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 vortices1 or flux tubes

1Abrikosov, 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 ∂µFµν = je

ν , ∂µ∗Fµν = jm ν

symmetric under exchange of fields and currents: Fµν → ∗Fµν = 1

2ǫµναβFαβ, ∗Fµν → −Fµν

E → B, B → −E, je

µ ↔ jm µ ,

jm

µ ↔ −je µ

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Dual Superconductor Picture:

• covariant Maxwell equations ∂µFµν = je

ν , ∂µ∗Fµν = jm ν

symmetric under exchange of fields and currents: Fµν → ∗Fµν = 1

2ǫµναβFαβ, ∗Fµν → −Fµν

E → B, B → −E, je

µ ↔ jm µ ,

jm

µ ↔ −je µ

• Idea: analogous mechanism responsible for quark confinement2
• 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 picture3
• 3e. 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 IR4
• 4Z. 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 rstT 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 Aµ in diagonal and off-diagonal components

Aµ = T rAr

µ = T iAi µ + T aBa µ

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Separating Abelian Parts:

• split gauge field Aµ in diagonal and off-diagonal components

Aµ = T rAr

µ = T iAi µ + T aBa µ

• the field strength tensor becomes F r

µν = F i µν + F a µν, with

F i

µν = ∂µAi ν − ∂νAi µ − g f iabBa µBb ν

F a

µν = Dab µ Bb ν − Dab ν Bb µ − g f abcBb µBc ν

Dab

µ = δab∂µ + g f abiAi 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

2

• dx Ba

µ(x)Ba µ(x) → min

• corresponds to Dab

µ Bb 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

2

• dx Ba

µ(x)Ba µ(x) → min

• corresponds to Dab

µ Bb µ = 0

• theory has remnant symmetry
• fixed to e. g. Landau gauge ∂µAi

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: S−1(p) = S−1 +

• d4k

(2π)4 Drs

µν(p − k)(g λr

2 γµ)S(k)gΓs

ν(k, p)

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 g2Drs

µν(p − k)Γs ν(k, p) → G((p − k)2)

(p − k)2 Tµν(p − k)δrs λs 2 γν

• where Tµν(q) = δµν − qµqν

q2 , 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(λr

2 λr 2 ) = 4 3

• in MAG: tr(λi

2 λi 2 ) = 1 3 and tr(λa 2 λa 2 ) = 1

• each with its own dressing function

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Truncated Quark-DSE in MAG: S−1(p) =ip / + m +

• d4k

(2π)4 1 3 Gdiag((p − k)2) (p − k)2 + 1Goff ((p − k)2) (p − k)2

• × Tµν(p − k)γµS(k)γν

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Parametrization of the Quark Propagator:

• parametrized in form of free quark propagator:

S(p) = 1 iA(p2)p / + B(p2)✶D = Z(p2) ip / + M(p2)

• project DSE onto basis elements p

/ and ✶D

• ⇒ integral equations for A(p2), B(p2)

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Ansätze for the Effective Coupling: Gdiag((p − k)2) (p − k)2 = 8πσ ((p − k)2 + µ2)2 Goff ((p − k)2) (p − k)2 = log(e + (p−k)2

λ2

)−1.5 (p − k)2 + λ2

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Preliminary results for the quark propagator5 with σ = 4, µ = 0.35

5compared 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 propagator5 with σ = 18, µ = 0.9

5compared 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 ⇒

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Quenched Diagramm: (ts)B′B(tr)BA(ts)AA′ = − 1 2Nc (tr)BA rs SU(2) SU(3) SU(4) ab aj −1

4ta

−1

6ta

−1

8ta

ib −1

2ti

−1

2ti

−1

2ti

ij

1 4ti 1 3ti 3 8ti

Sum −1

4(ti + ta)

−1

6(ti + ta)

−1

8(ti + ta)

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Unquenched Diagramm: i f rst(tt)BC(ts)CA = 1 2f rstf usttu = Nc 2 tr rst SU(2) SU(3) SU(4) abc

1 2ta

1ta abi

1 2ta 1 2ta 1 2ta

aic

1 2ta 1 2ta 1 2ta

ibc 1ti

3 2ti

2ti Sum 1(ti + ta)

3 2(ti + ta)

2(ti + ta)

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Further Plans:

• solve quark propagator and quark-gluon vertex DSEs self

consistently

• using Ansätze for the gluon propagator (cf. corresponding

lattice results6) and the 3-gluon vertex

6Bornyakov et al. Phys. Lett. B 559

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

Summary:

• introduced Dual Superconductor mechanism as illustrative

picture for confinement

• presented Maximally Abelian Gauge as a tool to investigate

QCD and Abelian dominance

• studies concerning quarks in MAG are currently ongoing

Dual Superconductor Picture of Confinement Maximally Abelian Gauge Quark Propagator Outlook

The End.