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Are narrow Σ hypernuclear states consistent with σ− atomic data?
Nuclear Physics A402 (1983) 349-365 @ North-Holland Publishing Company
ARE
NARROW
rf HYPE~U~LEAR Z-
ATOMIC
STATES CONSISTENT
WITH
DATA?
C.J. BATTY Rutherford Appleton
Laboratory, Chiifon, Didcol, Oxon, C/K A. GAL*
Weizmann Institute of Science, Rehovot, Israel Received
24 January
1983
are constructed by foIding a ftoite-range SW interaction into Abstract: E nuclear optical potentials the nuclear matt~r*distrjbnt~on and fitting the (compiex) potential strength to the available Z-atomic data. Energies and widths of Z hypernuclear single-particle unstable bound states (USS) in these optical potentkds are calculated for light nuclei and a substantial sensitivity of the calculated widths to the interaction range R is established. For example, the width of the lp Z* UBS drops from 6.6 to 1.5 MeV in i’c and from 11.9 to 6.8 MeV in :“O when R is increased from 0.8 to 1 fm, while the calculated widths of the 1s 2‘ LJBS in these hypernuclei also decrease with R, although remaining large (r 2 16 MeV). The effect of the Pauli principle on the nuclear IN -) ilN conversion is simulated by introducing an additiona density dependence of the fitted opticaX potential, but this also is found insufficient to quench the widths of 1s 2‘ UBS in this mass range to the point where they would give a clear signal in K, a) reactions.
1. Introduction Surprisingly
narrow
(f < 5 MeV) x hypernuclear
states have recently
been estab-
lished in studies of (Km, rr*) reactions on “C and 160 by the Heidelberg-Saclay Collaboration ‘) at CERN. These measurements augment earlier ones by the same group ‘) on 9Be as well as the measurement of the (K, r+) reaction on 6Li by the Brookhaven-Carnegie Mellon-Houston-MIT-Vassar Collaboration 3, at BNL. It is interesting, therefore, to expfore mechanisms which can yield relatively small 2: hypernuclear widths, at least for those excitations directly involved in the recent experiments. The approaches to this problem fall quite generally into two classes. The starting point in the first approach 4-6) is the 2I nuclear optical potential as determined from fitting the observed strong-interaction level shifts and widths in Z atomic X-rays 7‘8). The optical potential conventionally used in hadronic atoms * Supported in part by the US-Israei Binational Science Research of the Israel Academy of Sciences and Humanities. 349
Fo~ndatj~n
and by the Fund
for Basic
350 to
C.J. Batty
describe hadron-nucleus
et al. / iVurrow
2 hypernuclear
interaction is parameterized
V(r)=-$( 1+X_ >L+(r),
states
in the form
(1.1)
where II, is the hadron-nucleus reduced mass, m the nucleon mass, p(r) is the density distribution normalized to A and 2 is a complex “effective” hadronnucleon scattering length the imaginary part of which, for the present discussion, simulates true 2 absorption through the elementary conversion process ENjAN. The constant d is obtained by fitting the atomic data and is not necessarily to be identified with the zero-energy hadron-nucleon scattering amplitude in the nuclear medium as a first-order optical potential of the form (1.1) would imply. On the other hand, the form (1.11 does imply the use of a local potential. When used in the Z-nucleus Schriidinger equation it yields complex energy eigenvalues, E = -(B +$f'>, with values 6, .of the width satisfying r,,, 3 10 MeV and rls b 20 MeV for yC and YO. The second approach 9,‘o) consists of fitting separable YN coupled-channel interactions to the low-energy EN scattering and reaction data. This procedure provides a well-defined off-shelf extension of the two-body XN t-matrix which is then embedded in the nuclear medium to accommodate binding and Pauli effects. The resulting first-order E optical potential, schematically of the form tp, is a non-local one. It also is used in the Z-nucleus Schrtidinger equation to yield complex eigenenergies. In this way, e.g. ref. lo), widths F1,-4 MeV and rls> 10 MeV were calculated for ?C and “I;“O,considerably below the values quoted above for the first approach. Thus, it would appear that this second approach, but not the first one, is capable of reproducing the small widths observed in the recent experiment ‘) for C-levels tentatively assigned there to the lp orbit. In this paper we conclude that the observation ‘) of narrow _Xhypernuclear lp states is consistent with the Z- atomic data, provided a EN interaction with a range of about 1 fm is explicitly folded into the nuclear matter distribution in the construction of the optical potential. The previous analyses 4-6) in terms of (1.1) employed density shapes appropriate to the nuclear charge distribution, thus implicitly assuming a XN interaction range of about 0.8 fm. Refitting the atomic data, we find that 2% hypernuclear widths in light nuclei drop substantially when the range R is increased. Although no direct comparison exists between the ranges used in the two approaches, it appears from the parametrizations of refs. 9*10)that rather large values of R are common in fitting separable interactions to the YN low-energy data. These values are undoubtedly instrumental in reducing the calculated E-widths in light hypernuclei. The relationship between R and the rms radius of the XN interaction is discussed in sect. 4. The paper is arranged as follows: the refitting of the X atomic data in terms of the potential form (1. l), where p(r) is a properly defined distribution which accounts for the folding of a EN interaction of range R into the nuclear matter distribution,
C.J. Batty et ai. / Narrow
.Z hypernudear
351
states
is briefly described in sect. 2. Also discussed in that section is the sensitivity of the atomic data to modifications in the shape of p(r) and it is found that substantial departures from the conventional shape can be accommodated which still give a reasonable fit to the experimental data. We use this freedom to modify the form (1.1) in a way which mimics the suppression due to the Pauli principle, in anticipation of a corresponding reduction in C hypernuclear widths calculated with such a modification. The results of Z: hypernuclear calculations with various optical potentials that fit the E- atomic data are given in sect. 3 for light nuclei (A ~40). It is found that for XN interaction ranges R 3 1 fm it becomes possible to find ,X unstable bound states (UBS) embedded in the E-continuum whose width is considerably less than 10 MeV. Thus some, but not all, of the 2 hypernuclear states excited in (K-, ?r) experiments are likely to be relatively narrow (r s 5 MeV). E hypernuclear Is ground states are generally not expected to share this property. A general discussion of the results follows in sect. 4. 2. Optical potential analysis of 2T atoms X- atomic data 7.8) are limited at present to 5 shift measurements, 3 direct measurements of widths and 9 measurements of relative yields. In the analysis “) of these data in terms of the optical potential form (l.l), with p(r) of the same shape as that of the nuclear charge distribution “), the real (an) and imaginary (ar) parts of C?were adjusted by an automatic search routine to obtain a least-squares fit to the measurements. This yielded an = 0.363 f 0.048 fm ,
a~ = 0.202* 0.025 fm
(x2/F= 0.81) )
(2.1)
where the quoted errors were obtained from the least-squares variance-covariance matrix. There seems to be very little correlation between an and aI, as is shown in table 1.
Error analysis of d for II- atoms “)
0.363 f 0.048 0.348 i 0.054 0.358 i 0.045 f&33_? Q.U
0.202 f 0.025 P,_zzz 0.122 0.197 rt 0.026 0.195*0.023
12.2 13.3 12.3 12.6 13.3
“) This analysisfollows that of ref. I’). Underlined kept fixed.
0.81 0.83 0.77 0.79 0.83 values were
Parameters
of optical
potential
shapes
R =Q.O NUCi.
MHO
0 Mg Al Si S Ca Ti Ba
MHO 2PF 2pF 2PF 2pF 2pF 2pF 2pF
“1 The MHO
c
d
C
a
1.558 t.682 3.039
I.776 2.655 0.485 0.509 0.508 0.508 0.504 0.514 0.472
1.692 f XL5 2.990 2.840 2.930 3.165 3.510 3,750 S.?C)D
1 .+x32 1.517 0.548 0.569 0.569 0.569 0.563 0.567 Q.535
2.888 2.978 3.21 I 3.553 3.782 5.728
shape is itiai~i~~~fexpi,-ir!c)~)
fit (values given in fm)
R=X.O ~-“-^--1__
R -0.8 ~.~
Dist. “) ~.
c
used in the EM atomic
c I.761 l.870 2.945 2.818 2.922 3.141 3.487 3.728 5.685
and the 2pFshape
R = 1.2 .“.. a
c
n
0.885 1.220 0.582 0.600 0.593 0.602 0.596 if”S99 0.568
1.841 1.945 2.939 2.793 2.882 3,115 3.464 3.705 5.666
0.718 0.978 0.619 0.639 0.638 0.638 0.631 0.634 0.607
is (1 texp(&-c)/afi-‘.
