G2 gauge theories. Axel Maas. 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany

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1 G2 gauge theories Axel Maas 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany

2 Overview Why G2?

3 Overview Why G2? G2 Yang-Mills theory Running coupling [Olejnik, Maas JHEP'08, Maas MPLA'05, JHEP'11] Phase diagram [Danzer, Gattringer, Maas JHEP'09] Topological properties [Ilgenfritz, Maas PRD'12]

4 Overview Why G2? G2 Yang-Mills theory Running coupling [Olejnik, Maas JHEP'08, Maas MPLA'05, JHEP'11] Phase diagram [Danzer, Gattringer, Maas JHEP'09] Topological properties [Ilgenfritz, Maas PRD'12] G2 QCD [Maas, von Smekal, Wellegehausen, Wipf PRD'12, unpublished Lattice'13] Vacuum physics Phase diagram

5 Overview Why G2? G2 Yang-Mills theory Running coupling [Olejnik, Maas JHEP'08, Maas MPLA'05, JHEP'11] Phase diagram [Danzer, Gattringer, Maas JHEP'09] Topological properties [Ilgenfritz, Maas PRD'12] G2 QCD [Maas, von Smekal, Wellegehausen, Wipf PRD'12, unpublished Lattice'13] Vacuum physics Phase diagram Breaking G2 Summary

6 Mini-review: Maas & Wellegehausen, Lattice'12 proceedings, Overview Why G2? G2 Yang-Mills theory Running coupling [Olejnik, Maas JHEP'08, Maas MPLA'05, JHEP'11] Phase diagram [Danzer, Gattringer, Maas JHEP'09] Topological properties [Ilgenfritz, Maas PRD'12] G2 QCD [Maas, von Smekal, Wellegehausen, Wipf PRD'12, unpublished Lattice'13] Vacuum physics Phase diagram Breaking G2 Summary

7 G2

8 QCD as a gauge theory QCD is a gauge theory

9 QCD as a gauge theory QCD is a gauge theory L= 1 4 F a μ ν F a μ ν g F a μ ν = μ A a a ν ν A μ Gluons A a

10 QCD as a gauge theory QCD is a gauge theory L= 1 4 F a μ ν F a μ ν g g g F a μ ν = μ A a ν ν A a μ +gf a bc A μ b A ν c Gluons A a A coupling g Numbers f abc determined by the gauge group

11 QCD as a gauge theory QCD is a gauge theory L= 1 4 F a μ ν F a μ ν +Ψ i (id μ ij γ μ m) Ψ j g g g Gluons Quarks F a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ A a i A coupling g and mass m A μ b A ν c Numbers f abc determined by the gauge group u

12 QCD as a gauge theory QCD is a gauge theory L= 1 4 F a F a i id ij m j g g g Gluons Quarks F a = A a A a gf a bc D ij = ij iga a ij t a a A i A coupling g and mass m A b A c Numbers f abc and t a ij determined by the gauge group u g u

13 QCD as a gauge theory QCD is a gauge theory L= 1 4 F a F a i id ij m j g g g Gluons Quarks F a = A a A a gf a bc D ij = ij iga a ij t a a A i A coupling g and mass m A b A c Numbers f abc and t a ij determined by the gauge group u g u QCD: SU(3)

14 QCD as a gauge theory QCD is a gauge theory L= 1 4 F a F a i id ij m j g g g Gluons Quarks F a = A a A a gf a bc D ij = ij iga a ij t a a A i A coupling g and mass m A b A c Numbers f abc and t a ij determined by the gauge group u g u QCD: SU(3) Here: G2

15 Why G2? Conceptual

16 Why G2? Conceptual Quenched QCD

17 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important

18 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center?

19 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03]

20 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD?

