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Contribution of charge symmetry breaking forces to energy differences in mirror nuclei
Volume 198, number 1
PHYSICS LETTERS B
12 November 1987
C O N T R I B U T I O N OF CHARGE SYMMETRY BREAKING FORCES TO ENERGY DIFFERENCES IN MIRROR NUCLEI P.G. BLUNDEN TRIUMF, 4004 Westbrook Mall, Vancouver, B.C., Canada V6T 2A3
and M.J. IQBAL Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 3 June 1987
Contributions to energy differences in mirror nuclei and to the scattering length differences a p p - ann and a . . - a.p are evaluated from class III and class IV type charge symmetry breaking (CSB) potentials. We consider f - c o mixing, n°-rl mixing, one- and two-pion exchange contributions. Potentials which are consistent with these scattering length differences and with recent CSB p - n elastic scattering data are found to contribute about 150-250 keV to the energy differences in mirror nuclei.
A recent proton-neutron ( p - n ) elastic scattering experiment at T R I U M F [ 1 ] has confirmed the existence of class IV type [ 2 ] charge symmetry breaking (CSB) forces in the p - n system. Theoretically there are several sources for this CSB force - they include the electromagnetic spin-orbit interaction, p°-c0 mixing, and the one-pion exchange contribution arising from the p - n mass difference. For many years information about charge symmetry breaking in the nucleon-nucleon ( N N ) system has been inferred from a comparison of the binding energy differences of mirror nuclei AE= E o - En. The main contributions to AE are from the Coulomb interaction, the proton-neutron mass difference, and the electromagnetic spin-orbit interaction. Nolen and Schiffer [ 3 ] first pointed out that after these effects were subtracted there remains a discrepancy of typically several hundred keV, which is positive and which occurs throughout the periodic table. Despite some quite sophisticated theoretical calculations the anomaly persists, and it is now thought unlikely that the explanation is due to conventional nuclear structure [ 4,5 ]. However, class III (pp-nn) as well as class IV (pn) type CSB forces can give additional contributions to the binding energy 14
differences. Although there is theoretical work published on this subject [ 6,7 ], a systematic and detailed study of the contributions of both class III and class IV type forces is lacking. Specifically, to our knowledge there is no estimate of the contribution to AE from class IV type forces. Another source of information about CSB in the N N system is the difference in the scattering lengths Aa=a~,p-ann (Coulomb effects subtracted) and 8a=ann--anp. In fact it is very crucial that the CSB potentials used to calculate AE in mirror nuclei should also be used to calculate the above-mentioned scattering length differences. In this paper we have completed a detailed study of the effect of different class III and class IV type forces on energy differences in mirror nuclei, along with their contribution to Aa and 6a. We have used generally accepted values for coupling constants and particle mixing matrix elements. The parameters for the class IV type CSB potential are those which roughly reproduce the 477 MeV elastic p - n scattering results [ 8 ]. Although the energy involved in this case is large, in the absence of elastic scattering experiments at lower energies this is the best constraint one has at present.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 198, number 1
PHYSICS LETTERS B
12 November 1987
p°-rz mixing (Kp and Ko, arise from the tensor coupling):
P
gpgo, (P° [H[0))(T l "~-~'2) 4n
62
[exp(-mpr) Xk r
Fig. 1. p°-c0 mixing graph that leads to class III and class IV type charge symmetry breaking forces.
