The "landscape" of Pati--Salam heterotic superstring vacua
J. Rizos
备注：
4 pages, 2 figues, Presented at "School and Workshops on Cosmology and Strings: Theory-Cosmology-Phenomenology" (CORFU2009), September 6-13, Corfu, Greece
The main aspects of a recently developed method for classification of heterotic superstring vacua in the Free Fermionic Formulation are presented. Using these techniques we classify a big number of approximately $10^{15}$ heterotic string vacua with Pati--Salam, SU(4)x{SU(2)}_Lx{SU(2)}_R gauge symmetry with respect to their main phenomenological features as the number of families, Pati--Salam breaking Higgs, Standard Model Higgs doublets, additional triplets and exotic charge states. We identify an interesting subclass of these vacua, approximately one to one million, whose massless spectrum is completely free of fractionally charge states.
We extend the classification of free fermionic heterotic-string models to vacua in which the SO(10) GUT symmetry is broken at the string level to the Pati-Salam subgroup. Using our classification method we recently presented the first example of a quasi--realistic heterotic-string vacuum that is free of massless exotic states. Within this method we are able to derive algebraic expressions for the generalised GSO projections for all sectors that appear in the models. This facilitates the programming of the entire spectrum analysis in a computer code. The total number of vacua in the class of mo…
We extend the classification of fermionic $\mathbb{Z}_2\times\mathbb{Z}_2$ heterotic string orbifolds to non--supersymmetric Pati--Salam (PS) models in two classes of vacua, that we dub $\tilde S$--models and $S$--models. The first correspond to compactifications of a tachyonic ten--dimensional vacuum, whereas the second correspond to compactifications of the ten--dimensional tachyon--free $SO(16)\times SO(16)$ heterotic string. In both cases we develop a systematic method to extract tachyon--free four--dimensional models. We show that tachyon--free configurations arise with probability $\sim0…
We analyse the phenomenology of an exemplary exophobic Pati-Salam heterotic string vacuum, in which no exotic fractionally charged states exist in the massless string spectrum. Our model also contains the Higgs representations that are needed to break the gauge symmetry to that of the Standard Model and to generate fermion masses at the electroweak scale. We show that the requirement of a leading mass term for the heavy generation, which is not degenerate with the mass terms of the lighter generations, places an additional strong constraint on the viability of the models. In many models a top …
We construct and classify heterotic models where $\mathcal N=1$ supersymmetry is spontaneously broken à la Scherk-Schwarz. The theories contain chiral matter transforming under an observable Pati-Salam gauge symmetry and are classified according to a minimal set of phenomenological requirements, including the shape of their one-loop effective potential. The latter is essentially controlled by two parameters that are easily computable in terms of the defining string data. By imposing Bose-Fermi degeneracy at the massless level, known as the super no-scale condition, it is possible to suppress t…
Employing new scanning methods, we revisit the systematic construction of three-family $N=1$ supersymmetric Pati-Salam models in Type IIA orientifolds on ${\bf T}^6/(\mathbb Z_2\times \mathbb Z_2)$ with intersecting D6-branes. Arising from the stacks of D6-branes with $U(n)$ gauge symmetries, the Pati-Salam gauge symmetries $SU(4)_C\times SU(2)_L \times SU(2)_R$ can be broken down to the Standard Model via D-brane splitting as well as D- and F-flatness preserving Higgs mechanism. Also, the hidden sector contains $USp(n)$ branes, which are parallel with the orientifold planes or their ${\mathbb…
We analyze the running at one-loop of the gauge couplings in the spectral Pati-Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario's for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different scenarios and establish for all of them unification of the Pati-Salam gauge couplings. The boundary conditions are set by the usual RG flow for the Standard Model couplings at an intermediate mass scale at which the Pati-Salam symmetry is broken.