21 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes

22 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes Practical

23 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes Practical G2 QCD

24 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes Practical G2 QCD Fundamental representation real No sign problem

25 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes Practical G2 QCD Fundamental representation real No sign problem Full phase diagram accessible

26 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes Practical G2 QCD Fundamental representation real No sign problem Full phase diagram accessible Test of methods and models

27 Why G2? Conceptual Quenched QCD SU(3): Center is assumed to be important What happens with a trivial center? Simplest case: G2 instead of SU(3) [Holland et al. NPA 03] Resembles this (quenched) QCD? - Yes Practical G2 QCD Fundamental representation real No sign problem Full phase diagram accessible Test of methods and models Qualitative insights

28 G2 facts sheet G2 is an exceptional Lie group

29 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7)

30 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7) Can be written as factors SU(3) and the 6-sphere

31 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7) Can be written as factors SU(3) and the 6-sphere Macfarlane representation [Macfarlane IJMP 02] G2 element G=Z ( U U )

32 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7) Can be written as factors SU(3) and the 6-sphere Macfarlane representation [Macfarlane IJMP 02] G2 element S6 element ( U 0 0 ) G=Z U

33 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7) Can be written as factors SU(3) and the 6-sphere Macfarlane representation [Macfarlane IJMP 02] G2 element S6 element ( U 0 0 ) G=Z U SU(3) element Same, but conjugated SU(3) element

34 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7) Can be written as factors SU(3) and the 6-sphere Macfarlane representation [Macfarlane IJMP 02] G2 element S6 element ( U 0 0 ) G=Z U Fundamental representation 7 dimensional Adjoint representation 14 dimensional SU(3) element Same, but conjugated SU(3) element

35 G2 facts sheet G2 is an exceptional Lie group Rank 2 Subgroup of SO(7) Can be written as factors SU(3) and the 6-sphere Macfarlane representation [Macfarlane IJMP 02] G2 element S6 element ( U 0 0 ) G=Z U Fundamental representation 7 dimensional Adjoint representation 14 dimensional SU(3) element Same, but conjugated SU(3) element All representations are equivalent to real representations

36 G2 Yang-Mills Theory

37 G2 quenched QCD 14 gluons

38 G2 quenched QCD 14 gluons Asymptotically free, infrared strongly interacting

39 G2 quenched QCD 14 gluons Asymptotically free, infrared strongly interacting Asymptotic string tension zero

40 G2 quenched QCD 14 gluons Asymptotically free, infrared strongly interacting Asymptotic string tension zero Like full QCD

41 G2 quenched QCD 14 gluons Asymptotically free, infrared strongly interacting Asymptotic string tension zero Like full QCD Intermediate distance static quark-antiquark potential similar to SU(3) Yang-Mills theory [Wellegehausen et al. PRD 11, Liptak et al. PRD 08, Greensite et al. PRD 06] Used to set the scale

42 G2 quenched QCD 14 gluons Asymptotically free, infrared strongly interacting Asymptotic string tension zero Like full QCD Intermediate distance static quark-antiquark potential similar to SU(3) Yang-Mills theory [Wellegehausen et al. PRD 11, Liptak et al. PRD 08, Greensite et al. PRD 06] Used to set the scale Qualitatively similar glueball spectrum Similar gluon-gluon potential [Wellegehausen et al. PRD 11]

43 What about confinement? No Wilson confinement

44 What about confinement? No Wilson confinement Energy of a static, free quark finite Screened by three gluons [Holland et al. NPA 2003]

45 What about confinement? No Wilson confinement Energy of a static, free quark finite Screened by three gluons [Holland et al. NPA 2003] Heavy state, but string breaking can be observed [Wellegehausen et al. PRD 11]

46 What about confinement? No Wilson confinement Energy of a static, free quark finite Screened by three gluons [Holland et al. NPA 2003] Heavy state, but string breaking can be observed [Wellegehausen et al. PRD 11] Polyakov loop well-defined, but must be non-zero Measured to be very small [Pepe et al. NPA 2007] Non-trivial structure in the adjoint Polyakov loopfundamental Polyakov loop plane [Wellegehausen et al. PRD 09]

47 What about confinement? No Wilson confinement Energy of a static, free quark finite Screened by three gluons [Holland et al. NPA 2003] Heavy state, but string breaking can be observed [Wellegehausen et al. PRD 11] Polyakov loop well-defined, but must be non-zero Measured to be very small [Pepe et al. NPA 2007] Non-trivial structure in the adjoint Polyakov loopfundamental Polyakov loop plane [Wellegehausen et al. PRD 09] Asymptotic states still only glueballs

48 What about confinement? No Wilson confinement Energy of a static, free quark finite Screened by three gluons [Holland et al. NPA 2003] Heavy state, but string breaking can be observed [Wellegehausen et al. PRD 11] Polyakov loop well-defined, but must be non-zero Measured to be very small [Pepe et al. NPA 2007] Non-trivial structure in the adjoint Polyakov loopfundamental Polyakov loop plane [Wellegehausen et al. PRD 09] Asymptotic states still only glueballs Phase diagram?

