1. T=/=0 quark spectrum and crossover 1 Pedro Bicudo @ Bormio XLIX Pedro Bicudo CFTP, IST, Lisboa Motivation The finite T string tension  The quark mass gap equation and the chiral crossover at finite T and finite m0 Light and static-light meson spectra as a function of temperature T T=/=0 deconfinement, chiral crossover and meson spectrum

2. T=/=0 quark spectrum and crossover 2 Pedro Bicudo @ Bormio XLIX For ++ discussions, we also apply quark-gluon models and Lattice QCD to Temp=/=0, chiral symmetry, all sorts of exotics, molecules, excited hadrons and to surf tech Motivation

3. T=/=0 quark spectrum and crossover 3 Pedro Bicudo @ Bormio XLIX Motivation The main motivation is to contribute to understand the QCD crossover and the melting of the light hadron spectrum, for finite T, to be studied at LHC, RHIC and FAIR. But how to address the hadron spectrum, confinement and chiral symmetry?? - SU(3) lattice QCD, has difficulties with high excited spectra and finite ... - most models of QCD with chiral symmetry, say the Nambu and Jona-Lasinio model or the sigma models, have no explicit confinement, so this is a good job for the modern quark model that includes not only confinement but also chiral symmetry breaking and quark mass generation, notice we address here light quarks, not as easy as heavy quarks where, mb, mc >>  QCD, and mc >> Tc but mu, md, ms ~  QCD ~Tc ~ sqrt(  need the finite T propagator, the spontaneous chiral symmetry breaking, the relativisty of quarks and he hadronic coupled channels.

4. T=/=0 quark spectrum and crossover 4 Pedro Bicudo @ Bormio XLIX Motivation Illustration, thanks to FAIR SPS quarkyonic ? also RHIC, NA61/SHINE ?

5. T=/=0 quark spectrum and crossover 5 Pedro Bicudo @ Bormio XLIX NA60 data, thanks to Carlos Lourenço et al Motivation

6. T=/=0 quark spectrum and crossover 6 Pedro Bicudo @ Bormio XLIX Motivation Work program in progress: 0- Hadrons with heavy quarks at large T... I – Light quarks at large T and small  fit the string tension  at finite temperature from lattice data, compute dynamically the constituent quark running mass m(k) and the chiral condensate for any string tension  and current quark mass m 0. study the chiral crossover with the finite at finite T and m 0, compute the static-light spectrum at finite at finite T and m 0, compute the Polyakov loop at finite T and m 0, study the deconfinement crossover at finite T and m 0, compute the light meson spectrum at finite T and m 0, II – Small T and large  

7. T=/=0 quark spectrum and crossover 7 Pedro Bicudo @ Bormio XLIX Fits of the finite T string tension from the Lattice QCD energy F1 quark antiquark quantum confining flux tube or string The confinement, modelled by a string, is dominant at moderate distances V(r) ---->  + Vo +  r, at finite T log terms also occur 12 r At short distances we have the Luscher or Nambu-Gotto Coulomb due to the string vibration + the OGE coulomb, but since the Coulomb is important for UV renormalization but less important for chiral symmetry breaking, PB, PRD 2009 thus we will focus on the linear confinement  is the string tension

8. T=/=0 quark spectrum and crossover 8 Pedro Bicudo @ Bormio XLIX Fits of the finite T string tension from the Lattice QCD energy F1 For instance, this is the Lagrangian density E 2 -B 2 produced by three static hybrid colour charges computed in SU(3) lattice QCD Importantly we used no gauge fixing! Phys.Rev.D81:034504,2010 Lattice QCD computation of the colour fields for the static hybrid quark-gluon- antiquark system, and microscopic study of the Casimir scaling M. Cardoso, N. Cardoso, P. Bicudo

9. T=/=0 quark spectrum and crossover 9 Pedro Bicudo @ Bormio XLIX Profile of colour fields in the quark-antiquark system Fits of the finite T string tension from the Lattice QCD energy F1 For instance, this is the profile of the flux tube at T=0 computed in lattice QCD Importantly we used no gauge fixing! arXiv:1004.0166 Gauge invariant SU(3) lattice computation of the dual gluon mass and of the dual Ginzburg-Landau parameters and  in QCD N. Cardoso, M. Cardoso, P. Bicudo

10. T=/=0 quark spectrum and crossover 10 Pedro Bicudo @ Bormio XLIX Fits of the finite T string tension from the Lattice QCD energy F1 Actualy the string breaks at the energy of the first meson-meson decay channel, nevertheless the potential remains linear below the saturation energy, and that is what we now fit at finite T.

