LSZ-reduction, resonances and non-diagonal propagators: fermions and scalars

We analyze in details the effects associated with mixing of fermionic fields. In a system with an arbitrary number of Majorana or Dirac particles, a simple proof of factorizability of residues of non-diagonal propagators at the complex poles is given, together with a prescription for finding the"square-rooted"residues to all orders of perturbation theory, in an arbitrary renormalization scheme. Corresponding prescription for the scalar case is provided as well.


Introduction
Finding the correct form of renormalization conditions in gauge theories of particle physics is a nontrivial task, as the improper conditions can spoil the gauge symmetry even in the presence of a symmetry preserving regularization. For this reason, renormalization schemes with implicit renormalization conditions are often employed. A famous representative of this class is the MS scheme of dimensional regularization [1]. Zinn-Justin proved that non-Abelian gauge symmetries are preserved in minimal subtraction schemes based on a symmetry preserving regularization [2], what makes MS the most convenient choice, at least in non-chiral theories. In fact, the violation of gauge symmetry induced by regularization can be handled according to the general rules of "algebraic renormalization" [3,4,5] (see also [6] for a discussion in the context of dimensional regularization with the consistent 't Hooft-Veltman-Breitenlohner-Maison prescription for γ 5 ). This approach does not preclude the use of schemes with implicit renormalization conditions. For instance, in [7] a renormalization scheme for gauge theories in the cutoff regularization was constructed by carefully selecting a set of vertices to which non-minimal (cutoff-independent) counterterms are added. These non-minimal counterterms are completely fixed by investigating the Slavnov-Taylor identities for the gauge symmetry, what leads to a mass independent renormalization scheme with implicit renormalization conditions. In such a framework, renormalization can be considered as the first stage in the study of correlation functions, a stage which is independent of the physical interpretation of the studied model.
In this paper we are interested only in the second stage, i.e. in the extraction of scattering amplitudes from renormalized correlation functions in an arbitrary renormalization scheme. 3 More specifically, we are interested in amplitudes corresponding to the particles described by the system of mixed fields with non-diagonal propagators. In this context, the crucial objects are residues of the (renormalized) propagators; for instance in the case of scalar fields one finds the following expression for the full propagator non-pole part .
(1) The factorization of the residues at the (complex) poles is a well-known property [8]. For the mixed Z-photon system of the Standard Model it was explicitly demonstrated in [9]. For fermionic systems the factorization was shown in [10,11] (see also [12,13]). 4 The coefficients ζ k S[ℓ] associated with real poles m 2 S(ℓ) are crucial for obtaining correctly normalized (i.e. consistent with unitarity) transition amplitudes between the states of stable particles. On account of Cutkosky-Veltman rules, it is well-known [15] that the S-matrix is unitary provided that (1.) unstable particles appear only as internal lines, and (2.) asymptotic (free) fields appearing in the LSZ-reduced formula for the S-operator (see e.g. [16] and Eq. (37) below) are normalized so as to reproduce the behavior of full propagators about the poles associated with stable particles. Thus, the asymptotic field φ φ φ j associated with φ j can be written as φ φ φ j = ′ ℓ ζ j S[ℓ] Φ ℓ , where Φ ℓ is a canonically normalized free scalar field of mass m S(ℓ) , and the summation runs over indices labeling real poles.
For resonances the external lines, aka the "in" and "out" states, do not really exist. Nonetheless, coefficients ζ k S[ℓ] associated with the complex poles are useful in studying properties of unstable particles [8,9,10,11]. In this connection, the problem of finding a convenient prescription for coefficients ζ k S [ℓ] parameterizing the residues in Eq. (1) have gained renewed interest in recent years. In Ref. [17] the case of 3-by-3 mixing of scalar particles was analyzed in details. The factorization property (1) was demonstrated and explicit formulae for the coefficients ζ k S[ℓ] were given. These results were applied to the neutral Higgs sector of MSSM; it was shown that cross-sections obtained by neglecting the non-pole part in Eq. (1) agree to good accuracy with the cross-sections based on full propagators. Analysis of a generic nby-n mixing in fermionic systems was given in [18,19,20].