49 Phase diagram of G2 quenched QCD

50 Phase diagram of G2 quenched QCD T Phase line Temperature only external control parameter T=0 MeV

51 Phase diagram of G2 quenched QCD T Phase line Temperature only external control parameter [Danzer, Gattringer, Maas, JHEP09] T=0 MeV

52 Phase diagram of G2 quenched QCD T T c ~255 MeV Phase line Temperature only external control parameter First order transition [Pepe et al. NPA 07, Greensite PRD 07, Cossu et al. JHEP 07] Observed in thermodynamic observables Free energy, heat capacity T=0 MeV

53 Phase diagram of G2 quenched QCD T T c ~255 MeV Phase line Temperature only external control parameter First order transition [Pepe et al. NPA 07, Greensite PRD 07, Cossu et al. JHEP 07] Observed in thermodynamic observables Free energy, heat capacity Complicated by a bulk transition T=0 MeV Requires fine lattice [Cossu et al. JHEP 07] Remains with quarks

54 Polyakov loop [Danzer, Gattringer, Maas, JHEP'09 Pepe et al. NPA'07] Almost zero at low temperatures

55 Polyakov loop [Danzer, Gattringer, Maas, JHEP'09 Pepe et al. NPA'07] Almost zero at low temperatures First-order signal at the phase transition

56 Polyakov loop [Danzer, Gattringer, Maas, JHEP'09 Pepe et al. NPA'07] Almost zero at low temperatures First-order signal at the phase transition Significantly non-zero in the high-temperature phase Similar to full QCD

57 Polyakov loop [Danzer, Gattringer, Maas, JHEP'09 Pepe et al. NPA'07] Almost zero at low temperatures First-order signal at the phase transition Significantly non-zero in the high-temperature phase Similar to full QCD

58 Polyakov loop [Danzer, Gattringer, Maas, JHEP'09 Pepe et al. NPA'07] Almost zero at low temperatures First-order signal at the phase transition Significantly non-zero in the high-temperature phase Similar to full QCD

59 Polyakov loop [Danzer, Gattringer, Maas, JHEP'09 Pepe et al. NPA'07] Almost zero at low temperatures First-order signal at the phase transition Significantly non-zero in the high-temperature phase Similar to full QCD what about chiral symmetry?

60 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD

61 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD Reason: Real representation Quarks and anti-quarks are related

62 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD Reason: Real representation Quarks and anti-quarks are related Enlarged SU(2N)xZ 2 for N flavors

63 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD Reason: Real representation Quarks and anti-quarks are related Enlarged SU(2N)xZ 2 for N flavors Pauli-Gürsey symmetry, like 2-color QCD

64 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD Reason: Real representation Quarks and anti-quarks are related Enlarged SU(2N)xZ 2 for N flavors Pauli-Gürsey symmetry, like 2-color QCD More appropriate language: 2N Weyl flavors

65 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD Reason: Real representation Quarks and anti-quarks are related Enlarged SU(2N)xZ 2 for N flavors Pauli-Gürsey symmetry, like 2-color QCD More appropriate language: 2N Weyl flavors Baryon number is still well-defined (Unbroken) U(1) subgroup of chiral symmetry

66 Chiral symmetry in G2 [Holland et al. NPA 03, Pepe et al. NPA 07] Chiral symmetry is enlarged compared to ordinary QCD Reason: Real representation Quarks and anti-quarks are related Enlarged SU(2N)xZ 2 for N flavors Pauli-Gürsey symmetry, like 2-color QCD More appropriate language: 2N Weyl flavors Baryon number is still well-defined (Unbroken) U(1) subgroup of chiral symmetry Non-anomalous chiral symmetry breaking for 1 flavor possible

67 Quenched G2 QCD [Danzer, Gattringer, Maas, JHEP09] Chiral symmetry 'broken' in quenched G2 QCD

68 Quenched G2 QCD [Danzer, Gattringer, Maas, JHEP09] Chiral symmetry 'broken' in quenched G2 QCD 'Restoration' at the phase transition

69 Quenched G2 QCD [Danzer, Gattringer, Maas, JHEP09] Chiral symmetry 'broken' in quenched G2 QCD 'Restoration' at the phase transition

70 Quenched G2 QCD [Danzer, Gattringer, Maas, JHEP09] Chiral symmetry 'broken' in quenched G2 QCD 'Restoration' at the phase transition Like in QCD Unlike adjoint QCD [Bilgici, Ilgenfritz, Gattringer, Maas JHEP 09]

71 Topological properties Coincidence in SU(N) gauge theory possibly connected to topological properties Center vortices, monopoles, calorons,...

