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Hadron-nucleus scattering lengths derived from exotic atom data
Nuclear Physics A411 (1983) 399-416 4’) North-Holland Pubi~hing Company
HADRON-NUCLEUS SCATTERING LENGTHS DERIVED FROM EXOTIC ATOM DATA C.J. BATTY Rutherford Appleton Laboratory, Chilton, Didcor, Oxon, UK
Received 4 March 1983 (Revised 4 July 1983) Abstrad:
Hadron-nucleus S-wave scattering lengths for kaons, antiprotons and sigma hyperons have been derived from measurements of strong interaction effects in exotic atoms. The results are compared with the predictions of a simple square-well model.
1. Introduction
Measurements of strong interaction effects in exotic atoms have conventionally been analysed in terms of the optical model. Since strong interaction measurements for exotic atoms give information on the hadron-nucleus interaction at essentially zero energy, an alternative approach, which will be used in this work, is to analyse the data in terms of hadron-nucleus S-wave scattering lengths. Indeed, measurements on pionic, kaonic and antiprotonic atoms formed in hydrogen have already been attempted with the principal aim of determining the fundamental hadronnucleon S-wave scattering length. In this paper we consider nuclei heavier than helium and concentrate on measurements for stopping kaons, antiprotons and sigma hyperons. Analyses for pionic atom data have appeared elsewhere ‘). Very early and rather inaccurate data for kaonic atoms were used by Seki 2>to determine S-wave scattering lengths for kaons but no analysis of the considerable body of data now available has been published. Antiproton and sigma-hyperon data have not so far been analysed using a hadron-nucleus scattering length approach. 2. Theory and method 2.1. METHOD
For the exotic atoms considered in this paper, strong interaction measurements are only available for angular momentum states with 1~ 0. As a result it is not possible to use the relationship ‘) which exists between the measured strong interac399
400
C.J. Batty / Scattering lengths
tion shift and width for I=0 states and the S-wave scattering length. Instead it is necessary to use the optical model as an intermediate stage in deriving the S-wave scattering length from the measured strong interaction effects for I> 0 atomic states. This method has already been used by Seki *) for kaons and is only valid if the optical potential is local. However, it is known that a purely local potential gives an excellent fit 3-5) to the experimental data for kaons, anti-protons and sigma hyperons and there is no evidence from the data for a need to introduce non-local terms. This is in contrast to the situation for pions ‘) where there is good theoretical evidence that a non-local potential is required and where local potentials are unable to fit the experimental results. For the present work the use of a local potential would also not be valid if the derived scattering lengths were found to be particularly sensitive to the parameters of the model. This point will be discussed later in some detail. For kaonic, antiprotonic and sigma atoms, the optical model potential ‘> can be written in the form u=--
2?7 P (
1+‘1 &3(r), m>
where p is the hadron-nucleus reduced mass, m the mass of the nucleon, p(r) is the density distribution normalized to A and d is the (complex) “effective” scattering length for hadron-nucleon interactions. The relationship between d and the free hadron-nucleon scattering lengths has been discussed in many papers [see e.g. ref. 3)]. In the present work it plays the role of a purely phenomenological parameter which determines the strength of the optical potential U. An analysis of all presently available data in terms of the optical model has recently appeared for kaons “) and for antiprotons and sigma hyperons “). The model gives a good fit to the data. In the present work the optical potential given by eq. (1) is used to fit the strong interaction shift (E) and width (r) values for individual nuclei by adjusting the real (uR) and imaginary (a,) parts of the complex strength parameter d = an+ ia,. The form of p(r) was generally taken to be the same as that of the nuclear charge distribution and parameters from a compilation 6, of electron scattering measurements were used. The results of calculations in which a variety of alternative parameters were used for the density distribution are discussed later. Details of the methods used to calculate the strong interaction shift and width and to determine the best fit value of d are described in ref. “). The errors quoted for the fitted values of d were obtained from the least-squares variance-covariance matrix in the usual way. Using the value for d determined above the (complex) S-wave hadron-nucleus scattering length as was calculated by numerically solving the Schriidinger equation for 1= 0 with the optical potential iY but without the Coulomb potential and at an energy close to threshold. The error Aus on the calculated value of a, was obtained
401
CJ. Batty / Scattering lengths
using the usual least-squares expression
(2)
Ti+;~~Z&Z& I
I
where AXi AXj is the ijth element of the variance-covariance
2.2. ROLE
OF THE OPTICAL
matrix.
POTENTIAL
As we have already mentioned, the optical model provides an intermediate stage in going from the measured strong interaction shifts and widths, which are only available for 1> 0, to the derived 1= 0 S-wave scattering lengths. The use of the optical model in this intermediate stage is only valid if it is assumed to be local and if it can be shown that the calculated scattering lengths are essentially independent of the form of the potential. This latter feature was investigated in considerable detail for two exotic atoms, kaonic Si (1 = 2) and kaonic Sn (1 = 4). For these two cases calculations were first carried out using a variety of parameters for the potential distribution, p(r), in eq. (1). In terms of a folding model 6), p(r) is obtained by the convolution of a finite range hadron-nucleon interaction with the point nucleon matter distribution. In an analysis of kaonic atom data, Deloff and Law ‘) used a force with a rather large range corresponding to an rms radius of 0.7 to 0.9 fm. On the other hand, Atarashi et al. ‘) have used a much shorter-range force corresponding to the exchange of a heavy meson and are also able to obtain good fits to the data. In table 1, we show scattering lengths calculated for three different sets of parameters (c, a) for p(r), which was taken to be a Fermi distribution. These parameters correspond to three different choices for the rms radius of a gaussian interaction which is folded into the point nucleon density to obtain an effective p(r). For an rms radius of the force
TABLET Results for various forms of p(r)
(fy
c
a
0.0 0.8 1.2
2.978 2.926 2.882
0.508 0.569 0.638
0.0 0.8 1.2
5.329 5.300 5.267
0.527 0.583 0.648
a1
aSR
asI
Kaonic Si (I = 2) 0.947 f 0.070 0.752 f 0.091 0.826*0.051 0.387 f 0.058 0.678 f 0.037 0.164*0.038
-3.688 It 0.030 -3.669 f 0.034 -3.646 f 0.037
1.196*0.031 1.224 f 0.033 1.253*0.032
Kaonic 0.868 * 0.409 0.512+0.281 0.273*0.190
-6.022+0.147 -6.008*0.160 -5.989kO.178
1.344zto.134 1.370*0.131 1.404*0.137
aR
Sn (I = 4) 0.751*0.221 0.706*0.177 0.606*0.145
All values are in fm. p(r)=(l+exp((r-c)/a)-‘.
