A fresh look at neutral meson mixing

In this work we show that the existence of a complete biorthonormal set of eigenvectors of the effective Hamiltonian governing the time evolution of neutral meson system is a necessary condition for diagonalizability of such a Hamiltonian. We also study the possibility of probing the $CPT$ invariance by observing the time dependence of cascade decays of type $P^{\circ}(\bar{P^{\circ}})\to \{M_a,M_b\}X\to fX$ by employing such basis and exactly determine the $CPT$ violation parameter by comparing the time dependence of the cascade decays of tagged $P^{\circ}$ and tagged $\bar{P^{\circ}}$.


Introduction
In the Wigner-Weisskopf (W-W) approximation [1] the effective Hamiltonian which describes the P • − P • system is not Hermitian. Therefore the eigenkets of this Hamiltonian are indistinguishable (unless the Hamiltonian is normal, [Ĥ,Ĥ † ] = 0). The reason is that for such a system the orthogonality and completeness relations could not be written in terms of its eigenkets. In the presence of T violation in the P • − P • system, we are dealing with a nonhermitian Hamiltonian which is not normal. Therefore we can use the principles of non-hermitian quantum mechanics and reconsider the definition of diagonalizability of an operator. We use the biorthonormal basis for this propose. Here we emphasis that when a non-hermitian and non-normal operator is encountered, use of a complete biorthonormal basis is in order which reduces to orthogonal basis as soon as the operator is considered hermitian and normal. Therefore we conclude that we may use such a set of basis to describe the time evolution of neutral mesons in the presence of T violation.
As mentioned, in the presence of T violation the eigenkets of a non-hermition Hamiltonian does not satisfy the completeness and orthogonality relations. Therefore the eigenkets of such a Hamiltonian are not distinguishable. Due to this fact in writing down the transition amplitudes for cascade decays P • (P • ) → {M a , M b }X → f X, we use the biorthonormal basis when the intermediate states are eigenstates ofĤ.
We prove that the existance of a complete biorthonormal set of eigenvectors ofĤ is necessary condition for diagonalizability of the effective Hamiltonian governing the time evolution of the neutral meson systems and write down the spectral form of the Hamiltonian operator with this basis in section 2. In section 3 we discuss the time evolution of neutral meson system and introduce the T and CP T violation complex parameters and obtain the time evolution of flavor eigenkets. Finally in the last section we study the possibility of probing CP T invariance by observation of the time dependence of the cascade decays of type P • (P • ) → {M a , M b }X → f X by using the biorthonormal basis and introduce new ratios of decay amplitudes and exactly determine the CP T violation parameter by comparing the time dependence of the cascade decays of tagged P • and tagged P • .
2 Diagonalizability and the complete set of biorthonormal basis.
A linear operatorĤ acting in a separable Hilbert space and having a discrete spectrum is diagonalizable if an only if there are eigenvectors |ψ n ofĤ and |φ n ofĤ † that form a complete set of biorthonormal basis of {ψ n , φ n }, i.e. they satisfŷ and ψ m |φ n = δ mn , n |ψ n φ n | = n |φ n ψ n | = 1.
Where n is the spectral label and † and * denote the adjoint and complexconjugate respectively as usual. Moreover that δ mn is the Kronecker delta function and 1 represents the identity operator.
Nowhere in this definition it is assume that the operator is normal, i.e., [Ĥ,Ĥ † ]=0. A normal operator, in finite dimensions with no extra conditions and in infinite-dimensions with appropriate extra conditions, admit a diagonal metrix representation in some orthogonal basis. This is usually called diagonalizability by a unitary transformation. In view of (1) and (2) The spectral form ofĤ andĤ † may be written in the following form In order to see the equivalence of the existence of a complete biorthonormal set of eigenvectors ofĤ and its diagonalizability, we note that by definition a diagonalizable HamiltonianĤ satisfiesÂ −1ĤÂ =Ĥ • for an invertible linear operatorÂ and a diagonal linear operatorĤ • , i.e., there is an orthogonal basis {|n } in the Hilbert space and complex numbers E n such thatĤ • = n E n |n n|. Then letting |ψ n :=Â|n and |φ n := (Â −1 ) † |n , we can easily check that {|ψ n , |φ n } is a complete biorthonormal system forĤ. The converse is also true, for if such a system exists we may setÂ := n |ψ n n| for some orthogonal basis {|n } and by using equation (2) As long as T is invariant (no violation), the effective Hamiltonian is normal. In such a case the orthonormality relations between the basis ofĤ are valid and moreover that the eigenket ofĤ are discriminant and the biorthonormal basis turn into orthonormal basis automatically and |ψ n 's are the same as |φ n 's.

