# Presentation: 478256

**W.A. Zajc**

-  Exploring
- the Lower Limits
- of Perfection
- W. A ZajcColumbia UniversityDNP08 Workshop: *Quantifying the Character of the sQGP*

## Perfection  (No) Viscosity

**W.A. Zajc**

- Perfection  (No) Viscosity

- Isotropic in rest frame
-  No shear stress
-  no viscosity,  ****= 0
- Primer:
- Remove your organic prejudices
- Viscosity  ~ mean free path
- Small viscosity  Small  ******mfp**** **
*Zero*   viscosity  ******mfp** = 0 (!)

- X

## Pre-History 

**W.A. Zajc**

- Pre-History

- Landau (1955) significant extension of Fermi’s statistical model
- Considers fundamental roles of
- hydrodynamic evolution
- entropy
- “The defects of Fermi’s theory arise mainly because the expansion of the compound system is not correctly taken into account...(The) expansion of the system can be considered on the basis of relativistic hydrodynamics.”

## Landau on Viscosity

**W.A. Zajc**

**Use of hydro relies on R/******** >> 1**
**Negligible viscosity ******** equivalent to ****large Reynolds number ****Re**** ******** ********VR/******** >>1**** **
******VR/******** ~ V R / v****th**** ******** **
**so for relativistic system V ~ v****th**** and**
**Re****  >>1 ********  R /  ******** >> 1 ; see #1**

## RHIC Success

**W.A. Zajc**

## What Is The Viscosity at RHIC?

**W.A. Zajc**

- “Perfect fluid” (and/or “ideal hydrodynamics”) defined as “zero viscosity”.
**                     ****?**

## Why /s Matters

**W.A. Zajc**

- Why ****/s Matters

**Any engineer will tell you**
***Kinematic***** viscosity ********/ ******** ~ [Velocity] x [Length]       ****is what matters****(see Landau’s remark on Reynolds number)**
**Any relativist will tell you**
**  ******** ******** ********+ p**** **
**Any thermodynamicist will tell you**
** ****** **+ p = T s      ( at ********B**** =0 )**
**So**
** ********/******** ********/****** + p  (****/sT**)  =  **(****/s**) (1/T)****                               ~ (**damping  coefficient  x  thermal time**)**

## Numerical Value of RHIC Viscosity

**W.A. Zajc**

**Input :**
**Estimating s****QGP****  (per degree of freedom):**
**Method 1:**
**Method 2 (based on handy rule of thumb):**
**Number of (assumed massless) degrees of freedom: **
**                  **

## A Long Time Ago (1985)

**W.A. Zajc**

- Miklos Gyulassy and Pawel Danielewicz:
*Dissipative Phenomena in Quark-Gluon Plasmas*P. Danielewicz, M. Gyulassy _[Phys.Rev. D31, 53,1985. ](http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,D31,53)_
-      noted several restrictions on smallest allowed  :
- Most restrictive:
** ******** ****> h/<p>         ********     ******** > ~ n / 3**
- But recall s = 3.6 n for the quanta they were considering
-  ******/s  > 1 / (3.6 x 3) ~ 1 / (4 ********)                     !!**

## KSS Bound

**W.A. Zajc**

- This bound is  (now)  *very*  well-known in the nuclear physics community:
-  “*A Viscosity Bound Conjecture*”,  _[P. Kovtun](http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+Kovtun,+P.)_, _[D.T. Son](http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+Son,+D.T.)_, _[A.O. Starinets](http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+Starinets,+A.O.)_, _[hep-th/0405231](http://arxiv.org/abs/hep-th/0405231)_
- Where do “ordinary” fluids sit  wrt this limit?
- RHIC “fluid” *might*be at ~2-3 on this scale (!)

**T=10****12**** K**

**(**4

## AdS / CFT in Pictures

- _[Google Image ](http://images.google.com/imghp?hl=en&tab=wi)_search on “_[AdS/CFT hologram boundary](http://images.google.com/images?um=1&hl=en&rlz=1C1GGLS_en-USUS291&q=ads/cft+hologram+boundary&btnG=Search+Images)_”

**23-Oct-08**

**W.A. Zajc**

## AdS / CFT for Dummies

**23-Oct-08**

**W.A. Zajc**

- Graviton with 5-momentum  *k*  in bulk satisfies *k****k* = 0  described by *4* numbers

- Those *4*  numbers describe virtual gauge quanta on *4-d** * boundary

- ( Adopted from _[S. Brodsky figure ](http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.0514v1.pdf)_)

