The Lambda parameter of three flavor QCD · 2016. 7. 20. · Alpha Collaboration (DESY Zeuthen)...

The Lambda parameter of three flavor QCD

M. Bruno, M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Schaefer,H. Simma, S. Sint, R. Sommer

LPHAACollaboration

Alpha Collaboration (DESY Zeuthen) Lambda parameter 18.07.2016 1 / 42

Motivation

QCD

LQCD[A, ψ, ψ] =1

2g20

tr{FµνFµν}+∑

i={u,d,s,...}

ψi [γµDµ[A] + mi ]ψi

Very few parameters: bare coupling g0, bare masses mu,md ,ms, . . .later: replaced by renormalized quantities g, mu, md , ms, . . .or RGI quantities Λ,Mu,Md ,Ms, . . .Predictive power

I Hadron masses (→ most of the universe’s visible matter)I Decay constantsI Hadron structureI High-energy processes, cross-sectionsI Explains confinement and asymptotic freedomI Predicts a QCD phase diagram→ quark-gluon plasma etc.I . . .

Massless QCD: classical action is scale invariantSpontaneous breaking→ massive Λ, mN , mP etc.


Schemes and scalesThere is no such thing as the QCD coupling

Take some dimensionless quantity Φ

Renormalize it, e.g.

Φs(µ) = Zs(µ)Φ

I s: renormalization schemeI µ: renormalization scale

Bare perturbation theory (regulator still in place)

Φs(µ) = φ(0)s (µ) + φ

(1)s (µ) g2

0 + . . .

A renormalized coupling can be defined by

g2φ(µ) =

Φs(µ)− φ(0)s (µ)

φ(1)s (µ)

= g20 + . . .


Renormalization group

From here on: only massless renormalization schemes consideredA renormalized coupling satisfies the RG equation

µ∂gφ∂µ

= βφ(gφ)

Renormalized perturbation theory for β

βφ(gφ)g→0= −g3

φ

[b(0) + b(1)g2

φ + b(2)φ g4

φ + . . .]

I The first two coefficients are scheme independent

b(0) =11− 2

3 Nf

(4π)2 , b(1) =102− 38

3 Nf

(4π)4

I In MS-scheme: βMS known up to b(4)

MSF b(3): [T. van Ritbergen, J. Vermaseren, and S. Larin (1997)]F b(3): [M. Czakon (2005)]F b(4): [P. A. Baikov, K. G. Chetyrkin, J. H. Kühn (2016)]


Lambda parameterIntegration of the RG equation introduces a dimensionful constant Λ

Λφ ≡ µ[b(0)g2

φ(µ)]−b(1)/(2b(0)2

)

e−1/(2b(0)g2φ(µ))

×exp

− gφ(µ)∫0

(1

βφ(x)+

1b(0)x3 −

b(1)

b(0)2x

)dx

Λ is independent of renormalization scale µΛ is almost scheme independent

I Two schemes φ and φ′I Both possess an asymptotic expansion, then also

g2φ(µ) = g2

φ′(µ) + c(1)g4φ′(µ) + . . .

I Λφ = Λφ′ exp[c(1)/(2b(0))

]holds exactly

⇒ Λ is a much better “fundamental parameter” than some coupling at somescale.


World data

Many lattice resultsVery differentstrategiesSystematical effectsare difficult to control

0.6 0.7 0.8 0.9

==

==

El-Khadra 92UKQCD 92Bali 92Luscher 93Alles 96ALPHA 98Boucaud 98ABoucaud 98BBecirevic 99BBoucaud 00ABoucaud 01ASoto 01QCDSF/UKQCD 05Boucaud 05Boucaud 08Brambilla 10Sternbeck 10FlowQCD 15

FLAG estimate for =

Boucaud 01BALPHA 04QCDSF/UKQCD 05JLQCD/TWQCD 08CETM 10FSternbeck 10ETM 11CALPHA 12Karbstein 14

FLAG estimate for =

HPQCD 05AHPQCD 08AHPQCD 08BMaltman 08PACS-CS 09HPQCD 10HPQCD 10JLQCD 10Bazavov 12Bazavov 14

