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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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 2 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 3 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 3 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 4 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 5 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 5 / 13
Including The Sunset Diagram
1 Introduction
2 Including The Sunset Diagram
3 Summary and Outlook
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 6 / 13
Including The Sunset Diagram
Yang-Mills propagator DSE’s in Landau Gauge
= +
+ + +
+−1 −1
= +−1 −1
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 7 / 13
Including The Sunset Diagram
Yang-Mills propagator DSE’s in Landau Gauge
+ + +
=−1
+ +−1
= +−1 −1
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 7 / 13
Including The Sunset Diagram
Yang-Mills propagator DSE’s in Landau Gauge
=−1
+ +−1
+ + +
= +−1 −1
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 7 / 13
Including The Sunset Diagram
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 8 / 13
Including The Sunset Diagram
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 8 / 13
Including The Sunset Diagram
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)
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 9 / 13
Including The Sunset Diagram
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 10 / 13
Including The Sunset Diagram
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)
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 11 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 12 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 13 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 13 / 13
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
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 13 / 13
Thank Youfor Your attention!
Valentin Mader (KFU Graz) 4-gluon Interactions in DSEs confX 09.10.12 13 / 13
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