Two-colour QCD at T > 0 in the presence of a strong magnetic field M. M¨ uller-Preussker with collaborators E.-M. Ilgenfritz † , B. Petersson M. Kalinowski, A. Schreiber, (Humboldt-University Berlin, † JINR Dubna) JINR, BLTP, Seminar January 18, 2012
Outline: 1. Introduction 2. The lattice model 3. How to couple an external constant magnetic field B 4. The influence of an external magnetic field on the chiral condensate and on the critical temperature 5. The chiral condensate in the chiral limit 6. Conclusions and outlook
1. Introduction Very strong magnetic fields may exist (or have existed) √ during the electroweak phase transition ( eB ∼ 1 − 2 GeV), • √ in the interior of dense neutron stars (magnetons) ( eB ∼ 1 MeV), • √ in noncentral heavy ion collisions at RHIC ( eB ∼ 100 MeV) • √ and LHC ( eB ∼ 500 MeV), because antiparallel currents of the spectators create a strong magnetic field.
Non-central heavy ion collision Kharzeev, McLerran, Warringa, ’08 B y y x B z E z z
Such strong magnetic fields may lead to a strengthening of the chiral symmetry breaking • (increase of the chiral condensate, increase of F π , decrease of M π ), a change of the finite temperature chiral transition • both in temperature and in strength, the chiral magnetic effect (CME), leading to an event by event • charge asymmetry in peripheral heavy ion collisions.
Chiral model at T = 0 (Shushpanov, Smilga, ’97) � � �� ( eB ) 2 ( eB ) 4 < ¯ ψψ > B = < ¯ 1 + 1 ψψ > 0 π + O F 2 96 π 2 M 2 F 4 π M 4 π π √ In the chiral limit, M π ≪ eB ≪ 2 πF π ∼ Λ hadr : from J. Schwinger’s (’51) solution � � ( eB ) 2 π , ( eB ) 2 �� < ¯ ψψ > B = < ¯ ( eB ) log 2 1 + 1 ψψ > 0 + O F 2 16 π 2 F 4 Λ 4 π hadr � � ( eB ) log 2 1 − 1 M π 0 ( B ) = M π 0 (0) + . . . F 2 16 π 2 π � � ( eB ) log 2 1 + 1 F π ( B ) = F π (0) + . . . F 2 8 π 2 π √ M π + ( B ) = M π − ( B ) ∝ eB
√ Strong fields eB ≫ F π , M π , Λ hadr or deconfined phase ( T > T c ) < ¯ ψψ > B ∼ | eB | 3 / 2 = ⇒ e B the only scale Dyson-Schwinger equations suggest a selfconsistent quark mass: � � � � m q ( B ) ∼ | eB | exp − π/ ( α s c F ) � � ψψ > B ∼ | eB | 3 / 2 exp < ¯ − π � π/ (2 α s c F ) 2 where α s ≡ α s ( | eB | )
2. The lattice model Our simplified quark-gluon matter: - colour SU (2) , - staggered fermions without rooting of fermionic determinant, i.e N f = 4 flavours, - unique e.-m. charge. Why this model? - Very similar chiral behaviour as in SU (3) colour. - Can be extended to finite baryon chemical potential without sign problem. - Topology (important also for the CME) can be studied in a more simple case. - Much faster to simulate. Can take the chiral limit. Use a farm of PC’s (and recently GPU’s). - University requirement: nice model to be proposed for master students. Pioneering calculations (quenched SU(2)): Buividovich, Lushchevskaya, Polikarpov,... Full QCD (conflict): D’Elia et al ⇔ Bali, Bruckmann, Endrodi, Fodor, Sch¨ afer,...
Lattice gauge action: from elementary closed (Wilson) loops (“plaquettes”) µ,ν U † ν,µ U † U n,µν ≡ U n,µ U n +ˆ n,ν , U n,µ ∈ SU ( N c ) n +ˆ � 1 − 1 � β = 2 N c � S W = β Re tr U n,µν , G g 2 N c 0 n,µ<ν = 1 a 4 tr G µν G µν + O ( a 2 ) , � 2 n → 1 � d 4 x tr G µν G µν . 2 Continuum limit: − β 1 � � 1 1 ( β 0 g 2 2 β 2 (1 + O ( g 2 0 exp a ( g 0 ) = 0 ) − 0 )) . 2 β 0 g 2 Λ Latt 0 = ⇒ a → 0 for g 0 → 0 (or β → ∞ ), asymptotic freedom . For SU ( N c ) and N f massless fermions, independent on renormalization scheme: 3 N c N f − N 2 � 11 � � 34 � 1 3 N c − 2 1 c − 10 c − 1 3 N 2 β 0 = 3 N f , β 1 = N f . (4 π ) 2 (4 π ) 4 N c
Staggered fermion action Kogut, Susskind, ’75 • Use naive discretization and diagonalize action w.r. to spinor degrees of freedom. • Neglect three of four degenerate Dirac components. • Attribute the 16 fermionic degrees of freedom localized around one elementary hypercube to four tastes with four Dirac indices each. Chiral symmetry restored ⇐ ⇒ flavor symmetry broken. Naturally the mass-degenerated four-flavor case is described.
Path integral quantization for Euclidean time = ⇒ ’statistical averages’. Fermions as anticommuting Grassmann variables ⇒ analytically integrated ⇒ non-local effective action S eff ( U ) . = ’Partition function’ describing N f = 4 mass-degenerate staggered flavors: � ψ ] e − S G ( U ) + ¯ [ dU ][ dψ ] [ d ¯ ψM ( U ) ψ Z = � [ dU ] e − S G ( U ) Det M ( U ) = � [ dU ] e − S eff ( U ) , S eff ( U ) = S G ( U ) − log(Det M ( U )) = with M ( U ) ≡ D Latt ( U ) + m . Simulation on a finite lattice N t × N 3 s , with (anti-) periodic boundary conditions for gluons (quarks). Rooting prescription: for N f = 2 + 1 (+1) 4th-root of the fermionic determinant is taken. = ⇒ Locality violated (??)
