Experiments Confront Theory Unicorn: the QGP Hunters: Experimentalists. Leaving theorists as...

Experiments Confront Theory Theory: Effective theory for deconfinement, near T c. Today: only the “pure” glue theory (no dynamical quarks) Based upon detailed results from the lattice Experiment: No confrontation today... Soon: need to add quarks Present model may be the only real competition to AdS/CFT

The standard plot T→ ↑ T c SU(3) gauge theory without quarks, temperature T (Weakly) first order transition at T c ~ 290 MeV e/T 4 → ←3p/T 4 e(T)= energy density p(T)= pressure WHOT: Umeda, Ejiri, Aoki, Hatsuda, Kanaya, Maezawa, Ohno,

Another plot of the same Plot conformal anomaly, (e-3p)/T 4 : large peak above T c. ~ g 4 as T → ∞ ↑ T c T→ WHOT: Umeda, Ejiri, Aoki, Hatsuda, Kanaya, Maezawa, Ohno, ∞

Strong vs weak coupling at T c ? Resummed perturbation theory at 3-loop order works down to ~ 3 T c. Intermediate coupling: α s (T c ) ~ 0.3. Not so big... So what happens below ~ 3 T c ? Want effective theory; e.g.: chiral pert. theory: expand in m π /f π, exact as m π → 0. But there is no small mass scale for SU(3) in “semi”-QGP, T c → 3 T c. →→→ Andersen, Su, & Strickland,

Perhaps α s is not so big at T c Braaten & Nieto, hep-ph/ Laine & Schröder, hep-ph/ & From two loop calculation, matching original to effective theory: Pure gauge: α(7. T). 3 flavors of quarks: α(9. T). T c ~ Λ MS ~ 200 MeV. So α s eff (T) ~ α s eff (2 π T) ~ 0.3 at T c : not so big Grey band uncertainty from changing scale by factor 2. α(2πT)

Not strong coupling, even at T c QCD coupling runs like α(2πT), intermediate at T c, α(2πT c ) ~ 0.3 Braaten & Nieto, hep-ph/ , Laine & Schröder, hep-ph/ & HTL resummed perturbation theory, NNLO, good to ~ 8 T c : Andersen, Leganger, Strickland, Su, Assume: not a strongly coupled QGP near T c ! versus: AdS/CFT....

What to expand in? Consider SU(N) for different N. # perturbative gluons ~ N Scaled by ideal gas values, e and p for N = 3, 4 and 6 look very similar Implicitly, expand about infinite N. Explicitly, assume classical expansion ok N=3: Boyd et al, lat/ N = 4 & 6: Datta & Gupta, T→ N = 3, 4, 6 ↑ T c 4 T c ↑

Conformal anomaly ≈ N independent For SU(N), “peak” in e-3p/T 4 just above T c. Approximately uniform in N. Not near T c : transition 2nd order for N = 2, 1st order for all N ≥ 3 N=3: weakly 1st order. N = ∞: strongly 1st order (even for latent heat/N 2 ) Datta & Gupta, long tail? ↑ T c 4 T c ↑ T→ N = 3, 4, 6

Tail in the conformal anomaly To study the tail in (e-3p)/T 4, multiply by T 2 /(N 2 -1) T c 2 : (e-3p)/((N 2 -1)T 2 T c 2 ) approximately constant, independent of N Datta & Gupta, ↑ T c 4 T c ↑ T→ N = 3, 4, 6

Precise results for three colors From WHOT: 2 T c ↑ ↑ T c ↑ 1.2 T c T→ WHOT: Umeda, Ejiri, Aoki, Hatsuda, Kanaya, Maezawa, Ohno,

How to get a term ~ T 2 in the pressure? Expand pressure of ideal, massive gas in powers of mass m: Quasi-particle models: choose m(T) to fit pressure. Need m(T) to increase sharply as T → T c to suppress pressure. Inelegant... T→ m(T)↑ Above: Castorina, Miller, & Satz, Originally: Peshier, Kampfer, Pavlenko & Soff, PRD 1996 t = T/T c, a =.47, b =.13, c =.39

