We use the random phase approximation to describe the muon capture rate on
${}^{44}$Ca,${}^{48}$Ca, ${}^{56}$Fe, ${}^{90}$Zr, and ${}^{208}$Pb. With
${}^{40}$Ca as a test case, we show that the Continuum Random Phase
Approximation (CRPA) and the standard RPA give essentially equivalent
descriptions of the muon capture process. Using the standard RPA with the free
nucleon weak form factors we reproduce the experimental total capture rates on
these nuclei quite well. Confirming our previous CRPA result for the $N = Z$
nuclei, we find that the calculated rates would be significantly lower than the
data if the in-medium quenching of the axial-vector coupling constant were
employed.Comment: submitted to Phys. Rev. 

B. H. Wildenthal

C. D. Goodman

E. Caurier

E. K. Warburton

E. Kolbe

G. A. Rinker

G. F. Bertsch

G. Martinez-Pinedo

H. Primakoff

J. D. Walecka

J. Delorme

K. Langanke

M. Buballa

M. L. Goldberger

M. R. Plunket

N. C. Mukhopadhyay

O. Häusser

P. Vogel

T. Suzuki

T. W. Donnelly

English

arXiv

We use the random phase approximation to describe the muon capture rate on 44Ca, 48Ca, 56Fe, 90Zr, and 208Pb. With 40Ca as a test case, we show that the continuum random phase approximation (CRPA) and the standard RPA give essentially equivalent descriptions of the muon capture process. Using the standard RPA with the free nucleon weak form factors we reproduce the experimental total capture rates on these nuclei quite well. Thus, unlike the allowed β decays, with very small momentum transfer and exclusively the 1+ multipole, the muon capture rate, with momentum transfer of approximately muon mass magnitude, and a mixture of multipoles (dominated by 1-), does not require any quenching of the axial-vector coupling constant

Kolbe, E.

Langanke, K.

Vogel, P.

