Borel summability of the mass and theS-matrix in ϕ4 models

We show that in the ₂ ⁴ theory, the physical mass and the two-bodyS-matrix are Borel summable in the coupling constant at =0.     

Triviality of φ 4 4 and all that in a hierarchical model approximation

The infrared behavior of lattice _d ⁴ ,d 4, and dipole gases ind 1 is rigorously shown to be Gaussian within the context of a hierarchical approximation. Methods are developed to generalize the result beyond this approximation.On leave of absence from the Department of Mathematical Methods of Physics, University, 00-682 Warsaw, Hoza 74, Poland.     

Construction and Borel summability of infrared Φ 4 4 by a phase space expansion

We construct the thermodynamic limit of the critical (massless) ⁴ model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion. This expansion requires in a natural way the introduction of effective or running constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial rôle. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.     

The distributional Borel summability and the large coupling φ4 lattice fields

Following 't Hooft we extend the Borel sum and the Watson-Nevanlinna criterion by allowing distributional transforms. This enables us to prove that the characteristic function of the measure of anyg ^{–2 4} finite lattice field is the sum of a power series expansion obtained by fixing exponentially small terms in the coefficients. The same result is obtained for the trace of the double well semigroup approximated by then ^th order Trotter formula.     

Phenomenology of the hierarchical lepton mass spectrum in the flipped SU(5)×U(1) string model

A detailed phenomenological analysis of the lepton mass matrices and their implications in the low energy theory are discussed, within the recently proposed SU(5)×U(1) string model. The unification scale is highly constrained while the Yukawa couplings lie in a natural region. The flavour changing decays μ→eγ, μ→3e, μ→e are highly suppressed while the depletion in the in the flux of muon neutrinos reported by the Kamiokande is explained through ν_μ↔ν_τ oscillations.     

The hierarchicalφ 4-trajectory by perturbation theory in a running coupling and its logarithmby perturbation theory in a running coupling and its logarithm

We compute the hierarchical ⁴- in terms of perturbation theory in a running coupling. In the three-dimensional case we resolve a singularity due to resonance of power counting factors in terms of logarithms of the running coupling. Numerical data are presented and the limits of validity explored. We also compute moving eigenvalues and eigenvectors on the trajectory as well as their fusion rules.     

Local existence of the Borel transform in Euclidean φ 4 4

We bound rigorously the large order behaviour of ₄ ⁴ euclidean perturbative quantum field theory, as the simplest example of renormalizable, but non-super-renormalizable theory. The needed methods are developed to take into account the structure of renormalization, which plays a crucial role in the estimates. As a main thorem, it is shown that the Schwinger functions at ordern are bounded byK ⁿ n!, which implies a finite radius of convergence for the Borel transform of the perturbation series.     

On the appearance of the λ-point in a weakly nonideal Bose gas and the two-liquid Thiess-Landau model

Ultrasecondly quantized operators are defined, studied, and used to obtain asymptotic properties of the spectrum of a superfluidity problem with boundary. Dedicated to the memory of Vladimir Borovikov     

Convergence of perturbation series for renormalization constants in Kraichnan model with “frozen” velocity field

An instanton for the Kraichnan model with a ‘frozen’ velocity field is found. High-order asymptotics of the quantum-field perturbation expansion for the renormalization constant Z _ν are investigated. It is shown that this expansion is convergent, and the radius of convergence is calculated.     

Local existence of the Borel transform in euclidean massless Φ 4 4

We extend the methods of [1] to prove large order estimates on the renormalized Feynman amplitudes of massless ₄ ⁴ euclidean field theory, at non-exceptional momenta. The Borel transform of the perturbative series is analytic in a disk centered at the origin of the complex plane. This result is a step towards the rigorous investigation of the infra-red singularities in the Borel plane, for theories containing massless particles, like the gauge theories.     

