Polyakov Loop in a Magnetic Field

Transcription

1 Polyakov Loop in a Magnetic Field Kenji Fukushima (Department of Physics, Keio University) March 17, St.Goar 1

2 Talk Contents Relativistic Heavy-Ion Collision and Strong Magnetic Fields eb ~m ~118 gauss Fermion Propagator and Dimensional Reduction (+1) (1+1) in LLLA One Loop Polarization c.f. two-loop contribution to EH Magnetic-field Induced Screening Effect and Perturbative Polyakov-loop Potential March 17, St.Goar M g ~g eb

3 Relativistic Heavy-Ion Collision and Strong Magnetic Fields March 17, St.Goar

4 Conventional Starting... KF-Hatsuda (1) March 17, St.Goar 4

5 Relativistic Heavy-Ion Collisions Nucleus (Au,Pb) Collision Energy per nucleon-nucleon = LHC STAR ALICE March 17, St.Goar 5

6 Non-Central Collision Before Collision (seen from the upper hemisphere ) + + Looking for parity violation in heavy-ion collisions by Berndt Müller Physics, 14 (9) Centrality is to be determined event by event March 17, St.Goar 6

7 Non-Central Collision After Collision + Hot and Dense QCD Matter (Quark-Gluon Plasma) B March 17, St.Goar + 7

8 Estimated Magnetic Fields Classical (Pancake) Calcs (Kharzeev-McLerran-Warringa) eb=1[ MeV ] 14 B gauss UrQMD Calculations (Skokov-Illarionev-Toneev) March 17, St.Goar 8

9 Largest Magnetic Field in the Universe eb ~ m p 18 1 Gauss 6 1 ~1 Neutron Star (Magnetar) There should be some influence on QCD physics! March 17, St.Goar 9

10 Example Phase Diagram PNJL with a B Lattice Results Fukushima-Gatto-Ruggieri (1) D'Elia et al. (1) c.f. Fraga et al. (1) in PQM Chiral condensate is enhanced in accord with the magnetic catalysis. Gusynin-Miransky-Shovkovy Polyakov loop shows crossover at the same (pseudo-)critical temperature. March 17, St.Goar 1

11 Missing Coupling Discrepancy very similar to the problem at finite mb PNJL and PQM (in a mean-field approx.) tend to favor splitting between chiral and deconf. transitions. trl-dependent G Ruggieri-Gatto (1) March 17, St.Goar Diagrams missing in mean-field PNJL and PQM 11

12 Missing Coupling Origin of the PNJL (PQM) Coupling ln det[i g A m] dk /T /T ~ tr ln[1 L e ] tr ln[1 L e ] k k Missing Contribution to the trl-potential ln det[i D g A m] ~# g dx dy A x x, y A y c.f. renormalization of T Pawlowski, Schaefer, Wambach Vaccum-polarization b-function Two-loop Weiss potential? March 17, St.Goar 1

13 Example Chiral Magnetic Effect Classical Picture Left-handed Quark = momentum anti-parallel to spin B Right-handed Quark = momentum parallel to spin Kharzeev-McLerran-Warringa (7) Local P violation by instantons (QCD Physics) J if N 5= N R N L March 17, St.Goar 1

14 Key Equations Induced N5 by Topological Effects dn 5 g Nf = d x tr F F dt 8 QCD Anomaly Introduce m5 to describe induced N5 Induced J by the presence of N5 and B j = j = e 5 B i=flavor QED Anomaly q i 5 B in QCD March 17, St.Goar Metlitski-Zhitnitsky (5) Fukushima-Kharzeev-Warringa (8) 14

15 Derivation (naïve calculation) Thermodynamic Potential (UV divergent) q f B dp f = V N c n, s n, s T ln 1 e ] [ s=± n = f n,s = p q f B n sgn p s 5 m n,s Current (UV finite) Only surface terms! eb j =e n, s p = n, s p = ] n,s [ 4 s, n e B 5 eb =e s 5= n, s s,n March 17, St.Goar 15

16 Observable on Average What can be measured in heavy-ion collision experiments is not the current j ~ B (P -odd) but the current-current fluctuations <j j> (P -even). j j = j j j j connected Background As long as e is small; j j ~ Fukushima-Kharzeev-Warringa (9) March 17, St.Goar 16

17 Very naïve calculation Wrong!? e eb ~ ~ 1 eb A One is tempted to drop the UV divergence by hand. Dangerous calculation... but... L T = (UV fintie) But again, if this is accepted, the anomaly equation for <j> should receive a correction... Fukushima-Ruggieri (1) c.f. Miransky-Shovkovy Some confusions... March 17, St.Goar 17

