BI-TP 2006/27 Momentum Broadening in an Anisotropic Plasma Paul Romatschke1 1 Fakulta¨t fu¨r Physik, Universita¨t Bielefeld, D-33501 Bielefeld, Germany (Dated: February 2, 2008) The rates governing momentum broadening in a quark-gluon plasma with a mo- mentumanisotropyarecalculatedtoleading-logorderforaheavyquarkusingkinetic theory. It is shown how the problematic singularity for these rates at leading-oder is lifted by next-to-leading order gluon self-energy corrections to give a finite con- 7 tribution to the leading-log result. The resulting rates are shown to lead to larger 0 momentum broadening along the beam axis than in the transverse plane, which 0 2 is consistent with recent STAR results. This might indicate that the quark-gluon- n plasma at RHIC is not in equilibrium. a J 1 I. INTRODUCTION 1 2 Oneofthe hottest debated questions in thecontext of ultrarelativistic heavy ioncollisions v iswhethera“thermal”quark-gluonplasmahasbeencreatedatthehighestenergyrunsofthe 7 2 Relativistic Heavy-Ion Collider (RHIC). Onthe one hand, the success of idealhydrodynamic 3 fits to experimental data have been interpreted [1] to imply rapid thermalization of the bulk 7 matter at RHIC. On the other hand, perturbative estimates of the thermalization time [2] 0 6 result in much larger values, seemingly excluding the possibility of rapid thermalization (see 0 however [3]) . Plasma instabilities may help speeding up the equilibration process by rapidly / h isotropizing the system [4, 5, 6, 7, 8], probably by some cascade to the UV [9, 10, 11, 12, 13], p - although they might still be too slow to overcome the initially strong longitudinal expansion p rate [14]. Thus, it is currently unclear how (and if) the quark gluon plasma created at RHIC e h stays locally isotropic (let alone thermal) during most of its time evolution. : v Nevertheless, many theoretical calculations in the context of heavy-ion collisions assume i a homogeneous and locally isotropic system. However, the physics of anisotropic plasmas X (even if they are close to isotropy) can be quite different from that of isotropic plasmas, c.f. r a the presence of plasma instabilities in the former. Due to these potentially large differences, it might be interesting to reanalyze calculations of experimentally accessible observables in the context of anisotropic plasmas, wherever possible. In what follows, I will thus study momentum broadening in a homogeneous but locally anisotropic system. Since I will be interested in qualitative effects only, I limit myself to con- sidering momentum broadening for a heavy quark induced by collisions to leading logarithm (LL) accuracy. In section II, I will discuss the calculational setup for anisotropic plasmas. Section III will contain the calculation of the rates governing momentum broadening, which canbe doneanalytically inthe small anisotropy limit. In section IVfinally I willtry to apply these results to preliminary STAR data on jet shapes and discuss possible consequences. II. SETUP The setup of my calculation will be based on the work by Moore and Teaney [15]. They considered a heavy quark with momentum p moving in a static (thermal) medium with a 2 temperature T and I will quickly repeat some equations from their work which I want to use in the following. Due to collisions, the quark will loose momentum and the variance of its associated momentum distribution will broaden according to [15] d p = pη (p) D dth i − d (∆p )2 = κ (p) dth || i || d (∆p )2 = κ (p) dth ⊥ i ⊥ d (∆p )2 = κ (p), (1) z z dth i where I choose my coordinate