Some basic features of confinement are reviewed, in particular the symmetry
patterns of the dual dynamics. Open problems and possible directions of
progress are discussed.Comment: 6 pages, 1 figure. Invited talk at PANIC 05 Santa Fe NM October 24-28
  200

Di Giacomo, Adriano

English

arXiv

Adriano Di Giacomo

Crossref

Confinement of color: open problems and perspectives

arXiv:hep-lat/0602011v1  6 Feb 2006Confinement of color: open problems andperspectives.Adriano Di GiacomoDipartimento di Fisica Univ. PISA and INFN , 3 Largo B. Pontecorvo,56127 Pisa (ITALY)Abstract. Some basic features of confinement are reviewed, in particular the symmetry patterns ofthe dual dynamics. Open problems and possible directions ofprogress are discussed.Keywords: Confinement of Color, Deconfinement phase transition, Lattice QCDPACS: 11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.Mh, 64.60.FrEXPERIMENT.The experimental search for quarks was started in the 60’s , immediately after the conjec-ture of their existence as fundamental building blocks of hadrons[1]. The experimentalsignature is clear : they carry fractional chargesq =±1/3,±2/3.No quark has ever been observed, so that experiments only provide upper limits,which are summarized in Particle Data Group reports [2].For the ratio of the abundance of quarksnq to the abundance of protonsp in naturethe limit is{nqnp}observed ≤ 10−27 (1)to be compared to the expectation{ nqnp}expected ≃ 10−12 in the Standard CosmologicalModel [3].The reduction factor is≃ 10−15 !Similarly for the inclusive cross -sectionσp to produce a quark or an antiquark inhadron collisions one has [2]{σq}observed ≡ {σ(p+ p → q(q̄)+X)}observed ≤ 10−40cm2 (2)to be compared to the expected value in perturbation theory which is of the order of thetotal cross sectionσT ≃ 10−25cm2 . Again a reduction factor≃ 10−15 !The "natural " explanation is thatnq andσq are strictly zero , or that confinement isan absolute property of vacuum based on symmetry. This is similar to what happensin ordinary superconductivity: very small experimental limits exist for the resistivity ,which in fact proves to be strictly zero due to the Higgs-breaking of theU(1) symmetrycorresponding to the conservation of the electric charge.The above statement has far reaching consequences: the deconfining transition cannot be a crossover ifσq andnq are strictly zero, since by analyticity they would be zeroeverywhere in the phase diagram.An "unnatural" alternative also exists compatible with a crossover:nq andσq verysmall but not identically zero in the confined phase and thereis no order parameter. Ofcourse the choice is not a matter of opinions , but has been made by Nature. What wecan do is to investigate the question e.g. by numerical simulations on the lattice.THE DECONFINING PHASE TRANSITIONThe possible existence of a deconfining phase transition at atemperatureTc of the orderof the Hagedorn temperature was first discussed in Ref.[4] . Such a transition has beenobserved in numerical simulations of QCD on the Lattice.The partition function ofQCD at temperatureT can be written as a Feynman integralon a space- time compactified in time from 0 to1T with periodic boundary conditionsfor the gluon fields, antiperiodic for the quark fields. IfLs is the spatial extension of thelattice ,Lt the extension in time , withLs ≫ Lt ,T =1a(β ,m)Lt(3)wherea(β ,m) is the lattice spacing in physical units, which depends on the couplingconstantg2 (β = 2Ng2) and on the quark massesm , which are the inputs of the simulations.The real problem, both on the lattice and in experiment, is togive an operationaldefinition of confined and deconfined.In the quenched theory (no dynamical quarks) the Polyakov criterion is usuallyadopted. The static potential between aqq̄ pair , V (r) is related to the correlator V(r)of two Polyakov lines by the equationV (r) =−T lnD(r) (4)By cluster propertyD(r) =< L†(~r)L(0)>≈r→∞ |< L > |2+ k.exp(−σrT) (5)L(~r) is the Polyakov line , or the parallel transport along the time axis across the lattice.If < L >= 0 V (r)≈r→∞ σr (Confinement).If < L >6= 0 V (r)≈r→∞ constant (Deconfinement).< L > is an order parameter, the symmetry beingZ3. A phase transition is found onthe lattice at a temperatureTc ≃ 270Mev from a phaseT ≤ Tc where< L >= 0 to aphase atT > 0 where< L >6= 0 . Finite size scaling analysis shows that the transition isweak first order.In the presence of dynamical quarks the symmetryZ3 is explicitly broken and thestring breaks at large enough distances converting the energy into the formation of quarkpairs. No clear criterion exists to define confined and deconfined.Fig.1 shows the phase diagram forN f = 2 andu andd quarks with the same massm. At m = 0 the chiral transition occurs, and< ψ̄ψ > is the order parameter.Asm → ∞quarks decouple, the theory becomes quenched and< L > is an order parameter. Atgeneric values ofm neither< L > nor< ψ̄ψ > are order parameters. The transition linecorresponds to sharp variations of these quantities, or to peaks in their susceptibilitieswhose positions coincide within errors.