Determining the light-by-light contributions from Dyson ... · Determining the light-by-light...

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