system such that (∆p )2 = (p p )2 is the variance h || i || − h ||i of the momentum distribution in the direction parallel to the direction of the quark and (∆p )2 , (∆p )2 are the variances transverse to the direction of the quark (see also Fig.1). z h ⊥ i h i The functions η ,κ ,κ ,κ which encode average momentum loss as well as transverse D z || ⊥ and longitudinal fluctuations are calculated using kinetic theory. Schematically, they are given as [15] d p 1 h i = 2q0 f(k)(1 f(k q0)) f(k q0)(1 f(k)) dt 2v k,q|M| ± − − − ± Z h i d (∆p )2 = 2q2f(k) 1 f(k q0) dth || i Zk,q|M| || h ± − i d (∆p )2 = 2q2f(k) 1 f(k q0) dth ⊥ i Zk,q|M| ⊥ h ± − i d (∆p )2 = 2q2f(k) 1 f(k q0) , (2) dth z i k,q|M| z ± − Z h i where v = p/p0, f(k) is the gluon (quark) distribution function, is the scattering matrix and = d3k (similarly for q). If the transferred energy q0M v q is small, one can k (2π)3 ≃ · approximate R R q0 f(k) 1 f(k q0) f(k q0)[1 f(k)] f(k)[1 f(k)] ± − − − ± ≃ −T ± h i f(k) 1 f(k q0) f(k)[1 f(k)]. (3) ± − ≃ ± h i Thus, if the quark is non-relativistic or if q0 T, the coefficient κ can be related to the energy loss rate dp0 by κ = 2T dp0. This imp≪lies that also the oth|e|r coefficient functions − dt || −v2 dt can be calculated by the appropriate modifications in the integrand of the energy loss rate. A. Collisional Energy Loss Studies of collisional energy loss of a heavy quark in isotropic systems have a long history [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Assuming a perturbative expansion in powers (and logarithms) of the strong coupling α = g2, the leading-order logarithmic contribution to s 4π 3 the collisional energy loss can be obtained by calculating the matrix elements in Eqns. (2) with HTL propagators and restricting to soft momentum transfer q0 < T where Eqns. (3) can be used [15]. There is, however, an alternative way by Thoma and Gyulassy [18] to calculate the collisional energy loss which is equivalent to calculating the HTL matrix elements and doing the integrals in Eqns. (1). It is based on the expression for the energy loss of a quark by interaction with its induced chromoelectric field, dp0 = Re d3xJa (t,x) Ea (t,x). (4) dt ext · ind Z Here Ja = qavδ3(x vt) is the current of the quark with color charge qa and velocity v ext − and E is the induced electric field. In Fourier space Q = (q,ω), E can be written as ind ind Ei (ω,q) = iω Gij Gij Jj (Q), (5) ind − 0 ext (cid:16) (cid:17) where Gij and Gij are the full and free retarded gluon propagator, respectively. 0 The collisional energy loss for a theory with N colors thus becomes c dp0 g2(N2 1) d4Q = c − Im ωvi Gij Gij vj(2π)δ(ω v q) (6) dt − 2N (2π)4 − 0 − · c Z (cid:16) (cid:17) wherethemomentumexchangeisrestrictedto q < T andonlytheleadinglogarithmshould | | be kept. Because this restriction corresponds to Eqns. (3), the fluctuation coefficients may be glossed directly from Eqns. (1), finding g2(N2 1) d4Q 2Tq2 κ = c − Im ||vi Gij Gij vj(2π)δ(ω v q) || 2Nc Z (2π)4 ω (cid:16) − 0 (cid:17) − · g2(N2 1) d4Q 2Tq2 κ⊥ = 2Nc c− ImZ (2π)4 ω⊥vi(cid:16)Gij −Gi0j(cid:17)vj(2π)δ(ω −v·q) g2(N2 1) d4Q 2Tq2 κ = c − Im zvi Gij Gij vj(2π)δ(ω v q). (7) z 2N (2π)4 ω − 0 − · c Z (cid:16) (cid:17) A convenient feature of these Equations is that all the information about the medium resides in the gluon propagator G. Thus, in this form Eqns. (7) are valid both for isotropic as well as for anisotropic systems. B. Anisotropic Plasmas In the rest frame of a thermal system, particles move in all directions with equal prob- ability. However, in the rest frame of