Bjoern Petersen, Michael Ratz, Roland Schieren
We analyze patterns of remnant discrete symmetries that arise from U(1)^N theories by spontaneous breaking. We describe a simple, geometrical way to understand these patterns and provide methods for identifying the discrete symmetries and bringing them to the simplest possible form. Applications in GUT and string model building are briefly discussed.
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We revisit the residual symmetries that survive the orbifold projections, and find additional transformations that have been overlooked in the past. Some of these transformations are outer automorphisms of the downstairs continuous symmetry group. Examples for these transformations include the left-right parity of the Pati-Salam model and its left-right symmetric subgroup.
More about discrete symmetries in compactified string theories
我们讨论了几种弦紧化方案中的离散对称性。对于非$R$对称性，在各种情况下发现了与Gepner模型\cite{rosss}相同的光谱约束。对$R$对称性也建立了类似的约束，因此很自然地推测它们总是适用的。
Relying on Dirac neutrinos allows an infinity of anomaly-free discrete gauge symmetries to be imposed on the Supersymmetric Standard Model, some of which are GUT-compatible.
We review several geometric aspects and properties of the orbifold T6/Z(2)xZ(6')xOR with discrete torsion, that are crucial with respect to global model building and the search for discrete gauge symmetries in the context of intersecting D6-brane models. A global six-stack Pati-Salam model is used for illustration, and various characteristics of its effective field theory are discussed.
Stringy origin of non-Abelian discrete flavor symmetries
我们研究了超弦理论中非阿贝尔离散味对称性的起源。我们分类所有可能的非阿贝尔离散风味对称性，可以出现在杂种优势orbifold模型。这些对称性包括D_4和Delta（54）。我们发现耦合的对称性总是大于紧致空间的对称性。这是因为它们是轨道的几何结构与弦的空间群选择规则相结合的结果。我们还研究了可能的破坏模式。我们的分析产生了一个简单的几何理解的实现非阿贝尔味对称。
We study non-abelian discrete flavor symmetries, which can appear in magnetized brane models. For example, $D_4$, $Δ(27)$ and $Δ(54)$ can appear and matter fields with several representations can appear. We also study the orbifold background, where non-abelian flavor symmetries are broken in a certain way.
Discrete symmetries play a crucial role in particle physics. They appear abundantly in string model constructions. We focus here on the case of discrete $R$-symmetries which are intrinsically connected to the Lorentz group in extra dimensions and the appearance of $N$-extended supersymmetry. In that sense, discrete $R$-symmetries can be understood as "fractionally" extended supersymmetry. These symmetries reveal insight about the location of fields in extra dimensions (in particular the Higgs boson). Applications can be found in the solution of the $μ$-problem, suppression of proton decay and …
Constraints on discrete symmetries from anomaly cancellation in compactified superstring theories
紧化的弦论产生了离散对称性，如果它们要提供一个现实的低能理论，这些对称性是必不可少的。我们发现，在一类四维弦论中，这些对称性受到类似条件的约束，这些条件与在离散对称性是自发破缺的规范对称性的剩余物的情况下发现的离散反常抵消条件相似。这样的条件强烈地限制了低能有效理论的允许形式。
We study discrete flavor symmetries of the models based on a ten-dimensional supersymmetric Yang-Mills (10D SYM) theory compactified on magnetized tori. We assume non-vanishing non-factorizable fluxes as well as the orbifold projections. These setups allow model-building with more various flavor structures. Indeed, we show that there exist various classes of non-Abelian discrete flavor symmetries. In particular, we find that $S_3$ flavor symmetries can be realized in the framework of the magnetized 10D SYM theory for the first time.
Elements of Finite Order in Lie Groups and Discrete Gauge Symmetries
我们应用Kac的李群有限阶元理论来描述各种超对称大统一模型中的离散规范对称性。考虑到离散异常取消条件，我们确定的EFO产生的超对称$SO(10)$和$E_6$模型的上下文中的某些物质宇称。
The study of discrete gauge symmetries in field theory and string theory is often carried out by embedding them into continuous symmetries. Many symmetries however do not seem to admit such embedding, for instance discrete isometries given by large diffeomorphisms in compactifications. We show that in the context of string theory even those symmetries can be embedded into continuous ones. This requires extending the system to a supercritical string theory configuration with extra dimensions, on which the continuous symmetry acts. The extra dimensions are subsequently removed by closed string t…
We re-derive the R symmetries for the Z6-II orbifold with non-trivial Wilson lines and find expressions for the R charges which differ from those in the literature.