Caltech Authors - Main

PHYSICAL REVIEW C, VOLUME 62, 055502Muon capture on nuclei with NÌZ, random phase approximation, and in-medium value
of the axial-vector coupling constant
E. Kolbe
Physics Department, University of Basel, CH-4056 Basel, Switzerland
K. Langanke
Institute for Physics and Astronomy, University of Aarhus, DK-8000 Aarhus, Denmark
P. Vogel
Physics Department, California Institute of Technology, Pasadena, California 91125
~Received 6 June 2000; published 9 October 2000!
We use the random phase approximation to describe the muon capture rate on44Ca, 48Ca, 56Fe, 90Zr, and
208Pb. With 40Ca as a test case, we show that the continuum random phase approximation~CRPA! and the
standard RPA give essentially equivalent descriptions of the muon capture process. Using the standard RPA
with the free nucleon weak form factors we reproduce the experimental total capture rates on these nuclei quite
well. Thus, unlike the allowedb decays, with very small momentum transfer and exclusively the 11 multipole,
the muon capture rate, with momentum transfer of approximately muon mass magnitude, and a mixture of
multipoles~dominated by 12), does not require any quenching of the axial-vector coupling constant.
PACS number~s!: 24.30.Cz, 23.40.Hco
u
t
s
b
nd
ie
o
in
t
am
th
t
t
oo
s
ni
ra
se
re
tu
fre
er
in
nt.
e
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gle
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ner-
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tive
he
s
d
as
r
der
-
e
ex-
rity
uon
tudy
nt,
has
the
ma-
rs,The capture of a negative muon from the atomic 1s orbit,
m21~Z,N!→nm1~Z21,N11!* , ~1!
is a semileptonic weak process which has been studied f
long time ~see, e.g., the reviews by Walecka@1# or Mukho-
padhyay@2# and the earlier references therein!. The total cap-
ture rate has been measured for many nuclei@3#; in some
cases the partial capture rates to specific states in the da
ter nucleus have been determined as well.
The nuclear response in muon capture is governed by
momentum transfer which is of the order of the muon ma
The energy transferred to the nucleus is restricted from
low by the mass difference of the initial and final nuclei, a
from above by the muon mass. The phase space and
nuclear response favor lower nuclear excitation energ
There is an intimate relation between the inclusive mu
capture rate and the cross section for the antineutr
induced charged-current reactions; both are governed by
same nuclear matrix elements and proceed from the s
initial to the same final states.
Since the experimental data are quite precise, and the
oretical techniques of evaluating the nuclear response in
relevant regime are well developed, it is worthwhile to see
what extent the capture rates are theoretically underst
This is not only interesting per se, but should be viewed a
more general test of our ability to describe semilepto
weak charged-current reactions.
A step in this direction was undertaken by us seve
years ago in Ref.@4# where the continuum random pha
approximation~CRPA! was used to describe muon captu
on theN5Z nuclei 12C, 16O, and40Ca. We showed that the
method allows us to reproduce the experimental total cap
rates on these nuclei to better than 10% using the
nucleon weak form factors and two different residual int
actions. In particular, it was not necessary to apply the0556-2813/2000/62~5!/055502~4!/$15.00 62 0555r a
gh-
he
s.
e-
the
s.
n
o-
he
e
e-
he
o
d.
a
c
l
re
e
-
-
medium quenching of the axial vector coupling consta
This is contrary to various well-known indications that th
axial-vector coupling constantgA in nuclear medium is re-
duced from its free nucleon value ofgA51.26 to the value of
gA.1 when one analyzes the data onb decay between low-
lying states of the (sd) shell nuclei@5# and (p f) shell nuclei
@6#. In addition, the ‘‘missing Gamow-Teller strength’’ prob
lem, as revealed in the interpretation of the forward-an
(p,n) and (n,p) charge-exchange reactions@7#, is also often
quoted as evidence for quenching ofgA . Note that the
Gamow-Teller~GT! strength is concentrated in the giant G
resonance at excitation energies not very far from the e
gies involved in the muon capture, although this latter p
cess is usually dominated by the transitions to the nega
parity spin-dipole states. That is so in particular in t
double-magic nuclei16O and 40Ca where the GT strength i
strongly suppressed, while in12C it is essentially exhauste
by the transition to the analog of theT51 state at 15.11
MeV in 12C, whose contribution to the muon capture rate h
