Strong coupling and strange quark mass from lattice ... - IFSC - USP

Strong coupling and strange quark mass from
lattice QCD
Rainer Sommer
IFSC – USP, Sao Carlos, April 3, 2013
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

The talk is based on
◮ old work [
LPHA ACollaboration ]
◮ N f = 2 running
[Nucl.Phys. B713 (2005) 378, Michele Della Morte, Roberto Frezzotti, Jochen Heitger, Juri Rolf, RS, Ulli
Wolff ]
[Nucl.Phys. B729 (2005) 117, Michele Della Morte, Roland Hoffmann, Francesco Knechtli, Juri Rolf, RS,
Ines Wetzorke, Ulli Wolff ]
◮ Scale setting
[Strange quark mass and the Lambda parameter in two flavor QCD,
Fritzsch, Leder, Knechtli, Marinkovic, Schaefer, S, Virotta, 2012 ]
◮ Recent development
[Fritzsch & Ramos, arXiv:1301.4388 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Fascinating strong interactions
◮ jets at large energies
◮ hadrons at small energies
◮ nuclei at even smaller energies
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Fascinating strong interactions
THEORY
◮ jets at large energies
Q C D
◮ hadrons at small energies
Q C D
◮ nuclei at even smaller energies
Q C D
Believed to be described by a most beautiful theory: Q C D
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

QCD
L QCD = − 1
2g 0
2 tr{F µνF µν } +
◮ N f + 1 = 7 (bare) parameters
N f
∑
ψ f {D + m 0f }ψ f
f =1
◮ at low energy essentially 4 parameters
◮ in the following: u, d quarks mass-degenerate
s-quark quenched
3 parameters
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

. .
Strong force
A way to define the strong force is
x
y
F (r) = d dr V (r) , r = |x − y| Q _ Q
Perturbation theory (Feynman graphs)
F (r) = 4 1
3 4πr 2 g 2 + O(g 4 )
coulombic
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

. .
Strong force
A way to define the strong force is
x
y
F (r) = d dr V (r) , r = |x − y| Q _ Q
Perturbation theory (Feynman graphs)
F (r) = 4 1
3 4πr 2 g 2 + O(g 4 )
↓
r 2 0 F (r)
coulombic
r/r 0
Rainer Sommer
[r 0 ≈ 0.5 fm]
Strong coupling and strange quark mass from lattice QCD

Strong coupling
Perturbation theory (Feynman graphs)
F (r) = 4 1
3 4πr 2 g 2 + O(g 4 )
coulombic
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

1.2
1
0.8
0.6
0.4
0.2
r/r0
Strong coupling
Perturbation theory (Feynman graphs)
F (r) = 4 1
3 4πr 2 g 2 + O(g 4 )
A way to define the strong coupling is
coulombic
α qq (µ) = ḡ 2 qq(r)
4π
It is r-dependent: “it runs”
≡ 3 4 r 2 F (r) , µ ≡ 1/r
αqq(r) = r 2 F (r)/CF
β = 5.3, κ = 0.13638
β = 5.5, κ = 0.13671
3-loop Nf = 2
4-loop Nf = 2
Nf = 0, continuum
3-loop Nf = 0
4-loop Nf = 0
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Running and Renormalization Group Invariants
RGE:
µ ∂ḡ
∂µ
= β(ḡ) ḡ(µ) 2 = 4πα(µ) .
ḡ→0
β(ḡ) ∼ −ḡ 3 { b 0 + b 1 ḡ 2 + b 2 ḡ 4 + . . . }
b 0 =
1
(4π) 2 (
11 −
2
3 N f
)
Λ-parameter (ḡ ≡ ḡ(µ)) = Renormalization Group Invariant
= intrinsic scale of QCD
{ ∫ ḡ
}
Λ = µ (b 0 ḡ 2 ) −b1/2b2 0 e
−1/2b 0ḡ 2 1
exp − dg[
β(g) + 1
b 0g
− b1
3 b0 2 0
g ]
◮ there is a similar formula relating m i (µ) and M i : RGI quark masses
◮ Λ, M i have a trivial dependence on the “scheme”
scheme ↔ definition of ḡ, m
◮ they are the fundamental parameters of QCD
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Uses of α (or Λ)
◮ Λ is a fundamental constant of Nature, just like
α em = 1/137.0359997 for atomic physics
◮ α(µ) at high scale is needed for the search of new particles at the
LHC.
E.g. the Higgs.
◮ It is an important constraint for possible theories at higher energy
scales.
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