402
C.J. Batty / Scattering lengths
{r2)“* = 0.8 fm the charge distribution was used. For zero range, a Fermi distribution was used with the parameters adjusted so that when folded with a gaussian of rms radius 0.8 fm it gave a best least-squares fit to the charge distribution which was also taken to be of a Fermi form. For an rms radius of 1.2 fm a Fermi form was again used, with the parameters adjusted to give a best least-squares fit to the distribution obtained by folding a gaussian of rms radius ( 1.22-0.82)“2 = 0.894 fm into the charge distribution. The results are shown in table 1. We see that for both cases, increasing the rms radius of the interaction from 0 to 1.2 fm changes the real (usR) and imaginary (a& parts of the S-wave scattering length by only 1% and 5% respectively despite changes of 70% and 25 % in the real ( aR) and imaginary (ai) strength of the potential required to fit the strong interaction data. In a further series of calculations the effects of systematic changes in the radial parameter c, and the diffusivity a of the Fermi distribution were investigated for the two nuclei discussed earlier. In both cases it was found that changing c even by relatively large amounts, of the order of 20 %, gave only very small change of - 1% in either the real or imaginary parts of a,. On the other hand, there was considerably more sensitivity to the diffusivity parameter a. For values of a less than 0.5 fm for Sn and 0.3 fm for Si quite large and rapid changes were observed in the calculated values of as even when a was changed by relatively small amounts. On the other hand, values of a this small can probably be regarded as a rather unphysical choice for the diffusivity of the optical potential. For larger values of a more gradual and smaller changes in the scattering length were obtained. For example, in the case of Si, varying a from 0.3 to 0.8 fm caused the real part of us to vary by -3% and the imaginary part by 17%. For Sn, varying a from 0.5 to 1.0 fm gave changes of 5% in the real part and 34% in the imaginary part of us. Note, however, that these changes in CLare much larger than those expected for reasonable choices of the range of the hadronic interaction (see table 1). So far we have assumed a simple local optical potential (eq. (1)) which is a linear function of the nuclear density p(r). A limited number of alternative forms for the optical potential were also considered. For example, it has recently been suggested “) that a laplacian form
where a, is an additional complex parameter, gives an improved fit to the kaonic atom data. Using a potential of this form for kaonic Si with the charge distribution for p(r) gave a, = -3.414 f 0.101 + i( 1.284 f 0.045) fm to be compared with the value a, = -3.669 tfi0.034+ i( 1.224 f 0.033) fm obtained using the simple optical potential of eq. (1). For Z-atoms it has been pointed out lo) that effects which are non-linear in the density may be important. Since the Z-atom data is relatively inaccurate, kaonic Si
CJ. Batty / S~tte~~g
lengths
was again used as a test case with the optical potential of eq. (1) multipli~ additional factor
403
by an
For a value of /3 = 1, corresponding to complete suppression of the potential at r = 0, a value for the scattering length as = -3.88OAO.O28+ i(1.242kO.021) fm was obtained, again in reasonable agreement with the earlier values. Finally, we can compare our values with some very recent results of Dumbrajs et al. II). They calcufated S-wave scattering lengths for antiprotonic atoms using an optical potential derived from the basic RN interaction. For p- 0 they obtained us = -3.41+ il.12 fm in excellent agreement with the value as = -3.66 f 0.15 + i(1.08*0.18) fm obtained in this work. For P-S their value as= -4.43+i1.17 fm is again in remarkable agreement with the value as = -4.45 f 0.14 + i( I .27 f 0.11) fm given in table 4. We conclude that, providing reasonable choices are made for the form and parameters of the optical potential, it is possible to obtain quite reliable values for the S-wave scattering length.
2.3. SQUARE-WELL POTENTIAL
Before presenting the results obtained in this analysis, it is helpful to consider the predictions of a simple model assuming a square-well optical potential. Taking U in the form u_
-(V+iW),
IGR, r>R,
-{ 0,
(3)
where V and W are the real and imaginary strengths of the potential and R the potential radius, then if /3 =J2p(
V+ iW)fhc2,
(4)
the S-wave scattering length is given by tan @R %= P-R* Using eqs. (1) and (3), then
Scattering lengths calculated using the above formulae for the case of kaonic Si with R = 3.0 fm as a function of aR and for a series of values of aI are plotted in fig. 1. The marked oscillations in the value of as occur when a new bound state is formed in the potential. These oscillations are rapidly damped as the imaginary
404
C.J. Batty / Scattering lengths
of the potential increases, the reai scattering length approaches the value -R and the imaginary part varies rather smoothly as a function of aR. For large W, and hence relatively large imaginary part for p, the value of tan j?R + i and as can be written strength
@R+PI
for &R >>0 ,
“s=m-R
where p = &+ &. This simple formula gives values of as to good accuracy for &R > 2, which is true for most of the cases considered in this paper. The formula also shows that in the limit of a strongly absorbing sphere ( W +a), the real part of the scattering length ssR + -R and the imaginary part asI -+0. Values of p and of the scattering length for Si with R = 3.0 fm are given in table 2 for kaons, antiprotons and sigma hyperons. The values of d are taken from optical model fits to the exotic atom data4z5). For antiprotons, where the absorption is
0.0
-6.0
-8.0
I
-10.0 0.0
0.1
I
I
I
I
0.2
0.3
III
0.4
I 0.5
I
I
0.6
III 0.7
1 0.8
I
I
0.9 ReA
REAL
SCATTERING
LENGTH
FOR
SQUARE
WELL
POTENTIAL
Fig. 1. Real and imaginary S-wave scattering lengths calculated as a function of the strength of the real potential aR for kaons and Si using a square well of radius 3.0fm. Curves (a) to (e) correspond to imaginary potential strengths a, of 0.02, 0.04, 0.2, 0.4 and 0.84 fm, respectively.
405 12.0 r
10.0
(a)
IMAGINARY
SCATTERING
LENGTH
FOR
SQUARE
WELL
POTENTIAL
ReA
Fig. 1 (cont.)
strong, then usR is indeed close to the value -R and uSI is small. However, a rather similar value of asR is obtained for sigma hyperons where the absorption is much weaker, although in this case as1 is considerably larger. The statement that @n will be n~erically equal to the radius of the potential is only valid for very strong absorption and then us1 =O. Table 2 shows that even for antiprotons this situation is not fulfilled although the calculated real part of the scattering length does have a value which is numerically close to that of the potential radius. 3. Results A compilation of strong interaction shift and width measurements has been given previously 3, where references to the originai papers are given. In some cases there are several measurements for a particular nucleus and averaged values have been used. All currently available data have been included in the present analysis.
406
C.J. Batty / Scattering
lengths
TABLE 2 Calculations using square-well potential V+iW (MeV) K-
0.34+ i0.84 l.OO+i2.21 0.36+ iO.20
P P-
65+i159 131+ i291 42+i24
1.72+i1.16 3.24+ i2.09 1.63+ i0.42
- 2.73 + iO.40 -2.86+ i0.22 -2.87+ i0.67
Values of rI from refs. +‘). Potential radius R = 3.0 fm and A = 28.
