Spontaneous parity breaking and supersymmetry breaking in metastable vacua with consistent cosmology

0.16 MB·

by  Debasish Borah

Prepared for submission to JHEP
Spontaneous parity breaking and supersymmetry
breaking in metastable vacua with consistent
cosmology
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Debasish Borah and Urjit A. Yajnik
J

6 Department of Physics, Indian Institute of Technology Bombay,

Mumbai 400076, India
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p E-mail: [email protected], [email protected]
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e Abstract: We study the compatibility of spontaneous breaking of parity and successful
h
[ cosmology in a left-right symmetric model where supersymmetry breaking is achieved in

metastable vacua. We show that domain walls formed dueto this breakingcan beremoved
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v dueto Planck scale suppressedterms, providedtheparity breakingscale M is constrained
R
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to remain smaller than 1010−1011 GeV. Ensuring metastability is achieved naturally even
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4 if the entire mechanism operates at low scales, within a few orders of magnitude of the
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TeV scale. Taking M as high as permitted, close to the acceptable reheat temperature
. R
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after inflation, would require the magnetic phase of the Supersymmetric Quantum Chro-
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1 modynamics (SQCD) to have set in before the end of inflation.
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:
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Keywords: Supersymmetry Breaking, Beyond Standard Model, Cosmology of Theories
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beyond the SM, Supersymmetric Effective Theories
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Contents
1 Introduction 1
2 Domain Wall Dynamics 3
3 ISS Model: A Recap 4
3.1 The macroscopic model 5
3.2 SUSY breaking vacua 5
3.3 SUSY vacuum 6
4 D-Parity breaking with ISS type SUSY breaking 6
5 Domain Wall Removal 8
6 Longevity of metastable vacua 10
7 Results and Conclusion 10
1 Introduction
Left-Right Symmetric Models (LRSM) [1–5] with the gauge symmetry SU(3) ×SU(2) ×
c L
SU(2) ×U(1) provide a framework within which spontaneous parity breaking as well
R B L
−
as tiny neutrino masses [6–9] can be successfully implemented without reference to very
highscale physicssuchas grandunification. Thestructureofthegaugegroupalsosuggests
a discrete Z symmetry, so called D-parity [10, 11] under which the left and right handed
2
fields get interchanged and the gauge charges g , g are equal at some suitable scale.
2L 2R
In supersymmetric implementations of Left-Right symmetry [12–15], stability of gauge
hierarchy becomes natural and R-parity R = (−1)3(B L)+2S (where S is the spin,) is
p −
automatically a part of the gauge symmetry. A stable Dark Matter candidate is therefore
more natural than in the Minimal Supersymmetric Standard Model.
Spontaneousbreaking of exact discrete symmetries has cosmological implications since
they lead to frustrated phase transitions leaving behind a network of domain walls. These
domain walls, if not removed will be in conflict with the observed Universe [16, 17]. It
was pointed out [18, 19] that Planck scale suppressed non-renormalizable operators can
be a source of domain wall instability. Interestingly, this generic analysis needs a careful
revision when supersymmetry (SUSY) and gauge symmetries need to be incorporated. In
the Next to Minimal Supersymmetric Standard Model, the problem was found to persist
[20], in the sense that the gauge hierarchy problem does not get addressed if the operators
required to remove the domain walls are permitted. The problems encountered in that
– 1 –
model are however generic to introduction of a gauge singlet. In the Supersymmetric Left-
Right Models (SUSYLR) with all Higgs carrying gauge charges, it is possible to introduce
Planck scale suppressed terms that are well regulated. One can then demand that the
new operators ensure sufficient pressure across the domain walls that the latter disappear
beforeBigBangNucleosynthesis (BBN). Thisrequirementhas beendiscussedin[21]inthe
context of R-parity conserving SUSYLR models [12–14]. Similar analysis was shown to
place constraints also on R-parity violating SUSYLR models [22]. The upshot is that the
framework proposed in [21] gives rise to an upper bound on the D-parity breaking scale.
And this scale is found to be ∼ 1010GeV, far below the grand unification scale. The issue
of how supersymmetry breaking may be achieved in these models is open and deserves
special attention. In [23] gauge mediation mechanism was tailored to left-right symmetric
case. It was however found to be unnatural to expect the D-parity breaking operators to
also emerge from the hidden sector.
