Azimuthal Asymmetry of Direct Photons in High Energy Nuclear Collisions

TL;DR

This paper predicts a significant azimuthal asymmetry in direct photon production in non-central high energy nuclear collisions, which could serve as a signature of quark-gluon plasma formation.

Contribution

It introduces the concept that jet-plasma interactions produce a measurable photon azimuthal asymmetry with a phase shift, highlighting a new observable for quark-gluon plasma detection.

Findings

Predicted negative v_2 for direct photons in non-central collisions

Identified phase difference between photon and hadron anisotropies

Suggested photon asymmetry as a signature of quark-gluon plasma

Abstract

We show that a sizeable azimuthal asymmetry, characterized by a coefficient v_2, is to be expected for direct photons produced in non-central high energy nuclear collisions. This signal is generated by photons radiated by jets interacting with the surrounding hot plasma. The anisotropy is out of phase by an angle $\pi/2$ with respect to that associated with the elliptic anisotropy of hadrons, leading to negative values of v_2. Such an asymmetry, if observed, could be a signature for the presence of a quark gluon plasma and would establish the importance of jet-plasma interactions as a source of electromagnetic radiation.

Equations

$$\frac{dN}{p_{T}d{p}_{T}d\varphi }=\frac{dN}{2\pi p_{T}d{p}_T}\left[1+\sum _{k}2v_k(p_T)\cos (k\varphi )\right]$$

$$\frac{d{N}^{\mathrm{N}-\mathrm{N}}}{p_{T}d{p}_{T}d\varphi }=T_{AB}{f}_{a/A}\otimes {\sigma }_{a+b\to \gamma +c}\otimes f_{b/B}.$$

$$\frac{d{N}^{\mathrm{jet}-\mathrm{frag}}}{p_{T}d{p}_{T}d\varphi }=\sum _f{\frac{d{N}^f(\varphi )}{dq}|}_{q=p_T/z}\otimes D_{f/\gamma }(z,p_T)$$

$$\begin{gathered} \frac{d{N}^{\mathrm{jet}-\mathrm{th}}}{p_{T}d{p}_{T}d\varphi dy}=\int d^{4}x\frac{\alpha {\alpha }_s{T}^2}{8{\pi }^2}\sum _q{\left(\frac{e_q}{e}\right)}^2 \\ \times f_q(x;p_T,\varphi ,y)\left[2\ln \frac{4E_{\gamma }T}{m^2}-C\right] \end{gathered}$$

$$\frac{d{N}^{\mathrm{jet}-\mathrm{br}}}{d{p}_{T}d\varphi }=\int d^2{r}_{\perp }\mathcal{P}({\mathbf{r}}_{\perp }){\int }_0^{d}dtdk\frac{d{N}^q}{dq}\frac{d{\Gamma }^{q\to q\gamma }(q,k)}{dkdt}$$

$$v_2(p_T)=\frac{\int d\varphi \cos (2\varphi )dN/d{p}_{T}d\varphi }{dN/d{p}_T}.$$

Full text

At the Relativistic Heavy Ion Collider (RHIC) and soon at the Large Hadron Collider (LHC), nuclei are collided at ultrarelativistic energies in order to create a new state of matter: the quark gluon plasma (QGP) Harris:1996zx . Emitted particles that only interact through the electroweak interactions are very unlikely to interact again despite the dense medium which is created. Thus they are able to carry to the detectors information about the state of the system at the time they were created Feinb:76 . Photons and leptons thus constitute a unique class of penetrating probes.

The two most interesting sources of photons are those where the plasma is directly involved in the emission. These are the thermal radiation from the hot QGP KaLiSei:91 and the radiation induced by the passage of high energy jets through the plasma FMS:02 ; Zakharov:04 ; TGJM:05 . The thermal radiation is emitted predominantly with low transverse momentum $p_{T}$ and has to compete with photon emission from the hot hadronic gas at later times XSB:92 ; Turb:04 . Photons from jets are an important source at intermediate $p_{T}$, where they compete with photons from primary hard scatterings between partons of the nuclei Owens:86 . They probe the thickness of the medium: the longer the path of the jet, the more photons are emitted.

