November, 2008

Neutron-Anti-Neutron Oscillation: Theory

and Phenomenology
arXiv:0902.0834v1 [hep-ph] 5 Feb 2009

R. N. Mohapatra

Maryland Center for Fundamental Physics and Department of Physics, University of

Maryland, College Park, Maryland, 20742

Abstract

The discovery of neutrino masses has provided strong hints in favor of the possibility
that B-L symmetry is an intimate feature of physics beyond the standard model. I
discuss how important information about this symmetry as well as other scenarios for
TeV scale new physics can be obtained from the baryon number violating process,
n − n̄ oscillation. This article presents an overview of different aspects of neutron-anti-
neutron oscillation and is divided into the following parts : (i) the phenomenon; (ii)
the physics, (iii) plausible models and (iv) applications to cosmology. In particular, it
is argued how the discovery of n − n̄ oscillation can significantly affect our thinking
about simple grand unified theory paradigms for physics beyond the standard model,
elucidate the nature of forces behind neutrino mass and provide a new microphysical
view of the origin of matter in the universe.
1 Introduction

Particle oscillations are familiar phenomena in both classical and quantum mechanics and
have provided a wealth of information about the nature of matter and forces acting on them.

The most elementary example of a pendulum oscillation is caused by the force of gravity
and therefore provides information about the strength of gravity. In the domain of quantum
mechanics, two states close by in energy can transmute into one another if the Hamiltonian
for the system includes a force that connects them and if certain coherence conditions are

satisfied. Turning this question around, one can conclude that if experimentally an oscillation
between states, forbidden in the standard model is observed, it will then imply the existence
of hitherto unknown classes of interactions. Many such instances in the domain of particle

physics are known where study of particle oscillations have provided important clarification
about the fundamental nature of underlying physics. Kaon and neutrino oscillations are but
two glowing examples of this. The first in combination with the recently observed B − B̄
oscillations confirmed the nature of CP violation in standard model. The latter has for

the first time revealed the existence of new physics as well as new symmetries beyond the
standard model.

The question then arises as to whether there are other such possibilities that need to be
experimentally explored to probe physics beyond the standard model (SM). In this review,
I argue that neutron-anti-neutron (n − n̄) oscillation provides one such unique example and
is extremely timely in the aftermath of one of the fundamental discoveries in our field i.e.

the neutrinos are massive and can provide important clues to our understanding of new
symmetries and forces behind the neutrino mass generation.

The n − n̄ oscillation[1] is a unique kind of oscillation phenomenon compared to kaon

oscillations in that it breaks one of the once sacred conservations laws of physics i.e. con-

2
servation of baryon number which keeps all matter stable (just like kaon oscillation breaks
strangeness quantum number). It also effectively makes neutron a Majorana fermion (albeit
with a very tiny Majorana component), a point which is of some historical importance since

in the original paper of Majorana, he contemplated neutron being the particle which is its
own anti-particle.

The fact that n − n̄ oscillation breaks baryon number requires us to first ascertain that
the vast wealth of information available about the stability of matter from experiments and
general cosmological observations allow for this process to have a strength that will make it
observable with current technology. We show below that this is indeed the case.

Secondly as far as the physics implications go, another class of processes that probe
baryon number nonconservation is proton decay[2, 3]. They have been the focus of exper-

iment as well as theory for two reasons: first of all, simple grand unified theories[4] based
on SU(5) and SO(10) predict its existence at a level not far from experimental capabilities;
secondly, it used to be thought in the early 80’s that these theories may also explain the
origin of matter using proton decay as one of its key ingredients.

This situation however changed drastically with time due to two developements.

(a) The discovery of neutrino masses was confirmed in 1998 and a popular paradigm for
the origin of their smallness became the seesaw mechanism[5], where new symmetry, B-L
respected by then standard model forces had to be broken1 . Note that the dominant proton

decay modes of interest p → e+ π o respect the B − L symmetry and is therefore nor suited
for the study of forces responsible for neutrino mases. On the other hand, n − n̄ oscillations
break B-L symmetry by two unjits exactly like the seesaw mechanism for neutrino masses
does and therefore is uniquely suited for the study of the detailed nature as well as the
1
Without any new symmetry, the seesaw scale would naturally be expected to be the Planck scale, which
yields too small neutrino masses. A simple way to understand why the seesaw scale is lower than the Planck
scale is to assume that it is associated with a symmetry e.g. B-L .

3
scale of seesaw mechanism. There is a large class of interesting gauge models where indeed
neutrino Majorana masses via the seesaw mechanism leads directly to n − n̄ oscillations.

(b) The second development is that a new mechanism for understanding the origin of
matter was proposed in mid 80’s that did not use proton decay but rather used the seesaw
mechanism for neutrino masses[6] to generate matter-anti-matter asymmetry where again

B-L symmetry played a key role. Although grand unified theories such as SO(10) could
incorporate the B-L symmetry, these two developments took a lot of the wind from the
argument in favor of proton decay being necessarily the most theoretically relevant mode for
baryon nonconservation.

Furthermore, when the seesaw mechanism is embedded into GUT theories that include
B-L symmetry, they generally tend to fix the scale of B-L symmetry breaking to be at the

GUT scale, while there is no compelling reason for the seesaw scale to be that high. It could
even be at the TeV scale without making the theory too unnatural. The seesaw scale could
not therefore be probed by searching for proton decay whereas as already noted, in a large
class of interesting theories, n − n̄ oscillation does.

The question then arises as to whether all physics beyond the standard model is encap-
sulated in grand unified theories with neutrino masses being also a signal of the same GUT

scale or whether the neutrino masses indicate an alternate path that contains rich physics
far below the GUT scale. As we show below, n − n̄ oscillation provides an effective probe
of the second possibility which envisions a rich new layer of physics at intermediate scales
that involves low B-L breaking. Theoretical models indicate that a wide range of scales,

anywhere from a few TeV to 1011 GeV as well as the associated new physics could be probed
by pushing the n − n̄ oscillation oscillation time by three orders of magnitude over the cur-
rent experimental lower limit. In particular, its discovery will rule out simple grand unified
theories or in the very least have significant affect on their detailed nature. Furthermore

4
it will have profound effect on the nature of physics beyond the standard model as well as
provide a new microphysics view for the origin of matter, different from leptogenesis. It also
provides rich new physics that can be probed at the Large Hadron Collider.

The bottom line is therefore that it is urgent to pursue a dedicated search for for n − n̄
oscillation that can push the sub-GUT physics frontier to a new level.

This review is organized into the following parts: in the first part, I discuss the phe-
nomenon, its connection to nuclear instability ; in the next section, we discuss the broad

character of microphysics that would lead to n − n̄ oscillation and the kind of scales that can
be probed by this. In the next section, the nature of physics beyond standard model probed
by n − n̄ oscillation will be discussed including a discussion as to why B-L is an important
symmetry to be probed. In this section, we discuss specific and plausible gauge models for

n − n̄ oscillation both with and without supersymmetry. In the final section, we discuss the
implication of observable n − n̄ oscillation for baryogenesis and show how one can have a
new class of models for late baryogenesis after electroweak phase transition and sphaleron
freeze-out.

2 Phenomenology of n − n̄ oscillation

In order to discuss the main features of n − n̄ oscillation in various environments, it is

sufficient to consider a simple 2 × 2 Hamiltonian for the evolution of an initial beam of slow
moving neutrons and evaluate the probability of finding an anti-neutron in this beam after
a time t.      
∂ n En δm   n 
ih̄   =  (1)
∂t n̄ δm En̄ n̄
where δm denotes the n − n̄ mixing which parameterizes at the nucleon level the underlying
physics that breaks baryon number by two units. We will give examples of models where

5
this can happen at the appropriate level in a subsequent section. Note that we have left the
energies of the n and n̄ arbitray since they could be affected in different ways in the presence
of an external environment such as the nuclear field or a magnetic field etc.

The exact expression for the probability to find an antineutron at time given that at
t = 0, Pn (0) = 1 and Pn̄ (0) = 0 is:

4δm2 2
√
Pn̄ (t) = sin ( ∆E 2 + 4δm2 )t; (2)
∆E 2 + 4δm2

where ∆E = En − En̄ . The fact that neutron decays does not have any effect on this
formula as long as the neutron flight time is much smaller than the life time of neutron.

Let us consider two extreme cases of this formula which also happen to be interesting for
experimental purposes. For δm ≪ ∆E, this expression reduces to:
!2
δm
Pn̄ ∼ sin2 ∆En t (3)
∆En

There are two special cases of this realizable in nature:

 2
t
• Case (i): ∆Et ≪ 1: In this case we have, Pn→n̄ ∼ (δm · t)2 ≡ τn−n̄
This case
corresponds to free neutron oscillation in vacuum.
 2
1 δm
• Case (ii): ∆E · t ≫ 1: Pn→n̄ ∼ 2 2∆En
This for instance corresponds to bound

neutrons inside nucleus “oscillating” to anti-neutrons .

2.1 Stability of Nuclei and limit on τn−n̄:

As with all baryon number violating interactions, the existence of n − n̄ oscillation will lead
to nuclear instability. This will happen via a conversion of neutron to an anti-neutron inside

the nucleus which will then annihilate with the surrounding nucleons and lead to instability.
Clearly, if the nuclear transition probability inside the nucleus and in vacuum were equal,

6
this would cause all matter to disappear in a much shorter time than the age of the Universe
and would be unacceptable. However, it is an experimental fact that inside the nucleus the
neutron and the anti-neutron experience different nuclear potentials. In fact the difference

is so large (EN − EN̄ ∼ 100 MeV or so) that the transition probability formula relevant for
this case is that in case (ii) above and leads indeed to nuclear instability lifetimes of order
1032 years or so. Careful and detailed analysis of this question has been done by Dover, Gal
and Richards[11], Alberico et al[12] and more recently by Gal and collaborators[13]. Below

we present a crude representation of the above works to get the basic physics across:
!2
nuc. δm
Pn−n̄ ∼ Pn→n̄ ∼ . (4)
2∆En

where Pn→n̄ represents the free neutron oscillation probability in vacuum and δm is the
∆B = 2 off diagonal matrix element in the n − n̄ mass matrix in Eq. (1) and is related
h
to the oscillation time τn−n̄ = 2πδm
. Present ILL limit on τn−n̄ ≥ 108 sec. translates to
δm ≤ 6 × 10−33 GeV. A crude estimate of the nuclear instability life time in the presence
 −2
δm
n − n̄ oscillation from these intuitive considerations is τN ucl. ∼ 2∆En
10−23 sec.≃ 1034
years. More careful recent calculations that take into account the nuclear effects[13] are
stated in terms of the following relation:

τN uc. = R(τnn̄ )2 (5)

where R ∼ 0.3 × 1024 sec.−1 , which yields τN uc. ∼ 3 × 1032 yrs. Present experimental limits
on τN ucl. from various nucleon decay searches e.g. at Kamiokande, Sudan, IMB, Super-

Kamiokande, SNO are given below in table I with the value of τn−n̄ noted there[16].

Expt source of neutrons τN ucl. (yrs) τosc. (sec)

56
Soudan Fe 0.72 × 1032 1.3 × 108
56
Frejus Fe 0.65 × 1032 1.2 × 108
16
Kamiokande O 0.43 × 1032 1.2 × 108
16
Super-K O 1.77 × 1032 2.3 × 108

7
 Figure 1: Typical horizontal reactor set-up for n − n̄ oscillation search
.

Table Caption: Experimental lower limits on τn−n̄ and the nuclei used for deducing the
limit.

One can also anticipate a similar bound from the SNO experiment[20]. This implies → δm ≤
10−29 MeV. The nucleon decay searches for τn−n̄ become less efficient due to atmospheric

neutrino induced background as we move to higher precision goals for τn−n̄ . It would therefore
appear that in order to extend the τn−n̄ search much beyond the current limit, one needs to
focus more on free neutron oscillation experiments rather than than nucleon decay search.

We will see that present reactor neutron fluxes are precisely in the right range to probe
these values of τn−n̄ that is compatibitible with matter stability as well as latest limits on
proton decay.

8
2.2 Reactor Experiment to search for neutron oscillation

The basic idea behind a reactor search for n − n̄ oscillation is to have a cold neutron beam
from a reactor (neutron speed v ∼ 1 − 2 kilo meters/sec.) and have the beam pass through
a high vacuum, demagnetized region for as long as feasible to a detector at the end of this

“tunnel”. If one of the neutrons in the beam oscillates to an anti-neutron, this will annihilate
in the detector and create a multi-pion signal with total energy of ∼ 2 GeV. The need for
demagnetization comes from the fact that the magnetic field of the Earth (∼ 0.5 Gauss)
will split the neutron and anti-neutron energy levels by µn B ∼ 3 × 10−21 GeV. If we take a

typical flight time in the “tunnel” of about one sec., it will imply µn Btsim3 × 10−21 GeV
sec which about a 1000 times h̄ and hence does not satisfy the condition for free neutron
oscillation. It is therefore necessary to reduce the magnetic field to the level of 10−4 − 10−5
Gauss. This is known technology that uses µ-metal shielding. The typical horizontal set-up

for such an experiment is given in Fig. 1 above. Under these conditions, one gets for the
number of anti-neutrons for a given neutron beam the figure of merit of the experiment:
!2
t
#of events = Φn ×T (6)
τN −N̄
where Φn is the reactor flux; vn t= distance to detector; T running time. Maximum reactor
fluxes for a 100 MW reactor is about ∼ 1013 − 1014 neutrons per sec.; for t = 0.1 sec. and
T ∼ 3 years can yield a limit of 1010 sec. Using this technique, the first n − n̄ search was

carried out at the ILL laboratory in Grenoble, France and a lower limit of τn−n̄ ≥ 8.6 × 107
sec. was obtained[15]. New searches have recently been proposed which could improve this
limit by two orders of magnitude[17]. Fig. 2 compares the free oscillation time to the nuclear
instability life time (figure courtesy of Y. Kamyshkov Ref.[17]). It is clear that time of flight

t anf the flux are two crucial factors that determine how precisely one can measure the
n − n̄ transition time. Clearly, the slower the neutrons the bigger is the time of flight. A
possibility that is being pursued is to use ultracold neutrons with ∼ 100 nano-eV kinetic

9
 Figure 2: :
Comparision of free neutron oscillation time vs nuclear instability life time

10
energies which move with speeds of order 4-5 m/sec and are trapped in a bottle[18] and may
become competitive with cold reactor beam experiments in near future[19].

3 Operator analysis of ∆B = 2 processes and new physics

scale probed

Whenever a new process is used to probe physics beyond the standard model, a general

question that is essential to know is the mass scale probed by it. A simple way to get
a general idea about the scale probed by a physical process is to do an operator analysis
i.e. assuming a particular symmetry and spectrum of the low energy theory, write effective
higher dimensional operators for the process under consideration and see for what value of

the mass scaling the operator, the process is observable. This argument has been a useful
tool to probe physics at short distance scales probed by processes such as proton decay,
neutrino mass as well as corrections to standard model observables.

The method however has its limitations. For instance, often in these discussions, one
uses the SM fermion spectrum but if there is a SM non-singlet new particle with a 100 GeV

mass not discovered yet, the naive scale arguments can be misleading since new operators
may appear. Similarly, the presence of unknown higher symmetries could also invalidate
these arguments. We will apply this discussion to the ∆B = 2 operators below to see the
scale probed by neutron-anti-neutron oscillation.

3.1 Standard model operator

In the standard model, the effective operator responsible for ∆B = 2 and ∆L = 0 processes
is given by:

1
O∆B=2 = 5
[QQQQ(dc dc )∗ + uc dc dc uc dc dc ] (7)
M

11
 1
The strength of this operator G∆B=2 ≃ M5
. The n − n̄ mixing mass can be deduced from
this operator by simple dimensional analysis to be:

Λ6QCD
δmn−n̄ ≃ cG∆B=2 Λ6QCD =c (8)
M5

where we chooze c to be of order one. Attempts have been made in the past to estimate c
using bag as well as other phenomenological models for hadrons[21]. Taking the best current
lower bound on τn↔n̄ from ILL reactor experiment [15] which is 108 sec. and comparable

bounds from nucleon decay search experiments [16], one can then obtain a lower limit on M
to be around 10-300 TeV depending on other couplings in a theory. If proposals to improve
the precision of this search by at least two orders of magnitude [17] are carried out, , then
in the context of the standard model particles, the mass scale probed will be around a 1000

TeV. Thus we see that neutron-anti-neutron oscillation will probe physics at scales far below
the GUT scales that proton decay will probe. An important question to ask is whether the
scale probed can be higher in the presence of new physics around the TeV scale. We discuss
this below.

3.2 Supersymmetry and enhancement of ∆B = 2 operator

In the presence of supersymmetry at the TeV scale, there are new particle such as squarks
and sleptons with TeV scale mass. They can then enter the effective ∆B = 2 operator. This
kind of effect is familiar from discussions of supersymmetric grand unified theories where the
presence of super-partners with TeV scale masses can drastically alter the operator analysis

for nucleon decay. For instance the dimension six operator responsible for proton decay in
non-susy models goes like QQQL/M 2 whereas the presence of squarks with TeV or sub-TeV
masses, the dominant operator becomes QLQ̃Q̃/M, which has reduced power dependence
on mass and is well known to put severe constraints on grand unified theories.

12
 A typical leading operator for n − n̄ oscillation for supersymmetric case is:

1 c c ˜c c ˜c ˜c
O∆B=2 = u d d u˜ d d (9)
M3

Note that the power dependence on the seesaw scale (or B-L breaking scale) has now consid-
erably softened. The conversion of susy particles to SM particles brings some suppression;
but the the overall impact is that simple power counting arguments change. They imply
that if this is the leading operator one can probe the seesaw scale upto 108 GeV with n − n̄

oscillation times of 1010 − 1011 sec. The power dependence in fact softens even further if
there are new color sextet particles at or below the TeV scale. For instance, if there is a
scalar diquark coupled sextet field of ∆uc uc type, the leading operator becomes:

1 c c
O∆B=2 = d d ∆uc uc d˜c d˜c (10)
M2

and the scale reach of n − n̄ goes up to 1011 GeV. If on the other hand there is a field ∆uc dc ,
the leading order operator becomes:

1 c c
O∆B=2 = d d ∆uc dc ∆uc dc (11)
M

increasing the scale reach to the GUT scale. The question then arises whether there are
plausible models where this can happen. Below I give an example of models which have TeV
scale ∆uc uc fields naturally so that an 1/M 2 dependence on the seesaw scale is quite natural.

From this, we see that scale of observable n − n̄ physics can be anywhere from 300 TeV to
101 3 TeV

4 Neutrino mass physics probed by n − n̄ oscillation

To see what new physics is revealed by the search for n − n̄ oscillation, we give several
examples where low scale for seesaw manifests in the non-leptonic sector of the theory as

13
n − n̄ oscillation. This happens due to the connection of seesaw to local or global B-L
symmetry. There could also be other kind of beyond the standard model scenarioss where
extra space dimensions curled up to TeV−1 length scales and where additional dynamical

inputs can lead to observable n − n̄ oscillation. In the former case, it will throw important
light on the physics associated with neutrino mass and we discuss this below.

There are various reasons to think that B-L symmetry is an intimate feature of physics
beyond the standard model. Some of them are as follows:

• (i) The seesaw mechanism for understanding small neutrino masses requires the intro-
duction of right handed neutrinos[5]. In the presence of three right handed neutrinos it
is more natural for B-L symmetry which was a global symmetry of the SM Lagrangian

to become a local symmetry;

• (ii) An inherent aspect of seesaw is the Majorana mass of the right handed neutrino

that breaks the B-L symmetry providing one way to understand why the seesaw scale
is so much less than the Planck scale;

• (iii) A third reason appears once one admits the presence of supersymmetry at the
TeV scale, as is widely believed and requires the lightest SUSY particle (LSP) to be
naturally stable in order to be the dark matter candidate of the universe. The simplest
way to have SUSY LSP naturally stable is to have B-L as a symmetry of physics beyond

MSSM[7].

If indeed B-L symmetry is present in nature, there are several questions that immediately
come to mind:

• (a) Is it a global or local symmetry ?

• (b) is it a broken or unbroken symmetry ?

14
 • (c) if it is broken, what is the breaking scale and what is the associated physics ?

I will now argue that search for the process of neutron-anti-neutron oscillation[8, 9, 10] can
provide a partial answers to some of these important questions: in particular the question of
scale of B-L breaking and associated expanded gauge symmetry and hence physics associated
with the origin of neutrino mass. The point is that since a ∆B = 2 operator breaks B-L

by two units, it is natural to associate the scale associated with the ∆B = 2 operator also
with the scale of B-L breaking since neither B nor L are separate symmetries of the standard
model but rather the symmetry is B−L . Since seesaw mechanism also breaks B-L symmetry,
M could also be the seesaw scale. In fact, in the context of models such as those based on

SU(2)L × SU(2)R × SU(4)c [14], the same mass scale M is responsible for both processes as
was first shown in Ref.[10]. Another purely group theoretical way to see this is to note that
in left-right symmetric models, there is a relation between the electric charge and the B-L
generator as follows:

B−L
Q = I3L + I3R + (12)
2

For distance scales shorter than the electroweak scale, we have ∆I3L = 0 and electric charge
conservation then implies that ∆I3R = − B−L
2
implying that parity violation (or ∆I3R = 1)

implies not only that ∆L = 2 i.e. seesaw mechanism for neutrino mass but in the non-
leptonic sector it implies ∆B = 2 which is of course n − n̄ oscillation.

Thus the search for n− n̄ oscillation could not only illuminate the nature of the important
symmetry, B-L but could also be one way to unravel the mystery of the seesaw mechanism
that is expected to be major player in the physics of neutrino mass as well as the origin of
parity violation in Nature.

15
5 SU (2)L × SU (2)R × SU (4)c model with light diquarks

In this section we demonstrate by an explicit example how neutrino mass and n − n̄ are
intimately connected in this partial unification theory. We first recapitulate some elementary

facts about the model.

The quarks and leptons in this model are unified and transform as ψ : (2, 1, 4) ⊕ ψ c :

(1, 2, 4̄) representations of SU(2)L × SU(2)R × SU(4)c . For the Higgs sector, we choose, φ1 :
¯ c : (1, 3, 10)
(2, 2, 1) and φ15 : (2, 2, 15) to give mass to the fermions. The ∆c : (1, 3, 10) ⊕ ∆
to break the B − L symmetry. There are color sextet scalar diquark fields contained in the
∆c : (1, 3, 10) multiplet, which will play a crucial role in generating n − n̄ oscillation. We

first discuss the non-0supersymmetric version of the model in next subsection and follow it
up by the supersymmetric generalization where the scale reach of n − n̄ goes up all the way
to 1010 GeV or so.

5.1 Non-supersymmetric version

The Higgs potential for this model can be written as :

V (φ0 , φ15 , ∆R,L ) = V0 + Vm (13)

where

′ ′ ′ ′
Vm = λm ǫµνσρ ǫµ ν σ ρ ∆R,µµ′ ∆νν ′ ∆σσ′ ∆ρρ′ + (R ↔ L) + h.c. (14)

This term is crucial for generating baryon number non-conservation in this theory and with-
out it, the model cannot explain the origin of matter in the universe (as we see below). We
will see that the same term is responsible for the ∆B = 2 n − n̄ oscillation.

In order to discuss neutrino masses and n − n̄ oscillation, let us write down the Yukawa

16
couplings in the model:

LY = h1 Ψ̄L Φ1 ΨR + h15 Ψ̄L Φ15 ΨR + f ΨTR C −1 ∆R ΨR + (R ↔ L) + H.C. (15)

Using standard spontaneous breaking via the vev of (1, 3, 10) to the standard model gauge
group and Φ1,15 to give mass to both to charged fermions and Dirac mass to neutrinos,
we implement the seesaw mechanism for neutrinos. It was shown in ref.[10] that the same

(1, 3, 10) via the potential term Vm leads to n − n̄ oscillation via the diagram in Fig. 3. The

Figure 3: A Feynman diagram contributing to n − n̄ oscillation as discussed in Ref. 8.

Another class of graphs involves two ∆uc dc and one ∆dc dc scalar exchanges.

strength of n − n̄ oscillation is given by:

3
λf11 vBL
G∆B=2 = 6
(16)
M∆qq

We see the fifth power dependence on the seesaw scale since the diquarks are expected to be

17
at seesaw scale by the usual naturalness considerations. Also note that even after symmetry
breaking this model has a Z2 symmetry given by (−1)B .

It has recently been shown[22] that this model also explains the baryon asymmetry of
the universe via the mechanism of post-sphaleron baryogenesis to be discussed below[23].

5.2 Supersymmetric version

We now discuss the supersymmetric version of this model where the triplet fields doubled to

cancel the gauge anomalies. We also add a B − L neutral triplet Ω : (1, 3, 1) which helps to
reduce the accidental global symmetry of the model and hence the number of light diquark
states. The superpotential of this model is given by:

W = WY + WH1 + WH2 + WH3 (17)

where

¯ c − M 2 ) + λC ∆c ∆
WH1 = λ1 S(∆c ∆ ¯ c Ω + µi Tr (φi φi ) (18)

¯ c )2
(∆c ∆ ¯ c∆
(∆c ∆c )(∆ ¯ c)
WH2 = λA + λB (19)
MPℓ MPℓ

¯ c φ15 )
Tr (φ1 ∆c ∆
WH3 = λD , (20)
MPℓ
(21)

WY = h1 ψφ1 ψ c + h15 ψφ15 ψ c + f ψ c ∆c ψ c . (22)

Note that since we do not have parity symmetry in the model, the Yukawa couplings h1 and
h15 are not symmetric matrices. When λB = 0, this superpotential has an accidental global
symmetry much larger than the gauge group[25]; as a result, vacuum breaking of the B − L

18
symmetry leads to the existence of light diquark states that mediate N ↔ N̄ oscillation and
¯ ci =
enhance the amplitude. In fact it was shown that for h∆c i ∼ h∆ 6 0 and hΩi =
6 0 and all
VEVs in the range of 1011 − 1012 GeV, the light states are those with quantum numbers:

∆uc uc . The symmetry argument behind is that [25] for λB = 0, the above superpotential
is invariant under U(10, c) × SU(2, c) symmetry which breaks down to U(9, c) × U(1) when
h∆cν c ν c i = vBL 6= 0. This results in 21 complex massless states; on the other hand these
vevs also breaks the gauge symmetry down from SU(2)R × SU(4)c to SU(3)c × U(1)Y . This

allows nine of the above states to pick up masses of order gvBL leaving 12 massless complex
¯ cuc uc states. Once λB 6= 0 and is of order 10−2 − 10−3 ,
states which are the six ∆cuc uc plus six ∆
they pick up mass (call Muc uc ) of order of the elctroweak scale.

5.3 A new diagram for neutron-anti-neutron oscillation

To discuss n ↔ n̄ oscillation, we introduce a new term in the superpotential of the form[10]:

1 µ′ ν ′ λ′ σ′ µνλσ c
W∆B=2 = ǫ ǫ ∆µµ′ ∆cνν ′ ∆cλλ′ ∆cσσ′ , (23)
M∗

where the µ, ν etc stand for SU(4)c indices and we have suppressed the SU(2)R indices.
Apriori M∗ could be of order MP ℓ ; however the terms in Eq.(2) are different from those in

Eq. (4); so they could arise from different a high scale theory. The mass M∗ is therefore a
free parameter that we choose to be much less than the MP ℓ . This term does not affect the
masses of the Higgs fields. When ∆cν c ν c acquires a VEV, ∆B = 2 interaction are induced
from this superpotential, and n ↔ n̄ oscillation are generated by two diagrams given in

Fig. 3 and 4. The first diagram (Fig. 3) in which only diquark Higgs fields are involved
3 v
f11 BL M∆
was already discussed in [10] and goes like Gn↔n̄ ≃ Mu2c uc Md4c dc M∗
, Taking Muc uc ∼ 350 GeV,
Mdc dc ∼ λ′ vBL and M∆ ∼ vBL as in the argument [25], we see that this diagram scales
−3 −2
like vBL vwk . In ref.[24] a new diagram (Fig. 4) was pointed out which owes its origin to

19
supersymmetry. We get for its contribution to G∆B=2 :

g32 3
f11 vBL
Gn↔n̄ ≃ 2 2 2
. (24)
16π Muc uc Mdc dc MSUSY M∗
−2 −3
Using the same arguments as above, we find that this diagram scales like vBL vwk which is
therefore a significant enhancement over diagram in Fig.3 when vBL ≫ vwk . In order to

Figure 4: The new Feynman diagram for n − n̄ oscillation.

estimate the rate for n ↔ n̄ oscillation, we need not only the different mass values for which
we now have an order of magnitude, we also need the Yukawa coupling f11 . Now f11 is
a small number since its value is associated with the lightest right-handed neutrino mass.

20
However, in the calculation we need its value in the basis where quark masses are diagonal.
We note that the n − n̄ diagrams involve only the right-handed quarks, the rotation matrix
need not be the CKM matrix. The right-handed rotations need to be large e.g. in order to
(u,d) (u,d)† (u,d) diag.
involve f33 (which is O(1)), we need (VR )31 to be large, where VL Yu,d VR = Yu,d .
(u,d)
The left-handed rotation matrices VL contribute to the CKM matrix, but right-handed
(u,d)
rotation matrices VR are unphysical in the standard model. In this model, however, we
get to see its contribution since we have a left-right gauge symmetry. Let us now estimate

the time of oscillation. When we start with a f -diagonal basis (call the diagonal matrix
fˆ), the Majorana coupling f11 in the diagonal basis of up- and down-type quark matrices
can be written as (VRT fˆVR )11 ∼ (V31R )2 fˆ33 . Now fˆ33 is O(1) and V31R can be ∼ 0.6, so f11 is
about 0.4 in the diagonal basis of the quark matrices. We use MSUSY , Muc uc ∼ 350 GeV and
˜ dc dc i.e. Mdc dc is 109 GeV which is obtained from the VEV
vBL ∼ 1012 GeV. The mass of ∆
of Ω : (1, 3, 1). We choose M∗ ∼ 1013 GeV. Putting all the above the numbers together, we
get

Gn↔n̄ ≃ 1 · 10−30 GeV−5 . (25)

Taking into account the hadronic matrix element effect, the n − n̄ oscillation time is found to
be about 2.5 × 1010 sec which is within the reach of possible next generation measurements.
If we chose, M∗ ≃ MP ℓ , we will get for τn−n̄ ∼ 1015 sec. unless we choose the Mdd dc to

be lower (say 107 GeV). This is a considerable enhancement over the nonsupersymmetric
model of [10] with seesaw scale of 1012 GeV. We also note that as noted in [10] the model is
invariant under the hidden discrete symmetry under which a field X → eiπBX X, where BX
is the baryon number of the field X. As a result, proton is absolutely stable in the model.

Furthermore, since R-parity is an automatic symmetry of MSSM, this model has a naturally
stable dark matter.

21
6 Baryogenesis and n − n̄ oscillation

In the early 1980’s when the idea of neutron-anti-neutron oscillation was first proposed in
the context of unified gauge theories, it was thought that the high dimensionality of the

∆B 6= 0 operator would pose a major difficulty in understanding the origin of matter in

the Universe. The main reason for this assessment is that the higher dimensional operators
remain in thermal equilibrium until late in the evolution of the universe without contradicting
any low energy observations. This is because the thermal decoupling temperature T∗ for
vBL 1/9
 
such interactions goes roughly like vBL MP
which can be in the range of temperatures
where B+L violating sphaleron transitions are in full thermal equilibrium, say for example
vBL ≃ 10 − 100 TeV, as required for the case of observable n − n̄ oscillation. They will

therefore erase any baryon asymmetry generated in the very early moments of the universe
(say close to the GUT time of 10−36 sec. or so) in then prevalent baryogenesis models. Even
though GUT baryogenesis models are no more popular, the same argument will apply to
any other high temperature baryogenesis mechanism.

In models with observable n − n̄ oscillation therefore, one has to search for new mech-
anisms for generating baryons below the weak scale. In this section, we discuss such a

possibility[23] which was discussed in a recent paper. As we see below, it is ideally suited for
embedding into the G224 model discussed in Ref.[10] and leads to an interesting side “bonus”
that it puts an upper limit on the neutron-anti-neutron oscillation time τn−n̄ .

As an illustration of the way the new mechanism operates, let us assume that there is a
complex scalar field that couples to the ∆B = 2 operator as follows i.e. LI = Suc dc dc uc dc dc /M 6 ,
where the scalar boson has mass of order of the weak scale and B = 2. When < S >6= 0, this

interaction leads to baryon number violation by two units and observable n − n̄ transition
if M is in the few hundred to 1000 GeV range. Writing S = √1 (vBL + Sr + SI ), we see
2

22
that the direct decay of Sr involves both ∆B = ±2 final states and thereby satisfies the first
requirement for baryogenesis.

The first point to note is that the high dimension of LI allows the scalar ∆B 6= 0 decay
to go out of equilibrium at weak scale temperatures. This satisfies the out of equilibrium
condition given by Sakharov for generating matter-anti-matter asymmetry. To see this,

let us give some examples: if we chose the proton decay operators such as QQQL, the
decoupling temperature consistent with present experimental bounds on proton life time
would be around 1015 GeV or so. So to apply our mechanism, we need to consider higher
dimensional operators. A typical example of such an operator that we will focus on in this

paper is of type: uc dc dc uc dc dc /M 5 . As note earlier, this operator leads to n − n̄ oscillation

and for this to be in the observable range, the associated mass scale has to be in the few
TeV range. The decupling temperature can then be easily estimated from the formula
 1/9
10M
Td ∼ M MP
∼ 10−1.5 M. For appropriate values of M in the range of interest Td can be

below 100 GeV so that the sphalerons have gone out of equilibrium and baryogenesis follows.
To make these ideas more concrete, below we give an explicit example[23].

We consider an effective sub-TeV scale model that gives rise to the above higher dimen-
sional operator for n ↔ n̄ oscillation. It consists of the following color sextets, SU(2)L sin-
glet scalar bosons (X, Y, Z) with hypercharge − 34 , + 83 , + 23 respectively that couple to quarks.
These states emerge naturally if this low scale theory is embedded into the SU(2)L ×SU(2)R ×

SU(4)c (denoted by G224 ) model described in the previous section. The scalar field S can be
identified with the complex field ∆ν c ν c whose vev breaks B-L symmetry by two units as is the
case for us and Sr ≡ Re(∆0ν c ν c [22]. The X, Y, Z fields can be identified with the color sextet
partners of ∆ν c ν c in the (1, 3, 10) multiplet i.e. the ∆dc dc , ∆uc uc and ∆uc dc respectively. In

the X, Y, Z notation, one can write down the following standard model invariant interaction

23
Lagrangian:

LI = hij Xdci dcj + fij Y uci ucj + (26)

gij Z(uci dcj + ucj dcj ) + λ1 SX 2 Y + λ2 SXZ 2

The complex scalar field S clearly has B − L = 2. To see the constraints on the parameters

of the theory, we note that the present limits on τn−n̄ ≥ 108 sec. implies that the strength
Gn−n̄ of the the ∆B = 2 transition is ≤ 10−28 GeV−5 . From the above figure, we conclude
that

λ1 M1 h211 f11 2
λ2 M1 h11 g11
Gn−n̄ ≃ + ≤ 10−28 GeV −5 . (27)
MY2 MX4 MX2 MZ2

For λ1,2 ∼ h11 ∼ f11 ∼ g11 ∼ 10−3 , we require M1 ∼ MX,Y,Z ≃ 1 TeV to satisfy this

experimental bound. In our discussion of generic models of this type, we will stay close
to this range of parameters and see how one can understand the baryon asymmetry of the
universe. The singlet field will play a key role in the generation of baryon asymmetry. We
assume that < S >≫ MX and MSr ∼ 100 − 500 GeV. It can then decay into final states

with B = ±2 i.e. six quarks and six anti-quarks.

Note that the Lagrangian of Eq. (26), leads to tree level contribution to flavor changing

neutral current processes such as K − K̄, B − B̄ mixings etc and that will restrict the form of
the flavor structure of the coupling matrices suitably. These constraints can also be satisfied
in a realistic G224 embedded model which also explain the neutrino masses and mixing[22].
This is an important consideration since the magnitude of the baryon asymmetry depends

on this detailed flavor structure.

On the way to calculating the baryon asymmetry, let us first discuss the out of equilibrium

condition. As the temperature of the universe falls below the masses of the X, Y, Z particles,
the annihilation processes X X̄ → dc d¯c (and analogous processes for Y and Z) remain in
equilibrium. As a result, the number density of X, Y, Z particles gets depleted as e−MX /T

24
and only the Sr particle survives along with the usual standard model particles. One of the
primary generic decay modes of Sr is Sr → uc dc dc uc dc dc . There could be other decay modes
which depend on the details of the model and one has to ensure that the rates to other

modes are smaller than the six quark mode.

The generic chain of events leading to baryogenesis in this model is the following. At

T ∼ MSr , the decay rate is smaller than the Hubble expansion rate. As the Universe cools
below this temperature, the decay rate remains constant whereas the expansion rate of the
universe is slowing down. So at a temperature Td far below MS , S will start to decay when
the decay rate Γ ∼ H with Td is given by:

MP ℓ MS13 1/2
Td ≃ ( ) (28)
(2π)9 MX12

Since the corresponding epoch must be above that the QCD phase transition temperature,
this puts a constraint on the parameters of the model. Typically, we need to have MS /MX ∼

0.5 or so due to high power dependence of Td on this ratio. This implies that the X, Y, Z
masses cannot be arbitrarily high, since the heavier these particles are, the lower Td will be.
We expect this upper limit to be in the TeV range at most as we show below .

It is well known that baryon asymmetry can arise only via the interference of a tree
diagram with a one loop diagram. The tree diagram is clearly the one where S → 6q. There
are however two classes of loop diagrams that can contribute: one where the loop involves the

same fields X, Y and Z. A second one involves W-exchange, which involves only standard
model physics at this scale (Fig. 6). We find that the second contribution can actually
dominate. In fact, in the G224 embedding of the model, the first diagram vanishes. The
second diagram also has the advantage that it involves less number of arbitrary parameters

and the source of CP violation in this case is the same as the CKM CP violation present in
the standard model. The observed baryon asymmetry is related to the primordial baryon

25
 ∗
asymmetry ǫB as ηB ≃ ( ggrec
∗ )
3/4
ǫB ) where
Td

nS Γ(S → 6q) − Γ(S → 6q̄)

ǫB ≃ (29)
nγ Γ(S)

We find that

Figure 5: Tree level Feynman diagram for S decays.

Figure 6: One loop diagram for S decays.

ǫvertex
B α2 ImTr[gg T M̂u V Md g † Mu V ∗ M̂d ]
≃− 2
. (30)
Br 4 Tr(g † g)MW MS2
The magnitude of the asymmetry depends on MS as well as the detailed profile of the various

coupling matrices h, g, f and we can easily get the desired value of the baryon asymmetry by
appropriately choosing them. An important consequence of the above equation regardless
of the details is that if MS is much bigger than about 500 GeV, baryon asymmetry becomes

26
very small. This implies that MX,Y,Z in turn should not be much larger than a TeV, implying
that they can be accessible at the LHC. This also implies that the strength of n− n̄ transition
has a lower limit or an upper limit on τn−n̄ .

There is a dilution of the baryon asymmetry arising from the fact that Td ≪ MS since
the decay of S also releases entropy into the universe. Thius dilution factor is

(0.32g∗(Td )Td )3/4

dB ≃ ǫB (31)
(0.12MS + 0.32G∗ TD )3/4

Since the decay rate of the S boson depends inversely as a high power of MX,Y , higher X, Y
bosons would imply that ΓS ∼ H is satisfied at a lower temperature and hence give a lower
dB . For our preferred choice of parameters i.e. MX ∼ 2MS ∼ TeV, we find dB ∼ 20%. Also
for choice of the coupling parameters λ ∼ f ∼ h ∼ g ∼ 10−3 , and MS ≃ 200 GeV we find

τn−n̄ ≤ 1010 sec.

We also note that if the model is embedded into a G224 group, all three coupling matrices

h, g, f become equal; there are also some constraints on the gauge boson spectrum in this
model[22]. As noted above, an interesting consequence of adequate baryogenesis is that there
are new sub-TeV- TeV scale color sextet scalar bosons which can be observed at LHC[26].

