IL NUOVO CIMENTO

VOL. XL A, N. 2

21 Novembre 1965

An Application of the Absorption Model to the Reactions pp-~ u }=[. H/iiGAASEN a n d ~l. HOGAASEN (*)

C E R N - Geneva (ricevuto il 28 Maggie 1965)

Summary. - - We present a set of calculations of the reaction p~-~ YY in the antiproton laboratory momentum range from 3 GeV/e to 7 GeV/c. The process is assumed to take place through K* exchange only, and the calculations are performed within the framework of the absorption model (the distorted wave Born approximation). We obtain a good description of the angular dependence of the cross-section for the three ehamlels AA, AZ~ ~ and Z§ 9 in the whole energy range. However, the energy dependence is wrong, and appears the worst for the channel AA, where the angular dependence is best described.

1.

-

Introduction.

The process has in the last

a n t i p r o t o n - p r o t o n producing a pair a n t i h y p e r o n - h y p e r o n years been investigated in experimental (1.2) and theoret-

(*) On leave from Laboratoire de Physique Th~orique, Dgpartement de Physique Nucl~aire, 0rsay; supported in part by 0CDE-NAT0. (1) B. MUSGRAVE,G. PETREMAS, L. ]C~IDDIFORI),R. BOCK, ]~. FETT, B. R. FRENCII, J. B. KINSON, CH. PEYROU, M. SZEPTYCKA, J. BADIER, M. BAZIN, L. BLASKOVIC, B. EqUER, J. Hvc, S. R. BORENSTEIN, S. J. GOLDSACK, D. H~ MILLER, J. MEYER, D. ~R]~v~ru, B. TALL~NIand S. ZXLBE~AJCH: N~ovo Cimen$o, 35, 735 (1965); R. BOCK, e$ al. : YY production by 5.7 GeV/c antiprotons in hydrogen, report to the 1964 Dubna Conference, CERN/TC/Physics 64-18. ~%e corresponding data are referred in the text as (~CE RN data ~). (2) C. BALTAY, et al. : Interactions el high energy antiprotons in hydrogen, Stanford Nuclear Structure Meeting (1963); C. BALTAX, et al.: tlyperon production by 7 GeV/c antiprotons in hydrogen, report to the 1964 Dubna Con~ference. The corresponding data are referred in the text as (( Yale data ,),

I

AN APPLIC&TI{)N (}F THE ABSORPTION MODEL TO THE REACTIONS 1op--->~Y

561

ical (s.7) papers. E x p e r i m e n t s are characterized b y a strong forward peaking of the a n t i h y p e r o n relative to the direction of the incoming antiproton, indicating p r e d o m i n a n c e of higher p~rtial waves characteristic for w h a t ol~.e calls peripheral processes. Now it is well k n o w n t h a t by calculating a B o r n d i a g r a m for the process with the exchange of a K or a K* meson, one gets z c o n t r i b u t i o n f r o m low p a r t i a l waves exceeding the bounds set b y uniturity. This results m an angulai distribution for the produced particle which shows a too weak angular dependence. As, on t h e other hand, B o r n d i a g r a m s are easily calculable, a n d give a simple intuitive picture of w h a t is happening, it was t h e n n~tural to seek m e t h o d s b y which one would unitarize the B o r n a p p r o x i m a t i o n (s.9), or at least cut 4owu t h e c o n t r i b u t i o n of low p a r t i a l w~ves. The n m t h o d we stroll follow here b e a r s on this last point. Originating f r o m SOPKOWTCH (~), it was l a t e r e l a b o r a t e d (5.~,1o) and is usually called the absorption model. The d a m p i n g of the low p a r t i a l waves is o b t a i n e d b y including the elastic i n t e r a c t i o n in the initial a n d final states, which shows u p as an ((~bsorption~) factor m o d i f y i n g a\~ ~/% each p a r t i a l w a v e of the helicity a~lplitudes. This absorption factor is directly related to the elastic scattering , J ~ ' eK" data. F o r theoretical discussions of t h a t procedure we " ~ refer to refs. (9.J2). Earlier theoretical works, either based on the B o r n dinb" \d g r a m s (3) or on the Sopkovitch t y p e of calculations (4.7) indicate t h a t in order to fit the angular distribution, it is necesFig. 1. - The onesary to assume t h a t K*-meson exchange is p r e d o m i n a n t . particle exchange We therefore assume the process to t a k e place as in Fig. 1. diagram.

