Abstract
We present and prove a theorem of matrix analysis, the Flavour Expansion Theorem (or FET), according to which, an analytic function of a Hermitian matrix can be expanded polynomially in terms of its off-diagonal elements with coefficients being the divided differences of the analytic function and arguments the diagonal elements of the Hermitian matrix. The theorem is applicable in case of flavour changing amplitudes. At one-loop level this procedure is particularly natural due to the observation that every loop function in the Passarino-Veltman basis can be recursively expressed in terms of divided differences. FET helps to algebraically translate an amplitude written in mass eigenbasis into flavour mass insertions, without performing diagrammatic calculations in flavour basis. As a non-trivial application of FET up to a third order, we demonstrate its use in calculating strong bounds on the real parts of flavour changing mass insertions in the up- squark sector of the MSSM from neutron Electric Dipole Moment (nEDM) measurements, assuming that CP-violation arises only from the CKM matrix.
Highlights
We present and prove a theorem of matrix analysis, the Flavour Expansion Theorem, according to which, an analytic function of a Hermitian matrix can be expanded polynomially in terms of its off-diagonal elements with coefficients being the divided differences of the analytic function and arguments the diagonal elements of the Hermitian matrix
We prove a theorem in matrix analysis [11, 12], that we coin Flavour Expansion Theorem or FET, which says that an analytic function of a Hermitian matrix can be expanded polynomially in terms of its offdiagonal elements with coefficients being the divided difference of the analytic function and arguments the diagonal elements of the Hermitian matrix
In this article we have presented and proved a theorem of matrix analysis, the Flavour Expansion Theorem (FET), that remarkably translates any transition amplitude, obtained in terms of mass eigenstate basis parameters, into its corresponding amplitude in flavour eigenstate basis, purely algebraically, without the use of standard diagrammatic methods like the Mass Insertion Approximation (MIA) method
Summary
To set up a simple framework to introduce the standard techniques of flavour physics calculations, we consider a scalar field toy model composed of N -complex charged scalar fields ΦI , with family replication, and an extra neutral, real, scalar field η. Let us consider the “flavour” changing one-loop One-Particle-Irreducible (1PI) self-energy diagram of the mass eigenstates fields φi, shown in figure 1. Substituting in eq (2.10) the explicit algebraic expressions for the self-energies we obtain an interesting result - the flavour rotation of the mass eigenstates loop-function is an expansion in terms of mass insertions in flavour basis (no sum over K, L), UK. This result, can be obtained by a theorem of matrix analysis [cf eq (3.11)] stated, rendering diagrammatic calculations in flavour basis, similar to ones leading to eq (2.8), obsolete
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