IL NUOVO CIMENTO VOL. 85A, N. 4 21 Febbraio 1985

Coherent K~-K~ Regeneration in Nuclei.

FUMIYO UC~IYA1VIA

Institute o/ Applied Physics, University o/ Tsukuba- Ibaraki 305, Japan

(ricevuto il 20 Luglio 1984)

S u m m a r y . - We reinvestigate the high-energy KL-K s regeneration in nuclei taking into account the short interaction range of the vector meson exchange reaction compared to the pion exchange reaction. Different nuclear thickness functions for regeneration process from that for elastic process are used. We show that only a few percent change in the nuclear ~ skin ~> is allowed for heavy nuclear regenerators such as 2~

PACS. 25.80. - Meson- and hyperon-induced reactions and scattering.

1 . - I n t r o d u c t i o n .

Since PAIs and PIccIoNI (1) predicted the phenomenon of K s regeneration

which was observed by Good et al. (~) in 1961, K s regeneration by a nucleon

aroused interest in particle physics in conjunction with the determinat ion of the fundamenta l weak-interaction parameters . Hence a lot of data on

proton regeneration was accumulated and it has become known through the

phenomenological analysis t ha t the co contr ibut ion dominates over tha t from @

in hydrogen regeneration in the high-energy region, 30 GeV/c ~<PK ~< 130 GeV/c (3).

For the high-energy K s regeneration process in nuclei, the authors have

(1) A. PAIS and 0. PICCIONI: Phys. Rev., 100, 1487 (1955). (2) :R. H. GOOD, :R. P. MATSON, F. MULLER, 0. PICCIONI, W. M. POWELL, m. S. WHITE, W.B. FOWL~ and R.W. BIRGE: Phys. Rev. Lett., 124, 1223 (1961). (3) G.J. BECK, S.H. ARO:NSON, K. FREUDEI~,I'REICH, A. GSt)ONER, W.R. MOLZON, J. ROEHRIG, V.L. TELEGDI, B. WINSTEIN, H.G. KOBRACK, R.E. PITT and R.A. SwAnson: Phys. Rev. Lett., 42, 350 (1979).

315

316 F U M I Y O U C H I Y A M A

invest igated the regenerat ion phases, cross-sections and angular distr ibutions in te rms of both coherent product ion model (*) and optical model (5,6). In

the former, it is shown tha t the nuclear characteristics in nuclear regenerat ion most ly show up in the angular distr ibution. The authors showed t h a t the main features of the exper imenta l da ta may be deduced f rom the e lementa ry pro- cesses bu t failed to explain the fine s tructures around the dip. In the previous analysis (s) the same effective nuclear thickness functions is used for absorpt ion of K beam and K s regeneration. Namely we ignored the effect on the nuclear wave funct ion of the fact t ha t pion cannot pat ic ipate in regenerat ion while it has a impor tan t role in absorption. Therefore, the nuclear wave funct ion for absorpt ion m a y ex tend more t han t ha t for regenerat ion.

In this work we reinvest igate K~-K s differential cross-section from Pb at 4 GeV/c by use of the coherent product ion model and s tudy the dependence on nuclear interact ion range in terms of effective nuclear thickness function. The advantage of coherent product ion model being used in this paper over

optical model is the clear-cut separat ion of hadronic effects and nuclear ef-

fects (~).

2. - Modified coherent production model .

The coherent nuclear regenerat ion ampli tude f rom nucleus A of nucleon

number A is given (4) by

(1) ik ~ fd2bd3r 1 d3r~exp[iq.b]. ]~:~,~(q) = ~ . . .