The purpose of this paper is to generalize and to simplify the available in gauge symmetry. 4 We also mention in this context Ref. [14] where the factorization of the residues was shown for scalar, vector and fermionic systems, on the basis of the presupposed existence of the on-shell renormalization scheme. This approach makes the generalization to the complex poles rather difficult. the literature procedure of calculating coefficients ζ for fermions and scalars. Our analysis is closest in spirit to the one given in [18,19,20]; there are, however, some differences. First, we follow the philosophy of keeping the renormalization scheme as general as possible. In particular, we do not impose any concrete renormalization conditions on the two-point functions. Second, we offer a technical improvement in comparison with the analyzes of [8,18,19,20], where the cofactor matrix was used to get the formulae for ζ. By contrast, the coefficients ζ in our approach are expressed directly in terms of properly normalized eigenvectors of certain "mass-squared matrices", so that the case of degenerated eigenvalues is naturally covered by our prescription. Thus, the proposed prescription for finding ζ can be considered as a generalization of the standard procedure for finding tree-level mass eigenstates. The analysis of mixed vector fields along these lines will be given elsewhere [21].
The formulae presented below are valid in the generic case of mixed unstable Majorana or/and Dirac fermions (a scalar version is also given). For completeness, the expression for the Landau-gauge one-loop fermionic selfenergy of a general renormalizable model in the MS scheme is provided below. The paper is therefore intended to cultivate a long tradition by providing generic formulae that can be easily applied to (almost) any model at hand, especially in the present computer era, see e.g. [22], [23], [24], [25], [27], [28], [26], [7].
The remainder is organized as follows. In the next section the notation is specified, together with basic assumptions. In Sec. 3 the prescription for ζ matrices is given for massive Majorana particles (3.A), massive Dirac particles (3.B), generic spin-1/2 fermions (3.C) and scalars (3.D), together with the generic expression for fermionic one-loop self-energy (3.E). The correctness of the prescription is proved in Sec. 4 and the last section is reserved for conclusions.

Notation and assumptions
In most formulae given below indices are suppressed and matrix multiplication is understood. The summation convention is used only when an upper index is contracted with a lower one; whenever ambiguities may arise, sums are explicitly present. The Minkowski metric has the form Recall that a Majorana field [35] ψ a is a pair of a Weyl field χ a A , below referred to as the left-chiral Weyl field (LW), and its Weyl conjugate χ aȦ , alias the right-chiral Weyl field (RW) here a = 1, . . . , n is a generalized-flavor index, A andȦ are SL(2, C) indices, while a = (a, (A,Ȧ)). Take, for instance, (a toy version of) the Standard Model [31] in which all Weyl fields except for these that describe the electron-positron pair of states have been forgotten. Let λ A be a LW representing the charged component of the lepton (would-be) SU(2) L -doublet and let ρ A be a LW of the charged lepton SU(2) L -singlet. In this case n = 2 and the fields with the definite generalized-flavor (henceforth called flavor) can be chosen as though nothing (but common sense) prevents a more general choice with an arbitrary unitary matrix u, which off-diagonalizes the charge generator.
With chiral projections P L,R and the charge conjugation matrix C that expresses the Dirac conjugate of ψ in terms of ψ itselfψ = ψ ⊤ C , the renormalized (in some renormalization scheme) one-particle-irreducible (1PI) two-point function of Majorana fileds can be written in the following form where Clearly, matrices P R,L , C and / p carry only the SL(2, C) indices, while Z L,R and M L,R carry only the flavor indices; the tensor products ⊗ are not explicitly shown in Eq. (6). (Z L,R and M L,R are, essentially, the 1PI functions of different pairs of Weyl fields; Majorana fields are used here and below only for bookkeeping reasons.) The full propagator of Majorana fields is given by Inverting the two-point function in Eq. (6) one findŝ where (s ≡ p 2 ) and The antisymmetry of the fermionic two-point function, Eq. (6), yields and thus what gives hence the poles of both chiral partsD L,R of propagator in Eq. (9) appear at the same points s = m 2 (a) , obeying the condition In this paper three technical assumptions are made about the solutions to Eq. (15) and the matrices M 2 L (m 2 (a) ). First, it is assumed that each generalized eigenvector (see e.g. [29]) of M 2 L (m 2 (a) ) associated with the eigenvalue m 2 (a) is an (ordinary) eigenvector; in other words, it is assumed that in the Jordan basis for M 2 L (m 2 (a) ) the block corresponding to m 2 (a) is diagonal. This excludes standard pathologies associated with non-diagonalizable matrices (e.g. second order poles of gauge-field propagators in covariant non-Feynman gauges caused by pseudoHermiticity of the Hamiltonian [34]).
