We show that in minimal supersymmetric standard model (MSSM) with R-parity
breaking as well as in the left-right symmetric model, there are new observable
contributions to neutrinoless double beta decay arising from hitherto
overlooked diagrams involving the exchange of one W boson and one scalar boson.
In particular, in the case of MSSM, the present experimental bounds on
neutrinoless double beta decay lifetime improves the limits on certain R-parity
violating couplings by about two orders of magnitude. It is shown that similar
diagrams also lead to enhanced rates for $\mu^-\rightarrow e^+$ conversion in
nuclei, which are in the range accessible to ongoing experiments.Comment: Latex file; 9 pages; 3 figures available on reques

A. Halprin

J. Schecter

K. S. Babu

L. Hall

M. Doi

M. Schwarz

R. N. Mohapatra

V. Barger

W. C. Haxton

English

arXiv

Crossref

Physical Review Letters

New Vector-Scalar Contributions to Neutrinoless Double Beta Decay and Constraints on<i>R</i>-Parity Violation

We show that in minimal supersymmetric standard model (MSSM) with R-parity breaking as well as in the left-right symmetric model, there are new observable contributions to neutrinoless double beta decay arising from hitherto overlooked diagrams involving the exchange of one W boson and one scalar boson. In particular, in the case of MSSM, the present experimental bounds on neutrinoless double beta decay lifetime improves the limits on certain R-parity violating couplings by about two orders of magnitude. It is shown that similar diagrams also lead to enhanced rates for \mu^-\rightarrow e^+ conversion in nuclei, which are in the range accessible to ongoing experiments

Babu, K S

Mohapatra, Rabindra N

CERN Document Server

H
EP
-P
H
-9
50
63
54
hep-ph/9506354
BA-95-15
UMD-PP-95-117
April 1995
New Vector-Scalar Contributions to Neutrinoless
Double Beta decay and Constraints on R-Parity
Violation
K.S. Babu
1
Bartol Research Institute
University of Delaware
Newark, DE 19716
and
R.N. Mohapatra
2
Department of Physics and Astronomy
University of Maryland
College Park, MD 20742
Abstract
We show that in the minimal supersymmetric standard model (MSSM) with
R{parity breaking as well as in the left{right symmetric model, there are new ob-
servable contributions to neutrinoless double beta decay (
0
) arising from hith-
erto overlooked diagrams involving the exchange of one W boson and one scalar
boson. In particular, in the case of MSSM, the present experimental bounds on

0
lifetime improves the limits on certain R{parity violating couplings by about
two orders of magnitude. It is shown that similar diagrams also lead to enhanced
rates for 
 
! e
+
conversion in nuclei, which are in the range accessible to ongoing
experiments.
1
Address starting September 1995: School of Natural Sciences, Institute for Advanced Study,
Olden Lane, Princeton, NJ 08540; work supported by the Department of Energy Grant #DE-
FG02-91ER406267
2
Work supported by the National Science Foundation Grant #PHY-9119745
With the standard model of electroweak interactions brilliantly conrmed by
a host of experiments, the search is on for the next level of physics at TeV or higher
scales. There exist many interesting scenarios which address the various naturalness
problems of the standard model framework. A generic feature of many of these
scenarios is that conservation laws present in the standard model no longer remain
valid. A typical conservation law that breaks down is the one corresponding to
lepton-number symmetry. To make any theoretical headway, one needs to know the
strength of the lepton number non{conserving interactions as well as of any other
kind of interactions that may accompany them.
It has been known for a long time [1] that neutrinoless double beta decay (
0
)
is a very sensitive probe of lepton number violating terms in the Lagrangian such as
the Majorana mass of the light neutrinos [2], right{handed weak couplings involving
heavy Majorana neutrinos [3,4], as well as Higgs [5] and other interactions such as
those involving R{parity breaking in the supersymmetric model [6]. The reason why
this observation is interesting is that, the steadily improving experimental limits
[7] on 
0
life-time can then be translated into more stringent limits [8] on the
parameters of these new physics scenarios. This is an extremely valuable information
to have in our search for physics beyond the standard model.
It is the goal of this letter to point out another class of hitherto unnoticed
contributions to 
0
decay in the following two classes of theories: (i) Minimal
supersymmetric models (MSSM) with R-parity violation; (ii) Left-right symmetric
models with a low mass W
R
. These new contributions are of vector-scalar type in
that they involve the exchange of a W
L
together with a charged scalar boson with
a virtual light neutrino. These contributions do not involve a helicity ip of the
internal light neutrino, and therefore, their amplitudes are enhanced relative to the
ordinary contribution proportional to the light Majorana neutrino mass by roughly
a factor of p
F
=m

