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Electron capture to positon decay ratios and second-class currents
I
4-A
I
Nuclear Physics A232 (1974) 230-234; Not to be reproduced
by photoprint
@
North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
ELECTRON CAPTURE TO POSITON DECAY RATIOS AND SECOND-CLASS
CURRENTS
H. BEHRENS Institut
fiir
Experimentelle
Kernphysik
der
Universitiit
und des Kernforschungszentrrtms
Karlsruhe
and W. BiiHRING II. Physikalisches
Institut
Received Abstract: It is shown interaction.
that
electron
capture
der Universitiit 21 May
to B+ decay
Heidelberg
1974 ratios
are insensitive
to the induced
tensor
1. Introduction
The question of whether or not there is an induced tensor interaction in nuclear B-decay has received much attention during the last few years. Most of the relevant references can be found in the review article by Wilkinson ‘) and in the book by Blin-Stoyle “). There are some more recent papers by Eman et al. 3p4), Towner 5), and Kubodera et al. “). To obtain the theoretical expressions for the observables including induced terms is comparatively simple if the observables are given in terms of form factor coefficients. For it is the relation of the form factor coefficients to the nuclear matrix elements and coupling constants only which is modified by the induced interactions, while the observables in terms of the form factor coefficients remain unchanged. The papers 79“) which give the form factor coefficients in terms of nuclear matrix elements and coupling constants, however, pertain to p- and j3’ decay, while the LandoldBornstein tables g), which contain also formulas for electron capture, do not include induced interactions. To supply the lacking information for electron capture decays is the purpose of the present paper, in view of Vatai’s recent suggestion lo, 11) that information on the induced tensor interaction might be obtained by investigation of K/P’ ratios. Referring to the formulae for p- decay in the earlier papers *, ‘) we derive in sect. 2 the formal substitutions necessary to obtain the corresponding formulae for p’ decay and electron capture. A detailed discussion of K//I+ ratios in view of the induced tensor interaction follows in sect. 3. 230
K/p+ DECAY
231
RATIOS
2. Nuclear current and form factor coefficients The nuclear current on which our earlier paper “) (the notations and conventions
for the Dirac equation are given there) is based can be written
in case of p- decay?. For pi decay and electron capture we then need the hermitian conjugate current and obtain (assuming that the coupling constants are real)
The formulae [see refs. ‘*“)I exp ressing the form factor coefficients in terms of coupling constants and (reduced) nuclear matrix elements for p- decay, /3’ decay, and electron capture {EC) are therefore related as follows” p- decay
p* decay
EC
fM'fM-,
fM>
Pa)
fs
-As:,
WI
-+ --is --,
fTD --,-f,lA +
_fTlA,
134
f# z
+ fP/A --+ +-z+
(34
WQ
+
WI
wo
fP/A> -2, -+w; = w,+w,.
(34
{Terms with Z and W. result from the terms with A, and a/&,, respectively, in eqs. (I>, (2), and W, is the total energy of the bound state electron to be captured from the a+shell.) These are the necessary substitutions as far as the induced terms are concerned+“. In addition the well-known substitutions due to the differences in the lepton current as described in ref. “) for the pure V-A interaction have to be applied. Consequently the further substitution /3- decay 3,
/I” decay -+
-L
EC --f Iz,
t In the more recent literature this current is denoted by
(3g)
JM rather than Jp+.
tf There is a non-trivial typing error below table 6 on p. 129 of ref. 8): It should read& -+ -f, in place of _&+ -fT, in accordance with our earler publication ‘). ttt It should be noted that (3e) and (3f) only apply to the formulae expressing the form factor coefficients in terms of coupling constants and nuclear matrix elements, but not to the formulae expressing the observables in terms of form factor coefficients.
232
H. BEHRENS
AND W. Bi5HRINC
is necessary in the formulae for the form factor coefficients. But it should be noted that (3g) is to some extend an arbitrary definition as far as the signs in the relations between the form factor coefficients and the coupling constants and nuclear matrix elements are concerned. They have been chosen in accordance with ref. “) so that the substitutions given there for the formulae of the observables in terms of form factor coefficients are applicable. 3. K/j?” ratios and induced tensor interaction We here consider allowed Gamow-Teller transitions and derive formulae including correction terms, in particular those which contam contributions from the induced tensor interaction. Using the notation of ref.") we have for the K/P+ ratio 2, -= &?+
flC CK :> ffl+ q7’
(4)
and it is the possible deviation of CrJCP+ from unity which we are interested in. The bar denotes the appropriate average of the shape factor over the j3’ spectrum. We have from ref. “) CK = [n/r,& I>-m,(L 1>]” = JC(L I>, @a) 6
= A!@, l)i-m:(l,
l)= N(l,
+
i&(1, l.)wl,(l, 1) 11,
(5-b)
with?
