Chiral symmetry breaking in continuum QCD

Transcription

1 Chiral symmetry breaking in continuum QCD Mario Mitter Ruprecht-Karls-Universität Heidelberg Trieste, September 206 M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September 206 / 20

2 fqcd collaboration - QCD (phase diagram) with FRG: J. Braun, L. Corell, A. K. Cyrol, L. Fister, W. J. Fu, M. Leonhardt, MM, J. M. Pawlowski, M. Pospiech, F. Rennecke, N. Strodthoff, N. Wink,... Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> 0 crossover FAIR/NICA <ψψ> = / 0 vacuum hadronic fluid n B= 0 n B> 0 AGS SIS nuclear matter µ quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / 0 CFL neutron star cores M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

3 fqcd collaboration - QCD (phase diagram) with FRG: J. Braun, L. Corell, A. K. Cyrol, L. Fister, W. J. Fu, M. Leonhardt, MM, J. M. Pawlowski, M. Pospiech, F. Rennecke, N. Strodthoff, N. Wink,... Temperature early universe LHC quark gluon plasma RHIC SPS <ψψ> 0 crossover FAIR/NICA <ψψ> = / 0 vacuum hadronic fluid n B= 0 n B> 0 AGS SIS nuclear matter µ quark matter crossover superfluid/superconducting 2SC phases? <ψψ> = / 0 CFL neutron star cores large part of this effort: vacuum YM-theory and QCD M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

4 QCD with the FRG S[Φ] = Γ Λ [Φ]: use only perturbative QCD input α S (Λ = O(0) GeV) m q (Λ = O(0) GeV) M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

5 QCD with the FRG S[Φ] = Γ Λ [Φ]: use only perturbative QCD input α S (Λ = O(0) GeV) m q (Λ = O(0) GeV) gauge-fixed approach (Landau gauge): ghosts appear k Γ k [A, c, c, q, q] = 2 M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

6 QCD with the FRG S[Φ] = Γ Λ [Φ]: use only perturbative QCD input α S (Λ = O(0) GeV) m q (Λ = O(0) GeV) gauge-fixed approach (Landau gauge): ghosts appear k Γ k [A, c, c, q, q] = 2 vertex expansion: Γ[Φ] = n p,...,p n Γ (n) Φ Φ n (p,..., p n )Φ (p ) Φ n ( p p n ) M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

7 QCD with the FRG S[Φ] = Γ Λ [Φ]: use only perturbative QCD input α S (Λ = O(0) GeV) m q (Λ = O(0) GeV) gauge-fixed approach (Landau gauge): ghosts appear k Γ k [A, c, c, q, q] = 2 vertex expansion: Γ[Φ] = n p,...,p n Γ (n) Φ Φ n (p,..., p n )Φ (p ) Φ n ( p p n ) aim for apparent convergence of Γ[Φ] = lim k 0 Γ k [Φ] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

8 Landau gauge QCD two crucial phenomena: SχSB and confinement similar scales - hard to disentangle see e.g. [Williams, Fischer, Heupel, 205] quenched QCD: allows separate investigation: M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

9 Landau gauge QCD two crucial phenomena: SχSB and confinement similar scales - hard to disentangle see e.g. [Williams, Fischer, Heupel, 205] quenched QCD: allows separate investigation: quenched matter part [MM, Strodthoff, Pawlowski, 204] pure YM-theory [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] outlook: [Cyrol, MM, Strodthoff, Pawlowski, in preparation] unquenching YM-theory at finite temperature T > 0 M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

10 Landau gauge QCD two crucial phenomena: SχSB and confinement similar scales - hard to disentangle see e.g. [Williams, Fischer, Heupel, 205] quenched QCD: allows separate investigation: quenched matter part [MM, Strodthoff, Pawlowski, 204] pure YM-theory [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] outlook: [Cyrol, MM, Strodthoff, Pawlowski, in preparation] unquenching YM-theory at finite temperature T > 0 use results from lattice QCD to gauge truncation M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

11 Chiral symmetry breaking χsb resonance in 4-Fermi interaction λ (pion pole): M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

12 Chiral symmetry breaking χsb resonance in 4-Fermi interaction λ (pion pole): resonance singularity without momentum dependency t λ = a λ 2 + b λ α + c α 2, b > 0, a, c 0 t = t λ g = 0 g = 0, T > 0 g = g cr λ g > g cr [Braun, 20] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

