Why PeV scale left-right symmetry is a good thing

TL;DR

This paper explores the viability of PeV scale left-right symmetry models, showing they can be consistent with cosmology, supersymmetry, and leptogenesis, and may be observable in future experiments.

Contribution

It demonstrates that PeV scale left-right symmetry is compatible with cosmological constraints and supersymmetry, challenging the assumption that the scale must be very high.

Findings

PeV scale left-right symmetry remains viable within cosmological constraints.

The right-handed neutrino mass scale can be bounded from above, excluding some leptogenesis scenarios.

Models at the PeV scale could be detectable in collider and astroparticle experiments.

Abstract

Left-right symmetric gauge theory presents a minimal paradigm to accommodate massive neutrinos with all known conserved symmetries duly gauged. The work presented here is based on the argument that the see-saw mechanism does not force the new right handed symmetry scale to be very high, and as such some of the species from the spectrum of the new gauge and Higgs bosons can have masses within a few orders of magnitude of the TeV scale. The scale of the left-right parity breaking in turn can be sequestered from the Planck scale by supersymmetry. We have studied several formulations of such Just Beyond Standard Model (JBSM) theories for their consistency with cosmology. Specifically the need to eliminate phenomenologically undesirable domain walls gives many useful clues. The possibility that the exact left-right symmetry breaks in conjunction with supersymmetry has been explored in the context of gauge mediation, placing restrictions on the available parameter space. Finally we have also studied a left-right symmetric model in the context of metastable supersymmetric vacua and obtained constraints on the mass scale of Right handed symmetry. In all the cases studied, The mass scale of right handed neutrino $M_R$ remains bounded from above, and in some of the cases the scale $10^9$ GeV favourable for supersymmetric thermal leptogenesis is disallowed. On the other hand PeV scale remains a viable option, and the results warrant a more detailed study of such models for their observability in collider and astroparticle experiments.

Tables

Table 1. Table 1: Differences in values of soft supersymmetry breaking parameters for a range of domain wall decay temperature values T D subscript 𝑇 𝐷 T_{D} . The differences signify the extent of parity breaking.

$T_D/$GeV	$\sim$	$10$	${10}^2$	${10}^3$
$\left(m^2-m^{\prime 2}\right)/{\text{GeV}}^2$	$\sim$	${10}^{-4}$	$1$	${10}^4$
$\left({\beta }_1-{\beta }_2\right)/{\text{GeV}}^2$	$\sim$	${10}^{-8}$	${10}^{-4}$	$1$

Table 2. Table 2: Entries in this table are the values of the parameter f ​ ( γ , σ ) 𝑓 𝛾 𝜎 f(\gamma,\sigma) , required to ensure wall decay at temperature T D subscript 𝑇 𝐷 T_{D} displayed in the header row. The table should be read in conjunction with table 1 , with the rows corresponding to each other.

$T_D/$GeV	$\sim$	$10$	${10}^2$	${10}^3$
Adequate $\left(m^2-m^{\prime 2}\right)$		${10}^{-7}$	${10}^{-3}$	$10$
Adequate $\left({\beta }_1-{\beta }_2\right)$		${10}^{-11}$	${10}^{-7}$	${10}^{-3}$

Equations

$${\Phi }_i=(1,2,2,0),$$

$$\Delta =(1,3,1,2),$$

$$\bar{\Delta }=(1,3,1,-2),$$

$$\Omega =(1,3,1,0),{\Omega }_c=(1,1,3,0)$$

$$\begin{array}[]{ccc}{\langle\Omega_{c}\rangle=\left(\begin{array}[]{cc}\omega_{c}&0\\ 0&-\omega_{c}\end{array}\right),}&\hskip 20.00003pt{\langle\Delta_{c}\rangle=\left(\begin{array}[]{cc}0&0\\ d_{c}&0\end{array}\right),}&\end{array}$$

$$\delta {\rho }_{RD}\ge G{\sigma }^2\approx \frac{M_R^6}{M_{Pl}^2}\sim M_R^4\frac{M_R^2}{M_{Pl}^2}$$

