Charge dependence of one-boson exchange potentials

I

4.C

[

Nuclear Physics A153 (1970) 424--432; (~) North-HollandPublishino Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

CHARGE DEPENDENCE

OF ONE-BOSON

EXCHANGE POTENT/ALS

c. YALCIN t and C. T. YAP tt Department of Theoretical Physics, Middle East Technical University, Ankara, Turkey Received 30 April 1970 Abstract: The Fermi nuclear matrix elements MF for the AJ = 0, AT = =El E-decays of 41Ar, 4~Sc

and 52Mn are calculated theoretically in the jj-coupling shell model using in addition to the usual Coulomb potential one-boson exchange potentials with the mass splitting of the p-meson, ~p = p± _po, as a variable parameter. These results are compared with those obtained by using the Blin-Stoyle-Le Tourneux charge-dependent potential which contains parameters p and r that measure the deviation of nuclear forces from exact charge symmetry and q and s that measure the deviation of nuclear forces from exact charge independence. Comparison yields 3r --p ~ 0.1 Yowhereas 3s-- q is dependent on ~p. Using the well-attested relation of 3s-- q ~ 1~o, we obtain ~p in the range, --1½ to -- 5 MeV, which are consistent with experimental values todate. With 6p = -- 3 MeV it is found that the theoretical values of MF are consistent with the experimental ones for the decays under consideration.

1. I n t r o d u c t i o n

E x p e r i m e n t a l evidence for the charge independence o f nuclear forces is overwhelming. However, recent studies seem to indicate a few percent d e v i a t i o n f r o m exact charge independence 1 - 3 ) while d e v i a t i o n f r o m charge s y m m e t r y a p p e a r s to be a n o r d e r o f m a g n i t u d e smaller 4, s). By charge i n d e p e n d e n c e we m e a n the equality o f the pp, nn a n d p n forces in the same spin a n d o r b i t a l state whereas b y charge s y m m e t r y we m e a n the equality o f the p p a n d nn forces only. Such deviations are expected on theoretical g r o u n d s a n d are i n d i c a t e d b y experimental d a t a on nucleon-nucleon scattering lengths a n d on the energy differences o f various isobaric spin multiplets 1, 6). C h a r g e - d e p e n d e n t effects can be divided into two classes: direct a n d indirect, b o t h o f which are believed to be electromagnetic in origin 7). D i r e c t c h a r g e - d e p e n d e n t effects arise f r o m (i) the C o u l o m b force, (ii) the m a g n e t i c interaction, (iii) the finite charge a n d m a g n e t i c m o m e n t distributions o f the nucleons, (iv) the v a c u u m p o l a r i z a t i o n a n d (v) the p r o t o n - n e u t r o n mass difference. O f course, the m a i n c o n t r i b u t i o n comes f r o m the C o u l o m b interaction while the others can be neglected 7.8). Indirect c h a r g e - d e p e n d e n t effects t h a t m o d i f y the nuclear force giving rise to the specifically nuclear c h a r g e - d e p e n d e n t interaction m a i n l y arise f r o m (a) mass differences o f q u a n t a exchanged between the nucleons, (b) isobaric spin mixing o f m e s o n states o f the same spin a n d p a r i t y a n d (c) radiative corrections at the n u c l e o n - m e s o n vertices. t Work supported partly by the Turkish Scientific and Technical Research Council. tt On leave of absence from the Department of Physics, University of Singapore, Singapore 10, which is his present address. 424

BOSON EXCHANGE POTE1NWIAL

425

In this paper, we shall investigate the contributions of indirect charge-dependent effects to the Fermi nuclear matrix elements MF of the A J = O, A T = 1 fl-deeays of *lAr, ##Sc and S2Mn using one-boson exchange potentials described below.