a non-equilibrium system, particles might e.g. move predominantly in a plane, having an anisotropic probability distribution. Thus, dropping the assumption of isotropy, there will be at least one preferred direction in the system, which in the following I will take to be the z-direction. In the context of heavy-ion collisions, one can identify this direction with the beam-axis along which the system expands initially. As a model for an anisotropic system, I will assume that the anisotropic quark and gluon dis- tribution functions f(k) are related to the usual (isotropic) Fermi and Bose distributions f ( k ) as [26] iso | | f(k) = N(ξ)fiso k2 +ξ(k ez)2 , (8) · (cid:18)q (cid:19) 4 where ez denotes the unit-vector in the z-direction, ξ is a parameter controlling the strength of the anisotropy and N(ξ) is a normalization factor which in the following I will set to one. Note that ξ = 0 corresponds to the case of an isotropic system. Ananisotropicsystem bydefinitiondoesnothave onetemperature. However, itmayhave a dimensionful scale that is related to the mean particle momentum and which takes over the role of a “hard” scale in much the same way the temperature does in a thermal system. For a quark gluon plasma (even out of equilibrium), the natural choice seems to be the saturation scale Q . Since kinetic theory calculations remain valid even out of equilibrium s as long as there is a separation of scales between hard and soft momentum modes, one could replace T by Q times some coefficient when translating formulae from an isotropic to an s anisotropic system. For simplicity, however, I will refrain from doing that and keep T as a placeholder of the correct hard scale even for anisotropic systems. As outlined before, the main difference between calculating the fluctuation coefficients Eqns. (7) in isotropic and anisotropic systems, respectively, is the form of the gluon prop- agator G. While for isotropic systems the propagator in the HTL approximation is given by1 δij qiqj/q2 qiqj Gij (Q) = − + iso q2 ω2+Π (ω,q) ω2(Π (ω,q) q2) T L − − m2 ω2 ω2 q2 ω + q Π (ω,q) = D 1 − log | | T 2 q2 " − 2ω q ω q # | | −| | ω ω + q Π (ω,q) = m2 log | | 1 , (9) L D"2 q ω q − # | | −| | one finds for systems with an anisotropy of the form Eq. (8) within the same approximation [26] δij qiqj/q2 n˜in˜j/n˜2 (q2 ω2 +α(ω,q)+γ(ω,q))qiqj/q2 Gij(Q) = − − + − q2 ω2 +α(ω,q) (q2 ω2 +α(ω,q)+γ(ω,q))(β(ω,q) ω2) q2n˜2δ2(ω,q) − − − − (β(ω,q) ω2)n˜in˜j/n˜2 δ(ω,q)(qin˜j +qjn˜i) + − − , (10) (q2 ω2 +α(ω,q)+γ(ω,q))(β(ω,q) ω2) q2n˜2δ2(ω,q) − − − where here n˜ = ez qqz/q2 and the structure functions α,β,γ,δ are generalizations of ΠT − and Π to anisotropic systems. In the case of small anisotropy ξ, they are given as [26] L ωˆ2 1 α = Π (ωˆ)+ξ (5qˆ2 1)m2 qˆ2m2 T " 6 z − D − 3 z D 1 + Π (ωˆ) (3qˆ2 1) ωˆ2(5qˆ2 1)) , 2 T z − − z − # (cid:16) (cid:17) 1 ωˆ 2β = Π (ωˆ)+ξ (3qˆ2 1))m2 − L 3 z − D (cid:20) +Π (ωˆ) (2qˆ2 1) ωˆ2(3qˆ2 1) , L z − − z − (cid:16) (cid:17)i 1 Here and in the following (unless stated otherwise) I use the specific choice of gauge A0 = 0; physical observables are not affected by this choice as they are manifestly gauge-invariant. 