Renormalization Group Study of the Chiral Phase Transition

Transcription

1 Renormalization Group Study of the Chiral Phase Transition Ana Juričić, Bernd-Jochen Schaefer University of Graz Graz, May 23, 2013

2 Table of Contents 1 Proper Time Renormalization Group 2 Quark-Meson Model Truncation 3 Solution Strategies Taylor Expansion Potential on a Grid 4 Summary 1

3 Introduction to Renormalization Group The aim: full quantum action Γ Starting point: classical, bare action S at an ultraviolet scale Λ Step by step integration of fluctuations in momentum shells Scale dependent effective action Γ k Flow equation: the change of the average effective action with the scale k 2

4 Wetterich equation Here: one component scalar field ϕ(x) Introduce scale dependence with a regulator term in the generating functional Z k [J] = e W k [J] = Dϕ e S[ϕ] S k [ϕ]+ Jϕ (1) Regulator term is quadratic in fields S k [ϕ] = 1 2 R k (p) satisfies certain criteria (optimized regulator!) d D p (2π) D ϕ( p)r k(p)ϕ(p) (2) [D. F. Litim, J. M. Pawlowski (2002)] [D. F. Litim, J. M. Pawlowski (2002)] 3

5 Wetterich equation Effective average action via modified Legendre transform ( ) Γ k [φ] = sup J Jφ W k [J] S k [φ] Wetterich equation is tγ k = 1 2 Tr { tr k [Γ (2) k [φ] + R k] 1 } (3) (4) IR regularization: lim p 2 /k 2 0 R k (p) > 0 UV regularization: tr k (p) is peaked around p k [H. Gies (2006)] 4

6 Proper-time (PT) flow equation One-loop expression for effective action Γ 1 loop [φ] = S[φ] Tr ln S(2) [φ] (5) Schwinger proper-time representation of the logarithm and regulator function f (Λ, s) Γ 1 loop [φ] = S[φ] 1 ds 2 s f (Λ, s) Tr exp ( ss (2)) (6) scale dependence: f (Λ, s) f k (Λ, s) = f (sλ 2 ) + f (sk 2 ) Last step: RG improvement tγ k = ds s ( tf (sk 2 ) ) ) exp ( sγ (2) k (7) 5

7 PT flow as a approximation to Wetterich flow Class of optimized regulator functions tf (m) (sk 2 ) = 2 Γ(m) (sk2 ) m e sk 2 (8) Flow simplifies to [ ] m tγ k = Tr (Γ (2) k + k 2 ) 1 k 2 (9) Generalized proper time flows in the background formalism tγ Wetterich k = tγ sptrg k + tγ (2) k (10) Term tγ (2) k vanishes at the criticality In the leading order of derivative expansion it is possible to map Wetterich equation regulator PT flow regulator Mapping is injection: m d/2 + 1 (d is the number of dimensions) 6

8 Scales of QCD Above 2 GeV perturbative QCD At lower momentum scales: bound states, quark condensate, confinement... Possible hierarchy of the scales: Compositeness scale k φ 1 GeV - mesonic bound states are formed Chiral symmetry breaking scale k χ = 500 MeV Baryon formation 7

9 Motivation for Quark-Meson Model Start the RG evolution at Λ = k φ Initial effective action: two flavor quark-meson model { } Γ = d 4 1 x 2 ( µφ)2 + U(φ) + q [γ + g(σ + i π τγ 5)] q (11) Short-hand notation φ = (σ, π) U(φ) meson potential - Mexican hat potential Physical picture behind: integrate out all gluons non-local effective 4 and higher quark vertices are replaced with mesons Truncation: only scalar and pseudoscalar channel Most important for thermodynamics Baryons are omitted: good approximation for small quark chemical potentials 8

10 Flow for Quark-Meson Model Proper-time flow equation with optimized regulator function f 5/2 k (sk 2 ) Lowest order of derivative expansion { Γ[φ] = d 4 x U(φ) + 1 } 2 Z( µφ) (12) with Z = 1 Projection on constant fields φ(x) = (σ, 0) Yukawa g coupling is not running Temperature - imaginary time formalism And finite quark chemical potential β Γ = dτ 0 { } d 3 1 x 2 ( µφ)2 + U(φ) + q [γ + g(σ + i π τγ 5) + γ 0µ] q (13) Isospin symmetry is assumed Flow equation for the grand potential U k 9

11 Interpretation Flow equation: tu k (T, µ) = ( k5 1 coth Eσ 12π 2 E σ 2T + 3 coth Eπ E π 2T 2NcN f tanh Eq µ E q 2T ) 2NcN f tanh Eq + µ E q 2T quark/antiquark E q = k 2 + g 2 φ 2 pion E π = k 2 + 2U k σ-meson E σ = k 2 + 2U k + 4φ2 U k Additive contribution of relevant degrees of freedom to the equation T = 0: Silver Blaze property ( ) tu k = k NcN f Θ(E 12π 2 q µ) E σ E π E q (14) (15) Opposite roles of quarks and meson in RG evolution 10

12 Comparison with Mean-Field (MF) Approximation MF approximation: neglecting meson fluctuations replacing meson fields replaced with their expectational values (σ 0, 0) Grand potential for the MF d 3 p U = U B (σ 0, 0) + νt { βeq + ln[1 nq(t, µ)] + ln[1 (2π) 3 n q(t, µ)]}, (16) where n q(t, µ) = 1, n q(t, µ) = nq(t, µ) (17) 1 + exp[β(e q µ)] Cross out the meson potential, find scale derivative Flow in the MF approximation ( tu k (T, µ) = k5 2N cn f tanh Eq µ 12π 2 E q 2T ) Eq + µ + tanh 2T Flow for QM model enables studying the influence of the meson fluctuations (18) 11

