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arXiv:hep-ph/0612357v3 25 Mar 2007

UFIFT-HEP-06-20

OSU-HEP-06-14

Unified TeV Scale Picture of Baryogenesis and Dark Matter

K. S. Babua ‡, R. N. Mohapatrab †, and Salah Nasric §

aDepartment of Physics, Oklahoma State University, Stillwater, OK 74078, USA

bDepartment of Physics, University of Maryland,College Park, MD 20742,USA and

cDepartment of Physics, University of Florida, Gainesville, Florida 32611, USA

Abstract

We present a simple extension of MSSM which provides a unified picture of cosmological baryon

asymmetry and dark matter. Our model introduces a gauge singlet field N and a color triplet field

X which couple to the right–handed quark fields. The out–of equilibrium decay of the Majorana

fermion N mediated by the exchange of the scalar field X generates adequate baryon asymmetry

for MN∼ 100 GeV and MX∼ TeV. The scalar partner of N (denoted˜ N1) is naturally the lightest

SUSY particle as it has no gauge interactions and plays the role of dark matter.

˜ N1annihilates

into quarks efficiently in the early universe via the exchange of the fermionic˜ X field. The model

is experimentally testable in (i) neutron–antineutron oscillations with a transition time estimated

to be around 1010sec, (ii) discovery of colored particles X at LHC with mass of order TeV, and

(iii) direct dark matter detection with a predicted cross section in the observable range.

‡Email:kaladi.babu@okstate.edu

†Email:rmohapat@physics.umd.edu

§Email:snasri@phys.ufl.edu

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I. INTRODUCTION

. The origin of matter–anti-matter asymmetry of the Universe and that of dark matter

are two of the major cosmological puzzles that rely heavily on particle physics beyond the

standard model for their resolution. It is a common practice to address these two puzzles

separately by invoking unrelated new physics. For instance, a widely held belief is that either

the lightest supersymmetric particle (LSP) or the near massless invisible axion constitutes

the dark matter, while baryogenesis occurs through an unrelated mechanism involving either

the decay of a heavy right–handed neutrino (leptogenesis), or new weak scale physics which

makes use of the electroweak sphalerons. A closer examination of the minimal versions

of SUSY would suggest that to generate the required amount of dark matter density one

needs some tuning of parameters. The LSP should either have the right amount of Higgsino

component, or another particle, usually the right–handed stau, should be nearly degenerate

with the Bino LSP to facilitate dark matter co-annihilation. Similarly, the leptogenesis

mechanisms requires the heavy right–handed neutrino to have its mass in the right range to

generate the adequate amount of matter. Despite these possible problems, these ideas are

attractive since they arise in connection with physics scenarios which are strongly motivated

by other puzzles of the standard model, e.g., resolving the gauge hierarchy problem (in the

case of LSP dark matter), or generating small neutrino masses (in the case of leptogenesis).

In the absence of any experimental confirmation of these ideas, it is quite appropriate to

entertain alternate explanations which could be motivated on other grounds. Our motivation

here is to seek a unified picture of both these cosmological puzzles within the context of

weak scale supersymmetry without fine–tuning of parameters. We propose a class of models

where a very minimal extension of the MSSM resolves these puzzles in a natural manner

with testable consequences for the near future.

Our extension of MSSM involves the addition of two new particles: a SM singlet su-

perparticle denoted by N with mass in the 100 GeV range and an iso-singlet color triplet

particle X with mass in the TeV range. These particles, consistent with the usual R–parity

assignment, couple only to the right–handed quark fields. We discuss two models, one in

which the electric charge of X is 2/3 and another where it is −1/3. We show that in these

models, baryon asymmetry arises by the mechanism of post–sphaleron baryogenesis sug-

gested by us in a recent paper [1] involving the decay of the Majorana fermion N. The

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scalar component of N (denoted as˜ N1) has all the right properties to be the cold dark

matter of the universe without any fine–tuning of parameters. The purpose of the heavier

X particle is to facilitate baryon number violation in the interaction of N, and also to help

˜ N1annihilate into quarks. A very interesting prediction of these models is the existence of

the phenomenon of neutron-anti-neutron oscillation with a transition time in the accessible

range of around 1010sec. The TeV scale scalar X and its fermionic superpartner˜ X are de-

tectable at LHC. Furthermore, the model predicts observable direct detection cross section

for the dark matter.

II.OUTLINE OF THE MODEL

. As already noted, we add two new superfields to the MSSM – a standard model singlet

N and a pair of color triplet with weak hypercharge = ±4/3 denoted as X,X. The R–parity

of the fermionic component of N is even, while for the fermionic X it is odd. This allows

the following new terms in the MSSM superpotential (model A):

Wnew = λiNuc

iX + λ′

ijdc

idc

jX +MN

2

NN + MXXX .

(1)

Here i,j are family indices with λ′

ij= −λ′

jiand we have suppressed the color indices. An

alternative possibility is to choose X to have hypercharge −2/3 and write a superpotential

of the form (model B)

Wnew = λjNdc

jX + λ′

kluc

kdc

lX +MN

2

NN + MXXX .

(2)

In model B, additional discrete symmetries are needed to forbid couplings such as QLX

which could lead to rapid proton decay. In model A however, there are no other terms that

are gauge invariant and R–parity conserving. In particular, the X field of model A does

not mediate proton decay. We will illustrate our mechanism using model A although all our

discussions will be valid for model B as well.

The fermions N and˜ X have masses MN and MX respectively. As for the scalar com-

ponents of these superfields, the Lagrangian including soft SUSY breaking terms is given

by

− Lscalar= |MX|2(|X|2+ |X|2) + m2

X|X|2+ m2

X|X|2

+ (BXMXXX + h.c) + |MN|2|˜N|2+ m2

˜ N|˜N|2

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+ (1

2BNMN˜N˜N + h.c.) (3)

The 2 × 2 mass matrix in the (X,X∗) sector can be diagonalized to yield the two complex

mass eigenstates X1and X2via the transformation

X = cosθX1− sinθe−iφX2; (4)

X∗= sinθeiφX1+ cosθX2

where

tan2θ =

|2BXMX|

|m2

X− m2

X|; φ = Arg(BXMX)sgn(m2

X− m2

X) .(5)

Note that the angle θ is nearly 450if the soft masses for X and X are equal. The two mass

eigenvalues are

M2

X1,2=|MX|2+m2

X+ m2

2

X

±

?

?

?

?

?m2

X− m2

2

X

?2

+ |BXMX|2

(6)

The two real mass eigenstates from the˜N field have masses

M2

˜ N1,2= m2

˜ N+ |MN|2± |BNMN| .(7)

Here˜ N1is the real part of˜ N, while˜ N2is the imaginary part. (A field rotation on˜ N has

been made so that the BNMN term is real.) With these preliminaries we can now discuss

baryogenesis and dark matter in our model.

III. POST–SPHALERON BARYOGENESIS

. The mechanism for generation of matter-anti-matter asymmetry closely follows the

post–sphaleron baryogenesis scheme of Ref. [1]. As the universe cools to a temperature

T which is below the mass of the X particle but above MN, the X particles annihilate

leaving the Universe with only SM particles and the N (fermion) and˜ N1,2(boson) particles

in thermal equilibrium. The decay of N will be responsible for baryogenesis. We therefore

need to know the temperature at which the interactions of N go out of equilibrium. We

first consider its decay. Being a Majorana fermion, N can decay into quarks as well as

antiquarks: N → uidjdk, N → uidjdk. The decay rate for the former is

ΓN=C

128

(λ†λ)Tr[λ′†λ′)]

192π3

sin22θM5

N

?

1

M2

X1

−

1

M2

X2

?2

(8)

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Here the approximation MX1,2≫ MN has been made. C is a color factor, equal to 6. The

total decay rate of N is twice that given in Eq. (8) - to account for decays into quarks

as well as into antiquarks. As a reference, we take the contribution from X1exchange to

dominate the decay, and assume that the mixing angle θ ≃ 450. It is then easy to see that for

?

involving N such as q +N → ¯ q + ¯ q also go out of equilibrium at this temperature. Further,

(λ†λ)Tr[λ′†λ′)] ∼ 10−3, N decay goes out of equilibrium below its mass. Other processes

for T < MN, production of N in q + q scattering will be kinematically inhibited. Finally

there is a range of parameters in our model, e.g., M˜ X∼ 3 TeV, MN∼ 100 GeV, where the

rate for NN → uc¯ ucprocess which occurs via the exchange of the bosonic field in X also

goes out of equilibrium. We have checked that if N decay lifetime is ≤ 10−11sec., as it is

in our model, even if NN → uc¯ ucis in equilibrium, slightly below T = MN, the decay rate

dominates over this process and does not inhibit baryogenesis.

The decay of N, which is CP violating when one–loop corrections are taken into account,

can lead to the baryon asymmetry. Since the mass of the N fermion is below the electroweak

scale, the sphalerons are already out of equilibrium and cannot erase this asymmetry. The

mechanism is therefore similar to the post–sphaleron baryogenesis mechanism a la Ref. [1].

The only difference from the detailed model in Ref. [1] is that, there, due to the very

high dimension of the decay operator, the out of equilibrium temperature was above the

decaying particle (called S in Ref. [1]) mass giving an extra suppression factor of Td/MS

in the induced asymmetry (since generation of matter has to start when the temperature

is much below the S particle mass). In the present case, there is no suppression factor of

Td/MSin the induced baryon asymmetry.

In order to calculate the the baryon asymmetry of the universe, we look for the imaginary

part from the interference between the tree–level decay diagram and the one–loop correction

arising from W±exchange. These corrections have a GIM–type suppression, since the W±

only couple to the left–handed quark fields while the tree–level decay of N is to right–handed

quarks. Following Ref. [1], we find the dominant contribution to be

ǫB

Br≃

?

−α2

4

?Im[(λ∗ ˆ

Mu)TVˆ

M2

Mdλ′].[λ′∗ ˆ

WM2

MdVTλˆ

Mu]

N(λ†λ)Tr(λ′†λ′)

(9)

where Br stands for the branching ratio into quarks plus anitquarks, and

?

λ∗ ˆ

Mu

?T

=

(λ∗

1mu,λ∗

2mc,λ∗

3mt),

ˆ

Md = diag.{md,ms,mb} The interesting point is that as in Ref. [1]

the asymmetry is completely determined by the electroweak corrections. A typical leading

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term in Eq. (9) is of the form (−α2/4)(mcmtmsmb)/(m2

Wm2

N) which yields ǫB≃ 3 × 10−8

with only mild dependence on the couplings λi,λ′

ij. This can easily lead to desired value for

the baryon asymmetry.

IV. SCALAR DARK MATTER

. In a supersymmetric model, we expect every particle to have a super-partner. We

show below that in our extended MSSM the super-partner of N (denoted by˜ N1) has all the

properties quite naturally for it to play the role of scalar dark matter. In this context let us

recall some of the requirements on a dark matter candidate: it must be the lightest stable

particle and its annihilation cross section must have the right value so that its relic density

gives us ΩDM≃ 0.25. The desired cross section for a generic multi-GeV CDM particle is

of about 10−36cm2. In our model the presence of the TeV scale X particle, in addition to

playing an important role in the generation of baryon asymmetry, also plays a role in giving

the right annihilation cross section for˜ N1to be the dark matter.

Let us first discuss why˜ N1is naturally the lightest stable boson in our model. To start

with, in order to solve the baryogenesis problem, we choose the N superfield to have its mass

below that of the super-partners of the SM particles. In mSUGRA type models, generally,

one chooses a common scalar mass for all particles at the SUSY breaking scale (say MP), so

that scalar masses at the weak scale are determined by the renormalization group running.

