- Email: [email protected]
c for I = 0 and 1
Nuclear Physics B94 (1975) 413-430 © North-Holland Publishing Company
KAON-NUCLEON PARTIAL-WAVE AMPLITUDES BELOW 1.5 GeV/c FOR I = 0 AND 1 B.R. MARTIN Department o f Ph3'sies and Astronomy, University College London Received 30 April 1975
Kaon-nucleon partial-wave amplitudes for isospin I = 0 and 1 are presented from a simultaneous phase-shift analysis of K+p and K+d data in the region below 1.5 GeV/c kaon lab momentum,
1. Introduction The study o f KN two-body amplitudes at low energies has been o f considerable interest since the first suggestions [1,2] o f the possible existence of an exotic Z 1 resonance in the K+p system. Subsequent analysis has failed to settle this question definitively: on balance the evidence is not very compelling, although there is a residual doubt lingering over the P13 amplitude t . The fi)rm of the K+p phase shifts, however, is qualitatively well known, although there are still quantitative differences between even the latest solutions [4,5]. Interest has been reawakened in the question o f exotic resonances by the recent publication [6,7] o f K+n elastic and charge-exchange differential cross sections (d.c.s.) (obtained from K+d experiments), which supplement earlier measurements of the I = 0 total cross section (obtained from K+p and K+d total cross sections). The latter exhibits a large ( ~ 4 rob) enhancement around 1 GeV/c (see fig. 1), and had already raised the possibility o f an I = 0 exotic resonance, Z 0. Subsequent phase shift analyses of the 1= 0 KN system [8,9] produced solutions with suggested resonances in one or more o f the partial waves S01 , POt and D03. The published phase-shift analyses, however, suffer from one or more limitations: incomplete 1 = 0 data sets are used; either strictly model-dependent, or restrictive, parametrizations are employed; few partial waves are included; and the I = 1 amplitudes are kept fixed when fitting the K+n data. Therefore, in this paper, we present a simultaneous energy-dependent phase-shift analysis of both isospin channels, incor. porating the latest data, an improved parametrization, and higher partial waves, Mini reviews with a complete list of references may be found in ref. [3].
414
B.R. Martin / K N partial- wave amplitudes
thereby updating and extending previously published work. The resulting amplitudes should also be of use in helping to analyse KLP ~ Ksp data which are currently appearing.
2. Formalism
2.1. Observables We denote the spin-flip and non-flip amplitudes for a specific KN isospin I by
gl(w, O) and fl(w, O) respectively, (I = 0, 1), where w is the total centre-of-mass (c.m.) energy and 0 is the c.m. scattering angle. Amplitudes for the various two-body K+N processes are: f(K+p ~ K+p) =fp = f l ,
(2.1)
f(K+n "+ K+n) =fn = l ( f l +fO),
(2.2)
f(K+n -+ K°P) -=fx = ~(fl _ f 0 ) ,
(2.3)
and similarly for the spin-flip terms. The K+p elastic d.c.s, and proton polarization (unpolarized target) are given by
P
) = 2 Re(lp• gp)n, do Pp (-d--~p
(2.5)
where n - (q X q ' ) / l ( q
× q')l,
(2.5)
and q(q') is the initial (final) c.m. momentum of the kaon. In practice, data for reactions (2.2) and (2.3) are obtained from deuterium targets, and so account must be taken of the bound-state nature of the neutron. This may be done using the formalism of the impulse approximation, which leads to the following expressions for the d.c.s, off deutrons, where the bracketed particle is a spectator [ 10]: ( d ° ) ~[K+d - + ~K+n(P)] ) ~= (
~-~ [K+d ~ K0p(p)] =
n ={]fnl2+lgn[2}I ,
x
= ]fxl2(I - J) + [gxl2(I - ~J).
(2.6)
(2.7)
B.R. Martin / K N partial-wave amplitudes
415
The quantities I and J are form factors, depending on both w and 0, and may be calculated once a form is assumed for the deutron wave function. We have taken over the results of the authors of ref. [8] who used the (dominant) S-wave part of the Moravcsik-Gartenhaus wave function [11 ] with a cut on the spectator momentum 100 < qspec < 250 MeV/c. We return to this point in sect. 3 below. Measurements also exist of the total, and total inelastic cross sections, for both 1= 0 and 1 scattering, the former being obtained by combining K+p and K+d data. In addition to the 'break-up' measurements off deuterium, there is a limited amount of d.c.s, data for the coherent process K+d -+ K+d [ 12]. In the impulse approximation this can also be expressed in terms of free-nucleon amplitudes. If we again use only the S-wave part of the deuteron wave function, then [ 12] [K+d-+ K+d] =
[[fdl2 + ]
,
(2.8)
where fd ~ f p +fn '
gd =gp + g n '
S is the spherical deuteron form factor for an invariant momentum transfer (squared) t; and p is the c.m. momentum in the K+d system. The use of these data, and the calculation of the form factor, is discussed in subsect. 4.3 below. In the analysis which follows, Coulomb corrections were applied to the K+p and K+d elastic d.c.s, in the manner of the Lea et al. [13].
2.2. Parametrization The amplitudes f and g (dropping the isospin index) may be expanded in terms of dimensionless partial-wave amplitudes Fl+_(w) by
f(w, 0) = (l/q) ~
[(/+ 1)Fl+(W ) + 1Fl_(W)lPt(cos 0),
g(w, O) = (i/q) ~
[Fl+(W) -- Fl_(W)]Pil(cos 0).
l=0
(2.9)
(2.10)
The energy dependence OfFl+(W ) has been parametrized by the following form for the inverse amplitude:
CI(q)F~I(w ) = ~ a(l+-)(q/qmax)n - i[1 + O(q - qin)R2+_(q)] Cl(q) , (2.11) n=O
416
t3.R. Martin / KN partial-waveamplitudes
where
Cl(q ) = q2l+l /11
+
(q/qin)2l+l ] ,
(2.12)
M/
Rl+(q)_ ~ _,
b(l+)( q - q i n
m= 1 m
)m
\qmax_qin/
'
(2.13)
qin and qmax correspond to the inelastic threshold and maximum c.m. momentum of the analysis, respectively, and 0 is the usual (0, 1) step function. The parameters a(l+-)nand b(m/-+)are to be determined by fitting data. The advantages of this parametrization are: that it is explicitly unitary (by virtue of the theta function and the positivity of its coefficient); it has the characteristic q21 threshold behaviour (but switches this off smoothly above the inelastic threshold); and leads to very general rational function forms (ratios of polynomials) for the real and imaginary parts OfFl+_(w) which do not diverge as q -+ qmax" In practice, qin and qmax were taken to correspond to kla b = 0.52 and 1.6 GeV/c, respectively. The parametrization (2.11) was used for all waves with J ~ ~, with
(Ml, Nl)
= (3,5),
(3,5),
(2,4),
(2,3),
1,
2,
3,
for l=0,
respectively. (Less parameters are expected for the imaginary part o f F ~ l(w) than the real part because of the smooth behaviour of the inelasticity.) In addition, the F 7 and both G-waves were included with fixed values taken from the theoretical predictions of Alcock and Cottingham * [14]. The most extensive previous I = 0 analysis [8] included only S-, P- and D-waves.
3. Data
Since one of the main aims of this analysis is to extract 1 = 0 amplitudes, a reasonable balance must be maintained between the amount of data included from free nucleons (K+p, therefore pure I = 1), and from deuterons, the former being far more numerous. We have, in practice, included all available angular distributions for reactions (2.2) and (2.3) (off deuterons) below 1.5 GeV/c, and a subset of the available data for elastic K+p scattering in the same momentum range. The cut-off momentum was dictated by the availability of K+d data. The K+p data set includes all published polarization measurements, and those d.c.s, experiments having the fullest angular * An updated version of ref. [14] was-used which included A exchange terms.
B.R. Martin / K N partial- wave amplitudes
417
Table 1 Details of the data used in the phase-shift analysis Data type
Ref.
