Calculation of proton-deuteron phase parameters including the coulomb force

Nuclear Physics @ North-Holland

A445 (1985) 429-461 Publishing Company

CALCULATION PARAMETERS

OF PROTON-DEUTERON INCLUDING

THE COULOMB

PHASE FORCE

E.O. ALT

Institut Jir Physik, Vniversitd Mainz, Mainz, West Germany W. SANDHAS

Physikalisches Institut, Vniversitiit Bonn, Bonn, West Germany H. ZIEGELMANN

Znstitut ftir Theoretische Physik Vniversitiit Tiibingen, Tiibingen, West Germany Received

10 April 1985

Abstract: A previously proposed exact method for including the Coulomb force in three-body collisions is applied to proton-deuteron scattering. We present phase shifts for angular momenta up to L = 9, from elastic threshold to 50 MeV proton laboratory energy. Separable rank-one potentials are taken for the nuclear interactions. A charge-independent and a charge-symmetric choice, while leading to different neutron-deuteron and proton-deuteron phase parameters, nevertheless yield practically the same Coulomb corrections. We investigate, moreover, the question of P-wave resonances. A critical comparison of our results with those obtained in a coordinate-space formalism is performed. Furthermore, proposals for an approximate inclusion of the Coulomb potential are tested, and found unsatisfactory.

1. Introduction

It is a well-known fact that conventional integral equations of scattering theory cannot be applied directly, if the two-particle interactions contain long-ranged Coulomb forces. One way to circumvent this difficulty consists in working with screened Coulomb interactions, so that all methods of short-range collision theory can be used. The essential problem in such a formulation is to develop prescriptions, which allow one to extract from the screened relations the physical, unscreened observables. For two-fragment collisions of three particles a first step along these lines has been undertaken by Veselova lS2).A n intrinsically different approach is due to the present authors 3-9). Both treatments make use of the results obtained in refs. l”-12) for two-charged-elementary-particle scattering. In the method of Veselova lS2) the center-of-mass (c.m.) Coulomb part of the kernel of the Faddeev wave function equations 13) is separated off, and treated explicitly. The kernel of the resulting equations is then sufficiently well behaved so 429

430

E.O. Alt et al. / pd phase parameters

that, after renormalization,

they are amenable

they still suffer from two shortcomings. amplitudes

needed

and the transition

for the calculation between

Firstly,

to practical

calculations.

their solutions

of physical

these two quantities

observables, is non-trivial

However,

are not the scattering but wave functions, as soon as Coulomb

forces are involved. Secondly, above the break-up threshold, the kernel of these equations is still so singular that their applicability is not guaranteed r4). In the approach developed in refs. 3-9) neither of these problems shows up. The first one does not exist apriori, because the method is based on relations for transition operators rather than scattering states. The second one does not occur since, in contrast to the method of Veselova, the zero-screening limit is not performed within appropriately modified integral equations. Instead, the original relations of shortrange scattering theory are solved for increasing, but always jinite screening radii R. In order to investigate the behavior of the resulting on-shell two-fragment amplitudes for R going to infinity, a two-potential-type formula, based also on the splitting off of the c.m. Coulomb interaction, has been derived. This representation exhibits most clearly that the zero-screening limit does exist after diverging phase factors, evolving in the initial and in the jinal channels, are eliminated by a renormalization procedure. These phase factors originate in the Coulomb distortion of the relative motion of the centers of mass of the two fragments, and coincide, therefore, with those known from the genuine two-body Coulomb scattering. Consequently, in this way well-defined arrangement amplitudes for charged-particle systems interacting via unscreened Coulomb-like potentials are obtained. Let us add a few important comments. Firstly, this result has been established not only in the framework of integral equation theory 3-6), but also by means of resolvent identities 739), and by a time-dependent approach “). Secondly, all derivations manifestly demonstrate that the renormalization procedure is valid below and above the break-up threshold. Thirdly, the time-dependent formulation implies in addition that the unscreened scattering amplitudes defined in the above-mentioned way coincide

with those following

from Dollard’s

theory 15).

This screening and renormalization approach represents the basis of our practical method which correspondingly consists of the following steps: Calculate the interesting scattering amplitudes for Coulomb potentials by any method suitable for short-ranged interactions. [We, in particular, have used in refs. 3,4*16-‘8), and are using in the present paper, the quasiparticle method ‘9320).] Then renormalize the result with explicitly known renormalization factors. This procedure is to be repeated for increasing values of the screening radius until a stable, i.e. R-independent value is obtained. The latter is then interpreted as the desired physical charged-particle scattering amplitude. In other words, all numerical calculations are performed for short-ranged (nuclear and screened Coulomb) interactions and the unscreening limit is approached numerically in the resulting renormalized screened scattering amplitudes (in practice we have used a variant of this algorithm which is better suited for numerical reasons). Hence, when using the quasiparticle integral equations for

E.O. Alt et al. / pd phase parameters

431

this purpose, the compactness of their kernels is always trivially secured, in contrast to the erroneous claims made in refs. 14*21-24). For completeness we mention that in our preceding publications we have discussed various other procedures suggested by our formalism. Among them was also one based on modified integral equations with unscreened but smoothed kernels [see e.g. ref. 9), subsect. V, C]. This is closer to Veselova’s idea lT2),but formulated for scattering amplitudes rather than for wave functions. It, therefore, avoids the first of the shortcomings of her method discussed above.. [The same goal has been achieved in ref. 25) without the use of the screening technique.] But, of course, the second objection, which concerns the applicability beyond the break-up threshold, is still valid. For this reason, in our applications to proton-deuteron (pd) scattering 3,4,‘6-18)we have worked from the very beginning with the numerical renormalization procedure recalled above. And this is also done in the present investigation of pd phase parameters from the threshold up to 50 MeV. Beside these momentum space formalisms, an alternative approach, based on differential equations in coordinate space together with adequate boundary conditions, has been developed by Merkuriev 26*22*27). Applications to proton-deuteron scattering have been published in refs. 22-24).The main advantage of this method is that it is particularly adequate for the use of local interactions. Its main disadvantage lies in the fact that for practical calculations a partial-wave analysis of the two-particle subsystems must be performed, which necessarily has to be truncated. The error introduced by such a truncation is hard to estimate in view of the well-known lack of convergence of partial-wave series for Coulomb quantities 12*28). This point, and further objections against the afore-mentioned applications to pd scattering, will be dwelled upon later in the appropriate section. Notice that results for the 3He binding energy or pd scattering lengths will not be discussed. We only refer to a recent review *‘) for a list of references on these topics. Finally, two further approaches should be mentioned. In the method of refs. 30-32) the total Coulomb interaction is separated off in the three-body equations. However, these three-body pure Coulomb contributions cannot be calculated except under drastic approximations. The second approach renounces any attempt to solve differential or integral equations. Instead, a direct evaluation of the relevant quantities [full resolvents 33) or Moller operators ‘“)I by means of 8* discretization methods is performed. The paper is organized as follows. In sect. 2 we briefly recapitulate the most important aspects of our approach. Sect. 3 contains the specialization of the integral equations to the proton-deuteron system. The effective potentials occurring are described in sect. 4. Our method of solution of the integral equations is explained in sect. 5, which also contains a detailed exposition of the procedure used to numerically approach the zero-screening limit. The convergence questions which arise in this connection are discussed, together with a variety of convergence tests, in sect. 6. We present our results for the proton-deuteron scattering phase parameters

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E.O. Alt et al. / pd phase parameters

in sect. 7 and compare them in sect. 8 with the coordinate-space results of refs. 22-24). Here we also comment on the inadequacies of a similar comparison given by these authors and formulate our objections against the practical performance of their calculations. Finally, in sect. 9, we test the method proposed in refs. 30-32)and some further attempts for approximately calculating pd phase shifts, and show that they lead to unsatisfactory results. In the appendix we describe the spin structure of the various contributions to the effective potential. 2. The screening method As discussed in the introduction, our approach is based on the use of screened Coulomb potentials. The immediate consequence is that the integral equations of short-range three-body scattering theory, and in particular the quasiparticle formalism, can be applied.

2.1. QUASIPARTICLE

EQUATIONS

Let us briefly recapitulate the relevant equations already specialized to elastic proton-deuteron scattering, as given e.g. in refs. 5*9).Since we are primarily interested in the effects which arise from the incorporation of the Coulomb repulsion between the two protons, we choose for the nuclear interactions simple separable potentials of rank one. Denote the two protons as particles 1 and 2, each with charge e, and the neutron as particle 3. Then the pair interactions in the neutron-proton subsystems are v, = IXrr)&AXoll 2

cy=1,2,

(2-l)

VR

(2.2)

whereas for the two protons we have v, = l/&&(X3/ +

*

Here V, is the repulsive screened Coulomb potential, with R being the screening radius. In the calculations we choose exponential screening so that in coordinate space V, becomes V,(r)

=I exp (-r/R).

(2.3)

The strength parameters A, and the form factors 1~~) will be chosen to reproduce the low-energy nucleon-nucleon data (cf. sect. 5). Note that for convenience we suppress in this section the explicit indication of the spin dependence (see, however, sect. 3). The quasiparticle formalism developed in ref. 19) allows us quite generally to reduce the three-body integral equations to effective two-body Lippmann-

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E.O. Alt et al, / pd phase parameters

Schwinger-type

equations for effective two-body transition operators SK’: @E’(z)=

For the special interactions assume the closed form

Yb”)(z)+C “v-c,“,~(z)(e~~~(z)s~‘(z>.

(2.4)

(2.1) and (2.2) the effective two-body potentials

‘YE’

The effective free Green functions are given as for (~=1,2,

~~~~(z)={h,‘-(xu~Go(z)lx~)}-‘,

~~~~(z)={A;‘-(x~~G~(z)+G~(~)T,(z)Go(z)(x~)}-~.

(2.6) (2.7)

Here, spa = 1 - a,, denotes as usual the anti-Kronecker symbol, and G,,(z) is the free three-body resolvent. The screened Coulomb transition operator TR, associated with the potential (2.3), is defined by the Lippmann-Schwinger equation TR(z) = I’, + V&(z)

TR(z) .

(2.8)

We emphasize that, as pointed out in refs. 19,3-5),the above expressions for the effective potentials and Green functions are exact. This, in particular, means that all virtual excitations into the three-body continuum are correctly taken into account. The quantities appearing in (2.4) are still operators acting on the plane-wave states lqn) which describes the free relative motion of particle cy and the subsystem of the other two particles. For channels (Y,/3 in which the subsystem pairs are bound, ~E’(4j3, n) = (4&l~bR,W + Wlq,) represents the physical two-fragment condition E = q;/2M,

scattering amplitude, + I$, = q:/2M,

+ ,I?_

(2.9) provided the on-shell (2.10)

is fulfilled. Here, & and l$ are the binding energies of the incoming and outgoing two-body bound states, respectively, and M, (Mp) is the reduced mass in channel ff (P). 2.2. ZERO-SCREENING

LIMIT

For finite values of the screening radius R the scattering amplitudes (2.9) can be obtained from the integral equation (2.4) by standard techniques. It remains to clarify the relation to the physical amplitudes corresponding to unscreened Coulomb potentials.

434

E.O. Alt et al. / pd phase parameters

The solution of this problem, resentation of (2.9):

given in refs. 3-6*9), rests upon

the following

rep-

.$?(qb,qa)= &,&3t!(q&, a; q2n/2M,+iO)+(q~,~‘I~~~(E+iO)lq~~~). (2.11) In this two-potential scattering again

formula,

states associated

exponential

t: and 1qgi) are the transition amplitude and the with the center-of-mass Coulomb potential u:, for which

screening

is used, (2.12)

Here, pa denotes the relative coordinate between the two fragments of channel (Y, canonically conjugate to qa. Hence, U: acts between the charge of the free proton and the charge of the correlated neutron-proton pair concentrated in its center of mass. One important point to note is that the center-of-mass amplitudes and states in (2.11) are genuine two-body quantities. It is well known how the transition to the corresponding unscreened Coulomb amplitudes tz(q&, qa) and IqF&) is to be performed for them. Namely, after renormalization

kf(q&,qn; d/2K + iWi&(q,) z 14L$J~:,%2(%J The renormalization

factor

Z,,,(q,)

z

scattering we have

tz(q&.c) ,

of the form

Zddq~) =ew GWa,d4n)). For exponential

screening

the diverging

&,R(qn) = -+

phase

(2.13) (2.14)

I&L>.

is asymptotically

states

can be evaluated

[log (2qaR) - C] +O(log;T;R’)

(2.15) explicitly, ,

(2.16)

a

with C = 0.5712 . . . being the Euler number. A second essential observation concerns the amplitude 9::. Since, even after switching off the screening, it does not contain any long-ranged Coulomb interaction, the transition to infinite R can be performed without further precautions. From these facts we infer that after renormalizing eq. (2.11) by Z,,g’(qb) and ZL,%‘(qm), the limit R+co of its right-hand side and, therefore, also of the full amplitude exists,

(2.17)

E.O. Ali et al. / pd phase parameters

435

Let us add some remarks: (i) The above procedure, based on screening the Coulomb potential, represents one possibility for dejning the full on-shellamplitude To,(qi, qe) which descrhs proton-deuteron scattering under the influence of the nuclear interaction and the unscreened Coulomb potential. A first proof has been given in refs. 3-6) by starting from the quasiparticle formalism. Another proof, resting upon resolvent identities, has been published in refs. 7*9).W e emphasize, furthermore, that the right-hand side of eq. (2.17) can also be derived “) in the framework of Dollard’s time-dependent approach 15). This demonstrates that the above definition of Ta,(qb, qa) via the zero-screening limit (2.17) is identical to the one obtained from time-dependent scattering theory. (ii) The screening approach represents also a practical method, used already in for computing the full unscreened amplitude Tp,(qb, qa). It consists refs. 3T4*16-18) in calculating first the full screened amplitude Tga’(qb, qa) and the screened centerof-mass Coulomb amplitude tt(q&, qa) for finite R with methods well suited for short-ranged potentials. As follows from (2.11), their difference yields the Coulombmodified short-range amplitude (q$,2ITgt(E + iO)lq’,fih)which then has to be renormalized in the way discussed above. Repeating the calculations for increasing values of R leads to the corresponding unscreened amplitude (q~,~lT~~( E + iO)lqLTA),

(2.18) According to (2.17), we then only have to add the analytically known Coulomb amplitude tz(qh, qa) in order to obtain Tpa(qb, qu). Of course, the practicability of this procedure depends on how fast the zero-screening limit can be reached numerically. This will be discussed in detail later on. (iii) This strategy must not be confused with attempts to solve an integral equation directly for the unscreened amplitude Tsc, which is essentially the procedure proposed in ref. *‘) for energies below the break-up threshold [see also ref. ‘)I.

3. Effective equations for proton-deuteron scattering In the present section we give the explicit form of the equations pertaining to proton-deuteron scattering. Let us introduce the following notation for the various subsystems considered. The index “d” (=deuteron) denotes the neutron-proton subsystem with angular momentum I = 0 and spin 1, the index “s” (=singlet) the one with 2= 0 and spin 0, and “c” the proton-proton system with spin 0. The short-ranged pair interactions (2.1) and (2.2) between the three nucleons are chosen to act in S-waves only. Thus,

436

E.O. AIt et al. / pd phase parameters

we have

UP’,

P) = ~~(lP’l)~~X~(lPl)

m =

,

d, s,

(3.1)

whereas, due to the presence of the Coulomb force, ~~~P’,F~=Xc~lP’l~~,X~~lPI~+

VdP’-PI

(3.2)

acts in all partial waves. Only in subsystem “d” do we have a bound state, the deuteron, with binding energy &. As usual, we denote the possible two-fragment channels of the three-nucleon system by the same index ti as the correlated pair occurring in it. For instance, the incoming state consisting of a deuteron and a proton is labelled by d. Then taking into account the identity of the two protons, symmetrization leads to the following set of three coupled integral equations for the screened two-fragment amplitudes for a given value of total spin S, r=d,s

Here we have taken into account that S is conserved because of the simplicity of the interactions (3.1) and (3.2). The factor 2 in the third term of (3.3a) comes from antisymmetrization. For on-shell values of the momenta q=qf=qe=v$mN(E-Ed),

(3.4)

with mN being the nucleon mass, .TY$~(q’, q; E + i0) gives the exact screened scattering amplitude for elastic scattering of protons off deuterons with total spin S. Note that since only in subsystem “d” does a bound state (the deuteron) exist, only the effective free Green function %$:d has a pole proportional to (E + iO3q2/4mN - Ed)-’ = ~~3/4rn~)~q~ - q2+ iO)]-‘, the other Green functions Y&.., r=s or c, being non-polar. We, finally, mention that the above system of equations could be reduced to dimension 2x2 by eliminating 9~~‘” in (3.3a) by means of (3.3b). 4. The effective potential W”Lf?”