## The (Assumed) Connection

**W.A. Zajc**

*Exploit*    Maldacena’s “D-dimensional strongly coupled gauge theory  (D+1)-dimensional stringy gravity”
*Thermalize*   with massive black brane
*Calculate*   viscosity  = “Area”/16G
*Normalize* by entropy (density)  s = “Area” / 4G
*Dividing out*  the infinite “areas” :
*Conjectured * to be a lower bound “*for all relativistic quantum field theories at finite temperature and zero chemical potential*”.
*See* “*Viscosity in strongly interacting quantum field theories from black hole physics*”, P. Kovtun, D.T. Son, A.O. Starinets, Phys.Rev.Lett.94:111601, 2005, _[hep-th/0405231](http://arxiv.org/abs/hep-th/0405231)_

- Infinite “Area” !

- h

- A

- A

## Estimating /s 

**W.A. Zajc**

- Estimating /s

- Damping  (flow, fluctuations, heavy quark motion)  ~ /s
**FLOW: *****Has the QCD Critical Point Been ******Signaled by Observations at RHIC?,***** ****R. Lacey *****et al*****., ****Phys.Rev.Lett.98:092301,2007 ****(**_[nucl-ex/0609025](http://arxiv.org/abs/nucl-ex/0609025)_**)**
***The Centrality dependence of Elliptic flow, ******the Hydrodynamic Limit, and the Viscosity ******of Hot QCD*****, H.-J. Drescher *****et al*****., ****(**_[arXiv:0704.3553](http://arxiv.org/abs/0704.3553)_**)**
**FLUCTUATIONS:***** Measuring Shear Viscosity ******Using Transverse Momentum Correlations ******in Relativistic Nuclear Collisions*****, ****S. Gavin and M. Abdel-Aziz, ****Phys.Rev.Lett.97:162302,2006 ****(**_[nucl-th/0606061](http://arxiv.org/abs/nucl-th/0606061)_**)**
**DRAG, FLOW:***** Energy Loss and Flow of ******Heavy Quarks in Au+Au Collisions ******at √s******NN****** = 200 GeV***** (PHENIX Collaboration), ****A. Adare *****et al*****., ****to appear in Phys. Rev. Lett. (**_[nucl-ex/0611018](http://arxiv.org/abs/nucl-ex/0611018)_**)**

**C****H****A****R****M****!**

## Has the QCD Critical Point Been Signaled by Observations at RHIC?, R. Lacey et al., Phys.Rev.Lett.98:092301,2007 (nucl-ex/0609025)

**W.A. Zajc**

***Has the QCD Critical Point Been Signaled by Observations at RHIC?,***** ****R. Lacey *****et al*****., Phys.Rev.Lett.98:092301,2007 (**_[nucl-ex/0609025](http://arxiv.org/abs/nucl-ex/0609025)_**)**

**Signature: FLOW**
**Calculation:**
**Payoff Plot: **

- Fit v2 ~I1(w)/I0(w); w = mT/2T

- On-shell transport model for gluons**, **Z. Xu and C. Greiner, hep-ph/0406278.

- PHENIX v2/ data (nucl-ex/0608033) compared to R.S. Bhalerao et al. (nucl-th/0508009)

## Measuring Shear Viscosity Using Transverse Momentum Correlations in Relativistic Nuclear Collisions, S. Gavin and M. Abdel-Aziz, Phys.Rev.Lett.97:162302,2006 (nucl-th/0606061)

**W.A. Zajc**

*Measuring Shear Viscosity Using Transverse Momentum Correlations in Relativistic Nuclear Collisions*, S. Gavin and M. Abdel-Aziz, Phys.Rev.Lett.97:162302,2006 (_[nucl-th/0606061](http://arxiv.org/abs/nucl-th/0606061)_)

**Signature: FLUCTUATIONS**
**Calculation:**
**Payoff Plot: **

-      Diffusion eq. for fluctuations *g*

-     Compare to STAR data on centrality dependence of rapidity width  of pT fluctuations

-       Difference in correlation widths for central and peripheral collisions

## The Centrality dependence of Elliptic flow, the Hydrodynamic Limit, and the Viscosity of Hot QCD, H.-J. Drescher et al., (arXiv:0704.3553)

**W.A. Zajc**

***The Centrality dependence of Elliptic flow, the Hydrodynamic Limit, and the Viscosity of Hot QCD*****, H.-J. Drescher *****et al*****., (**_[arXiv:0704.3553](http://arxiv.org/abs/0704.3553)_**)**

**Signature: FLOW**
**Calculation:**
**Payoff Plot: **

-      Knudsen number K

-     Fits to PHOBOS v2 data to determine  for Glauber and CGC initial conditions

