Table Of Content

4 November, 1993 OSU{PP #93{1121 Table 1 Summaryof Lattice Results for OPE-mq-improvedcontinuummodel. Interpolating Fields MN (cid:21)N (cid:24) s0 3 (GeV) (GeV ) (GeV) (cid:0) (cid:1) abc a b c (cid:3) (cid:31)1 =(cid:15) (cid:0)u C(cid:13)5(cid:1)d u 0.938 0.016(3) 10.0(3) 1.97(12) abc a b c y (cid:31)2 =(cid:15) u Cd (cid:13)5u Not seen 0.0002(3) 2.43(12) 1.84(9) ((cid:31)1(cid:31)2+(cid:31)2(cid:0)(cid:31)1)=2 (cid:1) Fixed 0.0018(14) 2.46(15) 1.76(22) abc a b (cid:22) c (cid:31)SR =(cid:15) (cid:0) u C(cid:13)(cid:22)u(cid:1) (cid:13)5(cid:13) d 0.96(4) 0.035(6) 6.6(2) 2.18(12) abc a (cid:22) b c (cid:31)(cid:1) =(cid:15) u C(cid:13) u u 1.13(4) 0.027(7) 10.7(3) 2.39(10) (cid:3) y De(cid:12)nes the lattice spacing a. Inferred from(cid:31)1(cid:31)1 and (cid:31)1(cid:31)2 results. taminations in the correlation functions. How- is(cid:12)ed in a trivialmanner. The interpolating (cid:12)eld ever, interplay between the (cid:12)t parameters give (cid:31)2,whichvanishesinthenonrelativisticlimit,has rise to rather large uncertainties at tf =9. negligible overlap with the nucleon ground state. The results of Table 2 for (cid:21)N, evolved to a 2 4. ALTERNATE INTERPOLATORS scale of 1 GeV in the leading log approxima- tion, compare favorably with other approaches. These techniques are now used to investi- s0 is somewhat large as anticipated above. The gatenucleonproperties obtainedfromcorrelation smallervalueof(cid:21)N forRef.[2]maybedue tothe functionsofunconventionalnucleoninterpolating omissionoflarge(cid:11)s corrections totheWilsonco- (cid:12)elds which become noisyprior toa clear ground e(cid:14)cientoftheidentityoperatorusedtonormalize state domination. The results are summarizedin (cid:21)N. Such corrections could increase their result Table 1. by 20%. Thereisalonghistoryofargumentovertheop- timum nucleon interpolating (cid:12)eld to be used for REFERENCES QCDSRanalyses. Thesolutionisnowclear. This analysisindicatesthatthecontinuummodele(cid:11)ec- 1. M.A.Shifman,A.I.Vainshtein,andZ.I.Za- tively removes excited state contaminations and kharov, Nucl. Phys. B147 (1979) 385,448. allows the isolation of the ground state. Hence, 2. M.Chu, J. Grandy,S. Huang,and J. Negele, oneshouldchooseinterpolating(cid:12)eldssuchas(cid:31)SR PP MIT{CTP#2148,hep-lat/9306002,1993. [6] that give good convergence of the OPE, as 3. D. B. Leinweber, OSU-PP #93{1131,1993. the corresponding increase in continuum contri- 4. G. P. Lepage and P. B. Mackenzie, Preprint butions is tolerable. NSF-ITP-90-227,hep-lat/9209022,1992. Ground state nucleon properties are indepen- 5. D. B. Leinweber, R. M. Woloshyn, and dent of the interpolating (cid:12)eld used to excite the T. Draper, Phys. Rev. D 43 (1991) 1659. baryon from the vacuum. This invariance is sat- 6. B. L. Io(cid:11)e, Z. Phys. C 18 (1983) 67. 7. T. Schafer, E. Shuryak, and J. Ver- baarschot, Preprint SUNY-NTG-92-45A, hep-ph/9306220,1993. Table 2 8. D. B. Leinweber, Ann. Phys. (N.Y.) 198 Comparisonwith selected results for (cid:31)SR(cid:31)SR. (1990) 203. 2 Approach (cid:21)N(1 GeV ) s0 3 (GeV ) (GeV) The correlation functions used in this analysis This work 0.030(6) 2.18(12) were obtainedincollaborationwithTerryDraper Ref. [7] 0.032(1) 1.92(5) and Richard Woloshyn [5]. This research is sup- Ref. [8] 0.031(6) 1.69(15) ported in part by the Department of Energy and Ref. [2] 0.022(4) <1.4 the NationalScience Foundation. OSU{PP #93{1121 November, 1993 3 ratio of the continuum contributions as done in [2]. Our approach requires the use of tadpole- improvement[4] (cid:18) (cid:19)1=2 p 3(cid:20) 2(cid:20)! 