72 Topological properties Coincidence in SU(N) gauge theory possibly connected to topological properties Center vortices, monopoles, calorons,... What is the reason in G2?

73 Topological properties Coincidence in SU(N) gauge theory possibly connected to topological properties Center vortices, monopoles, calorons,... What is the reason in G2? G2 has instanton-like solutions of the classical equation of motions [Ilgenfritz & Maas, '12]

74 Topological properties Coincidence in SU(N) gauge theory possibly connected to topological properties Center vortices, monopoles, calorons,... What is the reason in G2? G2 has instanton-like solutions of the classical equation of motions [Ilgenfritz & Maas, '12] Direct product of two SU(3) instantons S6 part only rotates them

75 Topological properties Coincidence in SU(N) gauge theory possibly connected to topological properties Center vortices, monopoles, calorons,... What is the reason in G2? G2 has instanton-like solutions of the classical equation of motions [Ilgenfritz & Maas, '12] Direct product of two SU(3) instantons S6 part only rotates them Also other topological objects [DiGiacomo et al'08, Diakonovet al.'11] Change of topological properties at the phase transition?

76 Topological susceptibility [Ilgenfritz & Maas '12] Fewer topological lumps the higher the temperature

77 Topological susceptibility [Ilgenfritz & Maas '12] Fewer topological lumps the higher the temperature

78 Topological susceptibility [Ilgenfritz & Maas '12] [Ilgenfritz & Maas, unpublished] Fewer topological lumps the higher the temperature Topology reflects phase transition

79 G2 QCD

80 The QCD phase diagram a sketch T Vacuum Inside a nucleus Chemical potential (Density)

81 The QCD phase diagram a sketch T High-temperature phase Normal phase Experimental lower phase boundary Vacuum Inside a nucleus Chemical potential (Density) Experiment gives lower bounds for rapid changes

82 The QCD phase diagram a sketch T High-temperature phase Normal phase Experimental lower phase boundary Vacuum Inside a nucleus Chemical potential (Density) Experiment gives lower bounds for rapid changes Interesting areas are at high temperature

83 The QCD phase diagram a sketch T Crossover High-temperature phase Normal phase Experimental lower phase boundary Vacuum Inside a nucleus Chemical potential (Density) Experiment gives lower bounds for rapid changes Interesting areas are at high temperature Cross-over: High-temperature and low-temperature phase qualitatively identical!

84 The QCD phase diagram a sketch T Crossover (More or less) Accessible Normal phase High-temperature phase Experimental lower phase boundary Vacuum Inside a nucleus Chemical potential (Density) Experiment gives lower bounds for rapid changes Interesting areas are at high temperature Cross-over: High-temperature and low-temperature phase qualitatively identical!

85 The QCD phase diagram a sketch T Crossover (More or less) Accessible Normal phase High-temperature phase Experimental lower phase boundary High density phase (Color-)Superconducting? Crystal? Vacuum Inside a nucleus Chemical potential (Density) Experiment gives lower bounds for rapid changes Interesting areas are at high temperature and density Cross-over: High-temperature and low-temperature phase qualitatively identical!

86 The QCD phase diagram a sketch T Crossover (More or less) Accessible Normal phase Experimental lower phase boundary High-temperature phase No satisfactory firstprinciple results yet High density phase (Color-)Superconducting? Crystal? Vacuum Inside a nucleus Chemical potential (Density) Experiment gives lower bounds for rapid changes Interesting areas are at high temperature and density Cross-over: High-temperature and low-temperature phase qualitatively identical!