Obviously, measuring photons from either of the sources involving the quark gluon plasma would be an important step toward establishing its existence and would provide a stringent test of the reaction dynamics. We refer the reader to TGJM:05 for a discussion of the different photon sources. Recently, the PHENIX collaboration at RHIC published their first results on direct photons measured in Au+Au collisions at $\sqrt{s_{\rm{NN}}}=200$ GeV phenix:05gammaauau . Note that direct photons here are defined as the total inclusive photon yield minus the photons originating from the decay of hadrons like $\pi^{0}$ and $\eta$. The sources discussed above will contribute to the direct photon signal. The data covers the intermediate and high-$p_{T}$ range where thermal photons are not a leading source. This makes it particularly attractive to look for photons from jet-plasma interactions.

Nuclear collisions at finite impact parameter $b>0$ start out in an initial state which is not azimuthally symmetric around the beam axis. Instead, the initial overlap zone of the two nuclei has an “almond” shape. Therefore particle spectra measured in the final state are not necessarily isotropic around the beam axis. It has been argued that the translation of the original space-time asymmetry into a momentum space anisotropy can reveal important information about the system Olli:92 . Two different mechanisms are important here: hydrodynamic pressure for the bulk of the matter, at low- to intermediate-$p_{T}$, and a simple optical-depth argument for intermediate- to high-$p_{T}$ particles. We introduce a novel third mechanism in this Letter.

Let us define the reaction plane as the plane spanned by the beam axis and the impact parameter of the colliding nuclei. For the bulk of the matter, the initial space-time asymmetry leads to an anisotropic pressure gradient which is larger where the material is thinner, i.e. in the event plane. This translates into a larger flow of matter in this direction. The anisotropy is analyzed in terms of Fourier coefficients $v_{k}$ defined from the particle yield $dN/p_{T}dp_{T}d\phi$ as VoloZhang:94

[TABLE]

where the angle $\phi$ is defined with respect to the event plane. At midrapidity all odd coefficients vanish for symmetry reasons, leaving the coefficient $v_{2}$ to be the most important one. Its size determines the ellipsoidal shape of the anisotropy. It is clear that the elliptic asymmetry coefficient $v_{2}$ is always positive for hadrons at low and intermediate $p_{T}$ due to the hydrodynamic flow. On the other hand, jets lose more energy when they are born into a direction where the medium is thicker, i.e. out of the reaction plane. The stronger jet quenching leads to fewer hadrons at intermediate and high $p_{T}$ emitted into this direction. This “optical $v_{2}$” is not associated with flow but with absorption and implies positive $v_{2}$ for hadrons from jets. Measurements at RHIC for several hadron species confirm this behavior star:02v2 .

In this Letter, we discuss $v_{2}$ of direct photons. We concentrate on intermediate and high $p_{T}$ and the $v_{2}$ from all the relevant processes. We define a mechanism that works by absorption of particles or jets going through the medium as optical. It turns out that in some cases a new inverse-optical mechanism is in place for photons: there are more of them emitted into the direction where the nuclear overlap zone is thicker, thus leading to a situation where the anisotropy is shifted by a phase $\pi/2$. Correspondingly $v_{2}$ is negative in this case.

Let us now discuss the different contributions to the direct photon spectrum. Direct photons from primary hard Compton and annihilation processes $a+b\to\gamma+c$ are produced symmetrically with

[TABLE]

Here $\sigma_{a+b\to\gamma+c}$ is the cross section between partons, $f_{a/A}$, $f_{b/B}$ are parton distribution functions in the nuclei $A$ and $B$ and $T_{AB}$ is the overlap factor of the nuclei. The primary hard direct photons do not suffer any final state effect and do not exhibit any elliptic asymmetry.

Jets from processes ($a+b\to c+d$) are also produced symmetrically, however they are quenched once they start to propagate through the plasma. This is the optical mechanism that leads to positive $v_{2}$ for hadrons fragmenting from jets. We expect photons fragmenting from such jets in the vacuum ($c\to c+\gamma$, after $c$ propagated through the medium) to exhibit the same anisotropy. Their yield at midrapidity is given by

[TABLE]

where $dN^{f}(\phi)/dq$ is the distribution of jet partons $f$ with momentum $q$ traveling into the direction given by the angle $\phi$, and $D_{f/\gamma}$ is the photon fragmentation function.

