Pion absorption on the deuteron

Nuclear Physics @ North-Holland

A452 (1986) 644-668 Publishing Company

PION

ABSORPTION

J. VOGELZANG, Natuurkundig

Laborarorium

B.L.G.

ON THE DEUTERON BAKKER

der Vrue Universileit,

and

H.J.

de Boelelaan

BOERSMA

1081, Amsterdam,

The Netherlands

Received 1 July 1985 (Revised 2 September 1985) Abstract: The total and differential cross sections for the absorption of positive pions on deuterons have been calculated for pion kinetic energies between 3 and 220 MeV. The model is based on perturbation theory and includes the backward-moving pion. We varied several parts of the dynamical input, like the TN T-matrix and the absorption formfactor, and studied the effect of relativistic definitions of the relative momentum and of the choice of subsystem energies. Because of positive interference, the contribution of the backward-moving pion doubles the cross sections. The definition of the relativistic relative momentum according to Aaron, Amado and Young is to be preferred over that of Wightman and Girding. Also the off-shell behaviour of the TN interactions and the choice of subsystem energies influence the cross sections significantly. We need rather low values for the cutoff at the absorption vertex (0.4-0.5 GeV/c) in order to fit the experimental data.

1. Introduction The absorption process of pions on deuterons at low and intermediate energies has received much attention in the last few years. Both the theoretical treatment and the experimental information have been considerably improved recently. The theoretical approach can be roughly divided into two categories: (i) models based on perturbation theory 1-4), and (ii) three-body models 5-9). The latter have the advantage that they treat the coupling of two- and three-body channels unitarily. This has been achieved for the first time in a consistent way by Avishai and Mizutani 5). Blankleider et al. “) and Betz et al. 7, use relativistic kinematics for the pion only. Betz et al.. restrict their model to P33 rescattering; they omit the direct absorption. Rinat et al. *) and Perrot ‘) use relativistic equations and include p-meson rescattering. Perrot adds exchange of other heavy mesons (w, 7, A). In the perturbative calculations one has the opportunity to include four-body states, e.g. states with two nucleons and two pions. Furthermore, perturbative calculations can more easily be generalized to cases where more than two nucleons are involved than calculations based on exact N-body equations. Maxwell et al. ‘) include rescattering of a pion or a p-meson in TN S-waves and the TN P,, wave. Goplen et al. ‘) include S- and P-wave rescattering of a pion that can move both forward and backward in time. Grein et al. ‘) and Laget et al. “) treat the lowest-order contributions as Feynmann diagrams. They integrate correctly over the energy of the rescattered pion. 644

J. Vogelzang et al. / Ron absorption

645

Since we wish to apply our model to pion absorption on other nuclei, e.g. ‘He, we base our model on perturbation theory. Therefore our model does not meet unitarity. However, proper inclusion of unitarity would require the solution of the four-body problem in the case of 3He, which is beyond the scope of our investigation. There are several parts of a perturbative calculation that are subject to uncertainties. Some of them have been investigated already to some extent in the literature. It is the purpose of this work to study the sensitivities of the absorption of pions on deuterons to these parts extensively and systematically within one framework. To facilitate a comparison of our work with similar studies published before, we have constructed a model, denoted as standard-l, where the ingredients of the dynamics are chosen in such a way that a best fit to the experimental data on total and differential cross sections is achieved. We compare a perturbative calculation with the same input as used by Blankleider and Afnan “) to their complete calculation in order to estimate the importance of the multiple-scattering contributions neglected by us. On the other hand we varied the dynamical parts of the model one at a time in order to gauge their relative importance. Finally we combine some variations in the so-called standard-2 and standard-3 versions of the model. These models do not fit the experimental data as well as standard-l, but they incorporate some important features (backward-moving rescattered pions and the BlankenbeclerSugar definition of subsystem energies) and thus warrant separate study. Our model and the different choices for the dynamical input are described in sect. 2. The results are given in sect. 3, and sect. 4 contains our conclusions in which we also compare with other models. Here we want to make a remark on p-meson rescattering. It is known to be important in models with a strong p-nucleon tensor coupling ‘). As it is not clear whether such models are more realistic than models including weak tensor coupling for the p-meson, we feel that we are justified in neglecting p-meson rescattering in our calculations insofar as we attempt to estimate the effect of variations of the other dynamical ingredients of our model. 2. The model The contributions

retained in our perturbation-theory model are direct absorption and rescattering of a pion that can move both forward and backward in time. The contributions are depicted in fig. 1. We work in momentum space and in the rd c.m.s. The differential cross section is then given by

(2.1) where E,,., Ed and EN are the energies of the pion, the deuteron and an outgoing nucleon, and s is the square of the total energy of the system in the nd c.m.s. The summation is over the spins of the initial and final states. The element of the

J. Vogehng

646

et al./ Pion ubsor~tjo~

a

Fig. 1. Contributions included in our model, (a) direct absorption, (b) rescattering of a forward-moving pion, and (c) rescattering of a backward-moving pion.

transition matrix Tf, is the sum of the contributions abso~tion this cont~bution is

TFA(pr, k) = I

shown in fig. 1. For direct

d3p’ !P;;)’ (~‘1 Tab&p‘, k, &s) F&P’ - ;kI ,

(2.2)

@“@IO, 4) = (‘%-

(2.4)

and for rescattering it is

with WJ-’ ,

where q. is the off-shell energy of the forward-moving pion determined by fourmomentum conservation, and wq = m is the on-shell energy of the rescattered pion, m, is the pion mass and the + (-) sign denotes the forward (backward) moving pion. Tabs is the T-matrix describing the absorption of a pion on a single nucleon, and TnN the T-matrix describing the scattering of a pion off a nucleon. A partial-wave expansion has been performed and the angular integrations have been done analytically. The radial integrations are calculated numerically. The numerical accuracy of the final results is better than 1%.

