Scalar QCD. Axel Maas with Tajdar Mufti. 5 th of September 2013 QCD TNT III Trento Italy

Transcription

1 Scalar QCD Axel Maas with Tajdar Mufti 5 th of September 2013 QCD TNT III Trento Italy

2 Scalar QCD Bound States, Elementary Particles & Interaction Vertices Axel Maas with Tajdar Mufti 5 th of September 2013 QCD TNT III Trento Italy

3 Why Scalar QCD? Only confinement no chiral symmetry breaking Confinement independent of Lorentz structure

4 Why Scalar QCD? Only confinement no chiral symmetry breaking Confinement independent of Lorentz structure Simple(r) tensor structures

5 Why Scalar QCD? Only confinement no chiral symmetry breaking Confinement independent of Lorentz structure Simple(r) tensor structures Rich bound state spectrum

6 Why Scalar QCD? Only confinement no chiral symmetry breaking Confinement independent of Lorentz structure Simple(r) tensor structures Rich bound state spectrum Cheap lattice simulations Test case for functional equations

7 Why Scalar QCD? Only confinement no chiral symmetry breaking Confinement independent of Lorentz structure Simple(r) tensor structures Rich bound state spectrum Cheap lattice simulations Test case for functional equations Limited by (possible) triviality But triviality cutoff can be high enough

8 Scalar QCD Gauge theory

9 Scalar QCD Gauge theory L= 1 4 A a μ ν A a μ ν A a μ ν = μ A a a ν ν A μ WA Gluons A μ a

10 Scalar QCD Gauge theory L= 1 4 A a μ ν A a μ ν A a μ ν = μ A a ν ν A a μ +gf a bc A μ b A ν c WA WA A Gluons A μ a abc Coupling g and some numbers f

11 Scalar QCD Gauge theory L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k A a μ ν = μ A a ν ν A a μ +gf a bc A μ b A ν c WA WA A Gluons A μ a D μ ij =δ ij μ h Scalar quarks h i abc Coupling g and some numbers f

12 Scalar QCD Gauge theory L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a a A μ Gluons A μ b A ν c h WA A WA h WA Scalar quarks h i Coupling g and some numbers f abc and t a ij Gauge group SU(2)

13 Scalar QCD Gauge theory L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a a A μ Gluons A μ b A ν c h WA A WA h WA Scalar quarks h i Coupling g and some numbers f abc and t a ij Gauge group SU(2) No 'baryon number'

14 Scalar QCD Gauge theory L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k +λ(h a h a + v 2 ) 2 Gluons A μ a Scalar quarks A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a h i A μ b A ν c Couplings g, v, λ and some numbers f abc and t a ij h WA h h A WA h WA Gauge group SU(2) No 'baryon number'

15 Scalar QCD Gauge theory L= 1 4 A a μ ν Gluons A μ a Scalar quarks A a μ ν +( D μ ij h j ) + D ik μ h k +λ(h a h a + v 2 ) 2 A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a h i A μ b A ν c Couplings g, v=0, λ=0 and some numbers f abc and t a ij h WA h h A WA h WA Scalar self-interaction set to zero Gauge group SU(2) No 'baryon number'

16 Symmetries L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k +λ(h a h a + v 2 ) 2 A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a A μ b A ν c

17 Symmetries L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k +λ(h a h a + v 2 ) 2 A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a A μ b A ν c Local SU(2) gauge symmetry Invariant under arbitrary gauge transformations ϕ a (x) A a μ A a μ +(δ a b μ g f a bc A c μ )ϕ b h i h i +g t ij a ϕ a h j

18 Symmetries L= 1 4 A a μ ν A a μ ν +( D μ ij h j ) + D ik μ h k +λ(h a h a + v 2 ) 2 A a μ ν = μ A a ν ν A a μ +gf a bc D ij μ =δ ij μ iga a ij μ t a A μ b A ν c Local SU(2) gauge symmetry Invariant under arbitrary gauge transformations ϕ a (x) A a μ A a μ +(δ a b μ g f a bc A c μ )ϕ b h i h i +g t ij a ϕ a h j Global SU(2) quark flavor symmetry Acts as right-transformation on the quark field only A μ a A μ a h i h i +a ij h j +b ij h j