HereweareinterestedinmodelswhereSUSYbreakingisachievedwithoutanystrongly
coupled hidden sector. Such models are based on the idea that SUSY breaking vacuum is
a long lived metastable vacuum, as originally proposed by Intriligator, Seiberg and Shih
(ISS) [24]. This approach to SUSY breaking for the Minimal Supersymmetric Standard
Model was pursued in [25] and for SO(10) in [26]. Recently ISS type SUSY breaking was
considered for Left-Right symmetric model by Haba and Ohki [27]. However, D-parity is
broken only locally in this model, so that in the early Universe there would be formation of
patches corresponding to different vacua separated by a network of domain walls. We take
the model [27] as a specific realization and study the domain wall disappearance using the
framework proposed in [21].
The outcome of this study is that as in previous studies, an upper bound is required
on the scale of theD-parity breakingand therefore the scale M of SU(2) breaking. As a
R R
simple possibility, the entire program is successfully implemented if we treat all of the new
physics to be within a few orders of magnitude of the known scale ∼TeV, but in this case
gauge coupling unification would be problematic [28]. An alternative is to note that the
bound is tantalizingly close to the intermediate scale advocated in some string theoretic
unification models ( see for instance, [29] [30]). But this scale is numerically comparable
to the bound on the reheat temperature T after inflation required for the avoidance of
RH
gravitino overproduction. In the context of implementing direct supersymmetry breaking,
thishasimplicationsalsoforthescaleΛ belowwhichtheunderlyingSQCDcanbetreated
m
as a magnetic theory, and its relation to inflation, as discussed in the concluding section.
Before proceeding, it is worth mentioning that there are models where D-parity and
SU(2) gauge symmetry are broken at two different stages, at the cost of introduction of a
R
gauge singlet scalar field. Such models do not suffer from the problem of persistent domain
walls [10, 11, 15, 31–34]. If the model is descended from breaking of SO(10), there is an
interestingalternativetobestudied. FirstlyweobservethatthebreakingpatternofSO(10)
is model dependent and exact left-right symmetry may not be an effective symmetry at
any of the lower energy scales. However, in models where an exact left-right symmetry
occurs naturally in an intermediate energy regime, domain walls are bound to occur when
the D parity as an effective discrete symmetry gets broken. However, in this case Planck
– 2 –
suppressed terms ensuring the disappearance of the domain walls do not occur, because
being a gauge symmetry, quantum gravity effects do not naturally break it. This case
however is not pursued here.
This paper is organized as follows. In section 2 we briefly review the domain wall
dynamics. In section 3 we summarize the ISS model and in section 4 we discuss the model
originally proposed by Haba and Ohki [27] and then discuss how gravity can cure the
problem of domain walls in this model in section 5. We also discuss the longevity of the
metastable vacua in section 6 and summarize our results in section 7.
2 Domain Wall Dynamics
Discrete symmetries and their spontaneous breaking are both common instances and de-
sirable in model building. The spontaneous breaking of such discrete symmetries gives
rise to a network of domain walls leaving the accompanying phase transition frustrated
[16, 17]. The danger of a frustrated phase transition can therefore be evaded if a small
explicit breaking of discrete symmetry can be introduced.
Due to the smallness of such discrete symmetry breaking, the resulting domain walls
may be relatively long lived and can dominate the Universe for a long time. Since this
will be in conflict with the observed Universe, these domain walls need to disappear at a
very high energy scale (at least before BBN). Keeping this in mind, we summarize the two
cases of domain wall dynamics discussed in [21], one in which the domain walls originate in
radiation dominatederaandget destabilized alsowithintheradiation dominated era. This
scenario was originally considered by Kibble [16] and Vilenkin [35]. The second scenario
was essentially proposed in [36], which consists of the walls originating in a radiation
dominated phase, but decaying after the Universe enters a matter dominated phase, either
due to substantial production of heavy unwanted relics such as moduli, or simply due to a
coherent oscillating scalar field. In both the cases the domain walls disappear before they
come to dominate the energy density of the Universe.
