The φ4 lattice field theory as an asymptotic expansion about the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins, an asymptotic expansion of expectations about the Ising limit is established in inverse powers of the bare coupling constant λ. In the thermodynamic limit (N → ∞), the expansion is expected to be valid in the noncritical region of the Ising system.     

Revisiting Cardassian model and cosmic constraint

Boundary conformal field theory (BCFT) is the study of conformal field theory (CFT) in semi-infinite space-time. In the non-relativistic limit (x???x,t??t,???0), the boundary conformal algebra changes to boundary Galilean conformal algebra (BGCA). In this work, some aspects of AdS/BCFT in the non-relativistic limit were explored. We constrain correlation functions of Galilean conformal invariant fields with BGCA generators. For a situation with a boundary condition at surface x=0 ( $z=\overline{z}$ ), our result agrees with the non-relativistic limit of the BCFT two-point function. We also introduce the holographic dual of boundary Galilean conformal field theory.     

The :φ2: Field in theP(φ)2 model

Euclidean Field Theory techniques are used to study the Schwinger functions and characteristic function of the :φ²: field in evenP(φ)₂ models. The infinite volume limit is obtained for Half-Dirichlet boundary conditions by means of correlation inequalities. Analytic continuation yields Lorentz invariant Wightman functions. It is shown that, in the infinite volume limit, <:φ(x)²:>≧0 for both the Half and the Full-Dirichlet (λφ⁴)₂ model. This result also holds for a finite volume with periodic boundary conditions.     

Tensor representations of conformal algebra and conformally covariant operator product expansion

A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O(4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O(4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n-point functions from causality are described.     

The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type I × f N

In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type I × _f N, where I is an interval of the real line and N is a compact, d-dimensional Riemannian manifold either with or without boundary. Starting from an integral representation of the spectral zeta function, we find its analytic continuation by exploiting the WKB asymptotic expansion of the eigenfunctions of the Laplace operator on M for which a detailed analysis is presented. We apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion.     

Regge pole model for invariant functions. IV

TheNN→NN andγN→πN differential cross sections for polarized targets are discussed in terms of invariant functions. In former fits of high energy experimental data certain of these invariant functions came out to be small. Based on these fits thepn→np,p¯p→ n¯n, γp→ nπ ⁺ andγn→ pπ ^? differential cross sections are predicted to be independent of the target polarization.     

Exact one-point function of N=1 super-Liouville theory with boundary

In this paper, exact one-point functions of N=1 super-Liouville field theory in two-dimensional space–time with appropriate boundary conditions are presented. Exact results are derived both for the theory defined on a pseudosphere with discrete (NS) boundary conditions and for the theory with explicit boundary actions which preserves super conformal symmetries. We provide various consistency checks. We also show that these one-point functions can be related to a generalized Cardy conditions along with corresponding modular S-matrices. Using this result, we conjecture the dependence of the boundary two-point functions of the (NS) boundary operators on the boundary parameter.     

From N = 1 to N = 4 extended supersymmetric Yang-Mills fields in a covariant Wess-Zumino gauge

It is assumed that N = 1 supersymmetric Yang-Mills fields coupled to chiral matter fields can be renormalized in a covariant Wess-Zumino gauge with a minimal number of subtractions so that the Ward identities of supersymmetry, ordinary gauge invariance and matter-field-flavour symmetries are satisfied. The chiral Yukawa couplings are supposed to remain unrenormalized. I show that on the basis of these assumptions an N = 4 extended manifestly O(4) invariant theory can be constructed with finite Yukawa and φ⁴ couplings. A consequence of these non-renormalizations is the vanishing of the renormalization group β function.     

Singletons and logarithmic CFT in ADS/CFT correspondence

We discuss a possible relation between singletons in AdS space and logarithmic conformal field theories at the boundary of AdS. It is shown that the bulk Lagrangian for singleton field (singleton dipole) induces on the boundary the two-point correlation function for logarithmic pair. Bulk interpretation of mixing between logarithmic operator D and zero mode operator C under the scale transformation is discussed as well as some other issues.     

