Photon production from gluon mediated quark-anti-quark annihilation at confinement
aa r X i v : . [ nu c l - e x ] J un Photon production from gluon mediated quark-anti-quark annihilation at confinement
Sarah Campbell Columbia University, Nevis Labs, Irvington, NY, 10533, USA ∗ (Dated: June 23, 2015)Heavy ion collisions at RHIC produce direct photons at low transverse momentum, p T from1-3 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 . Theoretical models, including hydrodynamic models,struggle to quantitatively reproduce the large low p T direct photon excess and v in a self-consistentmanner. This paper presents a description of the low p T photon flow as the result of increasedphoton production from soft-gluon mediated q -¯ q interactions as the system becomes color-neutral.This production mechanism will generate photons that follow constituent quark number, n q , scalingof v with an n q value of two for direct photons. χ comparisons of the published PHENIX directphoton and identified particle v measurements finds that n q -scaling applied to the direct photon v data prefers the value n q = 1 . 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 simulationthat calculates the direct photon v while describing the shape of the direct photon p T spectra in aconsistent manner. The simulation, while systematically low compared to the data, is in agreementwith the Au+Au measurement at p T less than 3 GeV/c in both centrality bins. Furthermore, thisproduction mechanism predicts that higher order flow harmonics, v n , in direct photons will followthe modified n q -scaling laws seen in identified hadron v n with an n q value of two. PACS numbers: 25.75.Dw
I. INTRODUCTION
Direct photons are all of the photons produced in a col-lision excluding the products of hadronic decays. Theyare emitted throughout the evolution of the heavy ionmedium, and because they are color-neutral they donot experience subsequent interactions with the medium.As a result, their spectrum provides a time-integratedpicture of photon emission. Direct photons have var-ious sources, including prompt photons generated byearly hard parton interactions, photons produced inthe pre-equilibrium stage, and thermal photons radi-ated from either the quark gluon plasma (QGP) or thehadron gas stage (HG). In Figure 1, Feynman diagramsof prompt photon production mechanisms, quark-gluonCompton scattering, quark-anti-quark annihilation, andbremsstrahlung radiation, are shown. Prompt photonsare created in p + p collisions and dominate the yield athigh p T in heavy ion collisions. Prompt photon pro-duction rates can be calculated using perturbative QCD(pQCD); quark-gluon Compton scattering and quark-anti-quark annihilation have production rates of order α S α and bremsstrahlung radiation has a rate of order α S α . QCD thermal photons have the same productiondiagrams, shown in Figure 1, but with the partons ther-malized in the medium. In thermal photon pQCD calcu-lations, bremsstrahlung radiation is of order α S α and canexceed the production from the Compton scattering andannihilation processes. HG thermal photons have analo-gous production mechanisms to the Compton scattering ∗ Direct correspondence to:[email protected] and annihilation processes only with pions and ρ -mesonsinteracting instead of quarks and gluons. However, theproduction rates for thermal photons and other directphotons sources are not well constrained particularly inthe non-perturbative regime. This makes separating thecontributions of direct photons at low and intermediate p T difficult.The PHENIX experiment discovered a large di-rect photon excess at low p T , from 1-3 GeV/c, in √ s NN = 200 GeV Au+Au collisions at RHIC relative tothe yields of direct photons in p + p collisions scaled by thenuclear overlap factor, T AA [2, 3]. Subsequent analysesfound that these low p T photons, again from 1-3 GeV/c,have a large azimuthal anisotropy with respect to thecollision’s event plane [4]. Preliminary results from theALICE experiment at the LHC suggest similar behav-ior in 2.76 TeV Pb+Pb collisions [5, 6]. Hydrodynamicmodels are able to describe the direct photon yield withinitial temperatures of 300-600 MeV and thermalizationtimes between 0.15-0.5 fm/c [2]. Reproducing the largemeasured azimuthal anisotropies, v , at these early timeshas proven difficult for hydrodynamic models [7–9]. Thisis because the large azimuthal anisotropies generated byhydrodynamic pressure gradients need time to develop.To address this puzzle some theories introduce delayedQGP formation [10], new sources of photon productioninvolving strong magnetic fields [11, 12] and initial stateGlasma effects [13], while others consider increased con-tributions from the hadron gas stage due to baryon-baryon and meson-baryon interactions [14, 15].In this paper, the sources of identified hadron az-imuthal anisotropies are considered to understand theorigin of the similarly-sized direct photon v . At low p T ,bulk expansion dominates the hadronic v while at high FIG. 1: Feynman diagrams of prompt photon production by a) quark-gluon Compton scattering, b) quark-anti-quark annihi-lation, and c) bremsstrahlung radiation off of an outgoing quark [1].FIG. 2: A Feynman diagram of the quark-anti-quark anni-hilation interaction with a medium gluon producing a directphoton. p T , hadrons from jet fragmentation dominate. In theintermediate p T region, from 1-3 GeV/c, the measuredbaryon and meson v values split, with baryons reachinghigher values of v at higher values of p T [16]. When thebaryon and meson v values are scaled by their numberof constituent quarks, n q , a uniform behavior betweenbaryons and mesons is seen [17]. Coalescence models areable to reproduce quark number scaling by assuming thathadron production is dominated by the recombinationof flowing partons. They assume that thermalized co-moving quarks of a given p T will coalesce into mesons andbaryons with n q -times the p T and n q -times the v where n q = 2 for mesons and n q = 3 for baryons. In this frame-work, energy-momentum conservation is maintained bythe mean-field interaction resulting in soft gluon interac-tions with the medium [18].Similar mean-field or soft gluon interactions could me-diate quark-anti-quark annihilation as the system movestoward color neutrality, resulting in a large increase inphoton production. These interactions, a diagram isshown in Figure 2, would produce photons from partonicprocesses late in the system’s evolution when quarks areflowing. One consequence of this production is that these photons should reproduce constituent quark number scal-ing with the value n q = 2 for direct photons. Fur-thermore, this model provides a testable prediction thathigher order flow harmonics, v n , in direct photons shouldfollow the n q -scaling laws seen in identified hadron v n [19]again with n q = 2 for direct photons.Section II determines the n q for direct photons thatbest reproduces the quark number scaling seen in theidentified hadron v by using a χ analysis of existingdata [4, 20]. Section III details a coalescence-like MonteCarlo calculation that combined with the T AA -scaled p + p component is compared to the measured direct photon p T spectrum and v distribution. A two-component model isassumed where the low p T direct photon excess is primar-ily the result of quark-anti-quark annihilation mediatedby mean-field or soft gluon interactions as the systembecomes color neutral. II. THE n q -SCALING OF IDENTIFIEDHADRON AND DIRECT PHOTON v The elliptic flow of identified hadrons displays con-stituent quark number scaling in the 1-3 GeV/c p T re-gion [21, 22]. In the q -¯ q annihilation picture of directphoton production, this n q -scaling behavior should ex-tend to the direct photons with n q = 2. This is be-cause the n q -scaled v reflects the underlying anisotropyof the quarks and therefore is common for all hadronsand photons produced from these coalescing quarks. Athigh p T , this n q -scaling may breakdown as contributionsfrom hard processes begin to dominate in both the directphoton and identified hadron spectra. Figure 3 shows acomparison of the direct photon v [4] with the chargedpion, kaon and proton v [20] in the 0-20% and 20-40% √ s NN = 200 GeV Au+Au collisions. The n q -scaled v as a function of the n q -scaled p T and KE T are also pre-sented assuming that the n q value for direct photons istwo. The agreement between the scaled direct photon v and the pion, kaon and proton data is impressive despitethe large systematic error bars on the direct photon mea-surement. The scaled pions, kaons, protons and photonsagree at low KE T /n q in both centralities. At KE T /n q above 1.7 GeV, the direct photon’s scaled v drops belowthe pion values. This deviation can be understood as theresult of the increased photon production by initial hardprocesses [4]. Of particular note is how the direct photonand proton v /n q track together as they deviate from thepion values in the 20-40% centrality bin. This suggests asimilar transition to the high p T hard scattering regionfor the scaled protons and photons. While the 0-20%proton v does not extend high enough in KE T /n q , pro-tons in the 0-20% centrality are also expected to break n q -scaling at high KE T /n q and deviations are seen inthe 10-20% bin [20].A χ analysis is undertaken to determine if n q = 2best produces the agreement between the direct photonand the n q -scaled identified hadron v data. This is donein two ways. In Section II A, the datasets are compareddirectly. In Section II B, the n q -scaled identified hadron v are fit and the direct photon v are compared to thatfunction. A. χ comparison between the direct photon and n q -scaled hadron data A χ comparison is performed between the v for directphotons to the n q -scaled hadron data. The χ compar-ison of the direct photon and identified hadron data iscalculated according to χ = X Cent. X π,K,p X KE T /n q ( v γ /n qγ − v h /n q ) ( σ γ /n qγ ) + ( σ h /n q ) (1)where v γ is the direct photon v , v h is the identifiedhadron v for each of the summed hadrons, π , K and p.The χ is summed over the 0-20% and 20-40% centrali-ties comparing the n q -scaled pion, kaon and proton v /n q values to the direct photon v / n qγ where n qγ is the onlyparameter. Determining the photon and hadron uncer-tainties, σ γ and σ h , is complicated because the publishedsystematic errors for both the direct photons and identi-fied hadrons combine both point-to-point and correlatedsystematic errors [4, 20]. To address this the χ analysisis performed in two ways. In one case, the quadraturesum of the statistical and systematic errors for directphotons and the identified hadron uncertainties is used, σ = σ stat ⊕ σ sys . This assumes that the systematic er-rors are uncorrelated. Another χ analysis assumes thatthe systematic errors are fully correlated and the photonand hadron uncertainties are limited to their statisticalerrors, σ = σ stat . In both cases, the comparison of agiven pair of direct photon and hadron data points areincluded in the χ calculation only if the KE T /n q valuesare within 0.1 GeV/c of each other. An example of thisdata comparison over the full range in KE T /n q is shown T p - π + + π - +K + K pp+ γ (a) 1.6 for 0-20% × v0.000.050.100.150 1 2 3 4 5 (GeV/c) q /n T p 1.6 for 0-20% × q /n v(c)0.000.050.100.150 1 2 3 4 5 (GeV) q /n T KE 1.6 for 0-20% × q /n v(e) q / n v q / n v v T p Au+Au = 200 GeV NN s 20-40%(b) q /n T p(d) 20-40% q /n T KE(f) 20-40%
FIG. 3: (color online) The π (blue triangles), K (open ma-genta squares), p (red circles) and direct photon (open greencrosses) v as a function of p T in central 0-20% (a) and mid-central 20-40% (b) Au+Au collisions at √ s NN = 200 GeV.Panels (c) and (d) show the v /n q as functions of p T / n q for0-20% and 20-40% respectively. Panels (e) and (f) show the v scaled by the number of constituent quarks, n q , as a func-tion of KE T /n q , again for 0-20% and 20-40% centralities. Fordirect photons, n q = 2 is assumed. In panels (a), (c) and (e),the 0-20% v values are scaled by 1.6 for better comparisonto the 20-40% results. Error bars and shaded boxes aroundpoints represent their statistical and systematic uncertaintiesrespectively [4, 20]. in Figure 4 where the photon-to-identified hadron datacomparison plots with n qγ = 2 are presented. A χ of16 .
