Spontaneous compactification and CPN: SU(3) × SU(2) × U(1), sin2θW, g3/g2 and SU(3)-triplet chiral fermions in four dimensions

We examine the compactification of D=4+2N, Einstein-Maxwell-Dirac theory. It is shown that the manifold CP_N × M₄ is a solution of the equations of motion. The structure of the fermions, gauge bosons and their couplings in the four-dimensional effective theory is investigated. The scale of CP_N is quantized by a generalized Dirac condition. When the results are applied to the solution with internal space CP₁×CP₂, the weak mixing angle and the ratio of the couplings of SU(3) (g₃) and SU(2) (g₂) are defined by two integers and a hypercharge. An SU(3)-triplet chiral fermion can appear in four-dimensional effective theory.     

Strings in the CP2N−1 model

The CP₂^N?1 model is discussed using strings as collective variables in the hamiltonian formulation. The large N limit is obtained as a semiclassical approximation. The mass gap and β-function are computed. In the limit N → ∞, it is shown that the singlet spectrum contains both bound states and scattering states whose energies and wave functions are calculated.     

solitons in quantum Hall systems

We present here an elementary pedagogical introduction to CP_N solitons in quantum Hall systems. We begin with a brief introduction to both CP_N models and to quantum Hall (QH) physics. We then focus on spin and layer-spin degrees of freedom in QH systems and point out that these are in fact CP_N fields for N = 1 and N = 3. Excitations in these degrees of freedom will be shown to be topologically non-trivial soliton solutions of the corresponding CP_N field equations. We conclude with a brief summary of our own recent work in this area, done with Sankalpa Ghosh. Received 17 November 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: doug0700@mail.jnu.ac.in     

Classical CP N−1 crystal

Two-dimensional Euclidean CP ^N?1 ffields are conformal-transformed into fields with periodic spatial (crystal-like) structures, both at zero temperature and at finite temperature. Specific solutions of the crystal-like CP ^N?1 model are discussed     

The role of instantons in scale-invariant gauge theories

Instanton calculations in scale-invariant gauge theories, such as QCD, have long been plaqued by divergences at large distances where strong coupling effects are important. Furthermore, Witten has argued that quantum effects may cause the instanton gas to disappear and has displayed this phenomenon in the CP^(N?1) model at large N. It is argued here that instantons can play a role in calculations involving an inherent infrared cut-off, and this is demonstrated in the CP^(N?1) model for large N at a finite temperature. Some results on finite-temperature QED are also obtained in passing.     

Spontaneous CP violation in left-right symmetric gauge theories

Motivated by arguments in favour of spontaneous CP violation, we investigate the general conditions imposed by P and CP invariance on the Yukawa couplings in SU(2)_L × SU(2)_R × U(1) gauge theories. A complete discussion of these constraints is presented for two minimal Higgs sectors. In addition to the standard case of manifest CP invariance, we find two phenomenologically interesting models where the P and CP transformations are non-aligned in flavour and Higgs spaces. To a certain extent, P and CP invariance may be responsible for the existence of horizontal symmetries. An intimate connection between CP violation and the general flavour problem emerges.     

CPN−1 model: A toy model for QCD?

We examine the CP^N?1₂ models and discuss their relevance as toy models for QCD₄. Specifically, we study the role of instantons, θ vacua, and confinement in the $\text{1}\text{N}$ expansion. Our results, and comparisons with other two-dimensional models, suggest that most of the interesting features of these models are peculiarities of two-dimensional space-time and cannot be expected to reappear in QCD₄.     

CP violation andZ boson decays

We present a general discussion ofCP-violating effects in the following two and three body decays of theZ boson:Z→l ⁺ l ^?,Z→l ⁺ l ^?γ, wherel=e, μ, τ andZ→2 jets,Z→ 2 jets+γ, andZ→ 3 jets. Experimental observables sensitive toCP violation in these decays are discussed systematically for the case that polarizations of final state particles are not observed. It is shown that the standard model predicts only extremely smallCP-violating effects for the above decays. PossibleCP-violating interactions beyond the standard model are parametrized in terms of aCP-odd effective Lagrangian containing coupling constants proportional to Λ _P ^-1 and Λ _P ^-2 . Here Λ _CP is the mass scale associated with the assumed new interactions. We give estimates of the bounds obtainable for Λ _CP in experiments at LEP1.     

