Density dependence in kaonic atoms

Physics Letters B 308 ( 1993 ) 6-10 North-Holland

P H Y S I C S LETTERS B

Density dependence in kaonic atoms E. Friedman, A. G a l Racah Instttute of Physws, The Hebrew Unlverstty, Jerusalem 91904, Israel

and

C.J. Batty Rutherford Appleton Laboratory, Chtlton, Dtdcot, Oxon 0 X l l OQX, UK Received 4 March 1993 Editor: G.F. Bertsch

We study phenomenologlcally the density dependence of the K - optical potential Vopt(r) at zero kinetic energy by fitting ~t to a comprehensive set of atomic data across the periodic table. Two famihes of solutions offer improved fits over that for the standard tcffp(r) parametenzatlon of Vopt. One family contains solutions for Vovtwith RMS radn larger than those of the matter density p, as expected from fimte-range folding corrections. The other famdy offers substantially ~mproved fits to the data and gives rise to Re Vop~of a size smaller than that ofp. The new family contains solutions for which Voptis attractive reside the nucleus and on its surface, becoming repulsive at large d~stances The repulsion can be made to approximately agree with the low-density hmlt//opt--) tp where t is the free-space/~N t matrix. It is argued that this sign flip might be expected from the underlying nuclear propagation of the A ( 1405 )

Strong interaction effects in kaonic atoms are sizeable and, therefore, invalidate the application of perturbation theory. For example, the strong-interaction shifts of the lowest level observed in kaonic atoms, deduced from the energy measured for the corresponding X-ray transitions, are generally repulstve, whilst optical potentials fitted under various assumptions to these shifts plus the corresponding width values invariably come out attracUve in the nuclear surface and perhaps in the interxor [1 ]. The same holds true when using also yield values, equivalent to extracting the width of the "upper" levels. However, outside the nucleus, the K - nuclear optical potential must become repulstve by requiring it to satisfy the low-density limit [2] Vopt-,tp(r), where t is the freespace zero-energy/~N t matrix which is dominated by the I = 0 A ( 1405 ) sub-threshold resonance, and p is the nuclear density. This kind of situation, where Re Vopt(r) apparently contains both attracUve and repulsive components, naturally suggests a pronounced density dependence of the optical potential. Such a dependence is the topic of the present letter, 6

where we report on a new type of solutions for the K nucleus optical potential at zero kinetic energy, obtained by fitting a density dependent form specified below to a comprehensive set of measured shift, width and yield data m kaonic atoms across the periodic table. For these solutions, the RMS radius of the real part of the potential is found to be smaller than that of p, contrary to how the effect of a presumed underlying finite range and nonlocality is intuitively perceived. In particular, it becomes possible to reproduce the pattern mentioned above of attraction inside and repulsion outside of the nucleus. We will also outline how this sign fl~p m~ght be expected from the dynamics ofA ( 1405 ) propagation in the nucleus. The data used in this analysis of kaonic atom experiments were taken from the published literature and represent all currently available measurements from L1 to U inclusive, with the exception of the recent Brookhaven measurements [3 ] for W, Pb and U where two of the shift measurements have positive values. These positive values do not follow the systematic trends as a function of Z which are observed

0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V All rights reserved.

Volume 308, number 1,2

PHYSICS LETTERSB

for other nuclei and cannot be reproduced with a simple optical potential when it is also required to fit data for lighter nuclei [ 4 ]. The data set used in the present work is the same as that used in an earlier conventional analysis [ 1,4 ]. They consist of 24 measurements of energy shifts, 24 direct measurements of level widths together with 17 measurements of relative yields which can be used to give width values for the "upper" levels. The measurements cover a total of 7 atomic levels, from the 2p level in Li to the 8j level in U, and 24 different nuclei. For the density dependent ( D D ) optical potential we chose the form 2/ZVopt(r) = -4to

1+

lz

b+B

p(r)

p(r),

(1) where/1 is the K - nucleus reduced mass, m is the mass of the nucleon and p(r) is the nuclear matter density distribution, p=pp+p,, where pp is the point proton density distribution obtained from the experimentally determined charge distribution [ 5 ] by unfoldlng the finite size of the proton. The proton and neutron densities, normalized to Z and N respectively, were taken as 2- or 3-parameter Fermi distributions, except for light nuclei where harmonic oscillator densities were used. For N = Z nuclei we assumed Pn =Pp but for N > Z nuclei we increased slightly (by 0.1-0.3 fro) the radial parameters ofpn to qualitatively agree with predictions of Hartree-Fock calculations. The results of the present work do not depend on the precise choice ofpn, but we note that this increase in the radial parameters of p. improved slightly the fits to the data. For B = 0 , eq. (1) reduces to the standard t¢ffp parameterizatlon of the optical potential, for which we find a ;(2 per degree of freedom xZ/F= 1.85 with ( B = 0 ) b = (0.60 + 0.05) + I (0.89_+ 0 . 0 5 ) f m .

