Supersymmetric Pati-Salam Model Research Articles

Three-family supersymmetric Pati–Salam models with symplectic groups from intersecting D6-branes

We construct new three-family N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {N}}}=1$$\\end{document} supersymmetric Pati–Salam models from intersecting D6-branes with original gauge group U(4)C×USp(2)L×U(2)R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{U}(4)_C \ imes \ extrm{USp}(2)_L \ imes \ extrm{U}(2)_R$$\\end{document} on a Type IIA T6/(Z2×Z2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^6/({\\mathbb {Z}}_2\ imes {\\mathbb {Z}}_2)$$\\end{document} orientifold. We find that replacing a U(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{U}(2)$$\\end{document} with a USp(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{USp}(2)$$\\end{document} group severely restricts the number of three-generation supersymmetric models such that there are only five inequivalent models. Exchanging the left and right sectors, we obtain five dual models with gauge group U(4)C×U(2)L×USp(2)R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{U}(4)_C \ imes \ extrm{U}(2)_L \ imes \ extrm{USp}(2)_R$$\\end{document}. These ten models have different gauge coupling relations at string scale. The highest wrapping number is 4, and one of the models contains no filler O6-planes. Moreover, we discuss in detail the particle spectra, the composite particles through strong coupling dynamics, and the exotic particle decouplings. Also, we study how to realize the string-scale gauge coupling relations in some models.

String-scale gauge coupling relations in the supersymmetric Pati-Salam models from intersecting D6-branes

We have constructed all the three-family mathcal{N} = 1 supersymmetric Pati-Salam models from intersecting D6-branes, and obtained 33 independent models in total. But how to realize the string-scale gauge coupling relations in these models is a big challenge. We first discuss how to decouple the exotic particles in these models. In addition, we consider the adjoint chiral mulitplets for SU(4)C and SU(2)L gauge symmetries, the Standard Model (SM) vector-like particles from D6-brane intersections, as well as the vector-like particles from the mathcal{N} = 2 subsector. We show that the gauge coupling relations at string scale can be achieved via two-loop renormalization group equation running for all these supersymmetric Pati-Salam models. Therefore, we propose a concrete way to obtain the string-scale gauge coupling relations for the generic intersecting D-brane models.

The final model building for the supersymmetric Pati–Salam models from intersecting D6-branes

All the possible three-family {mathcal N}=1 supersymmetric Pati–Salam models constructed with intersecting D6-branes from Type IIA orientifolds on T^6/(mathbb {Z}_2times mathbb {Z}_2) are recently presented in arXiv:2112.09632. Taking models with largest wrapping number 5 and approximate gauge coupling unification at GUT scale as examples, we show string scale gauge coupling unification can be realized through two-loop renormalization group equation running by introducing seven pairs of vector-like particles from {mathcal N}=2 sector. The number of these introduced vector-like particles are fully determined by the brane intersection numbers while there are two D6-brane parallel to each other along one two-torus. We expect this will solve the gauge coupling unification problem in the generic intersecting brane worlds by introducing vector-like particles that naturally included in the {mathcal N}=2 sector.

The complete search for the supersymmetric Pati-Salam models from intersecting D6-branes

We construct a systematic method to build all the possible three-family mathcal{N} = 1 supersymmetric Pati-Salam models from Type IIA orientifolds on \U0001d54b6/(ℤ2× ℤ2) with intersecting D6-branes, in which the SU(4)C× SU(2)L× SU(2)R gauge symmetry can be broken down to the SU(3)C× SU(2)L× U(1)Y Standard Model gauge symmetry by the D-brane splitting and supersymmetry preserving Higgs mechanism. This is essentially achieved by solving all the common solutions for three generation conditions, RR tadpole cancellation conditions, and mathcal{N} = 1 supersymmetry conditions with deterministic algorithm. We find that there are 202752 possible supersymmetric Pati-Salam models in total, and show that there are only 33 independent models with different gauge coupling relations at GUTs scale after modding out equivalent relations, such as T-dualities, etc. In particular, there is one and only one independent model which has gauge coupling unification. Furthermore, one can construct other types of intersecting D-brane models utilizing such deterministic algorithm, and therefore we suggest a brand new method for D-brane model building.