$$\delta {\rho }_{MD}>M_R^4{\left(\frac{M_R}{M_{Pl}}\right)}^{3/2}$$

$${\mathcal{L}}_{soft}^1$$

$${\mathcal{L}}_{soft}^2$$

$${\mathcal{L}}_{soft}^3$$

$${\mathcal{L}}_{soft}^4$$

$${\mathcal{L}}_{soft}^5$$

$${\mathcal{L}}_{soft}={\mathcal{L}}_{soft}^1+{\mathcal{L}}_{soft}^2+{\mathcal{L}}_{soft}^3$$

$${\mathcal{L}}_{soft}={\mathcal{L}}_{soft}^1+{\mathcal{L}}_{soft}^4+{\mathcal{L}}_{soft}^5$$

$$V_{eff}^R\sim \frac{a(c_R+d_R)}{M_{Pl}}{M}_R^4{M}_W+\frac{a(a_R+d_R)}{M_{Pl}}{M}_R^3{M}_W^2$$

$$\delta \rho \sim {\kappa }^A\frac{M_R^4{M}_W}{M_{Pl}}+{{\kappa }^{\prime }}^A\frac{M_R^3{M}_W^2}{M_{Pl}}$$

$${\kappa }_{RD}^A>10^{-10}{\left(\frac{M_R}{10^6\mathrm{GeV}}\right)}^2$$

$${\kappa }_{MD}^A>10^{-2}{\left(\frac{M_R}{10^6\mathrm{GeV}}\right)}^{3/2},$$

$$X\leftrightarrow X,X^{\prime }\leftrightarrow -X^{\prime }.$$

$$W$$

$${\Lambda }_X=\frac{⟨F_X⟩}{⟨X⟩},{\Lambda }_{X^{\prime }}=\frac{⟨F_{X^{\prime }}⟩}{⟨X^{\prime }⟩}.$$

$$m_{f_L}$$

$$m_{f_R}$$

$$m_{{\varphi }_L}^2$$

$$m_{{\varphi }_R}^2$$

$$M_{a_L}=\frac{{\alpha }_a}{4\pi }\frac{\lambda ⟨F_X⟩+{\lambda }^{\prime }⟨F_{X^{\prime }}⟩}{\lambda ⟨X⟩+{\lambda }^{\prime }⟨X^{\prime }⟩}$$

$$M_{a_R}=\frac{{\alpha }_a}{4\pi }\frac{\lambda ⟨F_X⟩-{\lambda }^{\prime }⟨F_{X^{\prime }}⟩}{\lambda ⟨X⟩-{\lambda }^{\prime }⟨X^{\prime }⟩}$$

$$m_{{\varphi }_L}^2$$

$$\times$$

$$m_{{\varphi }_R}^2$$

$$\times$$

$$\delta m_{\Delta }^2=2({\Lambda }_X{)}^{2}f(\gamma ,\sigma )\left\{{\left(\frac{{\alpha }_2}{4\pi }\right)}^2+\frac{6}{5}{\left(\frac{{\alpha }_1}{4\pi }\right)}^2\right\}$$

$$f(\gamma ,\sigma )={\left(\frac{1+\text{tan}\gamma }{1+\text{tan}\sigma }\right)}^2-{\left(\frac{1-\text{tan}\gamma }{1-\text{tan}\sigma }\right)}^2$$

$$\text{tan}\gamma =\frac{{\lambda }^{\prime }⟨F_{X^{\prime }}⟩}{\lambda ⟨F_X⟩},\text{tan}\sigma =\frac{{\lambda }^{\prime }⟨X^{\prime }⟩}{\lambda ⟨X⟩}$$

$$\delta m_{\Omega }^2=2({\Lambda }_X{)}^{2}f(\gamma ,\sigma ){\left(\frac{{\alpha }_2}{4\pi }\right)}^2$$

Full text

left-right symmetry, neutrino mass, domain walls, supersymmetry, metastable vacua

Chirality seems to be an essential feature of fundamental physics, thereby allowing dynamical generation of fermion masses. However this did not require the world to be parity asymmetric, the way it is manifested in the Standard Model (SM). Indeed, the discovery of neutrino massesOkumura:2016zyh ; Fukuda:1998mi Davis:1986ts ; solar_data ; McDonald:2016xdg in the past two decades strongly suggests the existence of right handed neutrino states. The resulting parity balanced spectrum of fermions begs a parity symmetric theory and parity violation could then be explained to be of dynamical origin. Let us take stock of what principles we could put to use in a “down upwards” guesswork in energy scales.

