Non-perturbative renormalization of static-light four-fermion operators in quenched lattice QCD
Filippo Palombi, Mauro Papinutto, Carlos Pena, Hartmut Wittig
aa r X i v : . [ h e p - l a t ] J un Preprint typeset in JHEP style - HYPER VERSION
CERN-PH-TH/2007-097DESY 07-090MKPH-T-0709June 2007
Non-perturbative renormalization of static-lightfour-fermion operators in quenched lattice QCD
LPHA A Collaboration
Filippo Palombi
DESY, Platanenallee 6, D-15738 Zeuthen, GermanyE-mail: [email protected]
Mauro Papinutto
CERN, Physics Department, Theory Division, CH-1211 Geneva 23, SwitzerlandE-mail: [email protected]
Carlos Pena
CERN, Physics Department, Theory Division, CH-1211 Geneva 23, SwitzerlandE-mail: [email protected]
Hartmut Wittig
Institut f¨ur Kernphysik, University of Mainz, D-55099 Mainz, GermanyE-mail: [email protected]
Abstract:
We perform a non-perturbative study of the scale-dependent renormalizationfactors of a multiplicatively renormalizable basis of ∆ B = 2 parity-odd four-fermion op-erators in quenched lattice QCD. Heavy quarks are treated in the static approximationwith various lattice discretizations of the static action. Light quarks are described by non-perturbatively O( a ) improved Wilson-type fermions. The renormalization group runningis computed for a family of Schr¨odinger functional (SF) schemes through finite volumetechniques in the continuum limit. We compute non-perturbatively the relation betweenthe renormalization group invariant operators and their counterparts renormalized in theSF at a low energy scale. Furthermore, we provide non-perturbative estimates for thematching between the lattice regularized theory and all the SF schemes considered. Keywords:
B-Physics, Heavy Quark Physics, Lattice QCD, Non-perturbativerenormalization. . Introduction
Particle-antiparticle transformations of neutral B -mesons are currently being investigatedin the framework of major experimental programmes, aiming to constrain the top-quarksector of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Measurements of the oscil-lation frequencies ∆ m q ( q = d, s ) allow to extract | V tq | , once the quantum mechanicalamplitudes responsible for the elementary transitions are known. The latter are customar-ily represented in terms of the B -parameters B B q through an explicit factorization of thevacuum-saturation contribution, namely h ¯ B q |O ∆ B =2LL | B q i = 83 B B q f B q m B q . (1.1)A theoretical computation of the matrix element in Eq. (1.1) requires non-perturbativetechniques for a proper description of the low-energy dynamics of the B -mesons. Lat-tice QCD is the obvious methodology, insofar as all its systematic uncertainties can bereduced to an acceptable level. We refer the reader to [1, 2] for a review of recent latticedeterminations of the mixing parameters of the B -mesons.In a previous work [3], we have devised a novel strategy to compute the above matrixelements in lattice QCD, based on Heavy Quark Effective Theory (HQET) at leading orderin the heavy quark mass expansion in conjunction with twisted mass QCD (tmQCD) [4]in the light quark sector. This has the advantage of removing the unwanted mixings underrenormalization, which arise with ordinary Wilson-type lattice fermions. The main ideabehind the proposed approach originates from the general observation that in presenceof a chirality breaking lattice regularization, like the Wilson one, parity-odd four-fermionoperators can have simpler renormalization properties than their parity-even counterparts,as also shown in [5]. In the particular case where light quarks are described as Wilsonfermions and heavy quarks are treated in the static approximation, it is even possible todefine a complete basis (cid:8) Q ′ ± k (cid:9) k =1 ,..., of multiplicatively renormalizable parity-odd ∆ B = 2four-fermion operators, which is given in sect. 2 below (see also Eq. (2.12) of [3]). As de-rived in [3], this result is mainly due to the heavy quark spin symmetry and time reversal,which strongly constrain the chirality breaking pattern, especially in the parity-odd sec-tor. The adoption of tmQCD allows to take advantage of such properties by relating theparity-even operators of the effective static theory entering the computation of B B q to theaforementioned operator basis, viz. Q ′ + and Q ′ + . Accordingly, the additional mixing underrenormalization with Wilson-type lattice fermions is avoided, thus opening the way to adetermination of the B -parameters with reduced systematic uncertainties.This paper is devoted to a non-perturbative study of the renormalization group (RG)running of the operator basis (cid:8) Q ′ + k (cid:9) k =1 ,..., in the quenched approximation. Renormal-ization constants are defined in terms of Schr¨odinger functional (SF) correlators, whereperiodic boundary conditions (up to a phase θ for light-quark fields) are imposed alongthe spatial directions and Dirichlet boundary conditions are imposed in time. The SFformalism [6, 7], developed initially to produce a precise determination of the running cou-pling [8,9], has proved useful also in phenomenological contexts, like the RG running of the– 1 –uark mass [10–13], the computation of moments of structure functions [14], the evolution ofthe static-light axial current [15,16] and the computation of the Kaon B -parameter [17–19].In this framework, the operator running can be determined by computing the so-called stepscaling function (SSF) for a wide range of renormalized couplings, which extend from per-turbative to non-perturbative regimes. The SSF itself is determined through a recursivefinite-size scaling procedure, which provides a step-wise construction of the solution to theCallan-Symanzik equation. Through a sequence of Monte Carlo simulations at differentlattice spacings the latter is obtained in the continuum limit.The implementation of the non-perturbative renormalization programme in the frame-work of the SF is usually split into two parts. The first is the determination of the scaledependence of the relevant operators from low to high scales in an SF scheme, which yieldsuniversal, regularization-independent relations between renormalization group invariant(RGI) operators and their counterparts in the SF scheme. The second part is the matchingbetween the operators in the chosen SF scheme and the lattice-regularized theory. This isachieved by computing the relevant renormalization factors at a fixed low-energy hadronicscale µ had for several values of the lattice spacing. The combination of the renormalizationfactors with the regularization-independent part yields the total matching between the barelattice operators and the RGI ones. In this paper we report on the determination of thetotal renormalization factor in quenched QCD, with the heavy quarks treated in the staticapproximation and the light quarks discretized according to the O(a) improved Wilsonaction.In order to provide useful input for phenomenology, lattice determinations of B -parameters must have an accuracy at the level of a few percent. Thus, to avoid beingdominated by the numerical uncertainty in the renormalization factor, we aim for a targetprecision of the RGI constants within 1.5-2% in this work. It is well known that MonteCarlo simulations including static fermions are plagued by a deterioration of the numericalsignal. As shown in [20], this problem can be overcome through the adoption of statis-tically improved actions. An analysis of the signal-to-noise ratio shows that achieving arelative uncertainty around 1% in the continuum limit of the SSF is unattainable when thenaive discretization of the Eichten-Hill (EH) fermions is employed, especially in the deeplynon-perturbative regime. The use of different lattice discretizations allows to obtain inde-pendent determinations of the SSF at finite lattice spacing. Universality of the continuumlimit then imposes the constraint that results from different discretizations extrapolate to acommon value at vanishing lattice spacing. This fact can be exploited in order to constrainfits corresponding to different discretizations, so to reduce the systematic uncertainty.The paper is organized as follows. In sect. 2 we introduce the multiplicatively renor-malizable operator basis (cid:8) Q ′ + k (cid:9) k =1 ,..., . The RG equation, its formal solution and thestrategy used for the reconstruction of the operator scale evolution in various SF schemesare reviewed in sect. 3. Details concerning the lattice formulation and the Monte Carlosimulations are reported in sect. 4. Sect. 5 is devoted to the analysis of the numerical re-sults. Here we present a discussion of the noise-to-signal ratios observed in our simulations,– 2 –he continuum extrapolation of the SSF, the RG running in the continuum limit and theconnection to the hadronic observables. Conclusions are drawn in sect. 6. Tables and plotshave been collected in appendix A.
2. Static-light four-fermion operators
Here we briefly review the definition of the operator basis used in our calculation. For fulldetails, see sect. 2 of ref. [3].We consider a theory with a light quark sector consisting of two massless O( a ) improvedWilson-type fermions ( ψ , ψ ) and a heavy quark, represented by a pair of static fields( ψ h , ψ ¯ h ), which propagate respectively forward and backward in time. We are interestedin ∆ B = 2 static-light four-fermion operators. These are generically defined via O ± Γ Γ = 12 (cid:2) ( ¯ ψ h Γ ψ )( ¯ ψ ¯ h Γ ψ ) ± ( ¯ ψ h Γ ψ )( ¯ ψ ¯ h Γ ψ ) (cid:3) , (2.1)where Γ , are Dirac matrices, and the notation O ± Γ Γ ± Γ Γ ≡ O ± Γ Γ ± O ± Γ Γ (2.2)is adopted. Our attention will be restricted to the subset of the above operators which areodd under parity and are eigenvectors of the flavour exchange symmetry {S : ψ ↔ ψ } with positive eigenvalue. The operator basis commonly used in the literature is( Q + , Q + , Q + , Q + ) = (cid:0) O +VA+AV , O +PS+SP , O +VA − AV , O +SP − PS (cid:1) . (2.3)Note that the tensor structure T ˜T is redundant in the static approximation. The aboveoperator basis exhibits a non-trivial mixing pattern under renormalization, which makesit unsuitable to a non-perturbative numerical study of the RG running. As shown in [3],the mixing can be fully disentangled by taking appropriate linear combinations of the Q + k ’swith integer coefficients, namely (cid:0) Q ′ + , Q ′ + , Q ′ + , Q ′ + (cid:1) = ( Q + , Q + + 4 Q + , Q + + 2 Q + , Q + − Q + ) . (2.4)The operators Q ′ + k renormalize purely multiplicatively. The existence of a rearrangementof the standard operators, which yields multiplicative renormalizability without the needfor a fine tuning of the mixing coefficients with the bare coupling, is a consequence of theheavy quark spin symmetry characterizing the effective static field theory. It is thereforepeculiar to the ∆ B = 2 parity-odd four-fermion operators in HQET.