3.1. KAONS
Shift (E) and width (r) measurements are available for 24 nuclei ranging from Li to U These values and the calculated S-wave scattering lengths as obtained using the procedure described earlier are presented in table 3. Here it is the principal quantum number of the atomic state whose shift and width have been measured TABLE 3 Analysis of kaon data Nucleus Li Be ‘OB llB C 0 Mg Al Si P S Cl co Ni cu Ag Cd In Sn Ho Yb Ta Pb U
n 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 5 5 5 5 6 6 6 7 7
(k:V) 0.002 f 0.026 -0.079 * 0.021 -0.208 * 0.035 -0.167*0.035 -0.59 * 0.08 -0.025 +0.015 -0.027 f 0.015 -0.080 f 0.013 -0.139*0.014 -0.330 * 0.080 -0.494 + 0.038 -0.990+0.170 -0.099iO.106 -0.223 kO.042 -0.370 * 0.047 -0.18*0.12 -0.40*0.10 -0.53*0.15 -0.41*0.18 -0.30*0.13 -0.12*0.10 -0.27 f 0.50 -0.020 * 0.012 -0.26 * 0.04
(k;) 0.055 f 0.029 0.172*0.058 0.81*0.10 0.70+0.08 1.7310.15 0.017*0.014 0.214*0.015 0.443 *0.022 0.801 kO.032 1.44kO.12 2.19*0.10 2.91 kO.24 0.64*0.25 1.03*0.12 1.37*0.17 1.54*0.58 2.01 f 0.44 2.38hO.57 3.18*0.64 2.14+0.31 2.39+0.39 3.76+ 1.15 0.37*0.15 1.50*0.75
+.R
asI
(fm)
(fm)
-2.429 + 0.505 -2.521 kO.197 -2.561 kO.074 -2.496 + 0.057 -2.619iO.095 -4.533 f 0.224 -3.335 *0.077 -3.563 + 0.046 -3.669 * 0.034 -3.881*0.083 -3.903 f 0.036 -3.976 f 0.092 -4.59150.322 -4.826 f 0.087 -4.905 * 0.088 -5.836*0.261 -5.976kO.156 -6.005*0.154 -6.007*0.160 -7.258+0.177 -6.880*0.123 -6.920 * 0.296 -7.535 *0.290 -7.702*0.360
1.224kO.951 0.838 f 0.230 1.170*0.066 1.089+0.080 1.120*0.128 0.488 * 0.341 1.172*0.105 1.16410.046 1.224*0.033 1.206*0.093 1.245 ztO.037 1.094iO.082 1.181 kO.290 1.145*0.081 1.085 f 0.079 1.291 kO.210 1.187*0.130 1.16110.156 1.370*0.132 1.357kO.127 1.489+0.100 1.487*0.264 1.297*0.128 1.220*0.173
407
C.J. Batty / Scattering lengths
and the angular momentum 1= n - 1. Values of the potential strength d as a function of 2 are plotted in fig. 2. Fitting the real and imaginary parts of d by a least-squares method shows no evidence for 2 or A-dependent terms. The best fit for aR is obtained with aR = 0.35 f 0.03 fm with a x2 per degree of freedom (x2/F) equal to 0.86. For the imaginary part a, = 0.80 f 0.03 fm and x2/F = 1.25. These values are in good agreement with a R = 0.34 f 0.03 fm and a, = 0.84 f 0.03 fm obtained “) by fitting the shift and width data for all nuclei simultaneously. The fact that a simple optical potential of the form given by eq. (1) is able to fit data over a wide range of l-values with a constant value of d gives increased confidence in the use of a purely local optical model. Values of the derived S-wave scattering length are plotted as a function of A in fig. 3. A clear anomaly is the point for 0. The shift and width values for this nucleus are rather badly fitted by a global fit to all measurements simultaneously whilst the values of d obtained by a fit to the oxygen data alone have enormous uncertainties. The values of a, for 0 were therefore excluded from the following fits.
ReA
.--.
,
I 60
1 80
---
1
-l
100 2
REAL
KAON
POTENTIAL
Fig. 2. Strength of the real and imaginary optical potential for kaons as a function of Z
408
C.J. Batty / Scattering lengths ImA
1.50
1.00 t..,
~ 0.50
--_ 1
A
--
--_-
,. i
0.0
--__-
--_
1 I
_
I
20
0 I~AGINAffY
KAON
I
I
40
I
60
80
POTENTIAL
I
100 2
Fig. 2 (cont.)
The real part of the S-wave scattering length shows an approximate A”3 dependence which, as discussed in subsect. 2.3, is to be expected in the limit for a strongly absorbing sphere with radius CCA”~. Least-squares fits give a,,(fm)
= -(1.105~0.004)A0~3J9*0~0’0,
x2/F=
1.1 ,
which is plotted in fig. 3, asa
= -( 1.205 f 0.005)A”3
,
x2/F = 2.0 .
Eq. (7) suggests that an improved fit might be obtained using the form usR = co + c,AC2.No such improvement is obtained with the present results presumably because the expected magnitude of co is rather small (cO= 0.27 fm for the case considered in table 2) compared to R. The values of the imaginary part are moderately well fitted with as,= 1.203*0.016 fm for which x2/F = 1.3. A somewhat improved fit is obtained by introducing a term dependent on A with as1 = (1.157*0.023) + (0~0010~0.0004)A and x2/F = 1.0. Both these fits to the data are plotted in fig. 3.
C.J. Batty /
Scatteringlengths
2.00
1.00
/ 1 REAL
2
4
SCATTERING
6
10
20
40
60
100
205
LENGTH-KAONS
II
ImAs
1.50
1.00
0.50
1
0.0
I I II D 20 IMAGINARY
I I I I t I 40
60
SCATTERING
80
I
100
I
I
120
I
I
140
I
I
160
I
I
180
I
I
200
I
I
I
220
LENGTH-KAONS
Fig. 3. Values of the reai and imaginary S-wave scattering length for kaons as a function of A.
[ 240 A
410
C.J. Batty / Scattering lengths TABLE 4 Analysis of antiproton data
Nucleus
n
6Li ‘Li C N ‘60 ‘80 Si P S Fe Y Zr Yb
2 2 3 3 3 3 4 4 4 5 6 6 8
(kk) -0.07zto.17 -0.268 f 0.081 -0.004 * 0.010 -0.018*0.036 -0.111 f 0.032 -0.189*0.042 -0.038 * 0.039 -0.065 f 0.023 -0.06 * 0.04 -0.01 *to.31 -0.15*0.16 -0.45*0.10 0.26jzO.46
0.34*0.31 0.18*0.14 0.042*0.018 0.179*0.031 0.484+0.106 0.55kO.24 0.11*0.19 0.446 f 0.069 0.65*0.10 0.54 i 0.32 0.80 f 0.32 0.70*0.21 1.48*0.66
asR
as1
(fm)
(fm)
-3.97* 1.54 -2.77ztO.32 -3.14*0.45 -3.46*0.24 -3.66kO.15 -3.87*0.18 -4.04kO.21 -4.6OkO.12 -4.45*0.14 -5.16kO.17 -6.2OkO.32 -6.33*0.15 -7.81 kO.65
1.77* 1.19 0.37*0.31 1.04+0.69 1.11*0.17 1.08+0.18 0.97 * 0.30 0.79* 1.35 1.20*0.10 1.27kO.11 1.31 kO.82 1.07+0.32 0.59*0.20 1.5.5* 1.14
3.2. ANTIPROTONS
For antiprotons 13 pairs of measurements of strong interaction shifts and widths are available ranging over the nuclei from Li to Yb. However, the quality of the data is relatively poor. For example, 7 of the shift or width measurements are within one standard deviation of zero and only 5 shift and 7 width measurements differ by more than 3 standard deviations from zero. [See figs. 1 and 2 in ref. ‘).I The values of the shifts (E) and widths (r) and the calculated values of as are presented in table 4. Values of d are plotted as a function of 2 in fig. 4. Least squares fits to these values give a n = 1.67kO.20 fm with x2/F = 1.33 and aI=1.10*0.17fm with x2/F=2.31 to be compared with a,=1.00+0.32fm and a, = 2.21 f 0.26 fm obtained by a direct fit to the shift and width measurements for all nuclei simultaneously. The discrepancy between these two sets of values can largely be attributed to the rather poor accuracy of the experimental measurements. There are also strong indications that for some nuclei the values of an and a1 obtained by the search procedure are highly correlated. The S-wave scattering lengths are plotted in fig. 5. Despite the considerable uncertainties in uR and ur discussed previously the values of the real scattering length are seen to be quite well determined. Least-squares fits give usn(fm) = -(1.52*0.01)A(0~313*o~017’,
x2/F = 0.6 ,
which is plotted in fig. 5, or usR(fm) = -( 1.41 f 0.02)A1’3 ,
x2/F = 0.7 .
