t Hooft Determinant at Finite Temperature with Fluctuations

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1 t Hooft Determinant at Finite Temperature with Fluctuations Mario Mitter In collaboration with: Bernd-Jochen Schaefer, Nils Strodthoff, Lorenz von Smekal (former) PhD Advisers: Reinhard Alkofer, Bernd-Jochen Schaefer Universität Graz Universität Heidelberg Doktoratskolleg Graz Hadrons in Vacuum, Nuclei and Stars, funded by the Austrian Science Fund: FWF DK W1203-N16. Heidelberg, January 2013 M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

2 Table of Contents 1 Motivation 2 Axial Anomaly with Quarks and Mesons 3 Results M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

3 Chiral Symmetry and Axial Anomaly symmetry of massless QCD with N f flavors: U(N f ) L U(N f ) R = U(1)V /Z Nf SU(N f ) L U(N f ) R U(1) A /Z 2Nf M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

4 Chiral Symmetry and Axial Anomaly symmetry of massless QCD with N f flavors: U(N f ) L U(N f ) R = U(1)V /Z Nf SU(N f ) L U(N f ) R U(1) A /Z 2Nf spontaneously broken to U(N f ) L+R : quark condensate(s) qq one Nambu-Goldstone boson for every broken generator (pions,...) M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

5 Chiral Symmetry and Axial Anomaly symmetry of massless QCD with N f flavors: U(N f ) L U(N f ) R = U(1)V /Z Nf SU(N f ) L U(N f ) R U(1) A /Z 2Nf spontaneously broken to U(N f ) L+R : quark condensate(s) qq one Nambu-Goldstone boson for every broken generator (pions,...) explicit breaking by quark masses light pseudo-goldstone modes M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

6 Chiral Symmetry and Axial Anomaly symmetry of massless QCD with N f flavors: U(N f ) L U(N f ) R = U(1)V /Z Nf SU(N f ) L U(N f ) R U(1) A /Z 2Nf spontaneously broken to U(N f ) L+R : quark condensate(s) qq one Nambu-Goldstone boson for every broken generator (pions,...) explicit breaking by quark masses light pseudo-goldstone modes η -meson is no pseudo-goldstone boson N 2 f 1 +1 broken generators experiment Nf = 2 (N f = 3): 3 pions (+4 kaons and 1 η-meson) U(1)A anomalously broken [Adler, Bell, Jackiw, 1969], [Fujikawa, 1979] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

7 U(1) A at Chiral Transition Recent experimental results (PHENIX, STAR): [Csorgo et al., 2010] drop δm η 200 MeV at chiral transition temperature partial U(1) A -symmetry restoration at chiral crossover? Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> 0 crossover FAIR/NICA <ψψ> = / 0 AGS SIS quark matter hadronic fluid crossover superfluid/superconducting phases? n B= 0 n B> 0 2SC <ψψ> = / 0 CFL vacuum nuclear matter neutron star cores µ [CBM Physics Book, 2011] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

8 U(1) A at Chiral Transition Recent experimental results (PHENIX, STAR): [Csorgo et al., 2010] drop δm η 200 MeV at chiral transition temperature partial U(1) A -symmetry restoration at chiral crossover? Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> 0 crossover FAIR/NICA <ψψ> = / 0 AGS SIS quark matter hadronic fluid crossover superfluid/superconducting phases? n B= 0 n B> 0 2SC <ψψ> = / 0 CFL vacuum nuclear matter neutron star cores µ m s tric m s 0 0 2nd order O(4)? phys. point 1st N f = 2 2nd order Z(2) m u, md 2nd order Z(2) N f = 3 crossover 1st Pure Gauge N f = 1 [CBM Physics Book, 2011] [Laermann, Philipsen, 2003] Effects of anomaly in chiral limit [Pisarski, Wilczek, 1984] N f > 2: 1 st order N f = 2: 1 st 2 nd if insensitive to temperature for T < T c M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

