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**Viscosity & Dilepton Production**

Gojko Vujanovic Electromagnetic Probes of Strongly Interacting Matter: Status and Future of Low-Mass Lepton-Pair Spectroscopy ECT*: Trento, Italy May 22nd 2013

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**Outline Overview of Dilepton sources Low Mass Dileptons**

Thermal Sources of Dileptons 1) QGP Rate (w/ viscous corrections) 2) In-medium vector meson’s Rate (w/ viscous corrections) 3+1D Viscous Hydrodynamics Thermal Dilepton Yields & v2 Intermediate Mass Dileptons Charmed Hadrons: Yield & v2 Importance of viscous corrections to the QGP v2 Conclusion and outlook After a brief overview of the dilepton sources, I will spend the first half my time talking about the Low mass dileptons where thermal sources are important. These thermal evolution is done via our 3+1D… Then in second half of my talk, I will be saying a few words about the intermediate mass dileptons where charmed hadrons dominate and the importance of …

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**Evolution of a nuclear collision**

Space-time diagram Thermal dilepton sources: HG+QGP QGP: q+q-bar-> g* -> e+e- HG: In-medium vector mesons V=(r, w, f) V-> g* -> e+e- Kinetic freeze-out: c) Cocktail Dalitz Decays (p0, h, h’, etc.) Here is a space-time diagram of the heavy ion collisions. What we are really trying to characterize is the thermal radiation of the QGP. In the QGP phase, the qqbar annihilation is an important source of diletpons. The reason why we want to study dileptons is because they are able to escape the medium unscatted thus giving us clean information about the state of the matter at the moment of emission. In the hadronic sector, the main source of dileptons are vector mesons. Now since these vector mesons interact with the medium, their proprerties will be modified and these modification will be visible in the dilepton yield and v2. After freeze-out, we have a cocktail contribution which is dominated by Dalitz decays of various mesons here are a few examples. In the formation phase, the important sources are given here. In the intermediate mass range we have the decay of charmed hadrons, beauty hadrons, other vector mesons (charmonium, bottomonium) and drell-yan processes. For the purpose of RHIC phyiscs, the last 3 are subdominant and I will not be discussing about them any further. Other dilepton sources: Formation phase d) Charmed hadrons: e.g. D+/--> K0 + e+/- ne e) Beauty hadrons: e.g. B+/-->D0 + e+ /-ne f) Other vector mesons: Charmonium, Bottomonium g) Drell-Yan Processes Sub-dominant the intermediate mass region

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**Dilepton rates from the QGP**

An important source of dileptons in the QGP The rate in kinetic theory (Born Approx) More complete approaches: HTL, Lattice QCD. Here is a Feynmann diagram of an important source of diletpons in the QGP. - There are two ways to calculate dilepton rates from QGP, one is via kinetic theory and the other is FTFT. In the born approx they give the same result and that result is presented here. There are more sophisticated calculations of QGP rates using HTL or extracting them from the lattice QCD. However these more sophisticated calculations change the yields in the low mass region where the HG rates (I will soon present) dominate. In the intermediate/high mass region, where QGP rates dominate over HG, both HTL and lattice have already converged towards the born rate.

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**Thermal Dilepton Rates from HG**

Model based on forward scattering amplitude [Eletsky, et. al., Phys. Rev. C, 64, (2001)] Effective Lagrangian method by R. Rapp [Phys. Rev. C 63, (2001)] The dilepton production rate is : Resonances contributing to r’s scatt. amp. & similarly for w, f Self-energy is decomposed into two parts vacuum & thermal. The vacuum part is described via effective lagrangians. The thermal part is computed through the forward scattering amplitude. A list of resonances contributing to the rho FSA are on the right. In addition to results obtained through our model, I will also be showing results obtained through our collaboration with Ralf Rapp’s. He uses an effective lagrangian method to describe vector mesons in medium. In both cases, the goal is to compute the imaginary part of the retarded propagator, which then goes into the rate equation. ;

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**Energy-momentum conservation**

