Deconfinement and topological molecules

Transcription

1 CAQCD-2011 Deconfinement and topological molecules Mithat Ünsal, Stanford U. Based on work done in collaboration with Dusan Simic, arxiv: [hep-th], and brief mention of ongoing work with Erich Poppitz 1

2 Phases of pure Yang-Mills theory Z() = tr[e H R 3 ], Quark-Gluon Plasma phase (sqgp) high T = 1/T = radius of S 1 R3 S 1 Hadronic (confined) phase 3 low T 1 R S R 4 The phase transition occurs at strong scale of gauge theory and is non-perturbative. It is very hard to study by analytical methods. Lattice works, but it should be seen as numerical experiment. The problem is experimentally relevant and theoretically interesting. 2

3 Example 1 : Yang Mills on R 3 S 1 ZN center symmetry, order parameter = Wilson line ΩΩ circumference L g(x + L) = hg(x), h N = 1 trω(x, x + L) h trω(x, x + L) Aperiodic gauge rotations, h ZN Order parameter t Hooft L> Lc: unbroken center symmetry tr Ω n = 0 confined phase L < Lc: broken center symmetry tr Ω n 0 deconfined plasma phase A typical simulation result < P > < W > b 3

4 Experts assert (Gross, Pisarski, Yaffe 1981):, It is hardly surprising that we cannot explore the transition, as the temperature is lowered, from the unconfined to the confined phase using solely weak coupling techniques. Two ways to avoid this are found in the last 30 years: I) Gauge-string duality: Pro: Addresses strongly coupled gauge theory via semi-classical string theory Con: Comes with extra package: non-decoupling KK-tower of states, or the lack of asymptotic freedom. Useful at macro-level, microscopically (?). II) Compactification on S3 x S1, d=0 matrix models Can we possibly study this transition perturbatively? Not in infinite space, but maybe at finite small volume 1/R Λ with weak coupling at IR cutoff (if no zero modes, e.g. S 3 ), if transition persists to small volume. Aharony, Marsano, Minwalla, Papadodimas, Van Raamsdonk, hep-th/ Pro: Deconfinement can be pushed to weak coupling! (existence proof) Con: Thermo-dynamic limit requires N=infinity. Aharony et.al. consider but reject the possibility that a weak coupling deconfinement can be achieved in infinite space. 4

5 Things that one would like to understand in thermal YM: I) Can we find a calculable (semi-classical) deconfinement transition in an asymptotically free and confining gauge theory by using field theory techniques? 2) Is this even possible as the transition itself is non-perturbative? 3) Can we give a simple physical picture of the mechanism behind the deconfinement transition? 5

6 Calculable deconfinement on R2 x T2 lence Simic and I proposed a way to study the non-perturbative transition thorough semiclassical techniques. We use one circle (along which we guarantee absence of phase transition) as a control parameter, and the other circle is thermal. This set-up teaches us something new about the transition. Fixed N, arbitrary L Simic, Unsal, 2010, Anber, Poppitz, Unsal, QCD(adj) L We consider deformed YM theory on LN Λ>>1 M 4 (L, ) = R 2 S 1 L S 1 c YM c dym with double-trace deformation along the L-circle. 3 1 YM on R x S LN Λ< L L Ω deformed YM on R x S x S A semi-classical laboratory to keep non-perturbative effects under control: Similar philosophy used earlier in finite density QCD..T. Schafer and F. Wilczek, and M. G. Alford, K. Rajagopal and F. Wilczek, 6

7 Center-symmetry stabilization In YM, there are phase transition along both circles. We want to prevent the one on the L-circle, so that we can use it as a control device. adding explicit stabilizing terms: works Unsal, Yaffe/ Shifman,Unsal, 2008 adding (massless) adjoint fermions with p.b.c.: works Kovtun, Unsal, Yaffe 2007 Also see Ogilvie, Meisinger, Myers double-trace deformation prevents symmetry breaking (and at large-n, it has no effect on N= center symmetric dynamics!) S YM = S YM + R 3 S 1 P [Ω(x)] P [Ω] = A 2 π 2 L 4 N/2 n=1 1 n 4 tr (Ωn ) 2 A sufficiently positive, O(1) as N goes to. Deformation is O(N 2 ) perturbation. However, its effect on dynamics is sub-leading in N., not a small Veneziano referred to this deformation as a good samaritan. It does the good deed and sequesters itself. 7

