Coherent pion photoproduction on the deuteron in the Δ resonance region

NUCLEAR PHYSICS ELSEVIER

A

Nuclear Physics A 593 (1995) 435-462

Coherent pion photoproduction on the deuteron in the A resonance region* P. W i l h e l m ,

H. Arenhtivel

lnstitut fiir Kernphysik, Universitiit Mainz, D-55099 Mainz, Germany Received 4 April 1995; revised 3 July 1995

Abstract

Coherent pion photoproduction on the deuteron in the A resonance region is calculated adopting a nonrelativistic model based on time-ordered perturbation theory without rescattering. Results for cross sections and polarization observables corresponding to the polarized photon beam and/or the deuteron target are presented for which the influence of the nonresonant Born amplitudes and the sensitivity to several multipoles of the elementary photopmduction on the nucleon are studied in detail. Furthermore, the role of a two-body mechanism where photon absorption and pion emission take place at different nucleons is discussed.

1. Introduction

Electromagnetic pion production on light nuclei is of fundamental interest and thus constitutes a major topic in medium energy nuclear physics. One of the interesting aspects is that it can provide complementary information on the elementary pion production process on the nucleon. Among these reactions, coherent pion production on the deuteron is of particular interest because, first of all, its relative simplicity allows a more detailed microscopic treatment and, secondly, the deuteron may serve as a neutron target for the study of pion production on the neutron. For example, the coherent reaction is sensitive to the coherent sum of the ~,p ~ 7r°p and Tn ~ "nOn amplitudes. Since the extraction of elementary amplitudes essentially relies on an interpretation in terms of the plane-wave impulse approximation ( P W I A ) , it is necessary to study its validity, i.e., to have a quantitative estimate of all possible mechanisms going beyond the PWIA. * Supported by the Deutsche Forschungsgemeinschafi (SFB 201 ). 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 5 ) 0 0 3 4 8 - 7

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P Wilhelm, H. ArenhSvel/Nuclear Physics A 593 (1995) 435-462

Early calculations of the "yd ~ ~-°d reaction have suffered from an approximate treatment of the Fermi motion (see, e.g., Refs. [ 1,2] ). In a systematic study of the factorization approximation to treat the Fermi motion in PWIA, Rekalo and Stoletnii [ 3] have demonstrated its shortcomings. Rescattering of the produced pions has been investigated by Bosted and Laget [4] in lowest order. Pefia et al. [5] have considered a dynamical model for the NA interaction for various reactions presenting also some resuits for 9'd ~ 7r°d. Koch and Woloshyn [6] have emphasized the importance of charge exchange rescattering in lowest order for near threshold production, while Garcilazo and Moya de Guerra [7] investigated the influence of small relativistic components of the deuteron wave function and the dependence on pseudoscalar and pseudovector ~rN coupling. A recent PWIA calculation has been performed by Blaazer, Bakker and Boersma [8] with special emphasis on polarization observables. However, they do not present the energy dependence of the reaction over the whole A region and give no total cross section results. In the present paper we also want to restrict ourselves to the PWIA while a more realistic treatment including pion rescattering will be reported in a forthcoming paper. Our justification for such a procedure is the fact that the PWIA is the primary process against which all other effects will be gauged. Thus, our present aim of discussing the PWlA thoroughly is twofold. First of all, we want to study systematically the sensitivity to details of the elementary production amplitude, like, for example, the electric quadrupole excitation of the A resonance and to the nonresonant background amplitudes. This has not been done previously in such a detailed manner. For example in Ref. [8], the Born terms are taken from Dressier [9], the A term from Tiator et al. [ 10] and the to term from Ref. [4]. Thus, it is not clear for the reader how well for example the real part of the dominant elementary Ml+ multipole is described. Secondly, we want to go beyond the pure one-body process on which the PWIA is based by considering explicit two-nucleon propagation in the Born terms. This leads to additional two-body mechanisms which have not been considered before and which we want to analyze. Another new feature beyond PWIA is the occurrence of an imaginary part in the nucleon pole contribution as required by unitarity. We use a nonrelativistic model including pion, nucleon, and A degrees of freedom which is based on time-ordered perturbation theory. It describes the elementary production amplitude in a consistent way and provides a well-defined transition from the elementary process to the nuclear process. As already said above, this work will thus constitute a thorough basis for a forthcoming more realistic calculation. It will include a dynamical treatment of pion rescattering in the A resonance region via successive creation and decay of the resonance because the process ~/d ~ ~°d is an ideal tool to study the NA interaction within the two-nucleon system. Preliminary results of our model including rescattering amplitudes have already been published [ 11 ]. The paper is organized as follows. In Section 2 the multipole decomposition of the transition matrix is given and the observables in experiments with polarized photons and/or oriented deuterons are defined. The model for the elementary (7, ~r) amplitudes is developed in Section 3. Its predictions for the sum of (,y,~0) amplitudes on the

P Wilhelm, H. Arenh6vel/Nuclear Physics A 593 (1995) 435-462

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proton and neutron are compared with multipole analysis data. The formal yd ---, 7r°d amplitude is derived in Section 4. Finally, our results are discussed in Section 5.

2. C r o s s s e c t i o n a n d o b s e r v a b l e s

We consider the yd ~ ~-°d reaction in the photon deuteron center-of-momentum frame (c.m.). Using a coordinate system with the z-axis along the photon momentum k and the x-axis in the direction of maximal linear photon polarization [ 12], in which the pion momentum q has spherical coordinates 0 and ¢, the reaction matrix reads

Tin'am(q, k) = ei(a+m)¢ tm,am( W~,d, 0).

( 1)

Here, 3. denotes the circular photon polarization and m I (m) the spin projection of the outgoing (incoming) deuteron on q (k). Moreover,

W,d=o~+~+M~, :Wq+V/~+M

d,

o~-- Ikl,

COq=V/-~+m ~,

(2)

gives the invariant energy where Md and m,~ are deuteron and pion masses, respectively. Using a multipole decomposition, the t-matrix can be expressed as 1

tm'am(W,a,O) = --~ ~ E L[~)--' (g,~OIm'ljm')(lmLAljm+ A) g.j -

dm+am,

(0)

(3)

in terms of reduced matrix elements E~.j/Me.j L L of the electric/magnetic multipole operators for asymptotic pion deuteron angular momentum g,~ and total angular momentum j. The notation g = ~ 1 is used. For the d-matrix as well as for the normalization of reduced matrix elements the conventions of Ref. [ 13] are used. Angular momentum and parity conservation give the following selection rules:

[ j - II <~g~r,L<~j+l,

(_)L+t.

=

{-!

+1

for Ep.j L , for Me.j

(4)

and for the t-matrix the symmetry

t-m,-a-,, = (-)l+m'+A+m tin'am.

(5)

All observables have the general form

O_.CZ

Z

* Y d T~,]~nl2Cn,m, Tm,ampaapma~.

(6)

7nt~ m~Am

with

1 Iql~//--~2d+qZ~d +k2

c - -167rz Ikl

W2d

(7)

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Here, pr and pd are the density matrices for photon and initial deuteron polarization, respectively. A polarimeter for final deuterons would be formally described by the operator 12. However, in this work we will always set 12 = 1, i.e., we do not consider any polarization analysis of the outgoing deuteron. On the other hand, experiments with polarized photon beams and/or deuteron targets are planned, e.g., at MIT/Bates. Following Ref. [ 12], pr is characterized by the degree of linear and circular polarization P~ and Pc~, respectively, and pd is characterized by the degree of vector and tensor polarization P~ and pal, respectively, with angles 0d and ~bd defining an orientation axis. Choosing this axis as quantization axis, pd becomes diagonal. In general, the diagonalization of pd cannot be achieved by a mere rotation of the quantization axis. It is, however, possible for presently available sources of oriented deuterons. Finally, the differential cross section can be expressed in terms of the unpolarized cross section dtro/dO, the photon asymmetry Z, the vector target asymmetry Tll, the tensor target asymmetries T20, T21, T22, and various photon-target asymmetries T~M, TfM as

dodtr= dO-0dO[ 1 + Per2~cos(2~b)

1=1,2

M~>0

)]

+ Pcr~M sin [M(q~d -- ~b) + ~t1~r/2]}dlo(Od) i

+ P~ E

TetMc°s(~bM -- StlTr/2) dMo(0a)

'

(8)

M=--I

where CM = M(~bd - ~b) + 2~b. Explicit expressions for these asymmetries in terms of the t-matrix can be found in Appendix A.