2. One-boson exchange potentials The one-pion exchange potential represents the long-range behaviour of the nucleon-nucleon interaction. However, attempts have recently been made to represent the short-range behaviour of this interaction in terms of one-boson exchange potentials (OBEP) 9-11). The phase-shift analysis of the pp scattering data at 310 MeV performed by Stapp et al. 12) showed evidence on the existence of a strong L S force in the triplet odd state and a short range repulsion. The L S force obtained by the two-pion exchange potential was too weak compared with that indicated by experiment 13). This large L S force should be understood in terms of the exchange of a heavy vector or scalar meson instead of the exchange of two pions. Gupta 14) assumed the exchange of a neutral scalar meson, while Breit 1s) and Sakurai 16) considered the exchange of a neutral vector meson. However, these attempts were only successful in explaining certain properties of the nuclear force and did not contribute to a comprehensive understanding of the two-nucleon interaction. Nevertheless, it has been found that N-N scattering data for energies up to 300 MeV can be fairly successfully accounted for in terms of OBEP models, whose basic assumption being the exchange of virtual heavy mesons as well as pions 9-11). It is assumed that the two-nucleon potential is the sum of individual one boson exchange potentials. The exchange of various I = 0 and I = 1 heavy mesons produces a very short range interaction and in this way the short range behaviour of the two-nucleon potential can be explained theoretically. Each of the OBEP models assumes several heavy mesons as shown below: BS 9)

V = n(0-, 1 = l ) + p ( 1 - , 1 = I)+o9(1-, I = 0)+ao(0 +, I = 0) + r/(0-, I = 0 ) + a ( 0 +, I = 1),

$17-1s)

V=zr(O-,I=

1)+p(1-,I=

1)+co(1-,I=O)+o'o(O+,I=O) I = o),

SW lo)

V = Ir(O-,I = 1 ) + p ( 1 - , I = O ) + w ( 1 - , I = O)+ao(O + , I = O) + t/(0-, I = 0)+tp(1-, 1 = 0).

It is a well-established fact that there is a definite mass difference between the charged and neutral pions. Riazuddin a 9) has derived the formula for the chargedependent contribution due to this mass splitting. In OBEP models, charge-dependent contribution also arises from the exchange of p-mesons. It is a straight forward matter to calculate this charge dependence for the various OBEP models and we give below

426

c . YAL~IN AND C. T. YAP

vsas, and vSW,, the charge-dependent potentials of Sawada and of Scotti and Wong respectively, arising from the mass splitting of the isovector mesons. V~,as~ = ~z ~z

5p 0.21 +

e -p'

pr /

+ ' 1 " ' 2 {0.11 ( 1 - ~OOr) e-~°'+6p (3.79+ 1"26t e - " } l , (2.1) pr /

Vms,

p 4.23- 4.44 = ¢z z~

pr /

+-i"-2 {0.10 (1- 2-~r) e-'°'+@ (6.35-1"851 e-°'}], (2.2) pe / where ~Sp-- p + - p°. Recently, charge symmetry breaking contributions to the OBEP models have been calculated by Downs and Nogami 2o-2~). They have taken into account the mixing of pseudoscalar mesons with I = 0 and I = 1 as well as the mixing of vector mesons. For pseudoscalar mesons, they obtain (physical particles are designated by superscript N ) ~o = nO+c,rl, gl = t l - c ' n °, (2.3) with c' ~ 0.01. This corresponds to the following ~/- n ° mixing potential

V(~ nO) = (TI -[-'['z2)'l • if2

{0"00116n°e-"°' nOr

-0.01920q

e-"]

17r /

.

(2.4)

Also, Stevens 22) has shown that the ~ - n ° mixing corresponds in the static limit to the following charge-dependent potential

vS(,l:) = - ( 4 + , ,

(gO) {R, V ( : ) + R : V(~)},

where

v(:)=

\2M/

(. ; :

- -r,

(2.5)

r

Using (n°/2M) 2 g2/4n = 0.08 and ~ = (,,/-3]5)g, eq. (2.5) reduces to VS(r/n°) = - (¢,x+z~) (0.000139n ° e-'a" -0.00916t/ ~ } .

(2.7)

n°r

However, for vector mesons Downs and Nogami obtain ~o = p°

c,,m+d,,~p '

(5 = a " o ) - b " ~ o + e " p

°,

~p = a " q ) + b " o ) + h " p °,

(2.8a)

BOSON EXCHANGE POTENTIAL

427

with a" ~ 0.77,

b" ~ 0.64,

c" = 0.05,

d" ~ 0.04,

e" ~ 0.06,

h" ~ 0.005.

