RECENT RESULTS FORCHARGE OVERLAP MEASUREMENTS

Walter WILCOX

Department of Physics, Baylor University, Waco, TX 76798, USA

1 . Introduction

The subject of lattice charge-charge[I] andcharge-current overlap[2,3] is emerging as a sig-nificant technique for the investigation of lowto medium energy aspects of hadron physics . Inprinciple, all low energy hadron properties aremeasurable from these wavefunctions, from form.factors to structure functions. In order to movebeyond the qualitative stage, however, it is nec-essary to learr, about the systematics of theirbehavior and measurement on finite space/timelattices . In the case of the pion, we point out thero'e of G-parity in the Fourier transform of thesewavefunctions and the types of contributions ex-pected from G = t1 intermediate states . Ex-plicit lattice simulations on 103 x 20 and 163 x24lattices have been carried out to investigate lowenergy behaviors of these wavefunctions. In thecase of the pion, one expects the recovery of vec-tor dominance in the electric form factor andthe dependence ti e -mpr of the pion wavefunc-tion[4] at large-r. It is shown that both of thesebehaviors are present in a certain nonsummed(pd(r)pu(0)) charge density lattice probe . Sum-ming over all origins (Z. pd(r 4- a%)pu (2)) pro-duces a zero-momentum probe, but drasticallychanges the measured wavefunction .

A discussion of charge overlap in the context of the pion is given in which the importance of G-parity of intermediate

states is emphasized. Lattice results for summed (L.a pd(r + x)pu(x)) and nonsummed (pd(r)pu(0)) charge density

probes are contrasted .

2. Theory

0920-5632192/$05.000 1992- Elsevier Science Publishers B.V

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Nuclear Physics B (Proc. Suppl.) 26 (1992) 406-408North-Holland

The charge-charge overlap distribution func-tion for the pion (continuum formalism)

Theelectromagnetic current, Jim

J=q,, JA'+gdp,(qu = 2/3, qd = -1/3) is defined in terms of uand d flavor currents and J,, = (J, ip) where pis the charge density. Separating off the n = 7r

contribution and using

Of course, in present lattice implementations ofcharge overlap, one does not use the full electro-magnetic charge densities but uses the relativeprobe -pd(r,0)pu (0) in Eq. (1) to act on the

pem(~,)_ (7r+(0)1pem(T~

0)pem(0)17r+(0))

, (1)2m,

has a Fourier transform

Qem(g2) = d3r 8=q-rpem(r), (2)

given by

QCM(g2) 112= E (ir

+(0)ipem(0)1n+(q)) . (3)

n 4E4m,r

F,,(g2) = ( +(0)1(pu, -pd)17r+(q))_(4)m,r + E9

then shows us that

Qem(g2) > &(g2). (

separate u, d quark lines . Assuming exact SU(2)isospin and charge conjugation, we have

(7r+(0)1Pd(0)in+ (q))

=G. (ir+(0)1Pu(0)1n+(q» , (6)

where G� labels the G-parity of the state n+ .This results in the forms

Qem(g 2 ) --̀

Q(qu + G,,gd)21(~+(0)IP(0)1n+(g))12

4Eq mA (7)n

and

Qdu(g2 ) = J:(-G.) 1(r+(0)1P(0)in+(g))12 , (8)E4

q mA

where Qdu(g2 ) involves the probe -pd(r,0)Pu(0)and where p(0) stands for either e(O) or pd(O) .

Thus, there is no inequality like Eq . (5) betweenQdu (q 2 ) and F,r (q 2 ) . The only types of interme-diate contributions for Gn = 1 are isovector oddparity states . For example if we consider sin-gle particle states, the bi meson (in a relativeP-state) contributes negatively whereas radiallyexcited states of the pion contribute positivelyin Eq. (8) ; it is not possible a priori to predictif Qdu(g2 ) will approach F,r (g2) from above orbelow as q2 --+ 0 [5] . We now turn to lattice sim-ulations to try to answer this question .