3. The TN ~

~ N amplitude

In this section we discuss our model for the amplitude of the elementary pion photoproduction reaction on the nucleon. 3.1. Nonresonant amplitudes

First, we consider the nonresonant background amplitudes. The starting point is a 7rNN vertex v~. With outgoing pion momentum q, cartesian isospin index a, and in an otherwise obvious notation, it reads v~ = - f ~ iO'NN " qrr~,a. m~r

(9)

P Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462 7T

439

\ \

N

N

(~)

(b)

(d)

(o)

(~)

Fig. 1. Nonresonant amplitudes for vN ~ ~'N: (a) direct, (b) crossed nucleon pole, (c) and (d) two time-ordered contributions to the pion pole and (e) the seagull term.

The yNN vertex VrN is given by the nucleonic current JN which in the nonrelativistic limit consists of a convection and a spin part, i.e., U~,N= - ~ " jN(k)

=-~"

(~

~(t~+PN)+

~+k ) 2---M-~NiO'NNXk ,

(10)

with photon polarization vector a, nucleon mass MN, incoming (outgoing) nucleon momenta PN ( / ~ ) , and e = ~ (Xs "at-Xv'/'NN,3) , Xs=)(p-+-/~'n,

)(v = ) ( p - - , ~ n ,

,~ E { e , K } ,

(11)

where ep/n and Kp/n are the charge and anomalous magnetic moment of proton and neutron, respectively. Here we have assumed a covariant normalization for photon and pion states

(k'lk) = (2~r) 3. 2w~(k' - k),

(q'lq) = (27r) 3" 2Wq~(q' - q),

(12)

but (P~IPN) = 6(p~ - PN) for the nucleon. Time-ordered perturbation theory then leads to the direct and crossed nucleon pole amplitudes for yN ~ IrN shown in Fig. I. Introduction of the pionic current together with the yTr+Tr- vertex results in the pion pole amplitude in Figs. lc, d. Finally, the seagull amplitude in Fig. le is uniquely fixed by the gauge invariance constraint. Of course, pion pole and seagull amplitudes can contribute to charged pion production only. They belong to the isovector ( - ) amplitudes. Nevertheless, we will need the pion pole amplitude later on when we have to fix the yNA coupling, since it contributes to Ax3/: the background in the .,, 1+ and ~3/2 ~1+ multipole amplitude. The notation and explicit expressions for the amplitudes in Fig. 1 are given in Appendix B where the relevant formulas for the multipole decomposition as well as for the isospin decomposition of the yN ~ 7rN amplitude are also listed. As will be shown below, the model described so far predicts a nearly vanishing E~-+ (and also E~0+) multipole. Therefore, it does not satisfy the prediction of the low energy theorem (LET) for this s-wave amplitude [ 14]. In the ~rN pseudovector coupling scheme which fulfills the low energy theorem, the leading order O(m,~/My) of the low energy expansion of E~-+ arises from so-called Z-graphs with intermediate

P. Wilhelm, 1t. Arenh6vel/Nuclear Physics A 593 (1995) 435--462

440

nucleon-antinucleon states [ 14] which are not contained in our model. Due to the large mass of these states, their propagators can be treated as constants. Therefore, for all energies considered here, Z-graphs behave (also inside the nucleus) like contact graphs. Although it is not clear whether these Z-graphs represent the real physical mechanism for the E~-+, we add an effective current operator of contact type,

j ~ + ( k ) = -47riBa3 (1 + m~NN)Ed+ +,th OrNN,

(13)

in order to allow to study within our model the influence of the E++ multipole on the process 3M ~ zr°d. Denoting the invariant energy of the elementary reaction with wrN, Eq. (13) generates the multipole contribution

~E~+(wrN ) = m,~ + MN E~_th, wrN

(14)

with threshold value E~-+th.

3.2. Resonant amplitudes Next we consider the resonant part yN ~ A ~ 7rN of the amplitude for which adopt a conventional isobar model. Such a model assumes that the zrN interaction in P33 channel proceeds exclusively via the A resonance. It has been widely used in past (e.g., Ref. [ 19] ). Within this model the ~rN elastic scattering amplitude in the channel reads

t,,~( z ) = v*ag,, ( z )v,,.

we the the P33

(15)

The dressed A propagator in the A rest system is 1

ga(z) =

z - M°a -

,

(16)

-~A(z)

with the A self-energy

2a ( z ) = vago( z )vta,

(17)

which defines an energy dependent width Fa and mass Ma of the A due to X A ( E + ie) = MA(E) - M ° - ~i F A ( E ) .

(18)

We are aware of the fact that a more refined microscopic description of the self-energy would require the inclusion of the full spectrum of intermediate states in (17). The justification for the present model lies in the fact that it provides for our purpose a convenient parametrization for a fit of the P33 phase. It includes the essential physical mechanism for the imaginary part of the self-energy. Its real part we consider as purely phenomenological, which is fixed by the fit to the experimental phase. In fact, it turns

P Wilhelm, H. Arenh6vel/Nuclear Physics A 593 (1995) 435-462

441

out that its energy dependence is very weak, almost flat. In (17), go denotes the free propagator in the 7rN sector. For the ¢rNA vertex vta we use vat = - / va (q2~N) rNa,a o'Na' qrN, mr

(19)

with M N q -- Wq PN

q~rN =

MN + Wq

(20)

where the transition spin (isospin) operator o'Na = o'tan (~'Na = ~ ' ~ ) is normalized as

(3 II~a. (~-aN)ll ½) = -(½ liana (rNa)II 3)

= 2.

(21)

The hadronic form factor Fa together with a bare A mass ~ is fixed to fit the 7rN scattering phase shift in the P33 channel. In the actual calculation we take for Fa the dipole parametrization of Ref. [ 15 ], 2

F ~ ( q 2) = f a A2A ~m___~ AZa + q2 '

(22)

with f ] = 1.393, 47r

Aa = 287.9MeV,

M°a = 1315MeV.

(23)

This soft form factor might be surprising in view of the ones used in potential models of the NN interaction. However, it is known that the conventional A isobar model requires a soft form factor simulating the neglected nonresonant interactions in the P33 channel like the crossed two-pion graph. For the yNA vertex ~ra one has to take into account the magnetic dipole and a possible electric quadrupole excitation of the resonance, ~

~M1

(24)

~E2

UTA = Vya + V7a.

The M1 transition is known to be dominant. Following Weber and Arenh6vel [ 16], the ~MI/E2 .MUE2 corresponding currents v a = - 6 • JAN read •M~

JAn(E, k) =e

~I(E,

k)

iO'aN X krN ran,3,

(25)

•Z2 "E k" GE2(E,k) 1 Jan ~ , ) = e (o'aN ¢rnN • k~N + ~ a n • krn ~rcN) ran,3, 2MN 2

(26)

2MN

where M~ s = 1232 MeV and

kr N = MNk - ( M ~ s - MN) PN M~es

(27)

Unlike Ref. [ 16], where constant yNA couplings were used, effective energy dependent have to be taken here in order to and complex couplings GMar~(E,k ) and GaN(E,k) ~E2

442

P Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462

achieve a realistic description. They have been fixed in order to fit the experimental M3/2 1+ and ~3/2 ~1+ multipoles of pion photoproduction on the nucleon, respectively. For the elementary photoproduction amplitude t ~ the ansatz

b + otaga(z)~;ra( Z ) trr(Z) = trr(Z)

(28)

has been used in both channels, where t~r denotes the nonresonant background discussed 1:3/2 are concerned, the background is due to the pion pole above. As far as ~3/2 ""1+ and ~1+ graph (Figs. lc, ld) and the crossed nucleon pole graph (Fig. lb). From a microscopic point of view vra contains, besides a bare coupling OrA, the rescattering of pions (via the A mechanism) initially produced through the background mechanism, i.e., v r a ( z ) = ora + oago(z)t~r(z).