(2.8b)

The potentials corresponding to the above mixings are given below -- p r

V~'(r) = (%t +~2){0.482+0.148aa. a2}p e

pr

V~(r) = -(z~ + z2){0.457 + 0.142al • a2}to e - ~ " , (.or

V~(r) = _(%x +z~){0.0239+0.0082a~ • a2}q~

e -~r

¢pr

,

with O,o = - 5 ,

(2.9)

e --pr

V+(r) = - (z~ + z2){0.405 + 0.124at • a2}P - - , pr V+(r) = (z~ +z~2){O.379+O.117al • ~2}~ e-O', for

V+(r) = (z~+z~){0.0507+0.0174a 1 • 0.2)~0

e-¢'

~0r

,

with 9~, = 6.

(2.10)

The intrinsic breakdown of charge independence in the two-nucleon interaction is also expected because of the electromagnetic radiative corrections at the nucleonmeson vertices. The second order radiative corrections to the pion-nucleon and p meson-nucleon vertices have been calculated by Stevens 22) and putting the numerical values, we obtain Vra d =

~(1 +'c~)(1 +'C~) (0.000186, 0 e-~°" - +0.000369r/ e - , ' ] . .°r r/r y 3. F e r m i

nuclear matrix

(2.11)

elements

The Fermi nuclear matrix elements My for the AJ = O, AT = __+1 beta decays of 41Ar ~ 41K, 44Sc --, *4Ca and 52Mn ~ 52Cr are calculated theoretically in the jj-coupling shell model using the techniques described in ref. 2) but emplo)ing the charge-dependent potentials of the preceding section. For computational reasons, we shall re-write these potentials in the following forms a 2 B - } e -p'2,

(3.1)

Vm~xi.g = VOl,°)+V+(r)+V+(r)+V+(r) - (z~ +z2){A + +al " a2B+Je -at2,

(3.2)

vgix,.,

=

-

(%1 +%){A 2 - +tra

.

sv-="':'" = V'(t/rc°)+ Vy(r)+ Vf,(r)+ W(r) =- (zl +z=Z){A~" + a l " a2n~}e -arz, (3.3) s v+"`'*':` = V:(qn°)+ V+(r)+ V+(r)+ V+(r) - (z1=+'c2)(A=+ +a," a2B+}e -a':, (3.4)

428

C. YAL(~INAND C. T. YAP V,aa S

¼(l+z~)(l+z2)Ce-P

V ass --

1 2

(3.5)

"~,

t

(3.6)

SW 1 2 t Vmas, = z~z~{bpd + o 1 " o 2 ( e + f p J ' ) } e -at2. On equating the volume integrals of the Yukawa

and Gaussian

(3.7) functions we obtain

t h e v a l u e s o f t h e v a r i o u s c o n s t a n t s , w h i c h o f c o u r s e d e p e n d o n t h e c u t - o f f r a d i u s re, a s s h o w n i n t a b l e 1. TABLE 1 Constants dependent on the cut-off radius rc = 0.4 fm ABA+ B+ A~B~As + Bs + C

e

0.1236 0.0873 --0.0849 0.0338 0.1623 0.0330 --0.0461 --0.0206 0.0314 0.0046 0.0081 0.0431 0.0732 0.0073

f'

0.1441

D" E F'

d"

rc ~ 0.5 fm

MeV MeV MeV MeV MeV MeV MeV MeV MeV

0.1005 0.0972 --0.0725 0.0511 0.1310 0.0272 --0.0428 --0.0190 0.0305 0.0038 0.0117 0.0367 0.0694 0.0107

MeV

MeV

MeV MeV MeV MeV MeV MeV MeV MeV MeV MeV

MeV

0.1326

The Fermi nuclear matrix elements arising from the above potentials are calculated a n d t h e r e s u l t s s h o w n i n t a b l e 2. TABLE 2 The Fermi nuclear matrix elements Value o f Ms:x: l0 s *tAr Potentials V-mlxina V+mm.s