-       Decrease in flow due to finite size

## Closely Related Recent Development

- Application of _[Drescher et al. ](http://arxiv.org/abs/0704.3553)_ methodology to STAR flow data
- Yuting Bai, SQM08
- First extraction of ****/s versus centrality

**23-Oct-08**

**W.A. Zajc**

**10-Oct-08**

**STAR Preliminary**

## Energy Loss and Flow of Heavy Quarks in Au+Au Collisions at √sNN = 200 GeV (PHENIX Collaboration), A. Adare et al., Phys. Rev. Lett. 98:172301,2007 (nucl-ex/0611018)

**W.A. Zajc**

-  Moore and Teaney Phys.Rev.C71:064904,2005 (perturbative, argue ~valid non-perturbatively)

***Energy Loss and Flow of Heavy Quarks in Au+Au Collisions at √s******NN****** = 200 GeV***** (PHENIX Collaboration), A. Adare *****et al*****., Phys. Rev. Lett. 98:172301,2007** **(**_[nucl-ex/0611018](http://arxiv.org/abs/nucl-ex/0611018)_**)**

**Signature: FLOW, ENERGY LOSS**
**Calculation:**
**Payoff Plot: **

-  Rapp and van Hees Phys.Rev.C71:034907,2005, to fit ***both*** PHENIX v2(e) and RAA(e)

## Comparison of  estimates

**W.A. Zajc**

**S. Gavin and M. Abdel-Aziz: **
**PRL 97:162302, 2006**
** **

**p****T****fluctuations STAR**

**R. Lacey et al.: PRL 98:092301, 2007**** **

**v****2**** PHENIX **
**& STAR**

**Estimates of ********/s based on flow and fluctuation data**
**Indicate small value as well**
**Close to conjectured limit**
**Significantly below ********/s of helium (********s ~ 9)**

## Compare to the Competition

**W.A. Zajc**

**Damping of ****breathing mode in ****cold Fermi gas**
**(All figures courtesy of John Thomas, ****Duke University)**

- String-theory 1/4

- Schafer, cond-mat 2006

## Our Problem Is Much Harder

**W.A. Zajc**

- Non-relativistic:  Damping given by
- Relativistic: Causal, second-order expansion:
- Relativistic Fluid Dynamics: Physics for Many Different Scales
- Neglect various termsat your own risk:
- Baier et al., Relativistic viscous hydrodynamics, conformal invariance, and holography
- Natsuume and Okamura,Comment on “Viscous hydrodynamics relaxation time from AdS/CFT correspondence”

## Viscosity  Dissipation

**W.A. Zajc**

- Viscosity  Dissipation

- Non-relativistic:  Damping given by
- Where does this come from?
- Recall we’re interested in velocity gradients:
- Remove uniform rotation:
- Remove uniform (Hubble) expansion:

## For Further Details

**W.A. Zajc**

**See “Virtual Journal on QCD Matter”**
**Steffen A. Bass, ****Berndt Mueller,  ****William A. Zajc**
**qgp.phy.duke.edu**
**On the topic of****2****nd**** order hydro:**
**What a Difference ****a Term Makes **

## Quantifying  /s 

**W.A. Zajc**

*Quantifying*  /s

- : How?
- : By performing detailed and systematic hydrodynamic   simulations to understand sensitivity to ****/s   (and “everything” else)
*Viscosity Information from Relativistic Nuclear Collisions: **How Perfect is the Fluid Observed at RHIC?, **P. Romatschke and  U. Romatschke, **Phys. Rev. Lett. 99:172301, 2007** *
*Multiplicity Scaling in Ideal and Viscous Hydrodynamics, **H. Song and U. Heinz,  **Phys. Rev. C78, 024902,  2008** *
*Conformal Relativistic Viscous Hydrodynamics: **Applications to RHIC results at ****s**NN** = 200 GeV, **M.  Luzum and P. Romatschke,**Phys.Rev.C78:034915,2008.  *

## Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?, P. Romatschke and  U. Romatschke, Phys. Rev. Lett. 99:172301, 2007

**W.A. Zajc**

**Signature:  dN/dy, v****2****, <p****T****>**
**Calculation:****2****nd**** order causal viscous hydro:**
**(Glauber IC’s)**
**Payoff Plots:**

## Multiplicity Scaling in Ideal and Viscous Hydrodynamics, H. Song and U. Heinz,  Phys. Rev. C78, 024902,  2008

**W.A. Zajc**

- Multiplicity Scaling in Ideal and Viscous Hydrodynamics, H. Song and U. Heinz,  Phys. Rev. C78, 024902,  2008