J. Vogelzang et al. / Pion absorption

641

The differential cross section as a function of the angle 13between the incoming pion and a detected nucleon can be expanded into Legendre polynomials as follows:

g=x

(YziP*i(COS

0).

(2.5)

I

The coefficients (Ye, (Y*,and (Yeare known from experiments. The coefficient CQ equals the total cross section up to a factor 27r, (Yedetermines the slope and (Yethe curvature of the differential cross section as a function of cos* 13. We will now turn to the discussion ofthe various components of eqs. (2.2) and (2.3). 2.1. THE

DEUTERON

WAVE

FUNCTION

AND

THE

NN

INTERACTION

For the deuteron we used the Paris wave function given by Lacombe et al. lo). This widely used wave function describes the deuteron properties well. For the NN interaction in the final state we took the separable form given by Schweiger et al. I’). This interaction gives a good fit to the NN scattering data. Due to its separable form, the NN wave function Y:;‘+(p) can be easily calculated. Note that the NN propagator appearing in the NN wave function !I$.;‘+(p) is not renormalized. Therefore it contains some higher-order pion rescattering effects. This does not lead to inconsistencies if one omits the pole part of the P1, rescattering (see the subsect. 2.2). We include the NN interaction in the ‘So, 3P,, 3P2 and ‘D2 final states. Since the NN phase shifts are small in the higher partial waves in the energy range considered here, we neglected the NN interaction there. 2.2. THE

aN

INTERACTIONS

All three-body models for pion absorption on the deuteron contain separable TN interactions in order to make the computation feasible. To compare our model with those models and for computational convenience we also restricted ourselves to separable TN interactions. The T-matrix describing TN scattering is given by TwdP’, P, -&N) = Y’fndK’, K, %rdY.

(2.6)

In eq. (2.6), primed quantities pertain to the final state. The quantity p is the relative momentum of the pion and the active nucleon, and EIIN is the total TN energy in the ?rd c.m.s.; K and w,~ are the relative momentum and the total TN energy in the VN c.m.s. The factors y and y’, arising from the transformation of trN in the rrN c.m.s. to TmN in the md c.m.s., are discussed in subsect. 2.3. After partial-wave expansion, t,,, is given for each partial wave cr, (Y= LSJT, by f%(K’, ‘GC,)

= g~N(K’)T~N(W,N)S~N(K).

(2.7)

Several parametrizations of the TN interaction are given in the literature. We have chosen the interactions given by Thomas ‘*), Betz et al. ‘), Schwartz et al. 13) and

J. Vogelzang et al. / Pion absorption

648

Rinat

et al. “). These

and have different resonance

interactions

off-shell

have been

behaviour,

employed

notably

in three-body

calculations

in the P,, wave which dominates

at

energy.

The P,l wave requires

special

attention.

Since the quantum

numbers

of this wave

are the same as those of a nucleon, it may be split into a part consisting of a real nucleon (pole part) and a non-pole or background part. The pole part of Pi1 rescattering is already taken into account in the direct absorption. To avoid doublecounting, as already mentioned in the previous section, we only include the background part in Pi, rescattering. The formfactor of the pole part describes the reaction rN++N and can therefore be used in the absorption T-matrix. Mizutani et al. 14) and Blankleider et al. 6, give prescriptions for separating the pole and the background parts. It is well known that the resulting cross sections are extremely sensitive to the value of the cutoff of the absorption formfactor. To study this sensitivity more easily, we also used a monopole formfactor of the type given by Rinat et al. “):

(2.8) where K is the relative TN momentum, cutoff momentum. Note that Maxwell

mN the nucleon mass, and A the adjustable et al. ‘) use a formfactor of the type (2.9)

where K = ( K~, K) is the pion energy-momentum four-vector. This formfactor equals one when the pion is on-shell, and therefore has an off-shell behaviour that differs from eq. (2.8).

2.3. RELATIVE

MOMENTA

AND

RELATIVISTIC

T-MATRIX

TRANSFORMATIONS

The formfactors at the TN vertices are functions of the relative momentum. To study the importance of relativistic kinematics, we employed besides the well-known expressions for the relative momentum using non-relativistic or relativistic pion kinematics also those given by Aaron, Amado and Young (AAY) “) and by Wightman and G&-ding (WG) r6). For two particles with momenta p1 and p2, energies E, and E2 and masses m, and mZ, the relativistic relative momentum K is given by K=(~2+cj2)Pl-(~,+jl)P2

,

(2.10)

of the two particles

and w* = E2 - (p, +p2)*.