19 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] f(classical Higgs mass) g(classical gauge coupling)

20 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] f(classical Higgs mass) Confinement phase g(classical gauge coupling)

21 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] f(classical Higgs mass) Higgs phase Confinement phase g(classical gauge coupling)

22 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] (Lattice-regularized) phase diagram continuous f(classical Higgs mass) Crossover Higgs phase 1 st order Confinement phase g(classical gauge coupling)

23 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] (Lattice-regularized) phase diagram continuous Separation only in fixed gauges f(classical Higgs mass) Crossover Landau gauge Higgs phase 1 st order Confinement phase g(classical gauge coupling)

24 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] (Lattice-regularized) phase diagram continuous Separation only in fixed gauges f(classical Higgs mass) Crossover Coulomb gauge Landau gauge Higgs phase 1 st order Confinement phase g(classical gauge coupling)

25 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] (Lattice-regularized) phase diagram continuous Separation only in fixed gauges f(classical Higgs mass) Crossover Higgs phase 1 st order Confinement phase Same physical state space in confinement g(classical gauge coupling) and Higgs pseudo-phases, irrespective of couplings

26 QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07] (Lattice-regularized) phase diagram continuous Separation only in fixed gauges f(classical Higgs mass) Crossover Higgs phase Confinement phase Same physical state space in confinement g(classical gauge coupling) and Higgs pseudo-phases, irrespective of couplings Asymptotic states depend on whether ground states for given J PC are stable F 1 st order

27 Non-aligned gauges [Maas, MPLA'12] Explicit charge direction inconvenient beyond perturbation theory

28 Non-aligned gauges [Maas, MPLA'12] Explicit charge direction inconvenient beyond perturbation theory Define a gauge without preferred direction

29 Non-aligned gauges [Maas, MPLA'12] Explicit charge direction inconvenient beyond perturbation theory Define a gauge without preferred direction Local part fixed to Landau gauge by Gribov-Singer ambiguity fixed by minimal prescription Introduces usual Faddeev-Popov ghosts μ A μ a =0

30 Non-aligned gauges [Maas, MPLA'12] Explicit charge direction inconvenient beyond perturbation theory Define a gauge without preferred direction Local part fixed to Landau gauge by Gribov-Singer ambiguity fixed by minimal prescription Introduces usual Faddeev-Popov ghosts Global part fixed by h =0 Aligned Landau gauges also possible μ A μ a =0

31 Differentiating phases [Maas, MPLA'12, Caudy & Greensite'07]

32 Differentiating phases [Maas, MPLA'12, Caudy & Greensite'07] How to distinguish phases?

33 Differentiating phases [Maas, MPLA'12, Caudy & Greensite'07] How to distinguish phases? Relative orientation hdx hdx hdy is the magnetization

34 Differentiating phases [Maas, MPLA'12, Caudy & Greensite'07] How to distinguish phases? Relative orientation hdx hdy hdx is the magnetization But not so important anyway...

35 Typical spectra [Maas, Mufti PoS'12, unpublished] Higgs

36 Typical spectra [Maas, Mufti PoS'12, unpublished, Maas MPLA'13] Higgs Higgs W

37 Typical spectra [Maas, Mufti PoS'12, unpublished] Higgs QCD

38 Typical spectra [Maas, Mufti PoS'12, unpublished] Higgs QCD Rather different low-lying spectra 0++ lighter in (Landau gauge) QCD-like region 1-- lighter in (Landau gauge) Higgs-like region

39 Typical spectra [Maas, Mufti PoS'12, unpublished] Higgs QCD Rather different low-lying spectra 0++ lighter in (Landau gauge) QCD-like region 1-- lighter in (Landau gauge) Higgs-like region Use as operational definition of phase