“Experimental” determinations of α MS
0.5
α s (Q)
0.4
July 2009
Deep Inelastic Scattering
e + e – Annihilation
Heavy Quarkonia
0.3
0.2
0.1
QCD α s(Μ Z) = 0.1184 ± 0.0007
1 10 100
Q [GeV]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

“Experimental” determinations of α MS
0.5
α s (Q)
0.4
Deep Inelastic Scattering
e + e – Annihilation
Heavy Quarkonia
July 2009 2010
0.3
0.2
0.1
QCD α s(Μ Z) = 0.1184 ± 0.0007
1 10 100
Q [GeV]
α s(M Z )
There is a considerable spread. Errors seem often too agressive.
Theory uncertainties are usually dominating.
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Determination of Λ from lattice QCD
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Lattice determination of α qq
αqq(r) = r 2 F (r)/CF
1.2
1
0.8
0.6
0.4
β = 5.3, κ = 0.13638
β = 5.5, κ = 0.13671
3-loop Nf = 2
4-loop Nf = 2
Nf = 0, continuum
3-loop Nf = 0
4-loop Nf = 0
graph from [Leder & Knechtli, 2011 ]
N f = 0, continuum limit
[Necco & S., 2001 ]
N f = 2, small lattice spacing
[Leder & Knechtli, 2011 ]
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
r/r0
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Lambda parameter from α qq
1
0.95
0.9
2−loop
3−loop
4−loop
ALPHA
N f
= 0
1
0.95
0.9
2−loop
3−loop
4−loop
ALPHA
N f
= 2, O7
0.85
0.85
r 0
Λ MSbar
0.8
0.75
0.7
r 0
Λ MSbar
0.8
0.75
0.7
0.65
0.65
0.6
0.6
0.55
0.55
0.5
0 0.05 0.1 0.15 0.2 0.25
α qq
0.3 0.35 0.4
0.5
0 0.1 0.2 0.3 0.4
α qq
0.5 0.6 0.7
A realistic estimate of the uncertainty is impossible.
(Other members of the group come to a different conclusion
[Jansen, Karbstein, Nagy, Wagner, 2011 ])
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

The basic problem
1
L ≫
0.2GeV ≫ 1 µ ∼ 1
10GeV
≫ a
↑ ↑ ↑
box size confinement scale, m π spacing
⇓
L/a ≫ 50
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

The basic problem and its solution
1
L ≫
0.2GeV ≫ 1 µ ∼ 1
10GeV
≫ a
↑ ↑ ↑
box size confinement scale, m π spacing
⇓
L/a ≫ 50
LPHA ACollaboration
Solution: L = 1/µ
−→ left with
L/a ≫ 1
[Lüscher, Weisz, Wolff ]
Finite size effect as a physical observable; finite size scaling!
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

The step scaling function
It is a discrete β function:
σ(ḡ 2 (L)) = ḡ 2 (2L)
determines the
non-perturbative running:
u 0 = ḡ 2 (L max )
↓
σ(u k+1 ) = u k
↓
u k = ḡ 2 (2 −k L max )
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

