On Quarks in Maximally Abelian Gauge Richard Haider, Valentin Mader, - PowerPoint PPT Presentation

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook On Quarks in Maximally Abelian Gauge Richard Haider, Valentin Mader, Reinhard Alkofer Austria-Croatia-Hungary-Triangle workshop on Strong Interactions in Quantum Field Theory Szombathely, April 12, 2012

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Outline Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Dual Superconductor Picture of Confinement

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Type-I Superconductors: • expel weak magnetic fields • are penetrated by strong fields

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Type-I Superconductors: • expel weak magnetic fields • are penetrated by strong fields Type-II Superconductors: • additional intermediate phase • magnetic flux passes through cylindrical regions • so-called Abrikosov vortices 1 or flux tubes 1 Abrikosov, JETP 5:1174 (1957).

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Flux Tubes: • confined by super currents of condensed electric monopoles (Cooper-pairs) • field forced into tubes of discretized flux

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Flux Tubes: • confined by super currents of condensed electric monopoles (Cooper-pairs) • field forced into tubes of discretized flux • would also run between separated magnetic monopoles • static monopole potential would rise linearly with separation

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Dual Superconductor Picture: • covariant Maxwell equations ∂ µ F µν = j e ν , ∂ µ ∗ F µν = j m ν symmetric under exchange of fields and currents: F µν → ∗ F µν = 1 2 ǫ µναβ F αβ , ∗ F µν → − F µν j e µ ↔ j m j m µ ↔ − j e E → B , B → − E , µ , µ

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Dual Superconductor Picture: • covariant Maxwell equations ∂ µ F µν = j e ν , ∂ µ ∗ F µν = j m ν symmetric under exchange of fields and currents: 2 ǫ µναβ F αβ , F µν → ∗ F µν = 1 ∗ F µν → − F µν j e µ ↔ j m j m µ ↔ − j e E → B , B → − E , µ , µ • Idea: analogous mechanism responsible for quark confinement 2 • reversed roles of electric and magnetic fields ⇒ “dual” 2 ’t Hooft, talk given in Palermo, Jun 23-28, 1975 and Mandelstam, Phys. Rept. 23 (1976) 245-249.

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Dual Superconductor Picture: • color-electric charged particles confined by induced chromo-magnetic supercurrent • flux tubes ↔ hadrons • linear potential between quarks ⇒ confinement

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Dual Superconductor Picture: • color-electric charged particles confined by induced chromo-magnetic supercurrent • flux tubes ↔ hadrons • linear potential between quarks ⇒ confinement • investigations on the lattice support DSC picture 3 3 e. g. Bonati et al. Phys. Rev. D 85 , 065001 (2012)

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Maximally Abelian Gauge

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Magnetic Monopoles: • singularities in Abelian gauge field • but: QCD is non-Abelian

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Magnetic Monopoles: • singularities in Abelian gauge field • but: QCD is non-Abelian • monopoles are elements of the “Abelian” part of QCD

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Abelian dominance: • Dual Superconductor Picture implies confinement due to chromo-magnetic monopoles • IR dominated by monopoles • ⇒ Abelian part of the theory dominates IR 4 4 Z. F. Ezawa and A. Iwazaki, Phys. Rev. D25 (1982) 2681.

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Abelian Part ⇒ Cartan Subalgebra: • elements of an algebra [ T r , T s ] = i f rst T t , for which T i , T j � � = 0 • e. g. the N-1 diagonal elements of SU(N) → “diagonal part” • henceforth • i , j . . . for Abelian parts • a , b . . . for non-Abelian parts • r , s . . . for all together will be used

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Separating Abelian Parts: • split gauge field A µ in diagonal and off-diagonal components A µ = T r A r µ = T i A i µ + T a B a µ

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Separating Abelian Parts: • split gauge field A µ in diagonal and off-diagonal components A µ = T r A r µ = T i A i µ + T a B a µ • the field strength tensor becomes F r µν = F i µν + F a µν , with F i µν = ∂ µ A i ν − ∂ ν A i µ − g f iab B a µ B b ν F a µν = D ab µ B b ν − D ab ν B b µ − g f abc B b µ B c ν D ab µ = δ ab ∂ µ + g f abi A i µ

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Maximally Abelian Gauge: • fix gauge, such that off-diagonal part is minimized 1 � d x B a µ ( x ) B a µ ( x ) → min • 2 • corresponds to D ab µ B b µ = 0

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Maximally Abelian Gauge: • fix gauge, such that off-diagonal part is minimized 1 � d x B a µ ( x ) B a µ ( x ) → min • 2 • corresponds to D ab µ B b µ = 0 • theory has remnant symmetry • fixed to e. g. Landau gauge ∂ µ A i µ = 0

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Pros and Cons: • deeper insight in the different behavior of the separated gluons • investigate Abelian dominance • covariant ⇒ comparison to Landau gauge • discriminating diagonal and off-diagonal parts yields more terms • renormalizability demands four-ghost interaction

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Quark Propagator

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Dyson-Schwinger equation of the quark propagator in Landau gauge: d 4 k µν ( p − k )( g λ r � S − 1 ( p ) = S − 1 ( 2 π ) 4 D rs 2 γ µ ) S ( k ) g Γ s + ν ( k , p ) 0

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Rainbow Truncation: • approximate dressed gluon propagator and quark-gluon vertex by the bare structures with an effective coupling T µν ( p − k ) δ rs λ s ν ( k , p ) → G (( p − k ) 2 ) g 2 D rs µν ( p − k )Γ s 2 γ ν ( p − k ) 2 • where T µν ( q ) = δ µν − q µ q ν q 2 , the transversal projector

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook New in MAG: • treat diagonal and off-diagonal parts separately • instead of tr ( λ r λ r 2 ) = 4 2 3 • in MAG: tr ( λ i λ i 3 and tr ( λ a λ a 2 ) = 1 2 ) = 1 2 2 • each with its own dressing function

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Truncated Quark-DSE in MAG: d 4 k G diag (( p − k ) 2 ) + 1 G off (( p − k ) 2 ) � 1 � � S − 1 ( p ) = ip / + m + ( 2 π ) 4 3 ( p − k ) 2 ( p − k ) 2 × T µν ( p − k ) γ µ S ( k ) γ ν

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Parametrization of the Quark Propagator: • parametrized in form of free quark propagator: Z ( p 2 ) 1 S ( p ) = = iA ( p 2 ) p / + B ( p 2 ) ✶ D ip / + M ( p 2 ) • project DSE onto basis elements p / and ✶ D • ⇒ integral equations for A ( p 2 ) , B ( p 2 )

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Ansätze for the Effective Coupling: G diag (( p − k ) 2 ) 8 πσ = (( p − k ) 2 + µ 2 ) 2 ( p − k ) 2 = log ( e + ( p − k ) 2 ) − 1 . 5 G off (( p − k ) 2 ) λ 2 ( p − k ) 2 + λ 2 ( p − k ) 2

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Preliminary results for the quark propagator 5 with σ = 4 , µ = 0 . 35 5 compared to Alkofer, Watson, Weigel, Phys. Rev. D 65 094026

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Preliminary results for the quark propagator 5 with σ = 18 , µ = 0 . 9 5 compared to Alkofer, Watson, Weigel, Phys. Rev. D 65 094026

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Outlook

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook Self Consistent Solution: • dressed vertex needed ⇒