Non-perturbative contributions to the hot electroweak potential and the crossover

We discuss non-perturbative contributions to the three-dimensional 1-loop effective potential of the electroweak theory at high temperatures in the framework of the stochastic vacuum model. It assumes a gauge-field background with Gaussian correlations which leads to confinement. The instability of 〈F²〉 = 0 in Yang-Mills theory appears for small Higgs expectation value 〈φ²〉 in an IR regularized form. The gauge-boson propagator obtains a positive momentum-dependent “diamagnetic” effective (mass)² due to confinement effects and a negative one due to “paramagnetic” spin-spin interactions which are related to the 〈F²〉 = 0 instability. Numerical evaluation of an approximate effective potential containing these masses shows qualitatively the fading away of the first-order phase transition with increasing Higgs mass which was observed in lattice calculations. The crossover point can be roughly determined postulating that the effective φ⁴ and φ² terms vanish there.     

The effective potential and the bound states for a class of two dimensional gauge theories

The effective potential is calculated for a two dimensionalU(N) gauge theory with scalar quarks to leading order in the 1/N expansion. If there is noφ ⁴ interaction present, the potential is unbounded from below. If theφ ⁴ interaction is present, the potential is bounded from below and there is an unbroken and a spontaneously broken symmetry phase. The bound state spectrum of the unbroken phase is very similar to that of anU(N) gauge theory without theφ ⁴ term.     

Spontaneous symmetry breaking in the present and early universe

Spontaneous symmetry breaking in λφ⁴ theory is formulated in terms of the operator φ², and in a manner which requires no specific expectation value to be assigned to φ. At the one-loop order of perturbation theory, a renormalized effective action for a field ζ, linearly related to φ², is obtained as a gradient expansion. Potential advantages of this formulation in applications to phase transitions in the early universe are discussed. They include the possibilities (i) of obtaining a well-defined semiclassical equation of motion, and (ii) of following the evolution of a field theory from an initial symmetrical high temperature state without the introduction, ad hoc, of regions in which 〈φ〉 ≠ 0.     

Renormalization-group improvement of effective actions beyond summation of leading logarithms

Invariance of the effective action under changes of the renormalization scale μ leads to relations between those (presumably calculated) terms independent of μ at a given order of perturbation theory and those higher-order terms dependent on logarithms of μ. This relationship leads to differential equations for a sequence of functions, the solutions of which give closed form expressions for the sum of all leading logs, next to leading logs, and subsequent subleading logarithmic contributions to the effective action. The renormalization group is thus shown to provide information about a model beyond the scale dependence of the model's couplings and masses. This procedure is illustrated using the φ₆³ model and Yang–Mills theory. In the latter instance, it is also shown by using a modified summation procedure that the μ dependence of the effective action resides solely in a multiplicative factor of g²(μ) (the running coupling). This approach is also shown to lead to a novel expansion for the running coupling in terms of the one-loop coupling that does not require an order-by-order redefinition of the scale factor ΛQCD. Finally, logarithmic contributions of the instanton size to the effective action of an SU(2) gauge theory are summed, allowing a determination of the asymptotic dependence on the instanton size ρ as ρ goes to infinity to all orders in the SU(2) coupling constant.     

Dimensional renormalization of scalar field theory in curved space-time

The regularization and renormalization of an interacting scalar field φ in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, “vacuum” counter terms must also be introduced. These involve zero, first, and second powers of the Reimann curvature tensor R_αβψδ. Moreover, the renormalizability of the theory requires that the Lagrange function couple φ² to the curvature scalar R with a coupling constant η. The coupling η must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n ? 4. The coefficient functions of the counter terms determine the construction of φ² and φ⁴ in terms of renormalized composite operators 1, [φ²], and [φ⁴]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [φ²] and [φ⁴] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R² interaction which is of fifth order in the coupling constant λ.     

Familon emissivity of dense e − e + plasma in an external magnetic field

Emission of a familon due to the processes e ^? → e ^?+φ and e ^? → μ ^?+φ by dense magnetized plasma is analyzed as a possible mechanism of energy and momentum losses by astrophysical objects. The field-induced effective familon-photon coupling in the familon-emission process is taken into account. The contribution of these processes to the familon emissivity of a magnetized plasma in a supernova explosion is calculated. It is shown that there is asymmetry of familon emission in the process e ^? → μ ^?+φ.     

Soliton-like solutions of some nonlinear theories and transformations among them

The observation that the soliton-like solutions of a given second-order nonlinear differential equation define the separatrix of the equivalent autonomous system is used to obtain the one-soliton solutions for theφ ⁴ theories (the usual and the one with the wrong sign of the mass term), theφ ⁶, theφ ⁸, the sine-Gordon theories and the KdV equation. Transformations are given which transform the sine-Gordon equation into an equation belonging to theφ ²ⁿ class of theories. A procedure is evolved for obtaining the two-soliton solutions for the sine-Gordon theory without the use of Backlund transformations; it is suggested that this procedure may be useful for investigating the existence of similar solutions for theories of the polynomial type.     

