- Email: [email protected]
c
Nuclear Physics B42 (1972) 29-43. North-Holland Publishing Company
K+p ELASTIC
SCATTERING
BETWEEN
2.1 1 A N D 2 . 7 2 G e V / c
J.A. DANYSZ, B.K. PENNEY, B.C. STEWART, G. T H O M P S O N Imperial College of Seienee and Technology England
J.M. BRUNET, J.L. NARJOUX Coll~ge de France. France
J.E. ALLEN, N.J.D. JACOBS, P.H. LEWIS, P.V. M A R C H West.field College, England
Received 14 December 1971
Abstract: Final results are presented of the analysis of the elastic channel in an exposure of 40 000 pictures at each of the four incident K+ momenta 2.11, 2.31, 2.53 and 2.72 GeV/c taken in the 1.5 m British National Hydrogen Bubble Chamber at the 8 GeV/c proton synchrotron at the Rutherford High Energy Laboratory. Differential cross sections are presented and the results are compared with other published data. A Legendre polynomial analysis requires partial waves up to G wave at all momenta. For the backward peak, visible at each momentum, the slope and the intercept are calculated. A comparison of the forward peak is made with extrapolations from Regge models fitted at higher momenta.
1. I N T R O D U C T I O N The elastic channel in K+p scattering has been studied at incident K + m o m e n t a of 2.11, 2 . 3 1 , 2 . 5 3 and 2.72 GeV/c. The general details o f the exposure, data processing and partial cross section calculations are given elsewhere [1]. Preliminary results on the elastic scattering based on a p p r o x i m a t e l y one third o f the data were r e p o r t e d previously [2]. Interest in this m o m e n t u m region was stimulated by the report [3] o f structure in the total K+p cross section at 2.7 G e V / c incident K + m o m e n t u m . There were a t t e m p t s to explain this structure by the f o r m a t i o n o f an S = +1, B -- +I resonance which, in the quark model, could only be f o r m e d from at least five quarks. The results presented in this paper show no behaviour supporting the existence o f such a resonance. The values o f the total elastic cross section are given in table 1 ; they differ slightly from those reported previously [ 1 ].
30
J.A. Danysz et al., K+p elastic scattering Table 1 Elastic cross section. Momentum (GeV/c)
Cross section (mb)
2.11 2.31 2.53 2.72
6.73 5.87 5.69 5.27
-+ 0.25 -+ 0.20 -+ 0.20 + 0.20
2. D I F F E R E N T I A L CROSS SECTIONS
2.1. Preparation o f data Cuts were imposed on the incident K + momentum and on the fiducial volume. A study of the experimental biases [4] led to the imposition of further cuts on the experimental data. Only events with the cosine of the c.m.s, scattering angle, 0", less than 0.95 (corresponding to a proton range in the bubble chamber of 11 cm at 2.11 GeV/c and 21 cm at 2.72 GeV/c), and a dip angle (defined as the angle between the plane normal to the optical axes of the cameras and the plane of the tracks) of less than 72 ° were accepted. The accepted events were weighted to provide an isotropic distribution of the dip angle. The latter correction was only necessary for the forward part of the distribution (cos 0* > 0.85). A study [4] of the effects of the contamination of the beam by pions showed that pion and kaon elastic interactions are kinematically indistinguishable except for backwards scattering where the difference of m o m e n t u m of the scattered beam particles is sufficient to resolve the ambiguity. At 2.11 GeV/c this difference is 115 MeV/c at cos 0* = 1. It was found that the two reactions could always be distinguished for cos 0* < 0. The angular distributions, then, contained events in the forward hemisphere (cos 0* > 0) from both incident kaon and incident pion elastic interactions and in the backward hemisphere only from incident kaon elastic interactions. To correct for this effect, the distribution was modified by removing the estimated number of pion events from each interval of the forward part of the distribution. This number was calculated using known pion elastic differential cross sections [5] and the contamination of the beam by pions, which was about 6% at 2.3, 2.5 and 2.7 GeV/e and 12% at 2.1 GeV/c. A similar procedure was employed to derive the differential cross section as a function of t, the squared four m o m e n t u m transfer. The results are tabulated in tables 2 a - 2 d and plotted in figs. l a - l d .
2.2. Forwai'd peak At all momenta the forward part of the angular distribution is strongly peaked. The data in the forward region (t > - 1 (GeV/c) 2) are plotted in fig. 2 together with
J.A. Danysz et al., K+p elastic scattering.
31
Table 2a The d i f f e r e n t i a l cross s e c t i o n s at 2.11 G e V / c . No. o f events
d-2-° ( m b / s r ) d~2
0.95
672
4.34
± 0.17
0.90
522
3.37
± 0.15
0.80,
0.85
380
2.46
± 0.13
0.75,
0.80
252
1.63
± 0.10
No. o f events
do (mb/(GeV/c)2) at I
0.10, - 0 . 0 5
445
17.90
± 0.85
0.15, - 0 . 1 0
440
17.70
± 0.84
0.90,
0.20,
0.15
376
15.12
± 0.77
0.85,
-0.25, -0.20
296
11.90
±0.67
-0.30, -0.25
230
9.26
± 0.59
--0.35,
i t(GeV/c)2
cos 0 *
]
170
6.85
±0.50
0.70,
0.75
192
1.24
± 0.09
-0.40, -0.35
132
5.32
+0.45
0.65,
0.70
169
1.09
± 0.08
-0.45, -0.40
0.30
129
5.20
± 0.45
0.60,
0.65
130
0.84
± 0.07
-0.45
111
4.47
±0.42
0.55,
0.60
120
0.77
± 0.07
0.55, - 0 . 5 0
93
3.67
+ 0.38
0.50,
0.55
74
0.48
± 0.06
i 0.60, - 0 . 5 5
81
3.26
+0.36
0.45,
0.50
71
0.46
± 0.05 ± 0.04
-0.50,
-0.65, -0.60 --0.70, - 0 . 6 5
81
3.26
±0.36
0.40,
0.45
40
0.26
62
2.50
±0.32
0.35,
0.40
45
0.29
± 0.04
2.07
±0.21
0.30,
0.35
26
0.17
± 0.03 ± 0.02
-0.8
, -0.7
103
-0.9
, -0.8
66
1.33
±0.16
0.2 ,
0.3
40
0.13
-1.
, --0.9
54
1.09
±0.15
0.1 ,
0.2
27
0 . 0 8 7 ± 0.017
i-l.1
,-1.0
37
0.74
±0.12
0.
0.1
18
0 . 0 5 8 ± 0.014
, -1.1
22
0.44
+ 0.09
0.
18
0.058 ± 0.014
0.1
11
0.035 + 0.011
, -0.2
10
0.032 e 0.010
0.3
2
0 . 0 0 6 +- 0.005
-1.2
'-0.1
0.2 ,
-1.3
, -1.2
18
0.36
±0,09
-1.4
, -1.3
13
0.26
± 0.07
-0.3
± 0.06
~ 0.4,
, -1.4
9
-1.75, -1.5
18
0.145 ± 0 . 0 3 4
1-0.5
, -0.4
8
0.026 ± 0.009
-2.
17
0.137 + 0.033
'i 0.6 , - 0 . 5
4
0 . 0 1 3 ± 0.006
--2.25, - 2 .