We repeated the least-squares fit, using different ranges I3 for the ZN interaction, The quantity R is taken to be the rms radius of a gaussian distribution which is folded into the point nucleon density to obtain an effective p(r) to be used in eq, (1.1). For R = 0.8 Em, the charge d~s~ibutjo~ was retabed, as above. For zero range, a two-parameter Fermi (2pF) or modified harmonic oscilfator (MHO) 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 chaige distribution which was also taken to be of 2pF or MHO form. For ranges R = 1.0, 1.2 fm studied in the present work, a 2pF or MHO form was used with the parameters adjusted to give a best least-squares fit to the distribution obtained by folding a gaussian of rms radius (1.02 - 0.8zf1iz = 8.6 fm or (1.2’ - 0.82~1’2= 0.894 fm into the charge d~str~b~t~on. The parameters of the resulting densities p{r) which were used in the potential (1.1) are listed in table 2 for the set of nuclei for which data exist, The results of the fit to all E- atomic data are given in table 3 fox various choices of EN interaction range, for both the real part and the imaginary part of the optical potential; the corresponding ranges are denoted R, and RI. We note that the value of aR in the present fit for RR=R1 = 0.8 Em is slightly different from that of (2.1), for the older fit, due to an artefact of the fitting procedure; the best fit values obtained by the search routine depended very slightIy on the initial values used for the search. This effect is probably a reflection of the relatively poor quality of the experimental data. The x2 per degree of freedom is plotted in fig. 1, together with the resulting values of d as a function of R for R = RR = RI. It is clear from this figure that aithough all the fits are very satisfactory, the x2 quality of the fit improves slightly when the range R is increased, This latter operation enhances the overlap between the optical potential and the X wave fnn~t~o~ ~essent~~~~~ Coulomb),), so that smaller
C.J. Batty et al. /Narrow
2 hyper~~~~e~r states
TABLE
Results
RR (fm)
RI @ml
0.0 0.0 0.8 0.8 0.8 1.0 1.0 0.8 1.2 1.2
0.0 0.8 0.0 0.8 1.0 0.8 1 .o 1.2 0.8 1.2
3
of fits to all P- atomic
0.41 Ito. 0.39* 0.06 0.38*~,~6 0.35*0.06 0.33 j: 0.06 0.32=kO.O4 0.31+0.04 0.32 * 0.04 0.29* 0.04 0.27 f 0.04
353
0.23 * 0.04 0.18ztO.03 0.25 f 0.03 0.20*0.03 0.27 + 0.02 0.2OztO.03 0.17*0.03 0.14*0.02 0.20*0.03 0.14+0.02
data
X2
X2/F
17.5 13.8 13.7 11.7 11.3 11.5 10.9 11.3 11.0 11.0
1.17 0.92 0.91 0.78 0.76 0.77 0.73 0.75 0.73 0.73
strengths aR and aI are required for the optical potential in order to reproduce given values of the strong-interaction shift and width. Considering the results in table 3 for RR, RraO.8 fm we see that increasing RR affects only the value of an which decreases by about 17%. On the other hand, increasing RI decreases the value of aI by 30% and GR is slightly reduced. Thus, increasing RI from 0.8 to
E-AT~Ms
Fig. 1. The x2 per degree of freedom and the values of a = nn+iu, obtained in a least-squares fit to the +I-- atomic data in terms of the optical potential (1.1) as a function of the TN interaction range R used in the distribution p(r).
354
C,J. Etafryei aL f Numw X k~~g~~~~e~~ states
1.2 fm reduces at from 0.20 fm to 0.14 fm, roughly corresponding to a quenching of the width of 2’ hypernuclear states by 30%. The accompanying reduction in aK further enhances this quenching by pushing the X UBS more into the continuum. In order to compare X- atomic calculations with E hypernuclear calculations, it is useful to check the sensitivity of the various measured quantities to the radial distrib~t~u~ of the optical potential. We carried out ca~~u~at~ons where a focal perturbation or “notch” was introduced into the form p(r) by multipI~i~g the latter by
1 -d exp [-((r
-hJ/m>21,
(2.2)
where d measures the amount of the potential removed by the notch and lsN measures the radial extent of the perturbation. The values d = 0.2 and tzN= 0.5 fm were used, The calculated shift F (positive for attractions and width of the 4f level of .J- in ‘%i are plotted in fig. 2 as a function of Rx3 the radius of the notch, using
1
1
- 28 !$
cx2.93 fm, a4.569
fni
O-
P&l 0 [
028
r
o-
T(keV) 0.26 0
ti 0.24 0 t
t I
0.13 O&Ok&9 0,12 0
0 110 t 0
I
1
2
3
“d.--“-l
--4
NOTCH
5 RADIUS,
6
.“A..--
t
7
8
RN Cfm)
Fig. 2. Calculated shift (e) and width [r) for the ta =4 level in Z- Si as a function of the radius RN of the perturbing notch f2,2) in the opticat potent&f, The radid shape of the ~~~~~t~rbed potential is &a sho?m.
CY. Bntty et al. / Narrow E hypernuclear states
355
d = 0.35 + iO.20 fm and a 2pF distribution for p(r) with c = 2.93, a = 0.569 fm. The data for silicon was chosen for analysis since it is relatively accurate. It is evident from the figure that the energy shift is most sensitive to the density region 0.04~~ whilst for the width this region is 0.27~~. The result for the width may appear surprising in view of the common expectation that absorption takes place for observed hadronic atoms in the p/p0 region of few percent. Indeed, in kaonic silicon this was found to be the case 13). The difference here is caused by the considerably weaker imaginary part of the optical potential for sigmas which enables them to penetrate further into the nucleus before being completely absorbed. In order to study phenomenologically the suppression effects of the Pauli principle on hadron-nucleon collisions in the nuclear interior, we modified (for R = 0.8fm) the density p(r) appearing in (1.1) by including the multiplicative factor
1 -PtPwPm2’3,
(2.3)
with p = PR, p, when modifying the real or imaginary part of the potential, respectively, and 0 s @d 1. The two-third power law arises within a local density approximation because Pauli quenching starts as kg(r). However, the correction (2.3) arises also in other contexts. For example, the estimate @r=&(i(kF/q,)‘<< 1, where q0 = 290 MeV/c is the momentum release in Xc-p+ An, was derived r4) for a zero-range EN -+AN interaction. Even for ranges as large as provided by one-pionexchange (OPE), fir is still far from unity due to the moderating effect of the length scale defined by qO1. Indeed, calculations 15,16)which derive their main contribution from OPE arrive at values of pr about 0.3. On the other hand, if we choose to reproduce the 20% reduction due to Pauli effects claimed in ref. 17)for the imaginary part of the optical potential by using the form (2.3) then for p/p0 = 0.27, as obtained in the previous paragraph, a value for @r of about 0.5 is required. The situation is different for PR, for which the effect of the range of the ,ZN -+XN interaction should be more pronounced. In any realistic evaluation, the size of the OPE contribution reIative to the exchange of heavier mesons will introduce a considerable uncertainty in the actual value of PR. In the limiting case that OPE dominates the real part of the .ZN G-matrix, PR is expected to be closer to unity than to zero. In passing we mention that fitting cu-nucleus elastic scattering within a folding model for the real part of the optical potential, using the modification (2.3), leads Is) to a value /3n=0.65. Calculations with the modification (2.3) were carried out refitting the .LY atomic data and the results are shown in table 4. The quality of all the fits is excellent with no preference for any over the other. For a given value of ,@n, here chosen as either 0 or 1, increasing ,& from 0 to 0.5 gives only moderate changes in the values of uR and aI; aI increases by about lo%, hardly compensating for the much larger (50%) reduction in the potential shape at the origin, whilst an decreases by only about 5%. For a given value of Pr, however, increasing PR from 0 to 1 does
C.J. Batty et al. / Nurrow 2 ~ypernMcleur
356
TABLE
Fits to all Z- atomic
PR
Pl
0.0 0.0 0.0 1.0 I.0 1.0
0.0 0.3 0.5 0.3 0.5 1.0
“) Densities
4
data with the modification
(2.3) of the optical
0.19Lto.03 0.21 *to.03 0.22 zko.03 0.22 zko.03 0.23 f 0.03 0.26~~ 0.03
0.34kO.06 0.34*0.04 0.33*0.0s 0.49 * 0.07 0.48 i 0.07 0.46 k 0.07
corresponding
states
12.0 11.8 11.8 11.1 11.2 11.7
potential
“)
0.80 0.79 0.79 0.74 0.75 0.78
to R = 0.8 fm are to be used.
