Center-symmetric dimensional reduction of hot Yang-Mills theory

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1 Center-symmetric dimensional reduction of hot Yang-Mills theory Aleksi Kurkela, Univ. of Helsinki Lattice 2008 arxiv: , arxiv: with Philippe de Forcrand and Aleksi Vuorinen Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 1 / 11

2 Dimensional reduction At high T: For long distance properties ( x 1/T), the system looks 3d. Degrees of freedom are static modes φ 0 (x) φ(x,τ) = T n= exp(iω nτ)φ n (x) Effective action: Integrate out non-static modes Z = Dφ 0 Dφ n exp( S 0 (φ 0 ) S n (φ 0,φ n )) = Dφ 0 exp( S 0 (φ 0 ) S eff (φ 0 )) In practice: Need scale separation between static and non-static modes Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 2 / 11

3 Where Dimensional Reduction works? Scales in hot Yang-Mills: Perturbatively (g 1/ log(t)): Hard scale: 2πT Typical thermal momentum, non-static modes Soft scale: md gt Debye screening, static modes Asymptotic dimensional reduction Laermann, Philipsen Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 3 / 11 Non-perturbatively: m(t c ) 3T c? 2πTc m D ( gt+g 2 T ) m A0 ( gt ) m glue ( g 2 T ) M/T log 10 (T/T c )

4 Perturbative dimensional reduction Polyakov loop Ω = Tr [ P exp ( ig dτa 0 )], has Nc minima in the deconfined phase Z N center symmetry T < T c T T c T > T c Deep in deconfined phase: Expand fields around one minimum to get EQCD (= 3D Yang-Mills + adjoint Higgs): 1 S EQCD = d x[ 3 2 TrF ij + Tr(D i A 0 ) }{{}}{{} 2 m2 ETrA ] 4 λ ETrA }{{} adjoint kinetic spatial gluons interactions from integration out Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 4 / 11

5 Center-symmetric effective theories Goal: Want to construct an effective theory that Preserves the ZN center symmetry Reduces to EQCD at high T Is superrenormalizable Effective theory of Wilson lines not (super)renormalizable (Pisarski hep-ph/ ) Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 5 / 11

6 Center-symmetric effective theories Goal: Want to construct an effective theory that Preserves the ZN center symmetry Reduces to EQCD at high T Is superrenormalizable Effective theory of Wilson lines not (super)renormalizable (Pisarski hep-ph/ ) Idea: Construct effective theory for coarse grained Wilson loop (Yaffe+Vuorinen hep-ph/ ) Z(x) = T V Block V d 3 yu(x,y)w(y)u(y,x), / SU(N c ) Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 5 / 11

7 Center-symmetric theory for SU(2) For SU(2), sum of matrices proportional to SU(2) Z = λω, Ω SU(2), λ > 0 Z = 1 { } ( 1 Σ1 2 }{{} +i Π a σ }{{ a = 2 Σ + iπ 1 iπ 2 Π 3 } 1 iπ 2 + Π 3 2 Σ iπ 1 Singlet Adjoint scalar Transforms exactly like Wilson line Z λ 1 (x)zλ(x) gauge Z Z center Z 2 ) Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 6 / 11

8 Center-symmetric theory for SU(2) For SU(2), sum of matrices proportional to SU(2) Z = λω, Ω SU(2), λ > 0 Z = 1 { } ( 1 Σ1 2 }{{} +i Π a σ }{{ a = 2 Σ + iπ 1 iπ 2 Π 3 } 1 iπ 2 + Π 3 2 Σ iπ 1 Singlet Adjoint scalar Most general superrenormalizable Lagrangian with A i and Z: L Z(2) = 1 ( ) 2 Tr F ij 2 + Tr D i Z D i Z +V (Z) }{{}}{{} spatial gluons Adjoint Kinetic V (Z) = b 1 Σ 2 + b 2 Π 2 a + c 1 Σ 4 + c 2 (Π 2 a) 2 + c 3 Σ 2 Π 2 a }{{} interaction from integration out Higher order terms suppressed by scale difference m D /T. ) Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 6 / 11

