Scholarly article on topic 'Dilepton creation based on an analytic hydrodynamic solution'

Dilepton creation based on an analytic hydrodynamic solution Academic research paper on "Physical sciences"

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Academic research paper on topic "Dilepton creation based on an analytic hydrodynamic solution"

Cent. Eur. J. Phys. • 12(2) • 2014 • 132-140 DOI: 10.2478/s11534-014-0434-2


Central European Journal of Physics

Dilepton creation based on an analytic hydrodynamic solution

Research Article

Mate Csanad1*, Levente Krizsan1

1 Eötvös University, Department of Atomic Physics, Päzmäny P. s. 1/a, 1117 Budapest, Hungary

Received OS July 2013; accepted 24 December 2013

Abstract: High-energy collisions of various nuclei, so called "Little Bangs" are observed in various experiments of

heavy ion colliders. The time evolution of the strongly interacting quark-gluon plasma created in heavy ion collisions can be described by hydrodynamical models. After expansion and cooling, the hadrons are created in a freeze-out. Their distribution describes the final state of this medium. To investigate the time evolution one needs to analyze penetrating probes, such as direct photon or dilepton observables, as these particles are created throughout the evolution of the medium. In this paper we analyze an 1+3 dimensional analytic solution of relativistic hydrodynamics, and we calculate dilepton transverse momentum and invariant mass distributions. We investigate the dependence of dilepton production on time evolution parameters, such as emission duration and equation of state. Using parameters from earlier fits of this model to photon and hadron spectra, we compare our calculations to measurements as well. The most important feature of this work is that dilepton observables are calculated from an exact, analytic, 1+3D solution of relativistic hydrodynamics that is also compatible with hadronic and direct photon observables.

PACS C200B): 25.75.-q,25.75.Cj,25.75.Nq Keywords: hydrodynamics • quark-gluon plasma • dileptons

© Versita sp. z o.o.

1. Introduction

The Interest In relativistic perfect fluid hydrodynamics grew in the last decade due to the discovery of the strongly interacting quark gluon plasma: it is well known that the medium created in high energy heavy ion collisions is an almost perfect fluid, in particular the collective phenomena in soft hadron production can be most successfully described by hydrodynamic models [1]. Hydrodynamic mod-


els aim to describe the dynamics of the medium created In heavy ion collisions, and investigate the relation between the initial state, the final state and the equation of state. Many solve the equations of hydrodynamics numerically, but there exist a few exact, analytic solutions. The goal of this paper is to obtain analytic results on dilepton distributions based on an exact hydrodynamic solution. In case of numerical simulations, one may take more realistic effects into account, but one loses the analytic understanding of the connection between input parameters and final results. This understanding can however better be maintained when using exact, analytic solutions.

There was a long search for such exact, analytic solu-


tions of relatlvlstlc hydrodynamics, and only few applicable ones were found. Most of these are 1+1 dimensional, and few truly 1+3 dimensional solutions exist. In this paper we extract observables from the relativistic, ellip-soidally symmetric solution of Ref. [2]. Hadronic observables from this solution were calculated in Ref. [3] and compared to RHIC data successfully. Obtained model parameters describe the final state (the hadronic freeze-out), where hadrons are created. The time evolution of the medium can be probed by penetrating probes: leptons and photons, created throughout the evolution. Direct photon observables, based on the aforementioned model, were calculated in Ref. [4], and they were also successfully compared to RHIC data. In this paper we calculate transverse momentum distribution and invariant mass distribution of dileptons from the solution of Ref. [2]: this is the "missing link", the most important quantity not yet calculated via a model based on an exact, analytic 1+3D relativistic hydro solution.

2. Perfect fluid hydrodynamics

The hadronic observables can be extracted from a solution via this phase-space distribution at the freeze-out, based on the choice of an emission surface and Cooper-Frye like emission function. This distribution corresponds to the hadronic final state or source distribution (see details about such a calculation in Ref. [3]). In the case of dileptons or photons, one has to follow a different path. An important result for hydrodynamic models is, that because hadrons are created at the quark-hadron transition, hadronic observables do not depend on the initial state or the dynamical equations (equation of state) separately, just through the final state [6]. Thus same hadronic final state can be achieved with different equations of state or initial conditions. We may fix the final state from hadronic data, but we need penetrating probes, such as photon or lepton data to investigate the equation of state or the initial state. Photon creation is sensitive to the whole time evolution, thus both to initial conditions and equation of state as well, as discussed in Ref. [4], where direct photon observables were calculated and compared to data. Dilep-ton creation is also sensitive to the whole time evolution, as we will see in the next sections.