3. Renormalization group running
In order to prepare the ground for our study of the scale evolution of the operators Q ′ + k ,some basic concepts of the RG theory are briefly reviewed. A sketch of the computationalstrategy for the numerical reconstruction of the non-perturbative RG running in the SFscheme is then depicted. – 3 – .1 Callan-Symanzik equation The scale evolution of the operators provided by Eq. (2.4) is governed by a set of scalarCallan-Symanzik equations, µ ∂∂µ + β ∂∂g R + τ N f X j =1 m R ,j ∂∂m R ,j − γ ′ + k (cid:0) Q ′ + k (cid:1) R = 0 , (3.1)where k = 1 , . . . ,
4, and the renormalized operator is related to the bare lattice one through (cid:0) Q ′ + k (cid:1) R ( µ ) = lim a → Z ′ + k ( g , aµ ) Q ′ + k ( a ) . (3.2)Here g denotes the bare gauge coupling. If a mass-independent renormalization scheme isadopted, as assumed in the following, the RG functions β , τ and γ ′ + k depend only upon thecoupling. In particular, β ( g ) and τ ( g ) control the running of the renormalized parameters¯ g ( µ ) and m j ( µ ) through the RG equations µ ∂ ¯ g∂µ = β (¯ g ) , µ ∂m j ∂µ = τ (¯ g ) m j , (3.3)while the anomalous dimension γ ′ + k ( g ), which provides the radiative correction to the clas-sical scaling of Q ′ + k , is related to the renormalization constant Z ′ + k via a logarithmic deriva-tive, γ ′ + k (¯ g ( µ )) = lim a → (cid:18) µ ∂∂µ Z ′ + k ( g , aµ ) (cid:19) Z ′ + k ( g , aµ ) − . (3.4)We emphasize that β , τ and γ ′ + k are non-perturbatively defined functions. Their dependenceupon the coupling constant in the short-distance regime is expected to be asymptoticallydescribed by the first terms of the perturbative expansions β ( g ) = − g (cid:2) b + b g + b g + O ( g ) (cid:3) , (3.5) τ ( g ) = − g (cid:2) d + d g + O ( g ) (cid:3) , (3.6) γ ′ + k ( g ) = − g h γ ′ + ;(0) k + γ ′ + ;(1) k g + O ( g ) i . (3.7)The universality of the lowest order coefficients can be demonstrated by relating the Callan-Symanzik equations corresponding to different renormalization schemes. In particular, theleading order (LO) coefficients b and d , and the next-to-leading order (NLO) one b arefound to be b = (4 π ) − (cid:8) N − N f (cid:9) , (3.8) b = (4 π ) − (cid:8) N − (cid:0) N − N − (cid:1) N f (cid:9) , (3.9) d = (4 π ) − (cid:8) N − N − (cid:9) . (3.10)– 4 –n all renormalization schemes. The LO coefficients γ ′ + ;(0) k of the anomalous dimensionsof four-fermion operators are universal as well. Their values have been obtained in [3] byrotating the LO coefficient of the anomalous dimension matrix in the operator basis Q + k ,originally computed in [21, 22], to the diagonal basis Q ′ + k . These coefficients read γ ′ +;(0)1 = − (4 π ) − (cid:0) N − N − (cid:1) , (3.11) γ ′ +;(0)2 = − (4 π ) − (cid:0) N − − N − (cid:1) , (3.12) γ ′ +;(0)3 = − (4 π ) − (cid:0) N + 3 − N − (cid:1) , (3.13) γ ′ +;(0)4 = − (4 π ) − (cid:0) N − − N − (cid:1) . (3.14)The formal solution of the Callan-Symanzik equation relates the scheme-dependentRG running operator (cid:0) Q ′ + k (cid:1) R ( µ ) to the renormalization group invariant one (cid:0) Q ′ + k (cid:1) RGI , (cid:0) Q ′ + k (cid:1) RGI = (cid:0) Q ′ + k (cid:1) R ( µ ) (cid:20) ¯ g ( µ )4 π (cid:21) − γ ′ +;(0) k / b exp ( − Z ¯ g ( µ )0 dg γ ′ + k ( g ) β ( g ) − γ ′ + ;(0) k b g !) . (3.15)From a mathematical point of view, the RGI operator can be interpreted as the “integrationconstant” of the solution of the Callan-Symanzik equation. As such, it is uniquely definedup to an overall scale-independent factor. In Eq. (3.15) we have adopted the normalizationusually employed with four-fermion operators. The RGI operator can be easily shown to beindependent of the renormalization scheme. Note that all the scale dependence is carriedby a factor,ˆ c ′ + k ( µ ) = (cid:20) ¯ g ( µ )4 π (cid:21) − γ ′ +;(0) k / b exp ( − Z ¯ g ( µ )0 dg γ ′ + k ( g ) β ( g ) − γ ′ + ;(0) k b g !) , (3.16)which represents the integration of the RG functions β ( g ) and γ ′ + k ( g ) in the whole range ofrenormalization scales from µ to infinity. This integral receives perturbative contributionsin the region where ¯ g ( µ ) ≪
1. The total amount of non-perturbative contributions dependson how deeply in the non-perturbative regime the renormalization scale µ is placed and onthe rate of convergence of perturbation theory at the scale µ in the chosen renormalizationscheme. The computation of the evolution factor ˆ c ′ + k ( µ ) requires full knowledge of the RG functionsover a large range of scales. Numerical simulations can provide an insight into the non-perturbative region, but for that purpose Eq. (3.16) is of little practical use. We shall nowdescribe how the scale evolution can be determined non-perturbatively from low energies,corresponding to typical hadronic scales, to high energies, where the coupling is sufficientlysmall to make contact with perturbation theory.– 5 –ur task is to compute the proportionality factor between renormalized operators ata low-energy hadronic scale µ had and their counterparts at the scale µ , i.e. (cid:0) Q ′ + k (cid:1) R ( µ ) = U ′ + k ( µ, µ had ) (cid:0) Q ′ + k (cid:1) R ( µ had ) . (3.17)The renormalization is multiplicative, and hence U ′ + k is given by the ratio U ′ + k ( µ, µ had ) = ˆ c ′ + k ( µ had ) / ˆ c ′ + k ( µ ) . (3.18)Typically, we will think of the scale µ to lie in the ultraviolet, such that µ ≫ µ had . Sinceit is difficult to accommodate scales that differ by orders of magnitude in a single latticecalculation, it is useful to factorize the evolution and adopt a recursive approach. Theso-called step scaling functions (SSFs) σ + k and σ describe the change in the operators andthe gauge coupling, respectively, when the energy scale µ is decreased by a factor 2, i.e. σ ( u ) = ¯ g ( µ/ , u ≡ ¯ g ( µ ) ; σ + k ( u ) = ˆ c ′ + k ( µ/ c ′ + k ( µ ) . (3.19)In sect. 3.3 we shall sketch how σ + k and σ can be computed for a sequence of couplings u i , i =0 , , , . . . in lattice simulations. For the moment we simply state that the relation betweenoperators renormalized at scales µ had and 2 n µ had is obtained from the product of SSFs via U ′ + k (2 n µ had , µ had ) = ( n − Y i =0 σ + k ( u i ) ) − , u i = ¯ g (2 ( i +1) µ had ) . (3.20)If µ had is taken to be a few hundreds of MeV, it is safe to assume that 2 n µ had lies in aregime where perturbation theory can be applied, provided that one succeeds in computingthe SSFs for a sufficiently large number of steps. In our numerical determination describedin sect. 5 we have used n = 8, and thus we could trace the evolution non-perturbativelyover three orders of magnitude.Assuming that µ pt ≡ n µ had is large enough, one can evaluate ˆ c ′ + k ( µ pt ) by inserting theperturbative expressions for the anomalous dimensions and the β -function into Eq. (3.16).The relation between the RGI operators and their counterparts at the hadronic scale isthus given by (cid:0) Q ′ + k (cid:1) RGI = ˆ c ′ + k ( µ pt ) U ′ + k ( µ pt , µ had ) (cid:0) Q ′ + k (cid:1) R ( µ had ) . (3.21)It remains to specify the total renormalization factor ˆ Z ′ + k, RGI which links the RGI operatorto the bare operator Q ′ + k ( a ) on the lattice via (cid:0) Q ′ + k (cid:1) RGI = ˆ Z ′ + k, RGI ( g ) Q ′ + k ( a ) , (3.22)where the total renormalization factor is given by the productˆ Z ′ + k, RGI ( g ) = ˆ c ′ + k ( µ pt ) U ′ + k ( µ pt , µ had ) Z ′ + k ( g , aµ had ) . (3.23)– 6 –he factor Z ′ + k ( g , aµ had ) must be determined for each operator in a lattice simulation atfixed µ had for a range of bare couplings, using suitable renormalization conditions. Westress that the combination ˆ c ′ + k ( µ pt ) U ′ + k ( µ pt , µ had ) represents the universal, regularization-independent contribution to ˆ Z ′ + k, RGI . Finally, we note that all reference to the scales µ pt and µ had drops out in the total renormalization factor. The non-perturbative renormalization of local composite operators via the Schr¨odingerfunctional has become a standard method. The SF scheme is based on the formulation