The imaginary part can be fitted with a constant value and gives 1.11 f 0.06 fm with x2/F = 1.4. There is no clear evidence for any A-depen-
U s, =
C.J.Batty / Scattering
411
lengths
dence; as can be seen from fig. 5 this is partly because the majority available are for A < 32.
of values
3.3. SIGMA HYPERONS
The data available for sigma atoms is extremely limited and relatively inaccurate. Measurements of strong interaction shifts, widths and relative yields available ‘*) for 0 and for nuclei between Mg and S were analysed in the present work. The limited number of yield measurements available for other nuclei were not used since values of iI could not be determined for each nucleus separately. Values of the S-wave scattering length are given in table 5 and plotted in fig. 6. A least squares fit to the value of d gives uR = 0.335 kO.047 fm with x21 F = 0.28 and a1 = 0.167 f 0:027 fm with x*/F = 0.41, in good agreement with the values oR = 0.363 f 0.048 fm and a, = 0.202 f 0.027 fm obtained by fitting all the available data for the whole range of nuclei simultaneously.
4.00
3.00
2.00
1 .oo
0.0
REAL
PBAR
POTENTIAL
Fig. 4. Strength of the real and imaginary optical potential for antiprotons as a function of Z.
412
C.J. Batty / Scattering lengths
5.00
m
4.00
_
3.00
_
2.00
__
1.00
_:-.
0.0
* _ 0
I _i.---.
- - -
I
IMAGINARY
20 PEAR
-h-----_--__--_---_____--_
I~
60
40
00
100
POTENTIAL
Fig. 4
(cm.)
Least squares fits to the S-wave scattering lengths give with x2/F = 0.2 ,
asR(fm) = -(l.094~0.058)A1’3, as1= 1.72*0.15
fm
,
with x2/F = 0.5 .
These fits to the data are also plotted in fig. 6. 4. Discussion In this work we have shown that it is possible to obtain values for the hadron nucleus S-wave scattering length from measurements of strong interaction effects in exotic atoms. Since the strong interaction measurements are only available for I> 0 it is necessary to use the optical model as an intermediate stage in obtaining the scattering length from the measured shifts and widths for hadronic atoms. Assuming the potential is local it has been shown that the scattering lengths obtained are relatively insensitive to the parameters chosen for the form of the optical potential. A limited number of alternative forms for the potential have been considered.
C.J. Batty / Scattering lengths
413
A simple model using a square well potential, where the scattering length can be calculated analytically, shows that for kaons, antiprotons and sigma hyperons the real part of the scattering length will numerically have a value similar to that of the potential radius whilst the imaginary part will have a constant, but non-zero, value. Only in the case of extremely strong absorption will the real part of the scattering length be equal to the potential radius and then the imaginary part will be zero. Perhaps somewhat surprisingly the model shows that this will not be the case even for antiprotons. The real part of the S-wave scattering lengths obtained in this work shows an approximate A”3 dependence (see figs. 3,5,6) confirming a close relationship with the value of the nuclear radius. Values of the scattering length are presented in table 6 where the real part has been parameterized in the form asR = c1AC2used in the previous section. The values of a sR for kaons and sigma hyperons are seen to be rather similar and if, following the analogy with the square-well case, they are taken to correspond to an interaction radius give a value about 0.7 fm larger than the potential radius c. At this point the optical potential has a value of about 20% of its central value and it is interesting that this is the point where strong
10.00
6.00
4.00
I
1
2
REAL
SCATTERING
I
I
4
6
I
I
IO
20
1
40
I
60
,
100
I 200
LENGTH-PEARS
Fig. 5. Values of the real and imaginary S-wave scattering length for antiprotons as a function of A.
I
A
414
C.J. Batty / Scattering lengths ImAs
1.50
1.00
0.50
-
0.0
_.
I 40
IMAGINARY
-I
I
I 60
SCATTERING
80
I 100
I 120
I 140
I 160
I 180
I 200
I 220
240 A
LENGTH-PBARS
Fig. 5
(cont.)
interaction shift measurements for kaons are most sensitive “) to the form of the potential. For antiprotons the real part of the scattering lengths are about 1.25 fm larger than the potential radius, corresponding to a point at which the potential strength is about 10% of the central value. Values for the imaginary part of the scattering length, also given in table 6, are seen to be much larger than those given by the simple square-well model (table 2). TABLE 5 Analysis of sigma atom data Nucleus 0
Mg Al Si S
aSR
aSR
(fm)
(fm)
-2.73 * 0.22 -3.38 * 0.75 -3.28 kO.34 -3.58* 0.45 -3.10*0.59
1.62kO.49 1.55*0.26 1.71 kO.25 2.16hO.37 1.70+0.54
1 REAL
2
4
SCATTERING
6
10
20
40
60
100
200
A
LENGTH-SIGMAS
ImAs
0.0
I 0
I 20
IMAGINARY
I 40
I
I
SCATTERING
I
I
60
80
100
I
120
I
I
140
I
I
160
I
I
I
180
I
200
I
I
I
220
240 A
as a function
of A.
LENGTH-SIGMAS
Fig. 6. Values of the real and imaginary
S-wave scattering
length for sigma hyperons
C.J. Batty / Scattering lengths
416
TABLET Hadron-nucleus
S-wave
scattering
(2) K
1.105*0.004 1.52*0.01 1.094*0.058
lengths “) aSI
(fm) 0.359*0.010 0.313io.017 Ib J )
1.203*0.016 1.108*0.057 1.72kO.15
“) a,=a,,+ia,,; as,=-c,A% b, Fixed at this value.
This is almost certainly due to the need for a finite diffuseness for the shape of any realistic potential. For example, in the case of kaonic Si increasing the diffuseness parameter a of the Fermi distribution from a = 0 (square well) to a = 0.569 (charge distribution) causes the imaginary part of the scattering length to increase by a factor of 3. As we have already commented the value of as1 for antiprotons seems surprisingly large in view of their expected strong absorption by nuclei. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
J. Hiifner, L. Tauscher and C. Wilkin, Nucl. Phys. A231 (1974) 455 R. Seki, Phys. Rev. Lett. 29 (1972) 240 C.J. Batty, Sov. J. Part. Nucl. 13 (1982) 71 C.J. Batty, Nucl. Phys. A372 (1981) 418 C.J. Batty, Nucl. Phys. A372 (1981) 433 C.W. de Jager, H. De Vries and C. de Vries, Atom. Nucl. Data Tables 14 (1974) 479 A. Deloff and J. Law, Phys. Rev. Cl0 (1974) 1688 M. Atarashi, K. Hira and H. Narumi, Prog. Theor. Phys. 60 (1978) 209 C.J. Batty, Phys. Lett. 127B (1983) 162 C.J. Batty, A. Gal and G. Toker, Nucl. Phys. A402 (1983) 349; R. Brockmann and E. Oset, Phys. Lett. 118B (1982) 33 11) 0. Dumbrajs, A.M. Green and J.A. Niskanen, University of Helsinki preprint HU-TFT-83-22 12) C.J. Batty, SF. Biagi, M. Blecher, SD. Hoath, R.A.J. Riddle, B.L. Roberts, J.D. Davies, G.J. Pyle, G.T.A. Squier and D.M. Asbury, Phys. Lett. 74B (1978) 27
HADRON-NUCLEUS SCATTERING LENGTHS DERIVED FROM EXOTIC ATOM DATA C.J. BATTY Rutherford Appleton Laboratory, Chilton, Didcor, Oxon, UK
Received 4 March 1983 (Revised 4 July 1983) Abstrad:
Hadron-nucleus S-wave scattering lengths for kaons, antiprotons and sigma hyperons have been derived from measurements of strong interaction effects in exotic atoms. The results are compared with the predictions of a simple square-well model.
