The k2-dependence of the transverse pion electroproduction cross section in a generalized born term model

~.B.6~

Nuclear Physics B49 (1972) 461-474. North-Holland Publishing Company

THE k 2-DEPENDENCE

OF THE TRANSVERSE

ELECTROPRODUCTION IN A GENERALIZED

PION

CROSS SECTION BORN TERM MODEL

F. G U T B R O D Deutsches Elektronen-Synchroton DES Y, Hamburg

G. K R A M E R II. Institut fiir Theoretische Physik der Universitiit Hamburg

Received 14 July 1972 Abstract: Data on rr+-electroproduction off protons are analysed in a generalized Born term model, where the off-shell nucleon form factor Fl (k 2) is allowed to differ from its on-shell value. The longitudinal-transverse interference cross section is sensitive to the ratio of the nucleon form factor to the pion form factor, and the data require this ratio to be larger than one. The resulting transverse cross section, integrated over a small t-interval, follows the slow decrease of the total inelastic cross section as a function of the photon mass squared.

The presently available data [ 1 - 3 ] on positive pion electroproduction in the 2 GeV region have been analysed theoretically m a i n l y in terms of the vector dominance model [4, 5] (VDM) and the Born term model, including s-channel resonances [ 6 - 8 ] . Especially at small squared p h o t o n masses k 2 it has become clear that both m e t h o d s tend to underestimate the observed cross sections ~ at small values of m o m e n t u m transfer t. T w o possible sources of corrections have been investigated: (a) The electromagnetic pion form factor has been treated as an adjustable parameter in refs. [6, 7] resulting in higher values than VDM would predict. F r o m the work of S c h m i d t [6] it is apparent that this leads to a longitudinal-transverse interference term n o t quite consistent with the data. (b) In addition to the s-channel n u c l e o n and k-pole, more resonances have been taken into account by Devenish and Lyth [8] to saturate the dispersion integrals for the invariant D e n n e r y amplitudes [9]. With plausible assumptions on the i m p o r t a n t ratio of electric to magnetic couplings these authors were able to obtain the correct -{ There are possible discrepancies between the data of DESY tref. [1 ]) and CEA (ref. [2]) of about 15% which still lie inside the normalization errors [ 10l. Without particular justification we shall explore the consequences of the assumption that the higher cross sections of ref. [ 1 ] are correct within 10%.

462

F. Gutbrod, G. Krarner, Pion electroproduction

sign of the interference term and an increase of the unpolarized cross section (probably mainly the transverse one) into the right direction. Stimulated by the slow decrease of the total electroproduction cross section we wish to inquire into the possibility that the transverse single-pion electroproduction cross section is quite different from what is assumed in the VDM-like models. Especially we concentrate on a proper fit to the interference tenn. For this purpose we consider a generalized Born ternl model with the diagrmns of fig. 1, where all form factors are treated as free parameters. Since the exchanged pion is close to its mass shell in contrast to the nucleon, we shall not allow large deviations from the VDM-prediction in F~(k2), but shall try to explain the large cross sections by an increase of the proton form factor F 1(k 2) b e y o n d its mass shell value. The introduction of off-shell form factors is equivalent to adding terms into the dispersion integrals with k 2behaviour different from the nucleon pole. From the small t-behaviour alone it cannot be decided whether these terms appear in the same angular m o m e n t u m state as the nucleon or whether they correspond to higher spin resonances. The fits obtained indeed suggest a transverse cross section much in excess of the VDM-predictions. It is interesting to compare it to the total electroproduction cross section, and for this purpose the differential transverse cross section has been integrated over a small t interval. The model in its simplest form however does not properly describe the observed t-dependence both in photo- and electroproduction. We have therefore multiplied the Born tenn amplitudes by t (1)

j ( t) : e t/m2

which improves tire fit and allows a meaningful integration over ( - t ) . This may not

/!

//

ii

Fig. 1. Diagrams for the Born term model (BTM). The blobs represent off-shell form factors. I" Notation is explained in the appendix.