• •

Gauge principle provides massless force carriers

• •

Chirality provides massless matter

• •

Scalars can signal spontaneous symmetry breakdown

• –

provide masses through a universal mechanism

• –

provide naturally substantial amount of $CP$ violation.

• •

Supersymmetry provides a most comprehensive and elegant way of sequestering mass scales signalled by scalars.

Any extension we seek could first be guided by these principles. To be specific we assume the restriction that any exact internal symmetry should be gauged, and any scalar and fermionic fields introduced must be charged under at least one of these gauge symmetries. The left-right class of models are interesting from this point view and are explored as the first rung of a ”down upwards” unification. The two other alternatives not pursued by us are (1) $SO(10)$ unification with grand desert, as for instance proposed in Poh:2015wta ; Aulakh:2002zr ; Aulakh:2004hm or (2) the SM Higgs boson so instrumental to providing a perturbative description of weak interactions is secretly a member of a strongly coupled theoryLane:2016kvg ; Kaul:1981uk Csaki:2003si ; Kaplan:2003uc ; ArkaniHamed:2002qx .

As baryogenesis began to be ruled out in SMkrs.85 ; chem_potential_lepto and leptogenesis fukugita.86 became severely constrained in thermal $SO(10)$ scenariosdavidson&ibarra.02 Boltzmann_equations , our early observation sahu&yaj_prd.04 was that the see-saw mechanism GellMann:1980vs Yanagida:1979as Mohapatra:1979ia Schechter:1980gr Schechter:1981cv generically easily permits an $M_{R}$ scale as low as $10^{6}$GeV, considerably smaller than the scales of coupling constant unification. Secondly, that low scale non-thermal leptogenesis is consistent with the wash out constraints from low scale right-handed neutrinos. It is therefore appealing to look for left-right symmetry leftright_group as an intermediate stage in the sequence of symmetry breaking, and explore the possible range of masses $M_{R}$ acceptable for the heavy right handed neutrinos. In the hope that the new symmetries are within the reach of the LHC and several proposed future colliders, we may call it Just Beyond the Standard Model (JBSM).

A number of recent works have focused on this question and studied it in conjunction with correlated signatures such as with baryogenesis or leptogenesis, and also additionally Dark Matter production Davoudiasl:2015jja ; Dev:2015uca ; Chala:2016ykx ; Kim:2016xyi ; Zhuridov:2016xls ; Huang:2015izx ; Dhuria:2015xua ; Deppisch:2015yqa and gravitational wavesHuang:2016odd . There are also planned or ongoing experiments such as non-accelerator exploration of $N$-$\bar{N}$ oscillations Frost:2016qzt , role of right handed currents in the electric dipole moments of light nuclei Dekens:2014jka and hadronic CP violation Cirigliano:2016yhc , correlated signatures in neutrinoless double beta decay Drewes:2016lqo Asaka:2016zib on the one hand and collider signaturesDeSerio:2016ath Huang:2015izx Djouadi:2007ik on the other. Secondly a number of viable scenarios for low scale leptogenesis exist, such as resonant leptogenesis Pilaftsis:2005rv ; Dev:2014tpa as also the generically supersymmetric Affleck-Dine scenario Affleck:1984fy ; Allahverdi:2016yws . We considered specific models Cline:2002ia (non-supersymmetric), and Sarkar:2007er (supersymmetric version) of non-thermal leptogenesis, but it is presumably possible for the other scenarios such as referred in this paragraph also to be implementable within the parameter space of the left-right models to be considered here. Assuming that this is possible within the PeV scale, we have studied the consistency of some of the intrinsic features of the proposed left-right symmetric models with cosmology.

I Left-right symmetry : a supersymmetric revival

Left-right symmetric modelMohapatra:1980qe ; Mohapatra:1979ia needs a Supersymmetric extension as an expedient for avoiding the hierarchy problem. The minimal set of Higgs superfields required, with their SU(3)\otimes$$SU(2)_{L}\otimes $SU(2)_{R}\otimes$ $U(1)_{\scriptscriptstyle{B-L}}$ is,

[TABLE]

and further details of the model can be found in the references.

