Z. Phys. A - Hadrons and Nuclei 338, 197-204 (1991)
zo,,.o..,. fi~r Physik A Hadrons and Nuclei 9 Springer-Verlag 1991
Relativistic calculation for the reactions/i d and/id--+ 3r p at rest
5 np
C.G. Fasano and M.P. Locher
Theory Group, The Paul Scherrer Institute (PSI) formerly SIN, CH-5232Villigen, Switzerland Received August 17, 1990; revised version October 12, 1990 We present a fully relativistic calculation f o r / ~ d ~ 5~p and/~d --, 3 ~p that includes angular momentum and spin dynamics. We calculate the inclusive proton distributions from two diagrams: the leading "tree" diagram, and the diagram for pion rescattering. Pion-nucleon rescattering proceeds through the A, in the RaritaSchwinger formalism, thus preserving the correct angular dependence. We use realistic parameterizations of the deuteron, keeping both the S and D states. The loop integrations for the rescattering amplitude and the phase space integrations have been done numerically. We find that the combined effects of a correct treatment of the rescattered pion and relativity are small in comparison with a simple non-relativistic calculation. PACS: 13.75.Cs
1. Introduction
The bTN interaction at low energies is dominated by annihilation, a fundamental non-perturbative process of QCD. Reactions on the deuteron offer the possibility to probe the elementary reaction in various ways. In particular by choosing a suitable final state, the isospin, spin, energy, and strangeness dependence can be explored. An especially interesting case is the reaction ~d --* mrep at rest, where the inclusive spectrum is related to the sub-threshold elementary annihilation amplitude, since the proton spectator carries out energy. In order to establish the properties of the elementary annihilation, one must correct for trivial nuclear effects first. In particular, a proper description of the deuteron and of the rescattering mechanisms on the basis of color - singlet dynamics is required. The present paper is part of a systematic effort aiming at a reliable description of the nuclear part of these reactions on the deuteron. For the loosely bound deuteron, these aspects should be particularly well under control at low energies. We shall concentrate on the dominant
reaction channels/~d ~ 5 zcp and/Sd -~ 3 rip. Detailed data for the proton spectator spectra for these reactions at rest exist. They are partly from bubble chamber experiments [1] and partly from recent counter experiments at LEAR [2, 3]. We briefly review some of our previous results for the proton spectator distributions. In [4], Nozawa and Locher evaluated the Born term ("tree" diagram of Fig. 1), and the pion rescattering contribution (Fig. 2) in a non-relativistic formalism. The peak of the proton spectator distribution near 70 MeV/c is due to the tree diagram. For proton spectator momenta between 300 and 600 MeV/c simple estimates show that resonant reN rescattering should dominate over competing mechanisms (initial state interactions, pion absorption, pion pre-emission). Nozawa and Locher set the antiproton annihilation vertex to a constant, which was normalized to the data at the peak of the proton spectator spectrum. They used an S-state Hulth6n deuteron [-5], and described resonant re+ p rescattering with a scalar Breit-Wigner amplitude having no angular dependence. The spin of both the nucleons and the antinucleons was neglected. Normalizing at the peak, the authors found that the proton spectrum in the plateau region (300 to 600 MeV/c) where rescattering dominates, was underestimated by a factor of 5 to 10. In I-6], Fasano, Locher, and Nozawa improved the earlier non-relativistic model by considering the role of heavy mesons and pion multiplicity. The approximations of assuming a constant annihilation vertex and scalar rescattering amplitudes were not changed. They found that the rescattering and production of heavy mesons was unimportant. However, the proper counting of pion topologies resulted in enhancement factors for rescattering, removing the discrepancy in the plateau region for the pd--+ 5rep case, but not for the/~d--+ 3rep case. Fasano and Locher evaluated the tree diagram in a relativistic formalism including both S and D states for the deuteron for spectator momenta up to 200 MeV/c in [7]. The tree diagram dominates the spectator too-
198 mentum in this regime, so the spin and energy dependence of the elementary annihilation vertex can be explored directly9 The dispersion calculations of Kroll and Schweiger [8] suggested considerable structure for the corresponding energy range of the p/~ amplitude below threshold. Fasano and Locher found that a strong peak in the p/~ amplitude could be excluded, but a weak structure is consistent with the available data. At the same time, they concluded that the spin structure at the annihilation vertex is well described by an effective ~5 operator, excluding a large admixture of a unity operator9 In this paper, we continue our systematic investigation by considering the effects of relativity in full. We will treat the nuclear part of the dynamics with modern techniques, avoiding the simplifications of the previous discussions. For the first time, we retain and evaluate the full spin and momentum dependence of the rescattering part of the diagram without approximation. In Sect. 2, we present the formalism in detail In Sect. 3 we present the results. Section 4 contains a discussion and conclusions9