been subtracted in@4#. It is thus of interest to inquire whethe
a similar quenching applies in muon capture over a broa
range of nuclei.
There is a consensus that the quenching ofA in b decay
@and (p,n) reactions# is related to the neglect of configura
tions outside the usual 0\v shell model space and/or th
effect of meson-exchange currents. Thus, generally, one
pects that the quenching phenomenon might be multipola
and momentum transfer dependent. Evaluation of the m
capture rate makes it possible to extend the quenching s
to higher multipoles. Knowing whether, and to what exte
the quenching is needed to describe muon capture rates
practical application as well. For example, evaluation of
cross section for supernova neutrino detectors, or of the
trix elements for the neutrinoless doubleb decay involves
higher than 11 multipoles, and larger momentum transfe©2000 The American Physical Society02-1
n-
t
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n
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co
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.54
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hase
E. KOLBE, K. LANGANKE, AND P. VOGEL PHYSICAL REVIEW C62 055502than ordinaryb decay. Another application, already me
tioned, involves the neutrino~and antineutrino! induced
nuclear reactions, e.g. from the neutrinos associated with
pion and muon decays at rest.
In this paper we extend the previous calculation@4# to
heavier nuclei, in particular nuclei with the neutron exce
i.e., with a nonvanishing value of the initial isospin. In@4#
the continuum random phase approximation~CRPA! was
used, a method shown to be successful for the descriptio
the nuclear response to weak and electromagnetic probe@8#.
The method combines the usual RPA treatment with the
rect description of the continuum nucleon decay channel.
heavierN.Z nuclei that method is computationally qui
demanding. Moreover, it can describe only transitions
states in the daughter nucleus above the particle emis
threshold. In order to evaluate the total capture rate one
to include also the transitions to bound states, which cont
ute relatively more, and are typically not experimenta
separated in the heavier nuclei.
If one is interested in thetotal capture rate, the numeri
cally simpler standard RPA is just as good. As an additio
bonus, it avoids the distinction between the bound and
bound states. That the two methods, CRPA and stan
RPA, are equivalent for the present purpose is demonstr
in Fig. 1 for the case of40Ca. We show in the upper part th
differential capture rate as a function of the excitation ene
in the final nucleus. While the CRPA~for details see Ref.
@8#! is characterized by a continuous curve, nonvanish
everywhere above the threshold, the standard RPA is c
acterized by the ‘‘picket fence,’’ since there is a finite num
ber of discrete final states. The similarity of both methods
even better seen in the lower part of Fig. 1 which shows
integrated rate up to a given excitation energyE* . There
appears to be a slight systematic shift of a few MeV~caused
by the bound state contribution, presumably!, but the final
capture rates, and the typical excitation energies, are rem
ably similar. Thus, we use the standard RPA for the eva
tion of the muon capture in the selectedN.Z nuclei.
For the calculation in the present work we used the p
nomenological Landau-Migdal force with parameters t
has been shown to be applicable for a wide range of nu
@9#. All single-particle states below the Fermi level were i
cluded and two oscillator shells above it were taken i
account. Using again40Ca as a test case, we checked th
adding or subtracting in the calculation few subshells ab
the Fermi level does not visibly change the muon capt
rate. However, in208Pb enlarging the single particle spa
leads to an increase of the capture rate by about 5%. The
nucleon form factors were used to describe the weak nuc
current. In particular, the unquenched axial vector coupl
constantgA(0)51.26 was used.
Muon capture also depends on the induced pseudos
hadronic weak current. At the free nucleon level the cor
sponding coupling constant is determined by the Goldber
Treiman relation@10#
FP~q
2!5
2M pgA~0!
mp
2 2q2
, ~2!05550he
,
of
r-
or
o
on
as
-
l
n-
rd
ed
y
g
r-
-
s
e
rk-
-
-
t
ei
o
t
e
e
ee
ar
g
lar
-
r-
wheremp is the pion mass andgA(0)51.26.@In muon cap-
ture one often uses a dimensionless quantitygP
5mmFP(q
2) at the relevant momentum transferq2.
20.9mm
2 , such thatgP.8.4 for free protons.# In nuclear me-
dium FP can be renormalized again, and this renormalizat