Borel summability of the 1/N expansion for theN-vector [O(N) non-linear σ] models

We construct an analytic interpolation in 1/N for theN-vector [O(N) non-linear ] models withN-component fields on a lattice. This interpolation, valid at sufficiently high temperatures, extends over a large domain in the complex plane containing the half plane Re(1/N)>0. We use this result to show that the 1/N expansion of the free energy density and of the correlation functions is Borel summable in the thermodynamic limit and at high temperature.On leave from C.P.T., Ecole Polytechnique, F-91128 Palaiseau Cédex, France     

Study of ψ(4415) in the QPC model

Some authors consider the ψ(4415) to be the 4S or 5S excited tate of a cc pair. Starting from this assumption, we study he decays of the ψ(4415) to DD, D^*D^*, D_SD_S, D_S^*D_S^*, and get the corresponding branching ratios in terms of the Quark-Pair-Creation (QPC) model. Compared with the experimental data, we find that the results of 4S state agree much better than those of the 5S state. Therefore, it is more reasonable to assume the ψ(4415) to be a 4S state.     

Bound fermion states in the field of a soliton of the nonlinear O(3) σ model

The (2 + 1)-dimensional nonlinear O(3) σ model whose n field is coupled to the fermion field by the Yukawa interaction has been examined. The cases of the isosinglet and isodoublet fermion fields with respect to the internal symmetry group have been considered. It has been shown that bound states of the fermion in the n field of a soliton of the nonlinear O(3) σ model exist for some variants of the Yukawa interaction. The absence of zeroth fermion modes in the n field of the soliton has been established. The properties of the ground state of the fermion have been numerically studied. In particular, it has been shown that an increase in the spatial size of the soliton results in a decrease in the energy of the ground state. This leads to the instability of the soliton in a certain region of the parameters of the model.     

Study of hadrons using the Gaussian functional method in the O(4) linear σ model

We study properties of hadrons in the O(4) linear σ model, where we take into account fluctuations of mesons around their mean field values using the Gaussian functional(GF) method. In the GF method we calculate dressed σ and π masses, where we include the effect of fluctuations of mesons to find a better ground state wave function than the mean field approximation. Then we solve the Bethe-Salpeter equations and calculate physical σand π masses. We recover the Nambu-Goldstone theorem for the physical pion mass to be zero in the chiral limit.The σ meson is a strongly correlated meson-meson state, and seems to have a two meson composite structure. We calculate σ and π masses as functions of temperature for both the chiral limit and explicit chiral symmetry breaking case. We get similar behaviors for the physical σ and π masses as the case of the mean field approximation, but the coupling constants are much larger than the values of the case of the mean field approximation.     

A phase cell cluster expansion for a hierarchical Ø 3 4 model

The formalism developed in a previous paper is applied to yield a phase cell cluster expansion for a hierarchical ø ₃ ⁴ model. The field is expanded into modes with specific renormalization group scaling properties. The present cluster expansion for a vacuum expectation value is formally the natural factorization of each term in the perturbation expansion into the contribution of modes connected to the variables in the expectation via interactions, and that of the complementary set. The expectation value is thus realized as a sum of contributions due tofinite subsets of the modes. We emphasize the following additional features:

1. Partitions of unity are not used.
2. There areessentially no cut-offs.
3. The expansion is developed directly, without an initial need to prove an ultraviolet stability bound, the most difficult part of the traditional approach.

Our main interest in the present phase cell cluster expansion is founded in the belief that it may be the right vehicle for proving the existence of a nontrivial four-dimensional field theory.     

Renormalization of Dyson's hierarchical vector valued φ4 model at low temperatures

We investigate Dyson's hierarchical vector valued ⁴ model at low temperatures. The case is considered. The pure phase is constructed, and the existence of its large scale limit is proved. The limit is Gaussian, but an unusual normalization has to be chosen. In the direction of the spontaneous magnetization one has to divide by the square root of the volume, but in the orthogonal direction one has to divide by a different power of the volume for all low temperatures.     

A remark on higher orders of perturbation theory in Euclidean (super-φ3)4

Some aspects of the possible asymptotic behavior of Euclidean (super- ³)₄ (the Wess-Zumino model) are considered.Talk given at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.I wish to thank W.Lang for a useful conversation.     

Fermionic perturbation theory for the statistical mechanics of the nonlinear Schrödinger model

A fermionic perturbation theory is developed for the statistical mechanics of the nonlinear Schrödinger model. The theory is based on an interacting-fermion picture of the Bethe wave function. The inner product of the Bethe wave function is explicitly evaluated, and a simple graphical representation of it is given. The basic equations obtained for the free energy agree with those of Yang and Yang. In particular, the present theory gives a clear-cut meaning to the function of Yang and Yang: It represents a fermion energy at finite temperatures.