[PPT] - Non-perturbative results for n-point functions of Landau gauge PowerPoint Presentation

SLIDE 1

Non-perturbative results for n-point functions of Landau gauge Yang-Mills theory at (non-)vanishing temperature

Markus Q. Huber

in collaboration with Anton K. Cyrol, Lorenz von Smekal

Institute of Physics, University of Graz

Non-Perturbative Methods in Quantum Field Theory - Workshop in Theoretical Particle Physics

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 1 / 23

SLIDE 2

Correlation functions of QCD

Investigate:

◮ Confinement ◮ Dynamical symmetry breaking ◮ Bound states

(Non-perturbative) Determination of correlation functions:

◮ Lattice ◮ Effective theories, e.g., (refined) Gribov-Zwanziger, massive Yang-Mills ◮ Functional equations ◮ . . .

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 2 / 23

SLIDE 3

Functional equations and truncations

Functional equations. . .

◮ . . . are exact. ◮ . . . form an infinite system of equations.

Truncating the system: How close/far from exact solution?

◮ Calculate physical quantities. ◮ Compare to other methods. ◮ Modify the truncation.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

SLIDE 4

Functional equations and truncations

Functional equations. . .

◮ . . . are exact. ◮ . . . form an infinite system of equations.

Truncating the system: How close/far from exact solution?

◮ Calculate physical quantities. ◮ Compare to other methods. ◮ Modify the truncation.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

SLIDE 5

Functional equations and truncations

Functional equations. . .

◮ . . . are exact. ◮ . . . form an infinite system of equations.

Truncating the system: How close/far from exact solution?

◮ Calculate physical quantities. ◮ Compare to other methods.

If available.

◮ Modify the truncation.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

SLIDE 6

Functional equations and truncations

Functional equations. . .

◮ . . . are exact. ◮ . . . form an infinite system of equations.

Truncating the system: How close/far from exact solution?

◮ Calculate physical quantities. ◮ Compare to other methods.

If available.

◮ Modify the truncation.

Recent results indicate that primitively divergent correlation functions of Landau gauge Yang-Mills theory provide reasonable truncation for a quantitative, self-consistent and self-contained description.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 3 / 23

SLIDE 7

Outline

◮ Vacuum

◮ Propagators: Introduction ◮ Three-point functions: ◮ ghost-gluon vertex ◮ three-gluon vertex → talks by Blum, Senn ◮ Four-gluon vertex

◮ Non-vanishing temperature

◮ Ghost-gluon vertex ◮ Three-gluon vertex

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 4 / 23

SLIDE 8

Landau Gauge Yang-Mills theory

Gluonic sector of quantum chromodynamics: Yang-Mills theory

L = 1

2F 2 + Lgf + Lgh Fµν = ∂µAν − ∂νAµ + i g [Aµ, Aν]

Landau gauge

◮ simplest one for functional equations ◮ ∂µAµ = 0:

Lgf = 1

2ξ (∂µAµ)2,

ξ → 0

◮ requires ghost fields:

Lgh = ¯

c (− + g A ×) c

i

j

1

i

j k

i

j k l

j

k

1

i

j k

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 5 / 23

SLIDE 9

Propagators

i j

1

= +

i j

1 - 1

2

i j

1

2

i j

+

i j

1

6

i j

1

2

i j j i

1

= +

j i

1 -

i j

Models

r

results for vertices required.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

SLIDE 10

Propagators

i j

1

= +

i j

1 - 1

2

i j

1

2

i j

+

i j

1

6

i j

1

2

i j j i

1

= +

j i

1 -

i j

Models

r

results for vertices required.

(Tadpole vanishes perturbatively, but can contribute non-perturbatively [MQH, von Smekal ’14].)

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

SLIDE 11

Propagators

i j

1

= +

i j

1 - 1

2

i j

+

i j j i

1

= +

j i

1 -

i j

Typical truncation: no four-gluon vertex, bare ghost-gluon vertex, model for three-gluon vertex lattice data:

[Sternbeck ’06]

1

2 3 4 5 pGeV 1 2 3 4

Zp2

ne-loop truncation

Comparison with lattice results

→ missing strength in mid-momentum

regime; attributed to

◮ neglected two-loop diagrams? ◮ vertices?

