# Preparing for N_f=2 simulations at small lattice spacings

**ABSTRACT** We discuss some large effects of dynamical fermions. One is a cutoff effect, others concern the contribution of multi-pion states to correlation functions and are expected to survive the continuum limit. We then turn to the preparation for simulations at small lattice spacings which we are planning down to around a=0.04fm in order to understand the size of O(a^2)-effects of the standard O(a)-improved theory. The dependence of the lattice spacing on the bare coupling is determined through the Schr"odinger functional renormalized coupling.

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**ABSTRACT:**We study the critical slowing down towards the continuum limit of lattice QCD simulations with Hybrid Monte Carlo type algorithms. In particular for the squared topological charge we find it to be very severe with an effective dynamical critical exponent of about 5 in pure gauge theory. We also consider Wilson loops which we can demonstrate to decouple from the modes which slow down the topological charge. Quenched observables are studied and a comparison to simulations of full QCD is made. In order to deal with the slow modes in the simulation, we propose a method to incorporate the information from slow observables into the error analysis of physical observables and arrive at safer error estimates.Nuclear Physics B. 09/2010; - SourceAvailable from: Michele Della Morte[Show abstract] [Hide abstract]

**ABSTRACT:**We discuss kinematical enhancements of cutoff effects at short and intermediate distances. Starting from a pedagogical example with periodic boundary conditions, we switch to the case of the Schrödinger functional, where the theoretical analysis is checked by precise numerical data with Nf=2 dynamical O(a)-improved Wilson quarks. Finally we present an improved determination of the renormalization of the axial current in that theory.Physics Letters B. 03/2009; - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**Simulations of QCD are known to suffer from serious critical slowing down towards the continuum limit. This is particularly prominent in the topological charge. We investigate the severeness of the problem in the range of lattice spacings used in contemporary simulations and propose a method to give more reliable error estimates. Comment: 7 pages, 4 figures, talk presented at The XXVII International Symposium on Lattice Field Theory, July 26-31, 2009, Peking University, Beijing, China, v2: Correction in Sec. 4, results unchanged10/2009;

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Preparing for Nf= 2 simulations at small lattice

spacings

LPHA

A

Collaboration

DESY 07-159

SFB/CPP-07-55

MS-TP-07-35

CERN-PH-TH/2007-170

HU-EP-07/46

MIT-CTP 3873

M. Della Morte,

CERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerland

P. Fritzsch,

University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany

B. Leder, S. Takeda, O. Witzel, U. Wolff

Humboldt University, Newtonstr. 15, 12489 Berlin, Germany

H. Meyer,

MIT, Cambridge, MA 02139, U.S.A.

H. Simma∗, R. Sommer†

DESY, Platanenalle 6, 15738 Zeuthen, Germany

E-mail: rainer.sommer@desy.de

We discuss some large effects of dynamical fermions. One is a cutoff effect, others concern the

contribution of multi-pion states to correlation functions and are expected to survive the contin-

uum limit. We then turn to the preparation for simulations at small lattice spacings which we

are planning down to around a = 0.04 fm in order to understand the size of O(a2)-effects of the

standard O(a)-improved theory. The dependence of the lattice spacing on the bare coupling is

determined through the Schrödinger functional renormalized coupling.

The XXV International Symposium on Lattice Field Theory

July 30-4 August 2007

Regensburg, Germany

∗present address: University Milano Bicocca, Pz. della Scienza 3, 20126 Milano, Italy.

