The SU(2) quark-antiquark potential in the pseudoparticle approach Marc Wagner mcwagner@theorie3.physik.uni-erlangen.de http://theorie3.physik.uni-erlangen.de/ mcwagner 3 th March 26
Outline PP = pseudoparticle Basic principle. Building blocks of PP ensembles. PP ensembles. Quark-antiquark potential. Quantitative results. Summary. Outlook.
Basic principle (1) Pseudoparticle approach (PP approach): A numerical technique to approximate Euclidean path integrals (in this talk: SU(2) Yang-Mills theory QCD with infinitely heavy quarks): O = 1 DA O[A]e S[A] Z S[A] = 1 d 4 x F 4g µνf a a 2 µν, Fµν a = µ A a ν ν A a µ + ǫ abc A b µa c ν. A tool to analyze the importance of gauge field configurations with respect to confinement. A method, from which we can get a better understanding of the Yang-Mills path integral.
Basic principle (2) PP: any gauge field configuration a a µ, which is localized in space and time. Consider only those gauge field configurations, which can be written as a sum of a fixed number ( 4) of PPs: A a µ (x) = i ρ ab (i)a b µ (x z(i)). (i: PP index; ρ ab (i): degrees of freedom of the i-th PP, i.e. amplitude and color orientation; z(i): position of the i-th PP). Approximate the path integral by an integration over PP degrees of freedom: ( ) DA... dρ ab (i)... i ρab (1)a b µ (x z(1)) ρab (2)a b µ (x z(2)) ρ ab (3)a b µ (x z(3)) A a µ (x) x 1 x 1 x 1 x 1 + + = x x x x
Building blocks of PP ensembles Building blocks of PP ensembles: instantons, antiinstantons, akyrons (λ: PP size). x ν a a µ,instanton(x) = ηµν a x 2 + λ 2 x ν a a µ,antiinstanton (x) = ηa µν x 2 + λ 2 a a µ,akyron(x) = δ a1 x µ x 2 + λ 2. -4 4-2 2 2-2 4-4 a a µ x µ -4 4-2 2 2-2 4-4 Degrees of freedom: amplitudes A(i), color orientations C ab (i), positions z(i). A a µ (x) = A(i)Cab (i)a a µ,instanton (x z(i)) A a µ (x) = A(i)Cab (i)a a µ,antiinstanton (x z(i)) A a µ (x) = A(i)Cab (i)a a µ,akyron (x z(i)). Instantons, antiinstantons and akyrons form a basis of all gauge field configurations in the continuum limit.
PP ensembles (1) PP ensemble: a fixed number of PPs inside a spacetime hypersphere. Gauge field: A a µ (x) = A(i)C ab (i)a b µ,instanton (x z(i)) + i A(j)C ab (j)a b µ,antiinstanton(x z(j)) + j A(k)C ab (k)a b µ,akyron(x z(k)). k Choose color orientations C ab (i) and positions z(i) randomly. r spacetime 4-dimensional spacetime hypersphere region of strong boundary effects x A a µ is no classical solution (not even close to a classical solution)!!! r boundary Long range interactions between PPs. boundary of spacetime x region of negligibl boundary effects
PP ensembles (2) Approximation of the path integral: O = 1 ( ) da(i) O(A(i))e S(A(i)). Z i Solve this multidimensional integral via Monte-Carlo simulations. Exclude boundary effects: observables have to be measured sufficiently far away from the boundary. r spacetime 4-dimensional spacetime hypersphere region of strong boundary effects x r boundary boundary of spacetime x region of negligibl boundary effects
Quark-antiquark potential (1) Common tool to determine the potential of a static quark-antiquark pair: Wilson loops. Wilson loop (z: closed spacetime curve): W z [A] = 1 { ( )}) (P 2 Tr exp i dz µ A µ (z). Rectangular Wilson loop (R, T: spatial and temporal extension): W (R,T). Wilson loops quark-antiquark potential (R: quark-antiquark separation): 1 V q q (R) = lim T T ln W (R,T). Assumption: the potential can be parameterized according to V q q (R) = V α R + σr. V q q plotted against R V qq (R) = V - α / R + σ R V =., α = π / 12., σ = 1. 2 1.5 1.5 -.5-1 -1.5-2.5 1 1.5 2 2.5
Quark-antiquark potential (2) Method 1: Determine the string tension σ and the Coulomb coefficient α Guess the functional dependence of ensemble averages of Wilson loops: ) ( R ln W (R,T) = V (R + T α T + T ) + β + σrt. R Determine the string tension σ and the Coulomb coefficient α by fitting the Wilson loop ansatz to Monte-Carlo data for ln W (R,T). Several approaches: Area perimeter fits. Creutz ratios. Generalized Creutz ratios....