A simple solution Our model: generalization of Meisinger, Miller & Ogilvie, ph/ Dumitru, Guo, Hidaka, Korthals-Altes, & RDP, arXiv: ? Also: Y. Hidaka & RDP, , , , Assume there is some potential, V(q). The vacuum, q 0, is the minimum of V(q): Pressure is the value of the potential at the minimum: For T > 1.2 T c, a constant ~T 2 in the pressure, is due to a constant ~T 2 in V(q): Above 1.2 T c, 〈 q 〉 = 0. Except near T c, for most of the semi-QGP, the non-perturbative part of the pressure, ~T 2, is due just to a constant Region where 〈 q 〉 ≠ 0, and V(q) matters, is very narrow: T: T c → 1.2 T c Unexpected consequence of precise lattice data. Large N: makes sense to speak of classical 〈 q 〉 instead of fluctuations.

An effective (matrix) model for deconfinement Lattice: SU(N) gauge theories, without quarks. Simulations show: N = 3 close to N = ∞. Not just the pressure. Simple matrix model, valid in large N expansion (no small masses). Fit to pressure for all N with 2 parameters Good agreement with interface tensions Disagrees with the (renormalized) Polyakov loop - ? New: adjoint Higgs phase, with split masses, for T < 1.2 T c. Unexpected: transition region very narrow, < 1.2 T c Generalization of Meisinger, Miller, Ogilvie ph/ , MMO Dumitru, Guo, Hidaka, Korthals-Altes, & RDP, arXiv: Also: Y. Hidaka & RDP, , , ,

What the lattice tells us Weak dependence on # colors Not just the pressure...

Hidden Z(2) spins in SU(2) Consider constant gauge transformation: As U c ~ 1, locally gluons invariant: Nonlocally, Wilson line changes: L ~ propagator for “test” quark. SU(3): det U c = 1 ⇒ j = 0, 1, 2 SU(N): U c = e 2 π i j/N 1: Z(N) symmetry. Z(N) spins of ‘t Hooft, without quarks Quarks ~ background Z(N) field, break Z(N) sym. N = 3

Hidden Z(3) spins in SU(3) T >> T c T ~ T c T < T c Im l↑ Re l→ Lattice, A. Kurkela, unpub.’d: 3 colors, loop l complex. Distribution of loop shows Z(3) symmetry: z Interface tension: box long in z. Each end: distinct but degenerate vacua. Interface forms, action ~ interface tension: T > T c : order-order interface = ‘t Hooft loop: measures response to magnetic charge Korthals-Altes, Kovner, & Stephanov, hep-ph/ Also: if trans. 1st order, order-disorder interface at T c.

Usual spins vs Polyakov Loop T→ T c ↑ ↑ L = SU(N) matrix, Polyakov loop l ~ trace: Confinement: F test qk = ∞ ⇒ 〈 l 〉 = 0 Above T c, F test qk < ∞ ⇒ 〈 l 〉 ≠ 0 〈 l 〉 measures ionization of color: partial ionization when 0 < 〈 l 〉 < 1 : “semi”-QGP Svetitsky and Yaffe ’80: SU(3) 1st order because Z(3) allows cubic terms: Does not apply for N > 3. So why is deconfinement 1st order for all N ≥ 3?

Polyakov Loop from Lattice: pure Glue, no Quarks Lattice: (renormalized) Polyakov loop. Strict order parameter Three colors: Gupta, Hubner, Kaczmarek, Suggests wide transition region, like pressure, to ~ 4 T c. T → ↑ ↑ ~ 4 T c ←1.0 ← ~ 0.4 ↑ T c ↑T=0 ← Confined →← SemiQGP→ ← “Complete” QGP →

Polyakov Loop from Lattice: Glue plus Quarks, “T c ” Quarks ~ background Z(3) field. Lattice: Bazavov et al, quark flavors: weak Z(3) field, does not wash out approximate Z(3) symmetry. ↑“T c ”.8“T c ”↑ 2 “T c ”↑ ← 0.2 ← Hadronic →← “Semi”-QGP →←Complete QGP ↑ ↑T=0 ←1.0 T →