†Speaker.

c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

http://pos.sissa.it/

arXiv:0710.1263v1 [hep-lat] 5 Oct 2007

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Nf= 2 simulations at small lattice spacings R. Sommer

1. Introduction

The ALPHA collaboration has worked over the years on a determination of the QCD Λ-

parameter starting from experimental low energy hadronic input and using perturbation theory

in a renormalized coupling at sufficiently high energy scales. At these scales it was demon-

strated that perturbation theory is very accurate. The quoted results for the MS Λ-parameter are

Λ(0)

dynamical quarks. On the other hand the Nf= 5 value extracted by matching various exper-

imental data to perturbation theory in the (not always very) high energy region translates into

Λ(5)

the perturbative matching across the quark thresholds [5] yields Λ(4)

connect smoothly to the Nf= 0,2 numbers. In order to say more about this comparison, the low

energy scale r0should be replaced by an experimental observable and the continuum limit should

be evaluated with a better confidence than it was possible in [2]. Significant progress in the under-

standing of the continuum limit requires to simulate smaller lattice spacings with good accuracy.

We will motivate this further in Sect. 2. The difficult simulations are the ones in large volume

where for example the Kaon decay constant is to be determined to set the energy scale in GeV. We

will briefly explain in Sect. 3 that our previous approach of using Schrödinger functional bound-

ary conditions in that part of the calculation meets somewhat unexpected (practical) difficulties.

Since these are related to true dynamical fermion effects, they are theoretically interesting, but it

appears to be better to switch to (anti)-periodic boundary conditions in this part of the calculation.

In Sect. 4 we will finally discuss a determination of the dependence of the lattice spacing on the

bare coupling. This represents a useful piece of information for fixing the parameters of the large

volume simulations.

MSr0= 0.60(5) [1] in the quenched approximation and Λ(2)

MSr0= 0.62(4)(4) [2] with Nf= 2

MSr0≈ 0.55 [3], when r0= 0.5fm [4] is used. Superficially this suggests a nice agreement, but

MS/Λ(5)

MS≈ 1.4 which does not

Before entering our discussion we add a comment on the motivation. One might object to

the whole project of a determination of the Λ-prameter that very precise lattice determinations

for αMS(MZ) have already been published [6]. However, apart from the use of rooted staggered

fermions, in these determinations perturbation theory has been used at rather low renormalization

scales and for non-universal quantities (small Wilson loops). These are defined at the scale of

the (lattice)-cutoff. It is apparent from the discussion in [6] that the use of perturbation theory is

problematic. The known terms in the expansion either have large coefficients or, if one resums by

choosing a different scheme, the renormalization scale becomes even smaller and the expansion

parameter larger. In order to describe the data several higher order terms in the expansion are

fitted. Thus it appears that a computation following the ALPHA-strategy, where the continuum

limit is taken and perturbation theory is verified to apply for the considered renormalized coupling,

remains very well motivated. We do not see any alternative to this strategy if a full control of all

systematics is desired.

InthefollowingconsiderationsweusethestandardO(a)-improvedtheorywithWilson’sgauge

action and the non-perturbatively determined [7] coefficient cswof the Sheikholeslami-Wohlert

term [8].

2

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Nf= 2 simulations at small lattice spacings R. Sommer

2. Cutoff effects in ZA

In [9] we have presented evidence that cutoff effects tend to be larger in full QCD than they

are in the quenched approximation. Here we would like to draw the reader’s attention to the non-

perturbative determination of ZApresented in [10]. It uses a Ward identity in the Schrödinger

functional in an L3×9/4L geometry with L ≈ 0.8fm as in the quenched approximation [11]. It can

be shown that the quark-propagator disconnected diagrams which enter the Ward idenity vanish

in the continuum limit. They are of O(a2) at a finite lattice spacing. In contrast to the quenched

approximation where already at a = 0.1fm they were insignificant (in comparison to the numerical

precision), for Nf= 2 they contribute an about 15% effect in ZAat such a lattice spacing. Even if

this effect disappears very quickly at smaller a, it is unpleasantly large at the lattice spacings one

typically would like to include in a continuum extrapolation.

