The Neutralino Sector in the U(1)-Extended Supersymmetric Standard Model

Motivated by grand unified theories and string theories we analyze the general structure of the neutralino sector in the USSM, an extension of the Minimal Supersymmetric Standard Model that involves a broken extra U(1) gauge symmetry. This supersymmetric U(1)-extended model includes an Abelian gauge superfield and a Higgs singlet superfield in addition to the standard gauge and Higgs superfields of the MSSM. The interactions between the MSSM fields and the new fields are in general weak and the mixing is small, so that the coupling of the two subsystems can be treated perturbatively. As a result, the mass spectrum and mixing matrix in the neutralino sector can be analyzed analytically and the structure of this 6-state system is under good theoretical control. We describe the decay modes of the new states and the impact of this extension on decays of the original MSSM neutralinos, including radiative transitions in cross-over zones. Production channels in cascade decays at the LHC and pair production at $e^+e^-$ colliders are also discussed.


Introduction
Adding an extra U(1) X broken gauge symmetry to the gauge symmetries of the Standard Model is well motivated by grand unified theories [1]. The corresponding supersymmetric extension that generalizes the minimal supersymmetric Standard Model (MSSM) often appears as the low energy effective theory of superstring theories [2]. This U(1) X extended supersymmetric gauge theory shall henceforth be denoted as the USSM.
The Higgs sector associated with the broken U(1) X gauge symmetry provides an elegant solution to the µ problem in supersymmetric theories [3,4]. An effective µ parameter is generated by the vacuum expectation value of the new singlet Higgs field S, which breaks the U(1) X gauge symmetry. This is the same mechanism employed by the next-to-minimal supersymmetric standard model (NMSSM) [5].
However, the USSM possesses an additional advantage by avoiding the extra discrete symmetries of the NMSSM that, in the canonical version, result in the existence of domain-walls that are incompatible with the observed energy density of the universe. Moreover, the upper bound on the mass of the lightest Higgs boson of the MSSM is relaxed in the USSM due to contributions from the new singlet-doublet Higgs interactions and the U(1) X D-terms [6]. Various scenarios of this type have been discussed in the literature, see e.g. Refs. [7,8], in which the U(1) X gauge symmetry is embedded in the grand unification group E 6 (or one of its rank-five subgroups).
Including the extra symmetry, the gauge group is extended to G = SU(3) C ×SU(2) L ×U(1) Y ×U(1) X with the couplings g 3 , g 2 , g Y , g X , respectively. The matter particle content in the supersymmetric theory includes, potentially among others, the left-handed chiral superfieldsL i ,Ê c i ;Q i ,Û c i ,D c i , where the subscript i denotes the generation index, and the Higgs superfieldsĤ d ,Ĥ u ,Ŝ. The usual MSSM Yukawa termsŴ Y of the MSSM superpotential (i.e. without the µ term) are augmented by an additional term that couples the iso-singlet to the two iso-doublet Higgs fields: (1.1) literature [3,[8][9][10][11]. In this report we attempt a systematic analytical analysis of the neutralino system, based on the well-motivated assumption of weak coupling between the original MSSM and the new additional gaugino/higgsino subsystem. In contrast to the MSSM where exact solutions of the mass spectrum and mixing parameters can be constructed mathematically in closed form (see e.g. Ref. [12]), this is not possible anymore for the supersymmetric U(1) X model in which the eigenvalue equation for the masses is a 6th order polynomial equation. However, analogously to the NMSSM [13], Once the masses and mixings are determined, the couplings of the neutralinos to the electroweak gauge bosons and to the scalar/fermionic matter particles are fixed. Decay widths and production rates can subsequently be predicted for squark cascades at the LHC [14] and pair production in e + e − collisions at linear colliders [15]. Of particular interest are the radiative transitions between neutralinos in cross-over zones, where the masses of two neutralinos are nearly degenerate.
The report is organized as follows. In Sect. 2 we first describe the general basis of the neutralino sector in the USSM. Subsequently, for the naturally expected weak coupling between the MSSM and the new subsystem, the properties of the new higgsino and gaugino are derived in Sect. 3

. It is shown
to what extent the properties of the standard neutralinos are modified. The spectrum and the mixings are determined analytically in a weak-coupling perturbative expansion. The neutralino masses are determined to second-order, whereas the mixing matrix elements are determined to first-order in the weak coupling. The accuracy of the perturbative results will become apparent by comparing the analytic approximations with the numerical solutions, thereby demonstrating that a satisfactory understanding of the system can be achieved. As an illustration we will study the limit in which the gaugino mass parameters are significantly larger than the higgsino mass parameters, where both sets of parameters are assumed to be much larger than the electroweak scale. A general description of the neutralino couplings and decay widths is given in Sect. 4, including photon transitions. We also discuss production cross sections in e + e − collisions and cascade decay chains of squarks at the LHC that involve neutralinos.
Section 5 summarizes and concludes this report. Technical details of the analytical diagonalization procedures for the 6 × 6 neutralino mass matrix for non-degenerate and degenerate levels are given in three appendices.

Supersymmetric kinetic mixing
In a theory with two U(1) gauge symmetries, the two sectors can mix, consistently with all gauge symmetries, through the coupling of the kinetic parts of the two gauge bosons [16]. In the basis in which the couplings between matter and gauge fields have the canonical minimal-interaction form, the pure gauge part of the Lagrangian for the U(1) Y ×U(1) X theory can be written where the parameter sin χ is introduced to characterize the gauge kinetic mixing [17]. This Lagrangian generalizes to in a supersymmetric theory, whereŴ Y andŴ X are the chiral superfields associated with the two gauge symmetries. 1 The gauge/gaugino part of the Lagrangian can be converted back to the canonical form by the following GL(2,R) transformation of the superfields [16,17,19]: which acts on the gauge boson and gaugino components of the chiral superfields in the same form. The transformation alters the U(1) Y ×U(1) X part of the covariant derivative to The choice of the kinetic mixing matrix in the form given by Eq. (2.3) is motivated by the fact that the hypercharge sector of the Standard Model is left unaltered by this transformation, and the new effects are separated in the X sector (see, e.g., Ref. [20] for an alternative choice). Consequently, the effective U(1) X charge is shifted from its original value Q X to (2.6) gauge bosons and matter fields are generated even for matter fields with zero U(1) X charge originally.
In grand unification theories the two U(1) groups are orthogonal at the unification scale but small mixing [16] can be induced through loop effects when the theory evolves down to the electroweak scale.
In string theories, kinetic mixing can be induced at the tree level [19]; however, such mixing effects must remain small in order to guarantee the general agreement between SM analyses and precision data in a natural way [21].

