Spectral Continuity in Dense QCD Phys.Rev.D.78:011501,2008 Quark-Hadron continuity (in-medium) QCD sum rules Key words Different roles of vacuum condensates arXiv:0802.4143[ hep-ph ] M. T. (Saga) N. Yamamoto (Tokyo) T. Hatsuda (Tokyo) ( 有限密度QCDにおけるスペクトル連続性 ) m V 1/3 ∝ q q ∝ qq 2 ∝Δ µ μ µ c “ Spectral Continuity ” of hadrons
Plan of this talk 1. Introduction and motivations 2. In-medium QCD Sum Rules (QSR) 3. Flavor-octet vector mesons in QSR 4. Flavor-singlet vector meson in QSR 5. “Spectral Continuity” of vector mesons 6. Sum-mary and Per-spect-ives
1. Introduction and motivations Astrophysics new state Critical phenomena Many-body problem Physics Matter Condensed Physics Particle Nuclear Cosmology Exploring the QCD phase diagram is challenging QCD Observation QCD Phase Diagram Heavy Ion Collisions QCD coupling const. Compact Stars Big-Bang Cosmology ・Strongly-coupled Quantum Field Theory (QCD) ・Compact Stars (Neutron Stars, Quark Stars) ・Big-Bang Cosmology, Heavy Ion Collisions (Little-Bang) of matter
QCD phase transition and the phase diagram Strongly indicate the existence of such a ρ c T c (CSC) Color superconductor Hadron Quark-gluon plasma (QGP) T ρ [ Collins-Perry (1975) ] transition from hadron to quark-gluon phase ↓ Model calculations and numerical simulations ・ screening of color force ・ asymptotic freedom QCD @ high temperature( T ) / density(ρ) (roughly speaking) V ( r ) ~ α s ( T , µ ) − m sc r e r “ weakly interacting gas of quarks/gluons ” ∴ QCD vacuum undergoes a phase change at some values of T and ρ! “ Conjectured phase diagram ” 12 K T c ~ (150 − 200) MeV ~ 10 ρ c ~ several × ρ nm ~ 10 12 kg ⋅ cm − 3
Hatsuda-Yamamoto-Baym-M.T., Phys. Rev. Lett. 97 (2006) 122001 Possible New Critical Point in Dense QCD ＠μ≠０ ・Interplay b/w chiral & diquark condensates ・Presence of the U (1) axial anomaly New critical point & Crossover from hadron-to-CSC !! Ginzburg-Landau (GL) model Cartoon phase diagram in 2-light +1-medium flavors : critical point : crossover
Appearance of a new critical point Ginzburg-Landau free energy in massless 3 flavor quark matter anomaly -driven Possible phases σ : chiral condensate d : diquark condensate Ω 3 F = a 2 − c 3 + b + α 2 + β 4 4 2 σ 2 d 4 d − γ d 2 σ 3 σ 4 σ a , b , c , α , β , γ : GL parameters ‘ tHooft interaction ・ Mass term for d ・ external field for σ (equivalent to Ising Ferro-magnet )
Comments 1. All the lines and the points characterizing the whole phase boundaries can be determined analytically . 2. A similar critical point at low temperature has been derived by Kitazawa et al. [PTP108(2002)929], using the 2-flavor NJL model with scalar and vector type 4-fermion interactions. However since the axial anomaly does not produce a triple boson coupling in 2-flavors, the origin of their critical point will be different with that discussed here. 3. We performed the similar analysis in 2 flavor case and this case is found in an anisotropic anti-ferromagnet in reality Interplay between chiral and diquark condensates Intriguing!! such as GdAlO3 (e.g., see Chaikin-Lubensky ’ s textbook)
Excitation spectra Hatsuda-Yamamoto-Baym-M.T., Phys. Rev. D76 (2007) 074001 Generalized Gell-Mann-Oaks-Renner (GOR) relation Low energy excitations ～ Nambu-Goldstone (NG) bosons associated with chiral symmetry breaking (in both hadronic and CSC phases !) (Schafer-Wilczek) .. ・An example of ” spectral continuity ” of hadrons ・A concrete realization of “ quark-hadron continuity ”
Pion mass splitting mass splitting Pion unstable Taken from Gordon ’ s talk in QM08
Quark-Hadron (QH) continuity ρ,ω,φ,K* mismatch ? Nucl.Phys.B147 (1979) 385. Shifman-Vainshtein-Zakharov, .. baryons (8) quarks (9) vector mesons (8+1) gluons (8) QCD sum rules (QSR) Investigating some general aspects of hadron spectrum in medium T. Schafer and F. Wilczek, Phys. Rev. Lett. 82 (1999) 3956 (massive) gluons (gapped) quarks Baryons Fermions Vector mesons π(＆H ) NG bosons high μ (CFL) low μ(hadron) excitations π ’ ＆H However … N 3 = f
2. QCD sum rule (QSR) Operator product expansion (OPE): Current correlators continuum pole Dispersion relations i ikx RJ 0 AB ( ω ) = lim 4 xe A ( x ) J 0 B (0) ∫ r d Π L r 2 k k → 0 “ spectral function ” ∞ ρ ( u ) ∫ du 2 Π L ( ω ) = u 2 − ( ω + i ε ) 2 0 ρ ( u ) “ phenomenology ” ( Q 2 ) O n Π L ( Q 2 → ∞ ) ~ ∑ C n Q 2 n u n
