Monopole condensation in two-flavour Adjoint QCD
aa r X i v : . [ h e p - l a t ] M a r IFUP-TH/2008-03, IFUM-912-FT
Monopole condensation in two-flavour Adjoint QCD
Guido Cossu, Massimo D’Elia, Adriano Di Giacomo, Giuseppe Lacagnina, and Claudio Pica Scuola Normale Superiore and INFN, Pisa, Italy Dipartimento di Fisica and INFN, Genova, Italy Dipartimento di Fisica and INFN, Pisa, Italy INFN sezione di Milano, Italy Brookhaven National Laboratory, Upton, NY 11973-5000, USA
In QCD with adjoint fermions (aQCD) the deconfining transition takes place at a lower temper-ature than the chiral transition. We study the two transitions by use of the Polyakov Loop, themonopole order parameter and the chiral condensate. The deconfining transition is first order, thechiral is a crossover. The order parameters for confinement are not affected by the chiral transition.We conclude that the degrees of freedom relevant to confinement are different from those describingchiral symmetry.
PACS numbers: 11.15.Ha, 12.38.Aw, 14.80.Hv, 64.60.Cn
I. INTRODUCTION
Deconfinement and chiral symmetry restoration aretwo important features of QCD. Despite the fact thatthese two phenomena are in principle independent of eachother, finite temperature lattice simulations indicate thatthey occur at the same temperature within errors [1],making it hard to disentangle them. In particular, it isnot clear yet what is the interplay between the degreesof freedom relevant for the two transitions.Several proposals exist in the literature for the confine-ment dynamics, most of which are based on the presenceof topological excitations in the theory. A possible mech-anism for confinement is dual superconductivity of theQCD vacuum, which manifests itself as condensation ofmagnetic charges [2]. In order to investigate this propertyof the vacuum, one constructs an operator which carriesmagnetic charge and determines its vacuum expectationvalue, which is expected to be different from zero in theconfined phase and strictly zero in the deconfined sym-metric phase [3, 4, 5, 6, 7]. Contrary to what happens forthe Polyakov Loop, the symmetry described by this or-der parameter is not spoiled by the presence of dynamicalquarks. It can therefore be used as a good order param-eter for confinement also in full QCD. It has been shownindeed [6, 7] that in QCD with fermions in the funda-mental representation dual superconductivity disappearsat the same temperature where the chiral-deconfinementphase transition takes place.While there is much evidence that monopole conden-sation is strictly related to the dynamics of colour con-finement, a still unsolved issue concerns the relation be-tween dual superconductivity and the dynamics of chiralsymmetry breaking. As already pointed out, the coin-cidence of deconfinement and chiral restoration makesthis problem difficult in ordinary QCD. A system wheredeconfining and chiral transitions are distinct could pro-vide the framework to investigate this issue. One suchsystem is QCD with quarks in the adjoint representationof SU (3) (aQCD), in which the two transitions seemingly take place at different temperatures [8]. Furthermore, thecoupling to the adjoint quarks does not explicitly breakthe Z (3) center symmetry of the action, and therefore thetwo transitions can be characterized by two order param-eters, namely the Polyakov loop and the chiral conden-sate.The authors of ref. [8] performed lattice simulations ofaQCD with two flavours of staggered quarks, and foundtwo distinct phase transitions, with β dec < β chiral . Theyobserved a strong first order deconfinement transitionand a continuous chiral transition. They also checkedthat the Polyakov Loop, which is sensitive to deconfine-ment, is not significantly affected by the chiral transition.The nature of the chiral transition in N f = 2 aQCD hasbeen further investigated in [9], where the authors madean extensive analysis with the aim of determining theorder of the chiral transition. They found that the be-haviour of the magnetic equation of state was consistentwith the presence of a second order chiral transition inthe zero quark mass limit (see also [19]). For the purposeof this work, however, it is sufficient to know that a chiraltransition exists and is separated from the deconfinementone.The structure of the paper is the following. In SectionII we briefly review the basics of aQCD and of the or-der parameter for monopole condensation. In Section IIIwe report the results of our study of the deconfinementtransition both by use of the magnetic order parameterand of the Polyakov loop. We make a summary and drawconclusions in Section IV. II. ADJOINT QCD AND MAGNETIC ORDERPARAMETERA. aQCD
