The static-light matrix element needed to determine $f_B$ is studied on an
anisotropic lattice with $N_f=2$. The improvement in precision due to stout
links and all-to-all propagators is investigated.Comment: Lattice2004(heavy), Fermilab, June 21-26, 2004. 3 page

Alan Ó Cais

Foley

Jon-Ivar Skullerud

K. Jimmy Juge

Mike Peardon

Morningstar

Peardon

Richie Morrin

Sinéad M. Ryan

Nuclear Physics B - Proceedings Supplements

English

arXiv

The static-light matrix element needed to determine fB is studied on an anisotropic lattice with Nf = 2. The

improvement in precision due to stout links and all-to-all propagators is investigated

Juge, K. Jimmy

Morrin, Richie

Ó Cais, Alan

Peardon, Mike

Ryan, Sinéad M.

Skullerud, Jon-Ivar

NUI Maynooth Eprint Archive

Static-light matrix elements on a dynamical antisotropic lattice

Maynooth University ePrints and eTheses Archive

Submitted to Cornell University’s online archive www.arXiv.org in 2004 by Sinead Ryan.Post-print sourced from www.arxiv.orgK. Jimmy Juge, Richie Morrin, Alan O Cais, Mike Peardon, Sinead M. Ryan, Jon-Ivar Skulleru

Cais, Alan O.

Peardon, Michael J.

Ryan, Sinead M.