7
0 . 0 5 6 ± 0.021
i-0.7
, -0.6
8
0.026 ± 0.009
-2.5
,-2.25
9
0.073 ± 0.024
~-0.8
-0.7
15
0.048 ± 0.012 1
-2.6
, -2.5
7
0.141 ± 0.053
]-0.9
, -0.8
27
0.087 + 0.017
-2.7
, -2.6
11
0.221 ± 0 . 0 6 7
19
0.061 ± 0 . 0 1 4 |
12.8 , - 2 . 7
14
0 . 2 8 2 ± 0.075
-2.9
-1.5
, -1.75
0.18
,
,
2.8
18
0 . 3 6 2 ± 0.085
3.02,
2.9
16
0.315 + 0 . 0 7 0
i
1.0 ,
0.9
J
J.A. Danysz et al.. K+p elastic scattering
32
Table 2b The differential cross sections at 2.31 GeV/c.
t(GeV/c) 2
No. of events
do {mb/(GeV/c)2 ) dt
-0.10, -0.05
339
15.15
-+0.82
0.90,
--0.15, - 0 . 1 0 li-0.20, -0.15
305
13.64
_+ 0.78
238
10.64
-+ 0.69
-0.25, -0.20
182
8.13
-0.30, -0.25
173
7.73
-0.35, -0.30
150
-0.40, -0.45
116
-0.45, -0.40
93
-0.50, -0.45
92
No. of events
d o (mb/sr) d~2
0.95
492
3.50
± 0.16
0.85,
0.90
346
2.46
-+ 0.13
0.80,
0.85
291
2.07
± 0.12
±0.60
0.75,
0.80
190
1.35
-+ 0.10
+_0.59
0.70,
0.75
146
1.04
± 0.09
6.70
+-0.55
0.65,
0.70
106
0.76
± 0.08
5.18
+-0.48
0.60,
0.65
96
0.68
± 0.70
4.16
-+ 0.43
0.55,
0.60
83
0.59
-+ 0.06
4.12
±0.43
0.50,
0.55
54
0.38
± 0.05
-0.55, -0.50
55
2.46
-+ 0.33
0.45,
0.50
47
0.33
+0.05
-0.60, -0.55
69
3.09
+- 0.37
0.40,
0.45
31
0.22
_+0.04
0.65, - 0 . 6 0
52
2.32
±0.32
0.35,
0.40
20
0.14
+0.03
- 0 . 7 0 , -O.65
56
2.50
±0.33
0.30,
0.35
17
0.12
-+0.03
0.8,-0.7
90
2.01
-+0.21
0.2 ,
0.3
18
0.064 ± 0.015 [
O.9 ,
cos 0*
O.8
50
1.12
±0.16
0.1
0.2
12
0.043 ± 0.012
-1.0
-0.9
44
0.99
-+ 0.15
0.
0.1
12
0.043 ± 0.012
-1.1
1.0
26
0.58
±0.11
[-0.1
O.
6
0.021 -+ 0.009
!-1.2 , - 1 . 1
21
0.47
±0.10
i-0.2
-0.I
6
0.021 ± 0.009
-1.3 , -1.2
10
0.224 -+ 0.07
!-0.3
-0.2
5
0.018 +- 0.008
1.4 , - 1 . 3
[
-1.5
7
0.156 ± 0.06
-0.4
-0.3
1
0.004 -+ 0.003
, -1.4
6
0.134 ± 0.05
-0.5
-0.4
2
0.007 ± 0.005
1.5
13
0.116 ± 0.032
~-0.6
-0.5
5
0.018 ± 0.005
5
0.044 -+ 0.020
9
0.032 ± 0.009
-1.75, -2.
, -1.75
2.25, - 2 .
6
0.054 -+ 0.022
-2.5 ,-2.25
4
0.036 ± 0.018
-2.75, -2.5
8
0.072 ± 0.025
-3.0 ,-2.75
10
0.089 -+ 0.028
5
0.112 ± 0.050
!-3.1 , - 3 . 0 , -3.1
6
0.134 ± 0.054
-3.3 , -3.2
11
0.246 ± 0.074
14
0.391 ± 0.100
-3.2
3.38,
3.3
6
0.021 ± 0.009
, -0.8
10
0.036 + 0.011
, -0.9
25
0.089 ± 0.018
-0.7
-1.
!I
J.A. Danysz et al., K+p elastic scattering
33
Table 2c The differential cross sections at 2.53 GeV/c.
t(GeV/c)
No. of events
- 0 . 1 0 , ---O.05
323
14.10
-+ 0 . 7 9 |
~-0.15,
0.10
3O6
13.37
±0.76
i-0.20, - 0 . 1 5
261
11.40
±0.71
-0.25, -0.20
192
8.40
-0.30,
0.25
164
7.17
-0.35, -0.30
140
0.40, - 0 . 3 5
128
-0.45, -0.40
114
-0.50, -0.45
84
i-0.55, - 0 . 5 0 -0.60, -0.55
- (mb/(GeV/~
COS 0*
No. of events
do ~ (mb/sr)
0.90,
0.95
562
3.91
±0.17
0.85,
0.90
360
2.50
±0.13
0.80,
0.85
276
1.92
±0.11
+ 0.60
0.75,
0.80
195
1.36
+- 0.09
+ 0.56
0.70,
0.75
120
0.83
± 0.08
6.12
+_0.52
0.65,
0.70
105
0.73
-+ 0.07
5.60
± 0.49
0.60,
0.65
69
0.48
-+ 0.06
4.98 ++ 0.47
0.55,
0.60
61
0.42
+- 0.05
3.68
+ 0.40
0.50,
0.55
44
0.31
± 0.05
67
2.93
+ 0.36
0.45,
0.50
28
0.19
+- 0.04
62
2.71
+ 0.34
0.40,
0.45
25
0.17
± 0.03
-0.65, -0.60
44
1.92
+ 0.29
0.35,
0.40
18
0.13
±0.03
-0.70, -0.65
42
1.84
+- 0.28
0.30,
0.35
17
0.12
+-0.03
'-0.8 , -0.7
65
1.42
+ 0.18
0.20,
0.30
15
0.052 ± 0.013
-0.9 , -0.8
56
1.22
-+ 0.16
0.1
,
0.2
11
0.038 ± 0.12
-1.
l0
41
0.90
± 0.14
O.
,
0.1
4
0.014 -+ 0.007
24
0.52
+ 0.11
-0.1
,
0.
6
0.021 ± 0.009
-1.1
23
0.50
-+ 0.10
i-o.2
-o.1
8
0.028 ± 0.010
-1.3 , -1.2
17
0.37
± 0.09
-0.3
-0.2
4
0.014 -+ 0.007
!-1.4 , - 1 . 3
7
0.15
-+ 0.06
-0.4
-0.3
5
0.017 + 0.008
9
0.20
+ 0.07
-0.5
-0.4
3
0.010 ± 0.006
12
0.11
+ 0.03
-0.6
-0.5
3
0.010 +- 0.006
, -0.9
i-1.5
,-1.4
-1.75, -1.5 , -1.75
6
0.051 + 0.021
-0.7
-0.6
5
0.017 -+ 0.008
-2.25, -2.00
9
0.079 +- 0.026
-0.8
-0.7
5
0.017 ± 0.008
6
0.052 +_0.021
-0.9
-0.8
13
0.045 +_ 0.013
10
0.044 + 0.014
1.
-0.9
23
0.080 +- 0.017
-2.
-2.5 , -2.25 -3.0 , -2.5 -3.25, -3. -3.5
,-3.25
-3.6
,
3.5
i-3.7 , - 3 . 6 -3.78,
3.7
6
0.052 -+ 0.021
12
0.105 -+ 0.03
7
0.153 ± 0.06
14
0.306 ± 0.08
7
0.191 ± 0.07
J.A. Danysz et al., K+p elastic scattering
34
Table 2d The differential cross sections at 2.72 GeV/c.
G~V/c)U !-0.10 ,
--0.05
~No. of events
do (mb/(GeVfc)2)
cos 0*
at
No. of events
d o_o(rob/st) d~
397
14.36
-+0.72
0.90,
0.95
683
3.94
i-0.15 ,-0.10
363
13.14
-+0.69
0.85,
0.90
351
2.02
-+0.11
i-0.20 , -0.15 I i-0.25 , -0.20
321
11.62
-+ 0.65
0.80,
0.85
269
1.55
+- 0.09
236
8.55
-+0.56
0.75,
0.80
201
1.16
_+0.08
-0.30 , -0.25
187
6.77
-+ 0.49
0.70,
0.75
127
0.73
_+0.07
I
_+ 0.15
--0.35 , - 0 . 3 0
144
5.22
-+ 0.43
0.65,
0.60
92
0.53
-+ 0.06
-0.40
-0.35
125
4.53
+- 0.40
0.60,
0.65
65
0.38
-+ 0.05
! - 0. 45 , - 0 . 4 0
117
4.23
-+0.39
0.55,
0.60
47
0.27
-+ 0.04
0.45
90
3.26
-+ 0.34
0.50,
0.55
31
0.18
-+0.03
0.55 , - 0 . 5 0
58
2.10
-+0.28
0.45,
0.50
23
0.13
-+0.03
0.60 , - 0 . 5 5
75
2.72
-+0.31
0.40,
0.45
19
0.110 ± 0.025
~-0.50 ,
-0.65
, -0.60
41
1.48
±0.23
0.35,
0.40
8
0.046 ± 0.016
-0.70
, -0.65
39
1.41
+-0.23
0.30,
0.35
9
0.052±0.017
!