There is a U(4')\times U(4)-bundle on four-dimensional square root Lorentz manifold. Then a Pati-Salam model in curved space-time (Lagrangian) and a gravity theory (Lagrangian) are constructed on square root Lorentz manifold based on self-parallel transportation principle. An explicit formulation of Sheaf quantization on this square root Lorentz manifold is shown. Sheaf quantization is based on superposition principle and construct a linear Sheaf space in curved space-time. The transition amplitude in path integral quantization is given which is consistent with Sheaf quantization. All particle…
One or more scalar leptoquarks with masses around a few TeV may provide a solution to some of the flavor anomalies that have been observed. We discuss the impact of such new degrees on baryon number violation when the theory is embedded in a Pati-Salam model. The Pati-Salam embedding can suppress renormalizable and dimension-five baryon number violation in some cases. Our work extends the results of Assad, Grinstein, and Fornal who considered the same issue for vector leptoquarks.
We discuss the role of the pseudo-Riemannian structure of the finite spectral triple for the family of Pati-Salam models. We argue that its existence is a very restrictive condition that separates leptons from quarks, and restricts the whole family of Pati-Salam models into the class of generalized Left-Right Symmetric Models.
We search for the three-generation standard-like and/or Pati-Salam models from the $SO(32)$ heterotic string theory on smooth, quotient complete intersection Calabi-Yau threefolds with multiple line bundles, each with structure group $U(1)$. These models are S- and T-dual to intersecting D-brane models in type IIA string theory. We find that the stable line bundles and Wilson lines lead to the standard model gauge group with an extra $U(1)_{B-L}$ via a Pati-Salam-like symmetry and the obtained spectrum consists of three chiral generations of quarks and leptons, and vector-like particles. Green…
Motivated by orbifold grand unified theories, we construct a class of three-family Pati-Salam models in a Z6 abelian symmetric orbifold with two discrete Wilson lines. These models have marked differences from previously-constructed three-family models in prime-order orbifolds. In the limit where one of the six compactified dimensions (which lies in a Z2 sub-orbifold) is large compared to the string length scale, our models reproduce the supersymmetry and gauge symmetry breaking pattern of 5d orbifold grand unified theories on an S1/Z2 orbicircle. We find a horizontal 2+1 splitting in the chir…
We analyze the compatibility of the unified left-right symmetric Pati-Salam models motivated by non-commutative geometry and the TeV scale right-handed W boson suggested by recent LHC data. We find that the unification/matching conditions place conflicting demands on the symmetry breaking scales and that generating the required W_R mass and coupling is non-trivial.
We explore the potential of ultimate unification of the Standard Model matter and gauge sectors into a single $E_8$ superfield in ten dimensions via an intermediate Pati-Salam gauge theory. Through a consistent realisation of a $\mathbb{T}^6/(\mathbb{Z}_6\times \mathbb{Z}_2)$ orbifolding procedure accompanied by the Wilson line breaking mechanism and Renormalisation Group evolution of gauge couplings, we have established several benchmark scenarios for New Physics that are worth further phenomenological exploration.
We show how exotic stringy instantons can generate an effective interaction between color diquark sextets in a Pati-Salam model, inducing a Majorana mass term for the neutron. In particular, we discuss a simple quiver theory for a Pati-Salam like model, as an example in which the calculations of exotic instantons' effects are simple and controllable. We discuss some different possibilities in order to generate $n-\bar{n}$ oscillation testable in the next generation of experiments, Majorana mass matrices for neutrini and a Post-Sphaleron Baryogenesis scenario. Connections with Dark Matter issue…
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 $Γ_0(2)$ of the modular group to extract the leading asymptotics, we show that the super no-scale condition $n_B=n_F$ between the degeneracies of massless bosons $n_B$ and fermions $n_F$ in the full string spectrum is necessary but not sufficient, and n…
We show that it is possible to use gravitational wave detectors to observe the occurrence of a first order phase transition in Pati-Salam extensions of the Standard Model. We show that the peak frequency of the expected gravitational wave signals ranges within $0.1-10$ Hz. We find amusing that the next generation of gravity waves detectors are able to explore time honoured extensions of the Standard Model occurring at energy scales inaccessible by present and future particle physics accelerators.