f

(a) H. 1). 1). WATSoX: NUOVO Uimento, 29, 1338 (1963); C. H. CHAN: Phys. 1~ev., 133, B 431 (1964); D. BXSSlS, C. ITZYKSONa~d 1~[.JACOB: ~UOVOCimento, 27, 376 (1963). (4) N. J. SoeKOVlCg: 5~uovo Cimento, 26, 186 (1962). (5) A. [)AR, M. KUGLER, Y. DOTItAN and S. I~USSINOV: Phys. Bev. Lett., 12, 82 (1964): A. D;~R and W. TOBOCMAN: Phys. Bey. Lett., i2, 511 (1964); A. DhR: Phys. iiev. Lett., 13, 91 (1964). (s) L. DUI~AND and YAM TsI CnIu: a) Phys. Bey. Left., 12, 399 (1964); b) Phys. Bey., 137, B 1530 (1965). (7) (~. COHEN-TANNOUDJIand H. NAV~LET: ~'UOVO Cimento, 37, 1511 (1965). Is) K. ])IEWZ and H. PILKUHN: .Y~O?'O Cimento, 37, 1561 (1965); R. C. ARNOLD, Phys. Rev., 136, B 1388 (1964). (9) A. BIXLAS and L. VAN HOVE 1Vuovo Cimento, 38, 1385 (1965). (~o) K. GOTTFRIED and J. D. JACKSON: XUOVO Cimento, 34, 735 (1964); J. 1). JACKSON, J. T. DONOtlVE, K. (~OTTFBIED, R. KEYSER allClB. E. Y. SVENSSON: Phys. Bey., 139, B 428 (t965). (11) R. (~MN]~,~: Phys. Rev., 137, B 649 (1965). (l~.) H. HOGAAS~N and J. tt6(~AASE~: NUOVOCime~o, 39, 941 (1965).

562

H. H6GAASENand z. ~6GAASEN

Our calculations differ from the previous ones by using, together with the usual vector coupling of K* to the baryon vertex, a tensor coupling (magnetic moment type), and by the systematical investigation of the behaviour of the cross-sections with the variation of the incoming antiproton energy. In Sect. 2 we write down the most general form of the interaction necessary to obtain the diagram of :Fig. 1, show how we make the partial-wave expansion of the helicity amplitudes and how we correct them for the interaction in the initial and final states. For clarity the expressions for the amplitudes together with their partial-wave expansion are put in an Appendix. In Sect. 3 we present and discuss the numerical results of the calculations for pp ->AA, p~ ~ Z+Z ~: and p~ - > A Z ~ 2 4 7 ~ iwith antiproton laboratory momentum of 3 Geu 3.6 GeN/e (*), 5.7 Geu and 7 Geu (this last energy only for the AA channel). The results illustrate once more the failure of the model to describe correctly the energy-dependence of the cross-section when the exchanged particle has a spin different from zero (~). On the other hand, the angular dependence is very well described for p ~ - ~ A A , rather well for p ~ - > A Z ~ ~ and is in reasonably good agreement with experiment for p~--> Z+Z +.

2. -

Calculations.

To describe the process in Fig. 1 we start from the following effective Lagrangian which has the most general form (2.1)

Lf(x) =

igv~(x)?,~x(x) + mx g2 + my ~(~y(x)~.~x(x)) +

§ i-

gs a,(v~y(x)yJx(x))] K,(x) + Herm. conj.. ] mx + my

Here ~r(x) represents the hyperon field, V~c(x) the nucleon field, K~(x) the K* field (creating K *+ and destructing K*-), gr, gz and gs are the vector, tensor and scalar coupling constants of K* to the nucleon-hyperon vertex and mx and m r are the masses of the nucleon and hyperon. We shall further denote by a~,, b~,, % and ds the four-momenta for ~, p, Y and Y, respectively, as indicated in Fig. 1, and we define e~ = b~ -- d~ = c ~ - a , ,

t = -- el,e~,.

The calculations will be made in the centre-of-mass system and we shall assume the same mass for the hyperon and antihyperom (When we calculate (*) In order to have better statistics, the ar~gular distribution a~ 3.6 GeV/c includes also events from the same reaction at 4 GeV/c.

