The $\Lambda$-parameter in 3-flavour QCD and $\alpha_s(m_Z)$ by the ALPHA collaboration

0.53 MB·

by  M. Bruno

The Λ-parameter in 3-flavour QCD and α (m ) by the
s Z
ALPHA collaboration
M. Brunoa, M. Dalla Bridab,c, P. Fritzschd,e, T. Korzecf, A. Ramosd∗, S. Schaeferc,
H. Simmac, S. Sintg∗ and R. Sommerc,h∗
aPhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA
bDipartimentodiFisica,UniversitàdiMilano-Bicocca&INFN,sezionediMilano-Bicocca,
PiazzadellaScienza3,I-20126Milano,Italy
7 cNIC,DESY,Platanenallee6,D-15738Zeuthen,Germany
1
dPH-TH,CERN,CH-1211Geneva,Switzerland
0
2 eInstitutodeFísicaTeóricaUAM/CSIC,UniversidadAutónomadeMadrid,
C/NicolásCabrera13-15,Cantoblanco,Madrid28049,Spain
n
a fDepartmentofPhysics,BergischeUniversitätWuppertal,Gaußstr. 20,
J
D-42119Wuppertal,Germany

2 gSchoolofMathematics&HamiltonMathematicsInstitute,TrinityCollege,Dublin2,Ireland
1
hInstitutfürPhysik,Humboldt-UniversitätzuBerlin,Newtonstr.15,12489Berlin,Germany

]
t
a
E-mail:[email protected], [email protected],
l
- [email protected], [email protected],
p
e [email protected], [email protected],
h [email protected], [email protected], [email protected]
[

2 We present results by the ALPHA collaboration for the Λ-parameter in 3-flavour QCD and the
v
strongcouplingconstantattheelectroweakscale, α (m ), intermsofhadronicquantitiescom-
5 s Z
7 puted on the CLS gauge configurations. The first part of this proceedings contribution contains
0 a review of published material [1, 2] and yields the Λ-parameter in units of a low energy scale,
3
1/L . We then discuss how to determine this scale in physical units from experimental data
0 had
. for the pion and kaon decay constants. We obtain Λ(3) = 332(14)MeV which translates to
1 MS
0 αs(MZ)=0.1179(10)(2)usingperturbationtheorytomatchbetween3-,4-and5-flavourQCD.
7
1
:
v
i
X
r
a
CERN-TH-2016-262
DESY17-007
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
∗Speakers
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Λ(3) andα (m )bytheALPHAcollaboration S.Sint,A.Ramos,R.Sommer
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1. Introduction
ThestrongcouplinginQCDisafundamentalparameteroftheStandardModelanditsprecise
knowledge is both of principal importance and of practical relevance to LHC physics. We here
report on the ALPHA collaboration’s results for the Λ-parameter in 3-flavour QCD and α (m ).
s Z
TheprojectwasdesignedtomatchhadronicquantitiescomputedonCLSgaugeconfigurationsfor
3-flavourQCDatlowenergies[3,4]. ThestrategyfortheΛ-parameterwasexplainedin[5,6]and
reliesonthemethodsandtoolsdevelopedovermanyyears(see[7,8]andreferencestherein). As
a result we are able to defer the use of perturbation theory in 3-flavour QCD to high energies of
O(100 GeV). On the other hand, the determination of α (m ) requires the matching to the 4- and
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5-flavour theories across the charm and bottom quark thresholds which still relies on perturbation
theory [9, 10]. At all stages the continuum limit is taken and rather well controlled. Our strategy
combinestheperturbativeknowledgeofthestandardSFcouplingathighenergieswiththeadvan-
tageous properties of finite volume gradient flow couplings at low energies. A non-perturbative
matching between these 2 coupling schemes is performed at an intermediate scale 1/L ≈4GeV.
0
At the largest box size reached, L , the matching to the gradient flow time scalet∗ at the SU(3)
had 0
symmetric point is performed. Together with the recent results of ref. [4] (which rely on a new
highprecisiondeterminationoftheaxialcurrentnormalizationconstant[11],basedonthemethod
in[12]),thisallowsustoaccuratelyrelatetoalinearcombinationofpionandkaondecayconstants
andthusexpressallresultsinphysicalunits.