We investigate the construction of four-family ${\cal N}=1$ supersymmetric Pati-Salam models from Type IIA $\mathbb{T}^6/(\mathbb{Z}_2 \times \mathbb{Z}_2)$ orientifold with intersecting D6-branes. Utilizing the deterministic algorithm introduced in Ref. \cite{heCompleteSearchSupersymmetric2021}, 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 titled torus. In both cases, there exists a class of models with gauge coupling unification at string scale. In particular, for…
We demonstrate the existence of heterotic-string vacua that are free of massless exotic fields. The need to break the non-Abelian GUT symmetries in k=1 heterotic-string models by Wilson lines, while preserving the GUT embedding of the weak-hypercharge and the GUT prediction sin^2θ_w(M(GUT))=3/8, necessarily implies that the models contain states with fractional electric charge. Such states are severely restricted by observations, and must be confined or sufficiently massive and diluted. We construct the first quasi-realistic heterotic-string models in which the exotic states do not appear in t…
Classification of Left-Right Symmetric (LRS) heterotic-string vacua in the free fermionic formulation, using random generation of generalised GSO (GGSO) projection coefficients, produced phenomenologically viable models with probability $4\times 10^{-11}$. Extracting substantial number of phenomenologically viable models requires modification of the classification method. This is achieved by identifying phenomenologically amenable conditions on the Generalised GSO projection coefficients that are randomly generated at the $SO(10)$ level. Around each of these fertile cores we perform a complete…
All the possible three-family ${\cal N}=1$ supersymmetric Pati-Salam models constructed with intersecting D6-branes from Type IIA orientifolds on $T^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ 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 ${\cal N}=2$ sector. The number of these introduced vector-like particles ar…
Three-Family Supersymmetric Pati-Salam Flux Models from Rigid D-Branes
College of Physics and Optoelectronic Engineering, Shenzhen University（深圳大学物理与光电工程学院）；School of Physics, University of Electronic Science and Technology of China（电子科技大学物理学院）；School of Physics, Henan Normal University（河南师范大学物理学院）
This note aims to give a self-contain and detail explanation about U(k')times U(k) Pati-Salam model in curved space-time which derived from (1+n)-dimensional square root Lorentz manifold by self-parallel transportation principle. The concepts foundation of manifold from view point of category theory, fiber bundle and sheaf theories are reviewed. There are extra U(k')times U(k)-principal bundle and U(k)-associated bundle than (1+n)-dimensional Lorentz manifold. The conservative currents on square root Lorentz manifold are discussed preliminary. A detail proof of relation from sheaf quantization…
We explore the possibility of embedding the Pati-Salam model in the context of Type I brane models. We study a generic model with U(4)_C x U(2)_L x U(2)_R gauge symmetry and matter fields compatible with a Type I brane configuration. Examining the anomaly cancellation conditions of the surplus abelian symmetries we find an alternative hypercharge embedding that is compatible with a low string/brane scale of the order of 5-7 TeV, when the U(4)_C and U(2)_R brane stack couplings are equal. Proton stability is assured as baryon number is associated to a global symmetry remnant of the broken abeli…
A supersymmetric Pati-Salam model with wrapping number equal to 5 has been constructed in Type IIA orientifolds on $\mathbf{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ 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 mixing…
We construct a systematic method to build all the possible three-family ${\cal N}=1$ supersymmetric Pati-Salam models from Type IIA orientifolds on $\mathbb{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ with intersecting D6-branes, in which the $SU(4)_C\times SU(2)_L \times SU(2)_R $ gauge symmetry can be broken down to the $SU(3)_C \times SU(2)_L \times U(1)_Y$ 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 the RR tadpole cancellation conditions, ${\cal N}=1$ supersymmetry cond…
Motivated by possible scalar-leptoquark explanations of the recently reported $B$-decay anomalies, we investigate whether the required leptoquarks can be accommodated within models based on noncommutative geometry (NCG). The models considered have the gauge structure of Pati-Salam models, $SU(4)\times SU(2)_L\times SU(2)_R$, with gauge coupling unification at a single scale. In one of the models, we find a unique scalar leptoquark with quantum numbers $(3,1,-\frac{1}{3})_{321}$, originating from a complex multiplet $(6,1,1)_{422}$, which can potentially explain the $B$-decay anomalies if its m…
We provide an asymptotically safe Pati-Salam embedding of the Standard Model. Safety is achieved by adding to the theory gauged vector-like fermions and by employing recently developed large number-of-flavor techniques and results. We show that the gauge, scalar quartic and Yukawa couplings achieve an interacting ultraviolet fixed point below the Planck scale. The minimal model is a relevant example of a Standard Model extension in which unification of all type of couplings occurs because of a dynamical principle, i.e. the presence of an ultraviolet fixed point. This extension differs from the…
We report some recent progress towards classification of phenomenologically appealing heterotic string models in the Free Fermionic Formulation. We focus on a class of Z2xZ2 models with SO(10) space-time gauge symmetry and study their main phenomenological aspects. We further consider reducing the gauge symmetry to the Pati-Salam gauge group SU(4)XSU(2)_LxSU(2)_R and impose a series of phenomenological constraints including the existence of gauge symmetry breaking Higgs particles and the elimination of exotic fractionally charged states.