3+1D Hydrodynamics Viscous hydrodynamics equations for heavy ions: Initial conditions are set by an optical Glauber model. Solving the hydro equations numerically done via the Kurganov-Tadmor method using a Lattice QCD EoS [P. Huovinen and P. Petreczky, Nucl. Phys. A 837, 26 (2010).] (s95p-v1) The hydro evolution is run until the kinetic freeze-out. [For details: B. Schenke, et al., Phys. Rev. C 85, (2012)] (Tf =136 MeV) Energy-momentum conservation h/s=1/4p -To go from rates to yield, we use a hydrodynamic simulation so that we have a profile of the temperature and flow evolution of the thermalized system. We are using 2nd order viscous hydrodynamics satisfying the following equations. …. [- Where Tmunu has an ideal piece which we see here and a viscous correction prop to the shear-stress tensor. ] -The initial conditions of the hydro are set by a glaubber model, which gives the initial energy density profile of the system. Its normalization is chosen such that hadronic spectra are reproduced. -These sets of equations are solved via the KT method until kinetic freezeout. -The equations are evolved until a freeze-out chosen such that the hadron spectrum in particular the pion spectrum is reproduced.

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**Viscous Corrections: QGP rates**

Viscous correction to the rate in kinetic theory rate Using the quadratic Israel-Stewart ansatz to modify F.-D. distribution Dusling & Lin, Nucl. Phys. A 809, 246 (2008). ; Now, since we are using viscous hydrodynamics, there must be viscous corrections to the dilepton rates. For the QGP rates, the idea is to modify the distribution functions to include viscous corrections. For that purpose, we use the quadratic ansatz to modify the Fermi-Dirac distribution. This anzats comes from requiring that the Cooper-Frye formula in viscous hydrodynamics remains continuous across the freeze-out surface. Then the total rate separates into the ideal part and a viscous part. The correction can be written as qmu qnu pimunu and envelope function. The envelope has the analytic form … This was first derived back by Dusling and Lin 2008 and we were able to reproduce their results.

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**Viscous corrections to HG rates?**

Two modifications are plausible: Self-Energy Performing the calculation => these corrections had no effect on the final yield result! 1 2 ; ; Now should we have any viscous correction to the HG rates. Yes, of course, it is part of a very active field of Out-of-Equilibrium Field theory. It is not yet clear how these viscous corrections should be added in general. There are two plausible ways of doing this: by either modifying the self-energy or the Bose-Einstein distribution or both. For the imagyrary part of the retarded propagator, the self-energy was extended by modifying the thermal dist. Fct., since na is an actual thermal distribution of particle a. After applying the viscous correction, the total thermal self-energy can also be decomposed into an ideal part and a viscous correction, as before. Once again, we have the same form for the viscous correction namely an enveloppe function B2 and a tensor product. The other conceivable thing to modify is the Bose-Einstein factor. To do this properly one needs to modify the KMS relation which needs to be extended to include viscous corrections. What we have done until now is to treat is in a kinetic theory frame work (as vector mesons are bosons) and just attached a viscous correction to the BE dist. A more rigorous extension of the KMS relation is a work in progress. However, and this is important, after performing the calculation with all the viscous correction present in the hadronic rates, we found that the viscous correction had no effect on the final result: the yield.

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**Low Mass Dilepton Yields: HG+QGP**

For low M: ideal and viscous yields are almost identical and dominated by HG. These hadronic rates are consistent with NA60 data [Ruppert et al., Phys. Rev. Lett. 100, (2008)]. Here is the yield of the low mass dileptons as a function of the invariant mass at MIN. BIAS. The take-home message is that the invariant mass yield is almost insensitive to viscosity and are dominated by the HG. The hadronic rates used here are consistent with NA60 data as was demonstrated by Ruppert and collaborators back in 2008.

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**How important are viscous corrections to HG rate?**

Rest frame of the fluid cell at x=y=2.66 fm, z=0 fm 0-10%; h/s=1/4p Fluid rest frame, viscous corrections to HG rates: HG gas exists from t~4 fm/c => is small, so very small viscous corrections to the yields are expected. Direct computation shows this! Since the HG rates dominate in the low invariant mass, let’s investigate how important are the viscous corrections in the HG. This plot shows the evolution of the shear-stress tensor as a function of time. The hadronic gas lives in this region … so roughly from 4 fm/c and beyond. The viscous corrections which have the form … and because the shear-stress tensor over eta is small in that region the viscous corrections are very small. Our direct computation shows that the end result is not dependent on these correctios.

