Fortschritte der Physik / Progress of Physics: Band 14, Heft 5 1966 [Reprint 2021 ed.] 9783112500309, 9783112500293

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEM GESELLSCHAFT IN DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B A N D 14 • H E F T 5 • 1966

A K A D E M I E

- V E R L A G

•

B E R L I N

I N

HALT

L. D . R O P E R : Comparison of Recent Pion-Nucleon Phase Shift Analyses

305

K . - J . B I E B L a n d F . K A S C H L U H N : Dispersion T h e o r y and Impulse Approximation for Bound S t a t e Problems 317

Die „ F O R T S C H R I T T E D E R P H Y S I K " sind durch den Buchhandel zu beziehen. Falls keine Bezugsmöglichkeit durch eine B u c h h a n d l u n g v o r h a n d e n ist. wenden Sie sich bitte in der Deutschen Demokratischen Republik an den A K A D E M I E - V E R L A G , G m b H , 108 Berlin, Leipziger S t r a ß e

3-4

in der Deutschen Bundesrepublik an die Auslieferungstselle: K U N S T U N D W I S S E N , I n h a b e r Erich Bieber, 7 S t u t t g a r t ], Wilhelmstraße 4 - 6 bei Wohnsitz im Ausland an den Deutschen B u c h - E x p o r t u n d - I m p o r t , G m b H , 701 Leipzig, Postschließfach 276 oder d i r e k t an den A K A D E M I E - V E R L A G , G m b H , 108 Berlin, Leipziger S t r a ß e 3 - 4

Fortschritte der Physik 14, 3 0 5 - 3 1 5 (1966)

Comparison of Recent Pion-Nucleon Phase Shift Analyses L . DAVID ROPER

Kentucky Southern College, Louisville,

Kentucky

Introduction

Several pion-nucleon phase shift analyses in the 300—700 MeV energy range have recently been completed [1—5]. No two of the phase shift sets are in complete agreement. This work is an attempt to evaluate which of the available solutions best iilts the data [6]. The Saclay [4] and Hawaii [5] solutions were not yet available when the computer analysis for this report was done [7]. However, an attempt is made here to evaluate them in the light of this analysis. It is found that of the solutions examined the solution of AXJVIL et al. [2] (London solution) is the best solution, but the possibility exists that some of the features of the other solutions will improve the London solution. Apparently the best phase shifts currently available are the results of the 300—1000 MeV single-energy analyses of BAREYRE et al. [4] at Saclay. I t is feasible and desirable to check their results by an energy-dependent analysis similar to that used here. All of the solutions except the Hawaii [5] solution contain an absorptive [5] Z>13 resonance. The Livermore [1] and Saclay [4] solutions have an absorptive Pu resonance. The London solution has P u behavior compatible with a P n absorptive resonance with considerable background [9]. The two solutions given by BRANSDEN et al. [3] (Rutherford solutions) have large Pn phases (50—70°) and 8U resonances; one solution ( # 1 ) has a highly absorptive [5] Slx resonance and the other solution ( # 2 ) has an absorptive Slt resonance. Other differences among the Livermore, London, and Rutherford solutions are given in [6]. None of the Hawaii solution's T = 1/2 phase shifts exceed ¿ 4 5 ° up to 700 MeV. Apparently it is one of many solutions that single-energy analyses will yield in the 300—700 MeV energy range. Demanding continuity with lower and higher energies (up to 1000 MeV), the Saclay investigators [4] rejected phase shifts of the Hawaii type. Similar solutions were also rejected in the Livermore analysis [2]. The Saclay solution yields interesting features in the 700—1000 MeV range; viz., an absorptive *S'U resonance and a highly absorptive _D15 resonance as well as the usual F u resonance. Method of Comparison and Results

The London solution and the Rutherford solutions were fitted with the energy parameterization used in the Livermore analysis [1] in order to have a common basis of comparison. In general the fits were easily accomplished. There were 23

Zeitschrift „Fortschritte der Physik", Heft 5

306

L . DAVID R O P E R

some deviations between the reported phases and the resulting fits. After fitting the solutions, a selection of 937 data, including the late Berkeley [70] values of P~(6) at 365 MeV and Saclay [11] values of P+(6) at 410 and 492 MeV, was used Table 1 Comparison of recent solutions in a 300—700 MeV search using 937 data X2 expected

X2 input

X2

x2lf expected

1. Livermore 2. London 3. Rutherford # 1 4. Rutherford # 2

831 796 793 793

4943 5157 6333 7694

2385 2235 2795 2539

2.87 2.81 3.52 3.20

6. London with res. 7. London with Sn res. 8. London with P n and Slt res.

790 790 787

2258 2002 1947

2.85 2.53 2.47

Solutions

....