87 Spectrum of G2 QCD [Holland et al. NPA 03, Pepe et al. NPA 07]

88 Spectrum of G2 QCD Mesons [Holland et al. NPA 03, Pepe et al. NPA 07]

89 Spectrum of G2 QCD Mesons (zero baryon number) [Holland et al. NPA 03, Pepe et al. NPA 07]

90 Spectrum of G2 QCD Mesons (zero baryon number) Baryons (non-zero baryon number) [Holland et al. NPA 03, Pepe et al. NPA 07]

91 Spectrum of G2 QCD [Holland et al. NPA 03, Pepe et al. NPA 07] Mesons (zero baryon number) Baryons (non-zero baryon number) Diquarks - Goldstones Bosonic quark-quark bound states Like SU(2) QCD Fermionic 1 quark-3 gluon states Like adjoint QCD Fermionic 3 quark states Like fundamental QCD Different from SU(2) QCD Fermionic 5 and 7 quark states Glueballs

92 Spectrum of G2 QCD [Maas, von Smekal, Wellegehausen, Wipf unpublished] Heavy proton No parity doublets

93 Spectrum of G2 QCD [Maas, von Smekal, Wellegehausen, Wipf unpublished] Heavy proton No parity doublets Goldstones become light for light quarks Chiral symmetry breaking manifest in the spectrum

94 G2 QCD Phase Diagram

95 Finite density [Maas, von Smekal, Wellegehausen, Wipf '12] Quenched G2 QCD has the same phase structure as quenched QCD

96 Finite density [Maas, von Smekal, Wellegehausen, Wipf '12] Quenched G2 QCD has the same phase structure as quenched QCD G2 has only real representations No sign problem: Positive fermion determinant No sign problem for odd number of flavors Full phase diagram accessible

97 Finite density [Maas, von Smekal, Wellegehausen, Wipf '12] Quenched G2 QCD has the same phase structure as quenched QCD G2 has only real representations No sign problem: Positive fermion determinant No sign problem for odd number of flavors Full phase diagram accessible G2 has fermionic baryons Fermi effects at finite density correctly included Important for compact stellar objects

98 Finite density [Maas, von Smekal, Wellegehausen, Wipf '12] Quenched G2 QCD has the same phase structure as quenched QCD G2 has only real representations No sign problem: Positive fermion determinant No sign problem for odd number of flavors Full phase diagram accessible G2 has fermionic baryons Fermi effects at finite density correctly included Important for compact stellar objects 'Simplest' QCD with all these properties

99 Finite density [Maas, von Smekal, Wellegehausen, Wipf '12] Quenched G2 QCD has the same phase structure as quenched QCD G2 has only real representations No sign problem: Positive fermion determinant No sign problem for odd number of flavors Full phase diagram accessible G2 has fermionic baryons Fermi effects at finite density correctly included Important for compact stellar objects 'Simplest' QCD with all these properties Unquenched 1 flavor calculation

100 Phase diagram [Maas, von Smekal, Wellegehausen, Wipf '12]

101 Phase diagram [Maas, von Smekal, Wellegehausen, Wipf '12]

102 Phase diagram [Maas, von Smekal, Wellegehausen, Wipf '12]

103 Phase diagram [Maas, von Smekal, Wellegehausen, Wipf '12] Start of lattice artifacts - Discretization

104 Phase diagram [Maas, von Smekal, Wellegehausen, Wipf '12] Start of lattice artifacts Aoki phase? Start of lattice artifacts - Discretization

105 Low-density regime Lattice artifacts still hard to control

106 Low-density regime [Maas, von Smekal, Wellegehausen, Wipf unpublished] Baryon density aμ Chemical potential/diquark mass Lattice spacing/ diquark mass Lattice artifacts still hard to control

107 Low-density regime [Maas, von Smekal, Wellegehausen, Wipf unpublished] Baryon density aμ Chemical potential/diquark mass Lattice spacing/ diquark mass Lattice artifacts still hard to control Silver-blaze feature exists

108 Low-density regime [Maas, von Smekal, Wellegehausen, Wipf unpublished] Baryon density aμ Chemical potential/diquark mass Lattice spacing/ diquark mass Lattice artifacts still hard to control Silver-blaze feature exists Plateau at small density: Intermediate phase?

109 Identification of phases Different regimes identifiable Order given by the hadronic scales Transition to a phase dominated by fermionic baryons? [Light ensemble, Maas, von Smekal, Wellegehausen, Wipf unpublished]

110 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Quick rise into saturation

111 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Meson scale Baryon scale Quick rise into saturation

112 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Meson scale Baryon scale Quick rise into saturation Cannot be described by (LO) chiral perturbation theory

113 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Quick rise into saturation

114 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Quick rise into saturation Continuum (massless) Stefan-Boltzmann not good Lattice artifact?