In the mean time we have investigated the problem further, finding that qualitatively this effect

persists if one changes the angle θ in the spatial fermion boundary condition. Alternatively we

considered the Ward identity between static-light states in such a way that disconnected diagrams

are absent. Unfortunately, even when using HYP discretizations for static quarks [12] the statistical

errors in ZAbecome relatively large at the smaller lattice spacings. Still, we confirmed that ZA

defined in this way is rather close to the definition with light-light states but disconnected diagrams

dropped.

In general, cutoff effects are expected to be more prominent in correlation functions (and for

time separations) where excited state contributions are very important. We therefore investigate

at present whether the approximate ground state projection of [13] suppresses the disconnected

contribution to ZAthus accelerating the continuum limit. Whether this attempt is successfull or

not, these difficulties suggest that one most likely needs smaller a with dynamical fermions than

in the quenched approximation. We now turn to another strong effect of dynamical fermions – one

that is expected to persist in the continuum limit.

3. The large-volume Schrödinger functional

Apart from the non-perturbative evaluation of renormalization constants, the Schrödinger

functional also proved to be advantageous for the computation of hadron masses and matrix el-

ements such as FKin the quenched approximation [14]. A time extent of T = 3fm allowed to

clearly isolate ground state contributions. We have then attempted to compute the pseudoscalar

masses and decay constants for Nf= 2 with an L3×T Schrödinger functional, keeping L ≥ 2fm

and T ≈ 2.5fm. Indeed, at a quark mass around the physical strange quark mass (κ = 0.1355), the

effective mass of the pseudoscalar correlation functions fA, fP(see e.g. [15] for their definition)

exhibit short plateaux. An example is shown in the upper part of Fig. 1.

However, the plateaux disappear quickly when the quark mass is lowered. For a quark mass

of about half the strange quark mass (κ = 0.13605), excited state contaminations are strongly

present in both the vacuum channel and in the pion channel. The former yield contributions ∝

exp(−(T −x0)Evac

are included, fits to the correlation functions are still reasonable. We show a fit where we have

fixed Evac

1

1) and the latter ∝ exp(−x0(Eπ

1−mπ)). Once these two leading contaminations

= 2mπ, Eπ

1= 3mπ. These are the energies of the multi pion states with the correct

3

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Nf= 2 simulations at small lattice spacingsR. Sommer

Figure 1: The effective mass for the Schrödinger functional correlation function fPat β =5.3 on a 243×32

lattice for κ = 0.1355 and κ = 0.13605. The fit described in the text is extended outside the fit-range as a

dotted curve. The dashed line indicates the fitted pion mass.

quantum numbers, when the interaction of the pions is neglected. At sufficiently large L this is a

good approximation.

We may conclude that multi-pion states are observed, as expected in the full theory. Their

amplitude appears to be significantly stronger than with (point-to-point correlators and) periodic

boundary conditions [16]. The standard Schrödinger functional boundary operators have a strong

overlap with these states. Even though it is interesting to observe these strong effects of dynamical

fermions and a consistent description over a significant range of x0can be achieved in the form of a

fit, their presence hampers a reliable estimation of the systematic errors. We have hence decided to

switch to periodic boundary conditions for the purpose of computing large volume matrix elements.

4. The lattice spacing as a function of the bare coupling

As a first step towards such computations we now compute, in a massless renormalization

scheme, thedependencea(g0)ofthelatticespacingonthebarecouplingg0for0.04fm<

Of course the function a(g0) is not unique, but only defined up to cutoff effects, which depend on

the renormalized quantity that is held fixed. We employ a renormalization condition which is rela-

tively easily evaluated and which does not introduce artificially large a-effects. This has proven to

be the case for the standard Schrödinger functional coupling ¯ g2(L), defined in [17,18], at vanishing

quark mass.

We further specify a scale L∗by

∼a<

∼0.1fm.