FLAG estimate for =

ETM 11DETM 12CETM 13DHPQCD 14A

FLAG average for =

[FLAG Working Group (2016)]Alpha Collaboration (DESY Zeuthen) Lambda parameter 18.07.2016 6 / 42

World data

αS ≡g2

MS(MZ )

4π

PDG, Non-lattice:αNf =5

S = 0.1175(17)

Accuracy ∼ 1.5%

0.100 0.105 0.110 0.115 0.120 0.125

=,

==

El-Khadra 92Aoki 94Davies 94Wingate 95SESAM 99Boucaud 01BQCDSF/UKQCD 05

HPQCD 05AHPQCD 08AHPQCD 08BMaltman 08PACS-CS 09AHPQCD 10HPQCD 10JLQCD 10Bazavov 12Bazavov 14

ETM 11DETM 12CETM 13DHPQCD 14A

PDG nonlattice average

FLAG estimate

[FLAG Working Group (2016)]


Choosing a schemeFrom thousands of choices: how to choose a good scheme?(What does good mean?)

gφ needs to be defined non-perturbativelyAccurate at low energies

umax = g2φ(µ ∼ 200 MeV)

To make contact to hadronic physics (where the lattice shines)Accurate at high energies

umin = g2φ(µ ∼ 100 GeV)

Asymptotic freedom⇒ contact to PT. Needed to replace β → βPT in

gφ(µ)∫0

1βφ(x)

dx

Perturbative expansion should be known to at least 1 loop (better 2 loop)Alpha Collaboration (DESY Zeuthen) Lambda parameter 18.07.2016 8 / 42

Choosing a scheme

Control over finite volume effects L−1 � 200 MeVControl over scaling violations a−1 � 100 GeV

⇒We face a multi-scale problem

a� 1100 GeV

≤ 1µ≤ 1

200 MeV� L

⇒ La∼ 50000

Todays machines allow L/a < 100⇒ compromise?


Finite volume schemes

Finite-size-scaling: Identify 1µ ≡ L

Requires new simulations when µ is changedSeparates the scales. Now we only needa� 1

µ = L⇒ L

a ∼ 10Instead of continuous scale changes (β(g)), we can look atstep-scaling-function

σ(u) ≡ g2(µ

2

)∣∣∣u=g2(µ)


The Schrödinger FunctionalFinite volume needs boundary conditions. We use Schrödinger FunctionalGauge fields Uµ(x):

Uk (x)|x0=0 = exp(a Ck )

Uk (x)|x0=T = exp(a C′k )

Uµ(x + Lk) = Uµ(x)

We choose:

Ck =iL

diag(φ1, φ2, φ3), C′k =iL

diag(φ′1, φ′2, φ′3),

∑i

φi =∑

i

φ′i = 0

Matter fields ψ(x):

12

[1− γ0]ψ(x)|x0=0 = 0

12

[1 + γ0]ψ(x)|x0=T = 0

ψ(x + Lk) = eiθψ(x)


The gradient flow

Gradient flow ∼ (covariant) diffusion in “flow time” t[R. Narayanan and H. Neuberger (2006)]

∂tBµ(t , x) = DνGνµ(t , x), Bµ(0, x) = Aµ(x)

Dµ = ∂µ + [Bµ, ·]Gµν = ∂µBν − ∂νBµ + [Bµ,Bν ]

Correlators of B at t > 0 need no renormalization[M. Lüscher (2010)], [M. Lüscher and P. Weisz (2011)]

Most common use: definition of scales, e.g. t0

t20

4⟨Gaµν(t0, x)Ga

µν(t0, x)⟩

= 0.3


Gradient flow couplings

Finite-size-scheme couplings based on the GF[Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. Wong (2012)], [P. Fritzsch, A. Ramos (2013)]

g2GF = N−1(c)

t2

4

⟨Ga

ij (t , x)Gaij (t , x) δQ,0

⟩〈δQ,0〉

∣∣∣∣∣√8t=cL,x0=T/2

Our coupling:SF boundaries

I C = C′ = 0I θk = 1

2I L = T

Massless scheme mu = md = ms = 0c = 0.3, other values only for illustrationUse only “magnetic” components i , j = 1,2,3Evaluate the action density at x0 = T/2Project to trivial topological sector Q = 1

