Low-momentum direct-photon measurement in Cu + Cu collisions at

We measured direct photons for p T < 5 GeV /c in minimum bias and 0%–40% most-central events at midrapidity for Cu + Cu collisions at √ s NN = 200 GeV. The e + e − contribution from quasireal direct virtual photons has been determined as an excess over the known hadronic contributions in the e + e − mass distribution. A clear enhancement of photons over the binary scaled p + p ﬁt is observed for p T < 4 GeV /c in Cu + Cu data. The p T spectra are consistent with the Au + Au data covering a similar number of participants. The inverse slopes of the exponential ﬁts to the excess after subtraction of the p + p baseline are 285 ± 53(stat) ± 57(syst) MeV /c and 333 ± 72(stat) ± 45(syst) MeV /c for minimum bias and 0%–40% most-central events, respectively. The rapidity density, dN/dy , of photons demonstrates the same power law as a function of dN ch /dη observed in Au + Au at the same collision energy.


I. INTRODUCTION
Direct photons are excellent probes for understanding the time evolution of the hot and dense matter created in ultrarelativistic heavy-ion collisions [1,2]. Direct photons are produced throughout the collision and carry information about the medium at the time when the photons were emitted, because the only interaction is electromagnetic [3]. Direct photons are produced via interactions at partonic and hadronic levels in either initial hard scatterings of the collision or thermal radiation from the medium and, by definition, do not originate from hadron decays [4]. In particular, thermal * Deceased. † akiba@rcf.rhic.bnl.gov photons, which contribute dominantly at low momentum [5], are one of the most important probes because they allow us direct access to the thermodynamic properties of the created medium. However, photons from hadron decays account for a large fraction in the inclusive photon yield, typically more than 80% for heavy-ion collisions. The large number of decay photons makes the measurement challenging.
Two analysis methods, the virtual photon method [6] and the external conversion method [7], have been established to measure direct photons at low p T (p T < 5 GeV/c). Lowp T direct-photon measurements have been made in PHENIX and STAR experiments at the Relativistic Heavy Ion Collider (RHIC) for not only in Au+Au collisions [6][7][8] but also in p+p and d+Au [9] collisions. The virtual photon method makes it possible to measure direct photons even if the signal to background (S/B) is only a few percent, as in p+p and d+Au collisions, while in Au+Au collisions the S/B reaches 15%. The p+p measurement allows us to determine the hard photon yield from initial hard scatterings. No significant modification of the p T distribution of direct photons due to cold nuclear effects is seen in the d+Au data. Finally, an enhanced yield of low-p T direct photons, which is unexplainable by hard photon production and cold nuclear matter effects, has been discovered in Au+Au collisions in central and semicentral events at √ s NN = 200 GeV [6,7].
The ALICE experiment at the Large Hadron Collider (LHC) has also succeeded in measuring the low-p T direct photons with the external conversion method in Pb + Pb collisions at √ s NN = 2.76 TeV [10] and observed a larger yield and a higher inverse slope of the spectrum than at RHIC, implying that a larger and hotter thermalized medium is produced at the LHC energy. Further understanding of the thermal properties of the created hot medium can be realized through the systematic study of low-p T direct photon production within a wide range of system size and collision energy.
In this paper, we present the measurement of low-p T direct photons in Cu+Cu collisions at √ s NN = 200 GeV with the virtual photon method. This measurement may provide additional information on the system size dependence of low-p T direct-photon production. This paper focuses on two centrality classes, minimum bias (MB) and 0%-40% most-central collisions, for which the number of participants, N part , is similar to peripheral Au+Au (N part = 34.6 ± 1.2 [11] and 66.4 ± 2.5 [12]).

