PowerPoint Presentation

Thermal dileptons at RHICGojko Vujanovic

Thermal Radiation Workshop Brookhaven National Laboratory

December 7th 20121

1OutlineOverview of Dilepton sources Low Mass DileptonsThermal Sources of Dileptons 1) QGP Rate (w/ viscous corrections) 2) In-medium vector mesons Rate (w/ viscous corrections)

3+1D Viscous Hydrodynamics

Thermal Dilepton Yields & v2

Intermediate Mass DileptonsCharmed Hadrons: Yield & v2

Conclusion and outlook 2After a brief overview of the dilepton sources, I will spend most of my time talking about the Low mass dileptons where themal sources are important. The thermal diletpons have two important sources: HG and QGP. These two sources are subsequently evolved via 3+1D Viscous hydrodynamics to obtain the dilepton yields and v2. Then in the last few minutes of my talk, I will be talking about the intermediate mass dileptons where charmed hadrons are important. 2Evolution of a nuclear collisionThermal dilepton sources: HG+QGPQGP: 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.) 3Space-time diagram Other dilepton sources: Formation phased) Charmed hadrons: e.g. D+/--> K0 + e+/- ne e) Beauty hadrons: e.g. B+/-->D0 + e+ /-nef) Other vector mesons: Charmonium, Bottomoniumg) Drell-Yan Processes

Sub-dominant the intermediatemass regionHere 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.

3Dilepton rates from the QGPAn important source of dileptons in the QGP

The rate in kinetic theory (Born Approx)

More complete approaches: HTL, Lattice QCD.4

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. 4Thermal Dilepton Rates from HGThe dilepton production rate is :

Where,

Model based on forward scattering amplitude [Eletsky, et al., Phys. Rev. C, 64, 035202 (2001)]

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;

The key quantity to consider when dealing with the vector mesons is the imaginary part of the retarded propagator. The model we are using is based on the FSA that Eletsky and Kapusta devised back in 2001. 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. The FSA included all the resonances on the right. [ These are for the rho, and similarly they can be found for the omega & phi. ]In both cases, the goal is to compute the imaginary part of the retarded propagator, which then goes into the rate equation. 5Vacuum part is described by the following LagrangiansFor r:

6Vector meson self-energies (1)

Vacuum part is described by the following LagrangiansFor r:

Since w has a small width and 3 body state in the self-energy, we model it as

7Vector meson self-energies (2)

Vacuum part is described by the following LagrangiansFor f:

Since w has a small width and 3 body state in the self-energy, we model it as

8Vector meson self-energies (3)

Vector meson self-energies The Forward Scattering Amplitude Low energies:

High energies:

Effective Lagrangian method by R. Rapp [Phys. Rev. C 63, 054907 (2001)]

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Resonances [R]contributing to rs scatt. amp.& similarly for w, f

The low energy FSA is dominated by breit-wigner

The high energy FSA is described by the regge tail which is derived through general considerations of the behavior of the FSA as energy goes to infinity.

- In collaboration with Ralf Rapps we are exploring a different method based on the effective lagrangian, to describe vector mesons in medium.

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10Imaginary part of the retarded propagatorT=150MeVn0=0.17/fm3 Vujanovic et al., PRC 80 044907Martell et al., PRC 69 065206 Eletsky et al., PRC 64 035202rwfAnd here are the imaginary part of the vector meson propagators. Looking at the location of the peak and the width of these curves, we notice that there is a significant broadening compared to the vacuum values, and only a small mass shift.

This width broadening is expected and necessary if one hopes to have a phase transition from HG to QGP. Indeed, when the particle width is comparable to its mass, it is legitimate question to ask whether have a bound state anymore. Or rather are we observing a collective excitation quarks and gluons.

[This is one of the key ingredients required for a phase transition from HG to QGP. The other ingredient to fully go to a QGP, which is assumed to be chirally symmetric, is to have the mass of the bound state drop to essentially zero. The dropping mass is not something that can easily be obtained thorough Massive Yang Mills Lagrangian models and is something that is typically put in by hand.] 103+1D HydrodynamicsViscous hydrodynamics equations for heavy ions:

Initial conditions are set by the 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, 024901 (2012)] (Tf =136 MeV)11Energy-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. 11Viscous Corrections: QGP ratesViscous correction to the rate in kinetic theory rate

Using the quadratic ansatz to modify F.-D. distribution

Dusling & Lin, Nucl. Phys. A 809, 246 (2008). 12

;

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.