The time evolution of neutral meson system
In the Wigner-Weisskoff (W-W) approximation, which we shall use throughout, a beam of oscillating and decaying neutral meson system is described in its rest frame by a two component wave function where t is the proper time and |P • stands for K, D, B d or B s . The wave function evolves according to a Schrödinger like equation The matrixĤ is usually written asĤ =M − iΓ/2. WhereM =M † , and Γ =Γ † are 2 × 2 matrices called the mass and the decay matrices [1][2][3][4][5][6][7]. Decomposition ofĤ readŝ The flavor basis {|P • , |P • } satisfy orthogonality and completeness relations It is readily shown thatĤ is not hermitian. If [Ĥ,Ĥ † ] = 0 then, the orthogonality and completeness relations for eigenstates ofĤ are not satisfied, i.e., the eigenstates ofĤ are not discriminant states. Therefore we cannot diago-nalizeĤ or write its spectral form though its basis. To do this job we make the benefit of biortonormal basis. Such basis could be set up for neutral meson system. indeed the Hamiltonian is diagonalizable only with such set of basis.
According to section 2 the eigenvalues ofĤ are denoted by µ a = m a − iΓ a /2 and µ b = m b − iΓ b /2 corresponding to the eigenvectors |P a and |P b respectively. So that H|P a = µ a |P a , We also denote the eigenvalues ofĤ † by µ * a = m a + iΓ a /2 and µ * b = m b + iΓ b /2 corresponding to the eigenvectors | P a and | P b respectively. So that It is not difficult to check that the set {|P n , | P n }, n = a, b, is a complete biorthonormal system forĤ such that According to the definition given in the pervious section,Ĥ could be diagonalized such that which means and The signs in front of q a and q b in (13) and p a and p b in (14) are just a convention which may differ among different authors, or even from one neutral meson system to another within the same paper. Now we can write the spectral form of the Hamiltonian Ĥ The normalization conditions are We find from equations (8) and (13) that By considering the effect of discrete symmetries on the matrix elements ofĤ we have CP T conservation → H 11 = H 22 , T conservation → |H 12 | = |H 21 |, CP conservation → H 11 = H 22 and |H 12 | = |H 21 |.
The above conditions suggest the dimensionless complex CP and CP T parameter as [2] θ ≡ and the CP and T violation parameter as It is convenient to introduce If CP T violation is absent from the mixing, then q/p = q a /p a = q b /p b and √ 1 − θ 2 = 1. In that case one only needs to use q/p.
The time evolution of the neutral meson system is easily obtained using the spectral form of the HamiltonianĤ Using the eqs.

Cascade decay and CPT violation
It was conjectured by Azimov [7] that additional tests of CP T invariance (violation) could be performed by looking for involving the neutral B d and neutral kaon system in succession. This idea has been followed by Dass and Sarman [8]. In all these cases, an initial B • meson ( B • stands for both B • d and B • s ) can only decay to one of the kaon's flavor eigenstates. To the leading order in the Standard model, the decays B • d → K • + X and B • s → K • + X and respective CP conjugate decays are forbidden [9], [10]. The possibility of probing CP T invariance by observing the time dependence of the cascade decays of the type B • d (B • d ) → J/ψK • /K • → J/ψf was investigated more recently [11] by considering that in new physics and in higher order, one cannot neglect the processes B • d → J/ψK • and B • d → J/ψK • when considering such a radical possibility as CP T violation.
In such cases there are two times and two CP T violation parameters. The time t for B • d meson to oscillate before decaying into J/ψK • and time t ′ in which K • oscillates before decaying into f . The CP T violation parameter θ in the (B • d − B • d ) meson mixing and the CP T violation parameter θ ′ in the (K • − K • ) meson mixing. It is possible to determine θ by comparing the t dependence of the cascade decays of tagged P • and tagged P • . Indeed, θ is computed in much the same way as from the time dependence of non-cascade decays [12], [13]. The parameter θ ′ cannot be determined, because it always appears entangled with some undetermined ratios of decay amplitudes.
We study the possibility of probing CP T invariance by observation of the time dependence of the cascade decay of the type B • d (B • d ) → J/ψ{K L , K S } → J/ψf by employing the complete biorthonormal basis and introduce new ratios of decay amplitudes. In such cascade decays there are two times and one CP Tviolation parameter. The time t for the B • d meson to oscillate before decaying into J/ψ{K S , K L } and time t ′ in which the decay into final state J/ψf takes place. Since CP is violated, there is no final state that can be obtained only from K S ( or K L ) and not from K L ( or K S ). Therefore all calculations must involve the full transition chain We consider an experiment in which a tagged B • d evolves for time t and decays into an intermediate state J/ψK L or J/ψK S , which after time t ′ decays into the final state J/ψf . The amplitude for this process is By using eqs. (23) and (24) (time evolution of flavor eigenstates) it is easy to show that the above expression casts into the following Now we introduce the four parameters and Therefore we may write the amplitude as where We next consider the analogous experiment with an initial B • d . The amplitude is where It is easily checked that If we use the first order approximation for small parameters, i.e., the approximation of neglecting all products of θ and λ, then we find One concludes from eq. (37) that the CP T violation in B mixing ( the parameter θ) can in principle be determined either from the comparison of R and R, or from the comparison of S and S. Indeed, R = −R and S = −S unequivocally indicate the presence of CP T violation in the mixing of the B-meson.
One can measure The CP T violation in B-meson mixing by observation of the time dependence of the tagged cascade decays.

Conclusion
In this work we introduced the system of complete set of biorthonormal basis as a necessary condition for diagonalizability of non-hermitian Hamiltonian (regardless of being normal or not normal). We also noticed that in the presence of T violation, sinceĤ is not normal, the mass eigenstates do not satisfy completeness and orthonormal relations and therefore are indistinguishable and non-physical.
We also studied the possibility of CP T invariance in the cascade model of the type P • (P • ) → {P a , P b }X → f X by observation of the time dependence of the process. In this case there are two time parameters of t and t ′ and one CP T violation parameter. We showed that it is possible to determine the θ parameter by comparison of time dependence of the cascade models for B • and B • .
We conclude by remarking that in the cascade model of P • (P • ) → X(M • /M • ) → Xf there are two CP T violation parameter (θ and θ ′ ) where θ is the CP T violation parameter in P • −P • and θ ′ is the CP T violation parameter in M • −M • system. In this case the parameter θ ′ in indeterminable due to entanglement with other parameters [12]. However using the biorthonormal system of basis for P • (P • ) → {P a , P b }X → f X, there is only one CP T violation parameter which is quite determinable.