# Presentation: 523423

##     Deconfinement: Renormalized Polyakov Loops

-                           to Matrix Models

- Adrian Dumitru (Frankfurt), Yoshitaka Hatta (RIKEN & BNL), Jonathan Lenaghan (Virginia), Kostas Orginos (RIKEN & MIT), & R.D.P. (BNL & NBI)

- 1. Review: Phase Transitions at T ≠ 0, Lattice Results for N=3, nf = 0 → 3
-               (N = # colors, nf = # flavors)
- 2. *Bare* Polyakov Loops, ∀ representations R:  factorization for N→∞
- 3. *Renormalized *Polyakov Loops
- 4. Numerical results from the lattice:
- 	N=3, nf=0:  R = 3, 6, 8 (10?)
- 5. Effective (matrix!) model for *renormalized *loops

**Notes:**
Introduction, overview

## Review of Lattice Results: N=3, nf = 0, 2, 2+1, 3

- nf = 0:   T_deconf ≈ 270 MeV
- 	pressure *small *for T < T_d
- 	like N→∞: p ~ 1 for T<T_d, p ~ N^2 for T>T_d   (Thorn, 81)

- nf ≠ 0: as nf ↑, p_ideal ↑, T_chiral ↓
-       BIG change: between nf = 0 and nf = 3,
-                 p_ideal: 16 to 48.5 x ideal m=0 boson      Tc:   270 to 175 MeV!
- Even the order changes: first for nf=0 to “crossover” for nf = 2+1

- Bielefeld

- pressure/T^4 ↑

- T=>

**Notes:**
Review of Lattice: unscaled pressures

## Three colors, pure gauge: weakly first order

- Three colors, pure gauge: *weakly* first order

- (Some) correlation lengths
- grow by ~ 10!

- Bielefeld

- Latent heat:                 ~ 1/3 vs 4/3 in bag model.  So?
- Look at gauge-inv. correlation functions: (2-pt fnc. Polyakov loop)

- T < T_d : m = σ/T
- T > T_d: m = m_Debye

- .9 T_d ↑

- ↑ T_d

- T_d ↑

- ↑ 4 T_d

- T-dep.
- string
- tension ↑

- ↑Debye
-   mass

**Notes:**
Three colors: weakly first order?  string tension and debye mass

## Deconfining Transition vs N: First order ∀ N ≥ 4

- Deconfining Transition vs N: First order ∀ N ≥ 4

- Lucini, Teper, Wenger ‘03: latent heat ~ N^2 for N= 3, 4, 6, 8

*No* data for

- Gross-Witten -  First order *but:*

- N=>

- latent heat/N^2

- Is N →∞ Gross-Witten?

- x

- x

- x

- 3

- 4

- 6

- N=2: *second *order
-    => no latent heat
- N=3: “weakly” 1st
- N ≥ 4: *strongly *
*   *first order.

- 2

**Notes:**
Order of the deconf’g trans. for N=4,6...

## Flavor Independence

- Bielefeld

*Perhaps: *even for nf ≠ 0, “transition” *dominated *by gluons

- At T ≠ 0: thermodynamics dominated by Polyakov loops

**Notes:**
Flavor independence

## Wilson Lines at T ≠ 0

*Always: *“pure” SU(N) gauge, *no* dynamical quarks (nf = 0)

- Imaginary time formalism:

-  Wilson line in fundamental representation:

- = *propagator *for “test quark”at x, moving up in (imaginary) time

- = propagator “test anti-quark” at x, moving back in time

- (Mandelstam’s constraint)

**Notes:**
Definition of Wilson lines

## Polyakov Loops

- Wrap *all* the way around in τ:

- Polyakov loop = *normalized *loop = gauge invariant

- Confinement: test quarks don’t propagate

- Deconfinement: test quarks propagate

- Spontaneous breaking of global Z(N) = center SU(N)

- ‘t Hooft ‘79, Svetitsky and Yaffe, ‘82

**Notes:**
Def. of Polyakov loops

## Adjoint Representation

- Adjoint rep. = “test meson”

- Note: both coefficients ~ 1

- Check:

- = dimension of the rep.

- Adjoint loop: divide by dim. of rep.

- At large N,

*= factorization*

**Notes:**
Adjoint loop & factorization

## Two-index representations

- 2-index rep.’s = “di-test quarks” = symmetric or anti-sym.

