Symmetry, Confinement and the phase diagram of 9 0 QCD 0 2 n Adriano Di Giacomo a J Pisa Universityand INFN Sezione di Pisa 2 ] t a l - p Abstract e h Ageneraldiscussionispresentedofthepossiblesymmetriesresponsibleforconfinementofcolor [ and of their evidence in lattice simulations. The consequences on the phase diagram of QCD are also analyzed. 1 v 7 2 Key words: NonperturbativeQCD,Confinement, Deconfiningtransition,Duality. 2 PACS: 11.10Wx, 11.15Ha,12.38Mh,64.60Cn 0 . 1 0 1. Why symmetry? 9 0 : No free quark has ever been observed in Nature: the abundance of quarks nq com- v pared to the abundance of protons n has an experimental upper bound nq ≤ 10−27 Xi to be compared to the value 10−12 epxpected in the Standard CosmologicanlpModel in r absence of confinement. The cross section for inclusive production of quarks in hadron a −15 collisions, σ is also 10 times smaller than the perturbative expectation. The natural q explanation of these facts is that confinement is an absolute property, in the sense that n and σ are strictly zero due to some symmetry. As a consequence the deconfining q q transition is a change of symmetry, i.e. an order-disorder transition and can not be a cross-over.A similar situationexists in ordinarysuperconductivity:the resistivity in the superconducting phase has an exceedingly small experimental upper limit compared to the resistivity in the normalphase.The naturalexplanationis that the resistivity in the superconducting phase is strictly zero. A change of symmetry occurs at the transition from a Higgs broken U(1) symmetry (superconductor) in which Cooper pairs condense in the vacuum, to a normal phase in which the U(1) symmetry is exact. PreprintsubmittedtoElsevier 2January2009 2. What symmetry? Color symmetry is exact : it can not distinguish confined from deconfined. Center symmetry only exists in absence of dynamical quarks. Chiral symmetry only exists at zerom .MoreoverinsomecaseslikeQCDwithN =2adjointfermionschiralsymmetry q f restoration occurs at a different temperature than deconfinement [1][2], indicating that the relevantdegreesoffreedomatthe deconfiningtransitionarenotthe chiralones.The only way to get an extra symmetry is via duality [3][4], i.e. by looking at excitations with topologically non trivial boundary conditions. In (2+1)dim the homotopy is Π1 and the topologically non trivial excitations are vortices, in (3+1)dim the homotopy is Π2 and the excitations are monopoles [5][6]. For a generic gauge group G of rank r, r abelian field strength tensors (’t Hooft tensors) Fa , (a = 1,..r) can be defined[7] µν and in terms of them r magnetic currents ja ≡ ∂ Fa∗ . Non zero value of the currents ν µ µν ja is a violation of Bianchi identities, due to the presence of magnetic charges. The ν currents ja are conserved due to the antisymmetry of the dual tensor Fa∗ and define ν µν the dual symmetry. If the corresponding U(1) symmetries are Higgs broken magnetic charges condense in the vacuum and there is dual superconductivity (Confinement). If the symmetries are exact the vacuum is normal and chromoelectric charges deconfined. Anoperatorµcanbeconstructedcarryingnonzeromagneticcharge,anditsvev hµican be used as an order parameter for confinement [8][9][10], i.e. as a detector of monopole condensation. 