Quarks in the adjoint representation of SU (3) have 8color degrees of freedom and can be described by 3 × Q ( x ) = Q a ( x ) λ a (1)where λ a are the Gell-Mann’s matrices. In order towrite the fermionic part of the action for this model,the 8 − dimensional U (8) representation of the gauge links(which is real) must be used: U ab (8) = 12 Tr (cid:16) λ a U (3) λ b U † (3) (cid:17) (2)The full action is therefore given by: S = βS G [ U (3) ] + X x,y ¯ Q ( x ) a M ab (cid:0) U (8) (cid:1) x,y Q ( y ) b (3)where S G is the usual SU (3) gauge action with links inthe 3-dimensional representation and M is the staggeredfermions matrix. The Polyakov loop is defined as in thepure gauge case: L (3) ≡ L s h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ~x Tr L t Y x =1 ( U (3) ) ( x , ~x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i (4)where L s and L t are the spatial and temporal sizes of thelattice respectively and the trace is over color indexes.This quantity is an order parameter for the spontaneousbreaking of the center symmetry which is not broken byadjoint fermions (see Eq. 2). A well known result [10]relates L (3) to the free energy of an isolated static quarkin the fundamental representation L (3) ∝ e − F/T (5)in a gluonic bath at temperature T . In the center sym-metric phase, where L (3) = 0, the free energy is infinite,thus realizing confinement; L (3) can therefore be used asan order parameter for the deconfinement transition. B. Monopole condensation
The vacuum expectation value of a magneticallycharged operator, h µ i , was proposed in references [3, 4,5, 6] as an order parameter for the deconfinement tran-sition. The operator detects condensation of magneticcharges, i.e. Higgs breaking of the magnetic charge sym-metry. The procedure involves a gauge-fixing, the so-called Abelian Projection [11]. However, the particularchoice of the gauge is inessential as shown by numericalsimulations [5] and by theoretical arguments [12]. Theexplicit form of the vev of the magnetic charged operatoris given by h µ i = 1 Z Z [ dU ] e − e S = e ZZ (6) e S is obtained from the original action by the insertion ofa monopole field in the temporal plaquettes of a given time slice [3]. The measurement of a ratio of partitionfunctions is a difficult numerical task and so, to bettercope with fluctuations, one calculates the quantity ρ = ∂∂β ln h µ i = h S G i S − h f S G i e S (7)where S G is the ordinary gauge part of the action.Clearly, two simulations have to be run for each value of β , with and without the monopole insertion. The dropof the order parameter at the transition corresponds toa peak of − ρ . In the vicinity of the critical temperaturea scaling ansatz for the order parameter h µ i ≃ L − β µ /νs f ( L /νs ( β c − β )) (8)( β µ is the critical exponent associated to the order pa-rameter) implies ρ ≃ L /νs f ρ ( L /νs ( β c − β )) (9)where ν is the critical index of the correlation lenght and f, f ρ are universal scaling functions. In the case of a weakfirst order transition the critical exponent ν is equal to1 /
3, i.e. the − ρ peak is expected to scale with the spatialvolume: ρ/L s ≃ f ρ ( L s ( β c − β )) . (10) III. SIMULATIONS AND RESULTS
We simulated two flavours of adjoint staggeredfermions using the exact RHMC algorithm [13] for thesimulations with the monopole insertion and the Φ al-gorithm [14] for the other simulations. Trajectories hada length of N MD δt = 0 .
5, and typical integration steps δt = 0 . , .
005 depending on the mass (see below). Ac-ceptance rate was above 80% on average. Inversions ofthe fermionic matrix were performed using the ConjugateGradient Algorithm. We have run simulations mostlywith two different lattice sizes, L s × L t = 12 × , × am q = 0 . , .
04. We haveevaluated the average plaquette, the ρ parameter, thePolyakov loop, and the chiral condensate for several val-ues of β in the range (3 . , .
0) (smallest β s are not shownin graphs). In order to simulate the action ˜ S , C ∗ bound-ary conditions have been implemented [15]. Our codehas been run on the APEmille machine in Pisa and theapeNEXT facility in Rome. A. Results
The thermodynamical properties of the Polyakov loopwere the easiest to study. We used this observable as areference to investigate the confinement-deconfinementphase transition by means of the ρ parameter. ThePolyakov loop shows the typical behaviour of a sharpfirst order transition (see Fig. 1). The pseudocriticalvalue of β for the smallest mass was estimated, by in-spection of the data, to be at β = 5 .
25 (independently ofthe volume) where L (3) shows a clear discontinuity. Thisresult is in agreement with [8], where a smaller volume8 × β as an estimate for β dec inour finite size scaling analysis.For the magnetic order parameter, we find the ex-pected peak at values of β which coincide within errorswith those at which the Polyakov loop has its discontinu-ity. The ρ parameter is expected to approach zero inde-pendently of the volume as β →
0; it should also divergewith L s in the deconfined region [16]. We found a goodqualitative agreement of our data with the expectationsin both limits (see Figs. 2, 3, 6). Around the transitionfinite size scaling analysis shows that ρ has the scalingproperties of a first order transition for both values ofthe quark mass (Figs. 3, 4). In particular the height ofthe peak is proportional to L s within errors as expectedfrom eq. 10.We also looked for possible effects of the chiral transi-ton on the magnetic order parameter. The first step toaddress this issue was to locate the transition by meansof the natural order parameter, the chiral condensate h ¯ ψψ i . This parameter and its susceptibility (Fig. 5) werefound both consistent with a chiral transition in the re-gion around β chiral = 5 .
