- Email: [email protected]
Multiple scattering with three-body forces
ANNALS
OF PHYSICS
108,
401-407 (1977)
Multiple
Scattering
with
Three-Body
Forces
N. AUSTERN Department
of Physics, Deparfment
of Washington, Seattle, Washington 98195*, and Astronomy, University qf Pittsburgh, Pittsburgh, Pennsylvania 15260+
Unirersity
and
of Physics
Received March 18, 1977
The multiple-scattering formalism is generalized to account for simultaneous interactions of a projectile with two target particles. In the cluster expansion of multiple-scattering theory such three-body interactions combine with iterates of two-body interactions. Some comparisons of multiple scattering with the Faddeev theory are given.
1.
INTRODUCTION
This article presents a generalized multiple-scattering formalism that includes three-body interactions of the projectile with two different target nucleons at the same time. Such three-body interactions are the W(Ojk) terms in the overall Hamiltonian H = Ho + f
V@j) -t i
FV(O~~),
j
j=l
where 0 denotes the coordinates of the projectile, regarded as distinguishable from the particles 1 ... N of the target nucleus. Here Ha is the sum of the internal Hamiltonian of the target nucleus plus the projectile kinetic energy. By definition a three-body interaction W(Ojk) becomes zero if any one of the distances Toj , rOk, rjL becomes large. (Although W(Ojk) must be symmetrical under interchange of indices, ,jk f--t kj. for ease in counting pairs we use the convention that the smaller index is always placed first, j < k.) It is characteristic of the above Hamiltonian that the projectile retains its identity under the interactions V(Oj), W(Ojk), as is suitable for studies of elastic and inelastic scattering. However, actual physical projectiles, such as pions, can interact with nucleons and change into other, intermediate particles, such as p mesons. Or an actual projectile can be annihilated. The interaction W(Ojk) is a phenomenological device by which such processes are brought into Eq. (1). For example, in pion scattering the presence of an intermediate p must be associated with two nucleons; it is produced at one of them and converted back to a 7r at the other. This process is * Supported in part by the U. S. Energy Research and Development Administration. + Permanent address. Supported in part by the National Science Foundation. Copyright All rights
:n 1977 by Academic Press, Inc. of reproduction in any form reserved.
401 ISSN
0003-4916
402
N. AUSTERN
represented by W(Ojk), even though the p does not appear explicitly. Similarly, because pion annihilation proceeds primarily through pairs of nucleons, we expect to represent this process by an imaginary term in W(Ojk). One further motivation for this work is simply to observe how a multiple-scattering formalism is modified to accommodate three-body interactions. It is found possible to exploit one of the basic attitudes of the Faddeev theory, under which particular components of the wavefunction are put in association with particular terms of the interaction. The multiple-scattering formalism is developed in Section 2. A cluster development that relates three-body effects with iterations of two-body effects is presented in Section 3. Some relation to experiment is suggested in Section 4. Similarities to the Faddeev theory are explained in the Appendix.
2. MULTIPLE-SCATTERING FORMALISM
A multiple-scattering theory for the Hamiltonian of Eq. (1) is a procedure that replaces the interactions V(Oj), W(Ojk) by calculations with equivalent scattering operators, derived from simplified auxiliary scattering problems. In the absence of three-body interactions, the two-body interactions V(Oj) in H are eliminated in terms of scattering operators tj . In the original approach of Foldy and Watson [I], the analysis is begun by defining auxiliary wavefunctions #j with respect to each of the interactions V(Oj), #j SE Y - G,V(Oj) !?‘,
(2)
by subtracting from the full wavefunction the scattered wave caused by V(Oj). Here Go ES [E” - H&l
is the free outgoing Schwinger equation
Green’s function.
Substitution
(3)
of Eq. (2) in the Lippmann-
Y = 4 + G, 2 V(Oj) Y
(4)
j=l
gives
& = 4 + Go 5 Wj)
\k:
(5)
j+i
The incident wave I$ satisfies (E - Ho) + = 0.
(6)
It is a product of a projectile plane wave times the target nucleus ground-state wavefunction. Equation (5) is put entirely in terms of the #j by introducing scattering operators tj through the substitution V(Oj) Y = tjf)j .
