Observable sensitivities to scaling of K+-nucleon interactions within nuclei

NUCLEAR PHYSICS ELSEVIER

A

Nuclear Physics A 625 (1997) 261-271

Observable sensitivities to scaling of K+-nucleon interactions within nuclei R.J. P e t e r s o n a, A . A . E b r a h i m b, H . C . B h a n g c a Nuclear Physics Laboratory, University of Colorado, Boulder, CO 80309-0446, USA b Department of Physics, Assiut University, Assiut 71516, Egypt c Department of Physics, Seoul National University, Seoul 151-742, Korea Received 25 March 1997; revised 25 June 1997; accepted 1 July 1997

Abstract We have used an impulse approximation optical model to compute a range of K+-nucleus observables, considering how these vary with assumptions about alterations of the K+-nucleon amplitudes within complex nuclei. The observable elastic, inelastic, total and reaction cross sections are approximately linearly dependent upon the interior K+-nucleon amplitudes, not cross sections. Similar comparisons are made for quasi-elastic scattering, computing the number of nucleons contributing incoherently to the reaction through an eikonal model incorporating enhanced cross sections. All measured observables are consistent with enhanced in-medium K+-nucleon amplitudes. (~) 1997 Elsevier Science B.V. PACS: 25.80.Nv Keywords: K+-nucleus reactions; Optical model calculations; Medium modifications

1. Introduction Data for a number of observables for interactions of K + mesons with complex nuclei have been found to exceed the predictions of what should be very reliable theories. This has been shown to be quite independent of the specific model for the reaction and general enough that the excess has been attributed to a fundamental increase in the scattering probability of the K + from individual nucleons within complex nuclei, either by enhanced cross sections [ 1-4], by an increased range of the meson-nucleon interaction [ 5,6], or through access to internal meson exchange currents [ 7]. Since the small and almost entirely elastic cross section with nucleons that the K + enjoys up to a beam momentum of about 750 M e V / c allows this projectile to be the most penetrating of strongly interacting probes [ 8], it is unique in being able to sense strong interactions 0375-9474/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 5 - 9 4 7 4 ( 9 6 ) 0 0 3 8 1-3

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

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at hadronic densities near those of nuclear matter, making the alleged medium effect very significant. There are even hints that such medium alterations might be densitydependent, based on the example of 6Li, which has less than half the charge density of carbon and other heavier nuclei [9]. These conclusions on medium enhancements have arisen from case-by-case comparisons of data with calculations, such as those in Refs. [10-13]. We here provide a more general study within simple reaction models of the sensitivities of all common K+-nucleus observables to modifications of the interactions between K + and nucleons within nuclei.

2. Methods

2.1. Optical model methods We have used the great similarity between the pi and K + mesons to modify, without restructuring, an impulse approximation optical model code. Both mesons are pseudoscalar, and this limits their coupling with nucleons in the same way. We began with the distorted wave impulse approximation code DOPI [ 14,15] which uses the mesonnucleon coupling to derive a meson-nucleus optical potential of the form 2

4-

2~Un(X~In, = [A~p + A2~" pX~ + A3~ P]~m

(1)

together with a Coulomb potential, and solves the Klein-Gordon equation. Here, ~ =

E/hc with E the total center of mass pion energy, p is the nucleon distribution and coefficients A are determined through the impulse approximation. This model is limited to s- and p-waves in the meson-nucleon coupling. For pions across the resonance the L = 1 term is dominant, but the L = 0 term will be most important for our range of K + momenta. We modified the meson masses in the new code DOKAY, and changed the isospin couplings to the forms appropriate for interactions of isospin 1/2 K + mesons with nucleons. DOPI used a parameterization of the pi-nucleon coupling that was used anew for K+-nucleon scattering. Our K+-nucleon phase shifts were taken to be of the form tan 6e/q ~2~+1) = b + cq 2 + dq 4,

(2)

with q the meson center-of-mass momentum. The coefficients were obtained by fitting these forms to the SP92 solution of the phase shift compilation SAID [ 16]. We took these phase shifts to be purely real, and limited our coverage to meson lab momenta from 100 to 750 MeV/c. The K+-nucleon scattering begins to have a strong inelastic component above this momentum. Fig. 1 compares the phase shifts from SAID, obtained from an organized fitting process to the world data set, with the values from our fitting. The agreement is very good, especially for the important S11 amplitude. Phase shifts for all other partial waves were set to zero. As an example, Fig. 2 shows the K+-proton total scattering cross sections obtained from our parameterization compared to experimental