Momenta (GeV/c)
[17]
0.175 0.355 0.479 0.772 0.970 1.450 1.130
0.205 0.385 0.565 0.813 1.098 1.495
[19]
0.145 0.325 0.432 0.731 0.910 1.400 1.060
1201
1.259
1.336
1.373
I211
1.387
1.472
1.530
1221
0.778
123] [24] [251
0.870 1.320 1.330 1.370
0.910 1.370 1.430 1.450
0.970 1.450 1.540
°tot
[15,16]
mean values at 28 m o m e n t a
°inel Ref 1 (0=0 ° )
[ 15 ]
mean values at 25 m o m e n t a
this paper
values at 26 m o m e n t a
(a) K+p ~ K+p do/d~2
[4] [18]
P(O)
0.235 0.500 0.603 0.857 1.170 1.540
0.265 0.613 0.646 0.899 1.207
0.295 0.726 0.689 0.939 1.310
1.090 1.540
1.170
1.220
(b) K+n(p) ~ K ° p ( p ) do/d~
[26] [271
1281 1291
0.330 0.432 0.936 0.640 1.060 1.510 0.600 0.865
[281
0.600
[61
P(O)
0.529 0.524
0.641 0.604
0.811 0.688
0.771
0.853
0.720 1.130
0.780 1.210
0.850 1.290
0.900 1.350
0.980 1.420
0.970
1.210
1.365
0.524
0.604
0.688
0.771
0.853
0.720 1.130
0.780 1.210
0.850 1.290
0.900 1.350
0.980 1.420
(c) K+n(p) ~ K+n(p)
do/d~
[271 [71
0.432 0.936 0.640 1.060 1.510
atot
[15,161
mean values al 23 m o m e n t a
ainel Re f o (0 = 0 °)
I151
mean values at 18 m o m e n t a
this paper
values at 24 m o m e n t a
(d) 1 = 0 data
418
B.R. Martin / KN partial-wave amplitudes
coverage and greatest precision, chosen so that they form a fine momentum grid spanning the region analysed. Values of the total inelastic cross section, the total cross section, and the forward real part of the non-flip amplitude, Re f l ( w , 0 = 0°), were used for both isospins, also at a grid of momentum values. The cross sections were selected to lie on a smooth curve passing through the many measurements of these quantities [15,16], with errors reflecting the most accurate experiments. The amplitudes were taken from a new evaluation of forward dispersion relations, which is described in the appendix. Full details of all the data used in the analysis are given in table 1. The overall consistency of the data is generally good except for the K+n elastic d.c.s, in the forward hemisphere at some momenta below about 1 GeV/c, where there is a marked disagreement between the measurements of refs. [7] and [27]. We will return to this point in subsect. 4.2 below. A further point about the K+d breakup data from these two experiments is that the d.c.s, are extracted using different cuts on the spectator momentum: ref. [7] uses 100 < qspec < 250 MeV/c, whereas ref. [27] uses 0 < qspec < 280 MeV/c. Taking account of the errors on the data, the only form factor which changes significantly when using the latter cut is J, and then only in the extreme backward direction. In order to be able to include both chargeexchange sets of data we have therefore excluded points with cos 0 ~< -0.95.
4. Phase-shift analysis 4.1. Preliminaries
In the most extensive I = 0 analysis prior to the present work [8], three main types of phase-shift solution were presented: A, C and D. (Similar solutions have been published by Aaron et al. [9].) Solution A can be rejected at 600 MeV/c by its failure to reproduce the measured polarization of the non-spectator proton in the reaction K+d -->K0p(p) [28]. The other two solutions have the common feature of a large positive P01 amplitude, but differ in their other partial waves. A large positive P01 amplitude interfering with the (known) large negative S 11 is the simplest way of reproducing the observed large polarization (positive in our convention) in K+n charge exchange, but if P01 is the dominant I = 0 partial-wave amplitude, with Re P01 ~" - R e Sll > O, then this also provides a simple explanation of other prominent features of the angular distributions at 600 MeV/c: the backward dip in the d.c.s, for K+d ~ K0p(p); the maximum at 0 = 90 ° in the same d.c.s., and its ratio to the backward d.c.s.; and the shallow minimum at 90 ° in the d.c.s, for K+d -> K+n(p). Moreover, provided a scattering length approximation is valid at this momentum, then a Pot amplitude of the requisite size is predicted from dispersion relation sum rules [30]. Overall, therefore, the evidence for this type of solution looks strong. Away from the low-energy region, and particularly when inelasticity sets in, oth-
B.R. Martin / KN partial-wave amplitudes
419
er, qualitatively different, solutions are not ruled out, but none have yet been found which extrapolate to reproduce the observed charge-exchange polarization at 600 MeV/c. In this analysis, solutions are only explored which conform to this requirement, which in practice means that they have a large positive P01 amplitude in the vicinity of 600 MeV/c. 4. 2. Data fitting A three-step procedure was used to find the best phase-shift solution: firstly, the K+p (I = 1) data were fitted; secondly, the I = 1 parameters were kept fixed at the values found, and the I = 0 parameters varied to fit the K÷d data; finally, the parameters of both isospins were varied to fit all the KN data. Initially, we used the full KN data set (given in table 1) as published, with no corrections other than those described in sect. 3. In the final stage o f optimization, however, it was found that small renormalizations applied to some of the K+p data could improve the fit without significantly altering the amplitudes o f the solution. For completeness, these renormalization factors (which are usually within quoted normalization uncertainties) are given in table 2. No attempt was made to renormalize any deuterium data. The best solution (which we will call solution 1) has a normalized X2 (×2 per data point) o f 1.18, the breakdown o f which is shown in table 3. The poorest fit is to the K+n elastic d.c.s., and much o f the "excess" X2 is due to the inconsistencies, mentioned in sect. 3, between the K+n elastic data of refs. [7,27]. Removing the K+n elastic data of ref. [7] and reminimising produces virtually no change in the normal-
Table 2 Renormalization factors R for K+p data Data type
Ref.
Momenta (GeV/c)
do/ds2
[17]
0.145 0.265 0.500 0.565 0.731 0.936 1.098 1.373 1.472 1.387
0.175 0.295
1.170 1.450 1.540
[4] [181 [20] [211 P(O)
[23]
R 0.205 0.726
0.235
0.96 1.04 1.05
0.772
0.857
1.170 1.259 1.530
1.400 1.336
1.220
1.370
All other data used in the analysis have R = 1.
0.899
0.95 0.90 1.05
0.95 0.95 0.90 1.10 1.05
0.90
420
B.R. Martin / KN partial-wave amplitudes
Table 3 Details of the ×2 breakdown for the two solutions; N is the number of data points Solution 1
Solution 2
Channel N
X2
x2/N
N
X2
x2/N
K+p-~ K+p K+n(p) --, K°p(p) K+n(p) ~ K+n(p) I = 0 data a)
1715 470 311 65
1870.9 593.1 514.3 55.3
1.09 1.26 1.65 0.85
1715 470 225 65
1858.5 576.9 289.1 58.5
1.08 1.23 1.28 0.90
total
2561
3033.6
1.18
2475
2783.0
1.12
a) I = 0 total cross sections, total inelastic cross sections, and forward real parts for the non-flip amplitude. ized X2 values, and amplitudes which are also essentially unchanged from the former solution. However, removing the K+n elastic d.c.s, of ref. [27] and reminimising, while also producing little change in the overall normalized X2, does give a significantly better fit to the deuteron data. The ×2 breakdown for this solution 2 is shown in table 3. (The previous most extensive energy-dependent analyses in this region achieved normalized ×2 values of ~1.4 for K+p (ref. [4]), and ~1.6 for K+d (ref. [8]) compared with 1.08 and 1.22, respectively, obtained here.) While this seems to suggest that the data of ref. [27] are the ones in error, this conclusion is not without difficulties. Firstly, the discrepancies are at only three of the five momenta common to the two experiments. Secondly, the Birmingham-Rk group [27] have used their K+d data to extract d.c.s, for K+p(n) -~ K+p(n), and find good agreement with free-proton cross sections, except for the near forward region It] <~ 0.13 (GeV/c) 2, where the bound-state data are lower. This could be due to the onset of coherent K+d scattering, in which case the forward d.c.s, for K+n(p) -~ K+n(p) would be lower than the free-neutron d.c.s. [27]. Predictions of the latter, using the forward dispersion relations of the appendix and the optical theorem, are higher than the data of ref. [27]. However, these predictions are in reasonable agreement with the published data of ref. [7]. Lacking any further evidence on this point, solution 2 is preferred over solution 1 on the basis of its better ×2 values, although the designations '1 and 2' really only distinguish the data sets from which the two solutions were obtained; the amplitudes differ little from each other. This will be illustrated in subsect. 4.4 below. To illustrate the quality of the solution, we show in fig. 1 the fit to the total, and total inelastic cross sections for both I = 0 and I, from solution 2, and in figs. 2 and 3 the fit to some typical d.c.s, and polarization data from the same solution. Fig. 3 also gives predictions for K+n charge-exchange polarization at 1.2 GeV/c, and for K+n elastic polarization at 0.6 and 1.2 GeV/c.
B. R. Martin / K N partial- wave amplitudes I
F
[
I
I
4 21
]
25
17 I =1
"~]
]
20
i
T
i 0
11 I
L)
10
1
f
I
I
I
I
I
~
I 11
1
15
I
I
I
[=1 / ~inel
0 I
03
I
07
klab ( OeV/c }
[
11
[
[
15
~-
I
I
L
07 klab (OeV/c)
I
Fig. l. The I = 0 and 1 total, and total inelastic cross sections [15,16]. The points marked by circles are from the recent experiment of Carroll et al. [ 16]. The predictions are from solution 2 of the present analysis.