Inthiswrite-upwegothroughthedifferentstepsofthisstrategystartingfromthehighenergy
end(section2). TheconnectionbetweentheintermediatescaleL andL isdiscussedinsection
0 had
3, includingthematchingatscaleL . RelatingL toahadronicscaleisthesubjectofsection4.
0 had
This allows to quote the 3-flavour Λ-parameter in physical units. The perturbative connection to
5-flavour QCD is carried out in section 5, followed by our conclusions. Finally, a technical point
pertainingtotheinterpolationofthelowenergydataisrelegatedtoanappendix. Notethatthefirst
2 steps have been published in [1] and [2], respectively. Further details on the first step will be
given elsewhere [13]. The matching of L to a hadronic scale is currently being finalized. For a
had
recentaccountaimedatanon-latticeaudiencewereferto[14].
2. Thehighenergyregime
2.1 AfamilyofcouplingsintheSFscheme
UsingtheSchrödingerfunctionalinQCD[15,16],thespatialvectorcomponentsofthegauge
fieldatthetimeboundariesx =0,T aretakentobespatiallyconstantandAbelian[17],
0
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
(cid:12) i π 1 1 π
Ak(x)(cid:12)x0=0=Ck = Ldiag η− 3,η ν−2 ,−η ν+2 + 3 , (2.1a)
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
Ak(x)(cid:12)(cid:12)x0=T =Ck(cid:48) = Lidiag −π−η,η ν+12 +π3,−η ν−12 +23π , (2.1b)
for k=1,2,3. The parameters η and ν correspond with the existence of 2 abelian generators in
SU(3). Theabsoluteminimumoftheactionwiththeseboundaryvaluesisattainedfor[15]
B (x)=C +x0(cid:0)C(cid:48)−C (cid:1), B =0, (2.2)
k k T k k 0
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theinducedbackgroundfield,whichisuniqueuptogaugeequivalence. Theeffectiveactionofthe
SchrödingerfunctionalΓ[B]isthenunambiguouslydefinedanditsperturbativeexpansionstraight-
forwardinprinciple,
1
Γ[B]= Γ [B]+Γ [B]+O(g2), (2.3)
g2 0 1 0
0
withthefirsttermgivenbytheclassicalactionofthebackgroundfield,
Γ [B]=g2S[B]=2(π+3η)2. (2.4)
0 0
SettingallquarkmassestozeroandT =L,theonlyremainingscaleissetbyL. TheSFcoupling
atthisscaleisthendefinedasaderivativewithrespecttoabackgroundfieldparameter,
∂ηΓ[B] (cid:12)(cid:12) (cid:104)∂ηS(cid:105)|η=0 1
(cid:12) = = −ν×v¯(L). (2.5)
∂ Γ [B](cid:12) 12π g¯2(L)
η 0 η=0
Thederivativeproducesanexpectationvaluewhichcanbemeasuredinnumericalsimulations. In
fact,thereare2observables,theinversecoupling1/g¯2(L)andv¯(L)bothofwhicharemeasuredat
ν =0. Anewfeatureofourprojectisthere-interpretationoftheparameterν asindexofafamily
ofSFcouplings,
1 1
= −ν×v¯(L). (2.6)
g¯2(L) g¯2(L)
ν
Thishasfirstbeenenvisagedasamethodtoreducelargecutoffeffectsinstronglycoupledmodels
ofelectroweaksymmetrybreakingin[18].