When a scalar field φ acquires a vev at a scale M at some critical temperature T ,
R c
a phase transition occurs leading to the formation of domain walls. The energy density
trapped per unit area of such a wall is σ ∼ M3. The dynamics of the walls are determined
R
by two quantities, force due to tension f ∼ σ/R and force due to friction f ∼ βT4 where
T F
R is the average scale of radius of curvature prevailing in the wall complex, β is the speed
at whichthedomainwall isnavigating throughthemediumandT isthetemperature. The
epoch at which these two forces balance each other sets the time scale t ∼ R/β. Putting
R
all these together leads to the scaling law for the growth of the scale R(t):
R(t)≈ (Gσ)1/2t3/2 (2.1)
The energy density of the domain walls goes as ρ ∼ (σR2/R3) ∼ (σ/Gt3)1/2. In a
W
radiation dominated era this ρ is comparable to the energy density of the Universe
W
[ρ ∼ 1/(Gt2)] around time t ∼ 1/(Gσ).
0
The pressure difference arising from small asymmetry on the two sides of the wall
competes with the two forces f ∼ 1/(Gt2) and f ∼ (σ/(Gt3))1/2 discussed above. For
F T
– 3 –
δρ to exceed either of these two quantities before t ∼ 1/(Gσ)
0
M6 M 2
δρ ≥ Gσ2 ≈ R ∼ M4 R (2.2)
M2 R M
Pl (cid:18) Pl(cid:19)
Similar analysis in the matter dominated era, originally considered in [36] begins with
the assumption that the initially formed wall complex in a phase transition is expected
to rapidly relax to a few walls per horizon volume at an epoch characterized by Hubble
parameter value H . Thus the initial energy density of the wall complex is ρin ∼ σH .
i W i
This epoch onward the energy density of the Universe is assumed to be dominated by
heavy relics or an oscillating modulus field and in both the cases the scale factor grows as
a(t) ∝ t2/3. The energy density scales as ρ ∼ ρin /(a(t))3. If the domain wall (DW)
mod mod
complex remains frustrated, i.e. its energy density contribution ρ ∝ 1/a(t), the Hubble
DW
parameter at the epoch of equality of DW contribution with that of the rest of the matter
is given by [36]
Heq ∼σ3/4Hi1/4MP−l3/2 (2.3)
Assumingthatthedomainwallsstartdecayingassoonastheydominatetheenergydensity
of the Universe, which corresponds to a temperature T such that H2 ∼ GT4, the above
D eq D
equation gives
T4 ∼ σ3/2H1/2M 1 (2.4)
D i P−l
Under the assumption that the domain walls are formed at T ∼ σ1/3
8π σ4/3
H2 = Gσ4/3 ∼ (2.5)
i 3 M2
Pl
Now from Eq. (2.4)
σ11/6 M11/2 M 3/2
T4 ∼ ∼ R ∼ M4 R (2.6)
D MP3/l2 MP3/l2 R(cid:18)MPl(cid:19)
Demanding δρ > T4 leads to
D
3/2
M
δρ > M4 R (2.7)
R M
(cid:18) Pl(cid:19)
3 ISS Model: A Recap
TheISSmodelconsists of a pairof dualtheories related to each other throughthe“Seiberg
duality” [37]. There is a low energy theory which is referred to as the “macroscopic” or
“free magnetic theory” which is infra-red (IR) free. The high energy theory is known as
the “microscopic” or “free electric theory” and it is basically SU(N ) SQCD which is ultra
c
violet (UV) free. Seiberg duality says, SU(N ) SQCD (UV free) with N (> N ) flavors of
c f c
quarks is dual to a SU(N −N ) gauge theory (IR free) with N2 singlet mesons M and
f c f
N flavors of quarks q,q˜. Their approach revolves around studying the SUSY-breaking
f
dynamics of the microscopic SU(N ) SQCD in terms of the macroscopic, IR-free dual.
