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Exact renormalization group at finite temperature

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<strong>Exact</strong> renormalis<strong>at</strong>ion <strong>group</strong><br />

<strong>at</strong> <strong>finite</strong> temper<strong>at</strong>ure<br />

“Quarks, Hadrons, and the Phase Diagram of QCD”<br />

St Goar<br />

2 Sept. 2009<br />

Jean-Paul Blaizot, IPhT- Saclay


Outline<br />

- weak coupling techniques and why they are useful to<br />

understand hot QCD<br />

- insights from the exact Renormaliz<strong>at</strong>ion Group


Hot QCD


Thermodynamics of hot QCD<br />

(from M. Bazavov et al, arXiv:0903.4379)


Pressure for SU(3) YM theory <strong>at</strong> (very) high temper<strong>at</strong>ure<br />

(from G. Endrodi et al, arXiv: 0710.4197)


At T>3Tc Resummed Pert. Theory<br />

accounts for l<strong>at</strong>tice results<br />

Pressure<br />

Resummed Pert. Th.<br />

(SU(3) l<strong>at</strong>tice gauge calcul<strong>at</strong>ion from Karsch et al, hep-l<strong>at</strong>/0106019)<br />

(resummed pert. th. from J.-P. B., E. Iancu, A. Rebhan: Nucl.Phys.A698:404-407,2002)<br />

SB


Weak coupling techniques<br />

(non perturb<strong>at</strong>ive)<br />

-not limited to perturb<strong>at</strong>ion theory; in fact weak coupling<br />

techniques can be used to study non perturb<strong>at</strong>ive phenomena<br />

(many degener<strong>at</strong>e degrees of freedom, strong fields)<br />

Effective theories- Dimensional reduction<br />

Skeleton expansion (2PI formalism)


Perturb<strong>at</strong>ion theory is ill behaved<br />

A two naive conclusion:<br />

« weak coupling techniques are useless »


Similar difficulty in scalar theory<br />

The bad convergence of Pert. Th. is not rel<strong>at</strong>ed to non abelian fe<strong>at</strong>ures of QCD


Effective theroy<br />

Dimensional reduction<br />

Integr<strong>at</strong>ion over the hard modes<br />

In leading order<br />

D i = ∂ i − ig E A i<br />

g E ≈ g T<br />

Integr<strong>at</strong>ion over the soft modes<br />

(gT ≤ Λ E ≤ T)<br />

m E ≈ gT<br />

(g 2 T ≤ Λ M ≤ gT)<br />

Non perturb<strong>at</strong>ive contribution<br />

λ E ≈ g 4 T


Skeleton expansion<br />

Phi-derivable approxim<strong>at</strong>ions<br />

2PI formalism<br />

.......<br />

• J. M. Luttinger and J. C. Ward,<br />

PR 118, 1417 (1960)<br />

• G. Baym, PR127,1391(1962)<br />

• J. M. Cornwall, R. Jackiw, and<br />

E. Tomboulis, PR D10, 2428<br />

(1974)


Pressure in terms of dressed propag<strong>at</strong>ors<br />

St<strong>at</strong>ionarity property<br />

Entropy is simple!<br />

δP<br />

δG<br />

S = dP<br />

dT<br />

= 0<br />

= dP<br />

dT G<br />

P[G]


THE TWO-LOOP ENTROPY<br />

-effectively a one-loop expression; residual interactions start <strong>at</strong> order three-loop<br />

-manifestly ultraviolet <strong>finite</strong><br />

-perturb<strong>at</strong>ively correct up to order<br />

g 3


St<strong>at</strong>e of the art<br />

pure-glue SU(3) Yang-Mills theory<br />

• J.-P. B., E. Iancu, A. Rebhan: Phys.Rev.D63:065003,2001<br />

• G. Boyd et al., Nucl. Phys. B469, 419 (1996).<br />

from J.-P. B., E. Iancu, A. Rebhan:<br />

Nucl.Phys.A698:404-407,2002


Large N f


Pressure <strong>at</strong> Large N f<br />

Moore, JHEP 0210<br />

A.Ipp, Moore, Rebhan, JHEP 0301


Entropy in large N f<br />

Good agreement, even <strong>at</strong> large coupling<br />

J.-P. B., A.Ipp, A. Rebhan, U. Reinosa,<br />

hep-ph/0509052


Strong coupling<br />

Comparison with SSYM<br />

Hard-thermal-loop entropy of supersymmetric Yang-Mills theories<br />

J.-P. B, E. Iancu, U. Kraemmer and A. Rebhan, hep-ph/0611393


Leading order correction<br />

S<br />

S 0<br />

= 3<br />

4<br />

+ 45<br />

32<br />

Entropy formula<br />

At strong coupling (AdS/CFT)<br />

1<br />

ς(3) 3 / 2<br />

λ<br />

λ ≡ g 2 ( Nc )


Insights from the functional<br />

<strong>renormaliz<strong>at</strong>ion</strong> <strong>group</strong><br />

(scalar theory)<br />

J.-P. B, A. Ipp, R. Mendez-Galain, N. Wschebor<br />

(Nucl.Phys.A784:376-406,2007)<br />

and work in progress


Scales, fluctu<strong>at</strong>ions and degrees of<br />

freedom in the quark-gluon plasma


Non perturb<strong>at</strong>ive <strong>renormaliz<strong>at</strong>ion</strong> <strong>group</strong><br />

Courtesy, A. Ipp


Non perturb<strong>at</strong>ive <strong>renormaliz<strong>at</strong>ion</strong> <strong>group</strong><br />

(C. Wetterich 1993)


Local potential approxim<strong>at</strong>ion


Beyond the local potential approxim<strong>at</strong>ion<br />

(J.-P. B, R. Mendez-Galain, N. Wschebor, Phys.Lett.B632:571-578,2006)


Applic<strong>at</strong>ion to critical O(N) model<br />

(from F. Benitez, J.-P. Blaizot, H. Ch<strong>at</strong>e, B. Delamotte, R. Mendez-Galain, N. Wschebor,<br />

arXiv:0901.0128 [cond-m<strong>at</strong>.st<strong>at</strong>-mech])


Flow of coupling constant<br />

(LPA)<br />

weak coupling<br />

strong coupling


Flow of pressure<br />

(LPA)<br />

weak coupling strong coupling


Pressure versus m/T<br />

g^3<br />

g^3<br />

g^4<br />

g^4<br />

g^2<br />

g^2<br />

BMW (Exp.)<br />

LPA (Litim)<br />

LPA (Exp.)<br />

2PI<br />

LPA (Litim)<br />

2PI


1.02<br />

1.00<br />

0.98<br />

0.96<br />

0.94<br />

g^4<br />

g^6<br />

g^5<br />

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4<br />

g^3<br />

(g^6 from A. Gynther, et al ,hep-ph/0703307)


Summary<br />

•While strict perturb<strong>at</strong>ion theory is meaningless, weak coupling methods<br />

(involving resumm<strong>at</strong>ions) provide an accur<strong>at</strong>e description of the QGP for<br />

• The functional <strong>renormaliz<strong>at</strong>ion</strong> <strong>group</strong> confirms results obtained with various<br />

resumm<strong>at</strong>ions (<strong>at</strong> least in scalar case)

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