Meson Decay Constants from the Valence Approximation to Lattice QCD

a rXiv:h ep-la t/9319v17Oct1993Meson Decay Constants from the Valence Approximation to Lattice QCD F.Butler,H.Chen,J.Sexton 1,A.Vaccarino,and D.Weingarten IBM Research P.O.Box 218,Yorktown Heights,NY 10598ABSTRACT We evaluate f π/m ρ,f K /m ρ,1/f ρ,and m φ/(f φm ρ),extrapolated to physical quark mass,zero lattice spacing and infinite volume,for lattice QCD with Wilson quarks in the valence (quenched)approximation.The predicted ratios differ from experiment by amounts ranging from 12%to 17%equivalent to between 0.9and 2.8times the corresponding statistical uncertainties.1INTRODUCTIONIn a recent paper[1]we presented lattice QCD predictions for the masses of eight low-lying hardrons,extrapolated to physical quark mass,zero lattice spacing, and infinite volume,using Wilson quarks in the valence(quenched)approximation. The masses we found were within6%of experiment,and all differences between pre-diction and experiment were consistent with the calculation’s statistical uncertainty. We argued that this result could be interpreted as quantitative confirmation of the low-lying mass predictions both of QCD and of the valence approximation.It ap-peared to us unlikely that eight different valence approximation masses would agree with experiment yet differ significantly from QCD’s predictions including the full effect of quark-antiquark vacuum polarization.We have now evaluated the infinite volume,zero lattice spacing,physical quark mass limit of fπ/mρ,f K/mρ,1/fρ,and mφ/(fφmρ).To our knowledge there have been no previous systematic calculations of this physical limit of lattice meson decay constants.A review of earlier work is given in Ref.[2].The predicted mφ/(fφmρ) lies above its observed value by about its statistical uncertainty of approximately 15%.The predicted fπ/mρand1/fρare close to12%below experiment,equivalent to about0.9times the corresponding statistical uncertainties,and f K/mρis below experiment by17%,equal to2.8times its statistical uncertainty.Although overall the predicted values could be considered in fair agreement with experiment,the result that three of four decay constants range from0.9to2.8 standard deviations below experiment suggests the possiblity that the true,underlying values,if determined without statistical uncertainties,would fall somewhat below experiment.One possible source of such disagreement is that we have assumed exact isospin symmetry and ignored electromagnetic effects.In particular,although our prediction for fπlies below measured values determined from charged pion decays, reported numbers for the neutral pion decay constant[3]are quite close to our result. The significance of the closer agreement between our predicted fπand the neutral pion decay constant is unclear to us,however,as a consequence of the systematic uncertainties arising in the experimental determination of the neutral pion decay1constant.Another possible source of disagreement between our numbers and experiment may be the valence approximation itself.A simple physical argument tends to support of this alternative[4].The valence approximation may be viewed as replacing the momentum and frequency dependent color dielectric constant arising from quark-antiquark vacuum polarization with its low momentum limit[5].At low momentum, then,the effective quark charge appearing in the valence approximation will agree with the low momentum effective charge of the full theory.The valence approximation might thus be expected to be fairly reliable for low-lying baryon and meson masses, which are determined largely by the long distance behavior of the chromoelectric field.The valence approximation’s effective quark charge at higher momentum can be obtained from the low momentum charge by the Callan-Symanzik equation.As a consequence of the absence of dynamical quark-antiquark vacuum polarization,the quark charge in the valence approximation will fall faster with momentum than it does in the full theory,at short distance the attractive quark-antiquark potential in the valence approximation will be weaker than in the full theory,and meson wave functions in the valence approximation will be pulled into the origin less than in the full theory.Since decay constants are proportional to the square of wave functions at the origin,decay constants in the valence approximation could be expected to be smaller than in the full theory.The calculations described here were done on the GF11parallel computer at IBM Research[6]and use the same collection of gauge configurations and quark propagators generated for the mass calculations of Ref.