For the following analysis it is necessary to discuss in some detail the effective potential 7rfnRm)’ entering eq. (3.3). From the definition (2.5) it follows that the latter consists of a sum of five contributions:

The factors Afis contain spin recoupling coefficients, and also signs and the factor 2 occurring in eq. (3.3a), which result from antisymmetrization. They are collected

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E.O. Alt et al. / pd phase parameters

(0)

(1)

(2)

+d;E+q&= (3) Fig. 1. Diagrammatical indicate subsystem

representation form factors,

(4) of the contributions to the effective potential (4.1). Semicircles TR is the screened two-particle Coulomb transition operator.

in the appendix. Let us remark that these coefficients have been evaluated by treating the neutron and proton as distinguishable particles. This treatment leads, of course, to the same physical scattering amplitude as can be obtained by using the isospin formalism and symmetrizing all three nucleons. Comparison of both procedures represents a useful check. The spin-independent terms ‘V(niR)on which we omit from now on the superscript R whenever possible, are graphically represented in fig. 1. The corresponding matrix elements can easily be read off from this figure. V(O)represents the well-known nucleon exchange diagram, and is the only term which survives when the Coulomb potential is turned off. T(l) and V(‘) describe the Coulomb modifications of the proton-proton form factor, V (3) the direct , and Vc4) the exchange Coulomb contribution. As has been discussed in refs. 3-6), it is the direct Coulomb term y(3)(R) which develops, in the limit R + CO,that part of the effective potential which is of longest range. The latter can be isolated by the following decomposition (recall that, in our model 3 y(3)(R) has only non-vanishing diagonal elements): v;;Ryq’,

q; z) = &,uR(q’-q)F,(q’,

where yR is the center-of-mass function F, is given as

Fm(q’,4; z) =

q; z)+ P:ER)(qr,

Coulomb potential introduced

q; z) ,

(4.2)

in eq. (2.12). The

d’kdWxm(k -A) Lz

_3q,z,4

mN-k2/m,][z-3q2/4m,-(k-A)2/m,]’

(4*3)

with A = $(q’- q) denoting half of the momentum transfer. On the energy shell (3.4), F,(q, q; E + i0) is just the normalization integral for the deuteron wave function, which is taken equal to one. Thus, putting (4.4)

the decomposition

(4.2) can be simplified to v:;Ryq’,

q; z) = SmJyq’

- q) + +$yyqr,

q ; z) .

(4.5)

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E.O. AIt et al. / pd phase parameters

Thereby the point-Coulomb singularity of the effective potential is clearly displayed. This splitting has been used in refs. 3-6) to justify the renormalization and limiting procedure described in subsect. 2.2. For the numerical treatment employed in the following it is important to notice that, if we switch off all contributions to the effective potential V(,Rm), except the center-off-mass Coulomb potential &pR(q’q), and use the appropriate free two-particle propagator, the solution of eqs. (3.3) is center-of-mass Coulomb amplitude on-shell just the pure screened tR(q’, q; 3qaf4m,+ i0) appearing in eq. (2.11).

5. Method of solution

Our approach consists in solving the integral equations (3.3) for finite values of the screening radius R, and perfo~ing the zero-screening limit numerically. In the following we describe some of the problems encountered thereby. In order to reduce the required computer time, we have made one approximation, namely to replace in the effective potential ‘VCR)and in the effective free Green function %hR’the screened Coulomb amplitude TR by its Born term, TR =

v, .

(5.1)

For a calculation of the ‘He binding energy in which this approximation been made, see ref. 35).

5.1. TWO-BODY

has not

INPUT

We first describe the two-body input used in our calculations. As mentioned before, the Coulomb potential is screened with an exponential screening function. The S-wave form factors of the nuclear part of the interactions, eqs. (3.1) and (3.2), are taken to be of the Yamaguchi type,

x*(M) =--g$m 9

m =

d, s, c .

(5.2)

The range and strength parameters Pm and A, are fitted to the low-energy observables of the corresponding two-nucleon system. For m = d, s describing a system of one proton plus the neutron, the numerical values are collected in table 1. For protonproton scattering we considered two choices, a charge-independent and a chargesymmetric one. In the former the nuclear parameters BY’ and A’,‘)are taken to be identical to those for singlet neutron-proton scattering, & and A,, respectively. Then, using the explicit formulae of ref. 36) valid for a potential (3.2) with R = CO,we can calculate the corresponding Coulomb-modi~ed short-range scattering length azz and effective range r$. Of course, because of the assumed charge independence, the neutron-neutron scattering length an,, and effective range r,, coincide with a&,

E.O. Alt et al 1 pd phase parameters

439

TABLE 1 Physical input quantities and the resulting parameter values for the separable potentials, describing S-wave np scattering in the spin-l and spin-0 state

w

Input parameters

system d

I$ = akP = as = r%“,=

S

Parameters of the separable potentials

2.226 MeV 5.41 fm -23.78 fm 2.67 fm

Ad = Bd = A, = &=

-0.3817 fmm3 1.4055 fm-’ -0.1542 fm-3 1.1771 fm-’

and rt,,, respectively. In the second case, the formulae of ref. 36) are fitted to the experimental values for at; and rgz in order to extract the pure nuclear parameters py’ and A,(*I. These then allow us to calculate a,,” and r,, under the assumption of charge symmetry. Inspection of table 2, in which we have collected all the results, shows that arm and r,,, determined in this way, turn out to be quite close to their experimental values.

5.2. NUMERICAL

PROCEDURES

The integral equations (3.3) are, after angular momentum decomposition, onedimensional but with kernels of quite complicated singularity structure: part is present even for finite screening radius R, and part shows up in the limit R going to infinity. The treatment of the first class of singularities is explained to some extent in ref. 37). We, therefore, summarize them here only. For convenience we omit for the moment the superscript R. The ubiquitous pole singularity of the bound-state propagator 9&d is treated by subtraction. We have also investigated the symmetric integration around the pole, with no gain in numerical accuracy. For energies above the break-up threshold, the terms 7f$,L(q’, q), with i = 0, 1,2, develop the well-known TABLE 2 Parameters

PP system case 1 case 2

for the S-wave

separable

Input

nuclear potential which, describes pp scattering

parameters

A?’ = A, = -0.1542 fmm3 ~~‘=&=1.1771 fm-’ at: = -7.71 fm ri: = 2.86 fm

Case 1: charge-independent,

case 2: charge-symmetric

together

with the Coulomb

Calculated af$ = r:F = AL*’= pa) = choice.

-8.34 fm 2.51 fm -0.1140 fm-3 1.075 fm-’

potential,

quantities

arm = -23.78 fm r,, = 2.67 fm arm = -17.85 fm rnn = 2.985 fm

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E. 0. Aft et al. / pd phase parameters

moving singularities for q, 4’6 q. =$jrnNE, E being the three-body other contributions Vzz( q’, q), i = 3 and 4, are non-singular after decomposition. These moving singularities are separated off by means tion in the form of a Legendre polynomial of the second kind, QL( z). the full effective potential Vz:(q’, q) can be split into two parts,

K%;fs’,9; E+iO)=e(qo-q’)efq,-q)A~~,(q’,q;E+W?L +BZk(q’,

m,E

energy. The partial-wave of a subtracThus, finally,

-

qa - q”

44’

q; Ei-iO),

(5.3)

with AS and BsL being non-singular. Herein, t?(x) is the unit step function, e(x) = 1 for x > 0, e(x) = 0 for x < 0. In order to take care of this structure we divide the integration region into two intervals, from 0 to q,,, and from q. to infinity. In the first one the singular part of (5.3) is integrated by means of a product integration rule based on the Lagrange interpolation polynomials 38) (5.4) with prescribed quadrature points qi*For these we have investigated different choices. At low energies it was quite effective to choose equidistant Simpson 38) mesh points for 0s q d a < qo, and a denser distribution for a c q < q. in order to better exhaust the square-root singularity of the resulting amplitude. However, for higher energies convergence with respect to an increase in the number of qi became unsatisfactory. There it proved essential to revert to one of the “optimal” 39) quadrature rules, of which we use the one of Clenshaw and Curtiss. The nonsingular part of (5.3) was integrated with the same quadrature points in the usual way. For the second interval q * qo, we have chosen a Gauss-Legendre integration rule. An easy check that proves the reliability of this procedure is obtained by switching off the Coulomb potential completely. The nd phase parameters, obtained for the nucleon-nucleon parameter set 1, perfectly agree with the results of ref. “) to within the error estimates quoted therein. Next we discuss the problems which arise from the numerical unscreening of the Coulomb potential. For this purpose we introduce the screened total and center-ofmass Coulomb phase shifts, 2S+1SL(R) and &,, respectively, defined via

Y$)sL( qe,

qe;

E + i0) = - 4?iiiNq,

tLR(qe, qe; 3q,2/4mN+i0) = -

N

(ev

Pi2s+1~L(~)1- 1) ,

q text Ph,Lle

1) ,

(5.5) (5.6)

for a given value of total angular momentum L and spin S. As follows from the spin recoupling matrix Ac3”, given in the appendix, which multiplies the cm. Coulomb potential vR, the corresponding scattering amplitude and phase shift are

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E.O. Alt et al. / pd phase parameters

the same for S =z and S = 1, as it should be. Thus we omit the index S on these. quantities. Instead of discussing the full phase shift *‘+I&( R) it is customary and convenient to introduce the so-called (screened) Coulomb-modified short-range phase shift 2s+1 s sRL obtained by subtracting off the former the pure Coulomb part a,,, 2s+1

6 SR,L --

2s+18L(

R)

-

a,,

(5.7)

.