in the case of @’ decay and M,(I, 1) = -“Jq,,
~~~A~~~o~~~lz~-~~o~] -j~VF1011C~OI[ZI+3(Wof2~~)R],
(6bI
t It is sufficient for this investigation to use the simpler approximate formulae of ref. ‘1 rather than the more accurate expressions of ref. *), which are more complicated as far as the aZ terms are concerned.
K/p+ DECAY
-*Fy,l *F"110
233
RATIOS
=
+a [1+(j&) (wo+wK-: “$)]Jq
=
+nJi
y5
vF”111 =
4 x
-
2
;
+
6,
f*$JJ I
s
UXY -7
R
(fQ-4 Pb)
in the case of EC. Here we have used ]/3a = - ] b, where the old Cartesian notation for the Gamow-Teller matrix element ]g and for the other nuclear matrix elements is employed. In eqs. (7), (8) mainly correction terms from the induced tensor interaction have been retained. The vector form factor coefficient “F~,, does not contain induced tensor terms but has not been discarded since it gives the probably most important contribution of the conventional correction terms. From eqs. (5)-(9) we obtain finally
The quantities m, (1, 1) and some higher-order contributions omitted in eqs. (6) [see refs. **“)I would lead to additional terms in eq. (lo), some of which are of the same order. The simple eq. (lo), therefore, is not suitable to fit the experimental data, but has been derived to demonstrate the order of magnitude of the effects expected. Since W, s 1 and one knows from other experiments i, “) that (fr/L) s 3 x 10v3, the contribution of the induced tensor interaction to K/P+ ratios is very small and would be masked by the probably larger contributions of the conventional (the so-called second-forbidden) correction terms. Our eq. (10) is substantially different from the corresponding formula of Vatai and so is our conclusion. This is curious since Vatai’s derivation and ours proceed along the same lines. Although Vatai’s treatment is quite correct in principle, it seems to us that some sign errors occurred, unfortunately, which are responsible for the wrong result. Our conclusion that the K/p+ ratios are insensitive to the induced tensor interaction is not changed in view of some recent papers by Kubodera, Delorme, and Rho “) who point out that the nuclear current (1) is still too simple since off-shell and mesonic effects have been neglected. The off-shell effect in the simplest form corresponding to eq. (3) of ref. “) can easily be incorporated into our treatment. It is simply to replace? fr byfr +f; in eq. (20b) and in the second term of eq. (20d) of ref. “) [while fT remains unchanged in the third term of eq. (20d) of ref. “)I, and to perform the resulting substitutions in table 6 of ref. “). For the present paper this implies replacement of fT by fT +fi in t The coupling constants are denoted in ref. “) by gT, 9~’ in place of fT, J$‘. An additional, independent coupling constant 9 T’ = fT’ is needed for the off-shell effect.
234
H. BEHRENS
AND W. BiiHRING
eqs. @a) and (8b) while eqs. (7a) and (7b) and also the final formula (10) remain unchanged. As to the additional contribution of the induced tensor interaction caused by mesonic effects 6), we have the same situation as in eq. (10) that a small effect proportional to W, might occur. References 1) D. H. Wilkinson, Proc. Roy. Sot. 70 (1971/72) 307 2) 3) 4) 5) 6) 7) 8) 9)
R. J. Blin-Stoyle, Fundamental interactions and the nucleus (North-Holland, Amsterdam, 1973) B. Eman, D. Tadic, F. Krmpotic and L. Szybisz, Phys. Rev. C6 (1972) 1 B. Eman, B. Guberina and D. Tadic, Phys. Rev. C8 (1973) 1301 I. S. Towner, Nucl. Phys. A216 (1973) 589 K. Kubodera, J. Delorme and M. Rho, Nucl. Phys. B66 (1973) 253 W. Btihring and L. Schiilke, Nucl. Phys. 65 (1965) 369 H. Behrens and W. Btihring, Nucl. Phys. Al62 (1971) 111 H. Behrens and J. Janecke, Numerical tables for beta decay and electron capture, LandoltBornstein, New series, vol. I/4 (Springer, Berlin, 1969) 10) E. Vatai, Phys. Lett. 34B (1971) 395 11) E. Vatai, Atomki Kozlemenyek 14 (1972) 233
4-A
I
Nuclear Physics A232 (1974) 230-234; Not to be reproduced
by photoprint
@
North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
ELECTRON CAPTURE TO POSITON DECAY RATIOS AND SECOND-CLASS
CURRENTS
H. BEHRENS Institut
fiir
Experimentelle
Kernphysik
der
Universitiit
und des Kernforschungszentrrtms
Karlsruhe
and W. BiiHRING II. Physikalisches
Institut
Received Abstract: It is shown interaction.