~-0.80 ,
0.70
74
1.34
±0.16
0.20,
0.30
11
0.032+-0.010
i--0.90 ,
0.80
56
1.01
+-0.14
0.1 ,
0.2
8
0.023 ± 0.008
0.0 ,
0.1
3
0.009 -+ 0.005
-1.0
, -0.9
31
0.55
-+0.10
-1.1
, -1.0
29
0.53
-+0.10
-0.1
,-0.0
4
0.012±0.006
-1.2
, -1.1
21
0.39
+-0.08
-0.2
, -0.1
4
0.012 ± 0.006
1.3
-1.2
11
0.20
-+ 0.06
-0.3
, -0.2
4
0.012 +- 0.006
1.4
-1.3
8
0.15
± 0.05
-0.4
, -0.3
6
0.017 ± 0.007
1.4
5
0.094 _+ 0.040
-0.5
, -0.4
2
0.006 ± 0.004
-1.75
-1.5
-1.5
13
0.094 +_0.026
-0.6
, -0.5
2
0.006 ± 0.004
-2.0
-1.75
7
0.049 -+ 0.019
-0.7
, -0.6
2
0.006 ± 0.004
-2.5
2.0
11
0.040 _+0.012
-0.8
, -0.7
4
0.012 ± 0.006
3.0
-2.5
9
0.033 ± 0.011
, -0.8
4
0.012 ± 0.006
-3.5
, -3.O
3
0.011 + 0.006
, -0.9
27
0.078 -+ 0.017
-3.75
,-3.5
5
0.033 ± 0.015
-4.
, -3.75
4
0.029 ± 0.015
21
0.297 ~ 0.064
i
--4.128, - 4 . 0
-0.9 -1.0
J.A. Danj,sz et aL, K+p elastic scattering 100
100
10
>=
2.1
/
GeVlc
35 100
lOO
%
X
10
2-3
GeV/c
/]
J
10
i.Fl
# $+÷
-5-0
"..... ¢J
t
10
0-0
-2.5
(a) .
.
.
2.5
.
.
.
.
t
/
100
2.7
E
1
GeV/c
0.1
-5.0
(c)
-2,5 t
(GeV/c) =
'{ +
1
#
+ 0.1
0-1
-5.0
O0
(d)
1 10
10
e2,"* 10
¢ ]
(GeV/c) =
100
100
.
GeV/c
E
-2-5
-5-0
(b)
(GeV/c)a .
0.1
0-1
0.1
0.1
100
++ {f
1
0-1
-2.5
0-0
t (GeV/c) 't
Fig. 1. Differential cross sections at (a) 2.11 GeV/c, (b) 2.31 GeV/c, (c) 2.53 GeV/c,
(d) 2.72 GeV/c. data from the experiments of Whitmore et al. [6] at 2.53 GeV/e and 2.76 GeV/c and Albrow et al. [7] at 2.48 GeV/c and 2.74 GeV/c. The slopes observed in these two experiments are compatible with our results. The distributions are essentially exponential down to t = - 2 ( G e V / c ) 2. This fact contradicts the prediction o f Akheizer and Rekalo [8] who, on the basis o f a quark model, predict that K+p elastic scattering should show a second diffraction peak. To determine the value of do/dt at t = 0 and to parametrize the data a least squares fit of the function
J.A. Danysz et al., K+p elastic s'cattering
36
30
' 30
20
H~
T h ; s Expt
Whitrnore /
I0F
÷
--
~//I
20
ol
et
ALbrow e1 at
Exponential Fit
! i
~
,~ .s+
~10
A2o .~ i
u 0
t '°
I44I
i 12"5
~v+ 2.7
i
-1.o
|
-0,5
o.o
t (GeV/¢)l
Fig. 2. The d i f f e r e n t i a l cross s e c t i o n s for all m o m e n t a in the range t > - 1. I n c l u d e d are d a t a f r o m o t h e r e x p e r i m e n t s . The c o n t i n u o u s line r e p r e s e n t s the f i t t e d values o f the f u n c t i o n .
do _ da dt
"~ t =0 e
Bt "
J.A. Dan.vsz et al., K+p elastic scattering
da
37
{ 1)
d t - 7'°du' t = 0 e B t
to the forward part of the distribution was performed. Another fit was tried which included a quadratic term in the exponential, but it was found unnecessary as the coefficient of this term was always compatible with zero. The fit without a quadratic term was tried over ranges of t extending down to t = 0.6 (GeV/c) 2 without a systematic variation of B being found. The results of this fit are given in table 3. The values of the parameter B together with values from other experiments [6, 9, 10] for a similar range of t are plotted in fig. 3. The fitted values of d o / d t are shown superimposed on the data in fig. 2. Table 3 Results of a fit to the forward cross section of the function do _ d~t eBt dt t =0 "
IVlomentum GeV/c
B
2.11 2.31 2.53 2.72
3.65 ± 0.13 3.56 + 0.15 3.50 -+0.08 3.99 ± 0.14
(GeV/c) -2
Unlike K - p scattering, K+p elastic scattering exceeds the optical theorem value for the forward scattering amplitude by a substantial amount. The ratio of the real to the imaginary part of the forward scattering amplitude is computed from the relation i Re riO) 2 _ 16rr do t = 0
R--],mj 0 t
4 aV
1,
(2)
where o T is the total cross section [3] and the value of d o / d t at t = 0 is obtained from eq. (1). This value must be multiplied by a factor of 2.56 to convert from units of m b / ( G e V / e ) 2 to units of mb 2. The values of R are given in table 4. The value at 2.72 GeV/e (R2.72 = 0.37 -+ 0.05) is significantly different from that given by Whitmore [6] at 2.76 GeV/c (R2.76 = - 0 . 0 8 + 0.07) which reflects the systematic differences of the differential cross sections. 2.3. B a c k w a r d cross s e c t i o n The data show clearly the existence of a backward peak at all momenta. The backward parts of the angular distributions are plotted in fig. 4 together with those obtained in the spark chamber experiment of Carroll et al. [ 11 ] at 2.20, 2.33 and
38
J.A. D a n y s z et al., K+p elastic scattering
-_, . . . . . . . . .Model II ......
"7 "-d
¢ ....
~. . . .
f
This Expt.
2
3
5
7
9
15 20
Plab (GeVlc) Fig. 3. The value of the slope in the fit do _ d o
dt
Bt
h~ t =0 e
Data from the experiments described in refs. [ 6 - 1 0 ] are included. The solid line represents the prediction of model I and the dashed line that of model II.
2.45 G e V / c a n d the b u b b l e c h a m b e r e x p e r i m e n t o f W h i t m o r e et al. [6] at 2.53 and 2.76 G e V / c . It can be seen t h a t , w i t h i n the l i m i t e d p r e c i s i o n o f the p r e s e n t data, g o o d a g r e e m e n t is o b t a i n e d at 2.3, 2.5 a n d 2.7 G e V / c while at 2.1 G e V / c the backw a r d d i f f e r e n t i a l cross s e c t i o n s o f Carroll et al. are l o w e r t h a n the p r e s e n t values. Table 4 The square of the modulus of the ratio of the real part to the imaginary part of the forward scattering amplitude. Momentum (GeV/c)
R
2.11 2.31 2.53 2.72
0.60 0.28 0.25 0.37
-+ 0.06 -+ 0.06 -+ 0.05 -+ 0.05
3. L E G E N D R E P O L Y N O M I A L A N A L Y S I S A n s - c h a n n e l L e g e n d r e p o l y n o m i a l analysis was p e r f o r m e d o n the c o r r e c t e d angular d i s t r i b u t i o n s using the e x p a n s i o n
J.A. Danysz et al., K+p elastic scattering
39 !
0"1 ) ~'t~ , ,
Whitmore e~ o[ ]0.1 ~ Carroll et ol -Jr-- This Expt
t t
'ttt~t
1
f
0.01
o~
t 2'1 GeV/c ~'t't
I
, t,-t-' ttt
.,
2'5
I ~}"
GeVlc
' 0.01
0.0
,.2.7
.
.
-1.0
.
.
.
-O.?S COS
.
.
-O.S e*
Fig. 4. The differential cross sections for all momenta in the backward region. Included are data from the experiments of Carroll et al. [11] and Whitmore et al. [6].
do
d~
=~2 ~_j An P
(cosO*).
t,/
In this e x p a n s i o n the n o r m a l i s a t i o n i m p l i e d t h a t °el A
0
-
4~;~ 2
.
40
J.A. Danysz et al., K+p elastic scattering Table 5 Coefficients of the kegendre polynomial fit to the angular distributions.