Forces of the class IV type have the form [2]
F(r)(a~ - a 2 ) ' L ( z ~
K, 4M 2 '
(1)
(pO I H I o ) gn m~ - m p2 M 2
dfexp(-mer ) r
mj))
exp( r
'
and the proton-neutron mass difference in the onepion exchange (OPE) potential
G(r)-
g2 ~ 1 d ( e x p ( - ~ m ~ r ) ) 4n 2M 2 r drk . '
8=mn-mp
r ]
-- (3+2Kp+2Ko,)(at + a 2 ) ' r X p ] 1 ,
(3)
M,+Mp " Class III type forces have the isospin structure ( r l + z2), and therefore contribute only to the pp and nn systems. The important contributions are from
(4)
n°-q mixing:
g,~g,~( n ° I H I ~ ) ~ 1 4n --~5---2~..2 , , . n - m~ jaj "a2(zl +z2) 4 M 2 ×(m2eXP(rm':)
which arises from the interaction of the anomalous magnetic moment of the neutron with the proton. This contribution has already been included in shell model calculations of binding energy differences, and so we do not include it in our analysis. There are two other important contributions of this type, arising from p°-(o mixing (fig. 1 )
xl
exp(--mo, r)~
x[½(a~-a2)'rxe
angular momentum. Because of the isospin structure this interaction is only present in the p - n system. The simplest force of this kind is the electromagnetic spin-orbit interaction, with
4n
+ 4-_~(m~eXP(rmpr) _ m~eXP(rrn'r) ) 1 1 d(exp(-mpr) 4M 2 r d r \ r
G(r)(at × a 2 ) ' L ( z ~ ×z2) , where L=r×p is the relative
F(r)-gog'°
exp(-rn,,,r) r
× [(1 +Kp+Ko,) + 23a,"a2(l +Kp)(l +K~o) ]
-z2) ,
and
F(r) = ~
m~2 - - m a2
m2eXP(rm':)),
(5)
and one-pion exchange: g2 i rn2~ e x p ( - m ~ r ) ~-~30"1 "0"2('~'1 "~-~'2)4"--~ r
(6)
Here ~ ~ 0.00069 from the p - n mass difference. One can have a further contribution of this type due to a difference in the g : p p and g : , n couplings, which may arise from radiative corrections at the n Opp vertex. Morrison [9] has estimated the value ~ ~ -0.00023. In addition to one-pion exchange there are contributions to class III forces from two-pion exchange, illustrated in the box and crossed-box diagrams of fig. 2. Here the intermediate two-nucleon state will have a different energy threshold depending on whether it involves two protons or two neutrons, thus implying a difference in the pp and nn interaction strengths. However, there are discrepancies in the literature on the size and sign of this contribution. Riska and Chu [ 10 ] found an energy shift AE of several hundred keV in the wrong (negative) direction, and a contribution to the scattering lengths of 15
Volume 198, number 1
PHYSICS LETTERS B
I
I
71.o
71.o I
I I I
i
n p
g
K
m(MeV)
A (MeV)
13.55 3.46 15.58 5.09
6.1 0 -
134.96,139.57 769 782.6 548.8
1300 1300 1500 1500
/
7T-'- \ \
//71.~ ",z' / \ / \ / \ \
Fig. 2. Box and crossed-box two-pion exchange graphs that give rise to class III type charge symmetry breaking forces.
A a = - 2 . 7 fm. On the other hand, Noble [ 11 ] estimated the contribution to AE at low energy and zero momentum transfer and found a shift of several hundred keV in the positive direction. We have re-evaluated this contribution in a different manner, starting from the two-pion exchange potential of Partovi and Lomon [ 12 ]. The difference Fpo-I/n, has been calculated from their formulas (4.30), (4.31) and (4.36)-(4.38), but with the intermediate state masses appropriate to the box and crossed box diagrams of fig. 2. We were careful to evaluate only the difference Vpp- Fnn in any of the multi-dimensional integrals in order to minimize numerical errors. In contrast to the above mentioned results of refs. [ 10,11 ], we find that the two-pion exchange contribution is quite small, both to AE and to the scattering length difference Aa. In order to evaluate these CSB effects in finite nuclei we have used harmonic oscillator wave functions, with the oscillator parameter b chosen to fit elastic electron scattering data [ 13]. This gives a Coulomb contribution to AE which is within 10% of the best Hartree-Fock estimates [4], and so these wave functions should be accurate enough to evaluate the CSB effects. Hadronic form factors of the monopole type are included, which modifies the Yukawa functions in eqs. (2)-(6). In addition, short 16
Table 1 Boson exchange parameters used in our analysis. The hadronic form factors are of the monopole type, (A2-rn2)/(A2+q2). (n°lH[q)=-4200 MeV 2, ( p ° [ H l t o ) = - 3 8 5 0 MeV 2, M , =939.57 MeV, Mp=938.28 MeV.