463

F. Gutbrod, G. Kramer, Pion electroproduction

/

/

II

a)

b}

Fig. 2. Diagrams describing photoproduction off a bound nucleon. be a completely artificial procedure. For example, if we assume that the physical nucleon is a bound state of a point-like nucleon and a neutral boson particle, then photoproduction off the nucleon will proceed via diagram 2a, and f ( t ) will be determined by the square of the nucleon wave function. For t = 0 no change compared to the Born term model arises, and the nNN form factor in diagram 2b is also given by f ( t ) , consistent with current conservation, Before discussing our fits in detail, we want to stress the importance of the longitudinal-transverse term as a probe for amplitudes, due to the dominance of the pion pole. In terms of the s-channel helicity amplitudes we have [ 11 ] (c = kinematical factor independent of t)

= Re

o(L,++ - L _ + )

o(L,-+ + L ++)}

_ c sin 0 Re { ( F ~ ' + F 6 ) ( F 1 + F 2 + (1 + x ) ( F 3 + F 4 ) ) - (F~' - F ~ ' ) ( F 1 - F 2 - ( l - x ) ( F 3 - F4))} ,

(2)

where the F i are defined by the expansion of the matrix element T = ~-

X2+ [ i ~ ' n F 1 + ~ ' 0 ( n × k ) ' ~ F

2+in.k

0"~F 3 (3)

+ i n " ~ffl " E F 4 + i n " k I~ . ~ F 5 + i n • ~tk " ~ F 6 - i a "il e o F 7 - i n

" k eo F8] Xa .

Current conservation demands ko F'5 =-F 1 + x F 3 + F 5 : ~ F

, 8 ,

ko

F 6 =-xF 4 +F 6 :-~F

7.

(4)

In the electric Born term model we have, adding the usual gauge term proportional t o k . e,

F. Gutbrod, G. Kramer, Pion electroproduction

464

[ Fl(k2) Fn(k2) T = iegff(p2) "Y5 Ls _ M-~ ~ ( + - (2q t - /12

k)" e

F ( k 2) - F l ( k 2 ) ] + k2 k' e u(Pl),

(5)

where F l ( k 2 ) is the Dirac form factor of the nucleon and F~r(k2) in the pion form factor. This leads to the following expressions for the F i • Fl(k2) F 1 = - N (X/~-1 + M ) ( E 2 +M) ( W - M ) - s - M2 ' Fl(k2) F 2 = + N (x/~- 1

M)(E 2 - M ) ( W + ~ - -

F 3 = 2Nx/~ 1

F ( k 2) M)(E 2 - M)(E 2 + M ) - -

s - M2

'

t - #2

'

F ( k 2) F 4 = -2Nw/-~I + M ) ( E 2 +M) (E 2 - M) - t /12

"{-(W+M) Fl(k2-

F 7 = N (vC(E11 + M ) ( E 2 - M )

'

(6)

)

s - M2

~ ) F~r(k2 + ( 2 q 0 - k0) t - / 1 2

F . ( k 2) - Fl(k2 ) } + k0 k2 , '

- M)(E 2 +114) { - (W - M) Fl(k2) s M2

F 8 : -Nx/~I

FTr(k2) + (2q0 - k0) t -/12

FTr(k2) - Fl(k2 ) ] + k0

k2

J'

N - eg 47rW " Since for s >>M 2 VCffl +-M ~ ~ ( 1

k2--/12 ~ , 2UW~+ M) 2 ]

(w + M) 2

E 1 -+M~

2W

'

we see easily that the "high energy approximation"

(7)

465

F. Gutbrod, G. Kramer, Pion electroproduction

F2 ~-FI,

G ~-F3,

Fs ~-F7

(8)

works already well at I¢ ~> 2 GeV and - k 2 ~< 0.5 (GeV/c) 2 for all amplitudes except for the n u c l e o n pole c o n t r i b u t i o n s to F 7 and F 8. The nucleon poles in F 7 and F 8 however can be neglected compared to the pion pole for small - t . It is therefore natural to assume that o t is d o m i n a t e d by the spin flip terms, and the electric Born term yields for s >> M 2 oI

c sin 0 n

- ~ - . e

( F ~ - F 6'* ) ( F 1 - F 2 - 2 sin 2 5' O(F 3 - F 4 ) )

(F(k2)

-4csinOsX

F(k2)-Fl(k2)~

F

(9)

2tF(k2) ]

gb

o~v~

k 2 = -0. 26 (GeV/c) 2

flu (BTM)

t

02

0k J 06

0,8

q,

_;~ OT

O"I (BTM) G I (Fit)

S

.o,, o.,

Fig. 3. Fits to o u + e o L, o I at k 2 = -0.26 (GeV/c) 2. Two curves of the Born model with on-shell form factors are included (BTM). au(Fit) describes the result of the fit. All cross sections are in #b/(GeV/c) 2, k 2 and t are measured in (GeV/c) 2. Data are from ref. [ 1 ].