Adelaide Research & Scholarship

arXiv:hep-lat/0410013v1  11 Oct 2004 Static-light matrix elements on a dynamical anisotropic latticeK. Jimmy Jugea, Richie Morrina, Alan O´ Caisa, Mike Peardona, Sine´ad M. Ryana∗, Jon-Ivar SkullerudaaTrinLat Collaboration, School of Mathematics, Trinity College, Dublin 2, IrelandThe static-light matrix element needed to determine fB is studied on an anisotropic lattice with Nf = 2. Theimprovement in precision due to stout links and all-to-all propagators is investigated.1. INTRODUCTIONThe determination of heavy-light decay con-stants and form factors relevant to CKM phe-nomenology is a priority for lattice QCD. How-ever, to make a meaningful contribution the sta-tistical and systematic uncertainties in these cal-culations must be reduced to a few percent forsimulations carried out in the full theory. Thisis extremely costly, and new methods to improveprecision while keeping computational costs mod-est are important. We report preliminary resultsfrom our investigations of heavy-light physics onanisotropic lattices using stout links and all-to-allpropagators.2. SIMULATION DETAILSThe gauge action used in the simulations de-scribed here is Symanzik-improved with a nega-tive adjoint term. It has been used previously in astudy of glueballs [1] and in an exploratory studyof quenched heavy-quark physics on anisotropiclattices [2]. The fermion action is designed specifi-cally for large anisotropies when the temporal lat-tice spacing at is much finer than the spatical lat-tice spacing, as. It incorporates the usual Wilsonterm in the temporal direction and a Hamber–Wu term in the spatial directions to remove dou-blers. The action has been discussed in detail inRefs. [2,3] and is writtenS′ = ψ¯′Mrψ′−rat2ψ¯′(D2t −g2ǫiEi)ψ′+sa3sψ¯′∑iD4iψ′ (1)∗Talk presented by S.Ryanwhere the prime indicates that fields have beenrotated, Mr = µrγiDi + γ0Dt + µrm0, µr =(1 + 1/2atm0) and the spatial derivative is im-proved with a two-hop term. The usual Wilsonparameter is r = 1 and the Wilson-like parame-ter s = 1/8. To reduce the coupling to UV gluonmodes we use stout links [5]. The preservationof differentiability with respect to the link vari-ables of this smearing technique means that thehybrid Monte Carlo algorithm can be straight-forwardly applied. Quantum fluctuations renor-malise the anisotropy ξ = as/at, and the param-eters ξg and ξq in the gauge and quark actionsrespectively must be tuned such that the mea-sured anisotropy takes its required value. In thequenched theory the gluon and quark actions canbe separately tuned. However, in unquenchedsimulations this is no longer true and the quarksand gluons must have a common anisotropy torecover the correct continuum limit of the theory.The non-perturbative tuning of this action is un-der investigation. We require that the anisotropymeasured from the sideways potential agrees withthat determined from the low-momentum rela-tivistic pion dispersion relation. This simulta-neous tuning requires three initial simulations tofind the self-consistent point [4]. The results pre-sented here are based on one (untuned) simula-tion. The results do not therefore have a physicalinterpretation; the point of interest is rather theprecision which can be reached using anisotropiclattices and all-to-all propagators.3. RESULTSIn this exploratory simulation we use the hy-brid Monte Carlo algorithm with Nf = 2 and de-12Table 1Simulation parameters for one of three runsneeded to tune the anisotropy. us, ut are thetadpole-improvement coefficients in the spatialand temporal directions.Gauge Action Quark Actionβ 1.51 atm0 −0.057ξg 8.0 ξq 6.0us 0.752121 us, ut 1.0generate sea and valence quark masses. We havegenerated 250 gauge configurations on a 83 × 48lattice, with mpi/mρ ∼ 0.65. The parameter val-ues are given in Table 1.The combination of anisotropic lattices andstout links allows us to reach momenta, p >1GeV. Figure 1 shows the light pseudoscalar me-son dispersion relation obtained from point prop-agators and Jacobi smeared quark fields. Thequark mass is close to the strange quark and thepoint n2 = 6 corresponds to momentum (1, 1, 2)in units of 2π/asL or p ∼ 1.5GeV. This result isencouraging since it implies that high-momentumlight pseudoscalars can be reliably simulated onthese dynamical lattices. This is relevant forstudies of semileptonic decays such as B → πℓνwhich have in the past been restricted to the low-momentum regime. We have not explicitly disen-tangled the benefits of the anisotropic lattice fromthe stout links but experience in the quenchedcase without stout links, indicates that spatialmomenta higher than (2, 0, 0) are difficult to re-solve, implying that