There is an awkward impasse with this model, namely we would like to retain supersymmetry down to the TeV scale. So the first stage of gauge symmetry breaking has to respect supersymmetry. If we choose the parameters of the superpotential to ensure spontaneous parity breaking, then either the electromagnetic gauge invariance or the $R$ parity have to be sacrificedKuchimanchi:1993jg ; Kuchimanchi:1995vk . The first of these is unacceptable consequence, and the second entails a requirement of inelegant fixes. This problem was elegantly resolved in Aulakh:1997ba with further developments in Aulakh:1997fq ; Aulakh:1998nn . It contains the two additional triplet Higgs fields

[TABLE]

We refer to this briefly as ABMRS model. Supersymmetric minima breaking $SU(2)_{R}$ symmetry are signaled by the ansatz

[TABLE]

In this model, with an enhanced $R$ symmetry, we are lead naturally to a see-saw relation $M^{2}_{B-L}=M_{EW}M_{R}$. This means Leptogenesis is postponed to a lower energy scale closer to $M_{EW}$. Being generically below $10^{9}$ GeV, this avoids the gravitino mass bound but requires non-thermal leptogensisSahu:2004sb .

For comparison we also take an alternative model to this, considered in Babu:2008ep where a superfield $S(1,1,1,0)$ also singlet under parity is included in addition to the minimal set of Higgs of Eq. (1). This is referred to here as BM model.

II Transitory domain walls

A generic problem long recognised with the model with exact left-right symmetry is that of cosmological domain walls. The effective potential of the theory is unable to give preference to whether $SU(2)_{L}$ is broken first or $SU(2)_{R}$. There is therefore a competition to the SM vacuum from a “parity flipped SM” vacuum, with the low energy effective gauge group $SU(3)_{c}\otimes SU(2)_{R}\otimes U(1)_{\tilde{Y}}$ where $\tilde{Y}=T_{L}^{3}+\frac{1}{2}(B-L)$. Spontaneous parity breaking leads to formation of Domain walls which quickly dominate the energy density of the Universe. It is necessary for recovering standard cosmology that these walls disappear at least before the Big Bang Nucleosynthesis. In an intrinsically parity symmetric theory difference in the vacua resulting in destabilisation is not permitted. In the following we first obtain the quantitative requirement on the possible asymmetry so that standard cosmology is ensured. In the next two subsections we propose mechanisms that could be the source of such asymmetry without causing serious damage to the basic assumption of exact left-right symmetry.

There are several studies of wall evolution, and an estimate of the temperature at which the walls may destabilise, parametrically expressed in terms of the surface tension of the walls, in turn determined by the parity breaking scale $M_{R}$. By equating the terms leading to small symmetry breaking discussed in the previous para with this parametric dependence then gives a bound on $M_{R}$.

The dynamics of the walls in a radiation dominated universe is determined by two quantities : Kibble:1980mv , Tension force $f_{T}\sim\sigma/R$, where $\sigma$ is energy per unit area and $R$ is the average scale of radius of curvature, and Friction force $f_{F}\sim\beta T^{4}$ for walls moving with speed $\beta$ in a medium of temperature $T$. The scaling law for the growth of the scale $R(t)$ on which the wall complex is smoothed out, is taken to be $R(t)\approx(G\sigma)^{1/2}t^{3/2}$. Also, $f_{F}\sim 1/(Gt^{2})$ and $f_{T}\sim(\sigma/(Gt^{3}))^{1/2}$. Then the pressure difference required to overcome the above forces and destabilise the walls is

[TABLE]

The case of matter dominated evolution is relevant to moduli fields copiously produced in generic string inspired models Kawasaki:2004rx of the Universe. A wall complex formed at temperature $T_{i}\sim M_{R}$ is assumed to have first relaxed to being one wall segment per horizon volume. It then becomes comparable in energy density to the ambient matter density, due to the difference in evolution rates, $1/a(t)$ for walls compared to $1/a^{3}(t)$ for matter. For simplicity also demand that the epoch of equality of the two contributions is the epoch also of the onset of wall destablisation, so as to avoid dominance by domain walls. $T_{destab}$ is taken to be the temperature when the walls begin to be unstable and less dominanat. Thus we can set $M_{Pl}^{-2}T_{destab}^{4}\sim H^{2}_{eq}\sim\sigma^{\frac{3}{4}}H_{i}^{\frac{1}{4}}M_{Pl}^{-3}.$ The corresponding temperature permits the estimate of the required pressure difference,

[TABLE]

Thus in this case we find $(M_{R}/M_{Pl})^{3/2}$ Mishra:2009mk , a milder suppression factor than in the radiation dominated case above.

