arXiv subjects
Publications and source records attributed to Yves Brihaye.
We analyse the Witten-Woronowicz's type deformations of the Lie superalgebras spl(N,1) and obtain deformations parametrized by N(N-1)/2 independent parameters for N>2 and by three parameters for N=2. For some of these algebras, finite dimensional representations can be constructed in terms of finite difference operators, providing operators that are relevant for the classification of quasi exactly solvable, finite difference equations.
Let $P(N,V)$ denote the vector space of polynomials of maximal degree less than or equal to $N$ in $V$ independent variables. This space is preserved by the enveloping algebra generated by a set of linear, differential operators representing the Lie algebra $gl(V+1)$. We establish the counterpart of this property for the vector space $P(M,V) \oplus P(N,V)$ for any values of the integers $M,N,V$. We show that the operators preserving $P(M,V) \oplus P(N,V)$ generate an abstract superalgebra (non linear if $Δ=\mid M-N\mid\geq 2$). A family of algebras is also constructed, extending this particular algebra by $Δ-1$ arbitrary complex parameters.
We analyse the Witten-Woronowicz's type deformations of the Lie superalgebra osp(2,2) and obtain a deformation parametrized by three independent parameters. For some of these algebras, finite dimensional representations are formulated in terms of a finite difference operator, providing operators that are relevant for the classification of quasi exactly solvable, finite difference equations. Similar representations are also pointed out for the Lie superalgebra osp(1,2).
By taking a product of two sl(2) representations, we obtain the differential operators preserving some space of polynomials in two variables. This allows us to construct the representations of osp(2,2) in terms of matrix differential operators in two variables. The corresponding operators provide the building blocks for the construction of quasi exactly solvable systems of two and four equations in two variables. Some generalisations are also sketched. The peculiar labelling used for the generators allows us to elaborate a nice deformation of osp(2,2). This gives an appropriate basis for analyzing the quasi exactly solvable systems of finite difference equations.
We present the general ansatz, the energy density and the Chern-Simons charge for static axially symmetric configurations in the bosonic sector of the electroweak theory. Containing the sphaleron, the multisphalerons and the sphaleron-antisphaleron pair at finite mixing angle, the ansatz further allows the construction of the sphaleron and multisphaleron barriers and of the bisphalerons at finite mixing angle. We conjecture that further solutions exist.
In the electroweak sector of the standard model topologically inequivalent vacua are separated by finite energy barriers, whose height is given by the sphale\-ron. For large values of the Higgs mass there exist several sphaleron solutions and the barriers are no longer symmetric. We construct paths of classical configurations from one vacuum to a neighbouring one and solve the fermion equations in the background field configurations along such paths, choosing the fermions of a doublet degenerate in mass. As in the case of light Higgs masses we observe the level crossing phenomenon also for large Higgs masses.
We consider non-perturbative solutions of the Weinberg-Salam model at finite temperature. We employ an effective temperature-dependent potential yielding a first order phase transition. In the region of the phase transition, there exist two kinds of static, spherically symmetric solutions: sphalerons and bubbles. We analyze these solutions as functions of temperature. We consider the most general spherically symmetric fluctuations about the two solutions and construct the discrete modes in the region of the phase transition. Sphalerons and bubbles both possess a single unstable mode. We present simple approximation formulae for these levels.
We demonstrate the level crossing phenomenon for fermions in the background field of the sphaleron barrier, by numerically determining the fermion eigenvalues along the minimal energy path from one vacuum to another. We assume that the fermions of a doublet are degenerate in mass, allowing for spherically symmetric ansätze for all of the fields, when the mixing angle dependence is neglected.
We construct the sphaleron for several temperature dependent effective potentials. We determine the sphaleron energy as a function of temperature and demonstrate that the sphaleron energy at a given temperature $T$ is well approximated by the sphaleron energy at temperature zero scaled by the ratio of the vacuum expectation values of the Higgs field at temperatures $T$ and zero. We address the cosmological upper bound on the Higgs mass.