(Inverse) magnetic catalysis and phase transition in QCD

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1 (Inverse) magnetic catalysis and phase transition in QCD 1 Theory Seminar ITP Wien, Monday October In collaboration with Anders Tranberg (University of Stavanger), William Naylor and Rashid Khan (NTNU), PRD (2012), JHEP 1208 (2012) 002, JHEP 1404 (2014) 187.

2 Introduction Phase diagram of QCD

3 Introduction Hadronic matter in strong magnetic fields: Non-central heavy-ion collisions 2 2 Kharzeev, McLerran, and Warringa, Nucl. Phys. A 803 (2008) 227, Skokov, Illarianov, and Toneev, Int. J. Mod. Phys. A 24 (2009) 5925.

4 Introduction Hadronic matter in strong magnetic fields: Magnetars B = T 3 Electroweak phase transition B = T 4 3 Duncan and Thompson 4 Vachaspati, Enqvist and Olesen. Rummukainen et. al.

5 Lattice, effective theories, and models Lattice, effective theories, and models Lattice ( Buividovich et al, D Elia et al, Endrődi et al, Bali et al. Bag model (Fraga and Palhares) Chiral perturbation theory (Agasian and Shushpanov, Agasian and Fedorov, JOA...) (Polyakov-loop extended) Nambu-Jona-Lasinio model (Mizher, Chernodub, and Fraga, Fukushima, Ruggieri and Gatto, Kashiwa...) (Polyakov loop extended) Linear sigma model with quarks (Skokov, JOA+Tranberg+Naylor) Holographic models (Preis, Rebhan, and Schmitt) Schwinger-Dyson (Bonnet et al.)

6 Some important questions: Critical temperature as a function of B Chiral condensate increases with B at T = 0 Splitting of chiral and deconfinement transition Can be addressed by adding the Polyakov loop Phase diagram in a magnetic background Does the magnetic field change the order of the transition? Position of critical endpoint?

7 Lagrangian and symmetries [ ] L = ψ γ µ µ + g(σ iγ 5 τ π) ψ + 1 ] [( µ σ) 2 + ( µ π) [ ] 2 m2 σ 2 + π 2 + λ [ ] 2 σ 2 + π 2 hσ. 24 O(4)-symmetry broken to O(3) by chiral condensate. Three Goldstone bosons (pions). Magnetic field breaks SU(2) isopin symmetry. Only the neutral pion is a Goldstone mode. σ 2 + π 2 (σ 2 + π 2 0 ) + 2(π+ π ) with two separate couplings.

8 Mean-field approximation Background and quantum field σ φ + σ. Omit bosonic quantum and thermal fluctuations. Large-N c limit Effective potential F 0+1 = 1 2 m2 φ 2 + λ 24 φ4 hφ Tr log S 1, Tr log S 1 = f Tr log[iγ µ (P µ + q f A µ ) + m q ], m q = gφ.

9 Mean-field approximation One-loop contribution F 1 = B {P} = q f B 2π F T =0 1 = q f B 2π = q f B 2π s,k [ ] ln P0 2 + p2 z + MB 2 dpz [ ] 2π ln P0 2 + p2 z + mq 2 + q f B (2k + 1 s) s,k,p 0 s=±1 k=0 dpz pz 2 + mq 2 + q 2π f B (2k + 1 s) ( e γ E Λ 2) ɛ Γ( 1 + ɛ) [ m 2 + q f B (2k + 1 s)] 1 ɛ..

10 Mean-field approximation Hurwitz zeta function: ζ(s, q) = (k + q) s k=0 ) ɛ [ Γ( 1 + ɛ) ζ( 1 + ɛ, x f ) 1 ] 2 x f F1 T =0 = (q f B) 2 ( e γ E Λ 2 2π 2 2 q f B ( ) 1 Λ 2 ɛ [ ( 2(qf B) 2 = (4π) 2 + mq 4 2 q f B 3 ) ( ) 1 ɛ + 1 ] 8(q f B) 2 ζ (1,0) ( 1, x f ) 2 q f B mq 2 ln x f + O(ɛ), x f = m 2 q 2 q f B.

