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SUPPLEMENTSELSEVIER Nuclear Physics B (Proc. Suppl.) 119 (2003) 404406
www.elsevier.com/locate/np
Strange matrix elements of the nucleonRandy Lewisa, W. Wilcoxb and R. M. WoloshynC
aDepartment of Physics, University of Regina, Regina, SK, S4S OA2, Canada
bDepartment of Physics, Baylor University, Waco, TX, 76798-7316, U.S.A.
CTRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada
Results for the disconnected contributions to matrix elements of the vector current and scalar density havebeen obtained for the nucleon from the Wilson action at p = 6 using a stochastic estimator technique and 2000quenched configurations. Various methods for analysis are employed and chiral extrapolations are discussed.
1. MOTIVATION
Prediction, measurement and understanding ofstrangeness in the nucleon have proven to besignificant challenges for both theory and ex-periment. Of particular interest at present arethe strangeness electric and magnetic form fac-tors. Experimental results have been reported bySAMPLE[l] and HAPPEX[2], and other groupsare also planning experiments.[3]
A number of lattice QCD studies have been re-ported during the past few years[4,5], though theconclusions are somewhat varied. In the presentwork we report the results of a high statistics sim-ulation. Different analysis techniques are stud-ied and chiral extrapolations are discussed. Thestrangeness scalar density and electric and mag-netic form factors are all analyzed together, andresults are compared to experimental data.
2. SIMULATIONS AND ANALYSIS
The nucleon’s strangeness form factors arisefrom a disconnected strange quark loop that wecompute stochastically using real 2, noise[6]. Toreduce the variance, the first four terms of theperturbative quark matrix are subtracted.[7] (Forthe scalar density, the first five terms are sub-tracted.) The three matrix elements of interestare
M{S,M,E}(t,~ =
G(“)G$), -$+,Gg) (1)
P
where (N (Ss] N) = ZsG$), 2s x 1 - 3~~/(4n,),and the Mx (t, 3 can be extracted from the ratios
G$)(t t’ $IG(~)(~ 6)Rx(t’t’7q3 = (&&(2)(&) (2)
of 2 and S-point correlators by various methods:
t+1c [Rx(U’,q3 - ~x(t-W,41 + ~x(t,q37 (3)it’=1
C Rx (t, t’, 3 -+ constant + tMx (t, 3,t’=l
(4)
tfized
C Rx(t,t’,q3 -+ constant +tMx(t,$).t’=l
(5)
Any authentic signal should be visible with eachof these methods, and should be consistent amongall of them.[5]
Simulations have been performed on 203 x 32lattices using the Wilson action with /3 = 6.Dirichlet time boundaries are used for quarkswith the source five timesteps from the latticeboundary. Valence quarks use )E, = 0.152, 0.153and 0.154. The chirai limit is ~~ M 0.1571 and our)c, values are in the strange region. The mass ofthe quark in the loop will be held fixed at a valuecorresponding to ~1 = 0.152 and the analysis willbe based on 2000 configurations with 60 22 noisesper configuration. Consistent results (not shown
0920-5632/03/$ - see front matter 0 2003 Elsevier Science B.V. All rights reserved.doi: 10.1016/SO920-5632(03)01568-8
R. Lewis et al. /Nuclear Physics B (Proc. Suppl.) 119 (2003) 404-406 405
O&-W6I af -----------_--__-___-_______________ _ _ _,
I I I I0 10 2 0 3 0
Cd)2
0 10 20 304 I
(e)2 -
- - - - - - - - - - - - - - -I I I I I0 10 2 0 3 0
,
Figure 1. Ms(t,a from Eq. (3) with fit, = IE~ =0.152. From top to bottom, the plots show 422 =0 , 1 , 2 , 3 and 4, where & E (lOa/7r)$
in this brief article) have been obtained from theanalysis of 100 configurations with ~1 = 0.154 and200 22 noises per configuration.
Results for Ms(t, (13 from Eq. (3) are shown inFig. 1. For each q’considered, a clear plateau be-gins about 10 timesteps from the source. (Thesource is at t = 0 in the plots.) On the otherhand, results for M~(t,3 and M~(t,3 are con-sistent with zero for each $and K, as shown inTable 1.
3. CHIRAL FITS
Quark mass and momentum dependences canbe calculated analytically within quenched chiralperturbation theory from the diagrams of Fig. 2as well as all tree-level counterterms.