There are two kinds of contributions to the running of the soft SUSY breaking masses –

gauge contributions which increase masses as we move lower in scale, and Yukawa coupling

contributions which tend to lower the masses as we move lower in scale. As far as the scalar

˜N1particle goes since it has no gauge couplings, its mass naturally goes somewhat lower as

we move from the Planck scale to the weak scale and becomes naturally the lightest stable

SUSY particle. Furthermore since its couplings λiare in the range of 0.1-0.001, they are not

strong enough to drive m2

From Eq. (7), it is clear that of the two states˜ N1,2, the lighter one˜ N1is the LSP. The

˜ N2remains close in mass but above the LSP and can help in co-annihilation of the dark

˜ N1negative like the m2

Hu.

matter provided |BMN| ≪ M2

Dark matter annihilation. In the early universe, the LSP˜ N1 will annihilate into

quark-antiquark pair via the exchange of˜ X fermion. The annihilation cross section is given

N+ m2˜ N, if needed.

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by

σ(˜ N1˜ N1→ qq)vrel=C′(λ†λ)2

8πs

?a

btanh−1(b

a) − 1

?

(10)

where

a = 2E2− M2

˜ N1+ M2

X;b = 2E|− →p | . (11)

Here C′= 3 is a color factor, E and− →p are the energy and momentum of one of the˜ N1, s

is the total CM energy. For MX≫ E, the cross section reduces to

σvrel≃

1

8π(λ†λ)2|− →p |2

M4

X

. (12)

For the coupling λ3∼ 1, M?

We can now compare this with the dark matter in MSSM, which is usually a neutralino.

N1= 300 GeV and MX= 500 GeV, the cross section is of the

order of a pb as would be required to generate the right amount of relic density.

In MSSM, some tuning of parameters is needed, either to have the right amount of Higgsino

content in the LSP, or to have the right–handed stau nearly mass degenerate with the LSP

to facilitate co-anninhilation. In our model, there is no need for co-annihilation, but if

necessary, the mass of˜N2is naturally close to˜N1by a symmetry, viz., supersymmetry, if

|BMN| is small.

Dark matter detection. Due to the fact that˜ N1has interactions with quarks which

are sizeable, it can be detected in current dark matter search experiments. We present an

order of magnitude estimate of the˜ N1+ nucleon cross section. Even though the annihilation

cross section is of order 10−36cm2, the detection cross section on a nucleon σ˜ N1+pis much

smaller due to slow speed of the dark matter particle which limits the final state phase

space for the elastic scattering. Secondly, detection involves only the first generation quarks

whereas annihilation involves the second generation as well and thus if the N couplings are

hierarchical like the SM Yukawa couplings, it is easy to understand the smallness of detection

cross sections compared to σann. In our model the scattering of˜ N1(with momentum p) off

a quark (with momentum k) occurs via the s-channel exchange of the fermionic component

of X . The amplitude is given by M˜ N1+q= i

λ2

1

4M2

Xu(k′)γµu(k)Qµ, where, Q = k + p. At the

nucleon level , the time component of the vector current dominates (spin-independent) over

the spatial component (velocity dependent). The nucleon–˜ N1cross section is given by

σ˜ N1+p≃|λ1|4m2

p

64πM4

X

?A + Z

A

?2

,(13)

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where A,Z are the atomic number and charge of the nucleus and λ1is the coupling of N

to the first generation quarks. The sum |λ1|2+ |λ2|2is constrained by LSP annihilation

requirement but individually |λ1| is not. If we choose |λ1| ∼ 0.1 − 0.01, then the cross

section is around 10−43cm2− 10−47cm2which is in the range being currently explored [6].

