# Presentation: 233273

## Shear Viscosity of the “semi”-QGP

- 1. Deconfinement and Polyakov loops: possible phase transitions
- 2. SU(∞) on a small sphere (Sundborg ‘99, Aharony et al ’03, ‘05):
-         Matrix model; Gross-Witten point; semi-QGP
- 3. Lattice: pressure for SU(N)
-      With quarks: flavor independence.  Without quarks: N = 3 like N = ∞
-      Is the QCD coupling big at Tc?  Maybe *not*.
- 4. Renormalized Polyakov Loops & the semi-QGP
- 5. Shear viscosity of the semi-QGP
-     For heavy ions, is LHC like RHIC?
-                           Strong-QGP, N = 4 SUSY: *yes*.
-                                       Semi-QGP, *no*.

## 1. Some possible deconfining transitions

## Polyakov loops & deconfinement

- Polyakov loop: order parameter for deconfinement in SU(N):
- Ordinary magnetization:
-      
≠ 0 at low T,  
=0 at high T.
- Deconfinement: Polyakov loop “flipped”,
- Global Z(N) symmetry: *broken* at high T,
*                                       **restored* at low T.
- Classify possible deconfining transitions by change in < *loop *>.
*Assume *overall normalization of loop physical:
- Quarks act like background Z(N) field.
- Consider order parameter, *not* pressure, p(T); pressure always continuous.

- 
↑

- T→

## One possibility

- Logically possible, does *not* appear to arise in *any* context.  (Lattice, analytical...)
- General expectation before RHIC.

- Transition from confined phase to “complete” Quark-Gluon Plasma (QGP)
- Complete QGP: loop near 1, ≈ perturbative.
- Transition strongly first order.  Effect of quarks weak.

- confined

- “complete” QGP

## Another possibility

- Many quarks, *strong* background field.
- Loop increases gradually, probably no deconfining phase transition.

- Probably true for large number of flavors, completely wash out deconfinement.
- Also: perhaps no chiral transition?

## 1.→

- Tc↑

- T→

-  Hadronic →←     “semi”-QGP     →←Complete QGP→

- Analytic solution, and lattice, show: even with dynamical quarks, *three *regimes:
- Hadronic, <*loop*> ~ 0.
- “Semi”-QGP: <*loop*> nonzero, but *not* near one.  Matrix model.
- Complete QGP: <*loop*> near one.  Usual “perturbative” regime (resummed!)

## ×

- AdS/CFT: Can define <loop> = 1 at T = 0 (Polyakov-Maldacena, + scalars)
- Constant at T ≠ 0 (like pressure/T4) : value, vs g2 N?  *Not *a deconfining transition.

- 1.→

- T→

## 2. Deconfinement for SU(∞) on a small sphere

## SU(∞) on a small sphere: Hagedorn temperature

- Sundborg, hep-th/9908001
- AMMPV: Aharony, Marsano, Minwalla, Papadodimas, & Van Raamsdonk,
-                                                                                                    hep-th/0310285 & 0502149
- Consider SU(N) on a *very* small sphere: radius R, with g2(R) << 1.
- (Sphere because constant modes simple, spherically symmetric)
- At N = ∞, can have a phase transition even in a *finite* volume.
- When g2 = 0: by counting gauge *singlets*, find a Hagedorn temperature, TH :

- At N = ∞,  Hagedorn temperature is *precisely* defined, calculable at g2 = 0

## SU(∞) on a small sphere: effective theory

- Construct effective theory for low energy (constant) modes,
- by integrating out high energy modes, with momenta ~ 1/R:
- Consider (thermal) Wilson line:

- L is gauge dependent,
- Traces of moments gauge invariant,

- Effective theory for *l**j*: compute free energy in *constant* background A0 field:
- Q = diagonal matrix.

## SU(∞) on a small sphere & the Polyakov loop

- When g2 = 0:

- At the Hagedorn temperature, TH , *only* the first mode, *l*1, is unstable; all other modes are stable.  Concentrate on that mode, *l* ≡ *l*1.
- Vandermonde determinant in measure for constant mode gives “Vdm potential”:

- Vdm potential has discontinuity of *third* order at *l* = 1/2.
- Gross & Witten ’81; Kogut, Snow & Stone ’82....
- Sundborg, ’99....AMMPV ’03 & ‘05
- Dumitru, Hatta, Lenaghan, Orginos & RDP, hep-th/0311223 = DHLOP
- Dumitru, Lenaghan & RDP, hep-ph/0410294.

