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Exchange corrections to the axial vector decay of 15O and 17F
Volume 81B, number 3, 4
PHYSICS LETTERS
26 February 1979
EXCHANGE CORRECTIONS TO THE AXIAL VECTOR DECAY OF 150 AND 17F~ Chr. BARGHOLTZ Department of Physics, University of Stockholm, Stockholm, Sweden and Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11 794, USA Received 4 December 1978
A calculation is made of the mesonic exchange corrections to the axial vector decays 1 5 0 -* ISN and 17F --, 170. It is found that the large correction from pion exchange for A = 15 is cancelled by the exchange of p-mesons.
The properties of nuclei cannot be properly accounted for in the impulse approximation ascribing the properties of free nucleons to the constituent nucleons. Allowance has to be made for the internucleon interactions. The effects of such interactions, through the exchange of pions, on the axial vector operator have been studied in detail by Chemtob and Rho [1 ] who apply their results to tile magnetic moments of 3H and 3He and to the 3H -+ 3He 13-decay. It was shown by Blomqvist [2] and by Riska and Brown [3] that the dominant exchange correction to this decay stems from the tensor part of the isobaric exchange current between S- and D-state components of the wave function. Including corrections from oneand two-boson exchange, relativistic effects and wave function renormalization, the theoretical prediction for the strength of the GT matrix element between 3H and 3He agrees with the experimental value [4]. Blomqvist observed that applying the same exchange operator to the 13-decay of 150 and 17F one obtains contributions to the axial vector matrix element of approximately 20% and 0% of the value obtained in the impulse approximation using harmonic oscillator wave functions. In the case of 150 this removes the theoretical prediction still further away from the experimental value. Barroso and Blin-Stoyle [5] studied mirror el-decays around 160 and 40Ca. Their two-body operator included Work partially supported by USDOE contract number EY76-S-02-3001. 286
the diagrams in figs. l a - d . Using a phenomenological lagrangian they calculated the summed correction from the diagrams l a (only n-exchange) and lb (q2 ~ m 2) to be 21.9% and 0.0% for 150 and 17F, respectively. The present letter is a report on the effects of including p-meson exchange consistently. Besides being somewhat uncertain the contributions from pair excitation (1 c) and nucleon recoil (ld) added the correction from renormalization of the wave function turned out to be quite small in the calculation of Barroso and Blin: Stoyle and will be left out here. In the static quark model we get for diagram la: 2 = 1 6 f~NgAma Mh c~=.,; M~ - M N X ((*1 X'2)+-[~-(1 + 8c~p) (~ 1 X u : ) - ( 1 - 28~o) (i(~ 1 X , 2 ) ' i - ~ I X~2) X (1 +3liner+ 3/m2r2)]
-- 2(~1 + ~2)-+ [I (1 + 8c~p)(u 1 + u2) 1
+ (1 - 2a~o ) (i(a 1 + u2)" i - ~ (1I1 + O'2)) × (1 + 3/rn~r + 3/m2r2)]
-- 2(~1 -'2)+- [~ (1 + 15c~o)(*1 - ' 2 ) + (1 - 2 a ~ . )
(t(, 5 - a2).
i -
1 (o 1 _ q2))
× (1 + 3 / m , r + 3/m 2 r2)]} e-m"r/m,~ r , where bop = 8,r~r = 1 and 8 , o = 8p, =0. For diagram lb, using PCAC to connect p -+ 2ir to
Volume 81B, number 3,4
~
PHYSICS LETTERS
26 February 1979
__~_,_e___.
M --t
A
J
N
N
N
Y=O S= I
N (b)
(o)
I
7
Y=OS=l
or T=I S=O
T VI2
T VI2
y
T,P,w.-/1 ....
N
N
N
(e)
(a)
N
Fig. 1. Some one-meson exchange processes contributing to M12. (Only a and b are included in the present calculation.)
M~12-~fTrNfONm2 { (~rrt?- m2%
(I~1X'~ 2)-+ [2 (o.1 X o.2)
e-m
m; mp m~rr -- (i(o. i X 0"2)"i -- ½(o.i X o'2) )
3 +
T=I S=O
T=O S= or
T= I S=O Fig. 2. Two meson exchange processes correcting the expectation value of the GT-operator in even spatial states. Upper line contributes to M12, lower line to wavefunction renormalization.