(29)

It is energy dependent and complex such that (28) as a whole satisfies the unitarity constraint of Watson's theorem. In the literature (e.g., Refs. [ 17,18] ), the second term ra = vtAgafira is usually referred to as a dressed A amplitude and ~ra as a in (28) trr dressed vertex. With our convention for the couplings one finds for the ratio of the two dressed multipoles ~E2 1 GAN 2 ~M1 GAN

J~la+ _ /~/A+

(30)

~d3/2 ~3/2 multiFitting the on-shell matrix elements of (28) to the experimental .,. 1+ or ~l+ poles, one can fix the effective coupling only on-shell. That means, one can determine ~MI/E2t L-' I. "~ AN t~, '~eJ for E > mr + MN where the on-shell photon momentum ke is defined through

kg

E = ke + MN + 2M-----~"

(31)

However, within the two-nucleon system, one needs in principle the unknown off-shell behavior of Ora- Instead, we adopt the following approximation: /~MI/E2¢~,

vaN

,~, k) ,m

VAN

x~,ke)

,~.M1/E2. __ DAN ~mr -*- MN, kr~+M,)

if E > mr + MN else

(32)

To fix bra completely, one would have to assume a certain model for the half-off-shell extrapolation of tb~r. This has been done, for example, by Tanabe and Ohta [ 17] in order to extract bare TNA couplings. We believe that (32) is not a severe approximation for energies in the A region, on which we concentrate in this work. In order to fit the effective couplings, we write oMI/E2/~ AN

~ , kE) =/£MI/E2(E) e x p [ i ~ M 1 / E 2 ( E ) ].

(33)

First we consider the M1 transition using for the energy dependence the functional form

( qE ~ 2

/-LMI(E) =/-Z0+/tL2 \~--~/

( qE ~ 4

+/£4 \~-'~/

,

(34)

P Wilhelm, H. ArenhSvel/Nuclear Physics A 593 (1995) 435-462

40

60

(,o

E 20 o ~.+ "~-

443

::::iiiii ~

.

('f:+°)

-('T) _

2o

0

g: -20 2OO

500 k,~ b [ M e V ]

400

-20 200

300 k,8 b [ M e V ]

400

Fig. 2. The "~3/2 multipole (full curve) fitted to the analysis of Ref. [20] (circles). Squares represent the "1+ energy independent solution of the analysis FA93 of Ref. [21]. Separate background amplitudes (dotted): pion pole (~r) and pion plus crossed nucleon pole (~r+c). where qe is the on-shell pion momentum in the ~ N c.m. frame. For the phase ~M1 we follow Koch et al. [19], q~M1( E ) -

q3 al + a2q 2"

(35)

Jot3/2 The relevant formulas to calculate the ,,, 1+ multipole starting from (28) can be found in Appendix B. The fit, based on the data set of the multipote analysis of Berends and Donnachie [20], is shown in Fig. 2 and gives ~ = 4.16, /x2 = 0.542, /~4 = - 0 . 0 7 5 7 , and al = 0 . 1 8 5 fm -3, a2 = 4 . 9 4 fm - l . The recent multipole analysis of Arndt et al. [21] is also plotted in Fig. 2, in order to demonstrate the accuracy of the multipole analysis for this specific multipole. Furthermore, it shows that our fit is reliable also at lower energies, where no data from Ref. [20] are available. For the phase of the effective E2 coupling we take q~m ( E ) =

q3 bl + b2q6"

(36)

Trying to reproduce the energy dependent solution of Ref. [21 ], we found/XE2 = 0.55, and bl = 0.86 fm -3, b2 = 0.15 fm 3. The multipole El3/2 is shown in Fig. 3. It turns out that q~E2 is larger than q~M1 because the relative magnitude of the background is larger for ~,3/2 ~ + than for ~3/2 ,,, ~+. 3.3. Results for isovector (+) amplitudes As far as the direct contributions of the elementary amplitudes to coherent pion photoproduction on the deuteron are concerned, only the isovector ( + ) amplitudes A~-± are of interest. They are just given as the sum of the ~r° production amplitudes on the proton and neutron (see also Appendix B), i.e.,

444

P Wilhelm, H. Arenhfvel/Nuclear Physics A 593 (1995) 435-462

2

~"~" 0

"~'~ -

o"-

~,

t

,

i

-~-~- =.- . . . . . . . .

-4

'

200

'

300

'

'

i

,--

oooO o

°~ °'=

%

-.

,~

-2

400

,

1

"-' 0

2

,

~

200

k,~ b [MeV]

'

300

400

k,~ b [MeV]

Fig. 3. The ~1+ p3/2 multipole (full curve) fitted to the energy dependent solutions of the analysis [21 ] (dotted curve). In addition the energy independent solution from Ref. [21] (squares) and Ref. [20] (circles) are shown. The underlying background amplitude from the pion plus crossed nucleon pole ( ~ + c ) is shown by the dashed curve.

A~-+= ½(Ae~r,°p+ Ae~n).

(37)

In Fig. 4 the s- and p-wave multipoles E~-+, E~-+ and M +_ are plotted as predicted by our model. The higher partial waves with * > 1 are of minor importance for ~.0 production. This is not true for charged pion production because of the long range nature of the pion pole graph. Without the additional contribution of (13) our model gives a nearly vanishing E~-+ multipole which above the threshold region seems to be compatible with at least the energy independent solution from Arndt et al. [ 21 ]. In view of the experimental uncertainties we take as a realistic value E~-+th = - 2 x 10-3/m~ and obtain the dashed curve. For E++ we also get extremely small values (invisible on our scale), whereas the multipole analysis at about 300 MeV starts to show a nonvanishing but nevertheless small multipole. Taking into account the electric quadrupole excitation of the A leads to the dashed curve and clearly improves the description considerably.

8

.

.

.

.

.

.

,

.

8

8

ooOO ° ° ~

o 0 ;'~ [.1.1

~ o I o~:~_~.03~ so~..t

,

L

,

,

,

i

300 400 k~,b [MeV]

,

.

.

.

.

.

.

-4

-4

,

-%00

.

b0

-I-

-4

.

%00

300 kin b [MeV]

,o0

-800

300

400

kl. b [MeV]

Fig. 4. Real parts of the isovector ( + ) multipoles E++, El++ and M+_. Full curves: model prediction. For E++ the dashed curve corresponds to E+$th = --2 x 10-3/m~r, and for E++ to the inclusion of the resonant part. For M +_ the dashed curve shows the separate contribution of the crossed nucleon pole graph. Multipole analyses are as in Fig. 3.

P. Wilhelm, H. Arenhrvel/Nuclear Physics A 593 (1995) 435-462

445

40

40

2o 7 o

7+.t

+~ 0

o

q~

20 200

i 300

k~ b [MeV]

i 400

20 200

i 300

,

i _, 400

klab [MeV]

Fig. 5. Real and imaginary part of the M++ multipole. Full curve: model prediction; dash-dotted: separate contribution of the crossed nucleon pole graph in the real part. The multipole analyses are as in Fig. 3. The dashed curve is explained in the text at the end of Section 3.