In perturbation theory the β-function of the SF coupling family is known to 3-loops from
the 2-loop matching to the MS-coupling in [19, 20, 21, 22] combined with the knowledge of the
β-functionintheMS-scheme(cf.[23,24]andreferencestherein),
∂g¯ (L)
β(g¯ )=−L ν =−b g¯3−b g¯5−b g¯7+O(g¯9). (2.7)
ν ∂L 0 ν 1 ν 2,ν ν ν
In 3-flavour QCD the universal terms are b =9/(4π)2 and b =1/(4π4). The 3-loop coefficient
0 1
isthenfoundtobe
b =(−0.06(3)−ν×1.26)/(4π)3. (2.8)
2,ν
TheΛ-parameterintheSF schemeisdefinedby
ν
LΛSFν =ϕν(g¯ν(L))=(cid:2)b0g¯2ν(L)(cid:3)−2bb120 e−2b0g¯12ν(L)exp(cid:26)−(cid:90) g¯ν(L)dg(cid:20)β(1g)+b1g3 −bb21g(cid:21)(cid:27). (2.9)
0 0 0
When evaluating the Λ-parameter one would like to know from which scale L one can trust per-
turbation theory to evaluate the integral in the exponent and how one can quantify the associated
systematicerror. Weproceedasfollows: firstwefixareferencescale,L ,through
0
g¯2(L )=2.012, (2.10)
0
andusethestep-scalingfunction(cf.nextsubsection)tostepuptheenergyscalenon-perturbatively
byfactorsof2fromL toL =L /2n,forn=0,1,2,.... Wethenusetheν =0Λ -parameteras
0 n 0 SF
referencepointandconsider
L Λ = (Λ /Λ ) ×(L /L )×ϕ (g¯ (L )) (2.11)
0 SF SF SFν 0 n ν ν n
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
exp(−ν×1.25516) 2n
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1.22
1.20
u 1.18
/
)
L
/
a 1.16
u,
(
Σ
1.14
1.12
1.10
0 0.005 0.01 0.015 0.02 0.025 (a/L)2
Figure 1: Continuum extrapolation of the step scaling function. The leftmost points are the con-
tinuum values, whereas the stars are obtained from perturbative scale evolution using the 3-loop
β-function.
withϕ (g¯ (L ))evaluatedinperturbationtheory,byinsertingthevalueofthecouplingg¯2(L )ob-
ν ν n ν n
tainednon-perturbativelyfromthestep-scalingprocedureandthe3-looptruncatedβ-function. Up
to perturbative errors of order α2(1/L )=g¯4(L )/(4π)2, the result for L Λ must be independent
n ν n 0
ofthenumberofstepsnandthevalueoftheparameterν. Thisgivesusanexcellentcontrolover
the remaining systematic error stemming from perturbation theory. Before discussing the result
webrieflypresentsomekeyfeaturesofournumericalsimulations,themeasurementsandthedata
analysis.
2.2 Numericalsimulationdataandanalysis
For the simulation at high energies we choose the Wilson plaquette action in order to use 2-
loopperturbativeinformationatfiniteL/awhichisonlyavailableforthisregularization[19]. This
allowsformorecontrolofboundaryO(a)effectsandalsoforperturbativeimprovementofthenon-
perturbative data of the step-scaling function. We use the result for c from [25]. A very careful
sw
tuningofthebaremassparameterstotheircriticalvaluesin[26]reducesassociatedsystematicsto
negligible levels. All our simulations have been carried out with a modification of the openQCD
code[27]. ComparedtoearlierstudiesoftheSFcouplingin3-flavourQCDin[28](therewiththe
Iwasakigaugeaction),wehavesignificantlyreducedstatisticalerrors. Wehaveproduceddatafor
thestep-scalingfunctions
Σ (u,a/L)=g¯2(2L)| , (2.12)
ν ν g¯2ν(L)=u,m(L)=0
forlatticeresolutionsL/a=6,8,12andthecorrespondingdoubledlatticesizes1. Forν=0ourdata
correspondstou-valuesintheinterval[1.1,2.0]. CutoffeffectareO(a2)inthebulk,andO(a)from
thetimeboundaries. Thelatteraregovernedby2countertermsofdimension4withcoefficientsc
t
and c˜, known to 2-loop [19] and 1-loop order [29] respectively. Using these perturbative results
t
for the coefficients we expect that O(a) effects are strongly reduced. As a safeguard against any
1ForL/a=12wehavelimitedthedataproductiononthe24-latticesto3parameterchoices.
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O(a)contaminationofourcontinuumextrapolationsweincludeasystematicerrorasfollows. We
determine the c and c˜ derivatives of the coupling numerically by a variation in a few points and,
t t
together with the corresponding perturbative information arrive at a model for the sensitivity of
our data to such variations. We then use the last known perturbative coefficients for c and c˜ as
t t
an uncertainty and add the corresponding systematic error to the data. Note that this is likely an
overestimate of the true error [13]. Fig. 1 shows our data points for the step-scaling function at
ν =0. Wethenperformglobalfitsofthedata,forexample
a2 a2
Σ(2)(u,a/L)=u+s u2+s u3+c u4+c u5+ρ u4 +ρ u5 , (2.13)
0 1 1 2 1 L2 2 L2
where Σ(2) denotes the 2-loop improved non-perturbative data, such that cutoff effects appear, by
construction,firstatO(u4)[30]. Thecoefficientss ,s arefixedtotheirperturbativevalues,
0 1
s =2b ln2, s =s2+2b ln2, (2.14)
0 0 1 0 1
and we thus obtain the non-perturbative continuum step-scaling function σ(u) in terms of the fit
coefficientsc andc andtheircorrelation,asrequiredfortheerrorpropagation.