¯ g2(L∗) = 5.5,

(4.1)

4

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Nf= 2 simulations at small lattice spacingsR. Sommer

L/a

βκ

¯ g2(L)

am

8

8

8

8

10

12

16

5.3

5.3574

5.3574

5.3574

5.5

5.6215

5.8097

0.136197

0.13564

0.1367

0.136365

0.136712

0.136665

0.1366077

5.65(5)

5.59(5)

4.98(13)

5.26(6)

5.11(8)

5.62(9)

5.48(12)

0

0.024(1)

−0.011(1)

0

−0.0008(2)

0.0019(2)

0

[2]

interpolated

[2]

Table 1: Raw simulation results and interpolated values. Values of am = 0 indicate that |z| = |Lm| is

estimated to be at most 5×10−3.

which is known to lead to L∗/a>

shows a change of ¯ g2by about ∆¯ g2= 0.3 when the boundary O(a) improvement coefficient ct

is changed from its 1-loop to its 2-loop approximation. Using the non-perturbative beta-function

of [2],

∼8 for the planned range of a. For such a choice, table 7 of [2]

Ld

dL¯ g2= −2 ¯ gβ(¯ g) = 0.21(1) ¯ g4at ¯ g2≈ 5.5,

(4.2)

a value ∆¯ g2=0.3 translates into a 5% change in L∗and thus a. The definition eq. (4.1) is completed

byanexactdefinitionofthemasslesspoint. WechoosethePCACmassm(withnon-perturbativecA

[13]) with Schrödinger functional boundary conditions, with T = L = L∗, θ = 0.5 and a vanishing

background field.

Good guesses for the bare parameters g0,κ at a prescribed L/a are easily made starting from

table 11 of [2]. When the result of a determination of ¯ g2(L) is close to the target eq. (4.1) and m

is close to zero, we may correct by a first order Taylor expansion with derivatives eq. (4.2) and an

estimate of

s =1

L

∂

∂m¯ g2|L.

(4.3)

From the results at two different values of m and fixed β = 6/g2

0= 5.3574 in Table 1 we extract

β

log(L∗/a)

5.3000

5.3574

5.5000

5.6215

5.8097

2.056(08)

2.120(11)

2.368(14)

2.474(14)

2.776(19)

Table 2: Results for

L∗/a.

s = 2.2(5) at ¯ g2(L) ≈ 5.5.

(4.4)

The rest of the simulation results of that table are then corrected to match

the target with this value of s (including its error) and with eq. (4.2). We

arrive at Table 2 where a precision between 0.8% and 1.9% is seen. These

numerical values are very well described by the simple linear interpola-

tion formula

log(L∗/a) = 2.3338+1.4025(β −5.5)

as seen in Fig. 2 where a ±0.02 “error band” is shown.

(4.5)

5

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Nf= 2 simulations at small lattice spacings R. Sommer

Figure 2: The results for L∗/a as a function of β.

5. Outlook

Using the estimate a ≈ 0.08fm at β = 5.3 [16], we have estimated the pairs (β,L/a) =

(5.5,32) and (5.7,48) in order to remain in the large volume region L ≥ 1.9fm. We are currently

carrying out first simulations at these parameters. Quark masses on the L/a=48 lattice are initially

designed to be only slightly below the mass of the strange quark. The reason is that our first goal is

to carry out a precise scaling test, which is best done at not too small quark mass. Combining with

the results of [16,19] a significant range of a close to the continuum can be covered.

The simulations are currently being done with the DD-HMC algorithm [20]. Release 1.0 of

Martin Lüscher’s software [21] has been adapted for the BlueGene/L and an efficiency around

30% has been achieved. The simulations do thus run at a sufficient speed to expect results from the

BlueGene/L in Jülich rather soon. These efforts are part of coordinated lattice simulations (CLS)

carried out together with other lattice groups at CERN, Madrid, Mainz, Rome (Tor Vergata) and

Valencia.

Acknowledgements. We thank NIC for allocating computer time on the APE computers to this

project and the APE group for its help. This work is supported by the Deutsche Forschungsge-

meinschaft in the SFB/TR 09 and under grant HE 4517/2-1, by the European community through

EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”.

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