Relativistic model for nucleon-nucleon scattering including two-pion-exchange

Nuclear Physics 1349 (1972) 141-158. North-Holland Publishing Company

RELATIVISTIC MODEL FOR NUCLEON-NUCLEON SCATTERING INCLUDING TWO@ION-EXCHANGE * R.D. H A R A C Z Drexel University, Philadelphia, Pennsylvania 19104 R.H. T H O M P S O N ** Drexel University, Philadelphia, Pennsylvania 19104 and University o f California, Los Alamos Scientific Laboratory, Los Alarnos, New Mexico 8 7544 Received 15 May 1972 Abstract: A relativistic model for elastic nucleon-nucleon (N-N) scattering is constructed using the Blankenbecler-Sugar method and the two-pion exchange (TPE) contribution to the K-matrix. Two different models (field theory and dispersion relation) are considered to describe TPE. The inner region or core of the interaction, specified by S, P and 3D 1 phases, is treated using the Blankenbecler-Sugar equation, while the outer region or tail of the interaction is established by adding the TPE effect. Good agreement with the experimental phase parameters is attained, and the models are in good agreement with certain of the various phase-shift analyses considered where these show marked differences. A realistic e (J=T=-0 scalar meson) mass is used in the calculation, and the short range behavior of the interaction is fixed using pion-nucleon data and a simple phenomenological model for the meson-nucleon vertex function.

1. I N T R O D U C T I O N It has been k n o w n for some time that the one boson exchange (OBE) m e c h a n i s m can a c c o u n t for the general features o f the N - N interaction. The OBE m e c h a n i s m has been e m p l o y e d in a variety o f calculations all o f which yield reasonably good results. T h e various forms o f OBE include potential [ 1 , 2 ] , dispersion relation [3, 4] and pole [5, 6] models. The potential and dispersion m o d e l s a r e similar in that they use the meson resonances and their iterations to describe the long and m e d i u m range interaction while parameterizing the short range interaction by the use o f a c u t o f f function. The pole m o d e l does n o t use a short range parameterization and does n o t calculate S waves or the mixing parameter e 1. One serious criticism to OBE is that it * Work supported in part by the US Atomic Energy Commission (Los Alamos Scientific Laboratory). ** Present address: Los Alamos Scientific Laboratory.

142

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

does not include the irreducible TPE contribution to the scattering amplitude which is known to be important [ 7 - 9 ] . This problem has been circumvented in the past by using a fictitious, light scalar (J=T=-0) meson which is supposed to approximate TPE in that it gives a medium range attraction. In fact a scalar meson, the e with the proper quantum numbers is now thought to exist but with a mass of 650 to 900 MeV [10[. Recently, there have been two model calculations which include TPE and procedure phase shifts in good agreement with the phenomenological values for l >/2. One of the calculations [11] used a scalar meson with a mass of 715 MeV while the other [12] used no scalar meson at all. These calculations show the importance of TPE in providing a medium range attraction. In the present work a model is constructed to fit the full range of experimental phase parameters, and it preserves the good features of OBE and includes TPE to provide the necessary medium range attraction. The model is relativistic and is based on the Blankenbecler-Sugar (B-S) method [13] and the modifications necessary to include spin [14]. This method was used recently to construct a model for the N - N interaction [15], but TPE was not included directly (the iteration of the one-pionexchange (OPE) matrix element was included but the irreducible TPE matrix element was not). Thus a model is discussed here which employs the B-S equation to establish the lower angular m o m e n t u m states (S, P and 3D1), and the B-S equation + TPE treated in perturbation theory for all higher angular momentum states. This is analogous to the idea of using OPE to establish the higher partial-waves in phenomenological models. The e mass is fixed at 735 MeV from a recent p i o n - p i o n phase shift analysis [ 16], although this is recognized to be only a phenomenological treatment of the S-wave p i o n - p i o n interaction. It is not possible to include the effect of TPE in a totally unambiguous way due to the various methods employed in its calculation. In the present work two models for TPE which differ in spirit and result are considered. The two models are qualitatively similar (the shape of the TPE contribution as a function of energy is similar in the two models) but quantitatively quite different (the actual magnitudes are quite different), hence it is hoped that the data will prefer one or the other. The first TPE model is based on the exact relativistic TPE evaluation of Gupta, Haracz, and Kaskas [7] * and the corresponding phase parameters of Barker and Haracz [8] ** (model I). Model I computes TPE on the basis of an exact evaluation of all the fourth order diagrams arising from the pseudoscalar pion-nucleon-interaction. Thus, model 1 contains the fourth-order radiative corrections to OPE, the effects being relatively small. The second TPE model is that of Binstock [9] (model II) which is a dispersion relation model . Model II also includes the effect of N excitation, but this effect is found to be small. *

The two-pion-exchange phase shifts are presented by Haracz and Sharma, and by Wortman [71. ** In this reference the set of TPE phase parameters above the S state are presented at 18 energies between 10 and 350 MeV. ~** Footnote see next page.