Four-family supersymmetric Pati–Salam models from intersecting D6-branes

We investigate the construction of four-family supersymmetric Pati–Salam models from Type IIA orientifold with intersecting D6-branes. Utilizing the deterministic algorithm introduced in He W, Li T and Sun R (2021 arXiv: hep-th/2112.09632), we obtain 274 types of models with three rectangular tori and distinct gauge coupling relations at string scale, while 6 types of models with two rectangular tori and one tilted torus. In both cases, there exists a class of models with gauge coupling unification at string scale. In particular, for the models with two rectangular tori, one tilted torus and gauge coupling unification, the gaugino condensations are allowed, and thus supersymmetry breaking and moduli stabilization are possible for further phenomenological study.

The supersymmetry breaking soft terms, and fermion masses and mixings in the supersymmetric Pati-Salam model from intersecting D6-branes

A supersymmetric Pati-Salam model with wrapping number equal to 5 has been constructed in Type IIA orientifolds on T6/(ℤ2× ℤ2) with intersecting D6-branes recently. In particular, the string-scale gauge coupling unification can be achieved due to the intermediate-scale vector-like particles from mathcal{N} = 2 sector. We calculate the supersymmetry breaking soft terms, and study the Standard Model (SM) fermion masses and mixings. There are nine pairs of Higgs doublets from mathcal{N} = 2 sector. Interestingly, we can explain the SM quark masses and mixings, as well as the charged leptons masses from three point and four-point Yukawa interactions. Moreover, we calculate the supersymmetry breaking soft terms in a previous model with gauge coupling unification since we find a typo in the previous study.

Qualities of the axion and LSP in Pati-Salam unification with Z4R×ZN symmetry

In this paper we construct supersymmetric Pati-Salam (PS) models containing the minimal supersymmetric standard model and an invisible axion. The models include two discrete symmetries, ${\mathbb{Z}}_{4}^{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$, which maintain the quality of the accidental Peccei-Quinn (PQ) symmetry and thus the solution to the strong $CP$ problem. We require that the discrete anomaly conditions are satisfied for both ${\mathbb{Z}}_{4}^{R}\ifmmode\times\else\texttimes\fi{}{G}_{\mathrm{PS}}^{2}$ and ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{G}_{\mathrm{PS}}^{2}$. The vacuum expectation value of the PQ field spontaneously breaks all the discrete symmetries. R-parity is violated if any of the PQ field(s) has an odd charge under ${\mathbb{Z}}_{4}^{R}$. We present two explicit models which we refer to as a minimal model where R-parity violation is extremely suppressed, and a nonminimal model where R-parity violation is significant. In the latter model, the neutralino becomes unstable even if it is the lightest supersymmetric particle (LSP), and, in addition, there are new low-energy vectorlike states. In both examples, R-parity violation is sufficiently suppressed such that the proton is stable.

Inflation from a no-scale supersymmetric SU(4)C×SU(2)L×SU(2)R model

We study inflation in a supersymmetric Pati-Salam model driven by a potential generated in the context of no-scale supergravity. The Pati-Salam gauge group $SU(4)_{c}\times{SU(2)_{L}\times{SU(2)_{R}}}$ , is supplemented with a $Z_{2} $ symmetry. Spontaneous breaking via the $SU(4)$ adjoint leads to the left-right symmetric group. Then the $SU(2)_{R}$ breaks at an intermediate scale and the inflaton is a combination of the neutral components of the $SU(2)_{R}$ doublets. We discuss various limits of the parameter space and we show that consistent solutions with the cosmological data for the a spectral index $n_{s}$ and the tensor-to-scalar ratio r are found for a wide range of the parameter space of the model. Regarding the latter, which is a canonical measure of primordial gravity waves, we find that $r\sim{10^{-3}-10^{-2}}$. An alternative possibility where the adjoint scalar field $S$ has the role of the inflaton is also discussed.

The importance of being rigid: D6-brane model building on [formula omitted] with discrete torsion

Model building with rigid D6-branes on the Type IIA orientifold on T6/Z2×Z6′ with discrete torsion is considered. The systematic search for models of particle physics is significantly reduced by proving new symmetries among different lattice orientations. Suitable rigid D6-branes without matter in adjoint and symmetric representations are classified, and SO(2N) and USp(2N) gauge factors on orientifold invariant D6-branes are distinguished in terms of their discrete Wilson line and displacement parameters. Constraints on the non-existence of exotic matter prohibit global completions of local MSSM and left–right symmetric models, while globally defined supersymmetric Pati–Salam models are found. For the latter, only one particle generation possesses perturbative Yukawa couplings. Masses for the mild amount of exotic matter and the role of Abelian symmetries are briefly discussed.Last but not least, it is shown that for all three two-torus volumes of about the same order of magnitude, gauge coupling unification at one loop can be achieved, while for highly unisotropic choices a low string scale in the TeV range is compatible with the observed strengths of gauge and gravitational couplings.