E+w where E = E, + Ez is the total energy Aaron, Amado and Young define i

=E*+E;-E; 1

2E

’

E2=

E2-E;+E; 2E

’

(2.11)

649

_I. Vogelzang et al. / Pion absorption

G, =

w*+o:-w:

G*=

’

2w

w*-w:+w:

2w

(2.12)

’

w;=&-p;. Alternatively,

Wightman

and Girding

i, =

;,

=

define

E*+&E;

E*=

’

2E

(2.13)

w2+mf-mm:

I

2w

)

E*-,??;+E;

2E

(2.14)

’

_w*-m:+rn:

w2-

2w

(2.15)

’

Ef=pf+inf. When

both

particles

are on-shell,

(2.16)

the AAY and WG relative

momentum

are the

same and these formulae reduce to the expression for pi transformed to the two particle c.m.s. by a Lorentz boost with boost parameter B = -(pl +p2)/E. The rrN T-matrix and the absorption T-matrix are given in the TN c.m.s., while we need them in the rd c.m.s. For on-shell particles we write “) (2.17)

T(p’, P, E) = Y’t(K’, K, w)?‘, with the relativistic

T-matrix

transformation

Y =&wz/E,E,,

factors

y and y’ given by

$=Jw;w;IE;E;.

(2.18)

Unprimed (primed) quantities pertain to the initial (final) state, p is the relative momentum in the rd c.m.s., K that in the nN c.m.s., Ei the energy of particle i in the rd c.m.s. and wi that in the TN c.m.s. Giebink ‘*) has shown that eqs. (2.17) and (2.18) also can be used for off-shell T-matrix elements if the WG definition of the relative use the wi =m. relative

momentum

is employed.

For the energies

in eq. (2.18) one then should

on-shell expressions Ei = m, Ei = Jp’*+ mf, Wi = dK2 + rn: and We will use eqs. (2.17) and (2.18) also for the AAY definition of the momentum.

2.4. DEFINITION

In a complete

OF THE SUBSYSTEM

relativistic

calculation

ENERGIES

one should

integrate

over the energies.

We

here perform an approximate calculation where the energy of one of the particles is fixed. At the same time non-relativistic wave functions are used. These approximations are motivated by our wish to apply the same model as used for pion absorption calculations the spectator here to the case of other nuclei, e.g. 3He. In three-body particle is taken on-shell. This is the well-known Faddeev choice. In our model this corresponds to taking the lower nucleon in fig. 1 on-shell. To study the sensitivity of our model to this procedure, we also used the Blankenbecler-Sugar choice, where both nucleons in an NN intermediate state have equal energy. This choice is equivalent to that made in ref. ‘).

650

J. Vogelzang et al. / Pion absorption

We found it necessary to add an energy shift Eshitt to the total energy of the TN system in the PS3 wave in order to move the maximum in the total cross section to the correct position. This shift has a value of 20-30 MeV. This is of the same order of magnitude as the potential energy of a nucleon in deuterium and can be considered as a binding effect. Finally we note that our model is identical with the first two terms in a multiplescattering series expansion of the Blankleider and Afnan model if we use their rrN interactions and kinematics, and if we omit the energy shift Eshift, the backwardmoving pion and the final-state NN interaction. 3. Results Our model enables us to vary all ingredients discussed in sect. 2. We varied the TN T-matrix, the absorption formfactor, the relative momentum and the definition of the subsystem energies, and we studied the effect of including the NN interaction in the final state, the factors arising from the relativistic T-matrix transformation and the backward-moving pion. To present the effects of these variations in a systematical way, we start with a calculation called standard-l that describes the experimental data well. We compare variations in the input for each component separately with this standard-l calculation. In most cases variations in the input will worsen the agreement with experiment. This does not necessarily mean that the choice of this particular component is worse than the standard choice. Finally we will combine some variations in the.input and present them as the standard-2 and standard-3 calculations. 3.1. STANDARD-l

Figs. 2a-d show the results of the standard-l calculation. We used the Paris deuteron wave function, included NN interaction in the final state and took Thomas’ rrN T-matrix in all TN waves except Ptl, where we employed the background part of the PI1 T-matrix by Blankleider et al. We used a monopole formfactor with a cutoff of 510 MeV/c at the absorption vertex, the AAY definition of the relative momentum, included the relativistic T-matrix transformation factors, took the Faddeev choice for the subsystem energies, and omitted the backward-moving pion. The energy shift Eshin has a value of 24 MeV. The value of the cutoff of the absorption formfactor is not well known. Estimates by several authors lie between 600 and 1200 MeV/c. Our value of 510 MeV/c is not within this range. We discuss this problem in subsect. 4.1. The experimental points are taken from refs. 19-26).Lest the figures be obscured by too many data points, we included the most recent and accurate ones only. In fig. 2a we show the total cross section utot in mb as a function of T~.~.~.,the momentum of the pion in the rd c.m.s. in units of the pion mass. Besides’ the

14 -

-

s1mnmo1

---

DlRECT RESWTIOH 511 3wo Sill

.^”

3e

IO

I-

I-

4

I

,

>

,

*

-

STtroeRDl

---

MO Pll.