j
\
/
12 November 1987
range correlations were accounted for by introducing a lower cutoff in the relative wave function at 0.5 fm and renormalizing the wave functions. This is important since it reduces the s-wave contributions of the CSB potentials, which are large, and makes the results insensitive to the very short range behaviour of these potentials. Other choices for the form of the short range correlation function gave very similar results. The various coupling constants and form factors, summarized in table 1, are taken from the Bonn potential [ 14 ], with the exception of the q coupling constant. For the rlNN coupling constant the value used in N N potentials is a phenomenological fit, and is likely too large compared with the experimental value, as has been pointed out by Coon and Scadron [ 7]. We use the smaller value preferred by these authors, which has been multiplied by 1.31 to account for the n o mixing to the 12' as well as to the q. The n°-~q mixing matrix element is taken from McNamee et al. [15]. The meson mixing matrix element ( p ° l H I co ) can be evaluated from the co~2n partial decay width, using the formula of McNamee et al. [ 15 ] and the most recent value for the above mentioned branching ratio. There is some uncertainty in the magnitude of this matrix element. For example, Friar and Gibson [ 16] find a much larger value of - 6 0 0 0 MeV 2. Our results are given in table 2 for various orbital configurations near closed shell nuclei. The discrepancies between experiment and two of the best available theories are from the analysis of Sato [4], which is based on a Skyrme II Hartree-Fock calculation and the density matrix expansion (DME) method. The f - c o class III contribution is by far the largest, and
Volume 198, number 1
PHYSICS LETTERS B
12 November 1987
Table 2 Contributions to AE from class III and class IV type CSB forces in keV. The large class III p°-t0 contribution is shown separately. The Sk II and DME columns are the discrepanciesbetween theory and experiment given by Sato [4], A
b (fm)
Class III
(p°-o only)
ClassIV
Total
SklI
11
1,640
ps~
164
(126)
- 10
154
13
1.640
Pl/a
214
(178)
24
238
-
-
15
1.765
ps~ pu~
175 209
(138) (171)
-9 18
166 227
190 290
250 380
17
1.765
d5/2 lsln d3/2
129 200 175
(100) (164) (145)
-16 0 24
113 200 199
190 210 270
300 220 430
27
1.824
d~
191
(149)
- 14
177
-
480
29
1.824
1sl/2
248
(204)
0
248
-
290
31
1.879
ls~
255
(208)
0
255
-
540
33
1.879
d3n
244
(204)
21
265
-
360
39
1.950
lsv~ d~
266 257
(219) (212)
0 18
266 275
270 430
370 540
41
1.950
6n lp3/2 lPln
167 238 256
(132) (195) (213)
- 17 -6 11
150 232 267
350 340 330
440 380 410
we have shown it in a separate column. The next most i m p o r t a n t piece is the no_~ mixing. The two-pion exchange c o n t r i b u t i o n is almost zero because of a cancellation between direct a n d exchange pieces. Individually the direct pieces are about 2 0 - 4 0 keV. The o n e - p i o n exchange c o n t r i b u t i o n is also extremely small. Class IV type forces are of opposite sign for j=l+ 1/2 and j = l - 1/2 a n d therefore give an additional s p i n - o r b i t splitting which, for these nuclei, is about 3 0 - 4 0 keV. The contributions from p ° - o mixing and one-pion exchange are about equally important. Altogether, using these parameters, CSB effects can explain about 75% of the discrepancy between the Skyrme II theory a n d experiment. It also shows that the c o n t r i b u t i o n increases with increasing mass number, which is seen experimentally. Obviously because of the importance of the p ° - o c o n t r i b u t i o n our results would be significantly larger if we were to use a larger value for the particle mixing matrix element, such as that calculated by Friar a n d G i b s o n
DME -
-
[ 16 ]. We now t u r n to a discussion of CSB effects on the N N scattering lengths. Charge symmetry breaking of the N N force causes the ~So scattering lengths of the p r o t o n - p r o t o n a n d neutron-neutron systems to differ (Aa = app--ann¢O). The charge dependence of the N N force, on the other hand, causes the p r o t o n - n e u t r o n So scattering length anp to differ from ann. Only class III type CSB forces can contribute to the difference Aa, since class IV type forces vanish for p - p a n d n - n systems. We will first look at the difference Aa. The major c o n t r i b u t i o n in this case arises from p ° - o mixing. There are smaller contributions arising from n ° - r l mixing a n d the p - n mass difference in two-pion exchange ( T P E ) diagrams. We use the formalism of Ericson a n d Miller [ 17] to calculate Aa. For comparison we have also used the approximate method of Negele [ 5 ]. The ~So wave function is generated from the Reid soft-core potential [ 18 ]. Our results are s u m m a r i z e d in table 3. The total CSB effect is within the experimental uncertainty in Aa [ 19 ]. 17
Volume 198, number I
PHYSICS LETTERS B
12 November 1987
Table 3 Difference a~p-a,, (in fm) due to different CSB sources using the exact treatment of Ericson and Miller [ 17] and the approximate method of Negele [ 5 ]. The errors in the experimental number arise from the uncertainty in a,, and in the subtraction of Coulomb effects from app respectively [ 17,18 ].