11 Mean-field approximation Renormalized one-loop effective potential F 0+1 = 1 2 B2 [ K B 0 (βm) = 8 s,k 1 + N c f ] 4qf 2 3(4π) 2 ln Λ q f B 2 m2 φ 2 + λ 24 φ4 N c 2π 2 (q f B) [ζ 2 (1,0) ( 1, x f ) x f ln x f 1 4 x f 2 f 1 ] 2 x f 2 ln Λ2 N ct 2 2 q f B 8π 2 q f B K0 B (βm q). 0 dp z p 2 z p 2 z + M 2 B f 1 e β. pz 2 +MB 2 + 1

12 Magnetic catalysis at T = 0 F 0+1 /f 4 π φ [MeV] One-loop effective potential unstable - need bosonic contribution to stabilize. By subtracting one-loop effective potential at B = 0, one avoids the instability.

13 Phase transition Figure: Effective potential in the chiral limit. Order of phase phase transition depends on treatment of vacuum fluctuations First-order vs crossover at the physical point.

14 Functional renormalization group Flow equation for the effective average action Γ k [φ] that interpolates between the classical action S for k = Λ and full quantum effective action Γ 0 [φ] for k = Wetterich, Nucl. Phys. B 352 (1991) 529, Berges, Pawlowski, Schaefer and Wambach, Skokov...

15 Functional renormalization group S[φ] S[φ] + 1 d d q 2 (2π) d φ( q)r k(q)φ(q), Z[J] = Dφe S[φ] S k [φ]+ Jφ Γ k [φ] = ln Z k + jφ S k [φ] R k (q) satisfies various conditions to implement RG ideas: Suppress momenta pqk and leave momenta q > k unchanged. R k=0 (q) = 0 and R k=λ (q) =.

16 Flow equation for Γ k [φ] k Γ k [φ] = 1 [ ] ] [ 2 Tr k R k (q) Γ (2) 1 k + R k (q) + term for fermions. Flow equation is exact. Feynman diagram for flow equation k Γ k = 1 2

17 Functional renormalization group Cannot solve flow equation exactly. Derivative expansion: Γ k [φ] = β Z (2) k +Z (4) k dτ d 3 x { 1 2 Z (1) k [( σ) 2 + ( π) 2] [ ( 0 σ) 2 + ( 0 π) 2] + U k (φ) + Z (3) k ψγ 0 0 ψ } ψγ i i ψ + g k ψ [σ iγ 5 τ π] ψ +..., Local-potential approximation (LPA): Z (i) k = 1 yields equation for effective potential U k [φ]. Beyond LPA: calculate anomalous dimensions.

18 Functional renormalization group k U k = k 4 [ 1 12π 2 + qb 2π 2 ω 1,k ( 1 + 2nB (ω 1,k ) + 1 m=0 [ 1 + 2n B (ω 1,k ) ] N c 2π 2 θ s,f,m=0 Skokov, Phys. Rev. D 85 (2012) ] ( ) 1 + 2nB (ω 2,k ω 2,k k ( ) k ω 2 p 2 (q, m, 0) θ k 2 p 2 (q, m, 0) 1,k ( k 2 p 2 (q f, m, s) q f B k k ω 2 p 2 (q f, m, s) q,k ) [1 n + Flow equation depends on regulator. F (ω q f,k) n F (ω q f,k) ],

19 Functional renormalization group Boundary condition Γ k=λ [φ] = S classical, [ 1 S classical = d 4 x 2 m2 Λ φ2 + λ ] Λ 24 φ4. mλ 2 and λ Λ tuned to reproduce vacuum physics T = B = 0 for k = 0 mk,π 2 = Ũk ρ, φ=φk mk,σ 2 = mk,π 2 + Ũ k 2ρ 2 ρ 2. φ=φk Ũk φ = h, φ=φk

20 Figure: Effective potential in different approximations (left) and different t = ln k Λ. U 0 (φ)/λ φ min /MeV B qb 1/2 /Λ qb = 0 qb = 0.05 Λ 2 qb = 0.1 Λ 2 qb = 0.2 Λ 2 qb = 0.3 Λ 2 qb = 0.4 Λ φ/mev Figure: Effective potential for different values of B.

