Continuity of the Deconfinement Transition in (Super) Yang Mills Theory Thomas Schaefer, North Carolina State University 0.0 0.0 0.0 0.0 0.0 0.0 with Mithat Ünsal and Erich Poppitz
Confinement and the QCD string 4 m ps + m s 3 2 Π u 2 m ps [V(r)-V(r 0 )]r 0 1 0-1 Σ g + -2-3 quenched κ = 0.1575 1 1.5 2 2.5 3 r/r 0 Leinweber (2001) Bali (2001) Confinement well established numerically (and empirically)
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
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.
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.
SU(2) YM with n adj f = 1 Weyl fermions on R 3 S 1 Phase diagram in L-m plane
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.
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
Small S 1 : Effective Theory Consider small S 1 and Ω 1: A 4 is a Higgs field, theory abelianizes. Bosonic sector of effective 3d theory 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 Φ = b + iσ + 2θ α λ α
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
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 4 0 3. Magnetic charges Q α M = 1 4π d 2 S Tr [P α B]
Periodic instantons (calorons) Instanton solution in R 4 can be extended to solution on R 3 S 1 2 2 E + B r Q top = ±1 X=0 τ X=1 P = 1 Q α M = 0 SU(2) solution has 1 + 3 + 1 + 3 = 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
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
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.
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)
Non-perturbative effects at m = 0 from supersymmetry Monopoles contribute to superpotential: (λλ)e b+iσ d 2 θe Φ W = M3 PV L g 2 ( e b + e 2S 0 e b) Scalar potential V (b, σ) W Φ 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)
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).
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 ) 1 1728 ( 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
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
Outlook: higher rank gauge groups, θ dependence, pure gauge SU(N 3): First order transition Z N 0.0 0.0 0.0 0.0 0.0 0.0 Smooth N c limit (because Q top 1/N c ) µ 3 PV e 8π 2 g 2 Nc Λ 3
Large N c : Eigenvalues of P for N c = 4,5,6 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
G 2 : First order transition without change of symmetry. 0.0 0.0 0.0 1.5 1.5 0.0 1.5 1.5 0.0 1.5 1.5 0.0 Pure gauge theory: Find center stabilizing molecules from BZJ. But: Semi-classical approximation not reliable.