IL NUOVO CIMENT0
VOL. L V A , N. 2
21 5iaggio 1968
The Reaction p § p -+ d § ~:+. D. J . BROWN and M. K. SUNDARESA~ Physics Department,
Carleton U n i v e r s i t y - Ottawa
(riccvuto il 25 Marzo 1968)
A n u m b e r of recent studies, b o t h e x p e r i m e n t a l (1) and theoretical (2) h a v e been m a d e of the a b o v e reaction in the region of energies from a b o u t 1 GeV to several GeV. W c would like to report here on an effort m a d e to calculate the differential cross-section for the above reaction based on t h e m o d e l of a single-nucleon e x c h a n g e (3). The m a t r i x element for this process m a y be written in the form
(1)
T== X/2g('a)UT(p2)
\ u
M'-" ] Y~'u(Pl) "
~2 = ~ .
where M is the nucleon mass, g(u) the form f~tctor i n t r o d u c e d at the ~jVzh ) v e r t e x , n t h e f o u r - m o m e n t u m of the e x c h a n g e d n e u t r o n a n d / ~ involves the form factors in t h e dAoJT v e r t e x . F o r e v a l u a t i n g (1) explicitly, we h a v e to p u t in an expression for F. The dJ~?J~' v e r t e x for one off-mass-shell nucleon (n) and one on-mass-shell nucleon (P2) can bo w r i t t e n as
(2)
P(u)=
{ ] , ' ( ~ ) + H ( , ~ ) ( ~ . , , . - i ) ~ r " + [ ~ M + 2M (~ "
:lZ) (#~--,,)"
~*.
where e ~ represents the pohu'ization vector of t h e outgoing d e u t e r o n and F, G, H and i are i n v a r i a n t functions of ~t2 = u (a). Using (2) for P in (1) one can obtain in a straightforward m a n n e r an expression for the differential cross-section for the r e a c t i o n of interest. I t t u r n s out t h a t if one uses F, G, H and i e v a l u a t e d from t h e w o r k of GRoss (4) in calculating numerical values for t h e differential cross-section one obtains ridiculously large values in the energy region of interest and h t r t h e r the fall-off of the cross-section as a function of angle is too slew (s). (~) D. DEI~KER~ela1.: P:egs. L~tt., i I , 16[ ( i 9 ( ; [ ) ; F. T g a K o T e t a l . : Ph!is. l~ev. Lclt., 11, 4"[4 (1963); R . ~ . IIEI~z: Pie. D. Thesis, M i c h i g a n (1964). (~) It. M. ][I~.~[~z: Ph. D. Thesis, 5'IichigarL ([961); .l. ]~'IATHEWS ~Lo.d ]L DE(): Phys. Rev., 143, 1340 (1966). (a) T h e p r e s c n t cffort differs f r o m t h o s e i n r e f . (~) ill t h a t i t is f u l l y c o v a r i a n t . (4) F. L. GROss: Phys. Rev., 136, B 140 (1964). (~) A s i m i l a r r e m a r k h a s b e e n n m d e b y 5IATIIEWS a n d DEO w h o u s e d n o n r c l a t i v i s t i c d c ~ d c r o n w a v e functions.
THE
p+p-+d~-7: +
REACTION
347
Similar is t h e case in a c o v a r i a n t c a l c u l a t i o n of t h e p h o t o d i s i n t e g r a t i o n of t h e d e u t e r o n c a r r i e d o u t b y CvOURDIN et al. (6). I n view of this, GOURDIN et al. h a v e s u g g e s t e d t r e a t i n g ~he i n v a r i a n t f u n c t i o n s a t t,h e d,~N'U~~ v e r t e x ~.s f u n c t i o n s a p p r o x i m a % d p h e n o m enologieally b y a series of poles a n d t h e s e a d j u s t e d to o b t a i n a fit to t h e p h o t o disintegration data.
/ 4o /
I
A
\ \
b - 2o!
/
\
\
\ \ \
Iol
\\ ~
O L
-1.0
~i
L
--
.