P3t

,

lwl

3

I

I

PI3

Fig. 2. Standard-l results: (a) tatal cross section; (b) difierential cross sections; (c) a,; (d) a+ Expen’menral point: l - Rose I’?);0 - Richard-Serre et al. 20); v - Axen er ail *l); V - Preedom et al. **I; A - Ritchie et at. z3); A - Jioftiezer et at. ‘*I; q - Ritcbie ef af. *5); !Il - Aprile et al 26f. Curves: see text.

652

J. Vogelzang et al. / Pion absorprion

complete result (solid curve) this figure also contains the results for the direct absorption (dashed curve), the S,, and S3, rescattering (dotted curve), and the P3s rescattering (dot-dashed curve). The interference between the direct absorption and the Ps3 rescattering increases the total cross section and moves the maximum to lower energy. The behaviour at low energy is governed by the S-wave rescattering. The S3, wave dominates the total cross section in this region. Fig. 2b shows the differential ,cross section dm/dfl in mb/sr as a function of cos’ (%,.,), where %,.,. is the angle between the incoming pion and a detected nucleon in the rd c.m.s., at two energies: qc._ = 0.79 ( Tzb = 45 MeV, lower curves) and Q.,,,.~.= 1.42 ( TEb = 125 MeV, upper curves). We show the complete result (solid curves) and the result without the small P-waves P,,, P3, and P,, (dashed curves). Figs. 2c, d show the coefficients a2 and CX,in mb/sr as a function of qc.,_ for the complete result (solid curves) and the result without small P-waves (dashed curves). The influence of the small P-waves PI,, P31 and PI3 is negligible for the total cross section, but becomes more important for CQ and q. The agreement of the full calculation with experiment is good, except for (Yewhere the calculated values are positive while the experiments indicate negative values. Blankleider and Afnan find small and positive values of cyqat low energy while CY~ changes sign at q__. L- 1.60. 3.2. NN INTERACTIONS

AND

RELATIVISTIC

?--MATRIX

TRANSFORMATIONS

In figs. 3a-c we present the results without NN interactions in the final state (dashed curves) and without relativistic T-matrix transformation factors (dotted curves), compared with the standard-l result which includes both. The relativistic T-matrix transformation factors have almost no effect. The NN interactions in the final state decrease the total cross section at very low energy and increase it at resonance. Moreover, the NN interactions increase the slope of the differential cross section, especially at resonance, as can be seen from the plot of ty2 (fig. 3b). 3.3. RELATIVE

MOMENTUM

The results with different definitions of the relative momentum are shown in figs. 4a-c. Besides the AAY relative momentum (solid curves), we used the WC relative momentum (dashed curves), the relativistic pion kinematics relative momentum (dotted curves) and the non-relativistic relative momentum (dot-dashed curves). In the case of the relativistic pion kinematics and non-relativistic relative momentum, we omitted the relativistic T-matrix transformation factors which are unimportant (see subsect. 3.2). The total cross section obtained with the WG relative momentum is too small compared to the others at all energies. At low energy the use of the WG relative

J. Vogelzang et al, / Pion absorpGon

J

(

,

-

---

,

,

,

,

,

,

,

STRYOIYlO1 NO F.S.I. No

R.T.T.

653

I

-

ST-I

---

NO F.S.I.

,.,c -- _

NO R.T.T.

Fig. 3. NN interaction and relativistic T-matrix transformations: (a) total cross section; (b) cu,; (c) (Y+ Experimental points: as in fig. 2. Curves: see text.

momentum leads to a very small cross section, because the WG relative momentum differs much from the other relative momenta which have nearly the same value there. The results with the relativistic pion kinematics and the non-relativistic relative momentum differ little from each other, except for (Y,+where the result with the relativistic pion kinematics relative momentum is more in agreement with the experiments. For this choice of relative momentum, a4 changes sign at rfc._. = 1.55 and behaves like CK,as calculated by Blankleider and Afnan (see fig. 10~).

3.4. WN T-MATRIX

Fig. 5 shows the total cross section obtained with the aN T-matrices given by Thomas (solid curve), Betz et al. (dashed curve), Schwartz et al. (dotted curve) and Rinat et al. (dot-dashed curve). We use these TN T-matrices in all TN waves except

J. Vogelzang et al. / Pion absorption

654 IS

,

(

,

,I

-

--. _. ---

I2

(

,

1

,

STWRRO W.C.

,

,

,

,

,

1,

I,

1,

I

I

R.P.K. N.R.

-

a IO -

I-

6-

4-

3

---

N.R.

P ii

z s I

i’0

0. 00

0.40

L. 20

0.00

K-CHWPION

Fig. 4. Relative

momentum:

I.00

P.W

-04

f8lS.S

(a) total cross

considered

I 0.40

1

I 0.0

K-CNWPICN

section; (b) a,; Curves: see text.

