Single-Δ production contribution to proton-proton inelasticity parameters

Nuclear Physics A379 (1982) 349-368 Q North-Holland Publishing Company

SINGLE-~ PRODUCTION CONTRIBUTION TO PROTON-PROTON INELASTICITY PARAMETERS J. CÔTÉ', M. LACOMBE and B. LOISEAU

Division dt Physique Théoriquet, Institut de Physique Nucléain, 91406 Orsay, France and Laboratoire de Physique Théorique des Particules Élémentaires 75230 Paris Cedex OS, France

and W.N . COTTINGHAM Physics Department, University of Bristol, England

Received 5 November 1981 Abstract: We present here a meson exchange calculation of the nucleon-nucleon peripheral partial wave inelaaticity parameters below 1 GeV lab kinetic energy . These theoretical estimates if used as constraints in future NN phase-shift analyses should reduce the ambiguities in theinelastic region.

1. Introdnction The problem of determining the nucleon-nucleon elastic scattering matrix from scattering data has in the past been greatly aided by the theoretical constraints given by the one-pion l .z) and the two-pion a) exchange contributions to the high partial wave amplitudes. To-day, all these amplitudes are well determined at low energies (T,an < 330 MeV) where inelastic scattering is small. At higher energies meson production becomes more and more significant and for example at T,ab = 650 MeV this inelastic cross-section accounts for 40% of the total cross section 4). In this region, due to the proliferation of parameters needed to describe the inelasticity, phase-shift analysis becomes progressively more difficult and ambiguous s-s) . However, it happens that, up to energies of T,ab ~ 1 GeV the inelastic cross section is dominated by the quasi two-body delta . (d ) nucleon channel 9). Meson exchange calculations of the peripheral part of this d-production are as appropriate as for elastic scattering. Pion deuteron production is phenomenologically the next most important inelastic channel but with a cross section of at most 9% of the total 4). It has no direct contribution from meson exchange and is plausibly produced predominantly in low partial waves and is here neglected. In this paper we present meson exchange calculations of the high partial wave amplitudes for the reaction proton (p) + proton (p) -> d(1232) + nucleon (N) . ' Present address: Centre Météorologique Canadien, H9P1J3 Dorval, PQ, Canada . t Laboratoire Associé au CNRS.

May 1982

349

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J. CSté et al. / Single-d production

The ~ and the p are the only low-mass systems that can contribute to this reaction and are the only ones calculated here. We compute by unitarity the inelasticity parameters of the elastic partial wave amplitudes . These theoretical estimates if used as constraints in future phase-shift analyses should reduce the ambiguities in

the inelastic region . We are not the only group to have attempted such calculations . Starting with Ferrari and Selleri 1°) many authors have fitted meson exchange models of dproduction and decay directly with data on the energy spectra and angular distribui') tions of the three-body final state . The models generally contain parameters in form factors at the vertices . These can be chosen to give qualitative agreement iz) with the data . In another approach initiated by Amaldi et al. and followed by us partial-wave projections are obtained for the nucleon-nucleon partial wave inelasticity parameters is-16) . The meson exchange models are particularly reliable for the high partial waves (J r 4) and these waves are insensitive to the parameters

of the form factors. is) The work of Kloet and Silbar includes a- but not p-exchange in the dia) production amplitude ; however, as pointed out by Haapakoski the inclusion of the p with quark model coupling partly cancels the large tensor force of ~r-exchange. In the calculation presented here, this cancellation is even more complete than in ref.' 4) since we use the stronger p-coupling favoured by more recent ~rN [ref . l')] and NN [ref . 3)] analyses. Although the asymptotically large angular momentum amplitudes are unaffected by the p, in the energy range considered here, there is a strong suppression of the F- and, to a lesser extent, the G-wave amplitudes . d production through the F and G partial waves gives a significant contribution to the inelastic cross section and should be reasonably well . predicted by the singleboson exchange model. ib) The work of Green, Niskanen and Sainio' S) and of Green and Sainio does include both ~r- and p-exchange in the d-production amplitude. However, in the above work 13-16)' boson exchange amplitudes are considered to be the Born approximation to a multichannel potential scattering amplitude . Unitary partial wave amplitudes are then generated by solving the multichannel scattering equations. As pointed out in ref.' a), the complexity of the equations and the difference of the inputs make difficult the comparison of the results. Furthermore, these methods of unitarization depend both upon the ambiguous ofF-energy shell extrapolations of the amplitudes and on the necessity of introducing additional unknown Nd elastic potentials . Unitarity corrections should not be important for the high partial waves and they can not be reliably calculated . We believe that the unadorned Born terms should first be compared to experiment and this we do here before investigating possible unitary corrections through the K-matrix scheme . In sect . 2 of this paper, we give the kinematics and the definition of the partial wave helicity amplitudes of d-production . Sect . 3 shows how NN inelasticity parameters can be related to these partial wave amplitudes . The a- and p-exchange

