New Critical Point Induced by the Axial Anomaly in Dense QCD

1. New Critical Point Induced by the Axial Anomaly in Dense QCD N. Yamamoto (Univ. of Tokyo) T. Hatsuda (Univ. of Tokyo) M. Tachibana (Saga Univ.) G. Baym (Univ. of Illinois) hep-ph/0605018 RIKEN Workshop on "Frontiers in the physics of quark-gluon plasma" 2006.7.9

2. Outline • Introduction and motivation • The basic idea of Ginzburg-Landau approach • Application of GL approach to the chiral & diquark condensates • Results on QCD phase diagram • Summary and future problems

3. Introduction T RHIC & LHC Quark-Gluon-Plasma Hadrons ? Color superconductivity Neutron star & Quark star m What happens in this unknown region of the QCD phase diagram?

4. Motivation Quark-hadron continuity? Schäfer & Wilczek, Phys. Rev. Lett. (1999) Interplay between chiral & diquark order parameters by using model-independent approach. Hadron & CSC phases can be continuously connected because of the 1 to 1 correspondence of the elementary excitations between two phases. Ginzburg-Landau approach

5. The basic idea of Ginzburg-Landau approach

6. Ginzburg-Landau approach • Model-independent approach based only on the symmetry. • Free-energy is expanded in terms of the order parameter Φ (such as the magnetization) near the phase boundary. Ising model h=0 Z(2) symmetry : m ⇔－m

7. This system shows 2nd order phase transition. unbroken phase (T>Tc) broken phase (T<Tc) GL free-energy Z(2) symmetry allows even powers only. • This shows a minimal theory of the system. • b(T)>0 is necessary for the stability of the system. • a(T) changes sign at T=TC. → a(T)=k(T－Tc) k>0, Tc: critical temperature Whole discussion is only based on the symmetry of the system. (independent of the microscopic details of the model) GL approach is a powerful and general method to study the critical phenomena.

8. Application of GL approach to chiral & diquark condensates

9. GL approach for chiral & diquark fields • The chiral field Φ • Pisarski & Wilczek, Phys. Rev. D (1984) • The diquark field d • Iida, Matsuura, Tachibana & Hatsuda Phys. Rev. Lett (2004) We need to construct the coupling terms of Φ & d to study their interplay . In this GL approach, the guiding principle is the QCD symmetry G = SU(Nf)L×SU(Nf)R×U(1)B×U(1)A×SU(3)C. Axial-anomaly We consider two cases; 3-flavor (mu=md=ms=0), 2-flavor (mu=md=0, ms=∞)

10. GL approach for the chiral phase transition Chiral order parameter Axial anomaly(U(1)A→Z6) GL free energy of the chiral field Diagonal Ansatz for Φ: Reduced GL free-energy of σ The chiral phase transition shows 1st order by the cubic term.

11. GL approach for the CSC phase transition Diquark order parameter GL free energy of the diquark field Diagonal （CFL） Ansatz for d : Reduced GL free-energy of d The normal-CSC phase transition shows 2nd order.

12. GL approach for the interplay GL free-energy of the coupling (up to 4th order) Axial-anomaly (U(1)A→Z6) Reduced GL free-energy of σ‐d coupling Overall reduced GL free-energy for 3-flavors Axial-anomaly This term is neglected. Microscopic calculation (in weak coupling)

13. NG phase CSC phase Normal phase COE phase Reduced GL free-energy for 3-flavors The 4 possible phases We can locate phase boundaries & the order of the phase transitions by comparing the free energies.

14. COE phase : Z2 Crossover in terms of the QCD symmetry COE phase : Z2 γ-term : Z6 CSC phase : Z4 Once γ-term caused by the axial anomaly is present, the COE & CSC phases can’t be distinguished by symmetry and can be continuously connected!

15. Results –QCD phase diagram-

16. 1st order line terminates at a critical point A. A tricritical point D appears when 2nd order boundary splits into two. The area of the COE phase grows. : 1st order : 2nd order Phase diagram (3-flavor) The QCD axial anomaly induces a crossover between the COE & the CSC phases, and leads to a new critical point A!

17. mu,d=0, ms=∞ mu,d s=0 Critical point 0<mu,d<ms<∞ New critical point Speculative QCD phase diagram • A new critical point emerges in the real QCD phase diagram. • This crossover may be relevant to “quark-hadron continuity”. CSC

18. Summary • We construct the general GL free-energy of the σ & dincluding their coupling terms allowed by QCD symmetry. • The axial anomaly for 3-flavors acts as an external field for σ, and leads to a crossover between the hadron & CSC phases, and to a new critical point! • This new critical point may survive in the QCD phase diagram with realistic quark masses. →Relevance to “quark-hadron continuity”

19. Future problems • More direct connection to “quark-hadron continuity” • Elementary excitations such as meson, baryon… (in progress) • Effects of quark masses, β-equilibrium, charge neutrality…? • The effect of quark confinement • Where is the location of the new critical point? → phenomenological models & lattice QCD in future • Connection with the compact star

20. Back up slides

21. Phase diagram (2-flavor) b>0 b<0

22. stationary condition The emergence of the point A The effective free-energy in COE phase Modification by the λ-term

23. Coordinates of the characteristic points in the a-α plane 3-flavor 2-flavor (b>0)