: U(3) generators with with operators in OPE contribute to the correlation functions. q 2. The spectral function is just phenomenologically Comments at zero temperature and finite density with m = 0 . singlet vector mesons in 3 flavor quark matter In the following, we focus on flavor octet and 1. In medium, not only the Lorentz scalar but also the tensor QSR (cont ’ d) (Hatsuda-Lee ‘ 92, Hatsuda-Koike-Lee ‘ 93) ρ ( u ) given in terms of so called “ resonance parameters ” . A = q A γ µ q J µ τ q = ( u , d , s ) A [ ] = 2 δ A τ B AB tr τ τ ( A = 0, L ,8)
Warm-up --non-interacting quark matter@μ≠0 -- scattering of quarks on external current, i.e., the Laundau damping term (pole part) decay of the external current into qq-pair with Pauli blocking effect (continuum part) Spectral function in free quark matter the Fermi surface with 3 q q + i ∂ 0 q q + i ∂ 0 ( free ) ( Q ) = − 1 2 + 16 + 64 2 log Q Π L 4 6 2 π 9 Q 9 Q Lorentz non-scalar operators 2 S 0 = (2 µ ) 2 − S 0 ) F = AS 0 ( free ) ( u ) = F δ ( u 2 ) + A θ ( u ρ 2 ) A = 1/(2 π “ resonance parameters ” ( free ) ( u ) ρ “ pole + continuum ” F A 2 u S 0
Comment How the genuine nonpertubative effects such as the ? Corrections of the form and to condensates affect properties of hadrons from QSR? resonance parameters and the shape of the spectral function. and are compensated by the perturbative corrections to the this case can be taken into account in perturbation theory 2 n α s ln Q α s ( µ / Q ) ( free ) ( u ) ρ “ pole + continuum ” F A 2 u S 0
In-medium OPE (vector mesons) (8) ( x ) ≡ q a γ µ q ( x ), (1) ( x ) ≡ q 0 γ µ q ( x ) J µ ( x ) τ J µ ( x ) τ (8,1) = Π L ( free ) + δ Π L (8,1) , Π L 1 2 + 8 (8) = − πα s a λ a ' q ) a ' q ) 2 4 ( q 27 ( q δ Π L γ µ γ 5 τ γ µ λ 6 Q γ µ γ 5 τ 0 λ a ' q ) 2 + 8 (1) = − πα s γ µ λ a ' q ) 2 2( q 27 ( q δ Π L Q 6 8 (8) ≡ 1 (1) ≡ Π L 00 ∑ AA , Π L Π L Π L 8 A = 1 a : color SU(3) generators λ a : flavor SU(3) generators τ ( a = 1 ~ 8)
Comments in the same way. lead only to perturbative corrections to produce the chiral and diquark condensates and 3. The non-scalar operators like the quark-gluon mixed operators and the twist -4 quark operators do not if it exists, affects the octet and singlet mesons 2. Among others, the gluon condensate whose explicit forms are given in 1. There are several operators neglected up to 6 ), O (1/ Q Hatsuda-Lee ‘ 92, Hatsuda-Koike-Lee ‘ 93. α s 2 , π TrF µ ν ( free ) . Π L Ο d , τ 『twist(τ) ≡ cano. dim.( d )-spin』 α TrF { D µ , * G νλ } γ λ γ 5 q Ο 6,4 = q e.g.) 4 , 4 2 s Ο = µ ν π
Pairing patterns and factorization projection op. flavor After rewriting 4 quark ops. in chiral basis and making the Fierz rearrangement together with the factorization ansatz : , we obtain Let us consider here chiral condensate and diquark condensate color which are in the most attractive channels ( MAC ): q q qq α = diag ( σ , σ , σ ) i , j , k : α q j q i α , β , γ : 1 γ = diag ( ϕ , ϕ , ϕ ) β C γ 5 Λ + q k 4 ε ijk ε αβγ q j Λ + : positive energy 2 ∑ ∑ O = P l ⋅ P P ≅ l l l l Π σ = − 448 πα s 2 6 σ (8,1) ≅ Π σ + Π ϕ 81 Q (8,1) , where δ Π L (8) = − 5 (1) = − 320 πα s 27 Q 6 ϕ 2 Π ϕ 22 Π ϕ
Comments 1. The qualitative conclusion in the present work do not depend on the factorization ansatz. 2. Since the chiral condensate is flavor-diagonal , it does not distinguish between octet and singlet. 3. While, the diquark condensate has color-flavor structure so that it can smell flavors differently. This is why the flavor-octet and -singlet vector mesons, which are almost degene -rate at low density, tend to split at high density due to the appearance of diquark condensates.
sum rules ( FESR ) 3. Flavor-octet vector mesons in QSR Finite energy in terms of , and comparing the result with the OPE expression: Plugging this into the dispersion relation, carrying out the asymptotic expansion Spectral function 2 ) + A θ ( u 2 − m V 2 − S 0 ) (8) ( u ) = F δ ( u ρ 2 1/ Q F − AS 0 = 0 4 2 = − (2 µ ) 2 − AS 0 2 Fm V 2 2 π 3 = − (2 µ ) 6 4 − AS 0 (8) = − 448 πα s 2 + 15 (8) 2 3 Fm V 2 + O O < 0 σ 7 ϕ 27 2 π
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