Pati-Salam Research Articles

Pati-Salam models with A4 modular symmetry

The flavor structure of quarks and leptons and quark-lepton unification are studied in the framework of Pati-Salam models with A4 modular symmetry. The three generations of the left-handed and right-handed fermions are assigned to be triplet or singlets of A4. The light neutrino masses are generated through the type-I seesaw mechanism. We perform a systematic classification of Pati-Salam models according to the transformations of matter fields under the A4 modular symmetry, and the general form of the fermion mass matrix is given. We present four phenomenologically viable benchmark models which provide excellent descriptions of masses and flavor mixing of quarks and leptons, including neutrinos. In such models we find that the normal ordered neutrino mass spectrum is preferred over the inverted case, with neutrinoless double beta decay predicted to be too small to be observed by the next generation of experiments.

Revisiting a Realistic Intersecting D6-Brane with Modified Soft SUSY Terms

Because there are a few typos in the supersymmetry-breaking sfermion masses and trilinear soft term, regarding the current Large Hadron Collider (LHC) and dark matter searches, we revisit a three-family Pati–Salam model based on intersecting D6-branes in Type IIA string theory on a T6/(Z2×Z2) orientifold with a realistic phenomenology. We study the viable parameter space and discuss the spectrum consistent with the current LHC Supersymmetry searches and the dark matter relic density bounds from the Planck 2018 data. For the gluinos and first two generations of sfermions, we observe that the gluino mass is in the range [2, 14] TeV, the squarks mass range is [2, 13] TeV and the sleptons mass is in the range [1, 5] TeV. We achieve the cold dark matter relic density consistent with 5σ Planck 2018 bounds via A-funnel and coannihilation channels such as stop–neutralino, stau–neutralino, and chargino–neutralino. Except for a few chargino–neutralino coannihilation solutions, these solutions satisfy current nucleon-neutralino spin-independent and spin-dependent scattering cross-sections and may be probed by future dark matter searches.

Falsifying Pati-Salam models with LIGO

Falsifying Pati-Salam models with LIGO

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.

Finite naturalness and quark-lepton unification

We study the implications of finite naturalness in Pati-Salam models where SU(3)C is embedded in SU(4). For the minimal realization at low scale of quark-lepton unification, which employs the inverse seesaw for neutrino masses, we find that radiative corrections to the Higgs boson mass are at least δmh2/mh2∼O(104). The one-loop contributions to the Higgs mass are suppressed by four powers of the hypercharge gauge coupling. We find that for the vector leptoquarks the naively leading part of the two-loop corrections cancel. We assume the Dirac Yukawa couplings for neutrinos are equal to the up-type quark Yukawa couplings as predicted in the minimal theory for quark-lepton unification. Despite these findings, the two-loop corrections still dominate the finite naturalness bound. We mention a way to relax the lower bound on the vector leptoquark mass and have δmh2/mh2∼O(102). Published by the American Physical Society 2024

Gravitational waves from metastable cosmic strings in the Pati-Salam model in light of new pulsar timing array data

A series of pulsar timing arrays (PTAs) recently observed gravitational waves at the nanohertz frequencies. Motivated by this remarkable result, we present a novel class of Pati-Salam models that give rise to a network of metastable cosmic strings, offering a plausible explanation for the observed PTA data. Additionally, we introduce a hybrid inflationary scenario to eliminate magnetic monopoles that arise during the subsequent phase transitions from the Pati-Salam symmetry to the Standard Model gauge group. The resulting scalar spectral index is compatible with Planck data, and the tensor-to-scalar ratio is anticipated to be extremely small. Moreover, we incorporate a nonthermal leptogenesis to generate the required baryon asymmetry in our framework. Finally, the gravitational wave spectra generated by the metastable cosmic strings not only correspond to signals observed in recent PTAs, including NANOGrav, but are also within the exploration capacity of both present and future ground- and space-based experiments. Published by the American Physical Society 2024

Representation of Fermions in the Pati-Salam Model

Representation of Fermions in the Pati-Salam Model

The hidden sector variations in the 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 three-family Pati–Salam models from intersecting D6-branes

It is well-known that three-family supersymmetric Pati–Salam models from intersecting D6-branes, where either one or both of the 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} gauge factors are replaced by 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, are quite scarce. In order to construct all such kind of models with generic additional gauge symmetries, we fix the observable sectors and study all the possible hidden sectors. Thus, we are able to completely determine all types of such kind of the inequivalent models on a 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 from IIA string theory. We find the gauge coupling relations to be highly sensitive to the variations in the hidden sector. One of the models exhibits the gauge coupling unification for a particular solution at the string scale. In addition, we perform scan on the hidden sector variations for the models presented in arXiv:2112.09632, whose gauge coupling relations are still preserved. Thus, we fix the gap in the previous study and complete the model building for all the inequivalent 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.