We study the realization of non-Abelian discrete gauge symmetries in 4d field theory and string theory compactifications. The underlying structure generalizes the Abelian case, and follows from the interplay between gaugings of non-Abelian isometries of the scalar manifold and field identifications making axion-like fields periodic. We present several classes of string constructions realizing non-Abelian discrete gauge symmetries. In particular, compactifications with torsion homology classes, where non-Abelianity arises microscopically from the Hanany-Witten effect, or compactifications with …
Local discrete symmetries from superstring derived models
离散和整体对称性在标准模型的许多扩展中起着至关重要的作用，例如，保持质子寿命，防止味道改变中性电流等。一个重要的问题是，这样的对称性如何在量子引力理论中生存，如超弦理论。在一个特定的字符串模型，我说明了如何当地离散对称性可能会出现在字符串模型，并在防止快速质子衰变和口味改变中性电流发挥重要作用。局部离散对称性是由于超弦模型中威尔逊线对非阿贝尔规范对称性的破坏而产生的，它禁止了所有阶的非重整化项，例如介导快速质子衰变的五维算子。在规范和引力相互作用统一模型的背景下，为了确保给定的模型不与实验观测相冲突，必须找到的正是这种类型的局部离散对称性。
We study the presence of discrete gauge symmetries in D-brane semi-realistic compactifications. After establishing the constraints on the transformation behaviour of the chiral matter for the presence of a discrete gauge symmetry we perform a systematic search for discrete gauge symmetries within local semi-realistic D-brane realizations, based on four D-brane stacks, of the MSSM and the MSSM with three right-handed neutrinos. The systematic search reveals that Proton hexality, a discrete symmetry which ensures the absence of R-parity violating terms as well as the absence of dangerous dimensi…
Motivated by the necessity of discrete Z_N symmetries in the MSSM to insure baryon stability, we study the origin of discrete gauge symmetries from open string sector U(1)'s in orientifolds based on rational conformal field theory. By means of an explicit construction, we find an integral basis for the couplings of axions and U(1) factors for all simple current MIPFs and orientifolds of all 168 Gepner models, a total of 32990 distinct cases. We discuss how the presence of discrete symmetries surviving as a subgroup of broken U(1)'s can be derived using this basis. We apply this procedure to mo…
We comment on aspects of discrete anomaly conditions focussing particularly on $R$ symmetries. We review the Green-Schwarz cancellation of discrete anomalies, providing a heuristic explanation why, in the heterotic string, only the "model-independent dilaton" transforms non-linearly under discrete symmetries; this argument suggests that, in other theories, multiple fields might play a role in anomaly cancellations, further weakening any anomaly constraints at low energies. We provide examples in open string theories of non-universal discrete anomalies at low energies. We then consider the fact…
Gravity Cutoff in Theories with Large Discrete Symmetries
对于具有大周期性的精确量子数的理论，如Z_N离散对称性，我们给出了引力截止的一个上界。这个界限源于黑洞物理学。它类似于N粒子种类理论中出现的边界，尽管先验地，大的离散对称性并不意味着大量的种类。因此，出现了一个潜在的广泛的新理论，解决层次问题，降低引力截止由于存在大Z_{10^32}型对称。
Discrete R symmetries are interesting from a variety of points of view. They raise the specter, however, of domain walls, which may be cosmologically problematic. In this note, we describe some of the issues. In many schemes for supersymmetry breaking, as we explain, satisfying familiar constraints such as suppression of gravitino production, insures that the domain walls are readily inflated away. However, in others, they form after inflation. In these cases, it is necessary that they annihilate. We discuss possible breaking mechanisms for the discrete symmetries, and the constraints they mus…
Discrete gauge symmetries in global intersecting D-brane models constrain the exact form of the perturbative as well as non-perturbative superpotential. We derive the complete set of conditions on the existence of discrete Zn gauge symmetries on toroidal orbifolds, T6/Z(N) and T6/Z(2)xZ(2M}, with fractional or rigid D6-branes on tilted tori, for which global models of particle physics are known. Several examples of global left-right symmetric and Pati-Salam models are presented. Some discrete `stringy' Zn symmetries are trivial from the field theory point of view, while others have not been id…
We show that Abelian discrete symmetries Z_{N} and Z_{nR} from string orbifolds result by assigning vacuum expectation values(VEVs) only to specified singlets. These singlets obtaining VEVs carry the gauge charges (of the covering U(1) gauge group) as multiples of N for Z_N and n for Z_{nR}. We explicitly show this realization in a Z_{12-I} orbifold model.