It describes the effect of all shorter-ranged contributions of the effective potential VCR)including the interference with the Coulomb potential wR,and thus characterizes the screened Coulomb-modified short-range amplitude. If we denote the latter by .T~~sL(qe, qe; E + iO), it can be expressed, according to (2.11), as .!?-i?““( qe, qe; E + i0) = .Ti:)sL (qe, qe: E + i0) -tf(q,, 3 = - 4rimNq,

qe; 3qi/4mN+ i0)

e*‘QL(exp [2i2S+‘6ssL] - 1) ,

(5.8)

where we have used eqs. (5.5) and (5.6). The representation (5.8) allows us to write the zero-screening limit, explained in subsect 2.2 in an alternative, more detailed way. For large R the Coulomb phase shift 6,, is known ‘*) to separate into the unscreened Coulomb phase shift, given as the argument of the r-function, a,=argT(L+l+in),

(5.9)

with (5.10)

q = 2e2m,/3q,

being the Sommerfeld parameter, and the diverging phase factor +R (see eq. (2.16)), &Lx

(5.11)

uL+4R*

Similarly, the full screened phase shift “+‘SL( R) separates into an R-independent part plus the same diverging phase factor +R, *‘+‘SL( R) As a consequence, S,, exists,

R+CU

2s+1sL+ &.

@R cancels asymptotically

(5.12)

in (5.7) so that the R +co limit of

2s+1

*s-c1

2sc1 br

R

The latter quantities are called unscreened shifts. Thus, renormalizing eq. (5.8) by

s

(5.13)

SC,L.

Coulomb-modified

Z,r( qe) = e-2i+fi(%)

short-range

phase

(5.14)

442

E.O. A/t et al. / pd phase parameters

we obtain

the analogue

of eq. (2.18) for the partial-wave

-G1(q,){$lYL

(4.3 qe; E + io) -t&e, 3

=

- 4rrim,q,

R+CC Gp.gsL

projected

qe; 3qf/4m,+

amplitudes, if)))

ezirrL(exp [2i2s+16Sc,L]- 1)

(qe,qe;E + i0) .

(5.15)

That is, we arrive at the unscreened Coulomb-modified short-range amplitude expressed through the phase shifts 2s+‘&c,L. For the latter the summation of the partial-wave

contributions

(q’$‘I&~c(E

+ iO)lq’c+‘).Th us, the full unscreened

can be expected

to converge yielding the amplitude scattering amplitude for protondeuteron scattering can be obtained, as described at the end of sect. 2, by adding to it the explicitly known Coulomb amplitude .tc(q’, q).

6. Convergence tests Since

our calculational

asymptotia

is reached,

6.1. THE

procedure careful

CENTER-OF-MASS

consists

numerical

COULOMB

in increasing

convergence

PHASE

SHIFT

R step by step until

checks are necessary.

&,

As the simplest case we investigate numerically the large-R behavior of SR,L and compare it with the theoretically established result (5.11). We have already discussed at the end of sect. 4 that the c.m. Coulomb amplitude t: can be computed from the full set of integral equations (3.3) by switching off all terms in the effective potential

except

the cm.

of eq. (4.5). In addition,

Coulomb

the bound-state

free two-particle When solving

propagator. the resulting

phase

In table

shift

6,,.

potential,

single

3 we present

i.e. except

propagator equation,

Y&O;d has to be replaced

we obtain

its values

the first term on the r.h.s.

the screened

by the

Coulomb

for L = 0 and 8 at two different

energies. For each energy the number of mesh points has been kept the same for all screening radii R and angular momenta L. These phase shifts are compared with For large R we verify in this way the validity of the aR,L(wmpt.) = aL + 4R. asymptotic formula (5.11). As an additional independent check of accuracy, we give the phase shifts calculated with the help of the phase-function method 4’), and find excellent agreement.

6.2. THE

COULOMB-MODIFIED

SHORT-RANGE

PHASE

SHIFT

2S+‘SSR,L

The crucial question for the practicability of our method is how fast the zeroscreening limit is reached for the Coulomb-modified short-range phase shift 2s+‘&IzL, defined in eq. (5.7). Numerical problems may be expected when approaching the

E.O. AIt et

al. / pd phase parameters TABLE

443

3

Screened center-of-mass Coulomb phase shift 6K,L (in rad) as a function of the screening radius R, for two values of the proton laboratory energy and two values of the angular momentum L L=8

L=O [i%q 3

50

R [fml 10 100 1000 10 100 1000

sR,O

-0.14661 -0.35682 -0.56636 -0.067639 -0.11917 -0.17092

h&vvt.)

-0.14785 -0.35795 -0.56804 -0.067726 -0.11919 -0.17065

8

h&4

-0.14660 -0.35691 -0.56787 -0.067637 -0.11917 -0.17064

R,8

-0.002135 -0.11585 -0.31942 -0.012219 -0.058577 -0.11008

&,&ww.)

+0.099826 -0.11017 -0.32036 -0.006987 -0.058449 -0.10991

%d4

-0.002132 -0.11527 -0.31965 -0.012204 -0.058530 -0.10986

Compared are our resuhs from the same program, which is used later for the calculation of pd scattering, with those obtained 4’) by means ofthe phase function method (a). In addition, the predictions from the asymptotic formula (5.11) are shown.

c.m. Coulomb singularity -log [(q - q’)‘} as R goes to infinity. But this is under control, since it makes itself felt in both *S+1&(R) and SK, in the same way by producing here and there the diverging phase &, which cancels in their difference 2s+1

%R,L*

In this context we mention that, in order to minimize the influence of numerical inaccuracies arising from this nearby singularity, we subtract from the full phase shift 2S”6,(R) not the analytically known value of the screened Coulomb phase shift 6,, but the result calculated with the same accuracy from the same program in the way described above. In the zero-screening limit, ‘Vc3)develops another singularity, which may give rise to numerical problems as R increases. Its source can be traced back to the function F,(q’, q; E + i0) multiplying the center-of-mass Coulomb potential in the representation (4.2) of ‘P$,,, 3)(R1. As follows from the definition (4.3), F, is singular above the break-up threshold, provided E -3q12f4m, 3 0 and E -3q2/4m, 3 0. Its behavior can be elucidated by the following decomposition (recall that the form factors (5.2) act in S-waves only): F,(q’, q; E + i0) = e(E -3q’2/4m~)e(E

x x:(d%=@)/y,(Jm,E ’

I

-3q2/hN) -:q2)

d3k [E+i0-3q’2/4mN-k2/mN][E+i0-3q2/4m,-(k-A)2/m,]

with &q’, q; E + i0) being non-singular. It is worth mentioning that such a splitting can be introduced independently of our assumptions concerning the form of the nuclear interactions.

444

E.O. Alt et al. / pd phase parameters

The integral

occurring

in (6.1) can be evaluated . 2

explicitly

to give

w+JmN~-;q2+d

l=;log~~+Jm,E_aq’-P

(6.2)

’

Since this term multiplies (6.3) we see that for R equal to injinity the product

vRI contains, after partial-wave decomposition, singular terms proportional to {f, + f2 log (q’ - q)‘+ f3 log A}/A. Here, the fi are well-behaved functions, and A = (m,E -$q’2)1’2+ (mNE -iq2)1’2. This singularity, however, is less dangerous than the one from the c.m. Coulomb term in eq. (4.5), since, due to the condition E 3 3q2/4m, in (6.1), the free propagator 5&, multiplying this part of the effective potential, is restricted to a region where it is non-polar (see the remark at the end of sect. 3). In practical calculations it, indeed, turned out that, taking into account this part of the kernel of (3.3), the zero-screening limit is still attained for reasonable values of the screening radius R. Of course, some precautions appeared to be necessary in order to guarantee numerical stability. The convergence towards the zero-screening limit was demonstrated in ref. “) for the real part of the quartet S-wave phase shift 4SSR.O.The latter is plotted again in fig. 2 together with the phase shift for orbital angular momentum L = 8 (recall that the diverging phase 4R is independent of L). From these and related results we can draw the conclusion that the effective parameter, which governs the convergence rate, is (L+ l)/q,R, with qe being the on-shell momentum. The implication of this observation

is that for high angular

momenta

L and/or

low energies

the unscreening

sets in at much larger values of R. For E < 0, where the afore-mentioned

ij

E

I 1

10

Screening

100

radius

singularities

..I 1000

(fm)

Fig. 2. Convergence of the real part of the screened Coulomb-modified short-range pd phase shifts as function of the screening radius R, for total spin S = 3, and a proton laboratory energy of 50 MeV. Right-hand scale for total orbital angular momentum L = 0, left-hand scale for L = 8.