1. Introduction
Measurements of strong interaction effects in exotic atoms have conventionally been analysed in terms of the optical model. Since strong interaction measurements for exotic atoms give information on the hadron-nucleus interaction at essentially zero energy, an alternative approach, which will be used in this work, is to analyse the data in terms of hadron-nucleus S-wave scattering lengths. Indeed, measurements on pionic, kaonic and antiprotonic atoms formed in hydrogen have already been attempted with the principal aim of determining the fundamental hadronnucleon S-wave scattering length. In this paper we consider nuclei heavier than helium and concentrate on measurements for stopping kaons, antiprotons and sigma hyperons. Analyses for pionic atom data have appeared elsewhere ‘). Very early and rather inaccurate data for kaonic atoms were used by Seki 2>to determine S-wave scattering lengths for kaons but no analysis of the considerable body of data now available has been published. Antiproton and sigma-hyperon data have not so far been analysed using a hadron-nucleus scattering length approach. 2. Theory and method 2.1. METHOD
For the exotic atoms considered in this paper, strong interaction measurements are only available for angular momentum states with 1~ 0. As a result it is not possible to use the relationship ‘) which exists between the measured strong interac399
400
C.J. Batty / Scattering lengths
tion shift and width for I=0 states and the S-wave scattering length. Instead it is necessary to use the optical model as an intermediate stage in deriving the S-wave scattering length from the measured strong interaction effects for I> 0 atomic states. This method has already been used by Seki *) for kaons and is only valid if the optical potential is local. However, it is known that a purely local potential gives an excellent fit 3-5) to the experimental data for kaons, anti-protons and sigma hyperons and there is no evidence from the data for a need to introduce non-local terms. This is in contrast to the situation for pions ‘) where there is good theoretical evidence that a non-local potential is required and where local potentials are unable to fit the experimental results. For the present work the use of a local potential would also not be valid if the derived scattering lengths were found to be particularly sensitive to the parameters of the model. This point will be discussed later in some detail. For kaonic, antiprotonic and sigma atoms, the optical model potential ‘> can be written in the form u=--
2?7 P (
1+‘1 &3(r), m>
where p is the hadron-nucleus reduced mass, m the mass of the nucleon, p(r) is the density distribution normalized to A and d is the (complex) “effective” scattering length for hadron-nucleon interactions. The relationship between d and the free hadron-nucleon scattering lengths has been discussed in many papers [see e.g. ref. 3)]. In the present work it plays the role of a purely phenomenological parameter which determines the strength of the optical potential U. An analysis of all presently available data in terms of the optical model has recently appeared for kaons “) and for antiprotons and sigma hyperons “). The model gives a good fit to the data. In the present work the optical potential given by eq. (1) is used to fit the strong interaction shift (E) and width (r) values for individual nuclei by adjusting the real (uR) and imaginary (a,) parts of the complex strength parameter d = an+ ia,. The form of p(r) was generally taken to be the same as that of the nuclear charge distribution and parameters from a compilation 6, of electron scattering measurements were used. The results of calculations in which a variety of alternative parameters were used for the density distribution are discussed later. Details of the methods used to calculate the strong interaction shift and width and to determine the best fit value of d are described in ref. “). The errors quoted for the fitted values of d were obtained from the least-squares variance-covariance matrix in the usual way. Using the value for d determined above the (complex) S-wave hadron-nucleus scattering length as was calculated by numerically solving the Schriidinger equation for 1= 0 with the optical potential iY but without the Coulomb potential and at an energy close to threshold. The error Aus on the calculated value of a, was obtained
401
CJ. Batty / Scattering lengths
using the usual least-squares expression
(2)
Ti+;~~Z&Z& I
I
where AXi AXj is the ijth element of the variance-covariance
2.2. ROLE
OF THE OPTICAL
matrix.
POTENTIAL
As we have already mentioned, the optical model provides an intermediate stage in going from the measured strong interaction shifts and widths, which are only available for 1> 0, to the derived 1= 0 S-wave scattering lengths. The use of the optical model in this intermediate stage is only valid if it is assumed to be local and if it can be shown that the calculated scattering lengths are essentially independent of the form of the potential. This latter feature was investigated in considerable detail for two exotic atoms, kaonic Si (1 = 2) and kaonic Sn (1 = 4). For these two cases calculations were first carried out using a variety of parameters for the potential distribution, p(r), in eq. (1). In terms of a folding model 6), p(r) is obtained by the convolution of a finite range hadron-nucleon interaction with the point nucleon matter distribution. In an analysis of kaonic atom data, Deloff and Law ‘) used a force with a rather large range corresponding to an rms radius of 0.7 to 0.9 fm. On the other hand, Atarashi et al. ‘) have used a much shorter-range force corresponding to the exchange of a heavy meson and are also able to obtain good fits to the data. In table 1, we show scattering lengths calculated for three different sets of parameters (c, a) for p(r), which was taken to be a Fermi distribution. These parameters correspond to three different choices for the rms radius of a gaussian interaction which is folded into the point nucleon density to obtain an effective p(r). For an rms radius of the force
TABLET Results for various forms of p(r)
(fy
c
a
0.0 0.8 1.2
2.978 2.926 2.882
0.508 0.569 0.638
0.0 0.8 1.2
5.329 5.300 5.267
0.527 0.583 0.648
a1
aSR
asI
Kaonic Si (I = 2) 0.947 f 0.070 0.752 f 0.091 0.826*0.051 0.387 f 0.058 0.678 f 0.037 0.164*0.038
-3.688 It 0.030 -3.669 f 0.034 -3.646 f 0.037
1.196*0.031 1.224 f 0.033 1.253*0.032
Kaonic 0.868 * 0.409 0.512+0.281 0.273*0.190
-6.022+0.147 -6.008*0.160 -5.989kO.178
1.344zto.134 1.370*0.131 1.404*0.137
aR
Sn (I = 4) 0.751*0.221 0.706*0.177 0.606*0.145
All values are in fm. p(r)=(l+exp((r-c)/a)-‘.