�9 [~v(r~, r~, ra, ... r~)] 2 I-~ [1--F~,(b--S~l]FRo~(b--S,) I~ [1 - - l~s(b -- S~)], ZI~Zj ~/c~Z 1

where ~ is the ta rge t wave funct ion, b is the impact pa ramete r of the incident part icle of m omen tum k, and S~ is the impact pa ramete r of the i- th nucleon (the t ransverse pa r t of r~). The F ' s are profile functions which are defined in

terms of the regenerat ion and elastic-scattering ampli tudes of nucleon b y

(2)

_ 1 I ] FRo,(b) -- 2~ik K,~'-~Ksx(q) exp [-- iqb] d~q,

1 JIfK~X-*K~x(q) exp [-- iqb] d2q. /'KL(b) -- 2stik �9

S

(~) F. UCHIYAMA: Phys. Per. D, 9, 673 (1974). (5) H. FOETH, •. HOLDER, E. RADnRMACH~R, A. STAUD]~, P. DA.RRIULAT, J. DEUTSCH, K. KLEINKNI~CHT, C. RUBBIA, K. KITT]~L, M. I. FERRERO and C. G~osso: Phys. LetS. B, 31, 544 (1970). (6) W.L. WAnG and F. UCIt~YAMA: Nucl. Phys. B, 73, 23 (1974).

COHE~E~ K~.-K s R:EG~:N]~I~ATIO:N I]q :N'UCL:EI 317

We app rox ima te the nuclear wave funct ion by neglect ing correlat ions and using a p roduc t of a single nuclear dens i ty funct ion

A (3) ]yJ(ri, r2, ..., rA)p --> y~ e(r , ) .

i = l

We have t a k e n all of Chese A nuclear dens i ty funct ions equal in our pre-

vious analysis. Here we choose the i - th 4ens i ty funct ion different f rom the rest as it represents the nuclear densi ty responsible for regenerat ion as indicated

in eq. (1). There are droplet model calculations on neu t ron and p ro ton densi ty

dis t r ibut ions in nuclei predic t ing somewhat different nuclear densi ty radius (7). However , the p r i m a r y a im of this pape r is to see the effect due to the regenera-

t ion in teract ion rang% we use one average nuclear dens i ty for each for ab-

sorpt ion and for regenera t ion .

Excep t i =/= j we choose as before

(4)

and

(5)

Q(S,, z,) = ~o[1 -~ exp [ ( r , - e)/D]]-,

~ ( ~ i ' Z~) = ~Reg(Si , Zt) o

By assuming as before t h a t the nuclear dens i ty 0z var ies more slowly in com-

par ison w i t h / ' ~ as funct ions of Sj, we obta in in the l imit of large A

(6)

where

�9 fcxp [iq. b] g(b) cxp [-- �89 (1 - - i~) aT(b)] d~b,

(7)

and

(8)

{Np~ T ( G.\

(NdX) Re ]~p(O) + (1-- NdX) Re/~dO) (NF/A) I m t~:~(0) • (1 - - Np/A) Im/KLn(0) '

T(b) : AfQ(r) dz J

g(b) : A f e~,(r ) dz,

(7) W.D. ~ 4 : ~ s and W. J. SWIAT]~CHI: Nucl. Phys. A, 336, 267 (1980); W. D. MYeRs: LBL-14310 (1982).

318 FUI~IYO UCHIYAMA

which is the ve ry new thing in this analysis. Np is the number of protons in the nucleus. The coherent regenerat ion cross-section is then

(9) dq~ - - \dq~]o (b) [ J o ( a b ) e x p [ - - ~ ( 1 - - i ~ ) a T ( b ) ] b ]

where we h~ve defined an average nucleon regenerat ion forward cross-section by

N r

The clear-cut separat ion of hadronic effects and nuclear effects is seen in eq. (9).

3. - N u m e r i c a l c a l c u l a t i o n .

Aside from the effective nuclear thickness funct ion for K s regenerat ion, g(b), which is our main concern in this paper , we m a y use the rest of parameters in eq. (9) fixed to those used in the previous analysis of ref. (6). Expl ic i t numbers

and forms are given in table I. As for the effective regenerat ion nuclear thick- ness functions g(b), we may take its short-range characteristics into account by cut t ing the tai l (large impact parameters) par t of T(b) . The remaining major pa r t of the in tegrand of eq. (9) (the real pa r t of the bracket in eq. (9)) is p lo t ted in fig. 1. I t shows tha t the contr ibut ion to the cross-section m a y be large at the ta i l pa r t of b in the range of q~ of our interest . In fig. 2, we plo t ted the dif- ferent effective nuclear thickness functions g(b) which are used to calculate

cross-sections along with T(b).