The equations of hydrodynamics describe the local conservation of a given charge (n) and local conservation of energy-momentum density (T^v). The fluid is perfect if the energy-momentum tensor is diagonal in the local rest frame, i.e. viscosity and heat conduction are negligible. This can be assured if T^v is chosen as

= (e + p)u»uv - pg"

where e is energy density, p is pressure and is the metric tensor, diag(—1,1,1,1), while x^ = (t,rx,ry,rz) is a given point in space-time, t = Vt2 — r2 the coordinate proper-time, = d^ is the derivative versus space time, while p^ = (E, px, py, pz) is the four-momentum. The conservation equations are closed by the equation of state (EoS), which defines the relationship between energy density e and pressure p. Usually e = Kp is chosen, where k may depend on temperature T, and the first relativistic exact solutions with temperature dependent k were recently found in Ref [5]. In this paper we however use a solution with constant k. It is important to see that in this case k = 11c2, with cs being the speed of sound. Temperature can then be defined based on entropy density, energy density and pressure.

If a solution of the above partial differential equations is given, the phase-space distribution can be expressed by a (Boltzmann-)Juttner-distribution of

f (k, x) « exp

T (x )

3. Dilepton creation from hydro evolution

Many calculations exist on the nature of dilepton production In a hadron gas or in QGP, based on the thermal expectation value of the electromagnetic current-current correlators, e.g. those in Refs. [7, 8]. Dilepton emission from based on a hydrodynamic solution can be calculated similarly to Refs. [9-15]. Hydrodynamics in these works is utilized in form of a full-blown numerical simulation or approximative solutions. Our goal is, however, to obtain an analytic description of the dilepton distributions based on an analytic solution of hydrodynamics. In order to do this, we start from the dilepton source given as

= / d3k1dik2f(kt,x)f(k2,xHelCT

with k and k2 being the momenta of the two particles creating the dilepton pair, f(kt,x) the Jüttner-distribution, vrel is the relative velocity of the incoming pair, and a is the production cross-section of the given process. This is based on the well-known relation from nuclear physics, where the rate of a given process is proportional the density of the involved particles, the cross-section of the process and the average velocity: rate(A + B ^ X) = nAnB{aA+B^Xv). The assumption in this calculation is, that dileptons are created in annihilation-decay-like processes: qq is the incoming pair the quark-gluon phase,

and n+n— in the hadron gas phase. Then the dilepton creation happens through a quasireal photon or a vector meson.

The actual calculation of the creation rate goes as follows. First, we express the relative velocity in the pair-comoving frame in the following invariant form:

M2 L 4m2 (/U

= 2EE2V1 " M (4)

with M2 being the dilepton invariant mass squared and m

the mass of the two incoming particles (pions or quarks) and E12 their energy. Then we switch from the k1 and k2 to P = k1 + k2 and k = (k1 — k2)/2, and obtain in the pair center of mass system

d3ki d3k2 d3P 4d3k d3P 1 fA 4m2 kAlkAl^

-r1-^ = —^^T = — -TW MdMdQ,

E1 E2 EM E 16 V M2

with E = E1 + E2 and Q the solid angle corresponding to P; finally, the dilepton source is

dN f d'ip 1 L M2 L 4m2 1(Л

= J E VeV - M2MdMdQf ^Ш.*)^1 - m с. (6)

Next, we have to assume a cross-section for dilepton creation. The color-averaged qq ^ i+i— cross-section for a given flavor can be calculated as [16]

2 1 +2m2/M2

Ш2 ei

yi - 4m2/M2

with M2 being the dilepton invariant mass squared, eq the charge of the given quark (in units of e), and mq its mass. This then has to be summed up for the used flavors (u, d, s in this case).

