Hamiltonian approach to QCD: The Polyakov loop potential H. Reinhardt

Transcription

1 Hamiltonian approach to QCD: The Polyakov loop potential H. Reinhardt H. R. & J. Heffner Phys. Lett.B718(2012)672 PRD(in press) arxiv:

2 Phase diagram of QCD

3 non-perturbative continuum approaches Dyson-Schwinger equations FRG flow equations Hamiltonian approach

4 non-perturbative continuum approaches Dyson-Schwinger equations FRG flow equations Hamiltonian approach in Coulomb gauge

5 Outline introduction order parameter for confinement: Polyakov loop Hamiltonian approach to YMT in background gauge effective potential of the Polyakov loop deconfinement phase transition conclusions

6 recent work on the Polyakov-loop FRG F.Marhauser and J. M. Pawlowski, arxiv: J. Braun, H. Gies, J. M. Pawlowski, Phys. Lett. B684(2010)262 J. Braun,T.K. Herbst, arxiv J.Braun etal., Eur.Phys.J C70(2010) DSE C. Fischer, L. Fister, J. Lücker, J.M. Pawlowski, arxiv strong coupling lattice M. Fromm etal. JHEP1201(2012)042. lattice J. Greensite, Phys. Rev. D86(2012) J. Greensite and K. Langfeld, D87(2013) D. Smith etal Hamiltonian approach H. R. & J. Heffner, Phys. Lett.B718(2012)672 PRD(in press) arxiv:

7 Polyakov loop YMT at finite temperature T :compact Euclidean time P[A 0 ]( x) L = 1 d r trp exp i dx 0 A 0 (x 0, x) 0 order parameter for confinement: conf. phase: center symmetry deconf. phase: center symmetry-broken P[A 0 ]( x) exp[ F ( x)l ] P[A 0 ]( x) = 0 P[A 0 ]( x) 0 T 1 = L

8 Polyakov loop YMT at finite temperature T :compact Euclidean time P[A 0 ]( x) L = 1 d r trp exp i dx 0 A 0 (x 0, x) 0 order parameter for confinement: conf. phase: center symmetry deconf. phase: center symmetry-broken P[A 0 ]( x) exp[ F ( x)l ] P[A 0 ]( x) = 0 P[A 0 ]( x) 0 T 1 = L Polyakov gauge 0 A 0 = 0, fundamental modular region Jensen s inequality: A 0 = diagonal SU(2) : P[A 0 ]( x) = cos( A ( 0 x)l ) 2 0 < A 0 L / 2 < π P[A 0 ] unique function of A 0 P[A 0 ]( x) P[ A 0 ( x) ] alternative order parameters: P[A 0 ]( x) P[ A 0 ( x) ] A 0 ( x) F.Marhauser and J. M. Pawlowski, arxiv: J. Braun, H. Gies, J. M. Pawlowski, Phys. Lett. B684(2010)262 J. Braun,T.K. Herbst,arXiv

9 Effective potential of the order parameter for confinement background field calculation effective potential order parameter a 0 = A 0 ( x) const, diagonal (Polyakov gauge) e[a 0 ] min a 0 = a 0 P[A 0 ] P[a 0 ]

10 Effective potential of the order parameter for confinement background field calculation effective potential order parameter a 0 = A 0 ( x) const, diagonal (Polyakov gauge) e[a 0 ] min a 0 = a 0 P[A 0 ] P[a 0 ] 1-loop perturbation theory e PT [a 0 = x2π / L] Gross, Pisarski,Yaffe, Rev.Mod.Pys.53(1981) N.Weiss, Phys.Rev.D24(1981) P[a 0 = 0] = 1 deconfined phase

11 Effective potential of the order parameter for confinement background field calculation effective potential order parameter a 0 = A 0 ( x) const, diagonal (Polyakov gauge) e[a 0 ] min a 0 = a 0 P[A 0 ] P[a 0 ] 1-loop perturbation theory e PT [a 0 = x2π / L] Gross, Pisarski,Yaffe, Rev.Mod.Pys.53(1981) N.Weiss, Phys.Rev.D24(1981) P[a 0 = 0] = 1 deconfined phase aim of this talk: non-perturbative evaluation of e[ a 0 ] in the Hamiltonian approach

12 Polyakov loop potential in the Hamiltonian approach Hamiltonian approach assumes Weyl gauge A 0 = 0

13 Polyakov loop potential in the Hamiltonian approach Hamiltonian approach assumes Weyl gauge A 0 = 0 O(4)-invariance compactify (instead of time) one spatial axis to a circle of circumference L and interprete L 1 as temperature L compactify x 3 axis a = a e 3

14 Polyakov loop potential in the Hamiltonian approach Hamiltonian approach assumes Weyl gauge A 0 = 0 O(4)-invariance compactify (instead of time) one spatial axis to a circle of circumference L and interprete L 1 as temperature L compactify x 3 axis a = a e 3 YMT at finite length in a constant, color diagonal background field a calculate the effective potential e[a]