4.3. K+d ~ K+d As mentioned in subsect. 2.1, there is a limited amount o f data for coherent K+d scattering [ 12], and it was originally intended to include these in the phase-shift analysis. However, the data, as expected, are confined to the extreme forward directions, where the deuteron form factor of eq. (2.8) is varying most rapidly (almost exponentially) with t. Consequently, the predicted d.c.s, is extremely sensitive to the precise form factor (and hence deuteron wave function) used. (The poorly known small higher partial waves could "also be important.) Including these data, and using a range of wave functions, we were unable to find solutions with a nor-
422
B.R. Martin / K N partial-wave amplitudes I
2.0
1.0
K+n(p) =, K*n(pl 1.51 GeV/¢ ~
_
o.s
f
i.o
K+n(p) ~ K + n ( p ) 0.72 GeV/c N;illi 1.0
1.0 K+n(pl
pKOplp)
1.51 GeV/c
-C
t t
¢n
0.5
05
0 72 C-eV/c
3.0
K+ p
1.5
= K* p
1.3"- GeV/c
2O
1.0 w
1.0
0.5
K+ p
=,,'K+ p
081 GeV/c
I
0
-1
0 Cos O
0
1
0 Cos
Fig.2. Fit to sometypicaldifferentialCrosssectionsfromsolution2.
.I
B.R. Martin / KN partial-wave amplitudes
423
K°n ~ K*n 0 6 0 GeV/c
o
o
K*p ~
~
K* p
0.97 GeV/c
060 I
GeV/c
-I
-.t
-1
K*n ~K*n 1.20 GeV/c
~o K* p ~
K*p
I 20 GeV/c
1 • 54 G e W c
-1 -1
i 0
-1 -1
-I
COS ,~
[ 0
*1
I 0
-I
Cos 0 +
COS 0 ,
.
.
Fig. 3. Fit to K+p elastic and K n charge-exchange polarization from solution 2, and predictions for unmeasured polarization distributions from the same solution. malized X2 for all deuteron data better than about 2.5, which is significantly inferior to the values given in table 3. This was true even if a more exact formalism for K+d scattering was used [31 ] which included, in addition to the spherical form factor of eq. (2.8), quadrupole and magnetic form factors calculated using a modified HamadaJohnson wave function [32] due to Humberston [33]. We have, however, verified that our solutions are in semi-quantitative agreement with the distributions of ref. [ 12]. This is illustrated in fig. 4, where we show typical predictions of solution 2 for some of the data of ref. [ 12] using eq. (2.8), and calculating the spherical form factor from the S-wave part of the Moravcsik-Gartenhaus wave function [12]. 4. 4. Amplitudes The parameters for the variable amplitudes of solution 2 are given in table 4 (the F 7 and G-waves are fixed at the values from ref. [ 14], and Argand diagrams for these waves are shown in fig. 5. The qualitative behaviour of the I = 1 waves is similar to several published phaseshift solutions. Comparing in detail to the previous most extensive analysis of this region [4], shows that the S 11 amplitude bears a close resemblance to solution I1 of that paper, as, to a somewhat lesser extent, do the D-waves, whereas the P-waves are intermediate between solutions I and II. Turning to the I = 0 amplitudes, the solution obtained here has some features intermediate to solutions C and D of ref. [8]. For
424 I0
B.R. Martin / K N partial-wave amplitudes I
I
I
5
/,2 GeV/c
-'L
21 6eV/c
I
05
I
05
0
02
01
I 0 9
L O@
I
07
01
09
Cos 0
O@ Cos
Fig. 4. Predictions for K+d elastic d.c.s, from solution 2. example, the P01 wave is similar to that of solution D. However, the zeros present in the S01 and P03 of those solutions are not found here, and both these latter waves are uniformly negative and slowly varying. Of previously suggested resonance candidates [8,9] only P01 and D03 are possibilities, but in both cases observation of the Argand diagrams of fig. 5 shows that there are no significant velocity changes around the relevant loops, and so their status must be considered doubtful. As mentioned in subsect. 4.2, the amplitudes from solution 1 do not differ much from those of solution 2, and to illustrate this we show in fig. 5 the situation for the DO3 wave. The change shown is the maximum found in any partial wave, and in general the variation in the other waves is far less. This is particularly so for the I = 1 amplitudes, which are essentially unchanged between solutions. Finally, we have compared our solutions with several dispersion theoretic predictions. We find good agreement with Alcock and Cottingham [ 14] (recall that predictions from this reference for the F ~ and G-waves have been used as fixed input to the phase-shift analysis) for all partial waves ('correct signs' and semi-quantitative agreement in magnitudes) except for D05 , which is positive below ~ 7 0 0 MeV/c in ref. [14], but uniformly negative in our solutions. Likewise, our P-waves at threshold
-4.951 ( - 1 ) 3.532 ( - 1 ) 3.254 ( - 1 )
bI b2 b3
bI b2 b3
ao aI a2 a3 a4
(0) (0) (0) (0) (0)
4.681 ( - 2 ) 4.275(-1) 4.227 ( - 2 )
-4.411 8.227 -4.684 -5.622 6.058
Sll
(b) I = 1 parameters
-4.044 (1) 1.056 (2) - 1 . 0 9 2 (2) 4.915 (1) -7.685(0)
ao a1 az a3 a4
So I
(a) I = 0 parameters
(1) (0) (1) (2) (2)
2.141 (0) 1.010(0) - 2 . 4 6 3 (0)
-8.899 6.348 -1.796 3.854 -3.222
Pll
-4.178 (-1) 1.526 (0) -4.097 (-1)
3.279 (1) - 9 . 3 2 2 (1) 1.380 (2) - 1 . 4 1 2 (2) ~ 7.804(1)
Po I (2) (2) (3) (3) (3)
(2) (2) (3) (3) (3)
2.872 ( - 2 ) 6.975(0) - 5 . 4 5 2 (0)
1.318 9.225 -3.706 4.308 -1.656
PI3
3.956 (0) -1.911 (1) 1.687 (1)
-1.488 7.368 -2.391 3.150 -1.379
Po3 (2) (3) (3) (3)
(4) (4) (4) (4)
1.400 (0) -9.053(1)
-1.673 4.964 -5.160 1.833
D13
5.173 ( - 1 ) 1.623 (0)
9.469 1.145 -6.468 4.631
Do3 (3) (4) (4) (4)
(4) (4) (4) (3)
1.308 (1) -1.032(1)
-1.648 3.120 -1.103 -3.861
DIS
4.282 (0) - 4 . 0 0 3 ( 1)
-8.333 3.349 -4.681 2.100
Dos
4.692 ( - 1 ) - 5 . 6 7 0 (0)
- 8 . 6 1 3 (4) 1.169 (5) - 3 . 4 3 0 (4)
FlS
1.788 (0) 2.683 ( - 3 )
1.008 (5) 2.059 (5) 1.060 (5)
Fos
Table 4 Values of the parameters obtained for solution 2 in units such that m n -= 1 ; the numbers in brackets are the powers of ten which multiply the corresponding parameter
426
B.R. Martin / KN partial-wave amplitudes
\\
/ //~ls
\
[~111
I ~~8
I
P01 12~
l
7
~\\
.S01
T
I/)
0
0.5
I
-0.5
0
0.5
-05
/ ~ .D03
0.3-
/S
-0.1 ~D15
ts
\ to
D13
~~
,~1~ ~
005
" ?} '~
/
15 8
-0.1
~ F15
-0.1
0 / F05
0.3
Fig. 5. Argand diagrams of the SPD and F ~ waves of solution 2. The numbers indicate the lab momentum in 100 MeV/c units. The Do3 amplitude of solution 1 is shown by the broken line. are in good agreement with the sum rule estimates of Knudsen and Martin [30], and the partial-wave dispersion relation predictions of Hedegaard-Jensen et al. [34] except for the P03 amplitudes of the latter reference which is given as positive, in contrast to the predictions of refs. [30] and [ 14], and the present phase-shift solutions. I thank the members of the Birmingham-RHEL group, particularly F. Wickens and J. Homer, for access to their data prior to its publication; N. Cottingham for supplying numbers from the updated calculations of Alcock and Cottingham; G. Alberi for communicating details of the form-factor calculations of the BGRT group; and C. Wilkin for providing numerical values of the deuteron form factors of ref. [31].