## Deconfinement on a small sphere

- Have deconfining phase transition when m2 = 0: *first* order, < *l* > = 1/2 at Tc = TH.
- Obvious from potentials above and below Tc:

- m2 = -.1, deconfined phase =>

- <= m2 = +.1, confined phase

## Gross-Witten point

- At transition, order parameter <*loop*> jumps from 0 to 1/2.  Latent heat nonzero.
- DLP: masses vanish, asymmetrically: “critical” 1st order transition: “GW point”.
- At m2 = 0, <*loop*> jumps because of 3rd order discontinuity in Vdm potential
- GW point like tricritical point in extended phase diagram.

## Semi-QGP on a small sphere

- AMMPV ‘05: calculate free energy with Q ≠ 0 to *two* loop order at small R
- c3 > 0 ⇒  Tc = TH - O(g4).  Deconfinement first order, *below *TH

- Boundary btwn complete and semi-QGP *not* precise; <*loop*>  → 1 by T ~ # Tc?
- To higher order in g2 :

## 3. Lattice: pressure. N = 3 like N = ∞?

- Maybe αs is *not* so big at Tc

## T/Tc→

- _  __p(T)   _ _      _
- pideal(T)↑

- T/Tc→

- ←1.0

- Pure SU(3): *weakly* 1st order
- QCD: and 2+1 flavors: crossover
- Bielefeld: properly scaled,  ≈ *universal *pressure

- ← QCD:
- 2+1 flavors

- Tc↑

##  Lattice: SU(3) glue, no quarks 

- More sensitive than pressure: (e-3p)/T4, e = energy density, p = pressure
- Bielefeld, hep-lat/9602007.  Nt = # time steps: 6, 8 near continuum limit?
- Pressure: sum of ideal gas, T4, plus T2 , *then* “MIT bag constant”,  T0.

## Tc↑

- Lattice: SU(3) close to SU(∞)?

- Bringoltz & Teper, hep-lat/0506034 & 0508021:
- SU(N), no quarks, N= 3, 4, 6, 8, 10, 12.
- Deconfining transition first order, latent heat ~ N2.
- Hagedorn temperature TH ~ 1.116(9) Tc for N = ∞

- T/Tc→

- Tc↑

##  Maybe αs is not so big at Tc

-  Maybe αs is *not* so big at Tc

- Laine & Schröder, hep-ph/0503061 & 0603048
- Tc ~ ΛMS ~ 200 MeV.  But αseff(T) ~ αseff(2 π T) ~ 0.3 at  Tc: *not* so big
- Two loop calculation: grey band uncertainty from changing scale by factor 2.

## p/T4↑

- p/T4↑

- ←ideal gas

- “Helsinki” resummation: Di Renzo, Laine, Schröder, Torrero, 0808.0557
- Now to 4 loop, ~ g6.  *Works* to ~ 3 Tc, *fails* below.
*Why*, if αseff(Tc) is *not* so big?    Perhaps a *semi*-QGP near Tc?

- Non

- Grey: resummed
-           pert.theory.
- Red: lattice.

## 4. Renormalized Polyakov loops & semi-QGP

## Renormalized loops

- Polyakov ’80, Dotsenko & Vergeles ‘81...DHLOP ’03...
- Gupta, Hubner & Kaczmarek 0711.2251 = GHK
- Bare loop UV divergent.  At one loop  =>
- Like mass ren. of heavy quark.  In 3+1 dim.’s, linear div.
- Vanishes with dimensional regularization, but *not* on the lattice:

- Loop in representation R, Casimir CR.
- 1/(a T) = # time steps, Nt.  Renormalized loop:
- Can choose
- GHK: find approximate Casimir scaling:
- Like cusp anomalous dimension.

## Zero point energy & renormalized loops

- Renormalization valid for arbitrary Wilson loops:

- Two ambiguities:

- Overall scale trivial: Z0  = 1 by requiring < *loop* > → 1 as T →∞.
- E0 = ground state energy for potential from Wilson loop:  E0 = # √σ.  # ?
- E0 : can *define* = 0 order by order in perturbation theory with *any* regulator
-      (Obvious with dimensional reg..  Also true with higher derivatives...)
-      E0 = 0 also in string model: Nambu-Goto *plus* extrinsic curvature terms...
- Lattice provides *non*-perturbative way to *define* E0 = 0.  Still, its a choice...
- T = 0 potential with dynamical quarks: can define *energy* for string breaking?

## Renormalized loops at high T

- Gava & Jengo ’81:
- Renormalized loops approach unity from *above.*
- Can compute perturbatively, with dimensional regularization.
- Just fold Debye mass, mD , into propagator for A0:

## Lattice: ren.’d triplet loop, pure SU(3)

- GHK:  Lattice SU(3), *no* quarks.  Two ways of getting ren’d loop agree.
- <*triplet loop*> ~ 1/2 at Tc+!  N=3 close to Gross-Witten point?
- <*adjoint loop*> ~ 0.01 just below Tc . *Only* natural in matrix model.
- semi-QGP: from (*exactly*) Tc+ to 2 - 4 Tc (?).   <*loop*> ~ constant above 4 Tc.