lead to a substantial increase in the values of ffR (by about 50%), whereas aI increases by onIy about 7%. 3. Single-particle X unstable bound states In this section we report the results of calculations for 2 unstable bound states (UBS), defined as eigenstates of the X-nucleus Schrijdinger equation with the optical potential (1.1). Many general properties, as well as the systematics of such states, were studied in ref. “). Here, we recall only that the non-hermiticity of the optical potential may cause the appearance of UBS in the E-continuum (E? 0). We focus our calculations towards ‘ZC and ‘go, as these (and similarly ‘iBe, ‘,$J) are the hypernuclei for which some narrow states, tentatively assigned to the lp X-orbit, have been observed ‘f in (K-, rr-) reactions (and similarly in (K-, ‘rr+)) on “C and 160. The parameters of the MHO density distributions used for the A = 11, 15 and also for A = 8 core nuclei are collected in table 5 for various assumptions about the range R of the EN interaction (assumed to be the same for the real and imaginary parts). The parameters for R = 0.8 fm are taken from the charge distributions of llB, 15N and 9Be respectively as deduced 12) from electron scattering. This prescription, TABLE
Parameters
of MHO
shapes
for optical
R = 0.0 A,,,
8 11 15
potential
5
used in calculating
R = 0.8
UBS (values given in fm) R = 1.2
R-1.0
______C
a
C
a
C
a
C
a
1.668
0.820 1.213 1.998
1.791 1.690 1.810
0.611 0.811 1.250
1.857 1.760 1.875
0.534 0.684 1.033
1.934 1.841 1.952
0.463 0.574 0.852
1.559 1.688
3.57
C.J. Batty et al. / Narrow P hypernuclear states 7-i
t
I
I
18
’ 1 iI
16
4 2
-
R:0.8
fm
--R=l.Ofm
0 -2 -9
-8
-7
-6
-5
-4 a (FvkV)
-3
-2
-I
0
Fig. 3. Calculated I” binding energies II?) and widths (rj of the Ip UBS for A = 12 and 16 with ZN interaction ranges R = 0.8 fm (solid shapes) and R = 1 fm (dashed shapes) folded into the appropriate nuclear matter distributions. The central point in each one of the shapes central value of d given in table 3, and the edges of these shapes represent quoted for h, which according to table 1 are almost independent
whilst ignoring
drawn corresponds to the the effect of the error bars of each other.
A-’ effects,is su~ci~ntly accurate to enable a representative
study
of EN range effects to be made. Results of calculations for lpX” UBS in $C and $0 are shown in fig. 3 for R = 0.8 and 1.0 fm. In each of these hypernuclei, increasing the 2YNrange by this relatively modest amount has a drastic bearing on the calculated width which drops from 6.6 MeVt down to 1.5 MeV for ZKZ and from 11.9 MeV to 6.8 MeV for $0. The effect on the binding energies is significantIy weaker and to a first approximation the location of the UBS in the continuum remains unchanged due to a cancellation between the partial effects of the real part and the imaginary part of the optical potential both of which become shallower with R: the smaller well depth of the real part pushes the UBS further up into the continuum, whilst the smaller well depth of the imaginary part gives less repulsion and therefore induces ’ The value 10.1 or 9.9 MeV quoted in ref. 6, for $C was obtained for a potential form similar to (l.i), but where rnx replaces p within the brackets. The current calculation for 2~2, with the MHO distribution of table 2 and d of table 3, both for R = 0.8 fm, gives I-,, = 8.7 MeV; going from an A = 12 core nucleus to A = 11, as is done here, further reduces the width to 6.6 WV,
C.J. Batty
358
al. / Narrow C hypernu~l~~r
ef
slates
20
!6
8 6 4
2
4
I
-8
-E
-4
/
-2
I
/
I
/
/
0
2
4
6
8
9
IMeL’)
i IO
Fig. 4. Calculated .Z binding energies and widths of the various charged ip UBS and of the neutral 1s UBS for A = 16 as a function of the EN interaction range R (in fm). The nuclear core in the XL calculations is 15N.
a compensating attraction. For the width, combine constructively so as to diminish it.
in contrast,
these
two partial
effects
The full variation of B and r for the lp C UBS in this mass range is shown in fig. 4 for ranges between 0 and 1.2 fm and for the various ,Z charge states produced in i60(K-, r*). It is clear that R must be roughly between 1 to 1.2 fm in order for a narrow lp Z-state to be observed in these reactions. With such values for the range, the figure shows that the Is state is still too wide to be observed. When considering the dependence on the charge state we recall that the charge of the nuclear core is 7 for the case of .X* excitations on 160. The appropriate lp Coulomb energies were evaluated by adding a Coulomb potential for A’* due to the nuclear charge distribution and found to be 2.85 rtrO.15 MeV for “C and 3.6*0.2 MeV for I60 where the errors take into account the error bars in d and the uncertainty in the choice of R about 1 fm. Similar calculations for E UBS in Si and Ca are displayed in figs. 5 and 6, respectively. The Id X0 UBS in the nuclear field of **Si is predicted to be quite narrow for values of R between 1 to 1.2 fm, as shown in fig. 5. This state should be readily excited in the AL =0 low momentum transfer (K-, ?r ) reaction.
359
C.J. Batty et ai. / Narrow 2” hypernucfear stales
16
-4c
_ L--_L.i.._L...i 6
-12
Fig. 5. Calculated interaction ranges
-I I
-10
-9
-8 -7 E? (MeV)
-6
-5
-4
,E” binding energies and widths of the Id UBS in the nuclear core of “*Si for EN R = 0.8 fm (solid shape), R = 1 fm (dashed shape) and R = 1.2 fm (dot-dashed shape). The meaning of the shapes is the same as in fig. 3.
However, the Id state is not particularly narrow in 40Ca (rid = 11.8 MeV for the central point in the R = 1 fm calculation which is not shown in fig. 6) since it is more bound than in 28Si. In 40Ca it is the If X0 UBS that has a chance of appearing narrow for R = 1 fm, as shown in fig. 6. It would be interesting to look for the related AL = 1 excitation at intermediate values of the momentum transfer. All these states are predicted to be in the X-continuum. The effect of Pauli quenching, as described in the context of the atomic fit discussed in sect. 2, is displayed for the lp Z; UBS in light nuclei in fig. 7. Here using eq. (2.3), only the imaginary part of the optical potential was quenched, using the appropriate parameters from table 4 (& = 0) and table 5 (R = 0.8 fm). The widths go down as expected when pi is increased, which shows their sensitivity to the inner region of the nuclear density. Less pronounced is the change induced in the (real) binding energy, with these states becoming slightly more bound, but still in the continuum. The 1s Z UBS (not shown in the figure) also show quenching in their widths, from about 21 MeV to about 17 MeV for pi = 0.3 and further down to about 13 MeV for pi = 0.5.
360
-2/-
‘\
/
L.J-_-L_,._~. ..i__~~i._-i.__i -I
-4
-17
-16
-t5
-14
-13
-12
-II
-10
-8
-7
-6
-5
-4
B iMeW
Fig. 6. CafenIated .E:”and X- binding energies and widths of the If UBS in the m&ear core of 4*Ga for ZN interaction u~~ger R = 0.8 fm (solid shape) and R = 1 fm (dashed shapes). The meaning of the shapes is the same as in fig. 3.
r
12 I----_7 II
LLdUL__.L -8
-7
-6
-5 -4 B (Me’/1
-3
-2
C.J. Batty et al. / Narrow
If the real part of the optical the quenching
of the imaginary
potential
Z hypernuclear
361
states
is completely
quenched
part is kept to a realistic
extent
(PR= l), while (pi = 0.3), the lp
Xc0 UBS disappear for A = 12, 16. The real part of the potential now has an attractive pocket centered at the nuclear surface with a depth of onIy about 30% the depth of the pre-quenched potential and is capable of binding only the 1s UBS which occurs in the continuum. For A = 16 this situation is depicted in fig. 8, and
13
I
I
I
a
;;c*
12
II
x0
Is
IO Pauli
Quenching
: @=l.O+ i 0.3
I
I
-4
-3
-2
I
-1
0
B (MeV)
Fig. 8. 1” binding energies and widths of the 1s UBS for A = 16 and 41, with the real and imaginary parts of the optical potential quenched according to (2.3) with p = 1 and 0.3 respectively. The value of ci is taken from table 4 and the meaning of the shape for A = 16 is the same as in fig. 3.
the results (not shown) for A = 12 are very similar. Although under these extreme conditions the 1s UBS can become relatively narrow in light nuclei, such a conclusion does not hold for the heavier nuclei as shown for example by the Ca point in the figure. There, the 1s Z;” width decreases from 27 MeV for p = 0 down to 21 MeV for /3n = 0, pi = 0.3 and down further to 12.5 MeV for PR = 1, pi = 0.3. The effect of Pauli quenching on the imaginary part of the optical potential for the .X single-particle spectrum of UBS in a heavier nucleus, say ‘$Ca, is shown in fig. 9. It is clear that the If UBS, which is quite high in the continuum, is the one most likely to show up as a narrow excitation.
362
Pouli
Quenching
1
IP
Id i
Fig. 9. The spectrum of 6” UBS in the nuclear core of 4”Ca calculated with the imaginary part of the optical potential quenched according to (2.3) for values of p as marked in the figure.