28 is calculated using the quadrature sum of the sta-tistical and systematic errors for the photon and hadronuncertainties with 35 degrees of freedom,
N DF , and areduced χ , χ /N DF , of 0 .
47 is found. As a result of re-quiring photon-hadron matching in KE T /n q , the numberof degrees of freedom of the χ calculation changes as n qγ varies. This leads to a discontinuous χ distribution as afunction of n qγ , as seen in Figure 5.Figure 5 (a) shows the χ versus n qγ when statistical q / n v (a) = 2 γ q n sys σ ⊕ stat σ = σ γπ (b) 0-20% γ K (c) 0-20% γ p q /n T KE q / n v (d) 20-40% γπ q /n T KE (e) 20-40% γ K q /n T KE (f) 20-40% γ p FIG. 4: (color online) Example plots of the input data used for the calculation of χ comparing the v /n q vs KE T /n q foridentified hadrons (red circles) [20] and direct photons (black squares) [4] using the quadratic sum of the statistical and sys-tematic errors. Here, n qγ = 2 is assumed for direct photons. The 0-20% (top row) and 20-40% (bottom row) √ s NN = 200 GeVAu+Au results are shown. Pions (left column), kaons (middle column) and protons (right column) are separately plotted withthe direct photon data over the full KE T /n q range. The data are included in the χ calculation only if the identified hadronand direct photon KE T /n q values are within 0.1 GeV/c. The χ is calculated using the variation between direct photon andidentified hadron v /n q in all six plots. Error bars represent the statistical and systematic uncertainties summed in quadrature.A χ /NDF of 16 . /
35 = 0 .
47 is found using the full KE T /n q range available in the data. γ q n χ (a) sys σ ⊕ stat σ = σ < 1 GeV in 20-40% q /n T Range 1, KE < (1.7, 1) GeV q /n T Range 2, KE + 1 χ : σ
1 + 4 χ : σ
2 + 9 χ : σ γ q n χ (b) stat σ = σ < 1 GeV in 20-40% q /n T Range 1, KE < (1.7, 1) GeV q /n T Range 2, KE + 1 χ : σ
1 + 4 χ : σ
2 + 9 χ : σ FIG. 5: (color online) The χ distribution as a function of n qγ calculated using the quadrature sum of the statistical andsystematic errors for the hadron and photon uncertainties (a) and calculated using only the statistical errors (b). The χ calculation with an upper limit of KE T /n q < . ∗ marks; this is Range 2. Horizontal lines are drawn at the location of the χ min + 1 (solid), χ min + 4 (dashed) and χ min + 9 (dotted) for each calculation. and systematic errors are used to determine the χ andFigure 5 (b) shows the χ when only statistical errors areincluded. Open circles identify the χ values when an up-per limit of KE T /n q < n q -scaling [20].Another χ comparison, shown with ∗ marks and referredto as Range 2, restricts the KE T /n q range in both cen-trality bins with upper limits of 1.7 GeV and 1.0 GeVfor the 0-20% and 20-40% centralities respectively. Thisextends the KE T /n q cut to central collisions where the n q -scaling is expected to remain broken [20]. When the KE T /n q range is restricted the width of the χ distribu-tion increases reflecting the reduced resolving power ofthe χ comparison when fewer data points are included.The optimal n qγ values for n q -scaling are located at the χ minima, a value of 1.79 for all four χ data compar-isons. The error on the n qγ parameter is related to thewidth of the χ curve. It is determined from the rangeof n qγ values where the χ is below χ min + 1 for the 1 σ limit, χ min + 4 for the 2 σ limit, and χ min + 9 for the 3 σ limit. Horizontal lines are drawn at the χ min + n valuesin Figure 5 with solid lines for the 1 σ limits, dashed linesfor the 2 σ limits and dotted lines for the 3 σ limits. Whenthe systematic errors are assumed to be fully correlated,the σ = σ stat case, the n q ’s systematic error from thecorrelation must also be obtained. The systematic erroron the n qγ in the σ = σ stat case is found by shifting allof the photon and identified hadron v values to the ex-treme maximum or minimum values in their systematicerror ranges, re-calculating the χ in the n qγ -space, anddetermining the n qγ where χ reaches a minimum value.The optimal n qγ values and errors from this comparisonof data points are shown with their respective χ /N DF in Table II. B. χ analysis using fit to n q -scaled hadron data Here, a fit to the n q -scaled identified hadron data isused to describe the universal scaling distribution. The0-20% and 20-40% direct photon data is then comparedto this function and fit using TMinuit to find the optimal n qγ by minimize the χ , χ = X Cent. X KE T /n q ( v γ /n qγ − v fit ) ( σ γ /n q ) (2)where v γ is the direct photon v and v fit is the fit to the n q -scaled identified hadron v . The χ is summed overthe 0-20% and 20-40% centralities comparing the v fit tothe direct photon v / n qγ where n qγ is the only param-eter. Again, the χ minimization is performed in twocases to address how the direct photon uncertainty, σ γ ,relates to the direct photon systematic errors. One caseuses the quadrature sum of the statistical and systematicerrors for direct photons, σ = σ stat ⊕ σ sys . This assumesthe systematic errors are uncorrelated. The second case (GeV) q /n T KE(a) - π + + π - +K + K pp+Fit