c
– 4 –
3.1 The macroscopic model
The macroscopic model is a Wess-Zumino model with the following symmetry group out
of which SU(N) where N = N −N is gauged and the rest are global symmetries:
f c
SU(N)×SU(N )2×U(1) ×U(1) ×U(1) . (3.1)
f B ′ R
The matter content is as follows:
SU(N) SU(N ) SU(N ) U(1) U(1) U(1)
f f B ′ R
Φ 1 (cid:3) (cid:3) 0 −2 2
[Nf×Nf]
(3.2)
ϕ (cid:3) (cid:3) 1 1 1 0
[N×Nf]
ϕ˜ (cid:3) 1 (cid:3) −1 1 0
[Nf×N]
The field Φ is identified as the meson field. The fields ϕ and ϕ˜ are the dual quarks. The
Ka¨hler potential is
K = Tr ϕ ϕ +Tr ϕ˜ ϕ˜ +Tr Φ Φ (3.3)
† † †
h i h i h i
and the tree-level superpotential is
W = hTr[ϕΦϕ˜]−hµ2TrΦ. (3.4)
Denoting the scalar components of the superfields Φ,φ,φ˜ as M, q, and q˜respectively, the
bosonic part of the Lagrangian for macroscopic theory can be expressed as
1
L = Tr − F Fµν −D qDµq −D q Dµq −Tr ∂ M ∂µM −V (3.5)
c 2g2 µν µ † µ † f µ †
(cid:20) (cid:21)
h i
with a scalar potential V = VF +VD given by e e
2
V = |h2|Tr qq−µ21 2 + |h2|Tr |qM|2+ q M , (3.6)
F f Nf c † †
g2 h(cid:12) 2(cid:12) i g2 (cid:20) (cid:12)(cid:12) 2 (cid:12)(cid:12) (cid:21)
V = Tr (cid:12)qeq −q q (cid:12) − Tr qq −Tr q q(cid:12)e . (cid:12) (3.7)
D c † † c † c †
4 8
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
3.2 SUSY breaking vacua e e e e
SUSY is broken in the above model by rank condition if N < N . The most general SUSY
f
breaking vacuum is of the form
0 0 q q 1
M = N×N N×(Nf−N) , † = = N , (3.8)
0 M µ µ 0
(Nf−N)×N 0 ! ∗ (Nf−N)×N !
e
where M is an arbitrary N −N by N −N matrix. The vacuum energy is given by
0 f f
V = |hµ2|2(N −N) > 0, (3.9)
f
and the vacuum spontaneously breaks the N = 1 supersymmetry.
– 5 –
3.3 SUSY vacuum
From equations (3.6) and (3.7), the supersymmetric vacua lie at
qq = µ21 , qM = 0, q M = 0, qq −q q = 0. (3.10)
Nf † † † †
Forthesevacua,qandq˜arenon-zeroandhenceM mustbezero. HoweverifwegiveM some
e e e e
general non-zero VEV hMi as in equation (3.8), the dual quarks ϕ and ϕ˜ acquire a mass
hhMi. If we integrate out these massive flavors of quarks, the low energy superpotential
becomes
Wlow = N(hNfΛ−m(Nf−3N)detΦ)1/N −hµ2TrΦ (3.11)
Minimizing the above superpotential gives rise to the supersymmetric minima at
1
hhMi = Λmǫ2N/(Nf−N)1Nf = µǫ(Nf−3N)/(Nf−N)1Nf (3.12)
where ǫ ≡ µ . The value of ϕ and ϕ˜ or equivalently q and q˜for these minima is zero. Λ
Λm m
refers to the scale above which the magnetic theory is strongly coupled. For |ǫ| ≪ 1, the
SUSY preserving vacuum lies far away in the field space from the SUSY breaking vacuum
and hence can be parametrically long lived as was pointed out by Intriligator et al. [24].
4 D-Parity breaking with ISS type SUSY breaking
As suggested by the authors in [27], D-parity and spontaneous SUSY breaking can be
naturally achieved in an ISS type framework if a new strongly coupled gauge sector is
introducedtowhichtheleft-rightHiggsfieldsarecoupled. Theyproposedtheelectricgauge
theory to be based on the gauge group SU(3) ×SU(3) ×SU(2) ×SU(2) ×U(1)
L R L R B L
−
(in short G ) where SU(2) × SU(2) × U(1) is the gauge group of usual Left-
33221 L R B L
−
Right models and SU(3) is the new strongly coupled gauge sector introduced. Thedual
L,R
description similar totheoriginal ISSmodelgives risetoSU(2) broken meta-stable vacua
R
inducing spontaneous SUSY breaking simultaneously.
