A Model Study on Meson Spectrum and Chiral Symmetry Transition

1. A Model Study on Meson Spectrum and Chiral Symmetry Transition Da Huang @NTU

2. Content • General Overview • Dynamically Generated Spontaneous Chiral Symmetry Breaking In Chiral Dynamical Model At Zero Temperature • Chiral Thermodynamic Model of QCD and its Critical Behavior

3. QCD Lagrangian and Symmetry Chiral limit: Taking vanishing quark masses mq→ 0. QCD Lagrangian has maximum global Chiral symmetry :

4. The Reality 插入meson spectrum 图片 No so many symmetries in meson spectrum.

5. The Reality much heavier than other light pseudoscalars.

6. The Problem 1. Howare these (exact or approximate)symmetries in QCD broken (explicitly or spontaneously)? 2. Can we find some fundamental laws that govern the seemingly messy meson spectrum?

7. Some Hints 1. Pseudoscalars are much lighter than their chiral partners are Goldstone bosons of spontaneous chiral symmetry breaking 2. U(1)_A symmetry is anomalous, so there is no such symmetry at low energy. is not the Goldstone boson any more.

8. Previous Efforts • Lattice Calculations Advantage: first principle Shortcoming: lack transparency; Nielsen-Ninomiya No-Go theorem to limit the dynamical fermion • Model Calculation: NJL model Advantage: easy to understand chiral symmetry breaking Shortcoming: difficult to derive it from first principle • AdS/CFT Models: Model dependent

9. Finite Temperature Effects • Just as magnets When temperature is high enough, then the symmetry will be restored. -- general for any SSB system

10. Finite Temperature Effects • We expect that chiral symmetry should be restored at high enough T • Main Question: What is the nature of chiral symmetry restoration? ?Phase transition OR Crossover ? If phase transition, what is the order

11. Previous Efforts • In the chiral limit, general linear sigma-model predict: Pisarski & Wilczek 84' 2 flavors: 2nd Order; 3 or more flavors: 1st Order (no IR FP) • current quark mass term complicates the situation: Lattice QCD: Crossover Aoki et al 05' Nature Field Theory Model: mostly Crossover, NJL, PNJL, Quark-Meson model,... (but Model dependent) AdS/CFT model: model dependent

12. Big Bang Cosmology • Chiral symmetry breaking happens at 10^-6 s (T~200 MeV) after the Big Bang • The net baryon density in Universe is quite small(<< typical hadron mass), so • Effects: 1. Heat other particles due to entropy conservation; 2. If a strong first-order phase transition, then bubbles of hadron gas are formed. The bubble growth, collision and mergence can generate gravitational waves 3. The initial inhomogeneities have strong effects on BBN

13. Big Bang Cosmology However, if QCD transition is a crossover, as indicated by Lattice QCD and many model calculations, then it is difficult to find astrophysical evidence for this transition In summary, the strength of the QCD transition determines its significance in the evolution of our Universe.

14. Dynamically Generated Spontaneous Chiral Symmetry Breaking In Chiral Dynamical ModelAt Zero Temperature

15. QCD Lagrangian and Symmetry QCD Lagrangian with massive light quarks

16. Effective Lagrangian Based on Loop Regularization Y.B.Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004) Note that the field Φ has no kinetic terms, so it is an auxiliary field and can be integrated out.

17. The Derivation of Effective Action Instead, if we integrate out the quark fields, we can obtain the effective Lagrangian for mesons. Let us focus on the quark part of the Lagrangian The path integral method gives

18. The Derivation of Effective Action The Effective Action is Note that the imaginary part corresponds to the anomaly part, which we ignore here.

