Summary
The most striking characteristic of a recently proposed class of antisymmetrized optical potentials for elastic nucleus-nucleus scattering is their unconventional dependence on the choice of underlying connected-kernel scattering integral equations. The relationship of these two-cluster effective interactions to the more conventional, essentially formalism-independent, optical potential operator is established in detail. The interrelated attributes of formalism independence, Hermitian analyticity, and reality with respect to elastic unitarity are clarified. A concise wave function version of multiparticle scattering is developed as complementary to a transition operator approach and is used to show that a wave function starting point does not identity a preferred class of antisymmetrized effective interactions. The differences in the calculated elastic transition amplitudes that do arise between conventional optical potentials and those of the formalism-dependent variety when the same approximations are made to the auxiliary dynamical equations are identified.
Riassunto
La caratteristica piú saliente di una classe recentemente proposta di potenziali ottici antisimmetrizzati per lo scattering elastico nucleo-nucleo è la loro dipendenza non convenzionale dalla scelta sottostante delle equazioni integrali di scattering a nucleo connesso. Si stabilisce in dettaglio la relazione di queste interazioni efficaci a due cluster con l’operatore di potenziale ottico piú convenzionale, essenzialmente indipendente dal formalismo. Si chiarificano gli attributi correlati di indipendenza dal formalismo, analiricità hermitiana e realtà rispetto all’unitarietà elastica. Si sviluppa una versione concisa della funzione d’onda dello scattering a molte particelle come complementare ad un approccio all’operatore di transizione e si usa per mostrare che partendo dalla funzione d’onda non si identifica una classe preferenziale di interazioni efficaci antisimmetrizzate. Si identificano le differenze nelle ampiezze calcolate di transizione elastica che sorgono tra i potenziali ottici convenzionali e quelli della varietà dipendente dal formalismo quando le equazioni dinamiche ausiliarie si approssimano nello stesso modo.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
See ref. (2,6) for detailed calculations and a complete set of references to related work on the Dirac approach.
L. G. Arnold, B. C. Clark, R. L. Mercer andP. Schwandt:Phys. Rev. C,23, 1949 (1981).
B. C. Clark, S. Hama, R. L. Mercer, L. Ray andB. D. Serot:Phys. Rev. Lett.,50, 1644 (1983).
B. C. Clark, S. Hama, R. L. Mercer, L. Ray, G. Hoffman andB. D. Serot:Phys. Rev. C,28, 1421 (1983).
J. R. Shepard, J. A. McNeil andS. J. Wallace:Phys. Rev. Lett.,50, 1443 (1983).
B. C. Clark: talk atThe Conference on the Intersections Between Particle and Nuclear Physics (Steamboat Springs, Colo., 1984).
The discussion in ref. (8) is particularly useful in its delineation of some of the theoretical uncertainties. Outstanding among these is the apparently prominant role of virtualN - N states that seem to enter in with the Dirac modeling. A convincing field-theoretical explanation of this is still awaited. Other uncertain aspects include several questions more directly relevant to the present work, especially the definition of the OP, and we refer to these points when appropriate.
L. S. Celenza andC. M. Shakin:Phys. Rev. C,29, 1784 (1984).
We should keep in mind that the nuclear dynamics involved in nucleon-nucleon scattering,e.g., may not have a nonrelativistic limit in the usual sense.
For example, ref. (11) is a purely nonrelativistic investigation which compares approximation schemes using energy-dependent and energy-independent OPs neglecting, however, possible identity effects and therefore all of the nontrivial unitarity problems that enter into the realistic nuclear scattering situation. We ignore the energy-independent approach in the present work.
M. S. Hussein andE. J. Moniz:Phys. Rev. C,29, 2054 (1984).
S. K. Adhikari, R. Kozack andF. S. Levin:Phys. Rev. C,29, 1629 (1984).
S. K. Adhikari, R. Kozack andF. S. Levin:Phys. Lett. B,136, 5 (1984).
Brief outlines of some aspects of our results have been presented in the course of presentations of reviews of multiparticle scattering theory (ref. (15,16)).
K. L. Kowalski: inFew-Body Systems and Nuclear Forces, edited byH. Zingl, M. Haftel andH. Zankel,Lecture Notes in Physics, Vol.2 (Springer, Berlin, 1978), p. 393.