1 2
2.3 ResultsforL Λ
0
0.033
0.032
Λ
L0 0.031
0.030
0.029
0 0.005 0.01 0.015 0.02 0.025 0.03
α2
Figure 2: The extraction of the Λ-parameter using perturbation theory at different values of α,
plottedvs. α2(1/L ). Thedatapointsare,fromtoptobottom,forν =−0.5,0,0.3and,fromright
n
toleft,forn=0,1,...,5stepsbyafactor2inscale.
Giventhestep-scalingfunctionsinthecontinuumlimitourdataallowsustotakeupton=5
steps from L , thus covering a factor of 25 =32 in scale. For ν (cid:54)=0 one also needs to know the
0
valueofv¯atL [1],
0
v¯(L )=0.1199(10). (2.15)
0
We have considered O(10) ν-values around ν = 0. We present data for ν = 0.3 and ν = −0.5
besidesthereferencechoiceν =0. Fig.2showsthecorresponding3×6datapointsforL Λfrom
0
eq. (2.9), each corresponding to a determination of L Λ using non-perturbative data between L
0 0
and L and perturbation theory for L<L . As can be seen in the plot the data points nicely come
n n
togetheratsmallL(highenergyscales)whereα(1/L )≈0.1. Wearethereforeconfidenttoquote
n
ourresultwithanerrorof3%,
L Λ =0.0303(8) ⇔ L Λ(3) =0.0791(21), (2.16)
0 SF 0
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whichisthemainresultofourstudyofthelargeenergyregion. Inarecentletter[1]wehavealso
presentedaby-productofthisstudy,namelytheobservationthatrenormalizedperturbationtheory
in continuum QCD might be susceptible to larger systematic errors than often assumed. This is
apparentinfig.2andinthecontinuumresultforv¯(L)whereperturbationtheoryatα =0.19does
notlooktrustworthy. Thisisparticularlyworrisomegiventheveryadvantageouspropertiesofthe
SF -schemesinperturbationtheory[1,13].
ν
3. Connectingwithhadronicscales
Intheprevioussectionwehavedetailedourveryaccuratematchingwithperturbationtheory.
The result is given in eq. (2.16) and relates the Λ-parameter with L , defined implicitly through
0
g¯2 (L )=2.012. The corresponding energy scale, 1/L ≈4GeV, is still very large if one aims
SF 0 0
atusinglatticemethodstostudycontinuumpropertiesofQCDwhilehavingfinitevolumeeffects
undercontrol. Thereforewewillcontinueusingourfinitesizescalingtechniqueinordertorelate
thescaleL withascaleL characteristicofhadronicphysics.
0 had
In principle one could simply continue with the program explained in the previous section
until reaching the energy scale 1/L . Unfortunately the statistical precision of the SF coupling
had
deteriorates very fast when reaching large volumes (see for example the discussion in [31] and
references therein), making it very difficult to maintain the precision. In order to overcome these
problems we will continue to work with a finite volume renormalization scheme, but we will use
the gradient flow coupling in a finite volume with SF boundary conditions [32] for our running
coupling. Renormalized couplings based on the gradient flow have the nice property that their
varianceisroughlyindependentofthelatticespacing.
Themainresultofthissection[2]istheprecisedeterminationoftheratiooftwoscales
L /L =21.86(42) for g¯2 (L )=11.31. (3.1)
had 0 GF had
Note,however,thatintheusualstepscalingprocedure(seeprevioussection)theaimistodetermine
thevalueoftherenormalizedcouplingattwoscalesL andL thatareseparatedbyanintegerpower
1 2
ofthescalefactor(i.e. L =2nL withn∈Zwhenthescalefactoriss=2). Hereweneedtosolve
1 2
aslightlydifferentproblem: weknowthevalueofthecouplingattwoscales(L andL ),andare
0 had
interestedincomputingtheratioofthesescales. Withourconventions2,theβ-functionisdefined
by
∂g¯ (L)
GF
−L =β(g¯ ), (3.2)
GF
∂L
andratiosofscalessuchas(3.1)canbeeasilycomputed, oncetheβ-functionisknown, thanksto
therelation
L1 (cid:26) (cid:90) g¯GF(L1) dx (cid:27)
=exp − . (3.3)
L2 g¯GF(L2) β(x)
In order to determine the β-function, we will first determine the usual continuum step scaling
function,
(cid:12)
σ (u)=g¯2 (2L)(cid:12) , (3.4)
GF GF (cid:12)
g2 (L)=u
GF
2Werecallthatwealwaysworkinamass-independentrenormalizationscheme,andthereforetheβ-functiononly
dependsong.