TABLE I .

a = 20.76 mb ~ 0.112

~(r) = Qo[1 + e x p [r - c]/D]-~ c = 1.2A~ fm D = 0.6 fin

There are two types of <( cut ~> we adopt for presentat ion; continuous and sharp cuts. For the continuous cut we va ry the value of D in g(b) which is

the same expression as the T(b):

r

gl(b ) = 9~f[1 -b exp [ ( r - - c)/D]] -1 dz . - - c o

C 0 H E R E ~ T K L - K s REGEI~ERATION IN" NUCLEI 3 1 9

i 0 ~

10 -~

2 q = 0 . 0 1 2 5

\ \

\ \ \ \ / I I I I I I I

I I

/ /

/

J / =0. /

/ ' f

. . //

\

k \=1.4

\

I I

I I

Ii 10 -2 I I I I I I 1

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 b ( f m )

Fig. 1. - A par t of integrand in eq. (9), Re[Jo(qbhbe-�89 aT(b)], which is the weight of the g(b) contribution to the differential KL-K s regeneration cross-sections.

F o r the sha rp cut , we use

T(b) , b ~ b~,,~

(~2) g2(b) = 0 , b > br

I n fig. 3, t h e ca lcu la te4 cross-sect ions are p lo t t ed a long wi th t he experi-

m e n t a l da ta . N o t e t h e no rma l i za t ion in t he calcula t ions is a r b i t r a r y as can be

320 FUMIYO IYCHIYAMA

6

N,.

.o 5

c

~ 3

c 2

\ \

\ \

\ \

I I I I I 1 2 3 4 5

b ( f m )

\ \

\ \

\\ g2(b)

g1(b) ~ . . . _ I 6 7 8 9

Fig. 2. - The effective nuclear thickness functions T(b) for elastic and g(b)'s for regen- eration. The effect of the shorter interaction range on the regeneration reaction is ac- counted for by the cuts at large impact parameters. The dashed line is the nuclear thickness function T(b) used for absorption. The thin solid line is type-I continuous cut, with D = 0.5 fm and the thick solid line is type-II sharp cut with ben t = 8 fm.

absorbed in the factor , eq. (10), where the neu t ron regenera t ion power (which

is predic ted to be four t imes grea ter t h a n t ha t of p ro ton b y qua rk model) is

not well fixed. As is seen f rom the exper imen ta l da ta in fig. 3 bo th the locat ion

of the m i n i m u m and the ra t io of the first to the second m a x i m u m l imit t he

amoun t of allowed cut-off of the ta i l of g(b). For the cont inuous cut gl(b) the

m i n i m u m value of D which gives the m i n i m u m posi t ion q~ut less t h a n 1.0. �9 10-~(GcV/c) ~ a t fixed value c = 1 .2A+fm is D = 0.46. As D decreases,

the m i n i m u m location shifts to the higher value of q~ and the second m a x i m u m

increases in the case of continuous cuts. I n fig. 3a), a representa t ive differential

K s regenera t ion cross-section ob ta ined a t D = 0.5 is shown. W e have also calculated cross-sections for g2(b) which have chopped tails

as indicated in fig. 2. One of the typ ica l cross-section a t ben t = 8 fm is p lo t ted

b y the dot-dashed line in fig. 3. The m i n i m u m of bc~ t which gives q~in is less t han

1.0.10 -3 (GeV/c) 2 is 7.4 fm. The dependence on the cut location b~ of the

C O H E l C E N T KL-K s REGENERATION IN N U C L E I 321

o v

w-

I

~ I ' I ' I ' I '

lo ~ \ t \

1 0 2

b

I 1 I' I ' I ' I ' I

10o 0 , i I , I , , I , I , I , I , I , 0.4- 0,8 1.2 1.6 2.0

,7 ~G~v/~)~] ~I~

Fig. 3. - The differential KL-K s regeneration cross-section for 2~ The dot-dashed and solid lines are the results of the calculation using type I with D = 0.5 fm and type ]I with bcut = 8 fm cuts, respectively. The dashed line is the result of previous analysis where g(b) = T(b) is used. The experimental points ~re from ref. (5).

min imum location qm, n2 and the second max imum is similar r the case occured

in for the continuous cut ; as ben ~ gets smaller, the location of the min imum shifts to the higher q2 and the second max imum increases. The main different

characteristics arising in differential K s cross-section due to these two different

ways of cut t ing the tail arc tha t the differential K s regeneration cross-section

climbs up slower from the min imum for the continuous cuts than the sharp

cuts, while the second max imum is higher for the sharp cut than those from

the continuous cut. This lat ter tendency agrees wi th the common knowledge

from the diffraction scattering.