32π2

∫d4x εµνρσGa

µν(t , x)Gaρσ(t , x)


GF coupling on the latticeDiscretization:

ActionI Tree-level Symanzik improved gague action SLW : , ,

[M. Lüscher, P. Weisz (1985)]I O(a) improved Wilson fermions

Flow equationI Wilson-flow: use Wilson’s plaquette action SW :

a2 [∂tVµ(t , x)] V †µ(t , x) = −g20∂x,µSW [V ]

I Symanzik improved “Zeuthen-flow”[A. Ramos, S. Sint (2015)]

a2 [∂tVµ(t , x)] V †µ(t , x) = −g20

(1 +

a2

12∇∗µ∇µ

)∂x,µSLW [V ]

Action densityI Clover,

I Symanzik improved SLW , ,


Schrödinger Functional couplings

Family of boundary fields Ck = iL diag(φ1, φ2, φ3), C′k = i

L diag(φ′1, φ′2, φ′3),

parametrized by two angles: η and µ[M. Lüscher, R. Sommer, P. Weisz, U. Wolff (1993)]

φ1 = η − π

3

φ2 = η

(ν − 1

2

)φ3 = η

(−ν − 1

2

)+π

3

φ′1 = −φ1 −4π3

φ′2 = −φ3 +2π3

φ′3 = −φ2 +2π3

Define renormalized couplings as “susceptibility” of Γ to a change of theboundaries

g2ν = k

[∂Γ

∂η

∣∣∣∣η=0

]−1

Γ = − ln[∫

fields e−S[U,ψ,ψ]], k such that g2

ν = g20 + . . .


pros and cons

coupling NP def.accuracy

low µaccuracy

high µcutoffeffects PT

GF X X X big O(a2) norm onlySF X problematic X very small 2-loop

2-loop for g2SF with Wilson’s plaquette action only

relative precision ∆g2/g2

I ∼ constant for GF coupling→ O(1000) measurements necessaryI ∼ g2 for SF coupling→ O(100000) measurements when g large

Lattice artifactsI SF-coupling: L/a = 6, 8, 12 is enoughI GF-coupling: needs at least L/a = 8, 12, 16


Overall strategy

Large volume simulations: obtain 1/√

t0 in GeVLW-action, open boundaries in time

Compute√

t0Lhad

, Lhad defined by g2GF (Lhad) ≡ 11.31

Step scaling function of g2GF : given g2

GF (2L0) and g2GF (Lhad),

compute Lhad2L0

. L0 is an intermediate scaleLW-action, SF boundaries, no background field

Switch schemes: Find value of g2GF (2L0), where g2

SF (L0) = 2.012Step scaling function of g2

SF :g2

SF (L0)→ g2SF (L0/2)→ . . .→ g2

SF (Lperturbative)plaquette action, SF boundaries, background field A

Use PT to relate g2SF (Lperturbative) to Λ

(3)

MS

Λ(3)

MS=

1√t0︸︷︷︸

scale setting

×√

t0Lhad︸︷︷︸

connection to CLS

× Lhad

2L0︸︷︷︸GF running

× 2L0

L0︸︷︷︸change of schemes

× Λ(3)

MSL0︸︷︷︸

SF running


Algorithms

We use openQCD-1.2 and openQCD-1.4 for large volume simulations and amodified version of openQCD-1.0 for the SF simulations[M. Lüscher, S. Schaefer (2013)]

Very good solversI E/O preconditioningI SAP preconditioningI deflationI mixed precisionI optimized for intel and bluegene

Higher order integrators, multiple time scale integrationMass preconditioning à la HasenbuschRHMC for third quarkVery high degree of flexibilityaction→ product of pseudo-fermion actions


High energy running

Λ(3)

MS=

1√t0︸︷︷︸

scale setting

×√

t0Lhad︸︷︷︸

connection to CLS

× Lhad


× 2L0


× Λ(3)

