Including Four-Gluon Interactions Into Landau / Maximally ...€¦ · ˇ0:74 Abelian Dominance Two...

Including Four-Gluon Interactions IntoLandau / Maximally-Abelian Gauge

Dyson-Schwinger Studies

Valentin Mader∗, Reinhard Alkofer

Institut fur PhysikKarl Franzens Universitat

Graz

Confinement and the Hadron Spectrum X

Munich, 8.-12.10.2012

∗supported by FWF Doctoral Program on Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203-N16)

Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 1 / 13

Table of Contents

1 Introduction

2 Including The Sunset Diagram

3 Summary and Outlook


Introduction

Dyson-Schwinger Equations

equations of motion of Green functions of a quantum field theory

one possible (infinite) set of defining equations for the Green functions

by definition fully renormalized equations

practical calculations demand gauge-fixing and truncations

truncations can interfere with “self-renormalization”

= +−1 −1


Introduction

Maximally-Abelian Gauge [Kronfeld et al., PLB 4 (1987)]

discriminates between “Abelian” and “Non-Abelian” gauge fields

Aµ =

N2c−1∑r

ArµTr =

Nc−1∑i

AiµTi +

N2c−Nc∑a

BaµT

a

Lorentz-covariant, Ghost-Antighost-symmetric

separate gauge-fixing:

non-abelian: minimize∫d4xBaµB

aµ

abelian: maximize∫d4xAiµA

iµ

Landau gauge: maximize∫d4xArµA

rµ


Introduction

IR-Analysis of the MAG

use DSE’s and ERGE’s

IR power-law ansatze

no truncation involved

Dressing functions behave like [Huber, Schwenzer, Alkofer, EPJ C 68 (2010)]

[Huber, PhD Thesis, KFU-Graz (2010)]

Zd ∝(p2)−κ

, Zod ∝(p2)κ, God ∝

(p2)κ

κ ≈ 0.74

Abelian Dominance

Two loop terms are IR-leading


Including The Sunset Diagram

1 Introduction

2 Including The Sunset Diagram

3 Summary and Outlook



Yang-Mills propagator DSE’s in Landau Gauge

= +

+ + +

+−1 −1

= +−1 −1




+ + +

=−1

+ +−1

= +−1 −1




=−1

+ +−1

+ + +

= +−1 −1



Structure of Divergencies

+ + +

=−1

+ +−1

D(p2) =Z(p2)

p2 + Zmm2

D−1(p2) = c1Λ2 + c2 p2 log

[Λ2

p2

]+ something finite

quadratic divergences: modify gluon loop to cancel div’s of ghost loop

Zm(Λ2, µ2)m2(µ2) = 0 ⇒ c1 = 0

logarithmic divergences: MOM-scheme

Z(p2)p2→∞∝ αs(p

2)−γ



Including Sunset

Isun(p2) =Z4

6g4N2

c

∫ Λ

0d4q1

∫ Λ

0d4q2

Z(p21)Z(p2

2)Z(p23)

p21 p

22 p

23

DΓ(p, p1, p2, p3) T

T = T (z01, z02, z03, z12, z13, z23) zij =pi·pj√p2ip2j

Isun(p2) = c1Λ2 + c2 p2 log

[Λ2

p2

]+ something finite

different divergences appear in different combinations of zij ’s(overlapping divergences)



Including Sunset

Isun(p2) =Z4

6g4N2

c

∫ Λ

0d4q1

∫ Λ

0d4q2

Z(p21)Z(p2

2)Z(p23)

p21 p

22 p

23

DΓ(p, p1, p2, p3) T

T = T (z01, z02, z03, z12, z13, z23) zij =pi·pj√p2ip2j

solution: explicit counter term construction

T = T − (ζ − 4)(λ1 + λ2

(z2

12 + 3z201

))quadratic divergences entirely subtracted

Isun(p2) = c2 p2 log

[Λ2

p2

]+ something finite



Including Sunset

=−1

+ +−1

+ + +

D(p2) =Z(p2)

p2

0,0001 0,001 0,01 0,1 1 10 100 1000 10000 1e+05

p2[GeV

2]

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1Z(p2)

including sunsetone loop only

0,0001 0,001 0,01 0,1 1 10 100 1000 10000

p2[internal units]

1e-05

0,0001

0,001

0,01

0,1

1

10

100

1000

10000

1e+05Π(p

2)

oneloopsabs(oneloops)sunsetabs(sunset)


Summary

renormalization in non-perturbative functional methods demand novelappropriate techniques

in DSE’s one possibility is explicit counter-term construction

explicit counter-term construction can also be applied for two-loopterms (sunset)

in Landau gauge contributions of sunset-diagram not significant


Outlook: Maximally Abelian Gauge

proposed self-consistent system to solve:

= + +−1 −1

= + +−1 −1

! UV-behavior (given by one-loop terms)

! IR-behavior (given by two-loop terms)

! quadratic divergences subtracted

% numerical implementation


Thank Youfor Your attention!


Including Four-Gluon Interactions Into Landau / Maximally ...€¦ · ˇ0:74 Abelian Dominance Two...

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