1. CSué et al. / Single-d production

351

contributions are calculated in sect . 4. The results are presented and discussed in sect . 5 where we also draw some conclusions. Explicit expressions of the ~r- and p-exchange amplitudes are given in the appendix. 2. Kinematics and partial wave amplitudes In this section, we outline the kinematics and spin algebra of the reaction p+p->d+N . There are only two possible charge final states, d+p and d++ n, and isospin symmetry implies that d++ production cross section is three times that of the d+. We therefore need only to calculate the amplitude for the latter process. 2 .1 . KINEMATICS

Fig. 1 shows our labelling for the production reaction . The indices 1 and 2 refer to the incoming protons, 3 to the outgoing d and 4 to the outgoing proton . p, and A, are the 4-momentum and helicity of the particle i. For external particles on their mass shell and in the center-of-mass frame with

P3 = (Ee~ k)~

Pa = (Eq -k)

one has for the s and t Mandelstarr variables s g (Pi +Pz) z = 4Ez

z t ~ (Pz - Pa)z = 2(M - EEP + pk cos B) , also

(2 .2)

In the above M is the proton mass, Ma the d-mass, TL the laboratory kinetic energy of the incoming proton and B the angle between p and k. 2 .2. PARTIAL-WAVE AMPLITUDES

Denoting T(A3,11a, a,, Az, 8) the transition amplitude of the process ôf fig. 1, we define the partial-wave helicity amplitude of total angular momentum J as, Tr(~s~ ~a~ Ai~ ~z) =

+~

J i

d(cos e)dâw (e)T(~s, aa, ai~ Az. 9) ~

(2 .3)

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J. Côté et al. / Single-d production

Fig. 1. Labelling of the d-production reaction .

where the dÂw (e) are the usual rotation matrices A =Ai - Az .

i9),

with

W =A3 -Aa .

Parity conservation, which requires TJ(A3, Aa, Ai . Az)=-Ti(-As, -Aa~ -Ai~ -A2)

(2 .4)

reduces the number of independent amplitudes from 32 to 16 . For ease of application to the conventional inelasticity parameters of proton-proton scattering, we introduce the amplitudes T(J, L, S, A s, Aa), where the initial protons are coupled in the usual L, S scheme . The singlet (L = J) amplitudes are T(J, J~ ~ r A3~ Aa) = TJ(A3~ A4~ 2, 2) - TJ(A3r A4~ -2~ -2) ~

(2 .5)

the uncoupled triplet (L = J), T(J, J, 1, As, Aa) ° Ti(As. Aa~ z, -i) - Ti(As. Aa~ -~~ z) ,

(2 .6)

and the coupled triplet (L = J ~ 1), T(J, J -1,1, A3, A a) _

1

X

1 1 ~ TJ(A3r Aa, 2~ 2)+ TJ(A3r A4~ -2r -2)1 1 1 1 11

(2 .7)

Our normalization is such that the formula for the inelastic cross section for initial unpolarized protons into all d, nucleon final states is N~r

~= z ~ 2P J.L .S.A3.Aa

(2J+1)~T(J,L,S,A3,Aa)I2 .

(2 .8)

with M3Mapk N= ~- . 4 In formula (2 .8), T(J, L, S, A3, Aa) is the pp --' d +p amplitude but a factor of 4 is included in N to account for d++ production .