AN APPLICATION OF T I l E AI~SORPTION )dODEL TO T H E REACTIONS t ) p - r

563

the process p]5-->AE~ ~ we will then use as the hyperon mass the mean of the masses of A and )20 but we believe t h a t the error so introduced is unim,portant.) Then in the centre-of-mass system the four particles have the same energy E, and we denote by s the total energy of the incoming particles squared; s = 4 E 2. Still in the centre-of-mass system the magnitude of the three-momentum is denoted by q for the proton an4 antiproton, by q' for the hyperon and antihyperon while 0 is the angle between the proton and hyperon momenta. 7, and %,,--(1/2i)(7~7~--7W,) are the ordinary Dirac operators acting on the four-dimensional spinor space when we normalize the tree-particle spinors to u u = 2m where m~ is the mass of the baryon. The production amplitudes M for the process p ~ - ~ YY will be normalized so t h a t the differential crosssection is given b y d~ at

1 -

1

64 .q

12

t-

,

where the summation is over the 16 helicity amplitudes of the process. The mass of K* will be denote4 by m. With these notations the Lagrangian (2.1) gives the following expression for the amplitudes : (,.,)

211

~(-- a) gvy,

'/?t.W

gr

igs

@ Ttly0"/'~ -l- /~/,~ @ m,z

%} 'r(-- c)"

igs ,2i-4

e~} ~t(b)

Here ~ represent the negative-energy l)irac spinors. Using the properties (~3) of the charge ~-onjugation operator U (the superscript t denotes transposed):

r

k) = ut(k) ,

v(-- k) . . . . at(k) ('-~ and

!-~71J" =

-

-

Y/~,'

the exl)ression (2.2) is easily transfm'med into (2.3)

M =--u(c){gv71,--

g~

igz

%} u(a) ~v + et,ev/m 2

{ gr @ igz e~}u(b) " 9:d(d) gvY~@ mop@ m~ a~e~ m~-+ m~: (1~) J. HA~IILTON: ~/'h(; Theory o] Elem.eldarg Particles (Oxford. 1959).

564

~ . HOGAASEN aRd J, HOG&ASEN

We are going to make the assumption t h a t the baryons always can be t r e a t e d as being on the mass shell, and b y use of the Dirac equation, (2.3) can be written as (2.:t)

M

1

=

Gvv~(d)?~u(b)~(c)?~u(a).m2~_~ t § 1

+ GdS[~(d)~,~u(b)~(c) u(a) § ~(d) u(b)~(c)y, u(a)] m ~ t § 1

§ Gss ~(d) u(b) ~(c) u ( a ) ' - m~_t ; the quantities Gw, Ods and (~s are defined b y

Gv~ = - - (gv + g~)~,

Gds -- 4Egr(gv § gz) , m x § mr (2.5)

G~ = -- 2 (gv § g~) (gT + g~

m~:--my~ m~r § my/

]gz(q § q') § gs(q--q')I ~

49~E 2 (m~o§ mr) ~ i [ m-~

(rex + m~) ~

(m~v-- mr) gv

- k - !q--q']~]~ gz mov+ m~J "

One can see t h a t for a reasonable scalar coupling the terms containing gs are negligible. W e shall, therefore omit t h e m from now on, setting gz = 0. The calculations of the helicity amplitudes are now straightforward, b u t lead to r a t h e r long expressions which we give in an Appendix together with their expansion in terms of the r o t a t i o n function d~,,: E a c h helicity amplitude is expressed in the form ]mlnq-~

(2.6)

<4o4~1~14~4~>= 5 B'(8, z) eL(x) + A(s, z) ~ (2, + 1)eL,(x)cL,(z), it. train

~'mla

where x =cosO,

z= 1+

4 = 4a-- 4~,

m2+ (q--q')~ 2qq' '

4' = 4o-- L

aud

jm, o--max

(idl, 14'l) 9

The c~z are r o t a t i o n functions of the second kind

(12.14)

defined through the

(14) L. DURAND and YAM TsI CHtU: Spin e//eets in modi/ied single particle exchange models, Yale University preprint (1964); M. ANDREWS and J. GUNSON: Journ. Math. Phys., 5, 1391 (1964).