Momentum (GeV/c)
2.11
2.31
2.53
2.72
Probability
5
46
84
1
C1
2.33 + 0.02
2.36 ± 0.02
2.42 ± 0.02
2.51 + 0.02
C2
2.71 ~ 0.04
2.81 ± 0.05
3.03 +- 0.04
3.28 + 0.05
C3
2.15 ± 0.07
2.23 ± 0.09
2.62 ± 0.08
3.00 ± 0.08
C4
1.47 ± 0.10
1.55 + 0.12
1.99 ± 0.11
2.45 ± 0.11
C5
0.88 ± 0.11
0.86 ± 0.13
1.22 ± 0.12
1.62 ± 0.12
C6
0.44 ± 0.11
0.57 ± 0.12
0.73 ± 0.12
1.05 ± 0.12
C7
0.25 -+ 0.08
0.29 ± 0.09
0.36 ± 0.08
0.53 ± 0.11
C8
0.06 ± 0.05
0.21 ± 0.07
0.11 ± 0.06
0.36 ± 0.12
The coefficients are given as values of Cn, where A C - n n AO ' in table 5 and are p l o t t e d in fig. 5. At all m o m e n t a eight terms only are required to give a good fit, and all coefficients vary s m o o t h l y as a f u n c t i o n o f m o m e n t u m . This implies that waves higher than G wave are not required in the partial wave amplitudes, and that there is no evidence for an elastic s-channel resonance.
4. C O M P A R I S O N WITH R E G G E M O D E L S The forward part o f the elastic scattering angular distribution was c o m p a r e d with predictions made by extrapolating the results o f Regge-model fits, which had been successful at higher energies. Two representative models are discussed here; one o f the pure pole type [12] and one using poles and absorptive cuts [13]. The m o d e l described by Dass, Michael and Phillips [12] (hereafter called m o d el I) uses P, P', ¢o', p and A 2 poles; the p o m e r o n is treated as a normal Regge pole and given a slope of 0.33. Exchange degeneracy is n o t imposed, but finite energy sum rules are used to constrain the fit. The m o d e l given by B l a c k m o n and Goldstein [13] (model II) uses P, P', co, p and A 2 poles, P' being assumed exchange degenerate with A2, and co with p. The p o m e r o n is taken to be a fixed pole with squared-dipole m o m e n t u m transfer dependence. Absorptive cuts are i n t r o d u c e d by means o f the eikonal m o d e l orginated by Arnold [14], and a good fit to K±p elastic scattering from 5 G e V / c upwards is claimed by the authors [13].
t l..____
J.A. Danysz et al., K+p elastic scattering
1
2.6 )J
t ;
,+
,*
t
;
“:;’ 2.1
‘lab
Fig. 5. Coefficients
2.5
2.3 (
;
::
I’
in the Legendre
t
.I o.j :’ 2.7
2.1
Gev/c) analysis
+
t
,+ (
i 2.7
2.5
2.3
'lab polynomial
41
GeV/c)
of the angular
distributions.
Representative comparisons of both fits with the present data at 2.7 GeV/c are given in fig. 6. Model I is much the better fit. The same result is obtained at the other three incident momenta. The eikonal formulation on which model II is based is valid only fcr -t/s < 1.O [ 141, and it is questionable whether this condition is satisfied at 2.5 GeV/c; however, by restricting the comparison to the region I tI < 0.6 (CeV/c)* we ensure that -t/s < 0.1 at the upper two momenta of this experiment, and we therefore conclude that the poor agreement is a shortcoming of the model. The nature of the discrepancy between model II and the data can be seen more clearly in fig. 3. Here we plot data
42
J.A. Danysz et al., K+p elastic scattering .
I00"0 I
>~
,o.o t
E
f
b
I'01
-t3-~7
Model. l
~z¢-----"
i- i "
Model
II
//
0.1L
t
,
-0"6
,
,
-0"4
,
,
-0"2
,
0"0
MOMENTUM TRANSFER , 2 (OeVlc) Fig. 6. The forward differential cross section at 2.7 GeV/c. The solid line represents the prediction of model I and the dashed line that of model II.
from several experiments from 1 to 16 GeV/c [ 6 - 1 0 ] . The slope B is obtained by fitting the formula (1) to the distribution over the interval 0. < t < 0.6 [17]. It can be seen that neither fit is particularly good over the range of momenta shown in the figure, but that model I has the more satisfactory momentum dependence. Of the main differences between the two models, we consider it unlikely that the inclusion of absorption in model II is responsible for the poor fit, since this has been used with success elsewhere [ 16]. The alternative possibilities are the extra constraints on the trajectory parameters. The model has insufficient shrinkage to fit the data (see fig. 3), a feature which probably results from the assumption of a flat pomeron. This conclusion is in agreement with those of Frautschi, Hamer and Ravndal [15, 16], and Lovelace [17] who have also considered absorptive models.
5. CONCLUSIONS One of the main purposes of performing this experiment was to investigate the structure in the K+p total cross section at 2.7 GeV/c. This analysis has failed to find any evidence that favours the existence of an s-channel resonance strongly coupled to the elastic channel. The comparison of our data together with those from some other experiments with the predictions of the Regge exchange models of Dass, Michael and Phillips and of Blackmon and Goldstein gives the better agreement with the first of these models. We are grateful to S.J. Goldsack for useful discussions and to M. Spiro and A. Verglas for their contribution to this work in the early stages. The British groups acknowledge the financial support of the Science Research Council.
J.A. Danysz et al., K+p elastic scattering
43
REFERENCES [1] J.M. Brunet, J.L. Narjoux, J.A. Danysz, M. Spiro, A. Verglas, S.L. Baker, B.K. Penney, B.C. Stewart, G. Thompson, J.E. Allen, N.J.D. Jacobs, P.tt. Lewis and P.V. March, Nucl. Phys. B36 (1972) 45. [2] J.A. Danysz, M. Spiro, A. Verglas, J.M. Brunet, J.L. Narjoux, B.K. Penney, G. Thompson, P.H. Lewis, J.E. Allen and P.V. March, Nucl. Phys. B14 11969) 161. [3] R.J. Abrams, R.L. Cool, G. Giacomelli, T.F. Kycia, B.A. Leonti~, K.K. Li and D.N. Michael, Phys. Rev. Letters 19 11967) 259. [4] B.K. Penney, Ph.D. thesis RHEL report HEP/T/20. [5] F.E. James, J. Johnson and H.L. Kraybill, Phys. Letters 19 (1965) 72; C.T. Coffin, N. Dikmen, L. Ettlinger, D. Meyer, A. Saulys, K. Terwilliger and D. Williams, Phys. Rev. 159 (1967) 1169. [6] J. Whitmore, G.S. Abrams, L. Eisenstein, J. Kim, T.A. O'Halloran Jr. and W. Schufeld, Phys. Rev. D3 (1971) 1092. [7] M.G. Albrow, S. Andersson/Almehed, B. Bosnjakovic, C. Daum, F.C. Ern6, Y. Kimura, J.P. Lagnaux, J.C. Sens, F. Udo and F. Wagner, Nucl. Phys. B30 (1971) 273. [8] A.I. Akhiezer and M.P. Rekalo, Soy. J. Nucl. Phys. 8 11969) 469. [9] L.R. Price, N. Barash Schmidt, O. Benary, R.W. Bland, A.H. Rosenfeld and C.G. Wohl, A compilation of K+N reactions, UCRL 20000 K+N. [10] J.N. MacNaughton, L. Feinstein, F. Marcelja and H.G. Trilling, Nucl. Phys. B 14 (1969) 237. [ 11 ] A.S. Carroll, J. Fischer, A. Lundby, R.H. Phillips, C.L. Wang, F. Lobkowicz, A.C. Melissinos, Y. Nagashima, C.A. Smith and S. Tewkesbury, Phys. Rev. Letters 21 (1968) 1282. [12] G.V. Dass, C. Michael and R.J.N. Phillips, Nucl. Phys. B9 (1969) 549. [13] M.L. Blackmon and G.R. Goldstein, Phys. Rev. 179 (1969) 1480. [14] R.C. Arnold, Phys. Rev. t53 (1967) 1523. [15] S.C. Frautschi, C.J. Hamer and F. Ravndal, Phys. Rev. D2 (1970) 2681. [16] C.J. Hamer and F. Ravndal, Phys. Rev. D2 (1970) 2687. [17] C. Lovelace and F. Wagner, Nucl. Phys. B28 (1971) 141.