2. F o r m a l i s m for relativistic calculation
To proceed, we will define the formalism for the relativistic calculation. First we will consider the tree diagram of Fig. 1. The amplitude is [-7] : Tt= C ~r (p) F~Sv (n) FAv (/~)
(1)
where p is the external outgoing proton four-momentum, n is the neutron four-momentum, and p is the antiproton four-momentum. We use Bjorken and Drell conventions. The supercript c denotes the charge conjugation, F~ is the deuteron vertex, FA is the annihilation vertex, and C denotes the normalization constant. SF(n) is the neutron propagator:
5- - _
, p d b" Fig. 2. Diagram for the rescattering of a pion in the reaction pd ~ mnp. The cross marks the on-shellnucleon
energies were well reproduced by using a 75 times a constant for the annihilation vertex: FA=y5 Ta
(4)
where TAis the constant. With this choice, the amplitude for the tree diagram is: Tt(p, P)= CuC(p) [Tu G . ( t ) - 8 9 qu Gb(t)] 9~"(d,,~) SF(n) ~ TAvO)
(5)
The constant Ta will be determined by fitting the peak near 70 MeV/c in the spectator spectrum to the data. In this range, rescattering is negligible. For comparison with the rescattering diagram further below, we choose an equivalent expression for the tree amplitude by writing the outgoing Proton as a particle spinor, rather than a conjugate one: T*(p, fi)= Ca(p) [,~ Ga(t) + 89qu Gb(0] 9e u (d, 2) S~ (n) Ta u ~0 )
(6)
where: -O+m~ SOy(n)- n2 _ m2 Note that in this form, the ~s from the annihilation vertex has disappeared. The rescattering diagram is shown in Fig. 2. The general form for this amplitude is: Tr (p, fi) = ~ C
0+m~ SF (n) - n~ _ m~
--~
j d 4 p' a (p) F~S F (p') D (k) Fd S~e(n) 1F' A u c(p)
(2)
(7)
where mN is the nucleon mass, and the remaining conventions are the same as in [9]. We will use the multi-pole expansion of Oourdin [14] for the deuteron vertex functions:
where p' is the intermediate proton momentum inside the loop and is integrated, and F~ is the vertex for the pion rescattering. D (k) is the relativistic pion propagator for a pion with four-momentum k:
F~= [~. G , ( t ) - 8 9 qu Gb(t)] e"(d, 2)
(3)
Here q = n - p , and z~(d, 2) is the deuteron polarization vector for a deuteron with momentum d and helicity state 2, and t is the invariant momentum transfer to the deuteron. In [7], it was found that the data at low
d , P Fig. 1. Tree diagram for the reaction ~d-..+mnp
1 D ( k ) = k2 _ m~
where rn~ is the pion mass. In the spirit of the relativistic impulse approximation, we will require that the internal proton be on-shell, while the neutron is off-shell. The rescattered pion may be both on and of mass-shell. We describe the nN rescattering vertex in the A resonance region using the Rarita-Schwinger formalism [10-12] : r _f2 p~ + mA r-m~2 Pa2 -- ma2 + imA FA ,,, [ 1 . 2pAuPa~
P~UY~--P~,'YU~
199 Here ma is the A mass (1232 MeV), m= is the pion mass, f~NJ is the coupling constant 0cffNa/4rc=0.365), k' is the outgoing pion momentum, k is the incoming pion momentum, Pa is the A 4-momentum, and F~ is the A width. We use a phenomenological width that has the form: 2.0(q~m r) 3 G G'(qcm/qr) 1~ Ea-- 1.0+ 1.2(q~mr) 2 Jr 1.O+(qcm/q~)2
3. N u m e r i c a l
results
In the earlier non-relativistic calculation, analytic integration of the rescattering amplitude was possible as long as the rescattered pion was confined to its massshell and the rcN angular dependence was neglected. The phase space integration had to be done numerically. The present relativistic calculation is entirely numerical. This allows us to fully calculate the intermediate state geometrical factors, the S and D state deuteron contributions, the spin sums, and the contributions from the rescattering of an off-shell pion. Such an approach also allows us to easily disentangle portions of the calculation, as well as try different types of vertices with little difficulty (see [7]). The phase space integration was done using standard Monte Carlo techniques. However, because the loop integration for an off shell pion is a principal value integration, it was done using Gaussian quadrature. Thus the calculation of the rescattering diagram represents considerable numerical effort, since for each phase space configuration of pions and proton, a three dimensional integration must be done on the amplitude level. We found that convergence at the 10-30% level was still attainable with a reasonably small number of phase space integration points.