(2)

For B e 0 one has five parameters consisting of the complex b and B plus the exponent a which are to be determined by a least squares fit to the data. Since correlations exist between the various parameters, we usually held the value of a fixed while varying the four parameters given by b and B. Repeating this procedure over a range of values for a, we assembled the various fits into famihes of solutions defined by a smooth variation of the fitted parameters as a func-

24 June 1993

tion ofo~ for a given family. By starting from as many representative combinations of initial values as possible, and iterating these to get a (perhaps local) minimum ofz 2, w e found two families of solutions with lower zE/F values than for eq. (2) above. One family, with z2/F ~- 1.70, contains solutions for Vopt with RMS radii larger than those for the matter density p, as expected from finite range folding corrections [ 6 ]. However this value ofzE/Fis only marginally smaller than that obtained with the conventional approach and the resulting optical potentials violate strongly the low density theorem which here is of prime concern. We therefore do not discuss these potentials further. Results of fits to the data are summarized in fig. 1 for the other family, displaying in the lower part (a) a reduction of the RMS radii with respect to the RMS radius of the teffp potential (shown by the arrow on the left) for a representative nucleus (Ni). The middle part (b) shows the values ofz2/Fas a function of the DD parameter a. As the fit is based on 65 data points, the reduction ofx2/Ffrom 1.85 (eq. ( 2 ) ) to 1.25 is very substantial when only 4-5 parameters are being varied. The minimum obtained is quite shallow and although it is located near a = - 0. l, one may well accept any value up to about a = 1.0 if there are good reasons for doing so (see below). The upper part (c) of the figure shows the volume integrals per nucleon of the real (continuous) and imaginary (dashed) parts of the best fit potentials, again for Ni. These quantities are found to be nearly constant as a function of a, in analogy with other hadron-nucleus systems [ 7 ], suggesting that they are meaningful integral parameters representative of the K - nucleus interaction. Arrows on the left show the corresponding values for the best fit conventional density independent teffp potential equation (2). Also shown, with superscript (0), are the rapidly varying volume Integrals obtained for the linear part (bp, cf. eq. ( 1 ) ) of the best fit potentials. The arrows on the right show the implied volume integrals for a tp potential, i.e. eq. ( 1 ) with B = 0 , b=a, and a is the free-space zero-energy/(N scattering amplitude a = - 0 . 1 5 +i0.62 f m ,

(3a)

corresponding to the I = 0, 1 amplitudes [ 8 ] ao = - 1.70+i0.68 f m ,

(3b)

Volume 308, number 1,2 i

PHYSICS LETTERS B i

i

-

15oo

i

(c)

7000

/

/

i

i

i

- -

;Irn tm

~

Re

-

>( 1 ) <

~

/Reel

/

~- .,~

'

free

Im/.. 2"-_'///~ ......

\ 5ooL-

/ Irn(o)II // -500

free

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/ RetO)

I O F

I

3 8 -

(a)

I

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rm. - ' " - "

24 June 1993

plicitly how a density d e p e n d e n c e o f the K - nucleus optical potential is instrumental in reconcding the attraction (Re ( b + B) > 0 ) needed by Re Foot inside the nucleus a n d on its surface with the repulsion (Re b < 0) required by the low density theorem. In fig. 2 we c o m p a r e the D D potential (zZ/F = 1.37 ) o f eq. ( 4 ) , satisfying the low-density limit tp, to the best-fit teffp potential (xE/F-~ 1.85) specified by eq. ( 2 ) , both drawn here again for Ni. It is clear that a substantial departure o f the D D potential shape from that o f p ( r ) occurs for the real part which falls off considerably steeper than p in the surface region a n d well beyond. Due to the power-like steep rise o f the kaon atomxc wave function outside the nucleus, fairly large distances (e.g. 7 fm for N1) also contribute significantly to the strong-interaction effects in kaonlc atoms, in spite o f the exponential decrease o f Vopi. This is the radial range where Re Vopt flips its sign in order to satisfy the low-density limit constraint. However, the precise position o f the zero o f

~ g 6

i0 s __-

/x

~34v

__L_

r

-

I

I

I

I

I

I

,__

~-

0

tp

(1

ary

Fig 1. (a) RMS radii for real (continuous) and imaginary (dashed) parts of the best fit DD potentials for NI as a function of the DD parameter or. Arrow on the left indicates the RMS radius ofp. (b) x2/Fversus o~ Arrow on the left indicates the best fit value for a te~ppotential. (c) Volume integrals versus a for real (continuous) and Imaginary (dashed) parts of the potentials for NI Arrows on the left indicate the corresponding values for a t~p potential; arrows on the right indicate the values for the free space RNtp (eq. (3a)). Re (°) and Im (°) are for the linear parts of the best fit potentials (see text).

al = 0 . 3 7 + i 0 . 6 0 f m .

