QCD in an external magnetic field

Transcription

1 QCD in an external magnetic field Gunnar Bali Universität Regensburg TIFR Mumbai,

2 Contents Lattice QCD The QCD phase structure QCD in U(1) magnetic fields The B-T phase diagram Summary and Outlook GS Bali, F Bruckmann, G Endrődi, A Schäfer (Regensburg), Z Fodor, KK Szabó (Wuppertal), SD Katz (Eötvös Budapest), S Krieg (FZ Jülich) arxiv: [hep-lat], JHEP in print, arxiv: , PoS(Lattice 2011) 192. Gunnar Bali (Regensburg) Magnetic field 2 / 32

3 QCD (theory of strong interactions) L QCD = 1 16πα s FF + ψ f (D/ + m f )ψ f asymptotic freedom:α s (q) q 0? confinement? chiral symmetry breaking proton (artist s impression) Theoretically beautiful but analytical quantitative predictions are very difficult in the region of small momentum transfers (strong QCD)! = computer simulation Gunnar Bali (Regensburg) Magnetic field 3 / 32

4 Lattice QCD Na o o o o o o o a o o o o o o o o o o o o o o o o o o plaquette link site typical values: a 1 = GeV, Na = fm continuum limit: a 0, La fixed infinite volume: Na O = 1 [du] [dψ][d Z ψ] O[U]e S[U,ψ, ψ] Measurement : average over a representative ensemble of gluon configurations{u i } with probability P(U i ) [dψ][d O = 1 n n i=1 ψ]e S[U,ψ, ψ] O(U i ) + O O 1 n n 0 Gunnar Bali (Regensburg) Magnetic field 4 / 32

5 Input:L QCD = 1 16πα L FF + ψ f (D/ + m f )ψ f m lat p = m phys p mps/m lat p lat = mπ phys /mp phys a m u m d Output: hadron masses, phase diagram, decay constants etc... Extrapolations: 1 a 0: functional form known. 2 N : harmless but computationally expensive. 3 m lat q = mphys q has only very recently been realized. Gunnar Bali (Regensburg) Magnetic field 5 / 32

6 Pure gauge theory L YM = 1 16πα L FF Consider lattice of time extent N t a = 1/T and spatial volume V = (N s a) 3. Order parameter: Polyakov line P exp( F q /T) Low temperature: P = 0, confinement. High temperature: P z, z 3, deconfinement (V ). 0,25 N T =8 N T =4, disordered start N T =4, unit start 0,2 Im Poly. 0,15 0,1 0,05 0-0,2-0,1 0 0,1 0,2 0,3 Re Poly. Gunnar Bali (Regensburg) Magnetic field 6 / 32

7 Chiral symmetry (breaking) Global symmetry of the m = 0, n f = 3 QCDlite TM : Ignore U V (1) symmetry (baryon number conservation) U A (1) anomaly: µ j 5 µ = 1 16π 2 F F heavyη m = 0 χ-symmetry spontaneously broken at T< T c (order parameter ψψ ): SU L (3) SU R (3) SU V (3) 8 Nambu-Goldstone bosons! π K 0 + η π0 η K - + π K - K 0 Gunnar Bali (Regensburg) Magnetic field 7 / 32

8 QCD with masses Columbia plot FR Brown et al, PRL 65 (90) 2491 (borrowed from O Philipsen arxiv: ) Gunnar Bali (Regensburg) Magnetic field 8 / 32

9 Polyakov line and chiral condensate for physical quarks S Borsányi et al JHEP 1009 (10) 073 RenormalizedPolyakovloop 1.0 Continuum 0.8 N t 16 N t N t 10 N t T MeV ΨΨ R 0.4 Continuum N t 16 N t N t 10 N t T MeV T c ( ψψ) = 155(3)(3) MeV= 1.95(5) K. (Cross-over: other quantities may have different pseudocritical temperatures.) no order parameter but less than one in electrically charged particles differ by more than e/6 from a multiple of e! Gunnar Bali (Regensburg) Magnetic field 9 / 32

10 A possible phase diagram of QCD with chemical potential Gunnar Bali (Regensburg) Magnetic field 10 / 32

11 Gunnar Bali (Regensburg) Magnetic field 11 / 32

12 Noncentral heavy ion collision (100 MeV 2 ) eg = et. Gunnar Bali (Regensburg) Magnetic field 12 / 32

13 Gunnar Bali (Regensburg) Magnetic field 13 / 32

14 The QCD phase diagram in the B-T plane Low energy effective models of QCD predict(ed): increasing pseudocritical temperature T c (B) increasing strength 1/W (B) Mizher et al 10 Supported by NJL models, large-n c arguments, low-dimensional models, S-D equations Gatto et al 11, Johnson et al 09, Alexandre et al 01, Klimenko et al 92, Kanemura et al 98 T (MeV) Chiral transition Deconfinement transition Expected magnetic field generated in the LHC eb(m π ) T Χ, T P MeV D ΧSR C ΧSB TΧ TP T B MeV eb m Π 2 AJ Mizher et al PRD 82 (10) R Gatto, M Ruggieri PRD 83 (11) Gunnar Bali (Regensburg) Magnetic field 14 / 32

15 Other results (N f = 2): Decreasing T c (B) NO Agasian, SM Fedorov PLB 663 (08) 445 Almost constant (Lattice) M D Elia et al PRD 82 (10) T c (B) / T c (0) am= am=0.025 am= eb / T 2 Gunnar Bali (Regensburg) Magnetic field 15 / 32