The particle content of the electric theory is
Qa ∼ (3,1,2,1,1), Q˜a ∼ (3 ,1,2,1,−1)
L L ∗
Qa ∼ (1,3,1,2,−1), Q˜a ∼ (1,3 ,1,2,1)
R R ∗
wherea = 1,N andthenumbersinbracketscorrespondtothetransformationsofthefields
f
underthegaugegroupG . ThismodelhasN = 3andhencetohaveaSeibergdual[37]
33221 c
magnetictheory, numberofflavorsshouldbeN ≥ 4. ForN = 4thedualmagnetictheory
f f
will have the gauge symmetry of the usualLeft Right Models SU(2) ×SU(2) ×U(1)
L R B L
−
and the following particle content
φa(2,1,−1), φ˜a(2,1,1)
L L
φa(1,2,1), φ˜a(1,2,−1)
R R
– 6 –
1 (S +δ0) δ+
ΦL ≡ 1+AdjL = √2 δL L 1 (S L−δ0)
L− √2 L L !
1 (S +δ0) δ+
ΦR ≡ 1+AdjR = √2 δR R 1 (S R−δ0) (4.1)
R− √2 R R !
The Left-Right symmetric renormalizable superpotential of this magnetic theory is
W0 = hTrφ Φ φ˜ −hµ2TrΦ +hTrφ Φ φ˜ −hµ2TrΦ (4.2)
LR L L L L R R R R
The tree level Ka¨hler potential is
K0 = Trφ†LφL+Trφ˜†Lφ˜L +Trφ†RφR+Trφ˜†Rφ˜R+TrΦ†LΦL+TrΦ†RΦR (4.3)
The non-zero F-terms giving rise to SUSY breaking are
F = hφ φ˜ −hµ2δ andF = hφ φ˜ −hµ2δ (4.4)
ΦL L L ab ΦR R R ab
where a,b = 1,4 here and SUSY is broken by rank condition [24]. However depending on
the vev of the meson fields hΦ i,hΦ i, the chiral fields φ ,φ will acquire different masses
L R L R
proportional to hhΦ i and hhΦ i respectively. Suppose, Φ gets a non-zero vev and
L R R
accordingly φ acquire non-zero masses. Like in the ISS model (3.12) is we integrate out
R
these massive flavors, we arrive at the SUSY preserving vacuum with only the left handed
chiral fields φ . After integrating out the right handed chiral fields, the superpotential
L
becomes
W0 = hTrφ Φ φ˜ −hµ2TrΦ +h4Λ 1detΦ −hµ2TrΦ (4.5)
L L L L L − R R
which gives rise to SUSY preserving vacua at
1
hhΦ i= Λ ǫ2/3 = µ (4.6)
R m
ǫ1/3
where ǫ = µ . Thus the right handed sector exists in a metastable SUSY breaking vac-
Λm
uum whereas the left handed sector is in a SUSY preserving vacuum breaking D-parity
spontaneously. Soft SUSY breaking terms can also be induced in the right handed sector
as pointed out by the authors of [27]. However it is equally likely for Φ to acquire a vev
L
instead of Φ . In that case left handed fields will acquire a mass hhΦ i and get decou-
R L
pled. Integrating them out will give rise to a SUSY preserving right handed sector. Thus
D-parity is broken only locally and there will be formation of local patches containing left
and right handed sectors separated by a network of domain walls. Domain walls, being
extended objects have more energy density than matter and radiation and hence start
dominating the Universe very early. This will be in conflict with the observed Universe
and hence such walls should be removed at early times or at least before BBN. We discuss
one such possible way to get rid of them in the next section.
– 7 –
5 Domain Wall Removal
Although the electric theory is Left-Right symmetric, the magnetic theory breaks both
D-parity as well as SUSY spontaneously as discussed in the previous section. D-parity
is broken only locally giving rise to the formation of domain walls. Following [18, 19],
we know that Planck scale suppressed operators can break D-parity explicitly and can
make the domain walls disappear. However the magnetic theory where D-parity is broken,
has an UV cut-off Λ and hence all the gauge invariant higher dimensional terms will be
m
suppressedbyΛ andnotbyPlanckmassM . ButQCDeffects cannotberesponsiblefor
m Pl
breaking D-parity although quantum gravity effects can break global discrete symmetries
like D-parity explicitly. Therefore we assume that the differences in the left and right
sectors brought about by Λ suppressed operators are of the order 1 .
m MPl
We write the next to leading order terms allowed by the gauge symmetry in the su-
perpotential as well as Ka¨hler potential.
