Medium effects in K-nuclear interactions

NUCLEAR PHYSICS A Nuclear Physics A639 (1998) 485~492~

ELSEVIER

Medium effects in K-nuclear interactions A. Gal” aRacah Institute

of Physics, The Hebrew University,

Jerusalem

91904, Israel

A self-consistent derivation of K+ nucleus integral cross sections from transmission experiments at pi = 500 - 700 MeV/c is reviewed, including a table of updated total and reaction cross section values. The density dependence of the imaginary part of the self-consistent K-nucleus optical potential suggests that major in-medium reaction channels exist which are unaccounted for in dense nuclear media by the multiple-scattering V,, = tp optical potential and its straightforward, relatively minor modifications for kaon interactions. This observation is supported also by analyzing the recently measured K+ elastic scattering differential cross sections on 6Li and C, and it is not in conflict with the inelastic angular distributions measured in the same experiment. Previously proposed medium effects which are briefly recalled cannot be reconciled with the new data. 1. INTRODUCTION The low-energy interaction of K and K mesons with baryons is given in the chiral limit, where these mesons are perceived as Goldstone bosons of the spontaneously-broken sum x sum chiral symmetry for massless quark &CD, by the Tomozawa-Weinberg to the K* nuclear optical potential term [l] which gives rise to the tp approximation

Here, f = 93 MeV is the pseudoscalar meson decay constant and p = (i%y4N) is the vector density p = pP + p,, for symmetric nuclear matter. At the nuclear saturation density, po = 0.17fmm3, the depth of this potential is about ~t57 MeV. The inclusion of higher-order terms in the chiral perturbation expansion leads in most derivations, e.g. Refs. [2,3], to an additional attraction, associated with the KN sigma term. In order to gauge its significance, one may compare Eq. (1) to the threshold value of the tp optical potential for K+ mesons, which are expected to interact relatively weakly with nuclei:

v&y=-E(l+F)bp.

(2)

Here, M is the nucleon mass and b N -0.255 fm is the value of the isoscalar KN s-wave scattering length [4]. The depth of this potential in nuclear matter is about 33 MeV, leaving room for about -24 MeV attraction due to the KN sigma term. 0375-9474/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PI1 SO375-9474(98)003 15-7

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A. Gal/Nuclear

Physics A639 (1998) 485c-492~

The relative weakness of the K-nucleus interaction makes it particularly suitable for studying medium effects on the KN interaction. In this talk I will discuss departures from the VGt = tp form as deduced by fitting K+-nucleus total and reaction cross sections [5,6], and elastic scattering angular distributions [7] in the momentum range 500 - 700 MeV/c. As was argued in Refs. [6,8], the derived medium effects do not stem from any microscopic mechanism discussed so far in the literature. 2. INTEGRAL

CROSS

SECTIONS

Hadron-nucleus integral cross sections are derived in transmission experiments from the measured attenuation cross section ~~,tt(fi) for removal of the beam hadrons from a detector subtending a solid angle Q at the target:

u,(Q) = 642)

-&)&da ’

n =

u,ti (a) + i 0

0

lfN12 dfi -

ilfcl’-dfi 2 TRe [fNfJ dfi cl

,

(4

R

where the values of the reaction (on) and total (or) cross sections are obtained by extrapolating Eqs. (3, 4), respectively, to R = 0. The nuclear fN and the Coulomb fc elastic scattering amplitudes cannot be eliminated in Eq. (4) f or or(R) in favor of the measurable elastic differential cross section. It is clear that even if these elastic cross sections were experimentally known, so that on could be extracted reliably, the extraction of or for charged hadrons is always model dependent, since fN is never available experimentally. Traditionally, an optical potential is assumed for the purpose of extrapolation, even though it is not guaranteed that a particularly chosen V,, yields by calculation the same values of on and or which eventually are derived by using it throughout the extrapolation of the data. In this sense, most existing derivations of total cross sections in the literature do not satisfy the basic requirement of self-consistency. The question of self-consistency, posed for Vopt, is whether or not optical potentials which are constructed empirically to fit oR and or values derived from transmission experiments lead to the same values, within errors, when used in reanalyzing these same experiments. Friedman, Gal and Mares [8] have recently discussed this issue, showing that self-consistency for K+-nucleus cross sections can be achieved by using a density-dependent optical potential Vopt = &ff(p)p with Im t,ff(~) depending linearly on the average density in excess of a threshold value. In order to demonstrate the scope of the self-consistency problem we show in Fig. 1 ratios of ‘experimental’ to calculated cross sections [8], for calculations based on densityindependent V& = t,ffp fits to the 6Li cross sections. The ‘experimental’ cross sections were derived [5], from the attenuation cross sections measured in transmission experiments [9] at the AGS for K+ on 6Li, C, Si and Ca at 488, 531, 656 and 714 MeV/c, using the tp