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

SLIDE 12

Propagators

i j

1

= +

i j

1 - 1

2

i j

+

i j j i

1

= +

j i

1 -

i j

Typical truncation: no four-gluon vertex, bare ghost-gluon vertex, model for three-gluon vertex

2 4 6 8

pGeV

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Zp2

→ Using results from a three-gluon

vertex calculation, importance of two-loop diagrams shown.

[talk by Blum; Blum, MQH, Mitter, von Smekal ’13]

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 6 / 23

SLIDE 13

Three-point functions

Vertices (truncated):

i j k

= +

i j k

+

i j k

+

i j k

= +

i j k

+ 1

2

i j k

+ 1

2

i j k

+ 1

2

i k j

+

i j k

2

i j k

→ talk by Blum

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 7 / 23

SLIDE 14

The ghost-gluon vertex

To good approximation the ghost-gluon vertex can be taken as bare

[Ilgenfritz et al. ’07, Cucchieri, Maas, Mendes ’08, Schleifenbaum, Maas, Wambach, Alkofer ’05, Boucaud et al. ’11, Fister, Pawlowski ’12, MQH, von Smekal ’13, Aguilar, Ibáñez, Papavassiliou ’13, Pelaez, Tissier, Wschebor ’13].

Some influence on propagators [MQH, von Smekal ’13, Aguilar, Ibáñez, Papavassiliou ’13] and three-gluon vertex [Blum, MQH, Mitter, von Smeka ’14, Blum ’14, talk by Blum].

[MQH, von Smekal ’13]

2

4 6 8 10 pGeV 0.9 1.0 1.1 1.2 1.3 1.4

A0;p2,p2

lattice data:

[Sternbeck ’06]

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 8 / 23

SLIDE 15

Optimized effective three-gluon vertex

Lattice inspired model for three-gluon vertex with zero crossing:

1

2 3 4 5 pGeV 1.0 0.5 0.0 0.5 1.0 1.5 2.0

D A 3p 2,p 2,p 2

[Cucchieri, Maas, Mendes ’08, MQH, von Smekal ’13]

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 9 / 23

SLIDE 16

Optimized effective three-gluon vertex

Lattice inspired model for three-gluon vertex with zero crossing:

1

2 3 4 5 pGeV 1.0 0.5 0.0 0.5 1.0 1.5 2.0

D A 3p 2,p 2,p 2

[Cucchieri, Maas, Mendes ’08, MQH, von Smekal ’13]

Allows to effectively capture two-loop contributions.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 9 / 23

SLIDE 17

Propagator results

Dynamic ghost-gluon vertex, opt. eff. three-gluon vertex [MQH, von Smekal ’13]

1

2 3 4 5 pGeV 1 2 3 4

Zp 2

0.0

0.5 1.0 1.5 2.0 2.5 3.0 pGeV 1 2 3 4 5 6

Gp2

Good quantitative agreement for ghost and gluon dressings. ⇒ Input for further calculations.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 10 / 23

SLIDE 18

Propagator results

Dynamic ghost-gluon vertex, opt. eff. three-gluon vertex [MQH, von Smekal ’13]

1

2 3 4 5 pGeV 1 2 3 4

Zp 2

FRG results

[Fischer, Maas, Pawlowski ’08]

1 2 3 4 5 p [GeV] 1 2 Z(p

2)

Bowman (2004) Sternbeck (2006) scaling (DSE) decoupling (DSE) scaling (FRG) decoupling (FRG)

0.0

0.5 1.0 1.5 2.0 2.5 3.0 pGeV 1 2 3 4 5 6

Gp2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p [GeV] 2 4 6 8 10 12 14 G(p

2)

Sternbeck (2006) scaling (DSE) decoupling (DSE) scaling (FRG) decoupling (FRG)

Good quantitative agreement for ghost and gluon dressings. ⇒ Input for further calculations.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 10 / 23

SLIDE 19

Four-gluon vertex

◮ 20 one-loop, 39 two-loop diagrams

i j k l = + i j k l + (1 / 2) i j k l

i

j k l + (1 / 2) i j k l + (1 / 2) i k j l + (1 / 2) i l j k + (1 / 6) i j k l + i j k l + i k j l + i l j k

