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     

Strong coupling versus large N in σ-models

We show that the lattice CP^(N?1) model does not have a phase transition at N = ∞, of the type proposed for SU(N) gauge theories, despite having instantons. It does, however, have non-commuting large-N and strong coupling expansions. This rather unique situation is associated with the screening properties of the model.     

The (2+1)-dimensional supersymmetric CPN−1 model and the central charge

The supersymmetry algebra is examined for the (2+1)-dimensional supersymmetric CP^N?1 model, on the basis of the observation of Witten and Olive in (1+1) and (3+1) dimensions. We then demonstrate that also in this (2+1)-dimensional model the usual supersymmetry algebra is modified by the appearance of the topological numbers of the solitons, which are nothing but the instantons in (1+1) dimensions, as central charges. To obtain the model, we begin by constructing the supersymmetric model in (3+1) dimensions. Then it is reduced to (2+1) dimensions by means of the dimensional reduction technique. We observe that the (2+1)-dimensional supersymmetric CP^N?1 model thus obtained admits an O(2) extended supersymmetry.     

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.     

Non-linear σ-model and CP(n−1) at 2 + ε dimensions

Two systems, O(n) non-linear σ-model and CP^(n?1), are studied in the light of Elitzur's theorem, on the disappearance of infrared singularities at two dimensions. The consequences of the theorem are expressed in dimensional regularization, and issues like the proper analytic continuation to d = 2 + ε, the peculiarities of momentum-space Green functions near d = 2 and their renormalization, and the exponentiation of Green functions are clarified.The analysis is applied to compute the renormalization constants, and the gauge-invariant critical exponent η associated with the wave function of CP^(n?1) at one order higher than previously done. Finally, we conjecture on a possible connection between infrared finiteness and renormalizability.     

Large N classical equations and their quantum significance

We show that the large N limits of a wide variety of vector models may be obtained by studying the classical equations of motion. In particular, we derive a constraint which allows us to choose solutions of the classical field equations which directly give the correlation functions of N → ∞ quantum system. Models studied here include quantum mechanics on a sphere, two-dimensional linear and nonlinear O(N) field theories and the CP^N model.     

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.     

CPN−1 coupled to gauge fields: Mean field theory and monte carlo results

We define a two parameter lattice field theory which interpolates between the O (2N) Heisenberg model, pure U(1) gauge theory, and a lattice version of the CP^N?1 model. The phase diagram in space-time dimension d=4 is obtained by Monte Carlo simulation on a 4⁴ lattice, and the nature of the phases is discussed in mean field approximation.     

The CPN−1 model: A strong coupling lattice approach

We develop large N character-like expansion techniques for vector and vector-like models. These are used to compute the mass gaps and beta functions in the d = 2 CP^N?1 models. Surprisingly there is an intermediate region where neither strong coupling nor perturbation theory is applicable. This “unknown” region is a consequence of the non-commutativity of strong coupling and large N, an interesting mathematical effect not found in other models and due to an interesting physical phenomenon: superconfinement.     

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.     

Temperature Dependence of the CPN-1 Model and the Analogy with Quantum Chromodynamics

The two-dimensional CP^N-1 model — a simple field-theoretic analogue of four-dimensional quantum chromodynamics (QCD) — is analysed and reviewed. The major themes are the temperature dependence of the CP^N-1 model, and the analogy between CP^N-1 and QCD. A detailed treatment of the 1/N approximation of the CP^N-1 model is given. The main results emerging from this approximation are discussed at length. These are: asymptotic freedom, dimensional transmutation, confinement and topological charge nonquantization at zero temperature T = O, screening and topological charge quantization at finite temperature T. The analogy with QCD is explained in detail. A new, qualitative, analysis of the CP^N-1 model at finite temperature is introduced. This approach exploits the conformal invariance of the model to “heat” an arbitrary CP^N-1 field from T = O to finite temperature. This is achieved by conformal-transforming the flat Euclidean space-time of the T = O theory to the cylindrical space-time of the finite temperature theory.     

The transition from the lattice to the continuum: CPN−1 models at large N

We study in detail the large N solution for the CP^N?1 models on a euclidean lattice and their phase structure in D dimensions. In the two-dimensional case, we evaluate some physical quantities and discuss their behaviour in the continuum limit. In particular, for the internal energy, we found the analytic expression for the perturbative tail which, at large β, obscures the predicted renormalization group behaviour; such a tail, which, as we show, is reproduced by a mean field calculation at N=∞, must be subtracted in order to define a suitable scaling quantity. We also investigated the behaviour of the topological charge and we found no perturbative tail in agreement with previous Lüscher's results. Finally, some aspects of the supersymmetric case are discussed.     