AN A P P L I C A T I O N

OF T I l E

ABSORPTION

MODEL

T() T H E

REACTIONS

Iop->YY

565

relation

$min

Their properties and recurrence formulae for their calculations are discussed elsewhere (~). E a c h t e r m ill these expansions is now modified b y a factor k(j) which is determined b y the elastic scattering in the initial and final states. If one assumes the elastic scattering cross-section to be described b y do'el

(2.8)

o'2r

dt = 16~ e x p [ A t ] ,

where t is the m o m e n t u m t r a n s f e r squared, one is led (~o) to the f o r m (2.9)

k(j) = [ ( a - (', e x p [ -

(',

[-

Here t

s

(2.10)

C}" = 4 ~ A ~ '

7,

1 -- 2 q 2 ~ i ,

1 D = 2q"-A~'

!

A,(A~) represents the width of the diffraction p e a k for the elastic scattering in the initial (final) state, as shown in (2.8) and a~ ((rYr) represents the era'respending t o t a l cross-section. F r o m e x p e r i m e n t a l d a t a on p~ elustic scattering (15), we find C ~ = l , and the following values for A~: antiproton l~boratory momentmn

-P~b= 3 ,Geu

A~ = 16.

P~,~, = 3.6 Geu

.4, = 15,

5.7 Geu

A~ = 13,

PloD ~

P1~b = 7

Geu

A ~ = 12.

We know, of course, nothing a b o u t the elastic scattering of YY, and we choose in a first set of calculations the same values as in the initial state. A discussion a b o u t this a s s u m p t i o n will be m a d e in the n e x t Section. Let us note t h a t the values (', = C r = l, if t h e y do not ensure necessarily

(15) Proceedings el the 1963 Sienna International Con]ere~ce o.~ Elementary Particles

(Bologna, 1964), vol. 1, p. 259; "col. 2, p. 107.

566

~. H~3GAASENand J. ~ISGAASEN

unitarity, at least p r e v e n t a violation of the boundedness of partial waves dictated b y u n i t a r i t y when s goes to infinity (~-~).

3. - C o m p a r i s o n w i t h e x p e r i m e n t .

We performed numerical calculations of the processes p~ -~A~_, pp --> Z+Z + and p ~ - > A Z ~ ~ within the range 3 to 7 Geu for the incoming antip r o t o n l a b o r a t o r y m o m e n t u m . I t should be remembered t h a t in calculating the amplitudes we assumed equal masses for the two produced pal~icles, for the A Z ~+ A Z ~ cross-section we therefore made the approximation m,:o = ~nh = = 1.15 Geu The coupling constants of the K* to the nucleon-hyperon vertex are unknown, and at each energy we used t h e m as freeparameters to fit the angular distribution. We first tried different values for the ratio gz/gv----R. I f one believes in SU3, the ratio gz/gv should be the same for the K*J~Y vertex as for the p S ~ vertex. This gives from the p-photon analogy for the ps vertex (16) R = 3 . 7 . We f o u n d t h a t for values of R in the range f r o m 0 to 1 we could fit the angular distribution of A A events extremely well. whereas for R - - - - - 1 or R = 2 the forward peak became too broad. Thus within the K*-exehange picture S Ua s y m m e t r y appears seriously broken unless it is the p-photon analogy or the absorption model which cannot be trusted. On the other hand, the results are r a t h e r insensitive to variation ot' R in the range 0 to 1 and we fixed gr = gv over the whole energy range, leaving one multiptieative coupling constant ior each process to fit at each energy (*). W e shall call it the (~effective coupling constant ~. I f the model was able to describe well the energy variation of the cross-section the effective coupling constants would be the same at all energies. Thus their variation is a measure of the inability of the model to account for the energy-dependence of the crosssections. A f u r t h e r problem was to choose among the experimental data. As can be seen from Fig. 2, the data t a k e n from ref. (1) and ref. (2) are sometimes

(16) L. STODOLSKYall(]. J. J. SAKUItAI:Phys. Rev. T~ett., 11, 90 (1963); L. STODOLSKY: Phys. Rev., 134, B 1099 (1964). (*) Using g~ = 0 and (2A/q~)-1 = (2,~]q'2)-1 = 0.28 as in ref. (7) we find at 3.6 GeV/c g~.~vA/4z = 2.8 =k 0.3, in reasonable agreement with the value 3.05+~ of ref. (7). The lower value 2.2 obtained in ref. (~b)is due partly to the use of a somewhat smalle~ value for the A's and partly to the asymptotic approximation. The Kix_x, I Bessel functions appear to be a bad approximation of the e~, functions of ref. (6b) because z is relatively far from 1. Also the <,extraordinary ~ terms [the ones involving BJ(s, z) in (2.6)] are disregarded ia ref. (~b).