The USSM neutralino mass matrix
The Lagrangian of the neutralino system follows from the superpotential in Eq. (1.1), complemented by the gaugino SU(2) L , U(1) Y and U(1) X mass terms of the soft-supersymmetry breaking electroweak Lagrangian: where the W a (a = 1, 2, 3),Ỹ andX are the (two-component) SU(2) L , U(1) Y and U(1) X gaugino fields, and In parallel to the gauge kinetic mixing discussed in Sect. 2.1, the Abelian gaugino mixing mass parameter M Y X is assumed small compared with the mass scales of the gaugino and higgsino fields.
After breaking the electroweak and U(1) X symmetries spontaneously due to non-zero vacuum expectation values of the iso-doublet and the iso-singlet Higgs fields, the doublet higgsino mass and the doublet higgsino-singlet higgsino mixing parameters, are generated. The USSM neutral gaugino-higgsino mass matrix can be written in the following block matrix form 2 , where M 4 is the neutral gaugino-higgsino mass matrix of the MSSM, M 2 corresponds to the new sector containing the singlet higgsino (singlino) and the new U(1)-gauginoB ′ that is orthogonal to the binoB, and X describes the coupling of the two sectors via the neutralino mass matrix. More explicitly, in a basis of two-component spinor fields ξ ≡ (B,W 3 ,H 0 d ,H 0 u ,S,B ′ ) T , the full neutralino mass matrix is given by [10]: where the various gaugino mass parameters M 1 , M 2 , M ′ 1 and M K have been defined in Eqs. (2.7) and (2.8). Notice the absence of a diagonal mass parameter of the new singlino in contrast to the NMSSM where the cubic self-interaction generates this singlet mass term [13]. Two additional mass mixing parameters, are generated after gauge symmetry breaking and the effective charges Q ′ 1 , Q ′ 2 and Q ′ S are defined by (2.14) in terms of the Q i defined below Eq. (1.1). As usual, tan β ≡ v 2 /v 1 is the ratio of the vacuum expectation values of the two neutral SU(2) Higgs doublet fields, s β ≡ sin β, c β ≡ cos β, and s W , c W are the sine and cosine of the electroweak mixing angle θ W .
In general, the neutralino mass matrix M 6 is a complex symmetric matrix. To diagonalize this matrix, we introduce a unitary matrix N 6 such that Given the neutralino mass matrix M 6 , the physical neutralino masses m ph k , which are real nonnegative numbers, and the neutralino mixing matrix elements N 6 kℓ can be calculated. The mass term in the Lagrangian is given by: where the eigenvalues m k are real but not necessarily positive. The Takagi diagonalization of M 6 , which yields real non-negative diagonal mass elements, can then be achieved in a second step by taking m ph k = |m k | and defining the unitary matrix N 6 in Eq. (2.17) by N 6 = (P 6 V 6 ) * , where P 6 is a diagonal phase matrix with elements P 6 kℓ = ε is the sign of m k , which is also proportional to the CP-quantum number [28] of the neutralinoχ 0 k . More precisely, the relative CP-quantum numbers ofχ 0 k andχ 0 ℓ , which is the physical quantity of interest, is given by ε k ε ℓ .
Although the ordering of statesχ 0 k in ascending mass values is convenient, it is often useful to adopt an intermediate auxiliary convention. Note that the neutralino mass matrix is easily diagonalized in the limit of M K = v = 0 (i.e., before the coupling of the MSSM with the new gaugino/singlino block is introduced). In this limit, M 6 is real after rephasing the neutralino interaction-eigenstates (if necessary). That is, without loss of generality, we can choose M 1 , M ′ 1 , M 2 and µ to be real in this limit, in which case Eq. (2.18) yields the following mass eigenvalues: . Away from this limit, the mass-eigenstates χ 0 k ′ will be defined such that their masses are continuously connected to the masses of the corresponding states in the M K = v = 0 limit. This defines an alternative ordering of the statesχ 0 k ′ which will be indicated with primed subscripts.
We shall present a set of techniques for computing analytic approximations of the physical neutralino masses, m ph k ′ and the corresponding neutralino mixing matrix elements N 6 k ′ ℓ ′ . As previously indicated, the primed subscripts denote that these quantities refer to the physical neutralino statesχ 0 k ′ , whose ordering is specified above. Of course, at the end of the computation, one can convert to an ascending mass ordering convention by an appropriate relabeling of the states, masses and mixing matrix elements.

General analysis
It is well known that the MSSM neutralino mass matrix M 4 can be diagonalized analytically (see, e.g., Ref. [13]). In contrast, the diagonalization of the new USSM 6 × 6 neutralino mass matrix M 6 cannot be performed analytically in closed form. However, the case of physical interest is one in which both the couplings of the MSSM higgsino doublets to the singlet higgsino and to the U(1) X gaugino, and the coupling of the U(1) Y and U(1) X gaugino singlets are weak, i.e. the elements of the 4 × 2 submatrix X in Eq. (2.11) are small. Then, an approximate analytical solution can be found following the procedure given in Appendix B.
As an initial step, the 4×4 MSSM submatrix M 4 and the new 2×2 singlino-U(1) X gaugino submatrix M 2 are separately diagonalized: where the m k ′ are real and non-negative. Here we use primed subscripts to indicate that the neutralino states are continuously connected to the corresponding states in the M K = v = 0 limit, as discussed at the end of Sect. 2. The above procedure results in a partial Takagi diagonalization of the full neutralino mass matrix, M 6 : where O is a 4 × 2 matrix of zeros. The upper left and lower right blocks of M 6 are diagonal with real non-negative entries, but the upper right and lower left off-diagonal blocks are non-zero.
Performing a block-diagonalization of M 6 will remove the non-zero off-diagonal blocks while leaving the diagonal blocks approximately diagonal up to second order, due to the weak coupling of the two subsystems. That is, where

