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Determining the light-by-light contributions fromDyson-Schwinger equations
Christian S. Fischer
TU Darmstadt
25. October 2007
C.F., Journal of Physics G32, R253-R291 (2006).
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 1 / 34
Overview
1 The lbl contributions
2 The Dyson-Schwinger equations frameworkYang-Mills-theoryQuark-Gluon InteractionDχSB: Quarks and PionsQuark-Photon interaction
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 2 / 34
Overview
1 The lbl contributions
2 The Dyson-Schwinger equations frameworkYang-Mills-theoryQuark-Gluon InteractionDχSB: Quarks and PionsQuark-Photon interaction
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 3 / 34
lbl - total
atheoryµ = 116591793(68)× 10−11
alblµ = 100(39)× 10−11
F. Jegerlehner,arXiv:hep-ph/0703125.
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 4 / 34
lbl: PS-Meson-exchange and quark-loop
π0, η, η′-pole contributions
alblµ = 100(39)× 10−11
aπµ = 88(12)× 10−11
quark-loop contributions
alblµ = 100(39)× 10−11
aquarkµ = 21( 3) × 10−11
F. Jegerlehner,arXiv:hep-ph/0703125.
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 5 / 34
lbl: AV-Meson exchange and meson-loop
AV-pole contributions
alblµ = 100(39)× 10−11
aπµ = 10( 4) × 10−11
meson-loop contributions
alblµ = 100(39)× 10−11
aquarkµ = −19(13)× 10−11
F. Jegerlehner,arXiv:hep-ph/0703125.
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 6 / 34
Methodology
State of the art calculations done on basis of
Extended Nambu-Jona-Lasinio (ENJL) model
Vector meson dominance (VMD)
quark-hadron duality
short distance constraints
K. Melnikov and A. Vainshtein, PRD 70 (2004) 113006.M. Knecht and A. Nyffeler, PRD 65 (2002) 073034.M. Hayakawa and T. Kinoshita, PRD 57, (1998) 465.J. Bijnens, E. Pallante and J. Prades, NPB 474 (1996) 379.
Overall: Numerical agreement between different methods
Indiviual contributions: Disagreement
Consistent ab initio approach desirable...
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 7 / 34
Elements of an ab initio calculation
Need to determine:
quark propagator
quark-gluon interaction
quark-photon interaction
wave function of π, η, a0, ...
π − γ − γ form factor
π − π − γ form factor
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 8 / 34
Overview
1 The lbl contributions
2 The Dyson-Schwinger equations frameworkYang-Mills-theoryQuark-Gluon InteractionDχSB: Quarks and PionsQuark-Photon interaction
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 9 / 34
Motivation
QCD Green’s functionsare connected to confinement:
Gribov-Zwanziger/Kugo-Ojima scenariosPositivityQuark-antiquark potential
encode DχSBare ingredients for hadron phenomenology
Bound state equations:Bethe–Salpeter equation / Faddeev equation
The Goal:Ab initio description of hadrons as bound statesin terms of quark-gluon substructure
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 10 / 34
QCD Propagators of QCD: Covariant Gauge
Quarks, Gluons and Ghosts:
ZQCD =
∫
D[Ψ, A, c] exp{
−
∫
d4x(
Ψ̄ (iD/ − m)Ψ
−14
(
F aµν
)2+
(∂A)2
2ξ+ c̄(−∂D)c
)
}
Landau gauge propagators in momentum space:
DGluonµν (p) =
Z(p2)
p2
(
δµν −pµpν
p2
)
DGhost(p) = −G(p2)
p2
SQuark(p) =Zf (p2)
−ip/ + M(p2)
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 11 / 34
DSEs and BSE
−1
=−1
- + +
−1
=−1
-
−1
=−1
-
π, K...=
π, K...
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 12 / 34
Infrared Structure of YM-theory
DSE for the ghost-propagator
Selfconsistency in infrared:
Z (p2) ∼ (p2)2κ , G(p2) ∼ (p2)−κ
L. v. Smekal, A. Hauck, R. Alkofer, Phys. Rev. Lett. 79 (1997) 3591
κ > 0 ⇒ Well defined global colour charge!
P. Watson and R. Alkofer, Phys. Rev. Lett. 86 (2001) 5239
C. Lerche, L. v. Smekal, Phys. Rev. D 65 (2002) 125006.
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 13 / 34
General Infrared Structure
2n external ghost legs and m external gluon legs(one external scale p2; solves DSEs and STIs ):
Γn,m(p2) ∼ (p2)(n−m)κ
R. Alkofer, C. F., F. Llanes-Estrada, Phys. Lett. B 611 (2005)
Solution is unique!C.F. and J. M. Pawlowski, Phys. Rev. D 75 (2007) 025012.