Spontaneous compactification in generalized Candelas-Weinberg models

In their recent work on the dimensional reduction, Candelas and Weinberg considered a model which is compactified into a direct product space of the Minkowski space (M₄) and an N-dimensional sphere (S_N). In the present paper we investigate generalized models of their type which are compactified into M₄ × S_M × S_N and M₄ × S_M × CP₂. The compactification is caused by the quantum loop effect due to a large number of matter fields. The conditions for the vacuum stability are studied. Numerical computation of the loop effect is undertaken, and it is shown that some of the models of the type M₄ × S_M × S_N admit a stable solution which has finite circumferences of both of the extra spaces and positive coupling constants of the Einstein-Yang-Mills theory in four dimensions.     

Exactly solvable string compactifications on manifolds of SU(N) holonomy

A variety of heterotic string compactifications on the K3 surface, manifolds of SU(3) holomony, and higher holomony manifolds, are solved exactly. An example of the quintic hypersurface in CP₄ is worked out in detail. It is conjectured, and demonstrated in part, that any supersymmetric compactification of the heterotic string with an N=2 superconformal theory is equivalent to a compactification on a manifold of SU(N) holonomy, and in particular an arbitrary gluing of the discrete models with c=9 gives a solvable heterotic string compactification on some Calabi-Yau manifold. Calabi-Yau compactifications are seen to be exact vacua of string theory, retaining their topological and geometrical characteristics. Previously unknown enhanced gauge symmetries are found to arise for certain backgrounds.     

Spontaneous CP violation in extended technicolor models with horizontal U(2)L ⊗ U(2)R flavor symmetries (I)

The Eichten, Lane, Preskill mechanism for CP violation in extended technicolor (ETC) models is examined in models with “horizontal” U(2)_L ? U(2)_R flavor symmetries. We identify a class of chiral perturbations which generate spontaneous CP violation when the vacuum alignment is determined by Dashen's theorem, and show that, in general, a discrete symmetry must be invoked to avoid strong CP non-conservation. CP-violating phases appearing in the electroweak and broken ETC interactions may be naturally suppressed by ratios of ETC vector boson masses.     

Predictions of the CKM model for CP asymmetries in B decay

We discuss present-day uncertainties for the value of the CP-violating phase δ in the CKM matrix and point out how a knowledge of m_t and/or x_s could substantially reduce this uncertainty. A model-independent measurement of δ is, in principle, possible by studying certain CP-violating asymmetries, involving B⁰ mesons decaying into CP-conjugate hadronic final states. There exist three different classes of these asymmetries and we give estimates for their values, based on our present knowledge of the CKM matrix. Some comments on the experimental requirements for detecting these asymmetries are also presented.     

CP violation studies at Belle

CP-violation in the Standard Model (SM) is described by a single parameter and the pattern of possible CP-violating phenomena is thus extremely constrained. The Belle experiment at the KEK laboratory in Tsukuba (Japan) is testing if this pattern actually occurs by analyzing a sample of about 770 million B-meson decays. CP-violation observed at Belle is consistent with the Cabibbo-Kobayashi-Maskawa (CKM) mechanism, the theory of CP-violation within the SM. We briefly describe how the unitarity triangle allows to probe the CKM theory and review recent measurements of its angles and sides, namely determinations of the CKM angles ? ₁ and ? ₃, and the CKM magnitudes |V _cb | and |V _ub |.     

Two-photon decay of the B s meson in supersymmetric models

The two-photon decay mode B _s → γγ is studied within the supersymmetric extension of the Standard Model. The contributions of one-particle-reducible and one-particle-irreducible diagrams to the decay amplitude are analyzed in detail. Exact expressions for the CP-even and CP-odd amplitudes are obtained. The contributions of supersymmetric particles to the partial width with respect the decay mode B _s → γγ are estimated.     

Chiral symmetry breakdown,anomaly, and the PCAC relation on a lattice

Lattice fermion formulation is investigated using a solvable model which resembles quantum chromodynamics. CP₂^N?1 models with quarks are formulated on a lattice. For dynamical quarks, a generalized formulation of the Wilson and the Osterwalder-Seiler lattice fermion is used. In the $\text{1}\text{N}$ expansion, the spontaneous breakdown of chiral symmetry (which is softly broken by the quark mass) apparently occurs in this model, and the “pion” mass is calculated. From the above results, it is shown that the above lattice fermion formulations have the desired continuum limit. The axial-vector current is investigated and it is proved that the usual anomaly appears in the continuum limit and the PCAC relation is satisfied.     

Can the quark mass matrix be real?

We study the possibility of solving the strong CP problem in a theory where CP is broken spontaneously and the quark mass matrix, M_ij, not just its determinant, remains real to one part in 10⁸. We find that it is difficult to build natural models consistent with the known CP violation phenomenology. We find that such models require new fermions and possess a hierarchy problem. We present an SO(10) example which succeeds at the one-loop level, however, two-loop effects will introduce phases into M_ij.     

