infrared exponents and the strong coupling limit in...
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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).
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 33
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].
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 44
• Landau Gauge QCD Propagators
(DSEs, lattice LG,...).
• Strong Coupling Limit in Lattice Landau Gauge.
• Modified Lattice Landau Gauge.
• Summary and Conclusions.
ContentsContents
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 55
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].
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 66
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 Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 77
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 88
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 99
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1010
Ghost PropagatorGhost Propagator
L = 3fm
L = 4.6fm
inf. Vol.
Sternbeck et al., Adelaide—Berlin—Moscow, unpublished (preliminary).
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1111
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].
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1212
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].
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1313
SUSU(2) (2) PropagatorsPropagatorsvolume dependencevolume dependence
gluon
ghost
A. Sternbeck, L. von Smekal, D. B. Leinweber and A. G. Williams,PoS (LATTICE 2007) 340, arXiv:0710.1982 [hep-lat].
fixed β, unrenormalised.
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1414
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)
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1515
ββ = 0 = 0 ―― Gluon PropagatorGluon Propagator
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):
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1616
transverse mass m2 ∝ 1/a2
ββ = 0 = 0 ―― Gluon PropagatorGluon Propagator
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1717
ββ = 0 = 0 ―― Ghost PropagatorGhost Propagator
Not a finite volume effect!
G( a2p2)
κ = 0.595
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1818
0.1
1
0.01 0.1 1 10
ββ = 0 = 0 ―― Ghost PropagatorGhost Propagator
G−1fit (x) = c + d xκ
564
324244
x = a2p2
G−1( a2p2)
κ = 0.595
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 1919
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12 14 16
ββ = 0 = 0 ―― Ghost PropagatorGhost Propagator
G−1fit (x) = c + d xκ
564324244
x = a2p2
G−1( a2p2)
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2020
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2121
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2222
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 :
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2323
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2424
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].
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2525
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2626
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2727
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2828
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 . . .
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 2929
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
Infrared Exponents and the Strong Coupling Limit , LvSInfrared Exponents and the Strong Coupling Limit , LvS 3030
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: