infrared exponents and the strong coupling limit in...

Infrared Exponents and the Strong Coupling Infrared Exponents and the Strong Coupling

Limit in Lattice Landau GaugeLimit in Lattice Landau Gauge

Lorenz von SmekalLorenz von Smekal

St Goar, St Goar, MarchMarch 20082008

Wilhelm & Else Wilhelm & Else HeraeusHeraeus Seminar: Quarks and Seminar: Quarks and GluonsGluons in in StrongStrong QCDQCD

Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 22

Mτ MΖ

αS(Q) DSEs (N

f=0)

Running CouplingRunning Coupling• Determine αS from lattice Landau-gauge ghost/gluon propagators:

unquenched,

A. Sternbeck, K. Maltman, L. von Smekal, A. G. Williams, E. M. Ilgenfritz andM. Muller-Preussker, PoS (LATTICE 2007) 256, arXiv:0710.2965 [hep-lat].

αminMOMS (q2) =g2(a)

4πZL(q

2, a2)G2L(q

2, a2) + O(a2)

M. Schmelling, 1997

(QCDSF).


Running CouplingRunning Coupling

0.15

0.20

0.25

0.30

0 20 40 60 80 100 120

αs(p

2)

p2 [GeV2]

Nf = 2

(β, κ) = (5.40, 0.1366)fit: 4-loop using c1, c2, c3

c1, c2

c1

fit: 2-loop

0.65

0.70

0.000 0.005 0.010 0.015 0.020

r 0Λ

MS

2

amq

β = 5.255.295.40

• Minimom scheme αminMOMS known to 4 loops(Maltman, Sternbeck, LvS, in prep):

• Compare observables in PT:

4π2D0(Q2) = 1 +

αMSsπ

+ 1.64

(αMSsπ

)2+ 6.37

(αMSsπ

)3+ 49.1

(αMSsπ

)4+ . . .

= 1+αmMsπ

− 1.05

(αmMsπ

)2− 4.82

(αmMsπ

)3+ 3.13

(αmMsπ

)4+ . . . .

Baikov, Chetyrkin & Kuhn, arXiv:0801.182[hep-ph].


• Landau Gauge QCD Propagators

(DSEs, lattice LG,...).

• Strong Coupling Limit in Lattice Landau Gauge.

• Modified Lattice Landau Gauge.

• Summary and Conclusions.

ContentsContents


2

3

4

5 64321

1

0

[ ]GeVk

( )2kZ

0.664323 =× β lattice,

DSEs

Gluon PropagatorGluon Propagator

15.0,0

21)( 2

2

0

2

2 2

<<→

∝→

κ

κ

σ

k

kk

kZ k

DSEs:

Lerche & L.v.S., 2002Fischer & Alkofer, 2002

L.v.S. et al., 1997

Lattice:

Leinweber et al., 1998

GeV21 =−a

ERGE, J. Pawlowski

48^4 6.0

32^4 6.0

32^4 5.8

48^4 5.7

56^4 5.7

A. Sternbeck, E. M. Ilgenfritz, M. Muller-Preussker, A. Schiller andI. L. Bogolubsky, PoS LAT2006, 076 (2006) [arXiv:hep-lat/0610053].


1

0 1 2 3 4 5 6

Running CouplingRunning Coupling

0,

)()(4

)(

2

2222

2

→→

=

k

kGkZg

k

cα

πα

L.v.S. et al., 1997

2,...46.4)(

4=== c

c

c NIN κ

πα

Lerche & L.v.S., 2002

3,...97.2 == cc Nα

J. C. R. Bloch, A. Cucchieri, K. Langfeld andT. Mendes, Nucl. Phys. B 687 (2004) 76.

SU(2)163 × 32


Infrared Dominance of Infrared Dominance of GhostsGhosts

DSEs

κσ

−∝−

→

22

0

2

2 1)(2

kkk

kG k

Ghost Propagator

DSEs:

L.v.S. et al., 1997

15.0 << κ

Lattice:

Suman & Schilling, 1996

• DSEs: Lerche & L.v.S., PRD 65 (2002) 125006.

• Stochastic Quantisation: Zwanziger, PRD 65 (2002) 094039.

• ERGEs: Pawlowski, Litim, Nedelko, & L.v.S., PRL 93 (2004) 152002.