E.O. Alt et al. / pd phase parameters

of the effective potential

are missing,

of T(3) by means of a suitable

445

this can be simply taken care of by a smoothing

subtraction.

Above the break-up

threshold

more effort

is required. Another

test is provided

by the imaginary

Since #Jo as well as the pure Coulomb (5.7) and (5.12) that Im

2s+1

i.e. the imaginary

LSsRr

=

part

part of the full phase shift Im ZS+‘SL(R).

phase shift S,,

Im “+‘SL( R) -

R-W

are real, we conclude

from

Im “+‘SL = Im 2s+‘Ssc,L,

of the Coulomb-modified

short-range

phase

(6.4) shift,

which

equals that of the total phase shift, must be - at least asymptotically - independent of the screening radius R.This is, indeed, the case as inspection of table 4 shows where we present the corresponding inelasticity parameters 2s+1

= exp [ -2 Im 2s+‘BL(R)] ,

T=(R)

for L = 0 and 1, and S = i and g, for a large regime

(6.5)

of R-values.

TABLE 4

Inelasticity

parameters momentum

4qo(R) “q,(R) *v,,(R) k(R)

The numerical

for pd scattering in the quartet and the doublet state, both for total L = 0 and 1, as functions of the screening radius R, at E,,, = 50 MeV

angular

1

10

20

100

200

500

0.8840 0.8276 0.422 0.801

0.8853 0.8298 0.423 0.802

0.8854 0.8299 0.423 0.803

0.8852 0.8297 0.421 0.802

0.8846 0.8297 0.420 0.802

0.8840 0.8296

accuracy

achieved

for the phase

shifts

2S+1SL(R) and S,,

is

estimated from many tests to be better than 1% . (This was hardest to attain for the quartet S-wave and the doublet P-wave.) It is, therefore, of the same order of magnitude as the corrections expected from the neglected higher Born term contributions for TR.In general, this accuracy carries over to the Coulomb-modified shortrange phase shift 2S+1&L, except whenever “+l&( R) and 8R.L are nearly the same. In this case it is much harder to deduce precise results for their difference 2S+1SsR,L. Such a situation occurs, in particular, in the doublet channel for higher values of the total orbital angular momentum L.For example, at Era,, = 5 MeV the H-wave phase shift 2&S is of the order of 1% of each ‘&(R) and S,,; and at Elab = 100 MeV the same happens for La 8.Of course, reliable results also for these phase shifts can be obtained, albeit with more effort, and at the expense of a considerable increase in computer time.

E.O. Alt et al. / pd phase parameters

446

7. Results As explained in sect. 5, we calculate the Coulomb-modified short-range phase (pd) scattering for shift 2s+‘8s4L, defined in (5.7), for elastic proton-deuteron increasing values of the screening radius R until it has reached an R-independent value. In accordance with the discussion in sect. 5, cf. eq. (5.13), such a procedure yields 2s+1Ssc,k This quantity is to be compared with the phase shift 2S”&(nd) for neutron-deuteron (nd) scattering caused by the same separable nucleon-nucleon potentials (3.1) and (3.2). The latter can simply be obtained from the integral equations (3.3) after having switched off the Coulomb potential completely. The differences *‘+‘A, = Re [“s”S,c,,(pd)

-2S+*SL(nd)],

(7.1)

“+‘HL = 2S+1nsC,L(pd)- 2Sf’nL(nd) ,

(7.2)

customarily called the Coulomb corrections, arise from the interference between the nuclear and the Coulomb interaction. Here, for clarity, we have explicitly indicated the scattering systems (pd) or (nd). Note that the inelasticity parameters occurring in eq. (7.2) are given analogously to (6.5) by the imaginary parts of the co~esponding phase shifts. In figs. 3-6 we present the real parts of 2s+1&c,L and 2s+‘SL(nd) as well as the corresponding inelasticity parameters 2S*‘nsc,r and 2s+‘n,(nd) in the quartet (S = $) and the doublet (S = f) state, for angular momenta up to L = 2 and proton laboratory energies up to 50 MeV. In these calculations the charge-symmetric choice of the

0

10

30

20

ELob

LO

50

w

(MC%‘)

Fig. 3. Results for the zero-screening limits of the real parts of S-wave pd Coulomb-modified short-range phase shifts (sotid fines) for total spin 1 and i, as functions of the proton bombarding energy. For comparison, the corresponding results for nd scattering are shown (dashed lines).

E.O. Alt et at! / pd phase parameiers

441

Fig. 4. Same as fig. 3, but for orbital angular momentum L= 1.

nucleon-nucleon forces (case 2 of table 2) has been used. Let us discuss the results in some detail. (i) The overall energy dependence of the difference between pd and nd phase parameters can be characterized as follows. They are quite large at low energies, have in many cases a maximum between 10 and 20 MeV, and then decrease only slowly, if at all, with increasing energy. In fact, test calculations at Elab = 100 MeV show that even at these high energies they can be nearly as large as at 50 MeV. For the higher angular momentum states, not shown in the figures, they even increase in the energy range investigated. Such a behavior resembles the energy variation of the Coulomb corrections in the two-nucleon case [see e.g. ref. “‘)I. There is, however, one important distinction. In the latter case the S-wave Coulomb corrections die off very fast. For the threenucleon system, in contrast, the 2Sf1A0 for both S = f and S = $ remain fairly large even at the highest energy considered. G go.2

t

Fig. 5. Same as fig. 3, but for orbital angular momentum L = 2.

E.O. Ak et al. / pd phase parameters ELab

(MQV)

Fig. 6. Inelasticity parameters for pd (solid lines) and nd (dashed lines) scattering, for total spin S equal to $ and 3, and the lowest three values of the total orbital angular momentum L.

Let us now go into more detail. We first discuss the energy dependence of the Coulomb contribution to the inelasticity parameters displayed in fig. 6 since it is more uniform. The most striking fact is that the quartet S-state appears to be completely unaffected by switching on the Coulomb repulsion between the two protons, to the highest accuracy achieved. And in the 4P and the 4D state their effect is very small, too. Hence, in fig. 6 only one curve is drawn for each partial wave. In the doublet channel we find, however, sizeable corrections. A common feature of these is that the inelasticity parameters are increased when going over from nd to pd scattering. In other words, pd scattering is less absorptive than the nd reaction, in accordance with expectation. The real parts of the difference between nd and Coulomb-modified short-range pd phase shifts show a much more varied structure. First of all, they change their sign in the doublet P-wave around 2.2 MeV. The same happens in the quartet state in the P-wave around 30 MeV, and in the D-wave around 7.5 MeV. Furthermore, they are very small, in fact even compatible with zero within the claimed accuracy, for the doublet S-wave between approximately 4 to 10 MeV, and also for the quartet P-wave between 20 to 50 MeV. This latter fact will have some bearing on the differential cross section, the shape of which is strongly influenced by this partial wave. (ii) Next we discuss the behavior of the Coulomb corrections for varying angular momentum L. This can best be read off from table 5, where we have collected the percentage corrections to the real part of the phase shifts, “+‘Rt=

E.O. All e! al. / pd phase parameters

449

TABLES Real parts of the Coulomb corrections in percent of the real parts of the corresponding nd phase shifts in the quartet and the doublet state, at two values of the proton bombarding energy, as a function of the total orbital angular momentum; the same dependence for the Coulomb corrections to the inelasticity parameters, in percent of the corresponding inelasticity parameter for the nd scattering

0 10

50

4&. 2RL 4k *IL 4RL 2RL 4& 2k

5.1 0.7 8.8 5.1 4.0 1.7

1 -2.1 -39.6 0.26 4.8 0.8 -2.8 0.36 0.4

2 -1.3 -6.2 0.24 0.8 -3.9 -1 0.22 0.7

3

4

5

-4.6 -9.4

-8 -17

-20 -38

0.2 -2.2 -30 0.15 0.4

-8 -7 0.13 0.2

-12 -43 0.1

2Sf1A_.JIRe2Sf16L(nd)l, and to the absorption factors “+‘IL = 2S+1HL/2S+‘~L(nd), for proton laboratory energies of 10 and 50 MeV, and for L-values ranging from zero to five. Let us begin with the latter. Inspection of table 5 reveals that the important characteristics discussed for L < 2 retain their validity also for higher L-values. This concerns in particular the apparent absence of any noticeable correction to the quartet inelasticity parameters as obtained for nd scattering. From this fact we deduce that the quartet contribution to the total reaction cross section for pd scattering will practically coincide with the corresponding one for nd scattering. Furthermore, in the doublet channel, the corrections rapidly become small for increasing L but, nevertheless, always go in the direction of making pd scattering less inelastic than the nd reaction. For the real parts the picture is less uniform. In the doublet state we find a reduction of the nd phase shifts by the Coulomb corrections for all L investigated (L 5 8) except for the S-wave, and the P-wave for Elab< 2.2 MeV. That is, apart from this exception, we have Re 26sc,L(pd) < Re *&(nd) for L = 1,2, . . . ,8. A similar behavior is observed in the quartet channel, except for the P-wave above 30 MeV and the D-wave below 7.5 MeV. This implies that in all these cases the interference between the nuclear interaction and the Coulomb repulsion of the two protons leads to Coulomb-modified short-range contributions, which act effectively like a repulsion between the proton and the deuteron. This is to be contrasted with the situation for L = 0, where we find an effective attraction, except for S = 4 around and below the elastic threshold. The magnitudes of the real parts of the Coulomb corrections, both for spin f and $, increase strongly with increasing L. In fact, the percentage increase of the real parts of the pd phase shifts, as compared to the nd results, exceeds 40%