R.D. Haracz , R.H. Thompson, Nucleon-nucleon scattering

143

In order to account for the short range behavior of the N - N interaction a phenomenological vertex function for the N N-meson vertex is employed [17]. The free parameter in this vertex function has been determined using the low energy elastic pion-nucleon scattering data, and hence the vertex function introduces no new adjustable parameters. This feature of our model is quite different from other N - N models which treat the short range parameters as adjustable when fitting to the N - N phase parameters. Thus the only parameters in the model are the meson-nucleon coupling constants which are to be adjusted to obtain a best fit to the N - N phase parameters. The model is compared with experiment, and note is taken of how the model results compare with certain of the differing values in the Livermore [ 18] and Yale [19] phase shift analyses. In particular certain phase parameters (1PI, 3D 3 and 1F3 ) given by the two groups differ significantly in energy dependence beyond a lab energy (Elab) of approximately 200 MeV. Also, there is an ambiguity in the solution given for the mixing parameter e I between MAW-IX [20] and MAW-X [18] *

2. DESCRIPTION OF THE MODEL The derivation of the B-S equation used in this work is given in ref. [14]. The partial wave projection of the K-matrix (the B-S equation) in terms of the projection of the interaction kernel, WJS, is given by the following equation,

KJS(p,lf;g ,/i) = wJS(P, lf;g,/i) IJ+St oo + ~ f dkk 2 wJS(p, lf;k,l)g'(+,k,s)KJS(k,l;q, li ) I=IJ- SI 0

(1)

where c =fi = 1,

g ,(+, k , s ) = P {47r3mZ/E2(k)(E(k) - ½x/s)},

(2)

and P denotes the principal value is to be taken. The quantities p, q, and k are the c.m. momenta of a nucleon, m is the mass of the nucleon E(k) = (k 2 +m2), J is the total angular m o m e n t u m , Ii and If are the initial and final orbital angular momenta *** The result of ref. [9] used in model lI correspond to a cut-off of (3.5u) 2 in the dispersion relations for the TPE amplitudes, where u is the average pion mass. Cut-off values of (3~)2 and (4u)2 are also presented in this paper, and it is evident that the phase parameters are sensitive to these different values for the lower values of 1and the higher energies. The TPE phase parameters of ref.[8] are in better agreement with the results of ref. [9] for the cut-off (4u)2, and they are in good agreement with the results of the dispersion-theoretical model of Chemtob and Riska [34] which uses a cut-off of 50~ 2 to achieve cut-off independence. * Throughout this paper we refer to ref. [18] as MAW-X;ref. [19] as.Y IV and ref. [20] as MAW-IX.

144

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

and S is the total spin of the system. The contribution to WJS from single scalar, pseudoscalar and vector exchanges are given in ref. [15] as analytic expressions. In the present work the scalar meson is treated in the zero width approximation. This is done for the following reason. A scalar meson with a finite width has the same effect as shifting downward slightly the mass of a zero width meson [6, 11 ]. Therefore, the zero width meson used here is simply considered as an approximation to some heavier meson whose mass and width are not well known [10]. There is also a correction to the expression for the vector meson exchange contribution to WJs given in ref. [ 15]. This correction is given in the appendix of this article and arises because the B-S equation is not only off-energy-shell but also off-mass-shell. It turns out that this correction is small over the energy range of interest, but is certainly large enough to be included in the calculation. The matrix elements K JS are related to the Blatt-Biedenharn phase parameters in appendix B of ref. [21]:

KJO(k,J;k,J)

= tan

KJl(k,J+_l;k,J+l)

= (cos 2 ej tan 6j_+ 1 + sin2 ej tan 6j~ 1)/gO(k),

KJl(k;J+-I ,k, Jg 1)

= (tan 6j+ 1 - tan 6j_ 1) sin

J°/o(k), (3)

ej cos ej/nO(k),

and these are converted to the nuclear-bar phase parameters in the usual way. The density factor is given by

o(k) = 4n3km.

(4)

In the model the inner region of the interaction is identified with the S and P phase shifts, the e I coupling parameter and the 3D 1 phase shift which is coupled to the 3S 1 phase shift *. The outer region is identified with the remaining D-state phase parameters and all higher l states. The K-matrix for the inner region is determined from eq. (1) with WJs given by the OBE expressions of ref. [15] and the correction term given in the appendix. In the outer region the TPE contribution is added to the K-matrix ofeq. (1) as a perturbation. In order to do this the contribution to TPE generated by iterating OPE in e q. (1) is sub tracte d from the K-matrix of e q. (1), and the TPE whose largest contribution is from the box and crossed box diagrams is added directly to the K-matrix o f e q . (1) in the following way:

KJS(p, lf;P, li) = KJS(p, lf;P, li) + K{S1)(P,lf;P, li) - K[S)(p, lf;P, li),

(5)

where KJS(p, lf;P, li) is the final K-matrix with TPE included as a perturbation, KJ(S1)(p,lf;p, li) is the full TPE generated mainly from the box and crossed box diagrams, and K[S)(p,/f;p,/i) is the contribution to TPE obtained by iterating OPE and * The fact that TPE does not bring about agreement for the P or 3D l states is apparent from ref. 18] and more recently from the dispersion relation calculation of Ruska and Chemtob [22].

