Ofer Aharony Strings 2004, Paris, France Aharony, Marsano, Minwalla, Papadodimas and Van Raamsdonk, hep-th/ to appear

Transcription

1 The Decofiemet ad Hagedor Phase Trasitios i Weakly Coupled Large N Gauge Theories Ofer Aharoy Strigs 2004, Paris, Frace Aharoy, Marsao, Miwalla, Papadodimas ad Va Raamsdok, hep-th/ to appear

2 Outlie Review of the decofiemet ad Hagedor phase trasitios ad their relatio i large N gauge theories. Motivatios for studyig weakly coupled theories. The partitio fuctio of free large N SU(N) Yag-Mills theories at fiite volume as a uitary matrix model. Solutio of the matrix model ad the free Yag-Mills phase diagram. Geeralizatio to weakly coupled theories ad possible extrapolatios to strog couplig.

3 Cofiemet ad decofiemet Cofiemet (o charged fiite-eergy states) a experimetal property of QCD, believed to happe i may asymptotically-free gauge theories. Observed o lattice but hard to uderstad theoretically despite may models. Related to strog couplig at low eergies, so expected to disappear at high temperatures (compared to the QCD scale ~170 MeV) where the couplig is weak, i a decofiemet phase trasitio. (Also high desities) I geeral, hard to compute theoretically eve the order of this phase trasitio (except pure SU(3)).

4 Order parameters for decofiemet : θ At fiite temperature, Wilso lie aroud periodic 1 P = tr( P exp( A)) time directio N (i Euclidea path itegral) vaishes i cofied phase but ot i decofied phase (free eergy of exteral quark). Charged uder Z N global symmetry (for SU(N)) of large gauge trasformatios. θ Aother order parameter appears i the large N limit : i cofied phase the free eergy F( T ) ~ O(1) (glueballs, mesos) while i decofied phase 2 F( T ) ~ O( N ) (quark-gluo plasma). So, we ca use as a order parameter lim N F SU ( N )( T ) / which also vaishes i the cofied phase. N 2

5 Ca we possibly study this trasitio perturbatively? Not i ifiite space, but maybe at fiite small volume 1/ R >> Λ QCD with weak couplig at IR 3 cutoff (if o zero modes, e.g. S ), if trasitio persists to small volume. At fiite volume phase trasitios are geerally smoothed out (ad correspodigly thermal Wilso lie always vaishes by Gauss law). However, ca still have phase trasitios at large N. These ca be characterized by our secod order 2 parameter, or by < P >, eve though they are smoothed out for ay fiite N. Does the large N decofiemet trasitio persist to small R? This ca be aswered perturbatively, ad ca compute the order of the trasitio!

6 Decofiemet = Hagedor? I the t Hooft large N limit with fixed 2 λ = g YM N, gauge theories are believed to be equivalet to strig theories with g s ~ 1/ N. The cofiig phase is aturally described by strigs (Regge trajectories, etc.), while the decofied phase seems o-strigy. This is qualitatively similar to the Hagedor trasitio which could also occur i weakly coupled strig theories (Atick+Witte), at least i curved space; also there the order parameter wids aroud the temporal circle. Could these trasitios be related? There are ot may kow examples to study this issue

7 The best uderstood example is i the AdS/CFT correspodece betwee type 5 IIB strig theory o AdS 5 S ad the N=4 supersymmetric SU(N) Yag-Mills theory 3 o S. I this case we ca aalyze the strogly coupled gauge theory compactified o a 3-sphere of radius R usig strig theory at small curvature : θ I the microcaoical esemble, at large eergies (but small compared to 1/ ) there is Hagedor behavior with θ I the caoical esemble there is (Witte) a decofiemet-like phase trasitio (accordig to the order parameters) which is the Hawkig-Page trasitio, happeig at T = HP 2πR = So, o apparet relatio at strog couplig How about weak couplig? Naively o hope of strigiess T H g s. R λ 1/ 4. R

8 The partitio fuctio of free Yag-Mills theory (also derived by Sudborg, hep-th/ ) Free Yag-Mills theory at fiite volume is o-trivial, because we still have the Gauss law costrait the total charge must vaish, all states are siglets of the gauge group. Could this cause cofiemet? To compute the partitio fuctio we eed to cout the umber of iequivalet ways to put particles i some represetatio R, i such a way as to get a siglet of eergy E. This is a combiatorical problem whose oly iputs are the easily computable : z z B F ( T ) ( T ) = = exp( E bosoic oe particle states exp( E fermioic oe particle states / T ) / T )

9 We have two ways to compute the exact partitio fuctio : We ca sum over all states i the Fock space of our particles, ad the impose a projectio oto siglets. We ca explicitly compute the Euclidea path itegral of the free gauge theory (at oe-loop) with periodic time. Both computatios lead to the same result, which is a group itegral : Z( T ) = [ du ]exp{ R = 1 1 [ z R B T ( ) ( 1) T ( )] tr ( U I the path itegral, U is the holoomy of the gauge field alog the periodic time directio (averaged over space), which is the oly zero mode (all other fields appear quadratically ad ca be itegrated out). So, tr(u) is the aïve order parameter for decofiemet. z R F R )}