3.1. Tree diagram The results for the tree diagram for /3d --+=+ rc + rc ~z rc p are shown in Fig. 3. The contribution due to the S-state part of the deuteron is shown by the long-dashed line while the D-state contribution
I ....
/ ....
I'''
10 2
(9)
where qom is the m o m e n t u m of the pion in the rtN center of mass, and q~ is the center of mass m o m e n t u m of the resonance (q,=227.0MeV/c). The constants are r =0.78/m~, G = 7 1 . 7 MeV, and G'=20.0 MeV. The first term reproduces the well known q3 dependence at small momenta. We have added the second term by hand to damp the propagator (in lieu of putting in a form factor). It has a negligible effect at low momenta. The form of (9) is purely phenomenological. F o r our purpose a multiplicative damping factor would be equally suitable, see the discussion in [11], e.g. For both the tree and rescattering diagrams, we will use the deuteron parameters by Locher and Svarc [13] which fit the electron data. In our previous work, we noted only a very weak dependence on the deuteron parameters, so long as a " m o d e r n " deuteron (e.g. Gourdin [14], Locher-Svarc [13], or similar) is used.
....
10 3
q
101 > 10 0
i
~a Z
I0-i --r
/
/
/
\\ \ \ \
10-2
\/
tO-3
I .... I . . . . . 0 200 400 600 800 Speetator Proton Momentum (MeV/e)
]
p-d -~ ~+~r+~r-Tr-~-p ,
,
,
,
. . . .
Fig. 3. Tree contribution to the proton spectrum for the reaction /~d ~ 5~p. The data are defined in Sect. 3.1. The long-dashed line shows the contribution from S-state part of the deuteron for the relativistic calculation. The dot-dashed line shows the D-state contribution for the relativistic calculation, and the solid line shows the complete relativistic calculation of the tree that includes both the S and D states. The short-dashed line shows the comparable non-relativistic calculation that includes both S and D states. The deuteron parameters were taken fiom Locher-Svarc [1'3] for all calculations shown in this figure
is shown by the dot-dashed line, and the full calculation is shown by the solid line. The data from [1] are shown as squares; the data from [2] are shown as diamonds, and the data from [3] are shown as circles. Data from [1] and [2] have been normalized at the spectator momentum of 70 MeV/c to the data from [3]. Experimentally, the absolute normalization is more reliable in the plateau region, but since we are interested only in the ratio of peak to plateau, we have normalized to the peak where theory is more reliable. As one would expect, the low-momentum region is primarily due to the S-wave part of the deuteron. The contributions due to S and D are equal at proton spectator momenta of approximately 260 MeV/c. The strong dip structure is a zero that results from a zero in the S-wave deuteron vertex function of Locher and Svarc [13], typical for realistic deuteron wave functions. The D-wave contribution dominates in the region of this zero, but by spectator momenta of 600 MeV/c the S-wave contribution has recovered and is 20% of the D-wave. Thus the only region that the D-wave strongly dominates is near the S-wave zero. Note that the peak contribution due to the D-wave is approximately 70 times smaller than the peak S-wave contribution. We remark that the usual two term S-state Hulth6n wave function simulates
200
t
10 3 102
t
t
t
~~,~ "
~ 10 2
101
1oOi/
/
',
9
4~,,,
\
-_
10 0
10-1 L;
\
)"z
10-2
L!