5 ξ γ = (3Π (ωˆ) m2 )(ωˆ2 1)(1 qˆ2) , 3 T − D − − z ξ δ = (4ωˆ2m2 +3Π (ωˆ)(1 4ωˆ2))qˆ , (11) 3k D T − z where ωˆ = ω/ q and qˆ = q / q . For larger values of ξ one has to resort to numerical z z | | | | evaluations of α,β,γ,δ [26]. In all cases m is the isotropic Debye mass, D g2 m2 = ∞dp pf ( p ), (12) D π2 iso | | Z0 which for a thermal system with N colors and N light quark flavors becomes c f m2 = g2T2Nc 1+ Nf . D 3 6 (cid:16) (cid:17) III. MOMENTUM BROADENING As can be quickly verified, the general form of the propagator Eq. (10) reduces to the simple form Eq. (9) in the limit of vanishing anisotropy ξ 0. Thus, regardless whether → the system is isotropic or not, the fluctuation coefficients are found by inserting Eq. (10) into Eqns. (7). The frequency integration is trivial, whereas the integration over total momentum ex- change q contains a subtlety for non-vanishing anisotropy ξ: since the static limit (ω 0) | | → of e.g. α is real and negative, the propagator develops a singularity for space-like momenta q . Indeed, this singularity signals the presence of instabilities in the system [25, 26]. Since | | these instabilities correspond to soft gauge modes that grow nearly exponentially until they reach non-perturbative occupation numbers, one might dismiss any perturbative approach such as mine as futile. However, my calculation should be applicable up to the time where the soft mode occupation number has not grown non-perturbatively large yet. Even with fast growing instabilities, this can be a long physical timescale if either the initial fluctua- tions are tiny or there is a counter-acting effect such as the expansion of the system. In the latter case, one can even argue that unstable modes do not grow non-perturbatively large during plasma lifetimes at RHIC [14]. Therefore, even though for quantitative results one probably has to resort to numerical simulations, ignoring the effects of the non-perturbative soft gauge modes may be not such a bad approximation at least for some period of time. As a consequence, here the propagator singularity is mainly a practical obstacle to cal- culating physical observables since this singularity is in general non-integrable, as was first recognized by Arnold, Moore and Yaffe [25]. However, the fact that α has an imaginary part that vanishes only linearly in the static limit cures this problem for certain observables (namely those with numerators that also turn out to vanish in the static limit), which has been dubbed “Dynamical Shielding” [27]. In the formulation I have chosen, Eq. (7), there is only one power of the propagator and consequently the singularity is always integrable. However, the problem resurfaces in the next integration step, where for some of the fluctuation coefficients there is an uncanceled ω in the denominator of the integral. Unlike the collisional energy loss, which is finite to full leading-order (LO), dynamical shielding breaks for at least some of the fluctuation coefficients. I will show in the following how to calculate these to LL accuracy. 6 FIG. 1: Sketch of the coordinate system: the quark moves along the direction, perpendicular to || the beam axis which coincides with the anisotropy direction. The calculation is divided up into two parts: first I calculate the “regular” contribution within the LL approximation, which involves only quantities for which dynamical shielding works. Then I will deal with the contribution which would give the LO contribution (the constant under the log) in an isotropic system and show that in anisotropic systems, it gives a contribution to the LL. A. Regular contributions Limiting oneself to LL approximation, the q integration in Eq. (7) can be done as in | | Ref.