MSL0︸︷︷︸

SF running

[M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Sint, R. Sommer (2016)]

Find the critical line where mu = md = ms = 0→ κc(β,L/a)

Fix a set of reference coupling valuesu ∈ {1.10891,1.18446,1.26569,1.3627,1.4808,1.6173,1.7943,2.012}Find on L/a ∈ {4,6,8,12} β values such that the reference couplings aremeasuredAt same bare parameters: double the lattice size, measure the coupling⇒ Lattice step-scaling function Σ(u,L/a)

Fit: reliable description of σ(u) = lima→0

Σ(u,a/L) in u ∈ (1.10891,2.012)

g2SF (L0) ≡ 2.012, use σ(u) to compute g2

SF (2−nL0), use PT to obtainL0Λ

(3)

MS


Critical line

Definition of the (bare) PCAC mass m1

m(x0) =(∂∗0 + ∂0)fA(x0)

4fP(x0)+ cA

a ∂∗0∂0fP(x0)

2fP(x0)

With the usual SF correlation functions

fJ(x0) = −a6

3

∑yz

⟨Ja(x) ζ(y)γ5

τa

2ζ(z)

⟩ζ′ζ′

J(t)

x0 =T

x0 =0

t

ζζ

J(t)

Different choices of x0 and the boundary fields⇒ different lattice artifacts.A good choice is[M.Lüscher, S.Sint, R.Sommer, P.Weisz, U.Wolff (1997)]

m1 = m(x0 = T/2), BF point A⇔ η = 0, θ = π/5, T = L


Critical line

plaquette action with clover fermionsI cSW non-perturbative [CP-PACS JLQCD, N. Yamada et al (2005)]I ct 2 loop [A.Bode, P.Weisz, U.Wolff (1990)]I ct 1 loop [S.Sint, P.Weisz (1997)]I cA 1 loop [S.Sint, P.Weisz (1997)]

Measure forβ ∈ {6.00,6.25,6.50,6.75,7.00,7.25,7.50,7.75,8.00,8.50,9.00} andL/a ∈ {4,6,8,10,12,16}Lm1 for various am0 around a guessed amc

Lm1 ∼ Z (g2,a/L)× [Lm0 − Lmc ]

Z (g2,a/L) = const, works well at high energiesWe fit Lm1 vs Lm0 according to

amc(g20 ,L/a) = am2−loop

c (g20 ,L/a) + cL/a

1 g60 + cL/a

2 g80 + cL/a

3 g100

Lm1 = c0(Lm0 − Lm2−loopc )− L

a

[cL/a

1 g60 + cL/a

2 g80 + cL/a

3 g100

]Precision goal: Lm1 < 0.005


Critical line

L m

1

L/a=6

β=6.00−0.02

0

0.02

L m

1

β=6.10−0.02

0

0.02

L m

1

β=6.25−0.02

0

0.02

L m

1

β=6.50−0.02

0

0.02

L m

1

β=6.75−0.02

0

0.02

L m

1

Lm0

β=7.00

Lmc−0.01 Lmc Lmc+0.01

−0.02

0

0.02

β=7.25

β=7.50

β=7.75

β=8.00

β=8.50

Lm0

β=9.00



Critical line

L m

1

L/a=8

β=6.00−0.02

0

0.02

L m

1

β=6.10−0.02

0

0.02

L m

1

β=6.25−0.02

0

0.02

L m

1

β=6.50−0.02

0

0.02

L m

1

β=6.75−0.02

0

0.02

L m

1

Lm0

β=7.00


−0.02

0

0.02

β=7.25

β=7.50

β=7.75

β=8.00

β=8.50

Lm0

β=9.00



Critical line

L m

1

L/a=12

β=6.00−0.02

0

0.02

L m

1

β=6.10−0.02

0

0.02

L m

1

β=6.25−0.02

0

0.02

L m

1

β=6.50−0.02

0

0.02

L m

1

β=6.75−0.02

0

0.02

L m

1

Lm0

β=7.00


−0.02

0

0.02

β=7.25

β=7.50

β=7.75

β=8.00

β=8.50

Lm0

β=9.00



Direct vs global fit

0 0.5 1 1.5−0.03

−0.02

−0.01

0

0.01

g0

6

a[m

c −

mc2−

loop]