13 Taylor Expansion of the Effective Potential Start at UV scale: Mexican hat potential integrating out fluctuations generating new vertices couplings are scale dependent coefficients in expansion of U k notation: φ 2 = ρ Symmetric phase U k = N n=0 a n n! ρn (19) Broken symmetry phase U k = b 0 + N n=2 b n n! (ρ ρ0)n (20) U(0) ρ < 0 non-trivial minimum ρ 0 tu k N beta functions for {a 0,..., a N } tu k beta functions for {b 0, b 2,..., b N } additional tρ 0 Small number N of equations U k known only at its minimum 12

14 Initial ultraviolet parameters Initial effective potential in vacuum for UV scale k Λ U Λ (φ) = λ 4 (φ2 v 2 ) 2 cσ (21) Initial parameters give f π = σ 0 = 93 MeV for k = 0 and chiral symmetry breaking scale in vacuum Yukawa coupling: m q = gσ 0 Initial UV parameters fixed for finite T and µ Temperature range is limited T /Λ <

15 Influence of the Truncation Order f π [MeV] k χ [MeV] N =3 N =4 N = T [MeV] N =3 N =4 N =5 Results for µ = 0 Truncation orders N = 3, 4, 5 Convergence for N = 5 Note: k χ independent of the truncation order Second order phase transition at T c = MeV with N = T [MeV] 14

16 Mass Sensibility for Taylor Expanded Potential [MeV] m q m π m σ m q m π m σ T [MeV] Chiral limit: T c = MeV Pions - Goldstone bosons Explicit χsb: a crossover T > 200 MeV: pions and σ-meson acquire the same mass Meson masses rise linearly with T > 200 MeV - lowest quark Matsubara mode Influence of fixed initial potential parameters 15

17 Effective Potential on a Grid U k Problem: tu(σ i) = F (U (σ i), U (σ i)) Also, tu (σ i) = G(U (σ i), U (3) (σ i)) and so on... Find derivatives U (σ i) and U (3) (σ i) independently of the flow σ i σ i 1D grid with M σ-field points M > N flow equations global minimum of U k Taylor expansion of U(σ i) at each grid point to the third order Matching expansions at half distance between neighboring grid points 16

18 RG Evolution on Grid 4e9 (T, µ) =(100 MeV, 0) k =950 MeV 3e9 U [MeV 4 ] 2e9 1e σ [MeV] 17

19 RG Evolution on Grid 3e9 (T, µ) =(100 MeV, 0) k =850 MeV 2.5e9 2e9 U [MeV 4 ] 1.5e9 1e9 5e σ [MeV] 17

20 RG Evolution on Grid 2e9 (T, µ) =(100 MeV, 0) k =750 MeV 1.5e9 U [MeV 4 ] 1e9 5e σ [MeV] 17

21 RG Evolution on Grid 1.2e9 (T, µ) =(100 MeV, 0) k =650 MeV 1e9 U [MeV 4 ] 8e8 6e8 4e8 2e σ [MeV] 17

22 RG Evolution on Grid 7e8 (T, µ) =(100 MeV, 0) k =550 MeV 6e8 5e8 U [MeV 4 ] 4e8 3e8 2e8 1e σ [MeV] 17

23 RG Evolution on Grid 4e8 (T, µ) =(100 MeV, 0) k =450 MeV 3e8 U [MeV 4 ] 2e8 1e σ [MeV] Spontaneous symmetry breaking 17

24 RG Evolution on Grid 2.5e8 (T, µ) =(100 MeV, 0) k =350 MeV 2e8 U [MeV 4 ] 1.5e8 1e8 5e σ [MeV] Minimum of U k is fixed Inner part of U k flattens 17

25 RG Evolution on Grid 2e8 (T, µ) =(100 MeV, 0) k =250 MeV 1.5e8 U [MeV 4 ] 1e8 5e σ [MeV] Minimum of U k is fixed Inner part of U k flattens 17

26 RG Evolution on Grid 2e8 (T, µ) =(100 MeV, 0) k =150 MeV 1.5e8 U [MeV 4 ] 1e8 5e σ [MeV] Minimum of U k is fixed Inner part of U k flattens 17

27 RG Evolution on Grid 2e8 (T, µ) =(100 MeV, 0) k =100 MeV 1.5e8 U [MeV 4 ] 1e8 5e σ [MeV] Minimum of U k is fixed Inner part of U k flattens 17

28 RG Evolution on Grid 2e8 (T, µ) =(100 MeV, 0) k =50 MeV 1.5e8 U [MeV 4 ] 1e8 5e σ [MeV] Minimum of U k is fixed Inner part of U k flattens 17

29 Mass Sensitivity on the Grid [MeV] m q m π m σ m q m π m σ T [MeV] Chiral limit: T c = MeV Explicit χsb: crossover Comparison with the Taylor expanded potential... 18

30 Comparison Chiral limit: Explicit χsb [MeV] m π, TE m σ, TE m q, TE m π, grid m σ, grid m q, grid [MeV] m q, TE m π, TE m σ, TE m q, grid m π, grid m σ, grid T [MeV] T [MeV] 19

31 Summary Quark-meson model as a good approximation to QCD Flow equation Comparison of two solution strategies To do: map phase diagram for µ > 0 20