[27], and one finds e.g. g2(N2 1) dΩ 4Tqˆ2 1 T v2 ωˆ2 (n˜ v)/n˜2 κreg = c − q log Im − − · α ⊥ ⊥ − 2Nc Z (2π)3 ωˆ 1−ωˆ2 mD " 2(1−ωˆ2) (1 ωˆ2)2β +ωˆ2(n˜ v)2/n˜2(α+γ) 2ωˆ(1 ωˆ2)(n˜ v)δˆ + − · − − · , (13) 2ωˆ2(1 ωˆ2) # − where qˆ = q / q , δˆ= δ/ q , ωˆ = v q/ q . ⊥ ⊥ | | | | · | | I will consider the situation where the quark velocity is perpendicular to the beam axis, v e = 0. Thus I have ωˆ = vqˆ , n˜ v = ωˆqˆ and n˜2 = 1 qˆ2. From the explicit form of · z || · z − z the small ξ structure functions α,β,γ,δ in Eq. (11) it can be seen that their imaginary part always involves at least one power of ωˆ. All the ωˆ’s in the denominator of the integrand in Eq. (13) are thus canceled, showing indeed that there is no singularity to LO accuracy. For small anisotropies, the remaining integrations can be done analytically using Eqns. (9,11). I find dp g2(N2 1)m2 T 1 1 v2 = c − D log − Arctanh(v) dt − 2Nc 8π mD "v − v2 ξ + 3v 5v3 3(1 v2)2Arctanh(v) 6v4 − − − # (cid:16) (cid:17) g2(N2 1)Tm2 T 3 1 (1 v2)2 κreg = c − D log + − Arctanh(v) ⊥ 2Nc 4π mD "2 − 2v2 2v3 7 ξ + 3v +8v3 13v5 +3(1 v2)3Arctanh(v) 24v5 − − − # (cid:16) (cid:17) g2(N2 1)Tm2 T 3 1 (1 v2)2 κreg = c − D log + − Arctanh(v) z 2Nc 4π mD "2 − 2v2 2v3 ξ + 9v +24v3 7v5 +9(1 v2)3Arctanh(v) . (14) 24v5 − − − # (cid:16) (cid:17) As expected, these expression reduce to the known results [15] in the isotropic limit (ξ 0). → B. Anomalous Contribution Let us now concentrate on the naive “constant under the log” contribution of Eqns. (14). To this end, it is illustrative pick out the first term of the propagator Eq. (10), and pluck it into Eqns. (7), finding d3q qˆ2 q3 κ i v2 ωˆ2 qˆ2ωˆ2 Im , (15) i ∼ (2π)3 ωˆ − − z q2 ω2 +α(ω,q) Z (cid:16) (cid:17) − where the index i denotes i = , ,z and I have used the same notations as in the previous ⊥ || subsection. Denoting this part of the propagator as ∆ 1 = q2 ω2 +α(ω,q) allows me to −A − rewrite d3q κ qˆ2m2 f(qˆ ) v2 ωˆ2 qˆ2ωˆ2 q∆ ∆ , (16) i ∼ (2π)3 i D z − − z A ∗A Z (cid:16) (cid:17) where I introduced m2 ωˆf(qˆ ) = Imα and the means complex conjugation. This form now D z ∗ directly involves the squared matrix from Eqns. (2). As discussed before, the static limit of α is real and negative to leading order in g, which produces a non-integrable singularity in Eq. (16) (unless i = for which qˆ v = ωˆ cuts off the singularity). || || However, from the structure of Eq. (16), this singularity is reminiscent of so-called pinch- ing singularities, which are usually due to incomplete resummations of the propagator. In- deed, I find it is plausible that α has a non-vanishing imaginary part in the static limit at order2 O(g3), which allows one to integrate 1 Reα Reα+q2(1 ωˆ2) dqq3 ∆ 2 = Arctan − A | | 2(1 ωˆ2)2 "Imα Imα ! Z − 1 + log α 2+2Reαq2(1 ωˆ2)+q4(1 ωˆ2)2 . (17) 2 | | − − (cid:16) (cid:17)(cid:21) The second term in this equation gives a contribution to the LL for q = T m which D ≫ has been accounted for already in the previous subsection. The other term, however, now involves Imα rather than the ωˆ of Eq. (15) in the denominator, which because of the O(g3) contribution behaves as lim Imα m2 (ωˆ +cg), (18) ωˆ 0+ ∼ D → 2 See the appendix for details. 8 where c is just a number. Thus, the singularity at ωˆ 0+ is cut-off by this higher-order → resummation in the self-energy. However, this also entails that the naive LO correction gives a contribution to the LL instead, as I will show in the following. First note that for any nonzero ωˆ, the first term on the r.h.s. of Eq. (17) is finite also for c = 0 and thus only contributes to LO. In order to extract a potential LL contribution, one may thus take ωˆ 0+ everywhere except for the ωˆ in the denominator, which one has to → replace by ωˆ ωˆ +cg. For simplicity, I do this already in Eq. (15), using 3 → lim Im∆ +πδ q2 +Reα . (19) A ωˆ 