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           270 MeVCONFINED0 8mDECONFINEDT1ST ORDERµ170 MeV?        FIGURE 1. Picture to fixed heightThe region below the line is conventionally called confined,the region above itdeconfined. A renormalization group analysis [5] shows thatt e chiral transition is firstorder forN f ≥ 3 and can be second order or first order forN f = 2 , depending on thefact thatUA(1) symmetry is restored below or above the chiral transition.If the transition is second order the line atm 6= 0 is a crossover , implying that thedeconfining transition is not order-disorder (unnatural scenario) ; if the chiral transitionis instead first order also the line atm 6= 0 is first order and the transition can be order-disorder. Up to now lattice results were not conclusive on this point. New data [6]indicate that the transition is more compatible with first order . This is a fundamentalquestion for our understanding of confinement and deserves further study.DUALITYIn lattice QCD the dependence of the temperature [eq.(3)] on the coupling co stantis dictated by renormalization group .a(β ,m) ∝ 1ΛL exp(β2bo) with bo = − 14π2 (113 N −23N f )< 0 , which impliesT ∝ exp(N|bo|1g2) (6)Large values ofg (disorder) correspond to lowT , small values ofg (order) correspondto high T , a peculiar behavior due to the negative sign ofbo , ( asymptotic freedom),which naturally leads to Duality.Duality is a deep concept first developed in statistical mechanics [7][8], and thenextended to field[9] and string theory [10]. It applies to systems with non local, topolog-ically non trivial excitations.Such systems admit two equivalent complementary descriptions:(1) A direct description in terms of local fieldsΦ whose vacuum expectation values(vev) < Φ > are the order parameters. The topological excitationsµ are non local. Thisdescription is convenient in the weak-coupling regimeg < 1.(2) A dual description in whichµ are local fields,< µ > are the (dis)order parameters,andΦ are non local . This description is convenient in the strong coupling regimeg > 1,since the dual coupling constantgD is related to the direct one asgD ≈ 1/g.Duality maps the strong coupling limit of the direct description into the weak couplinglimit of the dual one and viceversa.The prototype theory is the 2d Ising model [8] which is defined on a 2d squarelattice in terms of a two-valued local fieldσi = ±1 with nearest-neighbor interactionhi j = Jσiσ j. The magnetization< σ > is the order parameter which is non zero belowa Curie temperatureTc, and vanishes above it. The partition function depends on theparameterβ ≡ JT . The model admits topologically non trivial 1d excitationsµ, whichare kinks with spin up in half space and down in the other half,or viceversa (antikinks).It can be shown that alsoµ is a two valued variableµ =±1 and that the equality holdsK[σ ,β ] = K[µ,β ′] (7)where K is the partition function andsinh(2β ) = 1/sinh(2β ′) or β ≈ 1/β ′. The systemcan be described equivalently in terms of the local variableσ or of the dual variableµand order -disorder are interchanged in the two descriptions. In particular forT ≥ Tc thedual is ordered ,< µ >6= 0 and the dual coupling is small.The idea is then to look at a dual description ofQCD [11], in terms of spacialconfigurations with non trivial topologyµ. In this description the low temperature phasewhich is disordered in the usual direct description should be ordered and< µ > shouldbe the order parameters.The problem is then to identify the dual symmetry and the corresponding topologicalexcitations. IfG is the gauge group topological excitations in(3+ 1)d are obtainedby breaking the symmetry to a subgroupH provided that the homotopyΠ2(G/H)is non trivial. Such excitations are monopoles [12] . The procedure is called abelianprojection[12]. ForSU(3) gauge group there are two independent monopole species.In (2+ 1)d the relevant homotopy isΠ1(SU(3)/Z3) and the natural topologicalexcitations are vortices. Vortices could be relevant to confinement also in(3+1)d[11].Understanding the relevant dual excitations and their symmetries amounts to under-stand the mechanism of color confinement.MECHANISM OF CONFINEMENT: RESULTS ANDPERSPECTIVESThe literature of the last ten years on the subject can be