stout links play an importantroˆle in these high-momentum simulations.3.1. All-to-all propagatorsA new algorithm to estimate all-to-all propa-gators was presented at this conference [6]. Withall-to-all propagators much more information canbe extracted from a gauge configuration thanwith point sources, which is of particular rele-vance for expensive dynamical simulations. Thealgorithm discussed in Ref. [6] uses low-modedominance corrected with a stochastic estimatorwhich yields the exact all-to-all propagator in afinite number of quark inversions. Once again weuse stout link smearing on the gauge fields and0 2 4 6n200.050.10.150.2(a tE(p) )2χ2/Nd.f. = 0.72/5Renormalised anisotropy, ξq = 4.99(6)Figure 1. Dispersion relation for the light quarkin the simulation; atmq = −0.057, close to thestrange quark mass.Jacobi smearing on the (light) quarks.3.2. Matrix elementsWe now study the effect of these all-to-all prop-agators on the static-light correlator. Althoughrelatively cheap to calculate, this is known to benoisy and difficult to determine with precision. Inaddition, using all-to-all propagators in the staticapproximation means that fB can be extractedfrom a combination of smeared-local (SL) andsmeared-smeared (SS) correlators rather than therestriction to local-local (LL) and SL correlatorswhen point sources are used. The advantages areclearly illustrated in Figure 2. The plot showsthe effective mass of a heavy-light meson deter-mined using both point and all-to-all propagators.The heavy quark was simulated in the static ap-proximation and the light quark is close to thestrange quark mass. It is important to note thatthe all-to-all dataset consists of 130 configurationswhereas the equivalent statistical precision usingpoint sources requires 250 configurations.Using point sources the effective mass and am-plitude (which yields fB) is determined from acombination of the LL and SL data in a regionwhere they overlap. Figure 2 shows that this istrue only for a few timeslices but also that thepurely local data in that region are becoming verynoisy, decreasing the reliability and precision withwhich fB can be extracted. In contrast, the all-to-all datasets agree over a much larger number oftimeslices reducing the uncertainties in the fitted30 10 20 30 40t/at0.150.20.250.3a tmeffpoint LL - 250 configspoint SL - 250 configsall2all SS - 130 configsall2all SL - 130 configsFigure 2. Effective mass of the static-light me-son. The light quark is close to the strange quarkmass. Open symbols are determined with pointpropagators and the solid points with all-to-allpropagators. The all-to-all data at large t/at havebeen hidden for the sake of clarity.mass and amplitude. Figure 3 shows a compari-son of the matrix-element amplitude determinedfrom point and all-to-all propagators. These areessentially double sliding window plots. The ver-tical spread represents the systematic error in theextracted value of Z2L as the fit range is variedfor the point LL and SL correlators and for theall-to-all SL and SS correlators respectively. Theplots are on the same scale and the all-to-all dataare very impressive especially since these requiredonly half the number of configurations comparedto the point sources.4. SUMMARYThe static-light matrix element was computedusing all-to-all propagators for the light quark onan anisotropic (but untuned) lattice with Nf = 2,using a stout-link background. The great advan-tage of being able to use smeared-smeared corre-lators was demonstrated by a direct comparisonwith the effective mass taken from conventionalpoint propagators. The tuning of the improved,dynamical anisotropic action is currently beingcarried out, and the resulting parameters will beused for a determination of fB. Work is also inprogress to study radially excited S-waves and P-15 20tmin(LL)0.10.20.30.40.5ZL2tSL=13tSL=14tSL=15tSL=16tSL=17tSL=18tSL=19tSL=20tSL=21tSL=2210 15 20tmin(SL)0.10.20.30.40.5ZL2tSS =   9tSS = 10tSS = 11tSS = 12tSS = 13tSS = 14tSS = 15tSS = 16tSS = 17Figure 3. A comparison of the stability of theamplitude with respect to tmin for point (upperpanel) and all-to-all (lower panel) propagators.waves; the preliminary results of this are highlyencouraging.5. ACKNOWLEDGEMENTSKJJ, RM and AOC are supported by the IITACPRTLI initiative. SR and JIS acknowledge sup-port from IRCSET.REFERENCES1. C. Morningstar and M. Peardon, Phys. Rev.D 60 (1999) 034509.2. J. Foley et al., hep-lat/0405030.3. J. Foley et al., Nucl. Phys. Proc. Suppl. 129(2004) 337–339.4. M. Peardon, Nucl. Phys. Proc. Suppl. 109A(2002) 212–217.5. C. Morningstar and M. Peardon, Phys. Rev.D 69 (2004) 054501.6. K.J. Juge et al., theseproceedings,hep-lat/0409069.

