LETTF-,P~ AL NUOVO CIMENTO
K~
~
VOL. 14, N. 12
22 N o v e m b r e 1975
Threshold Effect and 7rn Interaction. A Phenomenologieal Approach.
]~. BRENZIKOFER, M. B O G D A N S K I and E. J E A N N E T I n s t i t u t de P h y s i q u e de l' Universit~ - Neuchdtel, Suisse V. VUILLEMIN
I n s t i t u t de Physique nucl~aire de l' Universit~ - ~ausan~e, Suisse (ricevuto il 7 0 t t o b r e 1975)
A t h r e s h o l d e n h a n c e m e n t in t h e K 10 K 01 m a s s s p e c t r u m h a s b e e n o b s e r v e d in a large n u m b e r of e x p e r i m e n t s . T h i s effect is seen in t h e following r e a c t i o n s : I) II) III)
p p - ~ K ~ K ~ m ~ (m = 2, 3) (1.4), ~ : - p - ~ K ~ K ~ (5-7), K-JT->K~~
(8).
Several i n t e r p r e t a t i o n s h a v e b e e n g i v e n to e x p l a i n t h e s h a p e of t h i s effect. W e can m e n t i o n t h e a p p r o a c h e s b y a K~K ~ f i n a l - s t a t e i n t e r a c t i o n or an a b o v e t h r e s h o l d resonance. In this letter we propose a phenomenological model which predicts both the shape a n d t h e c r o s s - s e c t i o n v a l u e s of t h i s e n h a n c e m e n t . T h e p r o p o s e d m e c h a n i s m is b a s e d
(l) V . VUILLEMIN: Th~se, Uaiversitb de Lausanne (to be published). (a) l~I. _A_GUILAR-BEI%[ITEZ, Z. BARLOW, L. D. JACOBS, P. MALECKI, L. I%/[ONTAI%TET,CH. D'ANDLAU, A . ASTIER, J . COHEN-GANOUNA, M. DELLA I~EGRA a n d B. L6RSTAD: Phys. Leit., 2 9 B , 241 (1969); J . DUBOC, • . GOLDBERG, B. MAKOWSKI, A . M. TOUCHARD, R . A . DONALD, D. 1~. EDWARDS, J . GALLETLY
and N. WEST: NUC~. Phys., 46 B, 429 (1972); B. G. LORSTAD: Th~se, Universit6 de Paris (Orsay, 1969); J. E. GALLETLY: Thesis, University of Liverpool (1971). (8) J . BADIER, A. B O ~ E T , B. DRI~VILLON, M. l~AUBILLIER, ~R. GEORGE a n d M. RIVOAL: Nucl. Phys., 22B, 512 (1976). (a) R. BRENZIKOFER: Thi)se, Universit~ de NeuchAtel (1975); GLASGOW-LIVlr CH~kTEL-PARIS V I COLLABORATION. (*) R . I . HESS, O. I. DAHL, L. M. HARDY, J . KIRZ a n d D. H . MILLER: Phys. Rev. Left., 17, 1109 (1966). (6) W . BEUSCH, W . E. FISCHER, B. GOBBI, M. PEPn%r, E. POLGAR, P . ASTBURY, G. BRAUTTI, G. FINOCCHIARO, J . C. LASSALLE, A. MICHELINI, • . M. TERWILLIGER, D. WERSDALE a n d C. H . WEST: Phys. Left., 2 5 B , 357 (1967). (7) D . J . CRE~WEL, G. lc~. KALBFLEISCH, K . W u LAI, ft. M. SCARR, T. G. SCtIUI~IA1WN, I. O. SKILLICORN and M. S. WEBSTER: Phys. Re~). Lett., 16, 1025 (1966). (s) J . ALITTI, V. E . BARNES, D. J . CRE1WNEL, E. FLAMINIO, M. GOLDBERG, U. KARSHON, ]~. WU LAI, W . J . METZGER, J. S. O'NEALL, i~. P . SAMIOS, J . 1~. SCARR a n d T. G. SCHUMANN: Phys. Rev. Left.,21, 1705 (1968).
439
440
R . BRF.NZlKOFER, M. B O G D A N S K I , E . J E A N N E T a n d
v . WUILLF.Mllq
on the reaction IV)
~+~-~ KK
occurring as an I = 0, S-wave final-state interaction, as shown in fig. 1. ~
K
-
P
0
1
rn~
-
~
K
_1
o
_
f'N~
_~ - K
o1
P
F i g . 1. - D i a g r a m s i l l u s t r a t i n g t h e proposed K~K~~ thresho14 e f f ~ t m e c h a n i s m .
The u~ interaction shows a strong inelasticity from Kg, threshold up to about 1.25 GeV. We assume that this inelasticity is entirely due to reaction IV). I n particular the channels r:~:-->4~ and = ~ - + ~ are neglected. We write the reaction crosssection for IV) in a plane-wave approximation as 4 ~ ~.
where q is the m o m e n t u m of the pions in the r:7: system and */L the modulus of the elasticity parameter in a state of angular m o m e n t u m L. As ~ r , - 1 for L ~ 0 in the above-defined ~7: mass region (~), we are left with 4~
ae=F(1-~). I n order to compute the K~ ~ threshold effect in I) to I I I ) , we consider the following reactions : Ia) IIa) IIIa)
~p-+rz-T:+m~ (m = 2, 3), n-p-~=-~:+n, K-jW-~-7:+Y ,
in which IV) occurs as a final-state interaction. We do not consider here the pions produced b y p, f, g decays; for the other pions a phase-space distribution ~0' is assumed. The shape of the K K mass spectrum produced by this process is given by dN dmKK
_ 1--,/9 d~o' q*
dm,~,~
I n fig. 2 the full line shows the resulting K K mass distribution calculated for reaction I) with 7:r: data from (9). The dashed line indicates the pure phase-space K K mass distribution for reaction I). Let us mention that the shape of the threshold peak is mainly due to the 1 --V~ parametrization and that some minor differences may appear if we introduce other == data.
(,) B. HYAMS, C. JONES, P 9 WEILHAMMER, W . BLU~I, H . DIETL, G. GRAYER, W . KOCH, E . LORENZ, O. L~'TJEI~S, W . M~IWN'ER, .]-. MEISSBURGER, W . OCHS, U. STIERLI~ a n d F . WAGNER: Nucl. Phys., 6 4 B , 134 (1973).
O 0 T H R E S H O L D ~ F F ~ C T AND ~ K~K~
441
INTERACTION ~TC.
T h e c o m p a r i s o n of t h e calculated distributions w i t h e x p e r i m e n t a l K~K~ o o mass spect r a (i.s) shows t h a t t h e d i s t r i b u t i o n s can well be fitted w i t h such a m o d e l ; in p a r t i c u l a r t h e w i d t h of t h e e n h a n c e m e n t is c o r r e c t l y reproduced.
ck / f - U / \ 50 ~ev
1.1
1,0
1.2
1:3
F i g . 2. - K,K,oo m a s s d i s t r i b u t i o n of t h e t h r e s h o l d effect c a l c u l a t e d f o r ~ p - ~ KxK,~O * +~- a t 1.8 GoV/o (full line); t h e d a s h e d l i n e i ~ d i o a t e s t h e s h a p e of p u r e p h a s e - s p a c e d i s t r i b u t i o n .
T h e cross-section aT~ for t h e t h r e s h o l d effect is g i v e n b y 1.2~
f
1 ~ ~ ~.
a(~+~: - - ~ K K )
d~0~
din..
0.99
T h e i n d e x i runs o v e r t h e subchannels of reactions Ia), I I a ) or I I I a ) , characterized b y t h e p r o d u c t i o n cross-sections ~i. The e x p e r i m e n t a l a~ concern reactions in which t h e final-state i n t e r a c t i o n =r:-~=7: occurs instead of I V ) : =T:-->KK. s is t h e n u m b e r of u n c o r r e l a t e d =+~- pairs in subchannel i. F o r e x a m p l e in the reaction ~p-->r:+=+~:-=-, _~r1 : 4 for t h e s u b c h a n n e l w i t h four u n e o r r e l a t e d pious, hre : 1 for ::+~-po, / V 3 : 0 for pops, etc. 4~ 27r a(~+7:--~7:+~ -) : atot-- aa, w h e r e a~ot = ~
(PL + 1)(1 - - ~7~cos 2 ~ ) .
W e l i m i t t h e sum o v e r L to t h e first three terms. ~0~ is the L o r e n t z - i n v a r i a n t phase space r e l a t e d to subchannel i. The factor 88 is t h e r a t i o a ( T : r : - ~ K ~ F o r t h e calculation of aTE, only cross-sections a~ for reactions Ia) and I I a ) are available f r o m e x p e r i m e n t s (IO.li). T h e case of reactions I I I ) and I I I a ) is n o t clear as t h e
(,0) R . R . BURNS, P. E. COP~rDON,J . DO:brAHUE, ]~r A . I~ANDELKERN a n d J . SCItULTZ: Nucl. Phys., 27 B, 109 (1971); i t . A. DONALD, D. N. EDWARDS, R . S. MOORE, E. J . C. READ, S. REUCROFT, A . G. FRODESEN, T. JACOBSEN, S. SIRE, O. SKJEGGESTAD, P. SAETRE, H . TOFTE, 2~. BETTINI, S. LIMENTANI, IJ. PERUZZO, 1%. SXNTA~OE],O a n d S. SAR~ORI: Nucl. Phys., 6 B, 174 (1968); J . DAVIDSON, J . W . CHAPMAN, i t . W . GREEN, J . LYS a n d B. P . ROE: Phys. Rcv, 19, O, 77 (1974); Z. MINe MA, P. S. E A S ~ A N , B. Y. O ~ , D. L. PARKER, G. A . SMITH a n d R . J . S P R ~ A : Nucl. Phys., 51 B, 77 (1973); J . C I t Y , O N , t ). MASOI'~, H . MUIRHEAD, D. WALDREN, i t . RIGOPOULOS, P. TSILIMIGRAS a n d A. VAYAKI-SERAFIMIDOU: IVucl. Phys., 22 B, 85 (1970 ); A. ACCENSI, V. ALLES-BORELLI, ]~. FRENCH, 2~k.FRISK, J . M. HOWIE, W . ]~RISCHER, L . I~IICHEJDA, W . G. I~/~OORHEAD,B. W . POWELL, P. SEYBOTH a n d F. VILLEMOES : Phys. Left. ,20, 557 (1966). (11) D. H . MILLER, L. GUTAY, P . B. JOHNSON, F. J . LOEFFLER, R . L. ~r R . J . SPRAFKA a n d It. B. W I L L Y L r Phys. Rev., 153, 1423 (1967); R . L . EISNER, 1~ B. JOHNSON, P. It. KLEI~, R . E. PETERS,
442
R . BR~NZIKOF]~R, M. B O G D A N S K I , F,. JEAlffNET a n d
v . VUILLEMIlq
symbol Y represents an unanalysed missing mass (s). The rcr~data come from the energyindependent fits of HYA~S etal. (~). The results are presented in fig. 3 and 4 for reactions I) and II). The cross-sections a s predicted by our model are plotted as open circles. The full line is simply given to guide the eye. The error on these points, mainly due to the evaluation of ~SV,a~, m ay be estimated to 20%. In the same figures we have reported the experimental cross-sections, 0 0 defined as K1K~ threshold effect in (~-7).
102 -
101
o 3
~1~0
:I
vw101
::L ---= b~-10 o
b~
o 100
Fig. 3.
i
0.6
t
i
1
i
i
2
J
,
4
i
i
i
6 8
10-~
L ~ 'l'o
2'o
Fig. 4.
F i g . 3 . - E ~ K I t h r e s h o l d effect c r o s s - s e c t i o n s f o r p p - - ~ K [ K ~ + ~ -. data, + experimental cross-sections from (r,).
o c a l c u l a t e d p o i n t s f r o m pp--~4~
F i g . 4. - E~K~ t h r e s h o l d effect c r o s s - s e c t i o n s f o r Tc-p--~ K ~ K I n . o c a l c u l a t e d p o i n t s f r o m 7v-p * - ~ + ~ - n , + e x p e r i m e n t a l c r o s s - s e c t i o n s f r o m (~'~).
I t can be seen that the values of aTE are suitably reproduced on a wide m o m e n t u m range, for both reactions ~p~(K~ - and r~-p-+(K~K~ Let us point out 0 0 t h a t no normalization to the experimental K~K 1 data has been introduced.
R . J . SAHNI, W . L. YElV a n d G. W . TXUTFEST: Phys. Roy., 164, 1699 (1967); B. u OH, A . F. GARFrNKEL, 1%. MORSE, W . D. W~LKER, J . D. PRENTICE, E. C. WEST a n d T. S. YOON: Phys. Rev. D, 1, 2494 (1970); J . F. ALLARD, J . HENNESSY, R . H u s o ~ r , A. LLORET, J . SIX a n d J . J . VEILLET: Nuovo ~imento, 5 0 A , 106 (1967); G. BELLnCZ, B. DXUGERAS, D. FOUR~CZER, J . HEXNESSY, A . LLOEET, H . J . LUBATTL J . SIX, J . J . VEILLET, M. DI CORATO, E. FIOR127I, K:. I~IoRIYASU, P . NEGRI, M. ROLLIER, H . H . BII~GHAM, B. EQUER, C. W . FARWELL a n d W . B. FRETTER: N~ovo 0imento, 53 A, 798 (1968); J . BALLA~I, 6). B . CIL4_DWICK, Z. G. T. GUIEAGOSSIAN, W . B. JOHNSON, D. W . G. S. LEITZt a n d E . MORIYASU: Phys. LetL., 3 1 B , 489 (1970).