Non-zero temperature T ≡ 1 /L t = 1 / ( N t a ( β )) : T varied by changing β at fixed N t (changing N t at fixed β ). Order parameters: N c tr � N t 1 Polyakov loop: L ( � x ) ≡ x 4 =1 U 4 ( � x, x 4 ) , � L ( � x ) � = exp( − βF Q ) , F Q = free energy of an isolated infinitely heavy quark. = ⇒ F Q → ∞ , i.e. � L ( � x ) � → 0 within the confinement phase ( T < T c ). = ⇒ � L ( � x ) � order parameter for the deconfinement transition ( T = T c ). � ¯ Chiral condensate: ψψ � order parameter for chiral symmetry breaking ( T < T c ) and restoration ( T > T c ) ¯ Find critical T c (or β c ) from maxima of susceptibilities of L ( � x ) and/or ψψ .
Fixing the physical scale: 16 3 × 6 ( 24 3 × 6 ) T > 0 calculations done on lattices of size: 16 3 × 32 T = 0 calculations: The lattice unit scale a ( β ) fixed via scale parameter r 0 [R. Sommer, ’94], assumed to be the same as in real QCD: Compute static force F ( r ) = dV/dr phenomenologically well-known cc - or ¯ from ¯ bb -potential V ¯ QQ : F ( r 0 ) r 2 0 ≡ 1 . 65 ↔ r 0 ≃ 0 . 5 fm Then determine e.g. the pion mass m π . For T = 0 , ma = 0 . 01 , B = 0 we obtain at β = 1 . 80 ( ≃ β c for N t = 6 ). a = 0 . 17(1) fm m π = 330(25) MeV T c = 193(13) MeV preliminary
3. How to couple an external constant magnetic field B ¯ ¯ r ) = B B = (0 , 0 , B ) A (¯ 2 ( − y, x, 0) On the lattice we use the compact formulation. Constant magnetic field ≡ constant magnetic flux φ = a 2 ( eB ) through all ( x, y ) plaquettes. On the links define U (1) elements coupled to quark fields in lattice covariant derivative. r, τ ) = e − iφy/ 2 V x (¯ r, τ ) = e iφx/ 2 V y (¯ V x ( N s , y, z, τ ) = e − iφ ( N s +1) y/ 2 V y ( x, N s , z, τ ) = e iφ ( N s +1) x/ 2 φ = 2 πN b Flux will be quantized: N b = 1 , 2 , . . . DeGrand, Toussaint ’80 N 2 S � Typical field strength for β = 1 . 80 ≃ β c , N b = 50 ⇐ ⇒ ( eB ) ≃ O (1 GeV)
4. The influence of an external magnetic field on the chiral condensate and on the critical temperature Saturation behavior for various β ( V µ periodic in φ ): 0.2 0.2 1 . 70 1 . 70 1 . 78 1 . 78 0.18 0.18 1 . 80 1 . 80 0.16 0.16 1 . 82 1 . 82 1 . 90 1 . 90 0.14 0.14 0.12 0.12 ψψ � ψψ � a 3 � ¯ a 3 � ¯ 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 20 40 60 80 100 120 140 160 0 1 2 3 4 5 6 7 8 N b B [ GeV 2 ]
β -dependence ( ≡ T dependence) of the bare chiral condensate 0.25 0.3 N b = 0 N b = 0 10 10 20 20 50 0.2 0.25 50 120 120 0.15 0.2 ψψ � ψψ � a 3 � ¯ a 3 � ¯ 0.1 0.15 0.05 0.1 0 0.05 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 β β ma = 0 . 01 ma = 0 . 1 < ¯ ψψ > increases with B for all β = ⇒ T c increases
Polyakov loop 0.3 0.3 N b = 0 N b = 0 10 10 20 20 0.25 0.25 50 50 120 120 0.2 0.2 L L 0.15 0.15 0.1 0.1 0.05 0.05 0 0 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 β β ma = 0 . 01 ma = 0 . 1
Susceptibilities chiral condensate Polyakov loop 2.5 0.75 N b = 50 N b = 50 = 20 = 20 0.7 = 0 = 0 2 0.65 0.6 1.5 χ disc χ L 0.55 1 0.5 0.45 0.5 0.4 0 0.35 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 1.92 β β ma = 0 . 01 ma = 0 . 01 B ր ⇒ T c ր no splitting of the transition
Spatial anisotropy of plaquette averages: confined phase, β = 1 . 7 1.004 1.06 xy xz yz 1.003 xt 1.04 yt zt 1.002 plaquette anisotropy plaquette anisotropy 1.02 1.001 1 1 0.999 0.98 0.998 xy xz 0.96 yz 0.997 xt yt zt 0.996 0.94 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 0.12 am am N b = 0 N b = 50
transition region, β = 1 . 9 1.004 1.06 xy xz yz 1.003 xt 1.04 yt zt 1.002 plaquette anisotropy plaquette anisotropy 1.02 1.001 1 1 0.999 0.98 0.998 xy xz 0.96 yz 0.997 xt yt zt 0.996 0.94 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.02 0.04 0.06 0.08 0.1 0.12 am am N b = 0 N b = 50 Spacelike-timelike plaquette differences ∝ energy density
Recommend
More recommend