4. Conclusions We have demonstrated in the present work that the observation of narrow z’ hypernuclear states in (K-, a-) reactions on light nuclei is consistent with the available X- atomic data, provided the optical potential is constructed by folding a finite-range ;T;N interaction into the nuclear matter distribution. For ranges 1.0~ R c 1.2 fm the lp E UBS in the A = 12, 16 hypernuclei are calculated to be quite narrow, only a few MeV wide, in accordance with the recent experiment ‘). Wycech Edal. *‘) studied the scope and validity of the folding assumption for Catoms within a YN separable-interaction model, finding the resulting local representation of the first order optical potential to be valid to within 10%. The range recommended by these authors? corresponds to R = 1.33 fm, in our notation. Here we claim that the validity of folding for the imaginary part W(r) of the optical potential is assured by extending the arguments of ref. 14) according to which the relatively large momentum released in the E-p -An conversion justifies the use of semi-classical considerations, Thus, W(r)-$E~-p-t,~,p(r),
(4.1)
where p(r) is the folded nuclear density (normalized to A) and the bar refers to nuclear-medium corrections, e.g. Fermi averaging, binding and Pauli corrections, to the zero-energy X- conversion cross section. Fermi averaging of the measured cross sections turns out to be the most important correction, yielding ‘) a value ’ The misprints
in eqs. @a, b) of ref. “1 were corrected
in eqs. (8) and (9) of ref. 9).
C.J. Batty et ni. / Narrow X hypernuclear states vl+=
363
14 mb. Comparing (4.1) with (1.1) for large A, the following connection arises: ii3 = 8rraI/p.ZN,
(4.2)
leading to a value ar==0.15 fm consistent with the atomic fit presented in table 3 for ranges Rrc 1.2 fm. We note that this explanation of the notable small value for the imaginary part of the “effective” .EN scattering length appears to be rather different from that offered in ref. 17). Similar arguments, however, cannot be advanced for the case of the real part of the optical potential. The non-locality modifications are probably more significant and its calculation remains uncertain ‘*17),The E- atomic data alone do not provide, at present, a unique determination of the effective _XN scattering length 6, the fitted values being strongly correlated with the range R (fig. 1). For a local gaussian ZN + /lN conversion potential with rms radius (r’>&&, the rms radius used in the folding procedure (as described in sect. 2) is given by Ri = (f(r2L)i’*
,
(4.3)
fortuitously similar to the relationship holding within a separable interaction model 9,*7) for both real and imaginary parts of the optical potential provided (r”>!& stands for the rms radius of the separable gaussian form factors. The factor 2 in (4.3) arises because the conversion potential appears bitinearly in the construction of the imaginary part of the optical potential, once going out of the E-channel and the other back there from the n-channels. Pauli quenching of hypernuclear widths has also been considered in the present work, using the phenomenological representation (2.3). We conclude that the real part of the E optical potential cannot be fully Pauli-quenched in the nuclear interior since the resulting potential, refitted to the atomic data, is too shallow then to produce lp E UBS in the A = 12 and 16 hypernuclei, contrary to the situation suggested by experiment *>. Although in these cases the 1s z‘ UBS are calculated to lie just in the continuum with widths as small as 7 MeV, this relative narrowness of the Is UBS does not persist in heavier hypernuclei (fig. 8). Furthermore, we calculate only a moderate reduction in 2; hypernuclear widths for realistic Pauli quenching of the imaginary part of the optical potential. In our calculation, the quenching effect on the 1s UBS is not much larger than that on the lp UBS in light hypernuclei, contrary to the conclusions of ref. lo). It should be recalled, however, that a relatively large equivalent [in the sense of refs. ‘*17)].ZN interaction range R = 3”‘//3 (= 1.64 fm), which apparently enhances the effects of the Pauli principle, was employed in that work. In addition, the extent of the approximations in the calculation remains unclear, inasmuch as the density dependence was apparently considered only through the choice of considerably smaller values for the Fermi momentum k~ for lp states than for 1s states in these light hypernuclei. As for heavier hypernuclei, fig. 9 provides a useful overview of Pauli effects in our approach.
364
C.J. Batty et al. / Narrou: 2’ ~ypernuc~ear stufes
Within the framework of the present hypernudear calculation, narrow X UBS do appear in the continuum. However, UBS that are calculated to be narrow for a given mass range sink, as the mass number A is increased, into the potential well thus becoming wider and are occasionally replaced by new narrow UBS in the continuum. In the nuclear p-shell, the arena of present 2 hypernuclear experimentation, the potentially narrow X UBS are the lp ones (fig. 3), whereas the 1s E UBS are generally wide (fig. 4). In the nuclear sd-shell, the Id and 2s UBS are likely to appear narrow at the lower part of the shell (fig. 5), gradually becoming wider as the If UBS shows up as a narrow state at the end of the shell (fig. 6). We have already commented in ref. “) on the distinction between E UBS, which our calculations of lp states in the A = 12, 16 hypernuclei yield in the &continuum, and Z: Gamow states, which for real potentials signify resonances in the continuum (with wavefunctions exploding at infinity), but which for moderately strongly absorptive (complex) potentials emerge with slightly positive binding. In the absence of a model reaction calculation designed to determine which of the two alternative descriptions of .I$“states” is meaningful for (K-, 7~)excitations, the best circumstantial evidence in favour of UBS remains the lowest X hypernuclear excitation observed ‘) on 9Be. The obvious interpretation “) in this case is a lp ,X0 with a single-particle energy 8* 1 MeV in the continuum. In the present calculation,
6
Fig. 10. Cafcuiated
-9
-8
-7 -6 B (MeV)
-5
IL--l
I
I
I
-10
-4
-3
1” binding energy and width of the lp UBS in the nuclear core of ‘Be, assuming zero-range ZN interaction. The shape used is the same as in fig. 3.
C.J. Batty et al. f Narrow E hypernuclear states
365
employing fitted values of d (table 3) and density shapes (table 5) appropriate to the variation in R, we find a narrow ,X0 lp UBS only for R = 0 at 7 MeV in the continuum as depicted in fig. 10, and in nice agreement with experiment. The lp Gamow state appears in this calculation near E = 0, quite far from the observed excitation. We remark that the failure to obtain lp UBS in the 8Be core for R - 1 fm is not particularly disturbing since, as noted in sect. 3, our choice of nuclear densities ignored A-’ effects. The clearest interpretation of the most recent experiments ‘) probably holds for the “C(K-, 7~~) reaction since only one Z charge state and clue lp j-component is to be generally expected. Indeed, one clear signal is observed ‘) at 3 MeV in the continuum and tentatively assigned to the Z- lp orbit. Calculations similar to those reported in sect. 3 yield a lp ,Y’ UBS at B = -3 MeV, for R = 0.8 and 1.0 fm, with widths about 8 and 3 MeV, respectively. If Pauli quenching is superposed on the parametrization of the imaginary part of the optical potential for R = 0.8 fm, the binding changes between -3 to -2 MeV as /3i is increased from 0 to 0.5. The width decreases then from 7 to 4 MeV. Thus, in terms of UBS, our calculations reproduce the energy of the observed ‘) narrow excitation in a most natural way. On the other hand, the X lp “resonance” pole is calculated to be at B = 2.5 MeV and with r = 0.4 and 1.0 MeV for R = 0.8 and 1.0 fm, respectively. References 1) R. Bertini et al., Production of X hypernuclei, in Proc. Int. Conf. on hypernuclear and kaon physics, Heidelberg, 1982, ed. B. Povh, MPI H-1982-V20, p. 1 2) R. Bertini er al., Phys. Lett. 90B (19801 375 3) H. Piekarz et al., Phys. Lett. 1lOB (1982) 428 4) C.J. Batty, Phys. Lett. 87B (1979) 324 5) A. Gal and C.B. Dover, Phys, Rev. Lett. 44 (1980) 379,962(E) 6) A. Gal, G. Toker and Y. Alexander, Ann. of Phys. 13’7 (1981) 341 7) G. Backenstoss et al., Z. Phys. A273 (1975) 137 8) C.J. Batty et al., Phys. Lett. 74B (1978) 27 9) W. Stepien-Rudzka and S. Wyceeh, Nucl. Phys. A362 (1981) 349 10) J. Johnstone and A.W. Thomas, J. of Phys. G8 (1982) L105; Nucl. Phys. A392 (1983) 409 11) C.J. Batty, Nucl. Phys. A372 (1981) 433 12) C.W. de Jager, H. de Vries and C. de Vries, At. Nucl. Data Tables 14 (1974) 479 13) C.J. Batty, Nucl. Phys. A372 (1981) 418 14) A. Gal, in Proc. Int. Conf. on hypernuclear and kaon physics, Heidelberg 1982, ed. B. Povh, MPI H-1982-V20, p. 27 15) ‘J. Dabrowski and J. Rozynek, Phys. Rev. C23 (1981) 1706, and in the Proceedings quoted in ref. ‘) 16) R. Brockmann and E. Oset, Phys. Lett. 118B (1982) 33 17) S.Wycech, W. Stepien-Rudzka and J.R. Rook, Nucl. Phys. A324 (1979) 288 18) E. Friedman, H.J. Gils, H. Rebel and Z. Majka, Phys. Rev. Lett. 41 (1978) 1200
ARE
NARROW
rf HYPE~U~LEAR Z-
ATOMIC
STATES CONSISTENT
WITH
DATA?