K+p ELASTIC
SCATTERING
BETWEEN
2.1 1 A N D 2 . 7 2 G e V / c
J.A. DANYSZ, B.K. PENNEY, B.C. STEWART, G. T H O M P S O N Imperial College of Seienee and Technology England
J.M. BRUNET, J.L. NARJOUX Coll~ge de France. France
J.E. ALLEN, N.J.D. JACOBS, P.H. LEWIS, P.V. M A R C H West.field College, England
Received 14 December 1971
Abstract: Final results are presented of the analysis of the elastic channel in an exposure of 40 000 pictures at each of the four incident K+ momenta 2.11, 2.31, 2.53 and 2.72 GeV/c taken in the 1.5 m British National Hydrogen Bubble Chamber at the 8 GeV/c proton synchrotron at the Rutherford High Energy Laboratory. Differential cross sections are presented and the results are compared with other published data. A Legendre polynomial analysis requires partial waves up to G wave at all momenta. For the backward peak, visible at each momentum, the slope and the intercept are calculated. A comparison of the forward peak is made with extrapolations from Regge models fitted at higher momenta.
1. I N T R O D U C T I O N The elastic channel in K+p scattering has been studied at incident K + m o m e n t a of 2.11, 2 . 3 1 , 2 . 5 3 and 2.72 GeV/c. The general details o f the exposure, data processing and partial cross section calculations are given elsewhere [1]. Preliminary results on the elastic scattering based on a p p r o x i m a t e l y one third o f the data were r e p o r t e d previously [2]. Interest in this m o m e n t u m region was stimulated by the report [3] o f structure in the total K+p cross section at 2.7 G e V / c incident K + m o m e n t u m . There were a t t e m p t s to explain this structure by the f o r m a t i o n o f an S = +1, B -- +I resonance which, in the quark model, could only be f o r m e d from at least five quarks. The results presented in this paper show no behaviour supporting the existence o f such a resonance. The values o f the total elastic cross section are given in table 1 ; they differ slightly from those reported previously [ 1 ].
30
J.A. Danysz et al., K+p elastic scattering Table 1 Elastic cross section. Momentum (GeV/c)
Cross section (mb)
2.11 2.31 2.53 2.72
6.73 5.87 5.69 5.27
-+ 0.25 -+ 0.20 -+ 0.20 + 0.20
2. D I F F E R E N T I A L CROSS SECTIONS
2.1. Preparation o f data Cuts were imposed on the incident K + momentum and on the fiducial volume. A study of the experimental biases [4] led to the imposition of further cuts on the experimental data. Only events with the cosine of the c.m.s, scattering angle, 0", less than 0.95 (corresponding to a proton range in the bubble chamber of 11 cm at 2.11 GeV/c and 21 cm at 2.72 GeV/c), and a dip angle (defined as the angle between the plane normal to the optical axes of the cameras and the plane of the tracks) of less than 72 ° were accepted. The accepted events were weighted to provide an isotropic distribution of the dip angle. The latter correction was only necessary for the forward part of the distribution (cos 0* > 0.85). A study [4] of the effects of the contamination of the beam by pions showed that pion and kaon elastic interactions are kinematically indistinguishable except for backwards scattering where the difference of m o m e n t u m of the scattered beam particles is sufficient to resolve the ambiguity. At 2.11 GeV/c this difference is 115 MeV/c at cos 0* = 1. It was found that the two reactions could always be distinguished for cos 0* < 0. The angular distributions, then, contained events in the forward hemisphere (cos 0* > 0) from both incident kaon and incident pion elastic interactions and in the backward hemisphere only from incident kaon elastic interactions. To correct for this effect, the distribution was modified by removing the estimated number of pion events from each interval of the forward part of the distribution. This number was calculated using known pion elastic differential cross sections [5] and the contamination of the beam by pions, which was about 6% at 2.3, 2.5 and 2.7 GeV/e and 12% at 2.1 GeV/c. A similar procedure was employed to derive the differential cross section as a function of t, the squared four m o m e n t u m transfer. The results are tabulated in tables 2 a - 2 d and plotted in figs. l a - l d .
2.2. Forwai'd peak At all momenta the forward part of the angular distribution is strongly peaked. The data in the forward region (t > - 1 (GeV/c) 2) are plotted in fig. 2 together with
J.A. Danysz et al., K+p elastic scattering.
31
Table 2a The d i f f e r e n t i a l cross s e c t i o n s at 2.11 G e V / c . No. o f events
d-2-° ( m b / s r ) d~2
0.95
672
4.34
± 0.17
0.90
522
3.37
± 0.15
0.80,
0.85
380
2.46
± 0.13
0.75,
0.80
252
1.63
± 0.10
No. o f events
do (mb/(GeV/c)2) at I
0.10, - 0 . 0 5
445
17.90
± 0.85
0.15, - 0 . 1 0
440
17.70
± 0.84
0.90,
0.20,
0.15
376
15.12
± 0.77
0.85,
-0.25, -0.20
296
11.90
±0.67
-0.30, -0.25
230
9.26
± 0.59
--0.35,
i t(GeV/c)2
cos 0 *
]
170
6.85
±0.50
0.70,
0.75
192
1.24
± 0.09
-0.40, -0.35
132
5.32
+0.45
0.65,
0.70
169
1.09
± 0.08
-0.45, -0.40
0.30
129
5.20
± 0.45
0.60,
0.65
130
0.84
± 0.07
-0.45
111
4.47
±0.42
0.55,
0.60
120
0.77
± 0.07
0.55, - 0 . 5 0
93
3.67
+ 0.38
0.50,
0.55
74
0.48
± 0.06
i 0.60, - 0 . 5 5
81
3.26
+0.36
0.45,
0.50
71
0.46
± 0.05 ± 0.04
-0.50,
-0.65, -0.60 --0.70, - 0 . 6 5
81
3.26
±0.36
0.40,
0.45
40
0.26
62
2.50
±0.32
0.35,
0.40
45
0.29
± 0.04
2.07
±0.21
0.30,
0.35
26
0.17
± 0.03 ± 0.02
-0.8
, -0.7
103
-0.9
, -0.8
66
1.33
±0.16
0.2 ,
0.3
40
0.13
-1.
, --0.9
54
1.09
±0.15
0.1 ,
0.2
27
0 . 0 8 7 ± 0.017
i-l.1
,-1.0
37
0.74
±0.12
0.
0.1
18
0 . 0 5 8 ± 0.014
, -1.1
22
0.44
+ 0.09
0.
18
0.058 ± 0.014
0.1
11
0.035 + 0.011
, -0.2
10
0.032 e 0.010
0.3
2
0 . 0 0 6 +- 0.005
-1.2
'-0.1
0.2 ,
-1.3
, -1.2
18
0.36
±0,09
-1.4
, -1.3
13
0.26
± 0.07
-0.3
± 0.06
~ 0.4,
, -1.4
9
-1.75, -1.5
18
0.145 ± 0 . 0 3 4
1-0.5
, -0.4
8
0.026 ± 0.009
-2.
17
0.137 + 0.033
'i 0.6 , - 0 . 5
4
0 . 0 1 3 ± 0.006
--2.25, - 2 .