(7)
SCATTERING
WITH
THREE-BODY
FORCES
403
Then Eq. (5) becomes a set of coupled equations for the auxiliary wavefunctions,
Insertion of Eq. (7) in Eq. (2) suggests an alternative interpretation of Eq. (2) as a partial Lippmann-Schwinger equation for Y in terms of #,j . From this equation, the exact ti of Eq. (7) must obey tj = W?j){l
+ Gotjl,
(9)
or tj -= V(Oj){l + [Et - H, - V(Oj)]-r V(Q);,
(10)
integral equations for Oj scattering in the presence of the full target nucleus. These equations for tj contain only one of the projectile-nucleon interactions from the original H; the other interactions are brought in through the coupled equations (8). We recognize the basic structure of the multiple-scattering approach. Of course, all equations are defined in the full (N + 1)-particle Hilbert space. Calculation with partial wavefunctions is usually replaced by the introduction of partial scattering operators, by the substitution Ti$ = tj#j .
(11)
By this step the overall scattering operator (12)
reduces to a linear combination, T=
; Tj. j=l
(13)
The coupled equations (8) are converted to the form Ti = ti + tiGo c Tj
(14)
j+i
by left multiplication by ti and insertion of Eq. (11). Resemblances between Eqs. (2) and (13) and the Faddeev theory are discussed in the Appendix. In the presence of three-body forces, the two-body terms in the full Hamiltonian of Eq. (1) are still dealt with by the same steps outlined above. The previous equations of definition of partial wavefunctions $j and scattering operators tj never use the full H; therefore they are not affected by the three-body terms. The additional force only need be accounted for with additional partial wavefunctions and operators,
Sijxij
xij = Y - G, W(Oij) Y, TS W(Oij) Y, sij = W(Oij){l + G,s
(15) (16) (17)
404
N.
AUSTERN
defined in analogy with the previous ones. (As before, the smaller index in any double-index quantity is placed first.) Successive insertion of Eqs. (2), (15) in the Lippmann-Schwinger equation Y = # + Go c VOj> Y + Go c j=l
W(Ojk) Y
(18)
Kk
leads to an extended set of coupled equations for the auxiliary wavefunctions ?h
xij
=
d
$-
GO
1 j+i
t&j
+
Go
=
4
+
Go
$
tk#k
+
Go
C skZXkZ k
9
SkZXkZ
klfii
(19)
(20)
,
where the restriction on the second sum in Eq. (20) stands for k Z i and I # j. Once again the coupled equations for partial wavefunctions are converted into coupled equations for partial scattering operators, by Eq. (11) and by the corresponding equation for the three-body interaction, sikd
E
sjkxjk
(21)
-
Then Ti = ti + tiGo C Tk + Togo C SkZ3 k#i Sij
=
Sij
+
SijG,
(22)
k
Tk
-I
s&o
C kZ#ii
‘kZ
3
(23)
and T =
‘$J Tj + C S’ij .
j=1
i-3
(24)
This extended system of coupled equations is not essentially more complicated than Eqs. (13), (14) for two-body forces. However, the physical interpretation of sif is less straightforward than for ti ; it does not have the same close correspondence to some simple scattering experiment. It should be noted that Sij is an exact solution for the effects of W(Oij) in the absenceof two-body interactions. This further limits the physical interpretation of Sij . 3.
CLUSTERS
The scattering operators Sij express the interaction of the projectile with pairs of nucleons. On the other hand, iteration of the operators tj for the interactions of the projectile with individual nucleons also gives interactions with pairs, triplets, and higher clusters. Therefore the physical consequences of the sij need to be discussed together with the iterations of the tj . A cluster development is indicated, which distinguishes all interactions of the projectile with single nucleons, pairs, and so forth.
