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Non-Perturbative Renormalization Group approach to Dynamical Chiral Symmetry Breaking in extreme environments Kanazawa University Masatoshi Yamada with Ken-Ichi Aoki and Daisuke Sato arXiv:1404.3471 Introduction • Dynamical (DπSB) Chiral Symmetry Breaking ο The origin of Hadrons mass βͺ Current quarks masses: ππ = 3~6 MeV Constituent quarks masses: ππ = 300 MeV ο At finite temperature and density ο Restore chiral symmetry ο Phase diagram on π − π plane Lattice simulation Temperature LHC Quark-Gluon Plasma The lattice simulation has the sign problem. Our aims: RHIC The effective model analysis • Approach to DπSB at finite temperature and density. The mean field approximation • Beyond the mean field approximation. Schwinger-Dyson equation Hadrons 1/π expansion …etc. have difficulties in the further improvement of the approximation. Atomic nuclei Baryon density Neutron stars Talk plan NPRG 1. i. ii. The basic ideas DπSB via NPRG(π = π = 0) At finite temperature and density 2. i. Flow structures of the four-fermi coupling constant • • ii. iii. 3. large-N leading vs. non-leading Interpretation of the inverse four-fermi coupling constant Phase diagram on π − π plane How to investigate the order of phase transition in NPRG? At finite temperature and in external magnetic field i. ii. iii. Why external magnetic field? Magnetic catalysis vs. Magnetic inhibition Phase diagram on π − π΅ plane Part I Non-Perturbative Renormalization Group and Dynamical Chiral Symmetry Breaking Basic ideas Lower mode Infrared limit Wilsonian effective action Higher mode RG equation Shell modes Lower modes • The γ»γ»γ» γ» γ» γ» γ» F. Wegner and A Houghton. Phys. Rev. A8 (1973) 401 J. Polchinski Nucl. Phys. B231 (1984) 269 shell mode integration generates the change of effective action. Other formulation • RG eq. for the Legendre effective action IRοΌ UVοΌ C. Wetterch Phys. Lett. B301 (1993) 90 ο One-loop exact ο π Λ (π) is cutoff function. ο The propagator: ο Suppress the lower modes with π < Λ. ο The higher modes with Λ < π < Λ0 are integrated out. Approximation methods ο Ex: The scalar theory with π2 symmety at Λ 0 οDerivative expansion οLocal Potential Approximation(LPA) Non-Perturbative RG ο RG equation for the effective potential: The partial differential equation with the initial condition: οTruncation Need to truncate the expansion to some finite order γ»γ»γ» The infinite coupled ordinary differential equations with the initial condition: πΛπ , πΛ0 , πΛ0 = πΛ0 = β― = 0 DπSB • NJL model ο The chiral effective model ο Describe DπSB of QCD ο Invariant under chiral π(1)πΏ × π(1)π trans. ο The four-fermi coupling constant πΊ is fluctuation of order parameter: : the chiral susceptibility DπSB 2nd order transition DπSB via NPRG • The four-fermi interaction generates the effective four-fermi interactions by the shell mode integration. LPA + Truncation DπSB via NPRG • 1-loop diagram correction ο Distinguish Wick contractions Large-N leading part Large-N approximation(π → ∞) DπSB via NPRG • NPRG equation for π Canonical scaling Fixed Points: Ken-Ichi Aoki et al. Prig. Theor. Phys. 97 (1997) 479 Quantum correction DπSB via NPRG • The solution DπSB via NPRG • The flow structure of π DπSB The initial value: π0 ο The four-fermi coupling constant diverges at finite scale π‘c when π0 > π∗ . DπSB via NPRG • Solving RG eq. the large-N leading approximated γ»γ»γ» This is an infinite sum of the “tree” diagrams. The leading of 1/π expansion (mean field approximation) DπSB via NPRG • Solving RG eq. the large-N non-leading extended γ»γ»γ» γ»γ»γ» … … … γ»γ»γ» DπSB via NPRG • The large-N non-leading extended RG eq. contains the all 1/π orders. ο The leading order is included exactly. ο The non-leading orders are included partly. • In the next part, we investigate the nonleading effect at finite temperature and density. Talk plan NPRG 1. i. ii. The basic ideas DπSB via NPRG(π = π = 0) At finite temperature and density 2. i. Flow structures of the four-fermi coupling constant • • ii. iii. 3. large-N leading vs. non-leading Interpretation of the inverse four-fermi coupling constant Phase diagram on π − π plane How to investigate the order of phase transition in NPRG? At finite temperature and in external magnetic field i. ii. iii. Why