P,,, where in each case we took the background et al. The ?rN T-matrices

m

I

I Lao

1 L.0

I

*.oo

MSS

(c) ad. Experimenfal

P,, T-matrix

1

poinrs: as in fig. 2.

given by Blankleider

here differ in their off-shell behaviour,

in particular

in the Pj3 wave. In this wave, Betz ef al. use a dipole formfactor with a cutoff of 358 MeV/c, Schwartz et al. a monopole formfactor with a cutoff of 291 MeV/c and Rinat et al. a monopole formfactor with a cutoff of 355 MeV/c, while Thomas employs a formfactor consisting of two parts: a monopole with a cutoff of 291 MeV/c and a dipole with a cutoff of 671 MeV/c. These formfactors, normalized to one at momentum equal to zero, are plotted in fig. 6. At resonance the total cross section is dominated by the Pj3 rescattering. The most important amplitude at this energy is the one leading to the ID? final state. We integrate over the formfactors of the TN interactions. Apparently a formfactor that decreases slowly gives a larger contribution to the integrals than a form factor

X ta

,

,

(

Vogeizang

,

,

,

-

STRNOC#Ol

--_. *

BET2 ET AL. .XiiM?TZETAC. RINAT ET RL.

---

Fig. 5. nN T-matrix.

et al. / Pion ,

Experimental

,

,

points:

,

655

absorption ,

BS in

,

fig.

,

,

,

2. Curves:

,

see

,

,

1

text.

that decreases more rapidly, and the integrals contribute in a coherent way to the amplitude leading to the ‘D, final state. From fig. 5 we infer that the P,, contributions for the T-matrices given by Betz et al., Schwartz et al. and Rinat et al. are too small compared with the S-wave contribution. Only a Ps3 formfactor with a large cutoff, like Thomas’, produces a pronounced peak at resonance in the total cross section. We note that coupled-channels calculations for wd++pp using the T-matrices from Schwartz et al. and Rinat et al. also give too small values for the cross sections at resonance ‘“).

3.5. ABSORPTION FORMFACTOR

In fig. 7 we give the total cross section with the P,, pole formfactors from Blankleider et al. (dashed curve) and Mizutani et al. (model A, dotted curve and model B, dot-dashed curve), together with the standard-l result with a monopole formfactor with a cutoff of 510 MeV/c (solid curve). The Pi, pole formfactor from Blankleider et al. has two terms: a dipole with a cutoff of 553 MeV/c and a tripole with a cutoff of 789 MeV/c. The P,, pole formfactor from Mizutani et al. has a complicated structure: it is complex and energy dependent. The cutoff of the elementary pole formfactor is 893 MeV/ c for model A and 1109 MeV/ c for model B. The formfactors, normalized to one at momentum equal to zero are plotted in fig. 8. The aN total energy is taken equal to mN. In case of the formfactors of

656

RINRT

Fig. 6. Pzs formfactors namaiized

ET AL.

to one at zero momentum. Curues: see text.

Mizutani et al. we plotted the absolute values. From figs. 7 and 8 we see that the cross section increases with increasing cutoff of the absorption formfactor. The fact that the formfactor of Blankleider et at. does not follow this trend is probably due to a too low value of the TN coupling const~nt~~~~/4~, which is 0.068 compared to 0.080 for the other formfactors.

3.6. THE SUBSYSTEM ENERGIES

Figs. 9a-c show the results obtained with the ~lanke~beclcr-Sugar choice for the subsystem energies (dashed curves) together with the Faddeev choice (solid curves). The difference between the two results is due to the fact that in both the AAY relative momentum is used, which depends on the energies of the particles. When we employ an energy-independent relative momentum, the two definitions of the subsystem energies give almost the same results. Note that a4 becomes negative for the Blankenbecler-Sugar choice, in agreement with experiment. We will return to this paint in subsect. 3.10.

J. Vogelzang et al. / Pion absorption

---

tlIZUTAN1

657

B

12 -

,g

‘O-

E

6..

8

6-

0’ E

4-

2-

L”l”““““‘l”‘If oo.oo 0.20

0.40

1.46

0.60 iEs,F%

Fig. 7. Absorption

3.7. BLANKLEIDER-AFNAN

formfactor.

1.66

1.66

2.66

r&l6

Experimental points: as in fig. 2. Cuwes:

see text.

INPUT

In figs. lOa-c we show the results of the complete Blankleider-Afnan calculation (solid curve) and a perturbative calculation with the same input (dashed curves), where we omitted the NN interactions in the final state, the relativistic T-matrix transformation factors and the energy shift Eshirt, and used the relativistic pion kinematics relative momentum and Blankleider and Afnan’s pole PI1 form factor at the absorption vertex. The maximum of the total cross section is too far to the right for the perturbative calculation. The dotted curve is obtained when we include the same energy shift of 24 MeV in the PJ3 rrN total energy as in standard-l. Apparently, the multiple scattering effects, included in ref. 6), increase the total cross section and (Yedrastically in the energy region below resonance.

3.8. EACKWARD-MOVING

PION

Fig. 11 shows the total cross section including the backward-moving pion (solid curve) together with the contributions of the forward-moving pion only (dashed curve) and the backward-moving pion only (dotted curve). The contrbution of the backward-moving pion is smaller by a factor of about two than that of the forwardmoving pion, but the interference between the two contributions is strongly constructive. In order to fit the experimental data at resonance we had to decrease the cutoff of the monopole absorption form factor to 380 MeV/c, leaving the rest of the input

658

J. Vogelzang et al. / Pion absorption ABSORPT I ON

I.40

1 _

1.20 -

OaWo

I

1

------

I 2

I

I

I

~~HE~TuH Fig. 8. Absorption

formfactors

normalized

the same as in the standard-l standard-2.