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 14 / 34
Nonperturbative Running Coupling
αgh−gl(p2) = αµ G2(p2) Z (p2)
α3g(p2) = αµ [Γ3g(p2)]2 Z 3(p2)
α4g(p2) = αµ Γ4g(p2) Z 2(p2)
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 15 / 34
Running Coupling: IR-Universality
αgh−gl(p2) = αµ G2(p2) Z (p2) ∼ const /Nc
α3g(p2) = αµ [Γ3g(p2)]2 Z 3(p2) ∼ const /Nc
α4g(p2) = αµ Γ4g(p2) Z 2(p2) ∼ const /Nc
withΓ3g(p2) ∼ (p2)−3κ , Γ4g(p2) ∼ (p2)−4κ
G(p2) ∼ (p2)−κ , Z (p2) ∼ (p2)2κ
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 16 / 34
Running Coupling: IR-Universality
αgh−gl(p2) = αµ G2(p2) Z (p2) ∼ 8.92/Nc
α3g(p2) = αµ [Γ3g(p2)]2 Z 3(p2) ∼ const /Nc
α4g(p2) = αµ Γ4g(p2) Z 2(p2) ∼ 0.0086/Nc
C. Lerche and L. v. Smekal, PRD 65 (2002) 125006
C. Kellermann and C.F., in preparation
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 17 / 34
DSEs for Ghost and Glue
−1
=
−1
-1
2
-1
2-
1
6
-1
2+
−1
=
−1
-
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 18 / 34
Ghost and Glue
10-4
10-3
10-2
10-1
100
101
102
103
p2 [GeV
2]
10-4
10-3
10-2
10-1
100
101
102
G(p2)
Z(p2)
CF and Alkofer, PLB 536 (2002) 177.
IR: G(p2) ∼ (p2)−κ Z (p2) ∼ (p2)2κ κ ≈ 0.595
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 19 / 34
Ghost and Glue vs lattice
0 1 2 3 4 5 6 7 8p [GeV]
0
0.5
1
1.5
2
Z(p
2 )
Bowman et al. (2004)Sternbeck et al. (2005)DSE
0 1 2 3 4 5p [GeV]
1
2
3
G(p
2 )
DSELangfeld et al. (2004)Sternbeck et al (2005)
◮ Kugo-Ojima criterion satisfied: G(p2 = 0) → ∞
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 20 / 34
Running coupling
10-4
10-3
10-2
10-1
100
101
102
103
104
p2 [GeV
2]
10-3
10-2
10-1
100
101
α(p2 )
ghost-gluon-vertex4-gluon-vertex
CF and Alkofer, PLB 536 (2002) 177.
Kellermann and CF, in preparation
Ghost sector dominates deep IR
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 21 / 34
Infrared Structure of QCD
Quark-gluon vertex:
= + + + +
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 22 / 34
Infrared Structure of QCD
Quark-gluon vertex: lowest order in skeleton expansion
= + + ++
S(p) = ip/Zf
p2 + M2 +Zf M
p2 + M2
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 23 / 34
Infrared Structure of QCD
Quark-gluon vertex: lowest order in skeleton expansion
= + + ++
S(p) = ip/Zf
p2 + M2 +Zf M
p2 + M2
Γµ = ig4
∑
i=1
λiGiµ : G1
µ = γµ , G2µ = p̂µ , G3
µ = p̂/ p̂µ , G4µ = p̂/ γµ
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 23 / 34
Infrared Structure of QCD
Quark-gluon vertex: lowest order in skeleton expansion
= + + ++
S(p) = ip/Zf
p2 + M2 +Zf M
p2 + M2
Γµ = ig4
∑
i=1
λiGiµ : G1
µ = γµ , G2µ = p̂µ , G3
µ = p̂/ p̂µ , G4µ = p̂/ γµ
λ1,2,3,4 ∼ (p2)−1/2−κ↔ λ1,3 ∼ (p2)−κ
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 23 / 34
Running Coupling: IR-slavery
αqg(p2) = αµ [Γqg(p2)]2 [Zf (p2)]2 Z (p2) ∼
{
1p2 : DχSB
const : χS
0.01 0.1 1 10 100
Q [GeV]0
1
2
3
4
5
α(Q)YM-couplingsQuark-gluon coupling
R. Alkofer, C. F., F. Llanes-Estrada, hep-ph/0607293.
C. F., F. Llanes-Estrada, R. Alkofer, K. Schwenzer, in preparation.
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 24 / 34
DSEs and BSE
−1
=−1
- + +
−1
=−1
-
−1
=−1
-
π, K...=
π, K...