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

145

is given by the following expression, IJ+SI

KJ(S)(p, lf;P, li) =

~ f dkk 2 w[(S)(p, lf;k,l)g'(+,k,s)W[2S)(k,l;p, li), l=lJ-SI o (6)

where w~2S)(k,l;p,/i) is the OPE contribution to the interaction kernel. The full TPE matrix element is obtained from refs. [8, 9] using the following expressions

KJ(O)(p,J;pJ) J1 K(1)(p,g,p,J )

= Kj/rrp(p),

KJ(1)(p,J+l;p,J+l)

= 36J/rrp(p), = 30Jj+-l/rrp(p),

KJ(1)(p,J+l;p,J¥1)

-

(7)

½pj/rrp(p),

where K j, 36J, 30J-+l and Pj are the phase parameters in the Yale notation that are generated by TPE. The short range behavior of the N - N interaction in our model is given by a phenomenological vertex function whose only free parameter has been fixed from low energy pion-nucleon data. In the N - N channel this function has the following form, F(q 2) = (m 2 - A2)/(q 2 - A 2 ) ,

(8)

where A = 1730 MeV, m B is the mass of the exchanged boson and q is the four momentum transfer between the initial and final nucleons. The method of including F(q 2) in the calculation is the same as that given in ref. [15]. The above form for F(q 2) was suggested by Green [23] based on other considerations. The unique point here is that the mass parameter, A, has been fixed by considering the low energy pion-nucleon channel. The numerical methods used in obtaining solutions to eq. (1) are described in refs. [15,21 ]. The integral in eq. (6)is carried out numerically using the principal value subtraction method of ref. [15] and the Gauss quadrature scheme.

3. RESULTS The phase parameters corresponding to models I and II are shown in figs. 1 - 1 0 . The dashed lines correspond to model I while the solid lines correspond to model II. In cases where both models give almost identical results only the solid curves are shown. Also shown in figs. 1 - 1 0 are the energy independent phase parameters of MAW-X (the mixing parameter e 1 is taken from MAW-IX) and the Y-IV solutions for the 1P1, 1F 3 and 3D 3 phases. Generally, the model phase parameters compare quite well with the experimental phases as can be seen from the figures. The deuteron parameters are estimated * as a t = 6.24 fm (scattering length) and r t = 1.74 fm • Footnote see next page.

146

R.D. ttaracz, R.H. Thompson, Nucleon-nucleon scattering 80

I

I

I

~

I

@

IS 0 (n,p)

x

tS0 (p,p)

r

I

L

L x 400

6C

,@ 4O

i 2O F

-2~

~o

I I00

,

t

200

L 300

Eta b (MeV)

Fig. 1. The singlet phase shift 1SO. The dashed line refers to model I as described in the text, and the solid line refers to model 1I. The long-dashed line refers to model I with the coulomb corrections added as in ref. [151. The phenomenological phase parameters of ref. [ 18] with their uncertainties correspond to n-p and p-p scattering data, and they are shown as dots and crosses, respectively. The model results derive from fits to the n-p data alone. (effective range) for m o d e l I and a t = 6.04 fm and r t = 1.68 fm for m o d e l II. The experimental values [25] are given as a t = 5.425 fm and r t = 1.763 fm. The deuteron binding energy (B.E.) is e s t i m a t e d f r o m the effective range formula [26]. Model I gives B.E. = 1.5 MeV and m o d e l II gives B.E. = 1.6 MeV to be c o m p a r e d to the experimental value o f a p p r o x i m a t e l y 2.2 MeV. The d e u t e r o n parameters are hence only a p p r o x i m a t e , but this is n o t surprising in view of the fact that in the fitting procedure no data b e l o w Ela b = 50 MeV was considered. The masses o f the exchanged mesons are taken from the Rosenfeld tables [10] with the e x c e p t i o n of the e mass which is taken f r o m ref. [16] (the e mass is taken as 735 MeV which is consistant w i t h the range o f values given b y the Rosenfeld tables [10]). The vertex mass as discussed in the previous section is fixed at 1730 MeV from other considerations. The m o d e l parameters (the coupling constants o f the exchanged mesons) are given in table 1 for b o t h models 1 and II. In table 2 the q u a n t i t y KJ(Sl)(p, If;p,/i) - KJ(S)(P, If;p, li) is tabulated for various values of Ela b and a coupling strength, g2/47r = 14. These numbers are o f course necessary to reproduce our models.