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**Dilepton yields Ideal vs Viscous Hydro**

Since viscous corrections to HG rates don’t matter, only viscous flow is responsible for the modification of the pT distribution. Also observed viscous photons HG [M. Dion et al., Phys. Rev. C 84, (2011)] The presence of df in the rates doesn’t affect the yield! 0-10% M=mr Note the centrality class of the plot. It needs to be clear that viscous corrections to the HG rates are not important per centrality class. Hence the averaging procedure needed to compute the minimum bias HG yields is not the cause behind delta f being unimportant. -The difference between the ideal yields in red and the viscous yields solely comes from the viscous evolution of the medium. -Viscous flow tends to shift the low pt dileptons into higher pt. -A similar effect has been observed for photons when we apply viscosity.

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**Dilepton yields Ideal vs Viscous Hydro**

For QGP yields, both corrections matter since the shear-stress tensor is larger. Integrating over pT, notice that most of the yield comes from the low pT region. Hence, at low M there isn’t a significant difference between ideal and viscous yields. One must go to high invariant masses. M=mr -For the QGP rates on the other hand, the difference between the ideal and the viscous yields vs pt are larger since both viscous flow and viscous correction to the distribution function are contributing. -Once we integrate over pt, we realized that most of the contribution is coming low pt region. Therefore when we look at the yield as a function of invariant mass, the difference between ideal and viscous hydrodynamics, is not significant and can only be seen at high M.

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**Dilepton yields Ideal vs Viscous Hydro**

For QGP yields, both corrections matter since the shear-stress tensor is large. Integrating over pT, notice that most of the yield comes from the low pT region. Hence, at low M there isn’t a significant difference between ideal and viscous yields. One must go to high invariant masses. M=mr Notice: y-axis scale! - Now let’s look at the yield as a function of pt. The dotted dashed line is the HG and the QGP is the solid lines. The difference between the ideal yields in red and the viscous yields in blue solely comes from the viscous evolution of the medium. -Viscous flow tends to shift the low pt dileptons into higher pt. -A similar effect has been observed for photons when we apply viscosity. -The difference between these two curves is very small for the HF rates. -For the QGP rates on the other hand, it is larger since both viscous flow and viscous correction to the dist. Fct. Are needed. -Once we integrate over pt, we realized that most of the contribution is coming low pt region. Therefore when we look at the yield as a function of invariant mass, the difference between ideal and viscous hydrodynamics, is not significant and can only be seen at high M.

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**A measure of elliptic flow (v2)**

To describe the evolution of the shape use a Fourier decomposition, i.e. flow coefficients vn Important note: when computing vn’s from several sources, one must perform a yield weighted average. A nucleus-nucleus collision is typically not head on; an almond-shape region of matter is created. This shape and its pressure profile gives rise to elliptic flow. z x - Now let’s talk about v2. A typical collision between two nuclei is not head on. By looking at the overlap region, it is easy to see that an almond-shape region is created. The collision region is very hot and has greater pressure gradients in the x-direction than in the y-direction which leads to greater flow in the x-direction that in the y. This is the elliptic flow. It can be characterized by expanding the yield in a fourier series. Here is that expansion. The second Fourier coefficient v2 best describes elliptic flow. Now an important thing to note is that, when dealing with several sources of dileptons, v2 is a weighted average with the weight being the yield.

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**v2 from ideal and viscous HG+QGP (1)**

Viscosity lowers elliptic flow. Here is a quick look at v2 vs invariant mass. This v2 isn’t something you are used to seeing. You can actually isolate the flow from different contributions. The first bump comes from the flow of the rho omega complex, and the second bump is the flow of the phi. Now when we compared our results with the ones we obtained using Ralf Rapp’s rates we got very similar elliptic flow. Adding in viscosity has two effects, 1) it lowers v2 by making the velocity distribution more isotropic. The second effect is to broadens the v2 spectrum. So what I’ve done here is to take the blue curve and a constant so that the maxima’s match so that this effect becomes evident. This is coming from the fact that viscous evolution in the HG sector is associated with slightly higher temperatures thus broadening the spectrum.

