Determinant Representations For Scalar Products Of The XXZ Gaudin Model With General Boundary Terms

We obtain the determinant representations of the scalar products for the XXZ Gaudin model with generic non-diagonal boundary terms. In addition to the inhomogeneous parameters{zj}, the associated Gaudin operators{Hj}, depend on four free parame-ters{λi, λ2, ξ, Δ}. The common eigenstates (Bethe states) of the operators are con-structed by algebraic Bethe ansatz method. We have obtained the determinant repre-sentations of the scalar products for the boundary XXZ Gaudin model.In the last part, we study the relation of the symmetry group of a Feynman diagram and its reduced diagrams. We then prove that the counter terms in BPHZ renormaliza-tion scheme is in consistency with adding counter terms to interaction Hamiltonian in all cases, including that of Feynman diagrams with symmetry factors.