* It is only possible to give a rough estimate of the low energy deuteron parameters because a small numerical error in the phase shift, 6, causes a very large error in cot 6 near 6 = 180 °.

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

20

i

I

~

I

I

I

I

147

I

X Y-J3E •

M.~W - " ~

o

-20

-40

-6o

I

I

tO0

I

I

200

~

I

300

I

I

400

Eio b ( M e V )

Fig. 2. The phase shifts 3P o and ]P1. For 1Pt, the dashed line refers to model I and the solid line to model II, while for 3p0, where b o t h m o d e l s lead to almost identical results, the solid line refers to b o t h models. The p h e n o m e n o l o g i c a l phase parameter solutions MAW-X of ref. [ 18] and Y-IV of ref. [ 19] are s h o w n as d o t s and crosses, respectively, with their uncertainties.

Table 1 The m e s o n s used in the calculation are s h o w n along with their masses, spin and parity, isospin and effective coupling constants. Meson

Mass (MeV)

jP

T

Model I ge2ff/47r

138.7

0-

1

548.5

0

0

735

0+

0

6

960

0+

1

w

782.8

1-

0

p

763.0

1-

1

A = 1730 MeV

fig

14.45

fig

0.857

21.2

19.3

2.63

0.172

2 geff/4n 15.2

0.857

20.4

Model II

2.73 0.0 6.55

17.8 0.296

0.0 6.62

148

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering I00

I

I

I

r

80

I

X

3S t

•

3D I

60

40

20

-i

t

\° -20

_406

I

I

I00

l

I

200

i

I

300

i

I

400

Ela b (MeV)

Fig. 3. The triplet coupled phase shifts 3S 1 and 3D 1 and their coupling parameter e I. The notation for Models I and lI is as in fig. 2. The circles refer to the phenomenological solution MAW-IX for el of ref. [20], while the phenomenological points for 3S 1 and 3D 1 refer to MAW-X.

4. MESON P A R A M E T E R S . Tabulated in table 1 are the quantities g~ff/4rr =

g2/4rr (A2/(A 2 -

m2)) 2 ,

where m B is the mass of the exchanged boson. It is in fact this q u a n t i t y , g2ff/4~r, which is to be identified as the effective coupling strength since it is the residue of the OBE pole w h e n t (the four m o m e n t u m transfer) is equal to m 2. Let us n o w briefly compare the effective coupling strengths o b t a i n e d from models I and II with other estimates and measurements of these coupling strengths. In the following the model coupling strengths are given in parentheses (models I and II, respectively).

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

0

~

I

J

i

I

I

I

t

~

I

i

I

I

149

I

-tO~l os

-20

-30

-4.,'-,,

~

~o

I00

200

300

I

400

Eta b (MeV) Fig. 4. The triplet phase shift 3P l for models I and II with MAW-X. 4(3

i

20

[

i

I

r

I

i

I

I

I

i

I

~2,,~ ''~

2 r

r

I

I00

r

200

300

I

'400

Ela b (MeV) Fig. 5. The triplet phase shift 3D 2 for models I and II with MAW-X.

7r meson: (14.45 and 15.2}. The values given by Cutkowsky * from an N - N phase shift analysis are 14.1 (+ 1.4, - 0.8) and 13.9 (+ 1.6, - 1 . 7 ) . The smaller value given by model I is probably due to the fact that it has larger TPE corrections than model II. r~ meson: (0.85 7 and 0.85 7). There is no experimental evidence on the value of the 77 coupling strength, but SU(3) predictions [28] depending on the D / F ratio, place it at about one. e meson: (21.2 and 19.3). Again there is no direct experimental evidence as to the value of the e coupling strength. If one identifies the scalar particle in the o-model of Gell-Mann and Levy [29, 30] as the e, the coupling strength is given by the relation (gA/gv)g~ = gE where gA/gv is the ratio of the weak axial vector to vector coupling strength and g~r is the pion-nucleon coupling constant. This leads to a value ofg~/4~r -" 20, depending on the value used for gA/gV" Other values given for this coupling • Ref. [27]. A thorough analysis of the determination o f g ~ for p-p and n-p scattering is given in ref. [ 24 ].

R.D. Haracz, R.ft. Thompson, N u c l e o n - n u c l e o n scattering

150

Table 2 The quantity (K~(p,l';p,l)~l, - ~zK~'~"(P'I';p'I))is tabulated for a range ofEla b from 50 to 350 MeV for the singlet and-triplet states. ~l~e values for model I are given in the table, and where models I and lI differ significantly the values for model II are given in parentheses.