5)
A detailed derivation will be presented in Appendix B. The elements of the 4 × 2 mixing matrix Ω are given by: with i ′ = 1 ′ , . . . , 4 ′ and j ′ = 5 ′ , 6 ′ . After the block-diagonalization, the upper left 4 × 4 and the lower right 2 × 2 blocks need not be re-diagonalized up to second order in the small mixing X between the blocks, but the corresponding eigenvalues are shifted m k ′ → m ph k ′ to second order in the small mixing. The physical neutralino masses m ph k ′ are given by: The diagonal matrix of phases is chosen such that the m ph k ′ are real and non-negative, with the phases φ k ′ given by: (3.10) The (perturbative) Takagi diagonalization of the neutralino mass matrix M 6 has now been achieved, with the (real and non-negative) neutralino masses given by Eqs. (3.7) and (3.8), and the neutralino mixing matrix given by: The validity of the perturbative expansion relies on the assumption that 4 for all choices of i ′ = 1 ′ , . . . , 4 ′ and j ′ = 5 ′ , 6 ′ . That is, only degeneracies between the 4×4 block M D 4 and the 2 × 2 block M D 2 are potentially problematic. In particular, in the so-called cross-over zones in which the masses m i ′ ≃ m j ′ exhibit a near degeneracy and the corresponding residue Re (N 4 * XN 2 † ) i ′ j ′ = 0, mixing effects are enhanced and the analytical formalism in Appendix C must be applied.

The case of a real neutralino mass matrix
We shall present numerical case studies under the assumption that the parameters of the neutralino mass matrix are real. The general analysis then simplifies, since a real symmetric mass matrix can always be diagonalized by a similarity transformation, V MV T , where V is real and orthogonal. Since some of the mass eigenvalues of a real symmetric matrix may be negative, we complete the Takagi diagonalization, N * MN † , by introducing a suitable diagonal matrix of phases P and identifying the unitary neutralino mixing matrix by N = (P V ) * , as indicated below Eq. (2.18). In this case, the (perturbative) neutralino mass matrix diagonalization can be performed using the three-step procedure of Ref. [13]: In the first step, we diagonalize the (real symmetric) MSSM matrix M 4 : The mass eigenvalues, which are real but need not be non-negative, are denoted bym i ′ for i ′ = 1 ′ , .., 4 ′ .
The orthogonal diagonalization matrix V 4 is given explicitly in Ref. [12] for the most general choice of gaugino and higgsino mass parameters. Simple analytic forms for the neutralino mass and mixing matrix elements can be found in limits where either the gaugino parameters are much larger than the higgsino parameter or vice versa [29].
The exact analytic diagonalization of the new 2 × 2 submatrix M 2 singlet higgsino-U(1) X gaugino submatrix M 2 is straightforward. The matrix: is diagonalized by an orthogonal rotation V 2 as The eigenvaluesm 5 ′ ,6 ′ are given bỹ The orthogonal diagonalization matrix V 2 is given by: where the angle θ s satisfies the relations: and sin θ s = sign(x) Two limits are of particular interest: (i) If m s ≫ |M ′ 1 |, then the masses and the mixing parameters are approximately given bỹ corresponding to maximal mixing due to the large off-diagonal entries in the mass matrix M 2 .
(ii) In the opposite limit, |M ′ 1 | ≫ m s , and the mass eigenvalues and mixing angle are approximately given bym This is a typical see-saw type mixing phenomenon. The heavy 6th state is a U(1) X gaugino-dominated state, whereas the 5th neutralino state is a singlet-higgsino dominated state.
[2] Block-diagonalization of M 6 We can now perform a block-diagonalization of M 6 : 22) and the elements of the real matrix Ω are given by [cf. Eq. (3.6)]: with i ′ = 1 ′ , .., 4 ′ and j ′ = 5 ′ , 6 ′ . That is, the orthogonal matrix V 6 is conveniently split into the matrices V 4 and V 2 that diagonalize the 4 × 4 and 2 × 2 submatrices M 4 and M 2 respectively, and into the matrix V 6 B that performs the subsequent block-diagonalization [13]: After the block-diagonalization, the mass eigenvalues are shifted to second order in the perturbation X.
As expected, the eigenvalues fulfill the trace formula which is independent of the higgsino mass and the mixing parameters.
The perturbative results obtained above are valid if |( choices of i ′ and j ′ . In the regime of near-degeneracy,m i ′ ≃m j ′ , the perturbation theory breaks down, and the analytic approach of Appendix C must be employed. Note thatm i ′ = −m j ′ is not a case of masseigenvalue degeneracy, so that the perturbative results obtained above should be reliable. This may seem to be in conflict with results of the previous subsection, since the latter corresponds to the degenerate case of m i ′ = m j ′ , where we identify the positive masses m k ′ = |m k ′ | in the notation of Sect. 3.1.
However, a more careful analysis reveals that the condition given by Eq. (3.12) does not apply, since in the case of opposite sign mass eigenvalues, the residue Re ( The higgsino doublet-singlet and the higgsino doublet-U(1) X gaugino mixings generate additional singlino and U(1) X gaugino components in the wave functions of the original MSSM neutralinosχ 0 which is linear in the mixing parameter to first approximation as expected for off-diagonal elements.
Reciprocally, the MSSM gaugino/higgsino components and the singlino and U(1) X gaugino components in the wave functions ofχ 0 5 ′ andχ 0 6 ′ are reduced to with V 6 j ′ k ′ differing from V 2 j ′ k ′ only to second order in the mixing, as expected for diagonal elements.
[3] Ensuring that the physical neutralino masses are non-negative The diagonalization of a real symmetric matrix by an orthogonal similarity transformation produces a diagonal matrix with real but not necessarily non-negative elements. Hence, some of the eigenvalues m k ′ will typically be negative. Defining the unitary matrix N 6 = (P 6 V 6 ) * , where P 6 is a diagonal matrix , the Takagi diagonalization of the neutralino mass matrix is achieved with non-negative neutralino masses. In particular, the unitary neutralino mixing matrix N 6 * appears (instead of the real orthogonal matrix V 6 ) in the corresponding Feynman rules involving the neutralino mass-eigenstates.