The same cross-section for dilepton production in a pion gas can be calculated as

с = 4MF(M2)\W1 - 4ml/M2, (8)

where again M2 is the dilepton invariant mass squared, m„ the pion mass, and F(M2) is the electromagnetic form factor of pions, summed up for a range of exchangeable

particles (p and its excitations p' and p", as well as w or

\F (M2)\2 = £

(m2 - M2)2 + m2V2.

where Nt is a relative normalization factor (for which, in case of p, p' and p'' the values of 1, 8.02 x 10—3 and 5.93 x 10—3 are taken usually [10], and these factors have to be tuned to experimental data for other particles as well), mi is the mass of the exchanged particle and r2 its

width. One may use the vacuum mass and width of the given particles, or their in-medium value, if there is a hint at an in-medium spectral function modification. This is the case for high energy heavy ion collisions, however, we will stick to the vacuum values, taken from Ref. [17]. In order to obtain the mass- and average momentum dependence of dilepton creation, we assume a Juttner-like distribution of the incoming particles, as given also in Eq. (2):

f (k,x )d3k =

,-kfun (x)/T (x)

This means that we assume a thermalized medium containing quarks, or a (still) thermalized pion gas. Dileptons can be created in both media, the main difference will be the cross-section (the production mechanism described in the above sections), the temperature of the medium of the 'incoming' particles, and the duration of the emission (as well as the expanding size of the particle emitting region). In Eq. (6) f(k1,x)f(k2,x) appears, but this clearly equals f(P,x), if P = k1 + k2 is the total momentum (note, that the original formula of Eq. (6) neglects quantum-statistical and other correlations of the pairs, which is a good approximation, as this effect is smeared out by integrating on the relative momentum). Taking this into account, we get the following general result

dyMdMd2F>t 16(2n)

M2 1 -

-F^u,(x)/T (x)d4*.

(where m is the mass of the incoming particle, m„ or mq, so in fact the above is a separate formula for the QGP and the pion gas case). Next, we utilize this and a given solution of hydrodynamics to calculate dilepton distributions.

4. Dilepton creation in the analyzed solution

The analyzed 1+3D relativistic solution [2] assumes self-similarity and ellipsoidal symmetry. Thermodynamical quantities (in particular the temperature, as discussed below) are constant on the surface of expanding ellipsoids, given by constant values of

X(t)2 Y(t)2 Z(t)2'

where X(t), Y(t), and Z(t) are time dependent scale parameters (axes of the s = 1 ellipsoid), only depending on the time. Note that in this paper, we use X(t) = Y(t) in order to obtain a simple result. This is justified as az-imuthal asymmetry is integrated out in the investigated observables. The velocity-field describes a spherically symmetric Hubble-expansion:

and X(t),Y(t),Z(t) have to be constant to satisfy the equations of hydrodynamics.

The temperature distribution in this model T(x) is

T (x ) = To ( T )


where t0 is an arbitrary time when the temperature is T0. This is usually chosen to be the time of the freeze-out and thus T0 is the central freeze-out temperature (i.e. T0 = T|s=0 T=Tf)), while s is the above scaling variable and b is a parameter defining the temperature gradient. If the fireball is the hottest in the center, then b < 0. The above outlined solution may be applied to the time evolution of the strongly interacting QGP, but also to a thermalized hadron gas.

If we plug this solution in Eq. (11), we have to integrate the phase space distribution with respect to x. To perform this integration we will use a second order saddlepoint approximation.1 In this approximation the point of maximal

1 In case of a saddlepoint-type of integration is, we have

emissivity (the maximum of the f (k,x) Jüttner distribution in x) is

= Px 1—f- Fx (15)

ro.y = PytF' (16)

ro,z = pz-f (17)

while the widths of the particle emitting source distribution are

t \ -3/k+2 2 To

t \ -3/k+2 2 To

-) To F

t \ -3/k+2 2 To

-) To F

R2 = Pz

where we introduced the auxiliary quantities

Py = Pz =

k - 3 -

k - 3 -

k - 3 - '

(21) (22) (23)

where again k = c-2 is describing the EoS. The source widths depend clearly on time, as the system is expanding. Note that we assume X0 = Y0 in this paper, as noted after Eq. (12). The source then looks like

;\2 ir., -r

(x)lT(x) = e 2r2 2ry 2ry , with

F I t \ 3lK

C = - F - {F'2 - (PxPx + PyPy + PzPz )l2) . To \To I

an f fg type of expression, with f being a narrow distribution around x0 and g a slowly changing function. The second order saddle-point approximation can then be given as g(x0)f(x0)^fnAf with Af being the width of the narrow distribution f.

