The NΛK coupling constant

Volume 50B, number 3

PHYSICS LETTERS

10 June 1974

THE NAK COUPLING CONSTANT P. BAILLON, C. BRICMAN 1, M. FERRO-LUZZI, J.-M. PERREAU, ILD. TRIPP 2 and T. YPSILANTIS CERN, Geneva, Switzerland

Y. DECLAIS and J. SI~GUINOT University of Caen4, France Received 1 April 1974 A dispersion relation analysis has been performed using the real parts of the K±p forward scattering amplitudes measured over the incident momentum region from 1 to 3 GeV/c by the electronics experiment of the preceding letter. Two possible values are found for the NAK coupling constant. One of them is in very good agreement with the symmetry theory predictions.

The determination of the NAK coupling constant has remained an unsolved problem for more than a decade [1 ]. The methods employed- mainly based on the application of forward dispersion relations - have yielded a wide spectrum of values varying over the years and ranging from ~ 2 to ~ 14 [2]. The experimental data on which most determinations have been constructed are the real parts of the K±p forward scattering amplitude derived either from partial wave analyses or forward extrapolations of the elastic differential cross sections [3]. With a few exceptions at high energy, these values are not the results ofad hoc experiments and are frequently open to criticism. We present a determination of the NAK coupling constant based on the ratios of the real and imaginary parts for K±p measured by the experiment described in the preceding letter [4]. The K±p forward scattering amplitude in the laboratory is f± (6o) = D± (co) + iA ±(co) = ~ is the K± laboratory energy and k is the kaon momentum. The connection between K÷ and K- is given by the crossing relation f± (~) = f~ (-~o). We use a dispersion relation for f+ subtracted at ~ = 0 [5]

1 IISN, Bruxelles. 2 LBL, University of California, Berkeley. a Now at CERN. 4 Work supported by IN2P3.

D(w)=D(O)+coI(co)+g2

(D

Wy (w - Wy)

R.

(1)

Here g2 = (g2AK + 0.84 g2r.K)/41r is the effective coupling constant due to the A and ~ poles taken at an energy O~y = -0.11 GeV intermediate between ~aand w~ and R = [(mA- mp) 2 -m2K] [4m2pis the usual pole residue factor. Under the assumption that the contribution of the Y. pole is negligible, the value of g2 represents a good approximation of the A pole coupling constant. We use the same normalized units as in ref. [2]. The l(w) term represents the integral contribution and is separable into three parts W

I l(w) =

r

47r2 mJKLW(w+w)

o÷(6o') ]k' d~o'

~'(~'-~)J (2)

1 12 (w) = --~

mK f to rtA

+4_(~') 6o'(¢o' +~0) d e '

(3)

(4) 13 (co) = same as I 1 (w) but integrated from w to

oo.

The first integral covers the physical region up to w = 50 GeV: we have calculated it using a fit to the available K±p total cross sections (o±) for k > 0.25 GeV/c and the values predicted by the low energy phase shifts for the region near threshold. The second

383

Volume 50B, number 3

PHYSICS LETTERS

integral, over the unphysical region, has been obtained from the low energy solutions. The third integral, giving the a s y m p t o t i c c o n t r i b u t i o n in the region o f unk n o w n total cross sections, has been a p p r o x i m a t e d by the series expansion N 13 (co) "~ ~ b n co n, (5) n=0 where the coefficients b n and the limiting order N are d e t e r m i n e d by the fit described below. Eq. (1) has been used to calculate the K+p real parts at the energies where t h e y have been measured. A least-square fit was then p e r f o r m e d using the six K-p and three K+p real parts given in the previous letter [4]. The free parameters o f the fit are the coefficients of the expansion (5), the coupling constant g2 and the u n k n o w n q u a n t i t y D ( 0 ) . We find that two coefficients in (5) are necessary and sufficient to reach a good fit; a total o f four parameters has thus been d e t e r m i n e d ¢ using nine data points. As o t h e r authors have already noticed, the value o f the coupling constant thus o b t a i n e d depends on the choice o f the low energy solutions used to evaluate 12 (w). In particular, all the constant-K-matrix solutions yield a lower value o f g 2 than those including the effective-range terms. A m o n g the m a n y varieties o f low energy analyses we have chosen those o f refs. [6, 7] as representatives o f these two groups. Our solution using the constant-K-matrix o f ref. [6] yields g2 = 19.8 -+4.4 and using the effective-range K-matrix o f ref. [7] yields 27.3 -+4.4. In either case the four-parameter fit has a X2 o f 5.1 and predicts values of the real parts for the t w o solutions which are practically undistinguishable above k ~ 250 MeV/c. Table 1 lists the parameters and the values o f the ratio a = D/A predicted by the first solution as a f u n c t i o n o f the in-