1. CSté et al. / Singly-d production

353

3. Unitarlzation and inelsetldty parameters In our model, where we assume that single d-production saturates the inelasticities of proton-proton scattering, unitarity implies INN - SNNSNN = SdNS dN ~

where NN and dN label the final states . Eq . (3 .1) can be used to relate inelasticity parameters to the NN--> dN amplitudes . However, since our S, N will be calculated in the Born approximation, unitarity constraints might be violated in particular for low partial waves. Nevertheless, we shall first use the Born approximation for SdN , and then, consider a K-matrix unitarization procedure. 3.1 . INELASTICITY PARAMETERS IN BORN APPROXIMATION For a given J, we take the Livermore parametrization z) of the NN S-matrix . In the singet and triplet uncoupled states SNN

= cos R ez~

(3 .2)

and we obtain from eq . (3 .1)

with Y~=1V ~ IT(J~J~S~~3~aa)Iz . As.As

For the triplet coupled states cos R_ cos 2e e i sin 2e e

(3 .4)

e+ca~+a_+a~ i sin 2e cos R + cos 2e e

z+a-

Eq . (3 .1) gives for the diagonal matrix elements 1 with Yc..c .~ =N

~ I T(J. L. 1~ as, ~a)T (.T, L~~ 1, a3. aa)~ "

A3.A4

(3 .7)

We note that, if the right-hand side of eq . (3 .4) or (3 .7) becomes too large, unitarity is violated . The off-diagonal elements of the right-hand side of eq . (3 .1) are real in Born approximation, whereas those of the left-hand side are in general complex, which suggests that unitarity is generally violated in these off-diagonal terms. However, to the extent that the phase of the left-hand side which comes

354

J. CSré et al. / Single-d production

from higher orders in perturbation theory is small, we equate its magnitude with the right-hand side . This gives the equation for a 2 cos R1 cos R_[

sin44E

In our calculation we use the mixing parameter e given by the Paris potential 3) for energies above the pion production threshold. 3.2 . INELASTICTTY PARAMETERS BY K-MATRIX UNITARIZATION One way to ensure that the unitarity constraints are satisfied is to construct an explicitly unitary S-Matrix through the K-matrix SK

1+iK ~

(3 .9)

1-iK

= K+ - (KIVN~NN KdN~NN

KNN+dN, KdN~dN

(3 .10)

Our ansatz for the K-matrix is to set (1) KNN~NN = KNN ~ where KNN is the nucleon-nucleon elastic K-matrix calculated from the Paris potential extrapolated to high energies, (u) KNN~dN = SdN~2t ,

This is equivalent to the introduction, in the NN-sector of the S-matrix [eq. (3 .9)], of an effective complex K-matrix, the real part of which is KNN and the imaginary part is proportional to the transition probability to the dN channel. One has 1 + iKNN where for the uncoupled states KNI,,(J) = KNI,r (J) +4i Y~,

(3 .12)

KNN(L, L') =KNN(L, L')+4iY~,L " .

(3 .13)

and for the coupled states

If S are the NN Paris potential phases, we have for the uncoupled states KI,,N(J) = tan

s.

(3 .14)

J. CSté ~t al. / Single-d production

355

We then obtain, from eqs. (3 .2), (3.11) and (3.12), for the inelasticity parameters (3 .15) The corresponding formula for the coupled states can be obtained from eqs. (3.5), (3 .11) and (3.13) . It can be seen that, for a small NN-> dN transition amplitude and small S, eq . (3.15) leads to the same result as eq . (3.3); however, we here always satisfy tinitarity constraints. Had we not neglected the dN~dN transition, our results would be modified by an extra-term in the denominator of eq . (3.15) proportional to KAp~Ap . It could be possible to calculate the peripheral part of this amplitude arising from meson exchange and incorporate it in our scheme . Such calculations have been made, for example in ref. 14). This, however, goes beyond the scope of this paper in which, as stated in the introduction, we try to stay as close as possible to the simple Born approximation. 4. 0-prodadion heHdty amplitudes arising from ~r- and p-ezdumges We shall calculate the relativistic NN-~dN helicity amplitudes arising from ~rand p-exchanges (cf. fig. 2). We use the standard lagrangians for ~r(p)NN and aNd

~Z. A.

Fig . 2 . Feynman diagram for the ~(p)-exchange d-production amplitude .

vertices 2~. We give them below for particles of definite charge (other charge combinations can be constructed from isospin symmetry): ~a~pp

= ôaNNY~tYSY'~Y ~

-eP~ac = Brrx~G'YwV"tG-

and for the pNd : ~P+PQ++

='M+Me

T

4M ~~"(a"v" -a"V~)~G,

~G~.(a"v~ -a"v")y"rs~G+h .c.