3 . Results

Fig . 1 shows Qdu (g2 ) at the lowest lattice spa-tial momentum on 103 (,Q = 5.9, glow = 7r/5,o) and 163 (,(3 = 6.0, qt.,, = x/8, *) lattices forSU(3) Wilson fermions . The x-axis in this fig-ure represents the quantity q2/MP which takeson different values for the various i values used .(rc = .154, .152, .148, .140, and .130 for the 163and .154,150,.144, and .134 for the 103 lat-tices) . The solid line shown there is the vector

W. Wdcox/Recent resalts for charge overlap measurements

X

Fig . 1 : Qdu (q2 ) as a function of t = q2/m2,, as compasto vector dominance for summed and nonsummed latticedata.

dominance result F,r(g2)=1/(1-F q2/mp). Thesemeasurements are characterized by point pionsource and sink interpolation sources located atthe spatial origin . In addition, there is no sumon the spatial origin of the intermediate e(O)charge density ; it is positioned at the same spa-tial site as the source and sink fields . This tech-nique makes the overlap measurement as localas possible within the lattice spatial volume andminimizes spurious quark loop wraparound ef-fects due to the periodic quark boundary con-ditions . The overlap probe pd(r, t)pu(0, t) thenno longer projects solely on zero-momentum, butthe time extent of the Euclidean lattice can beused to damp out nonzero momentum contri-butions. (Measurements appear to show a reli-able plateau in the signal as a function of thetime postion, t, of the overlap probe.) The lowerpoints in Fig . 1 show the same measurement,

log P

-5 0

2

4

6

8

10

12

14r

Fig. 2: Contrast of the sununed (o) andnonsummed (O)pion wavefunctions at is = .152 on a loglo scale as afunction of relative separation r.

again using point interpolating fields, with a sumon the spatial position of the intermediate e(O)charge density for the 103 (0) and 163 (0) data.These measurements, as reported previously[2],are low compared with the vector dominanceline ; this is in contras with the nonsummed re-sults.

Fig. 2 shows a comparison of the chargeoverlap wavefunctions using summed and non-summed probes at x = .152 on the 163 lattices .As pointed out in [4], the large-r behavior ofthis wavefunction should be - e - mor . At thisvalue of ic, we have napa = .55(1), whereas theloge-slope of the nonsummed pion wavefunctionshown is j®sj = .56(3) . In fact, the slope ofthese wavefunctions agrees with the rho-mesonmass at all measured i values on both the 103and 163 lattices . In contrast, the summed datawavefunctions are significantly flatter and the

W. Wilcox/Recentrentsforcharge overlap measurements

expected large-r behavior is not explicitly seen .However, note that boundary corrections maybe crucial in revealing it in this case.[4] .

4. Discussion

We have discussed the role of G-parity in pionu, d overlap wavefunctions and contrasted twotypes of lattice measurements of these quantities .The local nonsummed wavefunction has the niceproperties ofexhibiting approximate vector dom-inance at the lowest spatial momentum and hav-ing the correct large-r asymptotic behavior, al-though it is momentum smeared. In contrast, thesummed zero-momentum version of this probe issignificantly flatter and it's Fourier transform islow compared to vector dominance. Further sim-ulations using the full electromagnetic currentdensity, pem, as well as time-separated measure-ments, will hopefully clarify the situation .

Acknowledgements

This work was partially supported by the Na-tional Center for Supercomputing Applicationsand utilized the NCSA CRAY 2 system at theUniversity of Illinois at UrbanaChampaign.

References

[1] K. Barad, M. Ogilvie andC. Rebbi, Plays. Lett . B143(1984) 222.

[2] W. Wilcox, Phys . Rev. D43 (1991) 2443.[3] W. Wilcox, Nucl. Phys . B (Proc. Suppl.) 20 (1991)

459.[4] M. C. Chu, M. Lissa and J. W. Negele, Nucl. Phys .

B360 (1991) 31 .[5] Discussions with J. Negele were helpful in clarifying

this point.