After integrating on the spatial coordinates, we get At midrapidity and after integration on the azimuthal an-

gle of the momenta our result is

(x)/T (X)d3x = 312 , . -2/k+2

I T T-2 \ 3'2 I t \ -2/k+2

eC (^J • (26)

T0 )3 (

Toy/M2 + Pj \ To T T2 \ 1 . ) - 3+3

M2 - Pj - PX+PLPj)

(2n)2 y/PxPyPz X

y/M2 + Pj \Toj \ Toy/M2 + Pj \Tol ' ' 4

. . 31k

M P2 Pl - px

Now we have to integrate on time, and assuming px = py (i.e. neglecting azimuthal asymmetry2), our result on the dilepton source is

MdMPtdPt 16(2n)2/2

M2 1 -


ka 3 - 4 r

4k 3-A¡3 T - 2'A^K



here we Introduced the following constant:

M2 - P2(1 + z+iL) ToVM2 + Pj

and £ = t/t0 is the time-variable divided by the t0, the "fixed point" of the time integration. We will choose this to correspond to the time of the quark-hadron transition (sometimes called the hadronic freeze-out), thus t0 = tfo, and T0 = Tfo is the temperature of this transition. The time limits of the integration for QGP are then [tini, tfo], while for the hadron gas phase they are [tfo, tfinai]. Here tn is the initial time of dilepton production, and Tini is the associated temperature, and similarly tfinai is the time when dilepton production in the hadron gas stops (one may call it a kind of kinetic freeze-out), and Tfinai is the corresponding temperature. Based on Eq. (14), one can also express £ as a function of the central temperature (since t = t here): £ = (Tfo/T(s = 0, t))K'3. We will use the temperature range to determine the integration limits when comparing the experimental data.

Given Eq. (28), one substitutes the given cross-section a in the above formula, and obtains the dilepton distribution based on the given production mechanism. Eq. (28) represents the main result of this paper: an analytic formula for the dilepton invariant mass and transverse momentum distribution3, for various production mechanisms. In the next section we analyze this result quantitatively, mainly its dependence on the model parameters.

5. Quantitative analysis of our results

Let us calculate the dilepton production from a thermal quark-gluon plasma type of medium, discovered at RHIC, and from a hadron gas, dominating dilepton production at SPS. Model parameters in such a calculation can be based on comparison of the model to hadron [3] and photon production [4]. In these fits to y/sNN = 200 GeV RHIC Au+Au data, we got Tfo = 204 MeV for the central (max-

2 Note, that the experimental dilepton observables are azimuthally integrated, so azimuthal asymmetry does not have to be taken into account.

3 The distribution given in Eq. 28 depends on both M and Pt, and it has to be integrated out numerically in order to obtain for example dN/dM

imal) temperature at the freeze-out, -0.34 or -0.84 (depending on centrality) for the transverse expansion over temperature gradient Xfilb and Yglb (similarly to Eq. (21), results don't depend on expansion or temperature gradient separately, just through their ratio), t0 = 1J fm|c for the freeze-out proper time. In this section we analyze the quantitative dependence of dilepton creation (invariant mass and transverse momentum distributions) on the most important parameters of k (the Equation of State) and the dilepton creation time interval, using values for the other parameters mentioned above. In the next section we will then compare our calculations to RHIC and SPS data.

EoS (k) dependence of invariant mass (M) and the transverse momentum (Pt) distributions for both QGP and pion gas are shown in Fig. 1 (note that in the pion gas case, we take into account the p channel and its excited states (p', p"), and for other contributions one may draw quite similar consequences). All distributions depend strongly on the Equation of State, in particular the Pt spectrum gets much steeper for increasing k values. The absolute magnitude of the M distribution changes with k, as for a large k, temperature changes are slower as a function of proper time. Hence (if the freeze-out temperature is fixed from hadronic data) the system spends more time near the freeze-out temperature if k is large. In fact the experimental data (of direct photons) supports large average k values (k = 1.1 [4]) in QGP created at RHIC. As for the EoS in a pion gas, one may use lattice QCD values [18], where a larger average is supported for subcritical temperatures. The used k values represent an average Equation of State. Clearly, it changes throughout the evolution, as the temperature changes as well, but to have an analytic description of dilepton creation, one has to assume a single average k. Recently, solutions with a QCD Equation of State (i.e. a k(T) function) were discovered [5], but those represent a constant temperature distribution, and are partly implicit, so we use the previously discussed solution here. Also note, that above k values of 5-6, dilepton invariant mass distributions are not very sensitive to the exact value of k in case of the hadron gas component, as shown on the lower right plot of Fig. 1.