ofQCD in a finite space-time volume T × L , with periodic spatial boundary conditions andDirichlet boundary conditions at Euclidean times x = 0 , T . [6, 7]. By imposing suitablerenormalization conditions at vanishing quark mass and by choosing a particular aspectratio T /L , the box size L remains the only scale in the formulation. The dependenceof composite operators and the gauge coupling on the renormalization scale can thus beprobed by changing the volume. In particular, the step scaling functions for a variety ofoperators can be computed via recursive finite-size scaling, ranging over several orders ofmagnitude in the physical box size.In order to fully specify our adopted finite-volume scheme, we have set the aspectratio to T /L = 1. Furthermore, as in ref. [10] we have imposed periodic spatial boundaryconditions up to a phase θ = 0 . Z ′ + k are given in terms of suitable correlation functions ofthe operators Q ′ + k (see Eq. (3.16) of [3]). Note that in [3] our notation for the renormal-ization constants and the corresponding SSFs is supplemented by two additional indices,e.g. Z ′ + ;( s ) k ; α . The index s = 1 , . . . , which can be used in order to probe the four-fermion operators Q ′ + k ; the index α = 0 , / . However, for notational clarity we shall drop the additional indices in the follow-ing. For each combination of s and α we compute the lattice SSFs of the operators Q ′ + k ,defined as Σ + k ( u, a/L ) = Z ′ + k ( g , a/ L ) Z ′ + k ( g , a/L ) (cid:12)(cid:12)(cid:12)(cid:12) m =0 , ¯ g ( L )= u , (3.24)i.e. the SSFs are evaluated in the chiral limit, m ( g ) = 0, (where m is the PCAC quarkmass defined following ref. [10]), for a given lattice size L/a and at fixed renormalized SFcoupling ¯ g SF ( L ) = u, µ = 1 /L . The lattice SSFs Σ + k depend not only on the definition See Eqs. (3.4)–(3.8) of [3]. See Eqs. (3.11)–(3.15) of [3]. – 7 –f the renormalization scheme, but also on the details of the lattice regularization. Theyhave, however, a well defined continuum limit, viz. σ + k ( u ) = lim a → Σ + k ( u, a/L ) . (3.25)Thus, at each fixed value of the renormalized coupling, the SSFs in the continuum limit areobtained by computing Σ + k ( u, a/L ) for several values of the lattice spacing and performingan extrapolation to vanishing lattice spacing.Our task is the determination of the scale evolution factor U ′ + k of Eq. (3.20) for µ had =1 / (2 L max ), where the scale L max is implicitly defined through¯ g SF ( L max ) = 3 . . (3.26)This value of the coupling corresponds to L max /r = 0 . r = 0 . µ had ≈
270 MeV. The sequence of couplings u i = ¯ g SF (2 − i L max ) , i = 0 , , , . . . (3.27)is computed by solving the recursion u = 3 . , σ ( u l +1 ) = u l . (3.28)The SSF of the coupling σ ( u ) has been calculated in the quenched approximation in [8,10].The SSFs of the four-fermion operators can then be evaluated for the sequence of couplings, u l , l = 0 , , , . . . , and Eq. (3.20) yields the RG evolution between the hadronic scale µ had =1 / (2 L max ) and the high-energy scale µ = 2 n − /L max .As described below in sect. 5.3, in practice we fit the data for each SSF to a polynomialand use the resulting fit functions in the recursions of Eqs. (3.28) and (3.20).
4. Lattice setup
As previously stated, light quarks are discretized in this work according to the Wilsonprescription with O( a ) Symanzik improvement. The general concept how to implementO( a ) improvement in the SF has been presented in refs. [6, 24]. As usual, the improvementof the Wilson action is achieved by adding the standard Sheikholeslami-Wohlert term [25].Field theories defined in finite volume with boundaries, such as the SF of QCD, requirethat suitable boundary counterterms be included as well, in order to fully cancel O( a ) latticeartefacts. The particular realization of the SF of refs. [6,24], which we adopt in this paper,lists two relevant counterterms, multiplied by the improvement coefficients c t ( g ) − c t ( g ) −
1, respectively.The improvement coefficient c sw has been computed non-perturbatively in the quenchedapproximation for a range of values of the bare coupling g [26] and is parameterized by– 8 –he interpolating formula c sw ( g ) = 1 − . g − . g − . g − . g . (4.1)By contrast, the coefficients c t and ˜ c t are known only in perturbation theory to NLO [27]and LO [28] respectively: c t ( g ) = 1 − . g − . g , (4.2)˜ c t ( g ) = 1 − . g . (4.3)Heavy quarks are treated in the static approximation. The original lattice action, firstderived by Eichten and Hill in [29], has been subsequently generalized in [20], in orderto improve the signal-to-noise ratio of static-light correlators at large time separations.Following this approach, we write it in the form S stat [ ψ h , ¯ ψ h , ψ ¯ h , ¯ ψ ¯ h , U ] = a X x (cid:2) ¯ ψ h ( x ) D W ∗ ψ h ( x ) − ¯ ψ ¯ h ( x ) D W0 ψ ¯ h ( x ) (cid:3) , (4.4)where the covariant derivatives are defined according to D W0 ψ ( x ) = 1 a (cid:2) W ( x ) ψ ( x + a ˆ0) − ψ ( x ) (cid:3) ,D W ∗ ψ ( x ) = 1 a h ψ ( x ) − W † ( x − a ˆ0) ψ ( x − a ˆ0) i . (4.5)In order to reproduce the original Eichten-Hill formulation, the generalized parallel trans-porter W ( x ) must be replaced by the temporal gauge link U ( x ). Moreover, the choiceof W ( x ) is constrained by the requirement of keeping the theory in the same universalityclass, to guarantee a unique continuum limit. In the following, we consider four differentchoices of W ( x ), all compliant with this requirement, i.e. W EH0 ( x ) = U ( x ) , (4.6) W APE0 ( x ) = V ( x ) , (4.7) W HYP10 ( x ) = V HYP0 ( ~α, x ) (cid:12)(cid:12) ~α =(0 . , . , . , (4.8) W HYP20 ( x ) = V HYP0 ( ~α, x ) (cid:12)(cid:12) ~α =(1 . , . , . . (4.9)In the above definitions V ( x ) denotes the average of the six staples surrounding the gaugelink U ( x ) and V HYP0 ( x ) represents the temporal HYP link of [30] with the approximateSU(3) projection of [20]. Two sets of HYP-smearing coefficients ~α are considered, leadingto two independent realizations of the HYP smeared parallel transporter. The static ac-tions so assembled are automatically O( a ) improved, without the need of time-boundarycounterterms, and differ among each other at finite cutoff by O( a ) terms.– 9 –he O( a ) improvement of correlation functions of composite operators is completedthrough the inclusion of the appropriate higher dimension counterterms in the lattice defi-nition of the local operators. We do not employ operator improvement here, and thereforewe expect that the dominant discretization effects are of O( a ). We note, however, that thecorrelation functions defined in Eq. (3.9) of [3] are O( a ) tree-level improved, implying thatall O( a ) counterterms to the local four-fermion operators vanish at this order. Thus we areleft with discretization errors of order g a . In our quenched simulations the bare coupling g (equivalently, β = 6 /g ) must be tunedfor a given lattice size, in order to produce a fixed value of the renormalized coupling ¯ g SF .Furthermore, renormalization conditions for four-quark operators are imposed at vanishingquark mass, expressed in terms of the critical hopping parameter, κ cr .The complete set of simulation parameters is reported in the first four columns ofTables 2 and 3. For each of the 14 values of the renormalized SF coupling mentioned,we have considered four different lattice resolutions, corresponding to L/a = 6 , , , β have been tuned at the various lattice spacings so to have ¯ g SF ( L ) = u i . Atfixed bare coupling we define κ cr as the value where the PCAC quark mass m ( g ) ofref. [10] vanishes. Following [10], the computation of κ cr is done at θ = 0. The amountof statistical samples generated in the course of the Monte Carlo simulations has beenfixed according to the value of the SF coupling and the lattice spacing, ranging fromO(1000 − − L/ a heatbath moves per overrelaxation). On eachindependent configuration the Dirac operator has been inverted via the BiCGStab solverwith SSOR-preconditioning [31, 32].