4. - Discussion and concluding remarks.

We have also calculated the differential K s regeneration cross-section, da/dq% using several other types of g(b) than the two shown in the previous

section bu t the results are similar. For instance, the most drastic and un-

3 2 2 F u ~ y o UCIIIYAMA

realist ic choice we have t r ied in our analysis is: g(b) = cons tant for b < 8 fm

and g(b) = 0 for b > 8 fm. E v e n in this case the dependence of d(r/dq ~ on bcu ~ is s imilar to the case of type I I , sharp cut, aside f rom the normalizat ion.

Needless to say t h a t the two shown are the best and others like this unrealist ic

one give overal l worse fit to exper iment . I t should be reminded tha t the authors have a l ready shown in our fo rmer analysis on regenera t ion in ref. (6), the fine

s t ruc ture a round the dip can be be t t e r f i t ted b y doubling the exper imenta l ly de te rmined e bu t not by adjust ing the sizes of c and D in the nuclear dens i ty

funct ion. l~einvcst igat ing the differcncial K s regenerat ion cross-section b y Pb, we

fu r the rmore find t h a t bo th the location of the m i n i m u m and the ra t io of the

first to second m a x i m u m in da/dq 2 are sensit ive to the behaviour of the nuclear

th ickness function g(b) at large impac t pa ramete r . This fact l imits the m a x i m u m

var ia t ion in the effective nuclear thickness funct ion for regeneration~ g(b)~ f rom

the ord inary elastic nuclear thickness funct ion T(b). The to ta l effective nucleus

in teract ion range for K s regenerat ion will be effectively smaller only a round the skin as depicted in fig. 4. In g~(b), the m a x i m u m permissible shift in D

~ 7 fm

Fig. 4. - SchematicM figure of a nucleus to display the change in the effective nuclear (( radius ~) due to co exchange dominance. The solid and dotted lines are for pion and interaction range, respectively.

is 0.14 fm for the fixed half radius c t h a t amoun t s to mak ing the ta i l fall

app rox ima te ly 1/4 fm fas ter for ~0spb, which is abou t 3 % of the nuclear radius.

This is quite small compar ing to a na ive es t imat ion of the magni tude of dif- ference between the in teract ion range of K s regenerat ion and elastic scat ter ing which is ro(1 - - m=/m~) _~ I fm~ where m= and m~ are pion and co meson mass

in the reaction~ respect ively. As a conclusion of this analysis~ though the overall shape of the differential

K s regenerat ion cross-section can be reproduced b y using sl ightly different

C O H e r E n T KL-K s R:EG~E]~ERATIO:I~ IN I~UCL:EI 3 2 3

n u c l e a r t h i c k n e s s f u n c t i o n for r e g e n e r a t i o n , t h e f i n e - s t r u c t u r e (s) a r o u n d t h e d ip

in t h e d i f f e r en t i a l c ross - sec t ion is s t i l l n o t we l l r e p r o d u c e d even if d i f fe ren t

n u c l e a r t h i c k n e s s f u n c t i o n s a rc u sed for K s r e g e n e r a t i o n .

(s) F. UenlYAMA: Phys. I~ett. B (1985), to appear.

�9 R I A S S U N T 0 (*)

Si rian,~lizz~ la rigenerazione ad ~]ta encrgiu K~-K s nei nuclei prendendo in considerazione il breve raggio d ' interazione della ceazione di scambio del mesone vet tor iale in confronto all~ reazionc di scambio pionico. Sono usate funzioni di spessore nucleate per il pro- cesso di rigcnerazione diverse da quelle per il processo elastico. Si mostra che soltanto un piccolo cambiamento percentu~le nello (~ st~to ~> nucleare ~ consen~ito per rigenera- tori nuclcari pesanti come Pb.

(*) Traduzione a cura della I~edazione.

Pe3ioMe He II0Yiyqert0.