does not necessarily obey the Goldberger-Treiman rela
@11#. We have shown in our previous work that the to
muon capture rates are not sensitive enough to the var
choices of FP renormalization. Consequently, througho
this work we use the Goldberger-Treiman relation.
The calculated total capture rates are collected in Tab
and compared with the data@3#. For comparison we also
show in Table I the earlier results@4# for the N5Z nuclei,
evaluated with the same residual Landau-Migdal for
Among the nuclei in Table I,56Fe is the only one with a
substantial contribution of the Gamow-Teller transition
which are known to be quenched. Hence the 11 part of the
capture rate was quenched by the empirical factor of 2
obtained by comparing the total experimental and calcula
GT strength. This reduced 11 rate was then combined with
the unquenched rate from the other multipoles in colum
of Table I. Clearly, a better agreement with experiment
all considered nuclei is achieved when the full value ofgA is
FIG. 1. The m2 capture rate as a function of the excitatio
energy in the final nucleus40K ~upper panel!. The continuous curve
is for the continuum random phase approximation~CRPA!, while
the dashed vertical bars are the results of the standard random p
approximation~SRPA!. In the lower panel the integratedm2 cap-
ture rate, up to the excitation energyE* , is shown.2-2
al
MUON CAPTURE ON NUCLEI WITHN.Z, RANDOM . . . PHYSICAL REVIEW C 62 055502TABLE I. m2-capture rates calculated within the standard~SRPA! and continuum~CRPA! models in
units of 103 s. The radiusr and diffusenessd of the extended nuclear charge distribution were set to (r ,d)
5(1.07 fm, 0.50 fm) for12C and (r ,d)5(1.07 fm, 0.57 fm) for all other nuclei. The Landau-Migd
force ~LM ! is used throughout.
Nucleus Expt.@3# SRPA SRPA CRPA~LM ! CRPA~LM !
gA(0)51.26 gA(0)51.0 gA(0)51.26 gA(0)51.0
12C 32.860.8 31.3c 22.9c
16O 102.6 0.6 103.2 75.8
40Ca 254467 a 2547 1846 2489 1800
44Ca 1793640 1722 1238
48Ca 1164b 1301 930
56Fe 44006100 4460c 3430c
90Zr 93506100 10288 7400
208Pb 134506180 16057 11436
aCorrected from the data for natural Ca.
bExtrapolated using the Primakoff formula fitted to40Ca and44Ca.
cCalculated with partial occupation of the single particle subshells; see@17#.m
i-
he
-
ic
-
er-
th
er-
dy a
on
the
a
te
nt
ec-
nce
ole
the-
the
re
eak
tion
rents.
od,
ar-
ed
d
e
it
ar
r-
theused. The quenched value leads to an obvious underesti
of the muon capture rate.
In Fig. 2 the fractional contributions of different mult
poles are shown for16O, 48Ca, 90Zr, and 208Pb. For the
closed shell nucleus16O the negative parity 12 and 22, and
to lesser extent 02, multipoles dominate, as expected. In t
intermediate mass nuclei,48Ca and 90Zr, the dominance of
these multipolarities is less pronounced. Finally, in208Pb the
positive parity 11 and 21 multipoles give the largest contri
bution, since the negative parity proton hole-neutron part
states are blocked. Note, however, that these 11 and 21
states correspond to the 2\v excitations involving different
major shells, not the usual 0\v Gamow-Teller transitions.
The dominance of dipole transitions for them2 capture
process in thep and s,d shell nuclei is a well-known phe
FIG. 2. Fractional contribution of different multipoles to th
total capture rate for four of the considered nuclei. The entries w
I p502,12,22 correspond to the 1\v excitations, while the entries
with I p511,21,31 correspond to the 2\v excitations. The data
points are connected by lines for better visibility and the nuclei
as indicated in the figure.05550ate
le
nomenon. It can be exploited in the generalized Goldhab
Teller model@12#, where all dipole and spin-dipole streng
is concentrated in a single collective state. The Goldhab
Teller model has been applied to the muon capture alrea
long time ago@12#. Here we have repeated the calculati
with one modification. Instead of using as the energy of
single collective state the energyED of the giant dipole reso-
nance in the initial nucleus, we placed the strength into
state at the excitation energyE* 5ED2EIAS in the final
nucleus, whereEIAS is the energy of the isobar analog sta
with T51 in the initial N5Z nucleus. Such an assignme
puts the strength close to the centroid of the excitation sp
trum obtained in the RPA. The method is rather crude, si
more detailed calculations clearly show that the spin-dip
strength is spread over a sizable energy interval. Never
less, the results displayed in Table II, obtained again with
full value of gA(0)51.26, support our conclusion that the
is no quenching in these nuclei of the operators of the w