8 Thermal Large-N equivalence Azeyanagi, Hanada, Unsal, Yacoby, 2010 and Simic, Unsal, 2010 Thermal large N equivalence Fixed N, arbitrary L L LN Λ>>1 dym c YM c c dy LN Λ< 1 Ω deformed YM on R x S x S L YM on R x S L Ω deformed YM on R L 2 Yang-Mills theory on R 3 S 1 is equivalent to deformed Yang-Mills theory on R 3 k (SL 1 )k S 1 2 for any finite value of L and for a given, up to 1/N corrections, provided that the [(Z N ) L ] k center symmetry is not spontaneously broken. 8

9 Deformed Yang-Mills and weak coupling confinement Start with = And make L-small. center-symmetric vacuum: tr ΩΩ = 0 eigenvalues repel compact adjoint Higgs field a) Attractive b)repulsive c)no force broken center unbroken center quantum moduli sp mkk = 1/NL, 2/NL,..., perturbative control when NLΛ << 1 4π/ L 4π/ L 4π/ L topological defects (instantons), mass gap, confinement (generalization of Polyakov s mechanism to R3 x S1) 2π/L 0 2π/L 2π/L 4π/ (LN) 2π/ (LN) 0 0 a)center broken b1)center symmetric b2)center symmetric finite or large N finite N large N 9

10 Semi-classical confinement Heavy W-boson particles m Wi = 2π LN, Q W i = gα i, i = 1,..., N. IR-theory abelianizes. Abelian duality in 3d F µν = g2 4πL ɛ µνρ ρ σ. Magnetic and topological charges for 3d instantons and twisted-instanton ( F, S π 2 tr F µν F µν ) = ( 4π g α i, 1 N The amplitude associated with these instanton events are: M αi N i=1 = e 8π 2 g 2 i θ N N e iα i σ M αi = e 8π 2 g 2 iθ ) i=1,...n types of instanton with identical action Extra monopole-instanton: Kraan, van Baal 98, and Lee, Yi, 97 usual 4d instanton amplitude, it may be viewed as composite. 10

11 Mass gap for gauge fluctuations Instantons generate mass gap and linear confinement by Debye mechanism. Generalization of Polyakov to R3 x S1 S dual = R 3 [ 1 2L ( g ) 2 ( σ) 2 ζ 4π ( πp ) ( ) 5/6 ( ) LNΛ LNΛ 8/11 m p = m σ sin, m σ = AΛ ln N 2π 2π N i=1 cos(α i σ) + ], p = 1,..., N 1, In general, the dynamics of center-symmetric theories in the NLΛ < 1 regime is quite similar to the one of supersymmetric gauge theories in the same domain. Similar tools can be used to solve both class of theories. Instantons generate mass gap and confinement by Debye mechanism. 11

12 Calculable deconfinement In the NLΛ < 1, semi-classical confinement domain, let us open a thermal circle and increase the temperature. How does deconfinement sets in? Related works (as well as part of inspiration) comes from 3d Georgi-Glashow model: Dunne, Kogan, Kovner, Tekin 2000, 2001, (Our analysis is parallel to this work) Agasian, Zarembo, 1997; Kovchegov, Son 2003 nce Fixed N, arbitrary L L c YM 3 1 YM on R x S LN Λ>>1 LN Λ< 1 c dym L L Ω deformed YM on R x S x S We consider deformed YM theory on M 4 (L, ) = R 2 S 1 L S 1 with double-trace deformation along the L-circle. 12

13 consider the regime where the monopole size is much smaller than in turn is much smaller than the inter-monopole separation. which By method of images from electrostatics, 1/ r potential enhances to ln r. Compactified 3d instantons map to 2d vortices. In 2d, vortices interact by logarithmic Coulomb interactions. r [U (Energy)] S 1 m 1 W c m 1 σ m 1 X3 W es 0/2 b) Equi potential surfaces of center symmetric D2 branes r [U (Energy)] S 1 m 1 W X 3 a) Equi potential surfaces of center broken D2 branes 13

14 Neglecting W-bosons momentarily, (will turn out to be wrong in a global sense) S dual = R 2 [ a 2 ( σ)2 ζ M N i=1 cos(α i σ) + ], a L ( g ) 2 4π This is the (generalized) Sine-Gordon model. It is also the well-known theory of 2d vortices. ( 3d instantons 2d vortices) The 2d theory is known to have a phase transition, called Berezinsky-Kosterlitz- Thouless (BKT)-transition! This sounds very similar to strong coupling LGT by Polyakov & Susskind from old days, (which maps to XY model), however, in our analysis, this BKT will turn out to be fake! 14

