Volume 172, number 1 PHYSICS LETTERS B 8 May 1986

CHARGE RADII FROM LATFICE RELATIVE CHARGE DISTRIBUTIONS

Walter WILCOX 1 and Keh-Fei LIU t

Department of Physics, University of Kentucky, Lexington, KY 40506, USA

Received 5 February 1986

Electromagnetic charge radii values are extracted from relative charge distribution data for pseudoscalars using lattice QCD techniques. This is done with Wilson fermions and SU(2) color at five values of the hopping parameter.

The extraction of charge radii matrix elements using lattice techniques [1-4] may ultimately afford the pos- sibifity of a detailed comparison of experiment and theory, thus yielding a direct test of QCD. It is important to develop computer time efficient methods to do the lattice calculations. Ref. [4] is an evaluation of the charge radii of SU(2) color Wilson pions using the source method [5] for computing the three point function involved. In this paper we will extract pseudoscalar charge radii values from relative charge distribution measurements pub- fished previously [6]. The results obtained here are in agreement with values for charge radii given in ref. [4]. The use of relative charge overlap in the context of Monte Carlo simulations was suggested in ref. [7].

Let us consider a euclidean continuum theory with nondynamical, conserved flavor currents J~, jd . We define

Pc(r) = (1/2m~r)(rt+(O)l-J~(O)Jd(r, 0) In+(0)). (1)

By inserting a complete set of states, this becomes

Pc(r)= 1 ~h f " d3p (2rt) ~ 4 n

m r exp ( - ir . p) (0)Is (0) I n+0,)) (n +(p) l - :d (0 ) 1 : ( 0 ) ) (2)

Now let

Q(q) =- fdar exp(iq, r )Pc(r) . (3)

Using (2) in (3) gives

Q(q)= ~ 1 Or+(O)lJ~ln+(q))(n+(q)l_jdl~r+(O)) " (4) n 4E~mr

Taking the derivative of Q(q) with respect to q2 at q2 = 0 then gives (we specialize to the case m u = rod)

~Q(q)[~l 2 [ qZ__o = 2 aF(q)[i~q 2 Iq2 =0 • (5)

We have used

(n+(O) l(J~, -J~)llr+(q)) = (m,r + Eq)F(q) (6) with F(0) = 1 and the fact that

Or+(O)lJ~,dln+(O))=O , n --/: Ir. (7)

Eq. (7) may be justified by considering the quantity (Tr+(0)IJ~'d(r, 0)In+(q)). By using the fact that we are

1 Present address: Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA.

62 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 172, number 1 PHYSICS LETTERS B

dealing with u, d charge eigenstates:

au ,d=fd3r j~ 'd ( r ,O) , QUlzr+)=lrt+), ad l r r+ )=-Dr+) ,

and by performing integrals over r and q, one can show that (7) holds. From (3) we know that

Oa(q)/i}q 2 [q2 =0 = - ~ f d3r r2?c(r),

assuming spherical symmetry. We define the continuum charge radius, Re, as

1 2 bF(q)/Oq2lq 2 =0 = - g R e "

Then using (11) and (12) in (5)gives

R2= fd3rr2ec(r). We transcribe this relation into lattice language using the correspondences

fd3roa 3 ~ , In(p))o[Nsa32E~]l/21n(p)), J~'d(z)oa-3j#'d(z) , r

as

R 2 = ~ ~ r 2 p ( r ) . r

8 May 1986

(8, 9, 10)

(11)

(12)

(13)

(14a, b, c)

(15)

In the above, a is the lattice spacing,N s is the number of spatial points in the finite lattice and

e(r) = N s (n+(O) l-j~(o)jd(r, 0) It r+(0)). (16)

It is essential that the charge densities/~,d give rise to conserved quantities in whatever lattice formulation is adopted. Fortunately, in either the Wilson or staggered fermion cases such quantities exist and are easily derived [2-4] . In order to unambigously define the charge radius, it is necessary to assume spherical symmetry which, however, will always be slightly violated in a lattice simulation due to statistical fluctuations and finite size effects. However, in the continuum limit we must have R -* R e, and this is all we demand. The comparison of data from other lattice definitions of R can be considered one check on the approach to the continuum limit.

The quantityP(r) in (16) has previously been measured using ten Monte Carlo gauge field configurations at 13 = 2.3 [6]. We employed a 103 X 16 lattice, SU(2) color, Wilson fermions and five values of the hopping parame- ter, K. The technique used in ref. [6] was to measure the quantity

with

< >-= fdUd, d , e x p ( - S g - S f ) ( ) , (18)

.for t = (t' - t) = Ate" 1. When this is done, one can show that

P(r, t , t ' ) ~ P(r). (19) A t ~ l

Once we have P(r), we use (15) to compute R 2. The amount of computer CPU time to extract the charge radius

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Volume 172, number 1 PHYSICS LETTERS B 8 May 1986

Table 1. Masses and charge radii for pseudosealar mesons as a function of the hopping parameter, K.

K m~r R

0.160 0.37 ± 0.03 2.83 ± 0.13 0.156 0.55 ± 0.03 2.74 ± 0.14 0.150 0.79 ± 0.02 2.49 ± 0.20 0.144 1.03 ± 0.02 2.17 ± 0.24 0.138 1.25 ± 0,02 1.88 ± 0.24

3£)

2,0

t I I I

I D I I I 6.5 7 .0 7.5

Fig. 1. Comparison of charge radii, in units of the lattice spac- ing a, between the data of ref. [4] (x) and present results (o).

for a given K value is essentially the same as in ref. [4] since both techniques require the calculation of two fermi- on propagators per configuration.

Our final results are presented in table 1. Fig. 1 shows this data along with the results of ref. [4]. The error bar estimates in our measurements are purely statistical. The agreement between these two very different techniques for extracting the charge radius is quite satisfactory.

We finish with some comments as to how these results may be improved. The matrix element of interest was isolated in the method of ref. [6] by performing a time-stretched measurement, At >> 1. We carefully examined there the consequence of a variation in At, and concluded the effect on our results was not severe. However, using a lattice that allows a longer time stretched measurement than the present case (we used At = 6a) would result in an improved projection of the desired matrix element. We would also recommend the use of non-local interpolat- ing fields in future charge radius measurements since the use of local fields will tend to bias the data toward small- er R values. Of course, the usual comments about performing the calculations with better statistics on larger lat- tices apply here as well.

Conversations with Bing-An Li and Yunlun Zhu are gratefully acknowledged. This research was partially sup- ported by Department of Energy Contract No. DE-FG05.84ER40154.

R eferen ces

[1] W. Wilcox and R.M. Woloshyn, Phys. Rev. Lett. 54 (1985) 2653. [2] W~ Wilcox and R.M. Woloshyn, Phys. Rev. D 32 (1985) 3282. [3] R.M. Woloshyn and A.M. Kobos, Phys. Rev. D 33 (1985) 222. [4] R.M. Woloshyn, TRIUMF Report TRI-PP-85-107. [5] S. Gotflieb, P.B. Mackenzie, H.B. Thacker and D. Weingarten, Phys. Lett. B 134 (1984) 346;

C. Bernard, in: Gauge theory on a lattice: 1984 (National Technical Information Service, Springfield, VA, 1984). [6] W. Wilcox and K.F. Lin, Relative charge distributions for quarks in lattice mesons, University of Kentucky Report. [7] K. Barad, M. Ogflvie and C. Rebbi, Phys. Left. B 143 (1984) 222.

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