- Di-quarks: two qks wrap once in time, or one qk wraps twice

- Again: both coeff’s ~ 1.  Subscript = dimension of rep.’s = (N^2 ± N)/2

- For *arbitrary* rep. R, if d_R = dimension of R,

- For 2-index rep.’s ±, as N →∞,

- corr.’s 1/N, *not *1/N^2:

**Notes:**
Two-index loops

## Loops at Infinite N

- “Conjugate” rep.’s of Gross & Taylor ‘93:

- If all test qks and test anti-qks wrap once and *only *once in time,

*Many *other terms:

- dimension R =

- As N → ∞, *if *#, #’... are all of order 1, *first *term dominates, and:

- Normalization: if

**Notes:**
Loops at N=inf

## Factorization at Infinite N

- Makeenko & Migdal ‘80: at N=∞, *expectation values *factorize:

- In the deconfined phase, the fundamental loop condenses:

- Phase trivial:

- = Z(N) charge of R, defined modulo N

- Magnitude *not *trivial: highest powers of              win.

- At infinite N, *any* loop order parameter for deconfinement:
*even *if e_R = 0!  E.g.: adjoint loop (Damgaard, ‘87 ) vs adjoint screening.

- N.B.:

**Notes:**
Factorization at N=inf

## “Mass” renormalization for loops

- Loop = propagator for infinitely massive test quark.
- 	Still has mass renormalization, proportional to length of loop:

- To 1-loop order in 3+1 dimensions:

- a = lattice spacing.  m_R = 0 with dimensional regularization, but so what?

- Divergences order by order in g^2.  *Only *power law divergence for
- 	straight loops in 3+1 dim.’s. ; corrections ~aT.
- In 3+1 dim.’s, loops with cusps *do* have logarithmic divergence.
- 	(Dokshitzer: *~ classical *bremstrahlung)
- In 2+1 dim.’s, straight loops also have log. div.’s. (cusps do not)

**Notes:**
Mass ren. for loops

## Renormalization of Wilson Lines

- Gervais and Neveu ‘80; Polyakov ‘80; Dotsenko & Vergeles ‘80 ....

- For *irreducible *representations R, *renormalized *Wilson line:

- = renormalization constant for Wilson line

- For straight lines in 3+1 dimensions, only:
*No** *anomalous dim. for Wilson line: *no* condition to fix        at some scale

- As R’s irreducible, different rep’s don’t mix under renormalization

- For *all* local, composite operators, Z’s *independent *of  T

- Wilson line = *non*-local composite operator:

- temperature *de*pendent: from 1/T, and

- (found numerically)

*Not *really different: a m_R = func. *renormalized *g^2, and so T-dependent.

**Notes:**
Ren. of Wilson lines

## Renormalization of Polyakov Loops

- Renormalized loop:

- Constraint for bare loop:

- For renormalized loop:

- as a → 0, no constraint on ren.’d loop

*If*

- E.g.: as T →∞, ren’d loops approach 1 from *above*: (Gava & Jengo ‘81)

- Smooth large N limit:

- => *negative “*free energy”
- for loop

**Notes:**
Ren’.g Polyakov loops

## Why all representations?

- Previously: concentrated on loops in fund. and adj. rep.’s, esp. with cusps.
- At T ≠ 0, natural for loops, at a given point in space, to wrap around in
- τ many times.  Most general gauge invariant term:

- By group theory (the character expansion):

*Only *this expression, which is linear in the *bare *loops, can be consistently ren.’d

- Note: If all Z_R => 0 as a=> 0,

- Irrelevant: physics is in the ren.’d, not the bare, loops.  Discovered num.’y:

**Notes:**
Why all rep.’s?

## Lattice Regularization of Polyakov Loops

- Basic idea: compare two lattices.  *Same *temperature, *different *lattice spacing

- If a << 1/T, ren’d quantities the same.

- N_t = # time steps = 1/(aT) changes between the two lattices: get Z_R

- N_s = # spatial steps; keep N_t/N_s fixed to minimize finite volume effects

- Numerically,

- Each f_R is computed at fixed T.  As such, there is *nothing *to adjust.