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**v2 from ideal and viscous HG+QGP (1)**

Viscosity lowers elliptic flow. Viscosity slightly broadens the v2 spectrum with M. Here is a quick look at v2 vs invariant mass. This v2 isn’t something you are used to seeing. You can actually isolate the flow from different contributions. The first bump comes from the flow of the rho omega complex, and the second bump is the flow of the phi. Now when we compared our results with the ones we obtained using Ralf Rapp’s rates we got very similar elliptic flow. Adding in viscosity has two effects, 1) it lowers v2 by making the velocity distribution more isotropic. The second effect is to broadens the v2 spectrum. So what I’ve done here is to take the blue curve and a constant so that the maxima’s match so that this effect becomes evident. This is coming from the fact that viscous evolution in the HG sector is associated with slightly higher temperatures thus broadening the spectrum.

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**v2(pT) from ideal and viscous HG+QGP (2)**

M=1.5GeV M=mr Now, unlike photons, dileptons have an extra degree of freedom the a invariant mass. Let’s look at v2(pt) for different invariant masses. One can isolate the HG from the QGP region by looking at two different invariant masses. [-Red Lines are from ideal hydro while blue lines include viscosity. Dotted lines means … , dashed-dotted lines are … solid lines are the sum of QGP + HG.] At low invariant mass M=755 MeV, the HG dominates over the QGP. At higher invariant mass M=1.5GeV the QGP dominates. This effect was first noticed by Chatterjee et al. back in 2007, however since we are using both thermal rates with viscous hydrodynamics, we can make a more quantitative comparison between the v2 of HG vs QGP. We can also make statements about what viscosity does to the pt distribution of v2. In the low invariant mass region viscosity tends to stop the growth of v2 at large pt. In the high invariant mass region, not only does viscosity stop the growth of v2 but it also shifts it to higher pt. This is because, in the QGP the viscous corrections to the rates are proportional to p^2 hence shifting the spectrum to higher pt. M is extremely useful to isolate HG from QGP. At low M HG dominates and vice-versa for high M. R. Chatterjee et al. Phys. Rev. C (2007). We can clearly see two effects of viscosity in the v2(pT). Viscosity stops the growth of v2 at large pT for the HG (dot-dashed curves) Viscosity shifts the peak of v2 from to higher momenta (right, solid curves). Comes from the viscous corrections to the rate: ~ p2 (or pT2)

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**Open Charm contribution**

Since Mq>>T, heavy quarks come from early times after the collision; Mq>> LQCD heavy quarks must be produced perturbatively. For heavy quarks, many scatterings are needed for momentum to change appreciably. In this limit, Langevin dynamics applies [Moore & Teaney, Phys. Rev. C 71, (2005)] Charmed Hadron production [C. Young et al. , PRC 86, , 2012]: PYTHIA -> Generate a c-cbar event using nuclear parton distribution functions. (EKS98) Embed the PYTHIA c-cbar event in Hydro -> Langevin dynamics to modify its momentum distribution. At the end of hydro-> Hadronize the c-cbar using Peterson fragmentation. PYTHIA decays the charmed hadrons -> Dileptons. Now in the last minute or so, I will focus on the Charmed hadron contribution. Charmed hadrons are important because they are affecting the shape of the spectrum over the entire M region. Of course they play a more prominent role in the high mass region. Since the typical mass of a heavy quark is much much bigger that the typical temperature of lambda qcd for that matter, heavy quarks must be produced perturbatively. Also, since they are heavy, many scatterings are needed for their momentum to change appreciably. In this limit Langevin dynamics applies (more about this topic can be found in the reference by Moore & Teaney). This is the set of equations that Langevin dynamics is solving. Get the pp sepctrum from PYHTIA and then modify it using langevin dynamics while the QGP phase still exists. The hadronize the quarks and decay them into dileptons.

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**Charmed Hadrons yield and v2**

0-10% 0-10% Here is the yield as a function of invariant mass. The STAR collaboration has provided us with both the Cocktail curve and the data. Our results show the effects of heavy quark energy loss. Langevin dynamics affects the invariant mass yield of charmed hadrons by increasing it in the low invariant mass region and decreasing it at high invariant mass. The total curves are here. The purple curve using langevin dynamics seems to describe the data better that without it, so heavy quark energy loss fit the data better. The second thing that energy loss does is to induce a v2 to the heavy quarks. Once we add the charm contribution we develop a v2 in the intermediate mass region whereas using pp alone doesn’t yield any v2. Heavy-quark energy loss (via Langevin) affects the invariant mass yield of Charmed Hadrons (vs rescaled p+p), by increasing it in the low M region and decreasing it at high M. Charmed Hadrons develop a v2 through energy loss (Langevin dynamics) so there is a non-zero v2 in the intermediate mass region.

