Mass Insertions vs. Mass Eigenstates calculations in Flavour Physics

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.


Introduction
QFT models defined by specifying the Lagrangian -choice of "field basis" not unique! Usual approach to construct QFT model: assume some symmetries, local or global choose particle content and their quantum numbers define interactions -add to Lagrangian all (or subset) of terms allowed by the symmetries of the theory and extra requirementsrenormalizability etc.
Result: Lagrangian in the "interaction basis" Fields in "interaction basis" Lagrangian may not represent the physical degrees of freedom! Further steps may be required: spontaneous symmetry breaking rediagonalization of mass matrices . . .

Result: Lagrangian in the "mass eigenstates" basis
Advantage: redefined fields represent physical degrees of freedom.
Disadvantage: couplings in "mass eigenstates" basis are usually complicated functions of the initial interaction basis parameters.
Toy example: self-energy in model with real scalar field η and complex scalar multiplet Φ I : Transition to mass eigenstates basis: Interaction basis: thick dots represent "mass insertions" -off diagonal elements of M 2 IJ .
infinite series of diagrams, complicated calculation and expression unphysical states on the external legs but: result in terms of the initial "interaction basis" parameters Y, M Transition to physical states U † jJΣ JI (p)U Ii = Σ ji (p):

Very particular relation!
Holds for 1-loop functions only?
Can it be generalized ? How? → Flavor Expansion Theorem Idea: Typical term in QFT mass-eigenstates amplitude: U Ii f (m 2 i )U * Ji can be expressed as an element of function of the matrix: Disadvantage: each power of mass insertion appears in infinite number of terms of Taylor series!
Can we derive another series (not Taylor) for f (M 2 ) in powers of the off-diagonal elements of M 2 only?
Answer -yes, on purely algebraic ground!

Flavor Expansion Theorem
Definition (Divided differences) Divided differences are defined recursively as Properties: symmetric functions of all arguments smooth degeneracy limit lim {x0,...,xm}→{ξ,...,ξ} Theorem (Flavor Expansion Theorem or "FET") Let's decompose and Hermitian matrix A as a sum of diagonal and non-diagonal part A = A 0 +Â: Then, matrix element f (A) IJ is given by (no sum over I, J): Series coefficients: divided differences of f (A 0 ) Expansion parameters: non-diagonal elements ofÂ. Common application of FET: for many processes leading order terms cancel and only higher ones are left -diagrammatic MI calculation tedious and error prone.

FET expansion of PV functions
Any 1-loop amplitude can be expressed in terms of Passarino-Veltman functions of the order n: Useful recursive relation: divided difference of 1-loop functions of the order n is a 1-loop function of the order n + 1!
FET formula for PV functions: immediately reproduces the relation discussed in the "toy model" applies to expansion with degenerate diagonal elements -PV functions not singular in this limit, no need to calculate derivatives explicit condition for FET series convergence Fermion propagator can be decomposed as Loop functions depend always on Hermitian matrices MM † or M † M.
Only some combinations of mixing matrices can appear in fermionic amplitudes: All can be expressed using FET formula, works for fermions as well! Applies also to Majorana fermions -then M = M T and U = V *

MassToMI Mathematica package
Great advantage of FET technique -can be easily automatized! Prescription: calculate amplitude in the mass eigenstates basis -less diagrams, more compact expressions, better suited for numerical computations expand result using FET implemented in MassToMI package -recover direct analytic dependence on "interaction basis" parameters (better suited for understanding of various effects) Avoids tedious and error-prone direct calculations of diagrams with mass insertions.
Example: H − → d IcJ decay in the MSSM, the triangle diagram with chargino (Dirac) C n , neutralino (Majorana) N j and down squark D i circulating in the loop.
Typical term (Z, O, U, V are scalar, Majorana and Dirac fermion mixing matrices): Control variables: FetMaxOrder=2. Only mass insertion products of the total order FetMaxOrder or lower are kept in the final result.