450

E.O. Alt et al. / pd phase parameters

in the doublet channel already at L = 5. Note that the large values for ‘Rz at 10 MeV and for 2R3 at 50 MeV reflect only the smallness of Re *6,(nd) and Re *S,(nd), respectively, at the corresponding energies. For nucleon-nucleon scattering the size of the Coulomb corrections decreases rapidly with increasing angular momentum of the state considered 42). The different situation found in the three-nucleon system is, however, not surprising. In fact, to a given total angular momentum L, various subsystem angular momenta couple, in particular also the low ones for which the two-body Coulomb corrections are large. (iii) Some indication of the sensitivity of the Coulomb corrections to the nuclear force can be obtained by comparison with results of a calculation with chargeindependent nuclear interactions (case 1 of table 2). This concerns, of course, only the doublet channel quantities. The first observation is that for higher angular momenta (L 2 2 at Elab = 10 MeV and L 3 4 at Elab = 50 MeV) the different twonucleon ‘So input yields identical phase shifts both for pd and for nd scattering, and, therefore, also identical Coulomb corrections. For low angular momenta, however, the two sets 1 and 2 of nucleon-nucleon forces produce quite different nd phase shifts. This is shown in table 6, which displays the differences in the nd phase parameters for L= 0 and 1 at two proton bombarding energies. A similar sensitivity is observed in the pd phase parameters. Nevertheless, their differences, i.e. the Coulomb corrections, which are also presented in this table, practically coincide within the estimated accuracy. The conclusion is that for all L the Coulomb corrections appear to be not sensitive to changes in the nucleon-nucleon force. (iv) Phase-shift analyses 43-45)of experimental pd scattering observables, and also the calculations in ref. 46), suggest the possibility of a 3He* resonance in the P-wave, preferentially in the quartet channel but S = 4 is not excluded. This question can best be answered by plotting the quantity 2Sfl

exp [2i2s+16sc,L]t SC,L

=

1

(7.3)

2i

TABLE

6

Coulomb corrections for a charge-independent (case 1) and charge-symmetric nuclear interaction in the ‘So channel, for two values of the proton laboratory orbital angular momentum L; also shown are the differences in the nd phase the cases 1 and 2 ZA!”

10 50

L=O 0.014 L= 1 -0.043 L=O 0.033 L= 1 -0.010

2Af’

0.015 -0.042 0.03 1 -0.010

Real parts are given in rad.

Re

‘6g’(nd)

- Re ‘Sy’(nd)

0.077 0.011 0.030 0.033

*HP)

2H’,2’

0.059 0.044 0.007 0.004

0.056 0.039 0.007 0.003

(case 2) choice of the energy and of the total parameters obtained for

Z~~‘(nd)-2q~‘(nd)

-0.008 -0.019 -0.03 1 -0.018

451

E.O. Alt et al. / pd phase parameters

in an Argand diagram. The results for I, = 1, and S = $ and S = 4, are presented in fig. 7. The quantity ztsc,l moves around the upper circle but in a clockwise fashion. This excludes a resonance inte~retation. In contrast, 4tsc,l clearly displays a resonant behavior. Under the assumption of a constant background phase shift, a fit to a Breit-Wigner form gives a resonance energy of about 15 MeV, and a half-width of about 10 MeV.

0.3 c&vi?

35.50 ‘2 20 . :%2 ‘b”/ l6 4fi:* ,

0.2.

’ I 1’ 1

Y’ - 0.3 ts,‘? 35

12 *

0.1 ^

8.

1%

'0. ,

0.1

0.2

0.3

0.4

l $J

20.

E -

*

0.1

0

-. - 0.2

0.1

,

0.2

0

/

/ ‘- 0.1

i7i CI N i

03

Fig. 7. Argand diagrams for the P-wave pd scattering amplitudes in the quartet and the doubiet channel. The numbers attached to the points indicate the proton laboratory energy.

8. Comparison with coordinate-space calculations Recently other calculations of pd-scattering phase shifts, based on Merkuriev’s coordinate-space formalism 26*22*27), have been published 22-24). Comparison with the results given by these authors should be very informative. First of all, since both approaches are considered to be exact any differences in the numerical answers can be ascribed to either the use of different nuclear forces, to approximations made when solving the equations, or possibly to numerical inao curacies. Moreover, additional hints on the important question of the nuclearpotential dependence of the Coulomb corrections may be obtained. For, in refs. 22-24) the S-wave part of the local potential of Malfliet and Tjon (MT) 47), rather than separable potentials, have been used.

8.1. COMPARISON OF NUMERICAL RESULTS

Such a comparison has already been attempted by Merkuriev “) and Pozdneev 23). There, in addition, the impression is given of a complete agreement between their results and the experimental phase shifts of Arvieux 4J). Unfo~unately this comparison suffers from the following drawbacks. Firstly, in the corresponding figures the results obtained with our method 3*16)are incorrectly drawn in. Secondly, also

E.O. Ah et al. / pd phase

452

parameters

the experimental data are misplaced and incomplete. A third uncertainty arises from the fact that the relationship between the Coulomb-modified short-range amplitude and the corresponding phase shift given in refs. 23S24*48) is incorrrect. The following discussion can, therefore, be made only under the proviso that the numerical results for the phase shifts of these authors, which are compared in refs. 22-24)with those of Arvieux and with ours, are, in fact, calculated with the correct formula. But apart from this, a more principal objection has to be made. It makes little sense to compare magnitudes of pd phase shifts in a situation, where already the nd phases cannot be considered reliable. This is, indeed, the case. For, first of all the nuclear potentials used in the calculations are not realistic. Moreover, the nd results of refs. 22*24)show noticeable discrepancies (which are even of the same order of magnitude as the Coulomb corrections) with those of refs. 49*20)obtained for the same MT interaction. This is illustrated in table 7 for the quartet channel.

TABLE?

Real part of the nd quartet S-wave phase shift (in rad) at three values of the proton laboratory energy, obtained for the S-wave part of the MT NN potential in refs 49.20.22 1

2.45 5.5 14.1

Ref. 49)

Ref. ‘Of

Ref. “)

1.995 1.663 1.265

1.983

2.05 1.64 1.23

The numbers quoted for ref. 22) are read off from a figure, and are, therefore, subject to a small unce~ainty.

What may be meaningful, however, is to compare the diferences between the pd and nd phase parameters, i.e. the Coulomb corrections 4A0, shown in the fourth and the seventh column of table 8. Inspection reveals that there is striking potential independence at low energy, which apparently gets lost for increasing energies, in contrast to the expectations roused in sect. 7, point (iii). Whereas 440 of refs. 22*24) becomes very small at 14.1 MeV, our value is relatively large, and according to fig. 3 it remains almost unchanged up to the highest energies investigated. Concerning the doubles state, there is a fairly pronounced disagreement between the nd phase shifts for the MT and Yamaguchi potentials, which has been known for a long time. According to table 8 the Coulomb corrections *A, found in refs. 22,24) and in the present approach differ considerably, which may indicate some sensitivity also of *A, to the choice of the potential. Whether this is a real effect is hard to decide in view of the approximations made in both approaches.

E.O. Ali et aL 1 pd phase parameters

453

TABLE 8 Real parts of the pd and the nd S-wave phase shifts in rad, and the corresponding Coulomb corrections Case 1 Re “+* 6SC.0 fpd) S=f

S=;

2.45 5.5 14.1 2.45 5.5 14.1

2.76 2.45 1.84 2.15 1‘75 1.24

Ref. ‘*) Re 2s+r& (nd) ’ 2.67 2.31 1.78 2.05 1.647 1.23

?.S+l~ 0

Re 2s+1Ssc0 (pd) *

0.09 0.08 0.06 0.10 0.10 0.01

2.812 2.639 2.213 2.126 1.763 1.320

Re “+’ S

WI 2.840 2.633 2.193 2.003 1.669 1.256

’

2s+‘A0 0.032 0.006 0.020 0.123 0.094 0.064

Compared are our results for charge-independent separable nuclear forces with those calculated in ref. 22) for the S-wave parts of the MT potentials, both for the doublet and the quartet channel, at three values of the proton laboratory energy. Note that the numbers quoted for ref. 22) are read off from a figure and are, therefore, subject to a small uncertainty. These pd values differ slightly from those of ref. r3).