Appendix
Forward dispersion relations We have included, as part of the data set for the phase-shift analysis, a grid of values of Re fI(w, 0 = 0 °) (I = 0, l) calculated from once - subtracted forward dis-
B.R. Martin / KN partial-wave amplitudes
427
persion relations. The latter are conveniently written in terms of the total kaon lab energy o0 as follows [30]: Re f / ( w ) ( M ~ M m )
I
CIRy(c°-m)
as- ~ ( 4 ; m-~-~
--
;-~
1
+e1(~)+i~': c[ vr(~),
(A.1)
where a/is the S-wave KN scattering length for isospin I; m ( M ) is the kaon (nucleon) mass; ~ y ~ ( M 2 - M 2 - m2)/(2M),
Rv
(Y = A, E),
- M) 2 - m2.1
= {G2yN~[(MY
~--X~]L
-2~S
'
and the coefficients are 1 co_-_c~, -- -c~ - 2,
co_- ~,
cO = _ c l
c o
=_cl
- ~,
,
-
.3
The subsidiary quantities V l ( w ) and P l ( w ) are defined by, VI(co) =
dco' (co' + w)(oo' + m) ' Wy~r
and pl(co) ~
co - m
p f dw'k' m
(w' - ¢o)(6o' - m) - (w' + co)(w' + m ) j '
where T 1 and ~I are given in terms of the K-*N total cross sections by T0 - 2 o n _ o
p,
T 1 =o+p,
T0=2o n__2op, ~1 ~op_ ,
and ~Yrr
= I ( M y + mrr) 2 2M
M 2 _
m2l"
B.R. Martin / KN partial-waveamplitudes
428
Table 5 Values of Ref I (0 = 0°) obtained from the forward dispersion relation (A.I) in units such that mn= 1 kla b (GeV/c)
Re f 0
Re f 1
0.2 0.3 0.4 0.5 0.6
0.033 0.066 0.109 0.155 0.197
-+ 0.051 ± 0.054 -+ 0.060 ± 0.069 ± 0.079
-0.333 -0.330 -0.330 -0.329 -0.324
÷ 0.017 -+ 0.018 + 0.020 ± 0.022 ± 0.024
0.7 0.8 0.9 1.0 1.1
0.224 0.228 0.213 0.187 0.143
± 0.086 ± 0.088 ± 0.086 -+ 0.083 -+ 0.076
-0.310 -0.280 -0.255 -0.246 -0.264
± 0.025 ± 0.027 • 0.030 ± 0.032 • 0.032
1.2 1.3 1.4 1.5
0.028 -0.033 -0.039 -0.034
-+ 0.060 ± 0.055 -+ 0.055 ± 0.056
-0.300 -0.345 -0.377 -0.399
-+ 0.033 ± 0.038 ± 0.043 ± 0.047
The input data used to calculate the forward real parts f r o m eq. (A. 1) were as follows. To evaluate VI, the S-wave K-matrix model o f Chao et al. [35] was used (solution B o f this paper) together with a Y~(1385) c o n t r i b u t i o n evaluated in a narrow-width model with broken S U ( 3 ) couplings. (The details are given in ref. [30].) The same input was used for the KN part ofP I for k' ~< 350 MeV/c. The remaining part ofP I for 0.35 < k' < 25 G e V / c was evaluated using tabulated cross sections [15,16], and above this m o m e n t u m we used models o f ref. [36], assuming equality o f K+n and K+p, and K - n and K - p total cross sections. The errors made by this ass u m p t i o n are within the total error on the o u t p u t real parts. For the KN couplings we used the values o f Knudsen and Pietarinen [37] (see also ref. [30]), i.e. g2KAN/4n = 11.0 and g2KN/4n = 1.4. (The latter is obtained by assuming that S U ( 3 ) is good for the ratio o f the A to 22 couplings.) These are probably among the most reliable estimates currently available, as they are deduced directly f r o m KN total cross sections and polarizatioh ,~easurements via forward dispersion relations. They do not use values o f Re f I (0 = 0 °) as input, and so there is no circularity here. Finally, there are the S-wave scattering lengths appearing in the subtraction constants. F o r I = 1 we use a~ = 0.30 - 0.02 fm, which is compatible with m o s t previous estimates. The I = 0 S-wave scattering length is not reliably k n o w n , but the I = 0 KN total cross section indicates that a s0 is very small. We have used a 0 = 0.0 -+ 0.04 fm, which covers previous estimates. The values used for a Is are b o t h consistent with those obtained in the present analysis. Since the quantities Re f l (0 = 0 °) are likely to be o f use in other calculations we give, in table 5, a brief tabulation o f the values calculated from eq. (A. 1). The errors
B.R. Martin / KN partial-wave amplitudes
429
are o b t a i n e d b y m a k i n g e s t i m a t e s o f the u n c e r t a i n t i e s in the various i n p u t s described above, a n d are, o f course, s o m e w h a t subjective.
References [1] A.T. Lea, B.R. Martin and G.C. Oades, Phys. Letters 23 (1966) 380. [2] R.L. Cool et al., Phys. Rev. Letters 17 (1966) 102. [3] Particle Data Group, Phys. Letters 50B (1974) 1; R.E. Cutkosky, Proc. 17th Int. Conf. on high-energy physics, London 1974, ed. J.R. Smith. [4] C.J. Adams et al., Nucl. Phys. B66 (1973) 36. [5] R.E. Cutkosky et al., Proc. Purdue Conf. on Baryon Resonances (1973); Proc. of 17th Int. Conf. on high-energy physics, London 1974, ed. J.R. Smith; N. Cottingham et al., Proc. London ConL, 1974; R. Arndt et al., Proc. London Conf., 1974; O. Fich et al., Proc. London Conf., 1974. [6] G. GiacomeUi et al., Nucl. Phys. B42 (1972) 437. [7] G. Giacomelli et al., Nucl. Phys. B56 (1973) 346. [8] B.C. Wilson et al., Nucl. Phys. B42 (1972) 445; G. Giacomelli et al., Nucl. Phys. BT1 (1974) 138. [9] R. Aaron et al., Phys. Rev. 7D (1973) 1401. [10] V.J. Stenger et al., Phys. Rev. 134B (1964) 1111. [11] M. Moravcsik, Nucl. Phys. 7 (1958) 113. [12] G. Giacomelli et al., Nucl. Phys. B68 (1974) 285. [13] A.T. Lea, B.R. Martin and G.C. Oades, Phys. Rev. 165 (1968) 1770. [14] J.W. Alcock and N. Cottingham, Nucl. Phys. B56 (1973). [15] G. Giacomelli, Prog. Nucl. Phys. 12 (1970) 77; D.V. Bugg et al., Phys. Rev. 168 (1968) 1466; R.L Cool et al., Phys. Rev. ID (1970) 1887; T. Bowen et al., Phys. Rev. 2D (1970) 2599; T. Bowen et al., Phys. Rev. 7D (1973) 22; G. Giacomelli et al., Nucl. Phys. B37 (1972) 577; A.A. Hirata et al., Phys. Rev. Letters 21 (1968) 1485; Nucl. Phys. B30 (1971) 157; G. Giacomelli et al., Nucl. Phys. B20 (1970) 301; See also ref. [13]. [16] A.S. Carroll et al., BNL report 17915 (1973). [17] W. Cameron et al., Nucl. Phys. B78 (1974) 93. [18] J.H. Goldman, Maryland thesis (1972), report no. 73-054. [19] G. Giacomelli et al., Nucl. Phys. B20 (1970) 301. [20J B.J. Charles et al., Phys. Letters 40B (1972) 289. [21] P.C. Barber et al., Nucl. Phys. B61 (1973) 125. [22] S. Focardi et al., Phys. Letters 24B (1967) 314. [23] M.G. Albrow et al., Nucl. Phys. B30 (1971) 273. [24] R.D. Ehrlich et al., Phys. Rev. Letters 26 (1971) 925. [25] J.D. Asbury et al., Phys. Rev. Letters 23 (1969) 194; B.A. Barnett et al., Phys. Rev. D8 (1973) 2751. [26] W. Slater et al., Phys. Rev. Letters 7 (1961) 378. [27] C.J.S. Damarell et al., Nucl. Phys. B94 (1975) 374. [28] A.K. Ray et al., Phys. Rev. 183 (1968) 1183.
430 [29] [30] [31] [32] [33] [34] [351 [36] [37]
B.R. Martin / KN partial-wave amplitudes A.A. Ftirata et al., Nucl. Phys. B30 (1971) 157. C.P. Knudsen and B.R. Martin, Nucl. Phys. B61 (1973) 307. C. Michael and C. Wilkin, Nucl. Phys. B l l (1969) 99. T. Hamada and I.D. Johnston, Nucl. Phys. 34 (1962) 382. J. Humberston, Private communication to the authors of ref. [31 ]. N. Hedegaard-Jensen et al., Phys. Letters 46B (1973) 46. Y.A. Chao et al., Nucl. Phys. B56 (1973) 46. C. Bourreley et al., Nucl. Phys. B67 (1973)452. C.P. Knudsen and E. Pietarinen, Nucl. Phys. B57 (1973) 637.