10 Solutio of the uitary matrix model As usual, to solve the matrix model we chage variables to the eigevalues of iα U, { e ad to the i }; i = 1, L, N; π αi π, eigevalue distributio ρ(α) which becomes cotiuous i the large N limit. For all fields i adjoit represetatio : Z( T ) = [ du ]exp{ = 1 [ T T ( ) ( 1) zf ( tr( U ) tr(( U B 1 + = [ dα ]exp{ i V ( αi α j )} V ( θ ) = l(si( θ / 2)) = 1 )] with a repulsive force comig from the measure, ad a attractive force growig with the temperature. z i< j 1 [ z B T ( ) ( 1) z F ) )} = T ( )]cos( θ )

11 This matrix model ca be solved exactly, by similar methods to those used to study the Gross-Witte phase trasitio of two dimesioal lattice gauge theories. The aalysis of the low-temperature phase is simplest i the (costraied) variables ρ ρ( α)exp( iα) dα = tr( U ) / N, i which the effective actio becomes S 2 = N ρ = [1 z T ( ) + ( 1) T ( )]. At low temperatures all coefficiets are positive, the large N saddle poit is a costat distributio ρ = 0, ad the quadratic itegratio gives : Z( T ) = T = 1 T 1 zb ( ) + ( 1) zf ( ) 1 B z F

12 This breaks dow at the temperature where zb ( TH ) + zf ( TH ) = 1 at which becomes tachyoic. Near this temperature we have (at large N) a Hagedor-like behavior Z T ) ~ 1/( T correspodig to a desity of states ρ( E) ~ exp( E / TH So, free large N Yag-Mills theories have a Hagedor spectrum! (ca also be see directly by coutig states) I the free N=4 SYM theory o (for example) we fid ). ( T ), T H = 1/ R l(7 4 3) = / R. H Above the Hagedor temperature, ρ 1 (which is precisely the temporal Wilso lie) becomes tachyoic ad codeses, ad at higher temperatures the higher modes of the distributio codese as well : S 3 ρ 1

13 We fid a (weakly) first-order phase trasitio at the Hagedor temperature, which is a decofiemet trasitio accordig to our two order parameters! For a give field cotet we ca compute all thermodyamical quatities (though the high-temperature expressios are complicated). For example, the free eergy :

14 Weak couplig behavior At weak couplig our matrix model is still valid, but the actio gets perturbative correctios. At k-loop order these take k k+ 1 the form λ tr( U ) tr( U ) L tr( U ) with computable coefficiets. The behavior ear the phase trasitio is domiated by the light mode ρ ~ tr( U ) / ad the effective Lagragia for this mode ear the trasitio takes the form Seff = N [ a( TH T ) ρ1 + bλ ρ1 1 N where a ad b ca be computed from two-loop ad three-loop diagrams i the theory. As usual, the order of the phase trasitio depeds o the sig of b : 4 ],,

15 b<0 : first order trasitio below T H b>0 : two cotiuous trasitios, first at T H

16 We must sum all relevat 3-loop vacuum diagrams i order to kow the order of the decofiig phase trasitio, ad whether it occurs at or below the Hagedor temperature. 3 I pure Yag-Mills theory o S, i a coveiet choice of gauge (ad after itegratig out some fields), the followig diagrams cotribute : So far we have summed 14 of the 15 3-loop diagrams; oe couter-term remais

17 Summary Weakly coupled large N gauge theories o compact spaces : * Exhibit a Hagedor spectrum (for fiite eergies i the large N limit), * Have a decofiemet phase trasitio at at a temperature iversely related to the size of the space. The decofiemet trasitio is either : * A first order trasitio below the Hagedor temperature, or * A secod order trasitio at the Hagedor temperature, followed by aother third order phase trasitio. The properties of the trasitio (ad of the strigy spectrum) ca be computed i perturbatio theory, through a uitary matrix model.

18 The details of cofiemet here are very differet from strog couplig (i pure YM ad i N=4 SYM), but the behavior of the order parameters is the same, so they could be cotiuously coected (by icreasig the radius / couplig) :

19 Future directios : Compute the value of b (= the order of the trasitio) i several iterestig cases (pure YM, N=4 SYM). Both sigs are possible! Whe addig a large umber of fudametals the trasitio is always smooth due to a potetial for sigle eigevalues (Schitzer). Try to uderstad the free strig theory correspodig to the free large N (supersymmetric) Yag-Mills theory. We kow the exact spectrum of siglestrig ad multi-strig states! Zero size AdS? Not tesioless strigs Uderstad better the itermediate phase which sometimes appears. Is this related to small black holes? Study itegrability properties of the large N limit. (Ifo about aom. dim.)

20 Geeralize to cases with zero modes (o the compact space). Curretly workig o torus (with Wisema). Here there is a more itricate phase structure ivolvig also spatial holoomies. We fid similar phase trasitios for the spatial holoomies, which at strog couplig become (via the AdS/CFT correspodece) the Gregory-Laflamme trasitio! Ca we extrapolate from weak couplig to strog couplig, ad get a good model for realistic decofiemet? Are the regimes cotiuously related?