3.9. INFLUENCE

OF THE

S,,

FORNFACTORS . 1

1

I

I

1

I 0

I a

HfflWOLE CUTOFF510 BLRNKLEIOER-AFNRN SOS HIZUTANI A HIZUTANI S

---

I I

I

I

I

to one at zero momentum.

calculation.

IO

“P IN” 1,~:

We will refer

Curves:

see text.

to this calculation

as

WAVE

The total cross section of the standard-2 result is too large at very low energy. Since in this region the total cross section is dominated by the S,, rescattering, we performed a calculation with a modified S3, interaction. The results are shown in fig. 12 for the standard-2 calculation (solid curve) and the calculation with the modified S3, interaction dashed curve). is given by The formfactor gmod of this modified Sjl interaction C

&d(P) = -p2+CY2’

(3.1)

where (Y= 3.0 fm-’ and C = 3.0 fm-‘. This is to be compared to the form factor of Thomas’ Sjl interaction used in the standard-l and standard-2 calculations:

g(p)=c,

c2

p2+p:+p2+gy

(3.2)

J. Vogelzang

L4 -

-

STRNDARO 1

---

B.B.S.

3

a

LP -

659

et al. / Pion absorption

LO -

ao,-

\ P-

\

‘. _____a--

/cc

/’

\

Fig. 9. Subsystem energies: (a) total cross section; (b) a,; (c) ad. see text.

Experimentalpoints:

as in fig. 2.

Curves:

with p, = 3.382 fm-‘, pZ = 1.107 fm-‘, C, = 6.0786 fm-’ and C2 = -0.1661 fm-‘. Both form factors, normalized to one at momentum equal to zero, are plotted in fig. 13. At small

relative

momenta,

our modified

S3, formfactor

is smaller

than Thomas’

formfactor, leading to a smaller total cross section and an improved agreement with the experimental data. The phase shifts of our modified S3r interaction are correct only for pion kinetic energies smaller than 40 MeV. Use of the modified S,, interaction has almost no effect on (Y*and CQ. 3.10. STANDARD ELEMENTS

In figs. 14a-d we present calculations of a different type that give the best agreement with the experimental data. The standard-l (solid curves) and standard-2 (dashed curves) calculations have already been discussed in the previous sections. They give similar results, except for the total cross sections at low energy.

J. Vogelzang et al. / Pion absorption

660 18 ,I

,

,

,

,

,

,

-

-

BLWLEIDER

-

--.__

PERTURBRTIVE

3,

,

-

B-R

,

,

B-R l4ITli

(

-

BLANnLElDER

--___

B-R

PERTLRRRTIVE INPUT

I,

I,,

(

,

I

,

COtPLETE

IN’UT

,

I,

I,

INPUT

SHIFT

,

,

,

,

,

0.40,

,

COMPLETE B-R WITH

INPUT

-

SHIFT

o.m-

,

1

,

,

-

ELFMLEIOER

---

PERTURBRTIVE

---

B-R

INPUT

,

,

,

,

,

COKPLETE B-R HITH

INPUT

SHIFT

C k P g

-0.20

,.I0 K-CllWPION

Fig. 10. Comparison

-

z.00

1.m

2.00

fWX5.5

with Blankleider and Afnan: (a) total cross section; points: as in fig. 2. Curves: see text.

(b) a,;

(c) u4. Experimental

In these figures we also show the results of a calculation we call standard-3, which includes the backward-moving pion and uses the Blankenbecler-Sugar subsystem energies.

We employed a monopole absorption formfactor with a cutoff at 470 MeV/ c and an energy shift in the PJ3 TN total energy of 32 MeV. The rest of the input is the same as in the standard-l and standard-2 calculations. The standard-3 calculation reproduces the total cross section well at low energy, but the width of the peak is too small. The Legendre coefficient cz2 is too small above resonance, while CQ agrees fairly well with the experimental data.

4. Discussion

and conclusions

Our results show that it is possible within our model to obtain a good fit of the experimental data to the total absorption cross section; the differential cross sections

J. Vogelzang

661

et al. / Pion absorption

00

K-CtlS/FI@4

Fig. 11. Backward-moving

pion. Experimental

MSS

points:

as in fig. 2. Curves:

see text.

\/I\, 8 -

I4 t

---

STANoAR 2 KWIFIED S31-WIVE

12

10

L”“““““““” oo.(‘o 0.20

0.40

0.0

0.10

140

K-CllS/PIoN

Fig. 12. SJ, wave. Experimenlcll points:

1.20

IMS

1.40

1.m

1-m

BS in fig. 2. Curves: see text.

2

J. Vogelzang et al. / Pion absorption

662

1.4a

_

I ---

1

53 1 FORHFRCTORS 1 I 1 1 I 1ms

I

1

tlODlFIED

I .20

O*rno

I1

I

11

II 2

hlE;T”H Fig. 13. S,, formfactors

normalized

11

11 0

9

I

IO

“P IN” I/F;

to one at zero momentum.