KAON-NUCLEON PARTIAL-WAVE AMPLITUDES BELOW 1.5 GeV/c FOR I = 0 AND 1 B.R. MARTIN Department o f Ph3'sies and Astronomy, University College London Received 30 April 1975
Kaon-nucleon partial-wave amplitudes for isospin I = 0 and 1 are presented from a simultaneous phase-shift analysis of K+p and K+d data in the region below 1.5 GeV/c kaon lab momentum,
1. Introduction The study o f KN two-body amplitudes at low energies has been o f considerable interest since the first suggestions [1,2] o f the possible existence of an exotic Z 1 resonance in the K+p system. Subsequent analysis has failed to settle this question definitively: on balance the evidence is not very compelling, although there is a residual doubt lingering over the P13 amplitude t . The fi)rm of the K+p phase shifts, however, is qualitatively well known, although there are still quantitative differences between even the latest solutions [4,5]. Interest has been reawakened in the question o f exotic resonances by the recent publication [6,7] o f K+n elastic and charge-exchange differential cross sections (d.c.s.) (obtained from K+d experiments), which supplement earlier measurements of the I = 0 total cross section (obtained from K+p and K+d total cross sections). The latter exhibits a large ( ~ 4 rob) enhancement around 1 GeV/c (see fig. 1), and had already raised the possibility o f an I = 0 exotic resonance, Z 0. Subsequent phase shift analyses of the 1= 0 KN system [8,9] produced solutions with suggested resonances in one or more o f the partial waves S01 , POt and D03. The published phase-shift analyses, however, suffer from one or more limitations: incomplete 1 = 0 data sets are used; either strictly model-dependent, or restrictive, parametrizations are employed; few partial waves are included; and the I = 1 amplitudes are kept fixed when fitting the K+n data. Therefore, in this paper, we present a simultaneous energy-dependent phase-shift analysis of both isospin channels, incor. porating the latest data, an improved parametrization, and higher partial waves, Mini reviews with a complete list of references may be found in ref. [3].
414
B.R. Martin / K N partial- wave amplitudes
thereby updating and extending previously published work. The resulting amplitudes should also be of use in helping to analyse KLP ~ Ksp data which are currently appearing.
2. Formalism
2.1. Observables We denote the spin-flip and non-flip amplitudes for a specific KN isospin I by
gl(w, O) and fl(w, O) respectively, (I = 0, 1), where w is the total centre-of-mass (c.m.) energy and 0 is the c.m. scattering angle. Amplitudes for the various two-body K+N processes are: f(K+p ~ K+p) =fp = f l ,
(2.1)
f(K+n "+ K+n) =fn = l ( f l +fO),
(2.2)
f(K+n -+ K°P) -=fx = ~(fl _ f 0 ) ,
(2.3)
and similarly for the spin-flip terms. The K+p elastic d.c.s, and proton polarization (unpolarized target) are given by
P
) = 2 Re(lp• gp)n, do Pp (-d--~p
(2.5)
where n - (q X q ' ) / l ( q
× q')l,
(2.5)
and q(q') is the initial (final) c.m. momentum of the kaon. In practice, data for reactions (2.2) and (2.3) are obtained from deuterium targets, and so account must be taken of the bound-state nature of the neutron. This may be done using the formalism of the impulse approximation, which leads to the following expressions for the d.c.s, off deutrons, where the bracketed particle is a spectator [ 10]: ( d ° ) ~[K+d - + ~K+n(P)] ) ~= (
~-~ [K+d ~ K0p(p)] =
n ={]fnl2+lgn[2}I ,
x
= ]fxl2(I - J) + [gxl2(I - ~J).
(2.6)
(2.7)
B.R. Martin / K N partial-wave amplitudes
415
The quantities I and J are form factors, depending on both w and 0, and may be calculated once a form is assumed for the deutron wave function. We have taken over the results of the authors of ref. [8] who used the (dominant) S-wave part of the Moravcsik-Gartenhaus wave function [11 ] with a cut on the spectator momentum 100 < qspec < 250 MeV/c. We return to this point in sect. 3 below. Measurements also exist of the total, and total inelastic cross sections, for both 1= 0 and 1 scattering, the former being obtained by combining K+p and K+d data. In addition to the 'break-up' measurements off deuterium, there is a limited amount of d.c.s, data for the coherent process K+d -+ K+d [ 12]. In the impulse approximation this can also be expressed in terms of free-nucleon amplitudes. If we again use only the S-wave part of the deuteron wave function, then [ 12] [K+d-+ K+d] =
[[fdl2 + ]
,
(2.8)
where fd ~ f p +fn '
gd =gp + g n '
S is the spherical deuteron form factor for an invariant momentum transfer (squared) t; and p is the c.m. momentum in the K+d system. The use of these data, and the calculation of the form factor, is discussed in subsect. 4.3 below. In the analysis which follows, Coulomb corrections were applied to the K+p and K+d elastic d.c.s, in the manner of the Lea et al. [13].
2.2. Parametrization The amplitudes f and g (dropping the isospin index) may be expanded in terms of dimensionless partial-wave amplitudes Fl+_(w) by
f(w, 0) = (l/q) ~
[(/+ 1)Fl+(W ) + 1Fl_(W)lPt(cos 0),
g(w, O) = (i/q) ~
[Fl+(W) -- Fl_(W)]Pil(cos 0).
l=0
(2.9)
(2.10)
The energy dependence OfFl+(W ) has been parametrized by the following form for the inverse amplitude:
CI(q)F~I(w ) = ~ a(l+-)(q/qmax)n - i[1 + O(q - qin)R2+_(q)] Cl(q) , (2.11) n=O
416
t3.R. Martin / KN partial-waveamplitudes
where
Cl(q ) = q2l+l /11
+
(q/qin)2l+l ] ,
(2.12)
M/
Rl+(q)_ ~ _,
b(l+)( q - q i n
m= 1 m
)m
\qmax_qin/
'
(2.13)
qin and qmax correspond to the inelastic threshold and maximum c.m. momentum of the analysis, respectively, and 0 is the usual (0, 1) step function. The parameters a(l+-)nand b(m/-+)are to be determined by fitting data. The advantages of this parametrization are: that it is explicitly unitary (by virtue of the theta function and the positivity of its coefficient); it has the characteristic q21 threshold behaviour (but switches this off smoothly above the inelastic threshold); and leads to very general rational function forms (ratios of polynomials) for the real and imaginary parts OfFl+_(w) which do not diverge as q -+ qmax" In practice, qin and qmax were taken to correspond to kla b = 0.52 and 1.6 GeV/c, respectively. The parametrization (2.11) was used for all waves with J ~ ~, with
(Ml, Nl)
= (3,5),
(3,5),
(2,4),
(2,3),
1,
2,
3,
for l=0,
respectively. (Less parameters are expected for the imaginary part o f F ~ l(w) than the real part because of the smooth behaviour of the inelasticity.) In addition, the F 7 and both G-waves were included with fixed values taken from the theoretical predictions of Alcock and Cottingham * [14]. The most extensive previous I = 0 analysis [8] included only S-, P- and D-waves.
3. Data
Since one of the main aims of this analysis is to extract 1 = 0 amplitudes, a reasonable balance must be maintained between the amount of data included from free nucleons (K+p, therefore pure I = 1), and from deuterons, the former being far more numerous. We have, in practice, included all available angular distributions for reactions (2.2) and (2.3) (off deuterons) below 1.5 GeV/c, and a subset of the available data for elastic K+p scattering in the same momentum range. The cut-off momentum was dictated by the availability of K+d data. The K+p data set includes all published polarization measurements, and those d.c.s, experiments having the fullest angular * An updated version of ref. [14] was-used which included A exchange terms.
B.R. Martin / K N partial- wave amplitudes
417
Table 1 Details of the data used in the phase-shift analysis Data type
Ref.