7 Extra dimensional models with observable n − n̄ os-

cillation

The possibility that there may be extra compact space dimensions has been under extensive
theoretical as well as experimental investigation inspired by the fact that superstring the-
ories do predict the existence of such models. In a large class of interesting models, these

dimensions are believed to be of order TeV−1 size[27, 28] making their manifestations in
many physical situations of experimental interest not only for colliders but also for low en-
ergy experiments. One class of such possibilities includes the breakdown of baryon number

27
by two units. We note two such discussions:

(i) In ref.[29], it was noted that if there are millimeter size extra dimensions, one way to
understand the observed fermion mass hierarchies would be to distribute the standard model
fermions in the bulk. In such a scenario, their Yukawa couplings would be suppressed if they
are further apart[30]. Thus in the context of a 5- or 6 dimensional scenario, observed mass

hierarchies pretty much fix the locations of the various standard model fermions in the bulk.
One can then ask the question as to what happens to higher dimensional operators such as
those giving proton decay and n − n̄ oscillation. Since these are low scale models, one has to
separate the quarks and leptons sufficiently to suppress these operators since their strength is

given by the overlap integral of the various wave functions in the extra dimension. The n − n̄
operator however involves only the right handed quarks and their locations is generally fixed
by the mass hierarchy considerations. So there is no freedom to suppress their strengths.
The strength of such operators was calculated in the context of six dimensional models in

Ref.[29] and found to be in the observable range for extra dimension sizes of order 45 − 100
TeV, which are of the same order as being contemplated for phenomenological viable models
of this kind. These considerations could also be applied to the case Randall-Sundrum models
where the SM fermions are also generally expected to be distributed in the bulk.

(ii) A second class of models were discussed in ref.[31], where general considerations of
global charge separation from the branes is used to argue that one cannot have proton decay

but observable n − n̄ oscillations due to the fact that neutrons are both color and electric
charge neutral. Such baryon number nonconservation occurs due to brane fluctuations which
can create baby branes that carry baryon number if they are color and charge neutral.

28
 uc dc dc uc
X
gluino
dc dc

Figure 7: neutron oscillation induced in R-parity breaking MSSM

8 R-parity breaking models

Finally, there have been explorations of n − n̄ oscilation to supersymmetry. As is well

known, in minimal supersymmetric extensions of the standard model (MSSM), a baryon

′′
number violating operator of the form λ uc dc dc is allowed in the superpotential. By itself it
can lead to ∆B = 2 operator of the form (uc dc dc )2 at tree level via gluino exchange (Figure
7). Typical strength of such operators arising from this new interaction is given[32] by:
2 ′′
4παs λ112
G∆B=2 = (32)
Mq̃4 MG̃

Note that due to color anti-symmetry, at the tree level, this interaction leads Λ− Λ̄ oscillation

rather than n − n̄ oscillation. However once radiative corrections are included, this operator
via flavor changing effects gives rise to neutron oscillation[33]. This puts constraints on
′′
the R-parity violating couplings of the λ type. It must be noted that if these couplings
are present in combination with R-P violating operators of type QLdc , then it leads to

catastrophic proton decay rates[34]; so in discussing ∆B = 2 transitions, we are assuming

that QLdc operators are absent.

9 Conclusion

In summary, there is now a widely held belief that neutrino mass provides a clue to the
existence of a new local B-L symmetry of nature. A key question in particle theory beyond
the standard model is then to find out the scale and detailed nature of physics associated

29
with this new symmetry. The situation is similar to the late 60’s when the hints in favor
of the existence of a fundamental local SU(2) symmetry of weak interaction were becoming
compelling that eventually led to the triumphant gauge revolution in the 1970’s. They were

subsequently confirmed by the discovery of neutral currents, followed by the discovery of the
W and Z boson. The two paths before us right now are (i) whether the B-L forces responsible
for neutrino mass represent physics close to the gravity scale of 1018 GeV as exemplified, say
by the grand unified theories or (ii) scales multi-TeV range far below the GUT scale with

signals visible to the Large Hadron Colliders. One direct way to explore this new physics is
to to carry out an experimental search for n − n̄ oscillation. As noted the discovery of n − n̄
oscillation will have far reaching implications not only for the nature of forces and matter
but will completely alter our thinking about such cosmological issues as the origin of matter,

nature of dark matter etc.

The main points stressed in this review are that: (a)there are a wide class of models

e.g. SU(2)L × SU(2)R × SU(4)c with type II seesaw, models based on TeV scale extra
dimension models where n− n̄ oscillation is in a range accessible to currently planned reactor
experiments and they will extend the existing limit on the strength of this process to an
exciting range; (b) Models that predict observable n − n̄ oscillation provide a new way to

understand the origin of matter where matter creation moment is “much” later than currently
contemplated models such as leptogenesis or GUT scale baryogenesis. More importantly,
unlike the other mechanisms for baryogenesis, this mechanism can be tested in current
collider experiments as well as neutrino experiments in some specific realizations.

On the experimental front, searches for n − n̄ transition using the proton decay type
set-ups are back ground limited[35] and cannot be used to push the limits for τn−n̄ too much

beyond the present limit. Also an interesting class of theories that gives rise to n− n̄ induced
baryogenesis lead to new coloe sextet particles observable at LHC[26].

30
 Finally, it is also worth noting that there is a related phenomenon involving neutrons
i.e. neutron-mirror neutron oscillation that can also be searched for in experimental set-
ups similar to that searching for n − n̄ oscillation[36]. This new class of oscillations have a

much weaker experimental limit on its strength and could be improved by longer “baseline”
searches being contemplated for n − n̄ oscillation.

I would like to thank K. S. Babu for many discussions on theoretical aspects of the topic
and Y. Kamyshkov, G. Greene, Mike Snow and Albert Young for many discussions on the
experimental prospects for n − n̄ oscillation. I am grateful to K. S. Babu and Y. Kamyshkov
for carefully reading the manuscript and suggesting improvements. This work is supported

by the National Science Foundation grant no. Phy-0652363.

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34