SCATTERING
WITH
THREE-BODY
405
FORCES
A cluster development is a linear reordering of the terms in an iterative expansion of one of the overall scattering operators, either of T itself or of some form of the optical potential U. Terms that refer to equal numbers of different target nucleons are grouped together. Analyses of such cluster developments are given in an article by Ernst et al. [2]. These authors show an essential arbitrariness in the definition of clusters: because the relation between T and U is not linear, clusters defined for one of these quantities do not correspond term by term with clusters defined for the other. Nevertheless, our present concern is only to see the relation between sij and iterations of fj , and for this purpose it is sufficient to remain within one version of the cluster development. Because integral equations for T are already in hand in Section 2, it is easiest to examine the cluster development that emerges from them. The coupled integral equations (22), (23) are solved for Ti , Sij in powers of ti and Sij by successive substitution of lower-order expressions for Tk and S,, on the righthand sides. Clusters of successively higher order are identified in these expansions by first collecting terms that have all particle labels the same, then terms that have all particle labels selected from any two digits, and so forth. Simple expressions are obtained if we retain only the parts that contribute to matrix elements between antisymmetric nuclear wavefunctions. Then the symmetric part of T is T symm = [T(l)
in which the cluster contributions T’l’
+ Tc2’ + Tt3’ -+ ...I symm
VW
,
are found to be
= Nt, )
Tc2’ = N(N
(26) -
l){t,G,t,
+
8~12
+
+
WdzGoslz
+
N,
+
$(t,
f2)
+ tlGot2Gotl +
12) G,s,, +
~~,,Gt,Got,
G&Z&
+
+ ...
+
i~l,G(rl
Q
+
+
+~,,G,(h
tz)
+
t2) Go.712
+
. ..>.
(27)
and so forth. The pair term is a complicated nonlinear composite of the operators t13 t2 7 s12. This composite has the same structure for all target nuclei; only its coefficient depends on N. An interpretation of the pair term is obtained [2] by applying the above equations to a nucleus with N = 2. Then T symm(iV=2)
= t,, = 2t, + 2{t,G,t,
+ . ..}.
(28)
where the expression in braces from Eq. (27) is copied in abbreviated form. Solution of Eq. (28) and substitution in (27) give the pair term as T(2) = N(N
-
1){$t12 - tI},
(29)
in terms of the supposedly known two-body amplitude t,, . We can imagine fitting t,, to experimental studies of scattering from the deuteron. Equation (29) is the same
406
N. AUSTERN
formal expression derived by Ernst et al. [2]. However, the present discussion shows that the definition of t13 can be extended to include the three-body operator slZ. This same generalization of tls has also been derived by Sicilian0 and Thaler [3].
4. CONCLUSIONS Immediate practical application of the above formalism does not seem likely. However, two conclusions can be drawn. First, we observe in Eq. (27) how slZ competes with second-order effects of tj . This may help in forming estimates of the importance of three-body effects, such as are mentioned in the Introduction. Second, Eq. (29) shows how data drawn from meson-deuteron scattering, for example, can be brought into analyses of scattering of mesons by heavier nuclei. Such a phenomenological procedure combines higher-order two-body effects with explicit three-body effects. APPENDIX:
COMPARISON
WITH FADDEEV
THEORY
Multiple-scattering theory and Faddeev theory each analyze a many-body system in terms of partial wavefunctions. However, in multiple-scattering theory only the motion of the projectile is considered to be unknown; all the wavefunctions of the target nucleus are considered to be available. Therefore, formal multiple-scattering theory works with complicated Green’s functions, such as G,, , which contain the entire Hamiltonian of the target nucleus. By contrast, Faddeev theory treats the projectile and the particles of the target nucleus on the same terms. This theory only uses simple Green’s functions, which contain no more than one interaction at a time. A perfect correspondence between these two methods is not possible. Nevertheless, there are interesting parallels, and these helped to organize Section 2 of the present article. The Faddeev analysis of a three-body system can be developed from the Schrijdinger equation [4, 51 (E--)Y=(V,,+~s/,,+V,,)Y. (Al) Component equations
wavefunctions are defined in terms of Y and the three interactions by the (E - K) $0) = V,,Y,
and cyclic permutations. boundary condition
Addition
WV
of the three equations (A2) restores (Al), with the
y = #(I) + pz, + pm
(A3)
Further development of the theory follows by substituting (A3) into Eqs. (A2) and rearranging for more convenient calculation. The equations so obtained can be recast as integral equations.
