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We present an $SO(10)$ model in which a dimension five operator induces kinetic mixing at the GUT scale between the abelian subgroups $U(1)_{B-L}$ and $U(1)_R$. We discuss in this framework gauge coupling unification and proton decay, as well as the appearance of superheavy quasistable strings with $G\mu \sim 10^{-8} - 10^{-5}$, where $\mu$ denotes the dimensionless string tension parameter. We use Bayesian analysis to show that for $G\mu$ values $\sim 4 \times 10^{-7} - 10^{-5}$, the gravitational wave spectrum emitted from the quasistable strings is in good agreement with the recent pulsar timing array data. Corresponding to $G \mu$ values $\sim 10^{-8} - 2 \times 10^{-7}$, proton decay is expected to occur at a rate accessible in the Hyper-Kamiokande experiment. Finally, we present the gravitational wave spectrum emitted by effectively stable strings with $G\mu\approx 10^{-8}$ that have experienced a certain amount of inflation. This can be tested with future detectors in the $\mu$Hz frequency range., Comment: 18 pages, 5 captioned figures

We discuss the appearance of topological structures from the spontaneous breaking of $E_6$ to the Standard Model via its maximal subgroup $SO(10) \times U(1)_\psi$. They include dumbbells, metastable strings, as well as domain walls bounded by necklaces. We provide a novel scenario for producing metastable strings based on the symmetry breaking $U(1)_\psi \longrightarrow Z_8 \longrightarrow Z_4$. The metastable string arises from the merger of $Z_8$ strings that bound a domain wall. An unbroken gauge $Z_2$ symmetry from $SO(10)$ breaking yields viable stable dark matter candidates as well as topologically stable strings. We discuss the gravitational wave emission from two varieties of cosmic strings, namely the superheavy metastable ones and the intermediate scale topologically stable cosmic strings., Comment: Minor comments included

We present a novel mechanism for producing topologically stable monopoles (TSMs) from the quantum mechanical decay of metastable cosmic strings in the early universe. In an $SO(10)$ model this mechanism yields TSMs that carry two units ($4\pi/e$) of Dirac magnetic charge as well as some color magnetic charge which is screened. For a dimensionless string tension parameter $G\mu \approx 10^{-9} - 10^{-5}$, the monopoles are superheavy with masses of order $10^{15} - 10^{17}$ GeV. Monopoles with masses of order $10^8 - 10^{14}$ GeV arise from metastable strings for $G\mu$ values from $\sim 10^{-22}$ to $10^{-10}$. We identify the parameter space for producing these monopoles at an observable level with detectors such as IceCube and KM3NeT. For lower $G\mu$ values the ultra-relativistic monopoles should be detectable at Pierre Auger and ANITA. The stochastic gravitational wave emission arises from metastable strings with $G\mu\sim 10^{-9}-10^{-5}$ and should be accessible at HLVK and future detectors including the Einstein Telescope and Cosmic Explorer. An $E_6$ extension based on this framework would yield TSMs from the quantum mechanical decay of metastable strings that carry three units ($6\pi/e$) of Dirac magnetic charge., Comment: 22 pages, 6 figures, 1 table, matches version accepted in JCAP

We show that, when connected with monopoles, the flat $D$-flat direction breaking the local $U(1)_{B-L}$ symmetry as an extension of the minimal supersymmetric standard model can be responsible for the signal of a stochastic gravitational wave background recently reported by NANOGrav collaborations, while naturally satisfying constraints at high frequency band. Thanks to the flatness of the direction, a phase of thermal inflation arises naturally. The reheating temperature is quite low, and suppresses signals at frequencies higher than the characteristic frequency set by the reheating temperature. Notably, forthcoming spaced-based experiments such as LISA can probe the cutoff frequency, providing an indirect clue of the scale of soft SUSY-breaking mass parameter., Comment: 10 pages, 2 figures, accepted for publication in JCAP