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anduseittoconstrainthefunctionalformofβ(g)byusingtheexactrelation
√
(cid:90) σGF(u) dx
log2=− . (3.5)
√u β(x)
3.1 Couplingdefinitionandchoicesofdiscretization
The gradient flow coupling with SF boundaries conditions has been studied in [32], and the
interestedreadershouldconsultthecitedreference. WeimposeSFboundaryconditionswithzero
background field (i.e. Ck =Ck(cid:48) =0 in eq. (2.1)). The gradient flow [33, 34] determines how the
gaugefieldB (t,x)evolveswiththeflowtimet (nottobeconfusedwiththeEuclideantimex )via
µ 0
the(non-linear)diffusion-likeequation
∂B (t,x) (cid:12)
µ =D G (t,x); A (x)=B (t,x)(cid:12) ; G =∂ B −∂ B +[B ,B ]. (3.6)
∂t ν νµ µ µ (cid:12)t=0 µν µ ν ν µ µ ν
The important point is that gauge invariant observables made out of the flow field B (t,x) are
µ
automaticallyrenormalized[35]fort>0. InparticularonecanusetheactiondensityTr(G G )
√ µν µν
to define a renormalized coupling at a scale µ ∝1/ 8t. There are many subtleties that lead us to
usethefollowingcouplingdefinition
t2 3 (cid:104)Tr(G (t,x)G (t,x))δˆ(Q)(cid:105)(cid:12)
g¯2GF(L)=N −14 i,∑j=1 ij (cid:104)δˆ(Qij)(cid:105) (cid:12)(cid:12)√8t=cL;x0=T/2;T=L. (3.7)
√
Note that the renormalization scale runs with the volume thanks to the relation 8t = cL (We
choosec=0.3inthiswork). Severalpointsrequiresomeexplanation
1. The SF breaks the invariance under translations in Euclidean time. Moreover full O(a) im-
provement with the SF setup requires to determine non-perturbatively two boundary im-
provement coefficients (i.e. c,c˜), which we only know to 1-loop order in perturbation the-
t t
ory(seebelow). Tominimizetheseboundaryeffects,wechoosetodefinethecouplingusing
the action density at x =T/2, and based only on the magnetic components (i,j =1,2,3)
0
ofthefieldstrengthtensor. WiththesechoicesboundaryO(a)effectscanbeestimatedfrom
ourdataset,andarefoundtobesmall.
2. Simulations with SF boundary conditions suffer from the common problem of topology
freezing [36]. In order to overcome this problem, we define the coupling only in the sec-
torwithzerotopologicalcharge(see[37]formoredetails).
3. FortheprecisedefinitionsofthenormalizationN andthedelta-functionδˆ(Q)projectingto
thezerochargesectorwereferto[2].
Sinceourfinalaimistodetermineourscalesinphysicalunitsfromthelargevolumesimulationsof
theCLSinitiative[3],weusethesamebarelatticeaction,exceptfortheEuclideantimeboundary
conditions. Inparticular,wesimulatethreemasslessflavoursofnon-perturbativelyO(a)improved
Wilson fermions and a Symanzik tree-level O(a) improved gauge action. There is some freedom
when defining this action near the Euclidean time boundaries, and we use option B of refer-
ence [38]. With this choice we know the 1-loop value of the boundary improvement coefficients
c,c˜ [39,40].
t t
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Therearemanypossibilitieswhentranslatingthecontinuumflowequation(3.6)tothelattice.
Differentdefinitionsleadtodifferentcutoffeffects, butthesecanbelarge. Followingref.[41]we
choosetheZeuthenflow. ThisparticulardiscretizationguaranteesthatO(a2)cutoffeffectsarenot
generated when integrating the flow equation. In [2] we performed a detailed study of the cutoff
effects offlow quantitiesand concludedthat, atleast forour data set, the scalingproperties ofthe
Zeuthenflowallowustoperformamoreaccuratecontinuumextrapolation.
