LETTERE

A:LLA

REDAZIONE

(La resparesab~litd scie~tifica degli scritti inseritl in questa rubrica ~ com'Pletamente lasciata dalla Direzione del periodico ai singoli aulori)

A Note on the Reaction = + + d - + p + p. ]~. G. BELTRAMETTI Istituto di t~isica ddl'Universitgt - Genova Istituto N a z i o n a l e di F i s i c a N u c l e a t e - Sezione di Genova

(ricevuto i l 14 D i e e m b r e 1961)

W e wish to m a k e some c o m m e n t s on the p r o b l e m of distinguishing the contrib u t i o n of the pionic s and p waves in t h e a b s o r p t i o n in flight b y d e u t e r i u m : (1)

~+÷d-+p+p

,

(or = - + d - ~ n + n ) ,

at low energies (less t h a n t h e energy of the ~, a ~ resonance). To be more precise, let us n o t e t h a t p a r i t y and angular m o m e n t u m conservation impose the following restrictions on the s t a t e of the two final, identical particles: in t h e usual spectroscopic notations, it is of the t y p e aP 1, if the pion is absorbed in an s-wave, and of the t y p e 1S0 o1' (and) 1D 2 if it is absorbed in a p - w a v e . Our p r o b l e m is to look for the p h e n o m e n o l o g i c a l d e t e r m i n a t i o n of t h e amplitudes of t h e three possible final states. The real p a r a m e t e r s of physical significance, which d e t e r m i n e such amplitudes, reduce to five due to an a r b i t r a r y overall phase factor. W e shall i d e n t i f y f u r t h e r one of these p a r a m e t e r s w i t h t h e t o t a l cross section of the process (at definite energy), or, in other words, we shall p u t e q u a l to unity, and real, the a m p l i t u d e of one of the three allowed states of the p r o d u c e d protons. E . g . t h e final state ] will be w r i t t e n in t h e f o r m : (2)

] = aP 1 ÷ ~/(1D2÷ Y 1,qo) ,

where two complex p a r a m e t e r s appear. This n o t a t i o n is s o m e w h a t imprecise since the t h i r d c o m p o n e n t of the t o t a l angular m o m e n t u m is n o t explicit b u t it can v e r y s i m p l y be m a d e precise (see e.g. the n o t a t i o n s of ref. (1)). I n a preceding w o r k (1) we showed t h a t the correlation degree P b e t w e e n t h e

(~) E. (¢. ]3ELTRAI~IETTI:i\Tuovo ('i~l¢~ltO, 21, 98 (1961). 42 $ - II Nuovo Cimento.

656

E. G. B]~LTICA~'VIETTI

proton spins can distinguish the s and p-contributions P:

i n p r o c e s s l i k e (1). W e d e f i n e :

i[~'(~),"(2), ]]12-

[[~'(])fl"(2), Ill 2

l[£(1)~"(2), ]3]~÷

I [ ~ ' ( D ~ " ( 2 ) , ]]1 -~ ,

where the square bracket stands for the integration over the spin co-ordinates and w h e r e a ' , (fi'), a n d ~", (fl"), a r e t h e s p i n - u p , ( s p i n - d o w n ) , s t a t e s i d e n t i f i e d b y t h e analysers A' and A" which measure the spin direction of protons 1 and 2 respecA' t i v e l y (see F i g . 1). I n t h e c a s e o* t r a n s v e r s e p o l a r i z a t i o n s , a n d if t h e q u a n t i z a t i o n a x e s , d e f i n e d b y the analysers A' and A", point in the same direction, the correlation degree P _,oion beam z ~ is e a s i l y s e e n t o v a n i s h ~or s - w a v e p i o n oa b s o r p t i o n , w h i l e i t is m a x i m u m ( e q u a l t o - - 1) Yor p - w a v e a b s o r p t i o n . n 2 T h e s i t u a t i o n w h i c h c o n c e r n s u s is however somewhat more complicated since we have to deal with simultaneus ~, s and p wave absorption and the various i n t e r f e r e n c e t e r m s will a p p e a r ( s e e e q .

(2)) (') (").