t
IfVr
',/
9
9
/ I
. . . . . . .
i0-1 i/
\, /
g
/
10-2
\/__ p - d -* 7r+Tr+71" 7r "iT p 10-3
10--3 0 200 400 600 800 Spectator Proton Momentum (MeV/c) Fig. 4. Tree contribution to the proton spectrum for the reaction /~d-* 3r~p. The data are defined in Sect. 3.1. The curves are defined as in the previous figure
the combined contributions of the S and D states rather well. The comparison of the fully relativistic calculation to the non-relativistic calculation of [6] is also shown in Fig. 3 (short-dashed line). As can be seen, the nonrelativistic and relativistic results are quite similar, even at high spectator momenta. Similar results for the reaction pd~rc+~z n are shown in Fig. 4. The data from [1] are shown as squares and the data from [3] are shown as circles. Data from [1] have been normalized at the spectator m o m e n t u m of 70 MeV/c to the data from [3], as has the calculation. As in the 5 pion production case, the S-wave contribution dominates at low momenta and the D-wave part only dominates in the region of the S-wave zero. In the high m o m e n t u m region in the 3 pion case, the data are flatter than in the 5 pion case, as is the calculation. As in Fig. 3, the short-dashed line in Fig. 4 shows the corresponding results from the non-relativistic calculation [6]. They are again close to the relativistic results. In the present calculations which use a constant annihilation vertex, the 5 n and 3 Tccases differ only by phase space. This is the origin of the flatter spectrum at high momenta for the 3 pion case. Note the ordinate scales are different in Figs. 3 and 4. We should of course not expect that the tree is appropriate for describing the high momentum regime.
3.2. Rescattering diagram The results for our fully relativistic calculation of the rescattering diagram for the reaction /~d
....
I ....
I ....
I ....
I,
0 200 400 600 800 Spectator Proton Momentum (MeV/c) Fig. 5. Single pion rescattering contribution to the proton spectrum for the reaction pd-~5~p. The data are defined in Sect. 3.1. The long-dashed line smoothly connects calculated points due to the S-state part of the deuteron for the relativistic calculation. The theoretical error bars shown are statistical (largest near proton spectator momenta of 400 MeV/c where the zero of the deuteron vertex function occurs). The dot-dashed line connects calculated points due to the D-state part of the deuteron for the relativistic calculation. The parameters used for the deuteron in the relativistic calculation are from Locher-Svarc [-13]. The short-dashed line shows the non-relativistic calculation fl'om [6] using a Hulth~n deuteron
n +•+n n n p are shown in Fig. 5. The data dispalyed are as in Fig. 3. The calculation includes rescattering through the A with an explicit Rarita-Schwinger propagator that preserves the correct angular dependence of the process according to (7) through (9). The normalization is relative to the tree for resonant A + + rescattering. The topological enhancement factor of Ref. 6 has not been included yet. The S-state contribution is indicated by the long-dashed line, and the D-state contribution is indicated by the dot-dashed line. The error bars on the theoretical curves are the statistical errors due to the Monte Carlo. The contribution of the rescattering diagram peaks at a spectator m o m e n t u m of approximately 300 to 400 MeV/c. The S-wave contribution is more than 10 times larger than the D-wave throughout the entire momentum range. The agreement with the non-relativistic calculation [-6] with a Hulth~n E5] deuteron for momenta above 500 MeV/c is deceptive. F o r a proper comparison, a Hulth6n deuteron must be used in the relativistic calculation as well. See the discussion in Sect. 4. Figure 6 shows the corresponding results from the relativistic rescattering calculation for /Sd --+/C+/C ~
p.
201 ' ' ' ' 1
1o2I"
....
I ....
I ....
I
10 2 -
101 3 l0 ~
i
.-
........................
>
_
\
i0 0
\
10_1
10-i
_
Z
t . . . . t--
--
Z
10-2 /
lO-3
f
f
,,,
0
p d -+ ~ + ~ I ....
200
I ....