SCATTERING
407
WITH THREE-BODY FORCES
The above Faddeev analysis defines its component wavefunctions by combining the interaction terms of the original H, one at a time, with the full wavefunction Y. The equations of definition can also be cast in integral form. For example, Eq. (A2) becomes lp”’ = $3) + (E - K)-1 V12Y, (A4) where (E - K) $3) = 0. (A5) This kind of association of component wavefunctions with individual interaction terms is used in the main text in a slightly altered fashion, in Eqs. (2) and (15). The latter equations define auxiliary wavefunctions that are dzfirences between the full !P and the scattered waves generated by individual interactions. That modification gives more convenient boundary conditions than Eq. (A4). The full Y of multiple-scattering theory can be expressed in terms of the auxiliary wavefunctions defined by Eqs. (2) and (15), in analogy to Eq. (A3) of the Faddeev theory. It is only necessary to add up the set of N(N -+- 1)/2 coupled equations derived by successive insertion of Eqs. (2), (15) in the Lippmann-Schwinger equation of Eq. (18). This procedure leads to y
=
] j=l c h
+
c
i
Xjk
a more complicated linear combination is given in [5, Eq. (3.17)].
-
bjita(zv
+
1) -
I}-1
,
646)
than in the Faddeev theory. A similar result
ACKNOWLEDGMENTS This work grew out of discussions with Dr. R. M. Thaler. I am grateful to Dr. G. A. Miller for reading the manuscript and for extensive discussions about Ref. [2]. I also acknowledge the kind hospitality of Dr. Louis Rosen and the Los Alamos Meson Physics Facility, where portions of this work were done. Conversations with Dr. Mikkel Johnson were helpful.
REFERENCES 1. L. L. FOLDY, Phys. Rev. 67 (1945), 107; K. M. WATSON, Phys. Rev. 89 (1953), 575; K. M. WATSON AND M. L. GOLDBERGER, “Collision Theory,” Wiley, New York, 1964; L. S. RODBERG AND R. M. THALER, “Introduction to the Quantum Theory of Scattering,” Academic Press, New York, 1967. 2. D. J. ERNST, J. T. LONDERGAN, G. A. MILLER, AND R. M. THALER, preprint, 1976. 3. E. R. SICILIANO AND R. M. THALER, preprint, 1977. 4. R. D. Ahlano, P&X Rev. 158 (1967), 1414, Eq. (A12). 5. Y. HAHN AND K. M. WATSON, Phys. Rev. A 5 (1972), 1718.
595/108/z-12
OF PHYSICS
108,
401-407 (1977)
Multiple
Scattering
with
Three-Body
Forces
N. AUSTERN Department
of Physics, Deparfment
of Washington, Seattle, Washington 98195*, and Astronomy, University qf Pittsburgh, Pittsburgh, Pennsylvania 15260+
Unirersity
and
of Physics
Received March 18, 1977
The multiple-scattering formalism is generalized to account for simultaneous interactions of a projectile with two target particles. In the cluster expansion of multiple-scattering theory such three-body interactions combine with iterates of two-body interactions. Some comparisons of multiple scattering with the Faddeev theory are given.
1.
INTRODUCTION
This article presents a generalized multiple-scattering formalism that includes three-body interactions of the projectile with two different target nucleons at the same time. Such three-body interactions are the W(Ojk) terms in the overall Hamiltonian H = Ho + f
V@j) -t i
FV(O~~),
j
j=l
where 0 denotes the coordinates of the projectile, regarded as distinguishable from the particles 1 ... N of the target nucleus. Here Ha is the sum of the internal Hamiltonian of the target nucleus plus the projectile kinetic energy. By definition a three-body interaction W(Ojk) becomes zero if any one of the distances Toj , rOk, rjL becomes large. (Although W(Ojk) must be symmetrical under interchange of indices, ,jk f--t kj. for ease in counting pairs we use the convention that the smaller index is always placed first, j < k.) It is characteristic of the above Hamiltonian that the projectile retains its identity under the interactions V(Oj), W(Ojk), as is suitable for studies of elastic and inelastic scattering. However, actual physical projectiles, such as pions, can interact with nucleons and change into other, intermediate particles, such as p mesons. Or an actual projectile can be annihilated. The interaction W(Ojk) is a phenomenological device by which such processes are brought into Eq. (1). For example, in pion scattering the presence of an intermediate p must be associated with two nucleons; it is produced at one of them and converted back to a 7r at the other. This process is * Supported in part by the U. S. Energy Research and Development Administration. + Permanent address. Supported in part by the National Science Foundation. Copyright All rights
:n 1977 by Academic Press, Inc. of reproduction in any form reserved.