X 1.6 for 0-20% q /n v q / n v (GeV) q /n T KE(b) 20-40%
FIG. 6: (color online) The 0-20% and 20-40% Au+Au v /n q vs KE T /n q for pions, kaons and protons are fit with a prob-ability density distribution of a Gamma function. High p T protons that deviate from n q -scaling in the 20-40% centralitybin are excluded from the fit and are not shown [20]. assumes that the systematic errors are fully correlatedand the photon uncertainties are limited to the statisti-cal errors, σ = σ stat .To obtain v fit , the n q -scaled identified hadron datais fit using a scaled probability density function of thegamma distribution, G ( x ) = A (( x − µ ) /β ) γ − e − x − µ ) /β β Γ ( γ ) (3)where x is KE T /n q , γ is the shape parameter, µ is thelocation parameter, β is the scale parameter, A is anoverall normalization scale and Γ ( γ ) is the gamma dis-tribution, Γ ( x ) = R ∞ t x − e − t dt . Figure 6 shows the fitresults when the 0-20% and 20-40% Au+Au identifiedhadron v /n q data are fit to Equation 3. In the 20-40%centrality bin, high KE T /n q protons that deviate fromthe n q -scaled pions are excluded from the fit and are notshown. Table I lists the parameters obtained from thefits for both centrality bins. TABLE I: Table of the results of a Gamma distribution fit tothe Au+Au v /n q vs KE T /n q .Parameters 0-20% 20-40% γ µ β A TMinuit fit is used to determine the n qγ where the χ from Equation 2 reaches its minimum value. This fit isperformed over two ranges. Range 1 removes the regionwhere the proton breaks the n q -scaling [20] by applyingan upper limit at KE T /n q < KE T /n q range inboth centrality bins with upper limits of 1.7 GeV and1.0 GeV for the 0-20% and 20-40% centralities respec-tively. This removes the region in the 0-20% bin where n q -scaling is expected to be broken [20]. TMinuit findsthe optimal n qγ value with statistical errors. When thedirect photon systematic errors are assumed to be fullycorrelated, the σ = σ stat case, the n qγ ’s systematic er-rors from this correlation must also be determined. Thisis done by shifting the direct photon v values to theextreme maximum and minimum of the systematic er-ror range and re-fitting with TMinuit to find n qγ at the χ minimum value. The resulting n qγ values and errorsfrom the TMinuit fits are shown in Table II with theirrespective χ /N DF .The low χ /N DF values under the σ γ = σ stat ⊕ σ sys heading reflect the over-estimation of the photon andhadron uncertainties when uncorrelated systematic er-rors are assumed. Under the σ γ = σ stat heading, whenonly the statistical errors are used in the χ determina-tion, the corresponding χ /N DF values are above one,a consequence of the underestimation of the uncertaintywhen the systematic errors are assumed to be fully corre-lated. The separation of the systematic errors into errorsthat are point-to-point independent and those that arecorrelated is needed to fully interpret the χ /N DF val-ues in these comparisons.The hypothesized value of n qγ = 2 is within the sys-tematic uncertainty region when the n qγ is determinedfrom the data with σ γ = σ stat in both Range 1 and Range2. The n qγ = 2 condition is inside of the 1 σ limit for the σ γ = σ stat ⊕ σ sys , Range 2 data comparison and withinthe 2 σ limit for the σ γ = σ stat ⊕ σ sys , Range 1 data com-parison. The n qγ values from the comparison to the fit ofthe n q -scaled hadron data are very similar to the directdata comparison results. A n qγ value close to 1.8 is foundover both ranges when σ = σ stat is assumed and in Range2 when σ = σ stat ⊕ σ sys is assumed. Only the TMinuit fitover Range 1 produces a n qγ value that differs from 1.8,however, it is within 2 σ of the n qγ = 2 hypothesis. Of theeight n qγ searches presented here, six are consistent with n qγ = 2 within 1 σ . The remaining two n qγ searches areconsistent with the n qγ = 2 hypothesis at the 2 σ level.These two comparisons both use the larger KE T /n q re-gion in the 0-20% centrality and σ = σ stat ⊕ σ sys . Thesecomparisons are affected by the