15 effective potential The effective potential in the Hamiltonian approach e( a) of a spatial background field a H a = min H A = a H a = (spatial volume) e( a) e( a) effective potential

16 Hamiltonian approach to Yang-Mills theory Weyl gauge: A 0 a (x) = 0 cartesian coordinates A i a (x) momenta Π i a (x) = δs / δ A i a (x) = E i a (x) H = 1 2 d 3 x(π 2 (x) + B 2 (x)) Π k a (x) = δ / iδ A k a (x) YM Schrödinger equation H Ψ[A] = EΨ[A] Gauss law DΠΨ = 0 more convenient: gauge fixing explicit resolution of Gauss law gauge invariant wave functionals: + a, A = 0 a background field Ψ[A] for a = 0 A = 0 H. Reinhardt Hamiltonian approach to QCD 8

17 Hamiltonian approach to YMT in background gauge [d,a]=0 H = 1 2 (J 1 Π JΠ + B 2 ) + H C Π = δ / iδ A J( A ) = Det( Dd) D = + A d = + a H C = 1 2 J 1 ρj( Dd) 1 ( d 2 )( Dd) 1 ρ color charge density: ρ = (A a)π Coulomb term Φ... Ψ = Λ DAJ(A) Φ (A)...Ψ(A) H. Reinhardt Hamiltonian approach to QCD 9

18 Hamiltonian approach to YMT in background gauge [d,a]=0 H = 1 2 (J 1 Π JΠ + B 2 ) + H C Π = δ / iδ A J( A ) = Det( Dd) D = + A d = + a H C = 1 2 J 1 ρj( Dd) 1 ( d 2 )( Dd) 1 ρ color charge density: ρ = (A a)π Coulomb term Φ... Ψ = Λ DAJ(A) Φ (A)...Ψ(A) Ψ H Ψ min Ψ A Ψ = a H. Reinhardt Hamiltonian approach to QCD 9

19 Variational approach trial ansatz Ψ a ( A) = 1 J(A,a) exp 1 dx dy(a(x) a)ω(x, y)(a(y) a) 2 gluon field gluon propagator A a = a A(x)A(, y) a = (2ω(x, y)) 1 variational kernel determined from H. Reinhardt Hamiltonian approach to QCD 12

20 The background field covariant derivative d = + a background field a = a k H k a H in the Cartan algebra [H k, H l ] = 0 adjoint representaion (T b ) ac = f abc roots H k σ = iσ k σ σ =(σ 1,...,σ r ) roots SU(2): H 1 = T 3 σ 1 =0, ± 1 SU(3): H 1 = T 3 H 2 = T 8 σ =(1,0), ( 1 2, 1 2 3), ( 1 2, 1 2 3) positive roots eigenvalues of id = i( + a) p σ = p σ a

21 Propagators in the background field propagators in presence of the diagonal background field in background gauge [ + a, A] = 0 a exact relation: G σ a, background gauge(p) = G a=0, A=0 (p σ ) p σ = p σ a ordinary Coulomb gauge propagators G a=0, A=0 (p) choice of background field: a = a e 3 compactify 3-axis: temperature p σ = p + (p n σ a) e 3, p n = 2πn / L T 1 = L

22 The effective potential energy density 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= χ background field p σ = p + (p n σa) p n = 2πn / L σ = 0 ± 1 periodicity e(a, L) = e(a + µ k / L, L) exp( iµ k ) = z k Z(N) ghost loop χ arises from the FP determinant in the kinetic energy input: ω(p), χ(p) from the variational calculation in Coulomb gauge at T=0 C. Feuchter & H. Reinhardt,Phys. Rev.D71(2005) D. Epple, H. Reinhardt, W. Schleifenbaum, Phys. Rev.D75(2007)

23 Variational approach in Coulomb gauge Numerical results for SU(2) D. Epple, H. R., W.Schleifenbaum, PRD 75 (2007) IR : ω(k) 1 / k UV : ω(k) k H. Reinhardt Hamiltonian approach to QCD 13

24 Static gluon propagator in D=3+1 D(k) = (2ω(k)) 1 Gribov s formula ω(k) = k 2 + M 4 M = 0.88GeV k 2 missing strength in mid momentum regime: missing gluon loop G. Burgio, M.Quandt, H.R., PRL102(2009) H. Reinhardt Hamiltonian approach to QCD 14

25 Variational approach to YMT with non-gaussian wave functional D. Campagnari & H.R, Phys.Rev.D82(2010) wave functional ψ A 2 = exp( S A ) ansatz S A = ω A γ (3) A 3 3! + 1 γ (4) A 4 4! exploit DSE H. Reinhardt Hamiltonian approach to QCD 15

26 non-gausssian wave functional D. Campagnari & H.R, Phys.Rev.D82(2010)

27 The effective potential energy density 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= χ background field p σ = p + (p n σa) p n = 2πn / L σ roots periodicity e(a, L) = e(a + µ k / L, L) exp( iµ k ) = z k Z(N)