12 Viscous corrections to HG rates?Two modifications are plausible: Self-Energy

Performing the calculation => these corrections had no effect on the final yield result! 13

;;

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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. 13For 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, 162301 (2008)].

Low Mass Dilepton Yields: HG+QGP 14

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.

14Fluid 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!

15Rest frame of the fluid cell at x=y=2.66 fm, z=0 fm0-10%

How important are viscous corrections to HG rate?

Since the HG rates dominate in the low invariant mass, lets 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. 15

Since viscous corrections to HG rates dont 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, 064901 (2011)]

Dilepton yields Ideal vs Viscous Hydro16M=mrThe presence of df in the rates is not important per centrality class!This is not a Min Bias effect.0-10%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. 16

Dilepton yields Ideal vs Viscous Hydro17M=mr

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 isnt a significant difference between ideal and viscous yields. One must go to high invariant masses. -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. 17

Dilepton yields Ideal vs Viscous Hydro18M=mr

Notice: y-axis scale!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 isnt a significant difference between ideal and viscous yields. One must go to high invariant masses. - Now lets 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. 18 A measure of elliptic flow (v2) Elliptic Flow

To describe the evolution of the shape use a Fourier decomposition, i.e. flow coefficients vn

Important note: when computing vns from several sources, one must perform a yield weighted average.19 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. xz

- Now lets 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. 19v2 from ideal and viscous HG+QGP (1)Similar elliptic flow when comparing w/ R. Rapps rates. 20

Here is a quick look at v2 vs invariant mass. This v2 isnt 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 Rapps 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 Ive done here is to take the blue curve and a constant so that the maximas 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. 20v2 from ideal and viscous HG+QGP (1)Similar elliptic flow when comparing w/ R. Rapps rates.

Viscosity lowers elliptic flow.

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Here is a quick look at v2 vs invariant mass. This v2 isnt 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 Rapps 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 Ive done here is to take the blue curve and a constant so that the maximas 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. 21v2 from ideal and viscous HG+QGP (1)Similar elliptic flow when comparing w/ R. Rapps rates.

Viscosity lowers elliptic flow.

Viscosity slightly broadens the v2 spectrum with M. 22

Here is a quick look at v2 vs invariant mass. This v2 isnt 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 Rapps 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 Ive done here is to take the blue curve and a constant so that the maximas 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. 22M 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 75 054909 (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) 23

M=1.5GeV

M=mrv2(pT) from ideal and viscous HG+QGP (2)Now, unlike photons, dileptons have an extra degree of freedom the a invariant mass. Lets 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.

23Charmed Hadron contributionSince Mq>>T (or LQCD), heavy quarks must be produced perturbatively; come from early times after the nucleus-nucleus collision.

For heavy quarks, many scatterings are needed for momentum to change appreciably.

In this limit, Langevin dynamics applies [Moore & Teaney, Phys. Rev. C 71, 064904 (2005)]

Charmed Hadron production: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.24

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. 24Charmed Hadrons yield and v2Heavy-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. 25

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 doesnt yield any v2. 25Conclusion & Future work26ConclusionsFirst 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.

v2(M) is reduced by viscosity and the shape is slightly broadened.

Studying yield and v2 of leptons coming from charmed hadrons allows to investigate heavy quark energy loss.

Future work Include cocktails yield and v2 with viscous hydro evolution.Include the contribution from 4p scattering. Include Fluctuating Initial Conditions (IP-Glasma) and PCE.Results for LHC are on the way.Read the slide. 26A specials thanks to:

Charles GaleClint YoungBjrn Schenke Sangyong Jeon Jean-Franois PaquetIgor Kozlov Ralf Rapp

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Born, HTL, and Lattice QCD28

Ding et al., PRD 83 034504 29Forward scattering amplitude results

Vujanovic et al., PRC 80 044907Here are the results for the FSA for the phi meson. The low and high energy pieces were joint together by a half-sided gaussian, whose parameters were tuned such that the dispersion relation is minimized. 29Dispersion relation30The f dispersion graphsThe dispersion relation

Vujanovic et al., PRC 80 044907V2 including charm at Min Bias31