p°-to mixing n°-rl mixing TPE OPE total
Exact
Approximate
0.99 0.15 - 0.21 0.19
1.46 0.00 - 0.20 0.19
1.12
1.45
The charge dependence of the N N force gives rise to differences in 8a=an,-a,p. The major source of 8a, as pointed out in ref. [ 17], is the difference in the masses of charged and neutral pions in one-pion exchange (OPE). The OPE contribution to ~ a, using a pseudoscalar n N N interaction, gives 5 a = 3 . 4 fm. The contribution from TPE due to pion mass differences arise from cancellations between box and crossed-box two-pion exchange diagrams. Using the Partovi-Lomon TPEP we get a contribution of 0.8 fm, in agreement with Ericson and Miller [ 17]. Since the theoretical calculations of the charge dependence of the n N N coupling constants are far from satisfactory we have not included this effect in our estimates of 5a. Thus we find that a treatment of CSB in terms of the meson exchange picture describes reasonably well the difference in ISo scattering lengths Aa and 5a. It also explains about 3/4 of the discrepancy between theory and experiment in binding energy differences of the mirror nuclei. The main contribution AE and Aa arise from the class III type CSB force due to p°-co mixing. Effects due to class IV type CSB forces are small. The sign of the class IV contribution to AE depends upon whether the configuration is a j = l + 1/2 or j=l-1/2 orbit. We find the contribution to AE and Aa from TPE to be small, contrary to other claims in the literature. The mass difference between charged and neutral pions is the main source of ha, in agreement with the findings of Ericson and Miller [ 17 ].
18
Experiment
0.55+0.45_+0.45
References [ 1 ] R. Abegg et al., Phys. Rev. Lett. 56 (1986) 2571. [2] E.M. Henley and G.A. Miller, in: Mesons in nuclei, Vol. 1, eds. M. Rho and D.H. Wilkinson (North-Holland, Amsterdam, 1977) p. 405. [3] J.A. Nolen Jr. and J.P. Schiffer, Annu. Rev. Nucl. Sci. 19 (1969) 414. [4] H. Sato, Nucl. Phys. A 296 (1976) 378. [5] J.W. Negele, Nucl. Phys. A 165 (1971) 305. [6] P. Langacker and D.A. Sparrow, Phys. Rev. C 25 (1982) 1194; Phys. Rev. Lett. 43 (1979) 1559. [7] S.A. Coon and M.D. Scadron, Phys. Rev. C 26 (1982) 562. [8] G.A. Miller, A.W. Thomas and A.G. Williams, Phys. Rev. Lett. 56 (1986) 2567. [9] L.K. Morrison, Ann. Phys. 50 (1968) 6. [ 10] D.O. Riska and Y.H. Chu, Nucl. Phys. A 235 (1974) 499. [ 11 ] J.V. Noble, in: AlP Conf. Proc. No. 97, ed. H.O. Meyer (AIP, New York, 1983) p. 83. [ 12 ] M.H. Partovi and E.L. Lomon, Phys. Rev. D 2 (1970) 1999. [ 13] M. Carchidi, B.H. Wildenthal and B.A. Brown, Phys. Rev. C 34 (1986) 2280. [ 14 ] R. Machleidt, lectures presented at the Workshop on Relativistic dynamics and quark-nuclear physics (Los Alamos, 1985). [ 15] P.C. McNamee, M.D. Scadron and S.A. Coon, Nucl. Phys. A249 (1975) 483. [16] J.L. Friar and B.F. Gibson, Phys. Rev. C 17 (1978) 1752. [ 17] T.E.O. Ericson and G.A. Miller, Phys. Lett. B 132 (1983) 32. [ 18] R.V. Reid, Ann. Phys. 50 (1968) 411. [19] O. Dumbrajs et al., 1982 Data compilation, Nucl. Phys. B 216 (1983) 277.