0.9
9
- 0.8 COS O~
-07
_
-0.6
Fig. |. Comparison of various theoretical calculations an(1 experiment for incident kinetic energy of 2.,~ GeV. The om;-pion exehartge graph ( - - - - - ) is from VAo, and the nom, elativistic deuteron graph ( . . . . . . . ) from ]IEINZ; - - , relativistic d(qlt('ron; ~,, HE1NZ; O, DEKKER~.
W e h a v e a t t e m p t e d a n e x a c t l y similar a p p r o a c h in t h e p r e s c n t c a l c u l a t i o n . W c find t h a t t h e e x p e r i m e n t a l d a t a a~ 2.8 GeV on t h e a n g u l a r d i s t r i b u t i o n c a n i n d e e d b e fitted (see Fig. 1) w i t h t h e i n v a r i a n t f u n c t i o n s i n v o l v i n g t h e s a m e set of p a r a m e t e r s w h i c h fit t h e p h o t o d i s i n t e g r a t i o n d a t a . F o r t h e ~A%V f o r m f a c t o r g(u), we h a v e used t h e f u n c t i o n used (7) in t h e fit to t h e b a c k w a r d =,~V d a t a :
(3)
I--D , g ~ ( u ) = g 2 ( : ] l 2) [ ( l ~ i u Z 3 i : ) / z 2
]~ ) ~D
,
D=0.14,
,=0.34GeV.
A slight m o d i f i c a t i o n of t h e p a r a m e t e r a f r o m 0.34 to 0.30 is f o u n d n e c e s s a r y in o b t a i n i n g t h e fit. W e believe t h a t this smaller v a l u e of ,~ s i m u l a t e s a b s o r p t i o n effects in our case
(d) 3I. GOURDIN et al.: Nuovo Cimento, 37, 524 l|965). (~) I~. ~'[. I][E[NZ artd 3~. It. Ross: Phys. Rev. Lelt., 14, 1091 (1965).
348
D.J.
BIIOWN
and
M. K .
SUNDARESAN
which h a v e n o t been explicitly p u t in. I n Fig. 1, the curves o b t a i n e d using Y a o ' s (s) single-pion e x c h a n g e calculation and the one from H e i n z ' s (9) calculation are also shown for comparison. In Fig. 2, the fit to the lower-energy d a t a at 2.5 GeV is shown. T h e
20 t
b
L
01.0
0.9
L--
0.8 cos 0~
L
0.7
0.6
F i g . 2. - C o n l p ~ r i s o n of c ~ p e r i m e n t a l D o i n t s ( f r o m ]]EINZ et ~ l l . ) a n d r e l a t i v i s t i c o n e - n e u t r o n e x c h a n g e c~tlculation a t i n c i d e n t k i n e t i c e n e r g y of 2.5 G c Y : z~, ][]'~]NZ el O1.; O, TL'I~KOT ('1 (~d.
fit is still reasonable a l t h o u g h not as good as at 2.8 GeV; the e x p e r i m e n t a l d a t a seem to h ~ v e a h u m p in t h e near b~ekward directions whereas t h e theoretical c u r v e seems to be going down monotonically. The reasonable fits o b t a i n e d in the present calculation at 2.8 GcV, lend credence to the possibility of single-nucleon exchange as a possible mechanism for the reaction p + p - > d + n +. :,k, ,
W e are t h a n k f u l to Dr. L. I~ESNICK for helpful discussions. W e would also like to t h a n k Mr. A. R I c c I and Mrs. V. SRI.nlVA~AN for help with c o m p u t e r p r o g r a m m i n g in this problem. This p r o j e c t was s u p p o r t e d by % g r a n t from the Natioual Research Council of Canada.
(~) T. YAo: Phgs. l~c~., 134, ~ tSl (1961). } (~) R. 3{. I{,,~.'~z et at.: l)ifJerential cro.~s-seelion ]or I~ ~I)--~d~ ,~, Michigan preprint (1967).