3. The phase diagram. A transition is a rapid change in physics at some value T of some parameter say c the temperature T. A transition shows up as a peak in susceptibilities, which are the derivativesofobservableswithrespecttoT.ForexampleapeakinthespecificheathC V isarapidchangeintheheathcontent.Atransitioniscalledacrossoverifnodiscontinuity develops at T as the volume V goes to infinity, it is named first order if some first c derivative of the free energy diverges , e.g. if the free energy itself has a discontinuity at T and C diverges as V →∞. c V Stating that a transition is a crossover is equivalent to verify that the free energy is analytic trough T , and this cannot be done on the basis of any numerical calculations c with a finite volume and a finite resolution. It can sometime be done with the help of sometheory.AclassicalexampleisthechiraltransitionatsmallquarkmassesinN =2 f QCD[11]. Assuming that the relevant degrees of freedom at the chiral-deconfinement transition are the chiral ones, on the basis of renormalization group arguments one can say that either the chiraltransitionis secondorder in the universalityclass of O(4), and then the transition is a cross-overat small non zero masses, or it is first order, and then itstaysfirstorderatsmallmasses.Inthefirstcaseatricriticalpointispredictedatfinite density, whose existence can be checked experimentally in heavy ion collisions[12] ; no tricriticalpointexists in the secondcase.Finite size scaling analysishas been performed by many groups[13], but none finds evidence for second order O(4). If the correlation lengths are large compared to lattice spacing scale invariance holds andoneexpectsforthevolumedependencee.g.ofthespecificheaththefollowingscaling law[14][15] 2 CV −C0 ≈LsανΦC(τLsν1,mLysh) (1) Here L is the spatial size of the lattice, τ =(1− T ) the reduced temperature and α, s Tc ν and y are critical indexes which are specific of the order and universality class of the h 1 transition. For second order O(4) α = −.24, = 1.34, y = 1.48. For weak first order ν h 1 α= 1, =3, y = 3. Eq.(1) can be tested on lattice data either by keeping the second ν h variable of the function ΦC fixed, by choosing m and Ls such that mLysh has a fixed value,sayK foragivenassumption(y )onthe universalityclass;orbykeepingthefirst h variable fixed and checking the dependence on the second one[14]. One has in the first case α 1 (CV −C0)/Lsν ≈ΦC(τLsν,K) (2) in the second case, at large values of mLyh[15], one has for second order O(4) s (CV −C0)≈m.13fC(τL1s.35) (3) Instead for weak first order 1 3 0 3 1 3 (CV −C0)≈LsfC(τLs)+ mfC(τLs) (4) Data onlattices L =4,L =16,20,24,32do not agreewith the scaling Eq.(2)with the t s choice O(4), they do with the choice weak 1st order [14][15]. Also Eq(3) is not satisfied by O(4). Eq.(4) instead is obeyed, but the first term, which is typical of first order looks to be negligible at present volumes, implying that the transition is too weak to observe a growth proportional to the volume at presently available volumes. Moreover, with L = 4 the lattice at the phase transition is rather coarse, and a check should be t done with smaller lattice spacings. Evidence for the existence of the first term of Eq.(4) is needed to make a definite statement on first order.No definite evidence exists by now for a cross-over. References [1] F.Karsch,M.LutgemeierNucl.Phys.B550,449(1999) [2] G.Cossu,M.D’Elia,A.DiGiacomo,G.Lacagnina,C.Pica;Phys.Rev.D77:074506,2008. [3] H.A,Kramers,G.H.Wannier;Phys.Rev.66,252(1941) [4] S. Gukov and E. Witten; Gauge theory, ramification, and the geometric Langlands program, [hep-th/0612073]. [5] G.’tHooft;Nucl.Phys.B79(1974) 276. [6] A.M.Polyakov; JETPLett.20(1974)194 [7] A.DiGiacomo,L.Lepori,F.Pucci;JHEP0810:096,2008. [8] A.DiGiacomo,G.Paffuti;Phys.Rev.D56:6816-6823,1997. [9] A.DiGiacomo,B.Lucini,L.Montesi,G.Paffuti ;Phys.Rev.D61:034503,2000. [10] A.DiGiacomo,B.Lucini,L.Montesi,G.Paffuti ;Phys.Rev.D61:034504,2000. [11] R.D.Pisarski,F.Wilczek;Phys.Rev.D29,338(1984) [12] M.A.Stephanov, K.Rajagopal,E.V.Shuryak;Phys.Rev.Lett.81,4816(1998) [13] See e.g. Owe Philipsen , Status of Lattice Studies of the QCD Phase Diagram. International Symposium Fundamental Problems in Hot and / or Dense QCD, Kyoto, Japan -2008. arXiv:0808.0672[hep-ph] 3 [14] M.D’Elia,A.DiGiacomo,C.Pica,Phys.Rev.D72:114510,2005. [15] G.Cossu,M.D’Elia,A.DiGiacomo,C.Pica,arXiv:0706.4470[hep-lat] 4