8. An unambiguous peak in thesusceptibility of the chiral condensate is visible only forthe lightest mass, am = 0 .
01. Comparison of resultsat different volumes shows that the chiral transition, atthis value of the fermionic mass, is compatible with acrossover. These results are in agreement with those al-ready contained in [8, 9]. The ρ parameter does not showany significant change at β ≃ β chiral (Fig. 2), the samehappens for the Polyakov Loop. Furthermore, the anal-ysis of [9] shows that a bare quark mass of am = 0 .
01 isclose enough to the scaling region of the chiral transition.It is therefore safe to conclude that the ρ parameter isnot affected by the chiral transition, i.e. that differentd.o.f. dominate at the two transitions. IV. SUMMARY AND CONCLUSIONS
We have studied deconfinement and the chiral transi-tion in lattice QCD with two flavors in the adjoint rep-resentation.Deconfinement is detected as a sharp jump at the crit-ical temperature β c of the Polyakov Loop (Fig. 1). It isalso seen as a sharp peak of the susceptibility ρ related tomagnetic charge condensation. The location of the jumpand of the peak concide within errors. Both parametersobey the scaling of a first order phase transition.The chiral order parameter h ¯ ψψ i has a drop at the de-confining transition, corresponding to a peak in its sus-ceptibility (Fig. 5) but does not vanish above it. h ¯ ψψ i drops to zero at an higher temperature where its suscepti- β L ( ) FIG. 1: The Polyakov loop, with am = 0 .
01 and 16 × β -1000-800-600-400-2000 ρ L s =12L s =16 FIG. 2: The ρ parameter, with am = 0 . L t = 4, for twodifferent spatial volumes. -300 -200 -100 0 100 ( β−β c )L s3 -0.25-0.2-0.15-0.1-0.05 ρ / L s L s =12L s =16 FIG. 3: Scaling of the ρ parameter, am = 0 . L t = 4. β c = 5 .
25, estimated from the Polyakov loop at L s = 16. -600 -400 -200 0 ( β−β c )L -4-3-2-1 ρ /L FIG. 4: Scaling of the ρ parameter, am = 0 . L t = 4. β c = 5 .
25, estimated from the Polyakov loop at L s = 16. β c h i r a l c ond . s u sc . L s = 16L s = 8 c h i r a l c ond . FIG. 5: Susceptibility of the chiral condensate, with am =0 .
01 and two different lattices: 8 ×
4, 16 ×
4. The data arecompatible with a crossover at the chiral transition. In theinset, the chiral condensate within the same range of β values.A clear jump is visible at the deconfinement phase transition. bility χ has a broad peak (chiral transition). The scalingof χ is compatible with a crossover. Neither the Polyakovline nor the magnetic order parameter show any changeat the chiral transition. From these observations we can conclude that:1. the magnetic order parameter detects deconfine-ment on the same footing as the Polyakov Loop.This again corroborates the mechanism of confine-ment by dual superconductivity of the vacuum, asin the case of pure gauge [2, 3].2. The degrees of freedom relevant to deconfinementare different from those relevant to chiral transition.The detectors of deconfinement ( L (3) and ρ ) areinsensitive to the chiral transition, and h ¯ ψψ i is nonzero above the deconfinement transition. L s -200-150-100-500 ρ ( β = ) FIG. 6: Behavior of the ρ observable varying the linear di-mension of the lattice at fixed β = 8. The last conclusion above can be relevant to the studyof ordinary QCD with N f = 2 in the fundamental repre-sentation [17]. There the two transitions for some reasonoccour at the same β c and the interplay of the two differ-ent kinds of degrees of freedom could be at the origin ofthe difficulties in determining the order of the transition.The analysis based only on chiral degrees of freedom [18]might prove to be inadequate.The work of C.P. has been supported by contract No.DE-AC02-98CH10886 with the U.S. Department of En-ergy. We wish to thank the apeNEXT staff in Rome forthe support. [1] F. Karsch, PoS(LATTICE 2007)015, arXiv:0711.0656;arXiv:0711.0661 [hep-lat] and references therein.[2] G. ’t Hooft, Nucl. Phys. B (1981), 455[3] A. Di Giacomo, B. Lucini, L. Montesi, G. Paffuti, Phys.Rev. D , 034503 (2000), hep-lat/9906024[4] A. Di Giacomo, B. Lucini, L. Montesi, G. Paffuti, Phys.Rev. D , 034504 (2000), hep-lat/9906025[5] J. M. Carmona, M. D’Elia, A. Di Giacomo, B. Lu-cini, G. Paffuti, Phys. Rev. D64, 114507 (2001), hep-lat/0103005[6] M. D’Elia, A. Di Giacomo, B. Lucini, G. Paffuti, C. Pica,Phys. Rev. D , 114502 (2005), hep-lat/0503035[7] J. M. Carmona, M. D’Elia, L. Del Debbio, A. Di Gia-como, B. Lucini and G. Paffuti, Phys. Rev. D , 011503(2002), hep-lat/0205025[8] F. Karsch, M. Lutgemeier, Nucl. Phys. B , 449(1999), hep-lat/9812023[9] J. Engels, S. Holtmann, T. Schulze, Nucl. Phys. B ,
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