IL NU()VO CIMENTO

VOL. XXXIV, N. 4

16 Novembre 1964

Relativistic Study of the Reaction p -k p -+ d q- W +. iV[, LE BELLAC (*), F. ~[. RENARD (*), D. SCHYFF (*) and J. TRA~ THA1NHVAN (*) Laboratoire de Physique Thgorique et Hautes Energies . Orsay

(rieevuto il 27 Giugno 1964)

Summary. - - A careful examination of the p ÷ p--~ d ÷ W + process is done in a covariant theory. The influence of different effects is studied in detail. The total production cross-scction, if the W + meson exists, is of the order of 10-a~ cm 2.

The s t u d y of the reaction p - ~ p - + d q - W has been proposed b y B E I ~ STEI~ (*) US being ~ possible way of detecting the vector meson W which is supposed to mediate the weak interactions. Despite the b a c k g r o u n d of strong interaction processes, this reaction has the a d v a n t a g e of allowing the use of large proton fluxes available from the accelerators. As BER~'STEIN pointed out, it is very easy to discriminate against unwanted bt's b y energy resolution (W width}. Furthermore, as the reaction p + p - ~ d - ~ W is a t w o - b o d y process, the analysis of final states m a y be easier t h a n for the reaction: -k J~" -+WZ-~z÷,V. The order of magnitude of the cross-section for this last reaction, as it was predicted b y LEE et al. (") and W~- et al. (a), lies between :tO-~8 cm 2 and 10 -3~ c m 2 - - t h e incident neutrino energy v a r y i n g from the threshold energy to 10 GeV. The comparison with the order of magnitude predicted for the reaction p + p - ~ d ÷ W is expected to be instructive.

(*) Postal address: Laborau)ire de Physique Th6,orique et Hautes Energies, Bgtimerit 211, Faculty4 des Sciences, ()rsay (Seine-et-Oise). (*) J. B~mxs~'eix: Pbys. Rec., 129, 2323 (19631. (~') T. D. LEe, P. 5[ARKSTEIN and C. N. Y.~x(~: Phys. lgev. Lett., 7, 429 (1961). (a) A. C. T. \Vu, C. t ). YAX(;, K. FUCHEL and S. HELLER: Phys. Rev. Lett., 12, 57 (196~).

RELATIVISTIC STUDY OF THE REACTION p q - p - * d + W

+

1097

The main contribution to this process is expected to be given b y the impulse approximation (neutron exchange diagram of Fig. 1); we calculate the cross-section for this reaction within this approximation and thus neglect meson-exchange contributions. To evaluate this diugram we need:

a) The n-p-d vertex. As expluined in ref. (~), the structure of this vertex is t a k e n to be = ~(n)[iyeA(t) -- ½ (n--p)eB(t)]$~u~(p),

(1)

where u~(p)=CYd(p), C is the charge conjugation matrix,

Fig. 1.- Impulse approximation diagram (the letters designate the fourmomenta).

Cy, C -1 = - - y ,T, Se is the deuteron spin vector, A(t) and B(t) are two invariant functions reluted respectively to a direct coupling and to a derivative coupling and which describe the deuteron structure.

b) The n-p-W vertex. Assuming the same weak-interaction-nucleon form factor (~) for both axial vector and vector coupling, this vertex can be written

~V(M2w) will be given b y the conserved vector current hypothesis, % being the polurizution vector of the W-meson, g is reluted to the Fermi coupling conscant G: g2=

G

M~

with

Gm~o "-- 10 -5 .

The m a t r i x element for the process p + p - ~ d + W

is t h e n given b y

M = g F ( M ) ~ {~(q) ~,,(1 + "fh) (~p --t --yd) m +~ i m [ i y , A ( t ) -- peB(t)]Uc(p) -- u ( P ) 7 ~ ( 1 ÷ yh) (7q--Yd)u--m ~'--im [irmA(u). -- q~B(u)]u~(q)} %~q, with

t = ( p - - d ) 2,

u = ( q - - d ) 2.