Ela b (MeV)

J

Singlet

Triplet

l'=l=J 50

110

170

230

290

350

l'=l=J+ 1

l'=l=J-1

l'=J+l,l=J - 1

2

0.00054 (0.00043)

0.00032 (-0.00034)

0.00003

0.0004

3

0.00003

0.00004

0.00000

4

0.00001

0.00001

0.00001

0.00006

0.00000

5

0.00000

0.00000

0.00000

0.00001

0.00000

2

0,0030 (0.0022)

0.0015 (-0.0022)

0.00025

3

0.00033

0.00040

0.00001

0.00548 (0.00341)

4

0.00009

0.00008

0.00001

0.00053

0.00000

5

0.00000

0.00001

0.00000

0.00017

-0.00001

2

0.0068 (0.00477)

0.0032 (0.00510)

0.00061

3

0.00098

0.00115

0.00001

0.0126 (0.00727)

-0.00097

4

0.00035

0.00025

0.00005

0.00152

-0.00001

5

0.00006

0.00007

0.00000

0.00058

-0.00003

2

0.00493 (-0.00870) 0.00217

0.00097

3

0.01140 (0.0077) 0.00187

0.00000

0.02136 (0.01161)

-0.00034

4

0.00076

0.00051

0.00012

0.00296

-0.00004

5

0.00015

0.00017

0.00010

0.00129

0.00004

2

0.00674 G0.01254) 0.00341

0.00121

3

0.01639 (0.01085) 0.00298

4

0.00133

5

0.00029

2

0.00846 (-0.01629) 0.00477

0.00125

3

0.02154 (0.01390) 0.00418

-0.00025

0.04118 (O.02055)

4

0.00201

0.00124

0.00035

0.00676

-0.00016

5

0.00049

0.00058

0.00032

0.00340

-0.00010

0.00097 (0.00068)

-0.00006

0.00034 -0.00008

-0.00089

-0.00165

-0.00258

-0.00007

0.03102 (0.01624)

-0.00216

0.00084

0.00024

0.00473

0.00002

0.00035

0.00002

0.00225

-0.00010 -0.00363 0.00018

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering I

~

I

r

I

;

151

I

iD

o~

I

I

~

ioo

I

I

20o

I

I

I

300

400

Etab (MeV)

Fig. 6. The singlet phase shift ID2 for models I and II with MAW-X. 30

i

1

I

I

~

I

r

~P2 20

,o

o

-112

I I00

i

I 200

i

I 300

I

I 400

Elob (MeV)

Fig. 7. The triplet coupled phase shifts 3P2 and 3F2 and their coupling parameter e 2 for models 1 and II with MAW-X. strength [30] are considerably larger than 20 and would be incompatible with the present model. 6 meson: (2.63 and 2. 73). No external estimates of the 6 coupling strength are known to the authors. In fact there is some uncertainty in the quantum numbers (spin and parity) of this meson [10]. However, because of its weak coupling and large mass (960 MeV) this meson plays a small role in our model affecting primarily S-waves. co meson: (20.4 and 1 7.8). The coupling strengths given by models I and II for the

15 2

R.D. Haracz, R.H. Thompson, Nucleon- nucleon scattering 20

[

I

i

I

I

r

i

303 y - T ~

16

0

3D3 MAW-T

•

3G3

X

4E3

// /,,,

/i //

B

/ /"

i//

12

la

/

i/....'

~D3/

i/I/lll/~ f.iI

4

°I

-4

_ ~z °

I

I

I00

t

I

200

]

I

300

L

I

400

Eiob (MeV) 3 3 Fig. 8. The triplet coupled phase shifts D 3 and G 3 and their coupling parameter e 3 for models I and II. For 3D3, the phenomenological solutions corresponding to Y-IV and MAW-X are shown as crossed circles and circles, respectively. The MAW-X solutions are shown as crosses for c 3 and dots for 3G 3.

co meson are considerably larger than those given by OBE pole + TPE models [11,12] The reason for this is the following. In the pole + TPE models TPE contributions have been included for P-waves where they are very large. In particular the TPE contribution to the 3P 0 phase shift is very large and negative. The contribution to the 3P0 from the co meson is also large and negative depending on its coupling strength. Therefore, since the pion coupling constant is fixed in the neighborhood of 1 3 - 1 5 , it is necessary to have a small co coupling constant in order to fit the data. In the present model full TPE corrections are not made to the 3P0, hence a much larger co coupling constant is necessary in order to fit the data. If the irreducible TPE diagram (crossed box) were correctly iterated in the kernel of the B-S equation, the co coupling strength would undoubtedly be reduced. A recent experiment [31 ] places g2/4rr = 3.5 +- 1.2. This contrasts with electroproduction calculations [32] which

R.D. Haracz, R.H, Thompson, Nucleon-nucleon scattering r

I

I

I

I

I

I

r

I

f

t

153

[

0

-2

~

-

_~

I

I

100

I

200

300

I

I

400

Etob (MeV)