P33 ••••

• *

.*•*

LONDON WITH Pu AND Su RES. X2=M7 '*

pn

/

/ •Ott

/ G 17 SH

F37 \ U1i

.. --•

• • •••

035

iati«

iff« HVf Iff/ft HHfa row SSW * • ••

P31 • • ••

g O

Ö -o u-1 O lo t< Ox t^ O

.... —

•4mm • • ••

S31

.....

ÎISÏÎ

....

T R .„.31 k-.k

933

....'

¡o c\i P io lo

^ ^ I X I O I O I O I O I O

ENERGY (MEV) Fig. 1. Solution with P u and S u resonances, (a) Phase shifts

K

307

Comparison of Recent Pion-Nucleon Phase Shift Analyses

in a determination of x 2 ' a > and then the parameters were allowed to vary to improve the fit. The results are given in the top part of Table I. The main difficulty encountered in fitting the Livermore parameterization to the solutions was with the London /S u phase and absorption parameter. The phase peaks (48°) at 600 MeY and drops to 60 at 700 MeY. The r? drops from 1.0 at 550 MeV to a m i n i m u m of 0.59 at 600 MeV and goes back up to 0.75 at 700 MeV. The phase behavior can be obtained by the momentum power series [J], but the 100

T

S31

095 0ß0 0.85

•

•

\

P31

X

Q80

• •• •

/

P3Î S »

\

0.75 0,70 Q6S Off) 055 0S0 0ft5 W 035 Q30 0,25 Q20 0.15

•'D1S and 033^

\

«* ••

\

..... P13

**

>

V 31

\ \

»

« •

,511

»

013' \

*

\

/

m 005

opo

IT, O lr> ® S C i?Vj iv} s c»l CrJ x

s °

if S •Ni ^

ENERGY

i? St

8 S lo

p P i> O P 5N lr> «© CVj ve ^i)} Wb(Pi,

1

p

=

, q ) d ( p ' 2

2

- p

2

)

cPp„

, p ) d*p

+

(7)

where rpB (pv p2; p) is the wave function of the bound state B in momentum space (pi,p2,q'\p,q)

=

d(q'

-q)y>B(PvP2-,p)-

(8)

The matrix element of the scattering operator Tt(E) in (6) depends on the momentum p2 of the "spectator" particle C2 only through its kinetic energy E2(p2) = pl/2M, the eigenvalue of K 2 , (Pi,

Pi,

q' I Tx(E)\Pl,

p2,

q)

=

(Pi,

q' I T i ( E -

E2(p2))

\Pl,

q)

d(p'2

-

p2).

(9)

Here T'x (E) is the two-particle scattering operator (6) without K2 for the scattering of the projetile particle P on the free particle Cv The first term on the right-hand side of (7) represents the off-energy-shell scattering of P on the free particle while C2 is staying as a spectator. The momentum distributions of the free constituent in the initial- and final state is given by the bound state wave functions ipB and y>% respectively. In the last term in (7) the role of the two constituents Cj is interchanged.

321

Dispersion Theory and Impulse Approximation

I t is convenient to separate the centre-of-mass motion and the relative motion of the constituents of the bound state according to Vb(PI, P 2 ; P ) = 12 3 J (2?t) fapïp«

yB(pr).

(27)

The factor arises from the definition of the relativistic field operators f j ( x ) of scalar particles, i.e. we have 3 ) (Olwfc)

I Pi) =

(¿jv; y =

-i{Pj°%j°-PjXj)

e

( 2 8 )

for the usual non-invariant one-particle states | Pj) used in the potential theory with normalization (p'j\pj) = à(jp'j — p j ) . Comparing (27) with (19) we obtain (P = Pi +

Pi) 3)

: [M 2 FBÌPÌ) 2p°2M*-p{

1 2p°

F b ( P Ì , Pi) p \ - ie] [ i f 2 FB{p\)

1

M i - p\

pi — ie] +B(pr).