Obviously, the model has a problem in the M+_. Carrying spin and parity of the nucleon, it is the only amplitude with a contribution from the nucleon pole diagram, in fact, a rather large contribution, which, together with a smaller one from the crossed nucleon pole graph, clearly falls to reproduce the data. Likewise it is not possible within effective lagrangian models to describe the M l - isovector multipoles with the Born terms alone. This drawback is usually repaired by the inclusion of the isoscalar vector meson w in the t channel (e.g., in the work of Laget [22] ). In the more recent effective lagrangian approach of Davidson, Mukhopadhyay and Wittman [23] quite a large contribution from the A in the u channel enters these multipoles in addition. However, it is by far not yet clear whether these two mechanisms represent the real physical ones. Contrary to the situation in the s channel, the A is always far offshell in the u channel irrespective of the photon energy. The same is true for the in the t channel. Thus, one would expect a strong suppression of their hadronic as well as electromagnetic vertices. The authors of Ref. [23] also state that it is fair to assume that these mechanisms may be mocking up the effects of the form factors. Furthermore, the tensor to vector ratio for the toNN vertex coupling, extracted by Davidson et al. when fitting the pion photoproduction data, does not agree with other SOUrCes.

In this paper we will not introduce the oJ exchange nor the A in the u channel. Instead we will consider an alternative version of our model where we switch off the isovector ( + ) background amplitudes completely. Looking at Fig. 4, this is certainly not a bad approximation. In doing so, also an appropriate modification of the yNA coupling becomes necessary which will be described below. In Fig. 5 the real and imaginary parts of the resonant multipole M++ are shown and compared with the analysis of Berends and Donnachie [20] as well as with the more recent one (FA93) of Arndt et al. [21]. With its large contribution from the A resonance, the M++ clearly dominates the isovector ( + ) multipoles. Here we obtain a good description of the data. Of course, this is mainly since we have fitted the A/t3/2 ,.. I+"

446

P Wilhelm, H. Arenh6vel/Nuclear Physics A 593 (1995) 435-462

But on the other hand, because of the decomposition /i//3/2 + ½~"1+ ,i,lrI / 2 , m~+ = 2...1+

(38)

t,,iU2 it demonstrates that the model is able to reproduce the ,,, t +, which is to a large extent generated by the pion pole graph. Now we consider our alternative approach where we assume that the isovector (-4-) background amplitudes due to the direct and crossed nucleon poles are strongly suppressed by form factors. This leads to a modified TNA coupling ~qMl which is fixed It'3/2 is given by the pion pole alone. The under the assumption that the background in ""l+ fit is nearly indistinguishable from the previous one. It gives/.t~ = 6.56,/z~ = -0.626, kt~ = 0.0127, and a~ = 9.62 fm -3, a~ = 7.98 fm - l . The resulting prediction for M++ is also shown by the dashed curve in Fig. 5. It gives even a slightly better description of the data. The real part of M++ is very close to the energy dependent solution of Arndt et al. This shows that the data do not at all contradict our assumption of a strongly suppressed crossed nucleon pole diagram.

4. The "yd ~

~r°d amplitude

A plane-wave impulse approximation (PWlA) usually serves as the starting point to calculate the amplitude for electromagnetic pion production on the deuteron [ 1,2,4,3] or on a nucleus in general. It corresponds to a direct embedding of the elementary amplitudes into the two-nucleon sytem. Our approach is somewhat different and we will point out the differences below. Nevertheless, it should be comparable with a PWlA. In our framework, the various electromagnetic and hadronic vertices which have been introduced above are treated in time-ordered perturbation theory. The amplitude contains contributions from the A resonance ( A [ 1 ] ) and nonresonant parts (NR) as represented in Fig. 8. The A amplitude reads Tma[ l = - ~ a " (q, m r [VQa Ga ( Wrd --~ ie)jaN[l] (Ea, k)[ m), 'an,

(39)

where we have introduced Vaa = v~(1) + v~ (2),

(40)

with arguments labeling nucleon 1 and 2. Moreover, 1 Ga(Wed + ie) = Ea - M a ( E a ) + ( i / 2 ) F a ( E a )

(41)

denotes the dressed propagator in the NA sector with an energy Ea = Wed -- MN

2MN

2M°a

(42)

available for the internal excitation of the resonant ¢rN subsystem. Since we always work in the c.m. frame, PN = --Pa holds. Finally, we have

P Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462

JaN I 11 = JaN ( 1) + JaN (2).

447

( 43 )

The multipole decomposition to rewrite the A[I] amplitude into the form of (3) is sketched in Appendix C. Due to the strong energy dependence of the resonant amplitude, it is important how the energy Ea of the resonant subsystem has been fixed. In fact, the energy dependence of the whole reaction is essentially determined by this. While within a PWIA some freedom remains, this is not the case in our framework. Actually, different approaches have been followed in the literature. Besides some possible minor differences depending on whether relativistic or nonrelativistic kinematics for the nucleon and A are used, our approach corresponds to the so-called spectator on-shell choice. It has also been applied by Blaazer, Bakker and Boersma [8]. On the other hand, Tiator et al. [10] and more recently Kamalov, Tiator and Bennhold [24] have chosen to take the active nucleon on-shell. Whether the nucleon before photon absorption or after pion emission is taken to be on-shell was thereby found to be less important [ 10]. Taking, for instance, the nucleon before photon absorption on-shell would lead to Ea =/~A with 2

/~2= [V/(PN_ k ) 2 + Mr~ + w ]

- p2,

(44)

where we have used the fact that the momentum of the active nucleon before photon absorption is just PN -- k. To compare both choices (42) and (44), we note the constraint E~- <~/~a ~< E+,

(45)

with 2

and remind the reader of (2) which relates the invariant energy Wyd tO the c.m. photon momentum k. In the A region the inequality Ea < E~- holds over the whole relevant region of the spectator momentum PN, which at the end has to be integrated over in order to reach a proper treatment of the Fermi motion. Thus, one would expect that the results calculated with the spectator on-shell choice are shifted towards higher energies on the energy scale in comparison with those when the active nucleon was taken on-shell. And indeed we could observe [25] this shift for the total cross section when comparing with a calculation of Kamalov and Tiator [26]. Now we turn to the nonresonant amplitudes. As for the elementary process, we have two contributions, T NR = T NP + T Nc,

(47)

namely direct and crossed nucleon pole amplitudes (NP) and (NC), respectively. Formally, they read (see also Fig. 6)

448

P Wilhelm,H. Arenhi~vel/NuclearPhysicsA 593 (1995)435-462

All] d

\\

' d

~Y

ci,l

\\

Nci21/ i/ / P2

J

/

j

Fig. 6. The yd ~ 7r°d amplitudes: A[1], nucleon poles NP[1], NP[2] and crossed nucleon poles NC[I], NC[2]. The momentumassignment for the evaluation of the amplitude NC is as used in Appendix C. For NC[ 1]: P l / 2 = -4-(p+ q/2) - k/2, and for NC[2]: P~/2 = d=(p- q/2) - k/2.

T~Pam(q,k ) = - ~ a . (q,m' VQN

1 (k) ) Wyd "~-ie - hN(1) -- hN(2) JNN[I1 m ,

(48)

TNCm( q, k ) =_,a.(q,m,

jNN[ll(k ) W r d _ W q _ w _ h N I (1)_hN(2)

VQN m ) , (49)

where

VaN = VtN(1) + vtN(2),

JNN[1]= J N ( 1 )

+ jN(2),

(50)

with v~ from (9), and hN denotes the free nucleon energy, i.e., hN = MN + ~ 2MN"

(51)

Again, some details concerning the multipole decomposition of NP and NC can be found in Appendix C. Both nonresonant amplitudes consist of a one-body (NP[ 1 ], NC [ 1 ] ) and a two-body part ( N P [ 2 ] , NC[2] ) as is shown in Fig. 6. For the latter ones, photon absorption and pion emission arise on different nucleons. Similar amplitudes for pion deuteron elastic scattering (replacing the incoming photon by a pion) have been discussed by Jennings [27]. They go beyond the embedding of the elementary yN ---*7rN amplitudes in a nucleus and are a consequence of antisymmetrization. Moreover, our treatment of the nucleon pole amplitude goes beyond the usual PWlA with respect to a further point. As a consequence of the proper treatment of nucleon propagation, NP gets an imaginary part. Of course, it has to be present because of unitarity. These two points are closely related because only the inclusion of NP[2] ensures that the pole in the full amplitude NP is exclusively due to physical intermediate nucleonic states, i.e., antisymmetrized states obeying the Pauli principle.