Transition region narrow: for pressure, < 1.2 T c ! For interface tensions, < 4 T c... Above 1.2 T c, pressure dominated by constant term ~ T 2. What does this term come from? Gluon mass m(T)? But inelegant... SU(N) in 2+1 dimensions: ideal ~ T 3. Caselle +...: also T 2 term in pressure. But mass would be m 2 T, not m T 2. T 2 term like free energy of massless fields in 2 dimensions: string? Above T c ? Need to include quarks! Can then compute temperature dependence of: shear viscosity, energy loss of light quarks, damping of quarkonia... Skipping to the punchline

Lattice: SU(N) in 2+1 dimensions SU(N) in 2+1 dim’s for N = 2, 3, 4, 5, & 6. Below plot of T c /T, not T/T c. Clear evidence for non-ideal terms ~ T 2, not ~ T T c /T→ ↑10 T c ↑1.1T c ↑ 2T c Caselle, Castagnini, Feo, Gliozzi, Gursoy, Panero, Schafer,

T→ ↑ T c deconf With quarks: “T c ” moves down: which T c ? Just glue: T c deconf ~ 290 MeV. Common lore: with quarks, one “T c ”, decreases. Matrix model: T c deconf does not change by addition of dynamical quarks. With light quarks, only chiral trans, when 〈 loop 〉 << 1. Hence T c chiral << T c deconf. ↑ T c chiral Lattice: Bazavov and Petreczky + HotQCD, Ren’d loop ↑ With quarks ↓ ←without quarks ↑

Shear viscosity changes with T In semi-QGP, η suppressed from pert. value through function R(q). Not like kinetic theory Log sensitivity, through constant κ R (q): Y. Hidaka & RDP, c pert : P. Arnold, G. Moore, and L. Yaffe, hep-ph/ , hep-ph/ T/T c →

“Bleaching” of color near T c. Roughly speaking, as 〈 loop 〉 → 0, all colored fields disappear. Quarks, in fundamental rep. as 〈 loop 〉. Gluons, in adjoint rep., as 〈 loop 〉 2. Bleaching of color as T → T c : robust consequence of the confinement of color QGP: quarks and gluons. Semi-QGP: dominated by quarks, by ~ 〈 loop 〉 Why recombination works at RHIC but not at LHC? (v 2 /# quarks vs kinetic energy/# quarks) Suppression of color universal for all fields, independent of mass. Why charm quarks flow the same as light quarks? (single charm vs pions) An effective theory can provide a bridge from lattice simulations to experiment

Lattice: order-order interface tensions σ Lattice: de Forcrand & Noth, lat/ σ ~ universal with N Semi-classical σ : Giovanengelli & Korthals-Altes ph/ ; / ; / : GKA ‘04 Above 4 T c, semi-class σ ~ lattice. Below 4 T c, lattice σ << semi-classical σ. Even so, when N > 3, all tensions satisfy “Casimir scaling” at T > 1.2 T c. ↑ T c 4.5 T c ↑ N = 4 Semi-classical↓ T→ ⇐ lattice

Other models Fit only the pressure, not interface tensions. Masses as T → T c + : some go up (massive gluons) some go down (Polyakov loops)

Models for the “s”QGP, T c to 4 T c 1. Massive gluons: Peshier, Kampfer, Pavlenko, Soff ’96...Castorina, Miller, Satz Castorina, Greco, Jaccarino, Zappala Mass decreases pressure, so adjust m(T) to fit p(T). Simple model. Gluons very massive near T c. 2. Polyakov loops: Fukushima ph/ Hell, Kashiwa, Weise Effective potential of Polyakov loops. Potential has 5 parameters... With quarks, at T ≠ 0, can go from μ = 0 to μ ≠ 0 3. AdS/CFT: Gubser, Nellore Gursoy, Kiritsis, Mazzanti, Nitti, Add potential for dilaton, φ, to fit pressure. Only infinite N. Relatively simple potential, 4. Monopoles: Liao & Shuryak, Vortices: Tuesday: Takuya Saito

Matrix model: two colors Simple approximation Two colors: transition 2nd order, vs 1st for N ≥ 3 Using large N at N = 2

Matrix model: SU(2) Simple approximation: constant A 0 ~ σ 3, nonperturbative, ~ 1/g: Point halfway in between: q = ½, l = 0. Confined vacuum, L c, Classically, A 0 cl has zero action: no potential for q. Single dynamical field, q Loop l real. Z(2) degenerate vacua q = 0 and 1: x x x Re l→ 1 0

Potential for q, interface tension Computing to one loop order about A 0 cl gives a potential for q: Gross, RDP, Yaffe, ‘81 Use V pert (q) to compute σ: Bhattacharya, Gocksch, Korthals-Altes, RDP, ph/ Balancing S cl ~ 1/g 2 and V pert ~ 1 gives σ ~ 1/√g 2 (not 1/g 2 ). Width interface ~ 1/g, justifies expansion about constant A 0 cl. GKA ‘04: σ ~... + g x x x

Symmetries of the q’s Wilson line L not gauge invariant, L → Ω † L Ω. Its eigenvalues, e ± i π q, are. Ordering of L’s eigenvalues irrelevant. Symmetries: q → q + 2 : q angular variable. Valid with quarks. Pure glue: also, q → q + 1, Z(2) transf., L → - L For pure glue, can restrict q: 0 → 1. Then Z(2) transf. q → 1 - q: Z(2) transf., plus exchange of eigenvalues Any potential of q must be invariant under q → 1- q

Potentials for the q’s Add non-perturbative terms, by hand, to generate ≠ 0 : By hand? V non (q) from: monopoles, vortices... Liao & Shuryak: ph/ , , , T < T c : 〈 q 〉 = ½ → 1 0 x x x 1 T >> T c : 〈 q 〉 = 0,1 → 0 x x x

Three possible “phases” Two phases are familiar:  〈 q 〉 = 0, 1: 〈 l 〉 = ± 1: “Complete” QGP: usual perturbation theory. T >> T c.  〈 q 〉 = 1/2: 〈 l 〉 = 0 : confined phase. T < T c Also a third phase, “partially” deconfined (adjoint Higgs phase) 0 T > T c x? Lattice: one transition, to confined phase, at T c. No other transition above T c. Still, there is an intermediate phase, the “semi”-QGP Strongly constrains possible non-perturbative terms, V non (q).

Getting three “phases”, one transition Simple guess: V non ~ loop 2, 1st order transition directly from complete QGP to confined phase, not 2nd Generic if V non (q) ~ q 2 at q << 1. Easy to avoid, if V non (q) ~ q for small q. Then 〈 q 〉  at all T > T c. Imposing the symmetry of q ↔ 1 - q, V non (q) must include x xx Term ~ q at small q avoids transition from pert. QGP to adjoint Higgs phase

Cartoons of deconfinement Consider: ⇓ a = ¼: semi QGP x x ⇓ a = 0: complete QGP x x a = ½: T c => Stable vacuum at q = ½ Transition second order x

Meisinger, Miller, Ogilvie ph/ : take V non ~ T 2 0-parameter matrix model, N = 2 Two conditions: transition occurs at T c, pressure(T c ) = 0 Fixes c 1 and c 3, no free parameters. But not close to lattice data (from ’89!) ↑ T c 3 T c ↑ T→ ⇐ 0-parameter model ⇐ Lattice Lattice: Engels, Fingberg, Redlich, Satz, Weber ‘89

1-parameter matrix model, N = 2 Dumitru, Guo, Hidaka, Korthals-Altes, RDP ‘10: to usual perturbative potential, Add - by hand - a non-pert. potential V non ~ T 2 T c 2. Also add a term like V pert : Now just like any other mean field theory. 〈 q 〉  given by minimum of V eff : 〈 q 〉  depends nontrivially on temperature. Pressure value of potential at minimum:

Lattice vs matrix models, N = 2 Choose c 2 to fit e-3p/T 4 : optimal choice Reasonable fit to e-3p/T 4 ; also to p/T 4, e/T 4. N.B.: c 2 ~ 1. At T c, terms ~ q 2 (1-q) 2 almost cancel. ↑ T c 3 T c ↑ T→ ⇐ Lattice ⇐ 0-parameter ⇐ 1-parameter Lattice: Engels, Fingberg, Redlich, Satz, Weber ‘89