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

263

%

30 L

(a)

S : ~ / /

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loaf

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--l_

t 200

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600

t~l~b [MeV/c] Fig. 1. Phase shifts in degrees for K+-N elastic scattering are shown for the present parameterization by the solid lines, and from the dala represented in SAID [ 16] by the dashed curves.

results [ 17]. Note the dominance of the s-wave scattering over the momentum range we cover. Our parameterization is fully adequate to account for free-space K+-nucleon scattering, and the coefficients from Eq. (2) are listed in Table 1. DOPI and thus DOKAY also allow us to compute non-spin inelastic scattering cross sections. We used the simple derivative coupling appropriate to first-order collective vibrational or rotational excitations of the target nuclei. Comparisons of these calculations to data are shown in Ref. [18]. Distributions of neutrons and protons within target nuclei were taken to be the same, and determined by unfolding the proton size from measured charge distributions [9]. For 6Li and lZc we used the distributions also used by Friedman et al. [12]. Our parameters are listed in Table 2. Externally read scaling factors fR and fI allowed us to vary separately the real and imaginary parts of all K+-nucleon amplitudes together in each calculation. We are then able to use DOKAY 1:o compute differential elastic scattering cross sections, inelastic scattering cross sections to sharply resolved states, total cross sections and reaction cross sections, each as a function of the scaling factors. For simple comparison to a very simple model appropriate to scattering of other strongly interacting particles, consider how these observables might vary with changes in the size R of a black disk target [ 19]. Total cross sections would vary as R 2, while

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

264

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Fig. 2. Total K + - p r o t o n cross sections c o m p u t e d f r o m the present parameterization are s h o w n for L = 0 only a n d for both L = 0 a n d 1. D a t a are f r o m Ref. [ 17]. O u r fit w a s restricted to m o m e n t a b e l o w 7 5 0 M e V / c . Table 1 Parameters b,c,d in p o w e r s o f ( M e V / c ) for Eq. ( 2 ) . The uncertainties ( b e l o w the values) were obtained a s s u m i n g a u n i f o r m 10% uncertainty in the p h a s e shift being fit, a n d are s h o w n only to indicate the relative uncertainty f o r e a c h partial w a v e Channel S01 SII P01 P03 PII P13 D03 D05 DI3 DI5

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x x x x x x x x x x

10 - 8 10 - 1 ° 10 - 9 10 -11 10 - 1 4 10 -15 10 -13 10 -15 10 -13 10 -15

6.408447 3.4 -6.170316 2.3 -3.693214 1.8 -4.162344 3.1 -4.790802 3.8

× x x × x x × x x x

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x x × x × x × x x x

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-6.598389 2.0 -3.555722 2.5 -8.338051 9.3 -3.428781 4.0 2.477331 1.6

x × × × × x × × × ×

10 - 1 4 10 - 1 5 10 -18 10 -19 10 - 1 9 10 - 2 ° 10 - 2 0 10 -21 10 -19 10 -20

1.948645 1.0 1.121378 9.8 2.354998 2.9 1.721274 2.2 -7.647003 6.5

x × × × x x × x x x

10 - 1 9 10 - 2 0 10 - 2 3 10 -25 10 - 2 4 10 -25 10 -25 10 - 2 6 10 -25 10 - 2 6

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

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Table 2 Parameters to describe the nucleon distributions for our samples are listed, together with several measures described in the text to estimate the radii reached by the K+ beams. A scale factor f = 1.0 was used for those calculations

Model c (fro) z w RR (fro) RI (fm) Aeff RQ (fm) p(RQ)/p(O)