Quark-antiquark potential (3) Method 1: Determine the string tension σ and the Coulomb coefficient α Results for PP ensembles containing 4 PPs: Coulomb coefficient α > attractive Coulomb-like interaction at small quark-antiquark separations. String tension σ > linear potential for large quark-antiquark separations, confinement. σ is an increasing function of the coupling constant g adjust the physical scale by choosing appropriate values for g..5 σ plotted against g σ(g) S 12 -ensemble: N = 35 12, n = 1., λ =.5.4 string tension σ.3.2.1 Marc Wagner, The SU(2) quark-antiquark potential in the pseudoparticle approach, 3 th March 26 1 2 3 4 5 6 coupling constant g
Quantitative results For quantitative results, including the string tension, we need other dimensionful quantities: Topological susceptibility χ = Q 2 V /V. Critical temperature T critical. Dimensionless quantities (physically meaningful): χ 1/4 /σ 1/2, T critical /σ 1/2, α. Consider different g = 2.... 5.5 (diameter of the spacetime hypersphere.9 fm... 2. fm). Results are in qualitative agreement with results from lattice calculations. Consistent scaling behavior of σ, χ and T critical. α should be constant. dimensionless ratio χ 1/4 /σ 1/2 dimensionless ratio T critical /σ 1/2 Coulomb coefficient α.6.5.4.3.2.1 1 2 3 4 5 6 coupling constant g.8.7.6.5.4.3.2.1.3.25.2.15.1.5 χ 1/4 /σ 1/2 plotted against g χ 1/4 /σ 1/2 (g) S 12 -ensemble: N = 35 12, n = 1., λ =.5 α(g) S 12 -ensemble: N = 35 12, n = 1., λ =.5 lattice results: α =.1....3 lattice result: χ 1/4 /σ 1/2 =.486 ±.1 S 12 -ensemble T critical /σ 1/2 plotted against g T critical /σ 1/2 (g) S 12 -ensemble: n = 1., λ =.5 lattice result: T critical /σ 1/2 =.694 ±.18 1 2 3 4 5 6 coupling constant g α plotted against g Marc Wagner, The SU(2) quark-antiquark potential in the pseudoparticle approach, 3 th March 26 1 2 3 4 5 6 coupling constant g
Quark-antiquark potential (4) Method 2: Calculate the quark-antiquark potential directly For large T: V q q (R)T ln W (R,T). From the slope of ln W (R,T) R=constant we can read off V q q (R). Results are in agreement with our previous results. ln W (R,T) R=constant plotted against T -ln W (R,T) R = constant (T) S 12 -ensemble: N = 35 12,... V q q plotted against R V qq (R) S 12 -ensemble: N = 35 12, n = 1., λ =.5, g = 4. -ln W (R,T) R = constant 2 1.5 1.5 R = a R = 2 a R = 3 a R = 4 a R = 5 a R = 6 a R = 7 a R = 8 a R = 9 a R = 1 a R = 11 a R = 12 a quark antiquark potential V qq in 1/fm 3 2.5 2 1.5 1.5 least squares fit of V - α/r + σr cutoff effects.5 Marc1 Wagner, 1.5 The SU(2) 2 quark-antiquark 2.5 potential 3 in the pseudoparticle.1 approach,.2 3.3 th March.4 26.5.6.7 temporal extension T quark antiquark separation R in fm
Summary The PP approach with 4 instantons, antiinstantons and akyrons is able to reproduce many essential features of SU(2) Yang-Mills theory: Quark-antiquark potential: Linear potential for large quark-antiquark separations (confinement). Coulomb-like attractive force for small quark-antiquark separations. Consistent scaling behavior of σ, χ and T critical. Dimensionless quantities χ 1/4 /σ 1/2, T critical /σ 1/2 and α are in qualitative agreement with results from lattice calculations.
Outlook Compare different PP ensembles to analyze, which gauge field configurations are responsible for confinement: Pure akyron ensembles (no topological charge density) deconfinement. Gaussian localized PPs (PPs of limited size) deconfinement for small PP size, confinement for large PP size. Overall picture: topological charge and long range interactions between PPs are important for confinement.