We also calculated the dilepton production dependence on the evolution time interval, as clearly this determines the absolute normalization of the results. Results on these are shown in Fig. 2. The curves are labeled with changing £, which represents the ratio of the time integration limits (and the quark-hadron transition time being always in the denominator, thus £ < 1 for QGP and £ > 1 for the pion gas). The results show that the longer the evolution lasts the more dileptons are produced.

The above two paragraphs represent one of the key points

of this analysis, since lepton production spans the whole evolution of the Little Bangs created in heavy ion collisions. Thus, with a time dependent model, one may extract information on the time evolution from dilepton observables. One of the most important pieces of information is the speed of sound, given through the Equation of State parameter k = 1/c2. In Ref. [4], by comparing direct photons to hydrodynamic calculations, we found a value of cs=0.36±0.02, and an initial temperature (determined based on the ratio of initial and final time, as well as on the freeze-out temperature) of ~ 500 MeV. However, dilepton production depends also on these numbers, and in Figs. 1,2 we analyzed this dependence. The Equation of State parameter k strongly influences the slope of the transverse momentum distribution, but also the shape of the mass distributions.

6. Comparision to data

We compared our results to dilepton measurements on d'sNN = 200 GeV RHIC Au+Au data of PHENIX [19]. As mentioned in the first paragraph of the previous section, we used the parameters from Ref. [3] (where results from this model were compared to hadronic data), in particular the freeze-out time of 7.7 fm|c. From tuning our parameters to the data, we got for the central initial QGP temperature (where thermal dilepton production starts) Tini = 210 MeV, while in the hadron gas, dileptons are produced until the central temperature drops to Tfinai = 110 MeV. These values correspond to £ = 0.49 - 1.0 for the QGP phase and £ = 1.0 — 1.6 for the hadron phase. Note that the length of the emissions correlates strongly with the weight of the given components. Also, these are values for the central temperature, as our fireball is cooler in the outer regions (and temperature gradient is determined by parameter b). The EoS was determined in Ref. [4] (where results from this model were compared to direct photon data), the value of k = 1J was used here as well. The results in this range of 300 MeV < M < 1800 MeV are not incompatible with the data, however, there is a small excess around M = 500 MeV, seen also if comparing to simulations based on p+p data [19]. Recently, it has been proposed that this excess may be attributed to the modification of the rj mass [20]. Transverse momentum spectra are analyzed there with a modified n' mass, and plugged in to the PHENIX dilepton cocktail. Present results can be used to analyze possible mass and width modifications to describe the dilepton excess at 500 MeV, based on modified momentum distributions;this is however outside the scope of present paper. Note also, that the production mechanisms we included go to zero at low invariant

200 600 1000 1400 1800 P, [MeV/c]

1000 1400 P, [MeV/c]

1000 1200

M [MeV/c2]

1 1 i i 6=0.1 -

QGP 6=0.2 -

6=0.3 ........

6=0.4 _

6=0.6 .........

6=0.7 ....... ■7,1. .,'. , . 1

M [MeV/c2]

1000 1200

MeV 100 o 1 1 hadron gas i i k=4 k=5 k=6 ........ k=7 ................ k=8 k=9 :

Pt N/P 10 u

1000 1400 Pt [MeV/c]

1000 1400 Pt [MeV/c]

Figure 1. Dilepton production for various equation of state parameters (k), in case of quark annihilation (top) and pion annihilation in a hadron gas through the p channel (bottom). Mass dependent curves are integrated out on Pt (between 100 and 2000 MeV), while transverse momentum dependent results are taken at M = 1000 MeV. These strongly depend on the Equation of State, the spectrum gets much steeper for increasing k values. The normalization changes as for a large k, temperature changes slower as a function of proper time. Note also that above k values of 5-6, dilepton invariant mass distributions are not very sensitive to the exact value of k, see the lowest plot.