16 Magnetic background field on the lattice Vector potential A ν = (0, Bx, 0, 0) = B = (0, 0, B) Lattice: multiply links U ν with u ν = e iaqaν U(1) u y (n) = e ia2 qbn x u x (n) = 1 n N x 1 u x (N x 1, n y, n z, n t ) = e ia2 qbn x n y u ν (n) = 1 ν x, y The magnetic flux through the x-y plane is constant: ( ) ( ) exp iq dσ B = exp iq dx ν A ν = e ia2 N x N yqb F Flux quantization due to the finite volume + boundary conditions: F a 2 N x N y qb = 2πN b N b Gunnar Bali (Regensburg) Magnetic field 16 / 32

17 Flux quantization Gunnar Bali (Regensburg) Magnetic field 17 / 32

18 Implementation and limitations B is invariant under N b N b + N x N y (periodicity) Lattice field is unambiguous if 0<N b < N x N y /4 Apply quantization for smallest charge q = e/3 Typical lattice spacings: Maximal B: qb max =π/(2a 2 ) eb 1 GeV e Gauss Typical aspect ratios: Minimal B: qb min = 2πT 2 (N t /N s ) 2 eb 0.1 GeV e Gauss Phenomenologically interesting region! Gunnar Bali (Regensburg) Magnetic field 18 / 32

19 Simulation and observables Partition function for three flavors (µ f = 0 in the simulation) Z = [du] e βsg [det M(q f B, m f,µ f )] 1/4 Observables ψψ f = T V f =u,d,s logz, χ f = T m f V Renormalization: cancel divergences by computing 2 logz mf 2, c2 s = T 2 logz V µ 2 s [ ψψ f r (B, T) = m f ψψf (B, T) ψψ ] 1 f (B = 0, T = 0) mπ 4 χ r f (B, T) = mf 2 [ χf (B, T) χ f (B = 0, T = 0) ] 1 mπ 4 Gunnar Bali (Regensburg) Magnetic field 19 / 32

20 Does B get renormalized? Does B induce any new divergencies? If so it has to be renormalized. Other renormalization factors may even become B-dependent. Quark masses for instance get renormalized. B breaks rotational symmetry and isospin symmetry. B modifies the free dispersion relation: E(B) = pz 2 + m2 + 2n qb So maybe something sinister or complicated is happening? Fortunately not: Protected by U(1) gauge invariance: (e B) r = e B due to Ward-Takahashi identity Z e Z3 = 1. B couples to a conserved current A ν ψγ ν ψ. For an external field there are no internal photon lines in Feynman diagrams no new type of divergent diagram. Gunnar Bali (Regensburg) Magnetic field 20 / 32

21 Simulation and analysis details Symanzik improved gauge action, N f = stout smeared staggered quarks at physical masses (Budapest-Wuppertal) action Simulate at various T and N b. Fit all points by a 2D spline function. Keep physical B fixed. Study finite volume effects with N s /N t = 3, 4, 5 Extrapolate to the continuum limit with N t = 6, 8, 10. Gunnar Bali (Regensburg) Magnetic field 21 / 32

22 Results: overview Gunnar Bali (Regensburg) Magnetic field 22 / 32

23 Chiral condensate ψψ decreases with B in the transition region. T c (B) decreases with B Gunnar Bali (Regensburg) Magnetic field 23 / 32

24 Previous study by D Elia et al Chiral cond. b= 0 Chiral cond. b = 8 Chiral cond. b = 16 Chiral cond. b = 24 Pol. loop b = 0 Pol. loop b = 8 Pol. loop b = 16 Pol. loop b = β ψψ always grows with B. T c (B) increases with B (largerβ= 3/(2πα): smaller N t a, higher T). The transition becomes narrower with bigger B. Gunnar Bali (Regensburg) Magnetic field 24 / 32

25 Comparison with D Elia et al D Elia et al Present study discretization errors N t = 4 N t = 6, 8, 10 naive staggered + Wilson stout + Symanzik quark flavours N f = 2 N f = light quark mass m π = 195 MeV m π = 135 MeV Gunnar Bali (Regensburg) Magnetic field 25 / 32

26 Quark mass dependence ūu(b, T) ūu(0, T): condensates are significantly different! Shift in theχ u peak positions! dramatic dependence on the light quark mass! Gunnar Bali (Regensburg) Magnetic field 26 / 32

27 Width of the transition At B = 0: broad crossover. What happens at B> 0? Height of the peak increases. However when normalized to the same height not much changes. Gunnar Bali (Regensburg) Magnetic field 27 / 32

28 Finite size scaling Comparison between N s = 16, 24, 32 on N t = 6 at eb/t 2 82 (Largest volume: V 7fm 3 ) The crossover persists up to eb = 1 GeV! Gunnar Bali (Regensburg) Magnetic field 28 / 32

29 Transition temperature Analyze B = const slices of the 2D surfaces Define transition temperatures by - inflection point for ψψ r and c s 2 - peak position forχ r 0.1 E.g. for the condensate: T (MeV) N b Then fit various N t results to T c (B, N t ) = T c (B) + b(b)/n 2 t. Gunnar Bali (Regensburg) Magnetic field 29 / 32

30 Phase diagram I Gunnar Bali (Regensburg) Magnetic field 30 / 32

31 Phase diagram II The effect is negligible for RHIC. The temperature reduces by less than 5 10% for LHC. The effect may be significant in the early universe. Gunnar Bali (Regensburg) Magnetic field 31 / 32

32 Summary and Outlook QCD with and external (electro-)magnetic field is interesting Lattice discretization & finite size effects are under control Phase diagram: decreasing T c (B) - complex, non-monotonic dependence in ψψ(b, T) - the crossover persists for large magnetic fields - no critical endpoint below eb 1 GeV Other questions are under investigation - magnetic susceptibility of the QCD vacuum - effects of magnetic fields on instanton shapes - the hadronic Zeemann/Paschen-Back/cigar-shape effect Gunnar Bali (Regensburg) Magnetic field 32 / 32