(3b c o n t ' d )

It xs easily seen from the figures that for oL=0.25 one m a y use the f r e e / ( N interaction in the linear part o f the potential and that it is associated with a " c o m p r e s s i o n " o f the optical potential relatwe to p(r). Moreover, if one imposes b=a on the linear part, the resulting fit getsz2/F= 1.37 with ( b = a ) B= (1.70+0.05) +i(-0.087+0.025)

fm,

Z >

10

i

102 L

This is a perfectly acceptable fit, d e m o n s t r a t i n g ex-

tp

1

01

L

10-2

10 -3

(4)

DD

,-

~ 0

a=0.25 +0.03.

DD

I 2

~

I 4

~

I 6

[ ~, t 8

\I 10

F' (fm)

Fig 2 Best fit potentmls for NI usmg the tetrpand DD models.

Volume 308, number !,2

PHYSICS LETTERS B

Re Vop,, or its existence, are not uniquely associated with the improved fit which is attributed to the modified shape. This modified shape of the potential is found also to partly solve a long-standing problem in kaonic atoms, namely, the inability to obtain simultaneously good fits to the strong interaction effects in the lower and upper levels [4]. For example, with the best fit t~rrppotential we have Z 2 per point of 1.49 when considering only the lower levels or 2.63 when considerlng only the widths of the upper levels. However, for the best fit DD potential (at a = - 0.1 in fig. 1 ) these values are 1.07 and 1.46, respectively, which is a significant step towards solving this problem. Finally, in order to accommodate theoretically the sign flip of Re Vopt, from repulsion in the very lowdensity regime as implied by the tp limit (cf. eq. (3a)) to attraction for nuclear matter density (cf. eq. ( 4 ) ) , it is instructive to follow a schematic approach motivated by the nuclear propagation ofA ( 1405 ). This I = 0 resonance lies only 27 MeV below the K-p threshold. Its decisive role in renormahzing the K - nucleus interaction near threshold was first suggested in ref. [9] and since then has been tested in several detailed calculations for specific kaonic atoms [ 10 ]. As summarized in ref. [ 11 ], by performing closure over the nuclear excitations which accompany the A(1405) propagation in the nucleus, the resulting nonlocal optical potential in the I = 0 channel is given by

2ltV(o°)t=K2( l + ~)aop(r, r ')

exp(iKIr-r'l )

Jr-r'l (5)

_

24 June 1993

opt=--4rc

1+

×aop( ~ (r+r'))g( - i V ) a ( r - r ' ) ,

(8)

where ~ IS the Fourier transform of

g(r-r') = -

K 2 exp(iKIr-r'[) 3jl(kp [r-r'] ) 4lrtr-r' I kplr-r' [ (9)

Eq. (8) reduces to a tp local form in the limit K-.oo since then g(x) -~3(x) and hence ~(t) -~ 1. We point out that, for a finite value of K, when ~ is not a constant, the nonlocahty of Vo(°t) in eq. (8) arises from both K dependence and kp dependence o f g in eq. (9). To approximate V opt (o) eq. (8), by an equivalent lo-

ca/DD potential of the form

2"V(°)~Fp(r)=-47r(l+~)a~)(P)P( r opt ) ~ .

,

(10)

one applies [ 6,13 ] a technique initiated by Perey and Buck [ 14 ], resulting in a transcendental equation

Fp(r)=q(r)~(-iFy2(r))

[Fpp~o > q ] ,

(11)

with

q( r) - - 4 n ( l + ~)aop( r) .

(12)

To study the renormahzatlon of q(r) inside the nucleus, short of exhaustively discussing the general solunon ofeq. ( 11 ), we consider the high-density limit, kp--,~, for which ~(t)=--

3K2( k--~- 1

K2 + ~t2

) + ....

(13)

where KZ=2(m+

mx- ) (E-B-Ere~ + ½IF) .