SV-mix~n* sV +mi ~ln~ Vrad

Vs mass

V sw mass

r¢=0.4fm 0.96 1.73 --0.82 -- 0.05

**Sc rc = 0 . 5 f m

--0.26 - - 1.34t~p

--0.38 -- 1.14tSp

--0.23 --1.17~p

--0.18 --2.92tSp

--0.27 --2.66t~p

--0.24 --3.58tSp

--0.34 --3.52tSp

--0.21 --3.23~p

--0.30 --2.95~p

0

-- 1.01 -- 1.98 1.00 0.04

r¢=0.5fm

--0.30 --0.89t~p

0

2.04 2.43 --0.54 -- 0.15

rc=0.4fm

--0.21 --1.05tSp

0

1.48 2.00 --0.69 -- 0.17

rc=0.5fm

-- 1.57 --2.32 0.79 0.03 0.18 --0.33 - - 1.00tSp

0

1.45 2.04 --0.65 -- 0.05

r,=0.4fm

S2Mn

0.t9

BOSON EXCHANGEPOTENTIAL

429

4. Deviation from charge symmetry In our previous calculation 2), we used the phenomenological charge-dependent potential Hn of Blin-Stoyle and Le Tourneux H , = E Vo{('cz")+'c:cJ))(p+raC'). ¢cJ))+.c¢o.c(J)(q+saCO.aeJ))}e-Pr2u.

(4.1)

i
This is, apart from the radial dependence, the most general, charge-dependent, static, central, two-body potential. Here Vo is the strength of the effective, charge-independent, two-body potential and was taken to be -51.9 MeV and fl-~r its effective range was taken as 1.73 fm. The parameters p and r measure the deviation from charge symmetry while q and s measure the deviation from charge independence. It is now possible to obtain values ofp and r either from the values of Fermi nuclear matrix elements obtained or by comparing the Blin-Stoyle and Le Tourneux potential with one-boson exchange potentials given in the preceding section. In either case, we obtain values o f p and r given in table 3. TABLE 3 Parameters o f the deviation f r o m charge symmetry Potential used

g¢~ = --5 V - talxtns-F Vrld

(fm)

0.4

0.5

p×lO 3

--2.5

r×lO 3

--1.7

(3r--p)×lO 3

--2.6

--3.6

g,~ = 6

s V - m t xl n8 + V~aa

V +mixes -~ Vrad

0.4

0.5

0.4

0.5

--2.1

--3.3

--2.7

1.5

--1.9

--0.6

--0.5

--0.7

1.5

1.2

--3.6

--4.3

S V +mixl,~g + V r . d

0.4

0.5

1.3

0.7

0.7

--1.0

0.4

0.37

0.5

0.4

From the table, we observe that using the mixing potential s r - ~ * * will yield values for 3r-p consistent with those obtained by Blin-Stoyle and Yal~in 4). Moreover, for this mixing potential we have used g~, = - 5 and so is consistent with the analysis of nucleon-isoscalar form factors 2a) indicating that go, is negative. The values of p and r are of the order of 10- 3 showing that deviation from charge symmetry is indeed very small.

5. Deviation from charge independence In exactly the same way, it is easy to obtain values for q and s and these are presented in table 4. However, these values depend on @, the mass splitting of the p-meson and also on the OBEP model used. If we now use the well-attested relation 1,2, 6)

3s-q ~ 0.01, we obtain values for 6p listed in the last row of table 4. We see that ¢5p is in the range

430

c. YAL~IN AND C. T. YAP

of - 1 ~ to - 5 MeV and this result is consistent with experimental values to-date in which the following averages 24) are given p - = 762.8_.+2.8 MeV, pO = 769.0_+2.7 MeV,

(5.1)

giving 6p = - 6.2_+ 5.5. We have not considered the p + value because it is the average of only two experiments, which are not quite consistent. TABLE 4 Parameters of the deviation from charge independence Sawada r . (fm)

0.4

Scotti-Wong 0.5

0.4

0.5

q × 104

- - 1.51-0.89~p

- - 1.47-0.73~p

-- 1.51-14.110p

- - 1.47-13.376p

$ x 104

- - 1.55-8.30~p

- - 2.26-7.07~p

-- 1.41-27.76~p

--2.05-25.55~p

( 3 s - - q ) × 103

--0.31-2.40~p

--0.53-2.05~p

- - 0 . 2 7 - 6.92t~p

- - 0 . 4 7 - 6.33t~p

~p ( M e V )