Second, it is assumed that each nonzero solution m 2 (a) is nonzero at the tree level, as is usually the case in the common seesaw models.
Third, it is assumed that, roughly speaking, the quantum corrections do not change the total number of solutions to Eq. and has a unique solution m 2 (a) = (m tree (a) ) 2 + O( ). In particular, the first and the third assumption are satisfied in this case. In general, assuming non-degeneracy of the tree-level masses is however not an option as physics is about symmetries. Therefore it is convenient (and desirable from practical point of view) to distinguish two special situations called below the Majorana case and the Dirac case.
Let G be the group of exact, linearly realized, internal global symmetries of the tree-level action that are respected by the renormalization conditions and let U(·) be the representation of G on the left-chiral flavor eigenfields χ a . The two-point function (6) obeys the following conditions Consider first a toy model in which fermions form three families, each one consisting of gluinos of the Minimal Supersymmetric Standard Model [35]. In this case the flavor index is a pair of an adjoint color index and a family index, and the most general matrices Z L,R and M L,R consistent with the SU(3) C symmetry have the form and thus where 1 k×k (with k = 8) is the identity matrix in the adjoint color space, while M fam L,R and Z fam L,R are 3 × 3 matrices in the family space. In particular, M 2 L (m 2 (a) ) are diagonalizable if e.g. the tree-level contribution to M 2 fam L (0) has non-degenerate eigenvalues. A situation in which the two-point functions have the form (17) with an arbitrary number f of "families", an arbitrary k, and with f different and nonvanishing eigenvalues of the tree-level contribution to M 2 fam L (0) is called below the Majorana case; the total number of flavors equals n = f × k. As far as the propagator and mixing are concerned, one can in this case restrict attention to a single color. 6 It is worth emphasizing that the Majorana case (as well as the Dirac case below) is defined here by demanding m tree (a) = 0 for all a, in order to make the corresponding 6 A more physical representative of the Majorana case is the type I seesaw mechanism with k = 1 and f = 3 + 3 neutrinos. prescription in Sec. 3.A (respectively, 3.B) as simple and practical as possible. 7 Vanishing masses require a separate treatment and they are dealt with in Sec. 3.C devoted to the generic case.
Consider next a more interesting example of three families of down-type quarks in the SM (clearly, the SU(3) C ×U(1) Q symmetry of the SM prohibits down-type quarks from mixing with other SM fermions). Without loss of generality, it can be assumed that the flavor eigenfields χ a have been chosen so that the anti-Hermitian generator of U(1) Q is diagonal where 1 ℓ×ℓ (with ℓ = 3) is the identity matrix in the color space. The most general matrices Z L,R and M L,R consistent with Eqs. (12) and the where µ L,R and I ± (s) are arbitrary 3 × 3 matrices and, in addition, I ± (s) are nonsingular. Thus with Using the relation (valid if, e.g. tree-level masses are non-vanishing, so that µ L (s) is nonsingular) one gets hence the determinant in Eq. (14) reads It follows from Eq. (23) that complex poles corresponding to the left-chiral flavor eigenfields with opposite charges are located at the same points s = m 2 (a) . A situation in which the two-point functions have the form (19) with an arbitrary number f of families, an arbitrary ℓ, and with f different and nonvanishing eigenvalues of the tree-level contribution to M 2 + (0) is called below the Dirac case (the total number of flavors equals n = 2 × f × ℓ). Once again, as far as the propagator and mixing are concerned, one can in this case restrict attention to a single color, i.e. one can effectively neglect color factors 1 ℓ×ℓ in Eqs. (19)- (20).
A simple prescription for the pole part of the propagator (9) is given in the next section for these two special cases. A generalization to arbitrary M 2 L (m 2 (a) ) matrices consistent with three assumptions stated above is provided as well.
It should be noted, however, that infrared problems (see e.g. [30]) are not discussed in this paper. In other words, it is assumed that an IR regulator was introduced (if necessary) so that the propagators do have simple poles at the points obeying Eq. (15).