2

i k j l

2

i l j k

2

i j k l + i j k l + i k j l + i l j k + 1

2

i j k l + 1

2

i j k l + 1

2

i k j l + 1

2

i j k l + 1

2

i j k l + 1

2

i k j l + 1

2

i l j k + i j k l + i j k l + i j k l

2

i j k l

2

i j k l

2

i j l k + i k j l + i l j k + 1

2

i k j l + 1

2

i l j k + i j k l + i j l k + i j k l + 1

2

i j k l + 1

2

i j k l + 1

2

i j l k + 1

2

i j k l + 1

2

i j k l + 1

2

i j l k + 1

2

i k j l + i k l j + 1

2

i k l j + 1

2

i k l j + 1

2

i l j k + i j k l + i j k l + i j k l + i j k l + 1

2

i j k l + 1

2

i j k l + i j l k + i j k l + 1

2

i j k l + 1

2

i j k l + 1

2

i j k l + i j l k + 1

2

i j l k

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 11 / 23

SLIDE 20

Four-gluon vertex

◮ 20 one-loop, 39 two-loop diagrams ◮ Keep UV leading diagrams

→ 16 diagrams

i j k l

= +

i j k l

+ 1

2

i j k l

+ 1

2

i k j l

+ 1

2

i l j k

+

i j k l

+

i k j l

+

i l j k

+

i j k l

+

i k j l

+

i l j k

+

i j k l

+

i j k l

+

i j k l

2

i j k l

2

i j k l

2

i j l k

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 11 / 23

SLIDE 21

Four-gluon vertex

◮ 20 one-loop, 39 two-loop diagrams ◮ Keep UV leading diagrams

→ 16 diagrams

◮ Calculate full momentum dependence.

→ Access to all permutations of this diagram. → 6 diagrams

i j k l

= +

i j k l

+ 3

2

i j k l

+ 3

i j k l

+ 3

i j k l

+ 3

i j k l

6

i j k l

+ s y m m e t r i z a t i

n
Oct. 8, 2014 | University of Graz | Markus Q. Huber | 11 / 23

SLIDE 22

Four-gluon vertex

◮ 20 one-loop, 39 two-loop diagrams ◮ Keep UV leading diagrams

→ 16 diagrams

◮ Calculate full momentum dependence.

→ Access to all permutations of this diagram. → 6 diagrams

i j k l

= +

i j k l

+ 3

2

i j k l

+ 3

i j k l

+ 3

i j k l

+ 3

i j k l

6

i j k l

+ s y m m e t r i z a t i

n

No dependence on unknown Green functions! → ’Truncation closes.’

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 11 / 23

SLIDE 23

Four-gluon vertex

◮ 20 one-loop, 39 two-loop diagrams ◮ Keep UV leading diagrams

→ 16 diagrams

◮ Calculate full momentum dependence.

→ Access to all permutations of this diagram. → 6 diagrams

i j k l

= +

i j k l

+ 3

2

i j k l

+ 3

i j k l

+ 3

i j k l

+ 3

i j k l

6

i j k l

+ s y m m e t r i z a t i

n

No dependence on unknown Green functions! → ’Truncation closes.’

◮ 6 external variables ◮ 4 integration variables

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 11 / 23

SLIDE 24

Tensor basis

Full calculation: Restriction to tree-level tensor:

Γabcd

µνρσ(p, q, r, s) = Γ(0),abcd µνρσ D4g(p, q, r, s).

Non-perturbative information in D4g(p, q, r, s). Derivation of DSE using DoFun [Braun, MQH ’11]: Mathematica package for derivation of functional equations

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 12 / 23

SLIDE 25

Tensor basis

Full calculation: Restriction to tree-level tensor:

Γabcd

µνρσ(p, q, r, s) = Γ(0),abcd µνρσ D4g(p, q, r, s).

Non-perturbative information in D4g(p, q, r, s). Derivation of DSE using DoFun [Braun, MQH ’11]: Mathematica package for derivation of functional equations Other DSE results (configuration A only):

◮ Box-only truncation: [Kellermann, Fischer ’08] ◮ Same truncation as here, semi-perturbative approximation (1 iteration): [Binosi, Ibáñez, Papavassiliou ’14]

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 12 / 23

SLIDE 26

Four-gluon vertex results

10

3

10 10

3

p

2[GeV 2]

0,5 1 1,5 2 2,5 3 D

4g, config. A

D

4g, config. B

D

4g, config. C

D

4g, config. C, fit

[Cyrol, MQH, von Smekal ’14]

2-parameter fit: D4g, dec

model (p, q, r, s) =

a tanh
b/¯

p2 + 1

D4g

RG(p, q, r, s)