7
0 . 0 5 6 ± 0.021
i-0.7
, -0.6
8
0.026 ± 0.009
-2.5
,-2.25
9
0.073 ± 0.024
~-0.8
-0.7
15
0.048 ± 0.012 1
-2.6
, -2.5
7
0.141 ± 0.053
]-0.9
, -0.8
27
0.087 + 0.017
-2.7
, -2.6
11
0.221 ± 0 . 0 6 7
19
0.061 ± 0 . 0 1 4 |
12.8 , - 2 . 7
14
0 . 2 8 2 ± 0.075
-2.9
-1.5
, -1.75
0.18
,
,
2.8
18
0 . 3 6 2 ± 0.085
3.02,
2.9
16
0.315 + 0 . 0 7 0
i
1.0 ,
0.9
J
J.A. Danysz et al.. K+p elastic scattering
32
Table 2b The differential cross sections at 2.31 GeV/c.
t(GeV/c) 2
No. of events
do {mb/(GeV/c)2 ) dt
-0.10, -0.05
339
15.15
-+0.82
0.90,
--0.15, - 0 . 1 0 li-0.20, -0.15
305
13.64
_+ 0.78
238
10.64
-+ 0.69
-0.25, -0.20
182
8.13
-0.30, -0.25
173
7.73
-0.35, -0.30
150
-0.40, -0.45
116
-0.45, -0.40
93
-0.50, -0.45
92
No. of events
d o (mb/sr) d~2
0.95
492
3.50
± 0.16
0.85,
0.90
346
2.46
-+ 0.13
0.80,
0.85
291
2.07
± 0.12
±0.60
0.75,
0.80
190
1.35
-+ 0.10
+_0.59
0.70,
0.75
146
1.04
± 0.09
6.70
+-0.55
0.65,
0.70
106
0.76
± 0.08
5.18
+-0.48
0.60,
0.65
96
0.68
± 0.70
4.16
-+ 0.43
0.55,
0.60
83
0.59
-+ 0.06
4.12
±0.43
0.50,
0.55
54
0.38
± 0.05
-0.55, -0.50
55
2.46
-+ 0.33
0.45,
0.50
47
0.33
+0.05
-0.60, -0.55
69
3.09
+- 0.37
0.40,
0.45
31
0.22
_+0.04
0.65, - 0 . 6 0
52
2.32
±0.32
0.35,
0.40
20
0.14
+0.03
- 0 . 7 0 , -O.65
56
2.50
±0.33
0.30,
0.35
17
0.12
-+0.03
0.8,-0.7
90
2.01
-+0.21
0.2 ,
0.3
18
0.064 ± 0.015 [
O.9 ,
cos 0*
O.8
50
1.12
±0.16
0.1
0.2
12
0.043 ± 0.012
-1.0
-0.9
44
0.99
-+ 0.15
0.
0.1
12
0.043 ± 0.012
-1.1
1.0
26
0.58
±0.11
[-0.1
O.
6
0.021 -+ 0.009
!-1.2 , - 1 . 1
21
0.47
±0.10
i-0.2
-0.I
6
0.021 ± 0.009
-1.3 , -1.2
10
0.224 -+ 0.07
!-0.3
-0.2
5
0.018 +- 0.008
1.4 , - 1 . 3
[
-1.5
7
0.156 ± 0.06
-0.4
-0.3
1
0.004 -+ 0.003
, -1.4
6
0.134 ± 0.05
-0.5
-0.4
2
0.007 ± 0.005
1.5
13
0.116 ± 0.032
~-0.6
-0.5
5
0.018 ± 0.005
5
0.044 -+ 0.020
9
0.032 ± 0.009
-1.75, -2.
, -1.75
2.25, - 2 .
6
0.054 -+ 0.022
-2.5 ,-2.25
4
0.036 ± 0.018
-2.75, -2.5
8
0.072 ± 0.025
-3.0 ,-2.75
10
0.089 -+ 0.028
5
0.112 ± 0.050
!-3.1 , - 3 . 0 , -3.1
6
0.134 ± 0.054
-3.3 , -3.2
11
0.246 ± 0.074
14
0.391 ± 0.100
-3.2
3.38,
3.3
6
0.021 ± 0.009
, -0.8
10
0.036 + 0.011
, -0.9
25
0.089 ± 0.018
-0.7
-1.
!I
J.A. Danysz et al., K+p elastic scattering
33
Table 2c The differential cross sections at 2.53 GeV/c.
t(GeV/c)
No. of events
- 0 . 1 0 , ---O.05
323
14.10
-+ 0 . 7 9 |
~-0.15,
0.10
3O6
13.37
±0.76
i-0.20, - 0 . 1 5
261
11.40
±0.71
-0.25, -0.20
192
8.40
-0.30,
0.25
164
7.17
-0.35, -0.30
140
0.40, - 0 . 3 5
128
-0.45, -0.40
114
-0.50, -0.45
84
i-0.55, - 0 . 5 0 -0.60, -0.55
- (mb/(GeV/~
COS 0*
No. of events
do ~ (mb/sr)
0.90,
0.95
562
3.91
±0.17
0.85,
0.90
360
2.50
±0.13
0.80,
0.85
276
1.92
±0.11
+ 0.60
0.75,
0.80
195
1.36
+- 0.09
+ 0.56
0.70,
0.75
120
0.83
± 0.08
6.12
+_0.52
0.65,
0.70
105
0.73
-+ 0.07
5.60
± 0.49
0.60,
0.65
69
0.48
-+ 0.06
4.98 ++ 0.47
0.55,
0.60
61
0.42
+- 0.05
3.68
+ 0.40
0.50,
0.55
44
0.31
± 0.05
67
2.93
+ 0.36
0.45,
0.50
28
0.19
+- 0.04
62
2.71
+ 0.34
0.40,
0.45
25
0.17
± 0.03
-0.65, -0.60
44
1.92
+ 0.29
0.35,
0.40
18
0.13
±0.03
-0.70, -0.65
42
1.84
+- 0.28
0.30,
0.35
17
0.12
+-0.03
'-0.8 , -0.7
65
1.42
+ 0.18
0.20,
0.30
15
0.052 ± 0.013
-0.9 , -0.8
56
1.22
-+ 0.16
0.1
,
0.2
11
0.038 ± 0.12
-1.
l0
41
0.90
± 0.14
O.
,
0.1
4
0.014 -+ 0.007
24
0.52
+ 0.11
-0.1
,
0.
6
0.021 ± 0.009
-1.1
23
0.50
-+ 0.10
i-o.2
-o.1
8
0.028 ± 0.010
-1.3 , -1.2
17
0.37
± 0.09
-0.3
-0.2
4
0.014 -+ 0.007
!-1.4 , - 1 . 3
7
0.15
-+ 0.06
-0.4
-0.3
5
0.017 + 0.008
9
0.20
+ 0.07
-0.5
-0.4
3
0.010 ± 0.006
12
0.11
+ 0.03
-0.6
-0.5
3
0.010 +- 0.006
, -0.9
i-1.5
,-1.4
-1.75, -1.5 , -1.75
6
0.051 + 0.021
-0.7
-0.6
5
0.017 -+ 0.008
-2.25, -2.00
9
0.079 +- 0.026
-0.8
-0.7
5
0.017 ± 0.008
6
0.052 +_0.021
-0.9
-0.8
13
0.045 +_ 0.013
10
0.044 + 0.014
1.
-0.9
23
0.080 +- 0.017
-2.
-2.5 , -2.25 -3.0 , -2.5 -3.25, -3. -3.5
,-3.25
-3.6
,
3.5
i-3.7 , - 3 . 6 -3.78,
3.7
6
0.052 -+ 0.021
12
0.105 -+ 0.03
7
0.153 ± 0.06
14
0.306 ± 0.08
7
0.191 ± 0.07
J.A. Danysz et al., K+p elastic scattering
34
Table 2d The differential cross sections at 2.72 GeV/c.
G~V/c)U !-0.10 ,
--0.05
~No. of events
do (mb/(GeVfc)2)
cos 0*
at
No. of events
d o_o(rob/st) d~
397
14.36
-+0.72
0.90,
0.95
683
3.94
i-0.15 ,-0.10
363
13.14
-+0.69
0.85,
0.90
351
2.02
-+0.11
i-0.20 , -0.15 I i-0.25 , -0.20
321
11.62
-+ 0.65
0.80,
0.85
269
1.55
+- 0.09
236
8.55
-+0.56
0.75,
0.80
201
1.16
_+0.08
-0.30 , -0.25
187
6.77
-+ 0.49
0.70,
0.75
127
0.73
_+0.07
I
_+ 0.15
--0.35 , - 0 . 3 0
144
5.22
-+ 0.43
0.65,
0.60
92
0.53
-+ 0.06
-0.40
-0.35
125
4.53
+- 0.40
0.60,
0.65
65
0.38
-+ 0.05
! - 0. 45 , - 0 . 4 0
117
4.23
-+0.39
0.55,
0.60
47
0.27
-+ 0.04
0.45
90
3.26
-+ 0.34
0.50,
0.55
31
0.18
-+0.03
0.55 , - 0 . 5 0
58
2.10
-+0.28
0.45,
0.50
23
0.13
-+0.03
0.60 , - 0 . 5 5
75
2.72
-+0.31
0.40,
0.45
19
0.110 ± 0.025
~-0.50 ,
-0.65
, -0.60
41
1.48
±0.23
0.35,
0.40
8
0.046 ± 0.016
-0.70
, -0.65
39
1.41
+-0.23
0.30,
0.35
9
0.052±0.017
!