Mean work functions effective for negative-ionic, electronic and positive-ionic emissions from polycrystalline surfaces

To elucidate the thermionic property of polycrystalline surfaces, a further study is made on the mean work functions (φ⁻, φ^e and φ⁺) effective for negative-ionic, electronic and positive-ionic emissions. Comparison between theoretical analyses and experimental data yields the conclusions as follows. (1) The equation of φ⁻ = φ^e holds always with both mono- and polycrystalline surfaces. (2) The relation of φ⁻ = φ^e < φ⁺ applies to polycrystalline surfaces because they bear the thermionic contrast (Δφ^* ≡ φ⁺ − φ^e > 0). (3) The value of Δφ^* ranges from ∼0.4 to 0.9 eV depending upon the surface species of polycrystalline metals (e.g., W, Re and Pt), whilst Δφ^* = 0 for monocrystalline surfaces. (4) When the degree of monocrystallization (δ_m) is less than ∼50%, the theoretical value of Δφ^* is virtually independent of δ_m and agrees well with experimental data, nearly the same within ±0.1 eV among the so-called “polycrystalline” surfaces of W. (5) As δ_m increases beyond ∼80 up to 100%, Δφ^* decreases rapidly down to 0 eV, showing again a good agreement between theory and experiment. (6) Our theoretical model is valid in evaluating the effective mean work functions, irrespective of the range of δ_m.     

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.     

On the algebraic consistency conditions in canonical λφ4 theory

The algebraic consistency requirements for the short distance behaviour of the operator product expansions φφ, φJ and JJ where (□ ? m²) φ = λJ, and J is local and trilinear (λφ⁴ theory) are studied. It is shown that in four dimensions (D = 4) under the assumption of a canonical scheme we cannot use “naive” Wilson expansions, abstracted from the short-distance behaviour of the products of free Wick powers. The inconsistency is also studied in an arbitrary and continuous number of dimensions and it disappears for D< 4.     

Phase transitions in De Sitter space

This paper uses zeta-function regularisation to calculate the one-loop functional determinants for fields of any spin in De Sitter space. As an example, we investigate the Coleman-Weinberg spontaneous symmetry breaking mechanism in massless scalar electrodynamics. The effective potential is calculated in Landau gauge. It depends upon the curvature, and upon the renormalised value of ζ (inζRφ²). The phase transition will be first or second order, and the critical curvature and mass are found. The methods can be applied to any gauge theory.     

Comment on azimuthal dependence in leptoproduction

It is pointed out that in the conventional calculation a large part of 〈cosφ〉 in leptoproduction comes from configurations in which the outgoing partons have small p_T. This indicates that the conventional QCD prediction for 〈cosφ〉 is not reliable in the experimentally available Q² regions at present.     

Barrier to internal rotation of the silyl group in cyclopropylsilane from combination band spectra

Combination band spectra arising in the region 2000–2300 cm^?1 of cyclopropylsilane can be attributed to the sum and difference bands of the internal rotation with an SiH stretching mode. Analysis of these spectra leads to a potential function for the internal rotation V(cm^?1) = 693.92(1 ? cos 3φ) ? 10.28(1 ? cos 6φ), where φ is the angle of internal rotation. The barrier to internal rotation is thus 1.98 kcal/mole.     

Thermodynamic properties of the φ4 lattice field theory near the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins with nearest-neighbor interactions, the partition function is transformed for large bare coupling constant λ into an Ising-like system with additional neighbor interactions. For d = 2 a mean field approximation is then used to estimate the difference in critical temperature between the lattice φ⁴ field theory and its Ising limit (λ = ∞). Expansions are obtained for the susceptibility and specific heat. The critical exponents are shown to be identical to the Ising exponents.     

Failure of the naïve loop expansion for the effective potential in φ4 field theory when there is “broken symmetry”

We determine the effective potential V[φ] in zero dimensional gφ⁴ field theory when there is broken symmetry classically $\text{[V}{}_{\mathrm{<mtext>cl</mtext>}}\text{=}\text{1}\text{4}\text{g(φ}{}^2\text{? a}{}^2\text{)}{}^2\text{]}$ as a power series in 1/g. We find that the usual loop expansion ignores a secondary minimum of the path integral and is only valid for |φ| ? a.When |ja| ? 1, where j is the external source, there is a secondary minimum of equal importance to the absolute minimum. In that region a kink solution to the differential equation for φ[j] exists, φ[j] = a tanh(ja) + O(ja²/ga⁴), which interpolates between the minima at φ = ?a and φ = +a. We obtain the 1/g expansion for V[φ] for φ < a and φ ? a. For all g, a mean-field approximation gives an expression for V[φ] which interpolates between these two regimes. In higher dimensions the kink solution becomes $\text{φ[j] = a}\text{tanh}\text{(jaΩ)}$, where Ω is the volume of space-time, suggesting a discontinuity in φ[j] in the infinite-volume limit.     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

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.     

Exact combinatorics of Bern-Kosower-type amplitudes for two-loop φ3 theory

Counting the contribution rate of a world-line formula to Feynman diagrams in (φ³ theory, we explain the idea of how to determine precise combinatorics of Bern-Kosower-like amplitudes derived from a bosonic string theory for N-point two-loop Feynman amplitudes. In this connection we also present a method to derive simple and compact world-line forms for the effective action.     

The vacuum structure of light-frontφ 1+1 4 -theory

We discuss the vacuum structure ofφ ⁴-theory in 1+1 dimensions quantised on the light-frontx ⁺=0. To this end, one has to solve a non-linear, operator-valued constraint equation. It expresses that mode of the field operator having longitudinal light-front momentum equal to zero, as a function of all the other modes in the theory. We analyse whether this zero mode can lead to a non-vanishing vacuum expectation value of the fieldφ and thus to spontaneous symmetry breaking. In perturbation theory, we get no symmetry breaking. If we solve the constraint, however, non-perturbatively, within a meanfield type Fock ansatz, the situation changes: while the vacuum state itself remains trivial, we find a non-vanishing vacuum expectation value above a critical coupling. Exactly the same result is obtained within a light-front Tamm-Dancoff approximation, if the renormalisation is done in the correct way.