 (100 MeV=1 eV )  10
8
(p
F
is the Fermi momentum of the
nucleons in the nuclei). Turning the argument around, the strength of such lepton
number violating scalar boson interactions can be constrained to be G
eff
 10
 8
G
F
from the ()
0
process (modulo nuclear matrix element uncertainties), which indeed
is a severe constraint. The eect turns out to be most dramatic on certain R-
violating couplings in the MSSM in that it directly involves the R-violating couplings
and superpartner masses. In the left-right models, although such contributions
involve unknown Higgs boson mixings, they do lead to interesting bounds for certain
range of values for these parameters. We also point out that similar enhanced vector-
scalar contributions exist in other rare processes such as 
 
! e
+
conversion in
nuclei which turn out to be in the experimentally accessible range.
The new contributions arise from the combination of two eective four-Fermi
interactions of the following type which as we will show later can arise in several
2
interesting classes of gauge models:
H
eff
=
G
F
p
2
(e

(1  
5
)
e
u

(1  
5
)d+ 
ee
1
d(1   
5
)u
T
e
C
 1
(1  
5
)e +

ee
2
d(1  
5
)
e
u
T
C
 1
(1   
5
)e ) + h:c: (1)
In the above, the rst term is the usual (V-A) interaction, the other two are eective
lepton number violating terms. In order to evaluate the matrix elements between
nuclear states, we need to do Fierz reordering of the 
ee
2
term, which casts it in the
form:
G
F
2
p
2

ee
2

d(1  
5
)ue
c
(1  
5
)
e
+
1
2
d

(1  
5
)ue
c
(1   
5
)


e

!:
The parameters 
ee
i
(i = 1; 2) characterize the new interactions that arise in a gauge
model so that any limit on them translates into limits on the parameters on the
theory leading to this interaction.
It is easy to see that the two new interaction terms in Eq. (1) give contributions
to 
0
decay which do not depend on the neutrino mass and involve a vector current
at one hadronic vertex and a scalar current in the other (hence the name vector-
scalar; of course as just mentioned, the 
ee
2
{type scalar interaction of Eq. (1) after
Fierz reordering generates also a tensor coupling). The resulting eective L = 2
Hamiltonian is then given in momentum space by:
H
L=2
= G
2
F
((
ee
1
+

ee
2
2
)u(1 + 
5
)du

(1  
5
)de

(1   
5
)
1
  q
Ce
T
+

ee
2
4
u

(1 + 
5
)du

(1  
5
)de



(1  
5
)
1
  q
Ce
T
) + h:c: (2)
In Eq. (2), q refers to the momentum of the internal light neutrino propagator.
This eective Hamiltonian of course has to be evaluated between nuclear states.
This gives rise to an eective neutrino potential as in the case of the neutrino mass
contribution to 
0
decay with the dierence that hm

=q
2
i
Nucl:
gets replaced by
h  q=q
2
i
Nucl:
(and a dierent operator for the tensor coupling). To the best of our
knowledge, such nuclear matrix elements for 
0
decay have not been evaluated in
the literature. We therefore resort to a crude estimate and assume an average value
of q to be equal to the Fermi momentum p
F
of the nucleons in the nucleus ( 100
3
MeV). The present upper limits on m

of about 1 eV then translates to an upper
limit on the new interaction parameter  as follows:

ee
1;2
 1 10
 8
: (3)
We realize that due to the crudeness of our estimate of the nuclear matrix
element, the above upper limit is likely to be uncertain by perhaps a factor of ve
or so. Nevertheless it is important to note that the limit on 
1;2
, barring unforeseen
nuclear suppressions, is rather stringent and will imply important restrictions on the
parameters of the gauge models leading to Eq. (1). Let us therefore, proceed to the
kind of gauge models where the last two terms in Eq. (1) can arise at low energies.
MSSM with R-parity violation:
As is well-known, the minimal supersymmetric standard model can have ex-
plicit [9] violation of the R-symmetry (dened by ( 1)
3B+L+2S
), leading to lepton
number violating interactions in the low energy Lagrangian. The three possible
types of couplings in the superpotential are [10]:
W
0
= 
ijk
L
i
L
j
E
c
k
+ 
0
ijk
L
i
Q
j
D
c
k
+ 
00
ijk
U
c
i
D
c
j
D
c
k
: (4)
Here L;Q stand for the lepton and quark doublet superelds, E
c
for the lepton sin-
glet supereld and U
c
;D
c
for the quark singlet superelds. i; j; k are the generation
indices and we have 
ijk
=  
jik
, 
0
ijk
=  
0
ikj
. The SU(2) and color indices in Eq.
(4) are contracted as follows: L
i
Q
j
D
c
k
= (
i
d

j
  e
i
u

j
)D
c
k
, etc. The simultaneous
presence of all three terms in Eq. (4) will imply rapid proton decay, which can be
avoided by setting the 
00
= 0. In this case, baryon number remains an unbroken
symmetry while lepton number is violated [11].
There are two types of vector{scalar contributions to 
0
and related L = 2
processes. These are shown in Figures 1 and 2. The dominant contribution to 
0
in this model arises from Fig. 1, where the exchanged scalar particles are the
~
b 
~
b
c
pair. This leads to a contribution to 
ee
2
given by

ee
2
'
0
@
(
0
113

0
131
)
2
p
2G
F
M
2
~
b
1
A
 
m
b
M
2
~
b
c
!
(tan +A
b
m
0
) : (5)
Here A;m
0
are supersymmetry breaking parameters, while  is the supersymmetric
mass of the Higgs bosons. tan is the ratio of the two Higgs vacuum expectation
values and lies in the range 1  tan  m
t
=m
b
 60. For the choice of all squark
4
masses as well as  and the SUSY breaking mass parameters being of order of
100 GeV, A = 1, tan = 1, we nd from Eq. (5) that 
0
113

0
131
 3  10
 8
,
which is more stringent limit on this parameter than the existing ones [12]. The
present limits on these parameters are 
0
113
 0:03; 
0
131
 0:26, which shows that
the bound derived here from 
0
is about ve orders of magnitude more stringent
on the product 
0
113

0
131
. If the exchanged scalar particles in Fig. 1 are the ~s   ~s
c
pair, b gets replaced by s in Eq.(5) and one obtains a limit 
0
121

112
 1  10
 6
,
which also is more stringent by about four orders of magnitude than existing limits
(
0
121
 0:26; 
0
112
 0:03). We note that the gluino exchange diagram [6,8] discussed
in the context of 
0
only constrains the parameter 
0
111
, while the vector{scalar
exchange graphs constrain several other couplings.
The diagram in Fig. 2, due to the antisymmetry of 
ijk
, does not contribute to

0
decay, but will be important for 
 
! e
+
conversion (see discussions below).
Let us also note that there exist indirect limits on the  and 
0
couplings
arising from the induced neutrino masses. The magnitudes of these masses are
given generically by m

ij
 
0
ikl

0
jkl
m
k
m
l
=(16
2
M
~q
), where m
k;l
stand for the masses
of the kth and lth down quarks (with a similar expression for the  couplings). The
couplings 
0
i33

j33
 610
 5
is the most severely constrained (being proportional to
the b{quark mass-squared) where the induced neutrino mass has been assumed to be
 100 eV , which is cosmologically safe. The corresponding limit on 
0
131

0
113
 :24
(using m

 1 eV ) is a trivial constraint to be compared with the 
0
decay limit
derived here.
Constraints on the Left-right symmetric model:
Let us consider the minimal left-right symmetric model with a see-saw mecha-
nism for neutrino masses [13]. The gauge group of the model is SU(3)
C
SU(2)
L