402
C.J. Batty / Scattering lengths
{r2)“* = 0.8 fm the charge distribution was used. For zero range, a Fermi distribution was used with the parameters adjusted so that when folded with a gaussian of rms radius 0.8 fm it gave a best least-squares fit to the charge distribution which was also taken to be of a Fermi form. For an rms radius of 1.2 fm a Fermi form was again used, with the parameters adjusted to give a best least-squares fit to the distribution obtained by folding a gaussian of rms radius ( 1.22-0.82)“2 = 0.894 fm into the charge distribution. The results are shown in table 1. We see that for both cases, increasing the rms radius of the interaction from 0 to 1.2 fm changes the real (usR) and imaginary (a& parts of the S-wave scattering length by only 1% and 5% respectively despite changes of 70% and 25 % in the real ( aR) and imaginary (ai) strength of the potential required to fit the strong interaction data. In a further series of calculations the effects of systematic changes in the radial parameter c, and the diffusivity a of the Fermi distribution were investigated for the two nuclei discussed earlier. In both cases it was found that changing c even by relatively large amounts, of the order of 20 %, gave only very small change of - 1% in either the real or imaginary parts of a,. On the other hand, there was considerably more sensitivity to the diffusivity parameter a. For values of a less than 0.5 fm for Sn and 0.3 fm for Si quite large and rapid changes were observed in the calculated values of as even when a was changed by relatively small amounts. On the other hand, values of a this small can probably be regarded as a rather unphysical choice for the diffusivity of the optical potential. For larger values of a more gradual and smaller changes in the scattering length were obtained. For example, in the case of Si, varying a from 0.3 to 0.8 fm caused the real part of us to vary by -3% and the imaginary part by 17%. For Sn, varying a from 0.5 to 1.0 fm gave changes of 5% in the real part and 34% in the imaginary part of us. Note, however, that these changes in CLare much larger than those expected for reasonable choices of the range of the hadronic interaction (see table 1). So far we have assumed a simple local optical potential (eq. (1)) which is a linear function of the nuclear density p(r). A limited number of alternative forms for the optical potential were also considered. For example, it has recently been suggested “) that a laplacian form
where a, is an additional complex parameter, gives an improved fit to the kaonic atom data. Using a potential of this form for kaonic Si with the charge distribution for p(r) gave a, = -3.414 f 0.101 + i( 1.284 f 0.045) fm to be compared with the value a, = -3.669 tfi0.034+ i( 1.224 f 0.033) fm obtained using the simple optical potential of eq. (1). For Z-atoms it has been pointed out lo) that effects which are non-linear in the density may be important. Since the Z-atom data is relatively inaccurate, kaonic Si
CJ. Batty / S~tte~~g
lengths
was again used as a test case with the optical potential of eq. (1) multipli~ additional factor
403
by an
For a value of /3 = 1, corresponding to complete suppression of the potential at r = 0, a value for the scattering length as = -3.88OAO.O28+ i(1.242kO.021) fm was obtained, again in reasonable agreement with the earlier values. Finally, we can compare our values with some very recent results of Dumbrajs et al. II). They calcufated S-wave scattering lengths for antiprotonic atoms using an optical potential derived from the basic RN interaction. For p- 0 they obtained us = -3.41+ il.12 fm in excellent agreement with the value as = -3.66 f 0.15 + i(1.08*0.18) fm obtained in this work. For P-S their value as= -4.43+i1.17 fm is again in remarkable agreement with the value as = -4.45 f 0.14 + i( I .27 f 0.11) fm given in table 4. We conclude that, providing reasonable choices are made for the form and parameters of the optical potential, it is possible to obtain quite reliable values for the S-wave scattering length.
2.3. SQUARE-WELL POTENTIAL
Before presenting the results obtained in this analysis, it is helpful to consider the predictions of a simple model assuming a square-well optical potential. Taking U in the form u_
-(V+iW),
IGR, r>R,
-{ 0,
(3)
where V and W are the real and imaginary strengths of the potential and R the potential radius, then if /3 =J2p(
V+ iW)fhc2,
(4)
the S-wave scattering length is given by tan @R %= P-R* Using eqs. (1) and (3), then
Scattering lengths calculated using the above formulae for the case of kaonic Si with R = 3.0 fm as a function of aR and for a series of values of aI are plotted in fig. 1. The marked oscillations in the value of as occur when a new bound state is formed in the potential. These oscillations are rapidly damped as the imaginary
404
C.J. Batty / Scattering lengths
of the potential increases, the reai scattering length approaches the value -R and the imaginary part varies rather smoothly as a function of aR. For large W, and hence relatively large imaginary part for p, the value of tan j?R + i and as can be written strength
@R+PI
for &R >>0 ,
“s=m-R
where p = &+ &. This simple formula gives values of as to good accuracy for &R > 2, which is true for most of the cases considered in this paper. The formula also shows that in the limit of a strongly absorbing sphere ( W +a), the real part of the scattering length ssR + -R and the imaginary part asI -+0. Values of p and of the scattering length for Si with R = 3.0 fm are given in table 2 for kaons, antiprotons and sigma hyperons. The values of d are taken from optical model fits to the exotic atom data4z5). For antiprotons, where the absorption is
0.0
-6.0
-8.0
I
-10.0 0.0
0.1
I
I
I
I
0.2
0.3
III
0.4
I 0.5
I
I
0.6
III 0.7
1 0.8
I
I
0.9 ReA
REAL
SCATTERING
LENGTH
FOR
SQUARE
WELL
POTENTIAL
Fig. 1. Real and imaginary S-wave scattering lengths calculated as a function of the strength of the real potential aR for kaons and Si using a square well of radius 3.0fm. Curves (a) to (e) correspond to imaginary potential strengths a, of 0.02, 0.04, 0.2, 0.4 and 0.84 fm, respectively.
405 12.0 r
10.0
(a)
IMAGINARY
SCATTERING
LENGTH
FOR
SQUARE
WELL
POTENTIAL
ReA
Fig. 1 (cont.)
strong, then usR is indeed close to the value -R and uSI is small. However, a rather similar value of asR is obtained for sigma hyperons where the absorption is much weaker, although in this case as1 is considerably larger. The statement that @n will be n~erically equal to the radius of the potential is only valid for very strong absorption and then us1 =O. Table 2 shows that even for antiprotons this situation is not fulfilled although the calculated real part of the scattering length does have a value which is numerically close to that of the potential radius. 3. Results A compilation of strong interaction shift and width measurements has been given previously 3, where references to the originai papers are given. In some cases there are several measurements for a particular nucleus and averaged values have been used. All currently available data have been included in the present analysis.
406
C.J. Batty / Scattering
lengths
TABLE 2 Calculations using square-well potential V+iW (MeV) K-
0.34+ i0.84 l.OO+i2.21 0.36+ iO.20
P P-
65+i159 131+ i291 42+i24
1.72+i1.16 3.24+ i2.09 1.63+ i0.42
- 2.73 + iO.40 -2.86+ i0.22 -2.87+ i0.67
Values of rI from refs. +‘). Potential radius R = 3.0 fm and A = 28.