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 13 / 23

SLIDE 27

Four-gluon vertex results

10

3

10 10

3

p

2[GeV 2]

0,5 1 1,5 2 2,5 3 D

4g, config. A

D

4g, config. B

D

4g, config. C

D

4g, config. C, fit

[Cyrol, MQH, von Smekal ’14]

2-parameter fit: D4g, dec

model (p, q, r, s) =

a tanh
b/¯

p2 + 1

D4g

RG(p, q, r, s)

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 13 / 23

SLIDE 28

Four-gluon vertex: Individual diagrams

10

4

10

3

10

2

10

1

10 10

1

10

2

10

3

p

2[GeV 2]

1

1 Z4

D

4g, config. A

ghost box gluon box static triangle dynamic triangle swordfish

◮ Swordfish ◮ Dynamic triangle ◮ Static triangle ◮ Gluon box ◮ Ghost box

In general the non-perturbative four-gluon vertex is very close to its tree-level. Note: Renormalization important to get the correct weights of all diagrams.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 14 / 23

SLIDE 29

Results for other dressing functions

◮ V1 close to tree-level. ◮ Logarithmic IR divergence ◮ For V2 and V3 individual diagrams

almost cancel each other.

◮ P small, but richer structure ◮ Not calculated self-consistently

10

4

10

3

10

2

10

1

10 10

1

10

2

10

3

p

2[GeV 2]

1

1 2 3 4 5 D

4g,V1 = D 4g, config. C

D

4g,V2, config. C

D

4g,V3, config. C

D

4g,P, config. C

[Cyrol, MQH, von Smekal ’14]

V1: tree-level V2, V3: only metric tensors, ∼ δµνδρσ P: four momenta, ∼ sµrνqρpσ

Note: Renormalization important to get the correct weights of all diagrams.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 15 / 23

SLIDE 30

Non-vanishing temperature

◮ Phase diagram: (dual) quark condensates, Polyakov loop potential, . . . ◮ Can be calculated from correlation functions! ◮ For now: Yang-Mills theory ◮ Goal: Calculate quantities that are difficult to obtain on the lattice.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 16 / 23

SLIDE 31

Propagators

Chromomagnetic and chromoelectric gluons from the lattice [Fischer, Maas, Müller ’10]:

→ Input for DSEs

Ghost dressing G(p2) from DSE [MQH, von Smekal ’13]:

i1 i2

1

i1

i2

1

i1 i2

i1

i2

Propagators from FRG: [Fister, Pawlowski ’11] Massive Yang-Mills: [Reinosa, Serreau, Tissier, Wschebor ’13]

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 17 / 23

SLIDE 32

Three-point warm-up: Ghost-gluon vertex

DSE calculation: self-consistent solution of truncated DSE, zeroth Matsubara frequency only

◮ Vertices quite expensive on lattice. ◮ Full momentum dependence from

functional equations. Vertex from FRG: [Fister, Pawlowski ’11]

preliminary

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 18 / 23

SLIDE 33

Ghost-gluon vertex: Continuum and lattice

1 2 3 4 p[GeV] 1.0 1.2 1.4 1.6 1.8

DA c

_ c(p2,p2,2π/3)

T=0.52Tc

1 2 3 4 p[GeV] 1.0 1.1 1.2 1.3 1.4

DA c

_ c(p2,p2,2π/3)

T=0.94Tc

1 2 3 4 p[GeV] 1.0 1.1 1.2 1.3 1.4 1.5 1.6

DA c

_ c(p2,p2,2π/3)

T=1.Tc

1 2 3 4 p[GeV] 1.0 1.1 1.2 1.3 1.4 1.5

DA c

_ c(p2,p2,2π/3)

T=2.Tc

Lattice: [Fister, Maas ’14]

preliminary preliminary preliminary preliminary

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 19 / 23

SLIDE 34

Three-gluon vertex: Continuum and lattice

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=0.52Tc

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=0.94Tc

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=0.98Tc

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=1.08Tc

Lattice: [Fister, Maas ’14]

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 20 / 23

SLIDE 35

Three-gluon vertex: Continuum and lattice

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=0.52Tc

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=0.94Tc

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=0.98Tc

1 2 3 4 p[GeV]