6Li

C

HO 1.77 0.327 0 1.57 1.58 4.08 1.62 0,55

HO 1.516 2,234 0 1.86 2.09 5.54 2.04 0.63

Ca 3pF 3.787 0.485 (fro) -0.139 3.08 3.59 12.9 3.62 0.49

Pb 2pF 6.583 0.506 (fin) 0 6.10 6.76 25.6 6.76 0,43

forward diffraction elastic scattering would vary as R 4. Diffraction minima would occur at angles dependent upon R -1. Inelastic cross sections, with a fixed nuclear deformation /3, would change as R 4. We will examine the changes on K ÷ observables with these dependencies in mind. Two measures o f the nuclear radius and density at which the scattering is most likely may be determined fi'om the scattering matrix elements generated in the solutions to the optical model scattering. These are the partial wave number of the maximum in the imaginary part o f the matrix and the partial wave where the real matrix element is half the central value. These are the same for the case of scattering from a black disk. We have converted these partial waves L for fR = fI = 1.0, with interpolation, to radii using R R and RI for the real and imaginary parts by R = L / k for 715 M e V / c scattering, and included these radii in Table 2 to indicate where the scattering is most important. Our results from these coherent scattering processes will be shown in Section 3.

2.2. Quasi-elastic scattering At momentum transfers above twice the Fermi momentum the K + projectile scatters incoherently from target nucleons, with energy losses near o~ = q2/2m. Since nucleons shield one another, only Aeff nucleons are sensed by this quasi-elastic scattering. We have used the eikonal methods widely used for quasi-elastic scattering to estimate Aeff at 705 M e V / c , similar to the study for pions [ 17]. We used the same nucleon distributions sampled in the optical model calculation, from Table 2, The K+-nucleon cross sections were changed, by the squares of fR and fl in comparison to the optical model method, in an eikonal model to compute Aeff. The quasi-elastic scattering differential cross section would then be given by do~(Q

E

) = f

2A

doeff~(free),

(3)

with the weighted average o f elastic differential neutron and proton cross sections for each target at the laboratory angle where the quasi-elastic scattering is observed and

266

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

f = f R = fI. This eikonal calculation also allowed us to estimate the density to which the K + projectile penetrates. We integrated the density out from some radius RQ to match the computed Aeff sensed by the K + probe. These radii and the nucleon density there are listed in Table 2 for a scale factor f of unity for 705 M e V / c beam. Quasi-elastic scattering of K + at this momentum occurs outwards from where the nucleon density is about one half of the central density. The radii for this scattering are near those estimated from the optical model, as seen in Table 2. Differential quasi-elastic scattering cross sections derived from this process will be shown in Section 3, where they will be compared in some cases to data obtained by integrating measured doubly differential cross sections [21].

3. Results We used a single set of nucleon distributions and a consistent set of scaled K +nucleon amplitudes or cross sections to generate observables such as those measured to investigate the effects of the scaling factors read into the optical model code to scale the amplitudes, or used as f2 to scale the cross sections for the incoherent eikonal calculation. Computed reaction and total K+-nucleus cross sections are nearly independent of alterations in the real part of the in-medium K+-nucleon interaction. A 20% change in both the real and imaginary parts of that interaction yields a 31% (35%) change in the formal elastic (2 + inelastic) calculation. Changing only the real amplitude by 20% changes the observable cross sections by 14% (18%), while an increase by 20% of only the imaginary part enhances these cross sections by 14% (22%). The angular dependence of elastic cross sections varying the real and imaginary enhancements independently is bounded by the predictions changing both together, showing little change due to interferences among these amplitudes. Calculations and discussions below use f = fR = fI, recognizing that some enhancements with fR are much less than with fI. Differential elastic scattering data of K + from carbon and 6Li [ 10] are shown in Fig. 3, with the solid curve from DOKAY. The dotted curves are from calculations using the model of Ref. [2] shown in Ref. [9], and the dot-dash curve from [ 10] uses the theory of Ref. [4]. In general, all three models agree with the data in the same fashion. The 6Li data at large angles include some unresolved counts from the low-lying 3 + inelastic transition, as discussed in Ref. [18]. Also shown in Fig. 3 are curves from DOKAY using a scale factor f = 1.3 for enhanced in-medium K+-nucleon cross sections, showing excellent agreement with the data. The need for this enhancement agrees with previous analyses of these data [ 10], but requires a somewhat larger scale factor for carbon. The calculation without enhancement is below the 6Li data. The forward differential elastic cross section shows a clear sensitivity to the scale factor. In Fig. 4 we plot the 15 ° (lab.) cross sections from DOKAY as a function of f . In contrast to the expectation of a quartic dependence from a diffraction model, these