We study the presence of discrete flavor symmetries in D-brane models of particle physics. By analyzing the compact extra dimensions of these models one can determine when such symmetries exist both in the context of intersecting and magnetized D-brane constructions. Our approach allows to distinguish between approximate and exact discrete symmetries, and it can be applied to compactification manifolds with continuous isometries or to manifolds that only contain discrete isometries, like Calabi-Yau three-folds. We analyze in detail the class of rigid D-branes models based on a Z_2 x Z'_2 toroi…
Discrete Symmetries and Localization in a Brane-world
研究了额外一维弯曲时空中的离散对称性。特别地，我们分析了五维和四维电荷共轭、宇称、时间反演和轨道对称性Z_2与四维膜世界上费米子局域化和洛伦兹不变性的相容性。然后，我们表明，当一个合适的拓扑标量场（"扭结“）被包括在内，费米子本地化是（五维）CPT不变性的后果。
We show that non-Abelian discrete symmetries in orbifold string models have a gauge origin. This can be understood when looking at the vicinity of a symmetry enhanced point in moduli space. At such an enhanced point, orbifold fixed points are characterized by an enhanced gauge symmetry. This gauge symmetry can be broken to a discrete subgroup by a nontrivial vacuum expectation value of the Kähler modulus $T$. Using this mechanism it is shown that the $Δ(54)$ non-Abelian discrete symmetry group originates from a $SU(3)$ gauge symmetry, whereas the $D_4$ symmetry group is obtained from a $SU(2)$…
Mapping out SU(5) GUTs with non-Abelian discrete flavor symmetries
我们构造了一类超对称SU（5）大统一理论模型，该模型从离散的非阿贝尔味对称性出发，产生了近三重最大轻子混合、观测到的夸克混合矩阵以及夸克和轻子质量. SU（5）大统一理论在平坦极限下的两个五维喉道上形成，由于I型跷跷板机制，中微子质量变小。离散的非阿贝尔味对称性是由循环群的半直积给出的，这些循环群在喉尖的红外膜处被破坏。结果，我们得到了SU（5）大统一理论，它提供了非阿贝尔味对称性和夸克-轻子互补性的组合描述。
We combine two popular extensions of beyond the Standard Model physics within the framework of intersecting D6-brane models: discrete Zn symmetries and Peccei-Quinn axions. The underlying natural connection between both extensions is formed by the presence of massive U(1) gauge symmetries in D-brane model building. Global intersecting D6-brane models on toroidal orbifolds of the type T6/Z2N and T6/Z2xZ2M with discrete torsion offer excellent playgrounds for realizing these extensions. A generation-dependent Z2 symmetry is identified in a global Pati-Salam model, while global left-right symmetr…
While non-abelian groups are undoubtedly the cornerstone of Grand Unified Theories (GUTs), phenomenology shows that the role of abelian and discrete symmetries is equally important in model building. The latter are the appropriate tool to suppress undesired proton decay operators and various flavour violating interactions, to generate a hierarchical fermion mass spectrum, etc. In F-theory, GUT symmetries are linked to the singularities of the elliptically fibred K3 manifolds; they are of ADE type and have been extensively discussed in recent literature. In this context, abelian and discrete sy…
The framework of compactified heterotic string theory offers consistent ultraviolet (UV) completions of the Standard Model (SM) of particle physics. In this approach, the existence of flavor symmetries beyond the SM is imperative and the flavor symmetries can be derived from the top-down. Such a derivation uncovers a unified origin of traditional discrete flavor symmetries, discrete modular flavor symmetries, discrete R symmetries of supersymmetry, as well as CP symmetry - altogether called the eclectic flavor symmetry. I will show a specific example of such a top-down derived eclectic flavor …
In this note, we discuss the consistency conditions which a discrete $Z_N$ symmetry should satisfy in order that it is not violated by gauge and gravitational instantons. As examples, we enlist all the $Z_N$ ${\cal R}$-symmetries as well as non-${\cal R}$ $Z_N$ symmetries (N=2,3,4) in the minimally supersymmetric standard model (MSSM) that are free from gauge and gravitational anomalies. We show that there exists non-anomalous discrete symmetries that forbid Baryon number violation up to dimension 6 level (in superspace). We also observe that there exists no non-anomalous $Z_3$ ${\cal R}$-symm…