On Weinberg's model of CP violation in gauge theories

The consequences of the Weinberg model of CP violation in gauge theories (ref. [1]) are discussed in detail. It is shown (using the valence quark hypothesis) that the contribution of the induced superweak interaction to the experimentally measured quantities η+? and η₀₀ are ≈m_K²/m_π² times larger than that of the direct milliweak decay K₂⁰→2π. The correction to the relation |η+?|/|η₀₀|=1 is also calculated in the model. We estimate the neutron dipole moment $\text{(D}{}_{\mathrm{<mtext>n</mtext>}}\text{??2.8·10}{}^{\mathrm{?25}}\text{e.cm.}\text{)}$ and note that the dipole moments of particles with strange quark constituents should be about three orders of magnitude larger $\text{(}\text{e.g.}\text{D}{}_{\mathrm{\Lambda }}\text{??1.15·10}{}^{\mathrm{?22}}\text{e·}\text{cm}$. The CP violating vertex d?s + gluon is also discussed.All physical quantities appear to depend crucially on the values of quark masses, so that CP violating effects in the models of the type considered may elucidate the nature of quark masses. We argue that the bare (or “mechanical”) values should be used in the calculations and this leads to a considerable difference in the estimates of various effects as compared to those of ref. [1].     

A simple solution to the strongCP-problem

We suggest soft breaking of Peccei-Quinn symmetry as a simple solution to the strongCP problem without the presence of the axion. In the context of theSU(2) _L ×U(1) model, it is shown how settingθ _tree=0 keeps θ calculable and, furthermore, extremely small: θ?10^?16. Unlike in the case of the axion the model is free from the cosmological domain wall problem. Possible extension of this idea to grand unification is discussed.     

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.     

Rotational properties of the odd-Z transfermium nucleus 255Lr by a particle-number-conserving method in the cranked shell model

Experimentally observed ground state band based on the 1/2-[521] Nilsson state and the first exited band based on the 7/2-[514]Nilsson state of the odd-Z nucleus ~(255)Lr are studied by the cranked shell model(CSM) with the paring correlations treated by the particle-number-conserving(PNC) method. This is the first time the detailed theoretical investigations are performed on these rotational bands. Both experimental kinematic and dynamic moments of inertia(J~(1)and J~(2)) versus rotational frequency are reproduced quite well by the PNC-CSM calculations. By comparing the theoretical kinematic moment of inertia J~(1) with the experimental ones extracted from different spin assignments, the spin 17/2~-→13/2~- is assigned to the lowest-lying 196.6(5) ke V transition of the 1/2~-[521] band, and 15/2~-→11/2~- to the 189(1) ke V transition of the 7/2~-[514] band, respectively. The proton N = 7 major shell is included in the calculations. The intruder of the high- j low-? 1 j_((15)/2)(1/2~-[770]) orbital at the high spin leads to band-crossings at ω≈0.20( ω≈0.25) Me V for the 7/2~-[514] α =-1/2(α = +1/2) band, and at ω≈0.175 Me V for the1/2~-[521] α =-1/2 band, respectively. Further investigations show that the band-crossing frequencies are quadrupole deformation dependent.     

Liquid–gas and other unusual thermal phase transitions in some large-N magnets

Much insight into the low temperature properties of quantum magnets has been gained by generalizing them to symmetry groups of order N, and then studying the large-N limit. In this paper we consider an unusual aspect of their finite temperature behavior—their exhibiting a phase transition between a perfectly paramagnetic state and a paramagnetic state with a finite correlation length at N=∞. We analyze this phenomenon in some detail in the large “spin” (classical) limit of the SU(N) ferromagnet which is also a lattice discretization of the CP^N−1 model. We show that at N=∞ the order of the transition is governed by lattice connectivity. At finite values of N, the transition goes away in one or less dimension but survives on many lattices in two dimensions and higher, for sufficiently large N. The latter conclusion contradicts a recent conjecture of Sokal and Starinets [Nucl. Phys. B 601 (2001) 425], yet is consistent with the known finite temperature behavior of the SU(2) case. We also report closely related first order paramagnet–ferromagnet transitions at large N and shed light on a violation of Elitzur's theorem at infinite N via the large-q limit of the q state Potts model, reformulated as an Ising gauge theory.     

N1−3H configurations in 4He

Wavefunctions for N¹?³H configurations in ⁴He are calculated using NN → N¹N transition potentials V and ³HN → N¹³H transition potentials from a two-nucleon exchange mechanism. The radial N¹?³H momentum distributions peak at ～ fm^?1 which is much lower than the short-ranged V. For (πN) S-wave isobars, e.g. N¹(1535) and N¹(1700), N¹?³H probabilities are estimated to be ～2‰ each, while those for N¹'s in higher (πN) partial waves are much smaller.     

CP violation at colliders

The prospects of experimental detection ofCP violation ate ⁺ e ^? and\(pp/p\bar p\) colliders are reviewed. After a general discussion on the quantities which can measureCP violation and on the implications of theCPT theorem, various possibilities of measuringCP violation arising outside the standard model are taken up.CP violation in leptonic processes, especially polarization effects ine ⁺ e ^?→l ⁺ l ^? are discussed next.CP violation in\(t\overline t \) andW ⁺ W ^? production and decay is also described.