Motivated by the NANOGrav 15 year data and other recent investigations of stochastic gravitational background radiation based on pulsar timing arrays, we show how superheavy strings survive inflation but the slightly heavier monopoles do not in a non-supersymmetric hybrid inflation model based on flipped $SU(5)$. With the dimensionless string tension parameter $G \mu\sim 10^{-6}$, the gravitational wave spectrum emitted by the strings, which are metastable due to breaking caused by monopole-antimonopole quantum mechanical tunneling, is compatible with the latest NANOGrav measurement as well as the advanced LIGO-VIRGO third run data. The string network undergoes about 30 $e$-foldings of inflation which suppresses the spectrum in the LIGO-VIRGO frequency range. With the symmetry breaking chain $SU(5) \times U(1)_X \to SU(3)_c \times SU(2)_L\times U(1)_Z \times U(1)_X \to SU(3)_c \times SU(2)_L \times U(1)_ Y$, the estimated proton lifetime is of order $10^{36}-10^{37}$ yrs., Comment: Revised version accepted for publication in JCAP

Composite topological structures such as superheavy "quasi-stable strings" (QSS) and "walls bounded by strings" (WBS) arise in realistic extensions of the Standard Model of high energy physics. We show that the gravitational radiation emitted in the early universe by these two unstable structures with a dimensionless string tension $G\mu\approx 10^{-6}$ is consistent with the NANOGrav evidence of low frequency gravitational background as well as the recent LIGO-VIRGO constraints, provided the superheavy strings and monopoles experience a certain amount of inflation. For the case of walls bounded by strings, the domain walls arise from the spontaneous breaking of a remnant discrete gauge symmetry around the electroweak scale. The quasi-stable strings, on the other hand, arise from a two step breaking of a local gauge symmetry. The monopoles appear from the first breaking and get connected to strings that arise from the second breaking. Both composite structures decay by emitting gravitational waves over a wide frequency range. The Bayes factors for QSS and WBS relative to the inspiraling supermassive black hole binaries are estimated to be about 60 and 30 respectively, which are comparable with that of metastable strings and cosmic superstrings., Comment: 6 pages, 1 table, 4 captioned figures, double column

We explore the gravitational wave spectrum generated by string-wall structures in an $SO(10)$ ($Spin(10)$) based scenario of pseudo-Goldstone boson dark matter (pGDM) particle. This dark matter candidate is a linear combination of the Standard Model (SM) singlets present in the 126 and 16 dimensional Higgs fields. The Higgs $126$-plet vacuum expectation value (VEV) $\left<126_H\right>$ leaves unbroken the $\mathbb{Z}_2$ subgroup of $\mathbb{Z}_4$, the center of $SO(10)$. Among other things, this yields topologically stable cosmic strings with a string tension $\mu \sim \left<126_H\right>^2$. The subsequent (spontaneous) breaking of $\mathbb{Z}_2$ at a significantly lower scale by the $16$-plet VEV $\left<16_H\right>$ leads to the appearance of domain walls bounded by the strings produced earlier. We display the gravitational wave spectrum for $G \mu$ values varying between $10^{-15}$ and $10^{-9}$ ($\left<126_H\right>\sim 10^{11}$ - $10^{14}$ GeV), and $\left<16_H\right>\sim 0.1$ - $10^2$ TeV range ($G$ denotes Newton's constant.) These predictions can be tested, as we show, by a variety of (proposed) experiments including LISA, ET, CE and others., Comment: 8 pages, 3 figures, version published in PLB

We discuss some testable predictions of a non-supersymmetric $SO(10)$ model supplemented by a Peccei-Quinn symmetry. We utilize a symmetry breaking pattern of $SO(10)$ that yields unification of the Standard Model gauge couplings, with the unification scale also linked to inflation driven by an $SO(10)$ singlet scalar field with a Coleman-Weinberg potential. Proton decay mediated by the superheavy gauge bosons may be observable at the proposed Hyper-Kamiokande experiment. Due to an unbroken $Z_2$ gauge symmetry from $SO(10)$, the model predicts the presence of a stable intermediate mass fermion which, together with the axion, provides the desired relic abundance of dark matter. The model also predicts the presence of intermediate scale topologically stable monopoles and strings that survive inflation. The monopoles may be present in the Universe at an observable level. We estimate the stochastic gravitational wave background emitted by the strings and show that it should be testable in a number of planned and proposed space and land based experiments. Finally, we show how the observed baryon asymmetry in the Universe is realized via non-thermal leptogenesis., Comment: 36 pages, 9 figures, version published in JCAP