Lattice vs 1-parameter model, N = 2 ↑ T c 3T c ↑ T→ ⇐ e-3p/T 4, lattice ⇐ e-3p/T 4, model ⇑ p/T 4, lattice ⇓ e/T 4, model ⇓ e/T 4, lattice ⇑ p/T 4, model

Width of transition region, 0- vs 1-parameter 1-parameter model: get sharper e-3p/T 4 because 〈 q 〉 -> 0 much quicker above T c. Physically: sharp e-3p/T 4 implies region where 〈 q 〉 is significant is narrow N.B.: 〈 q 〉 ≠ 0 at all T, but numerically, negligible above ~ 1.2 T c ; p ~ 〈 q 〉 . Above ~1.2 T c, the T 2 term in the pressure is due entirely to the constant term, c 3 ! ⇐ 0-parameter ⇓ 1-parameter ↑ T c 2 T c ↑T→

Polyakov loop: 1-parameter matrix model ≠ lattice Lattice: renormalized Polyakov loop. 0-parameter model: close to lattice 1-parameter model: sharp disagreement. 〈 l 〉 rises to ~ 1 much faster - ? Sharp rise also found using Functional Renormalization Group (FRG): Braun, Gies, Pawlowski, ; Marhauser, Pawlowski, ⇐ lattice ⇐ 0-parameter ⇓ 1-parameter ↑ T c T→ 2 T c ↑ Lattice: Cardoso, Bicudo, Can reconcile by (arbitrary) shift in zero point energy

Interface tension, N = 2 σ vanishes as T→T c, σ ~ (t-1) 2ν. Ising 2ν ~ 1.26; Lattice: ~ Matrix model: ~ 1.5: c 2 important. Semi-class.: GKA ’04. Include corr.’s ~ g 2 in matrix σ(T) (ok when T > 1.2 T c ) N.B.: width of interface diverges as T→T c, ~ √(t 2 - c 2 )/(t 2 -1). ↑ T c 2.8 T c ↑ T→ ⇐ matrix model Semi-classical  ⇐ lattice Lattice: de Forcrand, D’Elia, Pepe, lat/

Lattice: A 0 mass as T → T c - up or down? Kaczmarek, Karsch, Laermann, Lutgemeier lat/ μ/T goes down as T → T c m D /T goes up as T → T c Cucchieri, Karsch, Petreczky lat/ , Kaczmarek, Zantow lat/ Which way do masses go as T → T c ? Both are constant above ~ 1.5 T c. T→ Gauge invariant: 2 pt function of loops: Gauge dependent: singlet potential ↑ T c ↑ 2T c ↑ T c

Adjoint Higgs phase, N = 2 A 0 cl ~ q σ 3, so 〈 q 〉 ≠ 0 generates an (adjoint) Higgs phase: RDP, ph/ ; Unsal & Yaffe, , Simic & Unsal, In background field, A = A 0 cl + A qu : D 0 cl A qu = ∂ 0 A qu + i g [A 0 cl, A qu ] Fluctuations ~ σ 3 not Higgsed, ~ σ 1,2 Higgsed, get mass ~ 2 π T 〈 q 〉 Hence when 〈 q 〉 ≠ 0, for T < 1.2 T c, splitting of masses: ↑ T c T→ 1.5 T c ↑ ⇐ diagonal A 0 mode ⇐ off-diagonal A 0 modes At T c : m diag = 0, m off ~ 2 m pert. 1 → ↑ 1.2 T c m pert = √2/3 g T: m/m pert ~.56 at 1.5 T c, from V non.

Matrix model: N ≥ 3 Why the transition is always 1st order One parameter model

Path to Z(3), three colors SU(3): two diagonal λ’s, so two q’s: Z(3) paths: move along λ 8, not λ 3 : q 8 ≠ 0, q 3 = 0.

Path to confinement, three colors Now move along λ 3 : In particular, consider q 3 = 1: Elements of e 2π i/3 L c same as those of L c. Hence tr L c = tr L c 2 = 0: L c confining vacuum Path to confinement: along λ 3, not λ 8, q 3 ≠ 0, q 8 = 0.