We construct an MSSM-like model via Pati-Salam from intersecting D-branes in Type IIA theory on the $\Z_2 \times \Z_2'$ orientifold where the D-branes wrap rigid 3-cycles. Because the 3-cycles are rigid, there are no extra massless fields in the adjoint representation, arising as open-string moduli. The presence of these unobserved fields would create difficulties with asymptotic freedom as well as the prediction of gauge unification. The model constructed has four generations of MSSM matter plus right-handed neutrinos, as well as additional vector-like representations. In addition, we find th…
We provide a systematic construction of three-family N=1 supersymmetric Pati-Salam models from Type IIA orientifolds on $\IT^6/(\IZ_2\times \IZ_2)$ with intersecting D6-branes. All the gauge symmetry factors $SU(4)_C\times SU(2)_L \times SU(2)_R$ 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 ${\bf Z_2}$ images. The Pati-Salam gauge symmetry can be broken down to the $SU(3)_C\times SU(2)_L\times U(1)_{B-L} \times U(1)_{I_{3R}}$ via D6-brane splittings, and further down to …
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, $\mathbb{Z}_4^R \times \mathbb{Z}_N$, 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 $\mathbb{Z}_4^R \times G_{\rm PS}^2$ and $ \mathbb{Z}_N \times G_{\rm PS}^2$. The vacuum expectation value of the PQ field spontaneously breaks all the discrete symmetries. R-parity i…
We argue that the ordinary commutative-and-associative algebra of spacetime coordinates (familiar from general relativity) should perhaps be replaced, not by a noncommutative algebra (as in noncommutative geometry), but rather by a Jordan algebra (leading to a framework which we term "Jordan geometry"). We present the Jordan algebra (and representation) that most nearly describes the standard model of particle physics, and we explain that it actually describes a certain (phenomenologically viable) extension of the standard model: by three right-handed (sterile) neutrinos, a complex scalar fiel…
The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece express…
In the present work we perform a phenomenological analysis of the effective low energy models with Pati-Salam (PS) gauge symmetry derived in the context of D-branes. A main issue in these models arises from the fact that the right-handed fermions and the PS-symmetry breaking Higgs field transform identically under the PS symmetry, causing unnatural matter-Higgs mixing effects. We argue that this problem could be solved in particular D-brane setups where these fields arise in different intersections. We further observe that whenever a large Higgs mass term is generated in a particular class of …
We discuss the gravitino problem in contest of the Exotic see-saw mechanism for neutrinos and Leptogenesis, UV completed by intersecting D-branes Pati-Salam models. In the Exotic see-saw model, supersymmetry is broken at high scales $M_{SUSY}>10^{9}\, \rm GeV$ and this seems in contradiction with gravitino bounds from inflation and baryogenesis. However, if gravitino is the Lightest Stable Supersymmetric Particle, it will not decay into other SUSY particles, avoiding the gravitino problem and providing a good Cold Dark Matter. Gravitini are Super Heavy Dark Particles and they can be produced b…
We show that for supersymmetric AdS vacua on Type IIA orientifolds with flux compactifications, the RR tadpole cancellation conditions can be completely relaxed, and then the four-dimensional N=1 supersymmetry conditions are the main constraints on consistent intersecting D6-brane model building. We construct two kinds of three-family Pati-Salam models. In the first kind of models, the suitable three-family SM fermion masses and mixings can be generated at the stringy tree level, and then the rank one problem for the SM fermion Yukawa matrices can be solved. In the second kind of models, only …