(4 .1)

(4 .3)

(4.4)

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!. Côté et al. / Single-d production

In the above ~, V", ~ and ((<" represent the ~-, p-, N- and d-fields respectively . z° .zl) and ~"  _ The y" (W = 0, 1, 2, 3) and ys are the usual Dirac matrices 1.

zt (Y"Y~ - Y~Y")" We take the usual -rrNN coupling

z

$aNN

4a

=14.43 ,

(4 .5)

and the strong pNN coupling favoured by the more recent ~rN analyses l') and by the Paris potential a) (ôPNN

4~r

)2

=

(ôPNN - SPNN

32,

)2

4~r

=

0.6

(4 .6)

where ôPNN = öPNN + ôPIVN~

ôPNN - ôPNN

For the ~r(p)Nd coupling we take the SU(4) quark coupling modified by a symmetry breaking factor b which we introduce so as to reproduce the experimental zz), ~rNd coupling as computed from the d width (4 .7)

ô+rNd = b~ô+rNN e ôPNA = b "/zs gPNN .

(4 .8)

In the calculations presented here, we take l', =110 MeV which yields b =1 .21 . It should be noted that the relative sign from SU(4) symmetry is responsible for the cancellation between the ~r- and p-exchange amplitudes . As shown in the appendix, the sixteen independent helicity amplitudes corresponding to ~r- and p-exchange can be written as C+n(P)

t ~3~ ~4, ~1~ ~2) CC u (p) - COS B

n-0

n

(4 .9)

with

an(P)=

2 (EEP - Mz) + mÂ(P)

From eqs. (2 .3) and (A .25) one obtains 7i

(P)

(As~ Aa~ ~lr ~z) = 2(-1)A-"

2pk

z

"

(4 .10)

J+n

n=0 1-~l-n~ ~~

X C~(P) n (~3~ ~4~ ~1, ~2)Qf(~=a(P)) ~

where (,'; ,'nz~m) denotes the usual Clebsch-Gordan coefficient and Q,(a) are the Legendre functions of the second kind. The expressions for Cn (P) (.13, .la, ~ 1, az) are given in the appendix .

!. CSté et al. / Singlt-d production

357

The field theory formalism, used here, considers the particles to have definite mass which in particular ~s incorrect for both the p and d . The effects of the finite width of the p could be accounted for by including dispersion in the propagator as done in the Paris potential. We do not expect this effect to be important, we have not so far included it. On the other hand, the effect of the finite width of the d is very important since it allows for large pion production below the nominal d-production threshold. We account for this width by considering the d to have a dispersion in mass . This modifies YJ [eq. (3 .4)] and YtL [eq. (3 .7)] into ~

3

(f-M)z

duw(u)~k(u) Y,(s)=~~ ~ IT(J,J,S,a s ,aa )I z , 4E 7l J(M+m  ) Z as .Aa MsP

Y~,~-(s)=--~~J

(~-M)2

(4.12)

duw(u)~k(u)

x ~ IT(J, L~ 1, as, aa)T(J~ L~~ 1, as~ aa)I ~

(4.13)

~3.A4

The weight function w(u) is chosen to be the normalized square modulus of the I = Z, J p = (z) + aN-> aN partial wave amplitude f. We choose for f the parametrization given by Nagels et al. z~ in subsect. 5.8 : f

_

1 Mal'(u) q(u)Mâ-u-iMal'(u)'

(4.14)

with B(u =Mn) Meh(u)= 9 3 BnNa 4~r B(u) '

(4.15)

B(u)=1+R zg z (u) ,

z z 4 (u)=âu~1_ u (M +mn)+ u z(M z- m~) zJ .

(4.16)

In the above formulae we used R =1 fm and Me =1232 MeV. The weight function w(u) is w(u)=

~

1/

If(u)IZ

+(M+m~) 2

(4.17)

du' I f(u')1 2

For the other mass parameters we use M = 938.259 MeV ,

mn = 134.978 MeV ,

mP = 760 MeV ,

(4 .18)

J Cüté et al. / Single-d production

358 YO . ~

NN-+NNa ~AETtAI BRAVE CROSS SECTIONS

900. T 1~6~

M ~V)

Fig. 3. The individual and total contributions to the ~r-production cross section of the high partial waves of the rr- and p-exchange d-production amplitude calculated with the K-matrix formalism. The references for the experimental points can be found in the report of Flaminio et al. 4).