We classify discrete modular symmetries in the effective action of Type IIB string on toroidal orientifolds with three-form fluxes, emphasizing on $T^6/\mathbb{Z}_2$ and $T^6/(\mathbb{Z}_2\times \mathbb{Z}_2^\prime)$ orientifold backgrounds. On the three-form flux background, the modular group is spontaneously broken down to its congruence subgroup whose pattern is severely constrained by a quantization of fluxes and tadpole cancellation conditions. We explicitly demonstrate that the congruence subgroups appearing in the effective action arise on magnetized D-branes wrapping certain cycles of …
We study non-Abelian flavor symmetries on orbifolds, $S^1/Z_2$ and $T^2/Z_3$. Our extra dimensional models realize $D_N$, $Σ(2N^2)$, $Δ(3N^2)$ and $Δ(6N^2)$ including $A_4$ and $S_4$. In addition, one can also realize their subgroups such as $Q_N$, $T_7$, etc. The $S_3$ flavor symmetry can be realized on both $S^1/Z_2$ and $T^2/Z_3$ orbifolds.
Electron-positron annihilation into Dirac magnetic monopole and antimonopole: the string ambiguity and the discrete symmetries
本文讨论狄拉克磁单极子量子场论中的弦任意性问题。不同的处方会产生不同的物理结果。本文导出了电子正电子湮没为电子-反电子对的过程中由于离散对称性（C和P）所引起的约束。在通过单光子通道湮灭的情况下，绝对禁止自旋为0的单极子的产生;自旋为1/2的π和反π应该具有相同的螺旋度（或者，等价地，π-反π态应该是p波$^1P_1$）。
If nature exhibits low energy supersymmetry, discrete (non-$Z_2$) R symmetries may well play an important role. In this paper, we explore such symmetries. We generalize gaugino condensation, constructing large classes of models which are classically scale invariant, and which spontaneously break discrete R symmetries (but not supersymmetry). The order parameters for the breaking include chiral singlets. These simplify construction of models with metastable dynamical supersymmetry breaking. We explain that in gauge mediation, the problem of the cosmological constant makes "retrofitting" particu…
We study possible anomaly-free discrete R-symmetries that avoid the μ-problem and the dangerous D\leq 5 operators considering charge assignments that do not commute with the traditional grand unifying simple groups, such as SU(5) or SO(10), but commute instead with the so-called flipped-SU(5), with or without the operation of the GS mechanism. We find Z_N symmetries with N = 3, 6 in the anomaly-free case or N = 3, 4, 6, 8, 12, 24 in the case of anomaly cancellation through the Green-Schwartz mechanism. Non-unified cases (Z_5) have also been discussed. We also confront the construction of a 4D …
Gauge symmetries are known to be respected by gravity because gauge charges carry flux lines, but global charges do not carry flux lines and are not conserved by gravitational interaction. For discrete symmetries, they are spontaneously broken in the Universe, forming domain walls. Since the realization of discrete symmetries in the Universe must involve the vacuum expectation values of Higgs fields, a string-like configuration (hair) at the intersection of domain walls in the Higgs vacua can be realized. Therefore, we argue that discrete charges are also respected by gravity.
While non-linear realizations of continuous symmetries feature derivative interactions and have no potential, non-linear realizations of discrete symmetries feature non-derivative interactions and have a highly suppressed potential. These Goldstone bosons of discrete symmetries have a non-zero potential, but the potential generated from quantum corrections is inherently very highly suppressed. We explore various discrete symmetries and to what extent the potential is suppressed for each of them.