external magnetic fields? Magnetic catalysis vs. Magnetic inhibition Phase diagram on π − π΅ plane Part II DπSB at finite temperature and density DπSB at π ≠ 0, π ≠ 0 • The action • The shell mode integration at π ≠ 0, π ≠ 0. DπSB at π ≠ 0, π ≠ 0 • NPRG eq. for one-flavor and one-color NJL model No contribution No contribution Negative sign: suppress DπSB Thermal effect functions(threshold functions) DπSB at π ≠ 0, π ≠ 0 •π = 0 limit fermi surface ο πΌ1 (0, π) has the singularity at the fermi surface. DπSB at π ≠ 0, π ≠ 0 The non-leading effect becomes very large at low temperature. DπSB at π ≠ 0, π ≠ 0 • Solving RG equations ο Analysis method: Make the equation for the inverse four-fermi coupling constant. : large-N leading app. eq. : non-leading extended eq. DπSB: DπSB at π ≠ 0, π ≠ 0 • The flow structure(large-N leading app.) DπSB at π ≠ 0, π ≠ 0 • The phase diagram(large-N leading app.) DπSB at π ≠ 0, π ≠ 0 • The flow structure(non-leading extended) DπSB at π ≠ 0, π ≠ 0 • The phase diagram(non-leading extended) DπSB at π ≠ 0, π ≠ 0 • The phase diagram(leading vs. non-leading) DπSB at π ≠ 0, π ≠ 0 • The flow structure(large-N leading app.) The flow continues after DπSB. DπSB DπSB at π ≠ 0, π ≠ 0 • RG flow: π vs. π οΌ How to interpret the flows of π after the critical scale π‘c ? DπSB at π ≠ 0, π ≠ 0 • How 1. to interpret the flows of π ? Method 1: Weak solution method Ken-Ichi Aoki, Shin-Ichiro Kumamoto and Daisuke Sato. arXiv:1403.0174 ο We can define the flows of divergent π in a mathematically meaningful manner. 2. Method 2: Bosonization(H-S trans.) J. Hubbard, Phys. Rev. Lett. 3 (1959) 77 R. Stratonovich, Dokl. Akad. Nauk SSR 115 (1957) 1097 ο Investigate the bosonic effective potential DπSB at π ≠ 0, π ≠ 0 • Hubbard-Stratonovich transformation DπSB at π ≠ 0, π ≠ 0 • The inverse four-fermi coupling constant corresponds to the curvature of the effective potential at the origin. DπSB at π ≠ 0, π ≠ 0 • The 1st order transition case ο The 1st order transition keeps the positive curvature at origin. ο The four-fermi coupling constant doesn't diverge. ο We cannot judge whether 1st order transition or symmetric in this analysis. DπSB at π ≠ 0, π ≠ 0 • How to investigate the order of phase transition? 1. Method 1: Weak solution method : We don’t need to introduce the additional degrees of freedom to system. : We can use it in only the large-N leading approximation at present. 2. Method 2: Bosonization(H-S trans.) : We can investigate the beyond large-N effect. : We need to introduce the additional degrees of freedom to system. DπSB at π ≠ 0, π ≠ 0 • The bosonized action Bosonization Beyond LPA + Truncation Bosonic effective action DπSB at π ≠ 0, π ≠ 0 • RG equations: Complicate… ο The mesonic potential ο The Yukawa coupling constant ο The four-fermi coupling constant ο The field renormalization factors DπSB at π ≠ 0, π ≠ 0 • Consider only the mesonic effective potential running ο This is so-called Quark-Meson model in LPA. ο Symmetry breaking pattern: O(N)→O(N-1) ο Fix the Yukawa coupling constant for simplicity ο Solving this equation as a partial differential equation. ο The initial condition: B.J. Schaefer, Nucl. Phys. A757(2005) 479 2nd order transition 1st order transition Phase diagram Summary and Outlook • We analyzed NJL model ο The flow structures of the four-fermi coupling constant πΊ ο The inverse πΊ corresponds to the curvature of the effective potential at the origin. ο We cannot distinguish between the single well potential and the 1st order potential in this analysis. ο Bosonized NJL model ο Investigated the order of phase transition ο There are five RG partial differential equations ο Full evaluation is under calculation ο Study how the chiral phase structure changes at high density. Talk plan NPRG 1. i. ii. The basic ideas DπSB via NPRG(π = π = 0) At finite temperature and density 2. i. Flow structures of the four-Fermi coupling constant • • ii. iii. 3. large-N leading vs. non-leading Interpretation of the inverse four-fermi coupling constant Phase diagram on π − π plane How to investigate the order of phase transition in NPRG? At finite temperature and in external magnetic field i. ii. iii. Why external magnetic fields? Magnetic catalysis vs. Magnetic inhibition Phase diagram on π − π΅ plane Part III DπSB at