[1].The full set of mass and decay constant calculations took approximately one year to complete.GF11was used in configurations ranging from384to480processors,with sustained speeds ranging from5Gflops to7Gflops.With the present set of improved algorithms and480 processors,these calculations could be repeated in less than four months.22DEFINITIONSThe normalization we adopt for pseudoscalar and vector decay constants in continuum QCD is<0|Jµj(0)|V(p,ǫ,j)>=ǫµm j F j,<0|J5µj(0)|P(p,j)>=pµf j,for vector and pseudoscalar states,|V(p,ǫ,j)>and|P(p,j)>,respectively,normal-ized by<p|q>=(2π)3p0δ( p− q).Here j is aflavor-SU(3)octet index and vector and axial vectorflavor-SU(3)currents Jµj(x)and J5µj(x)are related to cannonical continuum quark and antiquarkfields,ψc(x)γµλjψc(x),J5µj(x)=.2Assuming exact isospin symmetry,we havef i=fπi=1,...3,f i=f K i=4,...7,F i=mρ√therefore,also been adopted for the present calculation.The smeared field φr ( x ,t )is related by the lattice antiquark field ψℓ( x ,t )byφr ( x ,t )= y G r ( x − y )ψℓ( y ,t ),G r ( z )=(√r 2).The fieldψℓ( x ,t ).We take the smeared fields φ0(x )andψ(x ),respectively.From these fields,define smearedcurrents by J 5jr (x )=φr (x )γµλj φr (x ),J 5µjr (x )=C P jr ′r (t )→Z A jr ′r ′′Measured in units of the lattice spacing a the decay constants f j a and F j a are then given,for any choice of smearing size r,by2(z A j Z AP j0r)2(f j a)2=.(2.9)m j aZ V jrrThe coefficients z A j and z V j arefinite renormalizations chosen so that the lattice cur-rents z A j a3J5µj0and z V j a3Jµj0approach the continuum currents J5µj and Jµj,respectively, as the lattice spacing approaches zero.These constants are often given the“naive”values=2k P j,z ANjz V N=2k V j.(2.10)jwhere k P j and k V j are the hopping constants corresponding to the mass of the quark and antiquark for a pseudoscalar or vector meson,respectively,withflavor j,assumed here to have m q=m,4k c3k V jz V0j=1−and z V1j z V0wherez A1=1−0.31α(1/a).(2.12)msDecay constants for mesons with m q=mFor the lattice83×32atβof5.70we calculated propagators only for source r and sink r′of size0.In all other cases we calculated propagators only for source size r of2.To determine decay constants,according to Eqs.(2.8)and(2.9),fits for the lattice83×32to propagators for the single sink of size0are sufficient,while for all other latticesfits are needed for sink sizes of both0and2.To determine the range of time separations to be used infitting for eachβand k value,we evaluated effective masses m P(t),m V(t)and m P′(t)byfitting C P r′r(t),C V r′r(t),and C A r′r(t),respectively, to Eqs.(2.4)-(2.6)at time separations t and t+1.The largest interval at large t showing an approximate plateau in an effective mass graph we chose as the initial trial range on which tofit each propatator to the corresponding asymptotic form of Eqs.(2.4)-(2.6).Similarly,the largest interval at large t showing an approximate plateau in a graph of C A r′r′′(t)/C P r′r(t)we chose as the initial trial range for afit to Eq.(2.7).Figures(1)-(12)show the plateaus at large t in the effective masses m P(t), m P′(t),the plateau at large t in the ratio C A02(t)/C P02(t),and the effective mass m V(t) for propagators with source r of2and sink r′of0.Figures(1)-(4)show results for the lattice163×32atβof5.70and the largest corresponding k value,0.1675. Figures(5)-(8)show results for the lattice243×36atβof5.93and the largest corresponding k,0.1581.Figures(9)-(12)show results for30×322×40atβof6.17 and the largest corresponding k,0.1532.Fits to data for a range of t were done by minimizing thefit’sχ2determined from the full correlation matrix for the data beingfit.An automaticfitting program repeatedly carried outfits on every connected interval of four or more points within the initial trialfitting range.Forfits to Eq.