F. Gutbrod, G. Kramer, Pion electroproduction

466 do

~.b

k2 =-0.55(GeV/c) 2

2O-

lo-

1oL

dff

gb

dt

~eV2

10-

i .... -10 t Fig. 4. Fits to ~u + e eL, o T and o I at k 2 = --0.55 (GeV/c) 2 . ou describes the result of the fit. Data are from ref. [ 1 ]. Table 1 Off-shell pion form factor Fn(k2), and off-shell proton Dirac form factor Fl(k2), determined to fit the data, for different values of k 2. For comparison we include the predictions from the vector dominance model for the pion form factor (F~r(VDM)), and the results for Fl(k 2) following from the dipole formula for G~E(k2) and G ~ (k2). GE(k2) follows from the fit value for Fl(k 2) assuming scahng between G E and G M.

k2 0.26 0.55 0.75

Fn(k 2)

FTr(VDM)

F I (k 2)

FI (dipole)

GE

G~(dipole)

0.746 0.522 0.394

0.69 0.52 0.44

0,98

0.672 0.48 0.39

0.78 0.48 0.40

0,536 0,318 0,237

0,72 0.66

F. Gutbrod, G. Kramer, Pion electroproduction

k2=-0.75 (GeV/c) 2

lab

da

c. 3d

467

GeV 2

20,

10-

: Cru*ECrC aT I

oio

0.1

0.12 t

0.1.(-, -cr T

dodt

10-

p.b GeV 2

(3"r

o.o2,

oo6 o.o8 olt, o12, -t

-10

Fig. 5. Same as fig. 4 at k 2 = -0.75 (GeV/c) 2.

Therefore the position of a possible zero t of o 1 is sensitive to the ratio F l ( k 2 ) / F~(k2). The experimental fact (see figs. 3-5), that this zero occurs at - t ~ 2/~2 for - k 2 = 0.26 (GeV/c) 2, and that at larger values of k 2 no zero is present, indicates that F 1 (k 2) > F~r(k2 ). If we abstract from the Born term model, b u t still assume pion pole d o m i n a n c e for - t ~ 2/12, then the interference term measures F 1 - F 2 i.e. the spin flip part of the perpendicularly polarized transverse cross section. In our fits we assumed that (a) the magnetic dipole N A ( 1 2 3 6 ) transition form factor G~l(k2 ) is proportional to GE(k2 ) [12], and (b) scaling between G~(k2) and G~d (k2) as well as G~-(k 2) = 0 holds. The b - c o n t r i b u t i o n has been calculated via fixed t-dispersion relation with a subtraction in the invariant amplitude A (5-) t The fact, that this zero is correlated with the zero in Oll as a consequence of eq. (8), was noted previously in ref. [6].

468

1~: G u t b r o d , G. K r a m e r , P i o n e l e c t r o p r o d u c t i o n

1.0 t ~(~IrbiN~rary Urbits)

0.75~

x

~.

x\\\\\\\

15-

,,\

-[\\

0.25-

"~"-~ .. + GeneraLizedBrlvi

--o'.1

~ ~ BIM

o;2 d.3 o'.~ o'.s o'.s o.'7 o18-.;-

Fig. 6. Comparison of the integrated transverse cross section oint with the total transverse electroproduction cross section [ 15 ]. The results of the fit are indicated as crosses, the values of the BTM with on-shell form factors is shown as dashed curve. according to Adler [13]. The agreement of this m o d e l with 7r+ p h o t o p r o d u c t i o n at W = 2.23 GeV is good, b u t this is n o t too meaningful, since the 7r-/~+ ratio is n o t reproduced for - t t> 0.05 (GeV) 2. In table 1 we show the resulting values of the foml factors compared with the predictions from VDM for FTr and those from the dipole formula c o m b i n e d with scaling between G ~ and G~I. The quality % of the fits is shown in table 2, 3 and 4 and in figs. 3, 4 and 5. The prediction of the model with on-shell form factors (and VDM for F,r ) are included in the tables and in fig. 3. One notices that F 1 lies substantially above F 1 dipole with a corresponding increase of or, b e y o n d the strict Born term model. The quality of the fit at k 2 = - 0 . 2 6 (GeV/c) 2 would be improved i f F ~ would be increased b y 5%, still below the value found by Schmidt [6], which is 0.87. Finally we have integrated the differential transverse cross section o u from - t m i n to - t m i n + 0.24 (GeV/c) 2, At k 2 = -0.26 a n d k 2 = -0.75 it seems difficult to fit the forward points adequately. It should be noticed however, that radiative corrections cause a smearing of the direction of the virtual photon, which might influence the separation of o u + ea L, a T and cq. This effect has not been taken into account in the data analysis [ 14].