8.2. THE CHARACTERISTIC

APPROXIMATIONS

OF BOTH APPROACHES

The above discussion concerned more technical aspects of the calculations. We now turn to a discussion of the more intrinsic approximations, thereby pointing to the characteristic advantages and disadvantages of both treatments. In our method we were forced by computer time and memory limitations to approximate the two-body Coulomb amplitude TR by its Born term V, (see eq. (5.1)) in the effective potentials flRj and effective propagator 3~~‘. Thereby we expect to introduce errors of the order e2 in these quantities. But all partial-wave contributions of the two-body Coulomb potential V, are automatically included. In the calculations of refs. 22-24),on the other hand, no approximation like (5.1) needs to be made. However, partial-wave expansions are necessary in all subsystems. The truncation to a finite number of angular momenta, required in practice, leads to an error, which is hard to estimate in view of the well-known lack of convergence of Coulomb partial-wave series 12,28).

8.3. COMMENT

ON THE PARTIAL-WAVE TRUNCATION

The consequences of subsystem partial-wave truncations can easily be investigated in our approach. To simplify the argument, we start with separable S-wave nuclear potentials for which the formalism, as summarized in sect. 2, is exucr. Let us first consider the quartet channel. There, the terms Y(*)(R) and ‘@*jtR)of the effective potential (4.1) do not contribute, due to the vanishing of the corresponding coupling matrices Atij3”, i = 1, 2 (see the appendix). The physical reason, of course, is that only the spin 1 of the deuteron, the form factor of which, however,

454

E.O. Alt et al. / pd phase parameters

does not experience any Coulomb correction, can couple to S =s. For the same reason the Coulomb-modified effective Green function %$,c)does not occur, either. In other words, Coulomb corrections can only originate from the remaining terms V(3)(R) and Y(4)(R). Their contribution to the effective potential can be written in detail as A ::3”V;;(R’(

q’, (I; z) + A ::3’2 Y$~,‘“‘( q’, q; z)

d3kx~(lk+~q’l)x,(lk+3ql) =

[z-(k2+(k+q’)2+q’2)/2m,,J[z-(k2+(k+q)2+q2)/2m,]

X{A Lz3”TR(q’+ik,

q+fk;

+n’“““T,(q’+$k; “??l

-q-;k;

z-3k2/4m,) z-3k2/4m,.,)}.

Introducing now a partial-wave decomposition of TR, and making use of the recoupling matrices given in the appendix, it is easily seen that only odd partial waves of the two-body Coulomb amplitude can contribute. This is again a consequence of the Pauli principle, which forbids two identical particles with parallel spin to occupy states with even relative angular momenta. Thus, we have demonstrated that for scattering in the quartet state the truncation of the Coulomb interaction to subsystem S-waves, for instance, is equivalent to the neglect of all Coulomb effects. That is, 4AL = 0, ‘HL = 0. This proof has been given for separable nuclear S-wave potentials. But the conclusion also holds for non-separable, in particular local, nuclear potentials acting in S-waves only, since such short-ranged interactions can always be expanded in a sum of separable terms. In the doublet channel the situation is less transparent. Here, truncation of the Coulomb potential to S-waves, e.g., does not lead to the vanishing of 2AL and 2HL. But also in this case the Pauli principle leads to partial cancellations of contributions to the effective potential. In view of these facts the truncation of subsystem partial-wave series, independently of questions of convergence, represents in the present case a delicate procedure which requires careful checks. 9. Comparison

with approximate calculations

of pd scattering

Most experiments in the three-nucleon system involve a proton as projectile (or target). It was, therefore, realized long ago that for a comparison between theoretical results and experimental data it is essential to include in the former the corrections expected from the Coulomb repulsion, at least in an approximate way. Thus, several approximate evaluations of pd scattering phase shifts, based on three-body integral equations, have been published. We are now in the position to test their validity by comparing them with our results.

E.O. Alt et al. / pd phase parameters

455

(i) The first of these proposals 3oT3’)[see also ref. 50)] has been applied (with an additional simplification) in ref. 51).It is based on the conjecture that in those states, in which the purely nuclear effective interaction Y(O)is repulsive, thus keeping the nucleon and the deuteron far apart, the major Coulomb effects might be of quasi-twobody nature. In other words, they might originate from the two-body center-of-mass Coulomb potential vR, i.e. from the first term on the r.h.s. of eq. (4.3, which is the interaction of longest range. All the shorter-ranged, genuine three-body Coulomb effects, which are contained in the contributions ‘T(l) + T(*) + $(3)+ ‘Tc4) to the effective potential (see sect. 4), may then be neglected. Hence, in this approximation, in the following termed “two-body”, the effective potential consists only of (9.1) a simplification which drastically reduces the numerical expense. The same idea has later been used in ref. “). In fact, calculations with such a simplified effective potential are only slightly more time-consuming than those for nd scattering. It is, therefore, important to evaluate the quality of the resulting approximate pd phase shifts, by comparing them with those for the full effective potential. For this purpose we have in addition performed calculations with the potential (9.1). In fig. 8 the resulting real parts of the Coulomb corrections *‘+‘AL, and in fig. 9 the Coulomb corrections to the absorption factors *‘+‘HL, as defined in (7.1) and (7.2), respectively, are presented for L=O, 1 and 2. For higher angular momenta we compare in table 9 the results for proton laboratory energies of 10 and 50 MeV. Inspection reveals that such an approximate treatment of the Coulomb interaction can be considered reasonable only at very low energies and for low angular momenta. The best agreement is obtained for quartet S-waves. A similar, though even less satisfactory situation prevails in the ‘P state below the break-up threshold. For energies larger than 5 MeV, and in particular for higher total orbital angular momenta, the “two-body Coulomb approximation” fails badly, producing Coulomb corrections of, in most cases, even the wrong sign. From this we can conclude that genuine three-body Coulomb effects are in general quite significant. At first sight this behaviour of the “two-body Coulomb corrections” appears puzzling. It may, however, be related to the fact that such an approximation strongly violates the Pauli principle. In order to understand this remark we have to go back to the discussion in sect. 8 of the effects of the Pauli principle. For simplicity we restrict ourselves to the clearcut quartet channel (for S = f the arguments are not so straightforward). In connection with eq. (8.1) we have pointed out that for S=$ contributions from even partial waves of the subsystem Coulomb amplitude are forbidden and thus have to cancel in the Coulomb correction terms Ac3)s”lr(3)+ A(4)sv4). But in the “two-body approximation” discussed above only one term, namely a part of A’3’solr’3)is taken into account and the other one is completely disregarded. Consequently, no cancellation of Pauli-forbidden states of the

456

E.O. Alt et al. 1 pd phase parameters

ELab (MQV) Fig. 8. Comparison of approximate calculations of the real parts of the Coulomb corrections with those using the full effective potential (4.1) (solid lines), as functions of the proton laboratory energies, for total spin S equal to t and s and total orbital angular momentum L = 0, 1, 2. The results obtained with the “two-body” approximation (9.1) are shown as dashed lines, those with the approximation (9.2) as dash-dotted lines.

proton-proton subsystem is possible, i.e. odd and even subsystem angular momenta contribute equally. In fact, such a defect is shared by any approximation which keeps e.g. only v3 or Yc4) or any unbalanced parts of them. We, finally, point out that there is some arbitrariness in the definition of the “two-body” Coulomb approximation (e.g. the second term on the r.h.s. of (9.1) may be modified off the energy shell). We have investigated several possibilities but without obtaining a substantial improvement of the model. (ii) Another approximate incorporation of Coulomb effects has been attempted in ref. 53). In our formalism it is equivalent to adding to the effective potential ‘YVco’,

E.O. Alt et al. / pd phase parameters

0

004

$002 zi

0

.Y’_._,

*p

,:A.

g am; 0.01B

.__-.--,C'-'

*D

Fig. 9. Same as fig. 8, but for the Coulomb

457

corrections

to the inelasticity

parameters.

TABLET Comparison of the Coulomb corrections obtained with the full effective potential (4.1), with those using the approximate expressions (9.1), (a), and (9.2), (b), for larger values of the orbital angular momentum, at two values of the proton bombarding energy, and, total spin 4 and 4

10

50

3 4 5 3 4 5

-4.6 -8 -20 -2.2 -8 -12

2 7 16 -3 12 8

Real parts are given in rad.

