Complex Saddle Points in Finite Density QCD

Transcription

1 Complex Saddle Points in Finite Density QCD Michael C. Ogilvie Washington University in St. Louis in collaboration with Hiromichi Nishimura (Bielefeld) and Kamal Pangeni (WUSTL) XQCD4 June 9th, 24

2 Outline Introduction Formalism Light and heavy quarks Conclusions

3 CK Symmetry QCD at μ= (and θ=) is invariant under charge conjugation (C) and complex conjugation (K) Finite-μ QCD: C is no longer a good symmetry but CK is C : A µ A t µ K : i i CK : P P (in Polyakov gauge) P (x) =P exp i dta 4 (x, t) det M ( µ) =[detm (µ)] follows from CK symmetry This property of finite density QCD is a generalized PT symmetry Bender and Boettcher physics/972; Meisinger and Ogilvie arxiv:9.745 Any approximation or algorithm should respect CK symmetry at μ Aarts arxiv:38.48; Fujii et al. arxiv: ; Cristoforetti et al. arxiv:32.52,

4 The SU(3) Polyakov Loop in a saddle point approximation μ = : Polyakov gauge P e i e i 2 e i 3 ei e i A TrFP is real and det(p)=: = + 2 = Confined Deconfined =2 /3 = htrp i = htrp i6= With this parametrization, we always have htrp i = htrp i μ P e i e i 2 e i 3 eiz e iz 2 e iz 3 A To the complex plane z j = j + i j

5 The Polyakov Loop: SU(N) The Polyakov loop is invariant under CK P = e i R dx 4 A 4 = diag e iz, e izj,e iz N C : A µ A µ K : i i A µ A µ The eigenvalues of Polyakov loop are real or in conjugate pairs: If P z j i = e iz j z j i then P CK z j i = CKP z j i = e iz j CK zj i Thus z j and zj are the eigenvalues.

6 The Polyakov Loop: SU(N) The eigenvalues can be uniquely specified: For an eigenvalue z j = j + i j, we have NX j= z j = & z k = z j For SU(N), there are [N/2] real parts, θj, and [(N-)/2] imaginary parts, ψj. For SU(3): (z,z 2,z 3 )=( i, i, +2i ) P eiz e iz 2 e iz 3 e i + e i + e 2 A trp = 2e cos + e 2 trp = 2e cos + e 2 Both real but trp trp + if ψ.

7 Perturbative calculation One-loop effective potential for Nf massless fermions: Z d 3 k h V f = 2N f T tr (2 ) 3 log +Pe ( = 4N f T 4 2 XN c j= (4) In general Vf is complex when zj is complex. µ)/t i (2) 2 (z j i µ T )2 + () (z j i µ 4 T )4 <Korthals Altes, Pisarski and Sinkovics 2> CK-symmetric background Polyakov loop makes Vf real: V f (,, T,µ) N f = Real µ T 2 µ µ 2 6µ T 3 2 µ + µ T

8 Saddle Point: st derivative Effective potential: V eff = V eff (A 4 ) A 4 T A 2 A Need to look for saddle points. First = hn red i hn green i = hn red i + hn green i 2 hn blue i = ) hn red i = hn green i = hn blue i Color neutral

9 Saddle Point: 2nd derivative Mass Matrix is no longer Hermitian, but is CK symmetric M ab V a 4 = T 4 i 4 p 2 V 2 Eigenvalues of the mass matrix m 2 ev = g2 T h (, ) g2 3 i 4 2 V @ 2 V eff 2 A A (,,T,µ) ± 2 p i B (,,T,µ) T µ2 apple N c T 2 + N f 2 When B<, the mass eigenvalues are complex. Stability 2 V 2 >, M has either two real eigenvalues or a complex eigenvalue 2 V 2 < Saddle points

10 The Models Effective potential: one-loop for quarks and gluons in a background Polyakov loop plus a term to give confinement at low T Gauge boson contribution V eff (P )=V g (P )+V f (P )+V d (P ) V g (P )= 2T 4 2 X n= n 4 Tr AP n Two phenomenological deformation terms, model A and model B Meisinger, Miller and mco, hep-ph/89 These forms clearly shows that the deformation potentials make TrFP very small at low T V A d = m2 T V B d = T R 3 X n= X n= n 2 Tr AP n n Tr AP n Tr A P = Tr F PTr F P

11 More on the deformation terms Both Model A and Model B have simple forms in term of Polyakov loop eigenvalues θj Model A is a quadratic function of the θj s and occurs naturally in the high-t expansion for massive particles V A d = NX j,k= ( 2 apple m 2 T 2 N jk ) 2 2 B 2 jk 2 Model B is obtained from Haar measure V B d = T R 3 ln 4 Y j<k sin 2 j k The parameters m and R are set to give a deconfinement transition in the pure gauge theory at 27 MeV Bielefeld Model A Model B Both models reproduce pure gauge thermodynamics fairly well.5..5 Ref T/T c

12 Quarks Basic formula has many variants: V f (P )= 2TN f Z d 3 k (2 ) 3 Tr F log +e µ k P + log +e µ k P Easy case: Mq=; ignore chiral symmetry breaking. Should be valid for light quarks at large T More realistic: PNJL model Heavy quark

13 TrP and TrP + : Model A, massless quarks TrP TrP* The Polyakov loop trp = 2e cos + e 2 trp = 2e cos + e 2 Crossover: no phase transition for fundamental fermions trp* > trp: it is easier to add antiquarks for μ > subtle effect: difference only significant in crossover region Fixed μ MeV 5 MeV 3 MeV 45 MeV T HMeVL MeV MeV.2 5 MeV 25 MeV m HMeVL Fixed T

14 Mass Matrix: Model A, massless quarks Mass matrix Complex mass leads to oscillatory behavior in color charge densities Possibly observable at FAIR Patel, arxiv:.77, T g µ2 2 + M ab V a 4 m 2 ev = g2 T 2 h 2 2 A ± 2 p i B Out[23]= T HMeVL g 2 m 2 9µ T 2 ± 2 T p 9 2 T 2 2µ (3µ T 2 ) T HMeVL m HMeVL B< s = m HMeVL

15 Boundary of oscillatory region: heavy quarks, model A 2 T HMeVL m HMeVL

16 Boundary of oscillatory region: heavy quarks, model A T 2 Quark Mass = 2 GeV 5 5 massless quark n = approximation numeric integration m

17 Boundary of oscillatory region: heavy quarks, model B THMeVL mhmevl

18 T vs Im(M): PNJL, models A & B Preliminary Model A 3 m=2 3 m=2 3 m= Out[34]= m= m= m= Model B 3 m=2 3 m=2 3 m= Effect discriminates between models of confinement cf. role of heavy quark endpoint Kashiwa, Pisarski and Skokov, arxiv: Out[72]= m= m= m=

19 Conclusions Complex saddle points in finite density QCD respect the CK symmetry of charge conjugation (C) and complex conjugation (K). trp trp* in the crossover region at finite μ Color neutrality is naturally achieved Complex mass eigenvalues: oscillatory behavior in color charge densities may occur Experimental consequences possibly observable at FAIR, but this is model-dependent. Potential to discriminate between models of confinement