current that change parity.
First forbiddenb decays, in particular those withDI p
502, have been often analyzed as a source of informa
on the enhancement caused by the meson exchange cur
This enhancement, whose origin is however well understo
is an example of the multipolarity dependence of the app
ent in-medium effects. In the context of the work report
here it is worthwhile to quote the work of Warburton an
Towner @13# who analyzed 18 first forbiddenb decays in
h
e
TABLE II. m2-capture rates calculated within the Goldhabe
Teller model in units of 103 s. The formulas of Ref.@12# are used.
In column 2 the original parameters are employed. In column 3
excitation energyE* is modified, as explained in the text.
Nucleus Expt.@3# Orig. values@12# Modified E*
12C 32.860.8 29.7 35.1
16O 102.6 0.6 79.2 123
28Si 87162 657 970
40Ca 254467 1490 27302-3
u
e
.
r-
lly
a
f
a
lu
in
c
rong
rity
re
PA
de-
so-
-
ny
tion
ul-
al
by
3-
E. KOLBE, K. LANGANKE, AND P. VOGEL PHYSICAL REVIEW C62 055502the lead region. They used a truncated shell model and fo
that theDI p512 transitions, which are unaffected by th
meson exchange currents, do not require any quenching
fact, their fit to an overall quenching factor for the first fo
bidden matrix elements results insq150.9860.05, compat-
ible with unity, i.e., with no quenching. That analysis, tota
independent of our evaluation of the muon capture, ag
supports the conclusion about the apparent absence o
appreciable quenching of the parity-changing~i.e., first for-
bidden! weak current operators.
The unique second forbiddenb decays are governed by
single operator 2@Y2s#
l53. For only a handful of them the
partial decay rates are known. When analyzed@14,15#, these
transitions do not allow one to draw any definitive conc
sion about the possible quenching of the correspond
strength. However, as pointed out above,m2 capture rate in
90Zr, and 208Pb, is strongly affected by the 2\v transitions,
whose operators are related to the second forbiddenb de-
cays. Since the agreement between the experimental and
culated rates in these two nuclei, and in particular in208Pb, is.
ys
,
05550nd
In
in
an
-
g
al-
not as good as in the other cases, we cannot make a st
statement regarding the quenching of the positive pa
‘‘second forbidden’’ multipoles based on the muon captu
calculations.
In conclusion, the present analysis shows that the CR
and SRPA methods are capable of describing the totalm2
capture rates quite well in a large range of nuclei. The
pendence of the muon capture rate on the isospin, the
called Primakoff rule@16#, is also reasonably well repro
duced. There is no indication of the necessity to apply a
quenching to the operators responsible for them2 process.
Thus our findings can be used as guidance in the evalua
of other semileptonic weak processes involving higher m
tipoles, i.e., transitions involving other than the 0\v spin
changing operators.
This work was supported in part by the Swiss Nation
Science Foundation, the Danish Research Council, and
the U.S. Department of Energy, Contract No. DE-F60
88ER-40397.s.@1# J. D. Walecka, inMuon Physics II, edited by V. W. Hughes
and C. S. Wu~Academic, New York, 1975!, p. 113.
@2# N. C. Mukhopadhyay, Phys. Rep., Phys. Lett.30C, 1 ~1977!.
@3# T. Suzuki, D. F. Measday, and J. P. Roalsvig, Phys. Rev. C35,
2212 ~1987!.
@4# E. Kolbe, K. Langanke, and P. Vogel, Phys. Rev. C50, 2576
~1994!.
@5# B. H. Wildenthal, Prog. Part. Nucl. Phys.11, 5 ~1984!.
@6# E. Caurier, A. Poves, and A. P. Zuker, Phys. Rev. Lett.74,
1517 ~1995!.
@7# C. D. Goodman and S. B. Bloom, inSpin Excitations in Nu-
clei, edited by F. Petrovichet al. ~Plenum, New York, 1983!,
p. 143; G. F. Bertsch and H. Esbensen, Rep. Prog. Phys50,
607 ~1987!; O. Häusseret al., Phys. Rev. C43, 230 ~1991!.
@8# M. Buballa, S. Drozdz, S. Krewald, and J. Speth, Ann. Ph
~N.Y.! 208, 346 ~1991!; E. Kolbe, K. Langanke, S. Krewald.
and F. K. Thielemann, Nucl. Phys.A540, 599 ~1992!.
@9# G. A. Rinker and J. Speth, Nucl. Phys.A306, 360 ~1978!; M.
R. Plunketet al., Phys. Rev. C56, 263 ~1997!.
@10# M. L. Goldberger and S. B. Treiman, Phys. Rev.111, 354
~1958!.
@11# J. Delorme and M. Ericson, Phys. Rev. C49, R1763~1994!.
@12# T. W. Donnelly, J. Dubach, and W. C. Haxton, Nucl. Phy
A251, 353 ~1975!.
@13# E. K. Warburton and I. S. Towner, Phys. Rep.243, 103~1994!.
@14# G. Martinez-Pinedo and P. Vogel, Phys. Rev. Lett.81, 281
~1998!.
@15# E. K. Warburton, Phys. Rev. C45, 463 ~1992!.
@16# H. Primakoff, Rev. Mod. Phys.31, 802 ~1959!.
@17# E. Kolbe, K. Langanke, and P. Vogel, Nucl. Phys.A652, 91
~1999!.2-4