~ig. ~. culation

produces

the ~ollowing result

A straightforward, but for the correlation degree:

t e d i o u s , cal-

g

~O = _ _ :

h gh :

I

3(u.z')(u.z") - - [~1 ~{(u.z)~ + ~ + ~ I~l = + 2 ~

[ ( u . ~ V - - 1] Re ~}(z'-z") +

+ ~ / 2 [ I m ~l -- %/2 Im (~r)](uAz").z -k ~/2 [Re q -- ~¢/2 Re ( ~ y ) ] ( u A z ' A z " ) . z ,

(3)

= 3 + I~1 ~((~'=)~ + ~ + ~1~12 + 2~/~ [ ( , - ~ t ~ -- .~] ~ e ~} -~ / ~ [ ~ m ~ -- ~/~ Im ( ~ r ) ] ( ~ ' ) ' z ,

(*) Relative to ref. (1) t h e situation is quite diffeient in respect to the energy of the ])ion. There, we h a d to deal with mesons bound in Coulomb orbits while here we neglect t h e Coulomb energy in comparison to the pion's kinetic energy. This fact and the presence of a preferred direction (the pion's momentmn) give rise to t h e interference t e r m between the various states appearing in eq. (2). (**) I n more detail, by choosing the pion b e a m direction as quantization axis, the final state can be written as an incoherent superpesition of t h e s t a t e s : m#Y

where t h e C's are Clebsoh-Gordan coefficients, the Y~(0~¢~) are spherical harmonic functions in the angular variables of proton 1, X~ttand X00are the s t a n d a r d spin-flanctions (N/'2Zo0= ~(1)fl(2)-- a(2)fl(1),' a n d fl being the spin-up and spin-down states respectively). This formula also provides a complete definition of V and y. The correlation degree P will be written: E 1[~'(1)~"(2), ~ ] l ~ - E

l[~'(1)~"(2), ¢~]i ~

Z [[#(1)~"(2), ~ ] 1 ~ + Z 1[~'(1)fl"(2), ~ ] ? " Y

A NOTE ON THE REACTION

~:+d-d->pd-p

657

where z and u are unit vectors in the momentum direction of the pion and of the proton 1 (that one detected by the analyser A') respectively while z ' and z" are unit axial vectors in the direction of the quantization axes defined by the analysers A ' and A ' , as shown in Fig. 1. We note explicitly ~hat formula (3) is calculated in the C.M. system. Going to the ]aboratory system small corrections (at the energies which concern us) are introduced, which could be easily evaluated. The experimental determination of P requires a coincidence measurement of the proton polarizations so that the product of the efficiencies of analysers A ' and A" will appear. It is to be noted, in this respect, that for proton energies above 100 MeV (pion's e n e r g y : ~ 60 MeV), these efficiencies are greater than 60°o (~). Of course there are simpler experiments than the one proposed which are able to give information about the complex parameters t~ and y but unable to determine them completely. Such experiments could involve the angular distribution of the emitted protons (~), their polarization degree perpendicular to the production plane (~) and eventually targets of polarized deuterons if, as it seems, these targets will be available in the near future. At this point, however, the experimental situation relative ~o the inverse reaction (4)

p ~ - p - ~ d+7:+ ,

must be taken into account. For this reaction the state (2) will now be the initial state. The phenomenological study of the four real parameters which are involved (~) has required four separate measurements, i.e.: (a) the angular distribution of the produced mesons (a), (b) the right-left asymmetry in the case of polarized incident protons (~,~), (c) the polarization of the produced deuterons (s,9), and (d) the energy dependence of the total cross section (~,~). Experiment (c) is difficult and rather unreliable because of the intrinsic difficulty in measuring the polarization of a deuteron (~0) and also because of the high energy ( > 141 MeV) and strong forward collimation of the deuterons from reaction (4). Experiment (d), compared to the first ones, is of a different kind; indeed, it cannot be interpreted in terms of the parameters ~ and ~, without some assumptions about the form of the cross sectiom such as the following: (5)