5. Results and condnsions The results of these calculations for T~ < 850 MeV are illustrated in figs . 3 to 9 . Fig. 3 shows the calculated contribution to the inelastic Tr production cross-sections of partial waves with 1, 2 calculated from the K-matrix formalism. Also included are experimental values 4). One can see the relative importance of the various partial waves inelasticities . There is some speculation in the literature as to possible

T~(MW)

Fig. 4 . The'DZ inelasticity parameter in the simple Born approximation for the a-exchange (" " ") and a +p exchange (- " - "). The full line (-) is the a+p exchange contribution computed with the more sophisticated K-matrix scheme . Also plotted for comparison are results of the empirical phase shift analysis of Arndt and Verwest s) (*) and of Hoahizaki e) (~) together with the theoretical calculations of Kloet and Silbar ") (x) and of Green and Sainio'6) (7) .

J. Côté et al. / Single-d production

359

T~~(MW) Fig. 5. As inTig. 4 but for'G, (upper curves and points) and lIs (lower curves).

dibaryon resonances in the tD2 and 3 F3 states t8 ). We find large cross sections for d-production in these states . Figs . 4 to 9 show inelasticity parameters R from a-exchange calculated in the Born approximation, and ~ + p exchange calculated in the Born approximâtion and also with the more sophisticated K-matrix formalism. For comparison, we also e give some results of the empirical phase-shift analysis of refs . .a) and the theoretical te). calculations of Kloet and Silbar (KS) ts) and of Green and Sainio (GS) One can see a strong suppression of the -rr-exchange inelasticity in t D2 and 3F3 and to a lesser extent in 3 FZ and tG4 partial waves when p-exchange is included . This cancellation is a consequence of our assumed SU(4) symmetry and of the strong pNN coupling. All higher partial waves are much less strongly affected. The difference between the simple Born approximation and the K-matrix calculation

Fig. 6. Aa in fig. 4 but for 3F3.

360

1. CSté et al. l Single-d production 3Q .

NN INELASTICITY 9N5

QQ . :

X

oc

Q.

300 .

~

400 .

_.

500 .

600 .

700 .

800 .

900 .

T Ie6tM~V ) Fig. 7 . As in fig . 4 but for'H s,

can be seen to be very small except for the 3PZ, 1 DZ, 3 FZ and 3F3 partial waves above 600 MeV. Since the unitarization scheme is model dependent our results are more reliable when this dependence is small. The 3PZ results given in fig. 8a illustrate how things go wrong with a peripheral estimate of a low partial wave . The shorter range p-contribution is bigger than that of the ~r, With the simple Born approximation, unitarity is violated above 600 MeV and, although the K-matrix treatment gives a unitary amplitude, the results are very model dependent. The relativity big effect of K-matrix unitarization on the 3FZ above 700 MeV is due to its coupling to the ill-determined 3PZ. Except for 3 PZ, our numbers compare well with the empirical analyses . For all but the 3PZ and 3Fs3H4 waves our inelasticity parameters are in reasonable agreement with those of GS . However, they differ from those of KS whose results are smaller than ours for all but the'DZ wave . Looking in detail at some individual partial waves. For the 'DZ we are in agreement with the empirical analysis . However, the large contribution of pexchange illustrates the sensitivity of this partial wave to the shorter range forces and at energies larger than 600 MeV, this very good agreement is model dependent. Below 600 MeV, the observed inelasticity appears to be in excess of ours and clearly deuteron production could be significant in this relatively low partial wave . The partial waves with J , 4 except for 1G4 at TL > 700 MeV are dominated by -rr-exchange and are the least model dependent. The l Ib and 3 H4 waves, which are very small in the empirical analysis of Arndt and Verwest, are not altogether negligible in our calculation . Some of the inelasticity, which they ascribe to the 3Hs wave and which is bigger than theory, could perhaps be assigned to these tIb and 3Ha waves.

1. Côté et al. / Single-d production

361

T I~b~MoV )

T I~b~ M oV ) Fig. 8. As in fig. 4 but for

(a)'PZ and (b) 3 FZ.