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

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Fig. 3. Data for differential elastic scattering of 715 MeV/c K + from C and 6Li Refs. [ 10,221 are compared to the optical model calculations, without "swelling", shown in that work. The dotted curve uses the model of Ref. [21 and the dot-dash curve uses the model of Ref. [4]. Calculations from DOKAY are shown by the solid curves for fR = fl = 1.0 and by the dashed curves for Ill = fl = 1.3. See Ref. 118] for a re-evaluation of the 6Li data.

cross sections are only approximately linear in f . The location of the first minimum in the 715 M e V / c K+-carbon elastic scattering is also shown in Fig. 4, and is only very weakly dependent on medium effects. By setting the charge of the meson to zero, we also were able to compute purely nuclear elastic scattering. Since there is significant nuclear- Coulomb interference in the angular range of the data, the different effects of this feature for 6Li and 12C targets could also influence the ratio of their cross sections, as shown in Refs. [ 10,18]. The computed ratio of carbon to 6Li elastic scattering with no nuclear Coulomb interference differs from that including this effect by only 4% at the smallest angle data point, and by about 3% at other angles. We conclude that conclusions on medium effects are not sensitive to this interference, and thus independent of the phase differences of K+-nucleon amplitudes in the two targets. Inelastic K + cross sections to the 4.4 MeV 2 + state of ~2C are shown in Ref. [ 18], and DWIA calculations are compared to the data there. Here we show the sensitivity of the inelastic maximum differential cross sections, near 20 °, to the common scale factor f . Results are shown in Fig. 5 at both 635 and 715 MeV/c. Data points from that work are shown, set at the point where they agree with the curves, to indicate the "swelling" they represent. This comparison indicates a need for f greater than 1.0, consistent with conclusions from the elastic data. Total and reaction cross sections from 500 to 700 MeV/c are shown in Fig. 6 [ 11,12],

268

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R.J. I'eterson et al./Nuclear Physics A 625 (1997) 261-271

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c o m p u t e d b y D O K A Y for 715 M e V / c K + on 12C

Larger enhancements are required for 1 2 C and heavier nuclei than for 6Li. Larger in-medium cross sections will shield more nucleons from the beam, so the magnitude o f quasi-elastic scattering cross sections shows two compensating effects [23] The sensitivity to the scaling factor f of the singly differential quasi-elastic cross section for 705 M e V / c beam on a range of targets at a laboratory angle of 42.6 °

270

R.J. Peterson et al./Nuclear Physics A 625 (1997) 261-271

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Fig. 8. The sensitivities of differential quasi-elastic scattering of 705 M e V / c K + at 42 °. (q = 479 MeV/c) are shown, computed from a simple eikonal model. Measurements yield 13.6(1.4), 33.7(3.6) and 69.5(7.4) m b / s r for C, Ca and Pb, respectively [21 ].

is shown in Fig. 8, with a momentum transfer of 479 MeV/c. The light targets show almost no dependence upon f . The data from Ref. [21] are 13.6(1.4), 33.7(3.6) and 69.5(7.4) mb/sr for C, Ca and Pb, respectively. These are not consistent with any of the calculations shown, and the eikonal approximation is certainly too simple to use for a quantitative analysis of these results. The ratios of observables for f = 1.3 to those for f = 1.0 at 715 MeV/c are 1.47 for 15 ° elastic scattering, 1.53 for 2 + inelastic scattering, 1.20 for reaction cross sections, 1.26 for total cross sections and 1.12 for quasi-elastic scattering. No observable for K+-nucleus scattering shows a dependence stronger than linear on the interior K +nucleon scattering amplitude. Our consistent calculations agree with other analyses of recent K+-nucleus data, with a persistent excess of data above calculations, using freespace K+-nucleon amplitudes, with 6Li perhaps anomalous because of its low density. As expected from diffraction models, forward differential elastic or inelastic scattering gives the strongest signals of K+-nucleon medium modifications.

Acknowledgements This work was supported in part by the US Department of Energy, by Assiut University and by the Korean Ministry of Education through BSRI of Seoul National University.

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