1(cid:0) (10) 4(cid:20)cr to account for otherwise large renormalization factors. Remaining renormalization associated with composite operators is argued to be small [3]. The(cid:20)dependence ofthese twowavefunction normalizationsis very di(cid:11)erent, and is crucial to recovering the correct mass independence of the Wilson coe(cid:14)cient of the identity operator. With this approach, the e(cid:11)ects of lattice Figure 1. The nucleon mass determined in each anisotropy maybe absorbed through a combina- analysis interval plotted as a function of tf. In tion of a larger continuum strength ((cid:24) > 1) and this and the following (cid:12)gure, bullets correspond marginally larger continuum threshold (s0). To to pole plus continuum (cid:12)ts fromt=6!tf, and test the QCDSR methodas closely as possible to open squares illustrate a simple pole (cid:12)t to the its actual implementation,we (cid:12)x the value for (cid:24) regiontf(cid:0)7!tf,whichisselectedtogivesimilar obtainedinafourparameter(cid:12)t fromt=6!20. uncertainties in the nucleon mass at t=20. This value is similar to that obtained by (cid:12)tting (7) to the (cid:12)rst few timeslices of the lattice data. The value of (cid:24) depends on the choice of nucleon interpolating (cid:12)eld and is largest for (cid:31)N of (2) at 10.0(3). This suggests the presence of signi(cid:12)cant anisotropy at the second time-slice following the source [3]. Infrared lattice artifacts are not a signi(cid:12)cant problemforthisapproach. Theultravioletlattice cuto(cid:11) is modeled in a manner similarto that for the continuum model. By the second time slice followingthesource,suchcorrectionsarefoundto be negligible. In the followingwe simplydiscard the source (t = 4) and (cid:12)rst (t = 5) time slices fromthe (cid:12)t of the correlation functions. Figure 2. The nucleon coupling strength deter- minedin each analysisintervalplotted as afunc- 3. NUCLEON CORRELATOR FITS tion of tf. Symbolsare as in Figure 1. Our purpose is to test whether the nucleon mass and coupling strength can be obtained ac- curately from a (cid:12)t considering only the (cid:12)rst few Figure 2 illustrates similarresults for the cou- pointsofthecorrelationfunction. Thelatticecor- pling strength. It is interesting to see that the relation functions [5] are (cid:12)t with (9) in a three simple pole determination of (cid:21)N fails to form parameter search of (cid:21)N, MN and s0, in analysis a plateau at large time separations. Similar re- intervals from t = 6 ! tf where tf ranges from sults are seen for the larger values of (cid:20) = 0:154 9 through 23. The nucleon mass determined in and 0.156. The plateau in MN and (cid:21)N for pole each of the intervals isplotted as a function oftf plus continuum(cid:12)ts indicates that the continuum in Figure 1. s0 displays a similarplateau. model e(cid:11)ectively accounts for excited state con- 2 November, 1993 OSU{PP #93{1121 interpolating (cid:12)elds. While there are formal (cid:12)eld The form of the spectral density used in the theoreticargumentsindicatingnucleonproperties continuum model is determined by the leading are independent of the interpolating (cid:12)eld, it re- termsoftheOPEsurvivinginthe limitt!0. In mainsto demonstratethat this isinfact the case Euclidean space and coordinate gauge the quark in