We discuss how in $SO(10)$ grand unification an observable number density of topologically stable intermediate mass ( $\sim 10^{14}$ GeV) monopoles survive inflation driven by a Coleman-Weinberg potential and non-minimal coupling of the inflaton field to gravity. The scalar spectral index $n_s$ is in excellent agreement with the current observations, and the tensor to scalar ratio $r\gtrsim 0.003$. The model also predicts the presence of intermediate scale topologically stable cosmic strings, and their gravitational wave spectrum reflects the amount of cosmic inflation experienced by the associated symmetry breaking. The discovery of these primordial monopoles and the stochastic gravitational wave background from the strings would provide important new insights regarding the symmetry breaking patterns in the early universe., Comment: 21 pages, 10 figures, matches version published in JCAP

Inspired by the recent determination of the $W$-boson mass by the CDF collaboration, we revisit an $SO(10)$ axion model in which a scalar $SU(2)_L$ triplet field with zero hypercharge is known to acquire a non-zero VEV through its mixing with the Standard Model Higgs doublet. The triplet VEV provides a sizable contribution to the $W$ mass, which helps in significantly lowering the $7\sigma$ discrepancy between the Standard Model prediction and the higher CDF value for $m_W$. We show that the relatively light triplet mass ($\sim (1-50)$ TeV) is compatible with gauge coupling unification and observable proton decay. An unbroken $Z_2$ gauge symmetry, coupled with the presence of two fermionic $10$-plets required to resolve the axion domain wall problem, means that both axions and a stable intermediate mass ($\sim 10^9-10^{10}$ GeV) fermion are plausible dark matter candidates. We also display the gravitational wave spectrum from the intermediate scale topologically stable cosmic strings predicted by the model., Comment: 20 pages, 6 figures, published version

We estimate the stochastic gravitational wave spectrum emitted from a network of cosmic strings in which the latter are effectively stable against breaking by monopole pair creation. The monopoles are produced at a higher scale from an earlier symmetry breaking and experience significant inflation before reentering the horizon. This gives rise to monopole-antimonopole pairs connected by string segments and the string loop formation essentially ceases. As a consequence, the lower frequency portion of the gravitational wave spectrum is suppressed relative to the no-inflation case with stable strings, which evades the stringent PPTA bound on the dimensionless string tension $G\mu$. We display the modified spectrum, accessible in the ongoing and future experiments, for $G\mu$ values in the range $10^{-10} - 10^{-15}$. We show how this `quasi-stable' string network is realized in realistic grand unified theories., Comment: 18 pages, 7 figures, published version

We investigate the impact of Coleman-Weinberg inflation on the stochastic gravity wave background spectrum emitted by intermediate scale cosmic strings. The string network is partially inflated and re-enters the horizon at later times after the end of inflation, such that the short string loops are not produced. This leads to a significant modification of the gravity wave spectrum that we explore in detail. We find that Coleman-Weinberg inflation can help to satisfy the Parkes Pulsar Timing Array (PPTA) bound for dimensionless string tension values in the range $G\mu > 1.1\times 10^{-10}$. We also identify the modified gravity wave spectra which, in the case of inflation, are compatible with the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) data. We then discuss the formation of monopoles and strings at the same breaking scale and the compatibility of the Monopole, Astrophysics and Cosmic Ray Observatory (MACRO) bound with the PPTA bound, and also with the NANOGrav data. Finally, an example of a realistic non-supersymmetric $E_6$ model incorporating successful Coleman-Weinberg inflation is presented in which monopoles and strings both survive inflation and are present at an observable level., Comment: 8 pages, Two columns, 12 figures, Version to appear in PRD