5. Numerical results
A compilation of renormalization factors Z ′ + k for all renormalized couplings, lattice spac-ings, schemes and discretizations of the static action would easily exceed the size of anordinary paper. For the sake of reproducibility, we report in Tables 2–9 those correspond-ing to the particular case of the HYP2 action and our preferred choice of the renormalizationschemes, i.e. ( s, α ) = (1 ,
0) for Q ′ + , , and ( s, α ) = (3 ,
0) for Q ′ + . A complete set of tablesand plots is available for download from the website [33]. A precise determination of the RGI renormalization constants can only be achieved if thestatistical error of the SSF at each simulated coupling and lattice spacing is kept under– 10 –ontrol. It is therefore important to monitor the noise-to-signal ratio R X (Σ + k ) = ∆Σ + k Σ + k , X = (EH , APE , HYP1 , HYP2) , (5.1)characteristic of the four chosen lattice discretizations of the static action. Here, ∆Σ + k denotes the statistical uncertainty of the SSF Σ + k , computed via the jackknife method.According to [20], R X is related to the value of the binding energy E stat ∼ a e (1) g + . . . of the static-light meson, which diverges linearly in the continuum limit. The leadingcoefficient e (1) depends upon the lattice discretization of the static action and its value setsthe rate of growth of the noise-to-signal ratio: a linear reduction of e (1) corresponds indeedto an exponential damping of the statistical fluctuations. From our results we deduce thegeneral trend R EH ≫ R APE & R HYP1 & R HYP2 . (5.2)As an example, we compare in Figures 1 and 2 the noise-to-signal ratio of the SSFof the operators {Q ′ + k } , k = 1 , . . . , R HYP2 isalmost constant against variations in the renormalized coupling and always lower than 1%.Moreover, it never increases by more than a factor 4 going from the coarsest to the finestlattice resolution at fixed coupling. This picture is completely reversed when looking at theEH discretization, as shown in the upper plots. Here, a clear increase of the noise-to-signalratio with the SF coupling and also with the lattice spacing is observed. In practice, theSSF has an acceptable uncertainty only in the perturbative region, i.e. for u . + , Σ + and Σ + are slightly noisier than Σ + with the HYP2 action, incontrast to the EH one. Since the simulations with the EH action are practically unusablefor our ultimate aims, the prevailing pattern is the one observed with the HYP2 discretiza-tion and will be reflected in the final statistical error of the RGI renormalization constantsof the various operators. The lattice SSFs Σ + k must be extrapolated to the continuum limit (i.e. to vanishing a/L )at fixed renormalized gauge coupling in order to obtain their continuum counterparts σ + k .Since the four-fermion operators have not been improved, we expect the dominant dis-cretization effects to be O( a ); thus, our data should exhibit a linear behaviour in a/L . Forevery combination of ( s, α ) we have therefore fitted to the ansatz Σ + k ( u, a/L ) = σ + k ( u ) + ρ ( u ) ( a/L ) . (5.3)Fits have been performed using either the whole available set of values of L/a or, al-ternatively, without taking into account the data at
L/a = 6, which may be subject tohigher-order lattice artefacts. – 11 –ollowing the spirit of refs. [11, 14, 17], one could perform a combined fit of the datacorresponding to the actions APE, HYP1 and HYP2, all of which have comparable noise-to-signal ratios in the range of lattice parameters covered in this work. However, dataobtained with the above actions differ noticeably only – if ever – at
L/a = 6, and arevery strongly correlated. As a consequence, a combined continuum extrapolation affectsonly marginally the result coming from the best choice of the action, i.e. HYP2, with areduction of the relative error of σ + k ( u ) at the level of a few percent. Thus, without anyloss, we will consider only the HYP2 data from now on.Fit results can be summarized as follows:(i) the typical statistical accuracy of our results for σ + k ranges from ∼ .
5% relative error,for the weakest couplings and fits that keep
L/a = 6, to ∼ .
5% relative error at themaximum value of u , for fits that discard L/a = 6. When
L/a = 6 is dropped, thefitted values of ρ are always essentially compatible with zero within the statisticaluncertainty for Q ′ + and Q ′ + , signalling a weak cutoff dependence of Σ + k . For Q ′ + and Q ′ + , on the other hand, they are zero only within two standard deviations at u & L/a = 6, non-zero values of ρ are usually obtained for the couplings u & . σ only;(iii) the goodness of fit, expressed by the value of χ per degree of freedom, is mostlyaround or below 1, although it reaches large values in some cases. This does notdepend systematically on the number of the fitted points or the value of the coupling.Anyway, given the small number of fitted data points, χ / d.o.f. for each single fit atfixed value of the coupling is a goodness-of-fit criterion of limited value; instead, thetotal χ / d.o.f. (summed over all values of the coupling at fixed operator and scheme)is always around or below 1, reaching maxima of the order of 1 . L/a = 6 datum to extract our final values of σ + k . The resulting continuumlimit extrapolations are illustrated in Figs. 3–6 for our reference schemes (chosen below).Complete tables with the results are available at [33]. The analysis described above yields accurate estimates of the continuum SSFs σ + k for awide range of values of the renormalized coupling. In order to compute the RG runningof the operators in the continuum limit as described in sect. 4, we need to fit these data,as well as those for the SSF of the renormalized coupling itself, to some functional form.Regarding the SSF of the coupling σ ( u ), we have followed the same procedure as in [10].This has been also adopted for the SSFs of the four-fermion operators, for which we have– 12 –ssumed the polynomial ansatz σ + k = 1 + N X n =1 s n u n , (5.4)motivated by the form of the perturbative series. In particular, the analytical expressionsof the first two coefficients are given in perturbation theory by s = γ ′ +;(0) k ln 2 , (5.5) s = γ ′ +;(1) k ln 2 + (cid:20)
12 ( γ ′ +;(0) k ) + b γ ′ +;(0) k (cid:21) (ln 2) . (5.6)While the first coefficient is entirely determined by the LO anomalous dimension and ishence universal, the second one, where the NLO anomalous dimension enters, is schemedependent. Contrary to the case of the fully relativistic four-quark operators consideredin [34], the NLO coefficient does not depend strongly on the chosen SF scheme.We have performed fits to the ansatz of Eq. (5.4) with N ranging from 2 to 4. Thecoefficient s is always kept fixed to the value in Eq. (5.5), and fits are performed either with s fixed to the value in Eq. (5.6) or keeping it as a free parameter. All fits are well behaved,with values of χ / d.o.f. ranging from 0 . .
6. It is worth mentioning that when s iskept as a free parameter, its fitted value lies in the ballpark of the perturbative predictionof Eq. (5.6), which can be taken as an indication that perturbation theory indeed describesthe data well within a large part of the range of scales covered by our simulations. However,our data are not accurate enough to allow for a more detailed check of the applicability ofperturbation theory beyond leading order.Once a definite expression for the fitted step scaling function is chosen, the solutionof the recursion relations provided by Eqs. (3.28) and (3.20) is unique. At that point, thevalue obtained for the RG running factor U ′ + k (2 n µ had , µ had ) of Eq. (3.20) is a function ofthe fit parameters only. We have checked that increasing the number of fit parametersprovides compatible results for U ′ + k (2 n µ had , µ had ) with slightly larger errors. The result isalso fairly insensitive to whether s is fixed to the perturbative prediction or not. Theconclusion is that, at the available level of precision, the bias induced by the choice of thefit function is not significant, which results in a numerically very stable determination ofthe SSFs. We quote as our best results those coming from a two-parameter fit with s fixed by perturbation theory.At this point it is useful to restrict the attention to a selected subset of renormalizationschemes. As discussed in [3], heavy quark spin symmetry provides a number of identitiesbetween the 10 SF schemes we have considered per operator . In practice, we have fourdifferent independent schemes for the operators Q ′ + and Q ′ + , and another eight for Q ′ + and Q ′ + . All these schemes should lead to the same RGI quantities, since the total renor-malization factors (see below) differ only by cutoff effects. This could be used potentially These identities have been verified explicitly for each combination of the labels α and s at all the levelsof our numerical analysis, which provides a check of the latter. – 13 –o improve continuum limit extrapolations by combining various schemes. However, thestrong statistical correlation between the different renormalization factors is likely to pro-duce only a small gain in precision. Therefore, we choose for each operator just one singlerepresentative scheme. This strategy has been seen to be justified in the fully relativisticcase, i.e. in the computation of B K [18, 19].As discussed in [17], the main criterion to define suitable schemes amounts to checkingthat the systematic uncertainty related to truncating at NLO the perturbative matchingat the scale µ pt ≡ n µ had in Eq. (3.18) is well under control. This in turn requires anestimate of the size of the NNLO contribution to ˆ c ′ + k ( µ pt ). To this purpose we have re-computed ˆ c ′ + k ( µ pt ) with two different values of the NNLO anomalous dimensions γ ′ +;(2) k :in the first case we set γ ′ +;(2) k /γ ′ +;(1) k = γ ′ +;(1) k /γ ′ +;(0) k ; in the second case, we guess γ ′ +;(2) k by performing a one-parameter fit to the SSF with s and s fixed by perturbation theory,and then equating the resulting value of s to its perturbative expression s = γ ′ +;(2) k ln 2 + h γ ′ +;(0) k γ ′ +;(1) k + 2 b γ ′ +;(1) k + b γ ′ +;(0) k i (ln 2) ++ (cid:20) (cid:16) γ ′ +;(0) k (cid:17) + b (cid:16) γ ′ +;(0) k (cid:17) + 43 b γ ′ +;(0) k (cid:21) (ln 2) . (5.7)For the operators Q ′ + , , , we find that in either case the central value of the combinationˆ c ′ + k ( µ had ) ≡ ˆ c ′ + k ( µ pt ) U ′ + k ( µ pt , µ had ) changes by a small fraction of the statistical error, of theorder 0 . σ –0 . σ . There is no systematic dependence on the choice of boundary operatorsor normalization factors in the renormalization condition. We thus conclude that thisparticular uncertainty is well covered by the statistical one and choose as our referenceschemes those labeled by ( s, α ) = (1 , Q ′ + , which carries relatively large NLO anomalous dimensions, theeffect can be as large as 0 . σ with s = 3, and of the order of 1 σ with the other values of s .There is no significant dependence on α . We therefore opt, conservatively, for ( s, α ) = (3 , c ′ + ( µ had ) a systematic uncertainty of 0 . B – ¯ B mixing amplitude is not particularly worrying, since the matrix elementof Q ′ + enters the latter only at O ( α s ) when the static theory is matched to QCD. It istherefore expected to contribute a relatively small fraction to the final uncertainty.The results for the operator RG running in these schemes are provided in Table 1.Those concerning the SSFs are collected in Table 10. The same results are illustrated byFigure 7. The RGI operator, defined in Eq. (3.15), is connected to its bare counterpart via a totalrenormalization factor ˆ Z ′ + k, RGI ( g ), as in Eq. (3.23). We stress that ˆ Z ′ + k, RGI ( g ) is a scale-independent quantity, which moreover depends on the renormalization scheme only viacutoff effects. Indeed, it depends on the particular lattice regularization chosen, though– 14 – s ˆ c ′ + ;( s ) k ;0 ( µ had ) a ( s ) k ;0 b ( s ) k ;0 c ( s ) k ;0 . . − . . ∗ . . − . . . . − . . . . − . . Table 1:
Running factor ˆ c ′ + ;( s ) k ;0 ( µ had ) and fit coefficients (see Eq. (5.8)) to the total renormalizationfactor ˆ Z ′ + k, RGI ( g ) introduced in Eq. (3.23). Here µ − = 2 L max . The schemes characterized by largersystematic uncertainties related to perturbation theory have been indicated with an asterisk. only through the factor Z ′ + k ( g , aµ had ), the computation of which is much less expensivethan the total RG running factor ˆ c ′ + k ( µ had ).We have computed Z ′ + k ( g , aµ had ) , µ had = 1 / (2 L max ) non-perturbatively at four valuesof β for each scheme and four-fermion operator, and for the four different static actionsunder consideration. The results for the HYP2 action and the reference renormalizationschemes defined in sect. 5.3 are given in Table 11. Upon multiplying by the correspondingrunning factors in Table 1, the total renormalization factors are obtained. These can befurther fitted to polynomials of the formˆ Z ′ + k, RGI ( g ) = a k + b k ( β −
6) + c k ( β − , (5.8)which can be subsequently used to obtain the total renormalization factor at any value of β within the covered range. We provide in Table 1 the resulting fit coefficients for theHYP2 action in our reference renormalization schemes. These parameterizations representour data with an accuracy of at least 0 .