Hard pions and axial meson exchange current effects in negative muon capture in deuterium

The contribution of the axial meson exchange current effects to the doublet transition rate in the reaction μ^? +d → 2n+ ν_μ is calculated by using the minimal, chiral and approximately gauge invariant Lagrangian model for the A₁ρπ system. The contribution from the ρ-π weak decay process current usually considered is found to be nearly cancelled by that from the A₁ pole graph which is prescribed by the underlying invariance principles. Correct treatment of the N¹ propagator in the N¹ excitation current of the pion range leads to ≈ 30 % suppression of the N¹ effect.     

Contributions of γd → πNN + π0 d channels to the spin asymmetry and the Gerasimov-Drell-Hearn integral for the deuteron

Contributions from the semi-exclusive channels γd → π^± NN + π⁰ d and γd → π⁰ X (X=pn or d) to the deuteron spin asymmetry and the Gerasimov-Drell-Hearn (GDH) integral are explicitly evaluated using an enhanced elementary pion photoproduction operator and a realistic, high-precision potential model for the deuteron wave function. The sensitivity of the results to the elementary pion photoproduction operator is also investigated and considerable dependence is found. Results for the deuteron GDH integral are compared with the measurements from A2 and GDH@MAMI Collaborations.     

Failure of length scaling in Wilson's ϵ expansion

We show that length scaling of the four-point scattering amplitude in Wilson's ? expansion is not consistent in the order of ?³. However, in conformity with conformal invariance at the critical point, momentum scaling in a given channel is consistent. This latter method permits us to calculate the dimension of the field φ² at the critical point without recourse to length scaling and one finds $\text{d}{}_{\mathrm{\phi }}\text{2 =}\text{d}\text{2}\text{?}\text{1}\text{ν}\text{to O}\text{(?}{}^2\text{)}$ as if length scaling were true. However, this does not imply Kadanoff's relation 2?α = νd which is predicted on length scaling. Indeed the above-mentioned inconsistency makes impossible the determination of α by these methods.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.     

A model of spontaneous symmetry breakdown in spatially flat cosmological spacetimes

This paper is an elaboration of a previous short exposition of a theory of spontaneous symmetry breaking in a conformally coupled, massless λø⁴ model in a spatially flat Robertson-Walker spacetime. Under the weakened global boundary condition allowing the physical spacetime to be conformal to only a portion of the Minkowski spacetime, the model admits a pair of degenerate vacua in which the ø → ? ø symmetry is spontaneously broken. The model is formulated as a euclidean field theory in a space with a positive-definite metric obtained by analytically continuing the conformal time coordinate. An appropriate time-dependent zero energy solution of the euclidean equation of motion representing the field configuration in the asymmetric vacuum is considered and the corresponding quantum trace anomaly 〈T_μ^μ〉 is computed in the one-loop approximation. The nontrivial infrared behavior of the model due to the singular nature of the classical background field forces a modification of the boundary conditions on the propagator. A general form for an “improved|DD one-loop trace anomaly is found by a simple argument based on renormalization group invariance. Via the Einstein equation, the trace anomaly leads to a self-consistent dynamical equation for the cosmic expansion scale factor. Some physical aspects of the back-reaction problem based on a simple power law model of the expansion scale factor are discussed.     

Mass generation in the O(3) non-linear σ model

We study the two-point function of the azimuthal angle, G^(φ)(x) = 〈e^iφ(x)e^?iφ(0)〉_inst [φ = arg (q¹ + iq²), where q^a is a three-component unit vector field], in the dense instanton gas approximation for the two-dimensional O(3) non-linear σ model. We find that G^(φ) (x) decreases exponentially as |x| → ∞. This suggests that the dense instanton gas may generate a mass gap in the O(3) non-linear σ model. The physical mechanism of this mass generation is also discussed.     

Wilson's operator expansion: Can it fail?

