We consider all possible scalar bilinears, which couple to two fermions of
the standard model. The various baryon and lepton number violating couplings
allowed by these exotic scalars are studied. We then discuss which ones are
constrained by limits on proton decay (to a lepton and a meson as well as to
three leptons), neutron-antineutron oscillations, and neutrinoless double beta
decay.Comment: 11 pages latex fil

ERNEST MA

H. V. KLAPDOR-KLEINGROTHAUS

Klapdor-Kleingrothaus H. V.

Leurer M.

Ma E.

Pati J. C.

UTPAL SARKAR

English

arXiv

We consider all possible scalar bilinears, which couple to two fermions of the standard model. The various baryon and lepton number violating couplings allowed by these exotic scalars are studied. We then discuss which are constrained by limits on proton decay (to a lepton and a meson as well as to three leptons), neutron-antineutron oscillations, and neutrinoless double beta decay

KLAPDOR-KLEINGROTHAUS, H. V.

MA, E.

SARKAR, U.

Max Planck Society eDoc Server

BARYON AND LEPTON NUMBER VIOLATION WITH SCALAR BILINEARS

Klapdor-Kleingrothaus, H.

Ma, E.

Sarkar, U.

MPG.PuRe

Baryon and lepton number violation with scalar bilinears

Klapdor-Kleingrothaus, H. V.

Ma, Ernest

Sarkar, Utpal

Publications of the IAS Fellows

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UCRHEP-T345
October 2002
Baryon and Lepton Number Violation
with Scalar Bilinears
H. V. Klapdor-Kleingrothaus1, Ernest Ma2, and Utpal Sarkar3
1 Max-Planck-Institut fu¨r Kernphysik, D-69029 Heidelberg, Germany
2 Physics Department, University of California, Riverside, California 92521, USA
3 Physical Research Laboratory, Ahmedabad 380 009, India
Abstract
We consider all possible scalar bilinears, which couple to two fermions of the stan-
dard model. The various baryon and lepton number violating couplings allowed by
these exotic scalars are studied. We then discuss which ones are constrained by lim-
its on proton decay (to a lepton and a meson as well as to three leptons), neutron-
antineutron oscillations, and neutrinoless double beta decay.
Grand unified theories (GUTs) appear to be the most natural extensions of the standard
model at the very high scale. In GUTs the gauge group is unified, so there is only one single
gauge coupling constant to explain all the forces of the quarks and leptons, which are also
treated at the same footing. However the scale of unification is much too high to be directly
tested in the laboratory. Thus indirect searches become very important. Many predictions
of GUTs were studied, but there is no supporting experimental evidence to date, with the
possible exception of neutrinoless double beta decay [1, 2]. These GUT predictions also
include new particles such as leptoquarks or diquarks and new interactions which violate
baryon and lepton numbers. The latter are also required for an understanding of the present
baryon asymmetry of the Universe. An observed Majorana neutrino mass also requires lepton
number violation. In the context of these issues we present in this paper a general analysis
of baryon and lepton number violation with scalar bilinears.
The scalar bilinears we consider are generalizations of the usual Higgs scalar doublet.
They couple to two fermions of the standard model and are present in many of its exten-
sions [3, 4, 5]. The quantum numbers of these particles are thus fixed, but their masses
and couplings could be arbitrary. Low-energy particle physics phenomenology (such as the
non-observation of baryon number violation) could be used to constrain their masses and
couplings [6, 7]. Baryogenesis with scalar bilinears has also been studied [8]. Recently a
general analysis of the neutrinoless double beta decay has been performed by constructing
effective higher dimensional operators [9]. Most of these operators may be realized in models
where scalar bilinears are included, a few examples of which were demonstrated [9].
In Table 1 we list all the scalar bilinears allowed in any extension of the standard model.
We then write down the different baryon and lepton number violating couplings of these
scalars and what kind of interactions these different couplings may induce, leading to a simple
classification of all the different possible baryon and lepton number violating interactions.
2
We include the usual Higgs scalar doublet of the standard model in these interactions.
Representation Notation qq q¯l¯ ql¯ ll
(1, 1,−1) χ− ×
(1, 1,−2) L−− ×
(1, 3,−1) ξ ×
(3∗, 1, 1/3) Ya × ×
(3∗, 3, 1/3) Yb × ×
(3∗, 1, 4/3) Yc × ×
(3∗, 1,−2/3) Yd ×
(3, 2, 1/6) Xa ×
(3, 2, 7/6) Xb ×
(6, 1,−2/3) ∆a ×
(6, 1, 1/3) ∆b ×
(6, 1, 4/3) ∆c ×
(6, 3, 1/3) ∆L ×
(8, 2, 1/2) Σ
Table 1: Exotic scalar particles beyond the standard model and their couplings to combina-
tions of two fermions.
For the scalar bilinears, it is not possible to define definite baryon or lepton number in
some cases. However, depending on their couplings to the combinations of two fermions, it
is possible to assign baryon and lepton numbers separately. The different assignments are
given in Table 2. For any particular interaction it will be possible to specify what processes
it would mediate and what is the change in baryon and lepton numbers. This will allow us
to classify them.
The quadratic terms with the scalars are trivial. For any quadratic term H1H2 to be
invariant under the gauge interactions, either both of them are singlets (i.e. neutral under
all gauge factors) or else H†1 = H2. Since all the scalars we are discussing are non-singlets,
3
Two-fermion combination qq q¯l¯ ql¯ ll
Baryon number 2/3 −1/3 1/3 0
Lepton number 0 −1 −1 2
B − L 2/3 2/3 4/3 −2
Table 2: Baryon and lepton numbers of the the scalars coupling to different combinations of
two fermions.
none of the quadratic terms can have any B − L quantum number, although some of them
may contribute to baryon or lepton number violation. Consider for example, Y †
a
Ya. Since
Ya can couple to q¯l¯ as well as to qq, this term may induce a ∆B = 1, ∆L = 1 process, but
B − L remains conserved.
The trilinear couplings are the most interesting ones. It was shown [8] that there are 38
possible trilinear couplings with the scalar bilinears. Although baryon and lepton numbers
are not always uniquely specified, all of them carry a B − L quantum number ±2. As we
shall see next, these trilinear operators can mediate four possible processes and hence can
be classified in four categories
(i) B = 1, L = −1: In this case both baryon and lepton numbers are changed by 1, so all
these couplings should give only proton decay, which violates (B − L) [10].
(ii) B = 0, L = 2: This gives lepton number violating processes, without affecting the
baryon number. So this contributes to the Majorana masses of the neutrinos. Some
of them will contribute to the neutrinoless double beta decay [11], but others would
4
involve neutrinos of different generations.
(iii) B = 1, L = 3: These operators give a three-lepton decay mode of the proton [10]. In
some models these processes could dominate over the usual proton decay modes.
(iv) B = 2, L = 0: In this case baryon number is changed by two unit with no change in
lepton number. This corresponds to neutron-antineutron oscillations.
Some of the operators may contribute to two process types. There are no trilinear couplings,
which conserve B − L (which includes of course the case B = 0 and L = 0).
There is only one trilinear coupling which can give proton decay into a lepton and a
meson (or mesons) and is not involved in any other process. It is thus possible to assign
B = 1 and L = −1 to this operator:
(B = 1, L = −1 )
O1 = µ1XaφYd (1)
In the following B − L quantum numbers for the trilinear couplings are not mentioned
explicitly since all of them have B − L = ±2. All other couplings which contribute to the
proton decay, can also mediate n− n¯ oscillations or the neutrinoless double beta decay.
There are only two operators which can give rise to the three-lepton decay mode of the
proton:
(B = 0, L = −2 ) (B = 1, L = −1 ) and (B = −1, L = −3 )
O2 = µ2YaY
†
c
χ+ O3 = µ3YbY
†
c
ξ† (2)
These operators will simultaneously allow the single-lepton decay mode of the proton
(B = 1, L = −1) and neutrinoless double beta decay (B = 0, L = 2). There is no trilin-
ear coupling which contributes to the three-lepton decay mode of the proton and no other
5
process.
There are four trilinear couplings which carry B = 2, L = 0 and contribute only to n− n¯
oscillations and not to any other process:
(B = 2, L = 0 )
O4 = µ4∆a∆b∆b O5 = µ5∆c∆a∆a
O6 = µ6∆cYdYd O7 = µ7∆L∆L∆a
(3)
Another four trilinear interactions contribute to both n− n¯ oscillations and proton decay:
(B = 2, L = 0 ) and (B = 1, L = −1 )
O8 = µ8YcYdYd O9 = µ9∆LYbYd
O10 = µ10∆bYaYd O11 = µ11∆aYdYc
(4)
There are several operators which contribute only to neutrinoless double beta decay:
(B = 0, L = 2 )
O12 = µ12φφχ
− O13 = µ13φφξ O14 = µ14χ
−χ−L++
O15 = µ15ξξL
++ O16 = µ16XbX
†
a
χ− O17 = µ17XbX
†
a
ξ
O18 = µ18∆bX
†
a
X†
a
O19 = µ19∆cX
†
a
X†
b
O20 = µ20∆LX
†
a
X†
a
O21 = µ21∆
†
a
∆bχ
− O22 = µ22∆
†
a
∆Lξ O23 = µ23∆
†
b
∆cχ
−
O24 = µ24∆
†
L
∆cξ O25 = µ25∆
†
a
∆cL
−−
(5)
There are few more operators contributing to neutrinoless double beta decay, which also
allow proton decay:
(B = 0, L = 2 ) and (B = −1, L = 1 )
O26 = µ26YaY
†
d
χ− O27 = µ27YbY
†
d
ξ O28 = µ28YcY
†
d
L−−
O29 = µ29XaXbY
†
c
O30 = µ30Xaφ
†Ya O31 = µ31Xaφ
†Yb
O32 = µ32XaXaY
†
a
O33 = µ33XaXaY
†
b
O34 = µ34XbYdφ
†
(6)
The proton decay constraints would then restrict the couplings of these operators making
6
them very much suppressed. However, if some of the couplings could be avoided with discrete
symmetries, then these operators could also contribute to the neutrinoless double beta decay
significantly. Finally there are four more operators which contribute to neutrinoless double
beta decay, proton decay, and n− n¯ oscillations simultaneously:
(B = 0, L = −2 ) , (B = 1, L = −1 ) and (B = 2, L = 0 )
O35 = µ35YdYaYa O36 = µ36YdYbYb
O37 = µ37∆aYaYa O38 = µ38∆aYbYb
(7)
The quartic terms belong to two broad classes, one with B − L = 0 and the other with
B − L = 4. Again they can be categorized under certain subclasses. Those which are
products of the usual quadratic couplings of the form Φ†Φ belong to the trivial category
with B = 0, L = 0 (and thus B − L = 0), which we shall not list here. All the quartic
couplings involving only the dileptons (χ−, L−−, and ξ), the usual Higgs doublet φ, and the
octet Σ also fall into this trivial category with B = 0, L = 0. There are also some other
quartic couplings involving the diquarks and leptoquarks, which can be classified under this
trivial category. These are
B − L = 0 (B = 0, L = 0 )
O˜1 = χ
+χ+ξξ O˜2 = φφχ
+L−− O˜3 = ΣΣχ
+L−−
O˜4 = φ
†φ†ξL++ O˜5 = ΣΣξ
†L−− O˜6 = φ
†φ†ΣΣ
O˜7 = ∆
†
a
XaXaχ
− O˜8 = ∆
†
a
XaXaξ O˜9 = ∆
†
b
XbXbL
−−
O˜10 = ∆
†
c
XbXbχ
− O˜11 = ∆
†
c
XbXbξ O˜12 = ∆
†
L
XbXbL
−−
O˜13 = ∆
†
a
XaXbL
−−
(8)
There are several B − L = 0 quartic couplings which mediate baryon number violating
processes. The simplest of them have B = 2 and L = 2. The hydrogen-antihydrogen
(p+ e− → p¯+ e+) oscillations and double proton decay into two positrons (p+ p→ e+ + e+)
are typical examples of such processes. There are 7 such interactions
B − L = 0 (B = 2, L = 2 )
7
O˜14 = ∆a∆c∆cL
−− O˜15 = ∆b∆b∆bχ
− O˜16 = ∆b∆b∆Lξ
O˜17 = ∆b∆L∆Lχ
− O˜18 = ∆b∆L∆Lξ O˜19 = ∆L∆L∆Lχ
−
O˜20 = ∆L∆L∆Lξ
(9)
There are other quartic couplings also, which allow these processes with B = 2 and L = 2.
They involve Ya, Yb, and Yc, which couple to two quarks as well as to an antiquark and an
antilepton. Hence some of these interactions can simultaneously allow B = 1 and L = 1
proton decay processes such as p → e+ or n → ν¯ associated with a neutral meson and a
lepton-antilepton pair. In all these interactions, the constraints from proton decay will not
allow processes with B = 2. These operators are
B − L = 0 (B = 2, L = 2 ) and (B = 1, L = 1 )
O˜21 = ∆cYaYdχ
− O˜22 = ∆cYbYdξ (10)
B − L = 0
(
B = 2
L = 2
)
,
(
B = 1
L = 1
)
and
(
B = 0
L = 0
)
O˜23 = ∆aYaYcχ
− O˜24 = ∆aYbYcξ O˜25 = ∆bYaYcL
−−
O˜26 = ∆bYaYaχ
− O˜27 = ∆bYaYbξ O˜28 = ∆bYbYbχ
−
O˜29 = ∆bYbYbξ O˜30 = ∆cYaYaL
−− O˜31 = ∆cYaYaL
−−
O˜32 = ∆LYaYaξ O˜33 = ∆LYaYbχ
− O˜34 = ∆LYaYbξ
O˜35 = ∆LYbYbχ
− O˜36 = ∆LYbYbξ O˜37 = ∆LYbYcL
−−
O˜38 = YaYcYdχ
− O˜39 = YbYcYdξ O˜40 = YcYdYdL
−−
(11)
B − L = 0
(
B = 2
L = 2
)
,
(
B = 1
L = 1
)
,
(
B = 0
L = 0
)
and
(
B = −1
L = −1
)
O˜41 = YaYaYaχ
− O˜42 = YaYaYbξ O˜43 = YaYaYcL
−−
O˜44 = YaYbYbχ
−
O˜45 = YaYbYbξ O˜46 = YbYbYbχ
−
O˜47 = YbYbYbξ O˜48 = YbYbYcL
−−
(12)
Another class of quartic couplings can give rise to B − L = 4 processes. Some of them
can produce only B = 2 and L = −2 processes like n + n→ e−e−pi+pi+ or n− n¯ oscillation
8
associated with charged leptons and charged pions. The couplings which can give rise to
these processes are
B − L = 4 (B = 2, L = −2 )
Ô1 = YdYdYdL
++ Ô2 = ∆aYdYdL
++
Ô3 = ∆a∆a∆aL
++ Ô4 = ∆a∆a∆bχ
+
(13)
The other subclass of B − L = 4 processes have B = 1 and L = −3, which give rise to
p→ νννpi+ for example. There are two such operators,
B − L = 4 (B = 1, L = −3 )
Ô5 = XaXaXaφ
† Ô6 = XaXaXaΣ
† (14)
There are also some quartic couplings with B − L = 4, which allow both of the above
two interactions. There are three such terms
B − L = 4 (B = 1, L = −3 ) and (B = 2, L = −2 )
Ô7 = YaYdYdχ
+ Ô8 = ∆aYaYdχ
+ Ô9 = ∆aYbYdξ
† (15)
In summary, we have made a general model-independent study of the possible baryon
and lepton number violations beyond the standard model with scalar bilinears. Depending
on the processes under consideration, they can be classified into only a few categories. We
list all the trilinear and quartic couplings and discuss the processes to which these operators
contribute.
Acknowledgement
Two of us (EM and US) thank the Max-Planck-Institut fu¨r Kernphysik for hospitality. The
work of EM is supported in part by the U.S. Department of Energy under Grant No. DE-
FG03-94ER40837.
9
References
[1] H. V. Klapdor-Kleingrothaus et al., Mod. Phys. Lett. A16, 2409 (2001).
[2] H. V. Klapdor-Kleingrothaus, A. Dietz and I.V. Krivosheina, Particle and nuclei, Letters
110, 57 (2002); Foundations of Physics 32, 1181 (2002).
[3] G. B. Gelmini and M. Roncadelli, Phys. Lett. 99B (1981) 411; C. Wetterich, Nucl.
Phys. B187 (1981) 343; G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys. B181
(1981) 287; R. N. Mohapatra and G. Senjanovic, Phys. Rev. D23 (1981) 165; E. Ma
and U. Sarkar, Phys. Rev. Lett. 80 (1998) 5716; G. Lazarides and Q. Shafi Phys. Rev.
D 58 (1998) 071702; W. Grimus, R. Pfeiffer and T. Schwetz, Eur. Phys. J. C 13 (2000)
125; E. Ma, T. Hambye and U. Sarkar, Nucl. Phys. B 602 (2001) 23; E. Ma, M. Raidal
and U. Sarkar, Eur. Phys. J. C 8 (1999) 301; Phys. Rev. D 60 (1999) 076005.
[4] A. Zee, Phys. Lett. B 93 (1980) 389; L. Wolfenstein, Nucl. Phys. B 175 (1980) 93.
[5] R.N. Mohapatra and J.D. Vergados, Phys. Rev. Lett. 47 (1981) 1713; M. Hirsch, H.V.
Klapdor-Kleingrothaus and S.G. Kovalenko Phys. Rev. D 54 (1996) R4207; Phys. Lett.
B 378 (1996) 17.
[6] M. Leurer, Phys. Rev. D 49 (1994) 333; S. Davidson, D. Bailey and B.A. Campbell,
Zeit. Phys. C 61 (1994) 613; F. Cuypers and S. Davidson, Eur. Phys. J. C2 (1998) 503;
J.P. Bowes, R. Foot and R.R. Volkas, Phys. Rev. D54 (1996) 6936.
[7] E. Ma, M. Raidal, and U. Sarkar, Eur. Phys. J. C8, 301 (1999).
[8] E. Ma, M. Raidal, and U. Sarkar, Phys. Rev. D60, 076005 (1999).
[9] K.Choi, K.S. Jeong and W.Y. Song, hep-ph/0207180.
10
[10] J.C. Pati, A. Salam and U. Sarkar, Phys. Lett. B 133 (1993) 330; J.C. Pati, Phys. Rev.
D 29 (1984) 1549; P.J. O’Donnell and U. Sarkar, Phys. Lett. B 316 (1993) 121.
[11] One of them was considered in, E. Ma and B. Brahmachari, Phys. Lett. B 536 (2002)
259.
11


Baryon and Lepton number violation with scalar bilinears

KLAPDOR-KLEINGROTHAUS, H.

Indian Academy of Sciences

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UCRHEP-T345
October 2002
Baryon and Lepton Number Violation
with Scalar Bilinears
H. V. Klapdor-Kleingrothaus1, Ernest Ma2, and Utpal Sarkar3
1 Max-Planck-Institut für Kernphysik, D-69029 Heidelberg, Germany
2 Physics Department, University of California, Riverside, California 92521, USA
3 Physical Research Laboratory, Ahmedabad 380 009, India
Abstract
We consider all possible scalar bilinears, which couple to two fermions of the stan-
dard model. The various baryon and lepton number violating couplings allowed by
these exotic scalars are studied. We then discuss which ones are constrained by lim-
its on proton decay (to a lepton and a meson as well as to three leptons), neutron-
antineutron oscillations, and neutrinoless double beta decay.
Grand unified theories (GUTs) appear to be the most natural extensions of the standard
model at the very high scale. In GUTs the gauge group is unified, so there is only one single
gauge coupling constant to explain all the forces of the quarks and leptons, which are also
treated at the same footing. However the scale of unification is much too high to be directly
tested in the laboratory. Thus indirect searches become very important. Many predictions
of GUTs were studied, but there is no supporting experimental evidence to date, with the
possible exception of neutrinoless double beta decay [1, 2]. These GUT predictions also
include new particles such as leptoquarks or diquarks and new interactions which violate
baryon and lepton numbers. The latter are also required for an understanding of the present
baryon asymmetry of the Universe. An observed Majorana neutrino mass also requires lepton
number violation. In the context of these issues we present in this paper a general analysis
of baryon and lepton number violation with scalar bilinears.
The scalar bilinears we consider are generalizations of the usual Higgs scalar doublet.
They couple to two fermions of the standard model and are present in many of its exten-
sions [3, 4, 5]. The quantum numbers of these particles are thus fixed, but their masses
and couplings could be arbitrary. Low-energy particle physics phenomenology (such as the
non-observation of baryon number violation) could be used to constrain their masses and
couplings [6, 7]. Baryogenesis with scalar bilinears has also been studied [8]. Recently a
general analysis of the neutrinoless double beta decay has been performed by constructing
effective higher dimensional operators [9]. Most of these operators may be realized in models
where scalar bilinears are included, a few examples of which were demonstrated [9].
In Table 1 we list all the scalar bilinears allowed in any extension of the standard model.
We then write down the different baryon and lepton number violating couplings of these
scalars and what kind of interactions these different couplings may induce, leading to a simple
classification of all the different possible baryon and lepton number violating interactions.
2
We include the usual Higgs scalar doublet of the standard model in these interactions.
Representation Notation qq q̄l̄ ql̄ ll
(1, 1,−1) χ− ×
(1, 1,−2) L−− ×
(1, 3,−1) ξ ×
(3∗, 1, 1/3) Ya × ×
(3∗, 3, 1/3) Yb × ×
(3∗, 1, 4/3) Yc × ×
(3∗, 1,−2/3) Yd ×
(3, 2, 1/6) Xa ×
(3, 2, 7/6) Xb ×
(6, 1,−2/3) ∆a ×
(6, 1, 1/3) ∆b ×
(6, 1, 4/3) ∆c ×
(6, 3, 1/3) ∆L ×
(8, 2, 1/2) Σ
Table 1: Exotic scalar particles beyond the standard model and their couplings to combina-
tions of two fermions.
For the scalar bilinears, it is not possible to define definite baryon or lepton number in
some cases. However, depending on their couplings to the combinations of two fermions, it
is possible to assign baryon and lepton numbers separately. The different assignments are
given in Table 2. For any particular interaction it will be possible to specify what processes
it would mediate and what is the change in baryon and lepton numbers. This will allow us
to classify them.
The quadratic terms with the scalars are trivial. For any quadratic term H1H2 to be
invariant under the gauge interactions, either both of them are singlets (i.e. neutral under
all gauge factors) or else H†1 = H2. Since all the scalars we are discussing are non-singlets,
3
Two-fermion combination qq q̄l̄ ql̄ ll
Baryon number 2/3 −1/3 1/3 0
Lepton number 0 −1 −1 2
B − L 2/3 2/3 4/3 −2
Table 2: Baryon and lepton numbers of the the scalars coupling to different combinations of
two fermions.
none of the quadratic terms can have any B − L quantum number, although some of them
may contribute to baryon or lepton number violation. Consider for example, Y †
a
Ya. Since
Ya can couple to q̄l̄ as well as to qq, this term may induce a ∆B = 1, ∆L = 1 process, but
B − L remains conserved.
The trilinear couplings are the most interesting ones. It was shown [8] that there are 38
possible trilinear couplings with the scalar bilinears. Although baryon and lepton numbers
are not always uniquely specified, all of them carry a B − L quantum number ±2. As we
shall see next, these trilinear operators can mediate four possible processes and hence can
be classified in four categories
(i) B = 1, L = −1: In this case both baryon and lepton numbers are changed by 1, so all
these couplings should give only proton decay, which violates (B − L) [10].
(ii) B = 0, L = 2: This gives lepton number violating processes, without affecting the
baryon number. So this contributes to the Majorana masses of the neutrinos. Some
of them will contribute to the neutrinoless double beta decay [11], but others would
4
involve neutrinos of different generations.
(iii) B = 1, L = 3: These operators give a three-lepton decay mode of the proton [10]. In
some models these processes could dominate over the usual proton decay modes.
(iv) B = 2, L = 0: In this case baryon number is changed by two unit with no change in
lepton number. This corresponds to neutron-antineutron oscillations.
Some of the operators may contribute to two process types. There are no trilinear couplings,
which conserve B − L (which includes of course the case B = 0 and L = 0).
There is only one trilinear coupling which can give proton decay into a lepton and a
meson (or mesons) and is not involved in any other process. It is thus possible to assign
B = 1 and L = −1 to this operator:
(B = 1, L = −1 )
O1 = µ1XaφYd (1)
In the following B − L quantum numbers for the trilinear couplings are not mentioned
explicitly since all of them have B − L = ±2. All other couplings which contribute to the
proton decay, can also mediate n − n̄ oscillations or the neutrinoless double beta decay.
There are only two operators which can give rise to the three-lepton decay mode of the
proton:
( B = 0, L = −2 ) (B = 1, L = −1 ) and (B = −1, L = −3 )
O2 = µ2YaY
†
c
χ+ O3 = µ3YbY
†
c
ξ† (2)
These operators will simultaneously allow the single-lepton decay mode of the proton
(B = 1, L = −1) and neutrinoless double beta decay (B = 0, L = 2). There is no trilin-
ear coupling which contributes to the three-lepton decay mode of the proton and no other
5
process.
There are four trilinear couplings which carry B = 2, L = 0 and contribute only to n− n̄
oscillations and not to any other process:
( B = 2, L = 0 )
O4 = µ4∆a∆b∆b O5 = µ5∆c∆a∆a
O6 = µ6∆cYdYd O7 = µ7∆L∆L∆a
(3)
Another four trilinear interactions contribute to both n− n̄ oscillations and proton decay:
( B = 2, L = 0 ) and ( B = 1, L = −1 )
O8 = µ8YcYdYd O9 = µ9∆LYbYd
O10 = µ10∆bYaYd O11 = µ11∆aYdYc
(4)
There are several operators which contribute only to neutrinoless double beta decay:
( B = 0, L = 2 )
O12 = µ12φφχ
− O13 = µ13φφξ O14 = µ14χ
−χ−L++
O15 = µ15ξξL
++ O16 = µ16XbX
†
a
χ− O17 = µ17XbX
†
a
ξ
O18 = µ18∆bX
†
a
X†
a
O19 = µ19∆cX
†
a
X†
b
O20 = µ20∆LX
†
a
X†
a
O21 = µ21∆
†
a
∆bχ
− O22 = µ22∆
†
a
∆Lξ O23 = µ23∆
†
b
∆cχ
−
O24 = µ24∆
†
L
∆cξ O25 = µ25∆
†
a
∆cL
−−
(5)
There are few more operators contributing to neutrinoless double beta decay, which also
allow proton decay:
( B = 0, L = 2 ) and ( B = −1, L = 1 )
O26 = µ26YaY
†
d
χ− O27 = µ27YbY
†
d
ξ O28 = µ28YcY
†
d
L−−
O29 = µ29XaXbY
†
c
O30 = µ30Xaφ
†Ya O31 = µ31Xaφ
†Yb
O32 = µ32XaXaY
†
a
O33 = µ33XaXaY
†
b
O34 = µ34XbYdφ
†
(6)
The proton decay constraints would then restrict the couplings of these operators making
6
them very much suppressed. However, if some of the couplings could be avoided with discrete
symmetries, then these operators could also contribute to the neutrinoless double beta decay
significantly. Finally there are four more operators which contribute to neutrinoless double
beta decay, proton decay, and n − n̄ oscillations simultaneously:
(B = 0, L = −2 ) , (B = 1, L = −1 ) and ( B = 2, L = 0 )
O35 = µ35YdYaYa O36 = µ36YdYbYb
O37 = µ37∆aYaYa O38 = µ38∆aYbYb
(7)
The quartic terms belong to two broad classes, one with B − L = 0 and the other with
B − L = 4. Again they can be categorized under certain subclasses. Those which are
products of the usual quadratic couplings of the form Φ†Φ belong to the trivial category
with B = 0, L = 0 (and thus B − L = 0), which we shall not list here. All the quartic
couplings involving only the dileptons (χ−, L−−, and ξ), the usual Higgs doublet φ, and the
octet Σ also fall into this trivial category with B = 0, L = 0. There are also some other
quartic couplings involving the diquarks and leptoquarks, which can be classified under this
trivial category. These are
B − L = 0 ( B = 0, L = 0 )
Õ1 = χ
+χ+ξξ Õ2 = φφχ
+L−− Õ3 = ΣΣχ
+L−−
Õ4 = φ
†φ†ξL++ Õ5 = ΣΣξ
†L−− Õ6 = φ
†φ†ΣΣ
Õ7 = ∆
†
a
XaXaχ
− Õ8 = ∆
†
a
XaXaξ Õ9 = ∆
†
b
XbXbL
−−
Õ10 = ∆
†
c
XbXbχ
− Õ11 = ∆
†
c
XbXbξ Õ12 = ∆
†
L
XbXbL
−−
Õ13 = ∆
†
a
XaXbL
−−
(8)
There are several B − L = 0 quartic couplings which mediate baryon number violating
processes. The simplest of them have B = 2 and L = 2. The hydrogen-antihydrogen
(p+ e− → p̄+ e+) oscillations and double proton decay into two positrons (p+ p → e+ + e+)
are typical examples of such processes. There are 7 such interactions
B − L = 0 ( B = 2, L = 2 )
7
Õ14 = ∆a∆c∆cL
−− Õ15 = ∆b∆b∆bχ
− Õ16 = ∆b∆b∆Lξ
Õ17 = ∆b∆L∆Lχ
− Õ18 = ∆b∆L∆Lξ Õ19 = ∆L∆L∆Lχ
−
Õ20 = ∆L∆L∆Lξ
(9)
There are other quartic couplings also, which allow these processes with B = 2 and L = 2.
They involve Ya, Yb, and Yc, which couple to two quarks as well as to an antiquark and an
antilepton. Hence some of these interactions can simultaneously allow B = 1 and L = 1
proton decay processes such as p → e+ or n → ν̄ associated with a neutral meson and a
lepton-antilepton pair. In all these interactions, the constraints from proton decay will not
allow processes with B = 2. These operators are
B − L = 0 ( B = 2, L = 2 ) and ( B = 1, L = 1 )
Õ21 = ∆cYaYdχ
− Õ22 = ∆cYbYdξ (10)
B − L = 0
(
B = 2
L = 2
)
,
(
B = 1
L = 1
)
and
(
B = 0
L = 0
)
Õ23 = ∆aYaYcχ
− Õ24 = ∆aYbYcξ Õ25 = ∆bYaYcL
−−
Õ26 = ∆bYaYaχ
− Õ27 = ∆bYaYbξ Õ28 = ∆bYbYbχ
−
Õ29 = ∆bYbYbξ Õ30 = ∆cYaYaL
−− Õ31 = ∆cYaYaL
−−
Õ32 = ∆LYaYaξ Õ33 = ∆LYaYbχ
− Õ34 = ∆LYaYbξ
Õ35 = ∆LYbYbχ
− Õ36 = ∆LYbYbξ Õ37 = ∆LYbYcL
−−
Õ38 = YaYcYdχ
− Õ39 = YbYcYdξ Õ40 = YcYdYdL
−−
(11)
B − L = 0
(
B = 2
L = 2
)
,
(
B = 1
L = 1
)
,
(
B = 0
L = 0
)
and
(
B = −1
L = −1
)
Õ41 = YaYaYaχ
− Õ42 = YaYaYbξ Õ43 = YaYaYcL
−−
Õ44 = YaYbYbχ
−
Õ45 = YaYbYbξ Õ46 = YbYbYbχ
−
Õ47 = YbYbYbξ Õ48 = YbYbYcL
−−
(12)
Another class of quartic couplings can give rise to B − L = 4 processes. Some of them
can produce only B = 2 and L = −2 processes like n + n → e−e−π+π+ or n − n̄ oscillation
8
associated with charged leptons and charged pions. The couplings which can give rise to
these processes are
B − L = 4 (B = 2, L = −2 )
Ô1 = YdYdYdL
++ Ô2 = ∆aYdYdL
++
Ô3 = ∆a∆a∆aL
++ Ô4 = ∆a∆a∆bχ
+
(13)
The other subclass of B − L = 4 processes have B = 1 and L = −3, which give rise to
p → νννπ+ for example. There are two such operators,
B − L = 4 (B = 1, L = −3 )
Ô5 = XaXaXaφ
† Ô6 = XaXaXaΣ
† (14)
There are also some quartic couplings with B − L = 4, which allow both of the above
two interactions. There are three such terms
B − L = 4 (B = 1, L = −3 ) and (B = 2, L = −2 )
Ô7 = YaYdYdχ
+ Ô8 = ∆aYaYdχ
+ Ô9 = ∆aYbYdξ
† (15)
In summary, we have made a general model-independent study of the possible baryon
and lepton number violations beyond the standard model with scalar bilinears. Depending
on the processes under consideration, they can be classified into only a few categories. We
list all the trilinear and quartic couplings and discuss the processes to which these operators
contribute.
Acknowledgement
Two of us (EM and US) thank the Max-Planck-Institut für Kernphysik for hospitality. The
work of EM is supported in part by the U.S. Department of Energy under Grant No. DE-
FG03-94ER40837.
9
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11


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Baryon and Lepton Number Violation with Scalar Bilinears

Baryon and Lepton Number Violation with Scalar Bilinears

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