II. THE PHENIX DETECTOR
The two PHENIX central arm spectrometers in conjunction with the beam-beam counters (BBCs) are used for this measurement. Figure 1 shows the beam view of the PHENIX detector configuration for the 2005 run. The BBCs, with rapidity coverage 3.1 < |η| < 3.9, are located at ±144 cm away from the nominal interaction point. They measure charged particles that are used to determine the z-vertex position, the centrality, and the event plane. They provide the MB event trigger with a trigger efficiency of 94%. The two central arms cover |η| < 0.35 and an azimuthal angle range of π/2 per arm. Each arm is instrumented with a drift chamber (DC) and pad chambers (PCs) that determine the trajectories and, together with a magnetic field, measure the momenta of charged particles. The material in front of the DC is minimal, 0.39% of a radiation length, to allow for a good momentum resolution of δp/p = 1% ⊕ 1.1% × p [GeV/c] above 0.2 GeV/c [13], and to minimize the amount of photon conversions. Eight separate sectors of electromagnetic calorimeters (EMCals) composed of two lead-glass (PbGl) calorimeters in the bottom sectors of the east arm and six lead-scintillator (PbSc) calorimeters for the remainder, provide an electromagnetic shower energy measurement with resolution E/E of 2.1% ⊕ 8.1%/ √ E for PbSc and of 0.8% ⊕ 5.9%/ √ E for PbGl (E in GeV) [13]. Requiring energy-momentum matching with an associated hit in the Ring ImagingČerenkov counter (RICH) provides a hadron rejection factor of better than 10 4 , thus providing good electron identification. The mass resolution for e + e − pairs is determined with a Monte Carlo simulation which is tuned to match the shape of the reconstructed e + e − mass distribution in the data below 90 MeV/c 2 [14], where e + e − pairs from π 0 Dalitz decays are dominant. The calculated e + e − mass resolution is σ ee = 3.1 MeV/c 2 for 1 < p T < 2 GeV/c, and it increases by about 1 MeV/c 2 per GeV/c as p T increases.

III. ANALYSIS
Low-p T direct photons, measured by using the virtual photon method, are the subject of this analysis. Any production process of direct photons has a higher-order process producing a quasireal virtual photon, which then produces a lowmass, high-p T e + e − pair. The relation between the photon emission (dN γ ) and associate electron pair rates (dN ee ) is expressed as where α, m e , m ee are the fine-structure constant and masses for the electron and the electron pair, respectively. S is introduced to factor out the difference between real and virtual photon emission. It is a process-dependent factor because it accounts for the effects of form factors, phase space, and spectral functions [15]. For direct virtual photons satisfying p T m ee , S is almost unity, while it drops to 0 as m ee approaches the parent hadron mass in case of hadron decays. As a result, S introduces a shape difference of the e + e − mass distributions for virtual photons and hadron decays. The key idea of this measurement is to utilize this shape difference. Therefore, the contribution of the e + e − pairs internally converted via virtual photons is determined as an excess yield over the known hadronic contributions in the mass region above the π 0 mass, typically 0.1 < m ee < 0.3 GeV/c 2 , by a template fit. The direct-photon fraction at m ee = 0 is then obtained by extrapolation of the template fit result. Finally, the obtained direct-photon fraction can be converted to the real direct photon yield by using the measured inclusive photon yield. A detailed description of the virtual photon method can be found in Ref. [15].
This measurement is based on a MB sample of 4.95 × 10 8 200 GeV Cu+Cu collisions with z vertex within 25 cm of  Figure 2 shows the E/p distribution for electrons from pairs with p T of 1−2 GeV/c, where E is measured with the EMCal, and p from the track radius in the magnetic field. All electron identification cuts except for E/p are applied for this figure. Because hadrons do not deposit their full energy in the EMCal, hadron contamination produces a tail in the negative region. This plot indicates the excellent purity of the electron sample. All electron candidates are required to have (E/p − 1)/σ E/p > −2, resulting in negligible hadron contamination.
Undesired pairs from several background sources contaminate the foreground pair distribution. The first source is fake pairs due to accidentally overlapping hits in various detectors. RICH ring-sharing and cluster overlaps in the PCs are the main sources for these fake pairs. They can be removed by geometric analysis cuts [15,16]. The RICH ring-sharing cut requires separation of ring centers for the two electrons of a pair to be greater than 25 cm, which is larger than the expected maximum diameter of a RICH ring, ∼16.8 cm. Tracks are also required to be separated by z > 0.5 cm and φ > 0.02 rad to remove overlap in the PCs.
The second background source is photon conversions in the detector material. These can be eliminated because the PHENIX tracking algorithm, which assumes all tracks come from the collision vertex, introduces an artificial opening angle of the conversion pairs with the decay plane perpendicular to the magnetic field.