PHYSICS LETTERS B
12 November 1987
C O N T R I B U T I O N OF CHARGE SYMMETRY BREAKING FORCES TO ENERGY DIFFERENCES IN MIRROR NUCLEI P.G. BLUNDEN TRIUMF, 4004 Westbrook Mall, Vancouver, B.C., Canada V6T 2A3
and M.J. IQBAL Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 3 June 1987
Contributions to energy differences in mirror nuclei and to the scattering length differences a p p - ann and a . . - a.p are evaluated from class III and class IV type charge symmetry breaking (CSB) potentials. We consider f - c o mixing, n°-rl mixing, one- and two-pion exchange contributions. Potentials which are consistent with these scattering length differences and with recent CSB p - n elastic scattering data are found to contribute about 150-250 keV to the energy differences in mirror nuclei.
A recent proton-neutron ( p - n ) elastic scattering experiment at T R I U M F [ 1 ] has confirmed the existence of class IV type [ 2 ] charge symmetry breaking (CSB) forces in the p - n system. Theoretically there are several sources for this CSB force - they include the electromagnetic spin-orbit interaction, p°-c0 mixing, and the one-pion exchange contribution arising from the p - n mass difference. For many years information about charge symmetry breaking in the nucleon-nucleon ( N N ) system has been inferred from a comparison of the binding energy differences of mirror nuclei AE= E o - En. The main contributions to AE are from the Coulomb interaction, the proton-neutron mass difference, and the electromagnetic spin-orbit interaction. Nolen and Schiffer [ 3 ] first pointed out that after these effects were subtracted there remains a discrepancy of typically several hundred keV, which is positive and which occurs throughout the periodic table. Despite some quite sophisticated theoretical calculations the anomaly persists, and it is now thought unlikely that the explanation is due to conventional nuclear structure [ 4,5 ]. However, class III (pp-nn) as well as class IV (pn) type CSB forces can give additional contributions to the binding energy 14
differences. Although there is theoretical work published on this subject [ 6,7 ], a systematic and detailed study of the contributions of both class III and class IV type forces is lacking. Specifically, to our knowledge there is no estimate of the contribution to AE from class IV type forces. Another source of information about CSB in the N N system is the difference in the scattering lengths Aa=a~,p-ann (Coulomb effects subtracted) and 8a=ann--anp. In fact it is very crucial that the CSB potentials used to calculate AE in mirror nuclei should also be used to calculate the above-mentioned scattering length differences. In this paper we have completed a detailed study of the effect of different class III and class IV type forces on energy differences in mirror nuclei, along with their contribution to Aa and 6a. We have used generally accepted values for coupling constants and particle mixing matrix elements. The parameters for the class IV type CSB potential are those which roughly reproduce the 477 MeV elastic p - n scattering results [ 8 ]. Although the energy involved in this case is large, in the absence of elastic scattering experiments at lower energies this is the best constraint one has at present.
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 198, number 1
PHYSICS LETTERS B
12 November 1987
p°-rz mixing (Kp and Ko, arise from the tensor coupling):
P
gpgo, (P° [H[0))(T l "~-~'2) 4n
62
[exp(-mpr) Xk r
Fig. 1. p°-c0 mixing graph that leads to class III and class IV type charge symmetry breaking forces.