-0.26

k2

33.3 ± 1.1

35.0 +- 1.2

29.4 +- 1.2

22.1 -+ 1.3

16.6 -+ 1.5

10.3 +- 4.6

0.010

0.020

0.032

0.050

0.075

0.105

13.6

17.1

21.7

26.7

30.9

34.2

11.1

13.7

17.0

20.2

22.7

23.6

- 5 . 3 ± 4.3

- 6 . 6 +- 2.4

- 2 . 6 +- 2.3

- 7 . 3 -+ 2.0

- 6 . 0 +- 2.1

+0.3 -+ 2.1

Exp

BTM

Exp.

Fit

oT

ou + e a L

6.62

-7.14

-7.40

-7.13

-6.15

-3.66

Fit

-2.48

-2.90

-3.31

-3.50

-3.27

-2.11

BTM

7.64

8.33

9.15

10.2

11.8

14.8

Fit

4.25

4.40

4.50

4.70

5.20

6.59

BTM

au

+5.3 -+ 4.4

+4.7 -+ 1.6

+0.55-+1.4

- 3 . 1 -+ 1.1

- 5 . 1 + 1.1

- 3 , 7 + 1.0

Exp.

+ 1.59

+1.08

-0.22

-2.48

-5.24

-7.75

Fit

oI

+3.89

+3.87

+3.16

+ 1.49

-0.87

-3.65

BTM

Table 2 Data and fits to the differential cross sections for 7r+ electroproduction at W = 2.2 GeV and - k 2 = 0.26 (GeV/c) 2. The experimental points are from ref. [1]. There are no data for o u . The columns BTM denote the predictions from a strict Born term model (A-pole included) with on-sheU form factors and VDM for F r r ( k 2 ) .

e~

Table 3 Data and fits to the differential cross sections for rr+ e l e c t r o p r o d u c t i o n at W = 2.2 GeV and - k 2 = 0.55 (GeV/c) 2. The experimental points are from ref. [ 11. There are no data for ou.The columns BTM denote the predictions from a strict Born term model (A-pole included) with on-shell form factors and VDM for Fir(k2). -k 2

0.55

t

ou + e e L

oT

ou

oI

Exp.

Fit

BTM

Exp.

Fit

BTM

Fit

BTM

Exp.

F it

BTM

0.020

29.3 ± 1.3

30.5

24.7

-

3.4 -+ 2.7

-0.87

-0.45

8.21

3.52

- 2 . 2 ± 1.3

-2.17

-0.19

0.032

26.4 ± 1.0

25.8

21.5

+ 3.2 ± 2.0

-2.01

-0.98

6.73

2.87

- 3 . 8 ± 0.9

-1.74

+ 1.0

0.050

20.0 + 0.8

20.4

17.5

-

0.34±1.8

-2.79

-1.18

5.63

2.50

- 1 . 0 ± 0.8

-0.64

+ 2.13

0.070

16.4± 1.0

16.3

14.2

+ 0.1±2.1

-3.04

-1.17

4.97

2.33

- 0 . 1 ± 1.0

+0.09

+2.62

0.095

15.1± 1.3

12.8

11.3

-

2 . 9 ± 2.7

-3.08

-1.09

4.44

2.20

-l.4e

1.4

+0.54

+2.77

0.130

6.3 ± 2.0

9.7

8.7

- 1 1 . 3 ± 4.6

-2.95

-0.96

3.94

2.06

1.7 ± 2.3

+ 0.79

+ 2.64

1:%

19.7-* 1.0

18.0 ± 1.0

14.4 ± 1.0

10.0 -+ 1.7

10.7 +- 2.9

7.6 ± 3.5

0.043

0.060

0.080

0.105

0.140

-0.75

Exp.