Muon capture on nuclei with N>Z, random phase approximation, and in-medium value of the axial-vector coupling constant

Crossref

Physical Review C

Muon capture on nuclei with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo></mml:math>random phase approximation, and in-medium value of the axial-vector coupling constant

PHYSICAL REVIEW C, VOLUME 62, 055502Muon capture on nuclei with NÌZ , random phase approximation, and in-medium value
of the axial-vector coupling constant
E. Kolbe
Physics Department, University of Basel, CH-4056 Basel, Switzerland
K. Langanke
Institute for Physics and Astronomy, University of Aarhus, DK-8000 Aarhus, Denmark
P. Vogel
Physics Department, California Institute of Technology, Pasadena, California 91125
~Received 6 June 2000; published 9 October 2000!
We use the random phase approximation to describe the muon capture rate on 44Ca, 48Ca, 56Fe, 90Zr, and
208Pb. With 40Ca as a test case, we show that the continuum random phase approximation ~CRPA! and the
standard RPA give essentially equivalent descriptions of the muon capture process. Using the standard RPA
with the free nucleon weak form factors we reproduce the experimental total capture rates on these nuclei quite
well. Thus, unlike the allowed b decays, with very small momentum transfer and exclusively the 11 multipole,
the muon capture rate, with momentum transfer of approximately muon mass magnitude, and a mixture of
multipoles ~dominated by 12), does not require any quenching of the axial-vector coupling constant.
PACS number~s!: 24.30.Cz, 23.40.HcThe capture of a negative muon from the atomic 1s orbit,
m21~Z ,N !→nm1~Z21,N11 !*, ~1!
is a semileptonic weak process which has been studied for a
long time ~see, e.g., the reviews by Walecka @1# or Mukho-
padhyay @2# and the earlier references therein!. The total cap-
ture rate has been measured for many nuclei @3#; in some
cases the partial capture rates to specific states in the daugh-
ter nucleus have been determined as well.
The nuclear response in muon capture is governed by the
momentum transfer which is of the order of the muon mass.
The energy transferred to the nucleus is restricted from be-
low by the mass difference of the initial and final nuclei, and
from above by the muon mass. The phase space and the
nuclear response favor lower nuclear excitation energies.
There is an intimate relation between the inclusive muon
capture rate and the cross section for the antineutrino-
induced charged-current reactions; both are governed by the
same nuclear matrix elements and proceed from the same
initial to the same final states.
Since the experimental data are quite precise, and the the-
oretical techniques of evaluating the nuclear response in the
relevant regime are well developed, it is worthwhile to see to
what extent the capture rates are theoretically understood.
This is not only interesting per se, but should be viewed as a
more general test of our ability to describe semileptonic
weak charged-current reactions.
A step in this direction was undertaken by us several
years ago in Ref. @4# where the continuum random phase
approximation ~CRPA! was used to describe muon capture
on the N5Z nuclei 12C, 16O, and 40Ca. We showed that the
method allows us to reproduce the experimental total capture
rates on these nuclei to better than 10% using the free
nucleon weak form factors and two different residual inter-
actions. In particular, it was not necessary to apply the in-0556-2813/2000/62~5!/055502~4!/$15.00 62 0555medium quenching of the axial vector coupling constant.
This is contrary to various well-known indications that the
axial-vector coupling constant gA in nuclear medium is re-
duced from its free nucleon value of gA51.26 to the value of
gA.1 when one analyzes the data on b decay between low-
lying states of the (sd) shell nuclei @5# and (p f ) shell nuclei
@6#. In addition, the ‘‘missing Gamow-Teller strength’’ prob-
lem, as revealed in the interpretation of the forward-angle
(p ,n) and (n ,p) charge-exchange reactions @7#, is also often
quoted as evidence for quenching of gA . Note that the
Gamow-Teller ~GT! strength is concentrated in the giant GT
resonance at excitation energies not very far from the ener-
gies involved in the muon capture, although this latter pro-
cess is usually dominated by the transitions to the negative
parity spin-dipole states. That is so in particular in the
double-magic nuclei 16O and 40Ca where the GT strength is
strongly suppressed, while in 12C it is essentially exhausted
by the transition to the analog of the T51 state at 15.11
MeV in 12C, whose contribution to the muon capture rate has
been subtracted in @4#. It is thus of interest to inquire whether
a similar quenching applies in muon capture over a broader
range of nuclei.
There is a consensus that the quenching of gA in b decay
@and (p ,n) reactions# is related to the neglect of configura-
tions outside the usual 0\v shell model space and/or the
effect of meson-exchange currents. Thus, generally, one ex-
pects that the quenching phenomenon might be multipolarity
and momentum transfer dependent. Evaluation of the muon
capture rate makes it possible to extend the quenching study
to higher multipoles. Knowing whether, and to what extent,
the quenching is needed to describe muon capture rates has
practical application as well. For example, evaluation of the
cross section for supernova neutrino detectors, or of the ma-
trix elements for the neutrinoless double b decay involves
higher than 11 multipoles, and larger momentum transfers,©2000 The American Physical Society02-1