(29)

The first term arises from the pole of the integrand at p\ = j/7lf2 + p\ — ie, the second one from the pole at p\ = — j/ikf2 -{-p\ + ie, and R(pr) from all other singularities of the integrand in the lower p\j half-plane. The contribution of the pole pi = M 2 to ipb(Pt) gives just (25), i.e. the proper impulse approximation (22) coincides with the corresponding formula (13) of the potential theory, if the second term in (29) and R(pr) can be neglected. The additional term R(pr) is originated from the two branch cuts of FB(p\, p\) starting at p\ = ]/(if + m)2 + p\ — ie and p\ = — }/(M + m)2 + pf + ie for physical p 2 ^ 0. The singularities of the integrand in (29) at the points with the negative sign of the square root arise from antiparticle states and can be disregarded in the nonrelativistic limit. The branch cut p% j !(M + wi)2 + p\ of FB(p\, p\) is due to the virtual dissoziation of the constituent G.l into other particles with the lowest mass (M + m). In the instantaneous limit [45, 62—65] of the Bethe-Salpeter equation (18) we have to neglect in the kernel F(pj, p2 \ p{, p'2) the dependence on the relative energies (Pi — PDI2 a n c l (pi0 — Pa0)/2 (in the centre-of-mass system of the constituents) leaving over a function V(pr, p'r) of the relative momenta pT = (p1 —Pi)j2 and p'r = (p{ — p\)j2 only. As a consequence, also the form factor does not depend on the relative energy (pi — pi)12 and ipB(pT) is given entirely by the first two terms in (29). If we drop furthermore antiparticle contributions, the Bethe-Salpeter equation goes over in the nonrelativistic limit into the Schrodinger equation (using (26))

(irf+ e°) ^ =4wJ§$ V(P" M

(30)

326

K . - J . BIEBL and F . KASCHLTTHN

where M 2 on the right-hand side of (30) arises from the nonrelativistic limit of y p o p l p ^ o 3 ) . In particular, the one-particle exchange kernel F j = g2 X X (to2 — (Px — pi)2)"1 leads to the Yukawa potential F x (r) = — (gr2/16 nM 2 ) er mr jr. We would like to mention that it is more appropriate within our approximation scheme to define generally the bound state wave function by the right-hand side of (25), i.e. essentially by the form factor with only one particle off the mass shell [45] which is directly involved in formula (22) for the scattering amplitude A. This definition should be used also in the nonrelativistic case, where the equaltime Bethe-Salpeter amplitude, not directly related to the scattering amplitude A, represents a different wave function, if retardation effects are included. We remark further that radiative corrections with respect to the spectator particles, which possibly may turn out to be important also in the nonrelativistic limit, must be included as corrections to the impulse approximation within the nonrelativistic potential theory. From this point of view the general form (21) of the relativistic impulse approximation defined in terms of Feynman diagrams is not the natural generalization of the impulse approximation considered in potential theory, but its on-mass-shell approximation for the spectator particles, i.e. the impulse approximation in the proper form (22) just discussed. However, from a more general point of view it is useful to define first the relativistic impulse approximation entirely in terms of Feynman diagrams. For these reasons we employ in the following not the equal-time Bethe-Salpeter amplitude, but the "wave function" (25), i.e. essentially the form factor with one particle off-the mass shell, as the proper quantity for comparison with potential theory (see chapter IV). This is quite important for the study of retardation effects. Finally we want to remark that the off-mass-shell amplitude A in (22) must be regarded as the off-energy-shell amplitude M in (13) according to (24). In many practical calculations it will be sufficient to approximate A by the onmass-shell amplitude p\ = p'2 = M2 corresponding to the on-energy-shell approximation (15) of M because of (14) and (26). This is certainly a reasonable approximation for loosely bound states since the singularities of A as a function of p\ (or p'j2) start at the normal threshold p\ = (M + m)2, whereas the form factor FB(p\) has anomalous thresholds below the normal one [20, 26, 45, 46]. 3. R e l a t i v i s t i c F o r m u l a t i o n w i t h i n Dispersion T h e o r y We may formulate the impulse approximation for the process considered also very easily in the framework of dispersion theory using, however, some off-massshell quantities (see e.g. [66]). A dispersion-theoretical study of the occuring form factors will be given in the next chapter. In the following we consider only the on-mass-shell approximation for the spectator particles, i.e. the impulse approximation in its proper form. The scattering amplitude obeys a dispersion relation for fixed t A(s,t)