Unification based on the mysterious cubic-structure grouping of quarks and leptons

We present a unification model based on the well-known but mysterious cubic-structure grouping of quarks and leptons that suggests an underlying symmetry connection deemed explainable by a unified theory. It results in an extension of the Pati-Salam model that consolidates the fermions into two sixteen-dimensional chiral representations of the gauge group. Moreover, the discrete cubic flavor symmetry arbitrarily conjectured in the literature is found here to be implied by the gauge symmetry. Furthermore, the gauge algebra also describes an eleven-dimensional spacetime decomposed into the usual spacetime and the seven-sphere, which is the manifold of unit octonions. This suggests such a spacetime is apt for embracing all elementary particles and their interactions.

Dimension-five baryon-number violation in low-scale Pati-Salam models

The gauge bosons of Pati-Salam do not mediate proton decay at the renormalisable level, and for this reason it is possible to construct scenarios in which $SU(4) \otimes SU(2)_{R}$ is broken at relatively low scales. In this Letter we show that such low-scale models generate dimension-5 operators that can give rise to nucleon decays at unacceptably large rates, even if the operators are suppressed by the Planck scale. We find an interesting complementarity between the nucleon-decay limits and the usual meson-decay constraints. Furthermore, we argue that these operators are generically present when the model is embedded into $SO(10)$, lowering the suppression scale. Under reasonable assumptions, the lower limit on the breaking scale can be constrained to be as high as $\mathcal{O}(10^{8})$ GeV.

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.

Implications of the zero 1-3 flavour mixing hypothesis: predictions for {theta}_{23}^{mathrm{PMNS}} and δPMNS

We revisit mixing sum rule relations in the lepton and quark sectors under the assumption that the 1-3 elements of the flavour mixing matrices ( {V}_L^u , {V}_L^d , {V}_L^e , {V}_L^{nu } ) are zero in the flavour basis. We consider the exact relations resulting from the validity of this “zero 1-3 flavour mixing hypothesis” and analyse their implications based on the current experimental data, including effects from RG running. In particular, we analyse how the existing precise measurement of {theta}_{13}^{mathrm{PMNS}} allows to derive predictions for {theta}_{23}^{mathrm{PMNS}} in models with constrained {theta}_{12}^{mathrm{e}} . As examples, we calculate the predictions for {theta}_{23}^{mathrm{PMNS}} which arise in 12 classes of Pati-Salam models and SU(5) GUTs that relate {theta}_{12}^{mathrm{e}} to {theta}_{12}^{mathrm{d}} . We also derive a novel “lepton phase sum rule”, valid under the additional assumption of small charged lepton mixing contributions. We furthermore point out that, in the context of GUT flavour models, the quark and lepton CP violating phases δCKM and δPMNS can both be predicted from a single imaginary element in the mass matrices.

