In the u- capture reaction (.I) the muon is captured in the 1s orbit of a nucleus, usually chosen to be heavy to ensure a high capture rate,. + and the final state e ...
I
SLAC-PUB-2301 April 1979 (T/E)
MAJORANALEPTONMEDIATED v- TO e+ CONVERSIONIN NUCLEI* A. N. Kamal+ Stanford Linear Accelerator Center Stanford University, Stanford, California
94305
and J. N. Ng TRIUMP, and Physics Department University of British Columbia, Vancouver British Columbia, Canada V6T lW5
ABSTRACT We estimate toe
+
in nuclei
muon capture
rate with
of the anomalous conversion Majorana
for a 0.5 GeV/c2 lepton
A sequential
and neutrino respect
ratio
via the exchange of a virtual
to be and v1 and
neutrino
lepton
Majorana particle. 13
and the dots
which are of no interest
ve and vu will give rise
to us. In parti11 and its be assumed small
to phenomena beyond the scope of this
for weak interactions.
Noc -= C?" where C is the charge conjugation
by two units.
doublets
by an
is given by the parameters lJ both of which are less than unity as we assume
gauge couplings
this
the CP
is generated
Next we assumed the No to be a Majorana particle,
term for
ignore
between No and ve and v
fi and y respectively universal
we shall
as
states
No will
The mixing
paper.
sector
In our
as v, and v . The latter two neutrinos are taken IJ with masses no larger than 2 eV. Thus we can now rewite
where No is now a mass eigenstate
cular
mass
weak eigenstates.
For the case of four
weak doublets
denote other
flavors
lepton
treat
to be very light our lepton
of the neutral
in the lepton
For 2n lepton
orthogonal
v2 which we will
phenomenon can occur for
among the corresponding
the mixing
treatment
a similar
-0 oc MoN N will
12 I.e., .
No =
The general
matrix.
mass
serve as a source or a sink for
This term obviously
violates
total
lepton
the
number
-5The model discussed
above can induce p- -t e+ conversion
via a second order weak process. 14 involving
the Majorana 9
particle
current
The W-boson field
unphysical field
processes
NR will
since it
density
is
where f is the gauge coupling.
The right-handed
Lagrangian
f BeL Y,, NL + Y;, yu NL)WP (
=
The terms involving
The interaction
in nuclei
(4) is denoted by W".
Higgs boson exchanges would be smaller. have no effect
on the charged weak
has weak hypercharge
Y=O and weak isospin
13=0. In the u- capture
reaction
of a nucleus,
usually
and the final
state
wave function
appropriately
ignore
(.I) the muon is captured
chosen to be heavy to ensure a high capture + e should in principle
such complexities
distorted
as the kinematics
by the nucleus
be described
state
the e+ .
are depicted
The generic
represented
virtual
No into
locally
to one proton
the conversion
Feynman diagram The u- is absorbed
l(a).
via one W-boson exchange and converts
The mass insertion
However, we shall
and estimate
in Fig.
rate,
by a Coulomb
by the nucleus.
of the final
rate by assuming plane waves for as well
in the 1s orbit
into
by the cross in Fig.
l(a)
a virtual turns
No.
the
a i" which scatters from the intermediate nucleus (A,Z-1) + and emerges as an e . Figure l(b) shows in detail the coupling of W-
For the initial 9, w
at a time (.single muon we will = =
$,& z3l2 34 Oa 0)
nucleon
approximation).
use the 1s state
wave function
given by
-iEPxo e - $ e
lb1 - iEPxo U
IJ
(54
and 4TT ao
where
the
Z is
m
conversion
more we will
rate we shall
only treat
recoils
as a unit,
The energy difference, is usually
less
ignore
i.e.,
incoherent
that
=
use the
correction.
capture
does not break up but can with
and final
pairs
+b4+4d-f$
from ordinary
(k
c
{ ik,x e
15
nuclear
states
the coherent than the
have a peak at m - AE. IJ also serves to cut down the back-
stated
radiative
capture.
above the matrix
(2) is given by second order perturbation
%
energy of -100 MeV.
about six times more often
the energy of e+ to be large
‘eke)
Further-
where the final
It has been argued that
-;)2-g
u X
kinematic
the e+ is emitted
With the model and assumptions
Jz
IJ
Then the e+ - spectrum will
ground from the Dalitz
reaction
Ii> and u
speaking
BE, between the initial
than 10 MeV.
effect.
this
the nucleus
is the dominant one occuring
Selecting
state
instead
system
the case of coherent
This implies
be excited.
effect
nuclear
of m . However such u are minor and since we are interested in an estimate of the
corrections
nucleus
(5b)
2
In Eq. (5b) we should strictly
reduced mass of the muon-nucleus
-+ FCe
u
e
number of the initial
atomic
the muon spinor.