13 (transverse) running couplings [Cyrol, MM, Pawlowski, Strodthoff, in preparation] running couplings α X qaq cac,prop cr p [GeV] agreement in perturbative regime required by gauge symmetry non-degenerate in nonperturbative regime: reflects gluon mass gap α qaq > α cr : necessary for chiral symmetry breaking area above α cr very sensitive to errors M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

14 4-Fermi vertex via dynamical hadronization [Gies, Wetterich, 2002] change of variables: particular 4-Fermi channels meson exchange efficient inclusion of momentum dependence no singularities calculation of model parameters from QCD k Γ k = (bosonized) 4-fermi-interaction h 2 π /(2 m2 π ) λ π RG-scale k [GeV] Yukawa interaction h π RG-scale k [GeV] [MM, Strodthoff, Pawlowski, 204] [Braun, Fister, Haas, Pawlowski, Rennecke, 204] [MM, Strodthoff, Pawlowski, 204] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

15 Vertex Expansion [MM, Strodthoff, Pawlowski, 204], [Cyrol, Fister, MM, Strodthoff, Pawlowski, 206] t = t = perm. t = 2 + perm. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

16 Vertex Expansion [MM, Strodthoff, Pawlowski, 204], [Cyrol, Fister, MM, Strodthoff, Pawlowski, 206] t = + t = t = t = + perm. t = t 2 = perm perm. t = perm. t = 2 + perm. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

17 Vertex Expansion [MM, Strodthoff, Pawlowski, 204], [Cyrol, Fister, MM, Strodthoff, Pawlowski, 206] t = + t = t = t = + perm. t = t 2 = perm perm. t = perm. t = 2 t = perm. t = perm. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

18 Quark propagator [MM, Pawlowski, Strodthoff, 204] Γ (2) qq (p) Z q(p) (/p + M(p)) quark propagator dressings Bowman et al., 05 /Z q M q p [GeV] FRG vs. lattice: bare mass, quenched, scale set via gluon propagator M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

19 Quark propagator [MM, Pawlowski, Strodthoff, 204] Γ (2) qq (p) Z q(p) (/p + M(p)) quark propagator dressings Bowman et al., 05 /Z q M q p [GeV] FRG vs. lattice: bare mass, quenched, scale set via gluon propagator agreement not sufficient: need apparent convergence at µ 0 lattice data: Bowman, Heller, Leinweber, Parappilly, Williams, Zhang, Phys. Rev. D7, (2005). M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

20 other 4-Fermi channels [MM, Pawlowski, Strodthoff, 204] (bosonized) 4-fermi-interactions RG-scale k [GeV] h 2 π /(2 m2 π ) λ η λ (S+P)- adj λ V-A λ V+A λ (V-A) adj λ (S-P)- λ(s-p)- adj λ (S+P)+ λ(s+p)+ adj bosonized only σ-π-channel momentum dependently sufficient other channels: quantitatively not important in loops M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

21 Quark-gluon interactions [MM, Pawlowski, Strodthoff, 204] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis, e.g. γ µ, i ( /p + /q ) γ µ [, 2 p, / /q ] γ µ M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September 206 / 20

22 Quark-gluon interactions [MM, Pawlowski, Strodthoff, 204] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis, e.g. γ µ, i ( /p + /q ) γ µ [, 2 p, / /q ] γ µ 2 () (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry 0. 0 p [GeV] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September 206 / 20

23 Quark-gluon interactions [MM, Pawlowski, Strodthoff, 204] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis, e.g. γ µ, i ( /p + /q ) γ µ [, 2 p, / /q ] γ µ 2 () (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry 0. 0 p [GeV] chirally symmetric tensors from operator q /D 3 q worsen result M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September 206 / 20

24 Quark-gluon interactions [MM, Pawlowski, Strodthoff, 204] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis, e.g. γ µ, i ( /p + /q ) γ µ [, 2 p, / /q ] γ µ 2 () (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry 0. 0 p [GeV] chirally symmetric tensors from operator q /D 3 q worsen result counteracted by tensor structures in Γ (4) AA qq and Γ(5) A 3 qq from q /D 3 q M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September 206 / 20