Non-abelian discrete gauge symmetries and inflation
当包括重力效应时，获得足够的势能平坦以提供缓慢的滚动膨胀通常是困难的。非阿贝尔离散规范对称性可以保证这种情况下暴胀子势的平坦性，并且还提供了暴胀可以结束的特殊场值。
Discrete Gauge Symmetries in Axionic Extensions of the SSM
我们研究离散规范对称性的SSM的轴子扩展提供了一个解决方案的$μ$ -问题。自动PQ对称性和质子的稳定性是由某些离散对称性保证。重点是L-违反离散对称性，我们讨论了两个来源的中微子质量和它们的相关性的太阳中微子问题。
Superstring Axion, Gaugino Condensation, and Discrete Symmetries
在这里，我提出的影响，高根诺凝聚和离散对称性的模型无关的轴子潜力，与Georgi和Nilles合作[hep-ph/9805510]。
The question whether global symmetries can be realized in quantum-gravity-matter-systems has far-reaching phenomenological consequences. Here, we collect evidence that within an asymptotically safe context, discrete global symmetries of the form $\mathbb{Z}_n$, $n>4$, cannot be realized in a near-perturbative regime. In contrast, an effective-field-theory approach to quantum gravity might feature such symmetries, providing a mechanism to generate mass hierarchies in the infrared without the need for additional fine-tuning.
Non-Abelian Discrete Symmetries and Neutrino Masses: Two Examples
本文讨论了最近两个非阿贝尔离散对称性（S_3和A_4）在理解中微子质量和混合中的应用。
The role of Symmetry Breaking mechanisms to search for New Physics is of highest importance. We discuss the status and prospects of the Discrete Symmetries CP, T, CPT looking for their separate Violation in LHC experiments and meson factories.
We discuss non-Abelian discrete R symmetries which might have some conceivable relevance for model building. The focus is on settings with N=1 supersymmetry, where the superspace coordinate transforms in a one-dimensional representation of the non-Abelian discrete symmetry group. We derive anomaly constraints for such symmetries and find that novel patterns of Green-Schwarz anomaly cancellation emerge. In addition we show that perfect groups, also in the non-R case, are always anomaly-free. An important property of models with non-Abelian discrete R symmetries is that superpartners come in dif…
In this note we show that the cosmological domain wall and the de Sitter quantum breaking problems complement each other in theories with discrete symmetries that are spontaneously broken at low energies. Either the symmetry is exact and there is a domain wall problem, or it is approximate and there exists an inconsistent de Sitter minimum. This leaves no room for many extension of the Standard Model based on such discrete symmetries. We give some examples that include NMSSM, spontaneous CP violation at the weak scale and some versions of the Peccei-Quinn scenario with discrete symmetries.
Non-Abelian Discrete Family Symmetries of Leptons and Quarks
本文综述了非阿贝尔对称性S_3、D_4和A_4在轻子和夸克族结构研究中的应用进展。
Many models of beyond Standard Model physics connect flavor symmetry with a discrete group. Having this symmetry arise spontaneously from a gauge theory maintains compatibility with quantum gravity and can be used to systematically prevent anomalies. We minimize a number of Higgs potentials that break gauge groups to discrete symmetries of interest, and examine their scalar mass spectra.
We derive effective actions for parity-violating fluids in both $(3+1)$ and $(2+1)$ dimensions, including those with anomalies. As a corollary we confirm the most general constitutive relations for such systems derived previously using other methods. We discuss in detail connections between parity-odd transport and underlying discrete symmetries. In (3+1) dimensions we elucidate connections between anomalous transport coefficients and global anomalies, and clarify a previous puzzle concerning transports and local gravitational anomalies.
Stabilizing the Axion by Discrete Gauge Symmetries
强CP问题的轴子解利用了一个整体的Peccei-Quinn U（1）对称性，这个对称性很容易受到量子引力效应的破坏。我们展示了离散的规范对称性如何保护轴子不受这种违反。PQ对称性是从离散的规范对称性发展而来的一种近似的整体对称性。基于N = 11，12等Z_N对称性，给出了实现DFSZ轴子和KSVZ轴子的简单模型。离散规范异常被取消的离散版本的格林-施瓦茨机制。在超对称扩展中，我们的模型提供了超对称破缺尺度、轴子尺度和超对称保持μ项之间的自然联系。
Discrete Symmetries from Broken $SU(N)$ and the MSSM
为了使引力效应不破坏离散对称性，有必要对它们进行规范。本文讨论了由高能$SU(N)$规范对称性破缺对$\Z_N$的规范化，并从母对称性反常抵消的要求出发，导出了由此产生的离散对称性的相容性条件。这些结果，然后应用到一个详细的分析可能的离散对称性，禁止质子衰变的最小超对称标准模型。
Mass hierarchy and flavour mixing from discrete symmetries
本文考虑一类基于群S（3）和A（4）的轻子的离散味对称性，它们具有混合破缺模式.我们的目的是构建模型，其中相同的flavon领域生产的混合模式也负责质量层次结构。
Discrete non-abelian flavor symmetries give in a natural way tri-bimaximal (TBM) mixing as showed in a prototype model. However neutrino mass matrix pattern may be very different from the tri-bimaximal one if small deviations of TBM will be observed. We give the result of a model independent analysis for TBM neutrino mass pattern.