Momenta (GeV/c)
[17]
0.175 0.355 0.479 0.772 0.970 1.450 1.130
0.205 0.385 0.565 0.813 1.098 1.495
[19]
0.145 0.325 0.432 0.731 0.910 1.400 1.060
1201
1.259
1.336
1.373
I211
1.387
1.472
1.530
1221
0.778
123] [24] [251
0.870 1.320 1.330 1.370
0.910 1.370 1.430 1.450
0.970 1.450 1.540
°tot
[15,16]
mean values at 28 m o m e n t a
°inel Ref 1 (0=0 ° )
[ 15 ]
mean values at 25 m o m e n t a
this paper
values at 26 m o m e n t a
(a) K+p ~ K+p do/d~2
[4] [18]
P(O)
0.235 0.500 0.603 0.857 1.170 1.540
0.265 0.613 0.646 0.899 1.207
0.295 0.726 0.689 0.939 1.310
1.090 1.540
1.170
1.220
(b) K+n(p) ~ K ° p ( p ) do/d~
[26] [271
1281 1291
0.330 0.432 0.936 0.640 1.060 1.510 0.600 0.865
[281
0.600
[61
P(O)
0.529 0.524
0.641 0.604
0.811 0.688
0.771
0.853
0.720 1.130
0.780 1.210
0.850 1.290
0.900 1.350
0.980 1.420
0.970
1.210
1.365
0.524
0.604
0.688
0.771
0.853
0.720 1.130
0.780 1.210
0.850 1.290
0.900 1.350
0.980 1.420
(c) K+n(p) ~ K+n(p)
do/d~
[271 [71
0.432 0.936 0.640 1.060 1.510
atot
[15,161
mean values al 23 m o m e n t a
ainel Re f o (0 = 0 °)
I151
mean values at 18 m o m e n t a
this paper
values at 24 m o m e n t a
(d) 1 = 0 data
418
B.R. Martin / KN partial-wave amplitudes
coverage and greatest precision, chosen so that they form a fine momentum grid spanning the region analysed. Values of the total inelastic cross section, the total cross section, and the forward real part of the non-flip amplitude, Re f l ( w , 0 = 0°), were used for both isospins, also at a grid of momentum values. The cross sections were selected to lie on a smooth curve passing through the many measurements of these quantities [15,16], with errors reflecting the most accurate experiments. The amplitudes were taken from a new evaluation of forward dispersion relations, which is described in the appendix. Full details of all the data used in the analysis are given in table 1. The overall consistency of the data is generally good except for the K+n elastic d.c.s, in the forward hemisphere at some momenta below about 1 GeV/c, where there is a marked disagreement between the measurements of refs. [7] and [27]. We will return to this point in subsect. 4.2 below. A further point about the K+d breakup data from these two experiments is that the d.c.s, are extracted using different cuts on the spectator momentum: ref. [7] uses 100 < qspec < 250 MeV/c, whereas ref. [27] uses 0 < qspec < 280 MeV/c. Taking account of the errors on the data, the only form factor which changes significantly when using the latter cut is J, and then only in the extreme backward direction. In order to be able to include both chargeexchange sets of data we have therefore excluded points with cos 0 ~< -0.95.
4. Phase-shift analysis 4.1. Preliminaries
In the most extensive I = 0 analysis prior to the present work [8], three main types of phase-shift solution were presented: A, C and D. (Similar solutions have been published by Aaron et al. [9].) Solution A can be rejected at 600 MeV/c by its failure to reproduce the measured polarization of the non-spectator proton in the reaction K+d -->K0p(p) [28]. The other two solutions have the common feature of a large positive P01 amplitude, but differ in their other partial waves. A large positive P01 amplitude interfering with the (known) large negative S 11 is the simplest way of reproducing the observed large polarization (positive in our convention) in K+n charge exchange, but if P01 is the dominant I = 0 partial-wave amplitude, with Re P01 ~" - R e Sll > O, then this also provides a simple explanation of other prominent features of the angular distributions at 600 MeV/c: the backward dip in the d.c.s, for K+d ~ K0p(p); the maximum at 0 = 90 ° in the same d.c.s., and its ratio to the backward d.c.s.; and the shallow minimum at 90 ° in the d.c.s, for K+d -> K+n(p). Moreover, provided a scattering length approximation is valid at this momentum, then a Pot amplitude of the requisite size is predicted from dispersion relation sum rules [30]. Overall, therefore, the evidence for this type of solution looks strong. Away from the low-energy region, and particularly when inelasticity sets in, oth-
B.R. Martin / KN partial-wave amplitudes
419
er, qualitatively different, solutions are not ruled out, but none have yet been found which extrapolate to reproduce the observed charge-exchange polarization at 600 MeV/c. In this analysis, solutions are only explored which conform to this requirement, which in practice means that they have a large positive P01 amplitude in the vicinity of 600 MeV/c. 4. 2. Data fitting A three-step procedure was used to find the best phase-shift solution: firstly, the K+p (I = 1) data were fitted; secondly, the I = 1 parameters were kept fixed at the values found, and the I = 0 parameters varied to fit the K÷d data; finally, the parameters of both isospins were varied to fit all the KN data. Initially, we used the full KN data set (given in table 1) as published, with no corrections other than those described in sect. 3. In the final stage o f optimization, however, it was found that small renormalizations applied to some of the K+p data could improve the fit without significantly altering the amplitudes o f the solution. For completeness, these renormalization factors (which are usually within quoted normalization uncertainties) are given in table 2. No attempt was made to renormalize any deuterium data. The best solution (which we will call solution 1) has a normalized X2 (×2 per data point) o f 1.18, the breakdown o f which is shown in table 3. The poorest fit is to the K+n elastic d.c.s., and much o f the "excess" X2 is due to the inconsistencies, mentioned in sect. 3, between the K+n elastic data of refs. [7,27]. Removing the K+n elastic data of ref. [7] and reminimising produces virtually no change in the normal-
Table 2 Renormalization factors R for K+p data Data type
Ref.
Momenta (GeV/c)
do/ds2
[17]
0.145 0.265 0.500 0.565 0.731 0.936 1.098 1.373 1.472 1.387
0.175 0.295
1.170 1.450 1.540
[4] [181 [20] [211 P(O)
[23]
R 0.205 0.726
0.235
0.96 1.04 1.05
0.772
0.857
1.170 1.259 1.530
1.400 1.336
1.220
1.370
All other data used in the analysis have R = 1.
0.899
0.95 0.90 1.05
0.95 0.95 0.90 1.10 1.05
0.90
420
B.R. Martin / KN partial-wave amplitudes
Table 3 Details of the ×2 breakdown for the two solutions; N is the number of data points Solution 1
Solution 2
Channel N
X2
x2/N
N
X2
x2/N
K+p-~ K+p K+n(p) --, K°p(p) K+n(p) ~ K+n(p) I = 0 data a)
1715 470 311 65
1870.9 593.1 514.3 55.3
1.09 1.26 1.65 0.85
1715 470 225 65
1858.5 576.9 289.1 58.5
1.08 1.23 1.28 0.90
total
2561
3033.6
1.18
2475
2783.0
1.12
a) I = 0 total cross sections, total inelastic cross sections, and forward real parts for the non-flip amplitude. ized X2 values, and amplitudes which are also essentially unchanged from the former solution. However, removing the K+n elastic d.c.s, of ref. [27] and reminimising, while also producing little change in the overall normalized X2, does give a significantly better fit to the deuteron data. The ×2 breakdown for this solution 2 is shown in table 3. (The previous most extensive energy-dependent analyses in this region achieved normalized ×2 values of ~1.4 for K+p (ref. [4]), and ~1.6 for K+d (ref. [8]) compared with 1.08 and 1.22, respectively, obtained here.) While this seems to suggest that the data of ref. [27] are the ones in error, this conclusion is not without difficulties. Firstly, the discrepancies are at only three of the five momenta common to the two experiments. Secondly, the Birmingham-Rk group [27] have used their K+d data to extract d.c.s, for K+p(n) -~ K+p(n), and find good agreement with free-proton cross sections, except for the near forward region It] <~ 0.13 (GeV/c) 2, where the bound-state data are lower. This could be due to the onset of coherent K+d scattering, in which case the forward d.c.s, for K+n(p) -~ K+n(p) would be lower than the free-neutron d.c.s. [27]. Predictions of the latter, using the forward dispersion relations of the appendix and the optical theorem, are higher than the data of ref. [27]. However, these predictions are in reasonable agreement with the published data of ref. [7]. Lacking any further evidence on this point, solution 2 is preferred over solution 1 on the basis of its better ×2 values, although the designations '1 and 2' really only distinguish the data sets from which the two solutions were obtained; the amplitudes differ little from each other. This will be illustrated in subsect. 4.4 below. To illustrate the quality of the solution, we show in fig. 1 the fit to the total, and total inelastic cross sections for both I = 0 and I, from solution 2, and in figs. 2 and 3 the fit to some typical d.c.s, and polarization data from the same solution. Fig. 3 also gives predictions for K+n charge-exchange polarization at 1.2 GeV/c, and for K+n elastic polarization at 0.6 and 1.2 GeV/c.
B. R. Martin / K N partial- wave amplitudes I
F
[
I
I
4 21
]
25
17 I =1
"~]
]
20
i
T
i 0
11 I
L)
10
1
f
I
I
I
I
I
~
I 11
1
15
I
I
I
[=1 / ~inel
0 I
03
I
07
klab ( OeV/c }
[
11
[
[
15
~-
I
I
L
07 klab (OeV/c)
I
Fig. l. The I = 0 and 1 total, and total inelastic cross sections [15,16]. The points marked by circles are from the recent experiment of Carroll et al. [ 16]. The predictions are from solution 2 of the present analysis.