~-0.80 ,
0.70
74
1.34
±0.16
0.20,
0.30
11
0.032+-0.010
i--0.90 ,
0.80
56
1.01
+-0.14
0.1 ,
0.2
8
0.023 ± 0.008
0.0 ,
0.1
3
0.009 -+ 0.005
-1.0
, -0.9
31
0.55
-+0.10
-1.1
, -1.0
29
0.53
-+0.10
-0.1
,-0.0
4
0.012±0.006
-1.2
, -1.1
21
0.39
+-0.08
-0.2
, -0.1
4
0.012 ± 0.006
1.3
-1.2
11
0.20
-+ 0.06
-0.3
, -0.2
4
0.012 +- 0.006
1.4
-1.3
8
0.15
± 0.05
-0.4
, -0.3
6
0.017 ± 0.007
1.4
5
0.094 _+ 0.040
-0.5
, -0.4
2
0.006 ± 0.004
-1.75
-1.5
-1.5
13
0.094 +_0.026
-0.6
, -0.5
2
0.006 ± 0.004
-2.0
-1.75
7
0.049 -+ 0.019
-0.7
, -0.6
2
0.006 ± 0.004
-2.5
2.0
11
0.040 _+0.012
-0.8
, -0.7
4
0.012 ± 0.006
3.0
-2.5
9
0.033 ± 0.011
, -0.8
4
0.012 ± 0.006
-3.5
, -3.O
3
0.011 + 0.006
, -0.9
27
0.078 -+ 0.017
-3.75
,-3.5
5
0.033 ± 0.015
-4.
, -3.75
4
0.029 ± 0.015
21
0.297 ~ 0.064
i
--4.128, - 4 . 0
-0.9 -1.0
J.A. Danj,sz et aL, K+p elastic scattering 100
100
10
>=
2.1
/
GeVlc
35 100
lOO
%
X
10
2-3
GeV/c
/]
J
10
i.Fl
# $+÷
-5-0
"..... ¢J
t
10
0-0
-2.5
(a) .
.
.
2.5
.
.
.
.
t
/
100
2.7
E
1
GeV/c
0.1
-5.0
(c)
-2,5 t
(GeV/c) =
'{ +
1
#
+ 0.1
0-1
-5.0
O0
(d)
1 10
10
e2,"* 10
¢ ]
(GeV/c) =
100
100
.
GeV/c
E
-2-5
-5-0
(b)
(GeV/c)a .
0.1
0-1
0.1
0.1
100
++ {f
1
0-1
-2.5
0-0
t (GeV/c) 't
Fig. 1. Differential cross sections at (a) 2.11 GeV/c, (b) 2.31 GeV/c, (c) 2.53 GeV/c,
(d) 2.72 GeV/c. data from the experiments of Whitmore et al. [6] at 2.53 GeV/e and 2.76 GeV/c and Albrow et al. [7] at 2.48 GeV/c and 2.74 GeV/c. The slopes observed in these two experiments are compatible with our results. The distributions are essentially exponential down to t = - 2 ( G e V / c ) 2. This fact contradicts the prediction o f Akheizer and Rekalo [8] who, on the basis o f a quark model, predict that K+p elastic scattering should show a second diffraction peak. To determine the value of do/dt at t = 0 and to parametrize the data a least squares fit of the function
J.A. Danysz et al., K+p elastic s'cattering
36
30
' 30
20
H~
T h ; s Expt
Whitrnore /
I0F
÷
--
~//I
20
ol
et
ALbrow e1 at
Exponential Fit
! i
~
,~ .s+
~10
A2o .~ i
u 0
t '°
I44I
i 12"5
~v+ 2.7
i
-1.o
|
-0,5
o.o
t (GeV/¢)l
Fig. 2. The d i f f e r e n t i a l cross s e c t i o n s for all m o m e n t a in the range t > - 1. I n c l u d e d are d a t a f r o m o t h e r e x p e r i m e n t s . The c o n t i n u o u s line r e p r e s e n t s the f i t t e d values o f the f u n c t i o n .
do _ da dt
"~ t =0 e
Bt "
J.A. Dan.vsz et al., K+p elastic scattering
da
37
{ 1)
d t - 7'°du' t = 0 e B t
to the forward part of the distribution was performed. Another fit was tried which included a quadratic term in the exponential, but it was found unnecessary as the coefficient of this term was always compatible with zero. The fit without a quadratic term was tried over ranges of t extending down to t = 0.6 (GeV/c) 2 without a systematic variation of B being found. The results of this fit are given in table 3. The values of the parameter B together with values from other experiments [6, 9, 10] for a similar range of t are plotted in fig. 3. The fitted values of d o / d t are shown superimposed on the data in fig. 2. Table 3 Results of a fit to the forward cross section of the function do _ d~t eBt dt t =0 "
IVlomentum GeV/c
B
2.11 2.31 2.53 2.72
3.65 ± 0.13 3.56 + 0.15 3.50 -+0.08 3.99 ± 0.14
(GeV/c) -2
Unlike K - p scattering, K+p elastic scattering exceeds the optical theorem value for the forward scattering amplitude by a substantial amount. The ratio of the real to the imaginary part of the forward scattering amplitude is computed from the relation i Re riO) 2 _ 16rr do t = 0
R--],mj 0 t
4 aV
1,
(2)
where o T is the total cross section [3] and the value of d o / d t at t = 0 is obtained from eq. (1). This value must be multiplied by a factor of 2.56 to convert from units of m b / ( G e V / e ) 2 to units of mb 2. The values of R are given in table 4. The value at 2.72 GeV/e (R2.72 = 0.37 -+ 0.05) is significantly different from that given by Whitmore [6] at 2.76 GeV/c (R2.76 = - 0 . 0 8 + 0.07) which reflects the systematic differences of the differential cross sections. 2.3. B a c k w a r d cross s e c t i o n The data show clearly the existence of a backward peak at all momenta. The backward parts of the angular distributions are plotted in fig. 4 together with those obtained in the spark chamber experiment of Carroll et al. [ 11 ] at 2.20, 2.33 and
38
J.A. D a n y s z et al., K+p elastic scattering
-_, . . . . . . . . .Model II ......
"7 "-d
¢ ....
~. . . .
f
This Expt.
2
3
5
7
9
15 20
Plab (GeVlc) Fig. 3. The value of the slope in the fit do _ d o
dt
Bt
h~ t =0 e
Data from the experiments described in refs. [ 6 - 1 0 ] are included. The solid line represents the prediction of model I and the dashed line that of model II.
2.45 G e V / c a n d the b u b b l e c h a m b e r e x p e r i m e n t o f W h i t m o r e et al. [6] at 2.53 and 2.76 G e V / c . It can be seen t h a t , w i t h i n the l i m i t e d p r e c i s i o n o f the p r e s e n t data, g o o d a g r e e m e n t is o b t a i n e d at 2.3, 2.5 a n d 2.7 G e V / c while at 2.1 G e V / c the backw a r d d i f f e r e n t i a l cross s e c t i o n s o f Carroll et al. are l o w e r t h a n the p r e s e n t values. Table 4 The square of the modulus of the ratio of the real part to the imaginary part of the forward scattering amplitude. Momentum (GeV/c)
R
2.11 2.31 2.53 2.72
0.60 0.28 0.25 0.37
-+ 0.06 -+ 0.06 -+ 0.05 -+ 0.05
3. L E G E N D R E P O L Y N O M I A L A N A L Y S I S A n s - c h a n n e l L e g e n d r e p o l y n o m i a l analysis was p e r f o r m e d o n the c o r r e c t e d angular d i s t r i b u t i o n s using the e x p a n s i o n
J.A. Danysz et al., K+p elastic scattering
39 !
0"1 ) ~'t~ , ,
Whitmore e~ o[ ]0.1 ~ Carroll et ol -Jr-- This Expt
t t
'ttt~t
1
f
0.01
o~
t 2'1 GeV/c ~'t't
I
, t,-t-' ttt
.,
2'5
I ~}"
GeVlc
' 0.01
0.0
,.2.7
.
.
-1.0
.
.
.
-O.?S COS
.
.
-O.S e*
Fig. 4. The differential cross sections for all momenta in the backward region. Included are data from the experiments of Carroll et al. [11] and Whitmore et al. [6].
do
d~
=~2 ~_j An P
(cosO*).
t,/
In this e x p a n s i o n the n o r m a l i s a t i o n i m p l i e d t h a t °el A
0
-
4~;~ 2
.
40
J.A. Danysz et al., K+p elastic scattering Table 5 Coefficients of the kegendre polynomial fit to the angular distributions.