C.J. BATTY Rutherford Appleton
Laboratory, Chiifon, Didcol, Oxon, C/K A. GAL*
Weizmann Institute of Science, Rehovot, Israel Received
24 January
1983
are constructed by foIding a ftoite-range SW interaction into Abstract: E nuclear optical potentials the nuclear matt~r*distrjbnt~on and fitting the (compiex) potential strength to the available Z-atomic data. Energies and widths of Z hypernuclear single-particle unstable bound states (USS) in these optical potentkds are calculated for light nuclei and a substantial sensitivity of the calculated widths to the interaction range R is established. For example, the width of the lp Z* UBS drops from 6.6 to 1.5 MeV in i’c and from 11.9 to 6.8 MeV in :“O when R is increased from 0.8 to 1 fm, while the calculated widths of the 1s 2‘ LJBS in these hypernuclei also decrease with R, although remaining large (r 2 16 MeV). The effect of the Pauli principle on the nuclear IN -) ilN conversion is simulated by introducing an additiona density dependence of the fitted opticaX potential, but this also is found insufficient to quench the widths of 1s 2‘ UBS in this mass range to the point where they would give a clear signal in K, a) reactions.
1. Introduction Surprisingly
narrow
(f < 5 MeV) x hypernuclear
states have recently
been estab-
lished in studies of (Km, rr*) reactions on “C and 160 by the Heidelberg-Saclay Collaboration ‘) at CERN. These measurements augment earlier ones by the same group ‘) on 9Be as well as the measurement of the (K, r+) reaction on 6Li by the Brookhaven-Carnegie Mellon-Houston-MIT-Vassar Collaboration 3, at BNL. It is interesting, therefore, to expfore mechanisms which can yield relatively small 2: hypernuclear widths, at least for those excitations directly involved in the recent experiments. The approaches to this problem fall quite generally into two classes. The starting point in the first approach 4-6) is the 2I nuclear optical potential as determined from fitting the observed strong-interaction level shifts and widths in Z atomic X-rays 7‘8). The optical potential conventionally used in hadronic atoms * Supported in part by the US-Israei Binational Science Research of the Israel Academy of Sciences and Humanities. 349
Fo~ndatj~n
and by the Fund
for Basic
350 to
C.J. Batty
describe hadron-nucleus
et al. / iVurrow
2 hypernuclear
interaction is parameterized
V(r)=-$( 1+X_ >L+(r),
states
in the form
(1.1)
where II, is the hadron-nucleus reduced mass, m the nucleon mass, p(r) is the density distribution normalized to A and 2 is a complex “effective” hadronnucleon scattering length the imaginary part of which, for the present discussion, simulates true 2 absorption through the elementary conversion process ENjAN. The constant d is obtained by fitting the atomic data and is not necessarily to be identified with the zero-energy hadron-nucleon scattering amplitude in the nuclear medium as a first-order optical potential of the form (1.1) would imply. On the other hand, the form (1.11 does imply the use of a local potential. When used in the Z-nucleus Schriidinger equation it yields complex energy eigenvalues, E = -(B +$f'>, with values 6, .of the width satisfying r,,, 3 10 MeV and rls b 20 MeV for yC and YO. The second approach 9,‘o) consists of fitting separable YN coupled-channel interactions to the low-energy EN scattering and reaction data. This procedure provides a well-defined off-shelf extension of the two-body XN t-matrix which is then embedded in the nuclear medium to accommodate binding and Pauli effects. The resulting first-order E optical potential, schematically of the form tp, is a non-local one. It also is used in the Z-nucleus Schrtidinger equation to yield complex eigenenergies. In this way, e.g. ref. lo), widths F1,-4 MeV and rls> 10 MeV were calculated for ?C and “I;“O,considerably below the values quoted above for the first approach. Thus, it would appear that this second approach, but not the first one, is capable of reproducing the small widths observed in the recent experiment ‘) for C-levels tentatively assigned there to the lp orbit. In this paper we conclude that the observation ‘) of narrow _Xhypernuclear lp states is consistent with the Z- atomic data, provided a EN interaction with a range of about 1 fm is explicitly folded into the nuclear matter distribution in the construction of the optical potential. The previous analyses 4-6) in terms of (1.1) employed density shapes appropriate to the nuclear charge distribution, thus implicitly assuming a XN interaction range of about 0.8 fm. Refitting the atomic data, we find that 2% hypernuclear widths in light nuclei drop substantially when the range R is increased. Although no direct comparison exists between the ranges used in the two approaches, it appears from the parametrizations of refs. 9*10)that rather large values of R are common in fitting separable interactions to the YN low-energy data. These values are undoubtedly instrumental in reducing the calculated E-widths in light hypernuclei. The relationship between R and the rms radius of the XN interaction is discussed in sect. 4. The paper is arranged as follows: the refitting of the X atomic data in terms of the potential form (1. l), where p(r) is a properly defined distribution which accounts for the folding of a EN interaction of range R into the nuclear matter distribution,
C.J. Batty et ai. / Narrow
.Z hypernudear
351
states
is briefly described in sect. 2. Also discussed in that section is the sensitivity of the atomic data to modifications in the shape of p(r) and it is found that substantial departures from the conventional shape can be accommodated which still give a reasonable fit to the experimental data. We use this freedom to modify the form (1.1) in a way which mimics the suppression due to the Pauli principle, in anticipation of a corresponding reduction in C hypernuclear widths calculated with such a modification. The results of Z: hypernuclear calculations with various optical potentials that fit the E- atomic data are given in sect. 3 for light nuclei (A ~40). It is found that for XN interaction ranges R 3 1 fm it becomes possible to find ,X unstable bound states (UBS) embedded in the E-continuum whose width is considerably less than 10 MeV. Thus some, but not all, of the 2 hypernuclear states excited in (K-, ?r) experiments are likely to be relatively narrow (r s 5 MeV). E hypernuclear Is ground states are generally not expected to share this property. A general discussion of the results follows in sect. 4. 2. Optical potential analysis of 2T atoms X- atomic data 7.8) are limited at present to 5 shift measurements, 3 direct measurements of widths and 9 measurements of relative yields. In the analysis “) of these data in terms of the optical potential form (l.l), with p(r) of the same shape as that of the nuclear charge distribution “), the real (an) and imaginary (ar) parts of C?were adjusted by an automatic search routine to obtain a least-squares fit to the measurements. This yielded an = 0.363 f 0.048 fm ,
a~ = 0.202* 0.025 fm
(x2/F= 0.81) )
(2.1)
where the quoted errors were obtained from the least-squares variance-covariance matrix. There seems to be very little correlation between an and aI, as is shown in table 1.
Error analysis of d for II- atoms “)
0.363 f 0.048 0.348 i 0.054 0.358 i 0.045 f&33_? Q.U
0.202 f 0.025 P,_zzz 0.122 0.197 rt 0.026 0.195*0.023
12.2 13.3 12.3 12.6 13.3
“) This analysisfollows that of ref. I’). Underlined kept fixed.
0.81 0.83 0.77 0.79 0.83 values were
Parameters
of optical
potential
shapes
R =Q.O NUCi.
MHO
0 Mg Al Si S Ca Ti Ba
MHO 2PF 2pF 2PF 2pF 2pF 2pF 2pF
“1 The MHO
c
d
C
a
1.558 t.682 3.039
I.776 2.655 0.485 0.509 0.508 0.508 0.504 0.514 0.472
1.692 f XL5 2.990 2.840 2.930 3.165 3.510 3,750 S.?C)D
1 .+x32 1.517 0.548 0.569 0.569 0.569 0.563 0.567 Q.535
2.888 2.978 3.21 I 3.553 3.782 5.728
shape is itiai~i~~~fexpi,-ir!c)~)
fit (values given in fm)
R=X.O ~-“-^--1__
R -0.8 ~.~
Dist. “) ~.
c
used in the EM atomic
c I.761 l.870 2.945 2.818 2.922 3.141 3.487 3.728 5.685
and the 2pFshape
R = 1.2 .“.. a
c
n
0.885 1.220 0.582 0.600 0.593 0.602 0.596 if”S99 0.568
1.841 1.945 2.939 2.793 2.882 3,115 3.464 3.705 5.666
0.718 0.978 0.619 0.639 0.638 0.638 0.631 0.634 0.607
is (1 texp(&-c)/afi-‘.