AN A P P L I C A T I O N O F T H E

ABSC)RPTION b I O D E L TO T I I E R E A O T I O N S p p - - > Y ~

567

in conflict. We usually c o m p a r e our calculations with the d a t a of ref. (~) except for P = 7 Ge~7/c where only the Yale d a t a are available. The error limits t h a t we p u t on the effective coupling constants are not to be t a k e n too seriously as t h e y are d e t e r m i n e d only f r o m the errors in the e x p e r i m e n t a l d a t a used. We recall also t h e u n c e r t a i n t y resulting f r o m the choice of the YY el:~stic

I

iooi

t

50

~:3_ 0 [ I00C tf

100 500 50

O'

u cl

l

2

P,o~in

I

.c

l

4 GeV/c

0

6

Fig. 2. - Experimental data for the total cross-sections, OERN {') data are marked with o, Yale ('-') data with =. a) a(AVn4-AV~ h) a(AA).

,n 1000 o u i:1

b

500

bl

p a r a m e t e r s (we took t h e m equal to the p}5 elastic p a r a m e t e r s which :flso have r a t h e r big uncertainties) and to our ignorance of the ratio

.GIg,..

0.5 cos 0

Figures 3 :tad 4 show the reFig. 3. The differeRtial cross-section (}ne sees t h a t fro' AA production, at a) I ' = 3 GeV/e and b) 3.6 GeV/c (CERN data). the angular distributiou is extrem e l y well fitted through the whole energy interval. Tile coupling constants we used to obtain this fit are p l o t t e d in Fig. 6, f r o m whi(;h it is evident t h a t the model is quite inadequate to describe the r a p i d fall of the e x p e r i m e n t a l cross-section w i t h el~.ergy. We now t u r n to the process p~ --~ Z + Z § As can be seen f r o m Fig. 5, the distribution for this rea(:tion is ratlmr different f r o m the distribution of the ATt events. A p a r t f r o m the p e a k at w;ry low m o m e n t u m transfer, which is at least as steep as the p e a k for the A}~ ease, there appears a n u m b e r of events s u l t s for p p -.~ Ask.

568

~. I[6GAASEN and J.

HOGAAS:EN

a t h i g h e r It I v a l u e s . W e call t h o s e <~w e a k p e r i p h e r a l e v e n t s ~>as t h e y a r e n o t of s t a t i s t i c a l n a t u r e , a p p e a r i n g o n l y f o r It] v a l u e s l o w e r t h a n ]tm~ [/2 (*).

,oo 200' >

-1

loo

400 3oo

b

23

o

05

,r]

.~ 200

10

-o 100

-t

0 1.0

08

.~ lOOOt 9o

~

500

b)

O,.o

o.,

cos

O

Fig. 4 . - The differential cross-section for AA production, at a) P = = 5.7 GeV/c and b) 7 GV/ce (CERN d a t a at 5.7 GeV/c, unpublished), [Yale d a t a at 7 GeV/c. ref. (2)].

O~

0.8 t

12

0'2

10C

J~

c

06 cos e

50

c;

0

04

rr], []7~ 16

Fig. 5. - The differential cross-section for ~+2~~: production (CERN data): a) P = 3 GeV/c; b) P = 3.6 GeV/c; c) P --- 5.7 GeV/c.

In the framework of the model, it appears very difficult to explain these weak peripheral gvents: increasing the tensor coupling will destroy the narrow peak. The additional exchange of a heavy particle with isospin ~- cannot be considered because this weak peripherality also shows up in p~-->AZ~176 A n o t h e r p o s s i b i l i t y w o u l d t o b e a s s u m e a m i x t u r e of K a n d K * e x c h a n g e ns

(*) They also cannot be compared to the events in the reaction ~p~V-~v-, which show a peak in the forward and in the backward direction.