11. T=/=0 quark spectrum and crossover 11 Pedro Bicudo @ Bormio XLIX Notice that at infinite m 0 we have a confining phase transition, and at finite m 0 we have a crossover, that gets weaker and weaker when m 0 decreases: r F T (r) m 0 =0 m 0 finite 2m 0 saturation at T<Tc 2 m 0 m 0 >> sqrt(  Fits of the finite T string tension from the Lattice QCD energy F1 our fit region

12. T=/=0 quark spectrum and crossover 12 Pedro Bicudo @ Bormio XLIX Potential:V(r) = - f d r Free Energy:F 1 (r)= - f d r – S d T OK for isotermic Internal Energy:E 1 (r)= - f d r + T d S OK for adiabatic qq With Temperature: V olume  r P ressure  f Fits of the finite T string tension from the Lattice QCD energy F1 Flux Tube picture of confinement

13. T=/=0 quark spectrum and crossover 13 Pedro Bicudo @ Bormio XLIX The Polyakov loop is a gluonic path, closed in the imaginary time t 4 / inverse temperature T direction in QCD discretized in a periodic boundary euclidian Lattice. It measures the free energy F of one or more static quarks P = N Exp[ - F /T ] tr{P(0)P + (r)} singlet = N Exp[ - F /T ] Fits of the finite T string tension from the Lattice QCD energy F1 T x y

14. T=/=0 quark spectrum and crossover 14 Pedro Bicudo @ Bormio XLIX If we consider a single solitary lonely quark in the universe, in the confining phase, his string will travel as far as needed to look for a partner antiquark, resulting in an infinite energy F. Thus the 1 quark Polyakov loop P is a frequently used order parameter for deconfinement. Fits of the finite T string tension from the Lattice QCD energy F1 q T P(T) Tc However, since we are interessed in appoaching the deconfinement transition from below Tc, we prefer here to use the string tension  as the order parameter, computed in the quark-antiquark colour singlet double Polyakov loop P. we have a crossover with a finite quark mass or a phase transition in pure gauge QCD!

15. T=/=0 quark spectrum and crossover 15 Pedro Bicudo @ Bormio XLIX Fits of the finite T string tension from the Lattice QCD energy F1 For instance, this is the mean average Polyakov Loop in quanched SU(2) lattices, computed by Nuno Cardoso, with our GPU. We can generate 500,000 96 3 X N t configurations Just in one day! arXiv:1010.4834 SU(2) Lattice QCD Simulations on Fermi GPUs Nuno Cardoso, Pedro Bicudo Our Lattice QCD SU(2) data

16. T=/=0 quark spectrum and crossover 16 Pedro Bicudo @ Bormio XLIX Lattice QCD data, thanks to Olaf Kaczmarek et al. PRD (2007) solid line : T=0 qq potential V Fits of the finite T string tension from the Lattice QCD energy F1 Q Free Energy F1 obtained with 2 polyakov lops

17. T=/=0 quark spectrum and crossover 17 Pedro Bicudo @ Bormio XLIX Q Free Energy F1 Fits of the finite T string tension from the Lattice QCD energy F1 solid line : T=0 qq potential V Lattice QCD data, thanks to Olaf Kaczmarek et al. PRD (2007) obtained with 2 polyakov lops

18. T=/=0 quark spectrum and crossover 18 Pedro Bicudo @ Bormio XLIX Linear fir of the longer distance part of the fee energy F. We cut the short distance in such a way that the fit is cutoff independent. r  1/2 F(r)/Tc obtained with 2 polyakov lops Fits of the finite T string tension from the Lattice QCD energy F1

19. T=/=0 quark spectrum and crossover 19 Pedro Bicudo @ Bormio XLIX Comparing the string tensions at T=0, with the cond mat magnetization curve The magnetization curve of a magnetic material is a text book curve well modelled by the statistics of spin 1/2 systems. Here we show that the same curve also models the  sting tension and the deconfinement curve!   T/Tc solid line : 2nd order phase transition approximation dashed line : 1st order phase transition Kaczmarek et al PRD (2000) Phys.Rev.D82:034507,2010 The QCD string tension curve, the ferromagnetic magnetization, and the quark-antiquark confining potential at finite Temperature. P. Bicudo, Fits of the finite T string tension from the Lattice QCD energy F1