(6)

Here, E - E ~ = 27 MeV, F is the A ( 1405 ) resonance width, and/~is a closure nucleon binding energy. The nucleon mixed density p(r, r ' ) is given for nuclear matter by [ 12 ]

p(r, r')=p(½(r+r')) 3Jl(kp Ir-r'l ) kplr-r'l '

(7)

Retaining only the leading term, we note that Fp = O (p t/3) and the following simple solution of eq. ( 11 ) is obtained:

a~(p)-

3K 2

k2 ao,

(14)

which demonstrates explicitly the sign reversal of Re V o(°) ,~(o) is neglected. Using the fully comp t i f I m Vopt plex values ofao, al quoted in eq. (3b), we plot in fig. 3 the departure of aerr(Po)-~~ t"eff r.(o) (P0) + 3al ] from a (cf. eq. (3a)), assuming that a] is not renormalized. Three representative values are chosen for -

where kp is the local Fermi momentum appropriate to the densityp. Eq. (5) may be rewritten as [6,13]

-

Volume 308, number 1,2

d

PHYSICS LETTERS B

E~2o

E ¢0

H

05

IIEIIllEIllrlEE O0 O5 10

15

24 June 1993

capable of motivating microscopically the observed density dependence. To summarize, we have introduced p h e n o m e n o logical density dependence into the K - nucleus optical potential which leads to greatly improved fits to all kaonlc atom data. The resulting potentials are found to be geometrically "compressed" with respect to the corresponding nuclear densities, and it becomes possible to satisfy the low density tp limit. Although we chose the K-matrix d e t e r m i n a t i o n [ 8 ] for this limit, it is quite possible to adopt other values [ 15] without strongly affecting (cf. fig. ( l c ) ) the conclusions of the present work.

Re aeff (fin) Fig. 3. The renormalized values aet'f(Po) of the free-space zero energyK'Nscattenng amplitude aeff(po= O) = a (eq. (3a)) due to A( 1405) propagation m the nuclear medium as gwen by eq. (14) and subsequent text Points b, c, d correspond to ( E - E , , s - B ) = 19, 0, - 1 9 MeV respectively, all with F=40 MeV Point c' evolves from c when Fls halved, and point c" ~sdue to the next order correction to c' K 2, eq. (6), viz. ( E - E r e s - B ) = 19, 0, - 19 MeV, all with F = 4 0 MeV and for p o = 0 . 1 6 fm -3 (kpo=263 M e V / c ) . The values aefr(Po), shown at the end points b, c, d of the c o n t i n u o u s lines drawn from the point a, display considerable sensitivxty to the assumed downward energy shift (/~= 8, 27, 46 MeV respectively). It is worth noting that the strongest renormalization of Re a, from a small a n d negative value to a large a n d positive value, occurs for the more conservative choice o f / ~ = 8 MeV; whereas by using a sizeable shift ( / ~ = 4 6 MeV) in order to cross below 1405 MeV, Re a becomes even more negative than it was. These results only indicate the trend which a full solution of eq. ( 11 ) should yield. Since eq. (13) is an expansion in terms of K 2 / k ~ , it is instructive to test the convergence by using a smaller value for this parameter. Thus we also show in fig. 3 ( i n dashed lines) the results of using ( E - E ~ s - / ~ ) = 0 a n d F = 20 MeV in both eq. (14), p o m t c', a n d on including the next order K2/k2o term, point c". We therefore conclude that the underlying propagation ofA ( 1405 ) is

10

References

[ 1] C.J Batty, Sov. J Part. Nucl. 13 (1982) 71 [ 2 ] C.B Dover, J Hiffnerand R.H Lemmer, Ann. Phys. (NY) 66 (1971) 248, J Hufner and C Mahaux, Ann. Phys (NY) 73 (1972) 525. [3 ] C.J. Batty et al., Phys Rev. C 40 ( 1989) 2154. [4] C.J Batty, Nucl. Phys. A 372 ( 1981 ) 418. [ 51 H De Vrles, C.W. De Jager and C. De Vnes, At Data Nucl. Data Tables 36 (1987) 495. [6 ] C J. Batty, E. Friedman, A Gal and G. Kalbermann, Nucl. Phys A 535 (1991) 548 [7 ] G W. Greenless, G J Pyle and Y C Tang, Phys Rev 17l (1968) 1115, E. Friedman, H.J Gds, H. Rebel and R. Pesl, Nucl Phys. A 363 (1981) 137 [81A.D. Martin, Nucl Phys B 179 ( 1981 ) 33. [9] W.A Bardeen and E W. Tongoe, Phys. Rev. C 3 (1971) 1785, Phys Len. B 38 (1972) 135 [ 10] M. Thles, Nucl Phys A 298 (1978) 334, R. Brockmann, W. Welse and L. Tauscher, Nucl. Phys A 308 (1978) 365. [ l l ] J M Elsenberg and D.S Koltun, Theory of meson interactions with nuclei (Wdey, New York, 1980) Ch. 5 ld, eq. (34). [ 12] J W. Negeleand D Vautherm, Phys Rev. C 5 (1972) 1472. [ 13] A Gal, Few-bodysystems, Suppl. 5 ( 1992) 290. [ 14] F. Perey and B Buck,Nucl Phys 32 (1962) 353 [ 15 ] K Tanaka and A Suzuki, Phys. Rev C 45 (1992) 2068