Forces of the class IV type have the form [2]
F(r)(a~ - a 2 ) ' L ( z ~
K, 4M 2 '
(1)
(pO I H I o ) gn m~ - m p2 M 2
dfexp(-mer ) r
mj))
exp( r
'
and the proton-neutron mass difference in the onepion exchange (OPE) potential
G(r)-
g2 ~ 1 d ( e x p ( - ~ m ~ r ) ) 4n 2M 2 r drk . '
8=mn-mp
r ]
-- (3+2Kp+2Ko,)(at + a 2 ) ' r X p ] 1 ,
(3)
M,+Mp " Class III type forces have the isospin structure ( r l + z2), and therefore contribute only to the pp and nn systems. The important contributions are from
(4)
n°-q mixing:
g,~g,~( n ° I H I ~ ) ~ 1 4n --~5---2~..2 , , . n - m~ jaj "a2(zl +z2) 4 M 2 ×(m2eXP(rm':)
which arises from the interaction of the anomalous magnetic moment of the neutron with the proton. This contribution has already been included in shell model calculations of binding energy differences, and so we do not include it in our analysis. There are two other important contributions of this type, arising from p°-(o mixing (fig. 1 )
xl
exp(--mo, r)~
x[½(a~-a2)'rxe
angular momentum. Because of the isospin structure this interaction is only present in the p - n system. The simplest force of this kind is the electromagnetic spin-orbit interaction, with
4n
+ 4-_~(m~eXP(rmpr) _ m~eXP(rrn'r) ) 1 1 d(exp(-mpr) 4M 2 r d r \ r
G(r)(at × a 2 ) ' L ( z ~ ×z2) , where L=r×p is the relative
F(r)-gog'°
exp(-rn,,,r) r
× [(1 +Kp+Ko,) + 23a,"a2(l +Kp)(l +K~o) ]
-z2) ,
and
F(r) = ~
m~2 - - m a2
m2eXP(rm':)),
(5)
and one-pion exchange: g2 i rn2~ e x p ( - m ~ r ) ~-~30"1 "0"2('~'1 "~-~'2)4"--~ r
(6)
Here ~ ~ 0.00069 from the p - n mass difference. One can have a further contribution of this type due to a difference in the g : p p and g : , n couplings, which may arise from radiative corrections at the n Opp vertex. Morrison [9] has estimated the value ~ ~ -0.00023. In addition to one-pion exchange there are contributions to class III forces from two-pion exchange, illustrated in the box and crossed-box diagrams of fig. 2. Here the intermediate two-nucleon state will have a different energy threshold depending on whether it involves two protons or two neutrons, thus implying a difference in the pp and nn interaction strengths. However, there are discrepancies in the literature on the size and sign of this contribution. Riska and Chu [ 10 ] found an energy shift AE of several hundred keV in the wrong (negative) direction, and a contribution to the scattering lengths of 15
Volume 198, number 1
PHYSICS LETTERS B
I
I
71.o
71.o I
I I I
i
n p
g
K
m(MeV)
A (MeV)
13.55 3.46 15.58 5.09
6.1 0 -
134.96,139.57 769 782.6 548.8
1300 1300 1500 1500
/
7T-'- \ \
//71.~ ",z' / \ / \ / \ \
Fig. 2. Box and crossed-box two-pion exchange graphs that give rise to class III type charge symmetry breaking forces.
A a = - 2 . 7 fm. On the other hand, Noble [ 11 ] estimated the contribution to AE at low energy and zero momentum transfer and found a shift of several hundred keV in the positive direction. We have re-evaluated this contribution in a different manner, starting from the two-pion exchange potential of Partovi and Lomon [ 12 ]. The difference Fpo-I/n, has been calculated from their formulas (4.30), (4.31) and (4.36)-(4.38), but with the intermediate state masses appropriate to the box and crossed box diagrams of fig. 2. We were careful to evaluate only the difference Vpp- Fnn in any of the multi-dimensional integrals in order to minimize numerical errors. In contrast to the above mentioned results of refs. [ 10,11 ], we find that the two-pion exchange contribution is quite small, both to AE and to the scattering length difference Aa. In order to evaluate these CSB effects in finite nuclei we have used harmonic oscillator wave functions, with the oscillator parameter b chosen to fit elastic electron scattering data [ 13]. This gives a Coulomb contribution to AE which is within 10% of the best Hartree-Fock estimates [4], and so these wave functions should be accurate enough to evaluate the CSB effects. Hadronic form factors of the monopole type are included, which modifies the Yukawa functions in eqs. (2)-(6). In addition, short 16
Table 1 Boson exchange parameters used in our analysis. The hadronic form factors are of the monopole type, (A2-rn2)/(A2+q2). (n°lH[q)=-4200 MeV 2, ( p ° [ H l t o ) = - 3 8 5 0 MeV 2, M , =939.57 MeV, Mp=938.28 MeV.