The step scaling function: σ(u) = ḡ 2 (2L) with u = ḡ 2 (L)
On the lattice:
additional dependence on the resolution
a/L
g 0 fixed, L/a fixed:
ḡ 2 (L) = u, ḡ 2 (2L) = u ′ ,
Σ(u, a/L) = u ′
continuum limit:
Σ(u, a/L) = σ(u) + O(a/L)
everywhere: m = 0 (PCAC mass defined in (L/a) 4 lattice)
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Continuum limit (N f = 4)
Σ (2) (u, a/L)
3.5
3
2.5
2
1.5
◮ Constant fit:
Σ (2) (u, a/L) = σ(u)
for L/a = 6, 8
◮ Global fit:
Σ (2) (u, a/L) = σ(u)+ρ u 4 (a/L) 2
for L/a = 6, 8
→ ρ = 0.007(85)
1
0 0.02 0.04 0.06 0.08
(a/L) 2
◮ L/a = 8 data:
σ(u) = Σ (2) (u, 1/8)
LPHA
[
ACollaboration (S., Tekin & Wolff, 2010); update: M. Marinkovic, 2013 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Non-perturbative running of α
LPHA
[
ACollaboration , 2005 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Non-perturbative running of α
LPHA
LPHA
[
ACollaboration , 2005 ] [
ACollaboration , 2001 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Non-perturbative running of α
T
T
time
time
0
space
0
space
LPHA
LPHA
[
ACollaboration , 2005 ] [
ACollaboration , 2001 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

time
T
0
space
Non-perturbative running of α
T
T
time
time
0
space
0
space
LPHA
LPHA
[
ACollaboration , 2005 ] [
ACollaboration , 2001 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

time
T
0
space
T
time
0
space
T
time
0
space
T
time
0
space
T
time
0
space
time
T
0
space
Non-perturbative running of α
T
T
time
time
0
space
0
space
LPHA
LPHA
[
ACollaboration , 2005 ] [
ACollaboration , 2001 ]
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

The master formula of the
LPHA ACollaboration
strategy
Λ (2)
MS
F K
=
1
F K L max
× L max
L k
× L k Λ (2)
SF × Λ(2) MS
Λ (2)
SF
Γ(K → µν µ )
❵ ♣ ❵ ♣ ❵ ♣ ❵
♣ ❵ ♣ ❵ ♣ ❵ ♣
aF ❵ ♣ ❵ ♣ ❵ ♣ ❵
♣ ❵ K L
♣ ❵ ♣ max /a
❵ ♣
❵ ♣ ❵ ♣ ❵ ♣ ❵
♣ ❵ ♣ ❵ ♣ ❵ ♣
❵ ♣ ❵ ♣ ❵ ḡ 2 ♣ (L❵
max )
❵
❵ ♣ ❵
❵ ♣ ❵
❵ ♣ ❵
non-perturbative ♣ ❵
♣
❵
♣
❵
♣
❵
❵ ♣
♣ ♣ ❵ ♣
♣ ❵ ♣
SSF’s
♣
❵❵❵❵
♣♣♣♣
✲
massless N f = 2 theory ḡ 2 (L k )
3-lp ✲
Λ (2)
MS
✻1-lp (exact)
Λ (2)
SF
µ = 0
µ
m b
M Z
✲
∞
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Setting the scale
Λ (2)
MS
F K
=
1
F K L max
× L max
L k
× L k Λ (2)
MS
◮ F K has been missing
◮
F K m K = 〈K(p = 0)|¯sγ 0 γ 5 u|0〉
◮ needs K-meson → “large” volume: L > 2 fm , 4/m π
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Setting the scale
Λ (2)
MS
F K
=
1
F K L max
× L max
L k
× L k Λ (2)
MS
◮ F K has been missing
◮
F K m K = 〈K(p = 0)|¯sγ 0 γ 5 u|0〉
◮ needs K-meson → “large” volume: L > 2 fm , 4/m π
◮ issues
◮ extracting ground state “plateau”
◮ autocorrelations (sufficient statistics)
and error analysis
◮ extrapolation to physical quark masses
◮ renormalization . . .
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Plateaux 64 3 128 lattice m π = 270 MeV
◮ m π
0.1
0.095
0.09
0.085
0.08
aMeff
0.075
0.07
0.065
0.06
0.055
◮ PCAC
mass
quark
0.05
2.3 x 10−3
10 20 x min
0 30 40 50 60
x0
2.2
aMpcac
2.1
2
10 20 30 40 50 60 70 80 90 100 110
t
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Error analysis, F K , 64 3 128 lattice m π = 270 MeV
◮ Done as proposed in
Critical slowing down and error analysis in lattice QCD simulations
[Schaefer, S, Virotta, 2011 ]
◮ Autocorrelation
function
0.25
0.2
ρfK(t)
0.15
0.1
0.05
0
−0.05
0 W u W l 50 100 150 200
t
◮ The tail contributes about 60% of the error.
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Extrapolation to physical light quark masses
Strategies
1) R = m2 K
F 2 K
= constant
2) M s = constant
Mstrange
M strange = const
R = const
M Phys
M light
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Extrapolation to physical light quark masses
0.08
0.075
0.07
β = 5.2
β = 5.3
β = 5.5
0.065
afK
0.06
0.055
0.05
0.045
0.04
0 y π
0.05 0.1 0.15
y 1
◮ systematic expansion including
M, M log(M), a 2 , not M × a 2 [there are some checks]
◮ agreement also with simple linear extrpolation (no M log(M))
y 1 =
m2 π
8π 2 F 2 K
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