p-~rr + e + v,
__{1+
Iii i i [
~r---
VI2
T=O S=
T=O S--I
3
]e -'n~r
We have included the full p-propagator (q2 + m 2 ) - I in both cases. For the pion-nucleon coupling we use f ~ 2 = 0.081 and for the p - n u c l e o n coupling a value based on the analysis by H6hler and Pietarinen [6] f 2 N = 5.28. Evaluating M~2 and Mt~12 numerically for A = 15 and A = 17 using pure one-hole and one-particle harmonic oscillator wave functions (hco = 14.2 MeV) multiplied by a correlation function [7] we get the results in column 2 of table 1. The numbers in parentheses are obtained by including only pion exchange in Mr2 and approximating the p-propagator by too2 in Mt~2. These latter results differ from those obtained
by Barroso and Blin-Stoyle using the phenomenological lagrangian (above) because of the weaker 7rNA and the stronger pNN coupling assumed in our case. As a result of the consistent inclusion of p-meson exchange the contribution from exchange currents in A = 15 vanishes almost exactly and the anomaly observed by Blomqvist and by Barroso and Blin-Stoyle when going from A = 15 to A = 17 has disappeared. Further corrections may arise from processes where more than one meson is exchanged in particular from the iteration of the comparatively strong tensor component. Such processes involve highly excited intermediate states (of the order several hundred MeV) and are of short range. Significant contributions are therefore expected only in even spatial states. Contributing processes involving the exchange of two mesons are depicted in fig. 2. Diagrams in the upper row contribute to M12, those in the lower row to the renormalization correction. We will assume that the summed contributions from the processes in fig. 2 is small. The cancellation is almost complete in the triton decay [8] and because of the very high excitation energy of the intermediate states the near cancellation probably survives in heavier nuclei. There are, however, contributions from 287
Volume 81B, number 3, 4
26 February 1979
PHYSICS LETTERS
Table 1 Corrections to the axial vector operator. Decay
8 (%) Theory
lSo--+ XSN lVF--, 170
Experiment d)
la + lb a)
cpb)
Rel c)
:£a)
-0.3 (7.4) --0.8 (1.0)
-2.3 -2.4
-4.1 -2.4
-6.7 (1.0) -5.6 (-3.8)
-11.8 +-0.9 -12.4 -+0.9
a) Numbers in parentheses correspond to pion-exchange alone. b) Ref. 19]. c) Ref. [10]. d) Quoted from ref. [5 ]. core polarization involving low energy excitations that we are not justified in ignoring. Adding to the correction from exchange currents (figs. la and b) the contribution from core polarization involving 2/fop excitations [9] and relativistic corrections to the GT operator [10] we obtain a total correction to the impulse approximation of - 6 . 7 % and -5.6%, respectively. Experimentally the corrections needed are for A = 15, - 1 1.8 -+0.9% and for A = 17, - 12.4-+0.8%. We have in our analysis left out any long range correlations described by the admixture of deformed components into the wave function. It was pointed out by Blomqvist that the presence of such deformed components could lead to a reduction of the order of 10% of the expectation value of the GT-operator. For A = 17 the inclusion of/9-exchange did not change the result significantly. For A = 15, including p-exchange cut down the discrepancy by approximately 60%. The remaining discrepancy is about equal in the two cases. This, we feel, points to the importance of including p-meson exchange consistently whenever estimating the role of exchange currents in axial vector 13-decay and related processes.
288
We are indebted to G.E. Brown for suggesting the problenr and for many enlightening discussions. We are also grateful for clarifying discussions with M. Rho and for the kind help by F.C. Khanna and I.S. Towner in eliminating a numerical error.
References [1] [2] [3] [4] [5] [6] [7]
[8]
[91 [10]
M. Chemtob and M. Rho, Nucl. Phys. A163 (1971) 1. J. Blomqvist, Phys. Lett. 32B (1970) 1. D.O. Riska and G.E. Brown, Phys. Lett. 32B (1970)662. W. Jaus, Nucl. Phys. A271 (1976) 495. A. Barroso and R.J. Blin-Stoyle, Nucl. Phys. A251 (1975) 446. G. H6hler and E. Pietarinen, Nucl. Phys. B95 (1975) 210. S.-O. B/ickman, private communication; see also fig. 2 in: G.E. Brown, S.-O. B~ickman,E. Oset and W. Weise, Nucl. Phys. A286 (1977) 191. M. Rho, Axial currents and pionic mode in nuclei, Lecture given at 1978 NATO Advanced Study Institute in Nuclear Theory (Banff, Canada, 1978). We are indebted to M. Rho for bringing these arguments to our attention. H. Shimizu, M. Ichimura and A. Arima, Nucl. Phys. A226 (1974) 282. H. Ohtsubo, M. Sano and M. Morita, Prog. Theor. Phys. 49 (1973) 877.
