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The influence of centre vortices on the quarkpropagator in lattice QCD
Daniel TrewarthaDerek Leinweber and Waseem Kamleh
CSSM, School of Chemistry and PhysicsUniversity of Adelaide
Monday, 7th April 2014
(University of Adelaide) Mon. 7/4/2014 1 / 14
Centre Vortices
Topological defect associated with centre group Z3 of SU(3)
Appear as surfaces of centre flux, loops U(C) linked to the surfaceacquire a centre flux
U(C)→ zU(c) (1)
with z a centre element
(University of Adelaide) Mon. 7/4/2014 2 / 14
Centre Vortices
Topological defect associated with centre group Z3 of SU(3)
Appear as surfaces of centre flux, loops U(C) linked to the surfaceacquire a centre flux
U(C)→ zU(c) (1)
with z a centre element
(University of Adelaide) Mon. 7/4/2014 2 / 14
Centre Vortices on the lattice
Transform to Maximal Centre Gauge, where links are broughtclose to centre elements
Zµ = exp[2πi
3mµ(x)
]I, mµ(x) ∈ −1, 0, 1 (2)
Require gauge transformation Ω(x) maximising overlap betweengauge links and centre elements∑
x,µ
Re Tr[UΩµ (x)Z†µ(x)
]→ Max (3)
(University of Adelaide) Mon. 7/4/2014 3 / 14
Centre Vortices on the lattice
Transform to Maximal Centre Gauge, where links are broughtclose to centre elements
Zµ = exp[2πi
3mµ(x)
]I, mµ(x) ∈ −1, 0, 1 (2)
Require gauge transformation Ω(x) maximising overlap betweengauge links and centre elements∑
x,µ
Re Tr[UΩµ (x)Z†µ(x)
]→ Max (3)
(University of Adelaide) Mon. 7/4/2014 3 / 14
Centre Vortices on the Lattice
Implemented through ’mesonic’ centre gauge fixing condition
Rmes =∑x,µ
|Tr Uµ(x)Ω|2 → Max (4)
Then we project onto Z3
13
TrUΩµ (x) = rµ(x) exp(iφµ(x)) (5)
So choose mµ(x) ∈ −1, 0, 1 closest to 3φµ(x)2π
Then a plaquette is pierced by a centre vortex if
Pµν = ZµZν(x + µ)Z†µ(x + ν)Z†ν(x) (6)= zI, z 6= 1 (7)
(8)
(University of Adelaide) Mon. 7/4/2014 4 / 14
Centre Vortices on the Lattice
Implemented through ’mesonic’ centre gauge fixing condition
Rmes =∑x,µ
|Tr Uµ(x)Ω|2 → Max (4)
Then we project onto Z3
13
TrUΩµ (x) = rµ(x) exp(iφµ(x)) (5)
So choose mµ(x) ∈ −1, 0, 1 closest to 3φµ(x)2π
Then a plaquette is pierced by a centre vortex if
Pµν = ZµZν(x + µ)Z†µ(x + ν)Z†ν(x) (6)= zI, z 6= 1 (7)
(8)
(University of Adelaide) Mon. 7/4/2014 4 / 14
Centre Vortices on the Lattice
Implemented through ’mesonic’ centre gauge fixing condition
Rmes =∑x,µ
|Tr Uµ(x)Ω|2 → Max (4)
Then we project onto Z3
13
TrUΩµ (x) = rµ(x) exp(iφµ(x)) (5)
So choose mµ(x) ∈ −1, 0, 1 closest to 3φµ(x)2π
Then a plaquette is pierced by a centre vortex if
Pµν = ZµZν(x + µ)Z†µ(x + ν)Z†ν(x) (6)= zI, z 6= 1 (7)
(8)
(University of Adelaide) Mon. 7/4/2014 4 / 14
(University of Adelaide) Mon. 7/4/2014 5 / 14
Centre Vortices on the Lattice
3 sets of configurations:Untouched configurations Uµ(x)Vortex only configurations Zµ(x) = exp
[ 2πi3 mµ(x)
]Vortex Removed configurations Z†
µ(x)UΩµ (x)
(University of Adelaide) Mon. 7/4/2014 6 / 14
Centre Vortices on the Lattice
3 sets of configurations:Untouched configurations Uµ(x)Vortex only configurations Zµ(x) = exp
[ 2πi3 mµ(x)
]Vortex Removed configurations Z†
µ(x)UΩµ (x)
(University of Adelaide) Mon. 7/4/2014 6 / 14
Centre Vortices on the Lattice
3 sets of configurations:Untouched configurations Uµ(x)Vortex only configurations Zµ(x) = exp
[ 2πi3 mµ(x)
]Vortex Removed configurations Z†
µ(x)UΩµ (x)
(University of Adelaide) Mon. 7/4/2014 6 / 14
Centre Vortices on the Lattice
3 sets of configurations:Untouched configurations Uµ(x)Vortex only configurations Zµ(x) = exp
[ 2πi3 mµ(x)
]Vortex Removed configurations Z†
µ(x)UΩµ (x)
(University of Adelaide) Mon. 7/4/2014 6 / 14
Centre Vortices
0 0.5 1 1.5 2 2.5 3 3.5 4
r σ1/2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
V(r
)/σ1/
2
Fit to Full SU(3) potentialVortex β=4.80
No-Vortex β=4.80
No-Vortex β=4.60
Vortex β=4.60
No-Vortex β=4.38
Vortex β=4.38
From P. O. Bowman et. al., Phys. Rev. D 84, 034501 (2011) [arXiv:1010.4624 [hep-lat]].