3.1. KAONS
Shift (E) and width (r) measurements are available for 24 nuclei ranging from Li to U These values and the calculated S-wave scattering lengths as obtained using the procedure described earlier are presented in table 3. Here it is the principal quantum number of the atomic state whose shift and width have been measured TABLE 3 Analysis of kaon data Nucleus Li Be ‘OB llB C 0 Mg Al Si P S Cl co Ni cu Ag Cd In Sn Ho Yb Ta Pb U
n 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 5 5 5 5 6 6 6 7 7
(k:V) 0.002 f 0.026 -0.079 * 0.021 -0.208 * 0.035 -0.167*0.035 -0.59 * 0.08 -0.025 +0.015 -0.027 f 0.015 -0.080 f 0.013 -0.139*0.014 -0.330 * 0.080 -0.494 + 0.038 -0.990+0.170 -0.099iO.106 -0.223 kO.042 -0.370 * 0.047 -0.18*0.12 -0.40*0.10 -0.53*0.15 -0.41*0.18 -0.30*0.13 -0.12*0.10 -0.27 f 0.50 -0.020 * 0.012 -0.26 * 0.04
(k;) 0.055 f 0.029 0.172*0.058 0.81*0.10 0.70+0.08 1.7310.15 0.017*0.014 0.214*0.015 0.443 *0.022 0.801 kO.032 1.44kO.12 2.19*0.10 2.91 kO.24 0.64*0.25 1.03*0.12 1.37*0.17 1.54*0.58 2.01 f 0.44 2.38hO.57 3.18*0.64 2.14+0.31 2.39+0.39 3.76+ 1.15 0.37*0.15 1.50*0.75
+.R
asI
(fm)
(fm)
-2.429 + 0.505 -2.521 kO.197 -2.561 kO.074 -2.496 + 0.057 -2.619iO.095 -4.533 f 0.224 -3.335 *0.077 -3.563 + 0.046 -3.669 * 0.034 -3.881*0.083 -3.903 f 0.036 -3.976 f 0.092 -4.59150.322 -4.826 f 0.087 -4.905 * 0.088 -5.836*0.261 -5.976kO.156 -6.005*0.154 -6.007*0.160 -7.258+0.177 -6.880*0.123 -6.920 * 0.296 -7.535 *0.290 -7.702*0.360
1.224kO.951 0.838 f 0.230 1.170*0.066 1.089+0.080 1.120*0.128 0.488 * 0.341 1.172*0.105 1.16410.046 1.224*0.033 1.206*0.093 1.245 ztO.037 1.094iO.082 1.181 kO.290 1.145*0.081 1.085 f 0.079 1.291 kO.210 1.187*0.130 1.16110.156 1.370*0.132 1.357kO.127 1.489+0.100 1.487*0.264 1.297*0.128 1.220*0.173
407
C.J. Batty / Scattering lengths
and the angular momentum 1= n - 1. Values of the potential strength d as a function of 2 are plotted in fig. 2. Fitting the real and imaginary parts of d by a least-squares method shows no evidence for 2 or A-dependent terms. The best fit for aR is obtained with aR = 0.35 f 0.03 fm with a x2 per degree of freedom (x2/F) equal to 0.86. For the imaginary part a, = 0.80 f 0.03 fm and x2/F = 1.25. These values are in good agreement with a R = 0.34 f 0.03 fm and a, = 0.84 f 0.03 fm obtained “) by fitting the shift and width data for all nuclei simultaneously. The fact that a simple optical potential of the form given by eq. (1) is able to fit data over a wide range of l-values with a constant value of d gives increased confidence in the use of a purely local optical model. Values of the derived S-wave scattering length are plotted as a function of A in fig. 3. A clear anomaly is the point for 0. The shift and width values for this nucleus are rather badly fitted by a global fit to all measurements simultaneously whilst the values of d obtained by a fit to the oxygen data alone have enormous uncertainties. The values of a, for 0 were therefore excluded from the following fits.
ReA
.--.
,
I 60
1 80
---
1
-l
100 2
REAL
KAON
POTENTIAL
Fig. 2. Strength of the real and imaginary optical potential for kaons as a function of Z
408
C.J. Batty / Scattering lengths ImA
1.50
1.00 t..,
~ 0.50
--_ 1
A
--
--_-
,. i
0.0
--__-
--_
1 I
_
I
20
0 I~AGINAffY
KAON
I
I
40
I
60
80
POTENTIAL
I
100 2
Fig. 2 (cont.)
The real part of the S-wave scattering length shows an approximate A”3 dependence which, as discussed in subsect. 2.3, is to be expected in the limit for a strongly absorbing sphere with radius CCA”~. Least-squares fits give a,,(fm)
= -(1.105~0.004)A0~3J9*0~0’0,
x2/F=
1.1 ,
which is plotted in fig. 3, asa
= -( 1.205 f 0.005)A”3
,
x2/F = 2.0 .
Eq. (7) suggests that an improved fit might be obtained using the form usR = co + c,AC2.No such improvement is obtained with the present results presumably because the expected magnitude of co is rather small (cO= 0.27 fm for the case considered in table 2) compared to R. The values of the imaginary part are moderately well fitted with as,= 1.203*0.016 fm for which x2/F = 1.3. A somewhat improved fit is obtained by introducing a term dependent on A with as1 = (1.157*0.023) + (0~0010~0.0004)A and x2/F = 1.0. Both these fits to the data are plotted in fig. 3.
C.J. Batty /
Scatteringlengths
2.00
1.00
/ 1 REAL
2
4
SCATTERING
6
10
20
40
60
100
205
LENGTH-KAONS
II
ImAs
1.50
1.00
0.50
1
0.0
I I II D 20 IMAGINARY
I I I I t I 40
60
SCATTERING
80
I
100
I
I
120
I
I
140
I
I
160
I
I
180
I
I
200
I
I
I
220
LENGTH-KAONS
Fig. 3. Values of the reai and imaginary S-wave scattering length for kaons as a function of A.
[ 240 A
410
C.J. Batty / Scattering lengths TABLE 4 Analysis of antiproton data
Nucleus
n
6Li ‘Li C N ‘60 ‘80 Si P S Fe Y Zr Yb
2 2 3 3 3 3 4 4 4 5 6 6 8
(kk) -0.07zto.17 -0.268 f 0.081 -0.004 * 0.010 -0.018*0.036 -0.111 f 0.032 -0.189*0.042 -0.038 * 0.039 -0.065 f 0.023 -0.06 * 0.04 -0.01 *to.31 -0.15*0.16 -0.45*0.10 0.26jzO.46
0.34*0.31 0.18*0.14 0.042*0.018 0.179*0.031 0.484+0.106 0.55kO.24 0.11*0.19 0.446 f 0.069 0.65*0.10 0.54 i 0.32 0.80 f 0.32 0.70*0.21 1.48*0.66
asR
as1
(fm)
(fm)
-3.97* 1.54 -2.77ztO.32 -3.14*0.45 -3.46*0.24 -3.66kO.15 -3.87*0.18 -4.04kO.21 -4.6OkO.12 -4.45*0.14 -5.16kO.17 -6.2OkO.32 -6.33*0.15 -7.81 kO.65
1.77* 1.19 0.37*0.31 1.04+0.69 1.11*0.17 1.08+0.18 0.97 * 0.30 0.79* 1.35 1.20*0.10 1.27kO.11 1.31 kO.82 1.07+0.32 0.59*0.20 1.5.5* 1.14
3.2. ANTIPROTONS
For antiprotons 13 pairs of measurements of strong interaction shifts and widths are available ranging over the nuclei from Li to Yb. However, the quality of the data is relatively poor. For example, 7 of the shift or width measurements are within one standard deviation of zero and only 5 shift and 7 width measurements differ by more than 3 standard deviations from zero. [See figs. 1 and 2 in ref. ‘).I The values of the shifts (E) and widths (r) and the calculated values of as are presented in table 4. Values of d are plotted as a function of 2 in fig. 4. Least squares fits to these values give a n = 1.67kO.20 fm with x2/F = 1.33 and aI=1.10*0.17fm with x2/F=2.31 to be compared with a,=1.00+0.32fm and a, = 2.21 f 0.26 fm obtained by a direct fit to the shift and width measurements for all nuclei simultaneously. The discrepancy between these two sets of values can largely be attributed to the rather poor accuracy of the experimental measurements. There are also strong indications that for some nuclei the values of an and a1 obtained by the search procedure are highly correlated. The S-wave scattering lengths are plotted in fig. 5. Despite the considerable uncertainties in uR and ur discussed previously the values of the real scattering length are seen to be quite well determined. Least-squares fits give usn(fm) = -(1.52*0.01)A(0~313*o~017’,
x2/F = 0.6 ,
which is plotted in fig. 5, or usR(fm) = -( 1.41 f 0.02)A1’3 ,
x2/F = 0.7 .
The imaginary part can be fitted with a constant value and gives 1.11 f 0.06 fm with x2/F = 1.4. There is no clear evidence for any A-depen-
U s, =
C.J.Batty / Scattering
411
lengths
dence; as can be seen from fig. 5 this is partly because the majority available are for A < 32.
of values
3.3. SIGMA HYPERONS
The data available for sigma atoms is extremely limited and relatively inaccurate. Measurements of strong interaction shifts, widths and relative yields available ‘*) for 0 and for nuclei between Mg and S were analysed in the present work. The limited number of yield measurements available for other nuclei were not used since values of iI could not be determined for each nucleus separately. Values of the S-wave scattering length are given in table 5 and plotted in fig. 6. A least squares fit to the value of d gives uR = 0.335 kO.047 fm with x21 F = 0.28 and a1 = 0.167 f 0:027 fm with x*/F = 0.41, in good agreement with the values oR = 0.363 f 0.048 fm and a, = 0.202 f 0.027 fm obtained by fitting all the available data for the whole range of nuclei simultaneously.
4.00
3.00
2.00
1 .oo
0.0
REAL
PBAR
POTENTIAL
Fig. 4. Strength of the real and imaginary optical potential for antiprotons as a function of Z.
412
C.J. Batty / Scattering lengths
5.00
m
4.00
_
3.00
_
2.00
__
1.00
_:-.
0.0
* _ 0
I _i.---.
- - -
I
IMAGINARY
20 PEAR
-h-----_--__--_---_____--_
I~
60
40
00
100
POTENTIAL
Fig. 4
(cm.)
Least squares fits to the S-wave scattering lengths give with x2/F = 0.2 ,
asR(fm) = -(l.094~0.058)A1’3, as1= 1.72*0.15
fm
,
with x2/F = 0.5 .
These fits to the data are also plotted in fig. 6. 4. Discussion In this work we have shown that it is possible to obtain values for the hadron nucleus S-wave scattering length from measurements of strong interaction effects in exotic atoms. Since the strong interaction measurements are only available for I> 0 it is necessary to use the optical model as an intermediate stage in obtaining the scattering length from the measured shifts and widths for hadronic atoms. Assuming the potential is local it has been shown that the scattering lengths obtained are relatively insensitive to the parameters chosen for the form of the optical potential. A limited number of alternative forms for the potential have been considered.
C.J. Batty / Scattering lengths
413
A simple model using a square well potential, where the scattering length can be calculated analytically, shows that for kaons, antiprotons and sigma hyperons the real part of the scattering length will numerically have a value similar to that of the potential radius whilst the imaginary part will have a constant, but non-zero, value. Only in the case of extremely strong absorption will the real part of the scattering length be equal to the potential radius and then the imaginary part will be zero. Perhaps somewhat surprisingly the model shows that this will not be the case even for antiprotons. The real part of the S-wave scattering lengths obtained in this work shows an approximate A”3 dependence (see figs. 3,5,6) confirming a close relationship with the value of the nuclear radius. Values of the scattering length are presented in table 6 where the real part has been parameterized in the form asR = c1AC2used in the previous section. The values of a sR for kaons and sigma hyperons are seen to be rather similar and if, following the analogy with the square-well case, they are taken to correspond to an interaction radius give a value about 0.7 fm larger than the potential radius c. At this point the optical potential has a value of about 20% of its central value and it is interesting that this is the point where strong
10.00
6.00
4.00
I
1
2
REAL
SCATTERING
I
I
4
6
I
I
IO
20
1
40
I
60
,
100
I 200
LENGTH-PEARS
Fig. 5. Values of the real and imaginary S-wave scattering length for antiprotons as a function of A.
I
A
414
C.J. Batty / Scattering lengths ImAs
1.50
1.00
0.50
-
0.0
_.
I 40
IMAGINARY
-I
I
I 60
SCATTERING
80
I 100
I 120
I 140
I 160
I 180
I 200
I 220
240 A
LENGTH-PBARS
Fig. 5
(cont.)
interaction shift measurements for kaons are most sensitive “) to the form of the potential. For antiprotons the real part of the scattering lengths are about 1.25 fm larger than the potential radius, corresponding to a point at which the potential strength is about 10% of the central value. Values for the imaginary part of the scattering length, also given in table 6, are seen to be much larger than those given by the simple square-well model (table 2). TABLE 5 Analysis of sigma atom data Nucleus 0
Mg Al Si S
aSR
aSR
(fm)
(fm)
-2.73 * 0.22 -3.38 * 0.75 -3.28 kO.34 -3.58* 0.45 -3.10*0.59
1.62kO.49 1.55*0.26 1.71 kO.25 2.16hO.37 1.70+0.54
1 REAL
2
4
SCATTERING
6
10
20
40
60
100
200
A
LENGTH-SIGMAS
ImAs
0.0
I 0
I 20
IMAGINARY
I 40
I
I
SCATTERING
I
I
60
80
100
I
120
I
I
140
I
I
160
I
I
I
180
I
200
I
I
I
220
240 A
as a function
of A.
LENGTH-SIGMAS
Fig. 6. Values of the real and imaginary
S-wave scattering
length for sigma hyperons
C.J. Batty / Scattering lengths
416
TABLET Hadron-nucleus
S-wave
scattering
(2) K
1.105*0.004 1.52*0.01 1.094*0.058
lengths “) aSI
(fm) 0.359*0.010 0.313io.017 Ib J )
1.203*0.016 1.108*0.057 1.72kO.15
“) a,=a,,+ia,,; as,=-c,A% b, Fixed at this value.
This is almost certainly due to the need for a finite diffuseness for the shape of any realistic potential. For example, in the case of kaonic Si increasing the diffuseness parameter a of the Fermi distribution from a = 0 (square well) to a = 0.569 (charge distribution) causes the imaginary part of the scattering length to increase by a factor of 3. As we have already commented the value of as1 for antiprotons seems surprisingly large in view of their expected strong absorption by nuclei. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
J. Hiifner, L. Tauscher and C. Wilkin, Nucl. Phys. A231 (1974) 455 R. Seki, Phys. Rev. Lett. 29 (1972) 240 C.J. Batty, Sov. J. Part. Nucl. 13 (1982) 71 C.J. Batty, Nucl. Phys. A372 (1981) 418 C.J. Batty, Nucl. Phys. A372 (1981) 433 C.W. de Jager, H. De Vries and C. de Vries, Atom. Nucl. Data Tables 14 (1974) 479 A. Deloff and J. Law, Phys. Rev. Cl0 (1974) 1688 M. Atarashi, K. Hira and H. Narumi, Prog. Theor. Phys. 60 (1978) 209 C.J. Batty, Phys. Lett. 127B (1983) 162 C.J. Batty, A. Gal and G. Toker, Nucl. Phys. A402 (1983) 349; R. Brockmann and E. Oset, Phys. Lett. 118B (1982) 33 11) 0. Dumbrajs, A.M. Green and J.A. Niskanen, University of Helsinki preprint HU-TFT-83-22 12) C.J. Batty, SF. Biagi, M. Blecher, SD. Hoath, R.A.J. Riddle, B.L. Roberts, J.D. Davies, G.J. Pyle, G.T.A. Squier and D.M. Asbury, Phys. Lett. 74B (1978) 27