E. KOLBE, K. LANGANKE, AND P. VOGEL PHYSICAL REVIEW C 62 055502than ordinary b decay. Another application, already men-
tioned, involves the neutrino ~and antineutrino! induced
nuclear reactions, e.g. from the neutrinos associated with the
pion and muon decays at rest.
In this paper we extend the previous calculation @4# to
heavier nuclei, in particular nuclei with the neutron excess,
i.e., with a nonvanishing value of the initial isospin. In @4#
the continuum random phase approximation ~CRPA! was
used, a method shown to be successful for the description of
the nuclear response to weak and electromagnetic probes @8#.
The method combines the usual RPA treatment with the cor-
rect description of the continuum nucleon decay channel. For
heavier N.Z nuclei that method is computationally quite
demanding. Moreover, it can describe only transitions to
states in the daughter nucleus above the particle emission
threshold. In order to evaluate the total capture rate one has
to include also the transitions to bound states, which contrib-
ute relatively more, and are typically not experimentally
separated in the heavier nuclei.
If one is interested in the total capture rate, the numeri-
cally simpler standard RPA is just as good. As an additional
bonus, it avoids the distinction between the bound and un-
bound states. That the two methods, CRPA and standard
RPA, are equivalent for the present purpose is demonstrated
in Fig. 1 for the case of 40Ca. We show in the upper part the
differential capture rate as a function of the excitation energy
in the final nucleus. While the CRPA ~for details see Ref.
@8#! is characterized by a continuous curve, nonvanishing
everywhere above the threshold, the standard RPA is char-
acterized by the ‘‘picket fence,’’ since there is a finite num-
ber of discrete final states. The similarity of both methods is
even better seen in the lower part of Fig. 1 which shows the
integrated rate up to a given excitation energy E*. There
appears to be a slight systematic shift of a few MeV ~caused
by the bound state contribution, presumably!, but the final
capture rates, and the typical excitation energies, are remark-
ably similar. Thus, we use the standard RPA for the evalua-
tion of the muon capture in the selected N.Z nuclei.
For the calculation in the present work we used the phe-
nomenological Landau-Migdal force with parameters that
has been shown to be applicable for a wide range of nuclei
@9#. All single-particle states below the Fermi level were in-
cluded and two oscillator shells above it were taken into
account. Using again 40Ca as a test case, we checked that
adding or subtracting in the calculation few subshells above
the Fermi level does not visibly change the muon capture
rate. However, in 208Pb enlarging the single particle space
leads to an increase of the capture rate by about 5%. The free
nucleon form factors were used to describe the weak nuclear
current. In particular, the unquenched axial vector coupling
constant gA(0)51.26 was used.
Muon capture also depends on the induced pseudoscalar
hadronic weak current. At the free nucleon level the corre-
sponding coupling constant is determined by the Goldberger-
Treiman relation @10#
FP~q2!5
2M pgA~0 !
mp
2 2q2
, ~2!05550where mp is the pion mass and gA(0)51.26. @In muon cap-
ture one often uses a dimensionless quantity gP
5mmFP(q2) at the relevant momentum transfer q2.
20.9mm
2
, such that gP.8.4 for free protons.# In nuclear me-
dium FP can be renormalized again, and this renormalization
does not necessarily obey the Goldberger-Treiman relation
@11#. We have shown in our previous work that the total
muon capture rates are not sensitive enough to the various
choices of FP renormalization. Consequently, throughout
this work we use the Goldberger-Treiman relation.
The calculated total capture rates are collected in Table I
and compared with the data @3#. For comparison we also
show in Table I the earlier results @4# for the N5Z nuclei,
evaluated with the same residual Landau-Migdal force.
Among the nuclei in Table I, 56Fe is the only one with a
substantial contribution of the Gamow-Teller transitions,
which are known to be quenched. Hence the 11 part of the
capture rate was quenched by the empirical factor of 2.54
obtained by comparing the total experimental and calculated
GT strength. This reduced 11 rate was then combined with
the unquenched rate from the other multipoles in column 3
of Table I. Clearly, a better agreement with experiment for
all considered nuclei is achieved when the full value of gA is
FIG. 1. The m2 capture rate as a function of the excitation
energy in the final nucleus 40K ~upper panel!. The continuous curve
is for the continuum random phase approximation ~CRPA!, while
the dashed vertical bars are the results of the standard random phase
approximation ~SRPA!. In the lower panel the integrated m2 cap-
ture rate, up to the excitation energy E*, is shown.2-2
MUON CAPTURE ON NUCLEI WITH N.Z , RANDOM . . . PHYSICAL REVIEW C 62 055502TABLE I. m2-capture rates calculated within the standard ~SRPA! and continuum ~CRPA! models in
units of 103 s. The radius r and diffuseness d of the extended nuclear charge distribution were set to (r ,d)
5(1.07 fm, 0.50 fm) for 12C and (r ,d)5(1.07 fm, 0.57 fm) for all other nuclei. The Landau-Migdal
force ~LM! is used throughout.
Nucleus Expt. @3# SRPA SRPA CRPA~LM! CRPA~LM!
gA(0)51.26 gA(0)51.0 gA(0)51.26 gA(0)51.0
12C 32.860.8 31.3 c 22.9 c
16O 102.660.6 103.2 75.8
40Ca 254467 a 2547 1846 2489 1800
44Ca 1793640 1722 1238
48Ca 1164 b 1301 930
56Fe 44006100 4460 c 3430 c
90Zr 93506100 10288 7400
208Pb 134506180 16057 11436