finite temperature and external magnetic field Temperature LHC Quark-Gluon Plasma RHIC Hadrons External magnetic field Atomic nuclei Baryon density Neutron stars Why external magnetic field? • Heavy-Ion Collision Elliptic flow • Neutron Magnetar star The strong magnetic field influence the phase structure of QCD matter. What’s happen in external B? • Dirac equation Landau gauge ο Energy eigenvalues Landau quantization Zeeman splitting π = 0: Lowest Landau level (LLL) What’s happen in external B? • The quark propagator(proper-time rep.) ο π·π ππ΅, π has the information of Zeeman splitting. ο LLL pole (1+1)-dimension What’s happen in external B? • The effective potential(mean field app.) LLL ο The potential curvature has a logarithmic singularity at π = 0. ο The curvature is always negative for small π. What’s happen in external B? • The gap eq. ο A finite π΅ induces always a non-vanishing dynamical mass. ο It is called “magnetic catalysis” • At finite temperature ο The critical temperature πc for chiral restoration should increase with increasing π΅. Magnetic catalysis V. Skokov, Phys. Rev. D85,034026 (2012) Polyakov-Quark-Meson model Recent lattice data(2012) G. S. Bali et al. JHEP 1202 (2012) 044 Inverse! • This behavior is called “Inverse Magnetic catalysis” or “Magnetic inhibition”. What is the mechanism which explains this phenomena? Mechanism • Many mechanisms are suggested. ο Dimensional Reduction(DR) of neutral pion K. Fukushima and Y. Hidaka, Phys. Rev. Lett. 110 (2013) 031601 ο DR of π 0 + Asymptotic free T. Kojo and N. Su, Phys. Lett, B726 (2013) 839 ο Polyakov loop F. Bruckmann et. al, JHEP 1304 (2013) 112 ο Sphalerons γ»γ»γ» J. Chao et. al, Phys. Rev. D88 (2013) 054009 The ideas • The quark system in strong magnetic field becomes almost like (1+1)dimension theory. • Include • Why the neutral pion effect: the neutral pion π 0 ? ο Charged pions π ± are as massive as ππ΅. ο π 0 is unaffected by a magnetic field if we treat it as a point particle. ο π 0 is constructed by two quarks which are have an electric charge. The ideas • π0 propagator: ο Tree level ο Loop correction π£⊥2 ∝π 1 ππ΅ → 0 for ππ΅ → ∞ The dimensional reduction occurs for also the neutral pion! The ideas N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133 S. Coleman, Math. Phys. 31 (1973) 259 Marmin-Wagner-Coleman theorem There cannot be spontaneous breakdown of continuous symmetries at (1+1)-dim. and (1+0)-dim. in (1+1)-dimension (ππ΅ → ∞) cannot exist according to the Marmin-WagnerColeman theorem. • π0 The ideas • The π 0 loop in the Self-Consistency eq. ο We introduce the ultra-cutoff for transverse momentum because the approximated propagator fail at π⊥ ~π(ππ΅). The ideas • At finite temperature for zero mode: π = 0 ο At high temperature, the quark loop effect is suppressed, therefore the pion loop effect becomes stronger than the quark loop effect. Inverse Magnetic Catalysis? Model • Quark-Meson model ο The transverse velocity is defined by π£⊥2 ≡ β₯ ⊥ ππ,Λ /ππ,Λ Proper-time RG equation • RG equations Cutoff function • This formulation makes the analysis easy. RG equations • The effective action • The field renormalization factors Phase diagram on π − π΅ plane The transverse velocity π£⊥ ππ΅ ↑ Summary • We investigated the so-called inverse magnetic catalysis. • We adopted the mechanism which is suggested by K. Fukushima and Y. Hidaka. ο The neutral pion plays important roles. • Unfortunately, we could not observe the inverse magnetic catalysis. Appendix Non-Perturbative RG γ»γ»γ» Wetterich equation • Partition function with the cutoff • Legendre effective action function The Cutoff function • The cutoff function must satisfy some conditions: ο ο ο • The 4-d optimized cutoff function • The 3-d optimized cutoff function D. F. Litim, Phys. Rev. D64 (2001) 105007 The Cutoff function • The cutoff function J. Berges, Phys. Rep. 363 (2002) 223 Local Potential Approximation LPA NJL model • Lagrangian • The partition function ο The generating function of connected diagram ο DπSB at π ≠ 0, π ≠ 0 • However… We cannot distinguish between the single well potential and the 1st order potential in this analysis. Effective Potential The gas-liquid transition Hokuriku-Shin-etsu Winter School 2014 @Hakusan city Hokuriku-Shin-etsu Winter School 2014 @Hakusan city