=
Yp(1-~5)(~nfMo)
-iE,,yO -ik,(x-y) e e
(k
Mo(1-y5)(kn+Mcr) -i(Ei-Ex)yo e
theory
:
n-
n
element for
)2-g
lM2)2
(k2
n-
e y,(l-y5)uP
u
I
-i(E,-Ef)xo e
xl
(6)
-7where If> is the final complete
set of intermediate
weak-current
of contour
with
states
energy Ef, and [X> is a
of energy Ex.
in k: can be performed
integration.
external
mation
state
The hadronic
We shall
using
since all
the usual
make the simplifying
momenta are small compared to 4.
for u- capture
external
techniques
assumption
the kz integration
momenta involved
are of the After
is done we have
-Ex)yo
-i(Ei
that
This is a good approxi-
order of 100 MeV whereas MW is in the range of 50 to 100 GeV.
-i(Ex-
Ef)xo
1
e
e
';(k,)
% Ye
(M$.$2
;;*iI + i(xO-y")(yowo-d
- 2
1
charged
is denoted by Ju(x).
The integration
all
nuclear
l
iCn) -
u
+ (Mu * %I Ii
4wu (YOo,J
M;-4
w
'Y"(~-Y~)
0(x0-Y") i
+
T
l
IQ
i
uP li>
(7)
4
CW
and
u2 = u
+ Mz
-8Similarly
with
(M, 2 MW),
We have kept the MW term which will of Mo from smaller
to larger
Next we invoke
the closure
the energy difference bably much smaller.
of ordinary
is usually
Hence, replacing
The closure
a gross error.
16
where we can approximate
EX by some average value
Ei-EX 17
approximation
Studies
as .
the range
than MW.
the energy of the intermediate the same order
enable us to investigate
which is of
muon capture
no greater
indicate
that
than 10 MeV and pro-
EX by should not introduce
approximation
then allows
us to use complete-
ness on IX> and obtain -i(Ei e
-Ex)y'
-i(Ei e
- CEpYO
-i(Exe
Ef)xo
X =
-b
The x",yo
Ef)xo
and kn integrations Performing
integrations. A=-
-i(e
can then be done by.contour
them in succession
leaves us with
if4ySM: 161~~ / 1
d3g d3G 6(EU+Ei-Ee-Ef)
;;(k,)
(4-M;)2
%'yp Lyv(l-~5)
e
0, (3)
u,,
(10)
.
where
L
=
c i=l
(114
I -9and
J1
J2
=
16~ E y"
=
8niT*c
co k2 j,(kr) s0 dk (E2-~;)2
(llc)
co
J3
=
-8vi
7.;
dk I
0
E2+w2 2 a w(3
03
J4
(.llb)
k3 jl(kr)
Old) (E2-wt)2
k2 jo(kr)
(114
= (E2-ti:-is)
co
J5
=
k3 jl(kr)
(110
(E2-ui-id
and E with (i>
5
We can now divide
r = I;1 = pq.
the mass of the Majorana E < Ma s s;
e.g.,
lepton
into
is in the intermediate
four cases: region
range of M(J>>MiJ, say MO cz 3 TeV/c2;
super heavy lepton
(iii)
Mo C*MW about 80 GeV/c2; and
(iv)
a very light
Ma. less than 1 MeV/c2.
The case of immediate the actual
the discussion
from 0.5 to 10 GeV/c2;
(ii)
note that
(llg)
Ef + Ee -
experimental value
interest
is the first
of MW is of less importance
one. although
We also in our
-lOnumerical
results
for
which is the value
cases (ii)
and (iii)
in the W-S model with
we will
use MWpy 84 GeV/c'
the weak mixing
angle,
Bw,
given by sin2eW = 0.25. Consider
first
both Mo and MW large
(cases (i)
to (iii)).
Then
we have 2
2.7~ Ey
J1=
J2
J4
o
Mo
z
=
J3
3
-Mar e
(124
2. + 6 -Mor -2a 1 y'r e
167~~E y"
-M r e o
CM; - 4)
r
16 2i$*?
e
(12b)
(12c)
and -Mar J5
=
- M;) Thus keeping
L
=
In the limit
only J2, J3 and J5 one gets
-4r2i
T*;
-Mar e 1+
+ (Ma : %'
i
(13)
Mo -CCMW, J5 drops out and (Mo 2 MW) term is small giving
L= In the limit
(12d)
r
-4v2if*;
-Mar e
E > 1 (21)
FL1 2MoR >> 1 -e G The "large
nucleus"
-MGrC
1 + rcMa +trEM:
limit
is easily
A= 64, Z=30,
medium heavy nuclei
+irzMz
(22)
met for Mo > 0.5 GeV; however,
for
ZR * 0.57 and one cannot use the limit aO
of Eq. (21). The capture
rate
into d3;
Rf(y-
+ e+,
However, since over all
=
IT /
e
1
-(2~)~
-2E
the final
the final
a particular
e
state
states
-+e+) R to@
final
to obtain =
-Ef)
6(Ell+Ei-Ee
nucleus
c
= 1.