3% (this comprises the point β = 6 .
0. The contri-bution from the error in the RG running factors of Table 1 has been neglected: since thesefactors have been computed in the continuum limit, they should be added in quadrature after the quantity renormalized with the factor derived from Eq. (5.8) has been extrapo-lated itself to the continuum limit.
6. Conclusions B − ¯ B mixing remains among the most important processes that are required to pin downthe elements of the CKM matrix precisely. However, in order to constrain the unitaritytriangle sufficiently well and to look for signs of new physics, theoretical uncertainties asso-ciated with hadronic effects must be further reduced. Non-perturbative renormalization offour-quark operators is an indispensable ingredient to enable lattice determinations of thecorresponding hadronic matrix elements with a total accuracy at the level of a few percent. Note that β = 6 . – 15 –n this paper we have described our fully non-perturbative calculation of the rela-tions between parity-odd, static-light four-quark operators in quenched lattice QCD andtheir renormalized counterparts. Our main results for the complete basis of operatorsare expressed by the interpolating formula of Eq. (5.8), in conjunction with the coefficientslisted in Table 1. In addition to the regularization-dependent, total renormalization factorsˆ Z ′ + k, RGI , we also list the universal running factors ˆ c ′ + k ( µ had ), which, if desired, can be com-bined with a different fermionic discretization, provided that the regularization-dependentmatching factor Z ′ + k ( g , aµ had ) (c.f. Eq. (3.23)) is re-computed.The bulk of the uncertainty associated with the renormalization originates from theuniversal running factors, which have been determined with an accuracy of 1 . − B -parameters in the continuum limit can be reached. The calculation of the bare hadronicmatrix elements in quenched twisted mass QCD is currently underway.Finally, we stress that our method can be straightforwardly extended to the un-quenched case, with the simulations to compute the step scaling functions for N f = 2dynamical quark flavours currently in progress [35]. Although Ginsparg-Wilson fermionsappear as the natural discretization to study left-left four-quark operators, the Schr¨odingerfunctional is far more complicated to implement than for Wilson-like fermions. In ourapproach, tmQCD serves to solve the intricate renormalization problem for four-quark op-erators, while the SF scheme is easy to implement and dynamical simulations with Wilsonfermions can be performed in an economical way. Acknowledgements
We thank R. Sommer for useful discussions, and J. Heitger for his kind technical help. F.P.acknowledges financial support from the Alexander-von-Humboldt Stiftung. M.P. acknowl-edges financial support by an EIF Marie Curie fellowship of the European Community’sSixth Framework Programme under contract number MEIF-CT-2006-040458. C.P. ac-knowledges partial financial support by CICyT project FPA2006-05807. We also thankthe DESY Zeuthen computing centre for technical support. This work was supported inpart by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”.– 16 – . Tables and figures β L/a ¯ g SF ( L ) κ cr Z ′ +;(1)1;0 ( g , a/L ) Z ′ +;(1)1;0 ( g , a/ L ) Σ +;(1)1;0 ( g , a/L )10.7503 6 0.8873(5) 0.130591(4) 0.9136(8) 0.8827(12) 0.9662(15)11.0000 8 0.8873(10) 0.130439(3) 0.9041(7) 0.8751(24) 0.9679(28)11.3384 12 0.8873(30) 0.130251(2) 0.8912(7) 0.8571(26) 0.9617(30)11.5736 16 0.8873(25) 0.130125(2) 0.8827(14) 0.8467(35) 0.9592(43)10.0500 6 0.9944(7) 0.131073(5) 0.9073(8) 0.8714(8) 0.9604(12)10.3000 8 0.9944(13) 0.130889(3) 0.8943(8) 0.8590(28) 0.9605(32)10.6086 12 0.9944(30) 0.130692(2) 0.8798(7) 0.8478(25) 0.9636(30)10.8910 16 0.9944(28) 0.130515(2) 0.8737(25) 0.8365(34) 0.9574(47)9.5030 6 1.0989(8) 0.131514(5) 0.8992(9) 0.8611(11) 0.9576(15)9.7500 8 1.0989(13) 0.131312(3) 0.8876(8) 0.8484(33) 0.9558(38)10.0577 12 1.0989(40) 0.131079(3) 0.8713(10) 0.8311(32) 0.9539(38)10.3419 16 1.0989(44) 0.130876(2) 0.8645(33) 0.8274(33) 0.9571(53)8.8997 6 1.2430(13) 0.132072(9) 0.8894(9) 0.8459(12) 0.9511(17)9.1544 8 1.2430(14) 0.131838(4) 0.8781(9) 0.8314(34) 0.9468(40)9.5202 12 1.2430(35) 0.131503(3) 0.8613(8) 0.8177(23) 0.9494(28)9.7350 16 1.2430(34) 0.131335(3) 0.8490(19) 0.8058(31) 0.9491(42)8.6129 6 1.3293(12) 0.132380(6) 0.8854(10) 0.8391(12) 0.9477(17)8.8500 8 1.3293(21) 0.132140(5) 0.8714(9) 0.8192(41) 0.9401(48)9.1859 12 1.3293(60) 0.131814(3) 0.8545(12) 0.8069(35) 0.9443(43)9.4381 16 1.3293(40) 0.131589(2) 0.8400(18) 0.7915(30) 0.9423(41)8.3124 6 1.4300(20) 0.132734(10) 0.8810(10) 0.8308(12) 0.9430(17)8.5598 8 1.4300(21) 0.132453(5) 0.8668(10) 0.8104(39) 0.9349(46)8.9003 12 1.4300(50) 0.132095(3) 0.8474(9) 0.7947(40) 0.9378(48)9.1415 16 1.4300(58) 0.131855(3) 0.8304(18) 0.7770(31) 0.9357(43)7.9993 6 1.5553(15) 0.133118(7) 0.8725(10) 0.8126(14) 0.9313(20)8.2500 8 1.5553(24) 0.132821(5) 0.8573(11) 0.8051(40) 0.9391(48)8.5985 12 1.5533(70) 0.132427(3) 0.8380(20) 0.7850(39) 0.9368(52)8.8323 16 1.5533(70) 0.132169(3) 0.8261(19) 0.7677(32) 0.9293(44) Table 2:
