Symmetry breaking by instantons in supergravity

The non-zero modes of different spin operators on the background of a self-dual gravitational instanton are all related by global supersymmetry transformations and completely cancel in the one-loop term, which is determined entirely by the zero modes. We derive the number of zero modes of each spin. In an asymptotically locally Euclidean self-dual instanton there are 2τ $\text{spin}\text{-}\text{3}\text{2}$ zero modes and 3τ spin-2 zero modes, where τ is the Hirzebruch signature. Up to 3 of the spin-2 zero modes (depending on boundary conditions) may correspond to global rotations. The $\text{spin}\text{-}\text{3}\text{2}$ zero modes break the U(1) chiral symmetry and give rise to helicity-changing amplitudes. Together with the spin-2 zero modes they determine the trace anomaly or scaling behaviour. We can compare our results with the perturbation theory predictions for the axial vector current and trace anomalies in K3, the unique compact self-dual gravitational instanton, because in this case there are no boundary terms. We obtain agreement.     

Classical solutions and the energy-momentum tensor

A hyperelliptic two-meron solution of the massless scalar φ^N theory in $\text{n =}\text{2N}\text{(N ? 2)}$ Euclidean dimensions is given. This solution (which interpolates between the two-meron solution and the instanton solution of this theory) is used to illustrate several theory-independent statements which can be made about the energy-momentum tensor for instanton, meron and elliptic meron solutions of all scale invariant classical field theories.     

Chirality,self-duality,and supergravity counterterms

The restrictions imposed by chirality invariance on higher-loop counterterms in supergravity are obtained. It is shown that their dependence on gravitational curvature and on spin-$\text{3}\text{2}$ field strength is such that they vanish when these quantities are self-or anti-self-dual. Implications regarding quantum corrections in the instanton sector are discussed.     

Quantum fluctuations in the one-instanton sector of the CPn−1 model

Within the two-dimensional CP^n?1 model we calculate the instanton density function in the dilute-gas limit by studying the quantum fluctuations in the one-instanton sector in the one-loop approximation. The result disagrees with the $\text{1}\text{n}$ expansion. Green functions in the one-instanton background are displayed.     

The role of instantons in quantum chromodynamics: (II). Hadronic structure

The “constituent quark” (or “valon”) substructure of ordinary hadrons is the consequence of small instanton radius $\text{?}{}_{\mathrm{<mtext>c</mtext>}}\text{?}\text{1}\text{3}\text{fm}$ in the QCD vacuum. The same parameter determines hadronic masses and other parameters, in particular the unusually large mass scale for “exceptional” hadrons with zero spin.     

Yang-Mills instantons and the S-matrix

We present a scheme for calculating gauge-invariant S-matrix elements in the presence of instantons. We exploit the conformal invariance of the zero-mass field equations. The asymptotic in and out states are defined by their values on null infinity $\text{J}$. We use this method to calculate to lowest-order S-matrix elements for scalar particles and fermions in a dilute gas of SU(2) instantons and anti-instantons. The scalar particles acquire an effective mass and an effective interaction of the form $\text{exp}\text{(?(?}{}^2\text{/16π) ?}\text{?}\text{)}$, where ? is the scale of the instanton, plus other interactions which cannot be presented by a local effective lagrangian. The fermions acquire the effective lagrangian obtained by 't Hooft. In the case of a single flavour of fermions, this corresponds to a mass term.     

A compact rotating gravitational instanton

A compact rotating gravitational instanton (a positive-definite metric solution of the Einstein equations with Λ term) is presented. The manifold is the nontrivial S² fibre bundle over S² and has χ = 4, τ = 0, but no spinor structure. The metric can be obtained from a special limit of the positive-definite analytic extension of the Kerr-de Sitter metric or alternatively from the Taub-NUT metric with Λ term. The action is about $\text{4}\text{1}\text{2}\text{\%}$ less negative than that of the Einstein metric on the trivial bundle S² × S².     

The role of instantons in generation of mesonic mass spectrum

It is suggested that instantons play the leading role in the mixing of $\text{s}\text{s and}\text{u}\text{u}\text{+}\text{d}\text{d}$ quark states in mesonic nonets and in the explanation of the Okubo-Zweig-Iizuka (OZI) rule. The non-diagonal polarization operator Π^su = 〈0|T{j^s(x)j^u(y)}0〉 is considered where $\text{j}{}^{\mathrm{<mtext>s</mtext>}}\text{=}\text{s}\text{O}\text{s}$ and $\text{j}{}^{\mathrm{<mtext>u</mtext>}}\text{=}\text{u}\text{O}\text{u}$ are the currents of the s and u quarks. It is proved that in the dilute instanton gas approximation for quarks in the external instanton field Π^su = 0 for the vector and tensor currents and Π^su≠0 for the axial current. Hence, after saturating Π^su by the low-lying mesonic states, we obtain the qualitative explanation of the OZI rule. The Q² dependence of the non-diagonal polarization operator of the axial currents, Π^su(Q²), is calculated and compared with the η′ meson pole term. Taking account of terms ～m_q² allows one, using the experimentally known ηη′ mixing angle, to find the η′ meson coupling constant with the axial current F_η′ ≈ 150 MeV and to estimate the η′π mixing angle.     

Lepton spectrum in semileptonic b-decay

We present some results from the computation of the charged lepton energy distribution in the decay b → c + $\text{l}$^? + $\text{ν}$ taking into account first order QCD corrections.     

Theory of the quantized Hall effect (III)

In the previous paper, we have demonstrated the need for a phase transition as a function of θ in the non-linear σ-model describing the quantized Hall effect. In this work, we present arguments for the occurrence of exactly such a transition. We make use of a dilute gas instanton approximation as well as present a more rigorous duality argument to show that the usual scaling of the conductivity to zero at large distances is altered whenever $\text{σ}{}_{\mathrm{xy}}{}^{\mathrm{(0)}}\text{≈}\text{1}\text{2}\text{ne}{}^2\text{/h}$, n integer. This then completes our theory of the quantized Hall effect.     

Band model parameters for the parallel bands of linear triatomic molecules—I. Theory

Generalized equations are developed for computing the band model parameters ($\text{S}\text{d}$), ($\text{S}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/D}$) and ($\text{1}\text{d}$) for the parallel bands of linear triatomic molecules. The convergence of these band-model parameters as a function of the number of quantum states included in a computation is discussed for the 4.3 μm CO₂ band at various temperatures and wavenumbers. Comparisons are made to previous theoretical calculations for the 4.3 μm CO₂ band, where it is shown that an insufficient number of quantum states was included in the earlier work to insure convergence of the band-model parameters.     

Another connection between the Λ parameters of the euclidean lattice and continuum QCD

We compute the relation between the lattice and the short-distance q$\text{q}$-force Λ parameters. The result of this perturbative computation is in striking agreement with the Creutz Monte Carlo estimate.     

Quantizing a classically ergodic system: Sinai's billiard and the KKR method

Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels E_n are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing E_n as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is $\text{O}$(S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution $\text{d}\text{?}\text{(E)}$ and an oscillatory contribution d_osc(E). $\text{d}\text{?}$ is $\text{O}$(?^?2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. d_osc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with $\text{O}$$\text{(?}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext><mtext>)</mtext>}}$ to d_osc, while isolated closed orbits (bouncing between discs) contribute terms with $\text{O}$(?^?1) to d_osc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual E_n (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum.