New ultraviolet operators in supersymmetric SO(10) GUT and consistent cosmology

TL;DR

This paper explores how adding a Planck-scale suppressed operator in supersymmetric SO(10) GUT models can eliminate domain walls, ensuring cosmological consistency and avoiding conflicts with big bang nucleosynthesis.

Contribution

It introduces a nonrenormalisable operator that breaks SO(10) symmetry to remove domain walls without disrupting standard cosmology.

Findings

Planck-scale operators can effectively remove domain walls.

The model remains consistent with big bang nucleosynthesis.

Low right-handed breaking scales pose cosmological challenges.

Abstract

We consider the minimal supersymmetric grand unified model (MSGUT) based on the group $\mathrm{SO}(10)$, and study conditions leading to possible domain wall (DW) formation. It has been shown earlier that the supersymmetry preserving vacuum expectation values (vev's) get mapped to distinct but degenerate set of vev's under action of $D$ parity, leading to formation of domain walls as topological pseudo-defects. The metastability of such walls can make them relatively long lived and contradict standard cosmology. Thus we are led to consider adding a nonrenormalisable Planck scale suppressed operator, that breaks $\mathrm{SO}(10)$ symmetry but preserves Standard Model symmetry. For a large range of right handed breaking scales $M_R$, this is shown to give rise to the required pressure difference to remove the domain walls without conflicting with consistent big bang nucleosynthesis (BBN) while avoiding gravitino overproduction. However, if the walls persist till the onset of weak (thermal) inflation, then a low $\sim 10 - 100$ TeV $M_R$ becomes problematic.

Tables

Table 1. Table 1: Property of various F 𝐹 F -terms under D 𝐷 D -parity flip

F-term		Original vev	$D$-Flipped vev
$F_{{\Phi }_{1234}}=F_{{\Phi }_{1256}}=F_{{\Phi }_{3456}}$	=	$2ma+2\lambda \left(a^2+2{\omega }^2\right)+\eta \sigma \overline{\sigma }$	$2ma+2\lambda \left(a^2+2{\omega }^2\right)+\eta \sigma \overline{\sigma }$,
$F_{{\Phi }_{1278}}=F_{{\Phi }_{3479}}=F_{{\Phi }_{5678}}$	=	$2m\omega +2\lambda \left(2a+p\right)\omega -\eta \sigma \overline{\sigma }$	$2m\omega +2\lambda \left(2a+p\right)\omega -\eta \sigma \overline{\sigma }$,
$F_{{\Phi }_{1290}}=F_{{\Phi }_{3490}}=F_{{\Phi }_{5690}}$	=	$2m\omega +2\lambda \left(2a+p\right)\omega -\eta \sigma \overline{\sigma }$	$-2m\omega -2\lambda \left(2a+p\right)\omega +\eta \sigma \overline{\sigma }$,
$F_{{\Phi }_{7890}}$	=	$2mp+6\lambda {\omega }^2+\eta \sigma \overline{\sigma }$	$-2mp-6\lambda {\omega }^2-\eta \sigma \overline{\sigma }$
Other $F_{{\Phi }_{ijkl}}$	=	$0$	$0$
$\left|F_{{\Sigma }_{a+1,b+3,c+5,d+7,e+9}}\right|$	=	$\frac{\left|\overline{\sigma }\right|}{2^{3/2}}\left|M+\eta \left(3a+p-6\omega \right)\right|$	$\frac{\left|\overline{\sigma }\right|}{2^{3/2}}\left|M+\eta \left(3a+p-6\omega \right)\right|$
$\left|F_{{\overline{\Sigma }}_{a+1,b+3,c+5,d+7,e+9}}\right|$	=	$\frac{\left|\sigma \right|}{2^{3/2}}\left|M+\eta \left(3a+p-6\omega \right)\right|$	$\frac{\left|\sigma \right|}{2^{3/2}}\left|M+\eta \left(3a+p-6\omega \right)\right|$
Other $F_{{\Sigma }_{ijklm}},F_{{\overline{\Sigma }}_{ijklm}}$	=	$0$	$0$

Table 2. Table 2: Constraints arising from different values of M R subscript 𝑀 𝑅 M_{R} in different cosmological eras

Era	$M_R$	$b\ge$
	${10}^7$	${10}^{-18}$
RD	${10}^9$	${10}^{-14}$
	${10}^{13}$	${10}^{-6}$
	${10}^7$	${10}^{-15}$
MD	${10}^9$	${10}^{-11.5}$
	${10}^{13}$	${10}^{-4.5}$

Equations

$$W_{\mathrm{ren}}=\frac{m}{4!}{\Phi }_{ijkl}{\Phi }_{ijkl}+\frac{\lambda }{4!}{\Phi }_{ijkl}{\Phi }_{klmn}{\Phi }_{mnij}+\frac{M}{5!}{\Sigma }_{ijklm}{\overline{\Sigma }}_{ijklm}+\frac{\eta }{4!}{\Phi }_{ijkl}{\Sigma }_{ijmno}{\overline{\Sigma }}_{ijmno},$$

$$\begin{array}[]{c}\Phi_{1234}=\Phi_{1256}=\Phi_{3456}=a,\\ \Phi_{1278}=\Phi_{129\,10}=\Phi_{3478}=\Phi_{3490}=\Phi_{5678}=\Phi_{5690}=\omega,\\ \Phi_{7890}=p,\\ \\ \Sigma_{a+1,b+3,c+5,d+7,e+9}=i^{a+b+c-d-e}\frac{\sigma}{2^{5/2}},~{}~{}~{}~{}a,b,c,d,e\in\{0,1\},\\ \\ \overline{\Sigma}_{a+1,b+3,c+5,d+7,e+9}=i^{-a-b-c+d+e}\frac{\overline{\sigma}}{2^{5/2}},~{}~{}~{}~{}a,b,c,d,e\in\{0,1\}.\end{array}$$

$$(L\leftrightarrow R)_{ij}=\begin{array}[]{r l}0&\mbox{$i\neq j$}\\ -1&\mbox{$i=j=0$}\\ 1&\mbox{$i=j=1,\ldots,9$}.\end{array}$$

$$D=\exp (-i\pi J_{23})\exp (i\pi J_{67})$$

$$D_{ij}=\begin{array}[]{r l}0&\mbox{$i\neq j$}\\ -1&\mbox{$i=j=2,3,6,7$}\\ 1&\mbox{$i=j=1,4,5,8,9,0$}.\end{array}$$

$$\begin{array}[]{c}\Phi_{1234}=\Phi_{1256}=\Phi_{3456}=a,\\ \Phi_{1278}=-\Phi_{1290}=\Phi_{3478}=-\Phi_{3490}=\Phi_{5678}=-\Phi_{5690}=\omega,\\ \Phi_{7890}=-p,\\ \\ \Sigma_{a+1,b+3,c+5,d+7,e+9}=i^{-a-b-c+d-e}\frac{\sigma}{2^{5/2}},~{}~{}~{}~{}a,b,c,d,e\in\{0,1\},\\ \ \overline{\Sigma}_{a+1,b+3,c+5,d+7,e+9}=i^{a+b+c-d+e}\frac{\overline{\sigma}}{2^{5/2}},~{}~{}~{}~{}a,b,c,d,e\in\{0,1\}.\end{array}$$

$$\begin{array}[]{c}\Phi_{1234}=\Phi_{1256}=\Phi_{3456}=a,\\ \Phi_{1278}=\Phi_{3478}=\Phi_{5678}=-\Phi_{129\,10}=-\Phi_{349\,10}=-\Phi_{569\,10}=\omega,\\ \Phi_{78910}=-p,\\ \\ \Sigma_{a+1,b+3,c+5,d+7,e+9}=i^{a+b+c-d+e}\frac{\sigma}{2^{5/2}},~{}~{}~{}~{}a,b,c,d,e\in\{0,1\},\\ \\ \overline{\Sigma}_{a+1,b+3,c+5,d+7,e+9}=i^{-a-b-c+d-e}\frac{\overline{\sigma}}{2^{5/2}},~{}~{}~{}~{}a,b,c,d,e\in\{0,1\}.\end{array}$$

$$D(⟨210⟩)=(L\leftrightarrow R)(⟨210⟩),D(⟨126⟩)=\frac{\sigma }{\overline{\sigma }}(L\leftrightarrow R)(⟨\overline{126}⟩),D(⟨\overline{126}⟩)=\frac{\overline{\sigma }}{\sigma }(L\leftrightarrow R)(⟨126⟩).$$

$$W=W_{\mathrm{ren}}+\frac{b}{M_{\mathrm{Pl}}}{W}_{\mathrm{nr}},$$

$$W_{\mathrm{nr}}=\frac{1}{(4!{)}^4}({\Phi }^T{M}^{\prime }\Phi {)}^2.$$

$$M=\left(\begin{array}[]{c | c}\mathbb{I}_{6\times 6}&0_{6\times 4}\\ \hline\cr\\ 0_{4\times 6}&K_{4\times 4}\end{array}\right)$$

$$J=j_1{J}_1^{+}+j_2{J}_2^{+}+j_3{J}_3^{+}\in \mathrm{so}(4),$$

$$J_{1}^{+}=\frac{1}{2}\left(\begin{array}[]{c c c c}0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0\end{array}\right),J_{2}^{+}=-\frac{1}{2}\left(\begin{array}[]{c c c c}0&0&1&0\\ 0&0&0&-1\\ -1&0&0&0\\ 0&1&0&0\end{array}\right),J_{3}^{+}=\frac{1}{2}\left(\begin{array}[]{c c c c}0&1&0&0\\ -1&0&0&0\\ 0&0&0&1\\ 0&0&-1&0\end{array}\right)$$

$$B-L=\left(\begin{array}[]{c | c}\begin{array}[]{r r r r r r}0&1&0&0&0&0\\ -1&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&-1&0&0&0\\ 0&0&0&0&0&1\\ 0&0&0&0&-1&0\end{array}&O_{6\times 4}\\ \hline\cr O_{4\times 6}&O_{4\times 4}\end{array}\right).$$

$$K=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c c c c}0&1&0&1\\ -1&0&1&0\\ 0&-1&0&1\\ -1&0&-1&0\end{array}\right).$$

$$M=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c | c}\sqrt{2}\,\mathbb{I}_{6\times 6}&0_{6\times 4}\\ \hline\cr\\ 0_{4\times 6}&\begin{array}[]{c c c c}0&1&0&1\\ -1&0&1&0\\ 0&-1&0&1\\ -1&0&-1&0\end{array}\end{array}\right).$$

$$J_{23}+J_{67}=\left(\begin{array}[]{r r r r r r r r r r}0&0&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&-1&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&-1&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\end{array}\right)$$

$$\frac{\partial W_{\mathrm{nr}}}{\partial {\Phi }_{ijkl}}=\frac{2({\Phi }^T{M}^{\otimes 4}\Phi )}{(4!{)}^4}\frac{\partial ({\Phi }^T{M}^{\otimes 4}\Phi )}{\partial {\Phi }_{ijkl}},$$

$$\begin{array}[]{rcl}\lx@intercol\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\left.V\right|_{\mbox{flipped vevs}}-\left.V\right|_{\mbox{original vevs}}$\mbox{}\hfil\lx@intercol\\ &&\\ &=&\frac{b^{2}}{M_{\mathrm{Pl}}^{2}}(4^{2}(3a^{2}+6\omega^{2}+p^{2}))((3a^{2}+6\omega^{2}+p^{2})^{2}-(3a^{2}+p^{2})^{2}).\end{array}$$

$$a=-0.67\frac{m}{\lambda },\omega =-0.21\frac{m}{\lambda },p=-0.27\frac{m}{\lambda },\sigma =\overline{\sigma }=0.51\frac{m}{\sqrt{\eta \lambda }},\lambda =0.12,\eta =0.21,$$

$$a\sim -10^{15},\omega \sim -10^{14},p\sim -10^{14},\sigma =\overline{\sigma }\sim 10^{15},$$

$$|{V|}_{\text{mbox}flippedvevs}-{V|}_{\text{mbox}orig.vevs}|\sim \frac{b^2}{M_{\mathrm{Pl}}^2}{M}_X^4{M}_R^2.$$

$$\delta \rho$$

$$\approx$$

$$b^2\ge \frac{M_R^4}{M_X^4}\mathrm{Radiation}\mathrm{dominated}\mathrm{case}.$$

$$H_{eq}\sim {\sigma }^{\frac{3}{4}}{H}_i^{\frac{1}{4}}{M}_{Pl}^{-\frac{3}{2}},$$

$$T_D^4\sim {\sigma }^{\frac{3}{2}}{H}_i^{\frac{1}{2}}{M}_{Pl}^{-1}$$

$$T_D^4\sim \frac{{\sigma }^{11/6}}{M_{Pl}^{3/2}}$$

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

$$b^2\ge \frac{M_R^{7/2}{M}_{\mathrm{Pl}}^{1/2}}{M_X^4}.\mathrm{moduli}\mathrm{dominated}\mathrm{case}$$

$${\rho }_{DW}(t_d)\sim {\rho }_{DW}(t_{eq})\left(\frac{a_{eq}}{a_d}\right)$$

Full text

New ultraviolet operators in supersymmetric $\mathrm{SO}(10)$ GUT

and consistent cosmology

**Piyali Banerjee111 banerjee.piyali3@gmail.com

and Urjit A. Yajnik222 yajnik@iitb.ac.in **

Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India

Abstract

We consider the minimal supersymmetric grand unified model (MSGUT) based on the group $\mathrm{SO}(10)$, and study conditions leading to possible domain wall (DW) formation. It has been shown earlier that the supersymmetry preserving vacuum expectation values (vev’s) get mapped to distinct but degenerate set of vev’s under action of $D$ parity, leading to formation of domain walls as topological pseudo-defects. The metastability of such walls can make them relatively long lived and contradict standard cosmology. Thus we are led to consider adding a nonrenormalisable Planck scale suppressed operator, that breaks $\mathrm{SO}(10)$ symmetry but preserves Standard Model symmetry. For a large range of right handed breaking scales $M_{R}$, this is shown to give rise to the required pressure difference to remove the domain walls without conflicting with consistent big bang nucleosynthesis (BBN) while avoiding gravitino overproduction. However, if the walls persist till the onset of weak (thermal) inflation, then a low $\sim 10-100$ TeV $M_{R}$ becomes problematic.

PACS Numbers:

I Introduction

All the elegant features of both $\mathrm{SO}(10)$ grand unified theory (GUT) and supersymmetric models can be incorporated in a Minimal Supersymmetric GUT (MSGUT). It naturally accommodates a heavy right handed neutrino in order to fill the $\mathbf{16}$ representation of $\mathrm{SO}(10)$ with matter fermions. Thus it naturally allows implementation of the seesaw mechanism Mohapatra:1980 which explains the small non-zero neutrino masses observed experimentally Fukuda:2001 . Supersymmetric models with left-right symmetric gauge groups ($\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$) naturally descend from $\mathrm{SO}(10)$ GUT. These models enjoy R-parity ($R=(-1)^{3(B-L)+2S}$) Mohapatra:1986 conservation, and so predict a stable Lightest Supersymmetric Particle (LSP) which is a promising dark matter candidate. While no signatures of low energy supersymmetry have been found at the LHC so far, supersymmetry continues to be an appealing mechanism for a consistent UV completion for the Standard Model (SM), and justifies the considerations here which relate to the effects of this model in the very early universe.

An important feature of many GUT’s is the occurrence of topological defects which can dynamically get formed in the early universe Kibble:1980 ; Kibble:1995 . Although the relevant covering group $\mathrm{Spin}(10)$ is simply connected, discrete symmetry arises at intermediate scales in several of the sequential breaking schemes of $\mathrm{SO}(10)$. The origin of this is the D-parity operation of the $\mathrm{Spin}(10)$ theory, which generates a discrete subgroup that becomes a $Z_{2}$ subgroup of the stability group in some breaking schemes. That such breaking schemes can give rise to metastable domain walls, and its implications for cosmology were studied by Kibble, Lazarides and Shafi Kibble:1982 . A wide class of possibilities for metastable defects of various dimensions were studied by Preskill and Vilenkin Preskill (1992). The main issue is that although the parent theory has a simply connected gauge group, the effective theory allows the formation of defects, which cannot unwind due to the large energy scale difference between the two sequential breakings. In all these cases quantum tunnelling remains a possible mechanism for the destabilisation of such defects due to simply connected nature of the original group.

In the supersymmetric case of $\mathrm{SO}(10)$ MSGUT a peculiar situation arises. The more conventional breaking scheme proceeds via Pati-Salam (PS) group times D-parity ($\mathrm{SU}(4)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}\times D$). This D-parity has the appealing consequence of allowing identical gauge couplings for the two $\mathrm{SU}(2)$’s allowing a chiral yet parity symmetric model of particle physics. The domain wall produced at the breaking of this $Z_{2}$ poses a challenge to cosmology but it could be addressed for example by adding a parity odd singlet Chang:1983fu or by linking the breaking of parity to the supersymmetry (SUSY) breaking Mishra:2009 . However the scheme of breaking $\mathrm{SO}(10)$ directly to SM proposed by Bajc et al. Bajc:2004 has an unusual feature that the unique electric charge conserving vacuum expectation value (vev) that preserves the SM gauge group and SUSY gives an alternate D-parity flipped vev that also conserves charge and preserves SUSY, but with $\mathrm{SU}(2)_{L}$ replaced by $\mathrm{SU}(2)_{R}$ as would be manifested by fermion charge assignment. The domain walls then separate not gauge equivalent vacua but subgroups that are mirror images of each other. Further, as was shown in some detail in Garg:2018 , there is no path that is F-flat or D-flat (in SUSY sense) that connects the two disjoint points of the coset space continuously. The resulting walls will be metastable more directly than those studied in Kibble:1982 and Preskill (1992), since the defects are subject to disintegration in the course of the same phase transition where they are formed. However depending on the nature of the phase transition and the rate of Hubble expansion, such defects begin to characterise robust local minima separated from the trivial vacuum by a substantial barrier which cannot be overcome once the Universe cools substantially below the scale of the phase transition. This class of defects was dubbed topological pseudo-defects Garg:2018 (TPD). Particularly, close to the Planck scale the expansion of the Universe would be too rapid and could compete with the time scale of their disappearance and lead to interesting consequences. Their persistence to later epochs of the Universe would be inconsistent with the current observed universe. Earlier works on the supersymmetric $\mathrm{SO}(10)$ GUT model studied here have not addressed this issue.

One way to remove these domain walls is to introduce Planck scale suppressed non-renormalizable operators Rai:1994 ; Lew:1993 that causes instability to domain walls. However this introduction of Planck suppressed operators may not work for all cases of SUSY and gauge symmetric models. In the Next to Minimal Supersymmetric Standard Model, the problem was found to persist Abel:1995 in the sense that the gauge hierarchy problem does not get addressed if the operators required to remove the domain walls are permitted. In the Supersymmetric Left-Right symmetric 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 before BBN. This requirement has been discussed in detail in Mishra:2010 in the context of R-parity conserving SUSYLR models Aulakh:1998 . Similar analysis was shown to place constraints also on R-parity violating SUSYLR models Borah:2011 .

We begin by discussion of discrete set of vacua degenerate with that signaled by the standard SM preserving vevs that are used in the first stage of breaking of $\mathrm{SO}(10)$ Bajc:2004 . Applying D-parity to them gives us a new set of supersymmetry preserving SM preserving vevs. The two sets of vevs are disconnected in the sense that they are separated by a potential barrier Garg:2018 . This fact leads to the danger of domain walls formed as topological pseudo-defects.

The aim of this paper is to study whether this degeneracy can be lifted using new operators in the superpotential. Since generic dimension 3 terms have all been used up in the superpotential, it is natural to search for non-renormalisable ones. We next argue that adding such $\mathrm{SO}(10)$ preserving terms to the superpotential will not lead to a pressure difference across any domain wall that can arise, and thus, cannot contribute to the required instability. This forces us to go beyond the cherished principle of exact $SO(10)$ gauge symmetry, in particular sacrifice some of its features related to ultraviolet (UV) completion. The mildest such violation could be considered along the lines of Gell-Mann GellMann:1961 and Okubo Okubo:1962 and add $\mathrm{SO}(10)$ breaking but SM preserving terms to the superpotential. In order to not touch the relevant and marginal operators, the leading new term in the superpotential could be of mass dimension $4$, suppressed by one power of the cutoff scale of the theory which we take to be the Planck scale $M_{Pl}$. We implement our idea by finding a coefficient matrix for the dimension four term drawn from the $\mathrm{SO}(10)$ group that commutes with SM gauge group embedded into $\mathrm{SO}(10)$. The systematically developed Pati-Salam embeddingAulakh:2005 proves to be a good computational aid for this purpose . This leads to a novel ultraviolet operator that can potentially cause sufficient instability and remove any domain wall that may arise.

Towards consistency check for such a scenario, we demand that the effects remain confined to UV limit approaching the Planck scale, and naturalness of such deviation from full $SO(10)$ is ensured by the coefficients of such terms remaining small. This naturalness test is carried out by considering the implications of such pressure difference to cosmology, specifically to ensuring that the DW disappear without conflicting with standard cosmology. This is done by testing these terms for three different scenarios of domain wall dynamics and their eventual removal during three possible kind of cosmological epochs, viz. radiation dominated era, early stage of matter dominated era and late stage of matter dominated era following a period of weak inflation. Domain wall removal during the same cosmological periods for the cases of left right symmetric supersymmetric models was earlier studied in Mishra:2010 and Borah:2011 . We calculate the conditions for successful domain wall removal during the three stated cosmological periods in our extended MSGUT model, leading to constraints on the scale of right hand symmetry breaking characterised by mass scale $M_{R}$. Generically it results in upper bounds on $M_{R}$, in fact forcing it to remain small compared to generic GUT scale.

The paper is organised as follows. In Sec. II we show the existence of $D$-flipped degenerate vacua, and the inadequacy of $SO(10)$ invariant terms in lifting this degeneracy. In sec. III we propose our strategy for overcoming the problem along the lines of octet dominance in old hadron flavour physics, and identify the required $SO(10)$ matrix. Section III.3 computes the effective potential terms arising from the new nonrenormalisable operator, with details in appendix A. In its subsection IV we impose the required conditions on the pressure difference terms to ensure consistent cosmology and obtain constraints on the scale of right handed symmetry breaking $M_{R}$. The final section V contains conclusions.

II MSGUT superpotential and the degenerate MSSM vacua

The heavy Higgs part of the renormalisable superpotential of $\mathrm{SO}(10)$ MSGUT is as follows Bajc:2004 :

[TABLE]

where $\Phi$ is the $4$-index anti-symmetric representation $\mathbf{210}$, $\Sigma$ is the $5$-index self-dual anti-symmetric representation $\mathbf{126}$, and $\overline{\Sigma}$ is the $5$-index anti-self-dual anti-symmetric representation $\mathbf{\overline{126}}$ of $\mathrm{SO}(10)$. In the above expression, all indices range independently from $1$ to $10$. Below, for brevity of notation, we shall use [math] to denote the index $10$ in our expressions.

In order to break $\mathrm{SO}(10)$ while preserving supersymmetry, we consider the following vevs. These vevs are uniquely determined by the requirement that they be singlet under the SM gauge group embedded into $\mathrm{SO}(10)$ via Pati-Salam and satisfy electric charge conservation Bajc:2004 .

[TABLE]

Remaining coordinates are set to zero in the vevs.

Let us consider the exchange of the $SU(2)_{L}$ and $SU(2)_{R}$ subgroups in this embedding. It amounts to the following element of $\mathrm{SO}(10)$ which achieves the desirable left-right (LR) flipping. (Aulakh:2005, , Equations 28,29):

[TABLE]

This flipping can be effectively obtained by utilising the $D$-parity operator defined to be,

[TABLE]

In the present embedding, it amounts to the following element of $\mathrm{SO}(10)$ (Aulakh:2005, , Equation 42):

[TABLE]

To see the detail, we note that the D-parity flipped vevs are as follows:

[TABLE]

On the other hand the LR-parity flipped vevs are :

[TABLE]

Observe that

[TABLE]

Thus D-parity mimics left right flipping in $\mathrm{SO}(10)$ MSGUT.

While this discrete symmetry is securely embedded in a compact group, the engineering of the superpotential required to obtain the MSSM encodes an accidental symmetry into the $F$ flatness conditions according to which for every choice of vev’s resulting in MSSM, there exists a set of $D$-flipped vev’s which satisfy the same $F$ flatness conditions. Further, as argued in Garg:2018 , there exist no flat directions connecting these mutually flipped set of vev’s. On the other hand a one parameter curve $U(1)_{D}$ generated by $D$ necessarily crosses a barrier in connecting the two sets of vev’s. Thus the vacuum manifold possesses accidental discrete symmetry, which can give rise to pseudo-topological defects due to causal structure of the evolving early Universe. Due to rapid cooling of the universe, such walls can remain metastable until destroyed by fluctuation and tunneling processes. The situation becomes analogous to transient DW arising from breaking of LR symmetry in certain SUSYLR models Mishra:2010 , although the walls are only metastable. In the LR case it was possible to assume Planck scale suppressed terms that break the discrete LR symmetry since gravity does not respect global symmetries. For a compact group $Spin(10)$ there is no consistent way to generate an energy imbalance between sectors related by a discrete symmetry operation which belongs to the group. This is why we shall need to investigate other methods for domain wall removal under the action of D-parity in $\mathrm{SO}(10)$ MSGUT.

We start this investigation by computing the F-terms evaluated at the original vevs as well as at the flipped vevs. D-flatness at the original and flipped vevs is ensured by utilising the condition $|\sigma|=|\overline{\sigma}|$ as in Bajc:2004 . The evaluations at the original vevs can also be found in (Bajc:2004, , Equations 6-9) and (AulakhGirdhar:2005, , Equations 20-23). Now we note that exactly the same values of $a$, $\omega$, $p$ ensure F-flatness at the original vevs as well as at the $D$-flipped vevs. The detail is shown in Table 1.

From table 1 we see that the $F$-terms evaluated at the original and $D$-parity flipped vevs are either the same, if the field value remains same after shifting, or negated if the field value gets negated after flipping. This is not a coincidence, but rather a consequence of the $SO(10)$-invariance of the superpotential where each index occurs exactly twice in a term. This is true even if we add higher degree $\mathrm{SO}(10)$ invariant non-renormalisable terms to the superpotential. So if $|\sigma|=|\overline{\sigma}|$, the scalar potential is the same whether evaluated at the original or the D-parity flipped vevs. Thus, a $\mathrm{SO}(10)$-invariant superpotential will never give a pressure difference. It becomes necessary to consider other alternatives.

III Non-renormalisable superpotential term

Since perfect $\mathrm{SO}(10)$ symmetry in the superpotential at and above GUT scale cannot destablise the non-topological domain wall, we need a ‘mild’ breaking of $\mathrm{SO}(10)$ symmetry. We are thus led to consider adding Planck suppressed degree four terms to the superpotential that depart from $\mathrm{SO}(10)$ invariance. Clearly, this must be done cautiously in order to affect only the UV end of the theory without sacrificing the desirable features of the low energy effective renormalisable theory. We do not speculate on the nature of the physics responsible for these higher order terms, but instead adopt a ‘minimal’ approach. Besides ensuring removal of the non-topological domain wall in the early universe, it will be nice if this mild SO(10) breaking provides some explanation also for the eventual disappearance of D-parity flipped particle physics in the low energy Universe. As will become clear below, our proposal for the non-renormalisable terms give the D-parity flipped vev slightly higher energy making them disfavoured. We show in this section how a ‘minimal’ $\mathrm{SO}(10)$ violating but SM preserving scenario can be systematically constructed as a generalisation of the Gell-Mann-Okubo “octet dominance hypothesis” from the eightfold way of flavour $SU(3)$. Indeed we take a more drastic step of adapting this to a true gauge symmetry. Thus, our full superpotential now mildly breaks SO(10) but still does not jeopardise SM.

III.1 Gell-Mann Okubo Formalism

We briefly recall the formalism of Gell-Mann GellMann:1961 and Okubo Okubo:1962 who independently gave an explanation for the mass splittings within several of the multiplets of flavour $SU(3)$, viz., the pseudoscalar mesons octet, baryon octet and baryon decuplet. (For the later more systematic developments see Gasser:1980sb ; Jenkins:1991ts ; Banerjee:1994bk ). Suppose one starts with exact flavour symmetry amongst the $u$, $d$ and $s$ quarks. In other words, we consider the global symmetry group $\mathrm{SU}(3)$. The pseudoscalar mesons are made up of these three quarks and can be arranged to form an octet $\Pi$ in the adjoint representation of $\mathrm{SU}(3)$. The Hamiltonian describing them at rest can be written as $H_{0}=\frac{\mu^{2}}{2}\mathrm{Tr}[\Pi^{\dagger}\Pi],$ where $\mu^{2}$ denotes their squared mass. This Hamiltonian is $\mathrm{SU}(3)$ symmetric and predicts that all the pseudoscalar mesons will have exactly the same mass $\mu$. However, this is not the case. This deviation from universal masses can be explained by postulating a perturbed Hamiltonian $H=H_{0}+H^{\prime},$ where the perturbation $H^{\prime}$ is given by $H^{\prime}=\frac{\alpha}{2}\mathrm{Tr}[\Phi^{\dagger}\Phi M].$ The $3\times 3$ ‘coefficient matrix’ $M$ is chosen so as to break $\mathrm{SU}(3)$ symmetry but preserve the lower energy isospin $\mathrm{SU}(2)$ symmetry. This is done by taking $M$ from the Lie algebra $\mathrm{su}(3)$ and requiring it to commute with $\mathrm{su}(2)$ embedded into $\mathrm{su}(3)$. This requirement fixes $M$ uniquely to be the Gell-Mann $\lambda_{8}$.

III.2 Choice of $SO(10)$ breaking matrix

Along similar lines we now consider the following extended superpotential:

[TABLE]

where

[TABLE]

Above $M^{\prime}$ is a linear operator acting on the column vector $\Phi$. We shall take $M^{\prime}$ to be the representation of a carefully chosen group element $M$ of $\mathrm{SO}(10)$. With a matrix $M$ in the notation of Sec. II, $M^{\prime}=M^{\otimes 4}$. We require that $M$ commute with all group elements $N\in\mathrm{SO}(10)$ that arise as images of the embedding of the SM group $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$ into $\mathrm{SO}(10)$. It is then clear that $W_{\mathrm{nr}}$ is SM-invariant but not necessarily $\mathrm{SO}(10)$ invariant.

We can now utilise the Pati-Salam embedding of SM group into $\mathrm{SO}(10)$ as given in (Aulakh:2005, , Section 2.2). It is easy to see that $\mathrm{SO}(10)$ matrices of the form

[TABLE]

commute with all $\mathrm{SO}(10)$ matrices $N$ that arise from the embedding of the SM group into $\mathrm{SO}(10)$. Above $K$ is an $\mathrm{SO}(4)$ matrix of the form $K=\exp(2\theta J)$, where

[TABLE]

$\theta$ is a real number between [math] and $2\pi$, $j_{1}$, $j_{2}$, $j_{3}$ are real numbers satisfying $j_{1}^{2}+j_{2}^{2}+j_{3}^{2}=1$, and

[TABLE]

are the three self-dual generators of $\mathrm{so}(4)$. Recall that the self-dual elements of $\mathrm{so}(4)$ form the embedding of $\mathrm{su}(2)_{R}$ into $\mathrm{so}(4)$. It can now be shown that $K$ is the image of the matrix $\exp(i\theta(j_{1}\tau_{1}+j_{2}\tau_{2}+j_{3}\tau_{3}))\in\mathrm{SU}(2)_{R}$ when embedded into $\mathrm{SO}(4)$.

The reason behind $M$ commuting with SM is as follows. Recall that $\mathrm{SU}(3)_{c}$ of SM gets embedded into the $\mathrm{SO}(6)$ part on the top left. The generator of $u(1)_{B-L}$ then maps to the $\mathrm{so}(10)$ matrix

[TABLE]

The $\mathrm{su}(2)_{L}$ of SM maps into the anti-self-dual generators and $\mathrm{su}(2)_{R}$ maps into the self-dual generators of $\mathrm{so}(4)$ in the bottom right. Since the self-dual generators of $\mathrm{so}(4)$ commute with the anti-self-dual generators, any linear combination of the three self-dual generators of $\mathrm{so}(4)$ will commute with $\mathrm{su}(2)_{L}$. We fix one such linear combination $J=j_{1}J_{1}^{+}+j_{2}J_{2}^{+}+j_{3}J_{3}^{+}\in\mathrm{so}(4),$ and map $\mathrm{u}(1)_{R}$ to it. This $J$ commutes with the embedding of $\mathrm{su}(3)_{c}\times\mathrm{u}(1)_{B-L}\times\mathrm{su}(2)_{L}\times\mathrm{u}(1)_{R}$ into $\mathrm{so}(10)$ and thus with the embedding of SM into $\mathrm{so}(10)$.

We shall take $\theta=\pi/2$, $j_{1}=j_{3}=\frac{1}{\sqrt{2}}$, $j_{2}=0$ to get

[TABLE]

This gives us

[TABLE]

Observe that the element $M\in\mathrm{SO}(10)$ does not commute with D-parity. In other words, the D-parity generator viz.

[TABLE]

is a broken symmetry generator in the non-renormalisable superpotential. This allows us the possibility of breaking the degeneracy between the two sets of vevs by means of the non-renormalisable term $W_{\mathrm{nr}}$ in the superpotential. If there exist basic reasons for the existence of such a term then there could be more terms of the same type of higher mass dimensions. However this leading term should suffice for our purpose.

III.3 The effective potential and domain wall removal

We next proceed to compute the effective potential in the two quasi symmetric vacua now split by the new non-renormalisable term. Observe that

[TABLE]

To keep the computation simple we note that the vev’s themselves will shift by $O(1/M_{\mathrm{Pl}})$, and thus the contribution to the effective potential from the change in the vev’s is $O(1/M^{4}_{\mathrm{Pl}})$ and can be ignored compared to the difference of $O(1/M^{2}_{\mathrm{Pl}})$ arising from the leading Planck suppressed terms. Thus we can use the vev’s already determined by Bajc et al. Bajc:2004 . In appendix A we have shown the evaluation of the resulting effective potential in this approximation. Since the scalar potentials evaluated at the $D$-flipped vevs are not the same, we get a pressure difference across the domain wall as follows

[TABLE]

From the above expression for the pressure difference, it is clear that $\omega$ is the controlling parameter for the difference in the scalar potential between the original and flipped vevs. For most ‘interesting’ regions of the parameter space, $\max\{a,p\}\gg\omega$ Bajc:2004 ; Aulakh:2004 ; AulakhGirdhar:2005 . For example, consider some ‘representative’ values

[TABLE]

taken from Equation 4 of Aulakh et al. Aulakh:2004 and Equation 57 of Aulakh and Girdhar AulakhGirdhar:2005 . This corresponds to $x=0.21$ and $\frac{\lambda M}{\eta m}=1.03$ in Equation 4 of Aulakh:2004 . The mass $m$ is set to the mass $M_{X}$ which is the lightest superheavy vector particle mediating proton decay, as mentioned in page 291 of AulakhGirdhar:2005 . For the above parameters, we take $m=10^{14}~{}\mathrm{GeV}/c^{2}$ using Fig. 4 and Equation 55 of Aulakh and Girdhar AulakhGirdhar:2005 . This gives us in units of $\mathrm{GeV}/c^{2}$

[TABLE]

leading to $a\gg p\geq\omega$. With these vevs, $\mathrm{SO}(10)$ MSGUT breaks down directly to MSSM. The assumption $\max\{a,p\}\gg\omega$ occurs for many other regions of parameter space breaking $\mathrm{SO}(10)$ MSGUT directly to MSSM, as well as for the case of one or two intermediate scales of symmetry breaking with both Pati-Salam and left-right symmetric MSSM as the intermediate gauge groups (Bajc:2004, , Section V).

In the following, we shall take $\max\{a,p\}\sim M_{X}$ and $\omega\sim M_{R}$, where $M_{R}$ is the (lower) energy scale at which a scalar field acquires a non-zero vev also setting the scale of domain wall tension. With this, the pressure difference becomes

[TABLE]

IV Constraint from cosmology

We now derive the constraint the parameter $b$ must satisfy if the non-renormalisable term were to be able to remove any domain walls that may have arisen in the course of $SO(10)$ breaking. Following Mishra and Yajnik Mishra:2010 , we consider the following three eras for domain wall appearance and removal.

IV.0.1 Evolution in radiation dominated universe

For most grand unified theories, barring the inflationary period, the universe is in a thermal state, and radiation dominated. The formation and evolution of domain walls in such a medium is a difficult problem, requiring numerical simulations. However, strong analytical arguments have been given and the essentials of this scenario was originally proposed by Kibble Kibble (1980) and Vilenkin Vilenkin (1985). We refer the reader to the original papers, or to a recapitulation in Mishra:2010 . Suppose the Domain walls arise at some temperature $T_{c}$, the critical temperature of a phase transition at which a scalar field $\phi$ acquires a non-zero vacuum expectation value. We shall be primarily interested in this scale being $M_{R}$, being the scale after which the Universe could accidentally have fallen into a $D$-flipped vacuum with $SU(2)_{L}$ of electroweak replaced by $\rightarrow$ $SU(2)_{R}$. The energy density trapped per unit area of the wall is $\sigma\sim M_{R}^{3}$. One can then argue that the required pressure across the walls to destabilise them must be

[TABLE]

where the second equation is obtained by substituting the relevant scales. combining this with Eqn. (19) gives

[TABLE]

IV.0.2 Evolution in matter dominated universe

The next two possibilities arise naturally in the context of removal moduli in superstring cosmology Coughlan et al. (1983); Banks et al. (1994); de Carlos et al. (1993) (see Kane:2015jia for a recent review). This issue has received fresh attention in Dutta:2014tya Cicoli:2016olq Bhattacharya:2017pws Bhattacharya:2017ysa Goswami:xyz , in view of the improved inputs from Cosmic Microwave Background data. When a modulus field begins to oscillate coherently it can be seen that near the minimum of the potential the oscillations are essentially harmonic, $\bar{KE}=\bar{PE}$, ie, making $p=T^{i}_{i}=KE-PE\approx 0$ where there is no summation on the generic space component index $i$ of the energy-momentum tensor. This is equivalent to a matter dominated era by the time the domain walls get destabilised.

The simpler possibility within this scenario is that the domain walls start decaying as soon as they dominate the energy density of the Universe. Thus no competition ensues between them and the oscillating moduli. This possibility was considered in Kawasaki and Takahashi (2005) by Kawasaki and Takahashi. The analysis begins by assuming that the initially formed wall complex in a phase transition rapidly relaxes to a few walls per horizon volume at an epoch characterized by Hubble parameter value $H_{i}$. Due to the assumption of almost simultaneous onset of degradation of domain walls, if the temperature at this particular epoch is $T_{D}$, then $H_{eq}^{2}\sim GT_{D}^{4}$ where

[TABLE]

from which we find that

[TABLE]

Putting together Eq.s (eq:Heq) and (24) we get,

[TABLE]

Now requiring $\delta\rho>T_{D}^{4}$ and substituting for $\sigma$ we get,

[TABLE]

Consequently, for $b$ this implies,

[TABLE]

IV.0.3 Evolution including weak inflation

The third possibility we consider is that both moduli fields as well the walls are present, and unlike the simplified case above, the latter do not disintegrate by the time they come to dominate the energy density of the Universe. A late period of exponential growth of the scale factor can occur due to some moduli frozen at zero value due to thermal corrections, later relaxing to their true minimum, when the universe also mildly reheats. This possibility, weak or thermal inflation, was considered Lyth and Stewart (1995),Lyth and Stewart (1996), with some recent developments reported in Hong:2015oqa ; Hayakawa:2015fga ; Cho:2017zkj . During the epoch when the weak inflation has ended and the moduli have begun to oscillate, they obey a matter equation of state. But the equation of state of for non-scaling domain walls is $p=-(2/3)\rho$, and the Friedmann scale factor evolves as $a(t)\propto t^{2}$. Then the walls dominate over the coherent moduli fields. Clearly this domination must last only for a limited epoch to result in acceptable cosmology.

Wall destabilisation mechanism in this case has not been widely pursued, and hence we recapitulate in detail the original reasoning of Mishra:2010 . The given situation is most likely in the case when the $\delta\rho$ is typically small, not large enough to destabilize the walls sufficiently quickly. The two kinds of dynamics interfering with the decay mechanism are rapid oscillations with large tension in the walls, of rapid bulk sweeping and friction with the medium. But eventually a small pressure difference will also win over the tension force because the walls straighten out, or the frictional force as the translational speed reduces drastically. Since we have no microscopic model for deciding which of these is finally responsible, we introduce a temperature scale $T_{D}$ at which the walls begin to experience instability. Note that unlike in the previous example, we will not be able to estimate $T_{D}$ in terms of other mass scales and will accept it as undetermined and consider a few reasonable values for it for our final estimate.