401 ISSN
0003-4916
402
N. AUSTERN
represented by W(Ojk), even though the p does not appear explicitly. Similarly, because pion annihilation proceeds primarily through pairs of nucleons, we expect to represent this process by an imaginary term in W(Ojk). One further motivation for this work is simply to observe how a multiple-scattering formalism is modified to accommodate three-body interactions. It is found possible to exploit one of the basic attitudes of the Faddeev theory, under which particular components of the wavefunction are put in association with particular terms of the interaction. The multiple-scattering formalism is developed in Section 2. A cluster development that relates three-body effects with iterations of two-body effects is presented in Section 3. Some relation to experiment is suggested in Section 4. Similarities to the Faddeev theory are explained in the Appendix.
2. MULTIPLE-SCATTERING FORMALISM
A multiple-scattering theory for the Hamiltonian of Eq. (1) is a procedure that replaces the interactions V(Oj), W(Ojk) by calculations with equivalent scattering operators, derived from simplified auxiliary scattering problems. In the absence of three-body interactions, the two-body interactions V(Oj) in H are eliminated in terms of scattering operators tj . In the original approach of Foldy and Watson [I], the analysis is begun by defining auxiliary wavefunctions #j with respect to each of the interactions V(Oj), #j SE Y - G,V(Oj) !?‘,
(2)
by subtracting from the full wavefunction the scattered wave caused by V(Oj). Here Go ES [E” - H&l
is the free outgoing Schwinger equation
Green’s function.
Substitution
(3)
of Eq. (2) in the Lippmann-
Y = 4 + G, 2 V(Oj) Y
(4)
j=l
gives
& = 4 + Go 5 Wj)
\k:
(5)
j+i
The incident wave I$ satisfies (E - Ho) + = 0.
(6)
It is a product of a projectile plane wave times the target nucleus ground-state wavefunction. Equation (5) is put entirely in terms of the #j by introducing scattering operators tj through the substitution V(Oj) Y = tjf)j .
(7)
SCATTERING
WITH
THREE-BODY
FORCES
403
Then Eq. (5) becomes a set of coupled equations for the auxiliary wavefunctions,
Insertion of Eq. (7) in Eq. (2) suggests an alternative interpretation of Eq. (2) as a partial Lippmann-Schwinger equation for Y in terms of #,j . From this equation, the exact ti of Eq. (7) must obey tj = W?j){l
+ Gotjl,
(9)
or tj -= V(Oj){l + [Et - H, - V(Oj)]-r V(Q);,
(10)
integral equations for Oj scattering in the presence of the full target nucleus. These equations for tj contain only one of the projectile-nucleon interactions from the original H; the other interactions are brought in through the coupled equations (8). We recognize the basic structure of the multiple-scattering approach. Of course, all equations are defined in the full (N + 1)-particle Hilbert space. Calculation with partial wavefunctions is usually replaced by the introduction of partial scattering operators, by the substitution Ti$ = tj#j .
(11)
By this step the overall scattering operator (12)
reduces to a linear combination, T=
; Tj. j=l
(13)
The coupled equations (8) are converted to the form Ti = ti + tiGo c Tj
(14)
j+i
by left multiplication by ti and insertion of Eq. (11). Resemblances between Eqs. (2) and (13) and the Faddeev theory are discussed in the Appendix. In the presence of three-body forces, the two-body terms in the full Hamiltonian of Eq. (1) are still dealt with by the same steps outlined above. The previous equations of definition of partial wavefunctions $j and scattering operators tj never use the full H; therefore they are not affected by the three-body terms. The additional force only need be accounted for with additional partial wavefunctions and operators,
Sijxij
xij = Y - G, W(Oij) Y, TS W(Oij) Y, sij = W(Oij){l + G,s
(15) (16) (17)
404
N.