Discrete and continuous symmetries in multi-Higgs-doublet models
我们认为希格斯部门的多希格斯偶极子模型中存在的简单对称性有关的各个领域。我们构建的基础不变的观测，原则上可以用来检测这些对称性的任何数量的双峰。分类的对称性到类是必需的，我们执行的情况下，两个和三个希格斯二重态的详细。
Inflationary models with a flat potential enforced by non-abelian discrete gauge symmetries
即使考虑到引力强度效应，非阿贝尔离散规范对称性也可以为暴胀子提供平坦的势。对称性的离散性也提供了特殊的场值，其中暴胀可以通过混合型机制结束。这种方法的一个有趣的特点是，它可以自然地导致非常平坦的潜力，因此，在原则上，在异常低的能量尺度的暴胀。两个例子的有效场论与这种机制，一个与混合出口和一个突变的混合出口。它们包括违反单字段一致性条件的明确示例。
Discrete symmetries and neutrino masses
利用具有$U(1) \times Z_3 \times Z_2$味对称性的Froggatt-Nielsen机制建立了中微子质量模型。该模型预测，$(2/3)m_2/m_3 \sim \sqrt{2}\sinθ_{13}$在轻子数违反尺度$M_1$。结果表明，$m_2/m_3$和$\sinθ_{13}$的小值是打破离散对称性的后果。
Breaking Discrete Symmetries in Broken Gauge Theories
研究了规范对称性破缺理论中离散对称性的自发破缺。预期的应用是CP打破理论与规范味对称性，但这里所描述的分析是初步的。我们省去了物质场，而把规范理论看作是弱耦合的，并被未指明的短距离力自发地破缺。我们开发了一个有效的场论描述所产生的低能量理论，并询问这个理论本身是否可以描述随后打破离散对称性。我们的结论是，这可以发生依赖于参数的有效理论，和内在的破坏是自然的秩序统一。
Non-Abelian Discrete Flavor Symmetries
这是一个不完整的调查，一些非阿贝尔离散对称性已被用于最近试图了解的味道结构的轻子和夸克。为了支持这种对称性，需要新的标量粒子。在某些模型中，它们的质量非常大，在这种情况下，在TeV尺度上可能没有太多它们存在的痕迹。在其他模型中，它们本身处于TeV尺度，在这种情况下，它们有合理的机会在欧洲核子研究中心的大型强子对撞机（LHC）上被发现。
Connect discreteness to continuousness in leptonic flavor symmetries
本文提出用群代数（GA）描述轻子质量矩阵对称性，通过对群$S_{4}$生成的198个非平凡元素进行分类，揭示了其唯象后果，将离散性与连续性联系起来。
Non-abelian discrete symmetries are of particular importance in model building. They are mainly invoked to explain the various fermion mass hierarchies and forbid dangerous superpotential terms. In string models they are usually associated to the geometry of the compactification manifold and more particularly to the magnetised branes in toroidal compactifications. Motivated by these facts, in this note we propose a unified framework to construct representations of finite discrete family groups based on the automorphisms of the discrete and finite Heisenberg group. We focus in particular in the…
Models of flavour with discrete symmetries
本文简要回顾了费米子质量、混合和具有离散非阿贝尔对称性的CP破坏理论模型的一些最新进展。然后，我们解释了最近提出的标准模型的最小S_3不变延拓的主要思想及其在轻子和夸克扇区中质量、混合和CP破坏的统一分析以及V_PMNS和V_CKM混合矩阵的显式计算中的应用。
Will discrete symmetries help solve the hierarchy problem ?
我们发现，在大统一理论尺度下，无质量的希格斯二重态可能是离散对称性的自然结果。这种机制不需要精细的微调或复杂的颗粒含量。同样的离散对称性也将防止质子衰变和味道改变中性电流。然而，该机制总是预测非最小标准模型。还包括一个明确的例子，说明这种机制是如何工作的。
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