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The two nucleon interaction and the Cohen approximation
Volume 44B, number 4
PHYSICS LETTERS
THE TWO NUCLEON INTERACTION
6 August 1973
AND THE COHEN APPROXIMATION
C.J. NOBLE and K.C. RICHARDS Bartol Research Foundation of The Franklin Institute, Swarthmore, Pennsylvania 19081, USA
Received 20 May 1973 The Cohen approximation to the scalar Bethe-Salpeter equation is used to study the exchange of mesons between nucleons. Numerically, the Cohen approximation is more accurate than the Blankenbecler-Sugar approximation in all partial waves.
The effect of the strong short range repulsion in the two nucleon interaction is to build high momentum components into the two nucleon wavefunction. In a simplified model, Fortes and Jackson [1] have shown that these high momentum components lead to relativistic effects even at low scattering energies. They considered a ¢3 theory, describing the S-wave interaction kernel as a superposition of the contributions due to the exchange of scalar mesons between scalar nucleons. Using this kernel, the ladder approximation Bethe-Salpeter (BS) equation is solved exactly by matrix inversion [2]. Fortes and Jackson concluded that relativistic effects are important especially when there are strong cancellations between attractive and repulsive terms in the interaction. A similar calculation taking into account the full complications of nucleon and meson spins has been carried out only for J = 1 partial waves [3]. It therefore seems appropriate to extend these calculations even approximately to determine the range of validity of the conclusions of ref. [1]. For this purpose, we have re-examined an off-shell unitary model originally proposed by Cohen [4]. This approximation has been claimed to provide a significant improvement on the numerical results of the Blankenbecler-Sugar (BBS) approximation [5]. The Cohen (C) model has the advantage that the integration over the relative energy variable may be explicitly carried out, leaving a simple one-dimensional integral equation (in the partial wave analyzed form) which is as easily solved as the Lippmann-Schwinger equation. The C method is easily extended to the spinor-spinor case and represents the zeroth approximation to the energy analytic model [6] through
which systematic improvements may be made. To determine the errors likely in the use of such models as that of Cohen, we have carried out a numerical comparison of the C and BBS approximations with the exact ladder BS results in the scalar theory.
u|/.
. . . . --.to)
~.(b)
0.9
""
--.~o) Sl -'i-
O~
....,. (b)
0.5
o,, / } / , , ~ ~ °.: // 0.1
•
0 $
Fig. 1. P-wave phase shifts versus total c.m.s, energy, s, for various coupling strengths, h. All masses unity. Curves a, b and c correspond to BBS, C and BS respectively. The exact ladder approximation BS results, c, are taken from ref. [7]. In the case of S-wave scattering for unit nucleon and meson masses, our results generally agree with those obtained by Cohen. We do find a small but systematic deviation between the C and BS phase shifts. Results obtained for P-wave scattering with unit mas315
Volume 448, number 4
PHYSICS LETTERS
shifts with energy but differs in magnitude from the BS predictions by up to about 10%. On the other hand, the BBS phase shifts differ markedly from the BS results especially for coupling strengths close to those corresponding to the appearance of new bound states. A similar situation is noted for scattering in relative D states. Some examples of these phase shifts are presented in fig. 2. For F and G waves the C and BS results differ by less than 2% for the highest energies in the unit mass case.
(a),. #
0.40
/
//
O,55 0.30
6 August 1973
(b)./
02' 82 Tr 0.2C
O,8lCXx\
0,1.' O.IC
/
,,~"
I
X: 5.0
t/
(lo-"
H
02 5
q
$
-O.P Fig. 2. D-wave phase shifts versus total c.m.s, energy, s, for various coupling strengths, h. All masses unity. Curves a, b, c correspond to BBS, C and BS respectively. Curve c taken from ref. [71.
ses are presented in fig. 1. For low coupling constants, the differences are less dramatic but show the same trends. The C model is seen to provide a good representation of the form of the variation of the phase
0
50
I00
150
200
250 ELABso0 Mev
0 -0,2 -0.4 8o -O.6 RADIANS -0.8 -I.0 -I.2
~
~
."<-.
co )
-I.4
Fig, 3. S-wave phase shifts versus laboratory energy for scalar nucleons with mass M = 4.758 fm -1 and the exchange of a boson with mass u = 4.9 fm -1 . Coupling strength h = - 3 3 8 . 2 7 . Curves a, b, and c correspond to BBS, C and BS respectively. Curve c taken from ref. [ 1 ].