18 Fermion Propagator and Dimensional Reduction March 17, St.Goar 18

19 Construction of the Propagator No Magnetic Field Background Particles d p e i x y i p x y x y = p m p p p s u s p u s p = p m Anti-Particles d p e i x y i p x y y x = p m p p p s v s p v s p = p m 4 d p i p x y i T x y = e 4 p m i March 17, St.Goar 19

20 Ritus' Method Landau Wave Function (A =Bx gauge) ip x ip x ip x id m P k x e [ f k x P k x = = Pk x e f k x f k x = k x 1 p /eb 1 f k x = k 1 x p / eb p = p,, eb k, p ip x ip x ip x f k x f k x p m ] Wave function of the harmonic oscillator Landau quantization March 17, St.Goar

21 Construction of the Propagator Magnetic Field Background Particles dp dp x y = k i p x y i p x y i p x y e p P k x p p m P k y Anti-Particles dp dp y x = k e i p x y i p x y i p x y p P k y p p m P k x March 17, St.Goar 1

22 LLLA Lowest Landau-Level Approximation P commutes with p p d p dp dp i p x y i p x y i p x y T x y = e 1 eb eb [ x p / eb y p /eb ] i e P p p m i 1 [ ] 1 P= 1 1 Landau zero-mode has only one spin state. s // B preferred. March 17, St.Goar The momentum conservation is highly non-trivial. A =Bx

23 One Loop Polarization March 17, St.Goar

24 Polarization (Self-energy) In configuration space i, ab a b F F x, y = ig tr [t t ] Tr [ S x, y S y, x ] Note that this is not a function of x y apparently In momentum space i, ab k, q = d x d y e 4 4 iq x ik y i, ab x, y, ab k, q = 4 4 k q ab q March 17, St.Goar 4

25 (1+1) Dimensional System Transverse Directions (to the Magnetic Field) 1 1 = = = = Longitudinal Directions (m,n either,) g eb d p p p q p q p g [ p p q m ] =i [ p m i ][ p q m i ] This is an ordinary expression for the one-loop polarization diagram in (1+1) dimensions March 17, St.Goar 5

26 Gauge Invariance Results from the Dimensional Regularization q q q = g q g eb 1 x 1 x dx 4 x 1 x m/ q c.f. Naïve Integration Fukushima-Kharzeev-Warringa (9) - component p p dp = = [ p ] p - component p m dp i m p = [ p p] g eb dp i #= = 4 March 17, St.Goar 6

27 Threshold Behavior in (1+1)-Dim ~ Schwinger Mass Strongly Dissipative Different from (+1)-dim in which no divergence appears. March 17, St.Goar 7

28 Magnetic-field Induced Screening Effect and Perturbative Polyakov-loop Potential March 17, St.Goar 8

29 Polyakov-loop Potential Background A4=diag(a,-a) Field (in Euclidean) 4 4 ±iga q ± 4=q 4± ga Haar measure 1 ij i j ln Z = Tr ln [q ± q q ] Tr ln [q ± 4 ] 1 = Tr ln[ q ± 4 q ] [q ± 4 ] Tr ln [q ± 4 ] Weiss Potential (in color SU()) [ V g a U Weiss = 1 mod g a mod ] March 17, St.Goar 9

30 Magnetic-field Induced Screening Effect of the Magnetic Field 1 ij i j Tr ln[ q± q q ] [ ] q 1 Tr ln q ± ij q i q j i j ± 4 M g q ± 1 = Tr ln [ q ± 4 q ][ q ± 4 q M g ][ q ± 4 ] Screened by the B-induced Gluon Mass g eb M g= I q / m 4 g eb m 4 Two-particle threshold at q =m March 17, St.Goar

31 Spectral Function Collective Excitation in a Strong B p = Im M g p 1 [ p Re M g p ] [Im M g p ] Massive and Dissipative Transverse Photon (Gluon) c.f. followking talks by Misha and Maxim (1+1)-dim Analogue to (+1)-dim Zero Sound March 17, St.Goar c.f. chiral magnetic wave Kharzeev-Yee (1) 1

32 Screened Weiss Potential Change in the Weiss Potential as Mg increases Calculation at m= Perturbative vacuum at gba = (trl = 1) is less stabilized (slightly). A barrier at the confining vacuum at gba = p (trl = ) is (slightly) suppressed. Small effect but correct direction to have a larger effective T March 17, St.Goar

33 Remarks and Conclusions Magnetic-field induced screening effects has a similar structure as the finite-t temperature. The one-loop polarization diagram is computed in the LLLA in a gauge-invariant procedure. Magnetic screening effect in (1+1) dimensions is embedded in the (+1)-dim transverse part in the gluon (photon) propagation collective excitation Physics with a high B ~ Physics at a high m B [effective (1+1)-dim systems] March 17, St.Goar