9 Effective Description The Quark-Meson Model (general N f ) [Jungnickel, Wetterich, 1996] L QM L M = q( µ γ µ iht a (σ a +iγ 5π a ))q +L M [ ] ( = tr µ Σ µ Σ +U ({ρ i},ξ) tr C (Σ+Σ )) Σ = t a (σ a +iπ a ), t a : group generators of U(N f ) ( ) ] i ρ i = tr[ Σ Σ, C = diag(c 1,...,c Nf ) U(1) A anomaly in QM-Model integrate instantons in QCD Lagrangian [ t Hooft, 1976] contribution to Lagrangian proportional (for QCD with θ = 0) det(q L q R )+det(q R q L ) axial anomaly in QM-model: ξ = det(σ)+det(σ ) M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

10 Wetterich Equation for Effective Potential Flow Equation in Leading Order Derivative Expansion k U k = k4 12π 2 E M/Q,i = [ 2N 2 f i=1 N f 2N c 1 E M,i coth 1 E Q,i i=1 k 2 +m 2 M/Q,i ( ) EM,i 2T { tanh ( EQ,i +µ i 2T ) ( EQ,i µ i +tanh 2T )} ] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

11 Wetterich Equation for Effective Potential Flow Equation in Leading Order Derivative Expansion k U k = k4 12π 2 E M/Q,i = [ 2N 2 f i=1 N f 2N c 1 E M,i coth 1 E Q,i i=1 k 2 +m 2 M/Q,i ( ) EM,i 2T { tanh ( EQ,i +µ i 2T ) ( EQ,i µ i +tanh 2T )} ] Two Approximation Methods Taylor expansion of U k in ({ρ i },ξ) discretize U k : grid ({ρi },ξ) splines,...for derivatives M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

12 N f = 2 effective potential (σ 0 ūu + dd, σ 3 dd ūu ): U k (Σ) = Ũk(ρ 1,ξ) c 0 σ 0 c 3 σ 3, first chiral invariant and t Hooft determinant: ρ 1 = σ2 0 +σ2 3 2, ξ = σ2 0 σ2 3 2, M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

13 Fix Parameters in Vacuum T = 0 initial potential: U Λ = a Λ,10 (ρ 1 ρ 1,0 )+a Λ,01 (ξ ξ 0 )+ a Λ,20 2 (ρ 1 ρ 1,0 ) 2 c 0 σ 0 c 3 σ 3 Yukawa coupling h m π, f π, m d +m u, m η and m σ at k 0 M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

14 Fix Parameters in Vacuum T = 0 initial potential: U Λ = a Λ,10 (ρ 1 ρ 1,0 )+a Λ,01 (ξ ξ 0 )+ a Λ,20 2 (ρ 1 ρ 1,0 ) 2 c 0 σ 0 c 3 σ 3 Yukawa coupling h m π, f π, m d +m u, m η and m σ at k 0 different isospin breakings c 3 M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

15 Mesonic Masses realistic (N f = 3) m η : m η (k 0) = 980 MeV explicit breaking: c 3 = c 0 masses [MeV] π σ a 0 η,a ± T [MeV] solid: Taylor RG, crosses: grid RG, short dashed: MF [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

16 Condensates realistic (N f = 3) m η : m η (k 0) = 980 MeV explicit breaking: c 3 = c 0 condensates [MeV] <σ 0 > 10*<σ 3 > T [MeV] solid: Taylor RG, crosses: grid RG, short dashed: MF [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

17 Influence of Explicit Symmetry Breaking realistic (N f = 3) m η : m η (k 0) = 980 MeV solid: c 3 = c 0, dotted: c 3 = 0.1 c 0 (dashed: 10 c 3 = 10 c 0 = c phys ) masses [MeV] π σ a 0 η,a ± condensates [MeV] <σ 0 > 10*<σ 3 > T [MeV] T [MeV] [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

18 Explicit Symmetry Breaking and Vacuum Alignment ρ 1 σ 2 0 +σ2 3 respects rotations in σ 0,σ 3 plane absence of determinant terms ξ: ( σ 0, σ 3 ) (c 0,c 3 ) M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

19 Explicit Symmetry Breaking and Vacuum Alignment ρ 1 σ0 2 +σ2 3 respects rotations in σ 0,σ 3 plane absence of determinant terms ξ: ( σ 0, σ 3 ) (c 0,c 3 ) physical: c 0 2 c 3 (or m d 2 m u ) but: σ 0 σ 3 (or dd +ūu dd ūu ) M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