difference between thepion v and direct photon v at KE T /n q > . v at high KE T /n q is the result of the increased direct photon contributionsfrom hard scattering at high p T , p T > . n qγ determination. Re-duced systematic errors on the direct photon v mea-surement and separating the systematic errors into er-rors that are point-to-point independent and those thatare correlated would reduce the uncertainty and improve the calculation of the χ in these comparisons. Proton v measurements that extend out to higher p T in the 0-20% centrality bin, and direct photon v measurementsin additional centrality bins and collision systems wouldprovide additional points for comparison benefiting thisanalysis by reducing the width of the χ distribution andimproving the resolving power of the n qγ parameter. Fur-thermore, direct photon azimuthal anisotropy measure-ments at higher orders, v n , will provide an additionaltests to this model. The model predicts that higher or-der direct photon v n will follow the higher-order modified n q scaling relation, with a universal curve in v n /n n/ q asa function of KE T /n q [19], with n qγ = 2 for direct pho-tons.Seven out of the eight χ comparisons shown here findan optimum n qγ value of approximately 1.8. In six cases,the n qγ = 2 condition is within 1 σ of the optimum value.In the remaining two cases, the n qγ = 2 condition iswithin 2 σ of the optimum value. These two cases arebiased by the hard scattering contributions at high p T .These results, in conjunction with the similarity in thedata seen in Figure 3, indicate that the direct photon v data are consistent with the hypothesis of n qγ = 2required by the q -¯ q annihilation production mechanism. III. SIMULATING THE DIRECT PHOTON v To further develop the ansatz of photon production atconfinement from coalescence-like quark-anti-quark an-nihilation, a data-driven Monte Carlo simulation is de-veloped. The crux of the direct photon puzzle is to rec-oncile the p T spectral shape with the large azimuthalanisotropy. In Section III A, the q -¯ q photon p T spectralshape and v are simulated with a Monte Carlo simu-lation. Rather than calculating the yields, a fit to themeasured p T distribution is performed in Section III Bto determine if the q -¯ q photon p T shape from the MonteCarlo is able to describe the large excess above the T AA -scaled p + p yield seen in the data. Then the direct pho-ton v is calculated by weighting the q -¯ q photon v bythe relative contribution of the q -¯ q photon component tothe total direct photon yield; the T AA -scaled p + p contri-bution is assumed to be azimuthally isotropic. A. Monte Carlo of coalescence-like q - ¯ q photon v production The Monte Carlo consists of randomly sampling quark m T values from a thermal Blast Wave distribution. Thequark flow is implemented by calculating the quark v from a fit of the measured n q -scaled identified hadron v and then sampling the quark φ from the v -modulated φ distribution. This process is repeated for three quarksand then co-moving requirements are applied.The quark’s m T is randomly sampled from a thermal TABLE II: Table of optimal n qγ values and errors with χ /NDFσ γ = σ stat ⊕ σ sys σ γ = σ stat n qγ ± ( stat ) χ /NDF n qγ ± ( stat ) ± ( sys ) χ /NDF Data, Range 1 1 . +0 . − . . /
20 = 0 .
24 1 . +0 . . − . − . . /
20 = 5 . . ± .
27 4 . /
17 = 0 .
27 1 . +0 . . − . − . . /
17 = 5 . . ± .
22 3 . /
13 = 0 .
26 1 . ± . +0 . − . . /
14 = 3 . . ± .
44 1 . / .
31 1 . ± . +1 . − . . / . Blast Wave distribution, d Ndm T dydφ ∝ m T r cosh( y ) × exp (cid:18) p T sinh( ρ ) cos( φ ) − m T cosh( ρ ) cosh( y ) T (cid:19) (4)where T is the temperature, m T = q p T + m q is thetransverse mass, ρ = tanh − ( β S ( r/R ) α ) is the boost an-gle, and φ is the azimuthal angle with respect to thereaction plane [23]. Further, β S is the surface velocity, R is the maximum radius in the region and m q is thequark mass. A β S value of 0.75 is assumed and is consis-tent with h β i = 0 . α set to one. A quark mass of300 MeV, temperature of 106 MeV and maximum radiusof 8.5 fm is used. The parameters of the Blast Wave dis-tribution are taken from Refs. [24] and [25]. These BlastWave parameters characterize the m T distribution of thelate-stage medium and therefore identical parameters areused for the Au+Au 0 −
20% and 20 −
40% centrality bins.The r , y and φ values that determine the Blast Wavedistribution are each chosen from flat distributions; r and y are the quark’s radius and rapidity respectively. Thequark’s y is chosen from ± .
50 and a ± .