Momentum (GeV/c)
2.11
2.31
2.53
2.72
Probability
5
46
84
1
C1
2.33 + 0.02
2.36 ± 0.02
2.42 ± 0.02
2.51 + 0.02
C2
2.71 ~ 0.04
2.81 ± 0.05
3.03 +- 0.04
3.28 + 0.05
C3
2.15 ± 0.07
2.23 ± 0.09
2.62 ± 0.08
3.00 ± 0.08
C4
1.47 ± 0.10
1.55 + 0.12
1.99 ± 0.11
2.45 ± 0.11
C5
0.88 ± 0.11
0.86 ± 0.13
1.22 ± 0.12
1.62 ± 0.12
C6
0.44 ± 0.11
0.57 ± 0.12
0.73 ± 0.12
1.05 ± 0.12
C7
0.25 -+ 0.08
0.29 ± 0.09
0.36 ± 0.08
0.53 ± 0.11
C8
0.06 ± 0.05
0.21 ± 0.07
0.11 ± 0.06
0.36 ± 0.12
The coefficients are given as values of Cn, where A C - n n AO ' in table 5 and are p l o t t e d in fig. 5. At all m o m e n t a eight terms only are required to give a good fit, and all coefficients vary s m o o t h l y as a f u n c t i o n o f m o m e n t u m . This implies that waves higher than G wave are not required in the partial wave amplitudes, and that there is no evidence for an elastic s-channel resonance.
4. C O M P A R I S O N WITH R E G G E M O D E L S The forward part o f the elastic scattering angular distribution was c o m p a r e d with predictions made by extrapolating the results o f Regge-model fits, which had been successful at higher energies. Two representative models are discussed here; one o f the pure pole type [12] and one using poles and absorptive cuts [13]. The m o d e l described by Dass, Michael and Phillips [12] (hereafter called m o d el I) uses P, P', ¢o', p and A 2 poles; the p o m e r o n is treated as a normal Regge pole and given a slope of 0.33. Exchange degeneracy is n o t imposed, but finite energy sum rules are used to constrain the fit. The m o d e l given by B l a c k m o n and Goldstein [13] (model II) uses P, P', co, p and A 2 poles, P' being assumed exchange degenerate with A2, and co with p. The p o m e r o n is taken to be a fixed pole with squared-dipole m o m e n t u m transfer dependence. Absorptive cuts are i n t r o d u c e d by means o f the eikonal m o d e l orginated by Arnold [14], and a good fit to K±p elastic scattering from 5 G e V / c upwards is claimed by the authors [13].
t l..____
J.A. Danysz et al., K+p elastic scattering
1
2.6 )J
t ;
,+
,*
t
;
“:;’ 2.1
‘lab
Fig. 5. Coefficients
2.5
2.3 (
;
::
I’
in the Legendre
t
.I o.j :’ 2.7
2.1
Gev/c) analysis
+
t
,+ (
i 2.7
2.5
2.3
'lab polynomial
41
GeV/c)
of the angular
distributions.
Representative comparisons of both fits with the present data at 2.7 GeV/c are given in fig. 6. Model I is much the better fit. The same result is obtained at the other three incident momenta. The eikonal formulation on which model II is based is valid only fcr -t/s < 1.O [ 141, and it is questionable whether this condition is satisfied at 2.5 GeV/c; however, by restricting the comparison to the region I tI < 0.6 (CeV/c)* we ensure that -t/s < 0.1 at the upper two momenta of this experiment, and we therefore conclude that the poor agreement is a shortcoming of the model. The nature of the discrepancy between model II and the data can be seen more clearly in fig. 3. Here we plot data
42
J.A. Danysz et al., K+p elastic scattering .
I00"0 I
>~
,o.o t
E
f
b
I'01
-t3-~7
Model. l
~z¢-----"
i- i "
Model
II
//
0.1L
t
,
-0"6
,
,
-0"4
,
,
-0"2
,
0"0
MOMENTUM TRANSFER , 2 (OeVlc) Fig. 6. The forward differential cross section at 2.7 GeV/c. The solid line represents the prediction of model I and the dashed line that of model II.
from several experiments from 1 to 16 GeV/c [ 6 - 1 0 ] . The slope B is obtained by fitting the formula (1) to the distribution over the interval 0. < t < 0.6 [17]. It can be seen that neither fit is particularly good over the range of momenta shown in the figure, but that model I has the more satisfactory momentum dependence. Of the main differences between the two models, we consider it unlikely that the inclusion of absorption in model II is responsible for the poor fit, since this has been used with success elsewhere [ 16]. The alternative possibilities are the extra constraints on the trajectory parameters. The model has insufficient shrinkage to fit the data (see fig. 3), a feature which probably results from the assumption of a flat pomeron. This conclusion is in agreement with those of Frautschi, Hamer and Ravndal [15, 16], and Lovelace [17] who have also considered absorptive models.
5. CONCLUSIONS One of the main purposes of performing this experiment was to investigate the structure in the K+p total cross section at 2.7 GeV/c. This analysis has failed to find any evidence that favours the existence of an s-channel resonance strongly coupled to the elastic channel. The comparison of our data together with those from some other experiments with the predictions of the Regge exchange models of Dass, Michael and Phillips and of Blackmon and Goldstein gives the better agreement with the first of these models. We are grateful to S.J. Goldsack for useful discussions and to M. Spiro and A. Verglas for their contribution to this work in the early stages. The British groups acknowledge the financial support of the Science Research Council.
J.A. Danysz et al., K+p elastic scattering
43
REFERENCES [1] J.M. Brunet, J.L. Narjoux, J.A. Danysz, M. Spiro, A. Verglas, S.L. Baker, B.K. Penney, B.C. Stewart, G. Thompson, J.E. Allen, N.J.D. Jacobs, P.tt. Lewis and P.V. March, Nucl. Phys. B36 (1972) 45. [2] J.A. Danysz, M. Spiro, A. Verglas, J.M. Brunet, J.L. Narjoux, B.K. Penney, G. Thompson, P.H. Lewis, J.E. Allen and P.V. March, Nucl. Phys. B14 11969) 161. [3] R.J. Abrams, R.L. Cool, G. Giacomelli, T.F. Kycia, B.A. Leonti~, K.K. Li and D.N. Michael, Phys. Rev. Letters 19 11967) 259. [4] B.K. Penney, Ph.D. thesis RHEL report HEP/T/20. [5] F.E. James, J. Johnson and H.L. Kraybill, Phys. Letters 19 (1965) 72; C.T. Coffin, N. Dikmen, L. Ettlinger, D. Meyer, A. Saulys, K. Terwilliger and D. Williams, Phys. Rev. 159 (1967) 1169. [6] J. Whitmore, G.S. Abrams, L. Eisenstein, J. Kim, T.A. O'Halloran Jr. and W. Schufeld, Phys. Rev. D3 (1971) 1092. [7] M.G. Albrow, S. Andersson/Almehed, B. Bosnjakovic, C. Daum, F.C. Ern6, Y. Kimura, J.P. Lagnaux, J.C. Sens, F. Udo and F. Wagner, Nucl. Phys. B30 (1971) 273. [8] A.I. Akhiezer and M.P. Rekalo, Soy. J. Nucl. Phys. 8 11969) 469. [9] L.R. Price, N. Barash Schmidt, O. Benary, R.W. Bland, A.H. Rosenfeld and C.G. Wohl, A compilation of K+N reactions, UCRL 20000 K+N. [10] J.N. MacNaughton, L. Feinstein, F. Marcelja and H.G. Trilling, Nucl. Phys. B 14 (1969) 237. [ 11 ] A.S. Carroll, J. Fischer, A. Lundby, R.H. Phillips, C.L. Wang, F. Lobkowicz, A.C. Melissinos, Y. Nagashima, C.A. Smith and S. Tewkesbury, Phys. Rev. Letters 21 (1968) 1282. [12] G.V. Dass, C. Michael and R.J.N. Phillips, Nucl. Phys. B9 (1969) 549. [13] M.L. Blackmon and G.R. Goldstein, Phys. Rev. 179 (1969) 1480. [14] R.C. Arnold, Phys. Rev. t53 (1967) 1523. [15] S.C. Frautschi, C.J. Hamer and F. Ravndal, Phys. Rev. D2 (1970) 2681. [16] C.J. Hamer and F. Ravndal, Phys. Rev. D2 (1970) 2687. [17] C. Lovelace and F. Wagner, Nucl. Phys. B28 (1971) 141.