j
\
/
12 November 1987
range correlations were accounted for by introducing a lower cutoff in the relative wave function at 0.5 fm and renormalizing the wave functions. This is important since it reduces the s-wave contributions of the CSB potentials, which are large, and makes the results insensitive to the very short range behaviour of these potentials. Other choices for the form of the short range correlation function gave very similar results. The various coupling constants and form factors, summarized in table 1, are taken from the Bonn potential [ 14 ], with the exception of the q coupling constant. For the rlNN coupling constant the value used in N N potentials is a phenomenological fit, and is likely too large compared with the experimental value, as has been pointed out by Coon and Scadron [ 7]. We use the smaller value preferred by these authors, which has been multiplied by 1.31 to account for the n o mixing to the 12' as well as to the q. The n°-~q mixing matrix element is taken from McNamee et al. [15]. The meson mixing matrix element ( p ° l H I co ) can be evaluated from the co~2n partial decay width, using the formula of McNamee et al. [ 15 ] and the most recent value for the above mentioned branching ratio. There is some uncertainty in the magnitude of this matrix element. For example, Friar and Gibson [ 16] find a much larger value of - 6 0 0 0 MeV 2. Our results are given in table 2 for various orbital configurations near closed shell nuclei. The discrepancies between experiment and two of the best available theories are from the analysis of Sato [4], which is based on a Skyrme II Hartree-Fock calculation and the density matrix expansion (DME) method. The f - c o class III contribution is by far the largest, and
Volume 198, number 1
PHYSICS LETTERS B
12 November 1987
Table 2 Contributions to AE from class III and class IV type CSB forces in keV. The large class III p°-t0 contribution is shown separately. The Sk II and DME columns are the discrepanciesbetween theory and experiment given by Sato [4], A
b (fm)
Class III
(p°-o only)
ClassIV
Total
SklI
11
1,640
ps~
164
(126)
- 10
154
13
1.640
Pl/a
214
(178)
24
238
-
-
15
1.765
ps~ pu~
175 209
(138) (171)
-9 18
166 227
190 290
250 380
17
1.765
d5/2 lsln d3/2
129 200 175
(100) (164) (145)
-16 0 24
113 200 199
190 210 270
300 220 430
27
1.824
d~
191
(149)
- 14
177
-
480
29
1.824
1sl/2
248
(204)
0
248
-
290
31
1.879
ls~
255
(208)
0
255
-
540
33
1.879
d3n
244
(204)
21
265
-
360
39
1.950
lsv~ d~
266 257
(219) (212)
0 18
266 275
270 430
370 540
41
1.950
6n lp3/2 lPln
167 238 256
(132) (195) (213)
- 17 -6 11
150 232 267
350 340 330
440 380 410
we have shown it in a separate column. The next most i m p o r t a n t piece is the no_~ mixing. The two-pion exchange c o n t r i b u t i o n is almost zero because of a cancellation between direct a n d exchange pieces. Individually the direct pieces are about 2 0 - 4 0 keV. The o n e - p i o n exchange c o n t r i b u t i o n is also extremely small. Class IV type forces are of opposite sign for j=l+ 1/2 and j = l - 1/2 a n d therefore give an additional s p i n - o r b i t splitting which, for these nuclei, is about 3 0 - 4 0 keV. The contributions from p ° - o mixing and one-pion exchange are about equally important. Altogether, using these parameters, CSB effects can explain about 75% of the discrepancy between the Skyrme II theory a n d experiment. It also shows that the c o n t r i b u t i o n increases with increasing mass number, which is seen experimentally. Obviously because of the importance of the p ° - o c o n t r i b u t i o n our results would be significantly larger if we were to use a larger value for the particle mixing matrix element, such as that calculated by Friar a n d G i b s o n
DME -
-
[ 16 ]. We now t u r n to a discussion of CSB effects on the N N scattering lengths. Charge symmetry breaking of the N N force causes the ~So scattering lengths of the p r o t o n - p r o t o n a n d neutron-neutron systems to differ (Aa = app--ann¢O). The charge dependence of the N N force, on the other hand, causes the p r o t o n - n e u t r o n So scattering length anp to differ from ann. Only class III type CSB forces can contribute to the difference Aa, since class IV type forces vanish for p - p a n d n - n systems. We will first look at the difference Aa. The major c o n t r i b u t i o n in this case arises from p ° - o mixing. There are smaller contributions arising from n ° - r l mixing a n d the p - n mass difference in two-pion exchange ( T P E ) diagrams. We use the formalism of Ericson a n d Miller [ 17] to calculate Aa. For comparison we have also used the approximate method of Negele [ 5 ]. The ~So wave function is generated from the Reid soft-core potential [ 18 ]. Our results are s u m m a r i z e d in table 3. The total CSB effect is within the experimental uncertainty in Aa [ 19 ]. 17
Volume 198, number I
PHYSICS LETTERS B
12 November 1987
Table 3 Difference a~p-a,, (in fm) due to different CSB sources using the exact treatment of Ericson and Miller [ 17] and the approximate method of Negele [ 5 ]. The errors in the experimental number arise from the uncertainty in a,, and in the subtraction of Coulomb effects from app respectively [ 17,18 ].