(2.7),which require only a singlefitting parameter,we looked at intervals of three or more points.Thefinalfitting range was chosen by the program to be the interval with the smallest value ofχ2per degree of freedom.Altough a variety of other criteria could be used to determine thefinal fitting range,an advantage of the method we adopted is that it can be implemented automatically thereby reducing the potential for biases.The reliability of our in our final extrapolated results depends to some degree on adopting a consistent choice of fitting ranges at different parameter values.The horizontal lines in Figures(1)-(12)show thefitted values of masses and7Z A02/Z P02,and the pair of vertical lines in eachfigure indicates the interval of t values inthefinalfitting range chosen.It is perhaps useful to mention that since the effectivemass at each t depends on data both at t and t+1,the effective mass shown at thehighest t within eachfinalfitting range depends on data outside thefitting range.Thus thefitted lines tend to approximate the average of the effective masses withinthefitting range but with the effective mass at highest t omitted.In all but onecase,the data shows clear plateaus at large t extending over more than four timevalues.These plateaus appear to befitted fairly reliably.For the rho propagator,C V02(t),on the lattice243×36,the plateau is more ambiguous.Thefit is made over the four t values from9to12,for which the three corresponding effective masses fallslowly.The effective mases at t of12to15then fall about one standard deviationbelow thefit.At t of16and17,the effective masses then return to the originalfittedvalue.This behavior is consistent with a statisticalfluctuation although it makes theidentication of the plateau at which tofit more parable ambiguitiesdo not occur elsewhere in our data.The interval chosen is approximately a rescalingof the rhofitting intervals of5to8,atβof5.70and13to16,atβof6.17.Also,therho decay constant obtained from thisfit in Section4is interpolates smoothly valuesobtained from less ambiguousfits at other values of quark mass and lattice spacing.If this point were simply elimated from our extrapolations of fρin quark mass andin lattice spacing,ourfinal continuum predictions would be nearly unchanged.Value ofχ2for thefits in Figures(1)-(12)are shown in Table2.Ourfitsfor sinks with size r′of2,fits at smaller k,andfits on the lattice243×32are of comparable quality to those shown and give comparableχ2.Statistical uncertainties of parameters obtained fromfits and of any function ofthese parameters were determined by the bootstrap method[7].From each ensembleof N gauge configurations,100bootstrap ensembles were generated.Each bootstrapensemble consists of a set of N gauge configurations randomly selected from theunderlying N member ensemble allowing repeats.For each bootstrap ensemble theentirefit was repeated,including a possibly new choice of thefinalfitting interval.The collection of100bootstrap ensembles thus yields a collection of100values of anyfitted parameter or any function of anyfitted parameter.The statistical uncertainty8of any parameter is taken to be half the difference between a value which is higher than all but15.9%of the bootstrap values and a value which is lower than all but 15.9%of the bootstrap values.In the limit of large N the collection of bootstrap values of a parameter p approaches a gaussian distribution and the definition we use for statistical uncertainty approaches the dispersion,d,given by√ms(1/a).For k c we used the values determined in Ref.[1].These are listed in Table(3).To determineαms(π/a)using the mean-field improved perturbation theory relation[8,4]1MS (π/a)=<T rU/3>ms(π/a)given by4π/g2ms(1/a).The corresponding values of z A1and z V1are shown in Table(4).9Values of the decay constant in lattice units fπa for the various lattices,βandk shown in Table(1)are listed in Tables(5)-(9).As a measure of the lattice spacingin each case,Table(10)gives the rho mass in lattice units mρ(m n)a,extrapolated tothe“normal”quark mass m n which produces the physical value of mπ(m n)/mρ(m n)[1].We list also values of m n itself.These parameters are not given for the lattice83×32since in this case we were not able to calculate propagators at small enough quark mass to perform the required extrapolation reliably.Thefinite renormalizationsfor the decay constants in Tables(5)-(9)all include both the leading term z A0ofEqs.(2.11)and thefirst order mean-field improved perturbative correction z A1ofEqs.(2.12).The second column in each table gives values of fπa found from Z A r′rdetermined from a directfit of C A r′r(t)to Eq.(2.6).The third column gives fπa foundfrom Z A r′r determined byfitting the ratio C A r′r(t)/C P r′r(t)to Eq.(2.7)and then usingthe value of Z P r′r found from afit of C P r′r(t)to Eq.