*Explicit *exp. of divergences to ~g^4 at a≠0:   Curci, Menotti, & Paffuti, ‘85

- N.B.: also finite volume corrections from “zero” modes; *to be computed.*

**Notes:**
Lattice reg. of Polyakov loops

## Representations, N=3

- Label rep.’s by their dimension:

- fundamental =

- adjoint = 8

- symmetric 2-index = 6

- special to N=3: anti-symmetric 2-index =

- “test baryon” = 10:

- Measured 3, 6, 8, & 10 on lattice

**Notes:**
Three color rep’s

## Lattice Results

- Standard Wilson action, three colors, quenched.

- Lattice coupling constant

- = coupling for deconfining transition:

- Non-perturbative renormalization:

- To get the same T/T_d @ different N_t, must compute at *different *

- Calculate grid in β, interpolate to get the same T/T_d at different N_t

- N.B. : Method same with dynamical quarks

- Measured

- (No signal for 10 for N_t > 4)

**Notes:**
Lattice results: details

## Bare triplet loop vs T, at different Nt

- Note scale=>
- ~ .3

- Tc

- Nt=4

- Nt=6

- Nt=8

- Nt=10

- Nt = # time
-          steps.
*Bare loop *
*vanishes as*
*Nt *→∞

- T/Tc=>

- Triplet loop↑

## Bare sextet loop vs T, at different Nt

- Note scale=>
- ~ .04

- Sextet loop↑

- Tc

- T/Tc=>

- Nt=4

- Nt=6

- Nt=8

- Nt=10

- Nt = # time
-          steps.
*Bare loop *
*vanishes more quickly as Nt *→∞

## Bare octet  loop vs T, at different Nt

- Octet loop↑

- Tc

- T/Tc=>

- Nt=4

- Nt=6

- Nt=8

- Nt=10

- Note scale=>
- ~ .06

- Very similar to
- sextet loop

## Bare decuplet loop vs T, at different Nt

- Note scale=>
- ~ .006

- Decuplet loop↑

- Tc

- T/Tc=>

- Nt=4

- Nt=6

- Nt=8

- Nt=10

*No *stat.’y
- significant
- signal for
- decuplet loop
- above Nt=4.

## Slide 23

**Notes:**
Bare triplet loop vs Nt

## a m_R ↑

- a m_R ↑

- a m_R looks like usual “mass”: smooth function of ren.’d g^2 =>smooth func.
- of T: except near T_d!  One loop: m_R ~ C_R; OK for T ~ 3 T_d.  Fails
- for T < 1.5 T_d

- T/T_d =>

- T_d ↑

- Casimir scaling for:  a m_R ↓

**Notes:**
Mass divergence m_R vs R, T.

## Renormalized Polyakov Loops

- No signal of decuplet loop at N_t > 4; C_10 big, so bare loop small

- Ren’d loop ↑

- T/T_d =>

- T_d ↑

- Find: ren.’d
- triplet loop, but
- also significant
- octet and sextet
- loops, as well.

**Notes:**
Ren’d Polyakov loops

## Results for Ren’d Polyakov Loops

- Like large N: Greensite & Halpern ‘81, Damgaard ‘87...

- (Similar to measuring adjoint string tension in confined phase)

- Transition first order →ren.’d loops *jump *at T_d:

- T > T_d:  Find ordering: 3 > 8 > 6.  But compute difference loops:

**Notes:**
Summary of results for ren.’d Polyakov loops

## Difference Loops: Test of Factorization at N=3

- Details of spikes
- near T_d?
*Sharp *octet spike
- Broad sextet spike

- Max. sextet diff. loop =>

- Max. octet diff. loop =>

- T_d  ↑

- <= 8

- <= 6

**Notes:**
Difference loops

## Bare Loops don’t exhibit Factorization

- Bare octet difference loop/bare octet loop: violations of factor.
- 50% @
- Nt =4
- 200% @ Nt = 10.

**Notes:**
Bare loops don’t exhibit factorization

## Bielefeld’s Renormalized Polyakov Loop

- Need 2-pt fnc at one N_t
- vs 1-pt fnc at several N_t

- Approx.
- agreement.