We repeated the least-squares fit, using different ranges I3 for the ZN interaction, The quantity R is taken to be the rms radius of a gaussian distribution which is folded into the point nucleon density to obtain an effective p(r) to be used in eq, (1.1). For R = 0.8 Em, the charge d~s~ibutjo~ was retabed, as above. For zero range, a two-parameter Fermi (2pF) or modified harmonic oscilfator (MHO) 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 chaige distribution which was also taken to be of 2pF or MHO form. For ranges R = 1.0, 1.2 fm studied in the present work, a 2pF or MHO form was used with the parameters adjusted to give a best least-squares fit to the distribution obtained by folding a gaussian of rms radius (1.02 - 0.8zf1iz = 8.6 fm or (1.2’ - 0.82~1’2= 0.894 fm into the charge d~str~b~t~on. The parameters of the resulting densities p{r) which were used in the potential (1.1) are listed in table 2 for the set of nuclei for which data exist, The results of the fit to all E- atomic data are given in table 3 fox various choices of EN interaction range, for both the real part and the imaginary part of the optical potential; the corresponding ranges are denoted R, and RI. We note that the value of aR in the present fit for RR=R1 = 0.8 Em is slightly different from that of (2.1), for the older fit, due to an artefact of the fitting procedure; the best fit values obtained by the search routine depended very slightIy on the initial values used for the search. This effect is probably a reflection of the relatively poor quality of the experimental data. The x2 per degree of freedom is plotted in fig. 1, together with the resulting values of d as a function of R for R = RR = RI. It is clear from this figure that aithough all the fits are very satisfactory, the x2 quality of the fit improves slightly when the range R is increased, This latter operation enhances the overlap between the optical potential and the X wave fnn~t~o~ ~essent~~~~~ Coulomb),), so that smaller
C.J. Batty et al. /Narrow
2 hyper~~~~e~r states
TABLE
Results
RR (fm)
RI @ml
0.0 0.0 0.8 0.8 0.8 1.0 1.0 0.8 1.2 1.2
0.0 0.8 0.0 0.8 1.0 0.8 1 .o 1.2 0.8 1.2
3
of fits to all P- atomic
0.41 Ito. 0.39* 0.06 0.38*~,~6 0.35*0.06 0.33 j: 0.06 0.32=kO.O4 0.31+0.04 0.32 * 0.04 0.29* 0.04 0.27 f 0.04
353
0.23 * 0.04 0.18ztO.03 0.25 f 0.03 0.20*0.03 0.27 + 0.02 0.2OztO.03 0.17*0.03 0.14*0.02 0.20*0.03 0.14+0.02
data
X2
X2/F
17.5 13.8 13.7 11.7 11.3 11.5 10.9 11.3 11.0 11.0
1.17 0.92 0.91 0.78 0.76 0.77 0.73 0.75 0.73 0.73
strengths aR and aI are required for the optical potential in order to reproduce given values of the strong-interaction shift and width. Considering the results in table 3 for RR, RraO.8 fm we see that increasing RR affects only the value of an which decreases by about 17%. On the other hand, increasing RI decreases the value of aI by 30% and GR is slightly reduced. Thus, increasing RI from 0.8 to
E-AT~Ms
Fig. 1. The x2 per degree of freedom and the values of a = nn+iu, obtained in a least-squares fit to the +I-- atomic data in terms of the optical potential (1.1) as a function of the TN interaction range R used in the distribution p(r).
354
C,J. Etafryei aL f Numw X k~~g~~~~e~~ states
1.2 fm reduces at from 0.20 fm to 0.14 fm, roughly corresponding to a quenching of the width of 2’ hypernuclear states by 30%. The accompanying reduction in aK further enhances this quenching by pushing the X UBS more into the continuum. In order to compare X- atomic calculations with E hypernuclear calculations, it is useful to check the sensitivity of the various measured quantities to the radial distrib~t~u~ of the optical potential. We carried out ca~~u~at~ons where a focal perturbation or “notch” was introduced into the form p(r) by multipI~i~g the latter by
1 -d exp [-((r
-hJ/m>21,
(2.2)
where d measures the amount of the potential removed by the notch and lsN measures the radial extent of the perturbation. The values d = 0.2 and tzN= 0.5 fm were used, The calculated shift F (positive for attractions and width of the 4f level of .J- in ‘%i are plotted in fig. 2 as a function of Rx3 the radius of the notch, using
1
1
- 28 !$
cx2.93 fm, a4.569
fni
O-
P&l 0 [
028
r
o-
T(keV) 0.26 0
ti 0.24 0 t
t I
0.13 O&Ok&9 0,12 0
0 110 t 0
I
1
2
3
“d.--“-l
--4
NOTCH
5 RADIUS,
6
.“A..--
t
7
8
RN Cfm)
Fig. 2. Calculated shift (e) and width [r) for the ta =4 level in Z- Si as a function of the radius RN of the perturbing notch f2,2) in the opticat potent&f, The radid shape of the ~~~~~t~rbed potential is &a sho?m.
CY. Bntty et al. / Narrow E hypernuclear states
355
d = 0.35 + iO.20 fm and a 2pF distribution for p(r) with c = 2.93, a = 0.569 fm. The data for silicon was chosen for analysis since it is relatively accurate. It is evident from the figure that the energy shift is most sensitive to the density region 0.04~~ whilst for the width this region is 0.27~~. The result for the width may appear surprising in view of the common expectation that absorption takes place for observed hadronic atoms in the p/p0 region of few percent. Indeed, in kaonic silicon this was found to be the case 13). The difference here is caused by the considerably weaker imaginary part of the optical potential for sigmas which enables them to penetrate further into the nucleus before being completely absorbed. In order to study phenomenologically the suppression effects of the Pauli principle on hadron-nucleon collisions in the nuclear interior, we modified (for R = 0.8fm) the density p(r) appearing in (1.1) by including the multiplicative factor
1 -PtPwPm2’3,
(2.3)
with p = PR, p, when modifying the real or imaginary part of the potential, respectively, and 0 s @d 1. The two-third power law arises within a local density approximation because Pauli quenching starts as kg(r). However, the correction (2.3) arises also in other contexts. For example, the estimate @r=&(i(kF/q,)‘<< 1, where q0 = 290 MeV/c is the momentum release in Xc-p+ An, was derived r4) for a zero-range EN -+AN interaction. Even for ranges as large as provided by one-pionexchange (OPE), fir is still far from unity due to the moderating effect of the length scale defined by qO1. Indeed, calculations 15,16)which derive their main contribution from OPE arrive at values of pr about 0.3. On the other hand, if we choose to reproduce the 20% reduction due to Pauli effects claimed in ref. 17)for the imaginary part of the optical potential by using the form (2.3) then for p/p0 = 0.27, as obtained in the previous paragraph, a value for @r of about 0.5 is required. The situation is different for PR, for which the effect of the range of the ,ZN -+XN interaction should be more pronounced. In any realistic evaluation, the size of the OPE contribution reIative to the exchange of heavier mesons will introduce a considerable uncertainty in the actual value of PR. In the limiting case that OPE dominates the real part of the .ZN G-matrix, PR is expected to be closer to unity than to zero. In passing we mention that fitting cu-nucleus elastic scattering within a folding model for the real part of the optical potential, using the modification (2.3), leads Is) to a value /3n=0.65. Calculations with the modification (2.3) were carried out refitting the .LY atomic data and the results are shown in table 4. The quality of all the fits is excellent with no preference for any over the other. For a given value of ,@n, here chosen as either 0 or 1, increasing ,& from 0 to 0.5 gives only moderate changes in the values of uR and aI; aI increases by about lo%, hardly compensating for the much larger (50%) reduction in the potential shape at the origin, whilst an decreases by only about 5%. For a given value of Pr, however, increasing PR from 0 to 1 does
C.J. Batty et al. / Nurrow 2 ~ypernMcleur
356
TABLE
Fits to all Z- atomic
PR
Pl
0.0 0.0 0.0 1.0 I.0 1.0
0.0 0.3 0.5 0.3 0.5 1.0
“) Densities
4
data with the modification
(2.3) of the optical
0.19Lto.03 0.21 *to.03 0.22 zko.03 0.22 zko.03 0.23 f 0.03 0.26~~ 0.03
0.34kO.06 0.34*0.04 0.33*0.0s 0.49 * 0.07 0.48 i 0.07 0.46 k 0.07
corresponding
states
12.0 11.8 11.8 11.1 11.2 11.7
potential
“)
0.80 0.79 0.79 0.74 0.75 0.78
to R = 0.8 fm are to be used.
lead to a substantial increase in the values of ffR (by about 50%), whereas aI increases by onIy about 7%. 3. Single-particle X unstable bound states In this section we report the results of calculations for 2 unstable bound states (UBS), defined as eigenstates of the X-nucleus Schrijdinger equation with the optical potential (1.1). Many general properties, as well as the systematics of such states, were studied in ref. “). Here, we recall only that the non-hermiticity of the optical potential may cause the appearance of UBS in the E-continuum (E?