F K L max
combine L max /a = L 1 /a with aF K
0.35
0.34
0.33
L1fK
0.32
0.31
0.3
0.29
strategy 1
strategy 2
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
(af K ) 2
x 10 −3
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Similar for the RGI mass of the strange quark
(Ms + ˆM)/fK
0.7
0.65
0.6
Ms/fK
0.7 strategy1
strategy2
linear
0.65
0.6
0.55
0.55
0 0.05 0.1 0.15
y 1
0 1 2 3 4
(af K ) 2 x 10 −3
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Results for the strange quark mass (no complete review of results)
N f m s (2GeV) Experiment Theory
0 97(4) m K + scale LGT,
LPHA ACollaboration
2 103(4) m K + scale LGT,
LPHA ACollaboration
3 96(3) m K + scale LGT, BMW
◮ Firm results
◮ Before lattice gauge theory, (still in the 90’s ) values between
100MeV and 200MeV were given
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

N f dependence of Λ MS and comparison to phenomenology
Λ MS [MeV]
N f : 0 2 3 4 5
Experiment Theory
M K , K → l2, l3
LPHA
SF [
ACollaboration ] 238(19) 310(20)
M K , M ρ SF [PACS-CS ] 362(23)(25) 239(10)
(6)(-22)
DIS, HERA PT, PDF-fits [ABM11 ] 234(14) 160(11)
DIS, HERA PT, PDF-fits [MSTW09 ] 285(23) 198(16)
“world av. ”[2011 ] PT 212(12)
e + e − → had (LEP) 4-loop PT 275(57)
◮ Non-trivial, non-perturbative N f -dependence.
◮ Small errors are cited, but overall consistency is not that great.
◮ More precision and rigor (PT only at high energy) will be very
useful.
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Towards even better precision: Gradient Flow and SF
◮ Gradient flow [Lüscher, 2010; Lüscher & Weisz, 2011 ]
new observables
◮ UV finite (proven to all orders of PT)
◮ excellent numerical precision
◮ renormalized coupling in finite volume with pbc [BMW, 2012 ]
◮ Flow in finite volume, SF [Fritzsch & Ramos, arXiv:1301.4388 ]
◮ lowest order PT to define a new coupling
◮ numerical investigation shows excellent precision
◮ General idea
x = (x 0, x) , t = flow time
∫
A µ(x) = quantum gauge fields : Z = D[A µ(x)] ...
B µ(x, t) = smoothed gauge fields ,
dB µ(x,t)
dt
= D νG νµ(x, t) + gauge fixing
B µ(x, 0) = A µ(x)
∼ − δS YM [B]
δB µ
correlation functions of B-fields at arbitrary points are finite
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow
dB µ(x,t)
dt
≡ Ḃ µ (x, t) = D ν G νµ (x, t) + D µ ∂ ν B ν (x, t)
} {{ } } {{ }
↑
↑
in PT: A µ (x) = g 0 Ā µ (x)
∼ − δS YM[B]
δB µ
B µ (x, t) = B µ,1 (x, t)g 0 + B µ,2 (x, t)g 2 0 + . . .
→ Ḃ µ,1 (x, t) = ∂ ν ∂ ν B µ,1 (x, t)
(∗)
eliminate by a t-dependent gauge trafo
G νµ = [∂ ν B µ,1 − ∂ µ B ν,1 ] g 0 + O(g 2 0 ) , D ν = ∂ ν + O(g 0 )
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow
◮ in PT: A µ(x) = g 0 Ā µ(x)
B µ(x, t) = B µ,1 (x, t)g 0 + B µ,2 (x, t)g0 2 + . . .
G νµ = [∂ νB µ,1 − ∂ µB ν,1 ] g 0 + O(g0 2 ) , Dν = ∂ν + O(g 0)
→ Ḃ µ,1 (x, t) = ∂ ν∂ νB µ,1 (x, t)
◮ heat equation
B µ,1 (x, t) =
∫
d D p e ipx b µ(p, t)
B µ,1 (x, t) =
ḃ µ = −p 2 b µ → b µ(p, t) = b µ(p, 0)e −p2 t
∫
d D y K t(x − y) Āµ(y) , Kt(z) = (4πt)−D/2 e −z2 /(4t)
◮ smoothing over a radius of √ 8t
◮ gaussian damping of large momenta
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow
◮ in PT: A µ(x) = g 0 Ā µ(x)
B µ(x, t) = B µ,1 (x, t)g 0 + B µ,2 (x, t)g0 2 + . . .
G νµ = [∂ νB µ,1 − ∂ µB ν,1 ] g 0 + O(g0 2 ) , Dν = ∂ν + O(g 0)
→ Ḃ µ,1 (x, t) = ∂ ν∂ νB µ,1 (x, t)
◮ heat equation
B µ,1 (x, t) =
∫
d D p e ipx b µ(p, t)
B µ,1 (x, t) =
ḃ µ = −p 2 b µ → b µ(p, t) = b µ(p, 0)e −p2 t
∫
d D y K t(x − y) Āµ(y) , Kt(z) = (4πt)−D/2 e −z2 /(4t)
◮ smoothing over a radius of √ 8t
◮ gaussian damping of large momenta
◮ all correlation functions of B µ are finite (t > 0) [Lüscher & Weisz, 2011 ]
in particular 〈E(t)〉 , E(t) = − 1 2 trGµνGµν
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow
For 〈E〉 , E = − 1 2 trG µνG µν
〈E〉 = E 0 g0 2 + E 0 g0 4 + . . .
E 0 = 〈tr[∂ µ B ν,1 ∂ µ B ν,1 − ∂ µ B ν,1 ∂ ν B µ,1 ]〉
∫
∼ e −p22t [p 2 δ µν − p µ p ν ] D µν (p) finite (also with cutoff reg’n)!
p
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow and SF-coupling
use the flow in finite volume (SF): T × L 3 world with Dirichlet BC in time, T = L
define
〈E(t)〉 ≡ − 1 2 〈trG µνG µν (x, t)〉 x0=T /2 = N t
g 2 MS(µ) (1 + c 2 1 g 2 MS + . . .)
g 2 GF(L) ≡ N −1 t 2 ∣
〈E(t)〉
∣
t=c2 L 2 /8
This is a family of schemes characterized by c (dimensionless)
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow and SF-coupling
use the flow in finite volume (SF): T × L 3 world with Dirichlet BC in time, T = L
define
〈E(t)〉 ≡ − 1 2 〈trG µνG µν (x, t)〉 x0=T /2 = N t
g 2 MS(µ) (1 + c 2 1 g 2 MS + . . .)
g 2 GF(L) ≡ N −1 t 2 ∣
〈E(t)〉
∣
t=c2 L 2 /8
This is a family of schemes characterized by c (dimensionless)
N (c) = c4 (N 2 −1)
128
∑
n,n 0
e −c2 π 2 (n 2 + 1 4 n2 0 )
× 2n2 s 2 n 0
(T /2)+(n 2 + 3 4 n2 0 )c2 n 0
(T /2)
n 2 + 1 4 n2 0
◮ the lattice version is known (and needed)
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow and SF-coupling
statistical precision: variance
relative variance = 〈E 2 〉 − 〈E〉 2
〈E〉 2
should be finite as a → 0, L/a → ∞
Numerically, Fritzsch & Ramos:
10 −1 0.02 0.05 0.1 0.2 0.4 0.6
rel. variance of E
10 −2
10 −3
10 −4
L/a = 6
L/a = 8
L/a = 10
L/a = 12
L/a = 16
10 −5
Rainer Sommer
c
Strong coupling and strange quark mass from lattice QCD