Large gaugino mass parameters
To illustrate the previous general discussion we shall first give a detailed parametric analysis in the limit in which all gaugino masses are much larger than the higgsino masses, and both sets much larger than the electroweak and the kinetic mixing scales, i.e.
All neutralino mass matrix parameters will be taken real.
[1] Starting again with the diagonalization of the MSSM submatrix M 4 , the diagonalization matrix V 4 defined in Eq. (3.24) can be parameterized up to second order according to standard MSSM procedure (see, e.g., Ref. [12]), as The effect of the 2 with the abbreviations c ± ≡ (c β ± s β )/ √ 2, removes the mixing between the blocks of the two gaugino and the two higgsino states. V G and V H rescale the gaugino and higgsino blocks themselves: to a diagonal form of the gaugino-higgsino mass matrix for large M 1,2 and µ. Their off-diagonal matrix elements are of second order and can be omitted consistently as they would only affect the eigenvalues at fourth order.
The 2 × 2 diagonalization matrix defined in Eq. (3.24) can be parameterized up to second order as The 2× 2 matrix V 2 generates a diagonal form of the singlino-U(1) X gaugino mass matrix for M ′ 1 ≫ m s .
After these steps are performed, the 4 × 4 and 2 × 2 mass submatrices are diagonal and the complete symmetric mass matrix M 6 takes the intermediate form where, in obvious notation, zero elements of the diagonal blocks are suppressed for easier reading, and The diagonal elementsm k ′ are given bỹ where c ± is defined below Eq. (3.31) and The parameter µ κ can be identified with the NMSSM-type singlino mass parameter [13]. Note that m 5 ′ = µ κ is small compared to all the other neutralino masses in the limit of large gaugino mass parameters considered in this subsection.
As a result of the block diagonalization of Eq. (3.34), the mass eigenvalues are shifted according to Eq. (3.25). The resulting mass eigenvalues to the desired order are given by: Note that the sum rule given by Eq. (3.27) is satisfied.
As expected, while the large SU(2) gaugino mass m 2 ′ is not affected by the singlino and the U(1) X gaugino, the MSSM U(1) mass m 1 ′ is affected by the U(1) kinetic mixing. All the higgsino states are modified by the interactions between the MSSM and the new subsystem. The value of m 5 ′ is raised by the interaction with the MSSM higgsinos, but remains small nevertheless.
The mixing in the wave-functions is described by the components of Ω alone since the 4 × 4 matrix V 4 and the 2 × 2 matrix V 2 deviate from unity only to second order in the small parameters of the order of the SUSY scales [i = 1 ′ , .., 4 ′ ]: The non-trivial mixing between two U(1) gaugino states, elements {1 ′ 6 ′ } and {6 ′ 1 ′ }, is generated by the non-zero Abelian gauge kinetic and mass mixing with non-zero M K . The analysis above fails when M 1 ≈ M ′ 1 ; this region of near degeneracy can be handled analytically using the results of Appendix C.
In Eqs. (3.38) and (3.39), perturbative corrections up to second order have been included for the masses and diagonal mixing matrix elements, whereas only the first order corrections have been given for the off-diagonal mixing matrix elements. This follows the usual procedure of stationary perturbation theory in quantum mechanics, which associates second-order corrections to the eigenvalues with the first-order corrections to the wave function. Consequently, the zeros that appear in some of the matrix elements of V 6 k ′ ℓ ′ , should be interpreted as approximate. For example, V 6 2 ′ 6 ′ and V 6 6 ′ 2 ′ are expected to receive higher order perturbative corrections and hence be shifted away from zero. Nevertheless, the fact that the magnitude of these matrix elements are so suppressed will have some dramatic consequences for the behavior of theχ 0 2 ′ andχ 0 6 ′ masses in regions of near-degeneracy.
[3] The final step is to identify . The physical masses m ph k are given by the absolute values of the m k given above. The neutralino states can then be reordered in ascending (non-negative) mass if desired.
The results of this subsection are easily generalized for the case of M 1 , M 2 , M ′ 1 , µ, v s ≫ v, M K . As long as the MSSM gaugino and higgsino parameters, M 1,2 and µ remain significantly larger than the electroweak scale v, the couplings between the MSSM and the new fields, generated by X, remain weak and the diagonalization of the mass matrix can still be performed analytically. However, instead of the approximate valuesm 5 ′ ,6 ′ in Eqs. (3.34) and (3.35) the exact solutions (3.16) must be used, and for V 2 the general rotation matrix (3.17) must be inserted. The approximation ceases to be valid at isolated points where X/(m i ′ −m j ′ ) is no longer a small perturbation, due to the degeneracy of mass eigenvalues m i ′ ≈m j ′ . In these cross-over zones the analysis described in Appendix C must be applied.

An illustrative example
To illustrate the properties of the two new neutralinos and the impact of the coupling of the two subsystems on the original MSSM neutralinos, we study, numerically and analytically, the evolution of the neutralino mass spectrum and representative examples for the mixing of the particles from a very light new U(1) X gaugino across typical MSSM mass scales up to very high scales. Gauge kinetic mixing has only a small impact on the spectrum and it will therefore be neglected in the illustrative example. Throughout the evolution, including all intermediate regions, the coupling between the new states and the MSSM states remains weak, apart from regions of mass degeneracy. The evolution affects primarily the spectrum of the two new neutralino states. In the initial limit, M ′ 1 small, two mediumheavy degenerate states, m 5 ′ ,6 ′ ∼ O(v s ), are realized in the spectrum. At the end of the chain, M ′ 1 large, the spectrum is of a see-saw type, including one heavy and one nearly zero-mass state.
As an illustrative example, we take M 2 = 1.5 TeV, m s = 1.2 TeV, µ = 0.3 TeV and M K = 0, and we assume the gaugino unification relation M 1 = (5/3) tan 2 θ W M 2 ≈ 0.5M 2 . Also, for the numerical analysis in this paper, we set tan β = 5. We adopt the N -model charge assignments [8], For definiteness we fix the gauge coupling at g X ≃ 0.46, evolved from its E 6 unification value of 5/3 g Y down to the electroweak scale; however the results are not very sensitive to this assumption. We could also choose to fix M X at its gaugino unification value under the assumption that all gaugino masses unify at the grand unification scale. This would correspond to a value of M ′ 1 ≈ M X = M 1 = 750 GeV (neglecting kinetic mixing effects). However, to illustrate the structure of the system in various scenarios, we shall be slightly more general by allowing M ′ 1 to vary over a large range of values (0 ≤ M ′ 1 ≤ 5 TeV).
To be specific, we choose the evolution with M ′ The evolution of the six (positive) neutralino masses 6 and the values of two typical V 6 mixing elements, 6 The eigenvalues 4 ′ and 5 ′ of the mass matrix [Eq.