=

— f M*'> n J s — s

0 + — f - ^ — n J u — u

0

(31)

where u = (p' — q)2 and As, Au are the absorptive parts in the respective channels. In (31) we have disregarded subtractions. In general we have to

Dispersion Theory and Impulse Approximation

327

deal with anomalous contributions in (31) [10—16], in particular if the impulse approximation actually is valid quantitatively, namely for loosely bound states. From unitarity we have. 2 Aa(s,t)=ZA*(P',q';n)A(n-,p.q)

(32)

and correspondingly for Au(u, t). The sum is to be extended over all possible intermediate states n. In the proper impulse approximation for A (s, t), where the spectator particles are on the mass shell, the amplitudes A(n;p, q) have obviously to be represented by diagrams shown in fig. 3, where the bold-faced lines indicate that the respective form factors and scattering amplitudes are extrapolated off the mass shell. Thus, considering only the first contribution according to the left-hand diagram + (1^2) in fig. 3, we have Atn-

F

B((p-Vt)%)A(n-V,p-

v a ) -

(33)

and from this we find £

A*(p',q';n)A(n;p,

Fig. 3.

q) =

(n)

X

x

M^-if'

x f s n X A*(p'

-J

A

-

*{P'

' ''

Pa

' ''

P n ) A{1>3

-

v*?)

m ¿ ( p + q - v* - 1 . . ., pn) A (p3, . . ., pn; p, q) =

py) x

— p2, q'\p3,

-

p2,q';p

-

P.?) Fb((P

X

Fb{(P'

M )

-

' "' ' Vn'V,q)

r

(2nf

x 2Alit(p'

P2 q

[ M > - (p' - ft)»] [JK* -

p2, q)

{p -

p2)*l

X

(34)

with the scattering amplitude Ai as defined previously in (22). Inserting (34) and the corresponding expression with (1 2) in (31) we arrive at an expression which is completely equivalent to (22).4) 4

) Actually we get mixed terms with respect to the indices 1,2 which, however, have to be neglected in the impulse approximation. This follows from the comparison of 2 Im A = A A* for A = A1 + A2 with 2 Im A^ = AfAf.

328

K . - J . BIEBL a n d F . KASCHLUHN

4. A p p l i c a t i o n t o P i o n - D e u t e r o n F o r w a r d S c a t t e r i n g I n the case of elastic pion-deuteron forward scattering the contribution of the unphysical region to the dispersion relations was calculated explicitly in ref. [72]. I t is only due to the intermediate two-nucleon state (i. e. to the absorption process). Here from the diagrams of fig. 3 only those shown in fig. 4 contribute. Explicitly they are given by the following formula, where gp(p2) is the proper pion-nucleon vertex function with one nucleon off the mass shell (not including (1'2) self-energy corrections), Fb(PÌ)9P(PÌ) M 2 - p \ - it

(35)

r i g . 4.

I t was possible to treat the calculation nonrelativistically with respect to the nucleon motion. Application of (25) leads directly to the deuteron bound state wave function which was approximated by the Hulthen function [60] fD(r) =

+ p)f2n{a,

a = ] / j f e = 0,33m,

-

p)2]1'* (er^ -

e~er)jr

(36)

0 = 6»

and the pion-nucleon vertex function was placed onto the mass shell (similar to the on-energy-shell approximation of the 71 + N —> n + N scattering amplitude in the elastic scattering process 7i + D - > 7 r + D, in order to get only physical

\

F0

F0 /

/

Y

P

1

/

/

9

/ r i g . 5.

quantities within the impulse approximation). We remember that the Fourier transform of each term in (36), the and the /5-term, is essentially (p^ + a 2 ) - 1 and (pi + P 2 )- 1 respectively. The first pole term, describing the asymptotic part of the wave function in ordinary space, corresponds to that part of the diagram in fig. 4 where the intermediate nucleon is on the mass shell, whereas the second pole approximates the whole contribution of the branch cut of the wave function or the from factor respectively as indicated in fig. 5 (here F0 = F{M2) and g = g{M2)). The branch cut part determines the behaviour of the wave function at small distances in ordinary space for which the many-pion exchange contributions will be quite important quantitatively. For not too large p2, i.e. for pr in the non-relativistic physical