A.N

APPLICATION

OF

THE

ABSORPTION

)[ODEL

TO

THE

REACTIONS

pF-~YY

569

i K exchange gives a flatter angular b i distribution. Such an a s s u m p t i o n could be tested easily f r o m the as y m p t o t i c b e h a v i o u r of the differen;~10 ~ L { tial cross-sections with increasing + energy. The a m p l i t u d e s originating f r o m the exchange of the K meson 4 5 6 7 3 decrease as 1Is relative to the amPLo~!n GeV/c plitudes f r o m K* exchange (") and Fig. 6. - The effective coupling conone should expect the weak periphstants as a function of energy plotted ia eral events to vanish with increaslogarithmic scale: 9 g~,a,A/4Jt with ing energy. This is not the case in A(AA) = A(pp); o denotes g~,vA/dz with the energy i n t e r v a l we study. As K 3A(AA) = A(pF) ; 9 derrotes g~,oV~:0/4~ with A(E+~,~) = A(pF) ; ~ denotes g~,v~d4u with exch~mge also would irLtroduce more A(E+Z ~) = 3A(pp). p a r a m e t e r s in our calculations we decided to disregard the weak pcripheral events assuming t h a t K* exchange was responsible for the m a r k e d peal; ill the forward direction a n d normalizing the theoretical curves to the 1000~ events in tile interval It[m~ to [ t ] ~ 0.5 (Geu ~. The results of the calculation are t, 500 o! cl shown in Fig. 5 and the a g r e e m e n t 2k between t h e o r y a n d ext)eriment is fairly good a t low m o m e n t u m transfer. The energy dependence of the effec~'1000 tive coupling constants is m u c h weaker thai1 for the AA production as can be seen f r o m Fig. 6. We examine n e x t the process pD-* -* Z ~+ E~ and determine the p r o d u c t of the effective coupling constants

ooi

05 cos 0

2

200

(3.1)

~-~

2

GAz = '2 gK*a~ g~*+VZ! 47r 47r

b y normalizing, as previously, the calculated curves to the e x p e r i m e n t a l curves.

X3

100 6

04

t

08

37 - 1l N a c r e Cimento A .

1.2

s 7. - The differential cross-section for (A~-2-~+E~ production (CERN data):a) P = = 3 GeV/c; b) 3.6 GeV/c; c) P = 5.7 GeV/c.

570

n . I-IOG/kAS]~N a n d J. It0GAASEN

The results are shown in Fig. 7. The angular distribution is reasonably fitted but at 5.7 GeY/c where the experimental data have the best statistics we are nevertheless far from 'the excellent agreement of the AA case. The calculated curves are normalized to the total cross-section although thel'e are some weak peripheral events. Having determined already gx*xA and gx.xr.~ from the two first processes, this last process should not involve any new parameter if the elastic interaction in the final state is the same. We are thus in a position to check the oneparticle exchange model by comparing, for example, the value g~,vm/4~r obtained from experiment and its value obtained from g~*JvA and GAm through (3.1); the following relation should hold 2

(3.2)

2

gx*5"Z+ - - 2 gK%VZ~

47~

GAm

4~

gx**,,A/47~"

T h e comparison is made in Table I. We note that, for /)----3 GeV/c and equal elastic scattering parameters, we get incompatibility between the two TA]3L~ I. -- Val~tes o/ the e]]ective coupling consta~ts and the ratio GA~/((g~,A~V)/4u). 2

2

Pla~

gK*.~A

in GeV/c

4~

3

(CERN)

3.7•

3.6

(CERN)

2.8~0.4

5.7

(CERN)

gK*5"Z+

AE/ 4~--

4z~

3.5• 1

1.6 -4- 0 . 1 5

6.8~1.9

2.5•

1.5 ~ 0 . 2 5

1.4~0.2

2.0~0.4

1.5 :k 0.5

1.2 =t= 0 . 1 5

3.25 ( Y a l e )

3.1-4- 0 . 2 5

6.3 :J: 1.2

2.1 :J: 0.6

3.69 ( Y a l e )

3.0 :j: 0 . 2 5

7.7 ~ 1.4

2.6 ~ 0.6

3

2.4 ~ 0.25

5.6 •

2.2 • 0.5

(CERN)