Γµ(p, k) ∼ γµ G2(k) A(k)
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 25 / 34
Partially unquenched quark
10-3
10-2
10-1
100
101
102
103
104
105
p2 [GeV
2]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
M (
p2 ) [G
eV]
strange, Nf=2+1
strange, Nf=0
up/down, Nf=2+1
up/down, Nf=0
10-3
10-2
10-1
100
101
102
103
p2 [GeV
2]
0.6
0.8
1.0
Zf (
p2 )
strange, Nf=2+1
strange, Nf=0
up/down, Nf=2+1
up/down, Nf=0
−(< qq >0)1/3 (MeV) Mch(p2 = 0) (MeV)
Nf = 0 266 416Nf = 2 + 1 271 388
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 26 / 34
Massive Quarks
10-2
10-1
100
101
102
103
p2 [GeV
2]
10-4
10-3
10-2
10-1
100
101
Qu
ark
Mas
s F
un
ctio
n:
M(p
2 )
Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit
10-3
10-2
10-1
100
101
102
103
p2 [GeV
2]
0.6
0.7
0.8
0.9
1.0
Zf(
p2 )
= 1
/A(p
2 )
Bottom quarkCharm quarkStrange quarkUp/Down quarkChiral limit
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 27 / 34
Partially unquenched light mesons I
0 0.05 0.1m
q (GeV)
0
0.2
0.4
0.6
0.8
1
1.2
Mps
(G
eV)
Nf=0, full SDE
Nf=3, full SDE
0 0.05 0.1m
q (GeV)
0.6
0.8
1
1.2
1.4
1.6
Mve
(G
eV)
Nf=0, full SDE
Nf=3, full SDE
mu ms mπ fπ mK fK mρ
Nf = 0 4.17 88.2 139.7 130.9 494.5 165.6 708.0Nf = 2 + 1 4.06 86.0 140.0 131.1 493.3 169.5 695.2
Nf = 3 4.06 139.7 130.8 690.0
PDG 3.0-5.5 80-130 139.6 130.7 493.7 160.0 770
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 28 / 34
Partially unquenched light mesons II
0 0.2 0.4 0.6 0.8 1 1.2M
ps [GeV]
0.6
0.8
1
1.2
1.4
1.6M
ve [
GeV
]N
f=3, full SDE
Nf=0, full SDE
Nf=3, phen. model
Nf=0, phen. model
Nf=2, JLQCD(2003)
Nf=0, JLQCD (2003)
Nf=2, CP-Pacs (2002/4)
C. F., P. Watson and W. Cassing, Phys. Rev. D 72 (2005) 094025
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 29 / 34
Quark-photon vertex
= +
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 30 / 34
Quark-photon vertex
= +
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 31 / 34
Quark-photon vertex
= +
Up to twelve independent tensor structures
Ward-Takahashi identity (WTI)
kµΓµ(p, q, k) = S−1(p) − S−1(q)
C.F. and Andreas Krassnigg, work in progress
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 31 / 34
Quark-photon vertex
10-4
10-3
10-2
10-1
100
101
102
103
104
105
p2 [ GeV
2]
0.0
0.5
1.0
1.5
2.0L
1 - vertex
L2 - vertex
L3 - vertex
L1 - WTI
L2 - WTI
L3 - WTI
Ward-Takahashi identity (WTI) satisfied!
kµΓµ(p, q, k) = S−1(p) − S−1(q)
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 32 / 34
Elements of an ab initio calculation
Need to determine:
quark propagator
quark-gluon interaction
quark-photon interaction
wave function of π, η, a0, ...
π − γ − γ form factor
π − π − γ form factor
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 33 / 34
Summary
Yang-Mills sector well under controlInfrared solution for any 1PI Green’s functionFixed point of running coupling
Progress in Quark sectorQuark-Gluon-Vertex: Confinement ↔ DχSBQuark-Photon-Vertex: WTI satisfied
Meson sectorGoldstone bosons under controlMissing pieces: axialvector mesons, π − γ − γ, π − π − γ
Christian S. Fischer (TU Darmstadt) lbl from DSEs 25. October 2007 34 / 34