RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-5. Hadron Physics (Theory)

Refraction of light in the quark-gluon plasma
A. Monnai∗1∗1

RIKEN Nishina CenterFig. 1. Schematic of medium refraction for (a) elliptic ﬂow
and (b) triangular ﬂow.

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Electromagnetic probes in high-energy heavy-ion
collisions provide us with important information on thequark-gluon plasma (QGP) because experimental data
indicate that the hot medium is color opaque but electromagnetically transparent. An important discovery
regarding such probes is the excessive yield of direct
photon elliptic (v2 ) and triangular ﬂow (v3 ).1) Direct
photons consist of prompt photons created in the initial hard processes and thermal photons emitted fromthe medium. Flow harmonics is deﬁned as a Fourier expansion coeﬃcient of particle spectra in the azimuthal
angle and is induced by geometrical anisotropy in thesystem via medium interaction. Since hadronic ﬂow
harmonics follows hydrodynamic description and isconsidered as an evidence for the existence of a strongly
coupled medium, quantitative understanding of the direct photon ﬂow harmonics is an important issue inheavy-ion phenomenology.
I have investigated the eﬀect of refraction on prompt
photons by the QGP medium in the transverse plane(Fig. 1).2) The emission rate for prompt photons isderived from p-p collision data.3) The path of a ray
in a medium with the refractive index n is given as
d2 x/dτ 2 = (1/2)dn2 /dx according to Fermat’s principle. The dynamical evolution of an inhomogeneous
medium should be considered, because the typical lifetime and size of the system are comparable. Here,I use a (2+1)-dimensional ideal hydrodynamic modelwith Monte-Carlo Glauber initial conditions and a lattice QCD equation of state. The initial time is 0.4 fm/cand the freeze-out temperature is 0.15 GeV. The temperature and frequency dependence of the refractive index is parametrized as n2 (T, ω) = 1−ωp2 /ω 2 , where theplasma frequency is parametrized as ωp2 = a2 T 2 . In thehigh-temperature limit, ωp2 ∼ m2D ∼ e2 T 2 is obtainedusing the Debye mass mD . This implies a2 ∼ 10−1
since e2 = 4παEM . The frequency ω is Doppler-shiftedfrom the original frequency as ω = ω0 /γ(1 + β cos ∆φ),
where ∆φ is the angle between the ﬂow and the direction of a ray. It should be noted that the phase velocity
in the QGP medium, vph = 1/n, exceeds the speed oflight, but causality is not violated because the groupvelocity remains smaller than unity. When n2 < 0, themedium does not bend a ray but partially absorbs it.
The elliptic ﬂow of prompt photons is shown inFig. 2. When there is no refraction, the quantity vanishes. For non-unity refractive indices, on the otherhand, positive v2 is observed above plasma frequencies. The magnitude, however, is not large enoughto account for the large photon v2 found in collider
experiments. Below the plasma frequency, the high-0.015
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Fig. 2. Diﬀerential elliptic ﬂow of prompt photons for different refraction parameters.

temperature region near the center of the medium becomes semitransparent and suppresses photons traveling horizontally, leading to negative v2 . Once theentire system becomes translucent at low momentum,
v2 again becomes positive because photons have betterchance of moving out of the medium in the directionof the minor axis. This implies that the QGP plasma
frequency can be constrained from measurements. Theabsorptive behavior has not been found in photon pT
spectra above 0.5 GeV at RHIC and 1 GeV at LHC.4)
Here, the former condition is more stringent and themaximum refraction parameter allowed is a2 ∼ 1-2.
Higher-order harmonics, v3 -v5 , are found to be positive but small. Note that the above argument depends
on the choice of refractive index. Numerical analyses
with diﬀerent indices and introduction of thermal photons will be performed in the future.
References
1) A. Adare et al. [PHENIX Collaboration]: Phys. Rev.
Lett. 109, 122302 (2012).
2) A. Monnai: arXiv:1408.1410 [nucl-th].
3) S. Turbide, R. Rapp and C. Gale: Phys. Rev. C 69,014903 (2004).
4) A. Adare et al. [PHENIX Collaboration]: arXiv:1405.3940[nucl-ex].

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