Within the framework of some simple models we discuss the status of the operator product expansion (OPE) in the presence of nonperturbative effects. We consider, in particular, the 4d Higgs model, 2d sigma model and the Schwinger model. The general formulation of OPE is presented and it is demonstrated that there exists a consistent procedure allowing one to define unambiguously both coefficient functions and matrix elements of composite operators. One of the key elements of the procedure is the introduction of an auxiliary parameter, the normalization point μ. For the simplest T-products discussed in the literature earlier we construct the corresponding OPE explicitly. Then we check its validity by comparing the results for the two-point functions with independent direct calculations of the same correlators. Although the general procedure is standard and does not vary from one theory to another, numerically the relative role of perturbative and nonperturbative contributions in vacuum condensates is different in different theories. The two extremes considered are the λ?⁴ theory with no spontaneous breaking of the symmetry and the O(N) sigma model in the limit N → ∞. In the former case there is only perturbative contribution to (?²), while in the latter case the perturbative pieces are suppressed by 1/N factors. Numerically QCD is much closer to the O(N) sigma model in the large-N limit. Comments on specific features of QCD are presented.     

The process γγ → X by parton model and prediction of equivalent-photon method for ee → ee + X

A parton model for the virtual photon process γ → X is examined. It is assumed that a scaling property applies and the total cross section is σ_γγ→X = s^?1φ(ξ), where the function ? is dependent on the scaling variable ξ defined by ξ = s/(q²₁q²₂). The cross section for the process ee → + X is calculated using the equivalent-photon technique.     

Central limit theorem in quantum field theory,generalized partons and application to deep-inelastic scattering

We prove the following elementary theorem. Ifφ ¹,...,φ ^N is a sequence of fields having identical, thougharbitrary, interactions but not interacting with each other and 〈φ _n 〉0,i=1,...,N then the generating functional of the «average» field φ^(N) may be explicitly obtained and may be written in terms of the two-point function of any of the fields φ _i . The theorem is then applied to define generalized parton fields \(\psi _j = \sum\limits_{i = 1}^N {\psi _{ij} } /\sqrt N \) as «averages» of basic fieldsψ _ij havingarbitrary interactions but not interacting with each other. We show that in the limitN→∞ Bjorken scaling, as observed at energies not too high, may be obtained if only quanta associated with generalized parton fields are excited in the hadron by the virtual photonwith no reference to the details of the underlying dynamics. ForN<∞, and the excitation of other quanta as well lead to a systematic breaking of scale invariance and the details of the dynamics are necessarily recovered which are expected to be applicable at higher energy regimes.     

Vacuum instability in scalar field theories

The class of scalar field theories with interaction gφ^2N?1, are studied using the semi-classical approximation. The imaginary part of the vertex functions generated by tunnelling out of the metastable ground state is calculated to first order. Using this result, the leading asymptotic behaviour of the renormalisation group β function for φ³ field theory is obtained in six dimensions. The validity of this result is discussed in view of the extra singularities which appear when the theory is just renormalisable. The structure of the perturbation expansion for n component φ³ theory is also discussed, and cases in which these theories yield perturbation expansions which are Borel summable, are pointed out.     

Dimensional regularization of massless theories in spherical space-time

The use of space-time curvature as an infra-red cut-off has been suggested for massless theories. In this paper we investigate the renormalization of massless theories in a spherical space-time (Euclidean version of de Sitter space) using dimensional regularization. Naive expectations are confirmed, namely that the coupling constant and wave-function renormalizations are independent of the curvature. Furthermore the curvature does not induce divergent mass terms or vacuum field values as would be possible on purely dimensional grounds. Although we have investigated only scalar field theories, φ⁴ theory in four dimensions and φ³ theory in six, these results are encouraging for an application of the method to gauge theories.Formally massless theories are conformally invariant so the formulation of the theory in a spherical space ought to be equivalent to its formulation in flat space. In fact the renormalization procedure breaks conformal invariance and removes this equivalence. We show that to achieve the flat space limit it is necessary to invoke the aid of the renormalization group. Thus the zero curvature limit can be achieved for infra-red stable theories (φ₄⁴) but not for infra-red unstable theories (φ₆³ as might be expected.