A spin-parity analysis of the Q enhancement

Nuclear Phys,cs B52 (1973) 383 -391 North-Ilolland Pubh,.hmg Compan}

A SPIN-PARITY ANALYSIS OF THE Q ENHANCEMENT G.T J O N E S I (.7 R'~. (,('neJa

Rct.mvcd 23 Novcmber 1972

Abstract ApplymgtheBerman-Jacobt,,so-stet) analvsp, to thc decay Q+-- K *01890) n +~,th subsequent K*-K+rr decay,,t ~as found that 1+,sthcdommantJP,isslgnment and, m contrast to decay-plane-normal angtilar distr,butmn studms, 2 Is strongly dlsfavoured Monte-Carlo techmqucs werc used to test the apphcabd,ty of the method

1. Introduction The Berman-Jacob two-step analysis I l] is a m e t h o d tor determining the spin and p a n t y of a resollance W]llch decays lilt() another resonance and a llon-resollant particle with subsequent decay ot tile intermediate resonance into non-resonant particles in tins paper the m e t h o d is applied to the decay O + ~ K*0(8c.~0)rr + ~K+rr where the Q was formed m the reaction K+p -+ K+prr+rr Sect 2 briefly describes the m e t h o d . Sect 3 discusses the results o f a p p l y n l g Monto-Carlo techniques to test the apphcablhty ot the m e t h o d In part,cular the tollowmg question are asked 0 ) how does the existence of the c o m p e t i n g K+p decay m o d e aftect the analysis, and (n) m vmw ot the tact that the analysis involves calculating the ratm ot t.orrelated quantltmS h o w sensltwe is the m e t h o d 9 Set.t 4 presents the results of applying the m e t h o d to data obtained trom 5 G e V / c and 10 G e V / c K+p interactions. Sect 5 summarlses the results

:I: CFRN 1 eUow from tile Umvers,ty of Birmingham

+, I /+)tit's. Sl)ttt paptt~ ol the Q

384

2. What does the Berman-Jacob two-step analy,,i.,, ~ y ? ('onslder the rea
4/l(O,dp,O',~5')cos2~ a ' s m 2 O d q d g 2 ' _ ___

fl(O,c~,O',(p')(3

5 cos 2 0 ' ) (3 cos 2 0

_=e(

1) df~d~2'

I)]/(]+1) 3 1(1 + 1)

K~+(K" )

/

by

x r

Ilg. 1

Defines angles 0,@,0 0

(2 1)

(; T Jones. Sptn-partO ~d the Q

385

where e Is the relative parity ot the Q and the K*,] is tile spin of tire Q and where d~2 and d ~ ' are the elements ot sohd angle for the unprinled and primed systems, respectively The light hand side ot this relation, henceforth called the BJ-ratlo. takes different values for different spin-parity assignments for the Q, m particular, l t i s + 2 t o r 1+ a n d - 2fol 2 spmparit) assigments ~ An experimental estimate of the BJ-rallo is obtained by evaluating BJ ratio =

---4 (cos2~' sin20)-

- --,

(2 2)

((3--5cos20') (3cos20 - 1)) where (J( O,(9,0',¢')) replesents the avelage over all events ot the functionf(O,q,O',q,) In deriving eq (2.2) it is assumed that the (,) Is m a detlnlte spin panty state and that it decays vta K*zr Under these condltmns the test is independent of tire density matrix of the Q (l e the production process) and of such tlungs as phase space factors and Brelt-Wlgner's for the Q and the K*. Perhaps it is worth recalhng that there Is evidence that even the 1+ part of the Q enhancement has a comphcated structure For example, apart from its well-known producmm vta a diffractive process, a low mass Kn'n"enhancement ot JP = l + has been seen m p-p anmhdations at rest [2] This analysis Is sensmve to the total amount of Q(JP = 1+) ~ K*(890)rr, irrespective of the production process In passing, It is worth mentioning that Chung [3] has developed a fomaahsm tor determining JP values of resonances which, like tile BJ-test, mw)lves taking tile ratio of IOmt "moments", see [4]

3. Investigation of the BJ-method by Monte-Carlo techniques 3.1. k'jJect oJ the presence oJ the competing Q--*K+p decay mode Tile Monte-Carlo phase-space program FOWL was used to generate two-step decays of Q's. JP values of 1+ and 2 were investigated. It was assumed that the I + case decayed by pure s-wave whde 2 - went by pure p-wave "I he matrix elements were respectively M(1 +) = s m 0 s m 0 ' c o s q ~ ' + c o s 0 c o s 0 , M(2 ) = 3 c o s 0 ( s m 0 s m 0 ' c o s q ' + c o s 0 c o s 0 ' ) - c o s 0 ' These were derwed by considering tile decay of Q's which are mmally in IJM) states I 10 ) and 120), respectwely. Since tile BJ-analysis Is independent of the density matrix of tire Q, this procedure IS justified

Maxmmm hkehhood fits t o the Q l)ahtz plot are consistent with these angul,,r momentum states being dominant [5]

386

(, I Ione~ Spm-palttr o] the Q

The t o l l o w m g tests were carried o u t w i t h large n u m b e r s o f g e n e i a t e d events: (a) | ' o r tile 1+ case, Q --, K * n + events were g e n e r a t e d a n d the BJ ratio was calculated. (b) I:-or the 1+ case, Q --, K+p e v e n t s were g e n e r a t e d a n d those tor w l u c h the K+n effective mass was in the range ( 0 . 8 4 0 . 9 4 ) G c V were analysed as ff they were Q ~ K * n + events. Tlus was d o n e to see to w h a t e x t e n t the presence o f an a d n u x t u r e ol Q--, K+p e v e n t s in an e x p e r i m e n t a l sample c o u l d be e x p e c t e d to attect the analysis. (c) Test ( a t was r e p e a t e d for the 2 - case (d) Test ( b ) was r e p e a t e d for the 2-- case Table 1 summarlse:, tile results I able 1 I I'

Test

Numerator

Denominator

average

dVCrd2e

Number of event',

BJ-ratio

I heory

1+

a b

-0 350 -0 558

-0 172 -0 362

2(1 ()00 42 447

2 035 1 542

2.0

2

t. d

-0453 -0714

0 227 0525

54 400 42403

-1 993 -1 359

-2 0

St) b e a r i n g m n u n d t h a t m a p p l y i n g the BJ test one assumes that tile Q has one d e l m l t e , l P value a n d that It decays vta K* ( 8 9 0 ) n -- one call see t h a t , m the h m l t o f large statistics, the presence ot an i n c o h e r e n t t a d m i x t u r e o f Q ~ K,o w o u l d n o t affect the a b l h t y o f the BJ m e t h o d to decide b e t w e e n 1+ a n d 2 a s s i g n m e n t s for tile JP value ot the Q However, for data samples t h a t one r e @ i t r e a s o n a b l y e x p e c t m one b u b b l e c h a m b e r e x p e r n n e n t , the errors t m g h t turn o u t to be t o o big for the test to bc ~tgmflcant. T h e p i e s e n c e ot some Q --* Kp decay -- b) r e d u c i n g the difference b e t w e e n the 1+ and 2- values of the BJ ratio w o u l d m a k e things even worse. In the n e x t s u b s e c t i o n the p r o b l e m ol errors IS discussed m detail 3 2 S t u d y ot the errors

As far as m r o r d e t e r m i n a t i o n s are c o n c e r n e d , the dlfllculty is that tile n u m e r a t o r a n d d e n o m m a t o l of tile BJ ratio are c o r r e l a t e d . T o o b t a i n a " f e e l " Ior the errors revolved, the cllrretll value of the BJ iatlo was p r i n t e d o u t tor tile first five t h o u s a n d events m test (a). S o m e o f the e x t r e m e values are gwen m the s e c o n d c o l u m n of table 2. The first c o l u m n c o n t a i n s the c o r r e s p o n d m g e v e n t n u m b e r T h e c o r r e s p o n d n l g ratios for a s e c o n d run ot tile M o n t e - C a r l o p m g l a m , differing from the first only in the s t a r t i n g value ot the r a n d o l n n u m b e r g e n e r a t o r , are given in the t h i r d c o l u m n . J- At 5 (_;eV/( and 10 (;cV/c, maxtmunl llkehhood llts to the O Dahtz plot yield a contnbutmn ol only a low percent to coherent mterfercnce between Kp and K*rr decay modes 151

G T Jones.Spm-partt.)o] the Q

387

Table 2 I vent number

BJ rauo for run I

BJ ratio for run 2

11 12 18 27 36 43 52 61 81 94 99 103

347 255 1.19 0 83 1 79 2 43 1 92 2 46 1 86 2,62 1 96 2,34 1 83 2 64 2 68 2 09 2 26 2 19 2 42 2 44

65 66 39 57 83 66 48 68 58 O2 02 2.24 2 10 176

20

20

116

201 236 380 561 659 957 999 (theory)

1 8(1

182 1 66 1 74 1.83 1 83

This procedure was t a m e d out for about 20 starting values ot tile random number generator, the two examples q u o t e d being selected because they showed particularly large variations In the value of BJ-ratlo it was tound that for samples of one to two hundred events (roughly what one has in tile experlnaental situation) the value o f the BJ-ratio for pure Q --* K*n decay should lie m the range ( I 3 3.0) wlule for the samples of between 50 and I00 events all the valves d e t e r m m e d were m the range(l.O 3.5). As far as these tests were concerned, the B J-ratios revolved m tests (b), (c) and (d) behaved similarly. These flndmgs are consistent with the result obtained t'rom applying the theory o f correlated errors. D e n o t i n g N determinations of the two quantities x I and x 2 by withk= 1,2andl = 1,2.. N the variance, var, of the ratio of these t w o q u a n t l t l e s is given [6] by

Xkt

var

x(_~2) L2(Xl){ var(xl) -

-

E2 (x2)

-

k,'2(x I )

where the e x p e c t a t i o n values

v a r ( x 2) +

.

.

L 2 ( x 2)

E(Xk) and

2cov(xl,x2) .

}

.

t:'(x I ) l ' ( x 2 )

the variances vat

(Xk) are

defined by

(,

388

I Jones. Spin parity

of

the Q

,~,

l:(X k ) = , ~ _ J XAt t=l

•

2

var(x/,)=7,.,l 1 z~l Ix/, ,

/:(x,)1

,

and where 1

cov (x I ,x 2) = ~ ; ~

~

Ix~, - tz (x I )1 1.,% - I,.' tx~ )l

t=l

defines tile covarlance ot v I and x 2 Tile Monte-Carlo p l o g l a m was lun for Q ~ K*lr with various staltmg values for tile r a n d o m - n u m b e r generator, and the c u r r e n t values of the t\)lk)wmg q u a n u t l e s of interest were pirated tile event number, the average value of the d e n o t m n a t o r of the B J-ratio, the B J-ratio and tile error on the BJ-latlo The error fluctuated qmte markedly from event to event but the long term trend was a steady drop In table 3 a lew results, c o r i e s p o n d l n g to e x t r e m e values ot the error are given for one run, followed by the results for every five-hundredth event Table 3 D e n o l n l r l d t o r dVerdge

BJ-ratio

Frror on BJ-ratio

100

-0 2 6 6

114

368 65O 449 583 606 880 899 2 O76 I 903

0 556 0 695

158 170 205 257 281 343 425

-0.232 -0 249 -0221 -0 260 4) 211 -0 221 -0 185 -0 216

500 1000 1500

-0 209 -0 193 4) 187

2 009 2 107 2 151

0 396 0 316 0 271

[ vent ftumbcr

0 511

0583 0 439 0 53O 0 487 0 56O 0 389

R c m e n l b e r m g that tile errors q u o t e d ale e x t r e m e values, it is reasonable to suppose that a typical " s t a n d a r d " error for about 200 events is o f the order ol 0.4 F r o m this ~t is leasonable to clam1 that tile B J-test call dlstmgtush between 1+ and 2 - asslgmnents for the spin-parity o f the Q tor samples as sm',dl as tile experunental ones For the reasons discussed earher, tile presence of s o n l e Q --* Kp conq)etulg with the Q ~ K*zr would not appreciably affect tins result. l:mally, the sensitwlt~ o f tile l~J-method to the presence of random background was tested by applying the test to 32 000 events generated according to phase space.

389

(; T Join's, &~m-parttv o} the Q

This was done for many dltferent starting values for the random generator. The value o f BJ-rano did not tend to any particular value, and tlus Is m h a r m o n y with the fact that the average values of the n u m e r a t o r and d e n o m i n a t o r of the ratio were veiy much smaller than those o b t a m e d from an analys,s ot Q's of defm,te spln-par,ty As an example, table 4 shows the results o f two long rt, ns Table 4 Run I

Nuiner,itor average

Denominator average

Number of events

}}J-ratio

1 2

0 02 -0 002

0 009 0 005

32000 32000

22 -0 4

Since these averages are small, the add,tlon of one event at any stage can 'alter the B J-ratio considerably. Consequently the errors are very large. it seems reasonable to speculate that. ,f the B J - m e t h o d y,elds an answer wh,ch clearly dlstmgu,shes between 1+ and 2 ass,gnments for the Q, then there can be very httle random background present. This ,s so because the effect o f random background (see table 4) Is to reduce the absolute value of both n u m e r a t o r and d e n o m i n ator averages, m a k m g it possible for one event to lntluence the ratio more - mid hence to increase the error.

4. Results of applying the BJ-metuod to tl,e decay Q -+ K*n ~ Knn Tile data used c o m e from exposures of tile CIzRN 2 m hydrogen bubble chamber to if-separated 5 G e V / c and 10 G e V / c K + beams. At 5 G e V / c the n u m b e r o f fits to the reactmn K+p --+ pK+rr+n was 6 719 while at 10 GeV/b there were 14 235 fits For those events which had more than one fit - arising from the mdlstmgu,sab,llty of fast K's and rr's m a bubble chamber the fits with the highest probablhty were selected. In order to obtain as pure a sample as possJble o f Q -+ K*n events, only those events for winch the K+rr - effecnve mass was m the range ( 0 . 8 4 - 0 . 9 4 GeV) were selected, and also an " A -cut" was made which rejected events for which the pn + effectwe mass was less than 1 5 GeV. Deflmng the Q region to be that for winch the K+rr+rr -- effectwe mass is m the interval ( I . 2 1.5 GeV), the cuts left 689 and 1623 events m tlus region for the 5 G e V / c and 10 G e V / c experiments respectively. The value of the BJ-ratm was calculated for varlot, s mass reg,ons m the Q as well as for " c o n t r o l regions" on either s,de. Table 5 summanses the results. The d o m i n a n t feature o f the results is that, after selecting the e f t e c u v e mass o f the K+rr system to correspond to that of the K*(890), the data strongly prefer a spm-panty ass,gnment o f 1+ to one o f 2 - for the Q

~, T. Jones, Sptn parttv o] the Q

390 Table 5 Incident momentum CGeV/c) _

I

Knn mass range (GeV)

Number of events

l)enomlnator average

BJ-ratm

t'rror

114-1.2 1 2 -1 3 1.3 -1 4 1.4 -I 5 1.14-1 2

94 279 235 175 44 248

-045 -(I 24 -0.31 -0 02 -0 41 -0 35

2~12 2 09 2 29 ~ 09 -0 78 2 69

083 0 97 0 78 56.96 0 93 0 80

1.2 -I .3

597

-0 40

1.98

(l 37

I 3 -1.4 1 4 -1 5 1 5 -I 56

603 422 104

-0 22 -0 18 -0.t) l

3 67 1.86 0.17

1.14 I 00 19 30

I

1.5 -1.56

10

I

_

_

F u r t h e r m o r e , one sees t h a t as the K + n + n - effective mass increases there is a tendency 0 ) for the average values o f the n u m e r a t o r aud d e n o m i n a t o r to decrease, a n d (11) for the e r r o r s to increase. This decrease in the sensitivity o f the test is n o t surprisnlg m view o f the fact t h a t one e x p e c t s tae K* ( 1 4 2 0 ) to begin to have an e t f e c t ( b e c a u s e It is a n e w s p i n - p a n t y state and the B J-lest assumes the presence o f only one JP state, d e c a y m g vta one particular m o d e ) as the K + n + n - mass increases. As discussed m sect 3, the a p p e a r ance o f m o r e r a n d o m b a c k g r o u n d w o u l d have a Sunllar effect. C o n v e r s e l y , the fact t h a t the test is fmrly s e n s m v e m the lower Q region suggests t h a t the data s~unple is especially clean m that region

5. C o n c l u s i o , s (1) M o n t e - ( ' a r l o i n v e s t i g a t i o n s suggest t h a t , even if the Q decays to some e x t e n t vta KO, the B J-test s h o u l d ( t o t e x p e r m a e n t a l l y a t t a i n a b l e data samples) be capable o f d i s t i n g u i s h i n g b e t w e e n 1+ and 2 - asslgments fol the JP ot the Q. (n) A p p h c a t u m ot the test to Q's p r o d u c e d m the r e a c t i o n K+p --+ p K + n + n - at 5 a n d 10 G e V / c suggests t h a t when one selects the K + n effectwe luass to c o r r e s p o n d to t h a t o f the K* ( 8 9 0 ) , one o b t a i n s a r e m a r k a b l y pure sample o f 1+ for the lower part o f the Q, w h e r e a s , at lugher mass, there seems to be m o r e c o n t m n l n a t i o n m the f o r m o f r a n d o m b a c k g r o u n d or ot o t h e r JP states Tile a u d l o r is grateful to Drs V T COCCOnl, D.C Colley, J.D. l l a n s e n , M. J a c o b , G. M c C a u l e y , D.R O. M o r n s o n a n d K. ZalewsM for helpful dlscusstons a n d suggestions lle also thaJlks the o p e r a t n l g crew o f the 2 0 0 cm CERN h y d r o g e n b u b b l e c h a m b e r , and the s c a n m n g aud m e a s u n n g s t a f f o f those l a b o r a t o r i e s w h i c h processcd the fihn.

O. F. Jones. Spin-parity o] the Q

References [11 121 [31 141 [5 [ 161

SM B e r m a n a n d M Jacob, S k A C 4 3 (19651 A A s t l e r e t a l , N u c l Phys B10(1969) 65 S U C h u n g , ( ' E R N 7 1 8(1971~ I1.1t Bmgham et al, Nucl Ph~,s. B48 11972) 589 K W J Barnharn ct al , Nut3 Phys. B25 (1970) 49. Kendall and Stuart, Advanced theory of statistic,,, vol. 1 p. 232

391