400
~
p
I ....
d -+ ~ + ~
10-3
I
600
Spectator Proton Momentum (MeV/c)
10 3
_ ' ' ' ' 1
....
I ....
I ....
i ,
I ....
I ....
~
I ....
0 200 400 600 S p e c t a t o r Proton M o m e n t u m (MeV/c)
800
Fig. 6. Single pion rescattering contribution to the proton spectrum for the reaction /~d~ 3)zp. The data are defined in Sect. 3.t. The curves are defined as in Fig. 5
....
p I,
800
Fig. 8. Tree plus rescattering contribution to the spectrum for the
reaction > d ~ + ~- = p. The data are defined in Sect. 3.1. The solid line shows the full relativistic calculation where the rescattering contribution has been multiplied by the topological factor N ~ = 12.0, which is n o t consistent with theoretical expectations. See [6] for a discussion. The dot-dashed line shows the tree contribution and the long-dashed line shows the rescattering without topological factors
102
101
3 100 \ \
,X3
\ \
Z
i0-I
--
\
10-2 p d ~
io-3
....
I ....
~+~+~
I ....
~ I,
~
p ,,
1
0 200 400 600 800 S p e c t a t o r P r o t o n M o m e n t u m (MeV/c) Fig. 7. Tree plus rescattering contribution to the proton spectrum
for the reaction /Sd--,~ + ~+ ~ - ~ - = - p . The data are defined in Sect. 3.1. The solid line shows the full relativistic calculation where the rescattering contribution has been multiplied by the topological factor N~,, = 4.0; see [6]. The dot-dashed line shows the tree contribution and the long-dashed line shows the rescattering without topological factors
Figures 7 and 8 show the complete results for the relativistic calculations for the 5 and 3 pion production processes, respectively. In Fig. 7, the solid line results from adding the tree and rescattering contributions incoherently. The topological enhancement factor N ~ = 4 from [6] has been included into the rescattering contribution. N o t e that the theoretical topological factor could be as large as N ~ = 7.5 in the case of full coherence for pions of equal charge. The long-dashed line shows the rescattering without topological factors, as in Fig. 5. The tree is shown by the dot-dashed line. Similarly, Fig. 8 shows the results for the 3 pion production channel. The curves are defined in the same way as in Fig. 7. The topological enhancement used to obtain the solid line is N3~= 12, which is much larger than the theoretical estimate of N3~ = 1.7 from [6]. In the 3 pion case therefore, the present calculation of rescattering does not improve the description of the data.
4. Discussion
and
conclusions
Several aspects of the results concerning the rescattering deserve discussion. First we shall explain why the D-state of the rescattering contribution is so small for all momenta. To this end, we rewrite the amplitude, (7), explicit-
202
ly putting the spectator proton propagator on the mass shell: C T ~(p,/~) = ~
....
1.0
I ....
I ....
I ....
'~I~..
_ ff+m~
t
j" d 4 p' ~i(p) 1;
I ....
m
N t
96 (p; - - E p , ) D (q) Fa S~ (n) r~ u ~(/~)
| t
0.8
(lO)
|
9 ,i
t
~eV/c
g00
i
m
i
|
| 89
i
|
Substituting the identity:
400 MeV/c
'
I
i~' + mu = 2 mN ~ ux, (P') t~, (p')
!
0.6
(14)
I
o o
At
!
_
200 MeV/c
I-
we obtain: 0.4
TA (p, D
C ~ d, p, 6 (p; - Ep,) D (q) 2 mN 2Ep, (2~) '~ 9a~(p) r~ u~,(p')[a~,(p') r~ S~(n) G u1(~)]
(12)
The part of (12) in square brackets may be identified with the amplitude for the tree diagram (6), allowing us to write:
, 6 (p; -- Ep,) D (k) Tfx(P' iO)= ( 2 ~ ) ~ ] d P 2E,, 9aa(p). ~ u~,(p') rLx(p', ~)
0.2 z
p d ~ 5zrp ....
0.0
2 mN - ,
(13)
We shall identify the pion rescattering T-matrix by T;a, (p, p, k) = aa(p) F~(k) uz,(p')
I ....
i ....
I ....
I ....