0+ → → (cid:16) (cid:17) As can be verified, for small anisotropies ξ the leading contribution to κ is entirely given i by this part of the propagator4. The integral over ωˆ then gives log(g) together with some finite LO contribution that I ignore. Thus, upon identifying log(g) = log T , I find for the anomalous contributions to κ − mD i g2(N2 1)Tm2 T ξv κanom = c − D log ⊥ − 2Nc 4π mD 12 g2(N2 1)Tm2 T ξv κanom = c − D log , (20) z − 2N 4π m 4 c D such thattogether withtheregularcontributions Eq. (14), thefullLLfluctuationcoefficients in the small anisotropy limit are given by g2(N2 1)Tm2 T 3 1 (1 v2)2 κ = c − D log + − Arctanh(v) ⊥ 2Nc 4π mD "2 − 2v2 2v3 ξ + 3v +8v3 13v5 2v6 +3(1 v2)3Arctanh(v) 24v5 − − − − # (cid:16) (cid:17) g2(N2 1)Tm2 T 3 1 (1 v2)2 κ = c − D log + − Arctanh(v) z 2Nc 4π mD "2 − 2v2 2v3 ξ + 9v +24v3 7v5 6v6 +9(1 v2)3Arctanh(v) . (21) 24v5 − − − − # (cid:16) (cid:17) C. Larger Anisotropies The relevant integrals of the previous subsections may also be evaluated numerically for arbitrary ξ using techniques from [26, 27]. The results are shown in Table I. For sufficiently small ξ, the results turn out to coincide with the analytic results Eqns. (21). From Table I, it can be seen that the anomalous contribution makes up only a few percent at small ξ, while it becomes more important for larger anisotropies. Indeed, at larger ξ, besides Eq. (19) then also the other parts of the propagator contribute, further 3 This δ-function is not to be confounded with the structure function δ! 4 This is because the structure function δˆ2 in the denominator may be dropped to leading order in O(ξ). However, this result is specific to choosing the quark velocity perpendicular to ez, so that n˜ v 0 in · → the static limit. 9 ξ =1 ξ =10 ξ =100 v κ (κreg) κ (κreg) κ (κreg) κ (κreg) κ (κreg) κ (kreg) z z z z z z ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ 0.05 0.50 (0.50) 0.98 (1.00) 0.23 (0.23) 1.45 (1.54) 0.08 (0.08) 1.72 (1.84) 0.15 0.50 (0.51) 0.97 (1.00) 0.22 (0.23) 1.44 (1.54) 0.07 (0.08) 1.69 (1.84) 0.25 0.51 (0.51) 0.97 (1.01) 0.22 (0.24) 1.40 (1.54) 0.07 (0.08) 1.63 (1.84) 0.35 0.52 (0.53) 0.96 (1.02) 0.22 (0.25) 1.35 (1.55) 0.07 (0.08) 1.55 (1.84) 0.45 0.53 (0.55) 0.96 (1.03) 0.22 (0.26) 1.31 (1.55) 0.07 (0.09) 1.47 (1.85) 0.55 0.54 (0.57) 0.96 (1.05) 0.23 (0.27) 1.26 (1.55) 0.07 (0.09) 1.40 (1.85) 0.65 0.56 (0.60) 0.97 (1.07) 0.24 (0.29) 1.22 (1.56) 0.07 (0.10) 1.32 (1.85) 0.75 0.58 (0.64) 0.98 (1.10) 0.25 (0.31) 1.17 (1.57) 0.07 (0.11) 1.24 (1.85) 0.85 0.60 (0.69) 1.00 (1.14) 0.27 (0.34) 1.13 (1.58) 0.08 (1.12) 1.16 (1.85) 0.95 0.64 (0.75) 1.02 (1.19) 0.30 (0.38) 1.09 (1.59) 0.09 (0.14) 1.08 (1.85) TABLEI:Thecoefficients multiplying g2(2NNc2c−1)T4mπ2D log mTD inκ⊥,κz forlargervaluesofthesystem anisotropy ξ, evaluated numerically. For convenience, also the regular contributions are given explicitly. enhancing the anomalous contributions. For the total κ , where the anomalous contribution z is stronger, this manifests itself by a decrease as a function of v for larger ξ, whereas the regular contribution would have had the inverse trend. Another trend that is apparent from Table I is that the ratio κ /κ for a given anisotropy z ⊥ ξ decreases for increasing velocity v. Put differently, the higher the quark’s momentum, the more circular its associated jet shape. IV. DISCUSSION Both from the analytic results Eqns. (21) as well from the numeric evaluation given in Table I, one finds that for anisotropic systems κ /κ is always larger than one. In other z ⊥ words, a charm quark jet in an anisotropic