25 Quark-gluon interactions [MM, Pawlowski, Strodthoff, 204] quark-gluon interaction most crucial for chiral symmetry breaking full tensor basis, e.g. γ µ, i ( /p + /q ) γ µ [, 2 p, / /q ] γ µ 2 () (4) quark-gluon couplings (7) (5) (6) (2) (3) (8) vertex strength: reflects gluon gap 8 tensors (transversally projected): classical tensor chirally symmetric break chiral symmetry 0. 0 p [GeV] chirally symmetric tensors from operator q /D 3 q worsen result counteracted by tensor structures in Γ (4) AA qq and Γ(5) A 3 qq from q /D 3 q expansion in BRST-invariant operators improves convergence? M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September 206 / 20

26 YM theory: modified STIs (mstis) regulator breaks BRST-symmetry of gauge-fixed action modification to Slavnov-Taylor identities R k ( 0 for k 0) [Ellwanger, Hirsch, Weber, 996] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

27 YM theory: modified STIs (mstis) regulator breaks BRST-symmetry of gauge-fixed action modification to Slavnov-Taylor identities R k ( 0 for k 0) [Ellwanger, Hirsch, Weber, 996] most immediate consequence for gluon propagator at Λ: ( ) [Γ (2) AA ] µν(p) = Z Λ p 2 δ µν pµ p ν + m 2 p 2 Λ δ µν mλ 2 Λ2 M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

28 YM theory: modified STIs (mstis) regulator breaks BRST-symmetry of gauge-fixed action modification to Slavnov-Taylor identities R k ( 0 for k 0) [Ellwanger, Hirsch, Weber, 996] most immediate consequence for gluon propagator at Λ: ( ) [Γ (2) AA ] µν(p) = Z Λ p 2 δ µν pµ p ν + m 2 p 2 Λ δ µν mλ 2 Λ2 msti determines mλ 2, but hard numerical problem due to Λ2 M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

29 YM theory: modified STIs (mstis) regulator breaks BRST-symmetry of gauge-fixed action modification to Slavnov-Taylor identities R k ( 0 for k 0) [Ellwanger, Hirsch, Weber, 996] most immediate consequence for gluon propagator at Λ: ( ) [Γ (2) AA ] µν(p) = Z Λ p 2 δ µν pµ p ν + m 2 p 2 Λ δ µν mλ 2 Λ2 msti determines mλ 2, but hard numerical problem due to Λ2 BRST-symmetry at k = 0 requires propagators of scaling type: lim p 0 [Γ(2) AA ] µν(p) (p 2 ) 2κ lim [Γ (2) cc ] p 0 µν(p) (p 2 ) +κ κ (0.5, ) scaling solution (BRST-symmetry) fixes m 2 Λ uniquely [Lerche, von Smekal, 2002] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

30 YM theory: modified STIs (mstis) regulator breaks BRST-symmetry of gauge-fixed action modification to Slavnov-Taylor identities R k ( 0 for k 0) [Ellwanger, Hirsch, Weber, 996] most immediate consequence for gluon propagator at Λ: ( ) [Γ (2) AA ] µν(p) = Z Λ p 2 δ µν pµ p ν + m 2 p 2 Λ δ µν mλ 2 Λ2 msti determines mλ 2, but hard numerical problem due to Λ2 BRST-symmetry at k = 0 requires propagators of scaling type: lim p 0 [Γ(2) AA ] µν(p) (p 2 ) 2κ lim [Γ (2) cc ] p 0 µν(p) (p 2 ) +κ κ (0.5, ) scaling solution (BRST-symmetry) fixes m 2 Λ uniquely [Lerche, von Smekal, 2002] lattice simulations: no scaling (decoupling solutions) [Cucchieri, Mendes, 2008] non-perturbative gauge-fixing on the lattice? [Maas, 2009] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

31 YM theory: results [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] Γ (2) AA (p) Z A(p) p 2 ( δ µν p µ p ν /p 2) running couplings gluon propagator dressing 3 2 FRG, scaling FRG, decoupling Sternbeck et al. running couplings α Acc α A 3 α A p [GeV] p [GeV] decoupling solutions: vary m 2 Λ slightly away from scaling value dashed-dot line: largest variation of m 2 Λ that leads to back-bending lattice data: A. Sternbeck, E. M. Ilgenfritz, M. Muller-Preussker, A. Schiller, and I. L. Bogolubsky, PoS LAT2006, 076. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

32 YM-theory: Truncation dependence [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] gluon propagator dressing 3 RG scale dep p [GeV] lattice data: A. Sternbeck, E. M. Ilgenfritz, M. Muller-Preussker, A. Schiller, and I. L. Bogolubsky, PoS LAT2006, 076. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