Fig. 9. The phase shifts 1G 4 and 3F 3 for models I and II with MAW-X. 6

I

I

i

p

r

I

Y-I~

fF3 I

t

I --

3F/ 4

i/l

_

I

I

100

i

T

P

200

~

I 300

i

aH 4

:

I ±

400

Eio b (MeV} Fig. 10. The triplet coupled phase shifts 3F 4 and 3H 4 and their coupling parameter e4, and the singlet phase shift IF 3. The solutions Y-IV and MAW-X are shown for 1F 3 as crossed circles and dots, respectively, while MAX-X is shown for the triplet phase parameters.

154

R.D. Haracz, R.H. Thompson, Nucleon nucleon scattering

predict a value ofg2/47r ~ 40. fw/g~o = 0 is consistent with experiment. p meson." (0.172 and 0.296, fig = 6.55 and 6.62). Experimental values [33] of o f g 2 / 4 ~ range from 0.3 to 0.95 and fo/gp from 3.70 to 6.82 *

5. DISCUSSION Models l and lI were fit to the energy dependent phase parameters of MAW-X. The fitting was carried out by adjusting the model coupling constants to minimize the function X2 which is defined by the equation

all phase parameters

i i i 2 {((5model -- (5phen )/A(5 phen}

(9)

i i where (5model are the model phase parameters, (sphen are the phenomenological energy dependent phase parameters of MAW-X and A(5~is the error assigned to the phase parameter 6~hen. The energy dependent phases used in evaluating the X2 in eq. (9) are at the energies Elab = 50, 170 and 290 MeV. It should be noted that the phase shift IS 0 is fit to n-p data alone. It happens that almost all of the contribution to the X2 comes from the p-p phase parameters. This is due to the fact that the error bars assigned to the p-p phases are very small compared to those assigned to the n-p phases. The result is that the model attempts to fit the p-p phases as precisely as possible and virtually ignores the n-p phases (the lower n-p angular momentum states have small enough error bars so that they do contribute significantly to the X2, but their contribution is still much less than the p-p phases; however, the higher n-p angular momentum states have such large error bars that they make no significant contribution to the X2). The result of this is that the model parameters are fixed by the p-p phases and the n-p phases with error bars which are small. The n-p phases with large error bars are hence predicted by the model since they have no influence in the fitting procedure. This freedom can be used to advantage in discussing the model by noting how the model compares with certain of the differing solutions in the phase shift analyses. Generally, the phase shift analyses of Y-IV and MAW-X in good agreement. However, certain n-p phases given as solutions by the two groups do differ significantly for Elab ~ 200 MeV. The differences exist in the 1P1, IF 3 and 3D 3 phases. There is also an ambiguity in the solutions given for the mixing parameter e 1 by the MAW-IX and MAW-X analyses. The solution given for e 1 by MAW-X is considerably larger than that given by MAX-IX. The Yale and Livermore solutions for the 1P 1 phase shift are shown in fig. 2. It is seen that the Y-IV solution is very different from the MAW-X solution for * The definition of ref. [32] forfo/g p differs from the definition in the appendix by a factor of 2. Hence for the purpose of comparison their values have been multiplied by a factor of 2 in the text of this article.

R,D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

155

E~b ~> 200 MeV. Also fig. 2 shows that both models I and II are in better agreement with the MAW-X solution. In order for the model solution to give results in agreement with Y-IV it is necessary for the N - N interaction to be much more repulsive for medium energies than either models I or II predict. The 3D 3 solutions o f Y-IV and MAW-X are shown in fig. 8. Again for energies greater than 200 MeV the respective solutions are quite different with the MAW-X solution actually going negative for higher energies. Here models I and II are in better agreement with the Y-IV solution. It turns out the the model 3D 3 is dominated by OPE + TPE with the other mesons making a comparatively small contribution. This phase shift is thus essentially fixed by the pion-nucleon coupling constant which is a relatively well known quantity. Hence neither models 1 or II are capable of giving a behavior similar to that of MAW-X. The IF 3 phases of Y-IV and MAW-X are shown in fig. 10. The error bars are very large for this phase shift, but the models seem to be in better agreement with the solution of MAW-X having the same general shape as a function of E.tab. Finally the mixing parameter e 1 is considered. The solution given by MAW-IX is shown in fig. 3. Both models l and II are in better agreement with this solution as opposed to that given by MAW-X which is also positive but considerably larger. The model solutions both fall within the error bars of the MAW-IX solution, and they are consistent with the Y-IV solution for Pl = sin 2e 1. Another point of some interest is that the 3D 2 phase shift is given quite well over the energy range considered by both models I and II. This is in contrast to previous OBE models. This is attributed to the fictitious, light scalar meson which was used to provide the mid range attraction in these previous models. This attraction is necessary in order to give agreement with the 1D 2 phase which is obtained from p-p data and hence has very small error bars. However, this attraction also causes the 3D 3 to drastically overshoot the phenomenological solution. This situation was tolerated due to the relatively large error bars assigned to the 3D 2 . When the scalar is given a heavier and more physical mass and TPE is included as in models I and If, it is possible to obtain a good representation of both phases. Finally, it does not seem possible based on the present work to determine which TPE model is preferable, i.e. both models I and II fit the data equally well with very similar values for the coupling constants.