3.A. Majoran case.
Consider first the Majorana case, Eqs. (17). In order not to obscure the notation it is assumed that k = 1 and fam superscripts are omitted; thus the total number of LWs is n = f . On the assumptions stated in Sec. 2, it is clear that Eq. (15) has, in the sense of formal power series, n different and non-vanishing solutions such that Re(m (a) ) > 0. Define It will be shown (in Sec. 4) that theD F (p) matrix in the full propagator of two Majorana fields, Eq. (8), has the following simple form matrices ζ L,R (as well as m) carry only flavor indices, while columns of ζ L and ζ R are given, respectively, by vectors ζ L[a] and ζ R[a] in the flavor space and obeying the following normalization condition with a normalizing factor where (Note that, on the assumptions stated above, Eqs. uniquely up to a sign; one could worry that the condition (30) cannot be imposed since e.g. [1, i] [1, i] ⊤ = 0, however such a pathology is impossible at the tree-level, and thus it is impossible for formal power series.) Moreover it will be shown that, if Feynman integrals contributing to Z L,R (p 2 ) and M L,R (p 2 ) do not acquire imaginary parts in a left neighbor- so that the following reality conditions are satisfied then all terms of a formal power series m (a) , Eq. (25), are real, and conditions (29)- (33) If, in particular, conditions (34) are satisfied for all a = 1, . . . , n, then matrices appearing in Eq. (28) obey ζ R = ζ L ⋆ and Eq. (27) has a simple interpretation: the Majorana field ψ in, e.g., the MS scheme of dimensional regularization can be expressed in terms of its on-shell scheme counterpart ψ OS (see e.g. [14]) as follows What if only some of the particles are stable? If Im(m (a S ) ) = 0, then one can introduce a free (interaction picture) Majorana field Ψ a S of mass m (a S ) with canonically normalized propagator and define (recall that b is the "total" index, cf. Eq. (3)) where the summation runs over all "stable indices". Clearly, Ψ red is a free quantum field and Eq. (27) implies that the chronological propagator of Ψ red reproduces the behavior of propagator in Eq. (8) about all poles located on the real axis. Thus, Ψ red is the field that appears in the LSZ-reduced formula for the S-operator describing the transitions between stable states [16] where Γ b c (x, y) is the Fourier transform of (6) (in the last three equations, ψ and Ψ represent not only fermions but also scalars and vectors). What about unstable particles? Consider, for instance, a theory in which heavy neutrinos described in terms of Majorana fields ψā N carrying a family indexā, interact with a Hermitian scalar field h and massless SM (anti)neutrinos, described in terms of Majorana fields ψb ν carrying a family indexb, through the following Lagrangian density (spinor indices are suppressed) The CP-asymmetry was calculated in [12,10,11,13] by looking at diagrams in which Nā is an internal (rather than an external) line, what leads to the following expression with where ζ R,L are ζ matrices for the ψā N fields, while the ellipsis indicates contributions of corrections to external lines of h and ψb ν fields, as well as loop corrections to the 1PI vertices (for simplicity it is assumed here that the mixing between light and heavy neutrinos is negligible, even though the present formalism is capable of describing quantum corrections to the mixing in the full 6×6 system).

3.B Dirac case.
Consider now the Dirac case, Eqs. (19). For simplicity of the notation it is assumed that ℓ = 1, thus the total number of LWs is n = 2f . On the assumptions stated in Sec. 2, it is clear that Eq. (15), cf. Eqs. (23)- (24), has f different and non-vanishing solutions such that Re(m (a) ) > 0. Define and It will be shown that theD F (p) matrix in the full propagator of two Majorana fields, Eq. (8), has the form while the ζ L,R matrices have a block-diagonal form with matricesζ X± built out of vectorsζ X[a±] (eigenspaces of M 2 ± (m 2 (a) ) are one-dimensional on the assumptions stated in Sec. 2), and obeying the following normalization condition Thenζ andζ with a normalizing factor where M 2 + ′ (s) ≡ dM 2 + (s)/ds , while c(a) ∈ C \ {0} is an arbitrary number which does not affect the pole part of the propagator.
Moreover it will be shown that, if the reality conditions (34) hold in a left neighborhood U a ⊂ R of p 2 = (m tree (a) ) 2 , then all terms of a formal power series m (a) , Eq. (44), are real, and there exists c(a) ∈ C \ {0} such that 3.C. Generic fermionic case. The above prescriptions can be generalized to the case constrained only by the three conditions discussed below Eq. (15). Recall that these conditions imply that the number of poles of the full propagator is equal to the total number n of LWs. The a label is assumed to distinguish different solutions m 2 (a) to Eq. (15); indices corresponding to this eigenvalue are labeled with a 1 , a 2 , etc.
Let m = m ⊤ be an arbitrarily chosen symmetric n × n matrix such that and Clearly, [ m 2 ] ar aq = m 2 (a) δ rq . It will be shown that theD F (p) matrix in the full propagator of two Majorana fields, Eq. (8), has the following form (the order of columns reflects the order of eigenvalues in Eq. (56)) obtained in the following way.
1. Nonzero m 2 (a) . Let ξ [a 1 ] , . . . , be a basis of the eigenspace obeying the following normalization conditions (recall that for each pair of nonsingular complex symmetric matrices S 1,2 there always exists a nonsingular matrix N such that S 1 = N ⊤ S 2 N , thus starting with an accidentally chosen basis of eigenspace one can always find vectors obeying Eq. (61); the non-singularity of the left-hand side of Eq. (61) is ensured by the assumptions listed below Eq. (15)). Define the following matrix where m(a) rq = m ar aq .
(Clearly, N (a) is determined only up to a complex orthogonal matrix.) Then and Moreover it will be shown that, if the m matrix is chosen to be diagonal obeying the following normalization conditions (for p 2 = 0 reality conditions (34) cannot be violated and thus Z L (0) is a Hermitian and positive matrix, cf. Eqs. (7) and (12)). Define the following matrix which is Hermitian (see Sec. 4) and find a matrix N (0) such that Then and It should be stressed that auxiliary normalization conditions (61)  3.D. Scalar case. Consider a set {φ ℓ } of n scalar fields. Without loss of generality it is assumed that φ ℓ are Hermitian. The renormalized 1PI two-point function where M 2 (s) = M 2 (s) ⊤ is a symmetric matrix, leads to the propagator and the gap equation with It is assumed that assumptions listed below Eq. (15) for fermionic solutions m 2 (a) and matrices M 2 L (m 2 (a) ), are satisfied also for their scalar counterparts, m 2 (ℓ) and M 2 (m 2 (ℓ) ). Let m 2 be a diagonal n × n matrix The ℓ label is assumed to distinguish different values m 2 (ℓ) ; indices corresponding to this value in Eq. (77) are labeled with ℓ 1 , ℓ 2 , etc.
It will be shown that the propagator (74) has the form where columns of ζ are given by vectors ζ [ℓr] (the order of columns reflects the order of eigenvalues in Eq. (77)) obtained in the following way. Let ξ [ℓ 1 ] , . . . , be a basis of the eigenspace obeying the following normalization conditions (starting with an arbitrary basis of eigenspace one can always find vectors obeying Eq. (81), just as in the fermionic case). Define the following matrix which is manifestly symmetric, and find a matrix N (ℓ) such that (Clearly, N (ℓ) is determined only up to a complex orthogonal matrix.) Then Moreover it will be shown that, if Feynman integrals contributing to M 2 (p 2 ) do not acquire imaginary parts in a left neighborhood U ℓ ⊂ R of p 2 = (m tree (ℓ) ) 2 , so that the following reality conditions are satisfied then all terms of a formal power series m 2 (ℓ) are real and there exists a N (ℓ) matrix obeying Eq. (83) and such that ζ [ℓr] = ζ ⋆ [ℓr] for all r. With fixed {ξ [ℓr] } eigenvectors this matrix is unique up to a real orthogonal matrix R(ℓ), i.e. N (ℓ) = N 0 (ℓ)R(ℓ).

3.E. Fermionic one-loop self-energy.