L/a=4

L/a=6

L/a=8

L/a=10

L/a=12

L/a=16

0.2 0.4 0.6 0.8 1−5

0

5x 10

−4

am

cdirect −

am

cglo

b.fit


Continuum limit

Measure Σ(u,a/L)

Improvement: Σ = Σ1+δ1u+δ2u2 , δ = Σ−σ

σ = δ1(a/L)u + δ2(a/L)u2 + . . .[B.Gehrmann, J.Rolf, S.Kurth, U.Wolff (2001)]

Error propagation ∆u → ∆Σ

Error propagation: systematical uncertainty in ct , ct → ∆Σ

0 0.005 0.01 0.015 0.02 0.025 0.031.08

1.1

1.12

1.14

1.16

1.18

1.2

1.22

1.24

(a/L)2

σ(u)

/u


Step-scaling

We tried many (global) fits, e.g.

Σ(u,a/L) = u + u2s0 + u3s1 + u4s2 + u5s3︸︷︷︸σ(u)

+(a

L

)2 (ρ1u3 + ρ2u4 + . . .

)︸︷︷︸

lattice artifacts

Construct sequence of couplings separated by factors of 2 in the scale:g2

SF (L0) = u0, uk = σ(uk+1)

Alternatively: instead of s3, fit effective b(3)eff coefficient of β-function

Final result

L0Λ(3)SF = 0.0303(8) ↔ L0Λ

(3)

MS= 0.0791(21)


Accuracy of perturbation theory

Ln = 2−nL0

Use 3-loop PT at α(1/Ln)⇒ Residual error O(α2)1

g2ν

= 1g2

SF− νv

g2ν withI ν = −0.5, blueI ν = 0, magenta

two fitsI ν = 0.3, red

0 0.005 0.01 0.015 0.02 0.025 0.03

0.029

0.030

0.031

0.032

0.033

α2(1/Ln)

L0Λ


Accuracy of perturbation theoryApplication of PT to the ν = −0.5 coupling at α ∼ 0.2⇒ wrong result.Much better visible in

ω(u) = v |g2SF =u,m=0 = v1 + v2u︸︷︷︸

known in PT

+O(u2)

No step-scaling: can use data set with L/a = 6,8,10,12,16,24fit1: ω(u) = v1 + v2u + d1u2 + d2u3 + d3u4

fit2: ω(u) = v1 + d1u + d2u2 + d3u3 + d4u4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180.11

0.12

0.13

0.14

α(1/Ln)

ω

d1, . . . , d4d1, . . . , d3two-loop PT


Low energy running

Λ(3)

MS=

1√t0︸︷︷︸

scale setting

×√

t0Lhad︸︷︷︸

connection to CLS

× Lhad


× 2L0


× Λ(3)

MSL0︸︷︷︸

SF running


Find the critical line where mu = md = ms = 0, this time: LW action, nobackground field→ κc(β,L/a)

Construct σ(u) as before, now with g2GF

I u ∈ {2.1257, 2.39, 2.7359, 3.2029, 3.8643, 4.4901, 5.3013, 5.8673, 6.5489}I L/a ∈ {8, 12, 16} and doubled lattices

Relate σ(u) to β(u) and use this non-perturbative β function to computeLhad2L0


Critical line II

La ∈ {4,6,8,10,12,16}β ∈ {3.3,3.4,3.5,3.6,3.8,4.0,4.3,4.6,5.0,6.0,8.0,16.0}Critical line well described by amc(g0,a/L) =

∑6k=0 µk g2k

0∑6i=0 ζi g2i

0

LW action with clover fermionsI cSW : non-perturbative

[J.Bulava, S.Schaefer (2013)]I ct = ct = 1 for mc

later: 1-loop [P.Vilaseca (2015)]I cA non-perturbative

[J. Bulava, M.Della Morte, J.Heitger,

C.Wittemeier (2016)]