As has been studied above, at $H_{eq}$ the energy density of the domain wall network dominates energy density of the Universe. The scale factor at this epoch is characterized by $a_{eq}$. Denoting the energy density of the domain walls at the time of equality as $\rho_{DW}(t_{eq})$, the evolution of energy density can be written as,

[TABLE]

where $a_{d}$ is scale factor at the epoch of decay of domain wall corresponding to time $t_{d}$. If the domain walls decay at an epoch characterized by temperature $T_{D}$, then $\rho_{DW}(t_{d})\sim T_{D}^{4}$. So from Eq.(28),

[TABLE]

In the matter dominated era the energy density of the moduli fields scale as,

[TABLE]

Substituting the value of $a_{eq}/a_{d}$ from Eq.(29) in the above equation,

[TABLE]

Since the energy density of the domain walls dominates the universe after the time of equality, $\rho_{DW}(t_{d})>\rho^{d}_{mod}$. So the pressure difference across the domain walls when they start decaying is given by,

[TABLE]

where we have used the relation $H_{eq}^{2}\sim G\rho_{DW}(t_{eq})$. Replacing the value of $H_{eq}$ from Eq.(23), and $H_{i}^{2}\sim G\rho^{in}_{DW}\sim M_{R}^{4}/M_{Pl}^{2}$,

[TABLE]

For the case of weak inflation, Equations (19) and (33) lead to the following constraint on the parameter $b$

[TABLE]

IV.1 Constraints on $b$

This completes the discussion of the three scenarios of wall degradation. We now tabulate the constraints arising from the above inequalities, assuming $M_{X}=10^{16}$ viz. of the order of the GUT scale, and $M_{Pl}=10^{19}$. We consider three candidate values for $M_{R}$: a ‘low’ value of $10^{7}$ corresponding to non-thermal or resonant leptogenesis, an ‘intermediate’ value of $10^{9}$ consistent with thermal leptogenesis without gravitino overabundance, and a ‘large’ value of $10^{13}$. For the radiation dominated and matter dominated eras, we obtain constraints on $|b|$ in order to ensure successful domain wall removal. For the case of weak inflation, we assume $|b|\leq O(1)$ and instead calculate constraints on the ratio $T_{D}/M_{R}$ in order to ensure successful domain wall removal. This is because the energy scale $T_{D}$ where the walls first experience instability is unknown to us.

From Table 2, we see that our model is easily capable of ensuring domain wall removal in the radiation dominated and matter dominated eras without conflicting with existing experimental data. This is because the constraints on $|b|$ are very easy to meet; the minuscule symmetry breaking term is enough to cause instability in domain walls formed during the two eras. For the case of weak inflation, assuming $|b|\leq O(1)$, we get constraints on the ratio $T_{D}/M_{R}$. The ratio is required to be less than $10^{-3}$. For $T_{D}>10$MeV as required by BBN, this comfortably accommodates the $\gtrsim 10^{10}$GeV scale expected of thermal leptogenesis. However an $M_{R}$ scale of $10-100$TeV scale that may be pursued at future colliders is marginally acceptable.

V Conclusion

In this paper, we take the $\mathrm{SO}(10)$ based MSGUT studied extensively in earlier works and examine it from a cosmological angle. We have identified the occurrence of topological pseudo-defects which arise from applying the D-parity operator to the SUSY preserving vevs breaking $\mathrm{SO}(10)$ MSGUT. In addition to the intended left handed MSSM this operation gives a new set of SUSY preserving vevs exchanging $SU(2)_{L}$ with $SU(2)_{R}$. These two sets of vevs are disconnected in the sense that there is no SUSY preserving path connecting them in parameter space. We then remark that this leads to a danger of long lived metastable domain wall formation in the early universe especially close to the Planck scale. A similar conclusion has recently been made in Garg:2018 .

We then investigate conditions under which domain walls, if formed, can go away early enough so as not to conflict with standard cosmology. We first observe that domain walls cannot go away even with dimension four operators in the superpotential as long as the theory is $\mathrm{SO}(10)$ symmetric. We then postulate a non-renormalisable term that breaks of $\mathrm{SO}(10)$ symmetry while still preserving SM symmetry. While we are unable to argue why Planck scale effects should distinguish the two mirror image subgroups, we have proposed the least disruptive way of avoiding undesirable cosmological consequences, and keeping in mind the open ended nature of UV completion of known physics, we could say the theory is unified up to Planck scale effects, or only quasi-unified. Armed with this, we investigate domain wall removal in three scenarios of the early universe viz. radiation dominated era, matter dominated era which is generic to string theory inspired models, and a weak inflation era following matter domination.

Our main conclusions are as follows:

If domain walls are formed in the radiation dominated era, they can easily disappear in that era itself in our model. This is true for a wide range of energy scales of domain wall formation. The same is true if domain walls are formed within the epoch dominated by moduli which make the universe approximately matter dominated. They can then disappear promptly with aid from the softly $SO(10)$ breaking operators. The parameter characterising the non-renormalisable term is not constrained for a wide range of values of the domain wall energy scale in both of these scenarios. 2. 2.

If domain walls are formed within the moduli dominated era and continue on during the coherent oscillation phase weak inflation, then domain walls can eventually completely disappear only under a certain condition. The condition required by our model is that the temperature at which the walls become unstable be at least three to four orders of magnitude smaller than the energy density at which the walls first appear. This still allows consistency with thermal leptogenesis subject to tackling gravitino overabundance. However a $10-100$ TeV scale right handed sector becomes only marginally acceptable.

Thus this work invites further careful analysis of other cosmological implications of $\mathrm{SO}(10)$ MSGUT.

Appendix A

We begin by evaluating the new term, expressing it in terms of components,

[TABLE]

Evaluating $W_{\mathrm{nr}}$ at the original vevs, we get

[TABLE]

Evaluating $W_{\mathrm{nr}}$ at the flipped vevs, we get

[TABLE]

We now evaluate the partial derivatives of $\frac{1}{(4!)^{2}}\Phi^{T}M^{\otimes 4}\Phi$ with respect to the various fields.

[TABLE]

Evaluating the partial derivatives at vevs determined in Bajc:2004 , we get

[TABLE]

Evaluating the partial derivatives at the flipped vevs, we get

[TABLE]

We can now compute the contribution to the F-terms arising from $W_{\mathrm{nr}}$ for both types of vevs.

[TABLE]

[TABLE]

Since both the original vevs as well as the flipped vevs are F-flat at the renormalisable level for suitably chosen values of $a$, $\omega$, $p$, the above expressions are the complete values, including renormalisable and non-renormalisable contributions, of the respective F-terms. In other words,

[TABLE]

The F-term contribution to the scalar potential evaluated at the original and flipped vevs becomes

[TABLE]

This gives us the following expressions for F-term contributions to the scalar potential.

[TABLE]

Since we take $|\sigma|=|\overline{\sigma}|$, both the original and the flipped vevs are D-flat. Thus, the scalar potentials evaluated at the two types of vevs arise solely from F-term contributions.