P. Wilhelm, 1-1.Arenh6vel/Nuclear Physics A 593 (1995) 435-462

449

Now we will consider the two-body amplitudes in some more detail since they have quite an interesting feature. Whereas beyond the threshold region all one-body amplitudes peak at forward angles, the two-body amplitudes contribute dominantly to backward angle pion production. The reason for this is that the pion and photon soon become very similar kinematically. Therefore, in order to minimize the momentum mismatch for the overlapping deuteron wave functions, the one-body amplitudes dominate the small pion angles. In other words, large angles corresponding to large momentum transfers are suppressed by the deuteron form factors. In case of the two-body amplitudes, the momentum mismatch becomes smallest if the absorbed and emitted momenta are opposite, since then the relative momentum of the two-nucleon system just remains unchanged. We would like to stress that the two-nucleon system plays an extraordinary role in this context, because if there are more than two nucleons present, the momentum mismatch cannot be minimized so effectively in this way. The fact that the two-body amplitudes exclusively originate from the nonresonant mechanisms makes this scenario particularly interesting. Any resonant pion production mechanism cannot lead to such kind of two-body amplitudes. Therefore, one can speculate whether an intensive experimental and theoretical study of the yd ~ 7r°d reaction mainly at large pion angles might help to disentangle resonant and nonresonant mechanisms in yN ~ 7r°N amplitudes. Obviously, this is a difficult task, mainly for two reasons. First, rescattering effects are supposed to be more important at larger angles. Second, a rather large cancellation between NP[2] and NC[2] occurs in this case. The reason for this can easily be understood. Both amplitudes have an identical vertex structure but essentially opposite signs in their propagators. However, this cancellation is not exact, since the absolute values of the propagators do not coincide. Moreover, NP[2] has an imaginary part but NC[2] always remains real. Thus, one might still hope to find a signature of these two-body amplitudes.

5. Results and discussions 5.1. Total cross section

We start the discussion with the total cross section which formally is given by an incoherent sum over the reduced matrix elements of (3), 2~-

L

O'tot---- T C ~

12

L

2

[[E~rj(W, yd). '~lM~j(W,d)l ] •

(52)

L£nj Our results, obtained with the deuteron wave function of the Bonn potential (OBEPR) [28], are shown in Fig. 7. The calculation includes the amplitudes A[ 1 ], NP and NC. According to the decomposition into partial waves and multipoles, we have separated the cross section into contributions from fixed multipoles or partial channels which are listed in Table 1.

450

P. Wilhelm, l-l. ArenhiJvel/Nuclear Physics A 593 (1995) 435-462

10 3

total

total 102 /

, :~,101

/

/

E1

H

/ /

1 =0

f

1=3 --"--

-- _

b

100

.

/ ''/

10 -1 2OO

./

M? - .

,/

z"

III

/,~(

300

400

I~" /

/i

. . .

200

kl. b [MeV]

/

/3 +

,

.~

/ //

-- -- =

~+--

dl

..... 400

1=4 /7----

//

~ ~. ----

/

4-

500

/

200

11/

I =5

300 400 k~. b [MeV]

kl, b [MeV]

Fig. 7. Separate contributions of various electromagnetic multipoles E L / M L (left panel), channels j~r (middle panel), and pion partial waves l,r (right panel) to the total cross section for ~d ~ ~r°d as a function of the photon laboratory energy.

The reaction is dominated by the 2 + channel. Its crucial importance can be traced back to the 5S2 (NA) partial wave with vanishing angular momentum barrier. Moreover, M1 transitions and P-wave pion production are the most relevant mechanisms. We would like to point out that a M1 transition in yd --+ 7r°d, for example, does not simply correspond to a M1 transition (a M l ± multipole) in the yN subsystem, since the Fermi motion leads to an admixture of other elementary multipoles. The next important multipole is M2 with its maximum position shifted to a somewhat higher energy, since the lowest contributing NA partial wave is a P-wave which requires a certain amount of rotational energy to be formed. Electric transitions contribute only a few percent to the total cross section. Although a few partial waves are sufficient to determine the total cross section, one has to take into account a larger number in case of other observables because of interference effects. In the following results we have included partial waves up to jmax = 10 in the A [ 1 ] and up to jmax = 5 in NP and NC amplitudes, respectively. Thus, we have obtained a relative accuracy of better than 1% for the differential cross section at extreme angles which was found to be rather sensitive to the number of partial waves. Table 1 Coupled ~rd, NN and NA partial waves with parity ~- and total angular momentum up to j = 3 and their possible multipole coupling to the yd system j~r

~'d

NN

NA

yd

0+ 01+

P0

ESO 3P0

5D0 3P0 3Sl, 3Db 5Dl 3Pl, 5P I, 5F1 5S2, 3D2, 5D2, 5G2 3p2, 5P2, 31=2, 5F2 3D3, 5D3, 3G3, 5G 3 5P3, 3F3, 5F 3, 5H 3

M1 El M1, E2 El, M2 MI, E2, El, M2, E2, M3, M2, E3,

1-

2+ 23+ 3-

P1 Sl, DI P2, F2 D2 F3 D3, G3

3pI ID2 3p2, 3F2 3F3

M3 E3 E4 M4

451

P. Wilhelm, H. ArenhSvel/Nuclear Physics A 593 (1995) 435-462

200

a_ 100 b

0

300

200

400

kLab [ M e V ] Fig. 8. Total cross section for "yd ~ 7rOd.The curves correspond to model D (dotted), DNP (short dashed), DNPNC (full), and Dt (long dashed). 5.2. The role o f the nonresonant amplitudes

We will first analyze the role of nonresonant amplitudes in ~,d ~ zr°d. Below we will consider a calculation including the A [ 1 ] amplitude alone, which will be called model D henceforth. Then we consecutively add the NP and NC amplitudes defining the models DNP and DNPNC, respectively. Finally, DNPNC has to be compared with our alternative description called D ~. The latter contains only a resonant amplitude, however, based on a modified coupling which has been fixed to fit the experimental M3/2 " - l + without any background contribution from the crossed nucleon pole graph as described in Section 3. The total cross section as predicted by the various models is shown in Fig. 8. It is clearly dominated by the A resonance. However, below and above the resonance region the relative importance of NP and NC grows. The amplitude NP always interferes constructively with A [ 1 ], whereas NC interferes constructively (destructively) for energies below (above) the resonance. This behavior of NC essentially reflects the role o f the background contribution from the crossed nucleon pole as shown in Fig. 5. Comparing D' and DNPNC, one observes a slight shift by about 10 MeV of the D ~ curve towards lower energies. The differential cross sections for fixed c.m. pion angles of 0 °, 90 °, and 180 ° are plotted in Fig. 9. At 90 ° the effect of NP and NC is rather small. At 0 °, adding NP ~¢L~0

. . . .

~"4o

,

. . . .

15

,

E

. . . .

,

. . . .

4

,

0 '~o~=180deg

0 { , = 90 deg

0Tr

~ =Odeg

3

30

;

~20

x

"

b I0 0 .... 200

i 500

k,~b

.

.

.