The interaction of jets with the medium can also produce photons in different ways: (i) scattering off plasma components can induce photon bremsstrahlung, (ii) hard leading partons may annihilate with thermal ones ($q+\bar{q}\to\gamma+g)$, or they can participate in Compton scattering ($q(\bar{q})+g\to q(\bar{q})+\gamma$). The latter case is also called jet-photon conversion, because the cross section is dominated by transfer of the entire jet momentum to the photon, $\mathbf{p}_{\gamma}\approx\mathbf{p}_{\rm jet}$. The jet-photon conversion yield at midrapidity is given by wong

[TABLE]

with $C=2.332$ and $m^{2}=4\pi\alpha_{s}T^{2}/3$. The distribution of jet partons $f_{q}(x;p_{T},\phi,y)$ at a space-time point $x$ is determined from the time dependent spectrum of jet partons propagating in the plasma, $dN^{q}/dq(\tau)$, as discussed in FMS:02 ; TGJM:05 . The time dependence is governed by the energy loss through induced gluon radiation, obtained with the complete leading order description by Arnold, Moore and Yaffe (AMY) AMY . It is clear that an anisotropy in $\phi$ is introduced by the different path lengths for jets traveling in and out of the event plane, leading to an increased probability for a jet-photon conversion in the direction where the medium is thicker. Such an inverse optical effect has not been observed before.

Medium induced bremsstrahlung ($q\to\gamma+q$) is implemented directly in the AMY formalism through splitting functions $d\Gamma^{q\to q\gamma}/dkdt$. The photon yield from this process is obtained by

[TABLE]

Here $\mathcal{P}(\mathbf{r}_{\perp})$ is the spatial distribution of hard processes creating jets in the transverse plane and $d=d(\mathbf{r}_{\perp},\phi)$ is the distance the jet has to travel from $\mathbf{r}_{\perp}$ into the direction of the angle $\phi$ to leave the fireball. Again it is obvious that the probability for induced bremsstrahlung to occur increases with the path length $d$ of the jet. Hence these photons are preferentially emitted into the direction where the medium is thicker, leading to negative $v_{2}$.

Finally, thermal photon emission constitutes another contribution to the direct photon spectrum. Thermal photons emitted by the medium are not prone to any optical effects, but the emitting matter experiences the anisotropic hydrodynamic push. However, the emission of intermediate and large $p_{T}$ photons peaks strongly at very early times where the temperature is highest kolb . We verified that at the time the system generates significant radial and elliptic flow the rate of thermal photons at intermediate and high $p_{T}$ is negligible.

Let us summarize what we have so far. We identified two processes, induced bremsstrahlung from jets and jet-photon conversion, that we expect to exhibit an inverse optical anisotropy ($v_{2}<0$). Photons from fragmentation show the regular optical anisotropy ($v_{2}>0$), while primary hard and thermal photons do not contribute to $v_{2}$ at intermediate and high $p_{T}$.

To quantify our arguments, we carry out a numerical calculation for Au+Au collisions at RHIC ($\sqrt{s}=200$ GeV). Photon spectra at midrapidity with their dependence on the azimuthal angle $\phi$ are calculated as described above for three different centrality classes. Our modeling of the nuclear collision is introduced in TGJM:05 . The initial conditions are constrained by $\tau_{i}T_{i}^{3}\sim dN(b)/dy/A_{\perp}(b)$, where the charged particle pseudorapidity densities $dN/dy$ can be found in phobos and $A_{\perp}$ is the overlap area of the nuclei in the transverse plane. With fixed initial time $\tau_{i}=0.26$ fm/c, the initial temperatures are $T_{i}$=370, 360 and 310 MeV for centrality classes $0-20\%,20-40\%$ and $40-60\%$ respectively. Comparing with measured photon spectra, a good agreement is obtained, for all centrality classes. Details will appear elsewhere santa_fe . The coefficients $v_{2}$ can then be calculated by using

[TABLE]