SU(2)
R
 U(1)
B L
. The Higgs sector of the model consists of the bi-doublet eld
  (1=2; 1=2; 0) and triplet Higgs elds: 
L
(1; 0; +2)
R
(0; 1; +2): The
Yukawa couplings which are invariant under gauge and parity symmetry can be
written as:
L
Y
= 	
L
h
`
	
R
+	
L
~
h
`
~
	
R
+Q
L
h
q
Q
R
+Q
L
~
h
q
~
Q
R
+
	
T
L

2
~  f
~

L
C
 1
	
L
+ L! R+ h:c: (6)
where h;
~
h are hermitian matrices while f is a symmetric matrix in the generation
space. 	 and Q here denote the leptonic and quark doublets respectively.
The gauge symmetry is spontaneously broken by the vacuum expectation val-
ues: h
0
R
i = V
R
; h
0
L
i ' 0 ; and hi = diag:(; 
0
): As usual, hi gives masses
to the charged fermions and Dirac masses to the neutrinos whereas h
0
R
i leads to
the see-saw mechanism for the neutrinos in the standard way [13].
5
The physics we are interested in comes from the left-handed triplet sector of
the theory through its mixing with the bidoublet eld which arise from the cou-
plings in the Higgs potential, such as Tr(
L

y
R

y
) after the full gauge symmetry
is broken down to U(1)
em
. Specically, there is a mixing between the singly charged
components of  and 
L
[14] (we denote this mixing term by an angle ). This will
contribute to the four-Fermi interaction of the form given by the 
ee
1
term through
the diagram shown in Fig. 3 with

ee
1
'
h
u
f
11
sin2
4
p
2G
F
M
2
H
+
; (7)
where we have assumed that H
+
is the lighter of the two Higgs elds. We get
h
u
f
11
sin2  6  10
 9
(M
H
+
=100 GeV )
2
, which is quite a stringent constraint on
the parameters of the theory. To appreciate this somewhat more, we point out that
one expects h
u
 m
u
=m
W
 5  10
 5
in which case, we get an upper limit for the
coupling of the Higgs triplets to leptons f
11
sin2  10
 4
(for m
H
+
= 100 GeV ).
Taking a reasonable choice of   M
W
L
=M
W
R
 10
 1
would correspond to a limit
f
11
 10
 3
. Limits on this parameters from analysis [15] of Bhabha scattering is
only of order :2 or so for the same value of the Higgs mass.

 
! e
+
conversion:
Another class of rare processes where the new vector-scalar contribution makes
an important impact is the process of 
 
! e
+
conversion in nuclei which arises
with an observable strength in the R violating MSSM. This involves the couplings

ijk
which were not constrained by the considerations of neutrinoless double beta
decay due to anti-symmetry of the Yukawa couplings. Another contribution involves
a dierent product of 
0
couplings than what appeared in the 
0
process. We
parametrize the eective four Fermi interaction for this process in analogy to Eq.
(1) with 
ee
1
replaced by two terms 
e
1
and 
e
1
and similarly for 
2
. Here 
e
1
is the
coecient of the termG
F
=
p
2[d(1 
5
)u
T
e
C
 1
(1 
5
)] etc. The eective strengths
of these couplings arising from Fig. 2 (as well as from Fig. 1 with e replaced by a
) are found to be

e
1
=

123

0
311
2
p
2G
F
m
2
~
m
2
~
c
m

(tan +A

m
0
)

e
2
=

0
213

0
131
2
p
2G
F
m
2
~
b
m
2
~
b
c
m
b
(tan +A
b
m
0
) (8)
6
with similar expressions for 
e
1;2
. The existing limits on the 
0
in Eq. (8) are 
0
213

0:09; 
0
131
 0:26, so that for the squark masses of order 100 GeV, 
e
2
can be as large
as 10
 1
or so. The branching ratio for 
 
! e
+
conversion relative to {capture can
be obtained by a naive scaling by the nucleon mass and we nd Br(
 