.

i

400

[MeV]

,

,

0

200

. . . .

i

300

. . . .

i

400

klab [MeV]

,

0

200

300

400

k,~b [MeV]

Fig. 9. Energy dependence of differential cross sections for 3,d ~ 7r°d at fixed pion angles Oc~m..Notation of curves as in Fig. 8.

452

P Wilhelm, H. Arenh6vel/Nuclear Physics A 593 (1995) 435-462

1.0

klab=4OOMeV

k~.b=3OOMeV

klab=240MeV 0.5 [,-1

0.0

-0.5 . . . . .

-1.0 1.0 kl=b=240MeV

,

. . . . .

,

. . . . .

.

.

.

.

.

,

.

.

.

.

.

,

.

.

.

.

.

kz=b=4OOMeV

klae=300MeV

0.5 0.0 -0.5 -1.0 1.0

. . . . .

. . . . .

,

. . . . .

kl~b=4OOMeV

kl,b=300MeV

klob=240MeV o

i

/

0.5

/ /

0.0

F-- 5

-0.5 -1.0 1.0

kla~ =300MeV/

ku, b=240MeV ~ ,, /

0.5

"l

kl.b=4OOMeV

\

//

E--

0.0

-0.5 -1.0

0

................. 60

0 ....

120

[deg]

180

0

60

120

oo= [ded

180

60

0 ....

120

180

[deg]

Fig. 10. Photon asymmetry ~', vector target asymmetry TII, and tensor target asymmetries /'2o and T2] as functions of pion angle 0~m" for various photon energies. Notation of curves as in Fig. 8.

and NC, leads to a strong reduction (enhancement) for energies below (above) 340 MeV. At backward angles one observes a very strong cancellation between NP and NC contributions which becomes stronger with increasing energy. For extreme forward and backward angles, the model D' lies between D and DNPNC. This is not valid around 90 ° where the situation is qualitatively similar to the total cross section; however, the shift is more pronounced. Fig. 10 provides an overview over single polarization observables for ~d --~ ~-°d and 7d ~ "nOd, respectively. The energy of 300 MeV is chosen just below the resonance position of the free NA system, i.e., W~d = MN + M~ s, which corresponds to klab ~ 320 MeV. The asymmetry 7"11 clearly differs in size between D' and DNPNC, being even

P. Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462 5O

4o

15

t:

0 c~m= 0 d e g lo

3O

e~.=

453

180 d e g

20 -0

\.,

"~ lo .... 0 2O0

i .... 500

i

400

~

0

200

kl~ b [ M e V ]

300

400

kl~b [MeV]

200

300 400 kl~ b [ M e V ]

Fig. 11. Influence of the electric multipoles on the differential cross section as function of photon energy for various fixed pion angles O~.m.The long (short) dashed curve includes the E++ (Eta_) as represented by the dashed curves in Fig. 4. The full curve is calculated without the resonant part of E++ and E~r+ without the additional background of (13). opposite in phase. This sensitivity of Tll to the Born terms has also been discussed by Blaazer, Bakker and Boersma [8]. The reason is that T1~ depends on the relative phase of the matrix elements as can be seen from (A.5) and (A.1). It would vanish for a constant overall phase of the t-matrix, a case which is approximately realized if only the A amplitude is considered. Except for E at lower energies, the other observables seem to be not particularly sensitive to the nonresonant contributions. In all relative observables, D and D r are very close to each other. One notes again a strong cancellation between NP and NC.

5.3. The role of the electric multipoles Now we want to investigate qualitatively the separate role of the electric multipoles E++ and E~-+. As discussed above, E~-+ is essentially generated by the contact current of (13) since our nonrelativistic model for the Born graphs gives only a tiny s-wave amplitude for 7r° production. On the other hand, in our model E++ is nearly exclusively due to the A. The sensitivity of the differential cross section on E~-+ and E++ is shown in Fig. 11. For Eft+ we used the multipole represented by the dashed curve in Fig. 4. For E~+ we take our parametrization (36) of the effective E2 excitation of the A. We would like to point out that E++ (at least at the Born graph level) contains only a very small nonresonant part. Thus, its extraction from Td -~ *rOd offers a direct access to the resonant part. Compared to TP --~ ~.0p the nonresonant isoscalar amplitudes are absent. In Fig. 11 the relative contributions of both electric multipoles are of the same order of magnitude. They are significant at extreme angles only. Below 320 MeV one observes at backward angles the strongest effect. Some asymmetries where the influence of the electric multipoles was found to be most pronounced are shown in Fig. 12. For energies below the resonance the photon asymmetry 2 shows some sensitivity on E++. Directly in the resonance region at 300 MeV photon energy and around pion angles of 30 °, Tll increases from about 30% to nearly 50% due to the resonant E++ part. The cross section for this kinematic is predicted to be of the order of 30/.tb. Finally, for forward angles, a qualitatively different shape arising from E++ is seen in the double polarization

454

P

0.5

.....

Wilhelm,

, .....

H.

ArenhOvel/Nuclear

1,0

i .....

Physics

.....

, .....

A

593

(1995)

435-462

0.5

r .....

kl~b=3OOMeV

klab=220MeV

kl~b=4OOMeV

0.5

[,q 0,0

0.0

[-~ 0 . 0

-0.5

-05

.... 0

' ..... 60

O ¢" m

i .... 120

.....

1.0 180

i ..... 60

0

0.5 i . . . . .

i ...... 120

180

0

O em. " [deg]

[deg]

i ..... 60

i ..... 120

180

Oe" m [ d e g ]

Fig. 12. Influence o f the electric multipoles on several polarization observables as function o f pion angle 8~.m. for various photon energies. Notation o f curves as in Fig. 11.

asymmetry T~I. However, at these angles this asymmetry is only of the order of 10% and thus may be difficult to measure. 5.4. The role of the two-body amplitudes NP[2] and NC[2]

In the discussion above we already noticed a rather strong cancellation between NP and NC in all observables in particular at backward angles. However, this does not yet imply that the net effect of the two-body amplitudes (TBA) NP[2] plus NC[2] is really small. In order to study this aspect, we have also performed a calculation without TBA. This can be achieved if one does not pay attention to the antisymmetrization condition for the intermediate NN system when evaluating NP and NC. It turns out that the feature of NP[2] and NC[2] to cancel each other is indeed extremely well realized at least for energies in the A region. Comparing differential cross sections with and without TBA, we always found relative differences smaller than 2% for energies between 200 and 400 MeV. Thus, it seems that the conventional PWIA approach which simply ignores the TBA is a rather good approximation. Furthermore, we emphasize that only the inclusion of both, NP[2] and N C [ 2 ] , is consistent. The inclusion of NP[2] alone, for example, would lead to quite different results as can be seen in Figs. 9 and 10. However, at lower energies where the process is less dominated by the A, the ne0.5

. . . . .

i

. . . . .

i

. . . . .

.....

1.0

i .....

0.3

i .....

. . . . .

i

. . . . .

r

. . . . .

klab=2OOMeV

kl.b=2OOMeV

kl~b=2OOMeV 0.5

f

0.0

\

0.0

0.0

-05

. . . .

' ..... 60

o "c,m.

' . . . . . . 120 180

[a~g]

.....

0.5 0

J ..... 60

i. . . . . 120

O c.m. " [deg]

180

0.3 0

60

120

180

e"c m . [deg]

Fig. 13. Influence o f the T B A on the tensor target asymmetries 7"20, T21 a n d T22 as functions o f pion angle 0~m" for a photon energy o f 2 0 0 MeV. The full (dashed) curve is calculated with (without) the contributions from N P [ 2 ] and N C [ 2 ] .

P Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462 6O

....

, .... ,~ H i l g e r et al.

455

20 15

~40

o H o l t e y e t al. BouqueL e[ al

I

o

o

2o

\

/~0,/ / \\

b

5 c in

o ~-"'<

0

o ,o, ey e, :,q

6 o H o l [ e y eL al.