(~) M. LE BELLAC, F . M. RENARD a n d J . TRAN THANH VAN: ~:~uOVO Cimento, 34, 450 (1964). 70 - II Nuovo Cimento.

1098

~I. L E B E L L A C ~ F . 5I. R]~NARD~ DI S C I t l F F

and

J.

T R A N T I I A N H VAN

The differential cross-section in t h e c e n t e r - o f - m a s s s y s t e m is g i v e n b y

da

d-~=

( m )2]IVI 1 ~:

Ipl 4,o,~, ]~. t M

i~ ,

where E is t h e n u c l e o n energT, IV a n d p t h e mon-ienta of t h e v e c t o r m e s o n a n d t h e nucleon, respectively. W e h a v e p e r f o r m e d t h e calculation of t h e differential a n d t o t a l cross-sections for several W m a s s e s f r o m 700 MeV to 1600 MeV a n d different f o r m s for t h e n - p - d v e r t e x f u n c t i o n s t e s t e d in p h o t o d i s i n t e g r a t i o n process. W e n o w discuss t h e influence of t h e following effects: d e u t e r o n s t r u c t u r e , inclusion of d e r i v a t i v e coupling, w e a k i n t e r a c t i o n n u c l e o n f o r m factors. 1) In]luencc o] the deuteron structure. I t is described b y t h e t-dependence of t h e v e r t e x f u n c t i o n s A(t) a n d B(t). As is e x p e c t e d f r o m t h e large v a l u e of t h e m o m e n t a t r a n s f e r ( n 2 ~ m 2 ~ - - 2 m 2 at t h r e s h o l d for t h e p r o d u c t i o n of a W - m e s o n h a v i n g a mass of ] .3 GeV) this effect is v e r y i m p o r t a n t . I n a first calculation, we ne~o'lect t h e d e r i v a t i v e coupling, i.e., we p u t B(t)-=0. I f we neglect t h e d e u t e r o n s t r u c t u r e A ( t ) = A ( m ~) (*), we o b t a i n ~ e a r i n g ' s result (5). I t is larger b y a f a c t o r 10 t h a n B e r n s t e i n ' s result (~) obt a i n e d in u similar n o n r e l a t i v i s t i c calculation in w h i c h t h e d e u t e r o n is described o n l y b y t h e a s y m p t o t i c S w a v e function: lo 3~ ~v(r) = c x p [

~r] r

10-~

i

/

~

~

I

_

_

~

i

.~o-~7 .%_?ZTL-LT.ZZ:Z_:-2

(~v(r) is t h e n o n r e l a t i v i s t i c l i m i t of t h e a b o v e n - p - d v e r t e x w i t h o u t structure). U s i n g t h e v e r t e x f u n c t i o n A(t), as given

lO-S~_

i

3

s

7

Eoo (Gev)

~

Fig. 2. - Comparison of total cross-sections from different theories for a X~T mass of l l 0 0 3 I e V : ---A(t)iA(m2), B(t)--0; Bernstein's non relativistic calculation; A(t) from ref. (~) B(t)=0; . . . . A(t) and B(t) from ref. (4) with 10~=5.4°/o; - - - - - A(t) and B(t) from ref. (~) with pg=6.50~&.

(*) A(m 2) depends only on the deuteron binding energy. (5) j. N~_~R~_~G: Phys. /~ct'., 132. 2323 (1963). The slight difference in Fig. 2 comes from a bad normalization used in Nearing's paper [eq. (3)].