AUSTERN
defined in analogy with the previous ones. (As before, the smaller index in any double-index quantity is placed first.) Successive insertion of Eqs. (2), (15) in the Lippmann-Schwinger equation Y = # + Go c VOj> Y + Go c j=l
W(Ojk) Y
(18)
Kk
leads to an extended set of coupled equations for the auxiliary wavefunctions ?h
xij
=
d
$-
GO
1 j+i
t&j
+
Go
=
4
+
Go
$
tk#k
+
Go
C skZXkZ k
9
SkZXkZ
klfii
(19)
(20)
,
where the restriction on the second sum in Eq. (20) stands for k Z i and I # j. Once again the coupled equations for partial wavefunctions are converted into coupled equations for partial scattering operators, by Eq. (11) and by the corresponding equation for the three-body interaction, sikd
E
sjkxjk
(21)
-
Then Ti = ti + tiGo C Tk + Togo C SkZ3 k#i Sij
=
Sij
+
SijG,
(22)
k
Tk
-I
s&o
C kZ#ii
‘kZ
3
(23)
and T =
‘$J Tj + C S’ij .
j=1
i-3
(24)
This extended system of coupled equations is not essentially more complicated than Eqs. (13), (14) for two-body forces. However, the physical interpretation of sif is less straightforward than for ti ; it does not have the same close correspondence to some simple scattering experiment. It should be noted that Sij is an exact solution for the effects of W(Oij) in the absenceof two-body interactions. This further limits the physical interpretation of Sij . 3.
CLUSTERS
The scattering operators Sij express the interaction of the projectile with pairs of nucleons. On the other hand, iteration of the operators tj for the interactions of the projectile with individual nucleons also gives interactions with pairs, triplets, and higher clusters. Therefore the physical consequences of the sij need to be discussed together with the iterations of the tj . A cluster development is indicated, which distinguishes all interactions of the projectile with single nucleons, pairs, and so forth.
SCATTERING
WITH
THREE-BODY
405
FORCES
A cluster development is a linear reordering of the terms in an iterative expansion of one of the overall scattering operators, either of T itself or of some form of the optical potential U. Terms that refer to equal numbers of different target nucleons are grouped together. Analyses of such cluster developments are given in an article by Ernst et al. [2]. These authors show an essential arbitrariness in the definition of clusters: because the relation between T and U is not linear, clusters defined for one of these quantities do not correspond term by term with clusters defined for the other. Nevertheless, our present concern is only to see the relation between sij and iterations of fj , and for this purpose it is sufficient to remain within one version of the cluster development. Because integral equations for T are already in hand in Section 2, it is easiest to examine the cluster development that emerges from them. The coupled integral equations (22), (23) are solved for Ti , Sij in powers of ti and Sij by successive substitution of lower-order expressions for Tk and S,, on the righthand sides. Clusters of successively higher order are identified in these expansions by first collecting terms that have all particle labels the same, then terms that have all particle labels selected from any two digits, and so forth. Simple expressions are obtained if we retain only the parts that contribute to matrix elements between antisymmetric nuclear wavefunctions. Then the symmetric part of T is T symm = [T(l)
in which the cluster contributions T’l’
+ Tc2’ + Tt3’ -+ ...I symm
VW
,
are found to be
= Nt, )
Tc2’ = N(N
(26) -
l){t,G,t,
+
8~12
+
+
WdzGoslz
+
N,
+
$(t,
f2)
+ tlGot2Gotl +
12) G,s,, +
~~,,Gt,Got,
G&Z&
+
+ ...
+
i~l,G(rl
Q
+
+
+~,,G,(h
tz)
+
t2) Go.712
+
. ..>.
(27)
and so forth. The pair term is a complicated nonlinear composite of the operators t13 t2 7 s12. This composite has the same structure for all target nuclei; only its coefficient depends on N. An interpretation of the pair term is obtained [2] by applying the above equations to a nucleus with N = 2. Then T symm(iV=2)
= t,, = 2t, + 2{t,G,t,
+ . ..}.
(28)
where the expression in braces from Eq. (27) is copied in abbreviated form. Solution of Eq. (28) and substitution in (27) give the pair term as T(2) = N(N
-
1){$t12 - tI},
(29)
in terms of the supposedly known two-body amplitude t,, . We can imagine fitting t,, to experimental studies of scattering from the deuteron. Equation (29) is the same
406
N. AUSTERN
formal expression derived by Ernst et al. [2]. However, the present discussion shows that the definition of t13 can be extended to include the three-body operator slZ. This same generalization of tls has also been derived by Sicilian0 and Thaler [3].