II.1 Soft terms as a source of lifting parity degeneracy

One source that could provide the required pressure $\delta\rho$ can be sought in a generic neutral scalar field $\phi$. It provides the higher dimensional operator that may break parity and which has the simple form Rai:1992xw $V_{eff}=\frac{C_{5}}{M_{Pl}}\phi^{5}$. The cosmological requirement then constrains the coefficient $C_{5}$. In realistic theories, there are several scalar fields entering such terms, and the structure of the latter is conditioned by gauge invariance and supersymmetry. Here we study a possible source of such asymmetric terms without sacrificing the symmetries of the superpotential, viz., in the soft supersymmetry breaking terms. The soft terms which arise in the two models, ABMRS and BM respectively, may be parameterised as follows

[TABLE]

For ABMRS model the relevant soft terms are given by,

[TABLE]

For BM model the soft terms are given by,

[TABLE]

It can be shown Preskill:1991kd that the adequate requirement to evade the domain wall problem is to demand $\delta\rho\sim T_{destab}^{4}$, where $\delta\rho$ is the difference in the effective potentials across the domain walls and $T_{destab}$ is the destablisation temperature, at which this difference becomes the dominant force acting on the walls. At a later epoch, the walls decay at temperature $T_{D}$. If the disappearnce of the walls is prompt, the two temperatures would be comparable. Even in the case when the epochs are separated in time, the entropy generated due to final wall disppearance may raise the temprature, bringing it closer to $T_{destab}$. In the absence of a detailed model we take $T_{D}\sim T_{destab}$. In order to obtain the observed Universe we need that $T_{D}$ remains higher than the Big Bang Nucleosynthesis (BBN) scale of a few MeV. This is the requirement imposed to constrain the differences between the soft terms in the Left and Right sectors Sarkar:2007ic ; Sarkar:2007er . In the BM model the $S$ field does not acquire a vacuum expectation value (vev) in the physically relevant vacua and hence the terms in eq.s (9) and (10) do not contribute to the vacuum energy. The terms in eq. (7) are suppressed in magnitude relative to those in eq. (8) due to having $\Omega$ vev’s to one power lower. This argument assumes that the magnitude of the coefficients $\alpha$ are such as to not mix up the symmetry breaking scales of the $\Omega$’s and the $\Delta$’s.

To obtain orders of magnitude we have taken the $m_{i}^{2}$ parameters to be of the form $m_{1}^{2}\sim m_{2}^{2}$ $\sim m^{2}$ and $m_{3}^{2}\sim m_{4}^{2}$ $\sim m^{{}^{\prime}2}$ Sarkar:2007er with $T_{D}$ in the range $10-10^{3}$ GeV Kawasaki:2004rx . For both the models we have taken the value of the $\Delta$ vev’s as $d\sim 10^{4}$ GeV. For ABMRS model additionally we take $\omega\sim 10^{6}$ GeV. The resulting differences required for successful removal of domain walls are shown in Table 1.

We see from table 1 that assuming both the mass-squared differences $m^{2}-m^{\prime 2}$ and $\beta_{1}-\beta_{2}$ arise from the same dynamics, $\Omega$ fields are the determinant of the cosmology. This is because the lower bound on the wall decay temperature $T_{D}$ required by $\Omega$ fields is higher and the corresponding $T_{D}$ is reached sooner. This situation changes if for some reason $\Omega$’s do not contribute to the pressure difference across the walls. The BM model does not have $\Omega$’s and falls in this category.

Between the epoch of destabilisation of the DW and their decay, leptogenesis occurs due to preferential motion of the DW to select SM as the low energy theory, as discussed in Cline:2002ia ; Sarkar:2007er . After the disappearance, i.e., complete decay of the walls at the scale $T_{D}$, electroweak symmetry breaks at a scale $M_{EW}\sim 10^{2}$ GeV and standard cosmology takes over.