316
--~ '
.L
50
.!_
)oo
.2_~¢~
,50
' .. ' 2o~B<~.Lso
---
~'~L',-ae
' 300
(O) Me V
-0.4
Fig. 4. S-wave phase shifts versus laboratory energy for scalar nucleons with M = 4.758 fm -] . Exchange kernel is a superposition of three bosons with masses and coupling constants tq = 0.7 fm -1 , h~ = 0.54583; ~2 = 2.8 fm -] , h2 = 86 109; /a3 = 4.9 fm -I , h 3 = - 3 3 8 . 2 7 . Curves a, b and c correspond to BBS, C and BS respectively. Curve c taken from ref. [1].
To further test the conclusions reached above, we also recalculated the simulated 1SO nucleon-nucleon phase shifts obtained by Fortes and Jackson to compare with the Cohen model predictions. Using their parameters, we obtain the results for simulated vector meson exchange and for the nucleon-nucleon interaction in the 1SO partial wave shown in figs. 3 and 4 respectively. Approximately the same degree of agreement between the various methods is found for simulated one-pion exchange. In all cases, the Cohen results lie between the BS and BBS phase shifts and are significantly better than those of the BS approximation. The extension of the C approximation to scattering in the spinor-spinor BS equation results in both a reduction of the two-dimensional integral equations to one dimension and a decrease in the number of coupled amplitudes. It appears feasible to carry out calculations even for J :/: 1 channels. The results of
Volume 44B, number 4
PHYSICS LETTERS
this procedure should be of considerable value in studying relativistic and off-shell effects in the two nucleon interaction via the spinor-spinor BS equation.
References [1] M. Fortes and A.D. Jackson, Nucl. Phys. A175 (1971) 449. [2] M.J. Levine, J. Wright and J.A. Tjon, Phys. Rev. 154 (1967) 1433.
6 August 1973
[3] J.L. Gammel, M.T. Menzel and W.R. Wortman, Phys. Rev. D3 (1971) 2175; T. Murota, M.-T. Noda and F. Tanaka, Progr. Theoret. Phys. 46 (1971) 1456. [4] H. Cohen, Phys. Rev. D2 (1970) 1738. [5] R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051. [6] J.D. Smith, Phys. Rev. D6 (1972) 2189. [7] C. Schwartz and C. Zemach, Phys. Rev. 141 (1966) 1454; B.C. Mclnnis and C. Schwartz, Phys. Rev. 177 (1969) 2621; C. Schwartz (private communication).
317
PHYSICS LETTERS
THE TWO NUCLEON INTERACTION
6 August 1973
AND THE COHEN APPROXIMATION
C.J. NOBLE and K.C. RICHARDS Bartol Research Foundation of The Franklin Institute, Swarthmore, Pennsylvania 19081, USA
Received 20 May 1973 The Cohen approximation to the scalar Bethe-Salpeter equation is used to study the exchange of mesons between nucleons. Numerically, the Cohen approximation is more accurate than the Blankenbecler-Sugar approximation in all partial waves.
The effect of the strong short range repulsion in the two nucleon interaction is to build high momentum components into the two nucleon wavefunction. In a simplified model, Fortes and Jackson [1] have shown that these high momentum components lead to relativistic effects even at low scattering energies. They considered a ¢3 theory, describing the S-wave interaction kernel as a superposition of the contributions due to the exchange of scalar mesons between scalar nucleons. Using this kernel, the ladder approximation Bethe-Salpeter (BS) equation is solved exactly by matrix inversion [2]. Fortes and Jackson concluded that relativistic effects are important especially when there are strong cancellations between attractive and repulsive terms in the interaction. A similar calculation taking into account the full complications of nucleon and meson spins has been carried out only for J = 1 partial waves [3]. It therefore seems appropriate to extend these calculations even approximately to determine the range of validity of the conclusions of ref. [1]. For this purpose, we have re-examined an off-shell unitary model originally proposed by Cohen [4]. This approximation has been claimed to provide a significant improvement on the numerical results of the Blankenbecler-Sugar (BBS) approximation [5]. The Cohen (C) model has the advantage that the integration over the relative energy variable may be explicitly carried out, leaving a simple one-dimensional integral equation (in the partial wave analyzed form) which is as easily solved as the Lippmann-Schwinger equation. The C method is easily extended to the spinor-spinor case and represents the zeroth approximation to the energy analytic model [6] through
which systematic improvements may be made. To determine the errors likely in the use of such models as that of Cohen, we have carried out a numerical comparison of the C and BBS approximations with the exact ladder BS results in the scalar theory.
u|/.