--4.17

--4.88

- - 1.45

- - 1.58

The negative value of 6p is also consistent with a calculation by Yalcin and Pojon [ref. 25)] on phase shifts. If there are no charge-dependent effects, then isotriplet phase shifts can be used for both pp and np scattering. However, the phase-shift analysis at 210 MeV indicates that the xS 0 np phase shift is 6 ° ___2 ° less than that of the pp value and this splitting increases with energy 26). This result may be attributed to the shortrange, charge-dependent effects in the two-nucleon potential dominated by =- and p-meson mass splitting and meson mixings discussed in this paper. In their calculation, consistency between the pp and np scattering phase shifts is obtained with 6,0 ~ - 3 MeV. 6. Results and discussion As the values of the Fermi nuclear matrix elements are dependent on 6p, we shall use tip = - 3 MeV. With this value of 6p, the contribution to MF from vSWs is roughly three times as large as from vSass. The reason is that Scotti and Wong have rather small value o f p ( = 591 MeV). H a d their p been close to the experimental value, we would have obtained roughly the same value as from vSas~. Table 5 gives the theoretical and experimental values of MF. F r o m table 5 we see that experimental data are not very satisfactory at the moment. However, for 44Sc, if we reject the experimental value of Boehm and Wapstra (and this is reasonable since their value is inconsistent with all the other later experimental values), then our theoretical values agree very well with the experimental values. For the other two decays, out theoretical values are certainly consistent with experimental

431

BOSON EXCHANGE POTENTIAL

d a t a b u t to say a n y t h i n g m o r e precise w o u l d require m u c h m o r e accurate experiments t h a n at present feasible. TABLE 5

Theoretical and experimental values of Mr Value of MF )< 10a 4tAr

Potential rc=0.4 fm

44Sc

52Mn

ro =0.5 fm rc =0.4 fm r~=0.5 fin

r~=0.4 fm re =0.5 fm

--0.82

--0.65

--0.69

--0.54

1.00

0.79

Vr,d

0

0

0

0

0.19

0.18

Vsm~s

2.94

2.37

3.76

3.04

3.28

2.67

Vcoulorab

0

0

0

0

10

10

VTotal

2.12

1.72

3.07

2.50

14.47

13.64

s V - raixlng

93

5:32

--0.985:22 Experimental values of MF X 10a

0

4-12

ref. 27)

31.24-8.7

ref. 3o)

0.44-0.7

ref. a3)

ref. 2s)

3.54-6.1

ref. 2s)

--20.74-8.3

ref. a4)

ref. 29)

2.44-6.0

ref. al)

6.64-0.6

ref. as)

8.74-3.5

ref. a2)

-- 6.64-5.5

ref. 2s)

11.04-4.1

ref. 36)

0 -4-1.8

O n e o f us ( C . T . Y . ) is grateful to M E T U

ref. 32)

for hospitality at the D e p a r t m e n t o f

T h e o r e t i c a l Physics, M E T U , A n k a r a , Turkey. H e is also grateful to Professor A. R a j a r a t n a m a n d Mr. Y. S. H o n for their e n c o u r a g e m e n t an d for m a k i n g this trip to A n k a r a possible. References 1) R. J. Blin-Stoyle, Charge dependent effects in the nucleus in Selected topics in nuclear spectroscopy, compiled by B. J. Verhaar (North-Holland, Amsterdam, 1964) pp. 213-225 2) C. T. Yap, Nucl. Phys. A100 (1967) 619 3) J. S. Jalbert, Ph.D. thesis, Virginia Polytechnic Institute (1967) 4) R. J. Blin-Stoyle and C. Yalqin, Phys. Lett. 15 (1965) 258 5) C. T. Yap, Nucl. Phys. A142 (1970) 161 6) C. T. Yap, J. Singapore Nat. Acad. Sci. 1 (1969) No. 2, pp. 71-76 7) E. M. Henley, Charge dependence of nuclear forces in isobaric spin in nuclear physics, ed. J. D. Fox et al. (1966) p. 3 8) R. J. Blin-Stoyle and J. LeTourneux, Ann. of Phys. 18 (1962) 12 9) R. A. Bryan and B. L. Scott, Phys. Rev. 135 (1964) B434 10) A. Scotti and D. Y. Wong, Phys. Rev. Lett. 10 (1963) 142; Phys. Rev. 138 (1964) B145 11) S. Ogawa, S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Prog. Theor. Phys. Suppl. 39 (1967) 140 12) H. P. Stapp, T. J. Ypsilantis and N. Metropolis, Phys. Rev. 105 (1957) 302 13) N. Hoshizaki and S. Machida, Prog. Theor. Phys. 27 (1962) 288 14) S. N. Gupta, Phys. Rev. Lett. 2 (1959) 124

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C. YALe:IN AND C. T. YAP

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