classified according to theguiding principles as follows.1) Search for dual symmetry. The basic idea is that the dual excitations carry mag-netic charge and condense in the confined phase producing dual superconductivity. Anoperatorµ carrying magnetic charge is constructed[13],[14],[15] and its vev is mea-sured by numerical simulations as a function of the temperatureT . If the basic idea iscorrect one expects< µ >= 0 for T > Tc in the thermodynamic limit ( infinite spatialvolume) ,< µ >6= 0 for T < TC. In principle it is not possible by direct measurement tostate that< µ >= 0 due to statistical error. To go around this difficulty a susceptibilityρ ≡ ∂ ln(<µ>)∂β is defined [13]. Since by construction< µ >β=0= 1,< µ >= exp(∫ β0ρ(β ′)dβ ′) (8)For T > Tc ρ →−∞ as the volume goes large , showing that< µ > is strictly zero. ForT < Tc ρ becomes volume independent at large volumes and< µ >6= 0. At T ≈ Tc ,ρ has a sharp negative peak corresponding to a rapid decrease of < µ > and obeys thefinite size scaling equationρL1νs= Φ(τL1νs ) (9)whereτ = 1− TTc is the reduced temperature, andν is the critical index describing thedivergence of the correlation lengthξ , ξ ∝ τ−ν . Eq(9) allows to extract the value ofν ,and to identify the order of the transition and the universality c ass.For quenched theory the location of the transition and the critical index obtained inthis way are consistent with the determinations made by use of the Polyakov line [14][16]. Moreover the result is independent on the choice of theab lian projection usedto define the monopoles[17]. Also in fullQCD with N f = 2 the method works and thecritical indexν turns out to be that of a first order transition [18][19].The net result of this approach is that the deconfining transitio s an order-disordertransition: the vacuum is a dual superconductor in the confined phase, normal in thedeconfined phase. Whatever the dual excitations they are magnetically charged in all theabelian projections.2) Approach choosing a special gauge, maximal abelian for monop les[20][21], max-imal central for vortices[22] to expose non local excitations."Surgical " analysis: eliminate somehow the non local excitations (monopoles or vor-tices )and check that physical properties like string tensio or alternative manifestationsof confinement like the infrared divergence of the ghost propagator disappear. Claimsfollow that the mechanism of confinement is understood.3) An interesting recent development is the exploration of the role of the center in theprocess of confinement[23] to check the claim that vortices ar the relevant excitations.In particular the gauge groupG2 which has no center does confine color.4) Phenomenology of monopoles in Maximal Abelian Gauge [24]: the idea is to guessfrom the observation of lattice configurations their dynamics and the possible existenceof excitations of different dimensionality.I think that a fair conclusion is that we do not know yet the dual excitations : we onlyknow that they carry magnetic charge in all abelian projections, and that the confiningvacuum is a dual superconductor in the confined phase , a property which disappearsabove the deconfining transition.CONCLUDING REMARKSA fundamental issue to understand confinement inQCD is whether the deconfiningtransition is order -disorder or can be a crossover , i.e. whether confinement is anabsolute property of the vacuum based on symmetry or is a dynamical change of physicalquantities likenq of Eq(1) or σq of Eq(2) by several orders of magnitude across acrossover. This question can possibly be settled by more extended simulations of thetheory on the lattice.Investigations using an order parameter which detects condensation of magneticcharges show that dual superconductivity is at work as a mechanism of confinement.Direct investigation of the role of the center by comparisonof different groups withthe same center can provide useful information, as well as the tudy of confinement withgauge groups likeG2 which have no center.We do not have yet a clear understanding of confinement.REFERENCES1. M. Gell-mann:Phys.Lett.8 (1964) 2142. Review of Particle Physics:EPJ15 (2000)3. L. Okun: Leptons and Quarks ,North Holland (1982)4. N. Cabibbo, G. Parisi:Phys. Lett.59B (1975) 675. R.D. Pisarski, F. Wilczek:Phys.Rev. D29, (1984) 3386. M. D’Elia, A. Di Giacomo, C. Pica:hep-lat/0503030, Phys.Rev.D ,in press.7. H.A. Kramers, G.H. Wannier:Phys. Rev.60 (1941) 2528. L.P. Kadanoff, H. Ceva:Phys. Rev. B3 (1971) 39189. N. Seiberg, E. Witten :Nucl. Phys. 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Confinement of color: open problems and perspectives

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