Non-minimal Higgs inflation and non-thermal leptogenesis in a supersymmetric Pati-Salam model

We consider a supersymmetric (SUSY) Grand Unified Theory (GUT)based on the gauge group GPS = SU(4)C × SU(2)L × SU(2)R, which incorporates non-minimalchaotic inflation, driven by a quartic potential associated withthe Higgs fields involved in the spontaneous breaking of GPS.The inflationary model relies on renormalizable superpotentialterms and does not lead to overproduction of magnetic monopoles.It is largely independent of the one-loop radiative correctionsand can become consistent with the current observational data onthe inflationary observables, with the symmetry breaking scale ofGPS assuming its SUSY value. Within our model, the strongCP and the μ problems of the minimal SUSY standard model canbe resolved via a Peccei-Quinn symmetry. Moreover baryogenesisoccurs via non-thermal leptogenesis realized by theout-of-equilibrium decay of the right-handed neutrinos, which areproduced by the inflaton's decay. We consider two versions of sucha scenario, assuming that the inflaton decays to the lightest orto the next-to-lightest right-handed neutrino. Both scenaria canbecome compatible with the constraints arising from the baryonasymmetry of the universe, the gravitino limit on the reheatingtemperature and the upper bound on the light neutrino masses,provided that the gravitino is somehow heavy. In the secondscenario, extra restrictions from the SU(4)C GUT symmetryon the heaviest Dirac neutrino mass and the data on theatmospheric neutrino oscillations can be also met.

Consistency of perturbativity and gauge coupling unification

We investigate constraints that the requirements of perturbativity and gauge coupling unification impose on extensions of the standard model and of the minimal supersymmetric standard model. In particular, we discuss the renormalization group running in several supersymmetric left-right symmetric and Pati-Salam models and show how the various scales appearing in these models have to be chosen in order to achieve unification. We find that unification in the considered models occurs typically at scales below ${M}_{\mathrm{B\ensuremath{\llap{\not\;}}}}^{\mathrm{min}}={10}^{16}\text{ }\text{ }\mathrm{GeV}$, implying potential conflicts with the nonobservation of proton decay. We emphasize that extending the particle content of a model in order to push the grand unified theory scale higher or to achieve unification in the first place will very often lead to nonperturbative evolution. We generalize this observation to arbitrary extensions of the standard model and of the minimal supersymmetric standard model and show that the requirement of perturbativity up to ${M}_{\mathrm{B\ensuremath{\llap{\not\;}}}}^{\mathrm{min}}$, if considered a valid guideline for model building, severely limits the particle content of any such model, especially in the supersymmetric case. However, we also discuss several mechanisms to circumvent perturbativity and proton decay issues, for example, in certain classes of extra dimensional models.

FLUX PARAMETER SPACES IN TYPE II VACUA

We study the flux parameter spaces for semirealistic supersymmetric Pati–Salam models in the AdS vacua on Type IIA orientifold and realistic supersymmetric Pati–Salam models in the Minkowski vacua on Type IIB orientifold. Because the fluxes can be very large, we show explicitly that there indeed exists a huge number of semirealistic Type IIA and realistic Type IIB flux models. In the Type IIA flux models, in the very large flux limit, the theory can become weakly coupled and the AdS vacua can approach to the Minkowski vacua. In a series of realistic Type IIB flux models, at the string scale, the gauge symmetry can be broken down to the Standard Model gauge symmetry, the gauge coupling unification can be achieved naturally, all the extra chiral exotic particles can be decoupled, and the observed Standard Model fermion masses and mixings can be obtained as well. In particular, the real parts of the dilaton, Kähler moduli, and the unified gauge coupling are independent of the very large fluxes. The very large fluxes only affect the real and/or imaginary parts of the complex structure moduli, and/or the imaginary parts of the dilaton and Kähler moduli. However, these semirealistic Type IIA and realistic Type IIB flux models cannot be populated in the string landscape.