4.3. K+d ~ K+d As mentioned in subsect. 2.1, there is a limited amount o f data for coherent K+d scattering [ 12], and it was originally intended to include these in the phase-shift analysis. However, the data, as expected, are confined to the extreme forward directions, where the deuteron form factor of eq. (2.8) is varying most rapidly (almost exponentially) with t. Consequently, the predicted d.c.s, is extremely sensitive to the precise form factor (and hence deuteron wave function) used. (The poorly known small higher partial waves could "also be important.) Including these data, and using a range of wave functions, we were unable to find solutions with a nor-
422
B.R. Martin / K N partial-wave amplitudes I
2.0
1.0
K+n(p) =, K*n(pl 1.51 GeV/¢ ~
_
o.s
f
i.o
K+n(p) ~ K + n ( p ) 0.72 GeV/c N;illi 1.0
1.0 K+n(pl
pKOplp)
1.51 GeV/c
-C
t t
¢n
0.5
05
0 72 C-eV/c
3.0
K+ p
1.5
= K* p
1.3"- GeV/c
2O
1.0 w
1.0
0.5
K+ p
=,,'K+ p
081 GeV/c
I
0
-1
0 Cos O
0
1
0 Cos
Fig.2. Fit to sometypicaldifferentialCrosssectionsfromsolution2.
.I
B.R. Martin / KN partial-wave amplitudes
423
K°n ~ K*n 0 6 0 GeV/c
o
o
K*p ~
~
K* p
0.97 GeV/c
060 I
GeV/c
-I
-.t
-1
K*n ~K*n 1.20 GeV/c
~o K* p ~
K*p
I 20 GeV/c
1 • 54 G e W c
-1 -1
i 0
-1 -1
-I
COS ,~
[ 0
*1
I 0
-I
Cos 0 +
COS 0 ,
.
.
Fig. 3. Fit to K+p elastic and K n charge-exchange polarization from solution 2, and predictions for unmeasured polarization distributions from the same solution. malized X2 for all deuteron data better than about 2.5, which is significantly inferior to the values given in table 3. This was true even if a more exact formalism for K+d scattering was used [31 ] which included, in addition to the spherical form factor of eq. (2.8), quadrupole and magnetic form factors calculated using a modified HamadaJohnson wave function [32] due to Humberston [33]. We have, however, verified that our solutions are in semi-quantitative agreement with the distributions of ref. [ 12]. This is illustrated in fig. 4, where we show typical predictions of solution 2 for some of the data of ref. [ 12] using eq. (2.8), and calculating the spherical form factor from the S-wave part of the Moravcsik-Gartenhaus wave function [12]. 4. 4. Amplitudes The parameters for the variable amplitudes of solution 2 are given in table 4 (the F 7 and G-waves are fixed at the values from ref. [ 14], and Argand diagrams for these waves are shown in fig. 5. The qualitative behaviour of the I = 1 waves is similar to several published phaseshift solutions. Comparing in detail to the previous most extensive analysis of this region [4], shows that the S 11 amplitude bears a close resemblance to solution I1 of that paper, as, to a somewhat lesser extent, do the D-waves, whereas the P-waves are intermediate between solutions I and II. Turning to the I = 0 amplitudes, the solution obtained here has some features intermediate to solutions C and D of ref. [8]. For
424 I0
B.R. Martin / K N partial-wave amplitudes I
I
I
5
/,2 GeV/c
-'L
21 6eV/c
I
05
I
05
0
02
01
I 0 9
L O@
I
07
01
09
Cos 0
O@ Cos
Fig. 4. Predictions for K+d elastic d.c.s, from solution 2. example, the P01 wave is similar to that of solution D. However, the zeros present in the S01 and P03 of those solutions are not found here, and both these latter waves are uniformly negative and slowly varying. Of previously suggested resonance candidates [8,9] only P01 and D03 are possibilities, but in both cases observation of the Argand diagrams of fig. 5 shows that there are no significant velocity changes around the relevant loops, and so their status must be considered doubtful. As mentioned in subsect. 4.2, the amplitudes from solution 1 do not differ much from those of solution 2, and to illustrate this we show in fig. 5 the situation for the DO3 wave. The change shown is the maximum found in any partial wave, and in general the variation in the other waves is far less. This is particularly so for the I = 1 amplitudes, which are essentially unchanged between solutions. Finally, we have compared our solutions with several dispersion theoretic predictions. We find good agreement with Alcock and Cottingham [ 14] (recall that predictions from this reference for the F ~ and G-waves have been used as fixed input to the phase-shift analysis) for all partial waves ('correct signs' and semi-quantitative agreement in magnitudes) except for D05 , which is positive below ~ 7 0 0 MeV/c in ref. [14], but uniformly negative in our solutions. Likewise, our P-waves at threshold
-4.951 ( - 1 ) 3.532 ( - 1 ) 3.254 ( - 1 )
bI b2 b3
bI b2 b3
ao aI a2 a3 a4
(0) (0) (0) (0) (0)
4.681 ( - 2 ) 4.275(-1) 4.227 ( - 2 )
-4.411 8.227 -4.684 -5.622 6.058
Sll
(b) I = 1 parameters
-4.044 (1) 1.056 (2) - 1 . 0 9 2 (2) 4.915 (1) -7.685(0)
ao a1 az a3 a4
So I
(a) I = 0 parameters
(1) (0) (1) (2) (2)
2.141 (0) 1.010(0) - 2 . 4 6 3 (0)
-8.899 6.348 -1.796 3.854 -3.222
Pll
-4.178 (-1) 1.526 (0) -4.097 (-1)
3.279 (1) - 9 . 3 2 2 (1) 1.380 (2) - 1 . 4 1 2 (2) ~ 7.804(1)
Po I (2) (2) (3) (3) (3)
(2) (2) (3) (3) (3)
2.872 ( - 2 ) 6.975(0) - 5 . 4 5 2 (0)
1.318 9.225 -3.706 4.308 -1.656
PI3
3.956 (0) -1.911 (1) 1.687 (1)
-1.488 7.368 -2.391 3.150 -1.379
Po3 (2) (3) (3) (3)
(4) (4) (4) (4)
1.400 (0) -9.053(1)
-1.673 4.964 -5.160 1.833
D13
5.173 ( - 1 ) 1.623 (0)
9.469 1.145 -6.468 4.631
Do3 (3) (4) (4) (4)
(4) (4) (4) (3)
1.308 (1) -1.032(1)
-1.648 3.120 -1.103 -3.861
DIS
4.282 (0) - 4 . 0 0 3 ( 1)
-8.333 3.349 -4.681 2.100
Dos
4.692 ( - 1 ) - 5 . 6 7 0 (0)
- 8 . 6 1 3 (4) 1.169 (5) - 3 . 4 3 0 (4)
FlS
1.788 (0) 2.683 ( - 3 )
1.008 (5) 2.059 (5) 1.060 (5)
Fos
Table 4 Values of the parameters obtained for solution 2 in units such that m n -= 1 ; the numbers in brackets are the powers of ten which multiply the corresponding parameter
426
B.R. Martin / KN partial-wave amplitudes
\\
/ //~ls
\
[~111
I ~~8
I
P01 12~
l
7
~\\
.S01
T
I/)
0
0.5
I
-0.5
0
0.5
-05
/ ~ .D03
0.3-
/S
-0.1 ~D15
ts
\ to
D13
~~
,~1~ ~
005
" ?} '~
/
15 8
-0.1
~ F15
-0.1
0 / F05
0.3
Fig. 5. Argand diagrams of the SPD and F ~ waves of solution 2. The numbers indicate the lab momentum in 100 MeV/c units. The Do3 amplitude of solution 1 is shown by the broken line. are in good agreement with the sum rule estimates of Knudsen and Martin [30], and the partial-wave dispersion relation predictions of Hedegaard-Jensen et al. [34] except for the P03 amplitudes of the latter reference which is given as positive, in contrast to the predictions of refs. [30] and [ 14], and the present phase-shift solutions. I thank the members of the Birmingham-RHEL group, particularly F. Wickens and J. Homer, for access to their data prior to its publication; N. Cottingham for supplying numbers from the updated calculations of Alcock and Cottingham; G. Alberi for communicating details of the form-factor calculations of the BGRT group; and C. Wilkin for providing numerical values of the deuteron form factors of ref. [31].