quark-gluon plasma will generically experience more broadening along the longitudinal direction than in the reaction plane. Assuming an initial jet profile with (∆p )2 = (∆p )2 , the ratio κ /κ can roughly be associated with h ⊥ 0i h z 0i z ⊥ the ratio of jet correlation widths in azimuth ∆φ and rapidity ∆η as h i h i ∆η κ /κ h i. (22) z ⊥ ≃ ∆φ h i Therefore, the calculation in the previous sections generically predicts this ratio to be larger than one in anisotropic systems. Due to the approximations in my calculation (heavy quark, leading-log, only collisional broadening), many effects that will change both κ and κ for real jets have been ignored. z ⊥ However, inclusion of other effects should not change my result on the ratio of κ and κ z ⊥ qualitatively, since κ = κ in isotropic systems. Thus, κ /κ > 1 in anisotropic systems z z ⊥ ⊥ should hold true also for real jets. Interestingly, recent measurements of the dihadron correlation functions by STAR [28] for Au+Au collisions at √s = 200 GeV indeed seem to implymuchbroadercorrelationsin∆η thanin∆φ. InFig.2,STARdataforthesecorrelation 10 FIG. 2: Dihadron correlation functions in azimuth ∆φ and space-time rapidity ∆η, for d+Au and central Au+Au at √s = 200 GeV (figure courtesy J. Putschke (STAR), Proceedings of Hard Probes 2006). For comparison, on the right plot I show an ellipse with eccentricity e √8 which ≃ 3 corresponds to a ratio κ /κ 3. z ⊥ ∼ is compared to a general ellipsoidal shape with eccentricity e √8, which according to ≃ 3 Eq. (22) I would associate with a ratio κ /κ 3. z ⊥ ∼ In other words, a charm quark jet with a “mean” momentum p 4.6GeV (or v 0.95) ≃ ≃ and a “mean” system anisotropy ξ 10 should experience broadening roughly consistent ≃ with the ellipsoidal shape in Fig.2, where “mean” here is referring to a mean over the whole system evolution. To get more quantitative, one would have to calculate the momentum broadening dynamically according to Eqns. (1), which can e.g. be done within a viscous hydrodynamics simulation [29, 30, 31]. However, notethatasageneralfeatureofanisotropic plasmas, one seemingly obtains jet shapes that can be much broader in rapidity than in azimuth. Other effects distorting theshape of jets inan isotropicplasma, namely flow effects [32], probably cannot account for such dramatic asymmetries unless invoking extreme flows. Interestingly, momentum-binned data of the dihadron correlations from STAR seem to show a trend towards more circular jet shapes for higher trigger momentum, which I also found in section III.C. Precise data could thus provide further tests for calculations, maybe helping to constrain the system anisotropy. In view of this, a finite system anisotropy could be a natural explanation for these broad rapidity correlations in central Au+Au collisions at √s = 200 GeV. Thus, turning the argument around, it may be that the sizable ratio ∆η / ∆φ is an indication that the h i h i plasma created atRHIC isafter allnot inequilibrium during a sizable fractionof itslifetime, calling into question the validity of the strongly advocated “perfect fluid” picture. Clearly, in order to shed more light onto this issue, many caveats on the application of my result to experimental data have to be addressed. Among other things, a calculation of the full LO correction as well as results for light quarks, gluons and the inclusion of radiation effects in anisotropic plasmas would be on the wish-list. Nevertheless, I hope that I was able to highlight the feasibility and potential value of re-calculating experimentally interesting