487~

A. Gal/Nuclear Physics A639 (1998) 48.5-492~

1.3

Si

Si

C

C

Ca %

1.0 i_ 400.0

500.0

600.0 lab. momentum

f

Ca

J

700.0

000.0

(MeV/c)

Figure 1. Ratios between experimental and calculated K+ nucleus integral based on fits to 6Li. Full squares represent (TR, open circles represent aT.

cross sections

potential. This procedure, however, suffers from a lack of self-consistency since the cross sections calculated from the tp or, according to Fig. 1, even those calculated from the t,ff~ potential, fall considerably short of the corresponding ‘experimental’ cross sections. The choice of the relatively dilute nucleus 6Li for a reference calculation is similar to the use of the deuteron in Ref. [9] for discussing ffT values. It was demonstrated in Ref. [8] that, in the present incoming momentum range and for a KN interaction strength as weak as here encountered, using an optical potential to describe scattering off as light a nucleus as “Li incurs errors of no larger than 2%. With respect to the free-space KN of 5 - 15% with scattering amplitude, the values obtained for Im t,ff show enhancement increasing momentum, in quantitative agreement with the estimates of meson-exchange current effects [lo]. The values obtained for Re t,ff, however, show a considerably larger departure. The KN in-medium t,ff is substantially less repulsive than the free-space t. It was pointed out in Ref. [8] that such a trend might be expected partly due to the proximity of the K*N and KA channels, but no quantitative estimates of this effect have been made. The ratios in Fig. 1 deviate considerably from the value of one in a way which is largely independent of the K+ beam momentum, but clearly depends on the target nucleus. Resolving phenomenologically these deviations, it was shown in Ref. [8] that very good self-consistent fits to the data could be obtained if the imaginary part of t,ff was made density dependent according to the empirical expression:

tdd

= Rekff + iIm teff[l + P(F- m$‘(ir - m)]

,

p=;

/

p2dr

,

(5)

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A. Gal/Nuclear Physics A639 (1998) 4&k-492~

where p is the average nuclear density. Values of p can be obtained from HF calculations or from the simpler single-particle calculations. It was found that the values j3 = (13.0 f 3.4) fm3

,

pth = (0.088 i 0.004) fmm3

:

(6)

provide excellent self-consistent fits at all four momenta for which transmission measurements on ‘jLi, C, Si and Ca were made at the AGS [9]. Attempting to avoid the use of a threshold density pth in Eq. (5) by using powers of the average or local density p(r) failed to produce fits to the data. The value of the threshold density (Eq. (6)) is considerably larger than the average density of ‘Li (0.049), while comfortably smaller than p of the other target nuclei, which are more than a factor of two denser than “Li. This reflects the failure to reconcile the ‘Li data with the data for the denser nuclei, unless the specific density dependence given by Eq. (5) is assumed. This density dependence, for the parameters of Eq. (6), amounts to 20 - 30% enhancement, depending on the nucleus. Table 1 summarizes the reaction and total cross sections obtained self-consistently, as described above [6], at. the four beam momenta. The uncertainties are the same as quoted in the earlier publication [5], reflecting statistical errors only. Systematic uncertainties are associated mostly with the optical model input which is inherent in the analysis [11,5]. These were estimated to be as large as 5-10%. Indeed it can be seen that the results in Table 1 differ from those derived earlier in Ref. [5] using V& = tp, and shown in parentheses, by about 5%, always exceeding these older values. For Re V,, it was not necessary to introduce as complex a density dependence as discussed above for Im V,,. In fact, Re t,jf could be kept independent of density, becoming progressively less repulsive with energy, so that Re V,, M 0 for pL N 700 MeV/c. It was also possible to respect the low-density limit, with a repulsive tp term of about 25 MeV,