We draw several conclusions from this work: The a-p cancellation which is important in 1D2 and 3F3 seems to be in agreement with experiment. This supports our choice of the pNN coupling strength and the SU(4) generalizations to the aNd and pNd couplings. The simple Born approximation is good for the calculation of the high partial wave inelasticities, a fact which gives confidence to the reliability of the results, since they are independent of ambiguous unitarization procedures and of the inclusion of form factors. We suggest that phase-shift analysis in the inelastic region would be improved by taking high partial wave inelasticities from meson exchange calculations . Our numbers are available for such use up to 1 GeV and beyond on request.

362

J. CBté et aJ. / Singlc-d production 15 .

~

NN INELASTICITY

a F~

Of~

O

Fig. 9. As in fig. 4 but for (a) 3Fa (upper curves and pointa) and as (lower curve), and (b) 3H4.

We thank Prof . R. Vinh Mau for his continuous interest in this work and for useful comments and discussions . One of us (W .N .C .) acknowledges the hospitality at the Division de Physique Théorique de l'IPN à Orsay and the support during this collaboration. Appendu In this appendix we give the detailed calculation of the relativistic helicity amplitudes Tac~>(A3~ ~La, a,, ~iz) . The Dirac spinor of helicity a is u(P, a)=6(P, ~)XÂ~2 (~) ~

(A .1)

1. CSté et al. / Single-d production

363

where b(P,

a)

2M(~

with

Z z

E

-p

2PMJ ,

(A.2)

2.

=M

It is a solution of the Dirgc equation: The 2-component spinor Xâ~ z (~) has spin projection a in the direction p. The relativistic spin z, d-state of helicity A, uv(p, A), can be constructed by coupling the relativistic spin-1 wave function eQ(p, A') of helicity A', with the Dirac spinor u (p, A ") as uP(P,

A) _

( 1 Az"I aZ )eo(P, a~)u(P, A~ ~

~  ' -i .o .-i .t ~

(A .4)

The polarization four-vector eQ(p, ~l) satisfies e~(P,

a )Pa = 0 ,

(A .5)

a) =0,

(A.6)

which implies the Rarita-Schwinger equations z3) P~uQ(P,

zl) With the momentum p along the z-axis one has eQ(P, a ) _ ~~AO + (0, _Ea (P)) ,

(A.8)

with

(

éo=lM,O,O,_ Eti(f)=

E -M

(A .9)

M ~,

~~(1, ti, 0),

Eo(p) _ (0,

(A .10)

0, 1)

Using the Feynman rules of subsect. 2.2 of ref. z°) one obtains for the ~r-exchange d+ production amplitude: 0

T~(as, ~a, ~i, ~s, g)= M suo(Ps, as)~(Pa, aa) m ~u(Pi, ~i)u(P2, Az) SarmrBxe_ ~ _

YsPi

364

1. Côté et aJ. / Single-d production

The initial state can be written as (A.12)

u(pl, ~1)u(Pz, ~z)=6(Pi, al)b(Pz, ~z)Xâ; z (~)Xlâz(P) =b(Pl,al)b(Pz,~z)a~ "

and the final state as

z

s

,

-azlAl-az/

Xa ' az(~),

(A.13)

.

(A.14)

3

1

uo~3,~3)u~!'a,~4)-~

z 1CA1

Â3IÂg/ev~3,~3)

\Âg

x b(Pa, A s)b (Pa, aa)(Xâ3z (~)) + (X 1â. (~))+

zlz~/z

~` /1 _ -~ a'=0 .1 As

as A

À3

i

I

s~

~ .

-Âq À3-A

xeP(ps, ~3)b(Ps, ~3)b(Pa, Aa)(Xa5-a.(~)) +,

where

(A.15) (A.16)

b (P, A) = b + (P, ~ )Yo

Eq. (A.11) with eqs. (A.13) and (A.15) give T~(~3, ~a, al, az, g)=

~ a .a' .A j

i

/i

_

A3

~1

I

s

_~z Al - Az 1

\/1 A3

ilz\

A3 as

Sr

_~a A"s - Aa A(~3, ~a, ~1, ~z) , a,~ - cos B

x eQ (ps, ~ s )P i

where A(~3, ~a, A1, Az) _ ~

8

N

~b(P3, ~3)b(Pl, al)b(Pa, ~a)Ysb(Pz, Az) " 2P~

(A.17) (A.18)