A. Background evaluation
After removing the detector-oriented fake pairs and conversions, the foreground distributions for unlike-sign (FG +− ) and like-sign pairs (FG −− , FG ++ ) can be expressed as Here FG refers to the data and BG refers to backgrounds whose shapes are calculated as described below, but whose normalization comes from a fit to the data (FG). S refers to the direct virtual photon signal and HD refers to correlated pairs from known hadron decays. It is notable that the like-sign pair distributions are composed of only random combinations (BG CM ), jet-induced correlations (BG JT ), and correlated fake pairs from double Dalitz decays of the π 0 , η (BG XC ). The sum of these backgrounds is referred to as BG SUM in this paper. Once compositions of these background contributions are known in the like-sign combination sample, the unlike-sign combination background, BG SUM +− , can be determined within the same analysis framework. 054902-5

Combinatorial background
The combinatorial background can be reproduced by the event mixing technique with event classification with respect to z-vertex position, event plane, and centrality. However, the modulation of the mass distribution by the elliptic flow, which is apparent in the real events, is not fully introduced in event mixing because of the limited reaction plane resolution. Thus, pairs in the mixed events are weighted by a factor based on the measured azimuthal anisotropy of single electrons [16] for given reaction plane classes. The weighting factor w, depending on the opening angle of a pair, is calculated as where φ, v a,b 2 are the pair opening angle and azimuthal anisotropy of each electron in a pair, respectively. The flow modulation makes at most a few percent difference in the mass shape.

Jet-induced correlation
Jet-induced correlations are pairs in which each electron is from a different parent, but both parents are from the same jet or back-to-back jets. Such events are simulated by PYTHIA8 [17,18] with CTEQ5L [19] parton distribution functions. The PYTHIA8-generated events are passed through a GEANT3-based [20] simulation of the PHENIX detector in which all detector effects such as the acceptance and efficiencies are taken into account. Uncorrelated combinations are evaluated by the event mixing technique within the simulated events. It is found that the shape of the like-sign mass distribution for the uncorrelated combinations is consistent with that for the foreground combinations in 0.6 < m ee < 1.1 GeV/c 2 .
Here, the true and other correlated pairs are removed from the foreground distribution before the comparison. Normalization of the uncorrelated combinations in a specific region of a pair opening angle, where opening angle distributions for correlated and uncorrelated pairs are consistent, gives a consistent result. A detailed description can be found in Ref. [16]. Finally the jet-induced correlations are obtained by removing uncorrelated combinations from the simulated mass distribution.

Correlated Dalitz and double Dalitz cross pairs
The other non-negligible source of correlated background is cross combinations from decays having two electron pairs in the final state, i.e., π 0 and η double Dalitz decays and Dalitz decays with a subsequent photon conversion. These cross combinations are localized at the very low mass region below the π 0 and η masses. The mass distributions of these cross combinations from π 0 and η are calculated by using the aforementioned GEANT3 simulation with the π 0 and η distributions measured by PHENIX.

Background normalization by BG SUM fit
The calculated BG −− and BG ++ distributions are the ingredients for a fit to FG −− and FG ++ , which then yields the contribution of each component to the background, BG SUM . Pairs from the same jet and back-to-back jets are separately included in the fit because they are influenced differently by jet quenching. The BG SUM fit to FG −− and FG ++ works very well. Figure 3 shows the like-sign and unlike-sign mass distributions of the data together with BG SUM normalized by the BG SUM fit for 1 < p T < 5 GeV/c where the virtual photon analysis is performed. The normalized BG SUM is in good agreement with the data for like-sign pairs. The contribution of the physically correlated pairs [S + HD +− in Eq. (4)] is significant in the foreground unlike-sign pair mass distribution below 0.3 GeV/c 2 .
A cross-check with the like-sign subtraction method [21] is done to demonstrate that the BG SUM +− properly accounts for all backgrounds. To infer the background in unlike-sign distributions, a correction must be made to account for the relative acceptance difference between like-and unlike-sign pairs. Thus, the acceptance-corrected like-sign pairs should The acceptance-correction factor α acc is calculated as the ratio of like-and unlike-sign pairs from mixed events. Figure 4 shows the background pair distributions of e + e − determined by the like-sign subtraction technique and the method used here for p T of 1-2 and 2−3 GeV/c, respectively. The two distributions are consistent within the statistical errors. The present method yields a smaller uncertainty, particularly at high p T . The combinatorial background [dashed (red) curves] has a much more significant contribution in BG SUM compared with those of the cross pairs [dotted (blue) curves] and jet-induced correlations [dashed-dotted (green) curves].