Since the soft terms are associated with supersymmetry breaking, they may be assumed to arise from the same mechanism that breaks supersymmetry. In Sec. III we discuss the implementation of gauge mediated supersymmetry breaking (GMSB) scenario for these models and treat the soft terms to have arisen from the hidden sector and communicated along with the messenger fields Mishra:2008be . Constraints on the hidden sector model and the communication mechanism can be obtained in this way.

II.2 Parity breaking from Planck suppressed effects

The soft terms studied above are tied to the scale of supersymmetry breaking. Another mechanism to look into, without incurring violence to the symmetries of the superpotential, is to assume that the parity breaking operators arise at the Planck scale Mishra:2009mk . Writing the lowest order terms in the superpotential suppressed by the Planck scale, and using the Kähler potential formalism, we obtain the expectation value of the effective potential of the scalar fields after substituting the vacuum expectation values of the relevant fields as follows, whose details can be found in Mishra:2009mk .

[TABLE]

and likewise $R\leftrightarrow L$. Here the constants $a$, $c_{R}$ etc. are dimensionless. Hence, with regrouping the above coefficients into coefficients $\kappa$ etc., which for naturalness should remain order unity,

[TABLE]

It is interesting to study this relation with $\kappa$ coefficients $O(1)$ and $\delta\rho\sim(10\mathrm{MeV})^{4}$, compatible with BBN. It then leads without any further assumptions to a lower bound on $M_{R}$ to be same as SM scale, leaving all higher scales open to being physically viable.

Now inserting these answers in the expectations derived from cosmological dynamics of DW, viz., $\delta\rho_{\scriptscriptstyle{RD}}$, $\delta\rho_{\scriptscriptstyle{MD}}$ derived in Eq.s (4) and (5), we obtain

[TABLE]

Thus only minuscule values of $\kappa$ coefficients suffice to allow PeV scale $M_{R}$. At the higher end, $M_{R}$ $\lesssim 10^{9}$GeV needed to avoid gravitino problem after reheating at the end of inflation remains viable with $\kappa_{RD}\sim 10^{-4}$, and also $M_{R}$ of the order of the intermediate scale $10^{11}$GeV remains marginally viable with $\kappa^{A}_{RD}$ $\sim O(1)$. But scales higher than that are not tolerated by naturalness.

Next, if we consider wall disappearance during moduli dominated regime, we find

[TABLE]

which can be seen to be a rather strict requirement barely allowing $M_{R}$ higher than our preferred PeV scale. In particular, taking $M_{R}\sim 10^{9}$GeV required to have thermal leptogenesis without the undesirable gravitino production, leads to unnatural values of $\kappa_{MD}>10^{5/2}$.

III Customised GMSB for Left-Right symmetric models

The differences required between the soft terms of the Left and the Right sector for the DW to disappear at a temperature $T_{D}$ are not unnaturally large. However the reasons for appearance of even a small asymmetry between the Left and the Right fields is hard to explain since the theory as adopted in Sec. I is parity symmetric. We now try to explain the origin of this small difference by focusing on the hidden sector, and relating it to supersymmetry (SUSY) breaking.

For this purpose we assume that the strong dynamics responsible for SUSY breaking also breaks parity, which is then transmitted to the visible sector via the messenger sector and is encoded in the soft supersymmetry breaking terms. We implement this idea by introducing two singlet fields $X$ and $X^{\prime}$, respectively even and odd under parity.

[TABLE]

The messenger sector superpotential then contains terms

[TABLE]

For simplicity, we consider $n=1$. The fields $\Phi_{L}$, $\bar{\Phi}_{L}$ and $\Phi_{R}$, $\bar{\Phi}_{R}$ are complete representations of a simple gauge group embedding the L-R symmetry group. Further we require that the fields labelled $L$ get exchanged with fields labelled $R$ under an inner automorphism which exchanges $SU(2)_{L}$ and $SU(2)_{R}$ charges, e.g. the charge conjugation operation in $SO(10)$. As a simple possibility we consider the case when $\Phi_{L}$, $\bar{\Phi}_{L}$ (respectively, $\Phi_{R}$, $\bar{\Phi}_{R}$) are neutral under $SU(2)_{R}$ ($SU(2)_{L}$). Generalisation to other representations is straightforward.

