QCD phase transition. One of the fundamental challenges in modern physics. Non-perturbative phenomena. No theoretical control at µ > 0 so far...

Transcription

1 QCD phase transition Ø One of the fundamental challenges in modern physics Ø Non-perturbative phenomena Ø No theoretical control at µ > 0 so far

2 Chiral Random Matrix (ChRM) model and U A (1) anomaly Before we started our study..., Hope: The simplest model for dynamical breaking of chiral symmetry no spacetime D.O.F. may reveal the most common features of chiral phase transition Problem: In mean-field models, the KMT term is responsible for U A (1) breaking; But it was unknown how to 2 implement it, and then no flavor dependence

3 Outline Introduction Chiral Random Matrix (ChRM) Model in brief Incorporating the UA(1) anomaly term Phase diagram at finite T/mu Meson condensation at finite mu at T=0 Conclusions & Outlook 3 3

4 2. Chiral Random Matrix Model in brief 4 4

5 QCD & Chiral Random Matrix Theory The QCD partition function Review: Verbaarschot-Wettig Chiral symmetry breaking (χsb) ρ(λ) λ χsb is characterized by low-lying Dirac modes, which are generated by complex YM dynamics 5

6 QCD & Chiral Random Matrix Theory The QCD partition func. ChRM theory Dirac operator matrix with constant elements distributed Gaussian-randomly Equivalence to QCD in the ε regime is known: m π << 1/L << m ρ Size of matrix ( 2N x 2N ) is proportional to box size, L 4 Application: (1) Universal spectral correlation (2) A model for QCD at N infinity 6

7 Model of QCD: effective Potential Ω Shuryak & Verbaarschot (1993) Integration over W Bosonization In thermodynamic limit & equal mass case Chirally broken ground state Nf is factorized Broken phase 7

8 Finite Temperature extension Jackson & Verbaarschot (1996) Introduce a non-random external field t respecting symmetry effective potential Symmetry restoration at finite T 2 nd order for any Nf Deteminant int is needed to study flavor dependence 8

9 Extension to Finite T & µ Halasz et al. (1998) t : respects symmetry µ : breaks hermiticity T m Independent of Nf 9 µ

10 3. Implementing the UA(1) anomaly term 10

11 Index theorem in ChRM model non-zero eigenvalues appear in pair with + chiralities Exception: zero eigenvalues Index theorem: N+ : #(eigenmodes), ν : topological # of gauge configuration Total partition fn is obtained by summing over ν (w/angle θ) In ChRM model, we deal with N x (N +ν) matrix W, which has ν exact-zero eigenvalues 11

12 Extension of Zero-mode Space Janik, Nowak & Zahed (1997) Sano, HF, Ohtani (2009) Divide low-lying modes into two categories N+, N- : Topological (quasi-) zero modes = associated with instantons(?) 2N : Near zero modes near-zero mode 12 The last term gives a phase e 2iNf ν θ when S S e 2iθ

13 Complete partition fn. Sum over ν (Ι) Janik, Nowak & Zahed (1997) If modes are associated to instantons, Poisson dist may be reasonable 't Hooft (1986) KMT-type UA(1) breaking term, appears Potential is unbound φ 3 term wins at large φ 13

14 Complete partition fn. Sum over ν (ΙΙ) T. Sano, HF, M. Ohtani (2009) With the Poisson,??? model restricted in IR modes in finite V The most natural regularization : Binomial dist 1-p p cells p: occupation prob Z is a polynomial (except for Gauss weight) KMT int. appears within the log. Stable ground state 14

15 15 4. Phase diagram at finite T & µ

16 Nf Dependent Thermal Phase Transition Chiral condensate Σ=1, α=0.3, γ=2 Nf=2 Nf=3 16

17 Nf Dependent Thermal Phase Transition Meson (curvature) mass Σ=1, α=0.3, γ=2 Singlet pseudo-scalar meson is massive by anomaly 17

18 Singularity at the critical point mud=0.01 & ms=0.2; α=0.5, γ=1 Only σ becomes massless Cf. Susceptibilities χ mm, χ µµ, χ TT all diverge at the CP 18

19 Topological susceptibility at finite T 19

20 Topological susceptibility at finite T Σ=1, α=0.3, γ=2 follows the chiral condensate 20

21 2+1 flavor phase diagram: µ=0 plane TCP crossover 1st order The stronger the KMT term, the wider the 1 st order region Expt 2/5 is understood in the Landau potential 21

22 Critical Surface 1 st order region expands as µ increses Familiar situation with constant KMT coupling α=0.5 & γ=1 O(4) criticality 22

23 If α depends on µ... 2 γ=1, α0=0.5, & µ0= Bending is possible, but model dependent

24 24 5. Meson condensation at T=0 at finite µ

25 Meson condensation at finite µ I HF, T. Sano, Cf. B. Klein, et al. (2003) α=0.5 & γ=1 Gap eq for ρ (m=0) At large µ I, a pion condensed phase appears At small µ I, a single chiral transition along µ q 25

26 Meson condensation at finite µ I & µ Y HF, T. Sano, Cf. Araki, Yoshinaga, (2008) Kaon condensed phases appear in some regions Competition among chiral restoration&meson condensations 26

27 5. Summary Chiral Random Matrix model with UA(1) anomaly is constructed as a mean-field model for QCD Chiral condensate, meson masses, topological susceptibility behave 'physically' as T&µ change Phase diagram in T-µ plane is qualitatively the same as other mean-field models Meson condensation is studied as a response to chemical potentials Chiral Random Matrix model is a useful toy model for QCD 27