Appendix
Forward dispersion relations We have included, as part of the data set for the phase-shift analysis, a grid of values of Re fI(w, 0 = 0 °) (I = 0, l) calculated from once - subtracted forward dis-
B.R. Martin / KN partial-wave amplitudes
427
persion relations. The latter are conveniently written in terms of the total kaon lab energy o0 as follows [30]: Re f / ( w ) ( M ~ M m )
I
CIRy(c°-m)
as- ~ ( 4 ; m-~-~
--
;-~
1
+e1(~)+i~': c[ vr(~),
(A.1)
where a/is the S-wave KN scattering length for isospin I; m ( M ) is the kaon (nucleon) mass; ~ y ~ ( M 2 - M 2 - m2)/(2M),
Rv
(Y = A, E),
- M) 2 - m2.1
= {G2yN~[(MY
~--X~]L
-2~S
'
and the coefficients are 1 co_-_c~, -- -c~ - 2,
co_- ~,
cO = _ c l
c o
=_cl
- ~,
,
-
.3
The subsidiary quantities V l ( w ) and P l ( w ) are defined by, VI(co) =
dco' (co' + w)(oo' + m) ' Wy~r
and pl(co) ~
co - m
p f dw'k' m
(w' - ¢o)(6o' - m) - (w' + co)(w' + m ) j '
where T 1 and ~I are given in terms of the K-*N total cross sections by T0 - 2 o n _ o
p,
T 1 =o+p,
T0=2o n__2op, ~1 ~op_ ,
and ~Yrr
= I ( M y + mrr) 2 2M
M 2 _
m2l"
B.R. Martin / KN partial-waveamplitudes
428
Table 5 Values of Ref I (0 = 0°) obtained from the forward dispersion relation (A.I) in units such that mn= 1 kla b (GeV/c)
Re f 0
Re f 1
0.2 0.3 0.4 0.5 0.6
0.033 0.066 0.109 0.155 0.197
-+ 0.051 ± 0.054 -+ 0.060 ± 0.069 ± 0.079
-0.333 -0.330 -0.330 -0.329 -0.324
÷ 0.017 -+ 0.018 + 0.020 ± 0.022 ± 0.024
0.7 0.8 0.9 1.0 1.1
0.224 0.228 0.213 0.187 0.143
± 0.086 ± 0.088 ± 0.086 -+ 0.083 -+ 0.076
-0.310 -0.280 -0.255 -0.246 -0.264
± 0.025 ± 0.027 • 0.030 ± 0.032 • 0.032
1.2 1.3 1.4 1.5
0.028 -0.033 -0.039 -0.034
-+ 0.060 ± 0.055 -+ 0.055 ± 0.056
-0.300 -0.345 -0.377 -0.399
-+ 0.033 ± 0.038 ± 0.043 ± 0.047
The input data used to calculate the forward real parts f r o m eq. (A. 1) were as follows. To evaluate VI, the S-wave K-matrix model o f Chao et al. [35] was used (solution B o f this paper) together with a Y~(1385) c o n t r i b u t i o n evaluated in a narrow-width model with broken S U ( 3 ) couplings. (The details are given in ref. [30].) The same input was used for the KN part ofP I for k' ~< 350 MeV/c. The remaining part ofP I for 0.35 < k' < 25 G e V / c was evaluated using tabulated cross sections [15,16], and above this m o m e n t u m we used models o f ref. [36], assuming equality o f K+n and K+p, and K - n and K - p total cross sections. The errors made by this ass u m p t i o n are within the total error on the o u t p u t real parts. For the KN couplings we used the values o f Knudsen and Pietarinen [37] (see also ref. [30]), i.e. g2KAN/4n = 11.0 and g2KN/4n = 1.4. (The latter is obtained by assuming that S U ( 3 ) is good for the ratio o f the A to 22 couplings.) These are probably among the most reliable estimates currently available, as they are deduced directly f r o m KN total cross sections and polarizatioh ,~easurements via forward dispersion relations. They do not use values o f Re f I (0 = 0 °) as input, and so there is no circularity here. Finally, there are the S-wave scattering lengths appearing in the subtraction constants. F o r I = 1 we use a~ = 0.30 - 0.02 fm, which is compatible with m o s t previous estimates. The I = 0 S-wave scattering length is not reliably k n o w n , but the I = 0 KN total cross section indicates that a s0 is very small. We have used a 0 = 0.0 -+ 0.04 fm, which covers previous estimates. The values used for a Is are b o t h consistent with those obtained in the present analysis. Since the quantities Re f l (0 = 0 °) are likely to be o f use in other calculations we give, in table 5, a brief tabulation o f the values calculated from eq. (A. 1). The errors
B.R. Martin / KN partial-wave amplitudes
429
are o b t a i n e d b y m a k i n g e s t i m a t e s o f the u n c e r t a i n t i e s in the various i n p u t s described above, a n d are, o f course, s o m e w h a t subjective.
References [1] A.T. Lea, B.R. Martin and G.C. Oades, Phys. Letters 23 (1966) 380. [2] R.L. Cool et al., Phys. Rev. Letters 17 (1966) 102. [3] Particle Data Group, Phys. Letters 50B (1974) 1; R.E. Cutkosky, Proc. 17th Int. Conf. on high-energy physics, London 1974, ed. J.R. Smith. [4] C.J. Adams et al., Nucl. Phys. B66 (1973) 36. [5] R.E. Cutkosky et al., Proc. Purdue Conf. on Baryon Resonances (1973); Proc. of 17th Int. Conf. on high-energy physics, London 1974, ed. J.R. Smith; N. Cottingham et al., Proc. London ConL, 1974; R. Arndt et al., Proc. London Conf., 1974; O. Fich et al., Proc. London Conf., 1974. [6] G. GiacomeUi et al., Nucl. Phys. B42 (1972) 437. [7] G. Giacomelli et al., Nucl. Phys. B56 (1973) 346. [8] B.C. Wilson et al., Nucl. Phys. B42 (1972) 445; G. Giacomelli et al., Nucl. Phys. BT1 (1974) 138. [9] R. Aaron et al., Phys. Rev. 7D (1973) 1401. [10] V.J. Stenger et al., Phys. Rev. 134B (1964) 1111. [11] M. Moravcsik, Nucl. Phys. 7 (1958) 113. [12] G. Giacomelli et al., Nucl. Phys. B68 (1974) 285. [13] A.T. Lea, B.R. Martin and G.C. Oades, Phys. Rev. 165 (1968) 1770. [14] J.W. Alcock and N. Cottingham, Nucl. Phys. B56 (1973). [15] G. Giacomelli, Prog. Nucl. Phys. 12 (1970) 77; D.V. Bugg et al., Phys. Rev. 168 (1968) 1466; R.L Cool et al., Phys. Rev. ID (1970) 1887; T. Bowen et al., Phys. Rev. 2D (1970) 2599; T. Bowen et al., Phys. Rev. 7D (1973) 22; G. Giacomelli et al., Nucl. Phys. B37 (1972) 577; A.A. Hirata et al., Phys. Rev. Letters 21 (1968) 1485; Nucl. Phys. B30 (1971) 157; G. Giacomelli et al., Nucl. Phys. B20 (1970) 301; See also ref. [13]. [16] A.S. Carroll et al., BNL report 17915 (1973). [17] W. Cameron et al., Nucl. Phys. B78 (1974) 93. [18] J.H. Goldman, Maryland thesis (1972), report no. 73-054. [19] G. Giacomelli et al., Nucl. Phys. B20 (1970) 301. [20J B.J. Charles et al., Phys. Letters 40B (1972) 289. [21] P.C. Barber et al., Nucl. Phys. B61 (1973) 125. [22] S. Focardi et al., Phys. Letters 24B (1967) 314. [23] M.G. Albrow et al., Nucl. Phys. B30 (1971) 273. [24] R.D. Ehrlich et al., Phys. Rev. Letters 26 (1971) 925. [25] J.D. Asbury et al., Phys. Rev. Letters 23 (1969) 194; B.A. Barnett et al., Phys. Rev. D8 (1973) 2751. [26] W. Slater et al., Phys. Rev. Letters 7 (1961) 378. [27] C.J.S. Damarell et al., Nucl. Phys. B94 (1975) 374. [28] A.K. Ray et al., Phys. Rev. 183 (1968) 1183.
430 [29] [30] [31] [32] [33] [34] [351 [36] [37]
B.R. Martin / KN partial-wave amplitudes A.A. Ftirata et al., Nucl. Phys. B30 (1971) 157. C.P. Knudsen and B.R. Martin, Nucl. Phys. B61 (1973) 307. C. Michael and C. Wilkin, Nucl. Phys. B l l (1969) 99. T. Hamada and I.D. Johnston, Nucl. Phys. 34 (1962) 382. J. Humberston, Private communication to the authors of ref. [31 ]. N. Hedegaard-Jensen et al., Phys. Letters 46B (1973) 46. Y.A. Chao et al., Nucl. Phys. B56 (1973) 46. C. Bourreley et al., Nucl. Phys. B67 (1973)452. C.P. Knudsen and E. Pietarinen, Nucl. Phys. B57 (1973) 637.