Static-light matrix elements on a dynamical anisotropic lattice

Crossref

MURAL - Maynooth University Research Archive Library

arXiv:hep-lat/0410013v1  11 Oct 2004Static-light matrix elements on a dynamical anisotropic latticeK. Jimmy Jugea, Richie Morrina, Alan Ó Caisa, Mike Peardona, Sinéad M. Ryana∗, Jon-Ivar SkullerudaaTrinLat Collaboration, School of Mathematics, Trinity College, Dublin 2, IrelandThe static-light matrix element needed to determine fB is studied on an anisotropic lattice with Nf = 2. Theimprovement in precision due to stout links and all-to-all propagators is investigated.1. INTRODUCTIONThe determination of heavy-light decay con-stants and form factors relevant to CKM phe-nomenology is a priority for lattice QCD. How-ever, to make a meaningful contribution the sta-tistical and systematic uncertainties in these cal-culations must be reduced to a few percent forsimulations carried out in the full theory. Thisis extremely costly, and new methods to improveprecision while keeping computational costs mod-est are important. We report preliminary resultsfrom our investigations of heavy-light physics onanisotropic lattices using stout links and all-to-allpropagators.2. SIMULATION DETAILSThe gauge action used in the simulations de-scribed here is Symanzik-improved with a nega-tive adjoint term. It has been used previously in astudy of glueballs [1] and in an exploratory studyof quenched heavy-quark physics on anisotropiclattices [2]. The fermion action is designed specifi-cally for large anisotropies when the temporal lat-tice spacing at is much finer than the spatical lat-tice spacing, as. It incorporates the usual Wilsonterm in the temporal direction and a Hamber–Wu term in the spatial directions to remove dou-blers. The action has been discussed in detail inRefs. [2,3] and is writtenS′ = ψ̄′Mrψ′−rat2ψ̄′(D2t −g2ǫiEi)ψ′+sa3sψ̄′∑iD4iψ′ (1)∗Talk presented by S.Ryanwhere the prime indicates that fields have beenrotated, Mr = µrγiDi + γ0Dt + µrm0, µr =(1 + 1/2atm0) and the spatial derivative is im-proved with a two-hop term. The usual Wilsonparameter is r = 1 and the Wilson-like parame-ter s = 1/8. To reduce the coupling to UV gluonmodes we use stout links [5]. The preservationof differentiability with respect to the link vari-ables of this smearing technique means that thehybrid Monte Carlo algorithm can be straight-forwardly applied. Quantum fluctuations renor-malise the anisotropy ξ = as/at, and the param-eters ξg and ξq in the gauge and quark actionsrespectively must be tuned such that the mea-sured anisotropy takes its required value. In thequenched theory the gluon and quark actions canbe separately tuned. However, in unquenchedsimulations this is no longer true and the quarksand gluons must have a common anisotropy torecover the correct continuum limit of the theory.The non-perturbative tuning of this action is un-der investigation. We require that the anisotropymeasured from the sideways potential agrees withthat determined from the low-momentum rela-tivistic pion dispersion relation. This simulta-neous tuning requires three initial simulations tofind the self-consistent point [4]. The results pre-sented here are based on one (untuned) simula-tion. The results do not therefore have a physicalinterpretation; the point of interest is rather theprecision which can be reached using anisotropiclattices and all-to-all propagators.3. RESULTSIn this exploratory simulation we use the hy-brid Monte Carlo algorithm with Nf = 2 and de-12Table 1Simulation parameters for one of three runsneeded to tune the anisotropy. us, ut are thetadpole-improvement coefficients in the spatialand temporal directions.Gauge Action Quark Actionβ 1.51 atm0 −0.057ξg 8.0 ξq 6.0us 0.752121 us, ut 1.0generate sea and valence quark masses. We havegenerated 250 gauge configurations on a 83 × 48lattice, with mπ/mρ ∼ 0.65. The parameter val-ues are given in Table 1.The combination of anisotropic lattices andstout links allows us to reach momenta, p >1GeV. Figure 1 shows the light pseudoscalar me-son dispersion relation obtained from point prop-agators and Jacobi smeared quark fields. Thequark mass is close to the strange quark and thepoint n2 = 6 corresponds to momentum (1, 1, 2)in units of 2π/asL or p ∼ 1.5GeV. This result isencouraging since it implies that high-momentumlight pseudoscalars can be reliably simulated onthese dynamical lattices. This is relevant forstudies of semileptonic decays such as B → πℓνwhich have in the past been restricted to the low-momentum regime. We have not explicitly disen-tangled the benefits of the anisotropic lattice fromthe stout links but experience in the quenchedcase without stout links, indicates that spatialmomenta higher than (2, 0, 0) are difficult to re-solve, implying that stout links play an importantrôle in these high-momentum simulations.3.1. All-to-all propagatorsA new algorithm to