6. CONCLUSIONS A model for the N - N interaction is constructed which is relativistic and includes the effects of TPE for the outer region of the interaction. By including TPE along with the iterations of OBE the need for a fictitious light scalar meson is eliminated. Thus the model has an improved theoretical foundation, and also it is found that the elimination of the light scalar brings certain phases (1D2, 3D 2 and e l ) w h i c h are not given well by pure OBE models more in line with experiment. Also the model has two fewer parameters than other OBE models since the mass is not treated as a free para-

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

156

meter, and the vertex or cutoff mass is fixed using a model for the N - N meson vertex and low energy pion-nucleon data. The model has good overall agreement with the full range of the experimental phases and gives reasonable deuteron parameters. The model provides a useful representation for the N - N interaction which can be used in other calculations such as nuclear matter, N - N bremsstrahlung and even nuclear structure calculations. Also since the model is relativistic, it should prove useful for extrapolating into the higher energy region. The authors wish to thank Mr. R. Reist for his patient and expert help with their numerical calculations. One of the authors (R. H. T.) would also like to thank Dr. Leon Heller for many profitable discussions.

APPENDIX. OFF-MASS-SHELL CORRECTION FOR THE VECTOR MESON The expression for the contribution to the kernel of eq. (1) due to the exchange of a single vector meson given in ref. [15]. There is a correction to this expression due to the off-mass-shell nature of the B-S equation which is given in this appendix. The interaction Hamiltonian is given by the expression gI

gV~N')' ~N~bv +0Cv/4m)t~NO v ~N fur,

(A.la)

whe re

Our = (TuT v - "yvTu)/2i,

(A. 1b)

fzv _

(A.lc)

ax~

ax~

;

~N is the nucleon field, ~b~ is the vector field, gv is the Dirac coupling constant, f v is the Pauli coupling constant, rn is the nucleon mass and the 3 , are the Dirac matrices with the convention T? = ")'07u3'0. Consider the N - N vector vertex generated by the Hamiltonian of eq. (A.Ula). The nucleon has a final four momentum p = (p,E(k)) and initial four momentum q = (q, E(k)) where E(k) = (k 2 + m2) 1/2 is the actual c.m. energy of an incident nucleon. Hence, the B-S equation is off-mass-energy-shell. The matrix element which results from the kUkV/rn 2 term in the vector meson propagator where m v is the vector meson mass and k = (p - q) is given by the expression

g2v M = (27r)6m2

(E(q)_E(p)) 2

[(P-q) 2+m2] (~(p)~00) co(q))(~(-p)')'0(2)

co(--q)), (A.2)

where co(p) is the usual Dirac spinor as given in ref. [15]. The term given by eq. (A.2) is to be added directly to eq. (31) in ref. [15] to obtain the correct expression for the exchange of a single vector meson.

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

157

The partial wave projections of eq. (A.2) are obtained using the formalism given in ref. [15]. The results are given in the expressions

wl'O(p,J;q,J)

= e(/3 + Z ) Q j ( Z ) ,

wJ'l(p,J;q,J)

= e{/3Qj(Z) + [(J+ 1)Q j _ 1 (Z) + JQJ+I (Z)]/(2J+l )},

(A.3a)

(A.3b)

wl'l(p,J±l;q,J±l)

= e{(2J+l )2 Q j ( Z ) + 2Qj+_ 1 (Z) [~+7)J(J+l ) +/31

+ 2Qj~I(Z)(/3-~f)J(J+I)}/(2J+I)21 wJ'l(p,J±l;q,J~l)

= e [J(J+l )] 1/2(/3_•)(Qj_

(a.3c)

1 (Z) - Q j+ 1 (Z))/(2J+l)2,

(A.3d) where

e =g2v(E(q ) - E(p))2/{2(21r) 5 m 2 m ~ } ,

(A.3e)

/3 = {(E(q) + m) (E(p) + m) + (E(q) - m) (E(p) - m)}/2qp,

(A.3f)

3' = {(E(q) + m) (E(p) + m) - (E(q) - m) (E(p) - rn)}/2qp,

(A.3g)

Z = {p2 + q2 + m 2 } / 2 q p ,

(A.3h)

and Q j ( Z ) is a Legendre f u n c t i o n of the second kind of order J. Eqs. (A.3a) - ( A . 3 d ) are the correction terms which should be added to the W JS terms o f ref. [15] for the vector meson. These terms have been included in the present calculation and are small due to the factor of (E(q) - E(p)) 2 in eq. (A.3e) for the energy range considered, b u t the corrections have a noticeable effect on the phase shifts and should be included *.