Mass [GeV]
Neutralino Masses 3' Outside the cross-over regions the approximate analytical mass spectra nearly coincide with the exact (numerically computed) solutions for the eigenvalues as demonstrated in Table 1 for three M ′ 1 values.
The mixing pattern is more directly reflected in the elements of the rotation matrix V 6 , as shown in Fig. 2. For zero kinetic mixing,χ 0 Their overlap with the MSSM higgsinos, V 6 5 ′ 4 ′ , is small except in the cross-over zone. The mixing V 6 5 ′ 6 ′ between the new U(1) X gaugino and singlino states is reduced from maximal mixing −1/ √ 2 for vanishing U(1) X gaugino mass parameter M ′ 1 to nearly zero mixing at asymptotically large M ′ 1 . The moderate change in the 5 ′ -4 ′ cross-over zone is a reflection of the 5 ′ 4 ′ variations by unitarity of the neutralino mixing matrix.

Neutralino Production and Decays
Neutralino production rates in various channels and decay properties in various modes are affected by the mixing of the neutralino states and by the mass and kinetic mixings of the gauge bosons associated with the broken U(1) X and SU(2)×U(1) Y gauge symmetries.
The Z and Z ′ bosons can mix through kinetic coupling, as analyzed before, and mass mixing induced by the exchange of the Higgs fields, for example, charged under both U(1) ′ s. The resulting Z and Z ′ mixing is described by the mass-squared matrix where the matrix elements are given by and where g 2 The eigenvalues of M 2 ZZ ′ and the Z and Z ′ mixing angle follow from The phenomenological constraints typically require this mixing angle to be less than a few times 10 −3 [21], although values as much as ten times larger may be possible in some models with a light Z ′ and reduced couplings [30].
For the neutralino production processes in e + e − annihilation it is sufficient to consider the neutralino- with i, j = 1, .., 6 and Explicitly, the couplings Z ij and Z ′ ij are given in terms of the USSM neutralino mixing matrix N by 7 Sfermion t/u-channel exchanges require the fermion-sfermion-neutralino vertices (with the fermion masses neglected): In Eq. (4.6) the coupling to the higgsino component, which is proportional to the fermion mass, has been neglected. These would have to be included if one were to study, e.g., the neutralino interaction with the top quark and squark.
For completeness, we also provide the fermion-fermion-Z 1,2 vertices: When switching from L → R in Eq. (4.7), the corresponding SU(2)×U(1) and U(1) X charges must be changed accordingly. I f 3 ≡ I f 3L is the SU(2) isospin component (note that I f 3R = 0), e f is the electric charge of the fermion f and Q ′ f L,R are the effective U(1) X charges of the left/right-handed fermions.
The neutralino production and decay properties in the USSM model with the additional gaugino and singlino states depend crucially on their masses with respect to the MSSM neutralino masses.
If they are much heavier than the other states, they will rarely be produced and so are practically unobservable. In contrast, if the singlino is lighter than the other states, a singlino-dominated state will be the lightest supersymmetric particle (LSP) into which the other neutralino states will decay, possibly through cascades.
In the following subsections, we present a brief description of the general formalism of neutralino production and the subsequent cascade decays of the neutralinos. Once charges and mixing matrices 7 For simplicity of notation, the USSM neutralino mixing matrix N 6 will be denoted by N in this section.
are generalized to the present U(1) X theory, the phenomenological infrastructure for cross sections and decay widths can be copied from the MSSM.

Singlino Production in e + e − Annihilation
The production processes of a neutralino pair in e + e − annihilation, 8 are generated by s-channel Z 1 and Z 2 exchanges, and t-and u-channelẽ L,R exchanges. The transition amplitudes, are built up by products of chiral neutralino currents and chiral fermion currents, coupled by bilinear "charges" Q LL , Q LR etc. The four generalized bilinear charges correspond to independent helicity amplitudes, describing the neutralino production processes for polarized electrons/positrons [12]. They can be parameterized by the fermion and neutralino currents and the propagators of the exchanged (s)particles as follows: (4.10) The first two terms in each bilinear charge are generated by Z 1 and Z 2 exchanges and the third term by selectron exchange; D Z 1,2 , D tL,R and D uL,R denote the scaled s-channel Z 1,2 propagators and the tand u-channel left/right-type selectron propagators with s = (p e − + p e + ) 2 , t = (p e − − pχ0 i ) 2 and u = (p e − − pχ0 j ) 2 denoting the Mandelstam variables for neutralino pair production in e + e − collisions. The couplings F iL,R of the gauge bosons Z i (i = 1, 2) to a fermion pair are given by where s ZZ ′ ≡ sin θ ZZ ′ , c ZZ ′ ≡ cos θ ZZ ′ , I f 3 = −1/2 and e f = −1 for the electron charges. Finally, the matrices Z 1,2ij and the vectors L i and R i are defined by (t W = tan θ W ) The e + e − annihilation cross sections follow from the squares of the relevant couplings, where S ij = (1+δ ij ) −1 is a statistical factor which is equal to 1 for i = j and 1/2 for i = j; Θ is the polar angle of the produced neutralinos; and λ PS ≡ λ PS (1, µ 2 i , µ 2 j ) denotes the familiar 2-body phase space function. The quartic charges Q i (i = 1, 2, 3) are given by products of bilinear charges: The integration over the polar angle Θ can easily be performed analytically.
The production cross sections for the three pairings of the two lightest neutralinos, {11}, {12} and {22}, are illustrated in Fig. 3 as functions of M ′ 1 . For the parameter set defined in Sect. 3.4, the corresponding Z ′ mass is M Z 2 = 949 GeV and the ZZ ′ mixing angle is θ ZZ ′ = 3.3 × 10 −3 ; these parameters are compatible with existing limits [21,30]. The center-of-mass energy of the e + e − collider is set to 800 GeV. Of course, if Z 2 is in the reach of the collider, running on the Z 2 resonance would be the most natural way to explore all the facets of the new particle sector in an optimal way.   Table 2. The production of light neutralino pairs, diagonal pairs in particular, are greatly enhanced although the light neutralino masses are nearly identical in the two models.