Gradient Flow and SF-coupling
statistical precision
autocorrelations scale as expected: τ int ∝ a −2
τint/τmeas
L/a = 6
L/a = 8
L/a = 10
1.5
L/a = 12
L/a = 16
1
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6
c
(a/L) 2 · τint/10 MDU
0.025
0.02
0.015
0.01
0.005
0
c = 0.3
c = 0.4
c = 0.5
5 10 15 20 25 30 35 40
L/a
Statistical precision is good and theoretically understood.
There will be no surprises on the way to the continuum limit.
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Gradient Flow and SF-coupling
systematic precision
keeping old SF-coupling ḡ SF(L) fixed (defines L), compute
[ 2
Ω(u; c, a/L) = ˆN −1 (c, a/L) · t 2 SF =u, m=0
〈E(t, T /2)〉]ḡ
t=c 2 L 2 /8
8
˜Ω(u; c, a/L)
7
ω(u; c)
7.5
7
Ω(u; c, a/L)
c = 0.5
6
5
6.5
6
c = 0.4
0.3 0.35 0.4 0.45 0.5
c
5.5
5
c = 0.3
0.2
0.1
R(u; 0.3, a/L)
R(u; 0.4, a/L)
R(u; 0.5, a/L)
4.5
4
0 0.005 0.01 0.015 0.02 0.025
(a/L) 2
0
0 0.005 0.01 0.015 0.02 0.025
(a/L) 2
◮ small cutoff effects −→ ready for applications −→ . . .
→ precise Λ-parameter
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Summary
The fundamental parameters of QCD
Λ, M i
◮ refer to the asymptotic high energy behavior of QCD
and therefore Nature
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Summary
The fundamental parameters of QCD
Λ, M i
◮ refer to the asymptotic high energy behavior of QCD
and therefore Nature
◮ nevertheless they can be connected non-perturbatively
through lattice simulations to hadron masses (and properties)
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Summary
The fundamental parameters of QCD
Λ, M i
◮ refer to the asymptotic high energy behavior of QCD
and therefore Nature
◮ nevertheless they can be connected non-perturbatively
through lattice simulations to hadron masses (and properties)
◮ firm and precise results can be obtained
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD

Summary
The fundamental parameters of QCD
Λ, M i
◮ refer to the asymptotic high energy behavior of QCD
and therefore Nature
◮ nevertheless they can be connected non-perturbatively
through lattice simulations to hadron masses (and properties)
◮ firm and precise results can be obtained
◮ the precision will be improved even further in the near future
Rainer Sommer
Strong coupling and strange quark mass from lattice QCD