Numerical values of the renormalization constant Z ′ +;(1)1;0 and the step scaling functionΣ +;(1)1;0 with HYP2 action at various renormalized SF couplings and lattice spacings. – 17 – L/a ¯ g SF ( L ) κ cr Z ′ +;(1)1;0 ( g , a/L ) Z ′ +;(1)1;0 ( g , a/ L ) Σ +;(1)1;0 ( g , a/L )7.7170 6 1.6950(26) 0.133517(8) 0.8664(11) 0.8024(10) 0.9261(17)7.9741 8 1.6950(28) 0.133179(5) 0.8503(9) 0.7836(40) 0.9216(48)8.3218 12 1.6950(79) 0.132756(4) 0.8292(10) 0.7684(37) 0.9267(46)8.5479 16 1.6950(90) 0.132485(3) 0.8154(21) 0.7553(41) 0.9263(55)7.4082 6 1.8811(22) 0.133961(8) 0.8571(12) 0.7891(24) 0.9207(30)7.6547 8 1.8811(28) 0.133632(6) 0.8399(11) 0.7705(38) 0.9174(47)7.9993 12 1.8811(38) 0.133159(4) 0.8192(15) 0.7493(35) 0.9147(46)8.2415 16 1.8811(99) 0.132847(3) 0.7983(28) 0.7356(30) 0.9215(50)7.1214 6 2.1000(39) 0.134423(9) 0.8478(13) 0.7687(17) 0.9067(24)7.3632 8 2.1000(45) 0.134088(6) 0.8304(12) 0.7512(33) 0.9046(42)7.6985 12 2.1000(80) 0.133599(4) 0.8040(15) 0.7241(34) 0.9006(46)7.9560 16 2.100(11) 0.133229(3) 0.7889(21) 0.7107(36) 0.9009(51)6.7807 6 2.4484(37) 0.134994(11) 0.8339(21) 0.7414(22) 0.8891(35)7.0197 8 2.4484(45) 0.134639(7) 0.8124(13) 0.7173(28) 0.8829(38)7.3551 12 2.4484(80) 0.134141(5) 0.7867(19) 0.7079(34) 0.8998(48)7.6101 16 2.448(17) 0.133729(4) 0.7734(25) 0.6899(34) 0.8920(53)6.5512 6 2.770(7) 0.135327(12) 0.8245(15) 0.7146(20) 0.8667(29)6.7860 8 2.770(7) 0.135056(8) 0.7995(14) 0.6946(36) 0.8688(48)7.1190 12 2.770(11) 0.134513(5) 0.7731(14) 0.6700(40) 0.8666(54)7.3686 16 2.770(14) 0.134114(3) 0.7546(24) 0.6531(34) 0.8655(53)6.3665 6 3.111(4) 0.135488(6) 0.8166(15) 0.6884(20) 0.8430(29)6.6100 8 3.111(6) 0.135339(3) 0.7902(14) 0.6688(32) 0.8464(43)6.9322 12 3.111(12) 0.134855(3) 0.7585(18) 0.6487(42) 0.8552(59)7.1911 16 3.111(16) 0.134411(3) 0.7399(24) 0.6291(36) 0.8503(56)6.2204 6 3.480(8) 0.135470(15) 0.8062(16) 0.6574(21) 0.8154(30)6.4527 8 3.480(14) 0.135543(9) 0.7791(14) 0.6458(44) 0.8289(59)6.7750 12 3.480(39) 0.135121(5) 0.7452(18) 0.6162(30) 0.8269(46)7.0203 16 3.480(21) 0.134707(4) 0.7243(22) 0.5951(39) 0.8216(60) Table 3:
Numerical values of the renormalization constant Z ′ +;(1)1;0 and the step scaling functionΣ +;(1)1;0 with HYP2 action at various renormalized SF couplings and lattice spacings (continued). – 18 – L/a ¯ g SF ( L ) κ cr Z ′ +;(3)2;0 ( g , a/L ) Z ′ +;(3)2;0 ( g , a/ L ) Σ +;(3)2;0 ( g , a/L )10.7503 6 0.8873(5) 0.130591(4) 1.0020(7) 0.9905(11) 0.9885(13)11.0000 8 0.8873(10) 0.130439(3) 0.9936(7) 0.9847(24) 0.9910(25)11.3384 12 0.8873(30) 0.130251(2) 0.9885(6) 0.9798(24) 0.9912(25)11.5736 16 0.8873(25) 0.130125(2) 0.9851(13) 0.9744(36) 0.9891(39)10.0500 6 0.9944(7) 0.131073(5) 1.0033(8) 0.9897(8) 0.9864(11)10.3000 8 0.9944(13) 0.130889(3) 0.9956(7) 0.9838(26) 0.9881(27)10.6086 12 0.9944(30) 0.130692(2) 0.9881(6) 0.9825(24) 0.9943(25)10.8910 16 0.9944(28) 0.130515(2) 0.9835(23) 0.9786(31) 0.9950(39)9.5030 6 1.0989(8) 0.131514(5) 1.0040(9) 0.9886(10) 0.9847(13)9.7500 8 1.0989(13) 0.131312(3) 0.9966(8) 0.9851(30) 0.9885(31)10.0577 12 1.0989(40) 0.131079(3) 0.9891(10) 0.9806(27) 0.9914(29)10.3419 16 1.0989(44) 0.130876(2) 0.9876(30) 0.9752(34) 0.9874(45)8.8997 6 1.2430(13) 0.132072(9) 1.0061(10) 0.9888(12) 0.9828(15)9.1544 8 1.2430(14) 0.131838(4) 0.9978(9) 0.9858(31) 0.9880(33)9.5202 12 1.2430(35) 0.131503(3) 0.9889(8) 0.9807(23) 0.9917(24)9.7350 16 1.2430(34) 0.131335(3) 0.9835(18) 0.9751(32) 0.9915(37)8.6129 6 1.3293(12) 0.132380(6) 1.0095(11) 0.9918(13) 0.9825(16)8.8500 8 1.3293(21) 0.132140(5) 0.9994(10) 0.9860(39) 0.9866(40)9.1859 12 1.3293(60) 0.131814(3) 0.9898(12) 0.9794(30) 0.9895(33)9.4381 16 1.3293(40) 0.131589(2) 0.9828(17) 0.9813(30) 0.9985(35)8.3124 6 1.4300(20) 0.132734(10) 1.0101(11) 0.9921(12) 0.9822(16)8.5598 8 1.4300(21) 0.132453(5) 0.9986(10) 0.9927(45) 0.9941(46)8.9003 12 1.4300(50) 0.132095(3) 0.9926(9) 0.9724(36) 0.9796(37)9.1415 16 1.4300(58) 0.131855(3) 0.9864(20) 0.9790(28) 0.9925(35)7.9993 6 1.5553(15) 0.133118(7) 1.0139(12) 0.9909(16) 0.9773(20)8.2500 8 1.5553(24) 0.132821(5) 1.0015(12) 0.9926(41) 0.9911(43)8.5985 12 1.5533(70) 0.132427(3) 0.9915(20) 0.9855(43) 0.9939(47)8.8323 16 1.5533(70) 0.132169(3) 0.9844(19) 0.9784(33) 0.9939(38) Table 4:
Numerical values of the renormalization constant Z ′ +;(3)2;0 and the step scaling functionΣ +;(3)2;0 with HYP2 action at various renormalized SF couplings and lattice spacings. – 19 – L/a ¯ g SF ( L ) κ cr Z ′ +;(3)2;0 ( g , a/L ) Z ′ +;(3)2;0 ( g , a/ L ) Σ +;(3)2;0 ( g , a/L )7.7170 6 1.6950(26) 0.133517(8) 1.0153(13) 0.9938(12) 0.9788(17)7.9741 8 1.6950(28) 0.133179(5) 1.0025(10) 0.9892(40) 0.9867(41)8.3218 12 1.6950(79) 0.132756(4) 0.9928(11) 0.9866(37) 0.9938(39)8.5479 16 1.6950(90) 0.132485(3) 0.9897(21) 0.9753(43) 0.9855(49)7.4082 6 1.8811(22) 0.133961(8) 1.0137(13) 1.0021(26) 0.9886(28)7.6547 8 1.8811(28) 0.133632(6) 1.0051(13) 0.9927(56) 0.9877(57)7.9993 12 1.8811(38) 0.133159(4) 0.9917(16) 0.9835(34) 0.9917(38)8.2415 16 1.8811(99) 0.132847(3) 0.9876(30) 0.9762(31) 0.9885(44)7.1214 6 2.1000(39) 0.134423(9) 1.0249(15) 1.0011(21) 0.9768(25)7.3632 8 2.1000(45) 0.134088(6) 1.0113(14) 0.9971(43) 0.9860(45)7.6985 12 2.1000(80) 0.133599(4) 0.9944(17) 0.9911(43) 0.9967(46)7.9560 16 2.100(11) 0.133229(3) 0.9857(22) 0.9841(39) 0.9984(46)6.7807 6 2.4484(37) 0.134994(11) 1.0276(25) 1.0153(30) 0.9880(38)7.0197 8 2.4484(45) 0.134639(7) 1.0158(15) 1.0013(35) 0.9857(37)7.3551 12 2.4484(80) 0.134141(5) 1.0005(21) 1.0035(43) 1.0030(48)7.6101 16 2.448(17) 0.133729(4) 0.9965(30) 0.9923(39) 0.9958(49)6.5512 6 2.770(7) 0.135327(12) 1.0371(19) 1.0174(27) 0.9810(31)6.7860 8 2.770(7) 0.135056(8) 1.0233(17) 1.0144(55) 0.9913(56)7.1190 12 2.770(11) 0.134513(5) 1.0054(17) 0.9972(54) 0.9918(56)7.3686 16 2.770(14) 0.134114(3) 0.9959(30) 0.9976(45) 1.0017(54)6.3665 6 3.111(4) 0.135488(6) 1.0494(21) 1.0317(31) 0.9831(35)6.6100 8 3.111(6) 0.135339(3) 1.0269(18) 1.0191(44) 0.9924(46)6.9322 12 3.111(12) 0.134855(3) 1.0156(24) 1.0200(55) 1.0043(59)7.1911 16 3.111(16) 0.134411(3) 0.9981(30) 1.0006(49) 1.0025(57)6.2204 6 3.480(8) 0.135470(15) 1.0544(22) 1.0390(33) 0.9854(37)6.4527 8 3.480(14) 0.135543(9) 1.0345(20) 1.0330(68) 0.9986(69)6.7750 12 3.480(39) 0.135121(5) 1.0186(24) 1.0279(46) 1.0091(51)7.0203 16 3.480(21) 0.134707(4) 1.0067(29) 1.0233(58) 1.0165(65) Table 5:
Numerical values of the renormalization constant Z ′ +;(3)2;0 and the step scaling functionΣ +;(3)2;0 with HYP2 action at various renormalized SF couplings and lattice spacings (continued). – 20 – L/a ¯ g SF ( L ) κ cr Z ′ +;(1)3;0 ( g , a/L ) Z ′ +;(1)3;0 ( g , a/ L ) Σ +;(1)3;0 ( g , a/L )10.7503 6 0.8873(5) 0.130591(4) 0.9377(5) 0.9039(8) 0.9640(10)11.0000 8 0.8873(10) 0.130439(3) 0.9250(5) 0.8913(20) 0.9636(22)11.3384 12 0.8873(30) 0.130251(2) 0.9093(5) 0.8758(21) 0.9632(24)11.5736 16 0.8873(25) 0.130125(2) 0.8987(12) 0.8578(36) 0.9545(42)10.0500 6 0.9944(7) 0.131073(5) 0.9313(6) 0.8926(6) 0.9584(9)10.3000 8 0.9944(13) 0.130889(3) 0.9180(6) 0.8785(23) 0.9570(26)10.6086 12 0.9944(30) 0.130692(2) 0.8993(5) 0.8647(20) 0.9615(23)10.8910 16 0.9944(28) 0.130515(2) 0.8886(19) 0.8504(30) 0.9570(40)9.5030 6 1.0989(8) 0.131514(5) 0.9258(7) 0.8830(8) 0.9538(11)9.7500 8 1.0989(13) 0.131312(3) 0.9114(6) 0.8693(23) 0.9538(26)10.0577 12 1.0989(40) 0.131079(3) 0.8921(8) 0.8473(25) 0.9498(29)10.3419 16 1.0989(44) 0.130876(2) 0.8833(24) 0.8364(28) 0.9469(41)8.8997 6 1.2430(13) 0.132072(9) 0.9182(7) 0.8687(10) 0.9461(13)9.1544 8 1.2430(14) 0.131838(4) 0.9034(7) 0.8523(25) 0.9434(28)9.5202 12 1.2430(35) 0.131503(3) 0.8825(7) 0.8346(18) 0.9457(22)9.7350 16 1.2430(34) 0.131335(3) 0.8677(14) 0.8180(29) 0.9427(37)8.6129 6 1.3293(12) 0.132380(6) 0.9145(8) 0.8626(10) 0.9432(13)8.8500 8 1.3293(21) 0.132140(5) 0.8968(7) 0.8435(33) 0.9406(38)9.1859 12 1.3293(60) 0.131814(3) 0.8763(9) 0.8206(29) 0.9364(35)9.4381 16 1.3293(40) 0.131589(2) 0.8588(14) 0.8093(26) 0.9424(34)8.3124 6 1.4300(20) 0.132734(10) 0.9093(8) 0.8534(10) 0.9385(13)8.5598 8 1.4300(21) 0.132453(5) 0.8918(8) 0.8405(31) 0.9425(35)8.9003 12 1.4300(50) 0.132095(3) 0.8702(7) 0.8058(33) 0.9260(39)9.1415 16 1.4300(58) 0.131855(3) 0.8535(16) 0.7962(26) 0.9329(36)7.9993 6 1.5553(15) 0.133118(7) 0.9035(9) 0.8395(12) 0.9292(16)8.2500 8 1.5553(24) 0.132821(5) 0.8846(9) 0.8284(34) 0.9365(40)8.5985 12 1.5533(70) 0.132427(3) 0.8600(16) 0.8018(35) 0.9323(44)8.8323 16 1.5533(70) 0.132169(3) 0.8426(15) 0.7830(27) 0.9293(36) Table 6:
Numerical values of the renormalization constant Z ′ +;(1)3;0 and the step scaling functionΣ +;(1)3;0 with HYP2 action at various renormalized SF couplings and lattice spacings. – 21 – L/a ¯ g SF ( L ) κ cr Z ′ +;(1)3;0 ( g , a/L ) Z ′ +;(1)3;0 ( g , a/ L ) Σ +;(1)3;0 ( g , a/L )7.7170 6 1.6950(26) 0.133517(8) 0.8969(9) 0.8296(9) 0.9250(14)7.9741 8 1.6950(28) 0.133179(5) 0.8773(7) 0.8111(30) 0.9245(35)8.3218 12 1.6950(79) 0.132756(4) 0.8515(8) 0.7881(33) 0.9255(40)8.5479 16 1.6950(90) 0.132485(3) 0.8356(16) 0.7693(38) 0.9207(49)7.4082 6 1.8811(22) 0.133961(8) 0.8868(10) 0.8210(20) 0.9258(25)7.6547 8 1.8811(28) 0.133632(6) 0.8674(9) 0.7913(44) 0.9123(51)7.9993 12 1.8811(38) 0.133159(4) 0.8400(13) 0.7706(30) 0.9174(38)8.2415 16 1.8811(99) 0.132847(3) 0.8192(25) 0.7497(27) 0.9152(43)7.1214 6 2.1000(39) 0.134423(9) 0.8809(11) 0.7993(15) 0.9074(21)7.3632 8 2.1000(45) 0.134088(6) 0.8594(10) 0.7731(30) 0.8996(37)7.6985 12 2.1000(80) 0.133599(4) 0.8261(13) 0.7430(32) 0.8994(42)7.9560 16 2.100(11) 0.133229(3) 0.8077(18) 0.7293(32) 0.9029(45)6.7807 6 2.4484(37) 0.134994(11) 0.8664(18) 0.7746(21) 0.8940(31)7.0197 8 2.4484(45) 0.134639(7) 0.8434(10) 0.7454(27) 0.8838(34)7.3551 12 2.4484(80) 0.134141(5) 0.8097(16) 0.7245(32) 0.8948(44)7.6101 16 2.448(17) 0.133729(4) 0.7931(22) 0.7036(30) 0.8872(45)6.5512 6 2.770(7) 0.135327(12) 0.8572(13) 0.7481(19) 0.8727(26)6.7860 8 2.770(7) 0.135056(8) 0.8303(11) 0.7221(33) 0.8697(41)7.1190 12 2.770(11) 0.134513(5) 0.7979(12) 0.6895(42) 0.8641(55)7.3686 16 2.770(14) 0.134114(3) 0.7753(22) 0.6707(33) 0.8651(49)6.3665 6 3.111(4) 0.135488(6) 0.8488(13) 0.7229(20) 0.8517(27)6.6100 8 3.111(6) 0.135339(3) 0.8182(12) 0.6960(31) 0.8506(40)6.9322 12 3.111(12) 0.134855(3) 0.7832(17) 0.6666(41) 0.8511(56)7.1911 16 3.111(16) 0.134411(3) 0.7593(22) 0.6458(36) 0.8505(53)6.2204 6 3.480(8) 0.135470(15) 0.8384(14) 0.6920(22) 0.8254(29)6.4527 8 3.480(14) 0.135543(9) 0.8074(13) 0.6668(45) 0.8259(57)6.7750 12 3.480(39) 0.135121(5) 0.7715(16) 0.6361(30) 0.8245(43)7.0203 16 3.480(21) 0.134707(4) 0.7445(21) 0.6115(39) 0.8214(57) Table 7:
Numerical values of the renormalization constant Z ′ +;(1)3;0 and the step scaling functionΣ +;(1)3;0 with HYP2 action at various renormalized SF couplings and lattice spacings (continued). – 22 – L/a ¯ g SF ( L ) κ cr Z ′ +;(1)4;0 ( g , a/L ) Z ′ +;(1)4;0 ( g , a/ L ) Σ +;(1)4;0 ( g , a/L )10.7503 6 0.8873(5) 0.130591(4) 0.9349(5) 0.9189(7) 0.9829(9)11.0000 8 0.8873(10) 0.130439(3) 0.9290(4) 0.9145(16) 0.9844(18)11.3384 12 0.8873(30) 0.130251(2) 0.9239(4) 0.9074(15) 0.9821(17)11.5736 16 0.8873(25) 0.130125(2) 0.9199(9) 0.9022(26) 0.9808(29)10.0500 6 0.9944(7) 0.131073(5) 0.9287(5) 0.9099(5) 0.9798(8)10.3000 8 0.9944(13) 0.130889(3) 0.9228(5) 0.9042(18) 0.9798(20)10.6086 12 0.9944(30) 0.130692(2) 0.9157(4) 0.9001(16) 0.9830(18)10.8910 16 0.9944(28) 0.130515(2) 0.9132(16) 0.8967(21) 0.9819(28)9.5030 6 1.0989(8) 0.131514(5) 0.9226(6) 0.9016(7) 0.9772(10)9.7500 8 1.0989(13) 0.131312(3) 0.9167(5) 0.8978(18) 0.9794(21)10.0577 12 1.0989(40) 0.131079(3) 0.9101(7) 0.8911(17) 0.9791(20)10.3419 16 1.0989(44) 0.130876(2) 0.9071(18) 0.8859(22) 0.9766(32)8.8997 6 1.2430(13) 0.132072(9) 0.9145(7) 0.8897(8) 0.9729(11)9.1544 8 1.2430(14) 0.131838(4) 0.9092(6) 0.8856(18) 0.9740(20)9.5202 12 1.2430(35) 0.131503(3) 0.9016(5) 0.8800(15) 0.9760(18)9.7350 16 1.2430(34) 0.131335(3) 0.8972(12) 0.8743(21) 0.9745(26)8.6129 6 1.3293(12) 0.132380(6) 0.9114(7) 0.8855(8) 0.9716(12)8.8500 8 1.3293(21) 0.132140(5) 0.9038(6) 0.8767(25) 0.9700(28)9.1859 12 1.3293(60) 0.131814(3) 0.8966(8) 0.8718(20) 0.9723(24)9.4381 16 1.3293(40) 0.131589(2) 0.8906(11) 0.8698(19) 0.9766(24)8.3124 6 1.4300(20) 0.132734(10) 0.9059(7) 0.8775(8) 0.9686(11)8.5598 8 1.4300(21) 0.132453(5) 0.8991(7) 0.8755(27) 0.9738(31)8.9003 12 1.4300(50) 0.132095(3) 0.8926(6) 0.8606(23) 0.9641(27)9.1415 16 1.4300(58) 0.131855(3) 0.8853(12) 0.8596(18) 0.9710(25)7.9993 6 1.5553(15) 0.133118(7) 0.9002(8) 0.8658(10) 0.9618(14)8.2500 8 1.5553(24) 0.132821(5) 0.8929(8) 0.8659(26) 0.9698(31)8.5985 12 1.5533(70) 0.132427(3) 0.8843(13) 0.8582(27) 0.9705(34)8.8323 16 1.5533(70) 0.132169(3) 0.8786(12) 0.8502(20) 0.9677(26) Table 8:
Numerical values of the renormalization constant Z ′ +;(1)4;0 and the step scaling functionΣ +;(1)4;0 with HYP2 action at various renormalized SF couplings and lattice spacings. – 23 – L/a ¯ g SF ( L ) κ cr Z ′ +;(1)4;0 ( g , a/L ) Z ′ +;(1)4;0 ( g , a/ L ) Σ +;(1)4;0 ( g , a/L )7.7170 6 1.6950(26) 0.133517(8) 0.8933(8) 0.8575(7) 0.9599(12)7.9741 8 1.6950(28) 0.133179(5) 0.8861(6) 0.8541(23) 0.9639(27)8.3218 12 1.6950(79) 0.132756(4) 0.8784(7) 0.8485(24) 0.9660(28)8.5479 16 1.6950(90) 0.132485(3) 0.8723(14) 0.8398(28) 0.9627(35)7.4082 6 1.8811(22) 0.133961(8) 0.8833(9) 0.8494(16) 0.9616(21)7.6547 8 1.8811(28) 0.133632(6) 0.8779(8) 0.8392(30) 0.9559(36)7.9993 12 1.8811(38) 0.133159(4) 0.8682(10) 0.8344(22) 0.9611(27)8.2415 16 1.8811(99) 0.132847(3) 0.8621(18) 0.8267(20) 0.9589(31)7.1214 6 2.1000(39) 0.134423(9) 0.8768(10) 0.8309(13) 0.9477(18)7.3632 8 2.1000(45) 0.134088(6) 0.8701(9) 0.8245(23) 0.9476(28)7.6985 12 2.1000(80) 0.133599(4) 0.8568(10) 0.8175(24) 0.9541(30)7.9560 16 2.100(11) 0.133229(3) 0.8526(14) 0.8146(24) 0.9554(32)6.7807 6 2.4484(37) 0.134994(11) 0.8617(16) 0.8114(17) 0.9416(26)7.0197 8 2.4484(45) 0.134639(7) 0.8554(9) 0.8016(21) 0.9371(26)7.3551 12 2.4484(80) 0.134141(5) 0.8451(13) 0.8032(24) 0.9504(32)7.6101 16 2.448(17) 0.133729(4) 0.8419(17) 0.7942(23) 0.9433(33)6.5512 6 2.770(7) 0.135327(12) 0.8530(11) 0.7872(16) 0.9229(22)6.7860 8 2.770(7) 0.135056(8) 0.8446(10) 0.7814(27) 0.9252(33)7.1190 12 2.770(11) 0.134513(5) 0.8344(10) 0.7730(29) 0.9264(37)7.3686 16 2.770(14) 0.134114(3) 0.8263(17) 0.7686(26) 0.9302(37)6.3665 6 3.111(4) 0.135488(6) 0.8449(12) 0.7654(16) 0.9059(23)6.6100 8 3.111(6) 0.135339(3) 0.8351(11) 0.7604(25) 0.9105(32)6.9322 12 3.111(12) 0.134855(3) 0.8241(14) 0.7551(30) 0.9163(40)7.1911 16 3.111(16) 0.134411(3) 0.8160(17) 0.7501(27) 0.9192(39)6.2204 6 3.480(8) 0.135470(15) 0.8334(13) 0.7384(17) 0.8860(25)6.4527 8 3.480(14) 0.135543(9) 0.8241(11) 0.7351(37) 0.8920(46)6.7750 12 3.480(39) 0.135121(5) 0.8144(13) 0.7319(24) 0.8987(33)7.0203 16 3.480(21) 0.134707(4) 0.8047(16) 0.7239(30) 0.8996(42) Table 9:
Numerical values of the renormalization constant Z ′ +;(1)4;0 and the step scaling functionΣ +;(1)4;0 with HYP2 action at various renormalized SF couplings and lattice spacings (continued). – 24 – σ +;(1)1;0 σ +;(3)2;0 σ +;(1)3;0 σ +;(1)4;0 Table 10:
Continuum extrapolations of Σ +;( s ) k ; α . Linear dependence on a/L is assumed. Data at L/a = 6 have not been taken into account. β L/a κ cr Z ′ +;(1)1;0 Z ′ +;(3)2;0 Z ′ +;(1)3;0 Z ′ +;(1)4;0 Table 11:
Results for Z ′ +;( s ) k ; α ( g , a/L ) at fixed scale L = 1 . r (corresponding to µ − = 2 L max ). – 25 – R E H ( Σ + ) / R H YP ( Σ + ) L/a = 6 L/a = 8 L/a = 12L/a = 16 0 0.002 0.004 0.006 0.008 0.01 1 1.5 2 2.5 3 3.5 R H YP ( Σ + ) u 0 15 30 45 60 1 1.5 2 2.5 3 3.5 R E H ( Σ + ) / R H YP ( Σ + ) L/a = 6 L/a = 8 L/a = 12L/a = 16 0 0.002 0.004 0.006 0.008 0.01 1 1.5 2 2.5 3 3.5 R H YP ( Σ + ) u Figure 1:
Comparison of the noise-to-signal ratio of the SSFs Σ + ;(1)1;0 and Σ + ;(3)2;0 computed usingthe EH and HYP2 lattice discretizations of the static action. R E H ( Σ + ) / R H YP ( Σ + ) L/a = 6 L/a = 8 L/a = 12L/a = 16 0 0.002 0.004 0.006 0.008 0.01 1 1.5 2 2.5 3 3.5 R H YP ( Σ + ) u 0 15 30 45 60 75 1 1.5 2 2.5 3 3.5 R E H ( Σ + ) / R H YP ( Σ + ) L/a = 6 L/a = 8 L/a = 12L/a = 16 0 0.002 0.004 0.006 0.008 0.01 1 1.5 2 2.5 3 3.5 R H YP ( Σ + ) u Figure 2:
Same as Fig. 1, but for the SSFs Σ + ;(1)3;0 and Σ + ;(1)4;0 . – 26 – igure 3: Continuum limit extrapolation of the SSF Σ + ;(1)1;0 at various SF renormalized couplingswith HYP2 lattice discretization of the static action. The SF coupling u increases from top-left tobottom-right, according to the first column of Table 10. – 27 – igure 4: Continuum limit extrapolation of the SSF Σ + ;(3)2;0 at various SF renormalized couplingswith HYP2 lattice discretization of the static action. The SF coupling u increases from top-left tobottom-right, according to the first column of Table 10. – 28 – igure 5: Continuum limit extrapolation of the SSF Σ + ;(1)3;0 at various SF renormalized couplingswith HYP2 lattice discretization of the static action. The SF coupling u increases from top-left tobottom-right, according to the first column of Table 10. – 29 – igure 6: Continuum limit extrapolation of the SSF Σ + ;(1)4;0 at various SF renormalized couplingswith HYP2 lattice discretization of the static action. The SF coupling u increases from top-left tobottom-right, according to the first column of Table 10. – 30 – igure 7: Left column: the step scaling function σ + k (discrete points) as obtained non-perturbatively. The shaded area is the one sigma band obtained by fitting the points to a polyno-mial. The dotted (dashed) line is the LO (NLO) perturbative result. Right column: RG runningof Q ′ + k obtained non perturbatively (discrete points) at specific values of the renormalization scale µ , in units of Λ. The lines are pertrubative results at the order shown for the Callan-Symanzik β function and the operator anomalous dimension γ . – 31 – eferences [1] M. Okamoto, PoS LAT2005 (2006) 013 [arXiv:hep-lat/0510113].[2] T. Onogi, PoS
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