9 Matching at T Parameters can be (almost) mached in perturbation theory (series in g 2 16π 2!): b 1 = 1 4 r2 T 2, b 2 = 1 4 r2 T g 2 T 2, c 1 /g 2 3 = r g 2, c 2 /g 2 3 = r g 2, c 3 /g 2 3 = r 2, g 2 3 = g 2 T Parameters functions of full theory parameters (g,t) and r rt: mass of fluctuation from SU(2) manifold r needs to be matched non-perturbatively Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 7 / 11

10 On the lattice: S a = S W + S Z + V (ˆΣ, ˆΠ), β = 4 S W = β x,i<j [ 1 1 ] 2 Tr [U ij], ag 2 3 ( ) 4 [ˆΠ2 S Z = 2 Tr β ˆΠ(x)U ] i (x)ˆπ(x + î)u i (x) x,i ( ) 4 (ˆΣ2 + (x) β ˆΣ(x)ˆΣ(x + î)), x,i ( ) 4 3 [ˆb1ˆΣ2 V = + β ˆb ) ] 2 2 ˆΠ2a + ĉ 1ˆΣ4 + ĉ 2 (ˆΠ2a + ĉ3ˆσ2ˆπ2a, x Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 8 / 11

11 Exact renormalization (2-loop lat-pt) AK [ /hep-lat] ˆb1 = b 1 /g (2ĉ 1 + ĉ 3 )β 4π + 1 { (48ĉ 2 16π ĉ ĉ } 3)[log 1.5β ] ĉ 3 +O(a), ˆb2 = b 2 /g (10ĉ 2 + ĉ 3 + 2)β 4π + 1 { 16π 2 (80ĉ ĉ ĉ 2 )[log 1.5β ] } ĉ O(a). Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 9 / 11

12 Results from simulations: Z 2 -restoring phase transition < Σ 2 > < Σ > 2 < Σ > /g 2 < Σ 2 > < Σ > < Σ > /g 2 left (β = 12,n = 64,r 2 = 5) right (β = 6,n = 64,r 2 = 5) Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 10 / 11

13 Results from simulations: N=64 N=96 N= N=64 N=96 N=128 B B (1/g 2 2 1/ν -1/g c )N (1/g 2 2 1/ν -1/g c )N left r 2 = 5, right r 2 = 10. 3d Ising universality class. B 4 = Σ 4 / Σ 2 2 = at criticality ν = 0.63 Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 10 / 11

14 Results from simulations: 1/g r 2 =10 r 2 =5 1/g r 2 =10, β=12 r 2 =5, β= /β N a 0 V (A + BL 1/ν ) large r short correlation length fine lattice small r long correlation length large volume r = 0: Σ decouples λφ 4 already done, X.P. Sun hep-lat/ Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 10 / 11

15 Results from simulations: 1/g Deconfined g 2 =5.1 Confined r 2 Phase diagram resembles the full theory (unlike in EQCD). Insensitive to r > 1 Phase transition at correct g! Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 10 / 11

16 Outlook Implementing center-symmetry to the effective theory gives correct phase transition in SU(2) Lots of simulations to do: Check accuracy near T c : Domain wall tension Spatial string tension Screening masses Make predictions: Heavy quarks: ZN breaking terms Finite chemical potential (Correct phase transitions?) Extension to large Nc apparent: At N c > 4 no more N c-dependent (super-renormalizable) operators. Can the theory accommodate fuzzy bag? p(t) = BMIT + B fuzzy T 2 + f pert T 4 Aleksi Kurkela,Univ. of Helsinki () Center-symmetric dimensional reduction of hot Yang-Mills Lattice 2008 theory 11 / 11