3,...972.2)(

4,...595.0 ==== c

c

c NIN κ

πακ


Finite Volume DSE SolutionsFinite Volume DSE Solutions

ghost

gluon

Fischer, Maas, Pawlowski & L.v.S., Ann. Phys. 322 (2007) 2916.

10-1

100

p [GeV]

1

10

G(p

2)

Infinite volumeL = 4.5 fmL = 5.1 fmL = 6.7 fmL = 7.9 fmL = 9.5 fmL = 10.7 fmL = 13.5 fm

0.0 0.4 0.8 1.2 1.6 2.0p [GeV]

1

2

3

4

D(p

2)

[1/G

eV2]

Inf. volumeL = 4.5 fmL = 5.1 fmL = 6.7 fmL = 7.9 fmL = 9.5 fmL = 10.7 fmL = 13.5 fm

0.2 0.43

4

need: π/L ≪ p ≪ ΛQCD


Gluon PropagatorGluon Propagator

0.0 0.4 0.8 1.2 1.6 2.0 2.4p [GeV]

0

1

2

3

4D

(p2)

[1/G

eV2]

L = 4.6 fmL = 9.7 fmSternbeck et al. (2005)Sternbeck et al. (2006)Infinite volume


Ghost PropagatorGhost Propagator

L = 3fm

L = 4.6fm

inf. Vol.

Sternbeck et al., Adelaide—Berlin—Moscow, unpublished (preliminary).


SUSU(3)(3) vsvs SUSU(2)(2)

gluon

ghost

A. Sternbeck, L. von Smekal, D. B. Leinweber and A. G. Williams,PoS (LATTICE 2007) 340, arXiv:0710.1982 [hep-lat].


SUSU(3)(3) vsvs SUSU(2)(2)

gluon

ghost



SUSU(2) (2) PropagatorsPropagatorsvolume dependencevolume dependence

gluon

ghost


fixed β, unrenormalised.


0.001

0.01

0.1

1

10

100

0.01 0.1 1 10

244

ββ = 0 = 0 ―― Gluon PropagatorGluon Propagator

324

564

κ = 0.595

a2p2

Z(a2p2) ∝ (a2p2)2κ ?

gluon mass: κ = 0.5

SU(2)



0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 16

564

324

244

Z( a2p2)

x = a2p2

κ = 0.656± 0.006

Zfit(x) = c x + d x2κ

d = 0.22± 0.01

c = 0.51± 0.01

Fit parameters (564):


transverse mass m2 ∝ 1/a2


c

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

20 25 30 35 40 45 50 55 60

d

L/a

x = a2p2Zfit(x) = c x + d x2κ,

c = c∞ + ccorra

L

fit finite volume effects:

d = d∞ + dcorra

L

c∞ = 0.57± 0.01 �= 0 ⇔

glu

on

fit p

ara

me

ters


ββ = 0 = 0 ―― Ghost PropagatorGhost Propagator

Not a finite volume effect!

G( a2p2)

κ = 0.595


0.1

1

0.01 0.1 1 10


G−1fit (x) = c + d xκ

564

324244

x = a2p2

G−1( a2p2)

κ = 0.595


0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12 14 16


G−1fit (x) = c + d xκ

564324244

x = a2p2

G−1( a2p2)


0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

20 30 40 50 60 70 80

infr

are

d e

xpo

nen

ts κ = κ∞ + κcorra

L

L/a

ghost

gluon

κ∞:

ββ = 0 = 0 ―― Infrared ExponentsInfrared Exponents

ghost: 0.678± 0.003

gluon: 0.684± 0.002


ββ = 0 = 0 ―― Infrared ExponentsInfrared Exponents

x = a2p2

Zfit(x) = c x (1 + dx)2κ−1,

G−1fit (x) = c (1 + d x)κ,

κ∞:

gluon: 0.571± 0.001

ghost: 0.569± 0.006

• alternative fit models:

Zfit(x) = c x + d x2κ,

G−1fit (x) = c + d xκ,

x = a2p2

κ∞:

ghost: 0.678± 0.003

gluon: 0.684± 0.002

• in either case : κgluon = κghost


Stereographic Projection Stereographic Projection

θ

• Consider SN

S

N

with

(x1, . . . , xN+1) →1

1 + xN+1(x1, . . . , xN )