practice. propagator has the expansion aa0 1 (cid:13)(cid:1)x aa0 1 mq aa0 2. THE CONTINUUM MODEL Sq = 2(cid:25)2 x4 (cid:14) + (2(cid:25))2 x2 (cid:14) Consider the following two-point function for 1 (cid:10) (cid:11) aa0 (cid:0) :qq : (cid:14) +(cid:1)(cid:1)(cid:1); (6) the nucleon 223 X h (cid:0)i~p(cid:1)~x G(t;~p) = e tr and G(t) has the followingOPE ~x (cid:10) (cid:12) (cid:12) (cid:11)i 3(cid:1)52 (cid:18) 1 28 mqa 14 m2qa2 (cid:0)4 0(cid:12)Tf(cid:31)N(x)(cid:31)N(0)g(cid:12)0 (1) G(t) ' 28(cid:25)4 t6 + 25 t5 + 25 t4 2 (cid:10) (cid:11) 3 (cid:19) where 56(cid:25) :qq: a (cid:0) +(cid:1)(cid:1)(cid:1) : (7) abc(cid:0) Ta b (cid:1) c 75 t3 (cid:31)N(x)=(cid:15) u (x)C(cid:13)5d (x) u (x); (2) The spectral density used in the continuum is the standard lattice interpolating (cid:12)eld for the model is de(cid:12)ned by equating (5) and (7). The nucleon, and (cid:0)4 = (1+ (cid:13)4)=4 projects positive continuummodelisde(cid:12)nedthroughtheintroduc- parity states for ~p=0. At the phenomenological tionofathresholdwhichmarksthee(cid:11)ectiveonset level, one inserts a complete set of states Ni and of excited states in the spectral density. de(cid:12)nes Z 1 (cid:10) (cid:12) (cid:12) (cid:11) (cid:0)st 0(cid:12)(cid:31)N(0)(cid:12)Ni;p;s =(cid:21)Niu(p;s); (3) (cid:26)(s)e ds (8) s0 where the coupling strength, (cid:21)Ni, measures the (cid:0)s0t X6 nX(cid:0)1 1 sk0 abilityof the interpolating (cid:12)eld (cid:31)N to annihilate = e k! tn(cid:0)k CnOn; the i'th nucleon excitation. For ~p = 0 and Eu- n=1 k=0 clidean time t!1, the ground state dominates where Cn and On are the Wilson coe(cid:14)cient and and normal ordered operator of the term t(cid:0)n in (7). 2 (cid:0)MNt The phenomenologyof G(t) is summarizedby G(t)!(cid:21)Ne : (4) Z 1 The spectral representation is de(cid:12)ned by G(t)=(cid:21)2N e(cid:0)MNt+(cid:24) (cid:26)(s)e(cid:0)stds: (9) Z 1 s0 (cid:0)st G(t)= (cid:26)(s)e ds; (5) Strictly speaking, (cid:24) = 1 but here is optimized 0 with(cid:21)N,MN,ands0 toaccountforenhancement 2 andthespectraldensityis(cid:26)(s) =(cid:21)N (cid:14)(s(cid:0)MN)+ of the correlator in the short time regime due to (cid:16)(s) where (cid:16)(s) provides the excited state contri- lattice anisotropy. Ref. [2] found the anisotropy butions. tobelargeforx(cid:0)x0<6forfreequarkcorrelators While it may be tempting to (cid:12)t the correla- and remain large in their interacting simulation tion function in the shorter time regime by in- at (cid:12) = 5:7. At (cid:12) = 5:9 there is some hope that cluding additional poles in the spectral density, anisotropy issues will be less problematic for the such an approach fails in a number of ways [3]. Fourier transformed correlators presented here. The correlation function is probably best de- However, at very short times the quarks are es- scribed by many states of diminishing coupling sentially free and the anisotropy must be accom- strengths and increasing widths. It may be that modated. the QCDSR inspired continuummodel is an e(cid:14)- The nucleon coupling strength, (cid:21)N, is deter- cient way of characterizing this physics. mined in absolute terms, without resorting to a OSU{PP #93{1121 November, 1993 1 A few points on point-to-point correlation functions a Derek B. Leinweber a Department of Physics, The Ohio State University, 174 West 18th Avenue, Columbus,OH 43210-1106 Theshort-time regime ofQCDtwo-point correlation functions is examined through aQCD-Sum-Rule-inspired continuum model. QCD Sum Rule techniques are tested and alternate nucleon interpolating (cid:12)elds are discussed. 