It is convenient to supplement the prescription for fermionic ζ L,R matrices by providing generic expressions for one-loop contributions in the MS scheme with anticommuting γ 5 to the two-point functions Z L,R and M L,R in Eq. (6). Consider an arbitrary renormalizable model, in which Majorana fields ψ a (spinor indices are suppressed for simplicity) interact with Hermitian scalar fields φ ℓ (already shifted if necessary, so that φ = 0) and Hermitian gauge fields A α µ via the following Lagrangian density Here f αab = −f ⋆ αba are matrix elements of ordinary anti-Hermitian gaugegroup generators (already containing the coupling constants), while Y ℓab = Y ℓba are matrix elements of symmetric Yukawa matrices. It is assumed that all fields are chosen to be the eigenfields of the tree-level mass-squared matrices, so that (clearly, without loss of generality one could assume that M F itself is diagonal; such a choice is however completely impractical for Dirac particles, as it implies that, for instance, the u matrix, Eq. (5), in the SM is non-diagonal). Functions Z L,R and M L,R can be parametrized in the following way where v ℓ (1) represents the one-loop contribution to the scalar vacuum expectation value (VEV), while Z L,R are produced by one-loop diagrams shown in Figure 1 (we work in the Landau gauge). Using the standard, minimally subtracted one-loop functions a R and b R 0 in the dimensional regularization [32] a R (m) = m 2 ln m 2 µ 2 − 1 , (hereμ is the renormalization scale of the MS scheme, related to the usual 't Hoot mass unit viaμ ≡ µ H √ 4π e −γ E /2 ), together with their combinations In particular, reality conditions (34) are violated whenever b R 0 has a nonvanishing imaginary part.
Clearly, in the expression for A Z , the limits m Vβ → 0 are to be taken for contributions of massless gauge bosons. On the other hand, the last term A B Figure 1: One-loop contributions to M L,R and Z L,R in the Landau gauge.
in A Z , even for spontaneously broken gauge symmetries, contains contributions of unphysical massless modes; as far as corrections to the pole masses are concerned, they cancel with similar contributions of would-be Goldstone bosons, as the gauge symmetry leads to the following relation where Y g is a Yukawa matrix of the (massless) would-be Goldstone boson φ g associated with a broken generator f γ , while s γ = −1 or s γ = +1. By contrast, contributions of unphysical modes to the ζ L,R matrices do not cancel completely, but Eq. (89) ensures that they do not contain resonant factors [12]. Thus, in the CP-asymmetry (41) these contributions cancel with similar "unphysical" corrections to the 1PI vertices (indicated by the ellipsis in Eqs. (42)-(43)).
Finally, the one-loop contribution v (1) to the VEV can be obtained from the tadpole cancellation condition in the Landau gauge Here V is the gauge-invariant tree-level potential of scalar fields, v (0) represents the tree-level VEV (i.e. 0) ), while T α is the generator of the gauge group on scalar fields; T α is normalized in such a way that the covariant derivative reads

4.A. Proof of generic fermionic prescription.
The proof is a simple exercise in linear algebra.
0 . Generalities. First of all, one has to calculate the following limits (cf. Eqs. (10) and (15)) It is convenient to start with something simpler On the assumptions stated in Sec. 2, this limit exists and gives a projection onto the eigenspace of M 2 L (m 2 (a) ) associated with m 2 (a) along the direct sum of remaining generalized eigenspaces of M 2 L (m 2 (a) ). To verify this statement, it is enough to calculate the action of the right-hand side of Eq. (91) on generalized eigenvectors of M 2 L (m 2 (a) ). Introducing the resolvent , for all eigenvectors ξ [ar] associated with m 2 (a) . Let λ θ = m 2 (a) be another eigenvalue of M 2 L (m 2 (a) ); the generalized eigenspace associated with it is spanned by (in general more than one) Jordan chain θ 1 , . . . , θ p (a subsequence of the Jordan basis for M 2 L (m 2 (a) ), see e.g. [29]) where θ 0 ≡ 0, i.e. θ 1 is an eigenvector. Let Q(s) = (s − λ θ ) −1 , the following identity can be easily checked by induction Thus as was to be shown. In particular P(a) 2 = P(a) .
Eq. (13) now yields Introducing another family { P (a)} of projections one can decompose the m 2 matrix in Eq. (56) as follows clearly a P (a) = 1 , and P (a) P (b) = δ ab P (a) .
