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
Outline Introduction Formalism Light and heavy quarks Conclusions
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:39.437 ; Cristoforetti et al. arxiv:32.52, 43.5637
The SU(3) Polyakov Loop in a saddle point approximation μ = : Polyakov gauge P = @ e i e i 2 e i 3 A @ ei e i A TrFP is real and det(p)=: + 2 + 3 = + 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 A @ eiz e iz 2 e iz 3 A To the complex plane z j = j + i j
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.
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 A @ e i + e i + e 2 A trp = 2e cos + e 2 trp = 2e cos + e 2 Both real but trp trp + if ψ.
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 4 2 2 + T 2 µ 2 + 2 2 µ 2 6µ 2 2 2 2 + 4T 3 2 µ + µ 3 2 + T 4 7 4 + 2 2 2 4 6 2 2 + 6 2 2 9 4 3 2
Saddle Point: st derivative Effective potential: V eff = V eff (A 4 ) A 4 T = @ A i @ 2 A Need to look for saddle points. First derivative @V eff @ = hn red i hn green i = @V eff @ = hn red i + hn green i 2 hn blue i = ) hn red i = hn green i = hn blue i Color neutral
Saddle Point: 2nd derivative Mass Matrix is no longer Hermitian, but is CK symmetric M ab = @2 V eff @A a 4 @Ab 4 = T 2 @ 4 i 4 p 3 @ 2 V eff @ 2 @ 2 V eff @ @ Eigenvalues of the mass matrix m 2 ev = g2 T 2 2 2 h (, ) g2 3 i 4 p @ 2 V eff 3 @ @ @ 2 V eff 2 @ 2 A A (,,T,µ) ± 2 p i B (,,T,µ) T 2 + 3 2 µ2 apple N c T 2 + N f 2 When B<, the mass eigenvalues are complex. Stability condition @ 2 V eff @ 2 >, M has either two real eigenvalues or a complex eigenvalue pair. @ 2 V eff @ 2 < Saddle points
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 2 2 2 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
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 2 3 5 3.5 The parameters m and R are set to give a deconfinement transition in the pure gauge theory at 27 MeV. 3. 2.5 2. Bielefeld Model A Model B Both models reproduce pure gauge thermodynamics fairly well.5..5 Ref. 2 3 4 T/T c
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
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.8.6.4.2..8.6.4 Fixed μ MeV 5 MeV 3 MeV 45 MeV 5 5 2 25 3 T HMeVL MeV MeV.2 5 MeV 25 MeV. 2 3 4 5 6 m HMeVL Fixed T
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, 2.597 4T g 2 2 3 + µ2 2 + M ab = @2 V eff @A a 4 @Ab 4 m 2 ev = g2 T 2 h 2 2 A ± 2 p i B Out[23]= T HMeVL g 2 m 2 9µ 2 2 2 T 2 ± 2 T p 9 2 T 2 2µ 2 4 2 (3µ 2 +4 2 T 2 ) T HMeVL 5 4 3 2 2 4 6 8 m HMeVL 5 4 3 2 B< s = 2 4 6 8 m HMeVL 5 4 3 2
Boundary of oscillatory region: heavy quarks, model A 2 T HMeVL 8 6 4 2 5 5 2 25 3 35 m HMeVL
Boundary of oscillatory region: heavy quarks, model A T 2 Quark Mass = 2 GeV 5 5 massless quark n = approximation numeric integration 5 2 25 m
Boundary of oscillatory region: heavy quarks, model B 3 25 2 THMeVL 5 5 5 5 2 25 mhmevl
T vs Im(M): PNJL, models A & B Preliminary Model A 3 m=2 3 m=2 3 m=3 25 25 25 2 2 2 5 5 5 5 5 5 Out[34]= 3 5 5 2 25 3 m=33 3 5 5 2 25 3 m=35 3 5 5 2 25 3 m=4 25 25 25 2 2 2 5 5 5 5 5 5 5 5 2 25 3 5 5 2 25 3 5 5 2 25 3 Model B 3 m=2 3 m=2 3 m=3 25 25 25 2 2 2 Effect discriminates between models of confinement cf. role of heavy quark endpoint Kashiwa, Pisarski and Skokov, arxiv:25.545 Out[72]= 5 5 3 25 2 5 5 5 5 2 25 3 m=33 5 5 2 25 3 5 5 3 25 2 5 5 5 5 2 25 3 m=34 5 5 2 25 3 5 5 3 25 2 5 5 5 5 2 25 3 m=35 5 5 2 25 3
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