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**Reassessing effects of viscosity on QGP v2 (1)**

M=1.5 GeV When we previously compared the differential v2(pT) going from ideal to viscous hydrodynamics (with viscous dR corrections), it seemed as though it behaved similarly to hadrons. h/s=1/4p Let’s now re-asses the validity of the viscous correction to the QGP rates. Put a plot of v2(pt) Notice that there is a large effect of the viscous correction to the ideal rates, but its nothing unexpected, as we see that in v2(pt) of hadrons. Now show v2(M) for b=8.85. Unlike charged hadrons, integrated v2 of dileptons is sensitive to deltaR.

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**Reassessing effects of viscosity on QGP v2 (2)**

M=1.5 GeV When we previously compared the differential v2(pT) going from ideal to viscous hydrodynamics (with viscous dR corrections), it seemed as though it behaved similarly to hadrons. h/s=1/4p Let’s now re-asses the validity of the viscous correction to the QGP rates. Put a plot of v2(pt) Notice that there is a large effect of the viscous correction to the ideal rates, but its nothing unexpected, as we see that in v2(pt) of hadrons. Now show v2(M) for b=8.85. Unlike charged hadrons, integrated v2 of dileptons is sensitive to deltaR. However, looking at v2(pT) with viscous hydro evolution alone, v2(pT) rises owing to the growth of the anisotropy (pmn) at early times; unlike hadrons! Also observed in viscous photons [M. Dion et al., Phys. Rev. C 84, (2011)].

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**Reassessing effects of viscosity on QGP v2 (3)**

M=1.5 GeV h/s=1/4p h/s=1/4p Let’s now re-asses the validity of the viscous correction to the QGP rates. Put a plot of v2(pt) Notice that there is a large effect of the viscous correction to the ideal rates, but its nothing unexpected, as we see that in v2(pt) of hadrons. Now show v2(M) for b=8.85. Unlike charged hadrons, integrated v2 of dileptons is sensitive to deltaR. v2(M) exhibits an even more pronounced effect coming from dR. Hence, the magnitude of the viscous correction must be further studied by improving dR, beyond its Israel-Stewart from. Note that v2(M) is very small (<1% for all centralities computed) : We start our hydro simulation by initializing our shear-stress tensor to zero In the QGP phase not enough time has passed for flow to build up Neglecting the effects of fluctuations in the initial conditions.

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**How important are viscous corrections to QGP v2?**

Reminder of viscous QGP rate: We will define viscous corrections to be large when |dR/Ro|>1. Frequent large corrections will yield unphysical results: calculations should not be trusted in the region of phase space where that occurs. So to assess the validity of our calculation, we will perform a test: All fluid elements (an element is 4-volume of size tDtDxDyDh) having |dR/Ro|>1 for a particular qm , are not considerer in the calculation. Now, since we are using viscous hydrodynamics, there must be viscous corrections to the dilepton rates. For the QGP rates, the idea is to modify the distribution functions to include viscous corrections. For that purpose, we use the quadratic ansatz to modify the Fermi-Dirac distribution. This anzats comes from requiring that the Cooper-Frye formula in viscous hydrodynamics remains continuous across the freeze-out surface. Then the total rate separates into the ideal part and a viscous part. The correction can be written as qmu qnu pimunu and envelope function. The envelope has the analytic form … This was first derived back by Dusling and Lin 2008 and we were able to reproduce their results.

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**Removing cells with |dR/Ro|>1**

v2(M) is not particularly sensitive to the cut in dR at low M region, however the situation worsens at higher M where large dRs increase v2(M) by as much as a factor of ~3 at M=2.5GeV. the delta R used is strongly dependant on the ansatz employed for the momentum dist. Of the quarks. So far in this talk I’ve only considerend the IS form of deltaf, however delta F should come from fundamental physics and there is no special reason that the ansatz proposed by IS applies here. Actually, it is useful to see how our results vary if choose another deltaF. In the following I shall consider a delta f derived… and has a momentum dependence that is closer to the underlying microscopic theory considered.