## T→

- Cheng et al, 0710.0354: ~ QCD, 2+1 flavors.  Tc ~ 190 MeV, crossover.
- <*loop*>: nonzero from ~ 0.8 Tc; ~ 0.3 at Tc; ~ 1.0 at 2 Tc.
- Semi-QGP from ~ 0.8 Tc (*below* Tc) to ~ 2-3 Tc (?). <*loop*> *small* at Tc .

- ←1.0

- ← 0.3

- Tc↑

- Ren.’d
- triplet
- loop ↑

- 2 Tc↑

- ←    Confined     →←           Semi-QGP*             *→←Complete QGP

- .8 Tc↑

## 4. Shear viscosity of the semi-QGP

##  Semi-QGP in weak coupling

- Hidaka & RDP 0803.0453.  Semi-classical expansion of the semi-QGP:

- Q ≠ 0: just like semi-classical calc. of ‘t Hooft loop.  Q = Qa , *diagonal* matrix.
- Work at large N, large Nf, use double line notation.  (Finite N ok, messy.)

- Perturbation theory in Bμ’s same as Q = 0, but with “shifted” p0’s.
- Amplitudes in real time: p0a → i ω, etc.  Furuuchi, hep-th/0510056
- Q (imaginary) chemical potential
- for (diagonal) color charge.
- e.g., for quarks:

## Z(N) interfaces = ‘t Hooft loop 

- Z(N) interface: Z(N) “twist” in z-direction. Atr = transverse area.

- tN = diag(1N-1, -N+1).  A0 ~ “coordinate” q(z).
*L*eff = classical + 1 loop potential, for *constant* A0

- Bhattacharya, Gocksch, Korthals-Altes & RDP, hep-ph/9205231
- Z(N) interface = ‘t Hooft loop: Korthals-Altes, Kovner & Stephanov, hep-ph/9909516
- Corrections ~ g3: Giovannangeli & Korthals-Altes hep-ph/0412322
-                     ~ g4:  Korthals-Altes, Laine, Romatschke 08...

##  How color evaporates in the semi-QGP

- AMMPV: simple trick.

- L = ei Q/T = Wilson line.  Obtain expressions in terms of moments of L, Lj.
- We *don’t* know (yet) effective theory for Q’s. *So we guess.*
- Take first moment, *l *=* <loop> *= < tr L>/N, from lattice for N = 3.
- For higher moments, given *l, *assume either: 1. Gross-Witten, or 2. step function.
- L ~ propagator of *infinitely* heavy (test) quark.
- In *this* semi-cl. expansion, for colored fields of *any* momentum and mass,
- As* l*→0*, all* quarks suppressed ~ *l *; *all* gluons, ~ *l**2** **: universal* color evaporation
- Smells right: *all *colored fields *should* evaporate as <*loop>* → 0.

##  Shear viscosity in the semi-QGP

##  Counting powers of <loop> = l → 0

-  Counting powers of <*loop*> = *l → *0

##  Small shear viscosity from color evaporation

*R *= ratio of shear viscosity in semi-QGP/complete-QGP at *same* g, T.
- Two different eigenvalue distributions give *very* similar results!
- When <*loop*> ~ 0.3, *R* ~ 0.3.

##  Shear viscosity/entropy

- Leading log shear viscosity/lattice entropy.  αs(Tc) ~ 0.3, “c” = 32.
*Large* increase from Tc to 2 Tc.*  Clearly* need results beyond leading log.
- Also need to include: quarks and gluons *below* Tc, hadrons *above* Tc.  *Not* easy.

##  Shear viscosity/entropy

- Leading log shear viscosity/lattice entropy.  αs(Tc) ~ 0.3.
*Large* increase from Tc to 2 Tc.*  Clearly* need results beyond leading log.
- Also need to include: quarks and gluons *below* Tc, hadrons *above* Tc.  *Not* easy.

## Strong- vs. Semi-QGP at the LHC

- At RHIC,  η/s ~ 0.1 ± 0.1
- Luzum & Romatschke, 0804.4015
*Close* to N = 4 SU(∞), η/s =1/(4 π).
- Strong-QGP: in N = 4 SU(∞),
*add* scalar potential to fit lattice pressure
- But η/s *remains* = 1/4π !
- Evans & Threlfall, 0805.0956
- Gubser & Nellore, 0804.0434
- Gursoy, Kiritsis, Mazzanti & Nitti 0804.0899
- So LHC nearly ideal, like RHIC.
- Semi-QGP, *and* non-relativistic systems →
*Large* change in η/s from Tc to 2 Tc.
- At *early* times, LHC *viscous,*
-                          *un*like RHIC

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