Parameters
of MHO
shapes
for optical
R = 0.0 A,,,
8 11 15
potential
5
used in calculating
R = 0.8
UBS (values given in fm) R = 1.2
R-1.0
______C
a
C
a
C
a
C
a
1.668
0.820 1.213 1.998
1.791 1.690 1.810
0.611 0.811 1.250
1.857 1.760 1.875
0.534 0.684 1.033
1.934 1.841 1.952
0.463 0.574 0.852
1.559 1.688
3.57
C.J. Batty et al. / Narrow P hypernuclear states 7-i
t
I
I
18
’ 1 iI
16
4 2
-
R:0.8
fm
--R=l.Ofm
0 -2 -9
-8
-7
-6
-5
-4 a (FvkV)
-3
-2
-I
0
Fig. 3. Calculated I” binding energies II?) and widths (rj of the Ip UBS for A = 12 and 16 with ZN interaction ranges R = 0.8 fm (solid shapes) and R = 1 fm (dashed shapes) folded into the appropriate nuclear matter distributions. The central point in each one of the shapes central value of d given in table 3, and the edges of these shapes represent quoted for h, which according to table 1 are almost independent
whilst ignoring
drawn corresponds to the the effect of the error bars of each other.
A-’ effects,is su~ci~ntly accurate to enable a representative
study
of EN range effects to be made. Results of calculations for lpX” UBS in $C and $0 are shown in fig. 3 for R = 0.8 and 1.0 fm. In each of these hypernuclei, increasing the 2YNrange by this relatively modest amount has a drastic bearing on the calculated width which drops from 6.6 MeVt down to 1.5 MeV for ZKZ and from 11.9 MeV to 6.8 MeV for $0. The effect on the binding energies is significantIy weaker and to a first approximation the location of the UBS in the continuum remains unchanged due to a cancellation between the partial effects of the real part and the imaginary part of the optical potential both of which become shallower with R: the smaller well depth of the real part pushes the UBS further up into the continuum, whilst the smaller well depth of the imaginary part gives less repulsion and therefore induces ’ The value 10.1 or 9.9 MeV quoted in ref. 6, for $C was obtained for a potential form similar to (l.i), but where rnx replaces p within the brackets. The current calculation for 2~2, with the MHO distribution of table 2 and d of table 3, both for R = 0.8 fm, gives I-,, = 8.7 MeV; going from an A = 12 core nucleus to A = 11, as is done here, further reduces the width to 6.6 WV,
C.J. Batty
358
al. / Narrow C hypernu~l~~r
ef
slates
20
!6
8 6 4
2
4
I
-8
-E
-4
/
-2
I
/
I
/
/
0
2
4
6
8
9
IMeL’)
i IO
Fig. 4. Calculated .Z binding energies and widths of the various charged ip UBS and of the neutral 1s UBS for A = 16 as a function of the EN interaction range R (in fm). The nuclear core in the XL calculations is 15N.
a compensating attraction. For the width, combine constructively so as to diminish it.
in contrast,
these
two partial
effects
The full variation of B and r for the lp C UBS in this mass range is shown in fig. 4 for ranges between 0 and 1.2 fm and for the various ,Z charge states produced in i60(K-, r*). It is clear that R must be roughly between 1 to 1.2 fm in order for a narrow lp Z-state to be observed in these reactions. With such values for the range, the figure shows that the Is state is still too wide to be observed. When considering the dependence on the charge state we recall that the charge of the nuclear core is 7 for the case of .X* excitations on 160. The appropriate lp Coulomb energies were evaluated by adding a Coulomb potential for A’* due to the nuclear charge distribution and found to be 2.85 rtrO.15 MeV for “C and 3.6*0.2 MeV for I60 where the errors take into account the error bars in d and the uncertainty in the choice of R about 1 fm. Similar calculations for E UBS in Si and Ca are displayed in figs. 5 and 6, respectively. The Id X0 UBS in the nuclear field of **Si is predicted to be quite narrow for values of R between 1 to 1.2 fm, as shown in fig. 5. This state should be readily excited in the AL =0 low momentum transfer (K-, ?r ) reaction.
359
C.J. Batty et ai. / Narrow 2” hypernucfear stales
16
-4c
_ L--_L.i.._L...i 6
-12
Fig. 5. Calculated interaction ranges
-I I
-10
-9
-8 -7 E? (MeV)
-6
-5
-4
,E” binding energies and widths of the Id UBS in the nuclear core of “*Si for EN R = 0.8 fm (solid shape), R = 1 fm (dashed shape) and R = 1.2 fm (dot-dashed shape). The meaning of the shapes is the same as in fig. 3.
However, the Id state is not particularly narrow in 40Ca (rid = 11.8 MeV for the central point in the R = 1 fm calculation which is not shown in fig. 6) since it is more bound than in 28Si. In 40Ca it is the If X0 UBS that has a chance of appearing narrow for R = 1 fm, as shown in fig. 6. It would be interesting to look for the related AL = 1 excitation at intermediate values of the momentum transfer. All these states are predicted to be in the X-continuum. The effect of Pauli quenching, as described in the context of the atomic fit discussed in sect. 2, is displayed for the lp Z; UBS in light nuclei in fig. 7. Here using eq. (2.3), only the imaginary part of the optical potential was quenched, using the appropriate parameters from table 4 (& = 0) and table 5 (R = 0.8 fm). The widths go down as expected when pi is increased, which shows their sensitivity to the inner region of the nuclear density. Less pronounced is the change induced in the (real) binding energy, with these states becoming slightly more bound, but still in the continuum. The 1s Z UBS (not shown in the figure) also show quenching in their widths, from about 21 MeV to about 17 MeV for pi = 0.3 and further down to about 13 MeV for pi = 0.5.
360
-2/-
‘\
/
L.J-_-L_,._~. ..i__~~i._-i.__i -I
-4
-17
-16
-t5
-14
-13
-12
-II
-10
-8
-7
-6
-5
-4
B iMeW
Fig. 6. CafenIated .E:”and X- binding energies and widths of the If UBS in the m&ear core of 4*Ga for ZN interaction u~~ger R = 0.8 fm (solid shape) and R = 1 fm (dashed shapes). The meaning of the shapes is the same as in fig. 3.
r
12 I----_7 II
LLdUL__.L -8
-7
-6
-5 -4 B (Me’/1
-3
-2
C.J. Batty et al. / Narrow
If the real part of the optical the quenching
of the imaginary
potential
Z hypernuclear
361
states
is completely
quenched
part is kept to a realistic
extent
(PR= l), while (pi = 0.3), the lp
Xc0 UBS disappear for A = 12, 16. The real part of the potential now has an attractive pocket centered at the nuclear surface with a depth of onIy about 30% the depth of the pre-quenched potential and is capable of binding only the 1s UBS which occurs in the continuum. For A = 16 this situation is depicted in fig. 8, and
13
I
I
I
a
;;c*
12
II
x0
Is
IO Pauli
Quenching
: @=l.O+ i 0.3
I
I
-4
-3
-2
I
-1
0
B (MeV)
Fig. 8. 1” binding energies and widths of the 1s UBS for A = 16 and 41, with the real and imaginary parts of the optical potential quenched according to (2.3) with p = 1 and 0.3 respectively. The value of ci is taken from table 4 and the meaning of the shape for A = 16 is the same as in fig. 3.
the results (not shown) for A = 12 are very similar. Although under these extreme conditions the 1s UBS can become relatively narrow in light nuclei, such a conclusion does not hold for the heavier nuclei as shown for example by the Ca point in the figure. There, the 1s Z;” width decreases from 27 MeV for p = 0 down to 21 MeV for /3n = 0, pi = 0.3 and down further to 12.5 MeV for PR = 1, pi = 0.3. The effect of Pauli quenching on the imaginary part of the optical potential for the .X single-particle spectrum of UBS in a heavier nucleus, say ‘$Ca, is shown in fig. 9. It is clear that the If UBS, which is quite high in the continuum, is the one most likely to show up as a narrow excitation.
362
Pouli
Quenching
1
IP
Id i
Fig. 9. The spectrum of 6” UBS in the nuclear core of 4”Ca calculated with the imaginary part of the optical potential quenched according to (2.3) for values of p as marked in the figure.