! e
+
) ' 10
 12
for this choice of 
e
2
. It is interesting that this is in the accessible range of the current
experiments. The sensitivity of these experiments is expected to improve by another
order of magnitude in the near future [16].
We note that the corresponding predictions for the left{right model is down
by several orders of magnitude.
In summary, we have discussed a new class of contributions to the double lepton
number violating processes which may arise in several extensions of the standard
model. In particular, we nd that the existing experimental limits on neutrinoless
double beta decay lead to very stringent constraints on the R{violating couplings
in MSSM as well as the lepton number violating Higgs couplings of the left{right
symmetric model. Furthermore, we nd the exciting possibility that for presently
allowed range of parameters in the MSSM, 
 
! e
+
conversion in nuclei is in the
observable range. While we have focussed on only two classes of models, our results
are more general and should apply to other schemes, such as L{violating leptoquark
models.
7
References
1. W. C. Haxton and G. Stephenson, Prog. in Part. and Nucl. Phys. 12,
409 (1984); H. Grotz and H. Klapdor, The Weak Interactions in Nuclear,
Particle and Astrophysics, Adam Hilger,Bristol, (1990); R. N. Mohapatra and
P. B. Pal, Massive Neutrinos in Physics and Astrophysics, World scientic,
Singapore (1991); M. Moe and P. Vogel, Ann. Rev. Nucl. and Part. Sc., 44,
247 (1994).
2. M. Doi, T. Kotani, H. Nishiura and E. Takasugi, Prog. Theor. Phys. Suppl.
83, 1 (1985).
3. A. Halprin, P. Minkowski, H. Primako and S.P. Rosen, Phys. Rev. D13,
2567 (1976).
4. R.N. Mohapatra, Phys. Rev. D34, 909 (1986).
5. R.N. Mohapatra and J. Vergados, Phys. Rev. Lett. 47, 1713 (1981); J.
Schecter and J.W.F. Valle, Phys. Rev. D25, 2951 (1982); W.C. Haxton, S.P.
Rosen and G.J. Stephenson, ibid., D26,1805 (1982); L. Wolfenstein, ibid.,
D26, 2507 (1982).
6. R. N. Mohapatra, Phys. Rev. D34, 3457 (1986).
7. H. V. Klapdor-Kleingrothaus, Prog. in Part. and Nucl. Phys. 32, 261 (1994);
A. Balysh et. al., Phys. Lett. B (to appear).
8. J. D. Vergados, Phys. Lett. B184, 55 (1987); M. Hirsch, H. V. Klapdor-
Kleingrothaus and S. G. Kovalenko, Heidelberg Preprint (1995).
9. L. Hall and M. Suzuki, Nucl. Phys. B231, 419 (1984).
10. In addition, one can write a term 
i
L
i
H
u
in W
0
. Although this term can be
rotated away by a eld redenition, the VEVs of the scalar neutrinos cannot
be set to zero in general then. In supergravity models, however, such VEVs
get induced only by the eect of large Yukawa couplings, in which case the
sneutrino VEVS will remain zero.
11. There is also the possibility of setting  = 
0
= 0, with 
00
6= 0, in which case
B is violated, but L is not. We shall not pursue this alternative here.
12. V. Barger, G. Giudice and T. Han, Phys. Rev. D40, 2987 (1989).
13. R.N. Mohapatra and G. Senjanovic , Phys. Rev. Lett. 44, 912 (1980); Phys.
Rev. D23, 165 (1981).
8
14. Note that in the limit 
0
= 0, this coupling does not induce a h
L
i. See K.S.
Babu and R.N. Mohapatra, Phys. Rev. Lett. 64, 9 (1990).
15. M. Schwarz, Phys. Rev. D40, 1521 (1989).
16. W. Bertl, private communication.
Figure Caption:
Fig.1: The dominant diagram contributing to 
ee
2
type Four-Fermi term (Eq. (1))
in the supersymmetric model.
Fig.2: The diagram contributing to 
e
1
type Four-Fermi terms relevant for 
 
! e
+
conversion in the supersymmetric model.
Fig.3: The vector{scalar exchange diagram for 
0
in the left{right symmetric
model.
9


New vector-scalar contributions to neutrinoless double beta decay and constraints on R-parity violation

New vector-scalar contributions to neutrinoless double beta decay and
  constraints on R-parity violation

New vector-scalar contributions to neutrinoless double beta decay and constraints on R-parity violation

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