Neutralino cascade decays and sfermion decays
If kinematically allowed, the two-body decays of neutralinos into a neutralino and an electroweak gauge bosons Z 1,2 are among the dominant channels. The widths of the decays,χ 0 i →χ 0 j Z k (k = 1, 2), are given by Two examples,χ 0 i →χ 0 1 Z 1 for i = 6, 3, illustrate the evolution of the widths with M ′ 1 in Fig. 4. The neutralinosχ 0 6 andχ 0 1 are identified with the MSSM SU(2) gaugino and the lighter of the MSSM higgsinos for small M ′ 1 , and with the U(1) X gaugino and the singlino for large M ′ 1 , respectively [cf. Fig. 1]. Even afterχ 0 6 crosses to the U(1) X gaugino at the 2 ′ -6 ′ cross-over zone, the width increases due to an increasing phase space factor (due to the increasing mass difference) and the fact thatχ 0 6 has a significant singlino component. However, once the stateχ 0 1 becomes singlino-dominated above the 5 ′ -4 ′ cross-over zone, the width of the decayχ 0 6 →χ 0 1 Z 1 drops dramatically as the mixing between the  U(1) X gaugino and the singlino state is strongly suppressed for large M ′ 1 . The stateχ 0 3 is the MSSM U(1) gaugino for small M ′ 1 , the singlino-dominated state for moderate M ′ 1 and the heavier of the MSSM higgsinos for large M ′ 1 . As theχ 0 3 mass drops, even only slightly, the two-body decayχ 0 3 →χ 0 1 Z 1 is kinematically forbidden for moderate M ′ 1 . However, the mode is kinematically allowed again and its magnitude increases when the mass of the singlino-dominatedχ 0 1 decreases sufficiently.
Similarly, the two-body decays of the charginos into a neutralino and the W ± gauge boson are expected to be among the dominant channels if kinematically allowed. The widths of the decays, ) and the W L,R are defined as The unitary matrices U L and U R diagonalize the chargino mass matrix via the singular value decomposi- Explicit formulae for the chargino masses and mixing matrices can be found in Refs. [31,32].
At the LHC, sfermion decays,f → fχ 0 i can produce complex cascades, as heavier neutralinos are often produced in the initial decay and subsequently decay through a number of steps before the lightest neutralino (which is presumably the LSP) is produced to end the chain. Thus, cascade decays are of great experimental interest at the LHC. The width of the sfermion 2-body decay into a fermion and a neutralino follows from where The rates for the reverse decays, neutralino decays to sfermions plus fermions,χ 0 i →ff ,f f are given by the corresponding partial widths 9 Supersymmetric particles will be analyzed at the LHC primarily in cascade decays of some initially produced squark or gluino. In the U(1) X extended model, the cascade chains may be extended compared with the MSSM by an additional step due to the presence of two new neutralino states, for example, At each step in the decay chain, we have placed the decay products from the previous step within brackets. The end result of the cascade above is the final state uZ 1 ℓℓχ 0 1 with visible particles/jet u, Z 1 ≃ Z, and two ℓ's.
For the parameter set introduced earlier, the partial widths involved in the cascade steps are shown for evolving M ′ 1 in Fig. 5. The sfermion decaysũ R →χ 0 6 u andl R →χ 0 1 ℓ are shown in the left panel and the neutralino decaysχ 0 6 →χ 0 5 Z 1 andχ 0 5 →l R ℓ are shown in the right panel. The first stepũ R →χ 0 6 u corresponds to the decay of the R-type u-squark toχ 0 6 , which coincides with the MSSM SU(2) gaugino for small M ′ 1 and, after the 2 ′ -6 ′ cross-over zone, with the U(1) X gaugino. The width increases dramatically before the decay is forbidden kinematically for M ′ 1 larger than 1.5 TeV. The second stepχ 0 6 →χ 0 5 Z 1 in this cascade chain corresponds to the decay of the MSSM SU(2) gaugino for small M ′ 1 , changing to the U(1) X gaugino decay thereafter. The dependence of this two-body decay mode on M ′ 1 is mainly of kinematic nature; the decay is not allowed for M ′ 1 between ∼ 0.8 TeV and ∼ 1.0 TeV. Just beyond the 2 ′ -6 ′ cross-over zone, it increases very sharply and keeps increasing moderately with M ′ 1 thereafter. The pattern for the third decayχ 0 5 →l R ℓ is mainly determined by the size of the U(1) Y and U(1) X gauge components of the stateχ 0 5 . Before the 2 ′ -6 ′ cross-over zone the state is a U(1) X gaugino so that the width is large. But the width is strongly suppressed for moderate and large M ′ 1 for whichχ 0 5 is a MSSM SU(2) gaugino with very small mixing with the two MSSM U(1) and U(1) X gauginos. The width for the final decayl R →χ 0 1 ℓ remains moderate as the U(1) components of theχ 0 1 state are small. Beyond the cross-over zone, the width decreases with the suppressed U(1) X component.
Conventional chains likeq → qχ 0 i → q[ℓl] → qℓ[ℓχ 0 1 ] may also be observed in the U(1) X extended model. However, the partial widths in the USSM can be very different from the MSSM. As an example, we consider the cascade chains, in which the intermediate neutralino stateχ 0 i is the second heaviest neutralino, i.e.χ 0 5 in the USSM andχ 0 3 in the MSSM. As demonstrated in Table 3, the width for the decay ofũ R to the second heaviest neutralino in the USSM is much smaller than in the MSSM.

Decays to Higgs bosons
The USSM Higgs sector includes two Higgs doublets H u and H d as well as the SM singlet field S [8, [33][34][35]. Their interactions are determined by the gauge interactions and the superpotential in Eq. (1.1).
Including soft SUSY breaking terms and radiative corrections, the resulting effective Higgs potential consists of four parts: where the F , D and soft-breaking terms V F , V D and V soft are given by The structure of the F term V F is the same as in the NMSSM without the self-interaction of the singlet field. However the D term V D contains a new ingredient: the terms proportional to g 2 X are D-term contributions due to the extra U(1) X which are not present in the MSSM or NMSSM. The soft SUSY breaking terms are collected in V soft . The tree-level Higgs potential is CP-conserving [34]. That is, one can rephase the Higgs fields to absorb the phases of the potentially complex coefficient λA λ . Thus, without loss of generality, we will assume that these parameters are real.
The term ∆V in Eq. (4.21) represents the radiative corrections to the Higgs effective potential [36].
The dominant contributions at one-loop are generated by top quark and scalar top quark (stop) loops due to the large Yukawa couplings; these terms are the same as in the MSSM. All the other modeldependent contributions do not contribute significantly at one-loop order [33]. Therefore, we will ignore these subdominant model-dependent radiative corrections in the following analysis.
The set of soft SUSY breaking parameters in the tree-level Higgs potential includes the soft masses Decays involving Higgs bosons can be quite different for different Higgs boson mass spectra. We first decompose the neutral Higgs states into real and imaginary parts as follows: The resulting Higgs mass spectrum consists of three CP-even scalars, one CP-odd scalar, and two charged Higgs bosons.
Generally, the width of a 2-body neutralino or charginoχ i decay to a neutralino or charginoχ j and a Higgs boson φ k (H 1,2,3 or A) is given by (i) For the decay of a neutralinoχ 0 i to a neutralinoχ 0 j and a scalar Higgs boson H k ,χ 0 i →χ 0 j H k , the couplings are given by,  Figure 6: The decay widths forχ 0 5,4 →χ 0 1 H 1 (left) andχ 0 5,6 →χ 0 1 A 1 (right) for the parameter set given in the text. For the purposes of example, the Higgs mass parameter M A is set to 1.25 TeV.
follows Eq. (4.25) with the left/right couplings given by The widths for the kinematically allowed decaysχ 0 5,6 →χ 0 1 A are shown in the right panel of Fig. 6 as a function of M ′ 1 in the benchmark scenario. In contrast to the scalar case, only a few decays are kinematically allowed since the CP-odd scalar A is heavy.
(iii) For completeness, we describe the decays of charginos to a neutralino and charged Higgs bosoñ 2, . . . , 6). These follow a similar pattern, but with the last index of the coupling removed: The same left/right couplings determine the decays of neutralinos to charginos and charged Higgs bosoñ   where λ PS ≡ λ PS (1, m 2 χ j /m 2 φ i , m 2 χ j /m 2 φ i ) and S jk = (1 + δ jk ) −1 is the usual statistical factor. The couplings C L/R ijk are related to their neutralino/chargino decay counterparts in the obvious way:

Neutralino radiative decays
In the cross-over zones of the neutralino mass eigenvalues, the gaps between the neutralino masses become very small. As a result, standard decay channels are almost shut and photon transitions between neutralino states [37] become enhanced. These photon transitions are particularly important in the cross-over zone 4 ′ -5 ′ at M ′ 1 ≃ 2.6 TeV [cf. Fig. 1]. The proximity of the two heavier states to the lightest neutralino dramatically reduces the rates of all other decay modes so that the radiative decays become non-negligible modes. Of course, also the γ transitions are phase-space suppressed in cross-over zones but less strongly than the competing standard channels due to the vanishing photon mass, even for 3-particle decays into a lighter neutralino and lepton-or light-quark pair.
The effective couplings gχ0 iχ 0 j γ in the partial decay widths so that decays to lepton pairs are allowed through couplings mediated by virtual Z-bosons which in fact are the dominant modes. Therefore, for illustrative comparison, the partial widths for standard electron-pair decaysχ 0 i →χ 0 j e + e − are also shown. Evidently the branching fractions for radiative decays in general drastically change in the complex 4 ′ -5 ′ cross-over zone near M ′ 1 ≃ 2.6 TeV compared with the side-band wings.

Summary and Conclusions
In this report we have investigated the neutralino sector in the U(1) X extension of the minimal Supersymmetric Standard Model, as suggested by many GUT and superstring models. The extended model has attractive features which solve several problems of the MSSM. It provides a natural solution of the µ-problem without creating cosmological problems. The upper limit of the light Higgs mass is somewhat increased and the growing fine tuning in this sector is reduced.
While the MSSM neutralino sector is already quite complex, the complexity increases dramatically in the extended model due to two additional degrees of freedom. However, the small coupling between the original four MSSM states and the two new states offers an elegant analytical solution of this problem within a perturbative expansion. We have worked out this solution in detail for the mass spectrum and the mixing of the states.
The expansion in the parameter v/M SUSY , the ratio of the electroweak scale v over the generic supersymmetry-breaking scale M SUSY , leads to an excellent approximation of the exact solutions. Even in the cross-over zones in which two mass eigenvalues are nearly degenerate, proper adaption of the analytical formalism provides an accurate description of the system. Thus, in the limit in which M SUSY is sufficiently above the electroweak scale, the neutralino system of the U(1) X -extended supersymmetric standard model is under good analytical control and its features are theoretically well understood.
A few examples of mass spectra, widths for cascade decays at LHC, decays to Higgs bosons, photon transitions and production cross sections in e + e − collisions illustrate the characteristic features of the model. and a constructive proof of Takagi's result, we provide, as a pedagogical example, the explicit Takagi diagonalization of an arbitrary 2 × 2 matrix. The latter will be particularly useful for considering cases in which there is a near-degeneracy in mass between two of the neutral fermions.

