Many faces of the Landau gauge gluon propagator at zero and finite temperature

Transcription

1 Many faces of the Landau gauge gluon propagator at zero and finite temperature positivity violation, spectral density and mass scales Pedro Bicudo 3, Nuno Cardoso 3, David Dudal 2, Orlando Oliveira 1, 1 1 Centro de Física Computacional, Universidade de Coimbra, Portugal 2 Department of Physics and Astronomy, Ghent University, Belgium 3 CFTP, Instituto Superior Técnico, Portugal September 5, 213

2 Outline 1 2 How-to First results 3

3 Landau T= ( Dµν ab (ˆq) = δ ab δ µν q ) µq ν q 2 D(q 2 ), Lattice computation of the gluon propagator: Large volume: access to the deep IR region, infinite volume limit SU(2): La = 27 fm, a =.22 fm SU(3): La = 17 fm, a =.18 fm A. Cucchieri, T. Mendes, PoS (LAT 27) 297 Small lattice spacing: large a also changes the propagator I. L. Bogolubsky et al., Phys. Lett. B676, 69 (29) O. Oliveira, P. J. S., Phys. Rev. D86, (212)

4 Spectral representation D(p 2 ) = + dµ ρ(µ) p 2 +µ 2 On the lattice: study the temporal correlator C(t) < C(t) = dp 2π D(p2 ) exp( ipt) = negative spectral density positivity violation gluon confinement C(t) > says nothing about ρ(µ) dωρ(ω 2 )e ωt

5 for the gluon propagator β= D(t) t (fm) Already observed in lattice simulations C. Aubin, M. C. Ogilvie, Phys. Rev D7, (24) A. Cucchieri, T. Mendes, A. R. Taurines, Phys. Rev. D71, 5192 (25)

6 Euclidean momentum-space propagator of a (scalar) physical degree of freedom G(p 2 ) O(p)O( p) Källén-Lehmann spectral representation G(p 2 ) = dµ ρ(µ) p 2, with ρ(µ) for µ. +µ spectral density contains information on the masses of physical states described by the operator O ρ(µ) = l δ(µ m 2 l ) O l 2,

7 G = L 2ˆρ = LL ˆρ where (Lf)(t) dse st f(s) is a Laplace transform inversion of Laplace transform: ill-posed problem Way out: Tikhonov regularization ill-posed problem y = Kx minimize Kx y +λ x 2 λ > is a regularization parameter x λ is the unique solution of the normal equation K Kx λ +λx λ = K y the operator K K+λ is strictly positive, hence invertible Morozov discrepancy principle: choose λ s.t. Kx λ y δ = δ δ: noise of input data A unique solution x λ,δ exists

8 Outline How-to First results 1 2 How-to First results 3

9 How-to First results Getting gluon spectral density L 2 ρ = D L 4 ρ+λρ = L 2 D dtρ(t) ln z t z t +λρ(z) = dt D(t) t + z consider 1-loop perturbative behaviour after p (latt) max integrals computed using Gauss-Legendre quadrature discretization leads to a linear system IR and UV cut-offs lattice data (8 4, β = 6.) interpolated using splines

10 Outline How-to First results 1 2 How-to First results 3

11 Results (preliminary) How-to First results Changing number of GL points Reconstructed propagator 14 Λ IR = GeV Λ UV = 5 GeV 1 Λ IR = GeV Λ UV = 5 GeV ρ(p²) GL GL GL GL GL GL GL no UV tail D(p²) [GeV -2 ] GL GL GL GL GL GL GL no UV tail p² [GeV²] p² [GeV²]

12 Results (preliminary) How-to First results Changing UV cutoff Reconstructed propagator Λ IR = GeV 1 Λ IR = GeV 1 Λ = 2 GeV Λ = 5 GeV Λ = 1 GeV Λ = 15 GeV no UV tail 1 Λ = 2 GeV Λ = 5 GeV Λ = 1 GeV Λ = 15 GeV No UV tail ρ(p²) D(p²) [GeV -2 ] p² [GeV²] p² [GeV²]

13 Results (preliminary) How-to First results Changing IR cutoff Reconstructed propagator 1 Λ UV = 5 GeV 1 Λ UV = 5 GeV ρ(p²) Λ IR = 1 MeV [S = 11.2] Λ IR = 2 MeV [S = 16.4] Λ IR = 3 MeV [S = 39.9] Λ IR = 4 MeV [S = 122.7] Λ IR = 5 MeV [S = 236.8] no UV tail [S = 12.9] D(p²) [GeV -2 ] 1.1 Λ IR = 1 MeV Λ IR = 2 MeV Λ IR = 3 MeV Λ IR = 4 MeV Λ IR = 5 MeV no UV tail p² [GeV²] p² [GeV²]