35 rapidity cutis applied to the resulting photons. The random choiceof φ ensures that each of the successive Blast Wave dis-tributions sample the full variation in azimuth.Rather than using this φ for the quark’s φ , the ther-mal quark’s φ is chosen from an data-driven procedure toreduce the simulation’s dependence on free parameters.This is done by using the m T obtained from the BlastWave to calculate the quark azimuthal anisotropy froma fit to the measured n q -scaled v of identified hadronsshown in Figure 6. Once the quark’s v , v q , is calcu-lated it is used to generate a 1 + 2 v q cos(2 φ ) probabilitydistribution to randomly select the quark’s φ . The v q iscalculated using a fit to the measured n q -scaled identifiedhadron v . A scaled probability density function of thegamma distribution, Equation 3, is fit to the n q -scaledidentified hadron v data as described in Section II B.This method effectively averages the φ variation withinthe Blast Wave distribution while still including radialboost effects. By choosing the φ from the 1 + 2 v cos (2 φ )distribution, the measured identified hadron v /n q isused to guide the modeled quark’s azimuthal anisotropy.This empirical approach to describe the quark’s az- imuthal anisotropy keeps the number of free parametersin the model to a minimum. One downside of this ap-proach is that the v q from the fit relies on the pion dataat high KE T /n q which has increasing contributions fromnon-thermal quarks either from hard processes and frag-mentation or hard thermal coalescence [26]. This mayunderestimate the amount of quark flow at high KE T /n q .The random determination of the quark’s m T and φ is repeated for the second and third quarks within theMonte Carlo event. The same rapidity and radius isassumed for subsequent quarks, and therefore the sameBlast Wave distribution. However, a new m T value issampled, v q is calculated and φ is sampled using the1 + 2 v q cos (2 φ ) distribution. The following co-movingrequirements, motivated by [18], are applied to all threequarks to produce a baryon and to the first and secondquarks to produce a meson, Mesons: | p − p | < p , | x − x | < ∆ x Baryons: | p − p | < √ p , | x − x | < √ x , | p + p − p | < √ p , | x + x − x | < √ x where p i and x i are the three dimensional momentumand position vectors of the various quarks, and ∆ p and∆ x are 0.2 GeV/c and 0.85 fm respectively [18]. Quarksand anti-quarks that annihilate to produce photons mustsatisfy the same co-moving requirements as mesons. Thefour-momenta of quark pairs and triplets that satisfythe co-moving requirements are summed to create pions,photons and protons respectively. The hadrons and pho-tons are brought on mass shell while maintaining kineticenergy conservation. Figure 8 shows the amount of en-ergy taken up by the gluon to bring the photon on massshell as a function of the direct photon’s KE T for the0-20% (left) and 20-40% (right) simulations. The z-axisis the number of counts and is shown with a logarithmiccolor scale. The gluon’s energy contribution is definedas E γ − E q − E q and has a value of approximately −
600 MeV. At photon KE T < E gluon extendsto lower energies of −
770 MeV, however, the majority ofthe contribution is located at −
600 MeV for all photon KE T values. This negative value means that the gluonremoves some of the energy from the quarks and passesit to the medium when the photon is produced. Addi-tional simulations maintaining momentum conservationand energy conservation are also performed, however, ki-netic energy conservation best reproduces the n q -scaling (GeV/c) T KE0 0.5 1 1.5 2 2.5 3 3.5 v × v (a) ) T (KE v ) T (2KE
2v ) T (3KE π γ p q (GeV/c) q /n T KE0 0.5 1 1.5 2 2.5 3 3.5 q / n v × q /n v (c) (GeV/c) T KE0.5 1 1.5 2 2.5 3 3.500.050.10.150.20.250.30.35 (GeV/c) q /n T KE0.5 1 1.5 2 2.5 3 3.500.020.040.060.080.1
FIG. 7: (color online) The v for pions, photons, protons andthrown quarks simulated using the fast Monte Carlo method.Plots (a) and (b) are the v vs KE T for the 0-20% and 20-40% respectively. Plots (c) and (d) are the n q -scaled resultsfor 0-20% and 20-40%. The 0-20% v values are scaled by 1.6to make the y-axis scales consistent. seen in the pion and proton v data. Figure 7 showsthe v for the thrown quarks and simulated pions, pro-tons and photons in 0-20% (a) and 20-40% (b) centralitybins. The v /n q vs KE T /n q , Figures 7 (c) and (d), showthat the n q -scaling is well reproduced in the simulation.Table III displays the inverse slopes of the Monte Carlo p T spectral shape when fit to an exponential in different p T ranges. These are consistent with the inverse slopesobtained from fits to the Au+Au data over similar p T ranges [2, 3]. TABLE III: Table of the inverse slope of the direct photon p T spectral shape in different centralities and p T ranges.Centrality p T range Monte Carlo Au+Au data [2, 3]0-20% 0 . . ± ± ± . . ± ± ± . . ± ± ± . . ±
10 217 ± ± B. Determining the yield of the q - ¯ q photoncomponent To find the total direct photon production, a two-component model consisting of the q -¯ q photon contribu- tion and the T AA -scaled p + p contribution is used. Whileadditional photon sources are expected, these are as-sumed to be negligible compared to the q -¯ q and T AA -scaled p + p components. The simulated q -¯ q photon con-tributions are normalized to the measured direct pho-ton yields. The normalization constant of the q -¯ q pho-ton component is determined from a fit to the measuredAu+Au [2, 3, 27] and T AA -scaled p + p data [2, 28, 29]using TMinuit. The normalization constant is the onlyparameter of the fit. The χ is calculated using the sta-tistical errors from the Monte Carlo simulation and thestatistical and systematic errors from the data summedin quadrature. At low p T where p + p reference data isscarce, the p + p yield is extrapolated from the power lawfit obtained from [3]. The normalization error on the q -¯ q photon component and the systematic error of the T AA -scaled p + p fit result in systematic error band onthe simulation.Figure 9 shows the resulting p T distributions for 0-20% and 20-40% Au+Au collisions. The various Au+Aumeasurements are shown in red circular symbols and the T AA -scaled p + p measurements are shown in blue squareand cross symbols. The p + p fit is shown with a greyband, the normalized q -¯ q photon contribution is shownwith a purple band and the total simulated yield is shownwith a cyan band. The error on the yield determinationresults in a systematic band on the q -¯ q photon contri-bution which is propagated to the total simulated yield.Below the main figures the ratio of the Au+Au data tothe simulation result is shown. This ratio is fit to a flatline and found to be consistent with one for both central-ities, a value of 0 . ± .
051 for 0-20% and 1 . ± . χ /N DF values for these flat line fitsare 22 . /
26 = 0 .
877 and 32 . /
26 = 1 .