19. The Derivation of Effective Action With the identities: and we can obtain: where

20. The Derivation of Effective Action In fact, we already obtain the desired form of the action. But a further step is need for generalization to finite temperature If we integrate chi-field back, we obtain

21. The Derivation of Effective Action In order to take into account the constitute quark mass contribution, we can split the Lagrangian into two parts Have choosing

22. The Derivation of Effective Action Integrating out the momentum integrals gives where

23. Dynamically Generated Spontaneous Symmetry Breaking

24. Dynamically Generated Spontaneous Symmetry Breaking By differentiating the effective potential w.r.t. the VEVs, we obtain the following Minimal Conditions/Generalized Gap Equations: If we take the limit of vanishing instanton effects, we can recover the usual gap equation

25. Dynamically Generated Spontaneous Symmetry Breaking Recall that the pseudoscalar mesons, such as π,η, are much lighter than their scalar chiral partners. This indicates that we should choose our vacuum expectation values (VEVs) in the scalar fields:

26. Scalars as Partner of Pseudoscalars According to their quantum numbers, we can reorganize the mesons into a matrix form. Scalar mesons: Pseudoscalar mesons :

27. Mass Formula Pseudoscalar mesons : Instanton interactions only affect mass of SU(3) singlet

28. Mass Formula

29. Predictions for Mass Spectra & Mixings

30. Predictions

31. Chiral Symmetry Breaking -- Answer to Our Primary Question • Chiral SUL(3)XSUR(3) spontaneously broken Goldstone mesons π0, η8 • Chiral UL(1)XUR(1) breaking Instanton Effect of anomaly Mass of η0 • Flavor SU(3) breaking The mixing of π0, η and η׳

32. Chiral Thermodynamic Model of QCD and its Critical Behavior

33. Motivation Provided that chiral dynamical model works so well to show chiral symmetry breaking and to predict the meson spectrum, we want to see what happens for this model at finite temperature --Chiral symmetry will be restored at high enough temperature

34. Effective Lagrangian by DH, Y-L Wu, 1110.4491 [hep-ph]

35. Closed-Time-Path Formalism Closed-Time-Path formalism makes the finite temperature effective Lagrangian rather simple by just replacing zero-temperature propagator with its finite temperature counterpart

36. Finite Temperature Effective Action

37. Dynamically Generated Spontaneous Symmetry Breaking

38. Dynamically Generated Spontaneous Symmetry Breaking By differentiating the effective potential with respect to the VEVs, we can obtain: After we take the limit of zero current quark masses, the gap equation can be simplified to

39. Critical Temperature For Chiral Symmetry Restoration In order to determine the critical temperature for chiral symmetry restoration, we need to make the following assumption: Note that μf2 (T) is the coupling of the four quark interaction in our model. In principle this interaction comes by integrating out the gluon fields. Thus, this assumption indicates that the low energy gluon dynamics have the same finite temperature dependence as the quark field.

40. Critical Temperature For Chiral Symmetry Restoration With the above assumption, we can determine the critical temperature for chiral symmetry restoration in the chiral thermodynamic model (CTDM)

41. Dynamically Generated Spontaneous Symmetry Breaking Recall that the pseudoscalar mesons, such as π,η, are much lighter than their scalar chiral partners. This indicates that we should choose our vacuum expectation values (VEVs) in the scalar fields:

42. Scalars as Partner of Pseudoscalars & Lightest Composite Higgs Bosons Scalar mesons: Pseudoscalar mesons :

43. Mass Formula Pseudoscalar mesons : Scalar mesons (Lightest Composite Higgs Boson) :

44. Predictions for Mass Spectra

45. Chiral Symmetry Restorationat Finite Temperature Vacuum Expectation Value (VEV)

46. Chiral Symmetry Restorationat Finite Temperature Pion Decay Constant

47. Chiral Symmetry Restorationat Finite Temperature Quark Condensate

48. Chiral Symmetry Restorationat Finite Temperature Mass for Pseudoscalar mesons

49. Chiral Symmetry Restorationat Finite Temperature This feature can be traced back to the fact that all of these quantities are proportional to the VEV near the critical temperature All of these quantities have the same scaling behavior near the critical temperature,

50. Implications of Our Study • As discussed before, if the QCD transition is a true 2nd phase transition, then there must be some dramatic changes in the physical quantities, thus it can leave some imprint in the early universe • gravitational waves • inhomegeneity in the BBN • ...