20 Explicit Symmetry Breaking and Vacuum Alignment ρ 1 σ0 2 +σ2 3 respects rotations in σ 0,σ 3 plane absence of determinant terms ξ: ( σ 0, σ 3 ) (c 0,c 3 ) physical: c 0 2 c 3 (or m d 2 m u ) but: σ 0 σ 3 (or dd +ūu dd ūu ) suppression of σ 3 requires other operator than ρ 1 M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

21 Vanishing Initial Anomaly m π m η = 2a 01 0 at initial scale Λ solid: 10 c 3 = c 0 masses [MeV] π σ a 0 η,a ± condensates [MeV] <σ 0 > 10*<σ 3 > T [MeV] T [MeV] [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

22 Z 2 Symmetric Limit initial potential: U Λ = a Λ,10 (ρ 1 ρ 1,0 )+a Λ,01 (ξ ξ 0 )+ a Λ,20 2 (ρ 1 ρ 1,0 ) 2 c 0 σ 0 c 3 σ 3 ρ 1 σ 2 0 +σ2 3 and ξ2n (σ 2 0 σ2 3 )2n respect Z 2 : σ 0 σ 3 M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

23 Z 2 Symmetric Limit initial potential: U Λ = a Λ,10 (ρ 1 ρ 1,0 )+a Λ,01 (ξ ξ 0 )+ a Λ,20 2 (ρ 1 ρ 1,0 ) 2 c 0 σ 0 c 3 σ 3 ρ 1 σ 2 0 +σ2 3 and ξ2n (σ 2 0 σ2 3 )2n respect Z 2 : σ 0 σ 3 U Λ with a 01 0 at Λ and c 3 = c 0 respects Z 2 corresponds to one massless bare quark M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

24 Z 2 Symmetric Limit initial potential: U Λ = a Λ,10 (ρ 1 ρ 1,0 )+a Λ,01 (ξ ξ 0 )+ a Λ,20 2 (ρ 1 ρ 1,0 ) 2 c 0 σ 0 c 3 σ 3 ρ 1 σ 2 0 +σ2 3 and ξ2n (σ 2 0 σ2 3 )2n respect Z 2 : σ 0 σ 3 U Λ with a 01 0 at Λ and c 3 = c 0 respects Z 2 corresponds to one massless bare quark physical infrared observables require Z 2 breaking via σ 0 σ 3 M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

25 Spontaneous Breaking of Z 2 ultraviolet: det(k = Λ) = 0, m η (k = Λ) = m π(k = Λ) U k (ρ 1,ξ) = a ij i!j! (ρ1 ρ0,1)i (ξ ξ 0) j i,j condensates [MeV] k [MeV] <σ 0 > <σ 3 > couplings [(100 MeV) 2 ] k [MeV] a 10 a 01 solid: c 3 = c 0, dashed: c 3 = 0.1 c 0, dotted: c 3 = 0.02 c 0, [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

26 New Phase transition: Z 2 Universality Class ultraviolet: det(k = Λ) 0, m η (k = Λ) m π(k = Λ) m 2 π a 10 +a 01 m 2 η a10 a01 condensates [MeV] <σ 0 > <σ 3 > T [MeV] β = 0.37 (Z 2 universality class: 0.326) [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

27 Z 2 Transition: Couplings ultraviolet: det(k = Λ) 0, m η (k = Λ) m π(k = Λ) m 2 π a 10 +a 01 m 2 η a10 a01 couplings [(100MeV) 2 ] T [MeV] a 10 a 01 couplings T [MeV] a 20 a 11 a 02 [MM, Schaefer, Strodthoff, von Smekal, in preparation 2013] M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

28 Summary and Outlook temperature dependence: m η t Hooft term isospin breaking new transition: Z 2 universality class M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18

29 Summary and Outlook temperature dependence: m η t Hooft term isospin breaking new transition: Z 2 universality class influence of chiral invariant ρ 2 rebosonization, determinant from QCD M. Mitter (KFU Graz) U A (1) Anomaly at T 0 Heidelberg, January / 18