p°-to mixing n°-rl mixing TPE OPE total
Exact
Approximate
0.99 0.15 - 0.21 0.19
1.46 0.00 - 0.20 0.19
1.12
1.45
The charge dependence of the N N force gives rise to differences in 8a=an,-a,p. The major source of 8a, as pointed out in ref. [ 17], is the difference in the masses of charged and neutral pions in one-pion exchange (OPE). The OPE contribution to ~ a, using a pseudoscalar n N N interaction, gives 5 a = 3 . 4 fm. The contribution from TPE due to pion mass differences arise from cancellations between box and crossed-box two-pion exchange diagrams. Using the Partovi-Lomon TPEP we get a contribution of 0.8 fm, in agreement with Ericson and Miller [ 17]. Since the theoretical calculations of the charge dependence of the n N N coupling constants are far from satisfactory we have not included this effect in our estimates of 5a. Thus we find that a treatment of CSB in terms of the meson exchange picture describes reasonably well the difference in ISo scattering lengths Aa and 5a. It also explains about 3/4 of the discrepancy between theory and experiment in binding energy differences of the mirror nuclei. The main contribution AE and Aa arise from the class III type CSB force due to p°-co mixing. Effects due to class IV type CSB forces are small. The sign of the class IV contribution to AE depends upon whether the configuration is a j = l + 1/2 or j=l-1/2 orbit. We find the contribution to AE and Aa from TPE to be small, contrary to other claims in the literature. The mass difference between charged and neutral pions is the main source of ha, in agreement with the findings of Ericson and Miller [ 17 ].
18
Experiment
0.55+0.45_+0.45
References [ 1 ] R. Abegg et al., Phys. Rev. Lett. 56 (1986) 2571. [2] E.M. Henley and G.A. Miller, in: Mesons in nuclei, Vol. 1, eds. M. Rho and D.H. Wilkinson (North-Holland, Amsterdam, 1977) p. 405. [3] J.A. Nolen Jr. and J.P. Schiffer, Annu. Rev. Nucl. Sci. 19 (1969) 414. [4] H. Sato, Nucl. Phys. A 296 (1976) 378. [5] J.W. Negele, Nucl. Phys. A 165 (1971) 305. [6] P. Langacker and D.A. Sparrow, Phys. Rev. C 25 (1982) 1194; Phys. Rev. Lett. 43 (1979) 1559. [7] S.A. Coon and M.D. Scadron, Phys. Rev. C 26 (1982) 562. [8] G.A. Miller, A.W. Thomas and A.G. Williams, Phys. Rev. Lett. 56 (1986) 2567. [9] L.K. Morrison, Ann. Phys. 50 (1968) 6. [ 10] D.O. Riska and Y.H. Chu, Nucl. Phys. A 235 (1974) 499. [ 11 ] J.V. Noble, in: AlP Conf. Proc. No. 97, ed. H.O. Meyer (AIP, New York, 1983) p. 83. [ 12 ] M.H. Partovi and E.L. Lomon, Phys. Rev. D 2 (1970) 1999. [ 13] M. Carchidi, B.H. Wildenthal and B.A. Brown, Phys. Rev. C 34 (1986) 2280. [ 14 ] R. Machleidt, lectures presented at the Workshop on Relativistic dynamics and quark-nuclear physics (Los Alamos, 1985). [ 15] P.C. McNamee, M.D. Scadron and S.A. Coon, Nucl. Phys. A249 (1975) 483. [16] J.L. Friar and B.F. Gibson, Phys. Rev. C 17 (1978) 1752. [ 17] T.E.O. Ericson and G.A. Miller, Phys. Lett. B 132 (1983) 32. [ 18] R.V. Reid, Ann. Phys. 50 (1968) 411. [19] O. Dumbrajs et al., 1982 Data compilation, Nucl. Phys. B 216 (1983) 277.