Z(3) paths in SU(3) gauge For SU(3), two diagonal generators, Z(3) paths: q 8 ≠ 0, q 3 = 0: Three degenerate vacua, for q 8 = 0, 1, and 2. Move between vacua along blue lines, Re l→ Im l↑ x x x

Confining vacuum in SU(3) Alternately, consider moving along λ 3. In particular, consider q 3 = 1: L c is the confining vacuum, X : “center” of space in λ 3 and λ 8 Move from deconfined vacuum, L = 1, to the confined vacua, L c, along red line: Re l→ Im l↑ x x x x Elements of e 2π i/3 L c same as those of L c. Hence LcLc

General potential for any SU(N) For SU(N), Σ j=1...N q j = 0. Hence N-1 independent q j ’s, = # diagonal generators. At 1-loop order, the perturbative potential for the q j ’s is As before, assume a non-perturbative potential ~ T 2 T c 2 : Ansatz: constant, diagonal matrix i, j = 1...N

Path to confinement, four colors Move to the confining vacuum along one direction, q j c : (For general interfaces, need all N-1 directions in q j space) Perturbative vacuum: q = 0. Confining vacuum: q = 1. Four colors: General N: confining vacuum = uniform distribution for eigenvalues of L For infinite N, flat distribution.

Cubic term for all N ≥ 3, so transition first order No term linear in φ. Cubic term in φ for all N ≥ 3. Along q c, about φ = 0 there is no symmetry of φ → - φ for any N ≥ 3. Hence terms ~ φ 3, and so a first order transition, are ubiquitous. Special to matrix model, with the q i ’s elements of Lie algebra. Svetitsky and Yaffe ’80: V eff (loop) ⇒ 1st order only for N=3; loop in Lie group Also 1st order for N ≥ 3 with FRG: Braun, Eichhorn, Gies, Pawlowski, Define φ = 1 - q, Confining point φ = 0

Cubic term for four colors Construct V eff either from q’s, or equivalently, loops: tr L, tr L 2, tr L N = 4: |tr L| 2 and |tr L 3 | 2 not symmetric about q = 1, so cubic terms, ~ (q - 1) 3. (|tr L 2 | 2 symmetric, residual Z(2) symmetry) Cubic terms special to moving along q c in a matrix model. Not true in loop model x ⇐ |tr L| 2 ⇐ |tr L 3 | 2 ⇑ q = 1 qc ⇒qc ⇒ ⇑.8 ⇑ 1.2

x ⇐ |tr L| 2 |tr L 3 | 2 ⇒ ⇑ q =1 qc ⇒qc ⇒ ⇑.8 ⇑ 1.2 Cubic term for four colors Asymmetric in reflection about q = 1 ⇔

Lattice vs 0- and 1- parameter matrix models, N = 3 Results for N=3 similar to N=2. 0-parameter model way off. Good fit e-3p/T 4 for 1-parameter model, Again, c 2 ~ 1, so at T c, terms ~ q 2 (1-q) 2 almost cancel. ↑ T c 3 T c ↑ T→ ⇐ 1-parameter ⇐ 0-parameter ⇐ Points: lattice Lattice: Bielefeld, lat/ Datta & Gupta,

Lattice vs 1- parameter model, N = 3 ↑ T c 3T c ↑ T→ ⇐ e-3p/T 4, lattice ⇐ e-3p/T 4, model ⇓ e/T 4, lattice ⇓ e/T 4, model ⇑ p/T 4, lattice ⇑ p/T 4,model

Polyakov loop: matrix models ≠ lattice Renormalized Polyakov loop from lattice does not agree with either matrix model 〈 l 〉 - 1 ~ 〈 q 〉 2 : By 1.2 T c, 〈 q 〉 ~.05, negligible. Again, for T > 1.2 T c, the T 2 term in pressure due entirely to the constant term, c 3 ! Rapid rise of 〈 l 〉  as with FRG: Braun, Gies, Pawlowski, ↑ T c T→2 T c ↑ ⇐ lattice ⇑ 0-parameter 1-parameter ⇓ Cannot reconcile by shift in zero point energy Lattice: Gupta, Hubner, and Kaczmarek,