If one chooses p and ~ to be in the xOz plane (~ = 0) the rotation which brings zl) ~ into ~ can be described in terms of d,'~w ~ (B) and one has xw~(~)=E dww~(B)xw(P), w

then

(A.20)

Using (A.8)-(A .10) with one obtains

(A.19)

eâs(~) _ ~ d,1nAi (9)e m(P) ,

(A.21)

~ 1 ~,, , ~eQ~3,~3)'P1 - a0sAj0+(il(À3)d0A3(e)

(A.22)

J. Côté et al. l Single-d production

365

with

(A.23) Eq. (A .17), with the help of eqs. (A.20) and (A.22), can be written as T~(~3,

~4, ~1, ~2, B) °

.3n ~ac03 1e~ ~3 A(~0 2

2

S

2)

\Â 3 2

,

2

S

x~,tl -.lzlAl- .lz/CA 3 -À4IÀ3-À4~ Using the relation 21)

(A.24)

X(ap~lAjp+a1(À3)d0A3(B))dAt-Az.A3_Aa(e) .

dv.t,nt (B)dwzms (B)

_

~ r3wama

Jl

Jz J3

Jl

Jz J3_

~1

~2 /~3

ml

m2 m3

I

r ~ dwsms (B) "

(A.25)

One can then see that eq . (A.24) can be written as eq . (4.9) with 1 Cn

(~3,

~4, ~1, ~2) =

A(~3,

~4, A1, ~2)

A3 x

l1

\as

xal(A 3

,

CO

~1 3 1A 3/~A 1

- ~4 ~3 - ~4

2 2, 2 A3IA3/~al )~

s

A1_Az

,

AzIÂI nAz/

a.A~ S 2 , 2 -Azlal-Az/CA3

11

n

0 al_~iz

~~

2 S , -AalA3- .la/

s

À"s_da

(A.26)

From the lagrangians (4.3) and (4 .4) one obtains forthe p-exchange d+ production amplitude T° (~s, ~a, al, ~z, 9) = ~g° xallv(Ps, ~s)ll(Pa, aa)~m ° -t

1(

)~

X l ôvNNy~ Q- ôPNN (P 2Ma

_M~

Mo

X So~YwY~°~-Bv~Yv. ( Pz +Pa)~ 2M J x u(pl, A1)u(Pz, ~z) "

(A.27)

J. Côté et al. l Single-d production

366

The prime on the y-matrices indicates that they refer to the 2-4 vertex. Using eqs . (A.12) to (A.15) and after some reductions, this amplitude can be cast into two terms : (A.28)

TP (As, Aa, A 1, Az, 6) = T i + TZ ,

where 7,P1 =

~ s.A 3 X

1 A r3

A3IA3/ \A1

LAO(A3, Aa, A1, Az,

and TzP -E a;

(

lr As

-AzIAI S)SA30

s

Az/CA3

-AalA3

+A1(A3, Aa, A1, Az,

s

Aa/

a -A2 . ai-Aa( B ) S) d0A3(e)~dA' , Cr cOS

p-

B

(A.29)

z NI >(xâ~Z(~))+(xl-â:(~))+eâ~(~) ~ Qxâ~z(~)xlâ~(P)

A3 A 3

xAz(A3, Aa, A1, Az) , ~P - cos 9

(A.30)

with E1 o(A3, Aa, A1, Az,

S)=

b(Pa, Aa)b(P3, As)Ys ~ 2k P

x J 8 ~ Yô ~k +2A a(Ea - Me)$s ~ (A.31) A1(As, Aa, A1, Az, s)

_ ~ 2k b(Pa, A3)b(Ps, As)Ysal(As)B(s)b(P1, A1)b(Pz, Az) , P

(A.32)

2

A z(A s, Aa, A1, Az) _ -~

$pNN$pNd -

2Pk

r

r

b(Ps, A s)Ysb(Pl, A1)b(Pa, Aa)YoYsb(Pz, Az) ,

(A.33)

where B(s)-M~$p~

2Ey Ma $oNNYoYô[1 - (4s - 3)Ysysl} ~ 1K

(A .34)

Eq. (A .29) has a reduction to (4.9) identical to that of eq. (A.24) . After some lengthy but similar manipulation eq. (A.30) can be also reduced to the form of eq.

