Non-perturbative Study of Chiral Phase Transition

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1 Non-perturbative Study of Chiral Phase Transition Ana Juričić Advisor: Bernd-Jochen Schaefer University of Graz Graz, January 9, 2013

2 Table of Contents Chiral Phase Transition in Low Energy QCD Renormalization Group Flows Application to Quark-Meson Model Summary and Outlook

3 Chiral symmetry of QCD Chiral symmetry restoration: a key feature of hot and dense QCD matter Fluctuations of QCD vacuum constituent quark mass In hot and dense QCD matter bare quarks Chiral phase transition Order parameter is the chiral condensate ψψ (0.24 GeV) 3 (in vacuum) Go beyond mean-field methods and include fluctuations Employ renormalization group methods

4 Scales of QCD Above 2 GeV perturbative QCD At lower momentum scales: bound states, quark condensate, confinement... Physics of scalar and pseudo-scalar mesons Figure: A possible hierarchy of the momentum scales of QCD, [The CBM Physics Book] Compositeness scale k φ 1 GeV - mesonic bound states are formed Chiral symmetry breaking scale k χ 500 MeV Confinement related to Λ QCD 200 MeV

5 Effective realization of low energy QCD For scales k χ < k < k φ relevant degrees of freedom - quarks and mesons Quark and meson interaction - constant Yukawa coupling g For k < k χ quarks will confine - cannot be neglected Beyond meson dynamics Constituent quark mass grows and quarks decouple Neglect influence of the confinement for studying meson physics Good truncation: quark-meson model

6 Motivation for Quark-Meson Model Left upper corner: SU(2) L SU(2) R O(4) symmetry restoration Figure: Columbia plot Two flavor QM model Initial effective action at compositeness scale { } Γ = d 4 1 x 2 ( µφ)2 + U(φ 2 ) + q [γ + g(σ + i π τγ 5)] q (1) O(4) symmetric meson field φ = (σ, π) Quark fields: q

7 Renormalization group flows: Wetterich equation Change of the average effective action with the scale Here: one component scalar field ϕ(x) Regulator term introduces RG scale k dependence Z k [J] = e W k [J] = Dϕ e S[ϕ] S k [ϕ]+ Jϕ (2) Regulator term is quadratic in fields S k [ϕ] = 1 d D q 2 (2π) ϕ( q)r k(q)ϕ(q) (3) D and R k (q) satisfies certain criteria Figure: Characteristic regulator function and its derivative, [H. Gies, 2006]

8 Wetterich equation Effective average action (modified Legendre transform) ( ) Wetterich equation is Γ k [φ] = sup J Jφ W k [J] S k [φ] (4) tγ k = 1 2 Tr tr k Γ (2) k [φ] + R k (5) Figure: Different regulator functions lead to different trajectories in the theory space, but starting and final point are fixed.

9 Regulator functions Regulator functions are sensitive to approximations and truncations Figure: [D. F. Litim, J. M. Pawlowski;2002] Stable flow equations and good convergence properties There is no a unique criterion for choosing the regulator function Alternative realization of flow: proper-time flow equations

10 Standard proper-time renormalization group flow One-loop expression for effective action Γ 1 loop [φ] = S[φ] Tr ln S(2) [φ] (6) Schwinger proper-time representation of the logarithm and regulator function f (Λ, s) Γ 1 loop [φ] = S[φ] 1 2 ds s f (Λ, s) Tr exp ( ss (2)) (7) k scale dependence is inserted replacing f (Λ, s) f k (Λ, s) = f (sλ 2 ) + f (sk 2 ) (8) Last step: RG improvement tγ k = ds s Class of optimized regulator functions ( tf (sk 2 ) ) ) exp ( sγ (2) k tf (m) (sk 2 ) = 2 Γ(m) (sk2 ) m e sk 2 (9) (10)

11 Comparison of Wetterich and PT flow equations Wetterich equation is derived from first principles Standard PT flow is not exact Physical significance of the flow equation depends on the systematic approximation method, and the power of its lowest order All approximations and expansions for Wetterich equation can be applied for standard PT flow Leading order of derivative expansion: a map R k f k Wetterich equation for effective potential standard PT flow equation for effective potential Inverse map does not exist in general

12 Generalized proper-time flows There is a way to map generalized PT flow to Wetterich equation in the background formalism Field ϕ is split into the background filed ϕ and fluctuations ϕ Generalized PT flow: Difference: terms tγ (2) k in flow equation Generalized PT flows are exact tf k (Λ, s) tf k (Λ, s, Γ (2) k ) (11)