Table 1 Integral cross sections (in mb) for K+ interaction with various nuclei from self-consistent analysis of transmission measurements. The values in parentheses are taken from a selfinconsistent analysis using V,t = tp. reaction total Si Ca 6Li p(MeV/c) bLi C C Si Ca 488 67.8 128.4 276.2 362.5 77.5 165.4 373.7 503.2 (65.0) (120.4) (265.5) (349.9) (76.6) (162.4) (366.5) (494.6) f 1.3 5 2.3 * 5.1 Zt 7.7 Zt 1.1 f 1.9 & 4.8 f- 7.7 531 73.2 136.8 299.1 384.0 80.7 168.9 391.7 521.6 (69.8) (129.3) (280.4) (367.1) (78.8) (166.6) (374.8) (500.2) It 0.8 f 1.4 f 3.4 Zt 4.5 f0.7 Ztl.3 Zt 3.3 f 4.4 656 79.0 148.2 311.8 408.6 86.4 179.5 403.2 548.8 (75.6) (141.8) (306.1) (401.1) (84.3) (174.9) (396.1) (531.9) f 1.1 * 1.5 & 3.4 Zt 5.0 & 0.7 f 0.8 •t 2.7 Z!Z4.2 714 82.2 152.8 320.2 417.1 88.5 183.8 411.3 550.4 (79.3) (149.3) (317.5) (412.9) (87.0) (175.6) (396.5) (528.4) * 1.2 f 1.5 f 3.6 Zt 5.5 f 0.6 f 0.9 f 2.3 f 2.8

A. Gal/Nuclear

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Physics A639 (1998) 485c-492~

by adding an attractive term depending on a higher power of the density overall Re V&t again becomes close to zero for pi N 700 MeV/c. 3. ELASTIC

AND

INELASTIC

such that the

SCATTERING

Differential cross sections for elastic scattering of K+, recently reported [7] for 6Li and C at pL = 715 MeV/c, were also studied. Whereas these data definitely require a significant in order to achieve reasonably amount of density dependence beyond the tp approximation good fits, the resultant potentials are incompatible with potentials obtained from fits to integral cross sections only. When fits to the combined data set at 714 MeV/c, consisting of both differential and integral cross sections, are made, Re V,, is weakly repulsive within the nucleus, turning into an attractive pocket at the surface which, however, violates the low-density limit. Examples for C are shown in Fig. 2, taken from Ref. [6]. The continuous

20.0

,

t

1 T

K+ C 714 MeVie

15.0

F

10.0

E C 1-

5.0 SF+ cc

0.0

1 .o

I 3.0

2.0

4.0

I 5.0

r (fm)

Figure 2. Real part for the K+-carbon potential differential cross sections (see text for notation).

at 714 MeV/c from fits to integral

and

curves are for unconstrained density dependence (DD) potentials and the dashed curves are for t,ffp potentials. Also shown are the uncertainties, obtained from a notch test [12], for the unconstrained DD potential baaed on macroscopic (MAC) densities. Replacing the MAC densities by single-particle (SP) densities does not change the overall picture, although there is a difference in the detailed shape of these potentials. As for the fitted imaginary part of the optical potential, the specific density dependence of Eq. (5) is upheld also by this extended set of data. The calculated differential cross sections are shown in Fig. 3, taken from Ref. [8].

A. Gal/Nuclear Physics A639 (1998) 485c-492~

49oc

K+ 715 MeV/c

-4.0

’ 0.0

10.0

Figure 3. Fits to differential and C.

20.0 cm. angle

30.0 (deg.)

cross sections for elastic scattering

of 715 MeV/c K+ by 6Li

New inelastic data for 6Li and C at 635 and 715 MeV/c have also been reported [7]. These inelastic excitations also require theoretical enhancement in order to fit the data. Even though the corresponding transition densities peak at the nuclear surface where p < &h, one cannot simply argue that these inelastic processes need not be strongly renormalized by the medium effect specified here, which is a global, not a local prescription, since it involves p, not p(r). Of course, more work is needed to decide whether or not these inelastic cross sections shed new light on the problem of medium effects in K+-nucleus interactions. 4.