Interface tension, N = 2 and 3 Order-order interface tension, σ, from matrix model close to lattice. For T > 1.2 T c, path along λ 8 ; for T < 1.2 T c, along both λ 8 and λ 3. σ(T c )/T c 2 nonzero but small, ~.02. Results for N =2 and N = 3 similar - ? Semi-classical  ⇐ matrix model, N = 2 ⇐ matrix model, N = 3 ↑ T c 5 T c ↑ T→ Lattice: de Forcrand, D’Elia, Pepe, lat/ de Forcrand, Noth lat/ lattice, N=3 

Adjoint Higgs phase, N = 3 For SU(3), deconfinement along A 0 cl ~ q λ 3. Masses ~ [λ 3, λ i ]: two off-diagonal. Splitting of masses only for T < 1.2 T c : Measureable from singlet potential, 〈 tr L † (x) L(0) 〉, over all x. T/T c → ⇐ 4 off-diagonal, K’s ⇐ 2 off-diagonal, π’s ⇐ 2 diagonal modes At T c : m diag small, but ≠ 0 m pert = g T, m/m pert ~.8 at 1.5 T c, from V non.

Matrix model: N ≥ 3 To get the latent heat right, two parameter model. Thermodynamics, interface tensions improve

Latent heat, and a 2-parameter model Latent heat, e(T c )/T c 4 : 1-parameter model too small: 1-para.: BPK: 1.40 ±.1 ; DG: 1.67 ±.1. 2-parameter model, c 3 (T). Like MIT bag constant WHOT: c 3 (∞) ~ 1. Fit c 3 (1) to DG latent heat Fits lattice for T < 1.2 T c, overshoots above. ↑ T c 3T c ↑ T→ ⇐ Lattice ⇐ 2-parameter ⇐ 1-parameter Latttice latent heat: Beinlich, Peikert, Karsch (BPK) lat/ Datta, Gupta (DG) Bag const ~ (262 MeV) 4 c 2 not near 1, vs 1-para.

Anomaly: 2-parameter model vs lattice ↑ T c 3T c ↑ T→ ⇐ Lattice ⇐ 2-parameter

↑ T c 3T c ↑ T→ ⇐ Lattice ⇐ 2-parameter Anomaly times T 2 : 2-parameter model vs lattice

Thermodynamics of 2-parameter model, N = 3 ⇐ e-3p/T 2 T c 2, lattice ⇓ e-3p/T 2 T c 2, model ⇑ p/T 4, lattice ⇓ e/T 4, lattice ⇓ e/T 4, model T/T c →

Interface tensions, 2-parameter model, N = 3 Order-order interface tension, σ, close to lattice. Order-order σ(T c )/T c 2 ~ st order transition, so can compute order-disorder σ(T c )/T c 2 ~.022, vs Lattice: Lucini, Teper, Wenger, lat/ ,.019 Beinlich, Peikert, Karsch lat/ ↑ T c 5 T c ↑ lattice, N=3 ⇓ ⇑ 2-parameter model, N=3 lattice, N=2 ⇓ Lattice: de Forcrand, D’Elia, Pepe, lat/ de Forcrand, Noth lat/ Order-order σ: T→

2-parameter model, N = 4 Assume c 3 (∞) = 0.95, like N=3. Fit c 3 (1) to latent heat, Datta & Gupta, Order-disorder σ(T c )/T c 2 ~.08, vs lattice,.12, Lucini, Teper, Wenger, lat/ T/T c → ⇑ p/T 4, lattice ⇑ p/T 4, model ⇓ e/T 4, lattice ⇓ ⇓ e/T 4, model ⇐ e-3p/T 2 T c 2, lattice ⇓ e-3p/T 2 T c 2, model

2-parameter model, N = 6 ⇑ p/T 4, lattice Order-disorder σ(T c )/T c 2 ~.25, vs lattice,.39, Lucini, Teper, Wenger, lat/ T/T c → ⇑ p/T 4, model ⇓ e/T 4, lattice ⇓ ⇓ e/T 4, model ⇐ e-3p/T 2 T c 2, lattice ⇓ e-3p/T 2 T c 2, model

Matrix model, G(2) gauge group G(2): no center, yet has “confining” trans: Holland, Minkowski, Pepe, Wiese, lat/ Very strong constraint on matrix model: we predict broad conformal anomaly/T 4. Pressure in G(2) not like pressure in SU(N), but interface tension