that
electron
capture
der Universitiit 21 May
to B+ decay
Heidelberg
1974 ratios
are insensitive
to the induced
tensor
1. Introduction
The question of whether or not there is an induced tensor interaction in nuclear B-decay has received much attention during the last few years. Most of the relevant references can be found in the review article by Wilkinson ‘) and in the book by Blin-Stoyle “). There are some more recent papers by Eman et al. 3p4), Towner 5), and Kubodera et al. “). To obtain the theoretical expressions for the observables including induced terms is comparatively simple if the observables are given in terms of form factor coefficients. For it is the relation of the form factor coefficients to the nuclear matrix elements and coupling constants only which is modified by the induced interactions, while the observables in terms of the form factor coefficients remain unchanged. The papers 79“) which give the form factor coefficients in terms of nuclear matrix elements and coupling constants, however, pertain to p- and j3’ decay, while the LandoldBornstein tables g), which contain also formulas for electron capture, do not include induced interactions. To supply the lacking information for electron capture decays is the purpose of the present paper, in view of Vatai’s recent suggestion lo, 11) that information on the induced tensor interaction might be obtained by investigation of K/P’ ratios. Referring to the formulae for p- decay in the earlier papers *, ‘) we derive in sect. 2 the formal substitutions necessary to obtain the corresponding formulae for p’ decay and electron capture. A detailed discussion of K//I+ ratios in view of the induced tensor interaction follows in sect. 3. 230
K/p+ DECAY
231
RATIOS
2. Nuclear current and form factor coefficients The nuclear current on which our earlier paper “) (the notations and conventions
for the Dirac equation are given there) is based can be written
in case of p- decay?. For pi decay and electron capture we then need the hermitian conjugate current and obtain (assuming that the coupling constants are real)
The formulae [see refs. ‘*“)I exp ressing the form factor coefficients in terms of coupling constants and (reduced) nuclear matrix elements for p- decay, /3’ decay, and electron capture {EC) are therefore related as follows” p- decay
p* decay
EC
fM'fM-,
fM>
Pa)
fs
-As:,
WI
-+ --is --,
fTD --,-f,lA +
_fTlA,
134
f# z
+ fP/A --+ +-z+
(34
WQ
+
WI
wo
fP/A> -2, -+w; = w,+w,.
(34
{Terms with Z and W. result from the terms with A, and a/&,, respectively, in eqs. (I>, (2), and W, is the total energy of the bound state electron to be captured from the a+shell.) These are the necessary substitutions as far as the induced terms are concerned+“. In addition the well-known substitutions due to the differences in the lepton current as described in ref. “) for the pure V-A interaction have to be applied. Consequently the further substitution /3- decay 3,
/I” decay -+
-L
EC --f Iz,
t In the more recent literature this current is denoted by
(3g)
JM rather than Jp+.
tf There is a non-trivial typing error below table 6 on p. 129 of ref. 8): It should read& -+ -f, in place of _&+ -fT, in accordance with our earler publication ‘). ttt It should be noted that (3e) and (3f) only apply to the formulae expressing the form factor coefficients in terms of coupling constants and nuclear matrix elements, but not to the formulae expressing the observables in terms of form factor coefficients.
232
H. BEHRENS
AND W. Bi5HRINC
is necessary in the formulae for the form factor coefficients. But it should be noted that (3g) is to some extend an arbitrary definition as far as the signs in the relations between the form factor coefficients and the coupling constants and nuclear matrix elements are concerned. They have been chosen in accordance with ref. “) so that the substitutions given there for the formulae of the observables in terms of form factor coefficients are applicable. 3. K/j?” ratios and induced tensor interaction We here consider allowed Gamow-Teller transitions and derive formulae including correction terms, in particular those which contam contributions from the induced tensor interaction. Using the notation of ref.") we have for the K/P+ ratio 2, -= &?+
flC CK :> ffl+ q7’
(4)
and it is the possible deviation of CrJCP+ from unity which we are interested in. The bar denotes the appropriate average of the shape factor over the j3’ spectrum. We have from ref. “) CK = [n/r,& I>-m,(L 1>]” = JC(L I>, @a) 6
= A!@, l)i-m:(l,
l)= N(l,
+
i&(1, l.)wl,(l, 1) 11,
(5-b)
with?
in the case of @’ decay and M,(I, 1) = -“Jq,,
~~~A~~~o~~~lz~-~~o~] -j~VF1011C~OI[ZI+3(Wof2~~)R],
(6bI
t It is sufficient for this investigation to use the simpler approximate formulae of ref. ‘1 rather than the more accurate expressions of ref. *), which are more complicated as far as the aZ terms are concerned.