0.0014 0.0013 0.001

+0.0015 -0.0021

-9.4 -11 -38 -30 -7 -43

10 15 34 82 6 41

-1.8 -0.6 -0.3 3 0.3 1.4

0.0016 0.0003

-0.0003

0.001

0.0037 0.0018 0.0013

-0.0050 -0.0009 -0.0009

0.0051 0.0020 0.0005

458

describing

E.O. Alt et al. / pd phase parameters

nd scattering,

‘T”@),but to neglecting

only the form factor modifications

represented

by Y(‘) and

Tc3) and Yc4), i.e. “Irnm Z y(O) nm+ oy-cl) nm + v*)nm *

(9.2)

Moreover, the Coulomb amplitude in the effective free Green function +Joo;,is ignored. We mention for completeness that in ref. 53) Coulomb-modified form factors exact to all orders of e* have been used, however, their off-shell variation has been neglected. It is a characteristic aspect of this model that in the quartet channel the contributions Y(l) and Y’(*) vanish identically. Therefore, the corresponding 4$c,L are completely unaffected by the Coulomb potential, i.e. they coincide with the nd phase shifts 4SL(nd). The doublet phase shifts, however, do get modified. In figs. 8 and 9, and in table 9 we have included also the Coulomb corrections obtained for this approximate effective potential. Inspection clearly reveals that also this simplified treatment of Coulomb effects produces unsatisfactory results. (iii) Motivated by the lack of success of these two approximations we have extensively searched for other approximate methods to include the major Coulomb effects without doing the whole calculation. This can be done systematically by switching on and off any one of the contributions Y(‘), i = 1, . . . ,4, to the effective potential, and also the Coulomb correction to the effective Green function Yoo:,. Similar to the situation encountered for nd scattering *Os3’),the various individual terms V(i) contribute to the pd phase parameters differently for different total spin S and total angular momentum L. Consequently, whereas in a given state one combination of the Yci’ may produce reasonable results, it will fail in most other states. The final conclusion is that, within the three-body theory employed by us, there does not seem to exist a subset of contributions which, while reducing the computational effort, would produce Coulomb corrections in at least semiquantitative agreement with the results of a complete calculation.

10. Summary In the preceding

sections

we have presented

numerical

results for proton-deuteron

scattering phase parameters from threshold up to 50 MeV proton laboratory energies. The method used is the screening and renormalization approach proposed in refs. 3-9). It consists in solving the three-body equations with screened Coulomb potentials, and letting the screening radius approach infinity in the suitably renormalized on-shell scattering amplitudes, obtained hereby. Despite the use of simple separable potentials of rank one for the nuclear part of the interaction, the effective potentials and Green functions in the resulting two-cluster equations, are still too complicated. Therefore, we resorted to one approximation, namely to replace the two-body Coulomb amplitude by its Born approximation. Then, the three-body equations could be solved exactly.

E. 0. Alt et al / pd phase parameters

459

The Coulomb-modified short-range pd phase parameters have been compared with the corresponding quantities describing neutron-deuteron scattering, which can becalculated from the same equations by switching off the Coulomb interaction between the two protons. Their differences, the so-called Coulomb corrections, have been discussed as functions of energy and total angular momentum. We have, moreover, compared our results with those calculated by means of a coordinate-space approach. Finally, we have tested two proposals for an approximate inclusion of the Coulomb interaction in pd scattering. Both of them have been found to give unsatisfactory results. Also an extensive search for further approximation schemes has not been successful. Thus, at present, it appears that reliable pd scattering phase parameters can be obtained only by solving the full complicated equations. Before ending we note that we have deliberately renounced making any comparison with the result of phase-shift analyses 43-45)of experimental data. The reasons are twofold. Firstly, we consider our model as being still too crude to provide a physically relevant description of pd scattering. Secondly, the unsplit “experimental” phase parameters of refs. 43-45)differ widely from each other. Thus, any progress in understanding the pd system requires further effort on both the theoretical as well as the experimental side. The authors appreciate discussions with many colleagues on various aspects of this work, among them in particular Gy. Bencze, W. Breunlich, C. Chandler, A. Gibson, B.F. Gibson, L.P. Kok, I.H. Sloan, and H.F.K. Zingl. One of us (E.O.A.) is grateful to the Rechenzentrum der Universitit Mainz for generous granting of computer time.

Appendix The spin recoupling coefficients can be calculated by standard procedures. Note that in the A$!,“, n, m = d, s, c, sign factors and the factor 2 of eq. (3.3a), both resulting from symmetrization, have been incorporated. (a) S = $. The only non-vanishing elements are

(b) S = 4. Here we obtain for the different contributions

460

E.O. Alt et al. / pd phase parameters

0 0 A(*)‘P=

i 00

0 0

4 -1 0

1 ,

,(4)1/2=4

1 0

0 1

0 0

0

0

0

11 1 1 A(3)1/2

=

1

)

ho

&

-1

0

$0

0

0

,

References 1) A.M. Veselova, Teor. Mat. Fiz. 3 (1970) 326 [Transl.: Theor. Math. Phys. 3 (1970) 5421 2) A.M. Veselova, preprint ITF-73-106P, Kiev (1973) 3) E.O. Alt, Proc. Int. Conf. on few body dynamics, Delhi, 1975, ed. A.N. Mitra et al. (North-Holland, Amsterdam, 1976) p. 76 4) E.O. Ah, W. Sandhas, H. Zankel and H. Ziegelmann, Phys. Rev. Lett. 37 (1976) 1537 5) E.O. Alt, W. Sandhas and H. Ziegelmann, Phys. Rev. Cl7 (1978) 1981 6) E.O. Ah, Proc. Int. Workshop on few-body nuclear physics, Trieste, 1978, ed. G. Pisent et al. (IAEA, Vienna, 1978) p. 271 7) E.O. Alt and W. Sandhas, Proc. Int. Conf. on few-body systems and nuclear forces, Graz, 1978, ed. H. Zing1 et al. (Springer, Berlin, 1978) p. 375 8) E.O. Ah and W. Sandhas, ibid. p. 373 9) E.O. Ah and W. Sandhas, Phys. Rev. C21 (1980) 1733 10) V.G. Gorshkov, ZhETF 40 (1961) 1481 [Transl.: JETP (Sov. Phys.) 13 (1961) 10371 11) S. Weinberg, Phys. Rev. 140 (1965) 8516 12) J.R. Taylor, Nuovo Cim. B23 (1974) 313; M.D. Semon and J.R. Taylor, ibid. A26 (1975) 48 13) L.D. Faddeev, ZhETF 39 (1960) 1459 [Transl.: JETP (Sov. Phys.) 12 (1961) 10141 14) A.M. Veselova, Proc. Int. Symp. on few particle problems in nuclear physics, Dubna, 1979 (Dubna, 1980) p. 326 15) J.D. Dollard, J. Math. Phys. 5 (1964) 729 16) H. Ziegelmann, Proc. Int. Conf. on few body systems and nuclear forces, Graz, 1978, ed. H. Zing1 et al. (Springer, Berlin, 1978) p. 236 17) E.O. Ah, Proc. Int. Conf. on the few body problem, Eugene, 1980, ed. F.S. Levin (Eugene, 1980) p. 1 18) E.O. Alt, Proc. IX European Conf. on few body problems in physics, Tbilisi, 1984, to be pusblished 19) E.O. Ah, P. Grassberger and W. Sandhas, Nucl. Phys. B2 (1967) 167 20) E.O. Ah and W. Sandhas, Phys. Rev. Cl8 (1978) 1088 21) Gy. Bencze and H. Zankel, Phys. Lett. 82B (1979) 316 22) S.P. Merkuriev, Acta Phys. Austriaca Suppl. 23 (1981) 65 23) S. Pozdneev, J. of Phys. G8 (1982) 1509 24) Yu.A. Kuperin, S.P. Merkuriev and A.A. Kvitsinskii, Yad. Fiz. 37 (1983) 1440 [Transl.: Sov. J. Nucl. Phys. 37 (1983) 8571 25) A.M. Veselova, Teor. Mat. Fiz. 35 (1978) 180 [Transl.: Theor. Math. Phys. 35 (1978) 3951 26) S.P. Merkuriev, Yad. Fiz. 24 (1976) 289 [Transl.: Sov. J. Nucl. Phys. 24 (1976) 1501 27) S.P. Merkuriev, Ann. of Phys. 130 (1980) 395 28) H. van Haeringen, Nuovo Cim. 34B (1976) 53 29) L.P. Kok, Proc. IX European Conf. on few body problems in physics, Tbilisi, 1984, to be published 30) J.V. Noble, Phys. Rev. 161 (1967) 945 31) Gy. Bencze, Nucl. Phys. Al96 (1972) 135 32) M.L. Zepalova, Yad. Fiz. 37 (1983) 1381 [Transl.: Sov. J. Nucl. Phys. 37 (1983) 8241 33) B.R. Johnson and W.P. Reinhardt, Phys. Rev. A28 (1983) 1930; A29 (1984) 2933 34) H. Kriiger, J. Math. Phys. 24 (1983) 1509; 25 (1984) 1875

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