I~ELATIVISTIC STUDY

OF T H E

REACTION

p+p-~d~

W+

1099

in ref. (4), we obtain a result smaller b y a factor 10 ~ (see Fig. 2). This is n o t surprising since the m o m e n t u m distribution in the deuteron falls off v e r y rapidly. The phenomenological form of A(t) and B(t) has been determined b y the static properties of the deuteron and the deuteron photo-disintegration up to t - - m ~= - - ~ m 1 - . We use these functions in this calculation for larger moment u m transfer, namely t - - m : ~ - - 2 m 2, although we have no means of checking whether t h e y are still valid. 2) In]luence o/ the derivative coupli~g. - I n the nonrelativistic limit, the derivative coupling which appears in eq. (3) leads to the D-state wave function; however, its role increases with energy. We have calculated the cross-section for two sets of structure functions corresponding to a 5.4% and to a 6.5% D-state percentage. I n both cases, the derivative coupling increases the crosssection approximately b y a factor 100 (see Fig. 2). At this stage, the result (including deuteron structure and derivative coupling but taking F ( M ~ ) = 1) is smaller t h a n ~Nearing's b y a factor 100 and t h a n Bernstein's b y a factor 10. Figure 3 is a plot of the total cross-section ¢r calculated for 4 values of W mass:

2L 3

M=700

MeV

L I J-

::L

2

M =tO00

9

~eV

o x~

bz

t M = 1600 MeV

0

2

Fig. 3.

4

-

6 8 E (GeV)

10

:,,.-

Total cross-section for several W masses.

0

2

4

6 8 Eo~ (GeV]

'

I'0

Fig. 4. - Forward cross-section for several \V masses.

m w = 700 MeV, 1000 MeV, 1300 MeV, 1 600 MeV. The choice of m w does n o t seem to affect very much the magnitude of a z (far enough b e y o n d threshold). Figure 4 is a plot of the differential cross-section in the forward direction where, as a consequence of the presumed dominance of Fig. 1 graphs, it is the largest. (da/d/2)0= o is plotted for the same values of m w as before. 3) Effects o/ weak interaction ]orm ]actors. As BER~STEI-~" pointed out, the vector form fa~.tor, which, according to the conserved current hypothesis,

1100

~[. LE BELL&C, F. 3[. RENARD, ]3. SCItlFF ~tIi£~ J. TRAIN T H A ~ H VAN

is proportional to the isovector e.m. nucleon form factor, m a y give an increase of the W production cross-section if m w is near an isovector multipion resonance. I t is possible to make a rough estimation of IFF(q ~= m2w)Is b y using: a) the ~ resonance with a mass of 750 MeV and a width of 150 MeV, b) another resonance (6) at 1300 MeV with a width of 300 MeV in a first ease, a n d 500 MeV in a second, and fitting the nucleon form factor for ~[*~ 0. W h e n the W + mass varies from 1 GeV to 1.6 GeV, we obtain an increase of the cross-section b y a factor 10 to 100. The effects mentioned above appear to be crucial in the d e t e r m i n a t i o n of the W + production cross-section. Although our aim was to give a more realistic estimation t h a n previous calculations did (~)(5), there remain m a n y imprecisions, some of t h e m could be removed b y a b e t t e r knowledge of: a) deuteron structure for larger m o m e n t u m transfer, which could b e obtained, for example, from the reaction p ÷ p - - > d d - ~ + ; b) e.m. isoveetor form factor for timelike m o m e n t u m transfers which m a y be estimated b y p ÷ ~ - * e + d - e -. Lacking such information for the time being, we think t h a t the value a ~ > 1 0 -s5 cm 2 for m w'~= 1.3 GeV is plausible. Comparison with the cross-section for , ÷ A ~ - - ~ W ÷ t x ÷ A " the order of magnitude of which is 10 -37 cm 2, will emphasize, if the W+-mesons exists, the interest for W production in association with deuterons, once the experimental difficulties are overcome.

We wish to t h a n k Prof. B. D'EsPAGNAT, Prof. M. GOURDIN and Prof. p m MEYER for instructive discussions and constant interest. (b) j .

S. ] , E V I N G E R :

_Y~tOVO

Cime~to, 26, 813 (1962).

RIASSUNTO

(*)

si f a u n attento esame del processo p + p - + d ÷ W + in una teoria covariante. Si studia dettagliatamente l'influenza di vari effetti. Se il mesone W+ esiste, la sezione d'urto di produzione totale ~ dell'ordine di 10-35 cm 2. (*) T r a d u z t o n e

a c u r a della R e d a z i o n e .