(2.4).The two sets of data inall cases are statistically consistent and in all cases,except for the lattice83×32 for k below0.1500,fπdetermined from ratiofits has a smaller statistical error fπdetermined from directfits.The ratio method tends to give less statistical noise,in effect,because it uses C P r′r(t),which is relatively less noisey,to determine m P and then extracts only Z A r′r from the more noisey propagator C A r′r(t).The direct method extracts both Z A r′r and m P from C A r′r(t)yielding an m P with greater noise,which is then multiplied by a possibly large t and exponentiated,feeding additional noise back into the value of Z A r′r.In the remainder of this paper we use only values of fπdetermined from ratiofits.Values of the decay constant in lattice units Fρa for the lattices shown in Table(1)are listed in the fourth column of Tables(5)-(9).Thefinite renormalizationsfor Fρshown in these tables all include both the leading term z V0of Eqs.(2.11)andthefirst order mean-field improved perturbative correction z V1of Eqs.(2.12).Thevalues of Z V r′r used to determine Fρwere all extracted from directfits of C V r′r(t)to Eq.(2.5).105VOLUME DEPENDENCEPercentage changes in decay constants going from83×32and163×32to 243×32,atβof5.70,are given in Table11.These changes are the same for all choices offinite renormalization.All of the differences appear to be of marginal statistical significance and may therefore best be viewed as upper bounds on the volume dependence of our results.A variety of different arguments suggest that, for the range of k,β,and lattice volume we have examined,the errors in valence approximation decay constants due to calculation in afinite volume L3are bounded by an expression of the form Ce−L/R.A simple non-relativistic potential model gives this expression with the radius of a hadron’s wave function for R.A more elaborate field theory argument gives for R the Compton wave length of a pair of pions,which is the lightest state that can be exchanged between a pair of identical pseudoscalar or vector mesons.Atβof5.70,R is thus very likely to be between3and5lattice units. Since the changes in decay constants shown in Table11between163and243are all less than5%for k≥0.1650,it follows that the differences between these values in 243and in infinite volume should be less than1.3%.For extrapolations to physical quark masses,we use only decay constants with k≥0.1650.6QUARK MASS EXTRAPOLATIONAt the largest k on each lattice,the ratio mπ/mρis significantly larger than its experimentally observed value of0.179.Thus to produce decay constants for hadrons containing only light quarks,our data has to be extrapolated to larger k or,equivalently,to smaller quark mass.We did not calculate directly at larger k both because the algorithms we used tofind quark propagators became too slow and because the statistical errors we found in trial calculations became too large.Define the quark mass in lattice units m q a to bem q a=12k c,(6.1)where k c is the critical hopping constant at which mπbecomes zero.We found fπa11and Fρa,for all three possible choices offinite renormalization in Eqs.(2.10)-(2.12) to be nearly linear functions of m q a over the entire range of k considered on each lattice.Figure13shows fπand Fρ,withfinite renormalizations including thefirst order mean-field improved perturbative correction,as functions of m q.Data is shown from the three lattices of Table1which we use to evaluate continuum limits,163×32, 243×36and30×322×40.For convenience,decay constants at eachβare shownin units of the central value of the rho mass mρ(m n),at the sameβ,extrapolated to the the“normal”quark mass m n.Quark masses m q for eachβare shown in units of the central value of the strange quark mass m s at the sameβ.The data Figure13has been scaled by the central values of mρ(m n)and m s taken as arbitrary external parameters,and the error bars shown do not include the effect of statistical fluctuations in mρ(m n)or m s.Table(10)gives values of m s found in Ref.[1]by requiring mπ[(m n+m s)/2]/mρ(m n)to be equal to the physical value of m K/mρ. The lines in Figure13arefits of decay constants measured in lattice units,fπa and Fρa,to linear functions of the quark mass in lattice units,m q a,at the three smallest quark masses in the data set at eachβ.Thesefits were obtained by minimizingχ2 obtained form the full correlation matrix amoung the data points.The correlation matrix was calculated by taking averages of data values and products of data values over bootstrap ensembles generated as described in Section(3).Theχ2for these fits,and correspondingfits on the lattice243×32atβof5.70,are given in Table 12.Thefits in Figure13appear to be quite good and provide,we believe,a reliable method for extrapolating decay constants down to light quark masses.With naive finite renormalization,Eq.(2.10),or zero order mean-fieldfinite renormalization, Eq.