- Kaczmarek, Karsch, Petreczky, & Zantow ‘02

**Notes:**
Bielefeld’s ren.’d loop

## Bielefeld’s Ren’d Polyakov Loop, N=2

- Digal, Fortunato, & Petreczky ‘02

- Transition second order:

**Notes:**
Bielefeld’s ren.’d loop, N=2

## Mean Field Theory for Fundamental Loop

- At large N, if fundamental loop condenses, factorization ⇒ *all *other loops

- This is a mean field type relation; implies mean field for

- General effective lagrangian for *renormalized *loops:

- Choose basic variables as Wilson lines, not Polykov loops:

- Loops automatically have correct Z(N) charge, and satisfy factorization.

- (i = lattice sites)

- Effective action Z(N) symmetric.  Potential terms (starts with adjoint loop):

- and next to nearest neighbor couplings:

- In mean field approximation, that’s it.  (By using character exp.)

**Notes:**
Mean field theory for fundamental loop

## Matrix Model (=Mean Field) for N=3

*Simplest *possible model: only

- (Damgaard, ‘87)

- Fit

- to get

- Find

- linear in T

- Now compute loops in other representations using this

**Notes:**
Graph of coupling vs T: linear!

## N=3: Lattice Results vs Simple Mean Field

*Approximate *agreement for 6 & 8.  Predicts signal for 10!

- <=10 ?

- Solid lines =
- matrix model.
- Points =
- lattice data for
- renormalized loops.

- 3=>

- 8=>

- <= 6

**Notes:**
Matrix model loops

## Difference Loops for Matrix Model, N=3

- Diff. loops: matrix model much broader & smaller than lattice data!  => new physics in lattice.

- <= Octet diff. loop

- Sextet diff. loop =>

- T_d ↑

- 4 T_d ↑

- Max. ~ - .03 matrix model  =>
-       vs	-0.20 lattice

- Curves =
- difference loops
- computed in
- matrix model
- Large N =>
- octet diff. loop <
- sextet diff. loop

- Max. ~ - .07 matrix model  =>
-       vs	-0.25 lattice

**Notes:**
Difference loops for matrix model.

## Matrix Model, N=∞, and Gross-Witten

- Consider mean field, where  the *only *coupling is

- Gross & Witten ‘80, Kogut, Snow, & Stone ‘82, Green & Karsch ‘84

- At N=∞, mean field potential is non-analytic, given by two *different *potentials:

- For fixed β, the potential is everywhere continuous, but its *third *derivative is not, at the point

- = confined phase

- = deconfined phase

**Notes:**
Matrix model, N=inf, Gross-Witten

## Gross-Witten Transition: “Critical” First Order

- Transition first order.  Order parameter jumps: 0 to 1/2.  Also, latent heat ≠ 0:

- But masses vanish, asymmetrically,  at the transition!

*If *β∼T, *and* the deconfining transition is Gross-Witten at N=∞, then
- the string tension and the Debye mass *vanish *at T_d as:

- But what about higher terms in the “potential”?

**Notes:**
Gross-Witten: Critical first order

## String related analysis @ large N

- hep-th/0310285: Aharony, Marsano, Minwalla, Papadodimas, Van Raamsdonk
- hep-th/0310286: Furuuchi, Schridber, & Semenoff

- By integrating over vev, one can show model with mean field same as
- model with *just *adjoint loop in potential.
- => Most general potential @ large N:

- Gross-Witten *simplest *model: c2 ≠ 0, c4=c6=...=0.
- AMMPR: consider c4 ≠ 0.  Work in small volume, => compute at small g^2.
- AMMPR:* either: *2nd order, or 1st order,
- Lattice for N=3: *close* to infinite N, with *small *c4, c6....?
- 	=> close to Gross-Witten?

## To do

- Two colors: matching critical region near T_d to mean field region about T_d?

- Higher rep.’s, factorization at N=2?

- Three colors: better measurements, esp. near T_d:

- For decuplet loop, use “improved” Wilson line?

- HTL’s

- “Spikes” in sextet and octet loops? Fit to matrix model?

- Four colors: is transition Gross-Witten?  Or is N=3 an accident?

*With dynamical quarks:  *method to determine ren.’d loop(s) *identical *

**Notes:**
Conclusions

## Bielefeld: Ren’d loop with quarks.  ≈ Same!

- Bielefeld: Ren’d loop *with *quarks.  ≈ Same!

- Kaczmarek et al: hep-lat/0312015

- Tc

- c/o quarks

- c quarks

- Ren’d
- triplet
- loop ↑

- T/Tc=>

## Slide 40