K/p+ DECAY
-*Fy,l *F"110
233
RATIOS
=
+a [1+(j&) (wo+wK-: “$)]Jq
=
+nJi
y5
vF”111 =
4 x
-
2
;
+
6,
f*$JJ I
s
UXY -7
R
(fQ-4 Pb)
in the case of EC. Here we have used ]/3a = - ] b, where the old Cartesian notation for the Gamow-Teller matrix element ]g and for the other nuclear matrix elements is employed. In eqs. (7), (8) mainly correction terms from the induced tensor interaction have been retained. The vector form factor coefficient “F~,, does not contain induced tensor terms but has not been discarded since it gives the probably most important contribution of the conventional correction terms. From eqs. (5)-(9) we obtain finally
The quantities m, (1, 1) and some higher-order contributions omitted in eqs. (6) [see refs. **“)I would lead to additional terms in eq. (lo), some of which are of the same order. The simple eq. (lo), therefore, is not suitable to fit the experimental data, but has been derived to demonstrate the order of magnitude of the effects expected. Since W, s 1 and one knows from other experiments i, “) that (fr/L) s 3 x 10v3, the contribution of the induced tensor interaction to K/P+ ratios is very small and would be masked by the probably larger contributions of the conventional (the so-called second-forbidden) correction terms. Our eq. (10) is substantially different from the corresponding formula of Vatai and so is our conclusion. This is curious since Vatai’s derivation and ours proceed along the same lines. Although Vatai’s treatment is quite correct in principle, it seems to us that some sign errors occurred, unfortunately, which are responsible for the wrong result. Our conclusion that the K/p+ ratios are insensitive to the induced tensor interaction is not changed in view of some recent papers by Kubodera, Delorme, and Rho “) who point out that the nuclear current (1) is still too simple since off-shell and mesonic effects have been neglected. The off-shell effect in the simplest form corresponding to eq. (3) of ref. “) can easily be incorporated into our treatment. It is simply to replace? fr byfr +f; in eq. (20b) and in the second term of eq. (20d) of ref. “) [while fT remains unchanged in the third term of eq. (20d) of ref. “)I, and to perform the resulting substitutions in table 6 of ref. “). For the present paper this implies replacement of fT by fT +fi in t The coupling constants are denoted in ref. “) by gT, 9~’ in place of fT, J$‘. An additional, independent coupling constant 9 T’ = fT’ is needed for the off-shell effect.
234
H. BEHRENS
AND W. BiiHRING
eqs. @a) and (8b) while eqs. (7a) and (7b) and also the final formula (10) remain unchanged. As to the additional contribution of the induced tensor interaction caused by mesonic effects 6), we have the same situation as in eq. (10) that a small effect proportional to W, might occur. References 1) D. H. Wilkinson, Proc. Roy. Sot. 70 (1971/72) 307 2) 3) 4) 5) 6) 7) 8) 9)
R. J. Blin-Stoyle, Fundamental interactions and the nucleus (North-Holland, Amsterdam, 1973) B. Eman, D. Tadic, F. Krmpotic and L. Szybisz, Phys. Rev. C6 (1972) 1 B. Eman, B. Guberina and D. Tadic, Phys. Rev. C8 (1973) 1301 I. S. Towner, Nucl. Phys. A216 (1973) 589 K. Kubodera, J. Delorme and M. Rho, Nucl. Phys. B66 (1973) 253 W. Btihring and L. Schiilke, Nucl. Phys. 65 (1965) 369 H. Behrens and W. Btihring, Nucl. Phys. Al62 (1971) 111 H. Behrens and J. Janecke, Numerical tables for beta decay and electron capture, LandoltBornstein, New series, vol. I/4 (Springer, Berlin, 1969) 10) E. Vatai, Phys. Lett. 34B (1971) 395 11) E. Vatai, Atomki Kozlemenyek 14 (1972) 233