PHYSICS LETTERS
26 February 1979
EXCHANGE CORRECTIONS TO THE AXIAL VECTOR DECAY OF 150 AND 17F~ Chr. BARGHOLTZ Department of Physics, University of Stockholm, Stockholm, Sweden and Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11 794, USA Received 4 December 1978
A calculation is made of the mesonic exchange corrections to the axial vector decays 1 5 0 -* ISN and 17F --, 170. It is found that the large correction from pion exchange for A = 15 is cancelled by the exchange of p-mesons.
The properties of nuclei cannot be properly accounted for in the impulse approximation ascribing the properties of free nucleons to the constituent nucleons. Allowance has to be made for the internucleon interactions. The effects of such interactions, through the exchange of pions, on the axial vector operator have been studied in detail by Chemtob and Rho [1 ] who apply their results to tile magnetic moments of 3H and 3He and to the 3H -+ 3He 13-decay. It was shown by Blomqvist [2] and by Riska and Brown [3] that the dominant exchange correction to this decay stems from the tensor part of the isobaric exchange current between S- and D-state components of the wave function. Including corrections from oneand two-boson exchange, relativistic effects and wave function renormalization, the theoretical prediction for the strength of the GT matrix element between 3H and 3He agrees with the experimental value [4]. Blomqvist observed that applying the same exchange operator to the 13-decay of 150 and 17F one obtains contributions to the axial vector matrix element of approximately 20% and 0% of the value obtained in the impulse approximation using harmonic oscillator wave functions. In the case of 150 this removes the theoretical prediction still further away from the experimental value. Barroso and Blin-Stoyle [5] studied mirror el-decays around 160 and 40Ca. Their two-body operator included Work partially supported by USDOE contract number EY76-S-02-3001. 286
the diagrams in figs. l a - d . Using a phenomenological lagrangian they calculated the summed correction from the diagrams l a (only n-exchange) and lb (q2 ~ m 2) to be 21.9% and 0.0% for 150 and 17F, respectively. The present letter is a report on the effects of including p-meson exchange consistently. Besides being somewhat uncertain the contributions from pair excitation (1 c) and nucleon recoil (ld) added the correction from renormalization of the wave function turned out to be quite small in the calculation of Barroso and Blin: Stoyle and will be left out here. In the static quark model we get for diagram la: 2 = 1 6 f~NgAma Mh c~=.,; M~ - M N X ((*1 X'2)+-[~-(1 + 8c~p) (~ 1 X u : ) - ( 1 - 28~o) (i(~ 1 X , 2 ) ' i - ~ I X~2) X (1 +3liner+ 3/m2r2)]
-- 2(~1 + ~2)-+ [I (1 + 8c~p)(u 1 + u2) 1
+ (1 - 2a~o ) (i(a 1 + u2)" i - ~ (1I1 + O'2)) × (1 + 3/rn~r + 3/m2r2)]
-- 2(~1 -'2)+- [~ (1 + 15c~o)(*1 - ' 2 ) + (1 - 2 a ~ . )
(t(, 5 - a2).
i -
1 (o 1 _ q2))
× (1 + 3 / m , r + 3/m 2 r2)]} e-m"r/m,~ r , where bop = 8,r~r = 1 and 8 , o = 8p, =0. For diagram lb, using PCAC to connect p -+ 2ir to
Volume 81B, number 3,4
~
PHYSICS LETTERS
26 February 1979
__~_,_e___.
M --t
A
J
N
N
N
Y=O S= I
N (b)
(o)
I
7
Y=OS=l
or T=I S=O
T VI2
T VI2
y
T,P,w.-/1 ....
N
N
N
(e)
(a)
N
Fig. 1. Some one-meson exchange processes contributing to M12. (Only a and b are included in the present calculation.)
M~12-~fTrNfONm2 { (~rrt?- m2%
(I~1X'~ 2)-+ [2 (o.1 X o.2)
e-m
m; mp m~rr -- (i(o. i X 0"2)"i -- ½(o.i X o'2) )
3 +
T=I S=O
T=O S= or
T= I S=O Fig. 2. Two meson exchange processes correcting the expectation value of the GT-operator in even spatial states. Upper line contributes to M12, lower line to wavefunction renormalization.