Curves:

see text.

are less well described, in particular freedom we have allowed ourselves

the coefficient CL,.This good fit is due to the with respect to the dynamical input. In this

section

of the choices

we will discuss

the relevance

we made.

4.1. T~NN VERTEX

The single most important parameter that determines the magnitude of the cross sections is the cutoff at the absorption vertex. This has been known for some time already I,*). Therefore it is important to examine the evidence on this parameter. The literature can be roughly divided into two parts: direct computations of the rNN form factor based on field theory, and indirect determinations based on models for the nucleon-nucleon interaction. Dominguez *‘) determines the pNN formfactor FnNN( t) from one-pion exchange fits to hadronic cross sections. If the formfactor is written as (4.1) where A, is the Goldberger-Treiman discrepancy (A, = 1 - FnNN(0) = 0.06), he finds A - 0.4 GeV/ c. This formfactor falls off rather steeply at very small r, but for larger

J. Vogelzang et al. / Pion absorption

663

I-

I

’

I

‘

1

’

r

1

s1Ruwto1 Sliwfma 2 s1mom?0 3

-

--_.-

b

Fig. 14. Standard-l-standard-3 results: (a) total cross section; (b) differential cross sections; (c) a,; (d) a+ ~x~ri~enlaI pints: as in fig. 2. Curves: see text.

664 t

J. Vogelzang et al. / Pion absorption

it can be approximated

of 0.8 GeV.

Kaulfuss

quite well by a monopole and

Gari 28) solving

TNA, pWN and pNA formfactors cutoff A = 1 GeV/c.

coupled

formfactor integral

arrive at a formfactor

Cass and McKellar

29) calculated

with a cutoff mass equations

for rNN

for rNN,

with a monopole

FrNN( t) with the help of an

extended dispersion relation. For -M; < t < 0 their result can be approximated by a monopole with a cutoff of roughly 0.7-0.8 GeV/c. The experimental data they show can be described in the same region by a much softer monopole with A = 0.55 GeV/c. Bag-model calculations of the 7rNN vertex also favour soft formfactors. The Virginia potential 30) employs PNN formfactors with a monopole cutoff of 0.64 GeV/ c. A more accurate determination of the formfactor favours an exponential form e-s*“‘Z with A = 0.5 GeV/c if the bag radius equals 1.1 fm, or A = 0.6 GeV/c for a bag radius of 0.9 fm [ref. “)I. In the framework of the cloudy bag model, Guichon et al. 32) determined the parameter R in the formfactor FrNN(q2) = 3j,(qR)/qR by the requirement that this formfactor gives the correct value for the Goldberger-Treiman discrepancy. The result can be approximated for 1ql-c 1 GeV/ c rather well by a monopole with a cutoff equal to 0.6 GeV/c. Semi-phenomenological nucleon-nucleon interactions are the other source of information on the nNN form factor. The Bonn group 33) uses monopole formfactors with cutoffs in the range 1.2-1.5 GeV/c. The Nijmegen potential 34) employs an exponential formfactor with a cutoff of 1.4 GeV/c. It is probably fair to summarize this situation by saying that if one insists on a monopole formfactor at the rrNN vertex, its cutoff should be in the region 0.61.2 GeV/ c. In our standard-l we needed a monopole agreement with the data. One way to explain

cutoff of 0.51 GeV/c in order to get this somewhat low value is to argue

that it is caused by neglecting p-meson rescattering. Maxwell et al. ‘) use monopole formfactors with cutoffs between 1.2 and 1.4 GeV/c. The p-meson rescattering contribution lowers the cross sections to the right magnitude, due to the strong pN tensor

coupling

employed

there. Their ratio

K

of the pN tensor

and the pN vector

coupling constant is equal to 6.6. By omitting p-meson rescattering we therefore underestimate the pionic rescattering contribution. One should keep in mind, however, that the reduction of the absorption cross section due to p-meson contributions will be less important in a framework where one uses for both the pion and the p-meson soft form factors like the Virginia potential. Use of a weaker pN coupling constant (K = 3.8) will also lower the effect of the p-meson. [Note that Perrot ‘) finds a small effect’for the p-meson contributions on the differential cross section at resonance using strong pN tensor coupling.] It is interesting to note that Grein et al. ‘) obtain agreement with the experimental data at resonance using a nNN formfactor with a monopole cutoff of 0.7 GeV/c. If the backward-moving pion contribution is included we need to lower the cutoff even more, to 0.38 GeV/c, because this contribution interferes constructively with that of the forward-moving (in time) pion. [This is in agreement with the results of

J. Vogelzang et aL / Pion absorption

665

Goplen et al. *), who need monopole rrNN formfactors with a cutoff of 0.3 GeV/c.] However, if the Blankenbecler-Sugar choice for the subsystem energies is made, the total cross section comes out with the correct magnitude in the resonance region if for A the value 0.47 GeV/c is adopted. We conclude that to achieve agreement with the experimental data the PNN vertex needs a soft cutoff; the value of A might be raised somewhat if p-meson rescattering is included and may get into agreement with values obtained from analyses independent of pion absorption.