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**Removing cells with |dR/Ro|>1**

M=1.5 GeV h/s=1/4p the delta R used is strongly dependant on the ansatz employed for the momentum dist. Of the quarks. So far in this talk I’ve only considerend the IS form of deltaf, however delta F should come from fundamental physics and there is no special reason that the ansatz proposed by IS applies here. Actually, it is useful to see how our results vary if choose another deltaF. In the following I shall consider a delta f derived… and has a momentum dependence that is closer to the underlying microscopic theory considered. Our v2(pT) results are robust at pT<1.5GeV: most of the fluid elements have |dR/Ro|<1. v2(pT) for pT above ~1.5GeV, is strongly dependent on the dR (or rather dn of the quarks) being used. Hence, it is instructive to go beyond I-S form of dn, by revisiting the fundamental physics it contains, and derive a new dn.

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**Viscous correction QGP rate: revisited**

To improve dR, a generalized distribution function is used: G is computed using Boltzmann equation for a gas of massless particles with constant cross-section. The viscous correction in I-S form is recovered when G=1. The new from of dn includes higher order corrections in k0/T, and we made sure that the series converges before truncating it. Let x=k0/T: the delta R used is strongly dependant on the ansatz employed for the momentum dist. Of the quarks. So far in this talk I’ve only considerend the IS form of deltaf, however delta F should come from fundamental physics and there is no special reason that the ansatz proposed by IS applies here. Actually, it is useful to see how our results vary if choose another deltaF. In the following I shall consider a delta f derived… and has a momentum dependence that is closer to the underlying microscopic theory considered. The low and the high energy G(x) were matched via a tanh function at x=11.2

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**Beyond Israel-Stewart: removing cells with |dR/Ro|>1**

v2(M) w/ and w/o the cut in |dR/Ro|>1, is not significantly different at all M (at most ~20%).

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**Beyond Israel-Stewart: removing cells with |dR/Ro|>1**

M=1.5 GeV h/s=1/4p v2(M) w/ and w/o the cut in |dR/Ro|>1, is not significantly different at all M (at most ~20%). v2(pT) is also better described and starts to breaking down for pT above ~2GeV. Key message: v2(M) is a better quantity to measure as it is less sensitive to the form of dR (or dn).

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**Effect of new dR on v2(pT)**

M=1.5 GeV h/s=1/4p h/s=1/4p New dR introduces additional terms E and 1/E terms which helps v2(pT) to peak at lower pT relative to the Israel-Stewart dR, thus increasing v2(pT ) for most M. Bottom line: dileptons, unlike hadron, are significantly more sensitive to the form of dR (viscous v2(M) decreases by ~2 compared to ideal case), so including the most accurate physics possible in dR is crucial.

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Conclusions First calculation of dilepton yield and v2 via viscous 3+1D hydrodynamical simulation. v2(pT) for different invariant masses has good potential of separating QGP and HG contributions. Modest modification to dilepton yields owing to viscosity. In the HG phase, v2(M) is reduced ~20%, (h/s=1/4p) by viscosity and the shape is slightly broadened. In the QGP phase, v2(M) is greatly affected by viscosity, more studies are on the way. Studying yield and v2 of leptons coming from charmed hadrons allows to investigate heavy quark energy loss. Read the slide.

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Future work Further investigate the effects of viscous corrections and temperature dependent h/s on elliptic flow of the QGP. Include cocktail’s yield and v2 with viscous hydro evolution. Include the contribution from 4p scattering. Include Fluctuating Initial Conditions (IP-Glasma). Results for LHC are on the way.

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**A specials thanks to: Charles Gale Clint Young Gabriel S**

A specials thanks to: Charles Gale Clint Young Gabriel S. Denicol Björn Schenke Sangyong Jeon Jean-François Paquet Igor Kozlov Ralf Rapp

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**Hadron Spectra from MUSIC**

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**Born, HTL, and Lattice QCD**

Ding et al., PRD

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**V2 including charm at Min Bias**

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**Reassessing effects of viscosity on QGP v2 (2)**

h/s=1/4p h/s=1/4p M=mr M=mr Let’s now re-asses the validity of the viscous correction to the QGP rates. Put a plot of v2(pt) Notice that there is a large effect of the viscous correction to the ideal rates, but its nothing unexpected, as we see that in v2(pt) of hadrons. Now show v2(M) for b=8.85. Unlike charged hadrons, integrated v2 of dileptons is sensitive to deltaR.

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