**Signature:  **Famous “hydro limits” plot of v2/****  vs (1/S)dNch/dy
**Calculation: **Full Israel-Stuart equations (Glauber IC’s)
**Payoff Plots:**

## Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?, P. Romatschke and  U. Romatschke, Phys. Rev. Lett. 99:172301, 2007

**W.A. Zajc**

**Signature:  dN/dy, v****2****, <p****T****>**
**Calculation:****2****nd**** order causal viscous *****conformal*****hydro:**
**(Glauber  and CGC IC’s)**
**Payoff Plots:**

## Comparison with other estimates

**W.A. Zajc**

**S. Gavin and M. Abdel-Aziz: **
**PRL 97:162302, 2006**
** **

**p****T****fluctuations STAR**

**R. Lacey et al.: PRL 98:092301, 2007**** **

**v****2**** PHENIX **
**& STAR**

**Estimates of ********/s based on flow and fluctuation data**
**Indicate small value as well**
**Close to conjectured limit**
**Significantly below ****/s ****of helium (********s ~ 9)**

## Beyond ‘t Hooft Limit(?)

**23-Oct-08**

**W.A. Zajc**

- Just for fun:
- Take “central value”of these estimates
- Apply it in the most naïve mapping  ****s= gYM2/4**** in
- “Coupling constant dependence of the shear viscosity in N=4 supersymmetric Yang-Mills theory”,Buchel, Liu, Starinets, hep-th/0406264
****S = 0.28 (!!)
- Amazingly consistent with estimates from AMY formalismof high pT energy loss (J. Nagle)
- Just for perspective:
- Allowed range

**4******** ********/s**

**“********s****”**

## The “Flow” Knows Quarks

**W.A. Zajc**

- The “fine structure” v2(pT) for different mass particles shows good agreement with ideal (“perfect fluid”) hydrodynamics
- Scaling flow parameters by quark content nq resolves meson-baryon separation of final state hadrons

**baryons**

**mesons**

## Quasi-particles and Minimal Viscosity

**W.A. Zajc**

- If you’re doing Boltzmann transport:
-  = energy density
-  = lifetime of quasiparticle
- Entropy density s ~ kB n 
- where last step
- follows from requirement that lifetime of quasiparticle must exceed ~ h / Energy
- establishes that the bound is from below
- But valuesnot compatible with quasiparticles with lifetimes > h / mass
- D. T. Son
- Levy et al. **Quasi-Particle Degrees of Freedom versus the Perfect Fluid as Descriptors of the Quark-Gluon Plasma.**
*How to reconcile with quark scaling of flow ? *

## Water  RHIC  Water  RHIC

**W.A. Zajc**

- Water  RHIC  Water  RHIC

- The search for QCD phase transition of course was informed by analogy to ordinary matter
- Results from RHIC are now “flowing” back to ordinary matter

**“*****On the Strongly-Interacting Low-Viscosity Matter Created in Relativistic Nuclear Collisions*****”,****L.P. Csernai, J.I. Kapusta and L.D. McLerran, Phys.Rev.Lett.97:152303,2006,  nucl-th/0604032 **

****** ****/ s**

## Is There a QCD Critical Point?

**W.A. Zajc**

- Here the analogy with phase transitionsin ordinary matter breaks down:
- Recall “ Properties of the medium are(at zero baryon number)uniquely determined by T ”
- Pressure = P(T) can’t vary independently(unlike water)
- But if baryon number is non-zero (intensive order parameter) baryon chemical potential B :
- To increase B :
*Lower*  collision energy
*Raise*  atomic mass
*Both*  part of RHIC II and GSI-FAIR

## Conclusions

- Can we say with confidence“the quark-gluon plasma displays less friction than any other known laboratory fluid”  ?
- Not yet: Cold atom competition
- Can we say with confidence“ ****/s for sQGP saturates the quantum bound” ?
- Not yet:
- Still must quantify contributions from
- Bulk viscosity
- Anomalous viscosity
- Eddy viscosity
- Hadronic viscosity
- Not to mention eccentricity fluctuations, EOS’s, freezeout schemes, ...
- What we can state with confidence:
*Tremendous progress has been made in understanding the fundamental role of ********/s and quantifying its value in the sQGP*
*This, more so than any other “observable”, has had the greatest impact on other fields in the history of relativistic heavy ion physics *

**23-Oct-08**

**W.A. Zajc**

## Summary

**W.A. Zajc**

- The near-simultaneous development of
- RHIC Discoveries
- Theoretical Synthesis
- Overthrow of “classical” QGP
- Emergence of AdS/CFT calculations
- represents a true *paradigm  shift*
- This workshop provides wonderful possibility to continue these compelling investigations