28 The effective potential energy density 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= background field p σ = p + (p n σa) p n = 2πn / L σ roots periodicity e(a, L) = e(a + µ k / L, L) exp( iµ k ) = z k Z(N) neglect ghost loop quasi-gluon gas χ(p) = 0 1 e(a, L) = L d 2 p ω(p σ ) σ n= n=

29 The effective potential energy density 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= background field p σ = p + (p n σa) p n = 2πn / L σ roots periodicity e(a, L) = e(a + µ k / L, L) exp( iµ k ) = z k Z(N) neglect ghost loop quasi-gluon gas χ(p) = 0 1 e(a, L) = L d 2 p ω(p σ ) σ n= n= limiting cases UV: IR: ω UV (p) = p ω IR (p) = M 2 / p Gribov: ω(p) = (p 2 + M 4 / p 2 ) ω IR (p) + ω UV (p)

30 The UV-effective potential χ(p) = 0 ω(p) = p 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= e(a, L) = 8 sin 2 (nal / 2) π 2 L 4 n 4 = 4π 2 3L 4 n=1 al 2π x 2 al 2π 1 N.Weiss 1-loop PT 2 Polyakov loop P P[a min = 0] = 1 deconfining phase

31 The IR-effective potential χ(p) = 0 ω(p) = M 2 / p 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= e IR (a, L) = 4M 2 sin 2 (nal / 2) π 2 L 2 n 2 = 2M 2 L 2 n=1 al al 2π 2π 1 x Polyakov loop P P[a min = π / L] = 0 confining phase

32 The IR-effective potential χ(p) = 0 ω(p) = M 2 / p 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= e IR (a, L) = 4M 2 sin 2 (nal / 2) π 2 L 2 n 2 = 2M 2 L 2 n=1 al al 2π 2π 1 x Polyakov loop P P[a min = π / L] = 0 confining phase deconfinement phase transition results from the interplay between the confining IR-potential and deconfining UV-potential

33 The IR+UV effective potential: χ(p) = 0 e(a, L) = e UV (a, L) + e IR (a, L) ω(p) = p + M 2 / p phase transition critical temperature: T C = 3M / π lattice : M 880MeV T C 485MeV

34 The IR+UV effective potential: χ(p) = 0 e(a, L) = e UV (a, L) + e IR (a, L) ω(p) = p + M 2 / p phase transition critical temperature: T C = 3M / π lattice : M 880MeV T C 485MeV χ(p) = 0 ω(p) = p 2 + M 4 / p 2 T C 432MeV

35 The effective potential-role of the ghost loop 1 e(a, L) = L d 2 p (ω(p σ ) σ n= χ(p σ )) χ n= neglecting the ghost loop in e(a, L) enhances the IR-contribution χ(p) confinement reduces the UV-contribution deconfinement pushes the deconfinement transition to a higher temparature

36 The full effective potential 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= variational calculation in Coulomb gauge SU(2) critical temperature: T C 270MeV

37 The effective potential for massive gluons χ(p) = 0 ω(p) = M 2 + p 2 x = al 2π 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= no phase transition Polyakov loop P P[a min = 0] = 1 deconfining phase

38 The effective potential for massive gluons χ(p) = 0 ω(p) = M 2 + p 2 x = al 2π 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= M = 0 ω(p) = p no phase transition N.Weiss 1-loop PT Polyakov loop P P[a min = 0] = 1 deconfining phase

39 The effective potential for massive gluons χ(p) = 0 ω(p) = M 2 + p 2 x = al 2π 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= ω(p) = M 4 p 2 + p 2 no phase transition phase transition

40 The effective potential for SU(3) SU(3)-algebra consists of 3 SU(2)-subalgebras characterized by the 3 non-zero positive rooots e SU (3) [a] = σ >0 e SU (2)(σ ) [a]

41 The full effective potential for SU(3) 1 e(a, L) = L d 2 p (ω(p σ ) χ(p σ )) σ n= n= variational calculation in Coulomb gauge T < T C T > T C x = a 3 L 2π, y = a 8 L 2π

42 Polyakov loop potential for SU(3) x = a 3L 2π, y = a 8L 2π = 0 input : SU(2) data : M = 880MeV T C = 283MeV

43 The Polyakov loop SU(2) SU(3)

44 critical temperature lattice : T C SU (2) = 295MeV T C SU (3) = 270MeV this work : T C SU (2) = 267MeV T C SU (3) = 277MeV FRG(Fister & Pawlowski) : T C SU (2) = 230MeV T C SU (3) = 275MeV

45 Conclusions effective potential of the Polyakov loop in the Hamiltonian approach input: vacuum propagators obtained in the variational calculation in Coulomb gauge neglect of ghost loop and use of the UV-gluon energy ω(p) = p : ` Weiss potential full potential: deconfinement phase transition SU(2): 2.order SU(3): 1.order T C 270MeV deconfinement phase transition is encoded in the vacuum wave functional on R 2 S 1 similar results: grand canonical ensemble J. Heffner, H. Reinhardt, D. Campagnari, Phys. Rev.D85(2012)

46 Thanks for your attention H. Reinhardt Hamiltonian approach to QCD 54