Gauge enhanced quantum criticality beyond the standard model

Standard lore ritualizes our quantum vacuum in the four-dimensional spacetime (4D) governed by one of the candidate Standard Models (SMs), while lifting towards some grand unification-like structure (GUT) at higher energy scales. In contrast, in our work, we introduce an alternative view that the SM is a low energy quantum vacuum arising from various neighbor vacua competition in an immense quantum phase diagram. In general, we can regard the SM arising near the gapless quantum criticality (either critical points or critical regions) between the competing neighbor vacua. In particular detail, we demonstrate how the $su(3)\ifmmode\times\else\texttimes\fi{}su(2)\ifmmode\times\else\texttimes\fi{}u(1)$ SM with 16n Weyl fermions arises near the quantum criticality between the GUT competition of Georgi-Glashow $su(5)$ and Pati-Salam $su(4)\ifmmode\times\else\texttimes\fi{}su(2)\ifmmode\times\else\texttimes\fi{}su(2)$. We propose two enveloping toy models. Model I is the conventional $so(10)$ GUT with a Spin(10) gauge group plus GUT-Higgs potential inducing various Higgs transitions. Model II modifies model I by adding a new 4D discrete torsion class of Wess-Zumino-Witten-like term built from GUT-Higgs field [which matches a nonperturbative global mixed gauge-gravity anomaly captured by a 5D invertible topological field theory ${w}_{2}{w}_{3}(TM)={w}_{2}{w}_{3}({V}_{\mathrm{SO}(10)})$], which manifests a beyond-Landau-Ginzburg quantum criticality between Georgi-Glashow and Pati-Salam models, with extra beyond the Standard Model excitations emerging near a hypothetical quantum critical region. If the internal symmetries were treated as global symmetries (or weakly coupled to probe background fields), we suggest a particular low-energy realization of model II as an analogous gapless 4D deconfined quantum criticality with new beyond the SM fractionalized fragmentary excitations of color-flavor separation, and gauge enhancement including a dark gauge force sector, altogether requiring a double fermionic Spin structure named DSpin. If the internal symmetries are dynamically gauged (as they are in our quantum vacuum), we suggest the model II's 4D criticality as a boundary criticality such that only appropriately gauge enhanced dynamical GUT gauge fields can propagate into an extradimensional 5D bulk. The phenomena may be regarded as SM deformation or ``morphogenesis.'' Although our derivation is based on kinematics and global anomaly matching constraints between UV parent and various candidate IR field theories, the constraints are still highly subtle and suggestive. Future verifications on the IR dynamics will be desirable.

Three-generation super no-scale models in heterotic superstrings

We derive the conditions for the one-loop contribution to the cosmological constant to be exponentially suppressed in a class of heterotic string compactifications with three generations of chiral matter, where supersymmetry is spontaneously broken à la Scherk-Schwarz. Using techniques of partial unfolding based on the Hecke congruence subgroup Γ0(2) of the modular group to extract the leading asymptotics, we show that the super no-scale condition nB=nF between the degeneracies of massless bosons nB and fermions nF in the full string spectrum is necessary but not sufficient, and needs to be supplemented by additional conditions which we identify. We use these results to construct three-generation Pati–Salam models with interesting phenomenological characteristics.

Importance of vector leptoquark-scalar box diagrams in Pati-Salam unification with vector-like families

We study lepton flavor violation (LFV) induced by one-loop box diagrams in Pati-Salam (PS) unification with vector-like families. The vector leptoquark (LQ) associated with the PS gauge symmetry breaking generally causes various LFV processes such as KL → μe and μ → e conversion at the tree-level, thereby driving its mass scale to be higher than PeV scale. The vector-like families are introduced to suppress such tree-level LFV processes, allowing the LQ to have TeV scale mass. In this paper, we point out that there are inevitable one-loop contributions to those LFV processes from the box diagrams mediated by both one LQ and one scalar field, even if the tree-level contributions are suppressed. We consider a concrete model for demonstration, and show that the vector-like fermion masses have an upper bound for a given LQ mass when the one-loop induced processes are consistent with the experimental limits. The vector-like fermion mass should be lighter than 3 TeV for 20 TeV LQ, if a combination of the couplings does not suppress KL → μe decay. Our findings would illustrate importance of the box diagrams involving both LQ and physical modes of symmetry breaking scalars in TeV scale vector LQ models.

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.

Sensitivity of lepton flavor violation process B+ → K+μ−τ+ in future experiments

The upcoming upgrades of Large Hadron Collider (LHC) to High Luminosity LHC (HL-LHC) and the proposed Future Circular Collider (FCC-hh) offer an opportunity of observing the Lepton Flavor Violating (LFV) process beyond the Standard Model. Monte Carlo simulation of generating and detecting an LFV process [Formula: see text] on CMS at LHC (Run-3), HL-LHC, and FCC-hh reference detector (FRD) is performed. This process is favored by the leptoquark models, particularly the three-site Pati–Salam model but disfavored by the gauged [Formula: see text] model, both of which are introduced to address the [Formula: see text] anomalies. We study possible backgrounds with similar final states and use Multi-Variable Analysis (MVA) to discriminate signal events from possible backgrounds. The MVA output is used to estimate Single Event Sensitivity (SES) of this process, yielding encouraging results on CMS at HL-LHC and FRD at FCC-hh with considerable SES values of about [Formula: see text] (CMS, LHC Run-3), [Formula: see text] (CMS, HL-LHC) and [Formula: see text] (FRD, FCC-hh), suggesting it’s promising to probe such process in the future high energy physics experiments and constrain the new physics models more rigorously.