estimate all-to-all propa-gators was presented at this conference [6]. Withall-to-all propagators much more information canbe extracted from a gauge configuration thanwith point sources, which is of particular rele-vance for expensive dynamical simulations. Thealgorithm discussed in Ref. [6] uses low-modedominance corrected with a stochastic estimatorwhich yields the exact all-to-all propagator in afinite number of quark inversions. Once again weuse stout link smearing on the gauge fields and0 2 4 6n200.050.10.150.2(atE(p) )2χ2/Nd.f. = 0.72/5Renormalised anisotropy, ξq = 4.99(6)Figure 1. Dispersion relation for the light quarkin the simulation; atmq = −0.057, close to thestrange quark mass.Jacobi smearing on the (light) quarks.3.2. Matrix elementsWe now study the effect of these all-to-all prop-agators on the static-light correlator. Althoughrelatively cheap to calculate, this is known to benoisy and difficult to determine with precision. Inaddition, using all-to-all propagators in the staticapproximation means that fB can be extractedfrom a combination of smeared-local (SL) andsmeared-smeared (SS) correlators rather than therestriction to local-local (LL) and SL correlatorswhen point sources are used. The advantages areclearly illustrated in Figure 2. The plot showsthe effective mass of a heavy-light meson deter-mined using both point and all-to-all propagators.The heavy quark was simulated in the static ap-proximation and the light quark is close to thestrange quark mass. It is important to note thatthe all-to-all dataset consists of 130 configurationswhereas the equivalent statistical precision usingpoint sources requires 250 configurations.Using point sources the effective mass and am-plitude (which yields fB) is determined from acombination of the LL and SL data in a regionwhere they overlap. Figure 2 shows that this istrue only for a few timeslices but also that thepurely local data in that region are becoming verynoisy, decreasing the reliability and precision withwhich fB can be extracted. In contrast, the all-to-all datasets agree over a much larger number oftimeslices reducing the uncertainties in the fitted30 10 20 30 40t/at0.150.20.250.3a tmeffpoint LL - 250 configspoint SL - 250 configsall2all SS - 130 configsall2all SL - 130 configsFigure 2. Effective mass of the static-light me-son. The light quark is close to the strange quarkmass. Open symbols are determined with pointpropagators and the solid points with all-to-allpropagators. The all-to-all data at large t/at havebeen hidden for the sake of clarity.mass and amplitude. Figure 3 shows a compari-son of the matrix-element amplitude determinedfrom point and all-to-all propagators. These areessentially double sliding window plots. The ver-tical spread represents the systematic error in theextracted value of Z2L as the fit range is variedfor the point LL and SL correlators and for theall-to-all SL and SS correlators respectively. Theplots are on the same scale and the all-to-all dataare very impressive especially since these requiredonly half the number of configurations comparedto the point sources.4. SUMMARYThe static-light matrix element was computedusing all-to-all propagators for the light quark onan anisotropic (but untuned) lattice with Nf = 2,using a stout-link background. The great advan-tage of being able to use smeared-smeared corre-lators was demonstrated by a direct comparisonwith the effective mass taken from conventionalpoint propagators. The tuning of the improved,dynamical anisotropic action is currently beingcarried out, and the resulting parameters will beused for a determination of fB. Work is also inprogress to study radially excited S-waves and P-15 20tmin(LL)0.10.20.30.40.5ZL2tSL=13tSL=14tSL=15tSL=16tSL=17tSL=18tSL=19tSL=20tSL=21tSL=2210 15 20tmin(SL)0.10.20.30.40.5ZL2tSS =   9tSS = 10tSS = 11tSS = 12tSS = 13tSS = 14tSS = 15tSS = 16tSS = 17Figure 3. A comparison of the stability of theamplitude with respect to tmin for point (upperpanel) and all-to-all (lower panel) propagators.waves; the preliminary results of this are highlyencouraging.5. ACKNOWLEDGEMENTSKJJ, RM and AOC are supported by the IITACPRTLI initiative. SR and JIS acknowledge sup-port from IRCSET.REFERENCES1. C. Morningstar and M. Peardon, Phys. Rev.D 60 (1999) 034509.2. J. Foley et al., hep-lat/0405030.3. J. Foley et al., Nucl. Phys. Proc. Suppl. 129(2004) 337–339.4. M. Peardon, Nucl. Phys. Proc. Suppl. 109A(2002) 212–217.5. C. Morningstar and M. Peardon, Phys. Rev.D 69 (2004) 054501.6. K.J. Juge et al., theseproceedings,hep-lat/0409069.

peer-reviewedThe static-light matrix element needed to determine fB is studied on an anisotropic lattice with Nf = 2. The
improvement in precision due to stout links and all-to-all propagators is investigated

Static-light matrix elements on a dynamical anisotropic lattice

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