~c

f

x

[] B a b a

eL a l

J

E

g2 b

o ~_12o o o c.m

0 20O

~

300 400 k,~ b [MeV]

14 j

\

1

~.~

0 ~ 200

' '

' ' ' ' ~ 300 400

kla b [MeV]

Fig. 14. Energy dependence of differential cross sections for various fixed pion angles 0 c.rll. *r calculated with the model D N P N C (full curve) and D ~ (dashed curve). Experimental data from Refs. [ 2 9 - 3 2 ] .

glecting of TBA becomes visible in the polarization observables. It leads to significant modifications in the tensor target asymmetries as is illustrated in Fig. 13 for k~ab = 200 MeV. The asymmetry T22 even shows a qualitatively different angular dependence becoming very small without TBA. However, it remains to check whether this rather clear signature of TBA is still of interest after rescattering amplitudes have been included in the model.

5.5. Comparison with experiment Experimental differential cross sections are available from Refs. [29-32]. Hilger et al. [29] provide the only data for small pion angles since they measured the ~r° decay photons. However, the separation of incoherent ,r°pn events was problematic. In measuring the recoiling deuteron, the other experiments are restricted to larger pion angles. Therefore it is up till now not possible to extract a total cross section from these experiments. Its energy dependence would test the leading multipoles of the model whereas differential cross sections are in addition sensitive to interference effects from smaller multipoles. Future experiments using gas targets may improve the situation. In Fig. 14 differential cross sections for the model DNPNC and D ~ are compared with experiment. We remind the reader that both models do not contain an extra E~-+ nor a resonant E++ contribution. Above 260 MeV at 90 ° and 120 ° the calculations clearly overestimate the cross section. This has been found also in Ref. [ 8]. As one can see from the results in Fig. 11, improvements with respect to the electric multipoles cannot remove this discrepancy which thus seems to clearly indicate the importance of

456

P Wilhelm, H. Arenh~vel/Nuclear Physics A 593 (1995) 435-462

rescattering effects. At 6 ° where rescattering effects can be assumed to be minimal, the model DNPNC is closer to the data.

6. Summary and conclusion We have investigated coherent pion photoproduction on the deuteron in the A (1232) resonance region using a nonrelativistic model which is based on time-ordered perturbation theory. As input the model contains yNN, ~rNN, 7rNA, and yNA vertices. The two A vertices are fitted to the pion-nucleon scattering phase shifts in the P33 channel and ~3/2 ~3/2 the resonant elementary pion photoproduction multipoles .,, 1+ and ~l +" The model predictions for the leading multipoles Ee+ and Me:i: with g ~< 1 of the elementary reaction have been compared with data of a multipole analysis, and very good agreement has been found for the multipoles M++ and E++, while for M +_ some discrepancy remained. However, this multipole is much less well known experimentally. In this comparison, those isospin combinations of multipoles which are accessible by the direct part of the yd --~ 7r°d amplitude, namely the isovector ( + ) multipoles, have been chosen in order to make the relation to the elementary process as transparent as possible. We have presented results for total and differential cross sections, and for various asymmetries corresponding to a polarized photon beam and/or an oriented deuteron target. In particular, we have studied in detail the interference of the nonresonant background amplitudes as given by the direct and crossed nucleon pole graphs with the dominant A excitation amplitude. In accordance with Ref. [8], the vector target asymmetry T~l has been found to be very sensitive to this interference. The dependence on the elementary input has been analyzed with respect to the effect of the small electric multipoles Eo+ and El+, where the latter one is connected to the electric quadrupole excitation of the A resonance. For Tll and the double polarization asymmetry T~l a remarkable sensitivity to E~-+ has been found. Furthermore, we have investigated the role of genuine two-body pion production amplitudes where photon absorption and pion emission take place at different nucleons. Such two-body mechanisms are quite interesting from a theoretical point of view. Due to a strong cancellation between the direct and the crossed two-body amplitudes, it is very important to consider them together, and one finds only a small net effect for energies in the A region. However, some observables, in particular the target asymmetry 7"22, seem to provide a signature of the two-body amplitudes at lower photon energies. Compared to experiment the theoretical cross sections clearly overestimate the data around and above pion angles of 90 °. This has also been found in previous studies [ 8,3 ] and points to the importance of rescattering effects. In the future, this model will serve as a starting point for a more realistic calculation of the yd --, ~r°d reaction with special emphasis on a dynamical treatment of the pion rescattering process via the successive creation and decay of the A resonance [ 35].

P Wilhelm, 14. ArenhOvel/Nuclear Physics A 593 (1995) 435-462

457

Appendix A In order to express the asymmetries of (8) in terms of the t-matrix, it is convenient to define, in complete analogy to the procedure for deuteron photodisintegration in Ref. [ 12], the quantities

,) tmqrntmqm' ') --m M t*m'le~t-m'l ....

c

VtM=--~

m' ma,

c

WtM=--~

~ --m M

m' mO~

(A.1)

(A.2)

Then one derives do-0 = V0o, d$2 do-0 = -Woo, ds2 do-0 TII ~ =2Im~j, dtro T2M--~=(2--6Mo)

(A.3) (A.4) (A.5) ReV2M,

do"0 T~'M--d~=(2--SMo)ReV1M, d °~' ° = 2 I m ½ M , T~'Md--do-o T(M - - ~ = iWIM, Ti M do-o_

W2M,

M = 0, 1,2,

(A.6)

M=0,1,

(A.7)

M - - 0 , 1 , 2,

(A.8)

M = 0, +1,

(1.9)

M=0,+1,-t-2.

(A.10)

Appendix B In this appendix formulas for the elementary amplitude y ( k ) N ( - k ) ~ 7 r ( q ) N ( - q ) with cartesian isospin index a of the pion are given. The nonresonant parts shown in Fig. 1 read t(a) = _

rr~, t(b)=

~rr

t(c+d) ~" t(e) ~z,

=

f~r r a o ' . q ( ~ + k ) ( o ' × k ) . e 2mTrMN W~,N -- MN

(B.I) '

f~ [-2~q.~+(~+k) i(trx k).~]7"aitr.q 2m~rMN wrN - MN -- Wq -- k - (q + k ) Z / 2 M N '

(B.2)

2f,r [Ta,~] itr" ( q - - k ) q. m~r (Wq _ k ) z _ tOq_ 2 k

(B.3)

= -f -l r Ira, ~] itr. ~. m#

'

(B.4)

P. Wilhelm, H. Arenh6vel/Nuclear Physics A 593 (1995) 435-462

458

The dressed resonant amplitudes corresponding to M1 and E2 excitation of the A read

i~lr = [ - q . kio'. ~ - ~ o" . q o ' . (k × ~) + io'. k q. ~] g'2Ml,

(B.5)

.-'E2_ t~, r - ½ (q. k io'. ~ + io'- k q. e) f f 2 ,

(B.6)

with f2M1/E 2 =

~, t, ~2 ~ f~MI/E2 l'Akt/ / ' a N

1

1

2m~.MN WrN -- Ma(WvN) + (i/2)Fa(wrN) "3 ([ra,r0] -- 6a0).

(B.7)

Now it is straightforward to calculate the CGLN amplitudes F/ [33] defined by MN 47rW t'~r = F1 io'. ~ +/72 o'. ~ w - (It × ~)

+ F3 io'. loll. ~ + Fnio'. ~lgl. ~.

(B.8)

They can be decomposed into an isoscalar ~ and two isovector amplitudes F,~, (B.9)

Fi = Fi- " ½ [ "l'a, 7"ol "4- Fi+ t~aO + l~Oi"l'a•

We further introduce as usual Vi 1/2 = F/+ -k- 2Fi-,

(B.10)

F,3/2 = F/+ - F i - .

(B.11)

The physical amplitudes are then given by F7 ± = v/2 ( ~ i + F/-),

(B.12)

FT°P/n = F ? 4-/V0/.