Tr(φ Φ φ˜ )TrΦ Tr(φ Φ φ˜ )TrΦ (TrΦ )4 (TrΦ )4
W1 = f L L L L +f R R R R +f L +f R (5.1)
LR L Λ R Λ L′ Λ R′ Λ
m m m m
K = −λ Tr(Φ†LΦL)2 −λ (TrΦ†LΦL)2 −λ Tr(Φ†RΦR)2 −λ (TrΦ†RΦR)2
1 1L Λ2 2L Λ2 1R Λ2 2R Λ2
m m m m
λ λ
− Λ′12L((φ†LφL)2+(φ˜†Lφ˜L)2)− Λ′12R((φ†RφR)2+(φ˜†Rφ˜R)2) (5.2)
m m
After the right(left) sector decouples we are left with the left(right) sector only. We find
the energy of these two sectors separately. The terms of order 1 are given by
Λm
h
V1 = S [f (φ0φ˜0)2+f φ0φ˜0S2 +(δ0 −S )2((φ0)2+(φ˜0)2)] (5.3)
R Λ R R R R R′ R R R R R R R
m
The minimization conditions give φφ˜= µ2 and S0 = −δ0. Denoting hφ0i = hφ˜0i = µ and
R R
hδ0i = −hS0i = M , we have
R R R
hf
V1 = R(|µ|4M +|µ|2M3) (5.4)
R Λ R R
m
where we have also assumed f ≈ f . For |µ| < M we have
R′ R R
hf
V1 = R|µ|2M3 (5.5)
R Λ R
m
If the scalar fields in the left sector also acquires similar vev, the terms of the order 1 in
Λm
the expression for energy are
hf
V1 = L|µ|2M3 (5.6)
L Λ R
m
Thus the effective energy density is
|µ|2M3
δρ ∼ h(f −f ) R (5.7)
R L
Λ
m
– 8 –
Assuming quantum gravity effects to bring such an explicit violation of D-parity we must
have
|µ|2M3 Λ5
R ≤ m (5.8)
Λ M
m Pl
Hereweareconsideringthedimensionlesscoefficientstobeoforderone. Theaboverelation
implies Λ6 ≥ M |µ|2M3. For matter dominated era, we must have
m Pl R
|µ|2M3 M 3/2
h(f −f ) R > M4 R (5.9)
R L Λ R M
m (cid:18) Pl(cid:19)
Assuming the dimensionless parameters to be of order one, the above relation gives
|µ|2M3/2
M5/2 < Pl (5.10)
R Λ
m
Using lower bound on Λ from equation (5.8), the above inequality gives the upper bound
m
on M
R
M < |µ|5/9M4/9 (5.11)
R Pl
Taking µ to be of same order as SUSY breaking scale which is TeV, we get
M < 1.3×1010 GeV (5.12)
R
Similarly for radiation dominated era, we have
|µ|2M3 M6
h(f −f ) R > R (5.13)
R L Λ M2
m Pl
Assuming dimensionless parameters to be of order one, we have
|µ|2M2
M3 < Pl (5.14)
R Λ
m
Using lower bound on Λ from equation (5.8), the above inequality gives the upper bound
m
on M
R
M < |µ|10/21M11/21 (5.15)
R Pl
Taking µ to be of same order as SUSY breaking scale which is TeV, we get
M <1011 GeV (5.16)
R
It is interesting to note that M as constrained in (5.12) is just above the upper bound
R
on the reheat temperature T after inflation required to avoid gravitino overabundance.
RH
The constraint (5.16) if saturated precludes the possibility of thermal leptogenesis [38] due
to the gravitino bound on T . But more importantly, the scale falls far short of either
RH
the scale of generic inflation ∼ 1015 − 1016 GeV [39–41] or the scale of SO(10) grand
unification ∼ 2 ×1016 GeV. Thus the successful completion of D-parity breaking phase
transition demands the introduction of this new scale lying below the grand unification
scale. Within the framework of SUSY, such a scale can be assumed to be protected from
mixing with higher scales.
– 9 –