115 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Quick rise into saturation Lattice artifacts only at very large potentials Too strong suppression lighter particles?

116 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Quick rise into saturation Better description at smaller potentials Rise too slow exponential?

117 Equation of state [Heavy ensemble Maas, von Smekal, Wellegehausen, Wipf unpublished] Quick rise into saturation Fermi-Dirac distribution of baryons works well Large range of potential dominates by baryons?

118 Breaking G2

119 Yang-Mills-Higgs SU(3) is subgroup of G2

120 Yang-Mills-Higgs SU(3) is subgroup of G2 Breaking by a Higgs effect possible [Holland et al. NPA 03] 6 massive gluons, 1 Higgs Can be made very massive

121 Yang-Mills-Higgs [Wellegehausen et al.'11]

122 Yang-Mills-Higgs [Wellegehausen et al.'11]

123 Yang-Mills-Higgs SU(3) is subgroup of G2 Breaking by a Higgs effect possible [Holland et al.'03] 6 massive gluons, 1 Higgs Can be made very massive Works well

124 Yang-Mills-Higgs SU(3) is subgroup of G2 Breaking by a Higgs effect possible [Holland et al.'03] 6 massive gluons, 1 Higgs Can be made very massive Works well G2 QCD more complicated

125 Yang-Mills-Higgs SU(3) is subgroup of G2 Breaking by a Higgs effect possible [Holland et al.'03] 6 massive gluons, 1 Higgs Can be made very massive Works well G2 QCD more complicated Requires complexification of fermions Locking of color and (Weyl)flavor required? Involves parity non-trivial Scalar vs. pseudoscalar Goldstones

126 Yang-Mills-Higgs SU(3) is subgroup of G2 Breaking by a Higgs effect possible [Holland et al.'03] 6 massive gluons, 1 Higgs Can be made very massive Works well G2 QCD more complicated Requires complexification of fermions Locking of color and (Weyl)flavor required? Involves parity non-trivial Scalar vs. pseudoscalar Goldstones Not yet solved

127 Strategy How to tackle the sign problem in QCD?

128 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD

129 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD possible?

130 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD, if possible, reintroduces sign problem

131 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD, if possible, reintroduces sign problem Test methods explicitly Analytic continuation, imaginary chemical potential, Taylor expansion, Lee-Yang zeros,...

132 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD, if possible, reintroduces sign problem Test methods explicitly Analytic continuation, imaginary chemical potential, Taylor expansion, Lee-Yang zeros,... Go off the lattice and combine

133 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD, if possible, reintroduces sign problem Test methods explicitly Analytic continuation, imaginary chemical potential, Taylor expansion, Lee-Yang zeros,... Go off the lattice and combine Derive effective theories

134 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD, if possible, reintroduces sign problem Test methods explicitly Analytic continuation, imaginary chemical potential, Taylor expansion, Lee-Yang zeros,... Go off the lattice and combine Derive effective theories Tests for truncation/approximations in continuum methods, e.g. functional methods

135 Strategy How to tackle the sign problem in QCD? G2 QCD is not QCD Breaking G2 QCD to QCD, if possible, reintroduces sign problem Test methods explicitly Analytic continuation, imaginary chemical potential, Taylor expansion, Lee-Yang zeros,... Go off the lattice and combine Derive effective theories Tests for truncation/approximations in continuum methods, e.g. functional methods Qualitative insights

136 Summary Conceptual insights Center is not relevant for: Phase transition Coincidence of chiral and 'deconfinement' phase transition Topological properties

137 Summary Conceptual insights Center is not relevant for: Phase transition Coincidence of chiral and 'deconfinement' phase transition Topological properties Quenched G2 QCD is almost the same as quenched QCD, up to the static potential

138 Summary Conceptual insights Center is not relevant for: Phase transition Coincidence of chiral and 'deconfinement' phase transition Topological properties Quenched G2 QCD is almost the same as quenched QCD, up to the static potential Practical insights: Phase diagram

139 Summary Conceptual insights Center is not relevant for: Phase transition Coincidence of chiral and 'deconfinement' phase transition Topological properties Quenched G2 QCD is almost the same as quenched QCD, up to the static potential Practical insights: Phase diagram Rough shape of the phase diagram of a gauge theory is similar to the expected one

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