aCorrected from the data for natural Ca.
bExtrapolated using the Primakoff formula fitted to 40Ca and 44Ca.
cCalculated with partial occupation of the single particle subshells; see @17#.used. The quenched value leads to an obvious underestimate
of the muon capture rate.
In Fig. 2 the fractional contributions of different multi-
poles are shown for 16O, 48Ca, 90Zr, and 208Pb. For the
closed shell nucleus 16O the negative parity 12 and 22, and
to lesser extent 02, multipoles dominate, as expected. In the
intermediate mass nuclei, 48Ca and 90Zr, the dominance of
these multipolarities is less pronounced. Finally, in 208Pb the
positive parity 11 and 21 multipoles give the largest contri-
bution, since the negative parity proton hole-neutron particle
states are blocked. Note, however, that these 11 and 21
states correspond to the 2\v excitations involving different
major shells, not the usual 0\v Gamow-Teller transitions.
The dominance of dipole transitions for the m2 capture
process in the p and s ,d shell nuclei is a well-known phe-
FIG. 2. Fractional contribution of different multipoles to the
total capture rate for four of the considered nuclei. The entries with
Ip502,12,22 correspond to the 1\v excitations, while the entries
with Ip511,21,31 correspond to the 2\v excitations. The data
points are connected by lines for better visibility and the nuclei are
as indicated in the figure.05550nomenon. It can be exploited in the generalized Goldhaber-
Teller model @12#, where all dipole and spin-dipole strength
is concentrated in a single collective state. The Goldhaber-
Teller model has been applied to the muon capture already a
long time ago @12#. Here we have repeated the calculation
with one modification. Instead of using as the energy of the
single collective state the energy ED of the giant dipole reso-
nance in the initial nucleus, we placed the strength into a
state at the excitation energy E*5ED2EIAS in the final
nucleus, where EIAS is the energy of the isobar analog state
with T51 in the initial N5Z nucleus. Such an assignment
puts the strength close to the centroid of the excitation spec-
trum obtained in the RPA. The method is rather crude, since
more detailed calculations clearly show that the spin-dipole
strength is spread over a sizable energy interval. Neverthe-
less, the results displayed in Table II, obtained again with the
full value of gA(0)51.26, support our conclusion that there
is no quenching in these nuclei of the operators of the weak
current that change parity.
First forbidden b decays, in particular those with DIp
502, have been often analyzed as a source of information
on the enhancement caused by the meson exchange currents.
This enhancement, whose origin is however well understood,
is an example of the multipolarity dependence of the appar-
ent in-medium effects. In the context of the work reported
here it is worthwhile to quote the work of Warburton and
Towner @13# who analyzed 18 first forbidden b decays in
TABLE II. m2-capture rates calculated within the Goldhaber-
Teller model in units of 103 s. The formulas of Ref. @12# are used.
In column 2 the original parameters are employed. In column 3 the
excitation energy E* is modified, as explained in the text.
Nucleus Expt. @3# Orig. values @12# Modified E*
12C 32.860.8 29.7 35.1
16O 102.660.6 79.2 123
28Si 87162 657 970
40Ca 254467 1490 27302-3
E. KOLBE, K. LANGANKE, AND P. VOGEL PHYSICAL REVIEW C 62 055502the lead region. They used a truncated shell model and found
that the DIp512 transitions, which are unaffected by the
meson exchange currents, do not require any quenching. In
fact, their fit to an overall quenching factor for the first for-
bidden matrix elements results in sq150.9860.05, compat-
ible with unity, i.e., with no quenching. That analysis, totally
independent of our evaluation of the muon capture, again
supports the conclusion about the apparent absence of an
appreciable quenching of the parity-changing ~i.e., first for-
bidden! weak current operators.
The unique second forbidden b decays are governed by a
single operator r2@Y 2s#l53. For only a handful of them the
partial decay rates are known. When analyzed @14,15#, these
transitions do not allow one to draw any definitive conclu-
sion about the possible quenching of the corresponding
strength. However, as pointed out above, m2 capture rate in
90Zr, and 208Pb, is strongly affected by the 2\v transitions,
whose operators are related to the second forbidden b de-
cays. Since the agreement between the experimental and cal-
culated rates in these two nuclei, and in particular in 208Pb, is05550not as good as in the other cases, we cannot make a strong
statement regarding the quenching of the positive parity
‘‘second forbidden’’ multipoles based on the muon capture
calculations.
In conclusion, the present analysis shows that the CRPA
and SRPA methods are capable of describing the total m2
capture rates quite well in a large range of nuclei. The de-
pendence of the muon capture rate on the isospin, the so-
called Primakoff rule @16#, is also reasonably well repro-
duced. There is no indication of the necessity to apply any
quenching to the operators responsible for the m2 process.
Thus our findings can be used as guidance in the evaluation
of other semileptonic weak processes involving higher mul-
tipoles, i.e., transitions involving other than the 0\v spin
changing operators.
This work was supported in part by the Swiss National
Science Foundation, the Danish Research Council, and by
the U.S. Department of Energy, Contract No. DE-F603-
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