Correlated pairs from hadron decays
The last e + e − background source (indicated as H D +− ) for the direct virtual photon signal is the known hadron decays. The invariant yields of π 0 in the 200 GeV Cu+Cu as measured by PHENIX [22] have been successfully parametrized by a modified Hagedorn fit: The resulting Hagedorn fit parameters for 0%-40% and MB samples are listed in Table I. Note that the large uncertainty of the absolute scale parameter A does not affect the direct-photon result because only the shape enters in determining the direct-photon fraction. A detailed description of this analysis appears in the next section, Sec. III B. m T scaling of the parametrized π 0 yield has been shown to accurately reproduce the invariant yields of other known hadrons [11]. All known hadron decays producing e + e − are simulated with this parametrization by a Monte Carlo event generator within the PHENIX framework [15] and passed through the PHENIX GEANT3 simulation. The simulated e + e − pair mass distributions for known hadrons are merged as a "cocktail" of the hadron decay contributions. The particle compositions in the hadronic cocktail are based on the measured yields. The particle ratios to the π 0 yield are identical to the p+p data [23].
An additional source of decay background is e + e − pairs from open heavy flavor decays. They hide behind the cocktail of photonic decays discussed previously in the mass region of interest below m ee = 0.3 GeV/c 2 . Their contribution becomes significant only around 0.6 GeV/c 2 , and then dominant in the high-mass region above 1 GeV/c 2 because of their large opening angle. Their low mass contribution can be extrapolated by using a model fit to the data in the highmass region [21]. PHENIX has reported that the low-mass distribution has a model dependence [16]. This model dependence results in a 100% uncertainty, particularly on the cc contribution. The open heavy flavor contribution is evaluated by binary scaling of the d+Au result [21]. However, the cc contribution is less than 0.1% at most in the mass region of interest, 0.3 GeV/c 2 , even if 100% uncertainty from the model dependence is taken into account.

B. Determination of direct-photon fraction
The direct virtual photon signal is now extracted as the remainder of the signal above the backgrounds described in the previous section, Sec. III A. A similar fitting procedure to the one described in Ref. [6] is employed, in which Eq. (8) is fit to the mass distribution, with the following difference: In 054902-7 the previous analysis only the hadronic cocktail was included in the fit. In the present measurement, the open heavy flavor and BG SUM contributions, which were subtracted before the fit in the previous measurements, are now included together with the hadronic cocktail as fixed contributions in the fit as Eq. (8). This is done in order for a log-likelihood fit to work properly even with limited statistics in the data, especially at higher p T : (8) where r γ is the only fit parameter and f c and f BG are the hadronic cocktail and the fixed contribution of a sum of the open heavy flavor and BG SUM pairs, respectively. The expected mass shape of the direct virtual photons, f dir , is calculated by a Monte Carlo simulation based on Eq. (1). It does not show the drop that appears in the mass shapes of e + e − pairs from π 0 , η Dalitz decays because of S ∼ 1 in Eq. (1). f dir , f c are normalized for m ee < 0.03 GeV/c 2 before the fit to ensure that the fit result matches the data at m ee = 0, where f dir and f c are identical. Finally, a loglikelihood fit is performed within a fit range of 0.1 < m ee < 0.3 GeV/c 2 to determine the direct virtual photon fraction for several p T bins separately [1 < p T < 1.5, 1.5 < p T < 2.0, 2.0 < p T < 3.0, 3.0 < p T < 5.0 GeV/c]. Figure 5 shows the e + e − pair mass distributions in Cu+Cu MB collisions for 1.5 < p T < 2.0 GeV/c. Figure 5(a) shows the data, the fit, the hadronic contribution, and the background BG SUM +− . Figure 5(b) shows the data and fit after BG SUM +− subtraction, the hadronic contribution, and cocktail components.

C. Systematic uncertainties
The major sources of systematic uncertainties of the directphoton fraction are (1) the background normalization, (2) the particle composition of the hadronic cocktail, (3) the e + e − mass range for the log-likelihood fit.
To evaluate the uncertainty of the direct-photon fraction, the fraction is recalculated by using the same procedure and varying each source within ±1σ of its uncertainty. The differences from the nominal value are quantified and taken as contributions to the uncertainty of the direct-photon fraction. An uncertainty of about 15%-40% comes from the fit mass range with different fit starting points from 0 to 0.15 GeV/c 2 . The particle compositions, dominantly π 0 /η add another 5%-15%. An additional 9.6% and 10% uncertainties are introduced from the MB trigger efficiency and e + e − pair acceptance when converting the direct-photon fraction to the yield. Total systematic uncertainties are calculated as a quadratic sum.