13 Flow for Quark-Meson Model Proper-time flow equation with optimized regulator function f k (x; m = 5/2) Lowest order of derivative expansion Projection on constant fields φ(x) = (σ, 0) and q = q = 0 Yukawa coupling is not running Temperature T is introduced by using Matsubara formalism Finite quark chemical potential by adding a term iµ qγ0q into the x effective action Isospin symmetry is assumed Flow equation for the effective potential U k

14 Flow and interpretation Flow equation: tu k (T, µ, φ 2 ) = ( 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 (12) E q 2T Energies for quark and antiquark E q = k 2 + g 2 φ 2, (13) pion and σ-meson Prime φ 2 -derivative E π = k 2 + 2U k, Eσ = k 2 + 2U k + 4φ2 U k (14) Correct dimensional factor Additive contribution of relevant degrees of freedom to the equation Three degenerate pions, one σ-meson Quarks and antiquarks: degeneracy factor (2s + 1)N cn f with s = 1/2 Quarks and antiquarks influence on mesons only through E π and E σ

15 Numerical solution The flow equation can be solved only numerically Two approaches: using the grid or expansion of the potential With Taylor expansion of the effective potential, a relatively small set of equations has to be solved But the potential is known only around its minimum With each order of the Taylor expansion, a new coupled equation is added Grid: potential for arbitrary φ 2 Important for first-order phase transition There is a coupled equation for each grid point

16 Taylor Expansion of the Effective Potential Minimum of the effective potential φ 2 = ρ is the order parameter for chiral symmetry In symmetric phase ρ 0 = 0 and we use U k = N n=0 The expansion is truncated at order N a n n! ρn (15) Set of differential equations for the expansion coefficients - beta functions If a 1 < 0 ρ 0 0 Broken symmetry phase: Additional equation for ρ 0 U k = b 0 + N n=2 b n n! (ρ ρ0)n (16) tρ 0 = 1 U k (ρ0) ( tu k(ρ 0) ) (17)

17 Initial ultraviolet conditions Initial effective potential in vacuum for UV scale k Λ U Λ (φ 2 ) = a 0(Λ) + a 1(Λ)φ 2 + a2(λ) φ 4 (18) 2 Initial parameters give f π = φ 0 = 93 MeV for k = 0 and chiral symmetry breaking scale in vacuum Yukawa coupling is given by the constituent quark mass in IR These initial UV parameters are used for finite T and µ Temperature does not influence dynamics at small scales for T /k < 0.1 For k Λ = 950 MeV, the results are valid up to T = 95 MeV System is set to (T = 40 MeV, µ = MeV) Runge-Kutta method with adaptive step size

18 Results for symmetric phase Taylor expansion coefficients (third order) 0.3 a 0 /Λ a 1 /Λ k [MeV] k [MeV] a k [MeV] a 3 Λ k [MeV] Figure: Running of the potential expansion coefficients in symmetric phase. Results for (T = 40 MeV, µ = MeV). Chiral symmetry breaking at k = MeV

19 Symmetric and broken symmetry phase Taylor expansion coefficients for both phases (truncated at third order) a 0 /Λ 4 b 0 /Λ k [MeV] a 1 /Λ 2 φ 2 0 /Λ k [MeV] 6.0 a k [MeV] b a 3 Λ 2 b 3 Λ k [MeV] Figure: Running of the potential expansion coefficients. Results for (T = 40 MeV, µ = MeV).

20 Minimum of effective potential Close to the phase transition φ 0 [MeV] k [MeV] Figure: Minimum of U k at (T = 40 MeV, µ = MeV). Pion decay constant f π = lim k 0 φ 0 = 8 MeV

21 Particle masses Chiral symmetry is spontaneously broken: three massless pions m [MeV] k [MeV] Figure: Particle masses for (T = 40 MeV, µ = MeV). Red line stands for quarks and antiquarks, blue lines for mesons. Full blue line are three degenerate pions, and dashed blue line is σ-meson.

22 Summary Aim: a more realistic description of chiral phase transition Wetterich, standard PT and generalized PT flow equations Comparison of flow equations Quark-meson model is an appropriate choice for the initial effective action Standard PT flow equation for effective grand canonical potential Numerical solution: expansion of the effective potential

23 Outlook Full phase diagram in T µ plane Scheme dependence: different truncations Influence of regulator functions (optimized and others) Comparison with alternative numerical techniques To include description of confinement phenomena Finite volume