1. Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density H. Ohno for WHOT-QCD Collaboration The 3rd A01 group workshop CCS, University of Tsukuba, July 7, 2010

2. H. Ohno for WHOT-QCD Collaboration Plan of this talk Introduction – QCD phase diagram – Scaling behavior of QCD Scaling study with Wilson quarks – Scaling behavior at finite μ – Derivative of chiral order parameter with respect to μ/T Numerical results Conclusion and future plan 1 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density

3. H. Ohno for WHOT-QCD Collaboration QCD phase diagram 2 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density   Hadron gas Quark-gluon-plasma critical end point Crossover χSB Early universe Heavy ion collisions Neutron star Important to understand

4. H. Ohno for WHOT-QCD Collaboration Critical behavior of QCD at zero chemical potential 3 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density m ud 0 0   Physical point 1 st order 2 nd order Crossover 1 st order msms QuenchedN f =2 N f =3 O(4 ) Tri-critical point m tc Z(2) Interesting subject: O(4) scaling at m s >> m tc Tri-critical point 1st order phase transition at m s << m tc

5. H. Ohno for WHOT-QCD Collaboration Scaling behavior of chiral order parameter in 2-flavor QCD 4 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density Chiral symmetry of 2-flavor QCD 3-d O(4) Heisenberg model [Wilczek, Intn.J.Mod.Phys. A7('92)3911; Rajagopal, Wilczek, NP B399('93)395] Recent lattice studies Wilson quarks Local for any N f Conserved flavor symmetry Explicit chiral symmetry breaking Higher computational cost KS quarks Non-local except for N f = 4 Explicit flavor symmetry breaking Partly conserved chiral symmetry Lower computational cost Consistent with O(4) scaling O(2) Simulation at physical point Subtraction and renormalization [CP-PACS, PRD63, 034502(2000)] O(N) scaling? [BNL-Bielefeld-GSI, arXiv:0909.5122]

6. H. Ohno for WHOT-QCD Collaboration Recent study with Wilson quarks 5/20 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density [CP-PACS, PRD63, 034502(2000)] Iwasaki improved gauge Clover improved Wilson quark N f = 2 16 3 ×4 lattice O(4) scaling function

7. H. Ohno for WHOT-QCD Collaboration Critical behavior of 2-flavor QCD at finite density 6 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density m ud  0 m ud =0   O(4) m ud Quenched 0 0   1 st order 2 nd order Crossover 1 st order  m ud   Interesting subject: O(4) scaling at small μ Curvature of critical or crossover line Tri-critical point at finite μ N f = 2

8. H. Ohno for WHOT-QCD Collaboration Our study 7 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density We investigate scaling behavior of 2-flavor QCD at finite chemical potential with Wilson quarks using derivative of chiral order parameter calculating curvature of 2nd order critical line in β-μ plane

9. H. Ohno for WHOT-QCD Collaboration Comparison with 3-d O(4) spin model 8 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density 3-d O(4) spin model magnetization: M external field: h reduced temperature: t 2-flavor QCD chiral order parameter: ud-quark mass: m q critical value of β: β ct Scaling function:

10. H. Ohno for WHOT-QCD Collaboration Chiral order parameter with Ward-Takahashi identity 9 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density [Bochichio et al., NPB262, 331 (1985)] Ward-Takahashi identities in the continuum limit Wilson quarks : explicit chiral symmetry breaking Ward-Takahashi identities : Tree level renormalization factor A proper subtraction and renormalization are required.

11. H. Ohno for WHOT-QCD Collaboration Scaling behavior for finite density 10 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density The critical line runs on m q = 0 axis in (m q,μ) plane t=0 μq/Tμq/T m q =0 β c is the curvature of the critical ine in the (β, μ q /T) plane. To confirm this scaling property, we calculate the second derivative of with respect to μ q /T.

12. H. Ohno for WHOT-QCD Collaboration Derivative of chiral order parameter 11 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density These operators can be calculated by random noise method

13. H. Ohno for WHOT-QCD Collaboration Simulation setup Action – Iwasaki improved gauge action – Clover improved Wilson fermion action – N f =2 Lattice size – 16 3 ×4 # conf. – 500-600 # noise vectors – 50 for each color and spin indices 12 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density βΚT/T pc 1.800.1411390.93 1.850.1400700.99 1.900.1388171.08 1.950.1377161.20 βΚT/T pc 1.800.1451271.07 1.850.1435021.18 1.900.1418491.32 1.950.1404721.48 m ps /m v = 0.65 m ps /m v = 0.80

14. H. Ohno for WHOT-QCD Collaboration Numerical result at zero μ (1) 13 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density Filled symbols: CP-PACS, PRD63, 034502(2000) Open symbols: New data in this study m q : measured in T=0 simulations. as a function of

15. H. Ohno for WHOT-QCD Collaboration Numerical result at zero μ (2) 14 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density β,δ: critical exponents of O(4) spin model. Filled symbols: CP-PACS, PRD63, 034502(2000) Open symbols: New data in this study Dashed line: O(4) scaling function [Toussaint,’97] Comparison with O(4) scaling function Scaling behavior is consistent with 3-d O(4) spin model.

16. H. Ohno for WHOT-QCD Collaboration Numerical result at finite density (1) 15 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density c =0.05 c =0.04 c =0.03 c =0.02 Second derivative of Solid line: O(4) scaling function with Scaling behavior is roughly consistent with our expectation.

17. H. Ohno for WHOT-QCD Collaboration Numerical result at finite density (2) 16 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density   To calculate the curvature of T c (μ q ), we need a(dβ/da)in the chiral limit. Curvature of β c (μ q )

18. H. Ohno for WHOT-QCD Collaboration Conclusion Scaling behavior of chiral order parameter is investigated in two-flavor QCD. At zero chemical potential, – it is consistent with 3-d O(4) spin model. At finite density, – chemical potential dependence for scaling variables is almost consistent with our expectation – and the curvature of second order critical line in β-μ plane is roughly calculated. 17 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density

19. H. Ohno for WHOT-QCD Collaboration Future plan More precise scaling study – increasing the statistics and the number of noise vectors Lighter m ud 2+1-flavor simulation 18 Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density