Semishifted hybrid inflation withB−Lcosmic strings

We discuss a new inflationary scenario that is realized within the extended supersymmetric Pati-Salam model which yields an acceptable $b$-quark mass for universal boundary conditions and $\ensuremath{\mu}g0$ by modestly violating Yukawa unification and leads to new shifted, new smooth, or standard-smooth hybrid inflation. Inflation takes place along a ``semishifted'' classically flat direction on which the $\mathrm{U}(1{)}_{\mathrm{B}\ensuremath{-}\mathrm{L}}$ gauge group remains unbroken. After the end of inflation, $\mathrm{U}(1{)}_{\mathrm{B}\ensuremath{-}\mathrm{L}}$ breaks spontaneously and a network of local cosmic strings, which contribute a small amount to the curvature perturbation, is produced. We show that, in minimal supergravity, this semishifted inflationary scenario is compatible with a recent fit to data which uses field-theory simulations of a local string network. Taking into account the requirement of gauge unification, we find that, for spectral index ${n}_{\mathrm{s}}=1$, the predicted fractional contribution of strings to the temperature power spectrum at multipole $\ensuremath{\ell}=10$ is ${f}_{10}\ensuremath{\simeq}0.039$. Also, for ${f}_{10}=0.10$, which is the best-fit value, we obtain ${n}_{\mathrm{s}}\ensuremath{\simeq}1.0254$. Spectral indices lower than about 0.98 are excluded and blue spectra are slightly favored. Magnetic monopoles are not formed at the end of semishifted hybrid inflation.

MinimalE6supersymmetric standard model

We propose a Minimal E6 Supersymmetric Standard Model (ME6SSM) which allows Planck scale unification, provides a solution to the ? problem and predicts a new Z'. Above the conventional GUT scale MGUT ~ 1016 GeV the gauge group corresponds to a left-right symmetric Supersymmetric Pati-Salam model, together with an additional U(1)? gauge group arising from an E6 gauge group broken near the Planck scale. Below MGUT the ME6SSM contains three reducible 27 representations of the Standard Model gauge group together with an additional U(1)X gauge group, consisting of a novel and non-trivial linear combination of U(1)? and two Pati-Salam generators, which is broken at the TeV scale by the same singlet which also generates the effective ? term, resulting in a new low energy Z' gauge boson. We discuss the phenomenology of the new Z' gauge boson in some detail.

Standard-smooth hybrid inflation

We consider the extended supersymmetric Pati-Salam model which, for $\ensuremath{\mu}g0$ and universal boundary conditions, succeeds to yield experimentally acceptable $b$-quark masses by moderately violating Yukawa unification. It is known that this model can lead to new shifted or new smooth hybrid inflation. We show that a successful two-stage inflationary scenario can be realized within this model based only on renormalizable superpotential interactions. The cosmological scales exit the horizon during the first stage of inflation, which is of the standard hybrid type and takes place along the trivial flat direction with the inflaton driven by radiative corrections. Spectral indices compatible with the recent data can be achieved in global supersymmetry or minimal supergravity by restricting the number of e-foldings of our present horizon during the first inflationary stage. The additional e-foldings needed for solving the horizon and flatness problems are naturally provided by a second stage of inflation, which occurs mainly along the built-in new smooth hybrid inflationary path appearing right after the destabilization of the trivial flat direction at its critical point. Monopoles are formed at the end of the first stage of inflation and are, subsequently, diluted by the second stage of inflation to become utterly negligible in the present universe for almost all (for all) the allowed values of the parameters in the case of global supersymmetry (minimal supergravity).

New smooth hybrid inflation

We consider the extension of the supersymmetric Pati-Salam model which solves the b-quark mass problem of supersymmetric grand unified models with exact Yukawa unification and universal boundary conditions and leads to the so-called new shifted hybrid inflationary scenario. We show that this model can also lead to a new version of smooth hybrid inflation based only on renormalizable interactions provided that a particular parameter of its superpotential is somewhat small. The potential possesses valleys of minima with classical inclination, which can be used as inflationary paths. The model is consistent with the fitting of the three-year Wilkinson microwave anisotropy probe data by the standard power-law cosmological model with cold dark matter and a cosmological constant. In particular, the spectral index turns out to be adequately small so that it is compatible with the data. Moreover, the Pati-Salam gauge group is broken to the standard model gauge group during inflation and, thus, no monopoles are formed at the end of inflation. Supergravity corrections based on a nonminimal Kaehler potential with a convenient choice of a sign keep the spectral index comfortably within the allowed range without generating maxima and minima of the potential on the inflationary path. So, unnatural restrictions onmore » the initial conditions for inflation can be avoided.« less