l. t.Sté et al. / Single-d production

367

(4 .9) and the sum of the two yields : CR(A3~ Aa~ At . A2)-

i ?,( Z A3IA3/ \At

1 C0

x

_i

t z n ~,t1o(As~ Aa, At . Az~ n) -AalA3-Aa/

~A3

1

,.a ;

A3 s

x~At -Az

x

z n , -AzIAt-Az/

2 12 A3 A3 11

2 At

A3 - Aa

2 A3

S 2 -Aa A3-A

n

~ At-Az

n

11

s

S 2 -Az At-Az

As A3-A

f1t(As . Aa~ At~ Az~ s)

A3IA3/

z x \ As - Aa z x~At

i I 1 ~~

Aa As

z

As -Aa

n \ i . -AzIAt-Az)

A3 As - A (A .35)

References 1) P. Cziffra, M.H . MacGregor, M.J. Moravcsik and H.P. Stapp, Phys. Rev. 114 (1959) 880 2) M.H . MacGregor, R.A . Arrdt and P.M. Wright, Phys . Rev. 169 (1968) 1149 3) M. Lacombe, B. Loiseau, J.M . Richard, R. Vinh Mau, J. CBté, P. Pirès and R. de Tourreil, Phys . Rev. C21 (1980) 861 4) V. Flaminio, LF. Graf, J.D . Herser, W.G . Moorhead and D.R.O . Morrison, CERN report CERN -HERA 79-03 (1979) 5) J. Byatricky, C. Lechanoine and F. L.ehar, Preprint D. Ph . P.E. 79-O1, CEN Saclay, Oct. 1979 6) R.A . Arrdt and B.J . Verwest, Texas A. & M. University preprint DOE/ER/05223-29, and contributed paper to the Polarvation Symp ., Santa Fe, Aug., 1980, ATP Proc . no. 69, p. 179 7) R. Dubois, D. Axer, R. Keeler, M. Comign, G.A . Ludgate, J.R . Richardson, N.M . Stewart, A.S. Clough, D.V . Bugg and J.A. Edgington, Queen Mary College preprint 8) N. Hoshizaki, Prog. Theor. Phys. 60 (1978) 1796 ; 61 (1979) 129, and private communication 9) S.J . Lindenbaun, R.H . Stemheimer, Phys . Rev. lOS (1957) 1874 ; S. Mandelatam Proc. Roy. Soc. A244 (1958) 491 10) E. Ferrari and F. Selleri, Nuovo Cim. 27 (1963) 1450 11) G. Wolf, Phys . Rev. 182 (1969) 1538 ; E. Boric, D. Drechsel, H.J. Weber, Z. Phys . 267 (1974) 393 ; G.N . Epatein and D.O . Riska, Z. Phys . A283 (1977) 193; B.J. Verwest, Phys. Irett . 83B (1979) 161 12) U. Amaldi Jr ., R. Biancsatelli and S. Francaviglia, Nuovo Cim. 57A (1967) 85

368

1. CSté et al. l Single-d production

13) W.M. Klcet and R.R. Silber, Nucl . Phys . A338 (1980) 281; R.R . Silber and W.M . Klcet, Nucl . Phys . A338 (1980) 317; R.R. Silber, private communication 14) P. Haapakoski, Phys. Lett. 48B (1974) 307 15) A.M. Green, J.A . Niskanen and M.E. Sainio, J. of Phys . G4 (1978) 1055 16) A.M. Green and M.E. Sainio, J. of Phys . G5 (1979) 503 17) G. Höhler and E. Pietarinen, Nucl . Phys . B95 (1975) 210; E. Pietarinen, Helsinki report (1978) ; P. Gauron, Paris report (no. IPNO/TH 78-07 (1978) 18) R. Vinh Mau, Comptes Rendus de la IXème Conférence Internationale sur la Physique à Hautes Energies et la Structure Nucléaire, Versailles 6-10 juillet 1981, Nucl . Phys. A374 (1982) 3c 19) M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) 20) M.M. Nagels et al., Nucl. Phys. B147 (1979) 189 21) S.U . Chung, CERN report CERN 71-8 (1971) 22) J.J.J. Kokkedee, The quark model (Benjamin, New York, 1969) p. 72 23) W. Rarita and J. Schwinger, Phys . Rev. 60 (1941) 60