. . . . --.to)
~.(b)
0.9
""
--.~o) Sl -'i-
O~
....,. (b)
0.5
o,, / } / , , ~ ~ °.: // 0.1
•
0 $
Fig. 1. P-wave phase shifts versus total c.m.s, energy, s, for various coupling strengths, h. All masses unity. Curves a, b and c correspond to BBS, C and BS respectively. The exact ladder approximation BS results, c, are taken from ref. [7]. In the case of S-wave scattering for unit nucleon and meson masses, our results generally agree with those obtained by Cohen. We do find a small but systematic deviation between the C and BS phase shifts. Results obtained for P-wave scattering with unit mas315
Volume 448, number 4
PHYSICS LETTERS
shifts with energy but differs in magnitude from the BS predictions by up to about 10%. On the other hand, the BBS phase shifts differ markedly from the BS results especially for coupling strengths close to those corresponding to the appearance of new bound states. A similar situation is noted for scattering in relative D states. Some examples of these phase shifts are presented in fig. 2. For F and G waves the C and BS results differ by less than 2% for the highest energies in the unit mass case.
(a),. #
0.40
/
//
O,55 0.30
6 August 1973
(b)./
02' 82 Tr 0.2C
O,8lCXx\
0,1.' O.IC
/
,,~"
I
X: 5.0
t/
(lo-"
H
02 5
q
$
-O.P Fig. 2. D-wave phase shifts versus total c.m.s, energy, s, for various coupling strengths, h. All masses unity. Curves a, b, c correspond to BBS, C and BS respectively. Curve c taken from ref. [71.
ses are presented in fig. 1. For low coupling constants, the differences are less dramatic but show the same trends. The C model is seen to provide a good representation of the form of the variation of the phase
0
50
I00
150
200
250 ELABso0 Mev
0 -0,2 -0.4 8o -O.6 RADIANS -0.8 -I.0 -I.2
~
~
."<-.
co )
-I.4
Fig, 3. S-wave phase shifts versus laboratory energy for scalar nucleons with mass M = 4.758 fm -1 and the exchange of a boson with mass u = 4.9 fm -1 . Coupling strength h = - 3 3 8 . 2 7 . Curves a, b, and c correspond to BBS, C and BS respectively. Curve c taken from ref. [ 1 ].
316
--~ '
.L
50
.!_
)oo
.2_~¢~
,50
' .. ' 2o~B<~.Lso
---
~'~L',-ae
' 300
(O) Me V
-0.4
Fig. 4. S-wave phase shifts versus laboratory energy for scalar nucleons with M = 4.758 fm -] . Exchange kernel is a superposition of three bosons with masses and coupling constants tq = 0.7 fm -1 , h~ = 0.54583; ~2 = 2.8 fm -] , h2 = 86 109; /a3 = 4.9 fm -I , h 3 = - 3 3 8 . 2 7 . Curves a, b and c correspond to BBS, C and BS respectively. Curve c taken from ref. [1].
To further test the conclusions reached above, we also recalculated the simulated 1SO nucleon-nucleon phase shifts obtained by Fortes and Jackson to compare with the Cohen model predictions. Using their parameters, we obtain the results for simulated vector meson exchange and for the nucleon-nucleon interaction in the 1SO partial wave shown in figs. 3 and 4 respectively. Approximately the same degree of agreement between the various methods is found for simulated one-pion exchange. In all cases, the Cohen results lie between the BS and BBS phase shifts and are significantly better than those of the BS approximation. The extension of the C approximation to scattering in the spinor-spinor BS equation results in both a reduction of the two-dimensional integral equations to one dimension and a decrease in the number of coupled amplitudes. It appears feasible to carry out calculations even for J :/: 1 channels. The results of
Volume 44B, number 4
PHYSICS LETTERS
this procedure should be of considerable value in studying relativistic and off-shell effects in the two nucleon interaction via the spinor-spinor BS equation.
References [1] M. Fortes and A.D. Jackson, Nucl. Phys. A175 (1971) 449. [2] M.J. Levine, J. Wright and J.A. Tjon, Phys. Rev. 154 (1967) 1433.
6 August 1973
[3] J.L. Gammel, M.T. Menzel and W.R. Wortman, Phys. Rev. D3 (1971) 2175; T. Murota, M.-T. Noda and F. Tanaka, Progr. Theoret. Phys. 46 (1971) 1456. [4] H. Cohen, Phys. Rev. D2 (1970) 1738. [5] R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051. [6] J.D. Smith, Phys. Rev. D6 (1972) 2189. [7] C. Schwartz and C. Zemach, Phys. Rev. 141 (1966) 1454; B.C. Mclnnis and C. Schwartz, Phys. Rev. 177 (1969) 2621; C. Schwartz (private communication).
317