Continuity of the Deconfinement Transition in (Super) Yang Mills Theory

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1 Continuity of the Deconfinement Transition in (Super) Yang Mills Theory Thomas Schaefer, North Carolina State University with Mithat Ünsal and Erich Poppitz arxiv: & arxiv:

2 Confinement and the QCD string 4 m ps + m s 3 2 Π u 2 m ps [V(r)-V(r 0 )]r Σ g quenched κ = r/r 0 Leinweber (2001) Bali (2001) Confinement well established numerically (and empirically)

3 Confinement and the QCD string Challenge: Understand confinement analytically Not just a problem in pure mathematics: Understand dynamics, suggest new observables,... Some successes (QCD-like gauge theories) Polyakov model (compact QED in 2+1) N = 2 SUSY YM softly broken to N = 1

4 Typical mechanism: Dual superconductivity Long distance description contains magnetic monopoles. Monopoles condense: Dual superconductivity. Landau-Ginzburg theory describes electric flux tubes: Confining strings. String tension determined by dual photon mass.

5 Confinement: Goals Mass gap in the pure gauge theory: m 0 ++ Tr[F 2 (x)]tr[f 2 (0)] f 2 exp( m 0 ++x) String tension, effective theory of the QCD string. [ ] W(C) = Tr exp i A µ dx µ exp( σa(c)) Polykaov line: Effective potential, correlation functions. [ ] β Ω( x) = Tr exp i A 4 dx 4 0 Critical temperature, center symmetry breaking C 0 Ω zω z Z N Theta dependence, d 2 E/dθ 2 0.

6 In this work we will pursue a more modest goal. We will study confinement and the deconfinement phase transition in a non-abelian gauge theory which is weakly coupled (by using a suitable compactification). We will argue that this theory is continuously connected (by decoupling an extra matter field) to pure gauge theory.

7 SU(2) YM with n adj f = 1 Weyl fermions on R 3 S 1 Phase diagram in L-m plane

8 Ingredients R 3 S 1 circle-compactified gauge theory. Small S 1 : Effective 3d theory involving holonomy and (dual) photon. Double expansion: Perturbative and non-perturbative effects (monopoles, topological molecules). Topological molecules: supersymmetry versus BZJ. Competition: Center stabilizing molecules, center breaking perturbative (and monopole) effects.

9 Gauge theory on R 3 S 1 SU(2) gauge theory, n f = 1 adjoint Weyl fermion L = 1 4g 2 F a µνf a µν i g 2 λa σ D ab λ b + m g 2 λa λ a A a µ(0) = A a µ(l) λ a (0) = λ a (L) Vacua labeled by Polyakov line Ω = exp [ i A 4 dx 4 ] x 4 Center symmetry Ω zω z Z 2

10 Small S 1 : Effective Theory Consider small S 1 : Effective theory in 3d Ω 1: A 3 4 is a Higgs field, theory abelianizes SU(2) U(1). Light bosonic modes: (dual) photon σ and holonomy b L = g2 32π 2 L [ ( i b) 2 + ( i σ) 2] + V (σ, b) Ω = ( e i θ/2 0 0 e i θ/2 ) b = 4π g 2 θ ǫ ijk k σ = 4πL g 2 F ij holonomy b dual photon σ Note: m = 0 effective theory can be super-symmetrized B = b + iσ + 2θ α λ α

11 Perturbation Theory Perturbative potential for holonomy (Gross, Pisarski, Yaffe, 1981) m2 V (Ω) = 2π 2 L 2 n=1 1 n 2 tr Ωn 2 = m2 L 2 B 2 ( θ 2π m = 0: Bosonic and fermionic terms cancel. m 0: Center symmetric vacuum tr(ω) = 0 unstable. ) Π Π 2Π 3Π

12 Non-perturbative effects Topological classification on R 3 S 1 (GPY) 1. Topological charge Q top = 1 16π 2 d 4 x F F 2. Holonomy (eigenvalues q α of Polyakov line at spatial infinity) [ ] β Ω( x) = Tr exp i A 4 dx Magnetic charges Q α M = 1 4π d 2 S Tr [P α B]