V. NEUTRON-ANTI-NEUTRON OSCILLATION

. One of the interesting predictions of our model is the existence of neutron-anti-neutron

oscillation at an observable rate. The Feynman diagram contributing to this process is given

in Ref. [2]. Since N is a Majorana fermion, it decays into udd as well as into ¯ u¯d¯d, which leads

to N −N oscillations. The strength for this process (taking into account the anti-symmetry

of λ′couplings) is given by:

G∆B=2 ≃(λ1λ′

12)2

MNM4

X

. (14)

The ∆B = 2 operator in this case has the form ucdcscucdcsc. The coupling λ1appearing

in this process is involves the first generations, the same coupling appears in the direct

detection of dark matter. It is reasonable to expect λ1to be somewhat smaller in magnitude

compared to the second generation counterpart λ2. Secondly, if we choose the strange

quark component in the nucleon to be about 1%, then choosing λ1λ′

12≃ 10−4, we find that

G∆B=2≃ 10−27GeV−5which corresponds to the present limit on τN−¯ N∼ 108sec. [3, 4].

There are proposals to improve this limit by two orders of magnitude [5] by using a vertical

shaft for neutron propagation in an underground facility e.g. DUSEL. It is interesting that

the expectation for the N − N transition is in the range accessible to experiments and this

can therefore be used to test the model.

It is important to point out here that there is no proton decay in this model due to the

fact that both the scalar and the fermionic parts of the singlet field N are heavier than SM

fermions.

N can be identified with the right–handed neutrino, but its couplings to the light neutri-

nos are forbidden. If this model is embedded into a seesaw picture, we are envisioning a 3×2

seesaw with two heavy right–handed neutrinos and a light one that is identified with the

N field that plays no role in neutrino mass physics. This can be guaranteed by demanding

that N and X fields are odd under a Z2symmetry whereas all other fields are even. The

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X¯ X mass term breaks this symmetry softly and does not affect the discussion. Note that

proton decay via the exchange of N is forbidden in this case.

We conclude by noting some interesting aspects of the model.

(i) The X particle in our model can be searched for at the LHC. Once produced, X

will decay into two jets, e.g., a b jet and a light quark jet. We point out that there is an

interesting difference in discovery of SUSY at LHC in our model. Consider up type squarks

pair produced at LHC. The squark will decay into a quark plus a neutralino. In our model,

the neutralino is unstable, it decays into ucdcdc˜ N. So one SUSY signal will be six jets plus

missing energy. The scalar up squark can also decay directly into˜ Ndcdc. In this case the

signature will be 4 jets plus missing energy.

(ii) It is also worth noting that the quantum numbers of N are such that it is a SM singlet

with B − L = 1 and therefore same as that of the conventional right–handed neutrino.

This model can therefore be used to understand the small neutrino masses via a low scale

seesaw mechanism provided there are at least two N’s and the Dirac masses for neutrinos

are suppressed. We do not dwell on this aspect of the model in this paper since it is not

pertinent to our main results. We however point out that our results are not affected by

the multi–N extension required for understanding neutrino masses. The RH neutrino which

plays the role in generating baryon asymmetry and dark matter is the lightest of the N

fields. This model is however different in many respects from some other suggestions of

right handed sneutrino dark matter in literature [8, 9, 10].

(iii) The models presented are compatible with gauge coupling unification, provided that

the X particle is accompanied by other vector–like states which would make complete 10+10

representations of SU(5). These extra particles will have no effect on baryogenesis and dark

matter phenomenology.

(iv) We also note that there is no one loop contribution to neutron electric dipole moment

in our model due to the λ′or λ couplings since they involve products of couplings of the

form λ†λ and similarly for λ′. We have also not found any two loop diagram involving the

X or N exchange that would contribute to neutron edm.

Acknowledgement: The work of KSB is supported by DOE Grant Nos.DE-FG02-

04ER46140 and DE-FG02-04ER41306, RNM is supported by the National Science Foun-

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dation Grant No. Phy-0354401 and S. Nasri by DOE Grant No. DE-FG02-97ER41029.

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