Nuclear Physics B (Proc. Suppl.) 34 (1994) 393-395 North-Holland

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PROCEEDINGS SUPPLEMENTS

Structure functions, form factors, and lattice QCD * Walter Wilcox a and B. Andersen-Pugh b

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

bDepartment of Physics, Luther College, Decorah, IA 52101, USA

We present results towards the calculation of the pion electric form factor mad structure function on a 16 3 x 24 lattice using charge overlap. By sacrificing Fourier transform information in two directions, it is seen that the longitudinal four point function can be extracted with reasonable error bars at low momentum.

1. I N T R O D U C T I O N 2. T H E C A L C U L A T I O N

The direct calculation of hadron structure func- tions by current overlap techniques is based on the simulation of the Euclidean hadronic ma- trix elements (h(0)]T[J~,(r, t)J,(0)]lh(0)), where J, = quJ~ + qdJ~ is the full electromagnetic cur- rent and .T d,u are the d, u quark flavor current densities. Such calculations are likely to be quite costly, so it seems worthwhile to check the valid- ity of this approach by less ambitious calculations utilizing the basic current overlap technique. The longitudinal piece (corresponding to p = v = 0 above, where J0 = ip) of structure functions for mesonic systems provides such a test[l]. We ex- pect for the pion that vector dominance should hold, resulting in a known elastic limit for both the p~f (same flavor) and pdpu (different flavor) sectors. This calculation serves to test whether the lattice size is large enough in space and time, the size of statistical errors, and other important issues. Since we use the conserved lattice cur- rent, there are also many useful numerical identi- ties which serve as checks on the calculation. In constructing these four point functions, we make multiple use of the sequential source technique[2] for quark propagators. Our work so far indicates that the full electromagnetic amplitude should be attainable for low momentum transfers; however, we see no indications of inelastic contributions in our results. In this brief report we will concen- trate on issues relating to systematics and statis- tical errors.

*Talk presented by B. Andersen-Pugh.

2.1. The Current Overlap Technique The structure function can be obtained from[3]

Q..(q2,t) = ~ e-iq'rp~,.(r,t) (i) r

where,

P.~(r, t) =

Ex(r+(0)lT[J~,(r + x, t)J~(x, 0)liar+(0)). (2)

It is the quantity P,v(r, t) which is directly calcu- lated in our simulations. We may study the form factor by using the relation [4]

Q00(q 2 , t) t>>l......

~ F2f q~e-( ~-,~, )t , , . (3)

For this calculation we utilized a 163 x 24 lat- tice and ~ = 62 in the quenced approxima- tion. We have omitted disconnected quark graph amplitudes because of the difficulty of simulat- ing the corresponding correlation functions. In constructing the P00(r, t) there are three distinct classes of connected diagrams which contribute; see Figure 1 where the effect of the currents is represented by an "X". The different flavor piece (Figure l(a)) is calculated by combining quark propagator lines from source and sink; this in- volves two quark inversions per configuration. In addition, the same flavor amplitude contains both "direct" (Figure l(b)) and "Z-graph" (Fig- ure 1(c)) contributions because of the indistin- guishability of the two currents. These are calcu- lated separately and the results added together;

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3 9 4 W. Wilcox, B. Andersen-Pugh /Structure functions, form factors, and lattice QCD

(a)

(b)

( c )

Figure 1. Three types of connected current over- lap diagrams: (a) different flavor; (b) same flavor direct; (c) same flavor Z-graph.

three additional propagators are needed to do this.

2.2. F o u r i e r R e i n f o r c e m e n t It is the x sum in equation (2) which is difficult

to perform when the currents involve the same fla. vor because of quark lines going from (r + x, t) to (x, 0) (r and x both summed). However, the sta- tistical error bars on Q~(q2 , t) were reduced by means of the following strategy. We use a charge density sheet operator,

~(z, t) = ~ p(x, y, z, t), (4)

Using this when q = qi, Q~' (q2, t) can be written (using translational invariance) as

2 t) O0 tq , --

• (~r+ (0)lT[P(z, t)~(0, 0)]1.+(0)), (5)

where Nz is the number of lattice points in the z direction.

Figure 2 illustrates the effect of using this tech- and ~ = .154 on 10 configurations. nique at q : g

The Fourier Reinforced ( "FR" ) result used the

o . ~

- 0 . 5 "

- l . ~ r

- 2 . ~

- 2 . r_,

I " 2 4 6 S I o ~2 14 16

t

Figure 2. Effect of using the extended operator r~uu/.2 t). Diamonds: ~(z,t) at ~ : .154 on w00~' l ,

Fourier Reinforced data; squares: nonreinforced data. The line here and in Figure 3 is the ex- pected vector dominance elastic limit.