200 400 600 800 Internal Proton Momentum (MeV/c)
1000
Fig. 9. Distribution of internal m o m e n t a of the spectator proton for the reaction fid~5np for outgoing proton momenta of 200, 400, and 800 MeV/c as marked. The histograms are normalized at the peak value
(14)
therefore (13) becomes: _, 2 mN 6 (P'o -- Ev,) D (k) T'(x(P' P)= ~ ~ d4p' 2E,, T~=ff'(P' p'' k) o 9 r.~,x(p, P)
(15)
1.0
,
~.
\'\ %',.,
When we decompose the tree into its S and D wave pieces, T~,X= T t s • TtD
(16)
~ 0.8 lii//:l!l:r/~/ \<\~, ';,, ~~ ,"',,
-
we see how the rescattering T-matrix is connected to the tree diagram: , 2 mN 5 (P'o -- Ev,) D (k) T,~x(p, ~) = (~4~)4 1 d'* p' 2 Ep,
9T~N(p, p', k) [ Tffx(p', t~) + ~'*'-.x,P , ~' D]
t
I~
(17) 400
The rescattering T-matrix is thus an integral over the tree amplitudes which contain the deuteron vertex functions. Let us examine (17) in terms of the contributions of S and D wave tree contributions shown in Figs. 3 and 4. The main contribution to the integral comes from on-shell pions that rescatter in the A resonance region as we have checked. The integral extends over all the internal proton momenta for any external proton momentum. In particular, it includes the peak from the Sstate wave function at small momenta. The D-state contribution, (which is substantial only in the narrow range of momenta from 200 to 500 MeV) is thus only a small
0.2
71 ~ l
~v/e ~",
\k " "\" "
2oo ~w/o
P-d ~ 5zrp 0.0
0
~
'
'
'
'
[ 50
~-':
""....
. . . . . . . 100
"1" ",--,: 150
Pion Scattering Angle (Degrees) Fig. 1@. Distribution of pion rescattering angle with on-shell kinematics for the reaction ~d~5rcp with outgoing proton momenta of 200, 400, and 800 MeV/c as marked. The histograms are normalized at the peak
203
part of the cross section even for large external proton momenta. This is very different from the reaction pp--,~d at the A resonance, where the two body kinematics forces the pion to be off-mass shell, and the dominant contribution comes from intermediate momenta of approximately 400 MeV/c (see [9]). Iri that case, the Dstate is more important. In Fig. 9, we demonstrate what the leading internal proton momenta generated by the external six-body phase space are. To this end, we have fixed the magnitude of the external proton three-momentum, and scanned the remaining phase space for events falling in the A resonance peak (E~N= t232 +__100 MeV). Figure 9 shows the resulting normalized distribution of internal proton momenta that can scatter into these states. It is clear that the momentum distribution is rather flat and always peaks at low internal momenta, which are completely dominated by the S-state. We obtained similar results for the 37zp case. To analyze the effect of the angular dependence of the 7zN vertex, we show histograms similar to Fig. 9 except now we analyze the distribution of pion rescattering angles. Figure 10 shows the pion rescattering angles for the l~d~51rp reaction. The histograms are marked by external proton momenta of 200, 400, and 800 MeV/c. w e see that the pion scattering angle of approximately 30 degrees is favored by phase space. It moves upward 10 3
10 2
101
"a
-}
10 0
9
~Q Z
lO-i
10-2
10-3
p d ~ Tr+Tr+Tr Tr Tr p
i
t ....
I ....
I ....
I ....