A. Direct-photon fraction
The direct-photon fraction as a function of p T is obtained for two different centrality classes, MB and 0%-40%.  [6]. Expectations from NLO pQCD calculations [24] are also shown by curves with different cutoff mass scales μ. from a next-to-leading-order (NLO) perturbative-quantumchromodynamics (pQCD) calculation [24] with different cutoff mass scales μ. While the p+p and d+Au results show agreements with the NLO pQCD calculation, an excess over the NLO pQCD calculation is seen in the Cu+Cu data as well as in Au+Au. The Cu+Cu excess is rather modest compared with Au+Au, possibly due to a smaller volume of the created medium.

B. Direct-photon spectra
The obtained direct-photon fractions are converted to direct-photon yields by using the inclusive photon yields calculated by the same Monte Carlo simulation used for the e + e − pairs of the hadronic cocktail. Figure 7 shows the directphoton spectra for Cu+Cu MB and 0%-40% most-central events. The p+p results [9] parametrized by a modified power-law function A pp (1 + p T 2 /B pp ) n pp and its T AA -scaled functions are shown as the dotted curves together with the data points. The modified power law is an empirical parametrization describing the p+p result well, especially at low p T . The same function has been employed in previous low p T direct-photon publications in heavy-ion collisions [6,7]. We have performed a least squares analysis in which p Tcorrelated and p T -uncorrelated errors are properly taken into account. A detail description of constraint parametrization can be found in Ref. [25]. The p+p data points measured by the EMCal in 4 < p T < 10 GeV/c are included in the fit in addition to the virtual photon measurement covering p T < 6 GeV/c. Here the lowest p T data point is just an upper limit. The best fit gives χ 2 /NDF = 18.9/17, which is the minimum obtained by variation of the p T -correlated errors. The uncertainty of the p+p fit is calculated by using the error matrix of the fit parameters and is indicated as bands on the scaled p+p fits. A different empirical parametrization, employed in Ref. [9], was tested as well. We treat the small deviation we find above 1 GeV/c as a maximum-extend error. We divide the deviation by √ 12 and add it in quadrature to the uncertainty of the fit.

C. Rapidity density
We further investigate the N part dependence of the directphoton yields as discussed in Ref. [7]. It has been reported that the Au+Au results [26] show an increasing trend for N part . The Cu+Cu data points help to have a closer look at the dependence in the small-N part region. The rapidity density for p T > 1 GeV/c at midrapidity, dN/dy(p T > 1 GeV/c), is calculated by summing the direct-photon yields in given p T 054902-9 where p T i , y i γ , and p T i are the mean p T , the direct-photon yield, and the p T -bin width for the ith p T bin. The bin-width correction C bw is evaluated based on the fit function f fit to the data shown in Fig. 7. C bw contributes an additional 3.5% uncertainty of dN/dy. Then, dN/dy for the binary-scaled p+p fit [26] is subtracted. Figure 8 shows dN/dy of the excess yield over the scaled p+p fit as a function of measured charged multiplicity dN ch /dη at midrapidity. A simple powerlaw fit with the fixed power of 1.25, (dN ch /dη) 1.25 , is done for both the Cu+Cu and Au+Au results as done in Ref. [26]. It works very well to describe the dN ch /dη dependence.
The inverse slope of the exponential fits and the rapidity density of the excess yield of direct photons over the scaled p+p fits for p T > 1 GeV/c are summarized together with dN ch /dη, N coll , N part corresponding to 0%-40%, MB Cu+Cu collisions in Table II.

V. SUMMARY AND CONCLUSIONS
Low-p T direct photons have been measured by using the virtual photon method for MB and 0%-40% most-central collisions in √ s NN = 200 GeV Cu+Cu collisions. A clear excess yield of direct photons over the binary-scaled p+p baseline is seen for Cu+Cu as in the previously reported Au+Au results. The Cu+Cu direct-photon p T spectra are consistent with the Au+Au data for similar N part . The exponential fits to the excess over the binary-scaled p+p baseline give inverse slopes of 285 ± 53(stat) ± 57(syst) MeV/c for MB and 333 ± 72(stat) ± 45(syst) MeV/c for 0%-40% centrality. The Cu+Cu data points improve our knowledge of the system size dependence of the excess yield of the direct photons, especially in the small-N part region. The Cu+Cu results on dN/dy for p T > 1 GeV/c follow the same dN ch /dη dependence as the Au+Au data as described by a simple power law.