $p+p,d+\mathrm{Au}$, and $\mathrm{Au}+\mathrm{Au}$ collisions. The results are compared with each other and to a previous result of $N_{\mathrm{qp}}$ calculations, without this explicit constraint, used in measurements of $\sqrt{s_{{}_{NN}}}=200$ GeV $p+p,d+\mathrm{Au}$, and $\mathrm{Au}+\mathrm{Au}$ collisions at the BNL Relativistic Heavy Ion Collider.

Figure 1

PHENIX2014 [6] method for ${E_T\equiv d{E}_T/d\eta |}_{y=0}$ distributions at $\sqrt{s_{{}_{NN}}}=200$ GeV. (a) Deconvolution fit to the $p+p E_T$ distribution for $E_T<13.3$ GeV with the corrected weights ${w^{\prime }}_i^{\mathrm{NQP}}$, with ${\epsilon }_{\text{NQP}}=1-p_{0_{\mathrm{NQP}}}=0.659$ calculated in the number of quark participants or $N_{\mathrm{qp}}$ model. Lines represent the properly weighted individual $E_T$ distributions for the underlying two, three, four, five, and six constituent-quark participants plus the sum. (b) $\mathrm{Au}+\mathrm{Au}$ compared to the NQP calculations using the central $1-p_0=0.647$ and $\pm 1\sigma$ variations of $1-p_0=0.582,0.712$ for the probability $p_0$ of getting zero $E_T$ on a $p+p$ collision with resulting ${\varepsilon }_{\mathrm{NQP}}=0.659,0.603,0.716$, respectively. (c) $d+\text{Au}$ calculation for the same conditions as in (b).

Figure 2

(a) Radial distribution $d\mathcal{P}/dr$ about the c.m. of the generated quark triplets as a function of $r$ (fm) for the three methods: (red) planar polygon (Sec. 3a); (blue) explicit joint (Sec. 3b); (green) empirical recentered (Sec. 3c); compared to $r^2{\rho }^{\mathrm{proton}}\left(r\right)$ from Eq. (4) (black), with a semilog plot shown as the inset (b). (c) Ratio of the generated distributions to Eq. (4). The planar polygon distribution is identical to $r^2{\rho }^{\mathrm{proton}}\left(r\right)$, so it obscures the black line.

Figure 3

Radial distribution $d\mathcal{P}/dr$ about the c.m. of the generated quark triplets as a function of $r$ (fm) for the PHENIX2014 [6] method compared to $r^2{\rho }^{\mathrm{proton}}\left(r\right)$ from Eq. (4).

Figure 4

$d{N}_{\mathrm{ch}}/d\eta /\left(0.5N_{\mathrm{qp}}\right)$ at midrapidity as a function of $N_{\mathrm{qp}}$ for $d{N}_{\mathrm{ch}}/d\eta$ measurements [26] in $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. The various methods and number of constituent quarks assumed per proton are given in the legend.

Figure 5

${E_T\equiv d{E}_T/d\eta |}_{y=0}$ distributions at $\sqrt{s_{{}_{NN}}}=200$ GeV calculated in the number of quark participants (NQP) model from the empirical recentered (Sec. 3c) method with ${\epsilon }_{\text{NQP}}=1-p_{0_{\mathrm{NQP}}}=0.690$. (a) NQP calculation of the $p+p E_T$ distribution for $E_T<13.3$ GeV with the parameters $p$ and $b$ for the $E_T$ distribution of one QP from the deconvolution fit to the same data (Table 5), where the thin lines shown are the $E_T$ distributions for two, three, four, five, and six QPs weighted by $w_n^{\prime }$ from Table 3 and the thick line is the sum. (b) $\mathrm{Au}+\mathrm{Au}$ compared to the NQP calculations using the central $1-p_0=0.647$ and $\pm 1\sigma$ variations of $1-p_0=0.582,0.712$ for the probability $p_0$ of getting zero $E_T$ on a $p+p$ collision [6] with resulting ${\varepsilon }_{\mathrm{NQP}}=0.690,0.636,0.745$, respectively. (c) $d+\text{Au}$ calculation for the same conditions as in (b) and for PHENIX2014 [6].

Figure 6

PHENIX2014 [6] $E_T$ distribution in $\mathrm{Au}+\mathrm{Au}$ at $\sqrt{s_{NN}}=200$ GeV compared to the number of quark participants (NQP) model calculations indicated, where ${\epsilon }_{\mathrm{NQP}}$ represents the probability for a constituent-quark participant to give nonzero $E_T$ in the detector. The PHENIX2014 NQP calculation is shown for the central value, ${\epsilon }_{\mathrm{NQP}}=0.659$, and the $\pm 1\sigma$ systematic variations in ${\epsilon }_{\mathrm{NQP}}=0.716,0.603$ [6] as detailed in Fig. 1.

Figure 7

(a) Plot of $\langle N_{\mathrm{qp}}\rangle$ vs ansatz, $\left[\right(1-x)\langle N_{\mathrm{part}}\rangle /2+x\langle N_{\mathrm{coll}}\rangle ]$, with $x=0.107$, from Table 7, compared to the PHENIX2014 result [6], with $x=0.08$. (b) Plot of ratio of $(\langle N_{\text{qp}}\rangle /3.524)$/ansatz, where 3.524 is the average of $\langle N_{\text{qp}}\rangle$/ansatz for the empirical recentered method over the entire centrality range 0%–92% in 5% bins, compared to the PHENIX2014 result [6] with average of $\langle N_{\text{qp}}\rangle /\text{ansatz}=3.894$.

Figure 8

Distribution of the number of quark participants in $\mathrm{Au}+\mathrm{Au}$ at $\sqrt{s_{{}_{NN}}}=200$ GeV scaled in $N_{\mathrm{qp}}$ by the indicated amounts so that they overlap the PHENIX2014 distribution [6]: (a) original weights $w_{\mathrm{NQP}}$; (b) corrected weights $w_{\mathrm{NQP}}^{\prime }$.

Figure 9

$E_T$ distributions at $\sqrt{s_{NN}}=200$ GeV calculated in the number of quark participants (NQP) model from the planar polygon method, with ${\epsilon }_{\mathrm{NQP}}=1-p_{0_{\mathrm{NQP}}}=0.707$. (a) NQP calculation of the $p+p$ data with the parameters $p$ and $b$ for the $E_T$ distribution of one QP from the deconvolution fit to the same data (Table 5), where the thin lines shown are the $E_T$ distributions for two, three, four, five, and six QPs weighted by $w_n^{\prime }$ from Table 3 and the thick line is the sum. (b) $\mathrm{Au}+\mathrm{Au}$ calculation with systematic uncertainties.

Figure 10

$E_T$ distributions at $\sqrt{s_{NN}}=200$ GeV calculated in the Number of Quark Participants (NQP) model from the Explicit Joint method, with ${\epsilon }_{\mathrm{NQP}}=1-p_{0_{\mathrm{NQP}}}=0.708$. (a) NQP calculation of the $p+p$ data with the parameters $p$ and $b$ for the $E_T$ distribution of one QP from the deconvolution fit to the same data (Table 5) where the thin lines shown are the $E_T$ distributions for two, three, four, five, and six QPs weighted by $w_n^{\prime }$ from Table 3 and the thick line is the sum. (b) $\mathrm{Au}+\mathrm{Au}$ calculation.