~,t~ ~ q(1 d- ~q~),

(2) L. ~VOLFENSTE[N: ,tlttl. [tel'. N~¢Cl. ~cI'., 6. 43 (1956); A. 1~. T.kYLOR: Ilep. Drog. Phys., 20, 125 (1957). (~) C. E. Co~i5": P h y s . I~)ev., 105, 1582 (1.957): A. M. SACHS, H. \\'I.xit'J~ a n d B. A. WOOTt~:N': P h y s . Rev., 109, 1733 (195S). (~) B. T. FELl): N ~ o v o ('imeldo, 12, 425 (1954). (~) A. II. ROSENFELI): P h y s . l~e~'.. 96. 139 (1954); M. GI,]LL-M,~,XN a n d K. 5I. WATSON: ~11~/~. lle~'. N~Wl. Nci., 4, 219 (195~). (~) F. S. CR:tWFORI) ¢~nd M. L. STI~]'~J~]NSON:P h y s . l~e~'., 97, 1305 (1955); T. 1[. FIELD, J. G. F o x , J. A. K~,N'~, 1~. A. S'I'AL[,WOOI) a n d I~. ]~. SUTTOA~: P h y s . ]?ev,, 109, 170~t (2958); L. G. ])OA'DROSI: P h y s . I,'er., 114, 1623 (1959); I'~. 1i. 5IARCI[: P h y s . ]~e~,., 120, 1S7~ (1960). (7) II. ]~]. 5IARSlIAK a n d A. M. L. 5{ESSIAn: N¢~OVO ('lmento, 11, 337 (1954). (8) K. 5f. ~V.~TSON a n d C. 1%icH)[_~A': P h y s . l~c~'., 83. 1256 (1951). (") R. TRIH,: P h y s . l{ev., 102, 862 (1956). (~e) L. 1'VoLI,'I,]h',~I']clN':Phy8, Ile~,., 98, 766 (1955): F. MAN'I*t, ttnd T. l~l,:(,(~l~;: Ph!l.s'. l~ei,., 99, ]47,~ (1955).

~58

~,. G. BELTRAM]~TTI

where q is the pion's momentum and the two terms in the brackets refer to pion emission in s and p waves respectively. A dependence like eq. (5) is expected to hold near the t-hereshold of reaction (4) but one must take into account the rapid increase of q with the energy of the incident protons. As a resuIt of the quoted experiments, the study of the parameters ~ and 7 for reaction (4) is to be regarded as not yet satisfactorily because of the lack of unicity in the possible sets of parameters, and also because every set is affected by strong experimental errors. To make further progress in this study the spin correlation effect, as we discussed before for reaction (1), could be useful. This implies, on the other hand, t h a t time reversal invariance holds for the reaction under discussion. Though no violation has been found, the problem of testing this invariance, in the field of strong interactions, is still of some interest, due to the low sensitivity of most of the experimental tests actually made (~). They could be divided into two groups: the first one contains those experiments which based on the so called ~ polarization-asymmetry equality >> for elastic scattering of protons by non-zero spin nuclei (~,~3); the second one, experiments in which a reaction such as ad-b--+cd-d is observed in both directions: this situation occurs particularly for some stripping-hke reactions (and their inverse pick-up processes) and for some charge-exchange process (1~). The limit placed by these e~periments upon the fraction of the Hamiltonian which is noninvariant wi~h respect to a time inversion is of the order of some percen~ but i t must be remembered t h a t the sensitivity of these tests is somewhat doubtful and often quite low (n). The comparison between reaction (1) and reaction (4) could be a typical test for time-reversal invariance relative to processes in which a pion is absorbed or created. I t has been shown (11), however, t h a t the comparison between ]Vl and !y] for the two reactions is not significant, but a comparison between the phases of ~ and y would be required. Also from this point of view a study of the reaction (1) along the lines discussed at t h e beginning of this note could produce useful results.

(1D E. M. H E N L E r a n d B. A. JACOBSO:UN: Phys. l~'ev., 113, 225 (1959). (1~) lo. HILLi,cIA:NT A. JOHANSSON, G. TIBELL: Phys. Rev., 110, 1218 (1958): A. A~.~S~IL~h~ a n d E. M. HAFNER: Phys. Rev. Left., 1, 255 (1958). (~) F. 1VJ[ANDL: Proc. Phys. Soc., 71, 686 (1958). (14) L. ROSEN a n d J. ]~. BROLLEY: P h y s . Rev. Left., 2, 9S (1959); D. ~O~)ANSI(Y, S. F. I~]CCLF,S, G. F. FARWELL, 5~. E. RICKEY 8~Ild P. C. ROBINSON: Phys. Rev. Lelt., 2, 101 (1959).