(n.13)

The projection formula to determine the various multipole amplitudes Ee:~ and Me~: from the Fi can be found in Ref. [34].

Appendix C This appendix gives some details for the multipole decomposition of the yd ~ ¢r°d amplitude. The electric and magnetic multipole operators TILl "eM and TILl "mM are defined as Lf

(k; j) =

d2~ [y[el(k ) x j[1](k)]M[L] ,

(c.1)

g

with see = - V ~

aline = 6gL.

~L-1 -

8eL+l,

(C.2) (C.3)

The reduced matrix elements in (3) are evaluated in momentum space. The contributions besides the one from NC have the form

P Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462

459

O0

E

i dpp2 ((g~d)jIIVQxHXp(gs)j)Gx(p)

_)_IZ

ML"j

gs b

x (Xp(gs)jiiT[ek2(JxNill)lld),

(C.4)

with X E {N,A}. For the NN pair the propagator is given by GN(p) =

( Wro - 2Mr~ - ~.:

+ ie

),

,

(C.5)

while Ga is given in (41). The antisymmetrized free NN and NA partial waves are denoted by INP(gs)j) and IAp(gs)j), respectively. In the following the momentum space deuteron wave function (isospin suppressed) [sal][Ja] (plmd) = Z Ut,(p)[ylM(p) x k" Jm, ' /d--0,2 .

with jd

=

1, Sd = 1,

(C.6)

is needed at a shifted argument. Using

(p+qlmd):Vr-~ E ~ ~w I,A'(P, q) ld=0,2 A=0 .~=0

i')lqlpld [

XL

~"AA

[y[lq](O) ×

y[lp] ( ~ ) ] [/d]

. [sd]] [J'd] × x j.~ ,

(C.7)

lqlt, with

(~),ti

+1

W t, AA"tP,

q) =

d ( p . 0)

qla ip+qlt" Ul,(lp+ql) PA(P" gl),

(c.8)

-1

fJqlpld

AA

1 ^2^^^

f

=(--)A • ~A lqlplo

/

(21d) ! (2/d - 2 A ) ! ( 2 A ) !

I

x

ld--A a A

0

0

0

'

(C.9)

one finds applying a general scheme which is described in detail in Ref. [25],

(Xp(gs)jlITleklm(JxYtil) lid) = v/4-~Sx E

V

TM

AaK'rS '

JK'PS's(P, k) ~ --KPgSL

K'PS'S

KS

X X--"wAa" 2.-,¢ Id tp, l k ) ~ ~_~(--) K 12Aa Klelan nXP~SL, laAA

le

(C.10)

P Wilhelm, H. ArenhOvel/Nuclear Physics A 593 (1995) 435-462

460

with AaK'~' _ ~ KPSSL -- ~

otL ( _ ~P+S+S+L+~'+g,/~ ~#t # ~ t a~", .' v ...... ,.,~, g

x RKPgSL

(

=

- -

0

SL1

PKK

t

(c.11)

'

) P+Sx+SN+S+I .3LjldSetpPS ~ ^ ~ ^ ^ ^ (sxllo'~llsN) S

(~ " lp){ X 0 0

S " K } ( s x S SN } lp Id g 1 SN S

{/dl

1}

' S L g S j

,

(C.12)

and SN = 2, SA = X/2, and in addition to (21) (1

0 1 2 [IO*NNI[ ~) = x/2,

(1 3

1

,

2

1

(C.13) (C.14)

II'~NNII=> = V~,

(= IIo-~N II=) = vq-6.

(C.15)

The nucleonic contributions are 1

p

(C.16)

J011o(P, k) = x/3 2MN' 1

Jlolo(P, k) -

k

(C.17)

4MN'

_ . / ~ ( 1 + Ks) k V~ 4MN ' and the A contributions are k Jloll (P, k) = - G~MI AN, 3MN

(C.18)

Jloll(p,k)

Jmll(p,k) -

P

3MN

M~ s - MN ~Ml -- ~ s G aN '

(C.20) (C.21)

2/~N ( ~ '

Jlo12(p,k) = - - V ~ Joll2(p,k) =

(C.19)

1~7 P

M~-

2M----N

MN ~E2

M~s

(C.22)

GAN"

Further, one finds

( ( g~d) JllVQxllXp( gs) j)

= x/4--~ s~x) E illa,,,(p, q) QP X Z

1 Z(_)lq wtAA a (p, ~q) laAA lqlp

with S~ = 2, S~ = 2/x/3,

~lqlpldo SdAA OQp,

(C.23)

P Wilhelm, H. Arenh6vel/Nuclear Physics A 593 (1995) 435-462

461

SQp=(--)lq+lp+s3~Irfq['p~dffgO~(SNl,o'lNsx) (~.lro lq 0 Q)(~.0 Ip 0 P ) {SNS SN 1 S1x }

XZ(_)x22(glrldX)(g~rlqQ}(s' x 1 j 1 lp x ld

j}{1P x 1 1

Q}

lp x ~

(C.24)

'

and HlO(P, q)

~ m

q

O)q

H01 (p, q) = --SXA ~TP

m~ '

r/ -- - - . MN -'}-tOq

m~ '

(C.25)

The calculation of waa ld in (C.23) in case of X = A has to be done with the following replacement in (C.8):

Utd(IP+ ql) ~

FA ( ( 2 q + r/p) 2) Uld(IP+ ql)"

(C.26)

The evaluation of the NC contribution is somewhat more involved. With the integration momentum p defined as in Fig. 6 one ends up with OO

(~,j)

=87r/dpp2 NC

Z

w~'a'(p'lk) Z w ~ a ( P ' ½ q) Z

ldAIA~

0

IdAA

AT ×ZGac(P'q'k) Z Ac,,

L ae/m

l~ p

"*A'M °t'd'Pt'~Z JdAA~lqlpld lqlp

, , pl , dlqlpldAG ) , lkK ( lkLg~rlkl

(C.27)

Ik

with GA~ given by the propagator expansion

,/ GA(p, q, k) = -~ 712

+1 d(~. k)

Pa(q" k)

W~,d-- 2MN -- k - tOq - p2/MN -- (q + k)2/4MN"

(C.28)

--1 The explicit form of K can be found in Ref. [25]. Because of the four-particle intermediate state the sum over A6 has to be truncated. A~ax = 2 turned out to be sufficient.

References

[ 1 ] P. Osland and A.K. Rej, Nuovo Cimento 32A (1976) 469. [2] C. Lazard, R.J. Lombard and Z. Maric, Nucl. Phys. A 271 (1976) 317. [3] M.E Rekalo and I.V. Stoletnii, J. Phys. G 17 (1991) 1643. [4] P. Bosted and J.M. Laget, Nucl. Phys. A 296 (1978) 413. [5] M.T. Pefia, H. Garcilazo, U. Oelfke and P.U. Saner, Phys. Rev. C 45 (1992) 1487. [6] J.H. Koch and R.M. Woloshyn, Phys. Rev. C 16 (1977) 1968. [7] H. Garcilazo and E. Moya de Guerra, Phys. Rev. C 49 (1994) R601. [8] E Blaazer, B.L.G. Bakker and H.J. Boersma, Nucl. Phys. A 568 (1994) 681. 19l E.T. Dressier, Can. J. Phys. 66 (1988) 279. I101 L. Tiator, A.K. Rej and D. Drechsel, Nuci. Phys. A 333 (1980) 343. [ 111 P. Wilhelm and H. Arenh6vel, Few-Body Syst. Suppl. 7 (1994) 235. 1121 H. Arenh6vel, Few-Body Syst. 4 (1988) 55. [ 13] A. Messiah, Quantum mechanics, vol. 2 (North-Holland, Amsterdam, 1964). [ 14] See, e.g., D. Drechsel and L. Tiator, J. Phys. G 18 (1992) 449.

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