REFERENCES [1] [2] [3] [4] [5] [6] [7]

T. Ueda and A.E.S. Green, Phys. Rev. 174 (1968) 1304. R.A. Bryan and B.L. Scott, Phys. Rev. 177 (1969) 1435. A. Scotti and D.Y. Wong, Phys. Rev. 138 (1965) B145. J.S. Ball, A. Scotti and D.Y, Wong, Phys. Rev. 142 (1966) 1000. S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Prog. Theor. Phys. 28 (1962) 991. R.A. Bryan and B.A. Arndt, Phys. Rev. 150 (1966) 1299. S.N. Gupta, R.D. Haracz and J. Kaskas, Phys. Rev. 138 (1965) B1500; R.D. Haracz and R.D. Sharma, Phys. Rev. 176 (1968) 2013; W.R. Wortman, Phys. Rev. 176 (1968) 1762. [8] B.M. Barker and R.D. Haracz, Phys. Rev. 186 (1969) 1624. [9] J. Binstock, Phys. Rev. D3 (1971) 1139. * After completing the present work the authors received a preprint from R.A. Bryan and A. Gersten. In this preprint the authors obtained a lower x 2 than has been previously obtained in fitting the model of ref. [15] to N-N data. However, they have used an incorrect form of the vector meson interaction. The correct form will modify their precise fit to the data significantly.

158

R.D. Haracz, R.H. Thompson, Nucleon-nucleon scattering

[10] A. Barbaro-Galtieri, S.E. Derenzo, L.R. Price, A. Rittenberg, A.H. Rosenfeld, N. BarashSchmidt, C. Bricman, M. Ross, P. Soding and C.G. Wohl, Rev. Mod. Phys. 42 (1970) 87. [ 11 ] J. Binstock and R. Bryan, Phys. Rev. D4 (1971 ) 1341. [12] B.M. Barker and R.D. Haracz, Phys. Rev. D4 (1971) 1336. [13] R. Blankenbecler and R. Sugar, Phys. Rev. 142 (1966) 1051. [14] R.H. Thompson, Phys. Rev. D1 (1970) 110. [15] A. Gersten, R.H. Thompson and A.E.S. Green, Phys. Rev. D3 (1971) 2076. [16] L.J. Gutay, Proc. of the Conf. on n - l r and K - n interactions, Argonne National Laboratory (1969). [17] Richard H. Thompson, unpublished. [18] H. MacGregor, R.A. Arndt and R.M. Wright, Phys. Rev. 182 (1969) 1714. This paper is referred to as MAW-X. [19] R.E. Seamon, K.A. Friedman, G. Breit, R.D. Haracz, J.M. Holt and A. Prakash, Phys. Rev. 165 (1968) 1579. This paper is referred to as Y-IV. [20] M.H. MacGregor, R.A. Arndt and R.M. Wright, Phys. Rev. 173 (1968) 1272. This paper is referred to as MAW-IX. [21] R.H. Thompson, A. Gersten and A.E.S. Green, Phys. Rev. D3 (1971) 2069. [22] D.O. Riska and M. Chemtob, Phys. Letters 36B (1971) 313. [23] A.E.S. Green, Phys. Rev. 73 (1948) 26; 75 (1949) 1926. [24] G. Breit, M. Tischler, S. Mukherjee and J. Lucas, Proc. Nat. Acad. Sci. (U.S.) 68 (1971) 897. [25] T.L. Houk and R. Wilson, Rev. Mod. Phys. 40 (1968) 672. [26] A. Gersten and A.E.S. Green, Phys. Rev. 176 (1968) 1199. [27] R.E. Cutkowsky and B.B. Deo, Phys. Rev. Letters 20 (1968) 1272. [28] E. Lomon and H. Feshbach, Rev. Mod. Phys. 39 (1967) 611. [29] M. Gell-Mann and M. Levy, Nuovo Cimento 16 (1960) 705. [30] F. Partovi and E.L. Lomon, Phys. Rev. DS, (1972) 1192. [31 ] P.J. Biggs, D.W. Braden, R.W. Clifft, E. Gabathuler, P. Kithcing and R.E. Rand, Phys. Rev. Letters 24 (1970) 1197. [32] P.L. Pritchett, J.D. Walecka and P.A. Zucker, Phys. Rev. 184 (1969) 1825. [33] G. Hohler (ed.), Springer Tracts in Modern Physiscs 55 (1970). [34] M. Chemtob and D.O. Riska, Phys. Letters 35B (1971) 115.