Now one sees that the formula that needs to be proven, Eq. (58), is equivalent to the following four sets of conditions (s a ≡ m 2 (a) ) Thus, one has to show that there exist matrices ζ L,R obeying, in addition to Eqs. (98)-(99), the following conditions It is enough to impose, instead of the last two equations, the following two (cf. Eq. (12)) Employing Eq. (66), and defining Eq. (107) can be further rewritten as where where this is nothing more than the condition (63). To prove that a matrix N (a) obeying Eq. (63) indeed exists, it is necessary to show that the Ξ(a) matrix, defined in Eq. (62), is symmetric. The following identity is easy to verify (cf. Eq. (11)); sandwiched between ξ ⊤ [aq] and ξ [ar] it gives since ξ [aq,r] are eigenvectors of M 2 L (m 2 (a) ) corresponding to the same eigenvalue.
Moreover, Eqs. (112) and (114) show that Hence, to complete the proof of the generalized prescription for m 2 (a) = 0, one has to show that the S(a) operator annihilates these generalized eigenvectors of M 2 L (m 2 (a) ) which correspond to eigenvalues different than m 2 (a) , so that S(a) = P(a). Because of Eq. (110) it is enough to prove the following property: let η be an eigenvector of M 2 L (m 2 (a) ) corresponding to the eigenvalue λ η and let θ be a generalized eigenvector of M 2 L (m 2 (a) ) associated with λ θ = λ η ; then η and θ are M L (m 2 (a) )-orthogonal This fact follows from the identity (cf. Eqs. (11) and (12)) Sandwiched between η ⊤ and θ 1 , i.e. the first element of a Jordan chain (92), it gives (λ η − λ θ ) × η ⊤ M L (m 2 (a) )θ 1 = 0 , while sandwiched between η ⊤ and θ r+1 yields where U(s) is unitary, while Λ(s) is diagonal (and positive, cf. Eq. (7)). On the other hand, a symmetric matrix can be written in the following form where V (s) is unitary, while µ(s) is diagonal, real and nonnegative. Hence and where Eq. (11) now reads where µ 2 (s) ≡ µ(s) 2 , and thus (cf. Eq. (14)) where C(a) is a square matrix. The normalization condition (61) reduces to (recall that m is now assumed to be diagonal) If Eq. (70) is satisfied, one finds and Eq. (68) gives im S(0) ⊃ ker M 2 L (0) ≡ im P(0) .

4.B. Proof of Majorana prescription.
The prescription for Majorana case follows immediately from the generalized prescription, since one can take m = m, with a diagonal matrix m, Eq. (26). In particular, the assumed non-degeneracy of the tree-level masses implies that N (a) is a 1 × 1 matrix and thus the freedom in Eq. (63)   Suppose that the reality conditions (34) are satisfied for s ∈ U a ⊂ R. Using the explicit form (19) of Z L,R and M L,R matrices as well as the fact that an arbitrary nonsingular complex matrix µ L (s) can be written as where V ± (s) are unitary, while µ(s) is diagonal and positive, one finds (similarly as in the generic case) the following local parametrization of µ L and where with unitary U ± (s), V ± (s) matrices and positive-diagonal µ(s), Λ ± (s) matrices. Hence M 2 ± (s) = ω ± (s) −1 µ 2 (s)ω ± (s) , where µ 2 (s) ≡ µ(s) 2 , and thus (cf. Eq. (23)) Thus, just like before, one sees that the solution m (a) = m tree (a) + O( ) to the gap equation (15) is real and that the eigenvectorsξ [a±] can be chosen as Comparing this with Eqs. (52) one sees that Eqs. (55) hold provided that c(a) is, for a particular choice (139), a phase factor, as was to be proved.
Following the same reasoning as for the fermionic case in Sec. 4.A, one finds that columns (associated with m 2 (ℓ) ) of the ζ matrix, defined according to Eq.
In particular, there exists a matrix C(ℓ) N (ℓ) which obeys Eq. (154) and is real, what ensures the reality of ζ [ℓr] .

Conclusions
We have analyzed in details the effects associated with mixing of scalar and fermionic fields. Presented results, together with their counterparts for vector fields [21], can be useful in the study of extensions of the Standard Model.
In particular, the prescription for "square-rooted residues" ζ is formulated entirely in terms of eigenvectors of certain matrices, and thus it can be efficiently employed in numerical calculations.