Table 1 The ratios a t = D+_/A+_expected from the fit using ref. [6] as low-energy input. The parameters are: g2 = 19.8 +-4.4. D (0) = - (2.16 -2_0.5) fm, b o =0.004 +-0.019 and b 1 = - 0.019 +- 0.009. The x2of this 5-constraint fit is 5.1. Values above 3 GeV/c are only given as an indication and should be taken with caution. Incident momentum

GeV/c

,1 It is worth pointing out that these data, obtained via a Legendre polynomial fit to the elastic differential cross sections, have been consistently ignored in most of the dispersion relation work because they were considered "too high". The values which can be found in their place in the latest compilations [ 3] bear no resemblance with those published in ref. [8]. Considering the importance of the low energy 384

a+

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.37 0.40 0.45 0.50 0.60 0.70 0.73 0.75 0.80 0.85 0.90 1.00 1.10 1.20 1.30 1.40 1.45 1.50 1.60 1.70 1.8o 1 . 9 0

cident m o m e n t u m . These are also shown in fig. 1 together with the measured values o f ref. [4] and some early results f r o m t w o bubble chamber e x p e r i m e n t s on K - p interactions [8] which are remarkably close to our predictions ,1. On the o t h e r hand it should be

10 June 1974

2.00

-21.45 -10.60 6.94 - 5.09 -

-

4 . 0 0

- 3.20 - 2.66 2.48 - 2.25 - 2.01 -

8.00 9.00 10.00 15.00 20.00

-0.317 -0.052 0.128 0.274 0 . 4 1 1

0.586 0.707 0.665 0.205 0.375

- 1.41 - 1.15 1.13 - 1.09

0.511 0.652 0.773 0.788 O.783

-

0.825 0.711 0.601 0.458 0.402

0.659 0.631 0.564 0.363 0.065

- 0.419 0.446 - 0.475 - 0.466 - 0.465

0.037 0.129 0.226 0.237 0.219

0.460 0.456 - 0.453 - 0.450 0.449

0.154 0.088 0.050 0.062 0.053

0.444 - 0.438 0.432 0.433 - 0.442 0.460

0.002 -0.027 - 0.085 --0.131 -0.174 -0.215

0.482 - 0.507 - 0.533 0.687 - 0.853

-0.256 -0.299 -0.339 -0.529 -0.712

-

1 . 7 8

-

-

-

-

-

2.50 3.00 4.00 5.00 6.00 7.00

a_

-

-

-

-

-

-

Volume 50B, number 3

PHYSICS LETTERS

T

'1

I"

10 June 1974

T

K'pr

i1,i

0,5

y

0

0.5

-0,5

o

K'p tt

~: -0.5

-0.5

it

-I .0

-I .0 0.5

I

2

3 4 5

Incident lab. momentum, P

-I.5 -2,0 0

o,5

I,o

1,5

2,o

incident lab, momentum, P

Z5

3,0

(GeV/c)

Fig. 1. The ratio o f the real and imaginary part o f the forward scattering amplitude as a function of the incident momentum. The full circles are from ref. [41, the crosses from ref. [8]. The curves represent the fit to the points o f ref. [4] alone.

noticed that the values ofa(K-p) predicted by the low energy solutions [6, 7] below ~- 200 MeV/c are different from those suggested by our fit .2. This is in contrast with the K+p situation where our fit agrees quite well with both the low energy predictions [9] and most of the other data available in this momentum region [3]. In an attempt to eliminate or reduce the dependence of the coupling constant on the low energy phase shifts, we have tried a conformal mapping procedure similar to that of ref. [10]. We have added to the value ofI2(co ) calculated via a low energy solution, a corrective term represented by a power expansion in the Continued , t region in the determination of the coupling constant, it is not surprising that the values o f g 2 found by earlier analyses using these lower values are smaller than those reported in the present study. , 2 The discrepancy between real parts near the KN threshold is best illustrated by a comparison in terms o f scattering lengths. Ref. [6] predicts a_ = - 0 . 8 6 fm for the real part of the scattering length whereas the corresponding dispersion relation wants a_ = - 0 . 3 9 fm. Ref. [7 ] predicts - 0.51 fm as compared to - 0 . 0 8 o f the dispersion relation.