1

1 2 3 4

DAAA(p2,p2,2π/3) T=1.08Tc

Lattice: [Fister, Maas ’14]

preliminary preliminary preliminary preliminary

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 20 / 23

SLIDE 36

Three-gluon vertex: Diagram contributions

Diagram contributions (T = 0: 6, T > 0: 22 diagrams):

◮ Only chromomagnetic propagators: dominant

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 21 / 23

SLIDE 37

Three-gluon vertex: Diagram contributions

Diagram contributions (T = 0: 6, T > 0: 22 diagrams):

◮ Only chromomagnetic propagators: dominant ◮ Both propagators contained: negligible

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 21 / 23

SLIDE 38

Three-gluon vertex: Diagram contributions

Diagram contributions (T = 0: 6, T > 0: 22 diagrams):

◮ Only chromomagnetic propagators: dominant ◮ Both propagators contained: negligible ◮ Only chromoelectric propagators: Importance depends on temperature,

especially for T < Tc.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 21 / 23

SLIDE 39

Three-gluon vertex

DSE calculation: semi-perturbative approximation (first iteration only)

p r e l i m i n a r y p r e l i m i n a r y

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 22 / 23

SLIDE 40

Summary

◮ Conjecture:

Primitively divergent correlation functions sufficient for a quantitative, self-consistent and self-contained description.

◮ Truncation naturally ’closes’ with four-gluon vertex. ◮ Two-point functions: Quantitative effects understood. ◮ Three-point functions: Good agreement with lattice. ◮ Four-point functions: Test four-gluon vertex in DSEs of lower n-point functions.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 23 / 23

SLIDE 41

Summary

◮ Conjecture:

Primitively divergent correlation functions sufficient for a quantitative, self-consistent and self-contained description.

◮ Truncation naturally ’closes’ with four-gluon vertex. ◮ Two-point functions: Quantitative effects understood. ◮ Three-point functions: Good agreement with lattice. ◮ Four-point functions: Test four-gluon vertex in DSEs of lower n-point functions.

Vertices at non-vanishing temperature:

◮ Qualitative agreement with lattice results ◮ Source of behavior of three-gluon vertex ∼ Tc elucidated ◮ Basis for model building ◮ Effects of dressed vertices, e.g., in Polyakov loop potential? ◮ Basis for extension to QCD and µ > 0

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 23 / 23

SLIDE 42

Summary

◮ Conjecture:

Primitively divergent correlation functions sufficient for a quantitative, self-consistent and self-contained description.

◮ Truncation naturally ’closes’ with four-gluon vertex. ◮ Two-point functions: Quantitative effects understood. ◮ Three-point functions: Good agreement with lattice. ◮ Four-point functions: Test four-gluon vertex in DSEs of lower n-point functions.

Vertices at non-vanishing temperature:

◮ Qualitative agreement with lattice results ◮ Source of behavior of three-gluon vertex ∼ Tc elucidated ◮ Basis for model building ◮ Effects of dressed vertices, e.g., in Polyakov loop potential? ◮ Basis for extension to QCD and µ > 0

Thank you for your attention.

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 23 / 23

SLIDE 43

One-momentum configurations

6 independent variables: 3 momentum squares, 3 angles

Configuration A B C Definition S2 = R2 = Q2 = p2 θr = θq = ψq = 0 S2 = R2 = Q2 = p2 θr = θq = ψq = π

2

S2 = R2 = p2, Q2 = 2p2 θr = π

2 ,

θq = π

4 ,

ψq = 0 Visualization s r q q r s s r q

Oct. 8, 2014 | University of Graz | Markus Q. Huber | 24 / 23

SLIDE 44

Other dressing functions

By transverse projection and orthonormalization construct from

V abcd

1,µνρσ = Γ(0),abcd µνρσ

V abcd

2,µνρσ = δabδcdδµνδρσ + δacδbdδµρδνσ + δadδbcδµσδνρ

V abcd

3,µνρσ = (δacδbd + δadδbc)δµνδρσ + (δabδcd + δadδbc)δµρδνσ + (δabδcd + δacδbd)δµσδνρ

the tensors V1, V2, V3. Another class (4 momenta):

Pabcd

µνρσ = (δabδcd + δacδbd + δadδbc)sµrνqρpσ + rµsνpρqσ + qµpνsρrσ

p2 q2 r 2 p2
Oct. 8, 2014 | University of Graz | Markus Q. Huber | 25 / 23