19
by using an average
Finally,
we obtain
(By)2(.Za>5 mUF2G2 M2 0
= $
(25)
where the gauge coupling via f2/4
= 4fi
f has been replaced
by the Fermi coupling,
GF and ke has been approximated
The capture
rate
The normal capture
rate
by m . v mass range (El < Mo introduced
to some other
twice and will
rate by
lower
states
of
the ratio
in Eq. (6) If>.
given in
Thus we can calculate
from Eq. (.27).
Thus
an
I -15The parameters Eq. (3) permits electrons muons.
neutrinoless
as well
will
20 shows that
analysis nuclei
of various
y can be as large
no neutrino
as is done in Ref.
two
for Ma > 0.5
hand, 8 can be determined of naturally
Taking the results
two
rare decay of kaons.
as unity
double-beta-decay
18.
into
of kaon into
of y2Gi for
On the other
The model of
of heavy nuclei
double-beta-decay
occur at level
GeV/c2 to tens of GeV/c2.
are fi and y.
double-beta-decay
as neutrinoless
The latter
Recent analysis
yet to be determined
of their
by
occuring
analysis
we
have B2 5
2.7x10
-3
f32 2
1.0 x 10-3
for
Mo =
1 GeV/c2
for
Mcr =
0.5 GeV/c2
Mcr=
1 GeV/c2
and
Hence the product (fw2
Alternatively,
(28)
(6~)~ has the limit 5
2.7 x 10-3
5
1 x 1o-3
for for
Mcf
we can use experimental
=
0.5 GeV/c2
information
(29)
from neutrino
hadron
reactions21 v V
Assuming this
P v
limit
+N+e-+N +N+p-+N for violation (f3Y)2
From Eqs. (25)-(27)
and (31),
for medium heavy nuclei
with
-3 2 x 10
5
2
(30)
of v-e universality 2x10
gives
-3
the branching Z= 30 and Zeff
22
(31) ratio
for
anomalous capture
taken from Ref.
23 is
i -16-
B.R
R - -t e+> to@
=
-14
$
0.7x10
I
2 x lo-l3
for Mo = 1 GeV/c2
R(P- + VJ
Using the value B.R
on (Byj2 from Eq. (29) we have instead
of the wavefunctions
(35)
Eq. (35) is
(API’N1o-2where Ap is the difference in MS in the nucleus and the intermediate nucleon.
momentum Hence we
I
-17estimate
this
mechanism to give a rate
25 therefore
mechanism; conversion
at least
smaller
we can neglect in the region
this
than the two nucleon
as a source for P- + e+
where the mass scale is set by Mo
or MW. Next we discuss
the effects
Mo >> MW. From perturbative
of a superheavy Majorana
Now we will
that Mo < 0.5 TeV/c2.
as a guide and the calculation
26
of gauge theories
treatment
use the value
follows
lepton,
of (6~)~ in Eq. (31)
as in the previous
given by (13b) and (13~).
Observe that
potential
is set by MW. The Mi4 behavior
has a range that
obtained
in the rate
under the interchange suppression effect
factor
ratio
capabilities.
now the effective
factors
is many orders
Yukawa
in the amplitude
is still are symmetrical
of magnitude below present
-22 10 for r C -f 0.
is a very conservative
knowledge of quarks and leptons For completeness
one obtains
experimental
a branching
However, we caution
based on extrapolation
estimate
the
-1 With rc = 2.5 GeV the
lepton.
Even when the core softens,
of the order
case with L
Mc $ MW. There is, however, an additional -2Morc e in the rate, which strongly suppresses
of a superheavy Majorana
branching
ratio
since other
one expects
that
this
of current
and is very model dependent.
we examine the case of a light
Majorana
lepton.
The mass, Ma, cannot be in the range between the mass of the kaon and the electron. detected.
Otherwise On the other
the kaon would decay into
hand for Mo < 50 KeV/c2 the No will
one will
not be able to distinguish
studying
the kaon or pion decays.
possibility. we return
p-No which will
To discuss to Eqs. (lo)-(llg>.
the effects
it
of such a light
In evaluating
be stable
or ve by just lJ data do not exclude this
from the usual
Current
be
v
Majorana
lepton
Ji one has to keep E2 in
and
-18One now has complex poles
the denominators. having
oscillations
of frequency
E being -100 MeV and a typical
E, dampened slowly nuclear
radius
are slow.
will
not depend on inverse
powers of Mo.
behave as Mz and since M,
.. __
p- (kp)
f(A,Z-2)
i (A,Z) 4-79
e+
e'(k,)
i(A,Z)
.
f (A, Z-2) (b)
(0)
Fig. 1
3580Al
e’
f (A, Z-2)
i(A,Z)
3580A2
4-79
Fig. 2