20. T=/=0 quark spectrum and crossover 20 Pedro Bicudo @ Bormio XLIX To study the chiral crossover we need to solve the mass gap equation. At the ladder/rainbow truncation of Coulomb Gauge QCD in equal time it reads, At finite T, one may replace the string tension,    and the mass gap equation is in units of . = - 0 The quark mass equation and the chiral crossover

21. T=/=0 quark spectrum and crossover 21 Pedro Bicudo @ Bormio XLIX Notice that the linear potential is infared divergent, and we need to regularize it when r  oo, equivalent to p  0, for instance we may use F. T. The quark mass equation and the chiral crossover

22. T=/=0 quark spectrum and crossover 22 Pedro Bicudo @ Bormio XLIX Also we use the Matsubara summation, in complex space, at finite T The quark mass equation and the chiral crossover

23. T=/=0 quark spectrum and crossover 23 Pedro Bicudo @ Bormio XLIX m 0 = 1.00 m 0 = 0.100 m 0 = 0.316 m 0 = 0.0316 m 0 = 3.16 may be measured in excited hadrons Phys.Rev.Lett.103:092003,2009 Probing the infrared quark mass from highly excited baryons P. Bicudo, M. Cardoso, T. Van Cauteren, F. Llanes-Estrada Solution of the mass gap equation at T=0, in units of  2 =0.19 GeV 2 =1 m(p) –m 0 The quark mass equation and the chiral crossover

24. T=/=0 quark spectrum and crossover 24 Pedro Bicudo @ Bormio XLIX Thus we get for the mass gap m(0) as a function of m 0 /  this curve. At finite T we just need to use the corresponding  arXiv:1007.2044 Chiral symmetry breaking solutions... with finite current quark masses P. Bicudo the mass gap m 0 /  m(0) The quark mass equation and the chiral crossover

25. T=/=0 quark spectrum and crossover 25 Pedro Bicudo @ Bormio XLIX At vanishing m 0 we get a chiral symmetry phase transition, and at finite m 0 we have a crossover, that gets weaker and weaker when m 0 increases: The quark mass equation and the chiral crossover m 0 = m u m 0 = m d m 0 = 0 This is the opposite behaviour of the confinement crossover with m 0 and thus the chiral and confinement critical points should NOT, in principle, be coincident the chiral crossover

26. T=/=0 quark spectrum and crossover 26 Pedro Bicudo @ Bormio XLIX We now compute the hadron spectrum, in particular the meson spectra. Notice that with T=/=0, the masses and wavefunctions do essentially scale with  (T), and thus also essentially vanish at T=Tc. The light hadron spectrum T light hadron mass M: dominated by the linear confining potential heavy hadron mass M: the Coulomb potential still binds above Tc T Tc 2m 0 melting point

27. T=/=0 quark spectrum and crossover 27 Pedro Bicudo @ Bormio XLIX The static-light is also interesting since, when dynamical quarks are included, the static quark-antiquark potential saturates when it reaches the energy for the creation of a quark-antiquark that produces a pair of static-light mesons. The lightest static-light meson has a mass M Sl larger than m 0. We also include hybrid, or flux tube excitations in the chiral confining quark model, and we get the Salpeter equation for the light quark with a static antiquark, Phys Lett B 127, 106 (1983), The exact qq potential in Nambu string theory, J. F. Arvis Eur. Phys. Jour. A 29, 343, Hybrid mesons with auxiliary fields, F. Buisseret, V. Mathieu Sq lqlq The light hadron spectrum

28. T=/=0 quark spectrum and crossover 28 Pedro Bicudo @ Bormio XLIX At Tc the string tension  (T) vanishes, the confining potential disappears, and thus all light hadrons decrease their mass until they melt at T=Tc. Chiral symmetry is not restored because m 0 / sqrt(  (T)) increases with T. The quark mass and Chiral symmetry and confinement crossovers T MM Tc MM MNMN M...

29. T=/=0 quark spectrum and crossover 29 Pedro Bicudo @ Bormio XLIX Conclusion & Outlook on light q with  symmetry and confinement We remark that the pure gauge string tension  (T) is well fitted by the condensed matter physics magnetization curve , and utilize it. We compute the dynamically generated quark mass m(p), solving the mass gap equation both for finite current quark masses m 0 and for finite T. The finite current quark masses turn both the confinement and the chiral symmetry phase transitions into two different crossovers. We study the full spectra of light hadrons at finite T a, including the excited spectra, and conclude that 1) the light hadron masses essentially vanish at T=Tc, 2) the light hadrons all melt at T=Tc, 3) the masses and wavefunctions essentially scale with  (T).