21 φ min /MeV T c /MeV T C T/MeV qb 1/2 /Λ Figure: Order parameter φ(t ) (left) and T c (B) (right).

22 Adding the Polyakov loop Polyakov loop Φ is an order parameter for confinement in (pure-glue) QCD Φ = 1 N c Tr L L = Pe i β 0 dτa 0(x,τ). Φ 0, confinenment at lowt, Φ 1, confinenment at hight. Adding the Polyakov loop by introducing a constant background A 0 6 µ µ iδ 0µ A µ. 6 Fukushima 2003.

23 Distribution function Limits N F N F N F Pure-glue potential 1 + 2Φe βeq + Φe 2βEq 1 + 3e βeq + 3Φe 2βEq + e 3βEq. 1 e 3βEq + 1, Φ = 0. 1 e βeq + 1, Φ = 1. U T 4 = 1 2 a(t )Φ2 + b(t) ln[1 6Φ 2 + 8Φ 3 3Φ) 4 ]. a(t ) and b(t) used to reproduce pure-glue QCD thermodyanmics. Other potentials with similar properties exist.

24 Transition temperature T 0 a(t ) = ( ) 2 ( ) 3 T0 T T T Make T 0 dependent of N f to include backreaction from fermions. Figure: Quark condensate and Polyakov loop two-color QCD.

25 Coupled equation for φ and Φ. QM and PQM model the same at T = 0. Less magnetic catalysis at finite temperature. Deconfinement transition hardly affected by B. T φ (B) / T φ (0) With Polyakov loop Without Polyakov loop Tφ, T Φ / 2, T Φ/ 2 [MeV] T φ T Φ /2 T Φ/ qb / m π µ =0MeV qb /mπ

26 Lattice calculations 0,30 0,25 u quark d quark u quark, quenched -<qq> R [GeV 3 ] 0,20 0,15 0,10 0,05 0, eh [GeV 2 ] Figure: Quark condensates at T = 0. Bali et al 2012 (left) and Bonnet 2014 (right).

27 Figure: T c as a function of the magnetic field B. D Elia et al 2010 (left) and Bali et al 2012 (right).

28 Z(B) = ψψ = due Sg det(d/(b) + m), 1 due Sg det(d/(b) + m)tr(d/(b) + m) 1, Z(B) Valence contribution: ψψ val 1 = due Sg det(d/(0) + m)tr(d/(b) + m) 1. Z(0) Sea contribution: ψψ sea 1 = Z(B) due Sg det(d/(b) + m)tr(d/(0) + m) 1.

29 Relative increment r = ψψ (B) ψψ (0) 1. Figure: Valence and sea contribution to the quark condensate. D Elia et al (2010).

30 Banks-Casher relation lim lim ψψ lim lim ρ(λ). V m 0 V m 0 Figure: Spectral density for different values of B. Endrodi et al 2013.

31 Reweighting S f (B) = log det(d/(b) + m) log det(d/(0) + m) ψψ = e S f (B) Tr(D/(B) + m) 1 0 e S f (B) 0, Figure: Scatter plot quark condensate as a function of S(B). Endrodi et al 2013.

32 Model calculations revisited B-dependent coupling constant - QCD inspired G(B) = 1 + α ln ( G(B, T ) = G(B) G 0 ( 1 + β qb Λ QCD ), 1 γ qb Λ 2 QCD T Λ QCD. Fit parameters to reproduce quark condensate at T = 0 and at high T. )

33 Model calculations revisited Figure: Farias (2014).

34 Conclusions and Outlook Order of phase transition depends on treatment of vacuum fluctuations - take them seriously. Functional RG predicts a second order transition Model calculations correctly predict magnetic catalysis at T = 0, but fail to reproduce inverse magnetic catalysis around the transition temperature. Some model calculations have been modified to reproduce lattice results - essentially curve fitting. Need to incorporate back-reaction from fermion determinant.