p-~rr + e + v,
__{1+
Iii i i [
~r---
VI2
T=O S=
T=O S--I
3
]e -'n~r
We have included the full p-propagator (q2 + m 2 ) - I in both cases. For the pion-nucleon coupling we use f ~ 2 = 0.081 and for the p - n u c l e o n coupling a value based on the analysis by H6hler and Pietarinen [6] f 2 N = 5.28. Evaluating M~2 and Mt~12 numerically for A = 15 and A = 17 using pure one-hole and one-particle harmonic oscillator wave functions (hco = 14.2 MeV) multiplied by a correlation function [7] we get the results in column 2 of table 1. The numbers in parentheses are obtained by including only pion exchange in Mr2 and approximating the p-propagator by too2 in Mt~2. These latter results differ from those obtained
by Barroso and Blin-Stoyle using the phenomenological lagrangian (above) because of the weaker 7rNA and the stronger pNN coupling assumed in our case. As a result of the consistent inclusion of p-meson exchange the contribution from exchange currents in A = 15 vanishes almost exactly and the anomaly observed by Blomqvist and by Barroso and Blin-Stoyle when going from A = 15 to A = 17 has disappeared. Further corrections may arise from processes where more than one meson is exchanged in particular from the iteration of the comparatively strong tensor component. Such processes involve highly excited intermediate states (of the order several hundred MeV) and are of short range. Significant contributions are therefore expected only in even spatial states. Contributing processes involving the exchange of two mesons are depicted in fig. 2. Diagrams in the upper row contribute to M12, those in the lower row to the renormalization correction. We will assume that the summed contributions from the processes in fig. 2 is small. The cancellation is almost complete in the triton decay [8] and because of the very high excitation energy of the intermediate states the near cancellation probably survives in heavier nuclei. There are, however, contributions from 287
Volume 81B, number 3, 4
26 February 1979
PHYSICS LETTERS
Table 1 Corrections to the axial vector operator. Decay
8 (%) Theory
lSo--+ XSN lVF--, 170
Experiment d)
la + lb a)
cpb)
Rel c)
:£a)
-0.3 (7.4) --0.8 (1.0)
-2.3 -2.4
-4.1 -2.4
-6.7 (1.0) -5.6 (-3.8)
-11.8 +-0.9 -12.4 -+0.9
a) Numbers in parentheses correspond to pion-exchange alone. b) Ref. 19]. c) Ref. [10]. d) Quoted from ref. [5 ]. core polarization involving low energy excitations that we are not justified in ignoring. Adding to the correction from exchange currents (figs. la and b) the contribution from core polarization involving 2/fop excitations [9] and relativistic corrections to the GT operator [10] we obtain a total correction to the impulse approximation of - 6 . 7 % and -5.6%, respectively. Experimentally the corrections needed are for A = 15, - 1 1.8 -+0.9% and for A = 17, - 12.4-+0.8%. We have in our analysis left out any long range correlations described by the admixture of deformed components into the wave function. It was pointed out by Blomqvist that the presence of such deformed components could lead to a reduction of the order of 10% of the expectation value of the GT-operator. For A = 17 the inclusion of/9-exchange did not change the result significantly. For A = 15, including p-exchange cut down the discrepancy by approximately 60%. The remaining discrepancy is about equal in the two cases. This, we feel, points to the importance of including p-meson exchange consistently whenever estimating the role of exchange currents in axial vector 13-decay and related processes.
288
We are indebted to G.E. Brown for suggesting the problenr and for many enlightening discussions. We are also grateful for clarifying discussions with M. Rho and for the kind help by F.C. Khanna and I.S. Towner in eliminating a numerical error.
References [1] [2] [3] [4] [5] [6] [7]
[8]
[91 [10]
M. Chemtob and M. Rho, Nucl. Phys. A163 (1971) 1. J. Blomqvist, Phys. Lett. 32B (1970) 1. D.O. Riska and G.E. Brown, Phys. Lett. 32B (1970)662. W. Jaus, Nucl. Phys. A271 (1976) 495. A. Barroso and R.J. Blin-Stoyle, Nucl. Phys. A251 (1975) 446. G. H6hler and E. Pietarinen, Nucl. Phys. B95 (1975) 210. S.-O. B/ickman, private communication; see also fig. 2 in: G.E. Brown, S.-O. B~ickman,E. Oset and W. Weise, Nucl. Phys. A286 (1977) 191. M. Rho, Axial currents and pionic mode in nuclei, Lecture given at 1978 NATO Advanced Study Institute in Nuclear Theory (Banff, Canada, 1978). We are indebted to M. Rho for bringing these arguments to our attention. H. Shimizu, M. Ichimura and A. Arima, Nucl. Phys. A226 (1974) 282. H. Ohtsubo, M. Sano and M. Morita, Prog. Theor. Phys. 49 (1973) 877.