Dispersion Theory and Impulse Approximation

329

region we expect that the /5-pole term approximates quite well the nearest anomalous branch cuts of fniPr) considered in this region as the dominant ones (compare for details chapter IV). Anomalous singularities of the izD forward scattering amplitude itself were determined explicitly by analytic continuation in the pion mass leading in this case to an anomalous pole and an anomalous logarithmic singularity at the same point situated 0.67 MeV below the normal two-nucleon threshold. Numerically the contribution from the anomalous pole turned out to be the most important one in the spin-flip amplitude, where the contribution from the unphysical region is large. The comparison with the experimental data was performed in such a way that in the observable region the standard impulse approximation was used in the form (15), where in addition the on-energy shell approximation is assumed. This approximation is satisfied by experimental data within 2 0 % even in the resonance region. We remark the important point that the on-energy-shell approximation in (15) is non-analytic in the sense that the pion-deuteron scattering amplitude shows in this additional approximation analytic properties different from the original impulse approximation. For instance, in the approximation (15) the pion-deuteron forward scattering amplitude is simply the sum of the two pion-nucleon forward scattering amplitudes (the form-factors (16) are equal to one for forward scattering). However, the last two amplitudes have, for instance, no branch cut in the unphysical region; consequently in this approximation there is no contribution corresponding to the absorptive process n + D - > iV + JV in the unitarity condition which indeed is small in this region. Consistency requires that the contribution from these pion-nucleon pole terms, as it arises in the physical region, must be equal to the whole contribution from the unphysical region of the pion-deuteron forward scattering amplitude calculated by explicit analytic continuation of the absorptive process contribution within the original impulse approximation without the on-energy-shell performance. This turned out indeed to be the case for the spin-flip amplitude within 1 0 % in the whole physical energy region. We stress that for this agreement the consideration of the /S-term is quite important which means that one has to include appropriately the off-mass-shell-properties of the form factors for loosely bound states even for small momentum transfer. 5 ) 5. N o n - E l a s t i c P r o c e s s e s In the preceeding sections we considered the elastic scattering by a composite particle. These considerations can be easily generalized to more complicated processes as already discussed in connection with (32). Let us consider in more detail the case, in which the bound state disintegrates in the course of scattering; as indicated in fig. 6. In the non-relativistic form of the impulse approximation we have only to replace in (13) the bound state wave function in the final state by the wave function of the continuous spectrum which takes into account final state interaction. Correspondingly in the relativistic case we have then to consider Feynman diagrams shown in fig. 7. We remark that in the unitarity relation ) For the process y + D -> N + JV the influence of the /5-term is explicitly studied in ref. [44].

5

K.-J. Biebl and P. Kaschluhn

330

(32) we should not include the final state interaction as considered in fig. 7 since it corresponds to higher approximations to the impulse approximation for the elastic scattering amplitude itself. This is shown in fig. 8 for a special contribution

r i g . 7.

Fig. 8.

Fig.

+

(1

genera] formulation of the impulse approximation for this case within the Hamiltonian formalism together with a necessary modification of the adiabatic condition was given in the non-relativistic limit of the relative motion of the constituents in ref. [10]. The corresponding Feynman diagrams describing this scattering process •of the general relativigtic case are drawn in fig. 10 whose non-relativistic instan-

Dispersion Theory and Impulse Approximation

331

teneous limit with respect to the constituents Cj of B leads to the results of ref. [10]. We remark once more that within the impulse approximation it is not necessary to consider the final state interaction in connection with the unitarity condition (32)6). Concluding this section we would like to mention that one may easily discuss on the same basis the impulse approximation for the general case, where the incoming particle is scattered on a bound state of mass MB which consists of n particles of mass Mj whose number is conserved. In the final state there may be an arbitrary number of particles besides the n constituent particles part of them, if not all, may be bound by each other. The general scattering diagram is drawn in fig. 11, the corresponding impulse approximation Feynman diagrams in fig. 12. The full amplitude is the sum of n scattering amplitudes, each of them describing the interaction of the incoming particle with one of the constituents of the bound state respectively, whereas the others are spectators. The explicit expressions for the scattering amplitude may be easily written down and are simple generalizations of the cases already studied. The appearing "form factors" are now, Fig. n.