33 YM-theory: Truncation dependence [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] gluon propagator dressing 3 2 RG scale dep. D mom. dep p [GeV] lattice data: A. Sternbeck, E. M. Ilgenfritz, M. Muller-Preussker, A. Schiller, and I. L. Bogolubsky, PoS LAT2006, 076. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

34 YM-theory: Truncation dependence [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] gluon propagator dressing 3 2 RG scale dep. D mom. dep. 3D mom. dep p [GeV] lattice data: A. Sternbeck, E. M. Ilgenfritz, M. Muller-Preussker, A. Schiller, and I. L. Bogolubsky, PoS LAT2006, 076. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

35 YM-theory: Truncation dependence [Cyrol, Fister, MM, Pawlowski, Strodthoff, 206] gluon propagator dressing 3 2 RG scale dep. D mom. dep. 3D mom. dep p [GeV] lattice data: A. Sternbeck, E. M. Ilgenfritz, M. Muller-Preussker, A. Schiller, and I. L. Bogolubsky, PoS LAT2006, 076. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

36 Outlook: unquenching extended truncation: [Cyrol, MM, Pawlowski, Strodthoff, in prep.] systematics of improving the truncation? M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

37 Outlook: unquenching extended truncation: [Cyrol, MM, Pawlowski, Strodthoff, in prep.] systematics of improving the truncation? BRST-invariant operators, e.g. ψ /D n ψ M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

38 Outlook: running couplings α cac = α AAA = α A 4 = ( ) Γ (3) 2 cac (p) 4π Z A (p)z c(p) 2 ( Γ (3) AAA (p) ) 2 4π Z A (p) 3 ( Γ (4) A 4 (p) ) 4π Z A (p) 2 running couplings α X α qaq = [Cyrol, MM, Pawlowski, Strodthoff, in preparation] ( Γ (3) qaq (p) ) 2 4π Z A (p)z q(p) 2 α cac,prop = 4π Z A (p)z c(p) 2 cac AAA A 4 qaq cac,prop p [GeV] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

39 Outlook: gluon propagator [Cyrol, MM, Pawlowski, Strodthoff, in preparation] gluon propagator dressing /Z A Sternbeck et al., 202 FRG p [GeV] lattice data: A. Sternbeck, K. Maltman, M. Müller-Preussker, L. von Smekal, PoS LATTICE202 (202) 243. M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

40 Outlook: quark propagator [Cyrol, MM, Pawlowski, Strodthoff, in preparation] quark propagator dressings Bowman et al., 05 M q, m π =40 M q, m π = p [GeV] comparison FRG with lattice: bare mass and scale setting lattice data: Bowman, Heller, Leinweber, Parappilly, Williams, Zhang, Phys. Rev. D7, (2005). M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

41 Outlook: (averaged) gluon propagator at T 0 [Cyrol, MM, Pawlowski, Strodthoff, in preparation] lattice data: magnetic component of gluon propagator from Silva et. al. Phys.Rev. D89 (204) M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

42 Summary and Outlook QCD with functional RG vertex expansion sole input α S (Λ = O(0) GeV) and m q (Λ = O(0) GeV) good agreement with lattice correlators M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

43 Summary and Outlook QCD with functional RG vertex expansion sole input α S (Λ = O(0) GeV) and m q (Λ = O(0) GeV) good agreement with lattice correlators Outlook QCD phase diagram: order parameters, equation of state and fluct. of cons. charges bound-state properties (form factors,pda... ) more checks on convergence of vertex expansion M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

44 Summary and Outlook QCD with functional RG vertex expansion sole input α S (Λ = O(0) GeV) and m q (Λ = O(0) GeV) good agreement with lattice correlators Outlook QCD phase diagram: order parameters, equation of state and fluct. of cons. charges. bound-state properties (form factors,pda... ) more checks on convergence of vertex expansion Poster by Anton K. Cyrol: fqcd: QCD with the Functional RG M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

45 Dynamical mass generation confined branch Higgs branch m 2 [GeV 2 ] m 2 min m 2 Λ - m2 Λ,scaling [GeV2 ] M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20

46 Gluon propagator maximum over UV mass parameter position of gluon prop. max. [GeV] confined Higgs fit points fit m 2 c m 2 Λ - m2 Λ,scaling [GeV2 ] (m 2 Λ -m2 c )/m2 c M. Mitter (U Heidelberg) χsb in continuum QCD Trieste, September / 20