4 The techniques presented here may be ofpractical use in determining heavy-light meson decay constants. 9 n a 1. INTRODUCTION [2] the focus is on space-like separated correla- J tionfunctions. ItwasconcludedthattheQCDSR 7 In the quest for an ab initio determination of inspired continuum model was su(cid:14)cient to de- the low-lying hadron spectrum, lattice QCD in- scribe the lattice correlation functions over the 5 vestigationshave focused on the large Euclidean- calculated range. Using the entire lattice corre- 0 time tails of three-momentum-projected two- lation function (including the deep nonperturba- 0 point correlation functions. As such, the short tive regime), ground state masses were extracted 1 timeregime,whereexcitedstatecontributionsare 0 andwere foundtoagreewithconventionallattice signi(cid:12)cant, has simplybeen discarded. 4 analyses. 9 The lattice approach to QCD allows the de- Here the emphasis is on determining ground / termination of correlation functions deep in the t state properties by examining only the (cid:12)rst few a nonperturbativeregime. However,theQCDSum points of the lattice correlation function. This l - Rule (QCDSR) method [1] is restricted to the is similar in spirit to QCDSR analyses. In this p near perturbative regime of the Operator Prod- e manner, the validity of the continuum model is h uct Expansion(OPE).In thisregime,onecannot rigorously tested. By extending the analysis in- ignorethecontributionsofexcited states inQCD terval of the correlation function deeper into the correlationfunctions. Toaccountforthesecontri- nonperturbative regime, the evolution of (cid:12)tted butions, the so-called \crude continuum model" ground state properties maybe monitored. Con- isintroduced [1]. Thecontributionsofthismodel ventional lattice results are recovered when the relativetothegroundstate,whosepropertiesone interval extends deep into the nonperturbative is really trying to determine, are not small. The regime. A sensitivity to the analysis interval in validityof this modelis relied upon to e(cid:11)ectively theextracted parameterswouldindicateafailure remove the excited state contaminations. of the QCDSR inspired continuum model. This This investigation examines the physics in approach also allows an examination of the im- the near perturbative regime of point-to-point portance of the higher order terms of the OPE correlation functions where QCDSR analyses in constructing continuum models. These terms are performed. The QCD Continuum Model were not investigated in [2]. is constructed for three-momentum-projected One would also like to evaluate whether these Euclidean-timetwo-point functions followingthe techniques are useful in analyzing lattice QCD techniques established in the QCDSR approach. correlation functions. Such techniques may be This model will then be used as a probe of the of practical use for analyzing two-point correla- physics represented in lattice QCD correlation tionfunctionswhichbecomenoisypriortoaclear functions, and as a test of QCDSR techniques. groundstate domination,such asheavy-lightme- Some attention has recently been given to son correlators. the behavior of lattice point-to-point correlation Finallythesetechniques willbeexploitedtoin- functions in the near perturbative regime. There vestigatenucleonpropertiesusingunconventional