15 BKT phase transition consider vortices in XY-model, with core size a in a 2d box with size R. Its energy is log divergent with system size. E = E 0 ln R a a It can be anywhere in the box. Thus, its entropy is ) 2 S ln ( R a Free energy, competition between energy and entropy: F = E T S = (E 0 2T ) ln R a R T < Eo/2 : Vortices have infinite energy, neutral pairs have finite energy, gapless phase. T > Eo/2 : unbound gas of vortices (Coulomb gas), finite correlation length, gapped phase. 15

16 0 Magnetic-BKT Conformal Dimension of vortex operator: < 2 : = 2 : > 2 : relevant, hence gapped marginal m = 2πL irrelevant, gapless ungapped m Δ [ cos α σ ] < 2 i Δ [ cos α i σ ] < 2 e gapped [e iα i σ ] = α2 i 4πa = 1 4πa = 4πL g 2 ; The effect described above is not the whole picture in deformed YM, due to the omission of W-bosons. The W-bosons are a small perturbation (with respect to topological defects) when e m W e S 0 or 4πL g 2. BKT does not occur in a domain where analysis is reliable... It is a fake! g 2 Magnetic or inverted-bkt 16

17 Electric-BKT Ignoring topological sectors momentarily: justified only when e S 0 e m W or 4πL g 2. The proliferation of the two-dimensional gas of W-bosons generates S p.t. = [ã N R 2 2 ( σ)2 ζ W Abelian duality in 2d 0 m gapped i=1 1 4πa ] cos(α i σ) +... d σ = dσ Δ [ cos α i σ ] < 2 gapless, ã 1 16π 2 a = L g 2 Δ [ cos α σ ] < 2 i e Electric-BKT e = 8πL g 2 17

18 0 Electric vs. magnetic competition We learned, in 2d long-distance theory, that phase transition is driven by a competition between electric and magnetic degrees of freedom. Our calculable field theory realizes the e&m competition scenario proposed by Liao&Shuryak,2006. m Δ [ cos α i σ ] < 2 magnetic-defects relevant Δ [ cos α σ ] < 2 i electric excitations relevant e e-vortices-free, m-vortices bound e-vortices bound, m-vortices-free Intermediate regime: Both magnetic and electric descriptions become strongly coupled, not easily calculable... The existence of phase transition can be shown analytically. (will skip this.) 18

19 Phase diagram (Inverse) critical temperature: dym c = { ( 11 ) ( 3 Λ 1 LNΛ ln LNΛ ) 2π 2π, LNΛ 1 cλ ( O(1/N 2 ) ) LNΛ 1 2 Confined M Β 1 0 Deconfined 1 2 LN M Deconfined, magnetic defects are still relevant. Deconfined, magnetic defects are irrelevant (or bound). T m = 2T c Could this have observable consequences? Possible.. 19

20 Since the topological defects are so crucial in the physics of deconfinement transition in our calculable deformation, their role should also be fully elucidated in thermal QCD. What is the microscopic mechanism of deconfinement phase transition in YM theory? What is the role of topological defects? Could we observe immediate consequence of topology? [One such effect discussed recently is chiral magnetic effect. (Kharzeev et.al.)] 20

21 Topological defects and molecules in SU(2) YM Consider Wilson line: Ω = Diag(e i θ/2, e i θ/2 ) θ Gross, Pisarski, Yaffe 1981: Only computes one loop-potential for Wilson line in perturbation theory. There are some very interesting non-perturbative effects neglected there... and becomes very important gradually. Leading monopole-instanton amplitudes M 1 = e 4π g 2 4 M 2 = e 4π g 2 4 θ+iσ, (2π θ) iσ, M1 = e 4π g 2 4 M2 = e 4π g 2 4 θ iσ, (2π θ)+iσ, At next-order, there are more interesting effects. M 1 M 1 = e 4π g 2 4 M 2 M 2 = e 4π g θ 2(2π θ) There are some supersymmetric theories in which we can reliably demonstrate that the analog of these molecules generate a center symmetric minimum! They also generate eigenvalue repulsion in pure YM theory. However, different from magnetic bions which Erich discussed in the previous talk, it is quite hard to establish these new molecules by just-semi-classical analysis. 21

22 We think these effects are also there even in pure YM theory and they generate a competing effect to the one-loop potential for Wilson line. This may give us the microscopic root cause of confinement/deconfinement transition. In rationalizing this, crucial earlier works which we benefitted are Bogomolnyi 80, non-susy QM Zinn-Justin 81, non-susy QM Balitsky-Yung 86, susy QM Yung 90. susy QFT. None-of-these are on R3 x S1, but the methods there transcend dimensions. On thermal R3 x S1, Poppitz & I concluded that these methods apply usefully. On some more exotic gauge theories, Argyres & I found similar topological molecules. They seem to be rather generic. Both of these works will appear during summer. 22

