Minimization of the scalar Higgs potential in the Finite Supersymmetric Grand Unified Theory

Exact mathematical solution of the minimization conditions of scalar the Higgs potential of the Finite Supersymmetric Grand Unification Theory is proposed and extremal field configurations are found. Types of extrema are investigated and masses of the new Higgs particles arisen after electroweak symmetry breaking are derived analytically. The conditions for existing of physically acceptable minimum are given. As it appears, this minimum is simple generalization of the analogous solution in the Minimal Supersymmetric Standard Model. Phenomenological consequences are discussed briefly.Comment: Latex, 18 pages, 1 postscript figure (included at the end

Difficulties of an Infrared Extension of Differential Renormalization

We investigate the possibility of generalizing differential renormalization of D.Z.Freedman, K.Johnson and J.I.Latorre in an invariant fashion to theories with infrared divergencies via an infrared $\tilde{R}$ operation. Two-dimensional $\sigma$ models and the four-dimensional $\phi^4$ theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. The consistent set of Fourier-transformation formulae is derived. However, the values for tadpole-type Feynman integrals in higher orders of perturbation theory prove to be ambiguous, depending on the order of evaluation of the subgraphs. In two dimensions, even earlier than this ambiguity manifests itself, renormalization-group calculations based on infrared extension of differential renormalization lead to incorrect results. We conclude that the extended differential renormalization procedure does not perform the infrared $\tilde{R}$ operation in a self-consistent way, as the original recipe does the ultraviolet $R$ operation.Comment: (minor changes have been made to make clear that no infrared problems occur in the original ultraviolet procedure of [1]; subsection 2.1 has been added to outline the ideas a simple example), 26 pages, LaTeX, JINR preprint E2-92-538, Dubna (Dec.1992

Comment on the ``θ\theta-term renormalization in the (2+1)-dimensional CPN−1CP^{N-1} model with θ\theta term''

It is found that the recently published first coefficient of nonzero $\beta$-function for the Chern-Simons term in the $1/N$ expansion of the $CP^{N-1}$ model is untrue numerically. The correct result is given. The main conclusions of Park's paper are not changed.Comment: 3 pages, LATE

Gluon self-interaction in the position space in Landau gauge

We propose a method to treat the three-gluon self-interaction vertex in the position space in D = 4 - 2e dimensions. As an example, we calculate a two-loop contribution to auxiliary Lcc vertex in the Landau gauge which contains the three-gluon vertex for SU(N) Yang-Mills theory. We represent the integral expression as a sum of separate contributions so that each of the contributions is a double finite integral or single integral (singular or finite) in the position space. In each double finite integral we use the freedom to shift exponents in powers in the denominator of integrands by some multiples of e, in order to perform at least one of the integrations by the uniqueness technique without corrupting the first term of the decomposition in e.Comment: 47 pages, 1 figure, minor corrections, Eq. for J_31 on page 11 is corrected, to appear in Int. J. of Mod. Phys.

Belokurov-Usyukina loop reduction in non-integer dimension

Belokurov-Usyukina loop reduction method has been proposed in 1983 to reduce a number of rungs in triangle ladder-like diagram by one. The disadvantage of the method is that it works in d=4 dimensions only and it cannot be used for calculation of amplitudes in field theory in which we are required to put all the incoming and outgoing momenta on shell. We generalize the Belokurov-Usyukina loop reduction technique to non-integer d=4-2e dimensions. In this paper we show how a two-loop triangle diagram with particular values of indices of scalar propagators in the position space can be reduced to a combination of three one-loop scalar diagrams. It is known that any one-loop massless momentum integral can be presented in terms of Appell's function F_4. This means that particular diagram considered in the present paper can be represented in terms of Appell's function F_4 too. Such a generalization of Belokurov-Usyukina loop reduction technique allows us to calculate that diagram by this method exactly without decomposition in terms of the parameter e.Comment: 6 pages, 3 figure

Non-renormalization theorems in softly broken SQED and the soft β\beta-functions

The renormalization of softly broken SQED is related to the one of supersymmetric QED by using the construction with a local gauge supercoupling and by taking into account softly broken anomalous axial U(1) symmetry. From this extended model one obtains the non-renormalization theorems of SQED and the counterterms of the soft breaking parameters as functions of the supersymmetric counterterms. Due to the Adler-Bardeen anomaly of the axial current an invariant regularization scheme does not exist, and therefore the $\beta$-functions of soft breaking parameters are derived from an algebraic construction of the Callan-Symanzik equation and of the renormalization group equation. We obtain the soft $\beta$-functions in terms of the gauge $\beta$-function and of the anomalous dimension of the supersymmetric matter mass. In particular, we find that the X-term of the scalar mass $\beta$-function as well as the gauge $\beta$-function in $l\geq 2$ are due to the Adler-Bardeen anomaly of the axial symmetry.Comment: 28 page

Softly broken supersymmetric Yang-Mills theories: Renormalization and non-renormalization theorems

We present a minimal version for the renormalization of softly broken Super-Yang-Mills theories using the extended model with a local gauge coupling. It is shown that the non-renormalization theorems of the case with unbroken supersymmetry are valid without modifications and that the renormalization of soft-breaking parameters is completely governed by the renormalization of the supersymmetric parameters. The symmetry identities in the present context are peculiar, since the extended model contains two anomalies: the Adler-Bardeen anomaly of the axial current and an anomaly of supersymmetry in the presence of the local gauge coupling. From the anomalous symmetries we derive the exact all-order expressions for the beta functions of the gauge coupling and of the soft-breaking parameters. They generalize earlier results to arbitrary normalization conditions and imply the NSVZ expressions for a specific normalization condition on the coupling.Comment: 24 pages, LaTeX, v2: one reference adde