As a result of the dynamical SUSY breaking we expect the fields $X$ and $X^{\prime}$ to develop nontrivial vev’s and $F$ terms and hence give rise to mass scales

[TABLE]

Both of these are related to the dynamical SUSY breaking scale $M_{S}$, however their values are different unless additional reasons of symmetry would force them to be identical. Assuming that they are different but comparable in magnitude we can show that Left-Right breaking can be achieved simultaneously with SUSY breaking being communicated.

In the proposed model, the messenger fermions receive respective mass contributions

[TABLE]

while the messenger scalars develop the masses

[TABLE]

We thus have both SUSY and parity breaking communicated through these particles.

As a result the mass contributions to the gauginos of $SU(2)_{L}$ and $SU(2)_{R}$ from both the $X$ and $X^{\prime}$ fields with their corresponding auxiliary parts take the simple form,

[TABLE]

and

[TABLE]

upto terms suppressed by $\sim F/X^{2}$. Here $a=1,2,3$. In turn there is a modification to scalar masses, through two-loop corrections, expressed to leading orders in the $x_{L}$ or $x_{R}$ respectively, by the generic formulae

[TABLE]

[TABLE]

The resulting difference between the mass squared of the left and right sectors are obtained as

[TABLE]

where,

[TABLE]

We have brought $\Lambda_{X}$ out as the representative mass scale and parameterised the ratio of mass scales by introducing

[TABLE]

Similarly,

[TABLE]

In the models studied here, the ABMRS model will have contribution from both the above kind of terms. The BM model will have contribution only from the $\Delta$ fields.

The contribution to slepton masses is also obtained from eq.s (25) and (27). This can be used to estimate the magnitude of the overall scale $\Lambda_{X}$ to be $\geq 30$ TeV based on Dubovsky:1999xc from LEP limits, which may not have changed significantly even in the light of LHC data Knapen:2016exe ; Allanach:2016pam . Substituting this in the above formulae (28) and (31) we obtain the magnitude of the factor $f(\gamma,\sigma)$ required for cosmology as estimated in table 1. The resulting values of $f(\gamma,\sigma)$ are tabulated in table 2. We see that the natural range of temperature for the disappearance of domain walls therefore remains TeV or higher, i.e., upto a few order of magnitudes lower than the scale at which they form.

IV Supersymmetry breaking in metastable vacua

The dilemma of phenomenology with broken supersymmetry can be captured in the fate of $R$ symmetry generic to superpotentials Nelson:1993nf . An unbroken $R$ symmetry in the theory is required for SUSY breaking. $R$ symmetry when spontaneously broken leads to $R$-axions which are unacceptable. If we give up $R$ symmetry, the ground state remains supersymmetric. The solution proposed in Nelson:1993nf , is to break $R$ symmetry mildly, governed by a small parameter $\epsilon$. Supersymmetric vacuum persists, but this can be pushed far away in field space. SUSY breaking local minimum is ensured near the origin, since it persists in the limit $\epsilon\rightarrow 0$. A specific example of this scenario Seiberg:1994pq ; Intriligator:2006dd ; Intriligator:2007py referred to as ISS, envisages $SU(N_{c})$ SQCD with $N_{f}>N_{c}$ flavours of quarks $q,\tilde{q}$ which is UV free, such that it is dual to a $SU(N_{f}-N_{c})$ gauge theory which is IR free, the so called magnetic phase, with $N^{2}_{f}$ singlet mesons $M$.

Thus we consider a Left-Right symmetric model with ISS mechanism as proposed in Haba:2011pr . They proposed the electric gauge theory to be based on the gauge group $SU(3)_{L}\times SU(3)_{R}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ (in short $G_{33221}$) where $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ is the gauge group of usual Left-Right models and $SU(3)_{L,R}$ is the new strongly coupled gauge sector introduced. The dual description similar to the original ISS model gives rise to $SU(2)_{R}$ broken meta-stable vacua inducing spontaneous SUSY breaking simultaneously.