K. L. Kowalski: inFew Body Problems in Physics, edited byB. Zeitnitz, Vol. I (North-Holland, Amsterdam, 1984), p. 465;Nucl. Phys. A,416, 465c (1984).
Reference (18) is an excellent source for the discussion of the rich variety of meanings of the OP concept in nuclear scattering. References (19,20) also contain descriptions of the state of the art a decade earlier.
Microscopic Optical Potentials, edited byH. V. von Geramb,Lecture Notes in Physics,89 (Springer, Berlin, 1979).
A. L. Fetter andK. M. Watson: inAdvances in Theoretical Physics, edited byK. A. Brueckner (Academic Press, New York, N. Y., 1965), p. 113.
R. Lipperheide:Nucl. Phys.,89, 97 (1966).
K. L. Kowalski:Phys. Rev. C,23, 597 (1981).
The antisymmetrization of theΓ effective interactions is carried out in ref. (21).
Although antisymmetrization is stressed in ref. (12,13) the implications of particle identity are irrelevant to the validity of the major proposals in these papers as well as to most of the points we investigate. Section4 contains an outline of the effects of identity.
It is stated in ref. (12) that a source of nonuniqueness in the OP is in the choice of an underlying connected-kernel formalism. The detailed structure of the OP need not depend on the structure of the many-particle theory one uses and indeed does not in the formalism of ref. (12) (cf. also ref. (26)). In general, a connected-kernel formalism provides, as with the transition operator, a means forcalculating an operator not a definition of it.
Thus, if the matrix elements of such an OP operator happen to besymmetric in a particular representation, they will also bereal. Typically, reality and symmetry are representation-dependent properties and refer to matrix elements. In accord with earlier usage (cf. ref. (26), footnote (22)), AKL mean by the reality of an OP-typeoperator its being free of elastic unitarity cuts.
R. Goldflam andK. L. Kowalski:Phys. Rev. Lett.,44, 1044 (1980);Phys. Rev. C,22, 949 (1980).
K. L. Kowalski:Ann. Phys. (N. Y.),120, 328 (1979).
A. Picklesimer andK. L. Kowalski:Phys. Lett. B,95, 1 (1980).
K. L. Kowalski andA. Picklesimer:Phys. Rev. Lett.,46, 228 (1981);Nucl. Phys. A,369, 336 (1981).
A. Picklesimer andR. M. Thaler:Phys. Rev. C,23, 42 (1981).
A. Picklesimer:Phys. Rev. C,24, 1400 (1981).
K. L. Kowalski:Phys. Rev. C,24, 1915 (1981).
K. L. Kowalski andA. Picklesimer:Ann. Phys. (N. Y.),139, 215 (1982).
K. L. Kowalski:Phys. Rev. C,25, 700 (1982).
This was not clearly understood until the global operator structure of the formalism was investigated. See ref. (28,29,31,33).
Here we are concerned solely with what are usually referred to asmicroscopic (orgeneralized) OPs as contrasted tophenomenological OPs. The simple structure that often appears in the latter type does not necessarily suggest a close relationship to any reasonably defined microscopic OP. The test of a microscopic OP is in reproducing the data and from this standpoint phenomenological OPs represent information of uncertain relevance. The significance of the energy averaged counterpart of a microscopic OP is, therefore, not very clear. The remarks ofP. E. Hodgson:Microscopic Optical Potential, edited byH. V. von Geramb,Lecture Notes in Physics, Vol.89 (Springer, Berlin, 1979), p. 459, are especially pertinent.
H. Feshbach:Ann. Phys. (N. Y.),5, 357 (1958).
The work of ref. (38) is confined to the case in which one cluster contains only a single particle. The generalization to arbitrary β is obvious.
References (21,27) are the first proposals of a framework for the systematic calculation of the OP (2.4).
E. O. Alt, P. Grassberger andW. Sandhas:Nucl. Phys. B,2, 167 (1967);P. Grassberger andW. Sandhas:Nucl. Phys. B,2, 181 (1967).
Gy. Bencze andC. Chandler:Phys. Rev. C,25, 136 (1982).