Figure 2. Dilepton production dependence on the time evolution interval in case of quark annihilation (top) and pion annihilation (bottom), calculated similarly to Fig. 1. In the first case, i is defined as the initial dilepton production time divided by the quark-hadron transition (hadronic freeze-out) time. In the second case, i is the ratio of the chemical freeze-out time and the hadronic freeze-out time.

masses, due to the vanishing phase-space of two-particle decays. However, three-particle decays of n°, and q or q' account for most of the dileptons produced at M < 200 MeV. In order to get a simple, analytic and comprehensive result, we stuck to the p, w and < contribution.

We also made a comparison to acceptance corrected thermal dilepton data from the SPS NA60 experiment [21] measured in 158 AGeV In+In collisions. Here we used similar parameters, however, we assumed a smaller radial flow (0.64 instead of 0.84) and a smaller central temperature freeze-out (Tfo = 140 MeV nstead of 204 MeV) (see e.g. Ref. [22] for a motivation for these values). The dilepton production starts also from a lower temperature, Tini = 200 MeV here, and ends at Tfinal = 130 MeV (corresponding to % = 0.4-1.0 in QGP and % = 1.0-12 in the hadron phase). It is important to see, that when looking at the comparison, there is clearly a difference between the two datasets: in case of PHENIX, quark annihilation plays the most important role, while for SPS, pion annihilation through p mesons is also important. Note though, that this contribution falls off faster than the QGP contribution, as the latter has a higher temperature, so dileptons at a high invariant mass have a higher probability of being created in QGP than in the hadronic phase. It is also clear, that even with the exchange of higher mass mesons, data above K 1 GeV/c cannot be reproduced. This may be attributed to other production mechanisms (not thermal production, as noted also in Ref. [23]), and the fact that we did not utilize in-medium hadron spectral functions. If medium effects were considered in our calculation (which we did not do, as our goal was to obtain a simple result and not go into the details of mass shifts and broaden-ings) then an enhancement would be expected around the p peak, and the data between Mp and M< would probably be explainable. This would then require a fine-tuning of the % parameter for both QGP and the hadron phase, in order to describe the low mass region as well. In such a detailed comparison, other processes and dilepton production channels shall be taken into account, which we plan to accomplish in a subsequent analysis.

600 800 M [GeV]

1000 1200

200 600 1000 1400 M [MeV]

Figure 3. Comparison with thermal dilepton production in 158 AGeV In+In collisions of NA60 [21] (0.4 to 0.6 GeV in p,, acceptance corrected) and in ^/sNn = 200 GeV Au+Au collisions of PHENIX [19] (with PHENIX acceptance). At SPS, the p contribution dominates dilepton production, but at some M values there might be a QGP part of the data. However, at RHIC, predominantly QGP is the source of dileptons. Note that we have used vacuum parameters, not in medium values for the vector mesons, which may be the cause for the excess at 800-1000 MeV at SPS. At RHIC, the low mass dilepton excess is also not incompatible with this analysis, and may be due to the large contribution from low mass n' production [20]. Note however that in this simple analysis, we did not take into account the dilepton production via n and n mesons, neither the n0 ^ ye+e- channel, which are dominant for dilepton production at M < 200 MeV.

7. Summary

In high energy heavy ion collisions, particles are created via different production mechanisms. While hadrons are created at the freeze-out of the medium, thermal photons and dileptons are constantly emitted from it. Hadrons thus reveal information about the final state, whereas thermally radiated photons and dileptons carry information about the whole time evolution. In this paper we calculated thermal dilepton production based on a hydrodynamical


The distinct feature of our paper is the analytic expression of the dilepton production based on an analytic 1+3d hydrodynamic solution, which depends only a few parameters, k, tn and finai (or the corresponding temperatures). The other parameters (such as transition time t0 or transition temperature T0, transverse flow ut) can be fixed from hadronic data. The dependence of the results on k (representing the EoS) lifetime ratio % has been analyzed. We found that both the equation of state and the production

time play an important role. Finally, we compared our calculations to RHIC and SPS data, and found according to expectations that in case of RHIC quark annihilation plays the most important role, while in case of SPS a large portion of dileptons are produced in a hadron gas. The RHIC dilepton data have been explained with the input parameters from earlier fits to RHIC direct photon and hadron distributions.


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