Planck scale unification in a supersymmetric Standard Model

We show how gauge coupling unification near the Planck scale Mp∼1019 GeV can be achieved in the framework of supersymmetry, facilitating a full unification of all forces with gravity. Below the conventional GUT scale MGUT∼1016 GeV physics is described by a Supersymmetric Standard Model whose particle content is that of three complete 27 representations of the gauge group E6. Above the conventional GUT scale the gauge group corresponds to a left–right symmetric Supersymmetric Pati–Salam model, which may be regarded as a “surrogate SUSY GUT” with all the nice features of SO(10) but without proton decay or doublet–triplet splitting problems. At the TeV scale the extra exotic states may be discovered at the LHC, providing an observable footprint of an underlying E6 gauge group broken at the Planck scale. Assuming an additional low energy U(1)X gauge group, identified as a non-trivial combination of diagonal E6 generators, the μ problem of the MSSM can be resolved.

Supersymmetric Pati–Salam models from intersecting D6-branes: a road to the Standard Model

We provide a systematic construction of three-family N = 1 supersymmetric Pati–Salam models from Type IIA orientifolds on T 6 / ( Z 2 × Z 2 ) with intersecting D6-branes. All the gauge symmetry factors SU ( 4 ) C × SU ( 2 ) L × SU ( 2 ) R arise from the stacks of D6-branes with U ( n ) gauge symmetries, while the “hidden sector” is specified by USp ( n ) branes, parallel with the orientifold planes or their Z 2 images. The Pati–Salam gauge symmetry can be broken down to the SU ( 3 ) C × SU ( 2 ) L × U ( 1 ) B − L × U ( 1 ) I 3 R via D6-brane splittings, and further down to the Standard Model via D- and F-flatness preserving Higgs mechanism from massless open string states in a N = 2 subsector. The models also possess at least two confining hidden gauge sectors, where gaugino condensation can in turn trigger supersymmetry breaking and (some) moduli stabilization. The systematic search yields 11 inequivalent models: 8 models with less than 9 Standard Model Higgs doublet-pairs and 1 model with only 2 Standard Model Higgs doublet-pairs, 2 models possess at the string scale the gauge coupling unification of SU ( 2 ) L and SU ( 2 ) R , and all the models possess additional exotic matters. We also make preliminary comments on phenomenological implications of these models.

Lepton flavour violation from Yukawa operators, supergravity and the see-saw mechanism

We investigate the phenomenological impact of different sources of lepton flavour violation arising from realistic models based on supergravity mediated supersymmetry breaking with Yukawa operators. We discuss four distinct sources of lepton flavour violation in such models: minimum flavour violation, arising from neutrino masses and the see-saw mechanism with renormalisation group (RG) running; supergravity flavour violation due to the non-universal structure of the supergravity model; flavour violation due to Froggatt–Nielsen (FN) fields appearing in Yukawa operators developing supersymmetry breaking F-terms and contributing in an non-universal way to soft trilinear terms; and finally heavy Higgs flavour violation arising from the heavy Higgs fields used to break the unified gauge symmetry which also appear in Yukawa operators and behave analogously to the FN fields. In order to quantify the relative effects, we study a particular type I string inspired model based on a supersymmetric Pati–Salam model arising from intersecting D-branes, supplemented by a U(1) family symmetry.

Global analysis of a supersymmetric Pati-Salam model

We perform a complete global phenomenological analysis of a realistic string-inspired model based on the supersymmetric Pati-Salam $SU(4)\times SU(2)_L \times SU(2)_R$ gauge group supplemented by a U(1) family symmetry, and present predictions for all observables including muon $g-2$, $\tau \to \mu \gamma$, and the CHOOZ angle. Our analysis demonstrates the compatibility of such a model with all laboratory data including charged fermion masses and mixing angles, LMA MSW and atmospheric neutrino masses and mixing angles, and $b \to s \gamma$, allowing for small deviations from third family Yukawa unification. We show that in such models the squark and slepton masses may be rather light compared to similar models with exact Yukawa unification.