Photon production from gluon-mediated quark–anti-quark annihilation at confinement

Heavy ion collisions at the BNL Relativistic Heavy Ion Collider produce direct photons at low transverse momentum $p_T$ from 1–3 $\mathrm{GeV}/c$, in excess of the $p+p$ spectra scaled by the nuclear overlap factor $T_{AA}$. These low-$p_T$ photons have a large azimuthal anisotropy $v_2$. Theoretical models, including hydrodynamic models, struggle to quantitatively reproduce the large low-$p_T$ direct photon excess and $v_2$ in a self-consistent manner. This paper presents a description of the low-$p_T$ photon flow as the result of increased photon production from soft-gluon-mediated $q-\overline{q}$ interactions as the system becomes color neutral. This production mechanism will generate photons that follow constituent quark number, $n_q$, scaling of $v_2$ with an $n_q$ value of 2 for direct photons. ${\chi }^2$ comparisons of the published PHENIX direct photon and identified particle $v_2$ measurements finds that $n_q$ scaling applied to the direct photon $v_2$ data prefers the value $n_q=1.8$ and agrees with $n_q=2$ within errors in most cases. The 0–20% and 20–40% Au+Au direct photon data are compared to a coalescence-like Monte Carlo simulation that calculates the direct photon $v_2$ while describing the shape of the direct photon $p_T$ spectra in a consistent manner. The simulation, while systematically low compared to the data, is in agreement with the Au+Au measurement at $p_T$ less than 3 $\mathrm{GeV}/c$ in both centrality bins. Furthermore, this production mechanism predicts that higher order flow harmonics $v_n$ in direct photons will follow the modified $n_q$-scaling laws seen in identified hadron $v_n$ with an $n_q$ value of 2.

Figure 1

Feynman diagrams of prompt photon production by (a) quark-gluon Compton scattering, (b) quark–anti-quark annihilation, and (c) bremsstrahlung radiation off of an outgoing quark [1].

Figure 2

Feynman diagram of the quark–anti-quark annihilation interaction with a medium gluon producing a direct photon.

Figure 3

The $\pi$ (blue triangles), K (open magenta squares), p (red circles), and direct photon (open green crosses) $v_2$ as a function of $p_T$ in (a) central 0–20% and (b) midcentral 20–40% Au+Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. Panels (c) and (d) show the $v_2/n_q$ as functions of $p_T/n_q$ for 0–20% and 20–40% respectively. Panels (e) and (f) show the $v_2$ scaled by the number of constituent quarks, $n_q$, as a function of ${\mathit{\text{KE}}}_T/n_q$, again for 0–20% and 20–40% centralities. For direct photons, $n_q=2$ is assumed. In panels (a), (c), and (e), the 0–20% $v_2$ values are scaled by 1.6 for better comparison with the 20–40% results. Error bars and shaded boxes around points represent their statistical and systematic uncertainties, respectively [4, 20].

Figure 4

Example plots of the input data used for the calculation of ${\chi }^2$ comparing the $v_2/n_q$ vs ${\mathit{\text{KE}}}_T/n_q$ for identified hadrons (red circles) [20] and direct photons (black squares) [4] using the quadratic sum of the statistical and systematic errors. Here, $n_{q\gamma }=2$ is assumed for direct photons. The 0–20% (top row) and 20–40% (bottom row) $\sqrt{s_{{}_{NN}}}=200$ GeV Au+Au results are shown. Pions (left column), kaons (middle column), and protons (right column) are separately plotted with the direct photon data over the full ${\mathit{\text{KE}}}_T/n_q$ range. The data are included in the ${\chi }^2$ calculation only if the identified hadron and direct photon ${\mathit{\text{KE}}}_T/n_q$ values are within 0.1 GeV/$c$. The ${\chi }^2$ is calculated using the variation between direct photon and identified hadron $v_2/n_q$ in all six plots. Error bars represent the statistical and systematic uncertainties summed in quadrature. A ${\chi }^2/\mathrm{NDF}$ of $16.28/35=0.47$ is found using the full ${\mathit{\text{KE}}}_T/n_q$ range available in the data.

Figure 5

${\chi }^2$ distribution as a function of $n_{q\gamma }$ calculated using (a) the quadrature sum of the statistical and systematic errors for the hadron and photon uncertainties and (b) only the statistical errors. The ${\chi }^2$ calculation with an upper limit of ${\mathit{\text{KE}}}_T/n_q<1.0$ GeV in the 20–40% centrality bin is shown with blue open squares; this is range 1. The calculation with upper limits of 1.7 and 1.0 GeV in the 0–20% and 20–40% centrality bins, respectively, is shown with red $*$ marks; this is range 2. Horizontal lines are drawn at the location of the ${\chi }_{\min }^2+1$ (solid), ${\chi }_{\min }^2+4$ (dashed) and ${\chi }_{\min }^2+9$ (dotted) for each calculation.

Figure 6

0–20% and 20–40% Au+Au $v_2/n_q$ vs ${\mathit{\text{KE}}}_T/n_q$ for pions, kaons, and protons fit with a probability density distribution of a $\mathrm{\Gamma }$ function. High-$p_T$ protons that deviate from $n_q$ scaling in the 20–40% centrality bin are excluded from the fit and are not shown [20].

Figure 7

Energy taken by the gluon as a function of the direct photon's ${\mathit{\text{KE}}}_T$ for the (a) 0–20% and (b) 20–40% simulations. The $z$ axis is the number of counts and is shown with a logarithmic color scale.

Figure 8

The $v_2$ for pions, photons, protons, and thrown quarks simulated using the fast Monte Carlo method. Plots (a) and (b) are the $v_2$ vs ${\mathit{\text{KE}}}_T$ for the 0–20% and 20–40% respectively. Plots (c) and (d) are the $n_q$-scaled results for 0–20% and 20–40%. The 0–20% $v_2$ values are scaled by 1.6 to make the $y$-axis scales consistent.

Figure 9

Direct photon yield versus $p_T$ for the (a) 0–20% and (b) 20–40% Au+Au data (red circles and asterisks) [2, 3, 27] on a ${\log }_{10}$ scale. The $T_{AA}$-scaled $p+p$ yields (blue squares and crosses) [2, 28, 29] are also shown including a power law fit to the $p+p$ data (black diagonal hatched band) [3]. The Monte Carlo yield from quark-anti-quark annihilation (cyan band) are fit to the data and are shown with the total fit yield (purple band) found by summing the Monte Carlo yield and the $T_{AA}$-scaled $p+p$ fit. The ratio of the Au+Au data over the total fit yield is shown in the lower plots. The thick black line is a flat line fit to this ratio with a value of $0.951\pm 0.051$ and $1.038\pm 0.065$ for the 0–20% and 20–40% ratios, respectively.

Figure 10

Direct photon $v_2$ versus $p_T$ for the (a) 0–20% and (b) 20–40% Au+Au data (blue circles) [4]. The Monte Carlo $v_2$ from quark–anti-quark annihilation (black open circles) and the total $v_2$ (small red closed circles) are also shown. The relative contribution of the quark–anti-quark annihilation component is shown in the lower plot of each figure.