The particle content of the electric theory is

[TABLE]

[TABLE]

where $a=1,N_{f}$ and the numbers in brackets correspond to the transformations of the fields under the gauge group $G_{33221}$. This model has $N_{c}=3$ and hence to have a Seiberg dual Seiberg:1994pq magnetic theory, number of flavours should be $N_{f}\geq 4$. For $N_{f}=4$ the dual magnetic theory will have the gauge symmetry of the usual Left Right Models $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ and the following particle content

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The Left-Right symmetric renormalisable superpotential of this magnetic theory is

[TABLE]

After integrating out the right handed chiral fields, the superpotential becomes

[TABLE]

which gives rise to SUSY preserving vacua at

[TABLE]

where $\epsilon=\frac{\mu}{\Lambda_{m}}$. Thus the right handed sector exists in a metastable SUSY breaking vacuum whereas the left handed sector is in a SUSY preserving vacuum breaking D-parity spontaneously.

We next consider Borah:2011aw Planck scale suppressed terms that may signal parity breaking

[TABLE]

The terms of order $\frac{1}{\Lambda_{m}}$ are given by

[TABLE]

The minimisation conditions give $\phi\tilde{\phi}=\mu^{2}$ and $S^{0}=-\delta^{0}$. Denoting $\langle\phi^{0}_{R}\rangle=\langle\tilde{\phi}^{0}_{R}\rangle=\mu$ and $\langle\delta^{0}_{R}\rangle=-\langle S^{0}_{R}\rangle=M_{R}$, we have

[TABLE]

where we have also assumed $f^{\prime}_{R}\approx f_{R}$. For $\lvert\mu\rvert<M_{R}$ Thus the effective energy density difference between the two types of vacua is

[TABLE]

Now we set $\mu\sim$TeV, the scale of supersymmetry breaking and since QCD cannot be expected to break parity, assume that the parity breaking terms are essentially of Planck scale origin. Using this self consistent requirement, for walls disappearing in matter dominated era, we get

[TABLE]

with $\mu\sim$TeV. Similarly for the walls disappearing in radiation dominated era,

[TABLE]

Thus in both cases we get an upper bound on the $M_{R}$ which can at best be an intermediate scale, while the PeV scale remains eminently viable.

V Conclusions

We have pursued the possibility of left-right symmetric models as Just Beyond Standard Models (JBSM), not possessing a large hierarchy. We also adopt the natural points of view that right handed neutrinos must be included in the JBSM with local gauge symmetric interactions and that the required parity breaking to match low energy physics arises from spontaneous breakdown. The latter scenario is often eschewed due to the domain walls it entails in the early Universe. We turn the question around to ask given that the domain walls occur, what physics could be responsible for their successful removal without jeopardising naturalness.

Seeking origins of parity breaking that are consistent with the principles outlined in the Introduction, we proceed to correlate this breaking to supersymmetry breaking. We have considered three models along these lines. One involves gauge mediated supersymmetry breaking, another assumes Planck scale breaking and the third relies on metastable vacua for supersymmetry breaking. The operators permissible in these scenarios are then constrained by the cosmological requirements on the dynamics of domain wall disappearance. In all the cases studied, The mass scale of right handed neutrino $M_{R}$ remains bounded from above, and in some of the cases the scale $10^{9}$ GeV favourable for supersymmetric thermal leptogenesis is disallowed. On the other hand PeV scale remains a viable option.

The possibility that such a low energy model maybe embedded successfully in the semisimple $SO(10)$ at a high scale has been explored separately Kopp:2009xt Borah:2010kk . The general message seems to be that the parity breaking scale is not warranted to be as high as the grand unification scale and further, several scenarios suggest that left-right symmetry as the Just Beyond Standard Model package incorporating the SM may be within the reach of future colliders.

VI Acknowledgment

This is a comprehensive resume of the work done jointly with Narendra Sahu, Anjishnu Sarkar, Sasmita Mishra and Debasish Borah. It is a happy duty to thank the collaborators. The coverage of other works along similar lines suffers from the limitation of space permitted here but also from limitation of my awareness of them. Most of the work was supported by grants from Department of Science and Technology. I thank the organisers of the Pheno1 conference IISER Mohali and the organisers and participants of the program Exploring the Energy Ladder of the Universe for enabling presentation and discussion of this work. The author would like to express a special thanks to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support.