There are six independent parameters and we
Table 1Fits to the matrix elements of Eq. (3) beginning10 timesteps from the source. $’ z (lOa/7r)$
GJ 422 Q = 0.152
w Gf 490.152 0 2.6(4) -0.009( 13)
1 1.7(2) 0.007(16) -0.008(S)2 1.2(2) -0.018( 14) 0.012( 10)3 1.1(5) -0.014(23) 0.008(17)4 0.7(6) 0.004(31) 0.026(40)
0.153 0 2.7(5) -O.OlO( 15)1 1.8(3) 0.012(22) -0.011(10)2 1.3(2) -0.021(20) 0.015(14)
1.2(6)ii 0.7(B)
-0.018(32) O.OOS(22)0.005(48) 0.029(56)
0.154 0 2.9(5) -0.013( 19)1 1.8(3) 0.019(33) -0.014(15)2 1.3(3) -0.022(31) 0.019(21)3 1.5(9) -0.029(53) 0.008(32)4 0.8(11) O.OlO(82) 0.021(81)
consider two extreme limits: either the quenched77’ is absent (y = 0), or y # 0 but the non-q’ pa-rameters are set to zero in the loops. The physi-cal situation must lie between these two extremes,and so we expect the true values to be somewherebetween the curves in Fig. 3. (For another recentdiscussion of chiral fitting, see Ref. [8].)
4. COMPARING TO EXPERIMENT
The results of Figure 3 compare to the existingexperimental results as follows:
Gz(q;) 0.14 f 0.29 f 0.31,=0.03 f 0.03,
;;. 6’] (6)
Gg’(q;) + 0.39Gtf(q2) 2
1
0.025 f 0.020 f 0.014,= Ref. [2]0.027 f 0.016, Fig. 3 (7)
where -4; = 0.1 GeV2 and -9; = 0.477 GeV2.For the scalar density, we find
(W/W) (N 15~1 w (0) = 0.15(2), (8)
which can be compared to the result of 0.195(g)reported in Ref. [9].
406 R. Lewis et al. /Nuclear Physics B (Proc. Suppl.) 119 (2003) 404-406
Figure 2. Leading loop diagrams from quenchedchiral perturbation theory. Dashed, solid anddouble lines denote octet mesons, octet baryonsand decuplet baryons respectively. A shaded boxdenotes a current insertion.
We have not attempted to estimate the sizesof systematic uncertainties in our results due, forexample, to quenching and to the use of chiralextrapolations for valence quarks in the strangeregion. The raw lattice data of Table 1 show,at best, only tiny strange quark effects over therange of momenta and quark masses that werestudied. Large strange-quark loop contributionsin the vector current matrix elements should beconsidered unlikely.
ACKNOWLEDGEMENTS
This work was supported in part by the Na-tional Science Foundation under grant 0070836,the Baylor Sabbatical Program, and the Natu-ral Sciences and Engineering Research Councilof Canada. Some of the computing was doneon hardware funded by the Canada Foundationfor Innovation with contributions from CompaqCanada, Avnet Enterprise Solutions and the Gov-ernment of Saskatchewan.
REFERENCES
1 . R. Hasty et. al., Science 290, 2117 (2000).2. K.A. Aniol et. al., Phys. Lett. B509, 211
(2001).3. For example, the A4 Collaboration at MAMI
and the GO Collaboration at Jefferson Lab.4. S.J. Dong, K.F. Liu and A.G. Williams, Phys.
Rev. D58,074504 (1998); D.B. Leinweber andA.W. Thomas, Phys. Rev. D62,07505 (2000);
” oL-----J0 0 . 2 0 . 4 0 . 6 0 . 8 10 . 2 I ' I ' I ' I '
09
5j-;--10 0 . 2 0 . 4 0 . 6 0 . 8 I
0 . 2 I ' I ' I ' I '(4
0 0 . 2 0 . 4 0 . 6 0 . 8 I-q* [GeVZl
Figure 3. Lattice results extrapolated to thephysical hadron masses.
5 .
6.
7 .
8 .9 .
N. Mathur and S.J. Dong, Nucl. Phys. (P.S.)94,311 (2001); W. Wilcox, Nucl. Phys. (P.S.)94, 319 (2001).R. Lewis, W. Wilcox and R.M. Woloshyn,hep-ph/0201190 (2002).S.J. Dong and K.F. Liu, Nucl. Phys. (P.S.)26, 353 (1992); Phys. Lett. B328, 130 (1994).C. Thron, S.J. Dong, K.F. Liu and H.P. Ying,Phys. Rev. D57,1642 (1998); W. Wilcox, hep-lat/9911013, in Numerical Challenges in Lat-tice Quantum Chromodynamics, edited by A.Frommer et. al. (Springer Verlag, Heidelberg,2000); Nucl. Phys. (P.S.) 83, 834 (2000); C.Micheal, M.S. Foster and C. McNeile, Nucl.Phys. (P.S.) 83, 185 (2000).J.W. Chen and M.J. Savage, hep-lat/0207022.S.J. Dong, J.F. Lagae and K.F. Liu, Phys.Rev. D54, 5496 (1996).