(University of Adelaide) Mon. 7/4/2014 7 / 14
Dynamical Mass Generation
Non-trivial topology gives rise to a non-zero topological charge, QIntersections and writhing points of centre vortices associated withtopological charge, centre of instanton-like objectsAtiyah-Singer index theorem;
Q = n+ − n− (9)
Linked to dynamical chiral symmetry breaking and dynamicalmass generation by Casher-Banks relation
< qq >= −πρ(0) as mq → 0 (10)
(University of Adelaide) Mon. 7/4/2014 8 / 14
Dynamical Mass Generation
Non-trivial topology gives rise to a non-zero topological charge, QIntersections and writhing points of centre vortices associated withtopological charge, centre of instanton-like objectsAtiyah-Singer index theorem;
Q = n+ − n− (9)
Linked to dynamical chiral symmetry breaking and dynamicalmass generation by Casher-Banks relation
< qq >= −πρ(0) as mq → 0 (10)
(University of Adelaide) Mon. 7/4/2014 8 / 14
Dynamical Mass Generation
Non-trivial topology gives rise to a non-zero topological charge, QIntersections and writhing points of centre vortices associated withtopological charge, centre of instanton-like objectsAtiyah-Singer index theorem;
Q = n+ − n− (9)
Linked to dynamical chiral symmetry breaking and dynamicalmass generation by Casher-Banks relation
< qq >= −πρ(0) as mq → 0 (10)
(University of Adelaide) Mon. 7/4/2014 8 / 14
Dynamical Mass Generation
Non-trivial topology gives rise to a non-zero topological charge, QIntersections and writhing points of centre vortices associated withtopological charge, centre of instanton-like objectsAtiyah-Singer index theorem;
Q = n+ − n− (9)
Linked to dynamical chiral symmetry breaking and dynamicalmass generation by Casher-Banks relation
< qq >= −πρ(0) as mq → 0 (10)
(University of Adelaide) Mon. 7/4/2014 8 / 14
Previous Studies
From P. O. Bowman et. al., Phys. Rev. D 84, 034501 (2011) [arXiv:1010.4624 [hep-lat]].
(University of Adelaide) Mon. 7/4/2014 9 / 14
Overlap Quark Propagator
Operator with lattice-deformed version of chiral symmetryAllows us to write momentum-space propagator as
S(p) =Z(p)
iq/+ M(p)(11)
(University of Adelaide) Mon. 7/4/2014 10 / 14
Overlap Quark Propagator
Operator with lattice-deformed version of chiral symmetryAllows us to write momentum-space propagator as
S(p) =Z(p)
iq/+ M(p)(11)
(University of Adelaide) Mon. 7/4/2014 10 / 14
Simulation Details
20 configurations using Luscher-Weisz mean-field improved action20× 20× 20× 40 lattice with a spacing of 0.125 fmSpacial extent of 2.5 fm, temporal 5fm
(University of Adelaide) Mon. 7/4/2014 11 / 14
Simulation Details
20 configurations using Luscher-Weisz mean-field improved action20× 20× 20× 40 lattice with a spacing of 0.125 fmSpacial extent of 2.5 fm, temporal 5fm
(University of Adelaide) Mon. 7/4/2014 11 / 14
Simulation Details
20 configurations using Luscher-Weisz mean-field improved action20× 20× 20× 40 lattice with a spacing of 0.125 fmSpacial extent of 2.5 fm, temporal 5fm
(University of Adelaide) Mon. 7/4/2014 11 / 14
Renormalization function
(University of Adelaide) Mon. 7/4/2014 12 / 14
Mass function
(University of Adelaide) Mon. 7/4/2014 13 / 14
Conclusion
Removal of centre vortices removes instanton-like objects anddynamical mass generationA centre vortex background alone cannot reproduce dynamicalmass generation
(University of Adelaide) Mon. 7/4/2014 14 / 14
Conclusion
Removal of centre vortices removes instanton-like objects anddynamical mass generationA centre vortex background alone cannot reproduce dynamicalmass generation
(University of Adelaide) Mon. 7/4/2014 14 / 14