(2.11),fπa and Fρafit straight lines in m q a about as well as thefirst order perturbatively renormalized data of Figure13.The linearfits of Figure13permit the determination of f K and Fφin addition to fπand Fρ.For a pion composed of a quark and antiquark with mass m q=mmqCharge conjugation invariance then givesαq=α=m n,will have the same decay constant as aqpion composed of a single type of quark and antiquark with m q=m=m s which,in the valence approximation,gives Fφ.Tables13andq14give the value of fπa,f K a,Fρa and Fφa obtained from thefits in Figure13.The statistical uncertainties in these quantities were obtained by a further application of the bootstrap method of Section(3).Bootstrap ensembles of the underlying gauge configurations were generated,and on each bootstrap ensemble the extrapolated decay constants were recalculated.The uncertainty in each decay constant was obtained from the resulting distribution.The correlation matrices used tofit bootstrap data to linear functions of m q a were taken to be the same as the correlation matrices for the full ensemble.To recalulate correlation matrices separately on each bootstrap ensemble by a further bootstrap would have been too time consuming.7CONTINUUM LIMITThe ratios fπ/mρ,f K/mρ,Fρ/mρand Fφ/mρfor physical quark masses we then extrapolated to zero lattice spacing.For Wilson fermions the leading asymp-totic lattice spacing dependence in these decay ratios is expected to be linear in a. On the other hand,as shown in Ref.[1],mρa follows the two-loop Callan-Symanzik scaling prediction inαcase is nearly the same.For lattice period L,the quantity mρL is respectively,9.08±0.13,9.24±0.19and,averaged over three directions,8.67±0.12[1].Thefits shown in Figure14were found by minimizingχ2obtained from the full correlation matrix among thefitted data.Since both the x and y coordinates of each of the threefitted points on each line have statistical uncertainties,we evaluated χ2among all six pieces of data and chose asfitting parameters the slope and inter-cept of the line along with the x coordinate of each point.The required correlation matrices were found by the bootstrap method as were the statistical uncertainties of the extrapolated predictions.The correlation matrices used infits for each bootstrap ensemble were again taken from the full ensemble and not recalculated on each boot-strap ensemble independently.Theχ2per degree of freedom for thefits in Figure14, are given in Table15.For the lattice spacing dependence of decay ratios found using zeroth order mean-fieldfinite renormalization and using naivefinite renormalization we also made fits to linear functions of mρa.The slopes with respect to mρa of the ratios fπ/mρ, f K/mρ,Fρ/mρand Fφ/mρalong withχ2for eachfit are also given in Table15.It is clear from these results,as menitioned in Section2,that naive renormalization leads to decay ratios with significantly stronger lattice spacing dependence than found for either zeroth orfirst order improved perturbative renormalization.There also appears to be some tendency forfirst order perturbative renormalization to lead to weaker lat-tice spacing dependence than zeroth order.It follows that the extrapolations we have done to zero lattice spacing are likely to be most reliable forfirst order perturbative renormalzation and least reliable for naive renormalization.8INFINITE VOLUME LIMITThe continuum ratios we found infinite volume were then corrected to infinite volume by an adaptation of the method used in Ref.[1]to correctfinite volume continuum mass ratios to infinite volume.From fπ/mρ,withfirst order perturbative renormalization,as a function of lattice spacing a and lattice period L,both measured14in physical units,define thefinite volume correction term∆(a,L)to be∆(a,L)=fπmρ(a,L).(8.1)The quantity which we would like to determine is∆(0,9/mρ).Now the ratio fπ/mρ, for L of9/mρ,undergoes a relative change of a bit less than20%as a goes from its value a5.7atβof5.70to0.Thus we would expect an error of about20%of∆(0,9/mρ) for the approximation∆(0,9mρ).(8.2)Moreover,from our earlier discussion of the exponential approach of decay ratios to their infinite volume limits,it follows that with an additional error of about20%of ∆(0,9/mρ)we have∆(a5.7,9mρ(a5.7,13.5mρ(a5.7,9mρ(0,∞)≈fπmρ)+fπmρ)−fπmρ).(8.4)The error in this approximation should be less than about40%of6%of fπ/mρ, which is2.4%of fπ/mρ.Table16lists estimates of the systematic uncertainties in equations corresponding to Eq.