Calculation of gauge couplings and compact circumferences from self-consistent dimensional reduction

We consider a system of gravity plus free massless matter fields in 4 + N dimensions, and look for solutions in which N dimensions form a compact curved manifold, with the energy-momentum tensor responsible for the curvature produced by quantum fluctuations in the matter fields. For manifolds of sufficient symmetry (including spheres, CP^N, and manifolds of simple Lie groups) the metric depends on only a single multiplicative parameter ?², and the field equations reduce to an algebraic equation for ?, involving the potential of the matter fields in the metric of the manifold. With a large number of species of matter fields, the manifold will be larger than the Planck length, and the potential can be calculated using just one-loop graphs. In odd dimensions these are finite, and give a potential of form C_N/?⁴. Also there are induced Yang-Mills and Einstein-Hilbert terms in the effective 4-dimensional action, proportional to additional numerical coefficients, D_N and E_N. General formulas are given for the gauge coupling g² in terms of C_N and D_N, and the ratio ?²/8πG in terms of C_N and E_N. Numerical values for C_N, D_N, and E_N are obtained for scalar and spinor fields on spheres of odd dimensionality N. It is found that the potential, g² and ?²/8πG can all be positive but only when the compact manifold has N = 3 + 4 k dimensions. (The positivity of the potential is needed for stability of the sphere against uniform dilations or contractions). In this case, solutions exist either for spinor fields alone or for suitable mixes of spinor and scalar fields provided the ratio of the number of scalar fields to the number of fermion fields is not too large. Numerical values of the O(N + 1) gauge couplings and 8φG/?² are calculated for illustrative values of the numbers of spinor fields. It turns out that large numbers of matter fields are needed to make these parameters reasonably small.     

Axions from chiral family symmetry

We investigate the possibility that family symmetry, G_F, is spontaneously broken chiral global symmetry. We classify the interesting cases when family symmetry can result in an automatic Peccei-Quinn symmetry U(1)_PQ and thus provide a solution to the strong CP problem. The result disfavors having two or four families. For more than four families, U(1)_PQ is in general automatic. In the case of three families, a unique Higgs sector allows U(1)_PQ in the simplest case of G_F = [SU(3)]³. Cosmological consideration also puts strong constraint on the number of families. For G_F = [SU(N)]³ cosmology singles out the three-family (N = 3) case as a unique solution if there are three light neutrinos. Possible implication of decoupling theorem as applied to family symmetry breaking is also discussed.     

CP violation with beautiful baryons

CP violation can be studied in modes of charmed or bottom baryons when a decay process is compared with its charge-conjugated partner. It can show up as a rate asymmetry and in a study of other decay parameters. Neither tagging nor time-dependences are required to observeCP violation with modes of baryons, in contrast to the conventionalB ⁰ modes. Numerous modes of bottom baryons have the potential to show largeCP-violating effects within the Standard Model. Those effects can be substantial for modes with aD ⁰, which is seen in a final state that can also be fed from a \(\bar D^0 \) . For instance, a comparison of theΛ _b→Λ _CP ⁰ with the \(\bar \Lambda _b \to \bar \Lambda D_{CP}^0 \) process can show sizeableCP violation. HereD _CP ^o denotesCP eigenstates ofD ⁰, which occur at a few percent. Six related processes, such asΛ _b→ΛD ⁰, \(\Lambda _b \to \Lambda \bar D^0 \) ,Λ _b→Λ _CP ⁰ , and their charge-conjugated counterparts, can extract ?, which is the most problematic angle of the unitarity triangle and which is conventionally probed with theB _s→ρ⁰ K _S asymmetry. HereD ⁰ andD ^?0 are identified by their charged kaon or lepton. We predictB(Λ _b→ΛD ⁰)～10^?5, thusB(Λ _b→Λ _CP ⁰ )～10^?7. Under favourable circumstances,CP violation can occur at the few tens of percent level. Thus 10²–10³ Λ _b→Λ _CP ⁰ decays start probing ?. Tables list many additional modes with typical branching ratios at the 10^?5–10^?6 level, with large detection efficiencies (in contrast to theD _CP ⁰ ), and with potentially largeCP-violating effects, such as Ξ _b ⁰ →ΛΨ, Λ?, ΛK^*0; Ξ _b ^? →ΛK^(*)?, Ξ^?K_s, Ξ^?K^*0, Ω _b ^? →Ξ^?φ, Ξ^?ρ⁰, ΛK^(*)?, ΩK_s, Ω^?K^*0.