23 Speculations in pure YM? Sum over the tower of monopole-instantons: [ e 4π g 2 4 ] θ+2πn w e iσ, a 5 a 4 n = 1 w 0 α 1 α 0 a) n m=+1 3d instanton tower n w=0 n w=1 n w=2 n w Z } {{ } F ( θ) n = 1 w n w=0 n =1 w n =2 w F ( θ + 2π) = F ( θ) F ( θ) = 1 π F (Ω) = 1 π n e Z n e Z ( 4π g 2 4 ( 4π g 2 4 4π g 2 4 4π g 2 4 V one loop (Ω) n Z ) 2 + n 2 e e in e θ ) 2 + n 2 e trω n e 2 1 n 4 trωn 2 b) n =+1 3d instanton tower at a =0 m Has interpretations as a sum over massless/light dyons wrapping the S1 cirlce! (1, n e ) dyons electrically coupled to background gauge field. Denominator is square of the flux of a dyon particle with (n m, n e )= (1, n e )! S 2 ( E + i B).d Σ = n e + i 4π g 2 4 Compare this with the GPY one-loop result. Functionally, similar, opposite in sign. The mathematics here is trivial. However, its suggestions are quite thought provoking. Discussion/criticism and suggestions are welcome! 5 n m 23

24 Supplementary material 24

25 Concluding remarks Formally, the deconfinement transition can be pushed to a semi-classical but non-perturbative domain. This provides a microscopic description. Deconfinement driven by a competition between electric and magnetic degrees of freedom, is, we believe the most interesting aspect of our formulation. This is a modest, as we only showed it in semi-classical regime, but noteworthy progress. Better use of compactification produces new methods to study gauge dynamics, non-perturbative aspects, confinement and deconfinement. Volume independence is a exact property of the large N limit in an interesting class of non-abelian gauge theories, but we still don t know how to benefit from it at its maximum. 25

26 Key players Outline (4) (5) (6) (1) (2) (3) 3d Polyakov model & monopole instanton -induced confinement Polyakov, 1977 monopole-instantons on R3 x S1 K. Lee, P. Yi, 1997, P. van Baal, 1998 Physics of vortices in 2d, Berezinsky, 1971; Kosterlitz, Thouless, 1973; Center-stabilization mechanisms on R3 x S1 Kovtun, MU, Yaffe, 2007, Ogilvie, Myers, 2007 Shifman, MU, 2008, MU, Yaffe, 2008, Large-N volume independence and semi-classical confinement MU, Yaffe 2008, (on lattice, by Bringoltz, Sharpe 2009) Thermal large-n equivalence and toroidal compactification Azeyanagi, Hanada, Yacoby, MU and Simic, MU,

27 tr N Ω = e F q Existence of phases Polyakov loop measures excess of free energy in the presence of an test charge in the gluonic heat bath. N tr N Ω 1 N i=1 e iν i eσ Low-temperature: The insertion of an electric charge into the medium may be viewed as a vortex in the sigma field theory. The vorticity is the electric charge: Q g = 1 dσ = ν j 2π C [ 1 N e iνi eσ 1 N ( = lim e S[in the presence of R )] 1 2π C dσ=ν j] S[in its absence ] = 0 N R N i=1 i=1 High-temperature: 1 N e iνi eσ = e +i 2π N k, k = 0, 1,..., N 1. N i=1 N thermal equilibrium states Low-temperature calculation is one of the few analytic demonstrations of vanishing Polyakov loop in microscopically 4d field theory. 27

28 Estimate for phase transition scale Refined picture of the phases and c [ m, e ] interval, where the transition must occur. Electric-magnetic competition: We conjecture that the transition should occur when e and m perturbations simultaneously become order one. 0 Δ m> 2 Δ e < 1/2 Δ m< 2 Δ e < 1 Δ m< 1 Δ e < 2 Δ m< 1/2 Δ e > 2 m c e e [e iα i eσ ] = Q2 W i 4πL = g2 4πL, m [e iα i σ ] = Q2 M i L 4π = 4πL g 2 e m = (Q W i Q Mi ) 2 (4π) 2 = 1 [no sum over i] Dirac quantization! Our calculable field theory realizes the e&m competition scenario proposed by Liao&Shuryak,2006 to explain the low viscosity and diffusion constant observed in sqgp at RHIC. 28