3.2 Determinationofthestepscalingfunction
Thebaremassm istunedtoitscriticalvaluem withexcellentprecision[26],forallvalues
0 cr
ofthebarecouplingβ =6/g2 usedhere. Anydeviationfromthecriticallineiswellbelowoursta-
0
tisticaluncertainties. Giventhefunctionm (β,L/a)(see[26]),oursimulationsdependessentially
cr
onlyononebareparameterβ.
In order to determine the step scaling function we perform 9 precise simulations at β ∈
{3.9,4.0,4.1,4.3,4.5,4.8,5.1,5.4,5.7} on a L/a = 16 lattice. These simulations provide our 9
targetcouplingsv
i
v ∈{2.1257,3.3900,2.7359,3.2029,3.8643,4.4901,5.3013,5.8673,6.5489}, (3.8)
i
for which we would like to obtain the continuum step scaling function σ(v). In order to do
i
this, we tune the bare coupling β on some L/a = 8,12 lattices to match these values of the
renormalized coupling. Once this tuning is satisfactory, we compute g¯2 on the doubled lattices
GF
(L/a = 16,24,32), with all bare parameters kept fixed, and thus obtain the lattice step scaling
functionΣ (v,L/a).
GF i
Wefindthatournonperturbativedatafollowsverycloselythefunctionalform
1 1
− =constant, (3.9)
Σ (u,L/a) u
GF
which allows us to propagate the uncertainties in u into Σ by using ∂Σ /∂u = Σ2 /u2. A
GF GF GF
(conservative) estimate of the boundary effects due to c is also propagated to Σ in a similar
t GF
fashion(see[2]forthedetails).
All in all we get lattice estimates of the step scaling function Σ (v,L/a) at approximately
GF i
9 values of the coupling. The very small mistuning can be corrected by shifting our data for Σ
GF
totheexacttargetvaluesbyusingtherelation(3.9). Theseshifteddatacanbeextrapolatedtothe
continuuminastraightforwardway. Figure3ashowstwotypesofcontinuumextrapolationsofthe
stepscalingfunction
Σ (v,L/a) =σ (v)+ρ ×(a/L)2, (3.10)
GF i GF i i
1/Σ (v,L/a) =1/σ (v)+ρ˜ ×(a/L)2. (3.11)
GF i GF i i
Note that the difference between these fit ansätze is O(a4). As it is apparent in Figure 3a, the
continuumextrapolatedvaluesforbothfitansätzeagreewithinonestandarddeviation,butthereis
asystematicdifferencebetweenthem. Despitethefactthatourdatashowsaverynicea2 scaling,
the large cutoff effects, specially at large values of u, induce a systematic effect in our continuum
extrapolations. We choose to include an estimate of this systematic uncertainty in the weights of
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2.2 FitΣ 2.2 FitΣ
Fit1/Σ Fit1/Σ
Continuum(fitΣ) Continuum(fitΣ)
Continuum(fit1/Σ) Continuum(fit1/Σ)
2 Data 2 Data
u 1.8 u 1.8
/ /
L) L)
/ /
a a
u, 1.6 u, 1.6
Σ( Σ(
1.4 1.4
1.2 1.2
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016
(a/L)2 (a/L)2
(a) Continuum extrapolation of the step scaling (b) Continuum extrapolation of the step scaling
functionatthe9targetvaluesofthecouplingus- functionatthe9targetvaluesofthecouplingus-
ingasweightsforthefittheuncertaintyinΣ.Note ingasweightsforthefittheuncertaintyinΣand
that different continuum extrapolations are sys- thesystematicestimateoftheO(a4)effects.
tematicallydifferent.
Figure 3: Continuum limit of the step scaling function with and without including the systematic
effect(3.12).
ourfits.
(cid:18) a(cid:19)4 u
∆sysΣ =0.05Σ 8 . (3.12)
GF GF
L u
max
This estimate comes from the size of the a2 cutoff effects at our largest value of the coupling
(around 20% for our coarser lattice with L/a=8), that suggests that the O(a4) effects are around
5%. This effect is added in quadrature to the uncertainty in Σ . The result of this procedure is
GF
apparent if one compares Figures 3a and 3b. The fit functional forms are less constrained by the
coarser lattices, resulting in a better agreement between the two extrapolation procedures. The
price to pay is an increased error in the extrapolations. All our final numbers follow this fitting
procedure. Theinterestedreadercanfindadetaileddiscussionin[2].