40 Phase diagram [Maas, Mufti, unpublished]

41 Phase diagram [Maas, Mufti, unpublished] Higgs QCD Complicated real phase diagram

42 Phase diagram [Maas, Mufti, unpublished] Higgs QCD Complicated real phase diagram QCD-like behavior even for negative bare mass

43 Phase diagram [Maas, Mufti, unpublished] Higgs QCD Complicated real phase diagram QCD-like behavior even for negative bare mass Similar bare couplings for both physic types

44 Phase diagram [Maas, Mufti, unpublished] Higgs QCD Complicated real phase diagram QCD-like behavior even for negative bare mass Similar bare couplings for both physic types

45 Propagators 3 propagators

46 Propagators 3 propagators Gluon D ab x y = A a x A b y

47 Propagators 3 propagators Gluon D ab x y = A a x A b y 1 scalar dressing function D μ ν ( p)=(δ μ ν p μ p ν p 2 )D( p)

48 Propagators 3 propagators Gluon D ab x y = A a x A b y 1 scalar dressing function Ghost D G ab x y = c a x c b y D μ ν ( p)=(δ μ ν p μ p ν p 2 )D( p)

49 Propagators 3 propagators Gluon D ab x y = A a x A b y 1 scalar dressing function Ghost D G ab x y = c a x c b y D μ ν ( p)=(δ μ ν p μ p ν p 2 )D( p) Negative semi-definite D G ( p)

50 Propagators 3 propagators Gluon D ab x y = A a x A b y 1 scalar dressing function Ghost D G ab x y = c a x c b y Negative semi-definite Both renormalize multiplicatively D μ ν ( p)=(δ μ ν p μ p ν p 2 )D( p) D G ( p)

51 Propagators 3 propagators Gluon D ab x y = A a x A b y 1 scalar dressing function Ghost Negative semi-definite Both renormalize multiplicatively Scalar D G ab x y = c a x c b y D H ij (x y)= <h i (x)h j+ D μ ν ( p)=(δ μ ν p μ p ν p 2 )D( p) D G ( p) ( y)>

52 Propagators 3 propagators Gluon D ab x y = A a x A b y 1 scalar dressing function Ghost D G ab x y = c a x c b y D μ ν ( p)=(δ μ ν p μ p ν p 2 )D( p) Negative semi-definite D G ( p) Both renormalize multiplicatively Scalar D H ij (x y)= <h i (x)h j+ ( y)> Requires more complicated renormalization D H (μ)=d H tl (μ) D H (μ)'=d H tl (μ)' D H tl ( p)=1/( p 2 +m r 2 ) μ=m r

53 Gluon propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished]

54 Gluon propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Significantly volume-dependent Decoupling-type

55 Gluon propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Significantly volume-dependent Decoupling-type Positivity violating

56 Gluon propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Significantly volume-dependent Decoupling-type Positivity violating Little impact when changing scalar sector

57 Ghost propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished]

58 Ghost propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Infrared enhanced But likely not divergent

59 Ghost propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Infrared enhanced But likely not divergent Derive a running coupling from p 6 D G 2 D

60 Ghost propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Infrared enhanced But likely not divergent Derive a running coupling from p 6 D G 2 D

61 Ghost propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] Infrared enhanced But likely not divergent Derive a running coupling from p 6 D G 2 D Not strongest at lowest bound state masses

62 Scalar quark propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] m r =1 GeV

63 Scalar quark propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] m r =1 GeV Requires mass renormalization Tree-level mass zero: Mass generation

64 Scalar quark propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] m r =1 GeV Requires mass renormalization Tree-level mass zero: Mass generation No sign (yet) of positivity violation

65 Scalar quark propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished] m r =0.25 GeV Requires mass renormalization Tree-level mass zero: Mass generation No sign (yet) of positivity violation

66 Vertices

67 Vertices Three 3-point vertices 4-point vertices too expensive

68 Vertices [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex < A a μc b c c > =D ad μ ν D be G D cf d e f G Γ ν

69 Vertices [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex < A a μc b c c > =D ad μ ν D be G D cf d e f G Γ ν G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl )

70 Vertices [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ

71 Vertices [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ Scalar-gluon vertex < A a μ h i h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A hh + > /(Γ tl D D H D H Γ tl )

72 Vertices [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) t F makes vertex flavor-conserving Flavor-violating vertex vanishes Flavor conserved

73 Vertices Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex Two momentum configurations < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished]

74 Vertices Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex Two momentum configurations < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished]

75 Vertices Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex Two momentum configurations < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) A p A p [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished]

76 Vertices Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex Two momentum configurations < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) A A p=0 p [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished]

77 Vertices Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex Two momentum configurations < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) A p=0 A p [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] q k with q=-k

78 Vertices Three 3-point vertices 4-point vertices too expensive Ghost-gluon vertex G A c c =Γ tl < A c c> /(Γ tl D D G D G Γ tl ) 3-gluon vertex G A3 =Γ tl < AAA> /(Γ tl DDD Γ tl ) Scalar-gluon vertex < A a μ c b c c > =D ad μ ν D be G D cf d e f G Γ ν < A a μ A b ν A c ρ > = D ad μα D be cf d e f νβ D ρ γ Γ αβ γ [Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished] < A a μ h i t F h j + > =D ad μ ν D ik H D jm dkm G Γ ν G A hh + =Γ tl < A ht F h + > /(Γ tl D D H D H Γ tl ) Two momentum configurations A A p=0 p p 2 =q 2 =k 2 q k with q=-k q k

79 Ghost-gluon vertex [Maas, Mufti PoS'12, unpublished] Only small deviations from tree-level Like in Yang-Mills theory Strongest effect at bound state mass scale

80 3-gluon vertex [Maas, Mufti PoS'12, unpublished] Infrared suppressed Sets in at bound state mass scale Absence of (supposed) sign change of Yang-Mills theory at small momenta?

81 Scalar-gluon vertex [Maas, Mufti PoS'12, unpublished] Essentially tree-level No indications for infrared effects (yet?)

82 Scalar-gluon vertex [Maas, Mufti PoS'12, unpublished] Essentially tree-level No indications for infrared effects (yet?) Different than a (pseudo-)confining 1-gluon exchange As in the quenched case [Maas, PoS'11, unpublished]

83 Scalar-gluon vertex [Maas, Mufti PoS'12, unpublished] Essentially tree-level No indications for infrared effects (yet?) Different than a (pseudo-)confining 1-gluon exchange As in the quenched case [Maas, PoS'11, unpublished]

84 Summary Scalar QCD a role model for QCD Scalar theory simpler...

85 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated

86 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated Test case for functional methods

87 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated Test case for functional methods Propagators in QCD-like region similar to Yang-Mills theory

88 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated Test case for functional methods Propagators in QCD-like region similar to Yang-Mills theory Positivity violating gluon Scalar quark not obviously positivity violating

89 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated Test case for functional methods Propagators in QCD-like region similar to Yang-Mills theory Positivity violating gluon Scalar quark not obviously positivity violating Vertices Yang-Mills-like

90 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated Test case for functional methods Propagators in QCD-like region similar to Yang-Mills theory Positivity violating gluon Scalar quark not obviously positivity violating Vertices Yang-Mills-like No infrared effects in the scalar-gluon vertex

91 Summary Scalar QCD a role model for QCD Scalar theory simpler......but distinction to Higgs-like physics complicated Test case for functional methods Propagators in QCD-like region similar to Yang-Mills theory Positivity violating gluon Scalar quark not obviously positivity violating Vertices Yang-Mills-like No infrared effects in the scalar-gluon vertex So far...no obvious confinement