(8.4)for other decay constants and other choices offinite renormalization.First order perturbative renormalization consistently gives the smallest systematic error in volume correction largely because the lattice spacing dependence of these decay ratios is smallest.The ratios fπ/mρ,f K/mρ,Fρ/mρand Fφ/mρ,for all three different choices of finite renormalization,extrapolated to zero lattice spacing with mρLfixed at9,and then corrected to infinite volume are shown in Table17.Forfirst order perturbative15renormalization we also give,in Table18,finite and infinite volume values of the ratios fπ/f K,Fρ/f K and Fφ/f K.The errors shown for infinite volume ratios are statistical only and do not include the estimates we have just given for the systematic error in our procedure for making infinite volume corrections.Forfirst order perturbative finite renormalization,a comparision of Tables16and17shows,however,that the systematic errors arising from our method of obtaining infinite volume results are much smaller than the statistical errors.For the range ofβused in our extrapolation to zero lattice spacing,first order mean-field theory improved perturbation expan-sions have been shown[8]to work quite well for a wide variety of different quantities. In addition,as we mentioned earlier,the extrapolation to zero lattice spacing should be most reliable for this renormalization scheme.Thus we believe that the num-bers in Table17obtained usingfirst order mean-field perturbative renormalization are significantly more reliable than those found using the other two renormalization methods.Zeroth order perturbative renormalization has been included,however,to provided some measure of the degree to which our results may be sensitive to the choice of renormalization.Half of the difference betweenfirst order and zeroth order mean-field perturbative renormalization appears to us to be a conservative estimate of the systematic uncertainty in thefirst order results arising from the missing sec-ond and higher order perturbative renormalization contributions.In all cases this uncertainty is significantly less than the statistical errors.Predictions obtained with naive renormalization have been included in Table17largely as a curiosity.It is interesting to notice,however,that the difference between thefinal,infinite volume results found with naive renormalization and those found withfirst order perturbative renormalization is still less than1.5times the naive renormalization statistical errors.The predicted infinite volume ratios in Table17are all statistically consistent with the correspondingfinite volume ratios.The main consequence of the correction to infinite volume is an increase in the size of the statistical uncertainty in each prediction.The experimental numbers shown in Table17for fπand f K are from charged particle decays and for fρfrom neutral decays.In all cases the uncertainties in the experimental values are0.001or less.As mentioned in the introduction,an16experimental value for the neutral pion decay[3]gives fπ/mρof0.110±0.005,which is quite close to our prediction.The systematic uncertainties in this experimental number,however,are larger than those for the charged pion decay.As a result the significance of the improved agreement of our prediction with the observed neutral pion decay constant is unclear to us.We would like to thank Paul Mackenzie for discussions,and Mike Cassera, Molly Elliott,Dave George,Chi Chai Huang and Ed Nowicki for their work on GF11. We are particularly grateful to Chris Sachrajda for calling our attention to an error in an earlier version of this paper.References[1]F.Butler,H.Chen,J.Sexton,A.Vaccarino and D.Weingarten,Physical ReviewLetters70,2849(1993).[2]D.Toussaint,Nucl.Phys.B(Proc.Suppl.)26(1992)3.[3]H.-J.Behrend et al.,Z.Phys.C49(1991)401.[4]A.X.El-Khadra,G.H.Hockney,A.S.Kronfeld and P.B.Mackenzie,Phys.Rev.Letts.69(1992)729.[5]D.H.Weingarten,Phys.Lett.109B,57(1982);Nuclear Physics,B215[FS7],1(1983).[6]D.Weingarten,Nucl.Phys.B(Proc.Suppl.)17(1990)272.[7]B.Efron,The Jacknife,the Bootstrap and Other Resampling Plans,Society forIndustrial and Applied Mathematics,Philadelphia,1982.[8]G.P.Lepage and P.B.Mackenzie,to appear in Physical Review D48(1993).1783×32 5.700.1400-0.165010002439163×32 5.700.1600-0.16752000219243×32 5.700.1600-0.1675400058243×36 5.930.1543-0.158******** 30×322×40 6.170.1500-0.153********latticeβC P02C A02C A02/C P02C V02Table2:Values ofχ2per degree of freedom forfits to propagators.83×32 5.700.169012±0.000102163×32 5.700.169405±0.000052243×32 5.700.169304±0.000035243×36 5.930.158948±0.00002630×322×40 6.170.153763±0.000018。