A.1 General analysis
Consider a system of n two component fermion fields ξ ≡ (ξ 1 , ξ 2 , . . . , ξ n ) T , whose physical masses are governed by the Lagrangian In general, the mass matrix M is an n × n complex symmetric matrix. In order to identify the physical masses m i and the corresponding physical fermion fields χ i , one introduces a unitary matrix U such that ξ = U χ and demands that ξ T M ξ = i m i χ i χ i . This corresponds to the Takagi diagonalization of a complex symmetric matrix, 12 which is governed by the following theorem [23,24]: where the m k are the singular values and the vectors v k are normalized to have unit norm. Following Ref. [25], the v k are called the Takagi  Finally, in the subspace corresponding to zero singular values, U is unique up to multiplication on the right by an arbitrary unitary matrix.
We shall establish the Takagi diagonalization of a complex symmetric matrix by formulating an algorithm for constructing U . A method will be provided for determining the orthonormal Takagi vectors v k that make up the columns of U . This is achieved by rewriting the n × n complex matrix equation Eq. (A.4) [with m real and non-negative] as a 2n × 2n real matrix equation [40]: converted into an ordinary eigenvalue problem for a real symmetric matrix. Since m ≥ 0, we solve the eigenvalue problem M S u = mu for the eigenvectors corresponding to the non-negative eigenvalues. 15 It is straightforward to prove that the total number of linearly independent Takagi vectors is equal to n.
Simply note that the orthogonality of ( Re v 1 , Im v 1 ) and the Takagi diagonalization of a complex symmetric matrix has recently been presented in Ref. [27] (see also Refs. [25,26] for previous numerical approaches to Takagi diagonalization).
A.2 Example: Takagi diagonalization of a 2 × 2 complex symmetric matrix The Takagi diagonalization of a 2 × 2 complex symmetric matrix can be performed analytically. 16 The result is somewhat more complicated than the standard diagonalization of a 2 × 2 hermitian matrix by a unitary similarity transformation. Nevertheless, the corresponding analytic formulae for the Takagi diagonalization will prove useful in Appendix C in the treatment of nearly degenerate states. Consider the complex symmetric matrix: where c = 0 and, without loss of generality, |a| ≤ |b|. We parameterize the 2 × 2 unitary matrix U in Eq. (A.2) by [41]: where 0 ≤ θ ≤ π/2 and 0 ≤ α , β , φ < 2π. However, we may restrict the angular parameter space further. Since the normalized Takagi 16 The main results of this subsection have been obtained, e.g., in Ref. [27]. Nevertheless, we provide some of the details here, which include minor improvements over the results previously obtained. 17 In the case of a zero singular value or a pair of degenerate of singular values, there is more freedom in defining the Takagi vectors as discussed below Eq. (A.4). These cases will be treated separately at the end of this subsection.
with |σ 1 | ≤ |σ 2 |. This ordering of the |σ k | is governed by the convention that 0 ≤ θ ≤ π/4 (the opposite ordering would occur for π/4 ≤ θ ≤ π/2). If det M = ab − c 2 = 0 (with M = 0) , then there is one singular value which is equal to zero. In this case, it is easy to verify that σ 1 = 0 and |σ 2 | 2 = Tr (M † M ) = |a| 2 + |b| 2 + 2|c| 2 . All the results obtained above remain valid, except that α is undefined [since in this case, the argument of arg in Eq. (A. 19) vanishes]. This corresponds to the fact that for a zero singular value, the corresponding (normalized) Takagi vector is only unique up to an overall arbitrary phase [cf. footnote 17].
We provide one illuminating example of the above results. Consider the complex symmetric matrix: The eigenvalues of M are degenerate and equal to zero. However, there is only one linearly independent eigenvector, which is proportional to (1 , i). Thus, M cannot be diagonalized by a similarity transformation [24]. In contrast, all complex symmetric matrices are Takagi not degenerate. Thus, all the formulae derived above apply in this case. One quickly determines that θ = π/4, φ = π/2, β = π/2 and α is indeterminate (so one is free to choose α = 0). The resulting Takagi diagonalization is U T M U = diag(0 , 2) with: This example clearly indicates the distinction between the (absolute values of the) eigenvalues of M and its singular values. It also exhibits the fact that one cannot always perform a Takagi diagonalization by using the standard techniques for computing eigenvalues and eigenvectors. 18 We end this subsection by treating the case of degenerate (non-zero) singular values, which arises when bc * = −a * c. Special considerations are required since not all the formulae derived above are applicable to this case [cf. footnote 17]. The condition bc * = −a * c implies that |a| = |b|, so that After noting that a/c = −b * /c * , Eq. (A.12) then yields: where c φ ≡ cos φ and s φ ≡ sin φ.
Note that separately, α and β depend on the choice of φ (although φ drops out in the sum). Explicitly, One easily verifies that Eq. (A.24) is satisfied. Moreover, using Eq. (A.9), α and β are now separately determined.
We illustrate the above results with the classic case of M = 0 1 1 0 . In this case M † M = 1 2×2 , so U cannot be deduced by diagonalizing M † M . Setting a = b = 0 and c = 1 in the above formulae, it follows that θ = π/4, σ 1 = −e −iφ and σ 2 = e iφ , which yields α = −(φ ± π)/2 and β = φ/2. Thus, Eq. (A.7) yields: which illustrates explicitly that in the degenerate case, U is unique only up to multiplication on the right by an arbitrary orthogonal matrix.
Appendix B: The small-mixing approximation In this appendix, we provide details of the neutralino mass matrix diagonalization in the small mixing approximation, in which the weak coupling between two off-diagonal matrix blocks can be perturbatively treated. For mathematical clarity, we present the solution for a general complex (N + M ) × (N + M ) symmetric matrix in which the N × N and M × M submatrices are coupled weakly so that their mixing is small: To obtain the corresponding physical neutralino masses, one must perform a Takagi diagonalization of where N N +M is a unitary matrix. 20 The Takagi  [1] In the first step, we separately perform a Takagi diagonalization of M N and M M : where the m k ′ are real and non-negative. The ordering of the diagonal elements above 21 is chosen according to some convenient criterion (e.g., see the discussion at the end of Sect.  Step [1] results in a partial Takagi diagonalization of M N +M : for all choices of i ′ = 1 ′ , . . . , N ′ and j ′ = N ′ + 1 ′ . . . , N ′ + M ′ . This condition will be an output of our computation below. The perturbative block-diagonalization is accomplished by introducing an (N + M ) × (N + M ) unitary matrix: where Ω is an N × M complex matrix that vanishes when X N M vanishes (and hence like X N M is perturbatively small). Note that N B N † B = 1 (N +M )×(N +M ) + O(Ω 4 ) which is sufficiently close to the identity matrix for our purposes. Straightforward matrix multiplication then yields: Re with i ′ = 1 ′ , .., N ′ and j ′ = N ′ + 1 ′ , .., N ′ + M ′ . Although the m k ′ are real and non-negative by construction, we see that the shifted mass parameters m k ′ are in general complex. Thus, to complete the perturbative Takagi diagonalization, we perform one final step.
[3] The diagonal neutralino mass matrix is given by: where P is a suitably chosen diagonal matrix of phases P = diag(e −iφ 1 ′ , . . . , e −iφ N ′ +M ′ ) , (B.20) such that the elements of the diagonal mass matrix M D N +M (denoted by m ph k ′ ) are real and non-negative. We identify the m ph k ′ with the physical neutralino masses. The unitary neutralino mixing matrix is then identified as: Starting from Eqs. (B.17) and (B.18), one can evaluate P to second order in the perturbation Ω. In particular, for ǫ 1,2 ≪ a, we have a + ǫ 1 + iǫ 2 ≃ (a + ǫ 1 )e iǫ 2 /a . From this result, we easily derive the second-order expressions for the physical neutralino masses m ph k ′ : (B.23) and the phases φ k ′ : We first interchange the first row and the k ′ th row of M N +M followed by an interchange of the first column and the k ′ th column, in order that m k ′ occupy the 1 ′′ 1 ′′ element of the matrix. 22 Next, we interchange the second row and the ℓ ′ th row followed by an interchange of the second column and the ℓ ′ th column, in order that m ℓ ′ occupy the 2 ′′ 2 ′′ element of the matrix. This sequence of interchanges has the effect of grouping the two nearly degenerate diagonal elements next to each other, resulting in two cases considered above-in the cross-over zone 2 ′ -6 ′ with very small δ, and in the cross-over zone 4 ′ -5 ′ with moderately small δ, as shown by the solid line trajectories of Fig. 9.
Thus, we have demonstrated that an analytic perturbative treatment of the neutralino mass matrix can be carried out, and all of its features understood, even in the case of a pair of near-degenerate states.