12.9 • 2.1

:

1.0

The values of g~,Ajr GA~ and g~s are determined at each energy by normalizing the calculated results to the experimental data. In t h e first five rows we have assumed equal shape of the different elastic diffraction peaks: A ( A X ) = A(A~-~) = A(p~) = A(Z+~), in the last row we have used 3.4(AA)=3A(AZ-~)=A(pp)=~A(~.+Z-~). The error limits indicated are based on uncertainties in the experimental cross-sections only.

values. However, the agreement is better at higher energies, thanks to the great uncertainty in the experimental results and to the fact t hat the three

AN P P L I C A T I O N

(~F

THE

A B S O R P T I O N M O D E L TO T H E

REACTIONS

p~-~YY

571

different cross-sections become less different. It is however easy in the framework of the model to make the experimental results compatible with the K* exchange, by assuming different elastic interactions iu the final states than in the initial states. As a specific example, we give in the last line of Table I the results of cMcul:~tions at 3 GeY, assuming 3A(AA) = 3A(AZ o) = A(p]3) = ~-A(Z+2+).

This affects very little the angular dependence compared to the first calculations. An assumption of this kind can also give the same energy-dependence for the effe(,t.ive coupling constants g~:,~z/dz and gK,a~A/4:~. To try, however, to explain Mong these lines the whole energy variation of the coupling consrants would most probably be meaningless. First, we would need to assume very great differences between the p~ and YVZ elastic scattering. Also, as the cross-section of YY production must fall at high energy, we would be forced to assume a strong shrinking of the elastic diffraction peak with increasing energy.

4.

-

Conclusion.

These calculations reflect both the satisfactory and unsatisfactory aspects of the absorption model: on one side a fair description of the angular dependence of the produced particles (except the weak peripheral events in the case of Z+Z + production); on the other side, a wrong energT-dependence of the cross-sections c~msed by the exchange of a particle with spin. In the framework of the model, this last result can be understood as a consequence of the assumption t h a t the effects of all She computing reaction channels can be represented by elastic scattering parameters only. t~egge pole theory would in principle give an easy solution to this difficulty, but in practice its predictive power is still very small. Thus, the absorption model and Igegge-pole model appear to have their strength on different aspects; t~egge poles make it easy to get the energy-dependence in the forward direction but are completely helpless in describing the variation with t, as the residue functions of the poles are, until now, beyond anybody's computational capacity. The absorption model, on the other hand, gives an intuitive understanding of the angular dependence but suffers from the well-known difficulties associated with the exchange of elementary particles with spin.

572

i~. H6~XAS]~ a n d a. ~SGAASES

We t h a n k Dr. B. FlCENCIt and Dr. B. SVENSSON for fruitful discussions. We are indebted to the C E R N experimental group for p e r m i t t i n g us to use their results at 5.7 GeV/c prior to publication, and to Dr. R. KEYSER for invaluable help with the numerical calculations. One of us (J.It.) would also like to t h a n k Prof. L. VAN HOVE for kind hospitality.

APPENDIX

The helieity Born amplitudes. We show in Table II how the sixteen amplitudes are expressed from six of t h e m : TABLE II. -

Relations betweenthe helicity amplitudes.

\\

++

+ --

--+

\..

I n order to write their complete expression it is useful to introduce the kinematical quantities

(A.1)

~•

(A.2)

~7~=

q q' ) mx~Eimy+~ ' ~/(E + m~)(E + m ~

qq,

)

1 ~: (mx + E)(my + E) "

The six independent amplitudes are now expressed in terms of ~_+, 9• the p r o d u c t i o n angle O, and the quantities Gvv, Gd~ and G~s, defined in (2.5). N o t e t h a t G~s depends on 0.

AN

PPI.IOATION

OF TIIE ABSORPTION MODEL T O THiE I~EACTIONS ~)p--)-'~Y

573

+ G,.2 cos ' ~ ~+~_ + ~,~ cos~~ ,

0

20

(m 2 - t)M2 = Gvv cos s O ( ~ + ~ ) + 2G,s cos ~ V+~- § Gss cos ~ ~_, 9 0

0

(m 2 - t) M~ = Grv sln~ eos~ (~+~_ + ~]+~_) -~G

. 0 0 ~ 0 0 4s sm~- cos~(~+ + ~ ) + Gss sin~- cos W~?+~?_,

(A.3) 0 0 (m 2 - t) M4 = Gvv s i n ~ cos ~ ($+ $_-- ~?+~_) --

0 0 ~ ~ G 0 0 § G~s sin ~ cos ~ (% § V - ) - - ~ sin-~ cos-~ ~7+~- , .=0

.