25 for the 0-20%and 20-40% ratios respectively. This confirms that thephotons generated by the gluon-mediated annihilation ofradially boosted quarks are able to describe the shapeof the direct photon p T spectra for both the 0-20% and20-40% centrality bins.The total direct photon v is the weighted average ofeach component’s v . The T AA -scaled p + p contributionis assumed to have no reaction plane dependence and,therefore, a v of zero. By weighting the simulated q -¯ q photon v by the relative contributions of the q -¯ q photonyield to the total simulated yield, the total low p T pho-ton v for each centrality can be calculated. Figure 10compares the simulated direct photon v to the measuredAu+Au v (solid blue circles) [4]. The open red circlesare the unweighted q -¯ q photon v generated in the MonteCarlo. The small black squares are the total direct pho-ton v assuming uniform azimuthal production from the T AA -scaled p + p source. The relative contribution of the q -¯ q photon component to the yield is shown below the v plots; this is the weight used to calculated the total sim-ulated v . The error in the q -¯ q yield normalization leadto the systematic error in this q -¯ q Monte Carlo weight.The systematic error in the modeled v is calculated fromthe quadrature sum of this normalization error and the E gluon (GeV)-0.8 -0.75 -0.7 -0.65 -0.6 ( G e V ) T KE (a) E gluon (GeV)-0.8 -0.75 -0.7 -0.65 -0.6 ( G e V ) T KE (b) FIG. 8: (color online) The energy taken by the gluon as a function of the direct photon’s KE T for the 0-20% (a) and 20-40% (b)simulations. The z-axis is the number of counts and is shown with a logarithmic color scale. - ( G e V / c ) T ) d N / dp T ( / p -6 -5 -4 -3 -2 -1 (a) × AA T PRL 104 132301 (2010)PRL 98 012002 (2007)PRD 86 072008 (2012)Model curves Monte Carloqq- p+p fit from × AA TPRC 91 064904 (2015)Total yield [GeV/c] T p0 1 2 3 4 5 6 7 8 9 10 M ode l ( A u + A u da t a ) -0.500.511.52 - ( G e V / c ) T ) d N / dp T ( / p -7 -6 -5 -4 -3 -2 -1 (b) × AA T PRL 104 132301 (2010)PRL 98 012002 (2007)PRD 86 072008 (2012)Model curves Monte Carloqq- p+p fit from × AA TPRC 91 064904 (2015)Total yield [GeV/c] T p0 1 2 3 4 5 6 7 8 9 10 M ode l ( A u + A u da t a ) -0.500.511.52 FIG. 9: (color online) The direct photon yield versus p T for the 0-20% (a) and 20-40% (b) Au+Au data (red circles andasterisks) [2, 3, 27] are shown on a log-scale. The T AA -scaled p + p yields (blue squares and crosses) [2, 28, 29] are also shownincluding a power law fit to the p + p data (grey band) [3]. The Monte Carlo yield from quark anti-quark annihilation (purpleband) are fit to the data and are shown with the total fit yield (cyan 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 aflat line fit to this ratio with a value of 0 . ± .
051 and 1 . ± .
065 for the 0-20% and 20-40% ratios respectively. n q -scaled v of identi-fied hadrons, with relative error values of 10% and 7%in 0-20% and 20-40% respectively. The model simula-tion of the total direct photon v extends out to a p T of 3.6 GeV/c in 0-20% and 3.2 GeV in 20-40%, abovewhich the simulation lacks sufficient statistics. For the0-20% centrality the total direct photon v agrees withthe measured results within error bars. However, abovea p T of 1.4 GeV/c, the simulated v is systematically atthe bottom of the error range. In the 20-40% centralitycomparison, the total simulated v agrees with the mea-sured results for p T less than 3 GeV/c, above 3 GeV/cit underestimates the measured v . In both centralities,the simulated direct photon v agrees with the measured v within errors. IV. CONCLUSIONS
Photon production from gluon mediated q -¯ q annihila-tion as the system becomes color neutral is proposed asa large additional source of direct photons. This wouldrequire direct photons follow n q -scaling with an n qγ = 2.The large direct photon flow measured in Au+Au colli-sions at RHIC is consistent with n q -scaling when n qγ = 2.Furthermore, in the 20-40% comparison where the high p T proton v /n q is seen to split from the n q -scaled pionresult, the direct photon v /n q follows the same trendas the proton. This suggests that direct photons andprotons may experience similar transitions from the re-combination dominated intermediate p T to the higher p T region dominated by hard processes. χ comparisons ofthe direct photon and identified hadron v in KE T /n q re-gions where n q -scaling is seen in identified hadron data,find that the direct photon v optimally agrees with theuniform n q -scaled curve when n qγ is near a value of 1 . χ comparisons are consistent with n qγ = 2, the remaining two comparisons are consistentat the 2 σ level. The two remaining comparisons includethe 0-20% high KE T /n q region where deviations from n q -scaling are expected. The χ comparisons would ben-efit from reduced systematic errors on the direct photon v measurement and the separation of the systematic er-rors into uncorrelated and correlated errors. Direct pho-ton and identified hadron v measurements in additionalcentralities, √ s NN and collisions systems as well as pro-ton v measurements that extend out to higher p T wouldprovide further points of comparison and thus improvethis analysis.A Monte Carlo simulation generates the q -¯ q annihi-lation photon component p T shape and φ modulationassuming a coalescence-like framework with quarks that follow a Blast Wave m T distribution and a data-driven v parametrization. The Monte Carlo is able to repro-duce the n q -scaling of pions and protons and determinethe q -¯ q photon v and the shape of its p T distribution.The resulting q -¯ q photon p T shape with the T AA -scaled p + p photon yield is able to describe the large direct pho-ton excess seen in 0-20% and 20-40% Au+Au collisions.The simulated direct photon v is consistent with themeasured v in the 0-20% centrality bin but systemati-cally low. In the 20-40% comparison the simulated directphoton v is able to reproduce the measured direct pho-ton v at p T less than 3 GeV/c but underestimates the v at higher p T as the T AA -scaled p + p contribution be-comes significant. The addition of thermal hard quarkpairs would likely contribute to additional yield and flowfor p T values above 3 GeV/c [26]. Future work wouldbenefit from a more robust hydrodynamic calculation ofthe flowing quarks near the phase transition with yieldestimates. This is particularly important as the deter-mination of the quarks v from the n q -scaled identifiedhadron v is expected to falter at high p T as non-thermalproduction mechanisms such as thermal hard coalescenceand fragmentation from hard interactions contribute tothe pion yield.This paper has focused on the published √ s NN = 200 GeV Au+Au 0-20% and 20-40% di-rect photon p T and v distributions. Future work tosimulate the q -¯ q photon contributions in more peripheralcollisions is promising. Additionally, the higher ordersof the direct photon flow presents a new quantity to dis-tinguish between the different photon processes. Giventhe soft-gluon mediated q -¯ q annihilation productionmechanism ansatz, the v n for direct photons is expectedto be similar to the pion v n at p T less than 3 GeV/c forhigher orders of n . This model predicts that higher-order v n n q -scaling laws seen with identified hadrons [19] willremain valid for the direct photon v n where the n qγ = 2. Acknowledgments
The author thanks John Lajoie and Paul Stankusfor many valuable conversations. Additional discussionswith Bill Zajc, Peter Steinberg, Anne Sickles, Rich Petti,Volker Koch and Che-Ming Ko are also recognized as arethe organizers and attendees at the 2013 ECT* work-shop on ”Electromagnetic Probes of Strongly InteractingMatter: Status and future of low-mass lepton-pair spec-troscopy”. This research was supported by US Depart-ment of Energy grants DE-FG02-86ER40281 and DE-FG02-92ER40692. [1] C. Klein-Boesing, ph.D. thesis, University of Muenster,2005. [2] A. Adare et al., Physical Review Letters , 132301(2010). v (a) = 200 GeV NN s0-20% Au+Au PRL 109 122302 (2012)Total flow Monte Carloqq- (GeV/c) T p M C w e i gh t qq - v -0.0500.050.10.150.20.250.3 (b) = 200 GeV NN s20-40% Au+Au PRL 109 122302 (2012)Total flow Monte Carloqq- (GeV/c) T p M C w e i gh t qq - FIG. 10: (color online) The direct photon v versus p T for the 0-20% (a) and 20-40% (b) Au+Au data (blue circles) is shown [4].The Monte Carlo v from quark anti-quark annihilation (red open circles) and the total v (black squares) are shown. Therelative contribution of the quark anti-quark annihilation component is shown in the lower plot of each figure.[3] A. Adare et al., Physical Review C , 064904 (2015).[4] A. Adare et al., Physical Review Letters , 122302(2012).[5] M. W. for the ALICE Collaboration, Nuclear Physics A , 573c (2013).[6] D. L. for the ALICE Collaboration, Journal ofPhysics:Conference Series , 012028 (2013).[7] R. Chatterjee, E. S. Frodermann, U. Heinz, and D. K.Srivastava, Physical Review Letters , 202302 (2006).[8] R. Chatterjee and D. K. Srivastava, Physical Review C , 021901 (2009).[9] F.-M. Liu, T. Hirano, K. Werner, and Y. Zhu, PhysicalReview C , 034905 (2009).[10] F.-M. Liu and S.-X. Liu, Physical Review C , 034906(2014).[11] B. Muller, S.-Y. Wu, and D.-L. Yang, arXiv:1308.6568.[12] K. Tuchin, Physical Review C , 024912 (2013).[13] M. Chiu, T. K. Hemmick, V. Khachatryan, A. Leonidov,J. Liao, and L. McLerran, Nuclear Physics A , 16(2013).[14] H. van Hees, C. Gale, and R. Rapp, Physical Review C , 054906 (2011).[15] O. Linnyk, W. Cassing, and E. L. Bratkovskaya, PhysicalReview C , 034908 (2014).[16] S. S. Adler et al., Physical Review Letters , 182301 (2003).[17] D. Molnar and S. A. Voloshin, Physical Review Letters , 092301 (2003).[18] V. Greco, C. M. Ko, and P. Levai, Physical Review C , 034904 (2003).[19] A. Adare et al., arXiv:1412.1038.[20] A. Adare et al., Physical Review C , 064914 (2012).[21] A. Adare et al., Physical Review Letters , 162301(2007).[22] A. Adare et al., arXiv:1412:1043.[23] E. Schnedermann, J. Sollfrank, and U. Heinz, PhysicalReview C , 2462 (1993).[24] Z. Tang, Y. Xu, L. Ruan, G. van Buren, F. Wang, andZ. Xu, Physical Review C , 051901(R) (2009).[25] F. Retiere and M. A. Lisa, Physical Review C , 044907(2004).[26] R. C. Hwa and C. B. Yang, Physical Review C , 024905(2004).[27] S. Afanasiev et al., Physical Review Letters , 152302(2012).[28] A. Adare et al., Physical Review D , 072008 (2012).[29] S. S. Adler et al., Physical Review Letters98