14 Outline 1 2 How-to First results 3

15 Getting hotter Gluon propagator at finite T splitted into two components transverse D T longitudinal D L D T (q 2 ) = D ab µν (ˆq) = δ ab( P T µν D T(q 2 4, q)+p L µν D L(q 2 4, q) ( 1 2V(Nc 2 A a i 1) (q)aa i ( q) q2 4 D L (q 2 ) = 1 V(N 2 c 1) ) ) q 2 Aa 4 (q)aa 4 ( q) ( ) 1+ q2 4 A a q 2 4 (q)aa 4 ( q) on the lattice: L t << L s T = 1 al t

16 finite T Temp. (MeV) β L s L t a [fm] 1/a (GeV) Simulations: use of Chroma and PFFT libraries keep a constant (spatial) physical volume (6.5fm) 3 all data renormalized at µ = 4GeV O. Oliveira, PJS, PoS(LATTICE212)216 Acta Phys.Polon.Supp. 5 (212) 139 PoS(Confinement X)45

17 Surface plots Longitudinal component Transverse component p(gev) 32 T(MeV) T(MeV) p(gev)

18 Outline 1 2 How-to First results 3

19 finite T - longitudinal component Below T C Above T c 5 3 C(t) T = T = 121 MeV T = 162 MeV T = 194 MeV T = 243 MeV T = 26 MeV T = 265 MeV T = 275 MeV C(t) 2 1 T = T = 275 MeV T = 285 MeV T = 29 MeV T = 35 MeV T = 324 MeV T = 366 MeV T = 397 MeV T = 428 MeV T = 458 MeV T = 486 MeV t [fm] t [fm]

20 finite T - transverse component Below T C Above T c T = T = 121 MeV T = 162 MeV T = 194 MeV T = 243 MeV T = 26 MeV T = 265 MeV T = 275 MeV T = T = 285 MeV T = 29 MeV T = 35 MeV T = 324 MeV T = 366 MeV T = 397 MeV T = 428 MeV T = 458 MeV T = 486 MeV t [fm] t [fm]

21 scale transverse component t pos vio [fm] T [MeV]

22 Outline 1 2 How-to First results 3

23 Longitudinal propagator spectral densities

24 Outline 1 2 How-to First results 3

25 Why gluon mass? 2 1 F 1 (r,t)/t c.8 (a).7.6 T=158 MeV, Boltzmann c= a=b=c rσ 1/2 T/T c (D) N k/n M g [MeV] At T = we have colour screening and flux tubes, J. M. Cornwall, Phys. Rev. D 26, 1453 (1982) N. Cardoso, P. Bicudo, Phys. Rev. D 87, 3454 (213) N. Cardoso, M. Cardoso, P. Bicudo [arxiv: [hep-lat]] at large T Debye screening, M. Doring, K. Hubner, O. Kaczmarek, and F. Karsch, Phys. Rev. D 75, 5454 (27) M. Bluhm, B. Kampfer and K. Redlich, Phys. Rev. C 84, 2521 (211) at T c a mass scale in the π and K multiplicities in heavy ions P. Bicudo, F. Giacosa, E. Seel Phys.Rev. C86, 3497 (212)

26 Gluon mass at finite T naive M L and M T function of T Interpretation The simplest ansatz for a massive propagator is, 1 D(p) = p 2 + M 2 M = 1/ D()

27 Longitudinal screening mass comparison 1.5 Longitudinal screening mass All data renormalized at 2GeV, comparison with PRD85(212)3437 Coimbra Maas et al (212) 1/sqrt(D L ()) 1.5 Data renormalized at 2 GeV T/Tc

28 Gluon mass at finite T Fits of the longitudinal propagator for a better IR ansatz, we fit D i using a Yukawa fit with mass M D i (p 2 ) = Z p 2 + m 2 and look for the largest fitting range p max this fits quite well D L the Yukawa does not fit D T T p max Z L M L χ 2 /d.o.f (16).468(13) (89).3695(73) (5).381(22) (67).374(21) (86).371(25) (45).31(14) (65).4386(83) (12).548(16) (5).595(85) (8).59(32) (24).5656(63) (55).78(13) (34).795(11) (24).822(89) (37).95(13) (24).9285(97) 1.55

29 Gluon mass at finite T Z(T) m g (T) [GeV] T [MeV]

30 Conclusions and outlook & spectral density Gluon unphysical for all T up to 5 MeV scale increases with temperature Gluons behave as quasi-particles for high T? Gluon mass scale at finite temperature 1/ D L () IR Yukawa fit Preliminary results, keep in touch... Supported by FCT via project CERN/FP/123612/211. D.D. acknowledges financial support from the Research-Foundation Flanders (FWO Vlaanderen). P.S. supported by FCT grant SFRH/BPD/4998/27.