7.4

9.7

12.2

15.1

18.8

22.0

Fit

ou + e o L

0.032

-t

k2

7.2

9.4

11.8

14.5

17.7

20.3

BTM

+0.9 +- 4.9

- 1 . 4 -+ 4.3

- 6 . 0 ± 3.2

- 2 . 5 -+ 2.1

- 1 . 2 -+ 2.2

- 2 . 8 ± 2.2

Exp.

oT

-2.10

-2.09

-1.96

-1.68

1,15

-0.50

Fit

-0.56

-0.61

-0.63

-0.59

-0.45

-0.215

BTM

3.34

3.90

4.41

5.02

5.84

6.68

Fit

ou

1.36

1.48

1.60

1.75

1.99

2.30

BTM

- 0 . 7 -+ 4.0

- 1 . 4 -+ 3.3

- 4 . 5 +- 2.0

-2.5 -+ 1.1

- 1 . 7 ± 1.0

4.7 ± 1.0

Exp.

o1

-0.13

-0.40

-0.72

-1.09

-1.42

-1.31

Fit

+1.95

+2.04

+1.97

+1.69

+1.12

+0.48

BTM

Table 4 Data and fits to the differential cross sections for n + electroproduction at W = 2.2 GeV and - k 2 = 0.75 (GeV/c) 2. The experimental points are from ref. [ 11. There are no data for a u . The columns BTM denote the predictions from a strict Born term model (/,-pole included) with on-shell form factors and VDM for Fn(k2).

-,..)

F. Gutbrod, G. Kramer, Pion electroproduction

472

t

Oint = --

• mln

f

+0.24

dt Ou(k2, t ) .

(10)

-tmin A possible artificial k 2-variation of Oint, which is due to the variation of tmin with k 2, is negligible. In fig. 6 we compare Oint and the k2-behaviour of the total transverse electroproduction cross section at W = 2.2 GeV, extracted [15] from a fit to uW2(W, k 2) under the assumption R = OL/OT = - k2/v 2. The predictions from the Born term model are also included in fig. 6. The agreement of the k 2 dependence of Oin t with the total transverse cross section is surprising. The deviation at k 2 = - 0 . 2 6 can be reduced at the expense of a larger increase o f b ~ b e y o n d the VDM value. If this similarity will be verified by direct measurement of % , our understanding of deep inelastic electron scattering may change appreciably, since both the dynamics and the experimental accessibility of single pion production is simpler than for most other reactions. Besides the channel discussed in this paper there are by now three other cases, where Born term models or VDM considerations predict a too strong decrease with k2: p-electroproduction [16], rr+A 0 [17] and r r - A ++ [18] production, and one may add the transverse photoproduction cross section [19] itself. This supports our main conclusion, that the slow decrease of Otot (electroproduction) might manifest itself already in simple channels like 7 P ~ rr+nAn equally good explanation for the interference cross section has been obtained by Kellet [5] within the framework of VDM applied to invariant amplitudes. His predictions for o u and o T are well above the BTM values, but are about 20% lower than ours which is compatible with the fact that his curves are closer to the CEA data than ours. Thus our calculations appear to confirm Kellets work, but it is stated in ref. [5] that in the invariant amplitude approach the transverse amplitudes follow the naive VDM expectations (which in turn agree with the BTM), whereas the longitudinal amplitudes have a more complicated behaviour at low energies than VDM predicts. It seems to us that the origin of the slow decrease of o u in VDM-like models needs more discussion. The authors are indebted to Dr. J. Gayler for providing the fit to the total electroproduction cross section based on ref. [15]. Thanks are due to Dr. K. Heinloth for numerous discussions. Useful remarks by Dr. B.J. Read are gratefully acknowledged.