"h

+

j=2,...,n) Fig. 12.

in general, many-particle transition amplitudes describing the virtual decay of the bound state into their constituents. III. Dispersion Relation Approach to the Deuteron-Two-Nucleon Form Factor

In this chapter we consider the form factor of the bound state with one constituent off the mass shell which occurs in the scattering amplitudes as described in the previous chapter. 6) Consequently in the calculation of the contribution from the intermediate two-nucleon states in the elastic pion-deuteron scattering amplitude one may use plane waves for these intermediate states as done in réf. \1I\.

332

K . - J . BIEBL a n d F . KASCHLTTHN

1. General R e m a r k s As a first step we may determine the form factor by the corresponding bound state wave function of potential theory using the relations (II, 25) and (II, 26) [45]. In particular for the Hulthen wave function of the deuteron (II, 36) the corresponding form factor F(T)7) has a pole at T = M} = M-1 + 2 ( 0 2 — ) is the phase-space factor 1

(25)

S=M,

Fig. 21.

and j dû the angular integration over the directions of the momentum of the intermediate nucleon N±. Inserting the pole approximation (23) with F(u) = F0 into the (N + 27î)-contribution to the unitarity relation of F (t) we obtain

(t '>-M)' Im F{t) = nfdw Q(t, m) F0K(t, M2, m)

M%\t),

(26)

4m*

where K(t, M2, co) is the S-wave projection of the denominator M2 — u(t, cos 6t, m) +i

JTitM Mi , ^ 2M + 2 M m . The anomalous branch cut extends from the "anomalous threshold" t = v+ (M2, co) = v ( M ; M ^ , M ; M , co) (compare (13) with m -> co) up to the "normal threshold" t = (M + co1/')2. Therefore F(t) has a continuum of "anomalous thresholds" for all co 22 4 m 2 up to the maximal value comax = {M% — 2 M ) \ M . The lowest one at t = v ( M , (2m)2) = t! arises from the lower endpoint co = 4m 2 of the co-integration in (29). For co r> comax the "anomalous threshold" t = v+{M, co) coincides with the corresponding „normal thershold" i m a x = (M + co^ax)2 = (M^ — M ) f M , which is found just below the physical deuteron-antinucleon threshold t = (Mb + M)2. This point is the upper limit of any anomalous singularity, since for co > comax a normal dispersion relation for the co-integrand in (29) holds. These anomalous thresholds lead to an additional term F' {t), to-be added to the right-hand side of (29) 2

+

2

2

2

2

2

2

2

+

2

2

2

2

2

2

t

aimai

(M +

ai1'2)'

), 1

4 m"

(30)

v+(M*,to)

where the relation Q

n

{t, co) disc\K{f, M 2 , co)] =

1

=

(31)

+

has been used (compare the analogous term Ft(t) in (18)). Interchanging the order of the co and t integrations we obtain •) + t,

(2m)' ¿max r dt'

+

/

• N -f- nn (equation (23) with F(u) replaced by F0). Therefore, the relation (26) is only an approximation for the absorptive part F(t) on the normal branch cut. However, the absorptive part on the anomalous branch cut, as determined in (32), is the correct one, since it depends only on the residuum of the nucleón pole of the production amplitude in the crossed channel u. If we take into account further the anomalous singularities of F(u) in (23) itself, the higher iterations of all anomalous singularities can be obtained. However, only a few ones of the higher iterative singularities are really in the physical sheet. They coincide with the Landau singularities of truss-bridge diagrams of fig. 17a with any number of pions on each dotted line. The position of these singularities can be determined iteratively by the function v^ (u) defined in (13), with the pion mass m replaced by the corresponding multiple thereof. IV. Comparison with Potential Theory The analytic structure of the deuteron-two-nucleon factor F(t) derived in the last chapter will now be compared with the analytic properties of the bound state wave function y>(p) of the deuteron determined by nonrelativistic potential theory with a suitable local, instantaneous potential V(r). This comparison is based on the relation (II 25) between F (t) and yi (p). After dropping the relativistic

345

Dispersion Theory and Impulse Approximation

square roots and numerical factors it reads [45] F(t) = (p2 + *2) y(p2),

t = M2 — 2{p2 + 0). This potential is the effective potential in the $-wave state for the energy -e, including also the contribution from the exchange potential. The Schrodinger equation for the bound state wave function rxp{r) = M V{r)rip(r),

(3)

at the energy |/s = MD. The comparison with (8) yields yi