0 0.5 1 1.5 20.124

0.126

0.128

0.13

0.132

0.134

0.136

0.138

0.14

g0

2

κc

1−loop PT

L/a=4

L/a=6

L/a=8

L/a=10

L/a=12

L/a=16

0 1 2

−1

0

1

x 10−5

κcd

ire

ct −

κcg

lob

.fit


Gradient flow step-scaling-function

Fit 1:1

Σ(vi ,aL ) = 1

σi+ ri ×

( aL

)2

Fit 2:Σ(vi ,

aL ) = σi + ri ×

( aL

)2

Various global fitsσ(umax) ∼ 11.31⇒ g2

GF (Lhad) ≡ 11.311.2

1.4

1.6

1.8

2

2.2

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Σ(u,a

/L

)/u

(a/L)2

Fit ΣFit 1/Σ

Continuum (fit Σ)Continuum (fit 1/Σ)

Data

−0.1

−0.095

−0.09

−0.085

−0.08

−0.075

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

1/σ(u)−

1/u

u

1-loopGlobal fit

Continuum


Step scaling to usual scaling

Lhad/L0 is not a power of 2, not even integer⇒ better to use β instead of σRelation between σ and β

log(2) = −

√σ(u)∫√

u

dxβ(x)

We make the ansatz β(x) = − x3

p0+p1x2+p2x4+...

(just the p0 term would be “effective 1-loop”)Fit:

log(2) + (ρ0 + ρ1u)(a

L

)2= +

√Σ(u,a/L)∫√

u

p0 + p1x2 + p2x4 + . . .

x3 dx

Weights for least-squares: contain statistical and systematical (O(a4))errors of Σ


Slow running

Result of a fit: p0 = 16.26(69), p1 = 0.12(26), p2 = 0.0038(211)

−0.08

−0.075

−0.07

−0.065

−0.06

−0.055

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

β(g)/g3

α

1-loop2-loop

nσ = nρ = 2nσ = nρ = 3

Final result

Lhad

L0= 21.86(42)


Matching GF and SF schemes

Λ(3)

MS=

1√t0︸︷︷︸

scale setting

×√

t0Lhad︸︷︷︸

connection to CLS

× Lhad


× 2L0


× Λ(3)

MSL0︸︷︷︸

SF running


We know: g2SF (L0) = 2.012

Here we compute: g2GF (2L0)

I Combine scheme-switch with one step-scaling step⇒ Couplings are used where they work well

I Step scaling of the GF coupling then allows to constructg2

GF (4L0), g2GF (8L0), . . ., g2

GF (Lhad)

Φ(u,a/L) = g2GF (2L)

∣∣∣g2

SF (L)=u,m=0

φ(u) = lima→0

Φ(u,a/L)

We already have a set L/a = 6,8,12 with g2SF = 2.012

I We added one more resolution L/a = 16I Same bare parameters, same action, doubled lattice, no BF: compute g2

GF


Matching GF and SF schemes

2.66

2.67

2.68

2.69

2.7

2.71

2.72

2.73

2.74

2.75

2.76

−0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

g2 GF

(2L

0)

(a/L)2

Zeuthen flowWilson flowContinuum

Final result

g2GF (2L0) = 2.6723(64)


Scale setting

Λ(3)

MS=

1√t0︸︷︷︸

scale setting

×√

t0Lhad︸︷︷︸

connection to CLS

× Lhad


× 2L0


× Λ(3)

MSL0︸︷︷︸

SF running

[M.Bruno, T.Korzec, S.Schaefer (2016)]

We use CLS ensembles[M.Bruno et al (2014)]+newer runs

LW action with clover fermions, open boundaries, Nf = 2 + 1Chiral trajectories with bare masses mu + md + ms = constantβ = 3.4, a ∼ 0.086 fmmπ/MeV ∼ 420, 350, 280, 220β = 3.46 a ∼ 0.077 fmmπ/MeV ∼ 420β = 3.55 a ∼ 0.064 fmmπ/MeV ∼ 420, 340, 280, 200β = 3.7a ∼ 0.050 fmmπ/MeV ∼ 420, 260


Observables

On these ensembles we measure (in lattice units)Pion and Kaon masses amπ, amK

Pion and Kaon decay constants afπ, afK , also afπK = 23 (afK + 1

2 fπ)

Need ZA, cA, bA, bA, amPCAC

Scale t0/a2

(Wilson flow + clover)derivatives of all these with respect to the bare quark masses

Form dimensionless ratios

φ2 = 8t0m2π, φ4 = 8t0(m2

K +12

m2π),

√t0fπK

CLS ensembles were tuned such that φsym4 ∼ 1.15

The measured derivatives allow us toI Change this valueI Change the whole chiral trajectory, e.g. φ4 = const instead of

mu + md + ms = const


Mass derivatives

A “derived observable” f (o1, . . . ,oN ,m), oi = 〈Oi〉Has the mass derivative

dfdm

=∂f∂oi

⟨∂Oi

∂m

⟩− ∂f∂oi

(⟨Oi∂S∂m

⟩− oi

⟨∂S∂m

⟩)+

∂f∂m

⇒ Small shifts f (m + ∆m) ≈ f (m) + ∆mdfdm

0.000 0.001 0.002 0.003 0.004 0.005a∆mq

2.8

3.0

3.2

3.4

3.6

3.8

4.0

t 0/a

2

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010a∆mq

0.052

0.054

0.056

0.058

0.060

0.062

0.064

af π


Scale setting strategy

Create a function that, given a t0 in fm2, computes fπK at the physicalpoint: F (tguess

0 ) = f resultπK

I Use physical mπ, mK to compute φguess2 , φguess

4I Shift all ensembles to φguess

4I Continuum + chiral extrapolation of shifted

√t0fπK to φguess

2 and a = 0

I return

√t0fπK

∣∣a=0,φ2=φ

guess2√

tguess0

Find numerically tphys0 by demanding

F (tphys0 )

!= f phys

πK

⇒We obtain tphys0 , φphys

2 , φphys4

Chiral extrapolation of t0/a2 to φphys2

⇒ Lattice spacing a =

√tphys0

t0/a2∣∣φ2=φ

phys2


Scale setting results

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8φ2

0.100

0.102

0.104

0.106

0.108

0.110

0.112

0.114

0.116

√t 0f π

K

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

a2/t0

0.100

0.102

0.104

0.106

0.108

0.110

0.112

0.114

0.116

(Almost) final result

ChPT ansatz: φphys4 = 1.11(2),

√8t0 = 0.412(4) fm

quadratic: φphys4 = 1.12(2),

√8t0 = 0.414(3) fm


Connecting to CLS runs

Λ(3)

MS=

1√t0︸︷︷︸

scale setting

×√

t0Lhad︸︷︷︸

connection to CLS

× Lhad


× 2L0


× Λ(3)

MSL0︸︷︷︸

SF running

Preliminary!

From scale-setting at φphys4 :

β tsym0 /a2

3.40 2.8619(96)3.46 3.662(16)3.55 5.166(18)3.70 8.596(28)3.85 xxx(xx)

Compute β values for same lattice spacing, when masses are zeroβ = β

1+0.216 amq/β

Use GF running to compute the (non-integer) Lhad/a values at β that leadto g2

GF = 11.31Continuum extrapolate: Lhad/a√

tsyma /a2√t0

Lhad≈ 0.1449(12)


Everything together

Preliminary result

Λ(3)

MS= 338(18) MeV

α(mZ ) = 0.1184(14)

Errors include a “safety margin”→ will still shrink

Error budget for Λ:High energy running: 2.6%scheme switching: 0.8%Low energy running: 1.74%connection to CLS: 0.81%scale setting: 0.87%


Conclusions and outlook

Conclusions

Connection to perturbation theory is a delicate issueworks at α ∼ 0.1 but maybe not at α ∼ 0.2With increased precision goals, systematical errors become more andmore difficult to controlSwitching to a gradient-flow scheme pays off: the usually expensivelow-energy running contributes a small error

Outlook

Very soon: final resultΛ(3) → Λ(4) → Λ(5)

perturbatively


The Lambda parameter of three flavor QCD · 2016. 7. 20. · Alpha Collaboration (DESY Zeuthen)...

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