APPENDIX. TABLE OF KINEMATICAL NOTATION k2 k0

= (virtual photon mass) 2 , = virtual photon energy in the 7rN c.m.s.,

F. Gutbrod, G. Kramer, Pion electroproduction k

= virtual p h o t o n m o m e n t u m in the n N c.m.s.,

k

= kllkl,

q0 q

= p i o n e n e r g y in the rrN c.m.s., = p i o n m o m e n t u m in the nN c.m.s.,

0

=q/[q[,

0 X E1 E2 M ~t W S

eZ/4n g2/4rr

= = = = = = = = = =

t

= (q0 - k0)2 - (q - k ) 2 '

473

angle b e t w e e n virtual p h o t o n and p i o n in the c.m.s., COS 0 , initial n u c l e o n energy in the rrN c.m.s., final n u c l e o n energy in fire nN c.m.s., n u c l e o n mass, p i o n mass, total e n e r g y in the rrN c.m.s., W2 , 1/137, 14.6,

The d i f f e r e n t i a l cross section for p i o n p r o d u c t i o n b y virtual p h o t o n s reads: do dtd4~-

°u + t a L e ° T c o s 2 ~ b +

2V/2~+ 1)o Icos~,

q~

= p i o n a z i m u t h a l angle versus the e l e c t r o n s c a t t e r i n g plane,

X1 , X2

= n u c l e o n Pauli spinors.

REFERENCES [1] C. Driver, K. Heinloth, K. H6hne, G. ttofmarm, P. Karow, J. Rathje, D. Schmidt and G. Specht, Phys. Letters 35B (1971) 77 and 81; Nucl. Phys. B30 (1971) 245. [2] C.N. Brown, C.R. Canizares, W.E. Cooper, A.M. Eisner, G.J. Feldman, C.A. Lichtenstein, L. Litt, W. Lockeretz, V.B. Montana and F.M. Pipkin, Phys. Rev. Letters 26 (1971) 987 and 991. [3] P.S. Kummer, A.B. Clegg, F. Foster, G. Hughes, R. Siddle, J. Allison, B. Dickinson, E. Evangelides, M. Ibbotson, R. Lawson, R.S. Meaburn, H.E. Montgomery, W.J. Shuttleworth and A. S of air, DNPL/P 67 Daresbury (1971 ). [4] H. Fraas and D. Schildknecht, DESY 71/72 (197l); Phys. Letters 35B (1971) 72; DESY 71/59 (1971). [5] B.H. Kellet, Nucl. Phys. B38 (1972) 573. [6] W. Schmidt, DESY 71/22 (1971). [7] F.A. Berends and R. Gastmans, Phys. Rev. Letters 27 (1971) 124. [8] R.C.E. Devenish and D.H. Lyth, Phys. Rev. D5 (1972) 47. [9] Ph. Dennery, Phys. Rev. 124 (1961) 2000. [10] K. Heinloth, in Proc. Daresbury study weekend no. 3, DNPL/R 15 (1971) and private communication. [1l] H.F. Jones, Nuovo Cimento 40A (1965) 1018. [12] A.B. Clegg, Proc. of the Int. Symposium on electron and photon interactions at high energies, Liverpool (1969).

474

F. Gutbrod, G. Kramer, Pion electroproduction

[13] S.L. Adler, Ann. of Phys. 50 (1968) 189. [141 K. Heinloth, private communication. [15] F.W. Brasse, E. Chazelas, W. Fehrenbach, K.H. Frank, E. Ganssauge, J. Gayler, V. Korbel, J. May, M. Merkwitz, V. Rittenberg and H.R. Rubinstein, DESY 71/68 (1971). [161 C. Driver, K. Heinloth, K. H~Shne, G. Hofmann, F. Janata, P. Karow, J. Rathje, D. Schmidt and G. Specht, DESY 71/56 (1971) and Nucl. Phys. B38 (1972) 1, [17] Analysis: A. Bartl, W. Majerotto and D. Schildknecht, DESY 72/4 (1972) and ref. [18]. Data: C. Driver, K. Heinloth, K. H~Shne, G. Hoffmann, P. Karow, D. Schmidt and G. Specht, Nucl. Phys. B32 (1971) 45 and ref. [18]. [18] Analysis: F.A. Berends and R. Gastmans, Phys. Rev. D5 (1972)204. Data: C.N. Brown et al., Contribution no. 274 to the 1971 Int. Symposium on electron and photon interactions at high energies, Cornell. [19] F. Bulos, R.K. Carnegie, G.E. Fischer, E,E. Kluge, D.W.G.S. Leith, H.L. Lynch, B. Ratcliff, B. Richter, H.H. Williams and S.H. Williams, Phys. Rev. Letters 26 (1971) 1457.