4. CONCLUSIONS Immediate practical application of the above formalism does not seem likely. However, two conclusions can be drawn. First, we observe in Eq. (27) how slZ competes with second-order effects of tj . This may help in forming estimates of the importance of three-body effects, such as are mentioned in the Introduction. Second, Eq. (29) shows how data drawn from meson-deuteron scattering, for example, can be brought into analyses of scattering of mesons by heavier nuclei. Such a phenomenological procedure combines higher-order two-body effects with explicit three-body effects. APPENDIX:
COMPARISON
WITH FADDEEV
THEORY
Multiple-scattering theory and Faddeev theory each analyze a many-body system in terms of partial wavefunctions. However, in multiple-scattering theory only the motion of the projectile is considered to be unknown; all the wavefunctions of the target nucleus are considered to be available. Therefore, formal multiple-scattering theory works with complicated Green’s functions, such as G,, , which contain the entire Hamiltonian of the target nucleus. By contrast, Faddeev theory treats the projectile and the particles of the target nucleus on the same terms. This theory only uses simple Green’s functions, which contain no more than one interaction at a time. A perfect correspondence between these two methods is not possible. Nevertheless, there are interesting parallels, and these helped to organize Section 2 of the present article. The Faddeev analysis of a three-body system can be developed from the Schrijdinger equation [4, 51 (E--)Y=(V,,+~s/,,+V,,)Y. (Al) Component equations
wavefunctions are defined in terms of Y and the three interactions by the (E - K) $0) = V,,Y,
and cyclic permutations. boundary condition
Addition
WV
of the three equations (A2) restores (Al), with the
y = #(I) + pz, + pm
(A3)
Further development of the theory follows by substituting (A3) into Eqs. (A2) and rearranging for more convenient calculation. The equations so obtained can be recast as integral equations.
SCATTERING
407
WITH THREE-BODY FORCES
The above Faddeev analysis defines its component wavefunctions by combining the interaction terms of the original H, one at a time, with the full wavefunction Y. The equations of definition can also be cast in integral form. For example, Eq. (A2) becomes lp”’ = $3) + (E - K)-1 V12Y, (A4) where (E - K) $3) = 0. (A5) This kind of association of component wavefunctions with individual interaction terms is used in the main text in a slightly altered fashion, in Eqs. (2) and (15). The latter equations define auxiliary wavefunctions that are dzfirences between the full !P and the scattered waves generated by individual interactions. That modification gives more convenient boundary conditions than Eq. (A4). The full Y of multiple-scattering theory can be expressed in terms of the auxiliary wavefunctions defined by Eqs. (2) and (15), in analogy to Eq. (A3) of the Faddeev theory. It is only necessary to add up the set of N(N -+- 1)/2 coupled equations derived by successive insertion of Eqs. (2), (15) in the Lippmann-Schwinger equation of Eq. (18). This procedure leads to y
=
] j=l c h
+
c
i
Xjk
a more complicated linear combination is given in [5, Eq. (3.17)].
-
bjita(zv
+
1) -
I}-1
,
646)
than in the Faddeev theory. A similar result
ACKNOWLEDGMENTS This work grew out of discussions with Dr. R. M. Thaler. I am grateful to Dr. G. A. Miller for reading the manuscript and for extensive discussions about Ref. [2]. I also acknowledge the kind hospitality of Dr. Louis Rosen and the Los Alamos Meson Physics Facility, where portions of this work were done. Conversations with Dr. Mikkel Johnson were helpful.
REFERENCES 1. L. L. FOLDY, Phys. Rev. 67 (1945), 107; K. M. WATSON, Phys. Rev. 89 (1953), 575; K. M. WATSON AND M. L. GOLDBERGER, “Collision Theory,” Wiley, New York, 1964; L. S. RODBERG AND R. M. THALER, “Introduction to the Quantum Theory of Scattering,” Academic Press, New York, 1967. 2. D. J. ERNST, J. T. LONDERGAN, G. A. MILLER, AND R. M. THALER, preprint, 1976. 3. E. R. SICILIANO AND R. M. THALER, preprint, 1977. 4. R. D. Ahlano, P&X Rev. 158 (1967), 1414, Eq. (A12). 5. Y. HAHN AND K. M. WATSON, Phys. Rev. A 5 (1972), 1718.
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