~(z, t) operator, whereas the nonreinforced result fixes x in (2) to a single location[5]. The two re- suits in Figure 2 are consistent with one another, although the FR result is systematically higher. Both are consistent with a single exponential be- havior (no inelastic part), even at small time sep- arations between the currents, t, although there is a mysterious bump in the F R result. This could be a statistical fluctuation or an effect of getting too near the sink interpolation fields. It is seen that the uncertainties in Q ~ ( q 2 , t ) are significantly reduced in the F R amplitude.

3. P R E L I M I N A R Y RESULTS

Zero momentum smeared pion interpolation fields are used at source and sink which are lo- cated on the time edges of the lattice in order to maximize the time extent available for measure- ments. Extensive numerical tests were performed on the Q0a~(q 2, t) amplitude to make sure nonva- cuum contaminations do not occur; no such ef- fects were seen. This quantity was reconstructed

W. Wilcox, B. Andersen-Pugh/Structure functions, form factors, and lattice QCD 395

0 .

- 1 .

- 2 .

- 2 . . . . . . . . . . 0 2 4 6 8 10 12 14 16 18 2 0 22

t

Figure 3. Diamonds: full electromagnetic ampli- tude, Qoo(q2,t), equation (6); squares: different flavor amplitude, Q0~(q 2, t).

so that the two nonlocal, conserved charge densi- ties were centered in time between the time edges. When measuring Q~(q2 , t), we used an extended source (FR case) or a point charge source (non FR) on time steps 8 and 9. Thus, the maximum time separation between the two charge operators is 15 in the same flavor case but 22 in the different flavor case.

We find that the same flavor spatiM correlation function is significantly smaller than the different flavor one. It is dominated by the direct graph; the Z-graph contributes only at small time sepa- rations and serves to make the correlation func- tion marginMly wider. At small t we see a spatial anisotropy similar to [6].

Figure 3 shows the full pion amplitude

Q00(q 2,t) = 4 du 2 5 -uu" 2 " Q00 (q ,t) + c200 (q, O, (6)

again at q = ~ for ~ = 0.154 compared with the different flavor piece Qd~ (q2, t) (for which the sum in (2) is easy). Using FR, the error bars on the full amplitude come under control. Both re- sults are apparently tending to the vector domi- nance elastic limit.

It is possible to measure both elastic and inelas- tic processes from our hadronic amplitudes. The various different flavor amplitudes are especially useful because of their small error bars. They con- tain the information to do a survey of the form factors of all groundstate nonsinglet mesons.

4. A C K N O W L E D G M E N T S

This work was partially supported by the Na- tional Center for Supercomputing Applications and the National Science Foundation under Grant PHY-9203306 and utilized the NCSA CRAY sys- tems at the University of Illinois at Urbana- Champaign.

R E F E R E N C E S

1. The longitudinM piece of the pion structure function vanishes at both low q2 (at fixed x) and high q2 (CMlan-Gross relation) and thus is almost purely elastic. The transverse correlation function, Qii(q2,*) (i = 1,2,3; q perpendicular to the ith direction), is purely inelastic.

2. See Ref. 12 in W. Wilcox, T. Draper and K.- F. Liu, Phys. Rev. 46D, 1109 (1992).

3. W. Wilcox, in Lattice 92, Proceedings of the International Conference on Lattice Field Theory, Amsterdam, The Netherlands, 1992, edited by J. Smit and P. van Baal [Nuel. Phys. B (Pro¢. Suppl.)] 30 (1993) 491.

4. W. Wilcox, in Lattice 91, Proceedings of the International Symposium on Lattice Field Theory, Tsukuba, Japan, 1991, edited by M. Fukugita et. al. [Nucl. Phys. B (Proc. Suppl.)] 26 (1992) 406. Please note that equation (5) of this reference should read Qern(q ~) > F2~(q ~) and that the Figure 1 ordinate actually rep- resents the extracted form factor, F~(q2), rather than Qdu(q2).

5. Fixing x to a single location does not mo- mentum smear the amplitude since we use zero momentum interpolation fields at the two ends.

6. M.-C. Chu, J. M. Grandy, S. Huang, and J. W. Negele, Phys. Rev. 48D, 3340 (1993).