I0

20

(GeVlc)

Fig. 2. High energy behaviour of c~. The continuous curve is the four-parameter fit to the data o f ref. [4] (triangles). The dotted curve is the two-parameter fit to the same data. The full circles represent measurements o f a both in magnitude and sign (ref. [11] ). The empty circles are measurements o f the magnitude o f a alone (ref. [3] ); they have been arbitrarily plotted as negative.

variable ~ = (~/co, m K - ~/co- co,ra)/(~/co - m K + ~/co- cO~rA). This variable maps the entire co-plane into the unit circle I~1 = 1 and its interior. The unphysical region co~rA~< co <~mK is projected on the circle and the analytic properties of the scattering amplitude allow an expansion in powers of ~. The conclusion of the study is that this procedure does not give a better fit to our data. The values of g2 found with the minimum order of the expansion are still de~ pendent upon the low energy input. Increasing the order of the expansion has the effect of introducing serious instabilities and a lack of accuracy in the determination o f g 2. Turning now to the high-energy behaviour of our solution, fig. 2 shows the values expected for c~ on the basis of the fit (solid curve) as compared to the data available between 3 and 20 GeV/c [3, 11] .3. Although ,3 These data belong to two distinct categories: (i) Those indicated by empty circles in fig. 2 refer to absolute determinations o f c~ derived from extrapolations o f elastic sections to t=O, sometime from as far away as t=0.2 GeV 2 . We have arbitrarily plotted them with a negative sign (most authors prefer to take a positive sign for the K - p and a negative one for the K+p). (ii) Those indicated by full circles are based on the Coulomb-nuclear interference and come with both the magnitude and the sign.

385

Volume 50B, number 3

PHYSICS LETTERS

predictions too far from the region o f the fit should not be taken seriously, we notice that the K+p curbe is in remarkable agreement with the data whereas the K-p curve disagrees with all three interference measurements (full circles) [11]. It should be pointed out that the fitting procedure described above does not depend on the validity of the Pomeranchuk theorem, In fact, the values given in table 1 for the coefficients of the expansion (5) are incompatible with a well behaved asymptotic limit to the K±p cross sections. On the other hand, we have also fitted our data using an exact calculation of 13 based on constant and equal K+-p cross sections (o+_(oo)= 19.2 mb). The result of this two-parameter fit is indicated by the dotted curve in fig. 2. The X2 increases to 14.8, the coupling constant remaining practically unchanged (g2 = 20.5 +- 2.2 for the solution using ref. [6] as low-energy input). The high-energy K+p data are now in strong disagreement with the predictions whereas the agreement with the K-p data is somewhat improved. In conclusion, neither fit is definitely discarded or accepted and the question of the asymptotic limit will have to wait until more data, particularly of the interference type, will be available in the high energy region. Of the two values obtained f o r g 2 when using the constant or the effective-range K-matrix parametrization of the unphysical region, the lower one agrees well with the SU(3) prediction. Assuming a pseudo-vector coupling and the values o f the NNTr and AZzr coupling constants given in ref. [2], one expects g2 = 19.2 to be compared with our g2 = 19.8 + 4.4. It should also be noticed that the value D/(D+F)= 0.612 used in the above calculation for the D and F octet couplings is remarkably close to the 3/5 independently predicted by SU(6). We conclude that: (a) the long-standing disagree-

386

10 June 1974

ment with SU(3) is no longer justified, (b) the bulk of the K-p data below 1 GeV/c is not as bad as it was claimed to be, (c) the present data at high energy are not sufficiently precise to arbitrate on the asymptotic limits proposed by our analysis and (d) there still remains an ambiguity in the determination of the cou. piing constant arising essentially from a lack of data near the K-p threshold and secondarily from the absence of an effective method to reduce the influence of the low energy region.

References [ 1 ] For a review of the status and methods see N.M. Queen et al., Fort. Phys. 17 (1969) 467 and its updating: Univ. of Rome internal report No. 460 (April 1973), unpublished. [2] A compilation of the coupling constants can be found in H. Pilkhun et al., Nucl. Phys. B65 (1973) 460. [3 ] A compilation of the real parts is given in O.V. Dumbrais and N.M. Queen, Fort. Phys. 19 (1971) 491 and its updating: N.M. Queen, Birmingham University report No. UB-KP-1-73 (June 1973), unpublished. [4] P. Baillon et al., Phys. Lett. 50B (1974) 377. [5] R. Perrin and W.S. Woolcock, Nucl. Phys. B4 (1968) 671. [6] B.R. Martin and M. Sakitt, Phys. Rev. 183 (1969) 1345. [7] D. Berley et al., Phys. Rev. D1 (1970) 1996. [8] R. Armenteros et al., Nucl. Phys. B21 (1970) 15; B. Conforto et al., Nucl. Phys. B8 (1968) 265. [9] S. Goldhaber et al., Phys. Rev. 9 (1962) 135. [10] O.V. Dumbrais, T.Y. Dumbrais and N.M. Queen, Nucl. Phys. B26 (1971)497. [11] K-p: T.H.J. Bellm el al., Phys. Letters 33B (1970) 438; J.R. Cambell et al., Nucl. Phys. B64 (1973) 1; R.W. Meijer, Ph.D. thesis (Amsterdam, 1973). K*p: T.H.J. Bellm et al., N. Cimento Letters 3 (1970) 389.