The free fermionic construction of the heterotic string in four dimensions produced a large space of three generation models with the underlying $SO(10)$ embedding of the Standard Model states. The $SO(10)$ symmetry is broken to a subgroup directly at the string scale. Over the past few years free fermionic models with the Pati-Salam and flipped $SU(5)$ subgroups have been classified. In this paper we extend this classification program to models in which the $SO(10)$ symmetry is broken at the string level to the $SU(4)\times SU(2)_L\times U(1)_R$ (SU421) subgroup. The subspace of free fermioni…
The ATLAS collaboration recently recorded possible excess in the di--boson production at the di--boson invariant mass at around 2 TeV. Such an excess may be produced if there exist additional $Z^\prime$ and/or $W^\prime$ at that scale. We survey the extra $Z^\prime$s and $W^\prime$s that may arise from semi--realistic heterotic string vacua in the free fermionic formulation in seven distinct cases including: $U(1)_{Z^\prime}\in SO(10)$; family universal $U(1)_{Z^\prime}$ not in $SO(10)$; non--universal $U(1)_{Z^\prime}$; hidden sector $U(1)$ symmetries and kinetic mixing; left--right symmetric…
The ${\cal N}=1$ supersymmetric Pati-Salam models with extra $SU(2)_{L_2/R_2}$ gauge symmetry from intersecting D6-branes
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Physics, Henan Normal University（河南师范大学物理学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
I discuss the existence of discrete properties in the landscape of free fermionic heterotic-string vacua that were discovered via their classification by SO(10) GUT models and its subgroups such as the Pati-Salam, Flipped SU(5) and SU(4) x SU(2) x U(1) models. The classification is carried out by fixing a set of basis vectors and varying the GGSO projection coefficients entering the one-loop partition function. The analysis of the models is facilitated by deriving algebraic expressions for the GSO projections that enable a computerised analysis of the entire string spectrum and the scanning of…
We construct supersymmetric Pati-Salam flux vacua in AdS from intersecting D6-branes on T^6/(Z_2 x Z_2'). The models constructed have three generations of MSSM matter plus right-handed neutrinos. Because the cycles wrapped by the D-branes are rigid there are no extra massless fields in the adjoint representation, arising as open-string moduli. However, we find that it is problematic to break the Pati-Salam gauge symmetry to the Standard Model (SM) while keeping the SM hypercharge massless.
The quest for the Standard Model among the huge number of string vacua is usually based on a set of phenomenological criteria related to the massless spectrum of string models. In this work we study criteria associated with interactions in the effective low energy theory and in particular with the presence of the coupling that provides mass to the top quark. Working in the context of the Free Fermionic Formulation of the heterotic superstring, we demonstrate that, in a big class of phenomenologically promising $Z_2\times Z_2$ compactifications, these criteria can be expressed entirely in terms…
We discuss the construction of heterotic--string models that allow for the existence of an extra $Z^\prime$ at low scales. One of the main difficulties encountered is that the desired symmetries tend to be anomalous in the prevailing three generation constructions. The reason is that these models utilise the symmetry breaking pattern $E_6\rightarrow SO(10)\times U(1)_ζ$ by GGSO projections. Consequently, $U(1)_ζ$ becomes anomalous. The spinor--vector duality that was observed in the fermionic $Z_2\times Z_2$ orbifold compactifications is used to construct a phenomenological three generation Pa…
The classification method of the free fermionic heterotic string vacua is extended to models where the $SO(10)$ GUT symmetry is broken directly at the string scale to the Left-Right Symmetric subgroup. Our method involves using a fixed set of basis vectors which are defined by the boundary conditions assigned to the free fermions before enumerating the string vacua by varying the Generalised GSO (GGSO) projection coefficients. It allows the derivation of algebraic expressions for the GGSO projections for each sector that generates massless states in the models. This enables a computerised anal…