Asstatedearlierournon-perturbativedataobeysverywelltherelation1/σ−1/u=constant
(the1-loopfunctionalform). Thissuggeststofitthecontinuumstepscalingfunctiontoafunctional
form
1 1 nsig−1
− =Q(v), Q(u)= ∑ c uk, (3.13)
i k
σ (v) v
GF i i k=0
wherethenumberoffitparametersn isvariedtochecktheconsistencyoftheresults.
sig
Finallyonecanalsocombinethecontinuumextrapolationsandtheparametrizationofthestep
scalingfunctionsbyfitting
1 1 nρ−1
− =Q(v)+ρ(v)(a/L)2, ρ(u)= ∑ ρ uk. (3.14)
i i k
Σ (v,L/a) v
GF i i k=0
Wefindgoodfitswhenn isatleast2. Notethatthisproceduredoesnotrequiretoshiftthedatato
ρ
constantvaluesofthecoupling. Allthesefittingproceduresproducearemarkableagreement,asis
discussedindetailin[2].
8
Λ(3) andα (m )bytheALPHAcollaboration S.Sint,A.Ramos,R.Sommer
s Z
MS
Fit n n u u u u s(g2,g2)
sig ρ 1 2 3 4 1 2
Σ 3 – 5.870(28) 3.954(22) 2.976(17) 2.385(15) 11.00(20)
1/Σ 1 3 5.843(20) 3.939(18) 2.971(16) 2.385(13) 10.96(18)
1/Σ 2 3 5.864(26) 3.944(19) 2.968(16) 2.378(14) 10.90(18)
1/Σ 3 3 5.864(27) 3.944(21) 2.968(17) 2.378(14) 10.90(19)
(3.17),P 2 2 5.872(27) 3.949(19) 2.971(16) 2.379(14) 10.93(19)
(3.17),P 3 3 5.874(28) 3.951(22) 2.972(17) 2.379(14) 10.93(20)
Table 1: Coupling sequence eq. (3.18) with u = 11.31 and scale factors s(g2,g2) for g2 =
0 1 2 1
2.6723,g2 =11.31 for different fits to cutoff effects and the continuum β-function. Fits are la-
2
beled by Σ or 1/Σ for continuum extrapolations according to eq. (3.10), eq. (3.11) or eq. (3.17).
Forglobalfitswespecifyn ,whileitsabsenceindicatesafitofdataextrapolatedtothecontinuum
ρ
ateachvalueofu=v.
i
3.3 Determinationoftheβ-function
Aswehavesaid,ourmainobjectiveistocomputearatioofscales,andinordertodothis,we
needtodeterminetheβ-function. Wechoosetheparametrization
g3 nsig−1
β(g)=− , P(g2)= ∑ p g2k. (3.15)
P(g2) k
k=0
The1-loopβ-functioncorrespondstothechoicen =1. Therelationbetweentheβ-functionand
sig
thestepscalingfunctionσ
GF
√ √
(cid:90) σ(u) dx (cid:90) σ(u) P(x2)
log(2)=− = dx
√u β(x) √u x3
(3.16)
(cid:20) (cid:21) (cid:20) (cid:21) n
p 1 1 p σ(u) sig p (cid:104) (cid:105)
=− 0 − + 1log +∑ k+1 σk(u)−uk ,
2 σ(u) u 2 u 2k
k=1
is used to fit the coefficients p . An alternative analysis consists in combining the continuum
k
extrapolationwiththedeterminationoftheβ-functionbyfitting
√
(cid:90) Σ(u,a/L) dx
log(2)+ρ(u)(a/L)2=− . (3.17)
(cid:101) √u β(x)
Notethatthisansatzprovidesyetanotherparametrizationofthecutoffeffects.
A quantitative test of the agreement between different functional forms consists in analyzing
thesequence
u =11.31, u =σ−1(u ),i=1,2,.... (3.18)
0 i GF i−1
Table 1 contains a sample of the different analysis considered in [2]. The agreement between
differentansätzeisremarkable. Thelastcolumnoftable1isthescalefactor
(cid:26) (cid:90) g2 dx (cid:27)
s(g2,g2)=exp − , (3.19)
1 2 g β(x)
1
9