I,
0 200 400 600 800 S p e c t a t o r P r o t o n M o m e n t u m (MeV/c) Fig. 11. Comparison of calculations using a Hutth~n S-state deuteron for the process / ~ d ~ 5 ~ p . The data are defined in Sect. 3.1. The long-dashed line shows the non-relativistic calculation of [fi] using a scalar Breit-Wigner for elastic ~N rescattering. The solid line smoothly connects points calculated using the same scalar Breit-Wigner in the relativistic formalism. The dot-dashed line connects points calculated in the relativistic formalism using the full Rarita Schwinger formalism that preserves the correct angular dependence of ~N scattering
only slightly with increasing momentum of the outgoing proton. We obtain similar results for the 37zp case (not shown). Since the rcN cross section is forward peaked above the resonance and backward peaked below the resonance, a correct treatment of the rcN angular dependence will lead to enhancement of the rescattering contribution at high momenta over the non-relativistic scalar Breit Wigner treatment of [6]. Such an enhancement can be seen in Fig. 11, where we compare the non-relativistic scalar Breit-Wigner (long-dashed curve), the relativistic calculation with the same scalar Breit-Wigner (solid curve), and our standard relativistic Rarita-Schwinger form with full angular and spin factors (dot-dashed curve). We use an S-wave Hulth6n deuteron in each case. We see that the relativistic and non-relativistic scalar Breit-Wigner curves are approximately the same, while the Rarita-Schwinger curve is higher at larger momenta as discussed. This enhancement is cancelled when the Locher-~varc deuteron [13] is used because of differences in the vertex functions, cp. Fig. 5 long-dashed line. This concludes our detailed discussion of the nuclear part of antiproton absorption on the deuteron. We have shown that the dominant reaction channels are not sensitive to the proper inclusion of the angular dependence of the 7~N rescattering amplitude, nor to the use of a realistic deuteron wave function. Moreover, the net effects of spin and relativity are surprisingly small. An open problem remains the proper description of the elementary NN annihilation amplitude. For the kinematics of the tree diagram, p absorption on the deuteron at rest shows some sensitivity to the energy dependence of the/3p interaction subthreshold, as has been discussed in [7]. That analysis also favors the spin independent transition operator at the N N vertex, which we have used throughout this paper. The energy dependence of the elementary NN amplitude further away form threshold into the unphysical region appears to be rather flat [8]. Even if it were not, this would not be of qualitative importance for the rescattering amplitude of i0d absorption, since it is dominated by low momentum transfers and hence by the threshold region of KrN, as we have shown. However, it may well be that some of the remaining discrepancies, particularly for the/Sd ~ 3 rcp channel, are do to the inadequate simplification of the spin-isospin structure of the elementary amplitude. A further investigation in this direction, however, will require the introduction of models for the elementary annihilation amplitude. We thank Satoshi Nozawa for helpful discussions. Numerous discussions with Joseph Riedelberger and Peter Tru61 from the Asterix collaboration are acknowledged. We thank Gerry Smith and Vladislav Sim/tk for information on the bubble chamber experiments. We would also like to thank Claude Amsler for additional discussions. Finally, we would like to thank the l~cole Polytechnique F6d6rale de Lausanne, Switzerland for providing us with Cray-2 computing time.
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204 2. Ahmad S. et al.: Proceedings of the Fourth LEAR Workshop 1987, p. 447, Physics at LEAR with low energy antiprotons. Harwood Academic Publishers 1988 3. Riedlberger, J. et al.: Phys. Rev. C40, 2717 (1989) 4. Nozawa, S., Locher, M.P.: Proceedings of the Fourth LEAR Workshop 1987, p. 763. Physics at LEAR with Low Energy Antiprotons, Harwood Academic Publishers 1988 5. Hulth6n, L., Sugawara, M.: Handbuch der Physik. Vol. 39: Structure of atomic nuclei. Flfigge, S. (ed.), p. 1. Berlin: Springer 1957 6. Fasano, C.G., Locher, M.P., Nozawa, S.: Z. Phys. A - Atomic Nuclei (in press)
7. Fasano, C.G., Locher, M.P.: Z. Phys. A - Atomic Nuclei 336, 469 (1990) 8. Kroll, P., Schweiger, W.: Nucl. Phys. A503, 865 (1989) 9. Grein, W., KSnig, A., Kroll, P., Locher, M.P., Svarc, A.: Ann. Phys. 153, 301 (1984) 10. Rarita, W., Schwinger, J.: Phys. Rev. 60, 61 (1941) 11. Oset, E., Toki, H., Weise, W.: Phys. Rep. 83, 281 (1982) 12. Williams, H.T.: Phys. Rev. 29, 2222 (1984) 13. Locher, M.P., Svarc, A.: Z. Phys. A - Atoms and Nuclei 316, 55 (1984) 14. Gourdin, M., Le Bellac, M., Renard, F.M., Tran Thanh Van, J.: Nuovo Cimento 37, 3240 (1965)