4.2. MULTIPLE-SCATTERING

AND FINAL-STATE

INTERACTIONS

If we calculate the two contributions eqs. (2.2)-(2.3) with the Blankleider-Afnan input we notice that the cross sections are too small. This is partly due to the neglect of final-state interactions of the outgoing nucleons and partly to the neglect of multiple-scattering corrections. The former are much less important than the latter, as can be seen if figs. 3a and 10a are compared. It is apparent that multiple-scattering corrections increase the cross sections beyond Q,.~. = 0.3. The shift we included in the nN subsystem energy is probably due to nucleonnucleon rescattering corrections (binding effect). 4.3. EFFECTS

OF RELATIVISTIC

KINEMATICS

Although our calculations employ non-relativistic dynamics, we have used besides the purely non-relativistic kinematics several commonly used choices for the relativistic definition of the relative momentum. These different choices affect the results significantly. The Wightman-G&ding choice i_sclearly ruled out. The relativistic-pion-kinematics relative momentum causes (Yeto go through zero around n,.,.,. = 1.6. This behaviour can be traced back to the amplitude of the 3Pz final state. This amplitude, which is strongly influenced by recoil corrections in the external rrN vertex, changes sign when the relativistic pion kinematics relative momentum is employed instead of the non-relativistic relative momentum. Because cy4is determined by the interference of several small amplitudes, it is very sensitive to the choice of relativistic kinematics. The Blankenbecler-Sugar choice for the subsystem energies produces significant changes if a relativistic relative momentum is used. With this choice, the width of the peak in the total cross section comes out too small, while a4 is negative. Therefore it would be interesting to see the results of a calculation where all the effects of multiple scattering are included in a model that incorporates the full relativistic pion propagator (i.e. DC+)+ DC-‘) and makes the Blankenbecler-Sugar choice. We remark that the results for the recoil corrections in the direct absorption obtained by Maxwell et al. ‘) are inconsistent with those of Goplen et al. ‘), who find that at low energy the deuteron S- and D-wave contributions tend to cancel

J. Voge~zang et al. / Picm adsorption

666

each other. This cancellation to be small. 4.4. THE PION-NUCLEON

causes the recoil corrections in the direct absorption

SCATTERING

AMPLITUDE

Within the framework of our model the aN off-shell T-matrix by Thomas 12) is strongly favoured. The main reason is that other models for the TN amplitudes have a much softer Ps3 formfactor which leads to a reduced value of the cross section in the resonance region. The remedy that comes to mind immediately (viz. raising the cutoff A) does not help, because it would spoil the agreement with the data at low energy where rrN S-waves dominate. For this reason it is not possible to obtain better agreement with the data for larger values of A than we have used, unless one adopts an unrealistic S-wave ~TNinteractions. As the present experimental data on the SI1 and S3, nN phase shifts leave very little room for alternatives to Thomas’ model for these waves, it is worthwhile to make an effort to redetermine these phase shifts at very low energies. The small P-waves, i.e. PI,, P3, and P,3, are relatively unimportant. They may be ignored, but for the P,, pole part which determines the PNN absorption vertex.

4.5. CONCLUDING

REMARKS

The coupled-channels ca~ulation by Blankleider and Afnan is at present the only calculation for the coupled NN-~TNN system that is in agreement with most of the experimental data. The results of Rinat and Starkand “) and of Perrot “) miss the absorption cross section by a factor 152.0, probably due to the off-shell behaviour of the Pj3 rrN formfactor. Betz and Lee ‘) find the correct magnitude for the total absorption cross section, but the maximum is at too high an energy. This might be caused by the neglect of the direct absorption. Our results show, in agreement with Goplen et al. ‘), that the full pion propagator, including the backward-moving (in time) pion must be included, because otherwise one misses a large part of the cross section. If the coupled-channels calculations are restricted to three-body models, they cannot incorporate this impo~ant effect. We are thus led to the conclusion that the absorption of pions on deuterons has not been fully understood yet. The rather soft absorption vertices used in the more sophisticated theories of pion absorption seem to be in agreement with existing estimates of the rNN vertex function. The results of Grein et al. 3), however, seem to indicate that a theory incorporating relativistic dynamics may use a harder rrNN vertex and still fit the data on pion absorption. This conclusion is corroborated by our standard-3 results. As their calculation treats multiple-scattering effects in an approximate way as distortion factors, it is hard to estimate the importance of the multiple-scattering corrections in a fully relativistic theory, looking at their results. Also an important

3. ~~el~ang et

at /

Ron absor~r~on

667

piece of info~atio~ is missing in their paper. Because they concentrated on the dibaryon-resonance region they did not give results for low pion energies. As our results show, including the full relativistic pion propagator increases the low-energy cross section relative to the cross section at resonance. It would be interesting to see whether this is also true in the model of ref. 3). We end in a speculative vein. An alternative to meson exchange models of nuclear forces at low interparticle distances is to invoke explicit quark degrees of freedom, We have seen already that bag models tend to predict softer meson-baryon formfactors than those extracted from the comparison of experimental data with meson field-theoretic models. Existing ideas 35)on the short-range behaviour of the deuteron wave function and on the meson-baryon and baryon-baton interactions may change our picture of pion absorption drastically. We wish to thank Dr. O.V. Maxwell for many stimulating discussions. This work has been supported by the “Stichting voor Fundamenteel Onderzoek der Materie” which is financially supported by the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek”.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

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