MEDIUM

EFFECTS

A brief discussion of several theoretical works is in order. Siege1 et al. [13] suggested that nucleon ‘swelling’ in the medium primarily affects the dominant Sii KN phase shift by increasing it in the nuclear medium. In the momentum range 500 - 700 MeV/c this amounts to decreasing (the negative) Im t, as required by the data, but increasing (the positive) Re t contrary to the trend suggested by fitting to the data, of decreasing it to become less repulsive. A similar remark holds against the mechanism of dropping vectormeson masses in the nuclear medium suggested by Brown et al. [14]. The meson-exchangecurrent effect considered in Refs. [15,10] is capable of producing the decrease necessary for Im t in order to fit the integral cross sections as function of energy, but it suggests no significant change in Re t. Finally, the most recent work by Caillon and Labarsouque [16], who consider medium effects for the mesons exchanged between the K+ and the bound nucleons, produces both modifications required in order to fit the data, namely decreasing Im t and Re t simultaneously to an extent which is comparable at 700 MeV/c

A. Gal/Nuclear Physics A639 (1998) 485c-492~

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with the values of t,ff found in Refs. [6,8]. However, these authors, as well as other works, are unable to produce the major medium effect which, in the phenomenological approach, is expressed by modifying Im t,ff by the [l +P(P-~th)@-~th)] factor. In this sense, no satisfactory theoretical approach yet exists to describe K+-nucleus interactions at intermediate energies. We point out two features which make it even harder for theory to explain the integral cross sections data: (i) the self-consistent values of or [6] are always larger, by up to 5% (see Table l), than the previous values [9] used in some of the theoretical works; (ii) there exist now o~ data [5,6,8], which indicate a similar problem for theory as the ar data do. This appears to suggest that the theoretically missing part of the cross section belongs to some in-medium major reaction channels mistreated by the tp optical model approach and its conventional, relatively minor modifications [17]; or that at present theory misses some unconventional in-medium effects that would strongly invalidate the tp starting point. ACKNOWLEDGEMENTS I would like to acknowledge a fruitful collaboration with E. Friedman and J. Mares. This research was supported by the U.S.-Israel Binational Science Foundation. REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. i2. 13. 14. 15. 16.

17.

Y. Tomozawa, N. Cim. 46A (1966) 707; S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. C.H. Lee, G.E. Brown, D.P. Min and M. Rho, Nucl. Phys. A 585 (1995) 401. N. Kaiser, P.B. Siegel and W. Weise, Nucl. Phys. A 594 (1995) 325. T. Barnes and E.S. Swanson, Phys. Rev. C 49 (1994) 1166. E. Friedman, A. Gal, R. Weiss et al., Phys. Rev. C 55 (1997) 1304. E. Friedman, A. Gal and J. Mares, Nucl. Phys. A 625 (1997) 272. C. Michael et al., Phys. Lett. B 382 (1996) 29; R.E. Chrien, R. Sawafta, R.J. Peterson, R.A. Michael and E.V. Hungerford, Nucl. Phys. A 625 (1997) 251. E. Friedman, A. Gal and J. Mares, Phys. Lett. B 396 (1997) 21. R. Weiss et al., Phys. Rev. C 49 (1994) 2569 and references cited therein. C. Garcia-Recio, J. Nieves and E. Oset, Phys. Rev. C 51 (1995) 237. M. Arima and K. Masutani, Phys. Rev. C 47 (1993) 1325. C.J. Batty, E. Friedman and A. Gal, Phys. Reports 287 (1997) 385. P.B. Siegel, W.B. Kaufmann and W.R. Gibbs, Phys. Rev. C 31 (1985) 2184. G.E. Brown, C.B. Dover, P.B. Siegel and W. Weise, Phys. Rev. Lett. 60 (1988) 2723. M.F. Jiang and D.S. Koltun, Phys. Rev. C 46 (1992) 2462. J.C. Caillon and J. Labarsouque, Phys. Rev. C 45 (1992) 2503; Phys. Lett. B 295 (1992) 21; Nucl. Phys. A 572 (1994) 649; A 589 (1995) 609; Phys. Rev. C 53 (1996) 1993. C.M. Chen and D.J. Ernst, Phys. Rev. C 45 (1992) 2011; M.F. Jiang, D.J. Ernst and C.M. Chen, Phys. Rev. C 51 (1995) 857.