Fig. 1 shows the coefficient $v_{2}$ as a function of $p_{T}$ for Au+Au collisions at RHIC and for the centrality classes 0-20%, 20-40% and 40-60%. The dotted lines give the results for primary hard direct photons and photon fragmentation. As expected photons from fragmentation lead to a positive $v_{2}$ which is diluted by adding primary hard photons The solid lines are the results when bremsstrahlung, jet-photon conversion and thermal photons are included. They meet our expectations for $v_{2}$ of direct photons including all source discussed above. The $v_{2}$ for induced bremsstrahlung and jet-photon conversion is indeed negative. Together they are able to overcome the positive $v_{2}$ from fragmentation, leading to an overall negative elliptic asymmetry for direct photons at moderate $p_{T}$. Only above 8 GeV/$c$ the direct photon $v_{2}$ is again positive, because the yield of photons from fragmentation is dominating over medium induced bremsstrahlung TGJM:05 in this range. The dashed lines in Fig. 1 show the $v_{2}$ for direct photons with no jet energy loss included. In this case, fragmentation photons do not exhibit an anisotropy and the elliptic asymmetry is only due to jet-photon conversion. Measurements of $v_{2}$ with sufficient accuracy could therefore constrain models for jet energy loss. The absolute size of $v_{2}$ is not large, about 2-3% for the 20-40% centrality bin around $p_{T}=4$ GeV/$c$ and up to 5% for the more peripheral bin. The reason is that the signal is diluted by isotropic photons (primary hard and thermal) and partially cancelled between the optical and inverse optical mechanisms. We also checked the dependence of $v_{2}$ on the temperature of the medium, by varying the initial temperature $T_{i}$ with $\tau_{i}T_{i}^{3}$ kept constant. The resulting changes are small: a change in $T_{i}$ of 40$\%$ generates a shift in $v_{2}$ of less than 20$\%$.

In Fig. 2 we show some $v_{2}$ signals that should be detectable at RHIC in the near future. The dotted lines show $v_{2}$ for inclusive photons before background subtraction. In this case the $v_{2}$ signal is dominated by contributions from decaying $\pi^{0}$ and $\eta$ hadrons. The resulting $v_{2}$ is positive and larger in magnitude. Only hadrons from fragmentation have been included. However, it has been shown that hadron production up to a $p_{T}$ of 4 to 6 GeV/$c$ receives significant contributions from recombination of quarks FMNB:03 . The dot-dashed lines show the $v_{2}$ of inclusive photons if decays of $\pi^{0}$ and $\eta$ from recombination are included, using FMNB:03 with parameters consistent with our jet fragmentation calculation. Data on $v_{2}$ for inclusive photons have been measured by the PHENIX collaboration Adler:2005rg . Our calculations for the total inclusive photons including decays from recombined hadrons agrees well with their results.

A very exciting option for the future is the possibility to experimentally distinguish between direct photons associated with jets and isolated direct photons. Photons from jet fragmentation and induced bremsstrahlung are in the former category, while the latter includes primary hard and thermal photons, and photons from jet-photon conversion. The dashed line in Fig. 2 shows the result for the fragmentation and bremsstrahlung processes only. They contribute with different signs and one notes a characteristic change of sign from negative values at low $p_{T}$ to larger positive values, up to 5%, at large $p_{T}$, where fragmentation dominates. The solid line shows the $v_{2}$ of all isolated direct photon processes, including primary hard direct photon, thermal and jet-photon conversion. Jet-photon conversion is the only source of an anisotropy, so that the resulting $v_{2}$ is relatively large and negative.

We briefly mention here that the inverse optical mechanism for photon $v_{2}$ could be also at work for jets going through hadronic matter. Possible examples are hadronic processes after hadronization of the jet, e.g. $\rho+\pi\to\gamma+\pi$, as well as Compton, annihilation and bremsstrahlung processes with partons in surrounding hadrons before hadronization. In the first case the $p_{T}$ of the interacting jet hadron will be smaller than the original $p_{T}$ of the jet due to energy loss and hadronization, shifting the emitted photons to smaller $p_{T}$. In the second case the yield will ultimately depend on the unknown parton content of the hot hadron phase. However the Compton and annihilation yields have a $T^{2}$ dependence at intermediate $p_{T}$, hence any reasonable assumption of the parton content and the temperature of the hadronic phase will lead to small yields at intermediate $p_{T}$. The admixture of photon $v_{2}$ from jet-hadron interactions is therefore suppressed.

To summarize, we present the first calculation of the lowest order azimuthal asymmetry coefficient $v_{2}$ for direct photons in high energy nuclear collisions. Jets interacting with a deconfined quark gluon plasma provide photons exhibiting an inverse optical anisotropy with characteristic negative values of $v_{2}$. An experimental confirmation would emphasize the existence of a quark gluon plasma and confirm jet-medium interactions as important sources of photons at intermediate $p_{T}$. The $v_{2}$ signal is generally of order 3-5% and should be experimentally accessible at RHIC. Even more promising would be a separation of direct photons emitted in a jet from isolated photons. Both sources carry their own characteristic $p_{T}$ dependence for $v_{2}$. The arguments presented here for direct photons immediately apply to production of lepton pairs as well. Dileptons from annihilation of jets in the medium and from medium-induced virtual photon bremsstrahlung SGF:02 should also exhibit negative $v_{2}$.