We consider magnetic monopoles and strings that appear in non-supersymmetric $SO(10)$ and $E_6$ grand unified models paying attention to gauge coupling unification and proton decay in a variety of symmetry breaking schemes. The dimensionless string tension parameter $G\mu$ spans the range $10^{-6}-10^{-30}$, where $G$ is Newton's constant and $\mu$ is the string tension. We show how intermediate scale monopoles with mass $\sim 10^{13}-10^{14}$ GeV and flux $\lesssim 2.8\times 10^{-16}$ ${\mathrm{cm}^{-2}\mathrm{s}^{-1}\mathrm{sr}^{-1}}$, and cosmic strings with $G\mu \sim 10^{-11}-10^{-10}$ survive inflation and are present in the universe at an observable level. We estimate the gravity wave spectrum emitted from cosmic strings taking into account inflation driven by a Coleman-Weinberg potential. The tensor-to-scalar ratio $r$ lies between $0.06$ and $0.003$ depending on the details of the inflationary scenario., Comment: 38 pages, 9 figures, Published version

We present a framework of dark- and visible-sector unification in the E6 embedding of the standard model. The demand for consistently getting the standard model leads to the existence of the dark-sector. We show that the hierarchy of vevs typifying unified models leads to multicomponent dark matter at the IR. The symmetry breaking itself categorises the matter content into dark- and visible-sector particles, the categorisation being uniform across different breaking chains. We discuss the stability of the dark matter particles and compare them to existing phenomenological models of dark matter. The central results follow from symmetry and hierarchy arguments. We present an indicative set of models of gauge coupling unification, to show that the framework can be embedded in realistic models of E6., Comment: Minor changes, typos fixed. 32 pages, 4 figures

We calculate the proton lifetime and discuss topological defects in a wide class of non-supersymmetric (non-SUSY) $SO(10)$ and $E(6)$ Grand Unified Theories (GUTs), broken via left-right subgroups with one or two intermediate scales (a total of 9 different scenarios with and without D-parity), including the important effect of threshold corrections. By performing a goodness of fit test for unification using the two-loop renormalisation group evolution equations (RGEs), we find that the inclusion of threshold corrections significantly affects the proton lifetime, allowing several scenarios, which would otherwise be excluded, to survive. Indeed we find that the threshold corrections are a saviour for many non-SUSY GUTs. For each scenario we analyse the homotopy of the vacuum manifold to estimate the possible emergence of topological defects., Comment: 56 pages, 12 captioned figures, 16 tables

We perform a detailed study of the grand unified theories $SO(10)$ and $E(6)$ with left-right intermediate gauge symmetries of the form $SU(N)_L\otimes SU(N)_R \otimes \mathcal{G}$. Proton decay lifetime constrains the unification scale to be $ \gtrsim 10^{16}$ GeV and, as discussed in this paper, unwanted cosmological relics can be evaded if the intermediate symmetry scale is $ \gtrsim 10^{12}$ GeV. With these conditions, we study the renormalization group evolution of the gauge couplings and do a comparative analysis of all possible left-right models where unification can occur. Both the D-parity conserved and broken scenarios, as well as the supersymmetric (SUSY) and Non-supersymmetric (Non-SUSY) versions, are considered. In addition to the fermion and scalar representations at each stage of the symmetry breaking, contributing to the $\beta$-functions, we list the intermediate left-right groups which successfully meet these requirements. We make use of the dimension-5 kinetic mixing effective operators for achieving unification and large intermediate scale. A significant result in the supersymmetric case is that to achieve successful unification for some breaking patterns, the scale of SUSY breaking needs to be at least a few TeV. In some of these cases, the intermediate scale can be as low as $\sim 10^{12}$ GeV, for SUSY scale to be $\sim 30$ TeV. This has important consequences in the collider searches for SUSY particles and phenomenology of the lightest neutralino as dark matter., Comment: 80 pages, 21 captioned figures, version accepted in PRD

In an earlier paper we have proposed a novel method to compute the decay width for a general $1\to n$ cascade decay where the propagators are off-shell and may be of different spins. Here, we extend our algorithm to accommodate those decays that are mediated by more than one such cascades. This generalizes our prescription and widens its applicability. We compute the three- and four-body toy decay chains where identical final states appear through different cascades. Here, we also provide the algorithm to calculate the interference terms. For four-body decays we discuss both symmetric and asymmetric cascades, providing the expressions for the detailed phase space structure in each case. We find that the results obtained with this algorithm have a very impressive agreement with those from standard softwares using a sophisticated Monte Carlo based phase space integration., Comment: 21 pages, 11 figures, pdflatex