Discrete space-time symmetries

Symmetries have always fascinated human beings; they are found in nature, art, and architecture. Physicists, like other scientists have often used symmetries as a basis of their understanding of nature. When the dynamics is unknown, symmetries serve to delineate and define it. When the dynamics is known, symmetries are used to study structure.In these two lectures, there are reviewed the theory and present understanding and status of two discrete space-time symmetries, namely parity (P) and time reversal (T).Lectures held at the Indian-Summer School on Electromagnetic and Weak Interactions of Particles and Nuclei, Sázava, Czechoslovakia, 6–11 September 1992     

Discrete symmetries in higher dimensions

We consider the constraints imposed by causality on the transformations of time reversal ?, charge conjugationC, and parityP in higher dimensional space-time.     

Discrete symmetries in Kaluza-Klein theories

We investigate discrete symmetries in theories of higher-dimensional (d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d ? 4 dimensional internal space. If the action of pure d-dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d-dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor.A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong-CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.     

Discrete symmetries and nonlocal reductions

We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.     

Discrete symmetries of Petrov type-I space-times

The arbitrariness in the definition of the canonical frame in a Petrov type-I space-time is examined and shown to imply that the Newman-Penrose formalism is invariant under a 24-element discrete symmetry group. The transformations of the N-P scalars under the operations of the group are obtained, and conditions are formulated which characterize space-times that maintain this symmetry globally.     

Discrete gauge symmetries in discrete MSSM-like orientifolds

Motivated by the necessity of discrete _ZN${\mathbb{Z}}_N$ symmetries in the MSSM to insure baryon stability, we study the origin of discrete gauge symmetries from open string sector U(1)$U(1)$?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)$U(1)$ factors for all simple current MIPFs and orientifolds of all 168 Gepner models, a total of 32 990 distinct cases. We discuss how the presence of discrete symmetries surviving as a subgroup of broken U(1)$U(1)$?s can be derived using this basis. We apply this procedure to models with MSSM chiral spectrum, concretely to all known U(3)×U(2)×U(1)×U(1)$U(3)\times U(2)\times U(1)\times U(1)$ and U(3)×Sp(2)×U(1)×U(1)$U(3)\times \mathit{Sp}(2)\times U(1)\times U(1)$ configurations with chiral bi-fundamentals, but no chiral tensors, as well as some SU(5)$\mathit{SU}(5)$ GUT models. We find examples of models with _Z2${\mathbb{Z}}_2$ (R-parity) and _Z3${\mathbb{Z}}_3$ symmetries that forbid certain B and/or L violating MSSM couplings. Their presence is however relatively rare, at the level of a few percent of all cases.     

Minimal violation of flavour and custodial symmetries in a vectophobic Two-Higgs-Doublet-Model

Tree-level accidental symmetries are known to play a fundamental role in the phenomenology of the Standard Model (SM) for electroweak interactions. So far, no significant deviations from the theory have been observed in precision, flavour and collider physics. Consequently, these global symmetries are expected to remain quite efficient in any attempt beyond the SM. Yet, they do not forbid rather unorthodox phenomena within the reach of current LHC experiments. This is illustrated with a vectophobic Two-Higgs-Doublet-Model (2HDM) where effects of a light, flavour-violating and custodian (pseudo)scalar might be observed in the _Bs→μ⁺μ⁻$B_s\to {\mu }^{+}{\mu }^{-}$ decay rate and in the diphoton invariant mass spectrum at around 125 GeV.     

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.     

Discrete symmetries and the einstein principle of relativity

It is possible to limit the field of the data exchanged by observers so that discrete symmetry principles of a new kind could be valid. The decay of the K⁰_L into two pions exhibits such a symmetry. Some other consequences, unfortunately difficult to test, are presented. One of the main results is that the weak interactions should possess symmetries which correspond to TP and TC in addition to CP, but TCP is not necessarily a symmetry.     

Leading particle production in light flavour jets

The energy distribution and type of the particle with the highest momentum in quark jets are determined for each of the five quark flavours making only minimal model assumptions. The analysis is based on a large statistics sample of hadronic decays collected with the OPAL detector at the LEP collider. These results provide a basis for future studies of light flavour production at other centre-of-mass energies. We use our results to study the hadronisation mechanism in light flavour jets and compare the data to the QCD models JETSET and HERWIG. Within the JETSET model we also directly determine the suppression of strange quarks to be by comparing the production of charged and neutral kaons in strange and non-strange light quark events. Finally we study the features of baryon production. Received: 19 November 1999 / Published online: 8 May 2000     

Discrete symmetries and the complex structure of Calabi-Yau manifolds

We show how the discrete symmetries, which may be present after Calabi-Yau compactification for specific choices of the complex structure, extend to the h_2,1 moduli — the scalar fields whose vacuum expectation values determine the complex structure. This allows us to determine much about the coupling of the moduli and hence the energetically favoured complex structure. The discrete symmetry transformation properties of the moduli are worked out in detail for a three-generation Calabi-Yau model and it is shown how minimization of the effective potential involving these fields selects the complex structure which leaves unbroken a set of discrete symmetries. The phenomenological implications of these symmetries are briefly discussed.     

The discrete family symmetries as the possible solution to the flavour problem

In order to explain the fermions’ masses and mixing parameters appearing in the lepton sector of the Standard Model, one proposes the extension of its symmetry. A discrete, non-Abelian subgroup of U(3) is added to the gauge group SU(3) _C × SU(2) _L × U(1) _Y . Apart from that, one assumes the existence of one extra Higgs doublet. This article focuses mainly on the mathematical theorems and computational techniques which brought us to the results.     

Discrete symmetries and 1/3-quantum vortices in condensates of F=2 cold atoms

In this Letter we study discrete symmetries of mean field manifolds of condensates of F=2 cold atoms, and various unconventional quantum vortices. Discrete quaternion symmetries result in two species of spin defects that can only appear in integer vortices while cyclic symmetries are found to result in a phase shift of 2pi/3 (or 4pi/3) and therefore 1/3- (or 2/3-) quantum vortices in condensates. We also briefly discuss 1/3-quantum vortices in condensates of trimers.     

Discrete symmetries,Cabibbo universality and flavor mixing angles

A model is proposed where deviations from exact Cabibbo universality are naturally small, and the correct value for the Cabibbo angle is obtained.