4. Conclusions We have demonstrated in the present work that the observation of narrow z’ hypernuclear states in (K-, a-) reactions on light nuclei is consistent with the available X- atomic data, provided the optical potential is constructed by folding a finite-range ;T;N interaction into the nuclear matter distribution. For ranges 1.0~ R c 1.2 fm the lp E UBS in the A = 12, 16 hypernuclei are calculated to be quite narrow, only a few MeV wide, in accordance with the recent experiment ‘). Wycech Edal. *‘) studied the scope and validity of the folding assumption for Catoms within a YN separable-interaction model, finding the resulting local representation of the first order optical potential to be valid to within 10%. The range recommended by these authors? corresponds to R = 1.33 fm, in our notation. Here we claim that the validity of folding for the imaginary part W(r) of the optical potential is assured by extending the arguments of ref. 14) according to which the relatively large momentum released in the E-p -An conversion justifies the use of semi-classical considerations, Thus, W(r)-$E~-p-t,~,p(r),
(4.1)
where p(r) is the folded nuclear density (normalized to A) and the bar refers to nuclear-medium corrections, e.g. Fermi averaging, binding and Pauli corrections, to the zero-energy X- conversion cross section. Fermi averaging of the measured cross sections turns out to be the most important correction, yielding ‘) a value ’ The misprints
in eqs. @a, b) of ref. “1 were corrected
in eqs. (8) and (9) of ref. 9).
C.J. Batty et ni. / Narrow X hypernuclear states vl+=
363
14 mb. Comparing (4.1) with (1.1) for large A, the following connection arises: ii3 = 8rraI/p.ZN,
(4.2)
leading to a value ar==0.15 fm consistent with the atomic fit presented in table 3 for ranges Rrc 1.2 fm. We note that this explanation of the notable small value for the imaginary part of the “effective” .EN scattering length appears to be rather different from that offered in ref. 17). Similar arguments, however, cannot be advanced for the case of the real part of the optical potential. The non-locality modifications are probably more significant and its calculation remains uncertain ‘*17),The E- atomic data alone do not provide, at present, a unique determination of the effective _XN scattering length 6, the fitted values being strongly correlated with the range R (fig. 1). For a local gaussian ZN + /lN conversion potential with rms radius (r’>&&, the rms radius used in the folding procedure (as described in sect. 2) is given by Ri = (f(r2L)i’*
,
(4.3)
fortuitously similar to the relationship holding within a separable interaction model 9,*7) for both real and imaginary parts of the optical potential provided (r”>!& stands for the rms radius of the separable gaussian form factors. The factor 2 in (4.3) arises because the conversion potential appears bitinearly in the construction of the imaginary part of the optical potential, once going out of the E-channel and the other back there from the n-channels. Pauli quenching of hypernuclear widths has also been considered in the present work, using the phenomenological representation (2.3). We conclude that the real part of the E optical potential cannot be fully Pauli-quenched in the nuclear interior since the resulting potential, refitted to the atomic data, is too shallow then to produce lp E UBS in the A = 12 and 16 hypernuclei, contrary to the situation suggested by experiment *>. Although in these cases the 1s z‘ UBS are calculated to lie just in the continuum with widths as small as 7 MeV, this relative narrowness of the Is UBS does not persist in heavier hypernuclei (fig. 8). Furthermore, we calculate only a moderate reduction in 2; hypernuclear widths for realistic Pauli quenching of the imaginary part of the optical potential. In our calculation, the quenching effect on the 1s UBS is not much larger than that on the lp UBS in light hypernuclei, contrary to the conclusions of ref. lo). It should be recalled, however, that a relatively large equivalent [in the sense of refs. ‘*17)].ZN interaction range R = 3”‘//3 (= 1.64 fm), which apparently enhances the effects of the Pauli principle, was employed in that work. In addition, the extent of the approximations in the calculation remains unclear, inasmuch as the density dependence was apparently considered only through the choice of considerably smaller values for the Fermi momentum k~ for lp states than for 1s states in these light hypernuclei. As for heavier hypernuclei, fig. 9 provides a useful overview of Pauli effects in our approach.
364
C.J. Batty et al. / Narrou: 2’ ~ypernuc~ear stufes
Within the framework of the present hypernudear calculation, narrow X UBS do appear in the continuum. However, UBS that are calculated to be narrow for a given mass range sink, as the mass number A is increased, into the potential well thus becoming wider and are occasionally replaced by new narrow UBS in the continuum. In the nuclear p-shell, the arena of present 2 hypernuclear experimentation, the potentially narrow X UBS are the lp ones (fig. 3), whereas the 1s E UBS are generally wide (fig. 4). In the nuclear sd-shell, the Id and 2s UBS are likely to appear narrow at the lower part of the shell (fig. 5), gradually becoming wider as the If UBS shows up as a narrow state at the end of the shell (fig. 6). We have already commented in ref. “) on the distinction between E UBS, which our calculations of lp states in the A = 12, 16 hypernuclei yield in the &continuum, and Z: Gamow states, which for real potentials signify resonances in the continuum (with wavefunctions exploding at infinity), but which for moderately strongly absorptive (complex) potentials emerge with slightly positive binding. In the absence of a model reaction calculation designed to determine which of the two alternative descriptions of .I$“states” is meaningful for (K-, 7~)excitations, the best circumstantial evidence in favour of UBS remains the lowest X hypernuclear excitation observed ‘) on 9Be. The obvious interpretation “) in this case is a lp ,X0 with a single-particle energy 8* 1 MeV in the continuum. In the present calculation,
6
Fig. 10. Cafcuiated
-9
-8
-7 -6 B (MeV)
-5
IL--l
I
I
I
-10
-4
-3
1” binding energy and width of the lp UBS in the nuclear core of ‘Be, assuming zero-range ZN interaction. The shape used is the same as in fig. 3.
C.J. Batty et al. f Narrow E hypernuclear states
365
employing fitted values of d (table 3) and density shapes (table 5) appropriate to the variation in R, we find a narrow ,X0 lp UBS only for R = 0 at 7 MeV in the continuum as depicted in fig. 10, and in nice agreement with experiment. The lp Gamow state appears in this calculation near E = 0, quite far from the observed excitation. We remark that the failure to obtain lp UBS in the 8Be core for R - 1 fm is not particularly disturbing since, as noted in sect. 3, our choice of nuclear densities ignored A-’ effects. The clearest interpretation of the most recent experiments ‘) probably holds for the “C(K-, 7~~) reaction since only one Z charge state and clue lp j-component is to be generally expected. Indeed, one clear signal is observed ‘) at 3 MeV in the continuum and tentatively assigned to the Z- lp orbit. Calculations similar to those reported in sect. 3 yield a lp ,Y’ UBS at B = -3 MeV, for R = 0.8 and 1.0 fm, with widths about 8 and 3 MeV, respectively. If Pauli quenching is superposed on the parametrization of the imaginary part of the optical potential for R = 0.8 fm, the binding changes between -3 to -2 MeV as /3i is increased from 0 to 0.5. The width decreases then from 7 to 4 MeV. Thus, in terms of UBS, our calculations reproduce the energy of the observed ‘) narrow excitation in a most natural way. On the other hand, the X lp “resonance” pole is calculated to be at B = 2.5 MeV and with r = 0.4 and 1.0 MeV for R = 0.8 and 1.0 fm, respectively. References 1) R. Bertini et al., Production of X hypernuclei, in Proc. Int. Conf. on hypernuclear and kaon physics, Heidelberg, 1982, ed. B. Povh, MPI H-1982-V20, p. 1 2) R. Bertini er al., Phys. Lett. 90B (19801 375 3) H. Piekarz et al., Phys. Lett. 1lOB (1982) 428 4) C.J. Batty, Phys. Lett. 87B (1979) 324 5) A. Gal and C.B. Dover, Phys, Rev. Lett. 44 (1980) 379,962(E) 6) A. Gal, G. Toker and Y. Alexander, Ann. of Phys. 13’7 (1981) 341 7) G. Backenstoss et al., Z. Phys. A273 (1975) 137 8) C.J. Batty et al., Phys. Lett. 74B (1978) 27 9) W. Stepien-Rudzka and S. Wyceeh, Nucl. Phys. A362 (1981) 349 10) J. Johnstone and A.W. Thomas, J. of Phys. G8 (1982) L105; Nucl. Phys. A392 (1983) 409 11) C.J. Batty, Nucl. Phys. A372 (1981) 433 12) C.W. de Jager, H. de Vries and C. de Vries, At. Nucl. Data Tables 14 (1974) 479 13) C.J. Batty, Nucl. Phys. A372 (1981) 418 14) A. Gal, in Proc. Int. Conf. on hypernuclear and kaon physics, Heidelberg 1982, ed. B. Povh, MPI H-1982-V20, p. 27 15) ‘J. Dabrowski and J. Rozynek, Phys. Rev. C23 (1981) 1706, and in the Proceedings quoted in ref. ‘) 16) R. Brockmann and E. Oset, Phys. Lett. 118B (1982) 33 17) S.Wycech, W. Stepien-Rudzka and J.R. Rook, Nucl. Phys. A324 (1979) 288 18) E. Friedman, H.J. Gils, H. Rebel and Z. Majka, Phys. Rev. Lett. 41 (1978) 1200