0

.

0

,

(m = - t) 315 = Gvv sin -~ (~_ -~ ~_) Jr 2G,s sin ~ ~+~_ + Gss sma~ ~+ ,

.~0 ,] .~0 0 + 2G4s sin ~ ~+~_ + Gs~ sin2~ ~ . For the partial-wave expansion we split Gzz (with gz = 0) in two terms G8 8 ~

0 G,~-4G~ sin ~ :~ ,

where

G~ = -- 2gr(g~ + gv)

(A.~) G~ --

g~ (4E~ + (q § q,)2) _ ( m ~ ' ~ my) 2g~ , (m~ + my) ~

g~ qq' , (rex + m~)"

and the expression is easily done in terms of the c functions defined in the text (2.7)

i I ~ B 1d~

! 1 k(O) + Bldoo(x)k(1) + AIi

e~oo(z)dJoo(x)k(j)(2J -~ 1),

j-O

(A.5)

M~ = B~a~,(x)

],'(1) ~-j Ao. i (2j ~- 1)c11(z)(~11(x)t J j=l

k(j)

,

1113 ~- Bad~o(x) k(1) + A, ~ (2i + ~)do(Z)dlo(x)k(i), 3--1

574

i~. H ( ) G A A S E N

D~D.d J . t t ( ) G A A S E N

M4 = B,dlo(x) k(1) + A, ~ (2] + 1)C~o(z)d~o(x)k(~), ~1 r

(A.~)

i~=

B~d~_~(x)k(l) + A~ ~ (2 i -~ l)r ~=1

Me = B.d~

1 k(O) + B.doo(X)~(1)+ A . ~ (2i + 1)cgo(z)dgo(x)k(~).

In the Born ~pproxim~tioa all the k(j)'s ~re identicul equal to 1, in the absorption model they ure given by (2.9) in the text. The expressions for the A's ~nd the B's ure given by Gvv, G~s, G~ ~nd G ~ as follows: --1

1

G

2

2

G1 2

GO ~

B __ G~s 2

Bs=Bd--

~/2 G1 2qq' ~+~G' --

qq,

1 _ _ 2 ~ § 2G~+~?_ ~- G o~]+2 -~- 2 ~~ss~+ 1 2 z ~~ , ~Bs = 4qq' (Gvv(~--~-)

A1--

_ _

o

2--,~,~

~'l--

2qq' 1 (G~(~+~_ + V+~-) + ~,~(~- + ~-) + G ~ _ 2qq'

A~ = - -

+ 2~L~+~_(i -- z)},

1 -- 2qq'

A~ = ~ 2qq'

2qq ~

(e~(~ ~_+ ~_) + ~ , ~ + ~ _ + ~ o ~ + ~oe~ ~ ( ~ _ z)},

.kN 2 P L I C k T I O N

(}F T H E

ABSOI~PTII}N MOI)JdI. TO T I I E R E . [ C T I O : ~ S pD->Yx--ir

RIASSUNT{)

575

(*)

Si presenta una serie di calcoli sulla reazione pD-> y ] s per u a iatervallo di impulsi dell'aatiprotone nel sistema del laboratorio da 3 GeV/e a 7 GeV/c. Si suppoae che il processo ~bbia luogo tmicameate trami~e lo seambio di K* e si effettuarto i e~lcoli hello schema del modello di assorbimeato (l'approssimazione d,i Born i a oada distorta). Si ottiene una buona descrizione della dipencleaza angol~re clella sezione d ' u r t o per i ire eanali AX, AZ~ ~ e ~A+E+ nelFintero intervallo di eaergia. Tuttavia, la dipendenza dall'eaergia b sbagliata e risulta pessima per il caaale AA, clove la dipendenza angolare 5 meglio deseritta.

(*) Tra~l~tz~ol~e a e u r a d e l l a R e d a z i o n e .