13 Periodic instantons (calorons) Instanton solution in R 4 can be extended to solution on R 3 S E + B r Q top = ±1 X=0 τ X=1 Ω = 1 Q α M = 0 SU(2) solution has = 8 bosonic zero modes dρ ρ 5 d 3 x dx 4 du e 2S 0 2S 0 = 8π2 g 2 4n adj fermionic zero modes d 2 ζd 2 ξ

14 Calorons at finite holonomy: monopole constituents KvBLL (1998) construct calorons with non-trivial holonomy BPS and KK monopole constituents. Fractional topological charge, 1/2 at center symmetric point. 2 (3 + 1) = 8 bosonic zero modes, 2 2 fermionic ZM. dφ 1 d 3 x 1 d 2 ζ e S 1 dφ 2 d 3 x 2 d 2 ξ e S 2

15 Topological objects (Q M,Q top ) = ( S 2 B dσ, R 3 S 1 F F) BPS KK BPS KK monopoles (1, 1/2) ( 1, 1/2) ( 1, 1/2) (1, 1/2) instantons (0, 1) (0, 1) bions (0,0) (0,0) Note: BPS/KK topological charges in Z 2 symmetric vacuum. Also have (2, 0) (magnetic) bions.

16 Topological objects: Coupling to low energy fields (Q M,Q top ) = ( S 2 B dσ, R 3 S 1 F F) BPS KK BPS KK e b e ±iσ (λλ) e b e iσ ( λ λ) monopoles instantons (λλ)(λλ) ( λ λ)( λ λ) bions e 2b e 2b Kraan, van Baal (1998); Lee, Lu (1998); Unsal (2008)

17 Non-perturbative effects at m = 0 from supersymmetry Monopoles contribute to superpotential: (λλ)e b+iσ d 2 θe B W = M3 PV L g 2 ( e B + e 2S 0 e B) Scalar potential V (b, σ) W B 2 M6 PV L3 e 2S 0 g 6 [ ( ) 8π cosh ( θ π) g2 ] cos(2σ) Center symmetric vacuum tr(ω) = 0 preferred Mass gap for dual photon m 2 σ > 0 ( confinement) Davies, Hollowood, Khoze, Mattis (1999)

18 Non-perturbative effects at m = 0 from BZJ Consider magnetically neutral topological molecules. Integrate over near zero-mode: S 12 V BPS,BPS e 2b e 2S 0 d 3 r e S 12(r) (BPS)(KK) S 12 (r) = 4πL g 2 r (q1 mq 2 m q 1 bq 2 b) + 4 log(r) (BPS)(BPS) r 12 Saddle point integral after analytic continuation g 2 g 2 (BZJ) V (b, σ) M6 PV L3 e 2S ( ) 0 8π g 6 cosh ( θ π) g2 Same for magnetically charged molecules: V cos(2σ).

19 Effective potential for m 0 Effective potential: molecules, monopoles, perturbation theory Ṽ = cosh 2b cos 2σ + m (cosh 2 L cos σ b b sinhb 2 3 log L 1 ) L > L c V(b ) ( m L 2 ) 2 1 log 3 L 1 (b ) 2. L < L c b L = LΛ, m = m/λ, b = 4π g2 ( θ π) Critical S 1 size L2 c = m 8 [ 1 + O ( 1 log L, m L 2 )], Corresponds to T c = 8 m Λ QCD

20 SU(2) YM with n adj f = 1 Weyl fermions on R 3 S 1 Phase diagram in L-m plane topological molecules monopoles, perturbation theory

21 Higher rank gauge groups, θ dependence SU(N 3): First order transition Z N Smooth N c limit (because Q top 1/N c ) µ 3 PV e 8π 2 g 2 Nc Λ 3

22 Large N c : Eigenvalues of P for N c = 4,5, θ 0: Get V k cos ( 2πk+θ N c ), k = 1,..., N 1. 2π periodicity + 1/N c scaling mulitiple branches Π 2Π 3Π

23 θ dependence of T c (Anber, arxiv: ) T cr Θ T cr Θ G 2 : First order transition without change of symmetry

24 Outlook Continuity of deconfinement transition on R 3 S 1 can be studied on the lattice (with presently available technology). Direct calculation in pure gauge theory: Find center stabilizing molecules from BZJ. But: Semi-classical approximation not reliable. Other topics: Fundamental matter, effective theories for the QCD string,....