ϕ(x) = tan(θ/2) x

SN−1

• Example SU(2):

1

2trUg → ln

(1 +

1

2trUg

)replace

suppresses South pole!

in sum over links ofgauge-fixing potential,

modified Landau gauge� −2π 2π

S S

N

ϕ :


χ(S1×#sites

) = χ(S1)#sites = 0#sites

Zgf =

∫ ∏

sites

d[θ, c, c, b] exp{−∑

i

s(ci(Fi(φ

θ) − iξ

2bi))}

= 0

• Compact U(1):

links U = eiφ, with g.t. φθ = φ + dθ

standard Morse potential: V [Uθ] =∑

links

cosφθ

Landau gauge: 0 = Fi(φθ) =

∂

∂θiV =

∑

µ

(sinφθi, µ − sinφθi−µ, µ

)

However,

Modified Lattice Landau GaugeModified Lattice Landau Gauge


θ

x(θ) = tan(θ/2)

• Use stereographic projection:

Morse potential: V [Uθ] =∑

links

ln(1 + cosφθ

)

and Fi(φθ) =

∑

µ

(tan(φθi, µ/2)− tan(φθi−µ, µ/2)

)

Explicitly worked out in 1-dim (eliminates all Gribov copies)and ≥ 2-dim (all but minima, 1st Gribov region) compact U(1).

Modified Lattice Landau GaugeModified Lattice Landau Gauge

L. von Smekal, D. Mehta, A. Sternbeck and A. G. Williams,PoS (LATTICE 2007) 382, arXiv:0710.2410 [hep-lat].


SUSU(2)(2) Gluon Propagator Gluon Propagator –– ββ = 0= 0

-3

-2

-1

0

1

2

3

4

5

0.01 0.1 1 10

D(a

2q2

)

a2q2

SU(2), β = 0

)

std: L/a = 243256

mod: L/a = 2432

(lnU: L/a = 32


SUSU(2)(2) Gluon Propagator Gluon Propagator –– ββ = 0= 0

∝ (a2p2)2κ,

κ = 0.6

terms

Zfit(x) = c x + d x2κ, x = a2p2

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16

D(a

2p2)

a2p2

324: stdmodlnU


SUSU(2)(2) Ghost Propagator Ghost Propagator –– ββ = 0= 0

1

10

0.01 0.1 1 10

G(a

2q2

)

a2q2

SU(2), β = 0

std: L/a = 243256

mod: L/a = 2432


SU(2)SU(2) Coupling Coupling –– ββ = 0= 0

αL =g2

4πZL(q

2)G2L(q

2)

0.01

0.1

1

10

0.01 0.1 1 10

αL(a

2q2

)

a2q2

SU(2), β = 0

αmaxc

std: L/a = 243256

mod: L/a = 2432

)(

4

κ

πα

INc

c =

Lerche & L.v.S., 2002:

,2,...46.4max === ccc Nαα

max for κ = 0.595 . . .


0

0.5

1

1.5

2

0.01 0.1 1 10

αL(a

2q2

)

a2q2

β = 2.5, L = 32

stdmod

SU(2)SU(2) Running Coupling Running Coupling –– finite finite ββ

αL =g2

4πZL(q

2)G2L(q

2)

1

0.01 0.1 1 10

αL(a

2q2

)

a2q2

β = 2.3, L = 56

stdmod


ConclusionsConclusions

• Strong Coupling Limit of Lattice Landau GaugeStrong Coupling Limit of Lattice Landau Gauge

• Modified Lattice Landau Gauge

π/L≪ p≪ ΛQCD →∞facilitate

However, this all happens for 1≪ a2p2

deviations at small momenta are not a finite-size effect!

solves Neuberger 0/0 problem of lattice BRST.

will allow to perform (Landau) gauge-fixed MC.

Dgluon ∼ (a2p2)2κ−1, Dghost ∼ (a2p2)−κ−1,

in particular, κgluon = κghost, and αS → αc ≈ 4 (SU(2)).

κ ≈ 0.6,

alternative gauge field definition.

observe infrared behaviour of functional methods at large momenta:

infrared exponents and the strong coupling limit in...

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