Such a generalization is often thought to reside in the paper byH. Feshbach:Ann. Phys. (N. Y.),19, 287 (1962) which investigates elastic nucleon-nucleon scattering via a wave function formalism with the full inclusion of the Pauli principle. This work does not contain a definition or a realization of an OP. A nondynamical transformation linking the AGS-defined OP to the Feshbach effective Hamiltonian is established in ref. (33) and reveals the common occurrence of the Feshbach resonances in both forms.
See ref. (8) and references cited therein.
The field-theoretical treatment of bound-state scattering problems byK. Huang andH. A. Weldon:Phys. Rev. C,11, 257 (1975), suggests that irreducibility classifications with respect to intermediate bound-state are likely to be ambiguous.
K. L. Kowalski:Lett. Nuovo Cimento,22, 531 (1978).
Gy. Bencze andP. C. Tandy:Phys. Rev. C,16, 594 (1977).
M. L’Huillier, E. F. Redish andP. C. Tandy:J. Math. Phys. (N. Y.),19, 1276 (1978).
F. S. Levin:Ann. Phys. (N. Y.),130, 139 (1980).
This derivation is not carried out in ref. (12) or in ref. (13). Had it been, it is possible that the case for the wave function point of view would not have been made so strongly by AKL.
The qualitative arguments of ref. (12,13) to the contrary are not convincing in this regard.
Gy. Bencze andE. F. Redish:Nucl. Phys. A,238, 240 (1975);J. Math. Phys. (N. Y.),19, 1909 (1978).
R. Goldflam andK. L. Kowalski:Phys. Rev. C,22, 2341 (1980).
K. L. Kowalski andA. Picklesimer:Phys. Rev. C,26, 1835 (1982).
Appendix B of ref. (34) contains a concise review of these techniques.
OnlyK( ^ β ) is considered in ref. (21).
The treatment of ref. (12,13) can be streamlined considerably by applying the standard (cf. ref. (59)) ansitymmetrization techniques to the integral equation forg t Ĝ -1.
D. J. Kouri andF. S. Levin:Phys. Lett. B,50, 421 (1974);Nucl. Phys. A,253, 395 (1975).
W. Tobocman:Phys. Rev. C,9, 2466 (1974).
AKL seem to have misinterpreted this procedure in that they appear to imply that antisymmetrized disconnected-kernel equations play a role in the treatment of PPAs in ref. (21). In fact, no attempt is made to construct antisymmetrized dynamical equations for Γ or Γt in this instance, nor is there any need to do this. Note, in particular, that not only does the antisymmetrized combination\(\Gamma _{\alpha ,\beta }^{KLT} \left[ {\hat \alpha } \right]\) which is defined by eq. (3.23) of ref. (21) not satisfy a connected kernel equation, it was not proposed that it satisfyany integral equation, not is any such equation ever required. The same comments apply for the method proposed in ref. (21) for constructing antisymmetrized transition operators from the PPAT-matrix equation. On the other hand, the integral equations proposed in ref. (21) for PPAs relatingT AGS a,b the KLT-type Γ and Γt operators have connected kernels, are label transforming and are antisymmetrized in the usual fashion. (65) One elementary consequence of this is the explicit appearance of the so-called nonorthogonality overlap matrix inV opt( ^ β ). It has been proven in ref. (29) that eigenvalue-unit problems associated with this matrix do not appear for correlated (and therefore all physical) ground-state wave functions (of both target and projectile). The explicit appearance of the overlap matrix is a crucial feature in exhibiting the Feshbach resonance structure inV opt( ^ β ), cf. ref. (33). The effective interactionK t( ^ β ) has been expressed in ref. (12) in terms of a formal Feshbach-type solution which may also exhibit such resonance-type signals. Whether these are formalism independent as they are inV opt( ^ β ) is questionable. Here Hermitian analyticity appears to be crucial.
Author information
Authors and Affiliations
Additional information
This work was supported, in part, by the U.S. National Science Foundation.
Rights and permissions
About this article
Cite this article
Kowalski, K.L. Formalism-dependent optical potentials. Nuov Cim A 92, 289–308 (1986). https://doi.org/10.1007/BF02724246
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02724246